High frequency acoustics in colloid-based
meso- and nanostructures by spontaneous
Brillouin light scattering
Dissertation
zur Erlangung des Grades
‘Doktor der Naturwissenschaften’
(Dr. rer. nat.)
im Promotionsfach Chemie
am Fachbereich Chemie, Pharmazie und Geowissenschaften
der Johannes Gutenberg–Universität Mainz
vorgelegt von
Dipl. Chem. Tim Still
geboren in Neuwied
Mainz, 2009
Die vorliegende Arbeit wurde im Zeitraum von März 2007 bis September
2009 am Max-Planck-Institut für Polymerforschung in Mainz unter der
Anleitung von Herrn Prof. Dr. XX XXX und Herrn Prof. Dr. X XXXX
angefertigt.
Tag der mündlichen Prüfung: 27.11.2009
Dekan: Prof. Dr.
Erster Berichterstatter: Prof. Dr.
Zweiter Berichterstatter: Prof. Dr.
Dritter Berichterstatter: Prof. Dr.
Der Klügere gibt nach! Eine traurige Wahrheit,
sie begründet die Weltherrschaft der Dummheit.
Marie von Ebner-Eschenbach
Contents
Abstract 1
1. Introduction 3
1.1. General Introduction . . . . . . . . . . . . . . . . . . . . . 3
1.2. Aims and Motivation . . . . . . . . . . . . . . . . . . . . . 10
1.3. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. Basics and Brillouin Light Scattering 13
2.1. Elastic Waves in Condensed Matter . . . . . . . . . . . . 13
2.1.1. Elasticity Theory Basics . . . . . . . . . . . . . . . 13
2.1.2. Elastic Waves in Isotropic Media . . . . . . . . . . 19
2.1.3. Spherical Waves . . . . . . . . . . . . . . . . . . . 22
2.2. Light Scattering Basics . . . . . . . . . . . . . . . . . . . . 23
2.3. Brillouin Light Scattering . . . . . . . . . . . . . . . . . . 29
2.3.1. BLS Basics . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2. BLS Instrumentation . . . . . . . . . . . . . . . . . 35
2.3.3. Vibrational Modes of Individual Particles . . . . . 45
3. Methods 49
3.1. Vertical Lifting Deposition . . . . . . . . . . . . . . . . . . 49
3.2. Melt Compression . . . . . . . . . . . . . . . . . . . . . . 51
3.3. Polymer and Colloid Characterization Techniques . . . . . 53
3.3.1. Photon Correlation Spectroscopy . . . . . . . . . . 53
3.3.2. Electron Microscopy . . . . . . . . . . . . . . . . . 54
3.3.3. Differential Scanning Calorimetry . . . . . . . . . . 56
3.3.4. Density Gradient Column . . . . . . . . . . . . . . 56
3.3.5. Wide Angle X-ray Scattering . . . . . . . . . . . . 57
I
Contents
3.3.6. UV/VIS Spectroscopy . . . . . . . . . . . . . . . . 58
3.3.7. Gel Permeation Chromatography . . . . . . . . . . 59
3.4. Theoretical Calculations . . . . . . . . . . . . . . . . . . . 60
3.4.1. Single-Sphere Scattering-Cross-Section Calculations 61
3.4.2. The Plane Wave Method . . . . . . . . . . . . . . 64
3.4.3. The Multiple Scattering Method . . . . . . . . . . 66
3.4.4. The Finite Element Method . . . . . . . . . . . . . 71
4. The Vibrations of Individual Colloids 73
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2. Elastic Vibrations in Homogeneous Polymer Colloids . . . 76
4.2.1. The ‘music’ of the spheres . . . . . . . . . . . . . . 76
4.2.2. The Influence of the Neighbors: Mixtures . . . . . 84
4.2.3. The Influence of the Rigidity: Copolymers . . . . . 87
4.2.4. The Influence of the Wave Vector: Suspensions . . 97
4.3. Elastic Vibrations in Nanostructered Colloids . . . . . . . 100
4.3.1. The Influence of the Temperature:
PS-SiO2 Core-Shell Particles . . . . . . . . . . . . 100
4.3.2. The Influence of the Components:
SiO2-PMMA Core-Shell Particles . . . . . . . . . . 107
4.4. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5. Phononic Behavior of Colloidal Systems 121
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1.1. The Phononic Band Diagram . . . . . . . . . . . . 124
5.1.2. The Effective Medium . . . . . . . . . . . . . . . . 128
5.1.3. Phononic Band Gaps . . . . . . . . . . . . . . . . . 132
5.2. Effective Medium Velocity in Defect Doped Opals . . . . . 137
5.3. Band Gaps in Polymer Opals and Disordered Systems . . 143
5.3.1. The Influence of the Order . . . . . . . . . . . . . 143
5.3.2. The Influence of the Composition . . . . . . . . . . 153
5.4. Band Gaps in SiO2 Colloidal Systems . . . . . . . . . . . 158
5.4.1. Phononic Behavior of Silica Suspensions . . . . . . 158
5.4.2. Silica–Poly(ethyl acrylate) Films (PhoXonics) . . . 163
II
Contents
5.5. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6. Characterization of Stable Organic Glass 169
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.2. BLS experiments on IMC . . . . . . . . . . . . . . . . . . 170
6.3. Materials and Methods . . . . . . . . . . . . . . . . . . . . 177
7. Concluding Remarks 179
7.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.2. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Acknowledgments 185
Abbreviations 189
Appendix 195
A. scattering geometry 195
A.1. transmission case . . . . . . . . . . . . . . . . . . . . . . . 195
A.2. reflection case . . . . . . . . . . . . . . . . . . . . . . . . . 197
Bibliography 201
III
Abstract
Materials that can mold the flow of elastic waves of certain energy in
certain directions are called phononic materials. The present thesis
deals essentially with such phononic systems, which are structured in
the mesoscale (<1 µm), and with their individual components. Such
systems show interesting phononic properties in the hypersonic region,
i.e., at frequencies in the GHz range. It is shown that colloidal sys-
tems are excellent model systems for the realization of such phononic
materials. Therefore, different structures and particle architectures are
investigated by Brillouin light scattering, the inelastic scattering of light
by phonons.
The experimental part of this work is divided into three chapters:
Chapter 4 is concerned with the localized mechanical waves in the indi-
vidual spherical colloidal particles, i.e., with their resonance- or eigenvi-
brations. The investigation of these vibrations with regard to the envi-
ronment of the particles, their chemical composition, and the influence of
temperature on nanoscopically structured colloids allows novel insights
into the physical properties of colloids at small length scales. Further-
more, some general questions concerning light scattering on such systems,
in dispute so far, are convincingly addressed.
Chapter 5 is a study of the traveling of mechanical waves in colloidal
systems, consisting of ordered and disordered colloids in liquid or elastic
matrix. Such systems show acoustic band gaps, which can be explained
geometrically (Bragg gap) or by the interaction of the acoustic band
with the eigenvibrations of the individual spheres (hybridization gap).
While the latter has no analogue in photonics, the presence of strong
phonon scatterers, when a large elastic mismatch between the composite
components exists, can largely impact phonon propagation in analogy
1
Abstract
to strong multiple light scattering systems. The former is exemplified
in silica based phononic structures that opens the door to new ways of
sound propagation manipulation.
Chapter 6 describes the first measurement of the elastic moduli in
newly fabricated by physical vapor deposition so-called ‘stable organic
glasses’.
In brief, this thesis explores novel phenomena in colloid-based hyper-
sonic phononic structures, utilizing a versatile microfabrication technique
along with different colloid architectures provided by material science,
and applying a non-destructive optical experimental tool to record dis-
persion diagrams.
2
1. Introduction
1.1. General Introduction
One of the most excitatory and technically promising new field in physics
in the last decades was for sure the emerging possibility to control and
manipulate the flow of light by engineering of macroscopic media with
periodic dielectric function, i.e., with periodic variation of the refractive
index in one, two, or three dimensions (Fig. 1.1). Since the first realiza-
tion of such photonic crystals, an abundance of publications on this topic
have been released, including several new high impact journals. Beside
some quantum mechanical effects, the striking capability of a photonic
crystal is its aptitude to prevent light from propagating in certain di-
rections with specified frequencies, while they can operate as omnidirec-
tional reflectors of light, independent from angle and polarization.[1] If
the propagation of electromagnetic waves is forbidden at certain frequen-
cies, one speaks about a photonic band gap.
The appearance of band gaps for distinct wavelengths in materials with
periodic changed dielectric constant ǫ can be rationalized as an interfer-
ence effect. The light is partially reflected at each layer interface. The
multiple reflection interfere, and if the periodicity spacing is commensu-
rate to the wavelength of the incident light, the destructive interference
eliminates the forward propagation of the electromagnetic wave. In the
simplest case of a one dimensional (1D) photonic crystal, i.e., with pe-
riodic modulation of ǫ only in z-direction, with normal incident light,
the gap occurs when the wavelength of the light λl is twice the crys-
tals spatial period a, λl=2a. In the concept of the reciprocal space that
is commonly used in solid state physics, this belongs to a wave vector
k = 2π/λl (in this case k is identical to its component in z-direction) at
3
1. Introduction
Figure 1.1.: Photonic or phononic crystals with periodicity in one, two, or three
directions. In photonic crystals, white and black segments have different refractive
index, while in phononic crystals the components must have different elastic moduli
or densities.
the edge of the first Brillouin zone, i.e., k = π/a. The problem was first
treated by Lord Rayleigh (1887), and such gap is usually called Bragg
gap. It can be shown that the width of such a photonic gap depends on
the dielectric contrast between the two components.
The principal concepts presented for the molding of electromagnetic
waves can, in principle, be transfered to mechanical waves, i.e., sound
waves. A material with periodic modulation of the elastic properties,
i.e., of the elastic moduli or the density, is called a phononic crystal after
the quantized mode of vibration in a rigid crystal lattice - the phonon
(in analogy to the photon in the photonic crystals).[2, 3] However, the
theoretical treatment of the propagation of mechanical waves is more
complicated. While the light propagation can be fully described by a
wave with distinct polarization and velocity, which depends on the re-
fractive index of the passed medium, a mechanical wave can, in principle,
propagate in two different ways.
Both mechanisms are schematically shown in Fig. 1.2. The longitu-
dinal sound wave on the left side shows displacement along the prop-
agation direction, while in the transverse sound wave the displacement
4
1.1. General Introduction
Figure 1.2.: Propagation mechanisms of mechanical waves: a longitudinal wave
with displacement along the propagation direction (given by the direction of the
wave vector k); b transverse sound wave with displacement perpendicular to the
propagation direction.
is perpendicular to the propagation direction symbolized by the phonon
wave vector k. Both waves propagate with different sound velocities cl
(longitudinal) or ct (transverse), respectively. The relevant propagation
mechanisms depend on the type of the material. In solid materials, both
kind of waves can propagate, while in liquid or gaseous media only longi-
tudinal waves are supported. Hence, the different mechanical waves are
often distinguished between elastic waves in solids and acoustic waves in
fluids. However, in this thesis the terms elastic wave, acoustic wave, or
sound wave will be used synonymously for all kinds of mechanical waves.
On an atomic scale, the concept of the phonon is used to describe
for example the heat capacity of solids. The frequencies of the allowed
mechanical waves are quantized as a function of the distance ℓ between
neighboring atoms. The atoms (the black points in Fig. 1.2) are dis-
placed, and the displacement of each atom is described by the displace-
ment vector u(r,t) as a function of position and time.
However, in this thesis the interest lays on much lower frequencies with
phonon wavelength Λ >> ℓ, thus the allowed mechanical frequencies can
be regarded as continuous. Fig. 1.2 holds also in this case, but now the
points represent volume elements containing many atoms. It should be
5
1. Introduction
noted that the spatial displacement of volume elements as a function of
time leads to areas of instantaneously increased or decreased pressure (or
local density), thus mechanical waves can be regarded as pressure waves.
This picture also rationalizes why there is no propagation of transverse
sound waves in fluids since fluids do not support shear (aside from very
viscous liquids).
Phononic crystals can be designed by a one-, two-, or three-dimensional
periodicity of their elastic properties in analogy to their electromagnetic
counterparts (Fig. 1.1). The geometry of the periodic structure has
strong influence on its mechanical properties. One of the first exam-
ples of a two-dimensional phononic crystal is a statue in Madrid, created
by the artist E. Sempere.[4] The piece of art consist of a large number of
hollow stainless-steel cylinders with diameter of 2.9 cm in a simple cubic
arrangement in a a=10 cm unit cell, the whole statue has a diameter
of 4 m. A simple transmission experiment with a sound generator and
detector in the range of audible frequencies (1-5 kHz) has shown the ex-
istence of phononic band gaps with strong sound attenuation depending
on the orientation of source and receiver relative to the statue, i.e., as a
function of the crystallographic direction.
Like in the case of photonic crystals, the Bragg gap appears at the
edge of the first Brillouin zone, i.e., the phonon wavelength Λ is twice
the lattice parameter a, Λ = 2a, and the corresponding wave vector is
again k = π/a. The frequency is given by 2πf = cl/Λ. Since typical
sound velocities are between 102 ms−1 (air) and 105 ms−1 (condensed
matter), attenuation in the audible range requires periodicities in the
cm to meter range, like in the example of the statue. Structures that
would attenuate seismic waves would have to be constructed even in km
scales, the Bragg frequency scales with 1/a. According to this, smaller,
technically handier structures with spacing in the mm, µm, or even nm-
scale correspond to frequencies in the MHz to THz scale.
Most of the realized phononic band gap systems are restricted to sonic
and ultrasonic crystals with macroscopic periodicity, e.g., some mm-sized
crystals assembled manually, which could be probed by simple acous-
tic transmission experiments in the commensurate frequency ranges.[5–7]
6
1.1. General Introduction
Only recently, Cheng et al. could show the realization of a hypersonic
phononic crystal based on the self-assembly of sub-micron colloidal poly-
mer crystals.[8, 9]
In fact, such mesoscopic structures have some advantages. They can be
easily prepared by self-assembly methods or by interference lithography[2]
on a (relative to the lattice spacing) large scale, they can show photo-
thermal or photo-acoustic effects, and their size is also commensurate
to the wavelength of visible light. Therefore, it is possible to create a
mesoscopic artificial crystal with periodic variation of refractive index
and mechanical properties, which would act as a photonic and phononic
crystal simultaneously. Sometimes, such ‘blind and deaf’ systems are
called phoXonic crystals.[10, 11] Due to their size, colloidal crystals from
mesoscopic spheres show nice colorful reflections of light depending on
the scattering angle relative to their crystallographic direction as it can
be found in natural opals. Therefore, such systems are often (also in the
present thesis) called artificial opals. Fig. 1.3 shows three photographs
of a polished natural opal, an artificial opal, which consists of dense
packed silica spheres (d=170 nm) in a organic liquid matrix, and a layer
of polystyrene spheres (d=550 nm) on a glass substrate.
These artificial opals are examples for colloid particle systems. In
general, a colloid is a particle or droplet of typical size between 1 nm and
10 µm that is dispersed in another gaseous, liquid, or solid medium, e.g.,
milk (fat droplets in water), fume (solid particles in gas), blood (solid
particles in liquid), etc. In this thesis the word colloid is used in a less
general way; here it means in particular the well defined colloidal solid
particles, which are fabricated by the methods of material chemistry.
Systems of these particles in a matrix material are referred to as colloidal
systems.
Such colloidal systems have many advantages. Due to the onward
development of colloidal and material science, nowadays it became pos-
sible to control a wide range of properties during the synthesis of colloids.
Monodisperse spherical particles can be produced in a size range from
few tens of nm to several microns, utilizing a wide range of materials
like metals, oxides (e.g., silica), or (organic) polymers. However, colloid
7
1. Introduction
Figure 1.3.: Natural and artificial opals: a Natural opal (Australia, Museum Idar-
Oberstein, Germany); b dense packed silica spheres (d=219 nm, DKI Darmstadt)
in organic liquid; c PS spheres (d=550 nm, M. Retsch, MPIP) on glass substrate.
science is not generally limited to spherical particle shape. For metals,
and with limitations for the form stability also for oxides, even smaller
nanoparticles can be created. Mesoscopic colloids can be additionally
structured in the nanoscale to manipulate or improve their physical and
chemical properties, leading, e.g., to spherical hybrid core-shell particles
investigated also in this thesis.[12] Another already mentioned advantage
is the possibility to realize well ordered single or multi-component sys-
tems by self-assembly (colloidal crystals). In Fig.3.1 several techniques to
achieve colloidal crystals will be presented, including sedimentation,[13]
electrodeposition,[14] centrifugation, filtering, vertical deposition,[15, 16]
and compression molding.[17] Especially the last two techniques play a
role in this thesis and will be introduced in more detail in the sections
3.1 and 3.2.
Since colloids exhibit outstanding physical and chemical properties
8
1.1. General Introduction
(e.g., large surface, optical properties, solubility, etc.) and tunability
(size, shape, softness, biodegradability, etc.),[18–20] this class of materials
found numerous technical applications, e.g., in suspensions as pigment or
sunscreen,[21] for drug-delivery applications,[22] or to improve the macro-
scopic mechanical properties of solids or polymers.[23]
This thesis is devoted to the investigation of the acoustic, i.e., phon-
onic, properties of colloidal particles and systems,which appear hetero-
geneous compared to the sound wavelength. Hence, rich unprecedented
behavior is expected due to the large number of parameters involved
and the sensitivity of elastic wave propagation to boundary conditions.
As already pointed out, the characteristic frequencies scale with the in-
verse size of the observed systems, for mesoscopic colloids hypersound,
i.e., GHz frequencies, become relevant. Such high frequencies are not
accessible anymore by ‘simple’ sound generation and subsequent detec-
tion with membranes (audible sound) or piezo crystals (ultrasound). In
fact, when measuring the sound propagating in a material, it is not abso-
lutely necessary to generate the hypersound artificially, since hypersonic
phonons are thermally excited at temperatures far away from 0 K.[24]
These phonons can be probed by inelastic light scattering, i.e., sponta-
neous Brillouin light scattering (BLS) in the GHz-range (related to the
mesoscopic length scale) and Raman scattering in the THz-range (for the
nanoscopic length scale).[25, 26] Another possibility is pump-probe spec-
troscopy, where a strong laser pulse impinges on a sample and a second
probe laser is used to measure the reflectivity of the sample as a function
of time.[27, 28] After Fourier transformation that approach also delivers
information about the frequencies of sound in the probed material, how-
ever, after an induced perturbation and not spontaneously.
In this thesis, BLS is utilized to probe phonon propagation (and lo-
calization) in micro- and nanostructures. The main principle of this
unique non-destructive and non-contact optical technique is the interac-
tion of photon and phonon. In fact, the photons of the probing single-
frequency laser light are scattered inelastically on the phonons in the
probed material propagating along a selected direction, which is given
by the scattering wavevector q that is a function of the scattering angle
9
1. Introduction
and the sample’s refractive index. In homogeneous systems the interact-
ing phonon is the one whose phonon wavevector k is identical with q as
a consequence of momentum conservation. Therefore, the BLS spectrum
at a given q consists of a Doppler shift at angular frequencies ω=±cl,tk
for longitudinal (l) or transverse (t) sound velocity c. With other words,
we measure the frequency shift of the probing light due to the creation or
annihilation of a phonon as a function of the momentum (the magnitude
of the wave vector) leading to the dispersion relation ω(q). The achieved
frequency shifts in the GHz-range are tiny compared to the frequency of
the laser light (1014 Hz), the required high resolution is attained by a
multi-pass tandem Fabry-Pérot interferometer.[29]
Of course, the use of BLS to investigate the mechanical properties of
matter is not limited to colloidal systems. The last chapter of this thesis
shows how BLS can be used to investigate the mechanics and kinetics
of so-called stable organic glasses, i.e., organic glasses that are created
by very slow vapor deposition on a temperate substrate.[30, 31] By doing
so, the molecules have more time to rearrange in an energetically more
favorable way, leading to elastic moduli comparable to normal glasses
that are aged for at least several hundred years.
1.2. Aims and Motivation
Colloids are very promising systems in many terms. They are auspi-
cious candidates to design hypersonic phononic systems that may be
the basis of elaborated devices that deal with the concurrent interaction
of phonons and light (functional phoXonic materials). However, so far
there is no large knowledge about the details of the nano- and mesome-
chanical behavior of individual colloids and the manipulation of sound
propagation in colloidal systems beyond the simple Bragg gap.[8]
In this thesis a systematic approach is presented that captures the
influence of different parameters on the mechanical waves localized in
spherical (as the most accessible and theoretically best to capture model
system), partially nanostructured, mesoscopic colloids as well as the
10
1.3. Outline
propagation of sound waves in systems composed of such particles.[12, 32]
The first realization of a so-called hybridization gap in polymer based
colloids will be presented, which originates from the interaction bet-
ween individual and collective colloid mechanical properties, leading to
crystalline as well as amorphous omnidirectional hypersonic band gap
systems.[33]
These developments, however, must not be regarded as the end of
the basic experimental investigation of such systems. It will be shown
that for silica based systems with increasing mechanical contrast new
effects occur, which will demand further strong efforts in the theoreti-
cal and experimental conquest of this young and promising field. Espe-
cially in consideration of the numerous appearing topics related to photo-
acoustics,[34] thermo-acoustics,[35] or nano- and micromechanics,[36, 37] it
seems worth to investigate colloidal model systems en detail and the inti-
mate relation of the heat conductivity in dielectric materials on phonons
can have impact on a directional heat flow in the future. This work may
be a humble contribution to this exciting development.
1.3. Outline
The structure of this thesis is as follows:
In the first chapter I introduce the theoretical and practical basics of
the propagation of elastic waves and of light scattering, in particular of
Brillouin light scattering and its applicability to colloidal particles and
colloid based systems.
The second chapter is about other methods utilized in this thesis,
including the theoretical calculations that are needed to quantitatively
describe the experimental findings. If these first two chapters might ap-
pear to extend some few details beyond the absolutely necessary length,
this is caused by the aim to present the important results in a common
notation, which is not the case in the underlaying literature.
Chapter 4 describes the work done on the vibrations of individual col-
loids that can be investigated in multiple scattering or thick transparent
11
1. Introduction
samples. Herein, the influences of packing in a sample consisting of many
spheres and that of the sphere material is reported as well as the dis-
cussion of general theoretical problems and their experimental answer
concerning the selection rules for eigenvibrations in BLS and the light
scattering intensity as a function of q.
The next chapter proceeds to the collective phononic behavior, i.e., the
propagation of sound waves, in colloidal systems embedded in a solid or
liquid matrix. Starting from a short description of the state-of-the-art,
the experimentally found effective medium velocities are discussed and
a new kind of phononic band gap, the hybridization gap, is presented.
It is demonstrated by the investigation of silica based colloidal systems
that there seem to be additional mechanisms to manipulate the phononic
band diagram.
Chapter 6 is the last experimental chapter and deviates from the scope
on colloids in the other parts of the thesis since it is about the charac-
terization of stable organic glasses.
Finally, this thesis concludes with main results and outlook in the
concluding remarks (chapter 7).
12
2. Basics and Brillouin Light
Scattering
In this chapter, first the basic expressions of elasticity will be introduced,
especially with regard to the propagation of elastic waves in isotropic me-
dia and their description as spherical waves. Then the general principles
of light scattering will be presented and applied to the special case of
Brillouin light scattering, including the discussion of its technical real-
ization.
2.1. Elastic Waves in Condensed Matter
2.1.1. Elasticity Theory Basics
In this part, fundamental concepts of the theory of elasticity are briefly
introduced, following mostly the notation of the excellent textbook of
Landau and Lifschitz.[38]
Essential terms are the strain and stress tensors. To introduce them,
we first define the displacement vector u that shifted a point P that can
be found by following the vector r (with the three components x1 = x,
x2 = y, and x3=z in a Cartesian system) from the origin of the coordinate
system to the point P ′ with coordinates given by the vector r′, i.e.,
u = r − r′. (2.1)
The distance between any two infinitesimally adjacent points dl is given
13
2. Basics and Brillouin Light Scattering
by
√ √
dl = dx2 2 21 + dx2 + dx3 = dx
2
i , (2.2)
using the Einstein summation convention behind the second equal. After
a deformation it becomes
√ √
′ ′2 ′2 2′ ′dl = dx1 + dx2 + dx3 = dx
2
i (2.3)
Therefore, it’s possible to write
′
dl 2
′2 ∂ui ∂ui ∂ui= dxi = (dxi+dui)
2 = dl2+2 dxidxk+ dxkdxl (2.4)
∂xk ∂xk ∂xl
substituting du = ∂uii ∂x dxk in the last step. With
∂ui
∂x dx dx =
∂uk
i k ∂x dxidxkk k i
and swap of i and l in the last term, finally we get
′
dl 2 = dl2 + 2uikdxidxk. (2.5)
The strain tensor uik is therein defined as
( ) ( )
1 ∂ui ∂uk ∂ul · ∂ul ≈ 1 ∂ui ∂ukuik = + + + . (2.6)
2 ∂xk ∂xi ∂xk ∂xi 2 ∂xk ∂xi
The last approximation is valid if second order terms can be neglected.
Obviously, uik is a symmetric tensor. Each symmetric tensor can be
diagonalized in any point. With the diagonal elements u(1), u(2), u(3) the
strain in any point can be written as the sum of three independent terms,
which give the strain in three orthogonal main directions.
′
dl 2 = (1 + 2u(1))dx21 + (1 + 2u
(2))dx22 + (1 + 2u
(3))dx23 (2.7)
For u(i) ≪ 1 and if higher order terms are neglected, the relative change
14
2.1. Elastic Waves in Condensed Matter
of elongation becomes
dx′i − dx √i = 1 + 2u(i) − 1 ≈ u(i). (2.8)
dxi
With that approximation we can write the relative volume change of an
infinitesimal small volume element dV → dV ′ as the sum of the diagonal
elements of the strain tensor.[38]
dV ′ − dV
= uii (2.9)
dV
The resulting force F on any partial volume of an elastic body can
be written as the integral over the forces on any element of the volume
∫
F dV . Because of the identity of actio and reactio for all forces between
any two points within the volume the resulting force can also be written
∫
as the sum over the integrals of the three components Fi, FidV , which
can be translated into an integral over the surface:
∫ ∫ ∮
∂σik
FidV = dV = σikdfk (2.10)
∂xk
Here, Fi is expressed as divergence of a second rank tensor, the stress
tensor σik:
∂σik
Fi = (2.11)
∂xk
In Eq. 2.10 dfi are the components of a vector that is always oriented in
the direction normal to the surfaces. The nature of the stress tensor is
visualized in Fig. 2.1. When placing the surfaces of the volume element
in the principal plane of the coordinate system (xy, yz, or xz), the com-
ponent σab of the stress tensor equals to the a-component of the force
that is normal to that xb-axis.
The work δw that is achieved to perform a deformation is
δw = −σikδuik. (2.12)
15
2. Basics and Brillouin Light Scattering
x
sxx
szx z
y
syx
Figure 2.1.: Illustration of the stress tensor for the σax-components working on
a surface normal to the x-axis. The picture is analogously valid for the other six
components when applied to the surfaces normal to the y- and z-axis.
For reversible, elastic deformations it follows for the free energy A (A =
U − TS, dU = TdS − δw, S: entropy) that
( )
∂A
σik = . (2.13)
∂uik T
It can be shown that the free energy after an elastic isothermal defor-
mation of an isotropic body can be expressed after series expansion and
neglecting higher order terms as
λ
A = A0 + u
2
ii + µu
2
ik, (2.14)2
introducing the Lamé coefficients λ and µ. Note that λ and µ are force
constants. Since it was already shown that the volume change during a
deformation is expressed by the sum of the diagonal elements uii, it is
obvious that, if the term containing the uii becomes zero, the last term
expresses a pure shear deformation. Therefore, µ is also called the shear
modulus, sometimes denoted by G.
16
2.1. Elastic Waves in Condensed Matter
By writing the following identity
( )
uik = uik −
1 1
δikull + δikull (2.15)
3 3
with uik = const · δik it follows directly that every deformation can be
written as a sum of a pure shear deformation and a homogeneous dilata-
tion. The term in brackets is surely a pure shear deformation, because
the sum of the diagonal elements vanishes (δii = 3) and the other term
is related to the homogeneous dilatation.
For a perfectly elastic body, Hook’s law can be generalized to state that
each component of the stress tensor is linearly related to each component
of the strain tensor:
σik = ciklmulm (2.16)
Here, c is the fourth-rank stiffness tensor with 34iklm = 81 components.
Taking into account the symmetry of the strain and stress tensors, it is
possible to reduce the number of coefficients to 62 = 36. However, these
36 components are not independent. In the general case the following
symmetry relations are valid:
ciklm = ckilm = cikml = ckiml (2.17)
I.e., in an anisotropic body there are at most 21 independent components,
which are usually noted in a matrix notation, which uses the itemized
identities to abbreviate the tensor notation and to replace the tensor by
a 6x6-matrix with only two indices per component. The following rules
are adopted for the indices:
tensor notation 11 22 33 23, 32 13, 31 12, 21
matrix notation 1 2 3 4 5 6
17
2. Basics and Brillouin Light Scattering
With that notation one can rewrite Eq. 2.16 as
    
σ1 C11 C12 C13 C14 C15 C16 u1
σ2 C12 C22 C23 C24 C25 C26u2
    
σ  3 C13 C23 C33 C  = 34
C35 C36 u3
     (2.18)
σ  4 C14 C  24 C34 C44 C45 C46 u4
    
σ  C  5 15 C25 C35 C45 C55 C56 u5
σ6 C16 C26 C36 C46 C56 C66 u6
While the discussion of the stiffness matrix for different crystal sym-
metries is done in specialized treatments, here only the isotropic case,
which is a good approximation for many cases, including also most poly-
mer systems, should be discussed in the rest of this section. For an
isotropic body, it can be shown that due to further symmetry consider-
ations the stiffness matrix in Eq. 2.16 has the following form:[39]
    
σ1 C11 C12 C12 0 0 0 u1
σ2 C12 C11 C12 0 0 0 u2
    
σ  C C  3 = 12 12
C11 0 0 0 u3
     (2.19)
σ   0 0 0 C 0 0 u 4 44 4
    
σ  5 0 0 0 0 C44 0 u 5
σ6 0 0 0 0 0 C44 u6
Because the three remaining constants are related by the following rela-
tion
C11 = C12 + 2C44, (2.20)
there are, indeed, only two independent constants remaining. These two
constants can be identified with the Lamé coefficients
λ = C12 and µ = C44, (2.21)
while C11 is also called the longitudinal modulus. Beside the shear mod-
ulus G = µ, some other elastic parameters are often encountered, namely
18
2.1. Elastic Waves in Condensed Matter
the bulk modulus, the Young’s modulus, or the Poisson’s ratio.
The bulk modulus K measures a substance’s resistance to uniform
compressions. It is defined as the ratio of the hydrostatic pressure to the
fractional volume change,
− ∂p 2K = = λ+ µ. (2.22)
∂uii 3
K should not be confused with the longitudinal bulk modulus
M = C11. (2.23)
The Young’s modulus E, also known as modulus of elasticity or tensile
modulus, is a measure of the stiffness of an isotropic elastic material. It
is given by the ratio of longitudinal stress (which has units of pressure)
and the dimensionless longitudinal strain,
σ11 µ(3λ+ 2µ)
E = = . (2.24)
u11 λ+ µ
The ratio ratio of the lateral strain to the longitudinal strain defines
the Poisson’s ratio,
−u22 λσ = = . (2.25)
u11 2(λ+ µ)
2.1.2. Elastic Waves in Isotropic Media
When a body is deformed, the deformation usually goes along with
changes in the temperature. However, the heat transport is normally
much slower compared to periods of vibrations in the body. Therefore,
one can regard the movements as (quasi) adiabatic. It can be shown
that in the adiabatic case the values for the Young’s modulus and the
19
2. Basics and Brillouin Light Scattering
Poisson’s ratio, given above for the isothermal case, change into
2Tα
2 Tα2
Eadiabatic = E+E and σadiabatic = σ+(1+σ)E , (2.26)
9Cp 9Cp
where α = (∂V/∂T )p and Cp is the specific heat at constant pressure.
[38]
In the following, σ and E mean their adiabatic values.
The general equation of movement can be written as
∂2ui ∂σik
̺ = (2.27)
∂t2 ∂xk
with mass density ̺. For the isotropic elastic medium the equation of
movement becomes
∂2ui E
̺ = ∇2 Eu+ − . (2.28)∂t2 2(1 + σ) 2(1 + σ)(1 2σ)grad divu
Regarding a plane elastic wave in x-direction in an infinite medium,
i.e., the deformation u depends only on the x-coordinate (all derivatives
with respect to y and z become zero), the components of the vector u
become
∂2ux − 1 ∂
2u 2 2 2 2x ∂ uy − 1 ∂ uy ∂ uz − 1 ∂ uz2 = 0; 2 = 0; 2 = 0. (2.29)∂x2 cl ∂t2 ∂x2 ct ∂t2 ∂x2 ct ∂t2
The equations 2.29 are one-dimensional wave equations, cl and ct are
their velocities of propagation. Obviously, the propagation in x-direction
is different from that in the other directions. If the displacement ux lies in
the direction of propagation of the wave, the wave is called a longitudinal
wave with longitudinal sound velocity cl:
√ √ √
E(1− σ) λ+ 2µ C11
cl = − = = (2.30)̺(1 + σ)(1 2σ) ̺ ̺
In the other directions, the displacement (uy, uz) lies in a plane normal
20
2.1. Elastic Waves in Condensed Matter
to the direction of propagation of the wave. Such a wave is called a
transverse wave with transverse sound velocity ct:
√ √
√
E µ C44
ct = = = (2.31)
2̺(1 + σ) ̺ ̺
Accordingly, from Eqs. 2.21 and 2.23 it is clear that we can express
longitudinal and shear modulus simply by
M = ̺c2l (2.32)
and
G = ̺c2t , (2.33)
respectively, while the bulk modulus K depends on the longitudinal as
well as on the transverse sound velocity, since after Eqs. 2.20 and 2.22 it
becomes
− 2 − 4 2 − 4K = M 2G+ G = M G = ̺(c c2
3 3 l 3 t
). (2.34)
The longitudinal sound velocity is always larger than the transverse
sound velocity. For σ ≤ 0.5,
√
cl > ct 2. (2.35)
One should note that the transverse wave has two possible polariza-
tions, which are orthogonal to each other. Furthermore, it is notable that
longitudinal waves involve changes in the volume of the medium, i.e., di-
latation or compression of a local volume element, while the transverse
waves cause no volume change.
21
2. Basics and Brillouin Light Scattering
2.1.3. Spherical Waves
Because this thesis treats in many parts experiments on colloids acting
as spherical scatterers, it is meaningful to introduce briefly the princi-
ple solution of the elastic wave equation for an isotropic medium. The
equations presented here will be required again in the discussion of the
vibrational eigenmodes of spherical particles.
In general, for a harmonic elastic wave of angular frequency ω, the
displacement vector u can be written as
u(r, t) = ℜ [u(r) exp(−iωt] . (2.36)
Using that one can write the general equation of motion in the following
time-independent form:[40]
(λ+ 2µ)∇(∇ · u)− µ∇×∇× u+ ̺ω2u = 0. (2.37)
In a spherical coordinate system, the displacement vector can be written
as the sum of three vectors u = l + m + n such that Eq. 2.37 can be
broken into three independent vector Helmholtz equations[41]
(∇2 + k2)l = 0, (∇2l + k2t )m = 0, (∇2 + k2t )n = 0, (2.38)
where l represents the displacement associated with longitudinal wave
and n and m represent the transverse displacements, which are or-
thogonal to each other. These three vectors can be related to scalar
functions ϕ, ψ, and χ that are solutions of a scalar Helmholtz equation
(∇2 + k2)f = 0:
1
l = ∇ϕ, (2.39)
kl
m = ∇× rψ, (2.40)
1
n = ∇×∇× rχ. (2.41)
kt
22
2.2. Light Scattering Basics
The transverse displacement vectors m and n are herein expressed as the
product of the constant position vector r and the scalar functions men-
tioned above.[41] In polar coordinates the solution of the scalar Helmholtz
equation is known. It is
flm(r, θ, φ) = Rl(kr)Ylm(θ, φ). (2.42)
Rl(kr) are n-th order spherical Bessel functions, which represent the
radial displacement. Ylm(θ, φ) are the n-th order spherical harmonics
(Legendre functions) with l = 0, 1, 2, 3, . . . and m being an integer −l ≤
m ≤ +l. Knowing this, the vector solutions of Eq. 2.38 can be written
down as
1
llm(R, kl) = ∇ [Rl(klr)Ylm(r̂)] ; l = 0, 1, 2, 3, . . . , (2.43)
kl
mlm(R, kt) = ∇× [rRl(ktr)Ylm(r̂)] ; l = 1, 2, 3, . . . , (2.44)
1
nlm(R, kt) = ∇×∇× [rRl(ktr)Ylm(r̂)] ; l = 1, 2, 3, . . . , (2.45)
kt
2.2. Light Scattering Basics
This section is a brief introduction into the basic principles of light scat-
tering. While the derivation of scattering theory on the basis of quantum
field theory is possible, herein the scattering medium as well as the light
are treated classically, leading to practically the same results within the
scope of this thesis. Of course there are light scattering effects, as for ex-
ample in the well-known Raman scattering technique, which deals with
rotational and vibrational transitions of single atoms or molecules, i.e.,
effects in a quantum length scale, that must be treated (at least partially)
quantum mechanically. However, this thesis deals with the investigation
of phonons by Brillouin light scattering (Chapter 2.3) in condensed mat-
ter. Thus the investigated phonons are classical waves with wavelengths
in the order of some nm up to µm, it is fully justified to apply a classic
theory.
23
2. Basics and Brillouin Light Scattering
The classical theory of light scattering in dense media developed by
Einstein[42] and Smoluchowski considers the sample as divided into small
volume elements large enough to contain many molecules (i.e., following
classical physics), but of linear dimension small compared to the wave-
length of light. An incident light wave induces a dipole moment in each
volume element, which becomes the source of scattered radiation. Pro-
vided that the induced polarization is constant through the medium,
the net scattered radiation in all directions but the forward will be zero
due to destructive interference, because the wavelets scattered from each
subregion differ only by a phase factor that depends on the relative posi-
tion of the small volumes. Therefore, neglecting small surface effects, it
is possible to pair each small volume with another small volume whose
scattered field is identical in amplitude but opposite in phase, thus they
cancel out.[41, 43, 44]
However, in real media there will always be small random fluctua-
tions in the local dielectric constant due to the thermal motion of the
atoms and molecules in the sample. Because these fluctuations should be
uncorrelated from one volume element to the next, these regions are op-
tically different, and therefore also the amplitudes of the scattered light
are uncorrelated. That means that now light is also scattered in other
directions than forward due to only partial interference.
The local dielectric constant at a point at position r and time t,
ǫ(r, t), is generally described by the dielectric constant fluctuation ten-
sor δǫ(r, t). It describes the relation between the local and the average
dielectric constant ǫ0,
ǫ(r, t) = ǫ0I + δǫ(r, t). (2.46)
I is the second-rank unit tensor.
When the incident light is a plane wave with field amplitude E0, angu-
lar frequency ωi, and incident propagation vector ki, the incident electric
field can be written as
Ei(r, t) = niE0 exp i(ki · r − ωit) (2.47)
24
2.2. Light Scattering Basics
with ni being the unit vector in the direction of the incident field.
At large distance R from the scattering volume, the scattered electric
field Es(R, t) can be obtained from the fact that the Maxwell equations
must hold for the total electric field, the incident electric field and as
well for the scattered electric field, where the total field is just the sum
of the two others:
E = Ei +Es. (2.48)
Indeed, the same should be valid for the electric displacement field D
and the magnetizing field H . The solution of the Maxwell equations is
lengthy and performed elsewhere.[44] In the end it follows for the com-
ponent of the scattered electric field Es(R, t) that
∫
E0
Es(R, t) = exp iksR d
3r
4πRǫ0 V
exp i(q · r − ωit [ns · [ks × (ks × (δǫ(r, t) · ni)]] , (2.49)
where ns is the polarization, ks is the propagation vector and ωs is the
frequency of the scattered plane wave field that reaches the detector,
while V indicates that the integral is over the whole scattering volume.
The scattering wave vector q is defined as the difference between incident
and scattered propagation vector,
q = ki − ks. (2.50)
The angle between ki and ks is called the scattering angle θ (see Fig. 2.2).
Fig. 2.2b makes clear that the magnitude of the scattering wave vector
q can be computed as
q2 = |q|2 = k2 + k2i s − 2ki · ks. (2.51)
When - like in most light scattering experiments - ki ≈ ks, i.e., the scat-
tering is quasi-elastic, the cosine rule becomes applicable and Eq. 2.51
25
2. Basics and Brillouin Light Scattering
Figure 2.2.: a At the detector, the total radiated field is the sum of all fields
radiated from any infinitesimal volume d3r at position r from the center O of the
illuminated volume. The detector is at positionR. b General light scattering setup:
Incident light of polarization ni, frequency ωi, and wave vector ki is scattered.
Although it is scattered in all directions (not shown for clarity), the figure shows
only the light with ns and wave vector ks that can reach the detector after passing
an analyzer. The wave vector q is shown in gray as ki − ks. (according to [44])
becomes
q2 = k2 + k2i s −
θ
2ki · k = 2k2s i (1− cos θ) = 4k2 2i sin . (2.52)2
Or, with ks = ki = 2πn/λ, where the length of the√incident wave vector
is written as function of the refractive index n (n = ǫ0) and the incident
wave length λ:
θ 4πn θ
q = 2ki sin = sin . (2.53)
2 λ 2
This is the Bragg condition.
Using the spatial Fourier transform of the dielectric fluctuation
∫
δǫ(q, t) = d3r exp iq · rδǫ(r, t), (2.54)
V
26
2.2. Light Scattering Basics
Eq. 2.49 can be rewritten as
E0
Es(R, t) = exp i(ksR− ωit) [ns · [ks × ks × (δǫ(q, t) · ni)]] .
4πRǫ0
(2.55)
This can be simplified:[44]
−k2E0
Es(R, t) =
s exp i(ksR− ωit)δǫis(q, t), (2.56)
4πRǫ0
where
δǫis(q, t) ≡ ns·δǫ(q, t)·ni (2.57)
is the component of the dielectric constant fluctuation tensor along the
initial and final polarization direction.
The time-correlated function of Es can then be written as
4 2
〈E∗ 〉 ks |E0|s (R, 0)Es(R, t) = 2 〈δǫis(q, t)〉 exp(−iωit). (2.58)16π2R2ǫ0
Then, the spectral density
∫
1 +∞
I (ω) = dτ 〈E∗E (t)E(t+ τ)〉 exp(−iωτ) (2.59)
2π −∞
of light reaching the detector with ns, ks, and ωs can be computed as
( )
I k4
∫ +∞
0 1
Iis(q, ωs, R) =
s dt 〈δǫis(q, 0)δǫis(q, t)〉 exp(i(ω)t)
16π2R2ǫ20 2π −∞
(2.60)
with I0 ≡ |E |20 and
ω ≡ ωi − ωs. (2.61)
27
2. Basics and Brillouin Light Scattering
The latter means that the spectral density (the intensity measured by
the detector) depends only on the difference between the incident and
the scattered frequency. Furthermore from Eq. 2.60 one can learn that
I ∝ k4is s (i.e., I −4is ∝ λ ) and I −2is ∝ R . The λ−4 dependence means
that electromagnetic radiation with short wavelengths is scattered more
than that with longer wavelength, as, e.g., apparent in the blue color of
the sky. The R−2 dependence is expected, because it just expresses the
attenuation of a spherical wave.
For a given experiment, the coefficient in Eq. 2.60 becomes a constant,
and the scattering intensity is then only affected by the spectral density
of the dielectric constant fluctuations, i.e., the integral.
∫ +∞
Iis(q, ω) ∝ dt 〈δǫ∗is(q, 0)δǫis(q, t)〉 (2.62)
−∞
The integral of Eq. 2.62 over frequency, the integrated intensity at all
frequencies, provides information about the q-dependent mean-square
fluctuations ǫ:
〈 〉
Iis(q) = |δǫis(q)|2 . (2.63)
Harking back on the definition of the scattering wave vector q (Eq.
2.50) and the frequency shift ω (Eq. 2.61), the scattering event can be
considered in terms of energy and momentum conservation. Most gener-
ally, during a scattering process, the scattered photon sustains an energy
change from ~ωi to ~ωs and a momentum change from ~ki to ~ks. This
must be related to the creation or annihilation of an excitation in the
scattering medium. It is:
~ω = ~ωs − ~ωi (2.64)
~k = ~ks − ~ki (2.65)
Note that in these equations the energy and momentum of the excitation
can have both positive and negative sign.
28
2.3. Brillouin Light Scattering
2.3. Brillouin Light Scattering
Brillouin light scattering (BLS) is the inelastic scattering of monochro-
matic laser light by phonons in the GHz frequency range. The required
high resolution is obtained by the use of multipass tandem Fabry-Pérot
interferometers. In BLS spectroscopy of transparent samples, the de-
sired dispersion relations are obtained by recording the phonon frequen-
cies as a function of the scattering wave vector q, which varies with the
scattering angle. However, in samples exhibiting strong multiple light
scattering, as for the dry colloidal crystals (opals), q is ill-defined and
hence q-dependent acoustic-like modes become inaccessible. Though, lo-
calized in space, i.e., q-independent modes can be recorded in the BLS
spectrum as it was demonstrated for sub-micron colloidal silica[45] and
polymer crystals.[46] These q–independent frequencies have been iden-
tified as the resonance modes of the individual colloidal particles, i.e.,
BLS can record numerous thermally excited elastic resonances in one
measurement (chapter 4). These eigenfrequencies are uniquely defined
by the geometrical and elastic characteristics of the particles. Based on
these data, the elastic properties of the materials can be calculated at the
nanoscale. Thus the combination of q–independent BLS spectroscopy of
multiply light scattering (opaque) samples and the dispersion relations
from the q–dependent BLS spectroscopy on transparent samples is a
powerful methodology to investigate the elastic behavior of nanostruc-
tured materials.
This section will introduce the basic principles of BLS following a sim-
ple approach. Further important results obtained by thermodynamical
considerations will be introduced very briefly. Then a description of the
BLS setup with selective attention on the main principles of the tandem
Fabry-Pérot interferometer is given.
2.3.1. BLS Basics
The main principle of BLS is the scattering of photons on sound waves
and the constructive interference of the multiply reflected light beam as
29
2. Basics and Brillouin Light Scattering
sketched in Fig. 2.3. A plane elastic wave creates a periodic change of
density and hence of the local dielectric constant in a medium, symbol-
ized by the black/white layers. The typical velocity of an acoustic wave
is between 103 and 104 ms−1, while the velocity of the probing light c is
≈ 3 · 108 ms−1. Because of this great discrepancy the dielectric inhomo-
geneities can be regarded as a quasi-static (i.e., ‘frozen’) lattice on which
the photons of the probing light are scattered. Thus, it is justified to
treat the medium as a periodic multilayer stack with periodicity Λ, the
wavelength of the phonon, as shown in Fig. 2.3. The probing laser light
is multiply reflected (under the scattering angle θ) on these layers and
the reflected light interferes on the detector, whose distant to the sample
R is much larger than the periodicity of the scattering planes (R ≫ Λ).
The reflected intensity reaches its maximum when the interference is
constructive, i.e.,
θ
2nΛ sin = λ (2.66)
2
with refractive index n and wavelength of light λ.
ki,wi
ks,ws
q
L
Figure 2.3.: BLS scattering process: The probing light beam is multiply reflected
on the planes of a quasi-static periodic multilayer stack of modulated local dielectric
constant with periodicity Λ. Only reflected beams that interfere constructively
account for the scattering intensity.
30
sound wave
2.3. Brillouin Light Scattering
Insertion of Eq. 2.66 in the Bragg condition (Eq. 2.53) leads to
2π 4πn θ
= sin (2.67)
Λ λ 2
and
2π
q = . (2.68)
Λ
That is to say the scattering wave vector q is equivalent to the wave
vector k of the sound wave (k = 2π/Λ), viz. by changing θ, q can be
chosen, too, and at every q the corresponding sound waves are probed
selectively. The identity q = k shows that the momentum conservation
(cf. Eq. 2.64) during the scattering process is realized by a momentum
transfer between the sound wave and the photon. From this point of
view it is clear that the scattered light changes its angular frequency
from ωi to ωs by inelastic interaction, in which a phonon, the acoustic
quantum, can be either created (Stokes process) or annihilated (anti-
Stokes process), depending on the direction of motion of the acoustic
wave. In terms of energy conservation considerations (Eq. 2.65) it is
obvious that the exchanged energy ~ω comes from the angular frequency
of the phonon, ωphonon = |ω| = ωs − ωi, which is related to the phonon
wavelength and the velocity c of the acoustic wave by
2πc
ωphonon = ω = . (2.69)
Λ
Therefore the frequency f = ω2π undergoes a Doppler shift:
cq c 4πn θ
fs = fi ± = fi ± sin , (2.70)
2π 2π λ 2
using the definition of q given in Eq. 2.53. Here, the minus corresponds
to a motion of the acoustic wave away from the detector leading to a fre-
quency decrease (or phonon creation; Stokes), and the plus corresponds
to a propagation direction of the acoustic waves towards the detector
31
2. Basics and Brillouin Light Scattering
leading to an increasing photon frequency (or phonon annihilation; anti-
Stokes). Obviously, it is
ω = ωs − ωi = ±cq. (2.71)
Therefore the BLS spectrum consists of doublets centered at the elastic
frequency with frequencies f = ±cq/2π (frequency shifts).
Eqs. 2.64 and 2.65 can now be rewritten for the Brillouin scattering
process as follows:
~ωs − ~ωi = ±~ω (2.72)
~ks − ~ki = ±~q (2.73)
Note that in these equations momentum and energy (frequency) of the
acoustic wave are positively defined and plus- and minus-signs corre-
spond again to anti-Stokes and Stokes process, respectively.
So far, this easy approach explains satisfactorily the appearance of the
doublet as well as the position of the frequency shifts. However, when
performing BLS experiments a central line always appears that is not
explained by the above considerations. Also the intensity of the signals
can not be predicted. To elucidate these features of a BLS spectra it is
necessary to take some ideas from thermodynamics into account.
The total scattering intensity of a Brillouin spectrum was first derived
by Einstein in 1910.[42, 43] He started with considering ǫ as a function
of density and temperature ǫ = ǫ(̺, T ). For ̺ and T are statistically
independent, one can write the total differential
( ) ( )
∂ǫ ∂ǫ
dǫ = d̺+ dT. (2.74)
∂̺ T ∂T ̺
Then
( )2 ( )
〈 〉
2 ∂ǫ 〈 〉2 ∂ǫ
2
〈 〉
(dǫ) = (d̺) + (dT )2 . (2.75)
∂̺ T ∂T ̺
32
2.3. Brillouin Light Scattering
Using the assumption
( ) ( )
∂ǫ ≫ ∂ǫ (2.76)
∂̺ T ∂T ̺
and
〈 〉
(d̺)2 kBTβT
= (2.77)
̺2 v
for the mean-square fluctuation in density in the volume element v, where
kB is Boltzmann’s constant and βT is the isothermal compressibility,
Eq. 2.75 becomes
( )2 ( )2
〈 〉 ∂ǫ 〈 〉 ∂ǫ kBTβT
(dǫ)2 ≈ (d̺)2 = ̺ . (2.78)
∂̺ T ∂̺ T v
Comparison with Eq. 2.63 shows the main result of these calculations,
I ∝ βT , (2.79)
i.e., the total scattering intensity depends on the isothermal compress-
ibility. Without going into details, the full vectorial expression for I(q)
is given by[44]
( )
∂ǫ 2
Iis(q) = (n
2 2
i · ns) V ̺ kBTβT (2.80)
∂̺ T
with scattering volume V . If V (θ) ≈ const, which may be valid de-
pending on the experimental details, the total scattering intensity is also
independent from the scattering angle.
Although Eq. 2.80 gives the total scattering intensity, experiments
showed that only a part of the intensity belongs to the Brillouin doublets
while the rest makes for a central line. These findings could be explained
by Landau and Placzek regarding ǫ(S, p) as function of entropy S and
33
2. Basics and Brillouin Light Scattering
pressure p.[38] In analogy to Eq. 2.75 one can write
( )2 ( )2
〈 〉 ∂ǫ 〈 〉2 ∂ǫ 〈 〉(dǫ) = (dS)2 + (dp)2 (2.81)
∂S p ∂p S
because fluctuations in S and p are independent, too.
The first term, also called the Rayleigh term, represents local entropy
fluctuations, which do not propagate in normal liquids and are the source
of the unshifted component of the scattered light with intensity IC . The
other term, the Brillouin term, represents the isentropic pressure fluctu-
ations, ergo, sound waves that contribute to the Brillouin doublet. With
〈 〉
2 kBT(dp) = , (2.82)
vβS
where βS is the adiabatic compressibility, and by writing
( ) ( ) ( ) ( )
∂ǫ ∂ǫ ∂̺ ∂ǫ
= = ̺βS (2.83)
∂p S ∂̺ S ∂p S ∂̺ S
the Brillouin term can be transformed into
( ) ( ) ( )
∂ǫ 2 2〈 〉 ∂ǫ kBT ∂ǫ kBTβS
(dp)2 = ̺2β2S = ̺ . (2.84)∂p S ∂̺ S vβS ∂̺ S v
It becomes clear that the intensity of the Brillouin doublet 2IB depends
on the adiabatic compressibility βS .
After further simplifications it can be shown that the ratio between
total scattering intensity (IC + 2IB) and the Intensity of the Brillouin
doublet is simply given by the ratio of isothermal and adiabatic compress-
ibility or of the specific heats at constant pressure or volume:[38, 43, 44]
IC + 2IB βT Cp
= = (2.85)
2IB βS CV
34
2.3. Brillouin Light Scattering
In the form
IC βT − βS Cp − CV
= = = γ − 1 (2.86)
2IB βS CV
it is known as the Landau-Placzek equation.
2.3.2. BLS Instrumentation
It was already pointed out that the relative shift of the photon frequency
in BLS spectroscopy(∼108–1011 Hz) is quite subtle compared to the ini-
tial frequency of the probing light (1014 Hz) or to the resolution obtained
by Raman spectroscopy, where frequency shifts in the order of 1013 Hz
are measured frequently by diffraction grating spectrometers. In order
to achieve such high resolution, Fabry-Pérot interferometers (FP’s) are
used in BLS. In combination with an highly monochromatic laser light
source a single FP or, even better, a design composed of two FP’s with
multiple light pass can give excellent results. Here the FP principle is
elucidated and the construction details of the used tandem FP as well
as the general features of the whole BLS setup are illuminated.
In principle, a single FP is not much more than an etalon as it is used
in laser resonant cavities. The FP consists of two plane mirrors with
reflectivity R mounted accurately parallel to one another. The spacing
between the mirrors is given by d and can be varied by moving one of the
mirrors. When the laser light enters the etalon through the first mirror,
which is usually a plane glass plate with a thin layer of metal on the
inner side, it is reflected numerously between the two mirrors. However,
the light with intensity II is already reflected when it enters the etalon,
so that we have only the intensity I0 = II(1 − R) inside the FP before
the first internal reflection. During the reflections within the FP, the
intensity decreases after x reflections to
I xx = I0R = II(1−R)Rx, (2.87)
because every time a part of the light is also transmitted; a typical value
35
2. Basics and Brillouin Light Scattering
for BLS experiments is R ≈ 0.93. Note that Eq. 2.87 is not fully true as
it does not take into account the dissipation of energy, which is usually
considered by the absorptance A, i.e., R+T +A = 1, with transmittance
T . Anyhow, it implies that with increasing reflectivity the number of
reflections inside the FP increases rapidly, and this is important for the
working principle of the system. When the light is reflected many times
between the mirrors, the reflected beams interfere. Only light that fulfills
the equation
mλ = 2nFPd cos θFP , (2.88)
with m being an integer and nFP being the refractive index inside the
FP, and θFP being the angle between the light in the FP and the normal
to the mirrors, is transmitted losslessly due to constructive interference.
Thus usually the interferometry is performed by moving one of the mir-
rors, there is just air between the mirrors (The other possibility would
be to vary n during a measurement.). With normal incidence (cos 0 = 1)
and n = 1 we can therefore simplify to
2d
λ = . (2.89)
m
That means that only light with wavelength 2d/m is transmitted, and
the transmitted λ can be easily scanned when scanning different d’s by
moving a mirror, e.g., by a piezo transducer. The separation between
two adjacent transmission maxima for a given d is called the free spectral
range (FSRλ = ∆λ). Differentiating Eq. 2.89 leads to
∆λ ∆m
= (2.90)
λ m
with initial wavelength λ. For m ± 1 it follows in the described setup
(normal incidence, nFP = 1)
λ2
FSRλ = , (2.91)
2d
36
2.3. Brillouin Light Scattering
or in the frequency domain
ν
FSR = , (2.92)
2d
with ν being the velocity of light in vacuum.
The linewidth of the transmitted line depends strongly on R. As de-
noted above, the more often the beam is reflected before it transmits the
second beam of the FP (or the first, but this light is uninteresting for the
further analysis) the more often it can interfere with itself. Therefore the
destructive interference will be enforced for wavelengths not satisfying
Eq. 2.89 with increasing reflectivity and the function I(λ) becomes more
narrow.
Indeed, the phase difference δp between each succeeding reflection is
2π
δp = 2d. (2.93)
λ
The transmission function of the etalon T (δ ) is found to be[47]p
(1−R)2 1
T (δp) = = , (2.94)
1 +R2 − 2R cos δ 2 δp 1 + c pF sin 2
where cF is the so-called coefficient of finesse F :
≡ 4RcF − . (2.95)(1 R)2
F is defined as the ratio between FSRλ and the full width at half
maximum (FWHMλ) of a transmission peak in the T (λ)-function, i.e.,
F gives the relative separation between nearest transmission peaks (cf.
Fig. 2.4) and is also the most important parameter for the practical
resolution of a spectrometer. As it is approximately
4
FWHMλ ≈ √ , (2.96)
cF
37
2. Basics and Brillouin Light Scattering
 
low F
FWHM
high F
FSR
wavelength 
Figure 2.4.: High and low finesse, FWHMλ at fixed FSRλ.
the finesse is found to be
√ √
FSRλ
F = ≈ π cF π R= − . (2.97)FWHMλ 2 1 R
However, in real experiments the ’practical finesse’ depends not only on
the reflectivity of the mirrors but also, e.g., on the mirror flatness. In
general, a high F is intended, but by practical limitations it cannot be
made much greater than about 100.[48]
Single-pass FP’s have been used to realized the first measurements of
traveling acoustic waves in solids and liquids. However, it turned out
quite early that the contrast was too low to resolve weaker signals and
also the interference between neighboring orders could create easily very
complicated spectra. Both problems were solved by John Sandercock
when he introduced his multi-pass tandem Fabry-Pérot interferometer
(multi-pass modes are also possible in single FP’s).
38
Intensity
 
2.3. Brillouin Light Scattering
The idea behind is quite simple: Two FP’s with slightly different FSR
are connected in series (tandem). An intelligent use of additional mir-
rors assures that the light passes each FP several times (multi-pass). In
the tandem operation, both FP’s must transmit the same wavelength
simultaneously by an appropriate scanning technique. Due to the dif-
fering FSR’s of both individual FP’s, always one of the FP’s blocks the
neighboring interferences. Let’s assume FP1 to have a free spectral range
FSRλ1 and FP2 FSRλ2. Then only wavelengths are transmitted that
simultaneously satisfy
λtrans = 2pd1 and λtrans = 2qd2, (2.98)
where d1 and d2 are the mirror distances in FP1 and FP2 and p and q
are integers. The effect of the tandem operation is sketched in Fig. 2.5b:
The next order of light transmitted through FP1 (i.e., ∆p = 1) is nearly
completely suppressed by FP2 and vice versa, depending on the reflec-
tivity of the mirrors. In our case the transmittance of wrong wavelengths
is about 10−3. According to that, the total FSR of the tandem interfer-
ometer is the distance between the common multiples of ∆λ1 and ∆λ2.
Therefore, with tandem FPI’s much higher finesses can be achieved.
Although the principle idea is not complicated at all, the technical re-
alization was problematic. Both FP’s have to be scanned synchronously,
and from Eqs. 2.89 and 2.98 it is clear that the ratio between the mirror
distances in both FP’s must be constant. With other words
∆d1 d1
= . (2.99)
∆d2 d2
Indeed, this condition was hard to satisfy as the total scanning distance is
usually below 1 µm and the scan is repeated many times during one mea-
surement, so that even very small asymmetries in the nm-range would
disable the experiment.
This problem was solved in 1987 by a new, elegant design, presented
in Fig. 2.5a.[29] The two FP’s used in J. Sandercock’s ‘parallelogram
geometry’ consist of one fixed mirror and one movable mirror, each.
39
2. Basics and Brillouin Light Scattering
The two movable mirrors are placed on a shared panel, the translation
stage, which is the part of the setup moved during the scan. Thus, both
FP’s change the mirror distance always at the same time. The FP’s are
mounted that the two mirrors of each would touch at the same time if
the translation stage would be moved to the very left. In order to satisfy
Eq. 2.99 the relative orientation of the two FP’s is chosen that way that
the angle ζ between FP1 and FP2 is fixed and the mirror distances have
then to be adjusted in a way that
d2 = d1 cos ζ. (2.100)
A movement of the translation stage to the right shifts the spacings
between the mirrors simultaneously by ∆d1 and ∆d1 cos ζ, i.e., the ratio
keeps constant.
In my experiments, a six-pass tandem Fabry-Pérot interferometer was
used. The path of the light inside the FP as well as the other details of
the applied BLS setup are sketched in Fig. 2.6. The sample is mounted
in the center of a goniometer (Huber) in a custom-made sample holder
with or without oven. A solid state pumped frequency-doubled Nd:YAG
laser (Coherence; 150 mW @ 532 nm) is fixed on the goniometer and
can be rotated so that scattering angles between 0◦ and ∼160◦ can be
chosen either in transmission or reflection geometry. In difference to
pure BLS backscattering techniques, i.e., θ = 180◦, this technique has
the advantage that not only the components of q but the whole wave
vector is changed according to Eq. 2.53.
Before the light reaches the sample, it passes a Glan polarizer (ex-
tinction ratio 10−5) with vertical polarization (V), i.e., perpendicular
to the scattering plane, to ensure fully polarized incident light. Behind
the sample the light scattered in the direction of the detector is col-
lected by an aperture and focused into the entrance pinhole of the tan-
dem Fabry-Pérot interferometer (JRS Scientific Instruments) by some
lenses. Before entering the FP, a Glan-Thompson analyzer (extinction
ratio 10−8) is passed that selects either vertically (V), i.e., perpendicular
to the scattering plane, or horizontally, i.e., parallel to the scattering
40
2.3. Brillouin Light Scattering
a d2=d1cosz b FP1
Dl1
FP2
FP2
Dl2
FP1
z tandem FP
translation
stage
d wavelength1
Figure 2.5.: a Basic mechanism of a Sandercock multipass tandem Fabry-Pérot
interferometer: The light passes FP1 and then FP2. Two mirrors (one of FP1 and
one of FP2) are moved simultaneously on a piezoelectric translation stage. The
angle ζ of the light path between the two FP’s is chosen in that way that the ratio
between d1 and d2 is constant. b Effect of the multi-pass through two FP’s: The
neighboring orders (viewed from the harmonized central signals) that can pass each
single FP are suppressed. Only the common multiples of ∆λ1 and ∆λ2 will be
transmitted gainlessly.
plane, polarized light (or, of course, everything in between). After pass-
ing the FP the transmitted light is detected by an avalanche photo diode
(APD) and processed by an multi-channel analyzer with 1024 channels.
The further processing is performed by a computer software.
The stability of the alignment is greatly enhanced by the use of a ref-
erence beam. Therefore, a small amount of the laser light is diverted via
the reflection (5%) of a parallel plate on a beam splitter and an optical
fiber from the incident laser beam and introduced as a reference beam
that gives the central line via an optical fiber. Therefore, a mechanical
shutter is used, which switches periodically the entrance to the FP be-
tween reference beam and scattered light and excludes the central elastic
line. In order to avoid mechanical disruptions the whole setup is placed
41
transmission
2. Basics and Brillouin Light Scattering
SAMPLE
TANDEM FABRY-PEROT
GONIOMETER
Figure 2.6.: BLS setup with six–pass tandem Fabry–Perot interferometer. Elec-
tronic stabilization over days to weeks is achieved by permanent compensation using
a diverted part of the unscattered light as reference beam. The goniometer allows
to record dispersion relations with continuous q–range. For the eigenmode spectra
the scattering angle is not relevant.
on an optical table with active vibration damping.
In some experiments dealing with temperature dependent effects, e.g.,
glass transition experiments (section 4.3.1) or measurements on kineti-
cally stable organic glasses (chapter 6), an oven is used in order to con-
trol the temperature of the sample in the range between ca. 10 ◦C and
200 ◦C. The oven is a metal cylinder with filament and coolant tubes in
the wall.[49] A small slit at the hight of the laser and a cylindrical quartz
glass insert in order to avoid heat transfer through the slit allow the
light to reach the sample and the detector afterwards. The temperature
is controlled electronically (built in-house) by two Pt-100 temperature
sensors inside the wall and inside the heat chamber, close to the sam-
42
POLARIZER
APERTURE
ANALZYERPDA
R
AS
E
L
2.3. Brillouin Light Scattering
ple. The temperature can be stabilized within better than ±0.2 K in the
given range.
Most experiments in this thesis deal with samples on a plane substrate,
i.e., ‘films’. When performing the BLS experiment on such samples, there
are in principle two different scattering geometries, the transmission and
the reflection geometry. In transmission geometry the light scattered
on the other side of the film than that of the incident laser beam is
investigated, while in reflection geometry the light scattered on the side
of the incident beam is regarded.
The two geometries are sketched in Fig. 2.7 and a full geometrical
derivation is given in the appendix.
In the transmission case, it is found that the magnitude of the scat-
tering wave vector becomes
[ ( ( ) ( ))]
4πn 1 1 1
q = sin sin−1 sin (θ − α) + sin−1 sinα (2.101)
λ 2 n n
with the angles α and θ given in Fig. 2.7a. It is shown in the appendix
that the length of the component of the scattering wave vector parallel
to the film, qpara, is
2π
qpara = (sinα+ sin (θ − α)) . (2.102)
λ
Obviously, in this equation the refractive index n of the sample is elimi-
nated.
A special transmission geometry in which q = qpara exists for θ = 2α.
In this case q is given simply by
4π θ
q = qpara = sin (for θ = 2α). (2.103)
λ 2
To use this special geometry is advantageous in several ways, indeed
nearly all the experiments in this thesis are performed using it. First
of all, it facilitates the calculation of q, as the refractive index of the
sample has not to be taken into account. Although n of most ’stan-
43
2. Basics and Brillouin Light Scattering
a
reflection
incident
n0=1 a laser
n ki ks qperp q
q qpara
n0=1 q
transmission SCATTERING
b
SCATTERING
incident
reflection laser
n0=1 a qperp
q
ks
n q q
ki
qpara
n0=1
transmission
Figure 2.7.: Principle BLS scattering geometries for a film sample (view atop the
scattering plane). a Transmission geometry b Reflection geometry. On the right
side the vector decomposition for the wave vector q in its components perpendicular
(qperp) and parallel to the film plane (qpara).
dard materials’ (e.g., typical polymers like polystyrene) is well-known,
the exact determination of n for unknown materials or, even more, for
composite materials, as they are widely used in this thesis, may become
complicated and time consuming. The second great benefit of this ge-
ometry is that in this case the direction of q is well defined parallel to
the film. When discussing colloidal crystals[8] it is important to know
44
2.3. Brillouin Light Scattering
which crystallographic direction is probed by the BLS experiment.
In the reflection geometry the general expression for q as a function of
n, θ, and α becomes
[ ( ( ) )]
4πn 1
q = cos sin−1
1 1
sinα + sin−1 ( sin (θ + α)) . (2.104)
λ 2 n n
Also for the reflection case there is a special scattering geometry. If
α = 180
◦−θ
2 the scattering wave vector is identical to its component per-
pendicular to the film, i.e., q = qperp in Fig. 2.7b.
In a cylindrical sample, e.g., a liquid in a NMR-tube, q has simply the
magnitude
4πn θ
qcylinder = sin . (2.105)
λ 2
2.3.3. Vibrational Modes of Individual Particles
In the foregone section it was pointed out that the result of a BLS experi-
ment is usually the plot of longitudinal or transverse (or mixed, especially
in thin films, e.g., the so-called Lamb waves[50]) sound waves as a func-
tion of the scattering wave vector q, while the frequencies f(q) are given
as a Doppler shift of the probing laser light around its elastic line. The
phase sound velocities are given by the slope of the f(q)-diagram. Thus,
in the case of a non linear function f(q), i.e., if the acoustic mode is
dispersive, the result of the BLS experiment is the dispersion relation in
the investigated q-range.
However, the appearance of a dispersion relation is only meaningful
if it is possible to determine q. If strong multiple scattering occurs in a
sample, q becomes ill-defined. In such samples f(q) is no longer accessi-
ble. Anyhow, now the inelastic scattering from localized modes, i.e., the
vibrational resonance modes, can lead to incoherent BLS in analogy to
the Raman scattering.
Indeed, in BLS experiments such samples show a spectrum where scat-
tering at all possible q’s between zero degree incident and backscattering
45
2. Basics and Brillouin Light Scattering
case contributes to the final result, independent from the experimental
scattering angle θ. Furthermore, also the polarization information is lost.
Nevertheless, the Brillouin spectrum of multiple scattering samples
with well defined shape still gives useful information, which are not ac-
cessible by other techniques. In chapter 4 of this thesis, several experi-
ments are discussed dealing with the detection of vibrational resonance
modes of spherical colloidal samples. BLS spectra of such samples de-
liver many useful information about the individual colloids and, in case
of hybrid materials, about the mechanical properties of the components,
too. This section will give a short introduction into the mathematical
description of the vibrations in spheres, their theoretical calculation, and
their detection by BLS.
Under stress-free boundary conditions, the vibrational modes are usu-
ally called eigenmodes. The eigenmodes for free homogeneous elastic
spheres have been derived by Lamb in the 19st century.[51] The modes
can be classified as torsional and spheroidal ones, both labeled by the
indices n, l, and m, which describe the radial (n) and the angular (l,m)
dependence of the displacement (cf. section 2.1.3). The torsional modes
are fully tangential, i.e., they involve only shear motions and do not cause
changes in the sphere volume - they don’t contribute to the BLS inten-
sity. The spheroidal modes involve usually both shear and stretching
motions, and they can be fully specified by two indices in analogy to the
atomic orbitals as the n-th order radial solution (n=1,2,3,...) for angular
momentum ‘quantum number’ l (l=0,1,2,...).[32, 41, 52] Only spheroidal
modes with l = 0 have purely radial displacement (breathing modes).[27]
By solving the wave equation for elastic and isotropic media (Eq. 2.37),
Lamb found the frequencies of the eigenmodes (n, l) to be (in a very
general expression):
√
( )
R(cl, ct)
f(n, l) = x(n, l) /d, (2.106)
̺
with x(n, l) being a constant for each individual mode, R(cl, ct) being
46
2.3. Brillouin Light Scattering
the rigidity, and d being the diameter of the sphere. The rigidity is a
function of the longitudinal and transverse sound velocities and has the
dimension of pressure, thus it is simply a modulus. Indeed, e.g., for the
(1,2)-mode, R is identical with the shear modulus µ (=C44).
The experiments discussed in chapter 4 are all related to the vibra-
tional modes of colloidal spheres in air. Because of the large density (and
elastic) mismatch between air and the colloids, the boundary-conditions
are quasi that of a stress-free, undamped vibration, thus these modes are
practically real eigenmodes and the values calculated by Lamb’s theory
are a good approximation. However, in the more general case of a elas-
tic vibrator embedded in an elastic matrix, some coupling should occur
between the eigenmodes and the propagating acoustic waves in the ma-
trix. The amount of coupling depends on the elastic mismatch between
spheres and matrix. While the limit case of large mismatch corresponds
to the free eigenmodes, where the elastic energy is totally localized in
the spheres, the other limit case is a zero-mismatch, which means that
all energy is coupled between sphere and matrix and an elastic wave can
travel the system like a homogeneous medium.
To calculate the distinct vibrational modes of elastic spheres with ra-
dius rs and density ̺s in an elastic matrix with density ̺m, one has to
introduce boundary conditions. At the surface of the sphere, the dis-
placement u has to be the same inside and outside the sphere:[41, 53]
ui| = umr=rs |r=rs (2.107)
The same must be fulfilled for the surface traction τ (τ = σ(r) · r̂, with
stress tensor σ and outgoing unit vector normal to the sphere surface r̂):
τ i| = τmr=rs |r=rs (2.108)
The field can be written as a linear sum of spherical waves as discussed
in section 2.1.3. The field inside the spheres becomes
∑
ui(r) = aL i M i N ilmllm(R, kl ) + almmlm(R, kt) + almnlm(R, kt) (2.109)
lm
47
2. Basics and Brillouin Light Scattering
with the coefficients aL , aMlm lm and a
N
lm. In the same way we can write for
the field in the matrix
∑
um(r) = bL m M m N mlmllm(R, kl) + blmmlm(R, kt ) + blmnlm(R, kt ) (2.110)
lm
with the coefficients bL M Nlm, blm and blm. With Eqs. 2.43–2.45 one can write
∑ 1
u(r) = xLlm ∇ [R Ml(klr)Ylm(r̂)] + xlm∇× [rRl(ktr)Ylm(r̂)] +kl
lm
xN
1
lm ∇×∇× [rRl(ktr)Ylm(r̂)] ,kt
(2.111)
where the x replace either the coefficients a inside or b outside the spheres
and kl and kt are the longitudinal wavenumbers (k = ω/c) inside (k
i) or
outside (km) the sphere, respectively.
With Eq. 2.111 it becomes clear that Eqs. 2.107 and 2.108 are equiv-
alent to three scalar equations, each, and the boundary conditions can
be expressed as a system of six homogeneous equations with an infinite
numbers of unknowns. Due to the orthonormality over the spherical sur-
face of the Legendre functions Ylm, it is possible to decompose each of
these equations into l equations, but independent from m. I.e., also the
coefficients a and b depend only on l, leading to a system of six equations
with six unknowns for each l. It can be broken into two smaller systems,
considering the orthogonality of mlm to nlm and llm, which contain
two equations involving the coefficients aMl and b
M
l and four equations
containing the coefficients aL Ll ,bl ,a
N
l ,and b
N
l , respectively. Non-trivial so-
lutions for both systems are only achieved if their determinants are zero,
leading to a set of discrete modes with angular frequencies ωnl for each l
and n-th order, corresponding to the expression given in Eq. 2.106.
The considerations presented in this section are the fundament of the-
oretical methods dealing with the scattering of plane waves on single and
multiple spheres presented in section 3.4.
48
3. Methods
3.1. Vertical Lifting Deposition and Colloidal
Crystals
The assembly of colloids into well defined coherent structures commonly
occurs under the influence of external fields, e.g., gravitational sedimen-
tation,[13] electrophoretic deposition,[14] or vertical deposition either by
evaporation[15] or lifting the substrate[16] (Fig. 3.1).
In the vertical lifting deposition method upon immersion of a hy-
drophilic substrate into a colloidal dispersion a meniscus at the sub-
strate is formed. Evaporation takes place at the three phase contact
lines (air, dispersion, and substrate), which causes a solvent flux to-
wards the meniscus, as shown in Fig. 3.2. Thus, colloidal particles are
constantly transported with the liquid to the crystallization front. An
interplay of long-range attractive and short-range repulsive forces causes
the self-assembly of the colloidal material into a face-centered cubic (fcc)
or hexagonally close packed (hcp) crystal .[54, 55] In order to control the
thickness of such a colloidal crystal – a critical parameter for further
use in ensuing applications – the substrate is withdrawn from the dis-
persion at a certain speed at given environmental parameters such as
temperature and humidity. Besides the fabrication of such ‘simple’ col-
loidal crystals also more complex systems like binary[56, 57] and ternary
colloidal crystals[58] were demonstrated. The colloidal crystals that are
topic of this article are produced using an enhanced vertical lifting ap-
paratus by M. Retsch.[59]
Alongside with this research, also the counterpart to the highly ordered
crystals - colloidal glasses [61, 62] - comprising of two distinct, monodis-
49
3. Methods
a c e
b d f
Figure 3.1.: Self assembly of colloidal crystals by a sedimentation, b electrode-
position, c centrifugation, d vertical deposition (cf. Fig. 3.2), e filtering, or f
compression molding.[60]
perse latex particles have already been developed.[33]. Doping of the
colloidal crystal with removable moieties (sacrificial templates) opens a
pathway to the designed introduction of defects, which are highly in-
teresting with respect to their contribution to phononic properties. At
the same time, colloidal crystals and glasses serve as templates for the
fabrication of so-called inverse opals.[63]. These materials exhibit an in-
terconnected 3D network with high surface area, since the constituent
spheres of the colloidal crystals have been removed. Inverse opals have
attracted strong interest in photonic crystal research, due to their full
photonic band gap,[64, 65] and are also promising but unexplored materi-
als for phononic experiments.
Complementary to the vertical deposition method a technique for the
fabrication of large area colloidal monolayers has been recently estab-
lished. This new method itself gives access to highly interesting new
materials. Multiple stacks of colloids of various diameters as well as de-
50
3.2. Melt Compression
climate chamber:
temperature, humidity
lifting direction
evapo-
ration
colloidal
dispersion
Figure 3.2.: Schematic draw of the colloidal crystallization process in a vertical
lifting deposition apparatus. The black arrow symbolizes the moving direction of
the substrate by an adjustable stepper motor with small step size (not drawn). The
surrounding climate chamber allows full control over air temperature and humidity,
which are important parameter in the crystallization process. (according to [60])
position on uneven or curved substrates have been demonstrated. Con-
trolled etching of such a colloidal monolayer allows for the first time
the fabrication of large areas of non-close-packed structures.[66] Finally,
colloidal monolayers are well known and widely used for surface pattern-
ing. Vertical lifting deposition, colloidal monolayer fabrication and the
combination of both are powerful tools in order to rationally design new
periodic functional materials with generic possibilities for defect tuning,
creation of multilayers, colloidal glasses, and inverse opals.
3.2. Melt Compression
Vertical lifting deposition is a method that delivers high quality colloidal
crystals several tens of microns thick and with cm2 areas. Although,
51
3. Methods
Figure 3.3.: a The flowing melt crystallizes along the plates of the press under
uniaxial compression.[17] b Melt compressed polymer opal film with PS core and
PEA shell.[67]
in principle it would be possible to upscale the method, it would be
unpractical and slow to produce much larger and thicker samples by
that. A technique that is much better suited if fast and large scale hybrid
material films of thickness up to the mm-range is needed is the melt
compression (or compression molding) technique that is schematically
shown in Fig. 3.1f and in more detail in Fig. 3.3a.
Melt compression can be utilized to prepare films out of hard core/soft
shell core-shell particles, where the soft shell becomes the matrix and the
hard cores form a crystalline (usually fcc) lattice, whose spacing depends
on the initial volume ratio of core and shell materials. The preparation
starts with the coagulation of the latex dispersion in a rubbery mass.
When the soft shell is an elastomer with low Tg, e.g., poly(ethyl acry-
late) (PEA), the shells form a continuous matrix in which the cores, e.g.,
harder polymers like polystyrene or oxides like silica, are dispersed. At
DKI the mass is shaped into a cylindrical sample, which is uniaxially
compressed at 170 ◦C in a Collin 300 press, with an initial pressure of
1 bar and a final pressure of 50 bar.[17] In the beginning of the pressing
process, the first crystalline layers appear on the plates of the press. As
52
3.3. Polymer and Colloid Characterization Techniques
shown in Fig. 3.3a, the pressure creates a horizontal flow perpendicular
to the direction of compression. The order increases with pressure by
the sequent formation of new crystalline layers above the already exist-
ing layers in order to achieve the densest packing. Finally, the whole
film should be crystalline. It could be shown only recently that in such
films the direction corresponding to the (111) plane of the fcc lattice
is radial from the center of the round press cylinder, i.e., multi-domain
ordering must be assumed, however, the crystallographic directions are
well defined.[68]
Fig. 3.3 shows a typical film prepared consisting of PS cores and PEA
matrix. It is obvious that in this case the particle size is chosen that way
that the film acts as a photonic crystal, since against the dark background
the film reflects the green light under Bragg conditions. Where it is
bended, the angle to the light changes and it is reflected on another
crystallographic plane, hence the color changes.
3.3. Polymer and Colloid Characterization
Techniques
The aim of this section is to introduce very briefly the different auxil-
iary techniques applied in this thesis to characterize polymer and colloid
samples, including spectroscopy, electron microscopy, calorimetry, den-
sity determination, and size exclusion methods.
3.3.1. Photon Correlation Spectroscopy
The photon correlation spectroscopy (PCS), also known as dynamic light
scattering, is a light scattering technique that can be used to obtain the
size distribution of small particles (d < λ) in diluted solution. When
laser light hits small particles, Rayleigh scattering occurs in all direc-
tions. Because the small particles undergo Brownian motion in the liq-
uid, the relative position of the scatterers within the scattering volume is
constantly changing. This results in a time-dependent scattering inten-
53
3. Methods
sity fluctuation, because the monochromatic and coherent laser light is
scattered on the moving scatterers. The light scattered from the individ-
ual scatterers interferes with the light scattered on all the other particles.
By using a correlator at a given scattering angle θ, i.e., at a given wave
vector q, one can record the time-correlation function
∫ ∞
G(q, τ) = 〈 1I(q, t)I(q, t+ τ)〉 = lim I(q, t)I(q, t+ τ)dt (3.1)
T→∞ T 0
with I(q, t) being the scattering Intensity at time t and wave vector
of magnitude q. Thus I(q, t) is randomly fluctuating with time due to
particle motion, G(q, τ) is decreasing exponentially from its initial value
〈 〉
G(q, 0) to zero for τ → ∞ - or, to be exact, from I2 to 〈I〉2. The so-
called normalized second order autocorrelation curve is generated from
the intensity trace as follows:
〈I(t)I(t + τ)〉
g2(q, τ) = 〈I(t)〉 (3.2)2
In the simplest case of monodisperse spheres, the autocorrelation func-
tion decays as a single exponential decay, which is directly related to
the diffusion coefficient D. When the viscosity η of the liquid is known,
D gives the radius of the spherical particles r by the Stokes-Einstein
relation
kBT
D = . (3.3)
6πηr
All PCS experiments in this thesis were performed by myself on a
setup built in-house at MPIP.
3.3.2. Electron Microscopy
The common characteristics of all types of electron microscopes is that
electron (particle) beams are used to illuminate a sample and to cre-
ate a magnified image. Fast electrons are generated by electron guns,
54
3.3. Polymer and Colloid Characterization Techniques
consisting of an electron source (cathode), an accelerating anode, and
electron lenses to collimate and focus the electron beam. The de Broglie
wavelength of a particle with particle momentum p is
h h
λ = = (3.4)
p γmev
√
where h is Planck’s constant and γ = 1− v2/ν2 the Lorentz factor for
very high velocities v. For an electron with mass me, the wavelength of
the electron particle beam depends on the velocity of the electrons, i.e.,
on the acceleration voltage (usually in the order of several kV). Usual
electron wavelengths are ≪1 nm, which is much smaller than the wave-
length of the light used in optical microscopes (200-1000 nm). Therefore
the theoretical resolution limit is extremely low for electron microscopes,
but in practical work resolution is limited by aberrations of the technical
components; however, still orders of magnitudes better than in an nor-
mal optical microscope. By the use of electron lenses (either magnetic
or electrostatic) with electronically adjustable foci, no movable objec-
tive system is needed in an electron microscope. Principally, there are
two main types of electron microscopes, scanning electron microscopes
(SEM) and transmission electron microscopes (TEM).
In the SEM a thin electron beam rasters the probed surface. In the
same time electrons that are emitted or scattered from the probed ob-
ject are detected and the registered intensity is correlated to the intensity
value of the scanned pixel. Typically, in a SEM setup secondary elec-
trons, back scattered electrons or, less often, cathode luminescence sig-
nals are detected. The most important strength of a SEM is the ability
in displaying surfaces in high resolution with great depth of view.
On the other hand, the transmission electron microscope probes not
only the surface but the whole gauge of a thin (some nm up to several
µm) sample using a transmission geometry. Here, the beam is usually
not scanning. A broader beam is illuminating a part of the sample and
the transmitted electron intensity is imaged onto a fluorescent viewing
screen and recorded by a CCD camera for further computing.
55
3. Methods
Electron microscope pictures shown in this thesis are recorded by
Markus Retsch, Gabrielle Schaefer, Gunnar Glasser, or Maren Mueller
at different setups at MPIP.
3.3.3. Differential Scanning Calorimetry
The differential scanning calorimetry (DSC) is a variation of caloric ther-
moanalysis techniques, as there are also differential thermal analysis
(DTA) or thermo gravimetry.[69] DSC is an isotherm technique. The
sample and a reference substance are heated simultaneously in a way
that both species have always the same temperature, i.e., ∆T = 0 (heat
compensation). Usually, the sample is heated (and cooled down) with a
well defined temperature rate (mostly 1-20 K/min). Therefore, a differ-
ent amount of heat ∆Q has to be added to the sample in comparison to
the reference substance, which shows in ideal case a linear dQ/dT . The
measurement records now the change of the sample with time d(∆Q)/dt
or with Temperature d(∆Q)/dT , respectively. In the resulting thermo-
gram exothermal phase transitions (e.g., crystallization) result in a peak
with relative maximum, while endothermal transitions (e.g., melting) re-
sult in a peak with a relative minimum in the d(∆Q)/dt or d(∆Q)/dT
curve. The Integrals over the transition peaks give the transition heat
exchanged in the respective transition.
DSC experiments in this thesis are performed by P. Raeder and M.
Droege at MPIP and in the group of Prof. M. Ediger at University of
Wisconsin, Madison, USA.
3.3.4. Density Gradient Column
In some cases the exact density of a sample is needed. While for poly-
mers or colloids the direct measurement of volume and weight might be
sophisticated, the use of a density gradient column is a popular method
to measure a sample’s mass density.[69] The column is a long vertically
standing glass, which is filled with a liquid whose density is increasing
gradually from top to bottom. This can, e.g., be realized by using a spe-
56
3.3. Polymer and Colloid Characterization Techniques
cial double-flask setup in which two aqueous solutions of a salt with two
different concentrations (and therefore different densities) are mixed to
a solution with continuously changing concentration and density, which
is carefully poured in the column.[70, 71] The density gradient is then
calibrated using several floaters of well known density. Drawing the cali-
bration curve of floater density against floating hight in the column, the
density of other samples can be determined by flotation.
The density gradient column technique is often used to measure the
crystallinity of a polymer. The degree of crystallization wc is defined as
̺c · ̺− ̺awc = (3.5)
̺ ̺c − ̺a
with ̺, ̺c, and ̺a being the density of the probed sample, the purely
crystalline polymer, and the purely amorphous polymer, respectively.
̺c is accessible from the crystal structure, ̺ and ̺a can be obtained
by flotation in the density gradient column. To obtain a completely
amorphous sample, the polymer has to be molten and quenched to low
temperatures with effectual celerity.
3.3.5. Wide Angle X-ray Scattering
Another method to determine the degree of crystallinity is the wide angle
X-ray scattering technique. X-ray structure analysis is in general the
most common method to measure crystal structures. When we took the
atoms in the crystal as points in a real space lattice and transform the
lattice into reciprocal space, the distance between two atoms in real space
is now the distance d between lattice planes. When a monodisperse,
coherent X-ray beam is scattered on parallel lattice planes, the scattered
beams interfere constructively when
2d sinφ = nλ (n = 1, 2, 3, . . .). (3.6)
φ is the glancing angle between the incident beam and the scattering
plane and λ the wave length of the X-ray beam. Eq. 3.6 is the Bragg
57
3. Methods
condition for X-Ray scattering. When scanning a range of scattering
angles at some distinct angles, where the Bragg condition is fulfilled,
the constructive interferences occur as sharp crystalline peaks in the
scattering spectrum. In partially crystalline samples the X-ray spectrum
at wide angles will consist of some distinct sharp peaks but also of an
underlaying broad halo, which originates from the amorphous parts of
the sample. Integrating these two areas in the spectrum gives access to
the degree of crystallization, which is
Ac
wc = (3.7)
Ac +Aa
With Ac and Aa representing the areas under the curve of the crystalline
signals and the amorphous halo. The discrimination of both parts is
highly subjective, which makes the results less accurate.[69]
3.3.6. UV/VIS Spectroscopy
Ultraviolet-visible spectroscopy (UV/VIS) is the spectroscopy of photons
in the UV and visible region, usually the probing light has wavelengths
between about 200 nm and 1000 nm. In this region of the electromagnetic
spectrum, molecules undergo electronic transitions. The use of UV/VIS
spectroscopy in polymer science lies mostly in the measurement of the
concentration of a polymer in solution. If the solution is diluted, the
Lambert-Beer law is valid:
A = log(I0/I) = ǫcL (3.8)
A is the measured absorbance, I0 is the intensity of the incident and
I the intensity of the transmitted light. ǫ is the extinction coefficient,
a material constant, L is the pathlength through the sample and c the
concentration.
Most UV/VIS spectrometers work with two cuvettes, one with the
sample, the other one as reference containing only the pure solvent, which
are compared for every wavelength. All spectrometers consist of a light
58
3.3. Polymer and Colloid Characterization Techniques
source, a dispersive element (lattice, prism, monochromator filters), the
sample, and a detector. Mostly avalanche photo diodes are used. To
obtain the spectrum the wavelength range of interest is scanned and
absorbance (or transmittance) is plotted against wavelength.
For structures with periodic pattern of the dielectric constant, e.g.,
colloidal crystals, light is diffracted on these pattern with subsequent
constructive and destructive interference. I.e., electromagnetic waves
of distinct wavelengths cannot pass such structures in distinct crystal-
lographic directions, dependent from the structure’s periodicity length.
This effect is referred to as photonic Bragg gap. The corresponding
wavelength can be calculated as
λBragg = 2sneff sinΘ, (3.9)
where s is the periodicity in the regarded crystallographic direction, neff
is the effective refractive index, and Θ is the angle between the incident
light and the scattering plane of the crystal. For a fcc colloidal crystal
with sphere diameter d, Eq. 3.9 becomes for the (111) plane of the crystal
λBragg = 1.63dneff sinΘ, (3.10)
with
√
neff = φsn2s + φmn
2
m, (3.11)
where φs and ns, or φm and nm are the volume fraction and the refractive
index of the spheres or the surrounding matrix, respectively.[72]
In this thesis UV/VIS is used to characterize colloidal mixtures. The
measurements have been performed by M. Retsch on the setup at MPIP.
3.3.7. Gel Permeation Chromatography
The gel permeation chromatography (GPC) is a size exclusion technique
to determine the distribution of the molar mass in polymers. The princi-
ple idea is that a diluted polymer solution travels a column containing a
59
3. Methods
macroporous gel. If the single molecules are large they cannot penetrate
the holes in the gel and therefore are not retarded, i.e., all molecules that
are larger as a certain value, dependent from the nature of the gel, eluate
together with the eluent. Smaller particles are retarded by entering the
holes. With decreasing particle size the retardation time increases up to
a certain point from which the eluation time is constant for all polymer
chains, i.e., only polymers within a certain range of dimension are sep-
arated. Behind the column the amount of polymer in the eluted liquid
as a function of time is recorded using different techniques as for exam-
ple refractometry or UV/VIS spectroscopy. The intensity of the signal
is proportional to the amount of eluted polymer. The easiest and most
usual way to obtain quantitative data from the elution time versus inten-
sity diagram is the comparison with monodisperse polymer standards, if
available. In principle it is also possible to use an universal calibration
for all polymers, because the size exclusion depends not really on the
molecular weight but on the hydrodynamic radius Rh of the polymer
molecule in solution. Rh correlates with the molecular weight M :
Rh = Φ[η]M (3.12)
with Φ being a constant and [η] being the intrinsic viscosity given by the
Mark-Houwink equation
( )
M αη
[η] = Kη , (3.13)
g/mol
where Kη and αη are parameters depending on the polymer, its shape,
and the solvent. The details of the sophisticated universal calibration is
given in standard textbooks of macromolecular chemistry and physics.[69]
3.4. Theoretical Calculations
The vibrations of individual spheres in different matrices as well as the
phononic band diagram in mesostructured media is widely discussed in
60
3.4. Theoretical Calculations
terms of experimental results mainly by BLS in this thesis. However, in
order to come to valueble conclusions, the most powerful approach is to
compare the experimental data with data calculated within the appro-
priate theories leading to either scattering cross sections when talking
about the vibrations of individual particles or phononic band structures
when talking about the phononic dispersion relation of the investigated
systems. Taking well-known parameters for the calculations allows cer-
tainty about the nature of the experimentally observed effects, while on
the other hand the use of theoretical important parameters as free fit
parameters can be utilized to extract (e.g., nanomechanical) information
from the measured spectra.
In this section I want to briefly describe the calculation methods used
in this thesis. Starting from the considerations in section 2.3.3, I will
introduce the scattering cross section obtained from the scattering of a
plane wave on a single spherical scatterer. From that I will describe the
related plane-wave method to calculate the band structure of periodic
composites. Anyhow, the more powerful method to do the latter and
more important for this thesis is the (layer) multiple-scattering approach,
which will be presented concludingly. Finally, the finite element approach
is discussed very briefly.
3.4.1. Single-Sphere Scattering-Cross-Section Calculations
To calculate the scattering-cross section of a single sphere,[53] one can
calculate the scattering of an incident plane longitudinal wave of the
form
( )
∑
uinc(r) = cL
1
lm ∇[Rl(klr)Ylm(r̂)] (3.14)ql
lm
or of a transverse wave of the form
( )
∑
uinc(r) = cM
i
lmRl(ktr)Ylm(r̂) + c
N
lm ∇×Rl(ktr)Ylm(r̂) (3.15)qt
lm
61
3. Methods
in analogy to Eq. 2.111 after decomposition into longitudinal and trans-
verse components. This plane wave is propagating in a homogeneous
medium. When the wave is incident on a solid homogeneous sphere, it
is scattered by it, and the wave field in the matrix um now consists of
the incident and the scattered field, uinc and usc. I.e.,
um = uinc + usc. (3.16)
The boundary conditions at the surface of the sphere, given above in
Eqs. 2.107 and 2.108, must still be fulfilled and can be written now as
ui| inc scr=rs = u |r=rs + u |r=rs (3.17)
and
τ i| = τ incr=rs | scr=rs + τ |r=rs , (3.18)
with
∑
usc(r) = dL ilmllm(R, kl) + d
M
lmmlm(R, k
i
t) + d
N i
lmnlm(R, kt) (3.19)
lm
It is shown that the coefficients for the incident wave cxlm (with x =
L,M,N) are known a priori.[53] As discussed in section 2.3.3, it is possible
to decompose this system into six scalar equations that can be separated
into an infinite number of equations that are functions of lm. When the
coefficients cxlm are known, there remain only the coefficients a
x
lm, given
in Eq. 2.109, and dxlm, given in Eq. 3.19 for the field inside the spheres
or the scattered field, respectively. Therefore exists for each lm a system
of six equations with six unknowns that can be solved. The detailed
algebraic analysis is done for example in Ref. [53].
The scattering cross-section σ is a measure for the likelihood of the
physical interaction between an incident particle and another particle -
in this case, the scattering of a wave by the single scatterer. It is defined
as the ratio of the scattered energy flux to the incident energy flux per
62
3.4. Theoretical Calculations
unit surface.[41]
For the discussed case of an incident longitudinal wave (subscript l)
the dimensionless scattering cross-section for a sphere with radius r is
shown to become:[73]
∞ [ | L 2 ( m )σ d N 2 ]∑l
= 4(2l + 1) lm
| k r |d |
| + l(l + 1)
l lm . (3.20)
πr2 kml r|2 kmt r |kmr|2l=0 t
Fig. 3.4 shows an example of a total dimensionless scattering cross-
section, calculated for a polystyrene (PS) sphere in a poly(dimethyl silox-
ane) (PDMS) matrix. The normalization over the frequency is carried
out by comparison with the longitudinal phase velocity in the matrix cml .
 
10
PS sphere in PDMS l=1
8
6
l=5
l=4
4
l=2 l=3
2
0
0.0 3.0 6.0 9.0
r/cml
Figure 3.4.: Total dimensionless scattering cross-section for a polystyrene sphere
in liquid PDMS matrix (cf. Fig. 5.11, cf. Fig. 3 in Ref. [33]) for longitudinal incident
wave. The peaks are labeled as spherical harmonics with l = 2, 3, 4, 1, 5, n = 1 for
all resonances. The elastic values used for the calculation are given in Tab. 5.4.
63
r2
 
3. Methods
3.4.2. The Plane Wave Method
As pointed out before (and more widely discussed in Chapter 5), in
phononic systems consisting of many elastic scatterers in a matrix with
different elastic properties, band gaps may occur in the band diagram,
originating from either the periodicity of scattering layers (Bragg gap,
BG) or from localized states in individual scatterers. The localized modes
in the individual scatterers are given by the maxima in the scattering
cross-section diagram discussed in section 3.4.1, and it seems plausible
to extend the method presented for the scattering cross-section to cal-
culate the band diagram of phononic systems by assuming a plane wave
scattered now by many scatterers. In order to obtain analytical solu-
tions, the plane wave (PW) method is restricted to periodic systems,
i.e., phononic crystals.[74]
The initial point of the PW approach is the general wave equation for
a medium locally isotropic
[ ( ) { ( )}]
∂2ui 1 ∂ ∂ul ∂ ∂ui ∂ul
= λ + µ + i, l = 1, 2, 3.
∂t2 ̺ ∂xi ∂xl ∂xl ∂xl ∂xi
(3.21)
Here, ui,l and xi,l are the Cartesian components of the displacement
vector u(r) and the position vector r, respectively; λ(r), µ(r), and ̺(r)
are the Lamé coefficients and the local mass density, respectively, which
are periodic functions of r with periodicity lattice vector R:
f(r +R) = f(r). (3.22)
As a result of the common periodicity of the three coefficients in
Eq. 3.21, its solutions can be chosen to satisfy Bloch’s relation
u(r) = eikruk(r), (3.23)
where k is a vector in the reciprocal lattice restricted within the first
Brillouin zone (BZ) and uk(r) is a periodic function. Hence, it is possible
64
3.4. Theoretical Calculations
to expand f(r) in a three-dimensional Fourier series:
∑
f(r) = f eiG·rG , (3.24)
G
where the summation is over all reciprocal vectors G, which can be
written as sum with integer coefficients of the orthonormalized vectors
that span the three-dimensional vector space in which R is the sum of
the basis vectors.
With that one can rewrite Eq. 3.23 after expanding uk in Fourier series
as
∑
u(r) = u ei(k+G)rk+G . (3.25)
G
The substitution of Eq. 3.24 (with f = λ, µ, ̺−1) and Eq. 3.25 into
Eq. 3.21 delivers finally the expression:

∑ ∑
[
ω2cui = ̺−1k+G λ ′′ ′ (k +G
′) (k +G′′)
G−G′′ G −G l i
G′ l,G
]
+µG′′−G′(k +G
′)i(k +G
′′) ll uk+G′+
  
∑ ∑
[ ]
̺−1 × µ ′ ′′ i  ′′ ′ (k +G ) (k +G ) u ′
G−G′′ G −G j j k+G
G′′ j
(3.26)
If the Fourier series in Eq. 3.25 is performed over M reciprocal vec-
tors (i.e., for M scatterers in the periodic medium), Eq. 3.26 is reduced
to a 3M × 3M matrix eigenvalue equation for the 3M unknown coef-
ficients uik+G (i = 1, 2, 3). Usually, M > 400 to achieve the desired
convergence.[74]
Note that for fluid systems, Eq. 3.26 can be simplified.[73] In liquids
µ = 0; by introducing the pressure p = −λ∇u Eq. 3.21 can be written
65
3. Methods
in the form
[ ]
λ(r)∇ ̺(r)−1∇p(r) + ω2p(r) = 0. (3.27)
Following the same considerations as for the (general) solid case one
obtains for M terms in the Fourier sum a M ×M system.
PW is a fast and easy-to-apply method to calculate the band diagram
in fluid/fluid or solid/solid systems, i.e., fluid or solid scatterers in ma-
trices of the same aggregate state. However, it fails when dealing with
solid/fluid systems, which are mostly discussed in this thesis.
When a solid scatterer is embedded in a liquid host, transverse waves
cannot propagate in the matrix and Eq. 3.27 becomes the appropriate
elastic wave equation. In that case the M eigenmodes for M scatterers
correspond to purely longitudinal waves. However, it is known that even
for a longitudinal incident wave, the field inside the scatterer will be both
longitudinal and transverse, i.e., there are localized transverse modes
inside the scatterer that cannot propagate.[73, 75]
The approach that uses the full (including µ =6 0) wave equation
(Eq. 3.21) fails because of the non-propagating character of these modes
and leads to no convergence at all. Using Eq. 3.27 would ignore com-
pletely the transverse component of the wave within the scatterer as it
would de facto replace the solid scatterer by the fluid scatterers of the
same λ and µ. Although it leads to mathematically reasonable results,
it was shown that these results are not suited to describe the reality.[73]
Another limitation of the PW method is that it can be only applied on
infinite periodic samples. This also means that it is unable to calculate
the transmission properties. An approach to overcome these limitation is
the multiple-scattering method that will be discussed in the next section.
3.4.3. The Multiple Scattering Method
The main idea behind the multiple scattering method is described sche-
matically in Fig. 3.5. A plane wave in a homogeneous medium impinges
on a system of N non-overlapping scatterers (n=1,2,...,N). The wave that
66
3.4. Theoretical Calculations
x x2
c dlm x3
lm dlm
t,x1
clm
x4
x6
d dlmlm
incident
x5
plane wave dlm (x=L,M,N)
Figure 3.5.: Schematic draw of the multiple scattering mechanism. A plane inci-
dent wave (with coefficients cxlm) is impinging an ensemble of scatterers, which can
have arbitrary shape. The total wave field acting on the n-th individual scatterer
(with coefficients ct,xnlm ), e.g., the red scatterer 1 in the middle, is a summation over
the impinging plane wave and the scattered waves from the other scatterers (with
coefficients dxnlm; in our example for n 6= 1).
impinges on each scatterer consists of N contributions, the incident wave
and the outgoing waves from any of the N-1 other scatterers.
With the coefficient notation used in the sections above, for a single
scatterer the coefficients for the scattered wave field, dxlm, are related to
the given coefficients (i.e., amplitudes) of the incident plane wave, cxlm:
∑
dxlm = T
x
xlm;x′l′m′clm (3.28)
x′,l′m′
The T -matrix is a 3 × 3-Tensor that can principally be calculated for
scatterers of every arbitrary shape with no need of homogeneity. The
calculation for homogeneous spheres can be found in literature[53] as
well as the solutions for inhomogeneous core-shell particles.[76]
Coming back to our system with N scatterers, the total wave field
incident on the n-th scatterer is the sum of the incident plane wave and
67
3. Methods
the wave field scattered from all scatterers n′ =6 n.
∑
uinc,n(r-r ) = uincn (r-rn) + u
sc(rn-rn′). (3.29)
n′ 6=n
I.e.,
∑
uinc,n( t,Lnr-rn) = clm llm(R, kl) + c
t,Mn
lm mlm(R, kt) + c
t,Nn
lm nlm(R, kt)
lm
∑
= cLlmllm(R, kl) + c
M
lmmlm(R, kt) + c
N
lmnlm(R, kt)
lm
∑
L,n′ M,n′ ′+ d N,nlm llm(R, kl) + dlm mlm(R, kt) + dlm nlm(R, kt)
n′ 6=n,lm
(3.30)
or
(
∑
uinc,n
1
(r-r ) = cLn lm ∇ [Rl(klr)Ylm(r-rn)]kl
lm
+ cMlm∇× [(r-rn)Rl(ktr)Ylm(r-rn)]
)
1
+cNlm ∇×∇× [(r-rn)Rl(ktr)Ylm(r-rn)]kt
( (3.31)
∑
+ dL,n
′ 1
lm ∇ [Rl(klr)Ylm(rn-rn′)]kl
lm,n′
+ dM,n
′
lm ∇× [(rn-rn′)Rl(ktr)Ylm(rn-rn′)]
)
+dN,n
′ 1
lm ∇×∇× [(rn-rn′)Rl(ktr)Ylm(rn-rn′)] .kt
In the last two equations the two sums on the right side belong to the
incident plane wave (with coefficients cxlm, x = L,M,N) and to the
′
scattered waves from the other scatterers (with coefficients dx,nlm ).
For the resulting total incident field the coefficients ct,xnlm are introduced
68
3.4. Theoretical Calculations
in Eq. 3.30. They are related to the scattering coefficients of the n-th
scatterer dx,nlm by the T -matrix (Eq. 3.28).
[
∑
Doing so for all N scatterers, one obtains a system of N lmaxl=0 (2l + 1)
]
∑
+2 lmaxl=1 (2l + 1) algebraic equations, in which the only known ampli-
tudes are those for of the incident plane wave. With the assumption that
terms with l > lmax don’t contribute significantly to the spherical wave
expansion, this system can be solved numerically to determine all dx,nlm .
So far, the method is not restricted to an equal shape of all scatterers
(however, for different shapes the T -matrix of all individual scatterers
would have to be known) or to any periodicity, as the relative position
of each pair of scatterers is considered in Eq. 3.31.
However, as the aim is to calculate the band structure in phononic
crystals as function of the reduced wave vector k in the first Brillouin
zone, the introduction of the crystal periodicity is needed. It was shown
that in this case the phononic band structure can be calculated in ana-
logy to the theory of Korringa[77], Kohn and Rostoker[78] (KKR theory)
developed to solve the Schrödinger equation for electromagnetic waves
in periodic lattices.[79]
After introducing the pressure field
p(r) = λ∇ · u(r), (3.32)
it is possible to write the wave equation for a periodic medium of solid
scatterers in a fluid matrix as follows:[79]
[ ] [ ]
∇2 ω
2 1 1 1
p(r)+ p(r)+ω2 p(r)+̺(r)× ∇ ∇p(r) = 0 (3.33)
c c2m (r) c2m ̺(r)
This can be written in analogy to the Schrödinger equation in the form
Hm(r)p(r) + U(r)p(r) = 0, (3.34)
where Hm(r)p(r) = 0 represents the wave equation for the matrix with-
out scatterer (Hm(r) = ∇2 + ω2/c2m). In a periodic system a Green’s
69
3. Methods
function approach can be chosen to reformulate Eq. 3.33.
∫
p(r) = G(r − r′)V r′pr′dr′ (3.35)
V
with Green’s functions
∑
G(r − r′) = eik·RnG0(r − r′ −Rn) (3.36)
n
and
1 ei|r−r
′ |ω/cm
G0(r − r′) = − | . (3.37)4π r − r′|
Rn is a Bloch’s vector, so that
p(r +R ) = eik·Rnn p(r), (3.38)
and the relation between the amplitudes of the scattered field at different
lattice sites n and n′ becomes[41]
dx,n
′
lm = e
ik·Rndx,nlm . (3.39)
The next step is the expansion of G(r − r′) and p(r′) into spherical
functions of r and r′. These calculations are performed in detail in
Ref. [79]. In the end the final multiple-scattering equation appears to be
( )
∑ ω
= A −1lml′m′ − ℑ(dl ′ ′ ′ ′′ )δll δmm al m = 0. (3.40)cm
l′m′
The coefficients Alml′m′ are the so-called structure constants as they
depend on k, ω, and the periodic lattice structure, the coefficients dl′
relate the incident to the scattered field.
Eq. 3.40 represents a linear homogeneous algebraic system. Its nontriv-
ial solutions give the eigenfrequencies of the periodic system and hence
the dispersion relation.
70
3.4. Theoretical Calculations
An enhanced variant of the multiple-scattering method calculates the
dispersion relation for samples consisting of different composites with
two-dimensional periodicity and is called layer-multiple-scattering ap-
proach (LMS).[40, 76, 80] For a slab parallel to a distinct crystallographic
plane, the reduced vector parallel to this plane, k‖, is usually a conserved
quantity. Therefore, LMS searches in each individual slab propagating
Bloch waves for given ω and k‖, which are the eigenmodes of the elastic
field in that slab. It is an on-shell method since it operates at a given
frequency. It was shown that LMS is a powerful method to determine
the dispersion relation as well as the transmittance of three dimension-
ally structured systems. E.g., 3-D phononic crystals can be regarded as
a succession of planes of scatterers parallel to a chosen crystallographic
plane. As already mentioned for the general MS approach, the LMS
technique takes the full vector nature of the acoustic field into account
and is therefore not limited to certain fluid/solid combinations, as for
example PW is.
Note that nearly all calculations in this thesis using the single-sphere
scattering cross-section and the multiple scattering algorithms have been
performed by my coworker Revekka Sainidou (Univ. Le Havre).
3.4.4. The Finite Element Method
The finite element method (FEM) is a numerical method to approxi-
mately solve partial differential equations. It is widely used in engineer-
ing and many other fields to simulate a wide range of different physical
problems.
The main principle of the technique is that the treated body is initially
discretized into a finite number of elements of finite size, which can be
described by a finite number of parameters. Within these finite elements,
basis functions are defined. Substitution of the linear combination of all
utilized basis functions into the differential equation to solve for each in-
dividual element (e.g., for mechanical problems the equation of motion)
in combination with the boundary conditions leads to a quadratic ma-
trix, whose dimension is given as the product of the number of elements
71
3. Methods
and the number of basis functions. This dimension is also the number
of degrees of freedom of the system. The boundary conditions at the
knots of neighbouring elements are often evident, e.g., in the case of a
mechanical deformation, the displacement (and even the first derivative
of the displacement) must be continuous at the boundary between two
elements. Finally, a system of ordinary differential equations is obtained
that can be solved directly, e.g., using Gaussian elemination. It is obvi-
ous that the quality of the approximation increases with the number of
elements, as well as the effort for the calculation does.
In this thesis, FEM is utilized only marginally to display some eigen-
vibrations. These simulations have been performed by myself using the
COMSOL Multiphysics software package.
72
4. The Vibrations of Individual
Colloids
4.1. Introduction
Nowadays, colloidal nano- and mesoscale particles with dimensions from
few nanometers up to one micrometer are used in a growing number
of applications, e.g., as fillers in polymer thin films to enhance thermo-
mechanical properties,[81] to improve coatings performance and as com-
ponents in nanocomposites operating as photonic,[1] plasmonic,[82] and
phononic structures.[8] For a wide range of such applications, information
on the mechanical properties and the stability of these colloidal compos-
ite materials are of paramount importance. Conventional rheological
measurements on the macroscopic system are often not sufficient to elu-
cidate the specific contributions of the nanostructured components. At
the nanoscale, forces negligible in macroscopic systems, such as deple-
tion, interfacial, and confinement effects often become significant, and
the behavior of the same materials in nanoscopic systems can consider-
ably deviate from the bulk. A fundamental understanding of transport
and thermomechanical properties of nanostructured materials is a pre-
condition to address a specific need by structural engineering. In the
case of colloidal composite materials, the vibrational modes confined to
the individual particles result from the elastic motion at the nanoscale
and should sensitively depend on the geometrical, architectural, interfa-
cial, and mechanical characteristics of the particles. However, there is a
paucity of non-destructive experimental techniques to probe this ‘music’
of particle vibrations since both high frequency resolution and sensitivity
are required to detect the numerous eigenmodes. Raman scattering[83, 84]
73
4. The Vibrations of Individual Colloids
has been utilized to measure few eigenfrequencies of nanoparticles with
dimension below 10 nm, whereas Brillouin light scattering (BLS)[45, 46, 85]
and optical pulse-probe techniques[27, 28, 86] can probe respectively the
spontaneous and stimulated vibrations confined in sub-micrometer parti-
cles. In the latter technique, the excited acoustic oscillations are observed
in the form of modulations of the transient reflectivity of the probe laser,
and hence the particles must possess good reflectance, e.g., by introduc-
tion of gold shells. In BLS, light is scattered inelastically by the density
fluctuations (phonons) associated with these particle localized modes at
thermal equilibrium and there are no further stringent conditions.
Self-assembly of colloidal particles in periodic structures[17, 87] (cf. sec-
tion 3.1) has received special attention as the resulting photonic and
phononic crystals have revealed the potential of manipulating the prop-
agation of electromagnetic and elastic waves. Their propagation is for-
bidden at ‘Bragg’ frequencies or wavelengths commensurate with the
lattice constant, which for sub-micrometer particles is comparable with
the wavelength of the visible light. Synthetic opals from these particles
can exhibit dual, i.e., hypersonic phononic and photonic[87, 88] band gaps
allowing for acousto-optical interactions.[89] Moreover, the design of sub-
micron particles that can act as strong localized resonant elements in an
appropriate matrix provides the possibility for additional gaps well be-
low the Bragg frequency, termed hybridization gaps (cf. chapter 5).[33, 90]
The opening of band gaps perturbs the phononic density of states which
impacts physical quantities such as group velocity, heat capacity and
heat conductivity in dielectrics being potentially useful for thermoelec-
tric devices.[56, 91] One of the pivotal concerns for these systems is the
phonon dispersion, which is essentially defined by the elastic parameters
of the constituent components and the spatial architecture of the com-
posite system. Colloid science can create novel materials that possess
spatial variation of density and elastic constants at the nanoscale but
their mechanical characterization remains difficult.
If vibrations are excited in a finite elastic body, it can act as an elastic
resonator. The elastic standing waves in stress-free boundary are referred
to vibration eigenmodes. In the case of elastic spheres, the analytical so-
74
4.1. Introduction
lutions were first derived by Lamb.[51] The eigenmodes of spheres can be
classified as torsional and spheroidal modes, both labeled by three in-
dices (nlm), which describe the angular (lm) and radial (n) dependence
of the displacement. Spheroidal resonance modes are fully character-
ized by the angular momentum l, imposed by the spherical symmetry of
the particle, and n, where n denotes the n-th order solution for a given
l (cf. section 2.3.3).[51, 83] The use of inelastic light scattering to mea-
sure vibration eigenmodes of small spherical particles was first performed
experimentally by low-frequency Raman scattering (RS).[83] Due to se-
lection rules only two distinct vibration eigenmodes contribute to the
RS of spherical particles with diameter (d) much smaller than the wave-
length of the probing light (d ≪ λ).[92] For bigger spheres with d ∼ λ,
Brillouin light scattering (BLS) in the GHz-range becomes the technique
of choice. [32, 45, 46] Due to the consideration of higher-order terms in the
electric multipole expansion and of retardation effects,[52, 93] BLS can
resolve a multitude of eigenmodes.
For homogeneous transparent systems, BLS measures the spectrum of
light inelastically scattered by the acoustic phonons with a selected po-
larization (longitudinal (l) or transverse (t)) and wave vector q, leading
to spectra consisting of doublets at ωB=±cq around the central elastic
Rayleigh line. Since c in soft materials like polymers is of the order of
103 ms−1 and q is in the range 1−30 µm−1, the frequencies f=ωB/2π fall
into the GHz range. This yields the two elastic constants C = ̺c211 l and
C 244 = ̺ct with cl and ct being the two phase sound velocities and ̺ be-
ing the mass density. For inhomogeneous turbid systems, e.g., powder of
mesoscopic (d ∼ λ) particles, q is ill-defined in the BLS experiment due
to strong multiple scattering. BLS can measure only localized in space
(and hence q-independent) vibrational modes. Each resonance mode ap-
pearing at frequency f(n, l) is characterized by the angular momentum
l of the n-th order. The frequencies of the individual vibrational modes
depend on their rigidity, mass density, and size dimensions of the parti-
cles. For the case of homogeneous spherical particles the frequencies are
given by Lamb in Eq. 2.106.
The frequencies can be theoretically obtained from the calculated den-
75
4. The Vibrations of Individual Colloids
sity of states (DOS) spectra of a single sphere as a function of the two
elastic constants and the inverse diameter. For polystyrene spheres, e.g.,
the constant in Eq. 2.106 becomes x(1, 2) ≈ 0.85 and R(cl, ct) = C44
with no adjustable parameter.[46]
In this chapter the state of the art for BLS measurements on ho-
mogeneous mesoscopic spherical particles is briefly summarized. Novel
studies dealing with mixtures (‘hybrids’) of different kind of spheres are
presented as well as a study on spheres prepared as copolymers with dif-
ferent compositions to elucidate the influence of the next neighbors and
of the rigidity on the mechanic vibrations in the mesoscale. The second
part of the chapter extends the scope on nanostructured colloids. Hybrid
material spherical core-shell particles are investigated as model systems.
Therein, especially the influence of heat on polymer cores contained in
a hard silica-shell and the influence of the composition in the vice versa
case of silica-PMMA core-shell particles with different ratio of core size
to total diameter on the vibrational eigenmodes is of interest.
Due to some theoretical support, these studies give an unprecedent-
edly comprehensive picture of the elastic properties of individual colloids,
although still numerous questions have to stay open - e.g., when deal-
ing with structures going beyond the investigated ‘easy’ spherical model
systems.
4.2. Elastic Vibrations in Homogeneous Polymer
Colloids
4.2.1. The ‘music’ of the spheres
As pointed out in section 2.3.3 and in the introduction of this chap-
ter, BLS can be utilized to measure the resonance modes of dry non-
transparent colloidal crystals. Due to the strong elastic form factor of
the individual spheres and the large elastical contrast with the surround-
ing air, the opals show strong multiple scattering. In such samples the
inelastic scattering from localized modes leads to incoherent BLS in ana-
76
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
logy to the Raman scattering.
Thus BLS can be utilized to analyze the particle eigenfrequencies, de-
scribing the spheroidal (n, l)–modes, with n as the n-th mode of the l-th
spherical harmonic. The first demonstration of the feasibility of the BLS
experiment was shown by Penciu et al. in the case of dilute suspensions
of giant core–shell micelles.[85] Few years later Kuok et al. have extended
this application to closely packed monodisperse silica nanospheres in
air.[45, 94] Up to six localized particle eigenmodes have been resolved out
of the numerous possible modes, probably due to the weak scattering
of moderately compressible silica. In a subsequent study, artificial soft
colloidal crystals, composed of monodisperse submicrometer polystyrene
(PS) spheres with diameter d between 170 nm and 856 nm have been
investigated.[46] Up to 21 q–independent eigenmodes have been resolved.
Fig. 4.1 shows exemplary three BLS spectra taken from colloidal PS
opals with diameters 180 nm, 360 nm and 550 nm.[32] The resonance
frequencies scale with 1/d (Fig. 4.2), which is in perfect agreement with
Lamb’s theory[51] and the theoretical predictions based on single-phonon
scattering cross-section calculations (cf. section 3.4.1).[76, 95] In the com-
putations, a plane sound wave propagating in air and impinging upon a
single PS sphere was considered and after subtracting the scattering am-
plitude for a rigid sphere of equal size, the sphere eigenmodes appear as
resonance peaks in the plot of scattering cross–section versus frequency.
Thereby the resonance frequencies f(n, l) can be identified as mode with
angular momentum quantum number l of n-th order.
Using the experimental values for the longitudinal sound velocity cl=
2350 ms−1, the transverse sound velocity ct=1210 ms
−1 and mass den-
sity ̺=1050 kgm−3 of bulk PS, all resolved frequencies are quantitatively
captured within 3% with no adjustable parameter. The product of fre-
quency and diameter is obtained to be a constant for every mode, as
is indicated by the solid lines in Fig. 4.2, which represent the computed
modes (n, l). It is shown that the first modes are the same for all samples
with the (1, 2)–mode being the lowest and most intense one.
From the longitudinal and transversal sound velocities the elastic con-
stants are directly accessible. These are the Poisson ratio σ, the Young
77
4. The Vibrations of Individual Colloids
Figure 4.1.: left SEM–images of colloidal PS crystals with d=180 nm, 360 nm
and 550 nm (top to bottom); right Corresponding q–independent BLS eigenmode
spectra of the PS opals. To capture all possible vibrations, two spectra recorded
at two different free spectral ranges are superimposed. Spectra are recorded at 20◦
(q=0.0041 nm−1). For the thickest particles (d=550 nm, bottom) there is a clear
cut–off bump around 15 GHz, which is the frequency of the acoustic phonon in PS
at backscattering geometry.
modulus E, and the shear modulus G (section 2.1.1).[96] Additionally to
the elastic constants, even more information can be extracted from these
spectra. The peculiar line shape can be regarded as a sensitive index
of the particle size distribution. An application with technological rel-
evance was the determination of the mechanical properties of spherical
glassy CaCO –particles.[96]3 The particles with mono-modal size distri-
bution and diameters in the range between 400 nm and 1500 nm were
made from amorphous CaCO3, with unknown elastic properties as yet.
Eigenmode acoustic spectra of the spheres were recorded and six or seven
resonance modes could be resolved. By calculating cl from the cut-off–
frequencies and taking ct and the density ̺ as floating fit parameters, all
78
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
 
15
7) ,6)
(1, (1 )
(1,
5
(1,0
)
)
(1,4
10
(1,3)
5 (1,2)
 FEM simulation for d=540 nm
0
0 1 2 3 4 5 6
d-1 / m-1
Figure 4.2.: Experimental resonance frequencies of the PS opals from Fig. 4.1
(solid symbols) versus the reciprocal particle diameter. The solid lines denote the
computed resonance frequencies.[32, 46] The open squares denote the frequencies for
the first seven modes simulated by the finite element methode for PS spheres with
d=540 nm.
signals could be assigned to the vibrational eigenmodes and a very good
agreement between theory and experiment was achieved. From the ob-
tained values for cl, ct and ̺ the Poisson ratio σ, Young modulus E and
shear modulus G were then reported. In conformity with the analysis
of the SEM–pictures all investigated colloidal samples in this thesis are
found to be quite monodisperse.
For the largest particle size in Fig. 4.1 (or Fig. 2 in [46]) there is
clearly a kind of cut-off frequency at about 14–15 GHz after which the
scattering intensity decreases rapidly towards zero. Because of strong
multiple scattering, q is not defined. The highest frequency contribution
79
f / GHz
(2,17)
(2,13)
(1,11
(2 ),10)
 
4. The Vibrations of Individual Colloids
corresponds to the acoustic phonon in bulk PS under backscattering
conditions, i.e., qBS=4πn/λ0. Using n=1.59 for the refractive index of PS
and for the laser wavelength λ0=532 nm, qBS=0.0376 nm
−1, and hence
a cut-off frequency for the longitudinal acoustic phonon (c =2350 ms−1l )
is indeed expected at 14.05 GHz.
A more detailed theoretical treatment of the Brillouin and Raman scat-
tering from the acoustic vibrations of spherical particles with diameters
in the order of magnitude of the wavelength of light was presented very
recently by Montagna.[52] While for very small particles (qR ≪ 1, with
R = d/2) only the ‘Raman–term’ that originates either from local field
changes due to dipole induced dipoles or from electronic polarizability
changes with the change of the atomic distances is important, for parti-
cles with diameters d ∼ λ0 also the ‘Brillouin–term’ becomes important.
The latter comes from the polarization fluctuations caused by spacial
displacement of scatterers by acoustic vibrations. Detailed calculations
are performed, which show that for a given l the intensity of a resonance
mode with a certain n strongly depends on qR, hence on the particle
size. However, a direct comparison with the experiment is not straight-
forward due to the ill-defined q. Nevertheless, it’s pointed out that for
larger particle sizes the range of qR also increases (0 ≤ qR ≤ qBSR). In
a given range of qR only the modes resonance frequencies that are close
to that of the acoustic phonon in the bulk, i.e., ω ≈ qcl, contribute con-
siderably to the Brillouin scattering. The total number of modes with
higher n and l that have their intensity maximum in the larger qR–range
also increases. This rationalizes the observation of the highest number of
modes for the better–formed cut-off for the largest particles in Fig. 4.1. It
should be noted that when calling the resonance modes ‘q-independent’,
in principle the frequency of the mode is indeed q-independent but the
intensity is theoretically not. The reason why in the BLS measurements
in this section the spectra are found to be independent from the angle
is that obviously all q’s 0 ≤ q ≤ qBS contribute equally to the multiple
scattering. In section 4.2.4 an experiment is presented that shows the
q-dependence of the scattering intensity of individual modes.
However, it was not settled whether selection rules apply for the BLS
80
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
from particles with diameters comparable to the wavelength of the inci-
dent laser light (qd ≈ 1). While Li et al.[93], based on group theory con-
siderations, claim that only particle resonances with even l’s contribute
to BLS eigenmode spectra, according to Montagna[52] resonances with
even and odd l’s should have comparable activities. In fact, the latter
is supported by a BLS study[46] of model PS spheres revealing up to 21
resonance modes theoretically captured with no adjustable parameter.
Moreover, utilizing a finite element method (FEM, cf. section 3.4.4), I
could validate the assignment for the first six modes with n=1 and l=2-7,
which are shown as open squares in Fig. 4.2 in perfect agreement with
the scattering cross–section calculations and the experimental frequen-
cies. Fig. 4.3 shows the FEM result for the lowest (1,2)-eigenmode of a
PS sphere with d=540 nm. Using the same parameters as in the cross–
section calculations (cl=2350 ms
−1, cl=1200 ms
−1, ̺=1050 kgm−3),[46]
the (1,2)-mode is found at 1.87 GHz, while the experimental mode was
measured at 1.94 GHz. Fig. 4.3a shows the deformated surface of the
sphere, while subfigures b-d show the deformated intersecting planes in
the planes of a Cartesian coordinate system. The color scale corresponds
to the total local displacement. In the intersecting planes, a displacement
symmetric to the center of the spheres is visible, which is also the only
knot of the displacement function, i.e., n=1. The shape of the deforma-
tion, which is amplified by a factor of 20 in order to clarify the direction
of the deformation, assigns the angular momentum quantum-number l;
in each intersecting plane the deformated sphere has two maxima and
two minima, i.e., l=2. (Note: The thin black lines denote the unde-
formated sphere.) Analog to the identification of the (1,2)-mode, also
the modes with higher l can be found from the FEM simulation. It is
noteworthy that there are a higher number of modes calculated following
the FEM approach than seen in the BLS experiment. However, most of
these modes are torsional modes, i.e., the displacement is fully tangential.
Such modes don’t cause inelastic light scattering, only spheroidal modes,
where the displacement is also (or fully) radial, contribute to the light
scattering spectrum.[52, 92, 93] By chosing only the spheroidal modes out
of the simulated eigenmodes, those with higher l are easily obtained from
81
4. The Vibrations of Individual Colloids
Figure 4.3.: FEM simultaion for the (1,2)-eigenmode of a PS sphere with
d=540 nm at 1.87 GHz; a total displacement (color scale) and deformation of
the surface, b-d displacement and deformation of intersecting planes in the three
planes of a Cartesian coordinate system. Deformations are amplified by a factor of
20.
the FEM results. Checking the displacement and the deformation in the
intersecting planes as in Fig. 4.3 allows the correct determination of n and
l. The shape of the surface for the modes with n=1 and l=3-7 is shown
in Fig. 4.4 together with the intersecting planes for l=3; for the higher l
the intersecting planes look analog to those for l=2 and l=3 with l defor-
mation maxima in each plane. Of course even higher order modes can be
found by FEM, however, with increasing order the assignment becomes
more and more complicated, since in the calculation mostly asymmetric
mixed modes appear. Theoretically, with much finer mesh, all modes
can be assigned clearly. Anyhow, even the six modes shown here (in
82
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
Figure 4.4.: FEM simulation for the higher l eigenmodes (n=1). Deformations are
amplified by a factor of 20.top: total displacement (color scale) and deformation of
a the surface and b-d the intersecting planes of the (1,3) spheroidal mode. bottom:
displacement and deformation of the surface for modes with l=4-7. For a PS sphere
with d=540 nm the frequencies are found to be 2.79 GHz (1,3), 3.58 GHz (1,4),
4.32 GHz (1,5), 5.03 GHz (1,6), and 5.74 GHz (1,7) in good agreement with the
experimental data (Fig. 4.2).
perfect accordance to another calculation method) proof that there is no
limitation on even l’s as claimed by Li et al.[93], since the assingnment
of the (1,3) and the (1,5)-mode with no fitting parameter is clear and
without any alternative. On the other hand, the FEM simulation of
the eigenvibrations of silica spheres with d=360 nm, whose eigenmode
spectrum is shown in Ref. [93], allows other assignments for the first
four modes than that chosen by the authors, who restricted themselves
on even l’s, utilizing the elastic parameters given by the same authors
experimentally (ct=2520 ms
−1, cl=3960 ms
−1, ̺=1960 kgm−3).[97]. The
mode at 8.76 GHz (assigned as (1,0)) could be the (1,3)-mode (8.60 GHz
by FEM), the mode claimed to be the (1,6)-mode at 14.4 GHz could be
a double signal containing the modes (1,5) and (1,6), found by FEM to
83
4. The Vibrations of Individual Colloids
appear at 13.14 GHz and 15.29 GHz, respectively. For higher orders, the
assignment is more difficult, however, the mode at 17.65 GHz (claimed
to be (2,4) or (3,2)) could be identified as the (1,7)-mode, calculated to
appear at 17.4 GHz.
It should be noted that deviating from Ref. [32] in Fig. 4.2 also the
breathing mode (1,0) is shown as a theoretical fit based on FEM cal-
culations, again in perfect agreement with the scatterin cross-section
method.[98] It is found that for the case of PS spheres the (1,0)-mode is
not far away from the (1,4). In fact, the third signal (from low to high
frequency) for the PS spheres in Fig. 4.1 seems slightly broadened, which
might be related to a weaker signal from the (1,0)-mode.
4.2.2. The Influence of the Neighbors: Mixtures
The assignment of the two phononic gaps is also corroborated by their
sensitivity on the disorder. In this context, non-crystalline colloidal films
were prepared by vertical lifting deposition of ‘hybrids’, binary mixtures
consisting of an equal number of two PS spheres with different diame-
ter (d=300 nm and 360 nm). The size polydispersity is then artificially
increased and no crystallization takes place, which is affirmed by SEM–
pictures. The eigenmode acoustic spectra of these dry hybrid films and
the dispersion relations in their infiltrated counterparts were measured
by BLS. The eigenmode spectrum of the 1:1 300:360 nm hybrid is shown
in Fig. 4.5 along with the eigenmode spectra of the individual one com-
ponent opals. Interestingly, the spectrum of the hybrid is a superposition
of the individual opals as is indicated by the vertical lines denoting some
exemplary resonance frequencies either from the small or the big spheres.
Moreover, hybrids of different relative ratios of the constituent spheres
were prepared. The intensity of the signals originating from a particu-
lar particle size relates to its composition in the mixture. In Fig. 4.5
this linear composition dependence is demonstrated by superimposing
the spectrum of the 1:1 300 nm:360 nm hybrid with the spectrum of
a 3:1 300 nm:360 nm hybrid, normalized to the peak intensity in the
360 nm spheres. For clarity a baseline correction was introduced into
84
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
Figure 4.5.: left Eigenmode acoustic spectra of the PS opals with d=300 nm
(blue), d=360 nm (red) and the symmetric PS hybrids 1:1 / 3:1 300 nm:360 nm
(black / green)(bottom to top). The vertical lines (dotted for d=300 nm, dashed
for d=360 nm) denote some exemplary eigenmodes appearing in the opals and in
the hybrids. In the top right corner the Stokes–sides of the hybrids’ spectra are com-
pared after baseline correction to better visualize the influence of the composition
on the relative intensity of the individual signals. right Corresponding SEM–images
of opals and hybrids shown on the left side.
the Stokes-side of the hybrids (inset to Fig. 4.5).
Apparently, but somewhat counterintuitive, the contact between the
spheres does not influence their eigenmodes – at least for non–infiltrated
dry films. We also examined only partially ordered samples of monodis-
perse nanospheres prepared by dropping a few drops of the suspension
of the spheres onto a glass substrate. In spite of the lack of any or-
85
4. The Vibrations of Individual Colloids
Figure 4.6.: left Comparison between the eigenmode acoustic spectra of the
550 nm PS single opal and the highly asymmetric 9:1 180 nm:550 nm PS hybrid.
right SEM–image of the 9:1 180:550 nm PS hybrid.
der, the eigenmode spectra of these samples are indistinguishable from
those of the colloidal crystals. To undermine the contact argument, the
consistency of the spectra’s features would be based upon the small dif-
ferences between the opals and the hybrids for 300 nm and 360 nm PS
spheres, also very asymmetric hybrids with a nine to one number ratio
of 180 nm and 550 nm PS beads were investigated. While for every
dense packing of spheres, the number of neighboring spheres and hence
the contact points as indicated by the SEM–images is twelve, this mean
number is different for the random packed hybrids depending on their
size disparity and their number ratio. For example, in the SEM image
for a 9:1 180 nm:550 nm hybrid in Fig. 4.6, the number of next neigh-
bors of the large spheres is increased dramatically by small beads that
accumulated around the bigger spheres. However, even in this extreme
case, the resonances from the 550 nm spheres are virtually unaltered as
demonstrated in the spectra of Fig. 4.6. A careful inspection, however,
of the two lowest frequencies reveals a very small shift of the hybrid
86
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
resonance modes towards higher frequencies as is shown in the inset of
Fig. 4.6 for the Stokes–sides of the spectra. Though the observed effect
is subtle (∼4% frequency shift) but significant. In this context it’s note-
worthy to mention that in Fig. 3 of Ref. [99], Li et al. compare the BLS
spectra from a 320 nm silica artificial opal and from a single sphere of
the same kind, measured by a micro–Brillouin light scattering setup. In
this figure it seems as if there is a shift between single spheres and opal.
Interestingly, the authors describe only the broadening in the opal due
to the polydispersity.
In summary, the number of next neighbors does not influence signif-
icantly the eigenvibrations of the individual spheres. The effect of the
disorder on the dispersion relation of our PS hybrids after infiltration is
discussed in chapt. 5.1.3.
4.2.3. The Influence of the Rigidity: Copolymers
While there are several BLS studies dealing with the size-dependence of
the eigenmodes of homogeneous spheres from different materials (section
4.2.1),[32, 45, 46] where the elastic properties are calculated from the ex-
perimental results, as well as two more elaborate studies on core-shell
particles, presented also in this thesis (section 4.3),[12, 100] a systematic
experimental investigation of the influence of the particle rigidity on the
eigenvibrations of mesoscopic copolymers is still missing. After Lamb
(Eq. 2.106)
√
( )
R(cl, ct)
f(n, l) = x(n, l) /d,
̺
with x(n, l) being a temperature independent constant for each indivi-
dual mode and R(cl, ct) being the rigidity, which is a function of longi-
tudinal and transverse sound velocity cl and c .
[51]
t The frequencies can
be theoretically obtained by single-phonon scattering cross section cal-
culations (section 3.4.1) or finite element modeling (section 3.4.4). Here,
BLS is utilized in a set of nearly uniformly sized (d ≈200 nm) polymeric
87
4. The Vibrations of Individual Colloids
colloids consisting of poly(methyl methacrylate) (PMMA) copolymerized
with poly(n-butyl acrylate) (PnBA) in different compositions in order to
investigate the development of the elastic constants as a function of the
copolymer composition. Therefore, bulk and individual colloidal proper-
ties are measured by BLS and consulted in combination to obtain insight
into the effect of copolymerization on the mechanical characteristics in
such small spheres. Due to the clearly differing rigidity of the homopoly-
mers at room temperature, which is confirmed by the large difference in
the glass transition temperature Tg (Tg(PMMA) ≈ 110 ◦C, Tg(PnBA) ≈
-43 ◦C), the softness of the copolymer increases rapidly with the amount
of n-butyl acrylate used in the synthesis.
Copolymerization of two distinct monomers permits the combination
of their individual properties and, hence, allows control over various
physical quantities, such as Tg. Random radical copolymerizations of
MMA and nBA was conducted by my colleague M. Retsch in order to
tune the Tg of the resulting colloid. The composition of nBA in the
copolymer was targeted to 10 wt%, 20 wt%, and 30 wt% through se-
lection of the weigth ratio of the two monomers in the initial emulsion.
A random copolymerization is indicated by the Q and e values of both
monomers in terms of the Q-e-schema of Alfrey and Price (nBA: Q=0.38,
e=0.85; MMA: Q=0.78, e=0.40).[101, 102] A homogeneous copolymeriza-
tion is indicated by the very similar r values (nBA: r=0.94; MMA:
r=0.91) not very different from r=1.
Differential scanning calorimetry (DSC) was used to determine the
glass transition temperature Tg. The results of the heating period of
the second run, i.e., after heating initially to 200 ◦C (and therefore bulk
properties), are shown in Fig. 4.7. Tg decreases rapidly with increasing
amount of nBA. Using the value measured for pure PMMA and the
literature value for nBA (-43 ◦C),[103] allows an excellent representation
of the glass transition temperature as a function of the composition by
the Fox-Flory equation[104]
1 φPMMA φPnBA
= + (4.1)
Tg Tg,PMMA Tg,PnBA
88
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
  
0.1  Gordon-Taylor 100 (K=0.82)
80  Fox-Flory
60
40
0.0 0 10 20 30
pure PMMA wt% nBA
10% nBA
-0.1
20% nBA
30% nBA
-0.2
0 20 40 60 80 100 120 140 160
T / °C
Figure 4.7.: DSC data for the four compositions at the second heating run, i.e.,
bulk properties. Tg’s are denoted by vertical lines. The inset shows Tg as a
function of the composition together with a Fox-Flory fit and a Gordon-Taylor fit
from Ref. [103].
with no fit parameter (inset). The Gordon-Taylor fit also shown in the
inset is borrowed from Ref. [103], including the empirical parameter
K=0.82, however, the small deviations originate partially in different val-
ues measured for pure PMMA. Our Tg’s could be perfectly represented
with T (PMMA)=113 ◦g C and K=0.65. In addition to the single Tg,
dielectric spectroscopy has been performed by my colleague K. Mpouk-
ouvalas, which showed a single α-relaxation for each sample, i.e., both
kind of segments feel the same energy landscape.
The density ̺ of the copolymers was determined by a density gra-
dient column using aqueous calcium nitrate solutions.[70] The experi-
mental results shown in Fig. 4.8 conform to a weight average density,
89
Heatflow / a.u.
Tg / °C
  
4. The Vibrations of Individual Colloids
 
1.20  
1.20
1.19
1.19  fit* samples
1.18 1.18
1.17 1.17
1.16
1.16 20 wt% nBA * =1.035 g cm-3
1.15 PBA
;
PMMA=1.195 g cm
-3
1.15 1.14
1.14 30 wt% nBA
0.0 0.1 0.2 0.3
wt% nBA
1.13
1.12
1.11
Calibration
1.10  samples
0 100 200 300 400 500 600 700 800 900
Column height / mL
Figure 4.8.: Determination of mass density as a function of the composition by a
density gradient column for the two lightest samples. The squares are well-known
calibration standards. The inset shows a fit for all four compositions.
using for the density of the homopolymers ̺ −3PnBA=1.035 g cm and
̺ −3PMMA=1.195 g cm .
The size of the spheres is ascertained by scanning electron microscopy
(SEM). The average molecular weight Mn and the polydispersity index
PDI of the polymers were measured relative to PMMA standards by
gel permeation chromatography (GPC). Crystallinity was excluded by
small angle X-ray scattering measurements. The particle properties are
itemized in Tab. 4.1.
By vertical lifting deposition[54] non-transparent films of the colloids
with thickness of about 30-50 µm were prepared by M. Retsch. The
BLS eigenmode spectra were recorded in transmission geometry at (non-
arbitrative, due to strong multiple scattering) scattering angle of 50◦.
Fig. 4.9 shows the experimental spectra. Because of the slightly differing
particle diameters given in Tab. 4.1 and the d−1-dependence of the modes
90
  / g cm-3
  / g cm-3
 
 
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
 
30 wt% nBA
20 wt% nBA
 10 wt% nBA
 0 wt% nBA
5 4 3 2 1
-4000 -3000 -2000 -1000 0 1000 2000 3000 4000
f*d
Figure 4.9.: Experimental BLS eigenmode spectra with frequency f normalized
with the particle diameter d. The dotted vertical lines on the Anti-Stokes side
denote the positions of the five resolved maxima for the pure PMMA spheres.
given in eq. 2.106 the frequency axis is normalized to the product of
frequency and diameter. The eigenmodes correspond to the maxima
of the spectra, all four spectra can be represented by five Lorentzian
signals, denoted by the dotted vertical lines on the Anti-Stokes side of
the spectra. However, the fourth spectral line is much broader than the
others.
Obviously, with increasing amount of nBA (bottom to top) the modes
are shifted towards lower frequencies. This correlates with the decreasing
rigidity in Eq. 2.106 caused by the softening in the copolymer. Notably,
the change in the density alone (ca. 4%) would lead to a frequency shift
91
 Intensity / a.u. (log scale)
   
    
4. The Vibrations of Individual Colloids
Table 4.1.: Particle and material properties of the investigated spheres obtained
by SEM (d), GPC (Mn, PDI), DSC (Tg) and density gradient column(̺).
wt% nBA d / nm Mn / g mol
−1 PDI T / ◦g C ̺ / g cm
−3
0 232 92500 4.2 113 1.195
10 214 89500 4.2 92 1.179
20 200 84200 3.2 68 1.163
30 204 110000 3.5 50 1.147
Table 4.2.: Experimental and theoretical bulk cl for homopolymers and copolymers
at room temperature.
wt% nBA cl(BLS)/ms
−1 cl(Wood)/ms
−1 cl(Wood)/ms
−1
c −1l,PnBA=2200 ms
0 2755 2755 2755
10 2678 2569 2663
20 2615 2421 2584
30 2498 2300 2515
100 1835 1835 2200
in the opposite direction.
According to Eq. 2.106, the frequencies of the eigenmodes are functions
of three parameters: cl, ct, and ̺ or, alternatively, Young’s modulus E,
Poisson’s ratio σ, and ̺. Since the mass density is experimentally known,
the adjustable parameters are the two sound velocities.
For the bulk copolymers, BLS can, in principle, deliver cl and ct by
recording the dispersion relations for polarized and depolarized light scat-
tering, respectively. Transparent samples were prepared by heating some
of the vertical lifted opals by a heat gun (T ≈100-200 ◦C) for a few
seconds, which results in a melting of the mesospheres. The polarized
spectra delivered cl, but the intensity of the theoretically accessible ct
in the depolarized spectra was not strong enough to determine a single
signal. The measured cl’s are presented in Tab. 4.2, together with the
value of a bulk PnBA sample.
92
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
Previous studies on homogeneous mesoscopic polymer and silica beads
show that there should be no significant change in the longitudinal sound
velocity that has to be taken into account in the colloids compared to
that in the bulk material.[32, 46, 97] Since the rigidity R for the lower
frequency eigenmodes is found to be a much stronger function of ct, i.e.,
shear modulus G = ̺c2t , than of cl, i.e., longitudinal modulus M = ̺c
2
l
(cf. section 2.1.2), it is meaningful to take the bulk longitudinal sound
velocities also as given in the description of the vibrational modes.
Tab. 4.2 shows the decrease of cl, i.e., softening, in the copolymers with
increasing amount of nBA. However, this decrease is small compared to
the experimental value of cl in the two homopolymers. In sections 5.1.2
and 5.2 of this thesis, several approaches to describe the sound propa-
gation in an effective medium composed of (at least) two mechanically
different materials are discussed. It turns out that the simple expression
1 φ1 1− φ1
= + , (4.2)
Meff M1 M2
which is known as Wood’s law,[105] is in many cases a good approxima-
tion to give an effective modulus Meff as a function of the components
individual moduli and the composition. If one assumes Wood’s law to be,
in principle, a good approximation also for copolymers, the theoretically
expected values for cl can be easily calculated, as shown in the second
column of Tab. 4.2. Obviously, the presented approximation strongly
overestimates the softening due to the nBA in the copolymers, leading
to a deviation of up to 8% for 30 wt% nBA. It will be shown in section
4.3.1 that the mechanical moduli are temperature dependent, strongly
decreasing at temperatures above Tg. At room temperature, all sound
velocities except for bulk PnBA are measured in the glassy state, i.e., for
purely elastic response. In contrast, bulk PnBA at room temperature is
in the rubbery regime, i.e., it behaves viscoelastically and hence its cl
is lower than for a glassy PnBA. In other words, the nBA segments in
the three glassy PMMA-r-PnBA copolymers assume a dense packing and
therefore should display elastic response due to the very slow (essentially
93
4. The Vibrations of Individual Colloids
frozen) dynamics. The success of Eq. 4.2 in representing the experimen-
tal cl values of the copolymers is optimized if a fictive cl=2200 ms
−1 is
used for the bulk PnBA, i.e., by introducing an ‘effective glassy longi-
tudinal sound velocity’. The third column in Tab. 4.2 shows the good
representation by Wood’s law, using this artificially increased value.
Returning to the eigenmode spectra in Fig. 4.9, the fixation of d, ̺,
and cl to their experimental values reduces the number of adjustable
parameters in the single phonon scattering calculations (section 3.4.1) or
FEM (section 3.4.4) into one (ct).
Scattering cross-section calculations were performed by my coworker
R. Sainidou.[73, 95] Utilizing reasonable values for ct, the first three modes
could be clearly assigned as the (1,2), (1,3), and (1,4)-mode, in analogy
to the results found in polystyrene and silica samples in section 4.2.1.
However, the fourth signal, which is much broader than the others, can-
not be assigned to the expected (1,5)-mode, since it appears at higher
frequencies. Indeed, its frequency is found between those of the (1,5)
and the (1,6)-mode, indicating that the signal is a superposition of both
modes. This explains also the broad appearance of the signal, although
a fit with two Lorentzians does not converge. Therefore, instead of pre-
tending a meaningful fit of two individual modes, one must be content
to show the signal in between these two modes. The ‘fifth’ mode, on the
other hand, can be assigned as the (1,7)-mode with acceptable accuracy,
by choosing ct given by the mean square fit of only the first three modes.
In fact, for this last mode, the deviation from the theory decreases with
increasing amount of nBA in the copolymer.
All experimental modes are shown in Fig. 4.10 together with the fit-
ted values for fd, using ct as the only floating parameter. The linear
fits have no theoretical meaning, however, they follow the trend of most
signals quite well, although the change between 0 wt% and 10 wt% does
not follow the linear trend for the first two modes. For the (1,7)-mode
(‘mode 5’), however, the change between 20 wt% and 30 wt% is quite
large regarding the virtually unaltered fd-values between pure PMMA
and 20 wt% nBA. Correspondingly, the relative deviation between exper-
iment and theory is largest (≈4%) for the (1,7)-mode and pure PMMA.
94
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
  
1350 2000
(1,2) (1,3)
1300 1900
1250 1800
1200
mode 1 1700 mode 2
1150   
2500 0 10 20 30 0 10 20 30
 (1,4) 3400  (1,5)
2400 3200 (1,6)
2300 3000
2200 mode 3 2800 mode 4
  
0 10 20 30 4000 0 10 20 30
3900 (1,7) 5
3750 3000 4
3600 32000
2
3450 mode 5
1000 1
0 10 20 30 0 10 20 30
nBA [wt%] nBA  [wt%]
Figure 4.10.: fd against the amount of nBA in the copolymer individually for all
four samples for each of the five observed signals and in the bottom right panel
together for all signals together. The number of the mode corresponds to the
number given in Fig. 4.9 next to the dotted vertical lines. The open blue circles
belong to the theoretically obtained frequencies for the (1,l)-modes with l=2-7 (full
circles for the (1,6)-mode), denoted in the top right corners of each panel. The
linear fits in red are guides for the eyes. Error bars are 2%.
On the other hand, a linear fit through only the 20 wt% and 30 wt%
points captures the calculated points with a deviation <2%.
From the experimentally obtained material parameters, in combina-
tion with the fitted values for ct, it is possible to calculate the elastic
95
f*d
    
   
4. The Vibrations of Individual Colloids
moduli of the mesospheres and by that to quantify their softening due
to the addition of nBA. The calculated values for the elastic moduli as
well as for Poisson’s ratio can be found in Tab. 4.3.
Table 4.3.: Fitted ct, elastic moduli (Young’s, longitudinal, and shear modulus),
and Poisson’s ratio.
wt% nBA c / ms−1t E / GPa M / GPa G / GPa σ
0 1530 7.14 9.07 2.80 0.277
10 1520 6.88 8.46 2.72 0.262
20 1466 6.35 7.95 2.50 0.271
30 1394 5.68 7.15 2.23 0.274
As expected, all moduli go down with increasing amount of nBA, in-
dicating the strong softening effect (by ≈20% between pure PMMA and
the softest copolymer). The more striking feature is the constance of
Poisson’s ratio σ. In a copolymer with increasing amount of a rubber-
forming monomer (and consequently strongly decreasing Tg), one could
expect a more rubber-like behavior of the copolymer, i.e., an increasing
σ. At room temperature, i.e., about 30 ◦C below the lowest Tg, however,
this is not the case. This finding means that the compressibility charac-
teristics at room temperature are virtually unaffected by the softening,
there is no shift towards rubber-like behavior below Tg. In turn, this
result corroborates the ‘trick’ applied in the beginning of this section,
when introducing the effective glassy sound velocity for PnBA at room
temperature to fit Wood’s law.
In summary, this section contains a systematic study on the softening
of mesoscopic PMMA spheres by copolymerization with nBA with the
help of BLS bulk and eigenmode measurements. The elastic moduli are
quantified. The Poisson effect is virtually unaffected by the softening.
This is described by an ‘effective glassy sound velocity’ of the homopoly-
mer PnBA at a temperature at which pure PnBA is a rubber.
96
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
4.2.4. The Influence of the Wave Vector: Suspensions
In Section 4.2.1 I pointed out that, theoretically, the scattering intensity
of the individual modes should be strongly q-dependent, however, due
to the strong multiple scattering all q’s corresponding to values between
zero and backscattering contribute equally to the spectrum and hence
the intensity distribution is the same independent from the scattering
angle θ. Based on symmetry arguments, Montagna showed that for a
given n the BLS intensity of modes with increasing l are only active
for increasing values of qd.[52] In fact, for a given system, the intensity
maximum of each mode (n,l) should appear at a different q, i.e., for a
transparent system that allows the observation of eigenmodes at different
scattering angles θ, while the angle of maximum intesity θmax increases
with l.
It was shown first in 2003 by Penciu that BLS can be used to measure
the eigenmodes of silica colloids suspensions in a refractive index match-
ing liquid (cyclohexane / decalin).[57, 95] By measuring the suspension in
an NMR-tube the sample can be regarded as quite thick, which allows
to see also eigenmodes that are not resolved anymore in the systems pre-
sented in Chapter 5. It is shown that the appearance of the eigenmodes
depends on the filling fraction of the silica spheres in the suspension. In
fact, quite high filling fraction are needed to resolve the eigenmodes at
all. Those have been achieved by centrifugation of the suspension at the
bottom of the NMR-tube. The graphs in Fig. 7 of Ref. [95] show the
resolved modes as a function of qd for two different silica particles with
diameter d=328 nm and 250 nm, respectively, at three different filling
fractions (φ=0.3, 0.62, and 0.67). For all systems, an effective medium
acoustic branch is observed that will not be further regarded in this sec-
tion. Besides that, there are up to four modes that appear at the same
frequency for each qd, resonance modes of the silica spheres. While at
φ = 0.3 only one such mode is found, in the polycrystalline or glassy
samples with significantly higher φ more modes can be distinguished.
Although in Ref. [95] some exemplary spectra at different qd give evi-
dence that the q-dependence of the BLS intensity can be shown by these
97
4. The Vibrations of Individual Colloids
Figure 4.11.: Exemplary BLS spectra of polycrystalline silica colloids (d=375 nm)
in SR256 for seven different scattering angles, corresponding to 3.3≤ qd ≤11.6, at
two different free spectral ranges (a 40 GHz, b 15 GHz). The vertical lines denote
the approximate frequencies of the eigenmodes.
samples, a systematic investigation is still missing. Therefore, in this sec-
tion the results of BLS measurements on silica particles with d=375 nm in
an index matching matrix of liquid ethoxy-ethoxyethyl acrylate (SR256)
are presented in order to bridge this gap of knowledge. The sample was
synthesized by my coworker D. Kiefer at DKI, Darmstadt, with an initial
filling fraction φ=0.35. The dispersion relation for this filling fraction is
presented in section 5.4.1. Here, I will focus on the polycrystalline sam-
ple that was obtained from the dispersion by centrifugation for 90 min
at 4000 rpm (φ ≈0.70-0.74).
Fig. 4.11 shows BLS spectra taken from the polycrystalline sample
at seven different angles (3.3≤ qd ≤11.6) for two different free spectral
ranges. The eigenmodes are marked by vertical lines, the dispersion
relation is shown in Fig. 4.12. While for the two lowest eigenmodes the
deviation from the mean varies by about ±1 GHz, the distribution of
frequencies for the other resonance modes is very narrow.
Though, when carrying out a more detailed investigation of the spectra
in Fig. 4.11, it becomes clear that the relative intensities of the individual
98
4.2. Elastic Vibrations in Homogeneous Polymer Colloids
 
25
20
15
10
5
0
0.0 2.0 4.0 6.0 8.0 10.0 12.0
q*d
Figure 4.12.: Eigenmode frequencies as a function of qd extracted from the spectra
in Fig. 4.11. The dashed lines represent the mean frequency for each eigenmode.
The solid line shows a linear fit of the longitudinal acoustic branch (open circles)
(cf. section 5.4.1).
resonances vary systematically with the scattering angle θ. While the
lowest frequency eigenmode around 5 GHz is strongest at low θ, the
highest frequency modes appear strongest at high angles, qualitatively
conforming the theoretical predictions.[52] The lowest mode is strong at
θ=30◦ (qd=3.3), weak at 90◦ (qd=9.0) and nearly disappears at 110◦
(qd=10.4). The second mode around 9 GHz only appears between 50◦
and 90◦, while it is invisible at lower and higher qd. Also the higher
frequency modes could be only foreboded in the spectrum at 30◦. At
50◦ and 70◦ the third and fourth mode start to appear weakly, the fifth
mode quite feeblishly at 70◦. Especially for the fourth and fifth mode,
99
f / GHz
 
4. The Vibrations of Individual Colloids
intensity increases clearly up to 110◦, while the difference between 110◦
and 130◦ is only small.
Although a full theoretical description of the q-dependence is missing
for the present system, the qualitative agreement of the experiment with
the predictions is a striking hint of their correctness.
4.3. Elastic Vibrations in Nanostructered Colloids
4.3.1. The Influence of the Temperature: PS-SiO2 Core-Shell
Particles
Core-shell particles exhibit properties that may be substantially different
from those of the templated core. These include their density, mechanical
stability, and optical characteristics. Proper design of core and shell
material provides the possibility to optimize the properties of core-shell
particles according to their applications, thus making them attractive
from both a scientific and a technological point of view. Silica shells
are desirable because of excellent biocompatibility and easily tunable
surface functionality.[106, 107] Furthermore, silica shells are much stiffer
compared to most organic shells.[108, 109] Here the rigidity of thin glassy
silica shells is utilized for the realization of shape persistent polymer cores
at temperatures well above the polymer glass transition temperature Tg.
In this section BLS is employed to investigate the shape-persistence
of spherical core-shell colloids consisting of a soft polystyrene (PS) core
with d=400±12 nm and a hard silica (SiO2) nanoshell with thickness
L=37±3 nm against heating; this size was determined by scanning elec-
tron microscopy, averaging over about 100 spheres. The synthesis of the
core-shell particles by soap-free emulsion polymerization of the PS core
(slightly copolymerized with divinylbenzene) and subsequent coating by
silica is described elsewhere.[109] Here, it is shown that a relatively thin
SiO2 shell acts as a ‘nano-armor’ of the polymer core even in the molten
state above its glass transition temperature while the particles retaining
remarkably high mechanical strength.
Herein, the strong localization of the vibration modes in core-shell
100
4.3. Elastic Vibrations in Nanostructered Colloids
 
 PS 400 nm
 PS-SiO2 CSP
-10 -8 -6 -4 -2 0 2 4 6 8 10
f / GHz
Figure 4.13.: Eigenmode spectra of the uncoated d=400 nm PS spheres (black,
top) and of the PS-SiO2 core-shell particles (grey, bottom) at room temperature.
The (1,2)-mode (arrow) is the strongest.
particles with high elastic contrast is utilized to demonstrate the shape
robustness of polymer based colloidal particles upon heating well within
the fluid phase of the core. Both sound velocities (cl, ct) are sensitive in-
dex of T ,[36, 49, 110, 111]g due to the rapid increase of compressibility (C
−1
11 )
and decrease of shear rigidity (C44). Hence, tracking the frequency of a
distinct vibration mode (Eq. 2.106) with increasing temperature senses
the decrease of the mechanical strength and marks the glass transition in
the f−T diagram if the particle shape is preserved. Thus, BLS measures
single sphere mechanical properties.
For the BLS experiment, an opaque dry film of randomly packed
PS-SiO2 with a total thickness of several tens of microns was used.
[32]
Fig. 4.13 shows high resolution BLS eigenmode spectra of the PS-SiO2
and the parent plain PS spheres (d=400 nm) recorded at room temper-
ature. Both spectra exhibit several clearly distinguishable modes, with
the lowest frequency mode being the strongest spectral line. The com-
parison of the line shapes of the two spectra shows that the spectrum
101
Intensity / a.u.
 
4. The Vibrations of Individual Colloids
Figure 4.14.: a Exemplary eigenmode spectra during heating and after cooling
(side arrows). The solid lines in the Stokes side of the spectra indicate the repre-
sentation of the two low frequency modes by Lorentzian spectral lines (green, sum
red). The dashed line indicates the position of the (1,2)-mode at room tempera-
ture. b SEM pictures of the core-shell particles before and after heating to 165 ◦C
and cooling back to room temperature.
of the core-shell particle is very akin to that of the bare PS spheres.
The low frequency eigenmodes are therefore core-like, i.e., the signals
are related to vibrations localized in the PS core. In the inverted case of
core-shell particle with hard silica core and soft polymer shell only shell-
like modes were identified.[12] Both cases can be explained by the fact
102
4.3. Elastic Vibrations in Nanostructered Colloids
that the BLS signal increases with the compressibility of the medium in
which the elastic energy of the mode is localized, i.e., the scattering from
a hybrid particle structure is dominated by the contribution of the soft
component.
To verify that the softening of the core is monitored while the shape
of the particle remains unaltered, I followed the T -dependent eigenmode
spectra, focusing on the first (and strongest) signal of the core-shell par-
ticles (arrow in Fig. 4.13). This is assigned to the (1,2)-mode by compar-
ison with the resonance modes of the computed DOS spectra as reported
for soft (PS, PMMA) and hard (silica) spheres.[32, 45, 46] Due to the small
expansion coefficient of fused silica (5.5·10−7 K−1), the diameter d in
Eq. 2.106 is virtually temperature independent.
The heating experiment was performed at a non-arbitrative scattering
angle of 50◦ while the sample was encased in a glass windowed oven (cf.
Fig. 2(b) in Ref. [49]). The sample was heated from room temperature
to 165 ◦C in steps between 30 K (low T ) and 10 K (high T ) within a
few minutes for each step. After equilibrating the sample for 15 min
at each temperature, the eigenmode spectrum was recorded for about
15 min probing always the same spot (∼50 µm). This ensures that even
the intensities of the main modes can be traced as a function of tem-
perature. Fig. 4.14a shows eigenmode spectra at different temperatures
upon heating and after cooling over a narrower frequency range than in
Fig. 4.13, in order to further boost the resolution. All recorded spectra
have been represented in the range of the (1,2)-signal by a Lorentzian
using its frequency, line width, and amplitude as adjustable parame-
ters. The (1,2)-mode is red shifted by ≈0.2 GHz at 165 ◦C relative to
its frequency at room temperature either before (bottom) or after (top)
heating, indicating a small decrease of shear modulus (C44). Note that
the same mode undergoes a much smaller shift at 70 ◦C.
The experimental frequencies f(1,2) shown as a function of tempera-
ture in Fig. 4.15 are well represented by two distinct straight lines, ex-
pectedly both lines with negative slopes due to decreasing rigidity. The
characteristic kink in the f −T diagram signifying the transformation of
a glassy into a rubbery state occurs at 107±3 ◦C, which is slightly higher
103
4. The Vibrations of Individual Colloids
a) b)
Figure 4.15.: a The variation of the frequency of the (1,2) vibration eigenmode
(arrow in Fig. 4.13) of the PS-SiO2 core-shell particle with temperature indicating
two linear regimes (solid lines). The shaded area embraces the glass transition
temperature of the PS core. b Experiment analogue to a, but with no crosslinker
(DVB) in the PS core. (b: dPS=480 nm + 60 nm SiO2-shell)
than the Tg of bulk amorphous PS (100
◦C) due to the crosslinking of the
PS core with divinylbenzene (DVB).[112] The presence of a single Tg in
Fig. 4.15 indicates a homogeneous, over few hundreds nanometers, core
environment and hence this value of Tg is further confirmed by differ-
ential scanning calorimetry (DSC) measurements on the bare PS cores
and the PS-SiO2 particles. In both systems Tg was found to amount to
107.5±3 ◦C, whereas in the absence of crosslinking Tg lowers to ≈100 ◦C
(see Fig. 4.15b), in agreement with DSC. The clear detection of the glass
transition affirms the validity of the obtained findings.
The relative drop of f(1,2) between room temperature and Tg is less
than 2% in the PS-SiO2 particles and more than 6% in the bare PS cores
(cf. Fig. 4.16). In the rubbery state of the core, f(1,2) at the highest
examined temperature (165 ◦C) is decreased relatively to its value at Tg
only by about 6%. Assuming a constant density, this f(1,2) value corre-
sponds to a shear modulus c44 ≈1.45 GPa in the confined polymer melt at
about Tg+60 K. For comparison, a corresponding decrease in the trans-
verse sound velocity of PS films is about 40% (i.e., c <1 GPa).[49]44 This
104
4.3. Elastic Vibrations in Nanostructered Colloids
remarkable small temperature effect on the rigidity of the present three-
dimensionally confined PS-SiO2 particles as a result of the increased
pressure in the core is unprecedented.
Even at temperatures far above T , the spectra (Fig. 4.14a at 165 ◦g C)
still possess the typical shape for T<Tg with minor changes in the in-
tensity. This proofs that even in the rubbery regime the PS core keeps
its spherical shape. The thermal expansion of the PS core is expected
to increase by more than a factor two above Tg and hence the pressure
on the silica shell.[113] Especially at temperatures well above Tg, it is
this pressure that could, in principle, break the shell. However, when
heating up to 165 ◦C, the intensity of the observed mode changes only
very little within the experimental error. After cooling back to room
temperature, not only the frequency (dashed line in Fig. 4.14) but also
the intensity and the linewidth (≈0.39 GHz full width half maximum)
assume their initial values before heating. Thus the thin silica shell can
stand the higher pressure under these conditions and prevents the PS
core to change its shape. This notion is corroborated by the scanning
electron microscope (SEM) photographs of the core-shell particles before
and after the heating cycle to 165 ◦C shown in Fig. 4.14b. The lack of
visible defects after the heating procedure confirms the results obtained
by the BLS. Actually, the silica shell acts as an armor for the PS core.
In Fig. 4.16, the corresponding BLS experiment on bare PS parti-
cles is presented. These eigenmode spectra are very sensitive to the
particle shape. For the bare PS particles f(1,2) displays stronger de-
crease with temperature than for the PS-SiO2 core-shell particles (inset
to Fig.4.16) due to the absence of the confinementand partially to the
now unsuppressed expansion of the PS spheres, i.e., decrease of c44 and
increase of d in Eq.2.106. Already below Tg the line shape of the (1,2)-
mode changes and above Tg the signal vanishes completely within a few
minutes. Starting at room temperature, for increasing temperature a
shift to lower frequencies and a broadening/splitting of the signal is ob-
served. The spectral shape of the (1,2)-mode (around 2.5 GHz) in the
BLS spectra of the bare PS core in Fig. 4.16 severely changes as the
temperature increases towards Tg. At temperatures above 85
◦C, the
105
4. The Vibrations of Individual Colloids
 
2.70
108 °C (ns)
2.55
2.40
30 60 90 106.5 °C (ns)
T / °C  
 103.7 °C
101 °C
 97 °C
 85 °C
 53 °C
 23 °C
-4 -3 -2 2 3 4
f / GHz
Figure 4.16.: Eight exemplary spectra for BLS eigenmode measurements on the
bare d=400 nm PS particles as a function of temperature: The two lowest reso-
nances are fitted by Lorentzians. The dotted line denotes the center of the lowest
eigenmode at room temperature. Measurements above 105 ◦C are performed with
a new sample (ns) rapidly heated to these temperatures. The inset in the top left
shows the f -T diagram for T≤Tg in analogy to Fig. 4.15.
single Lorentzian shape initially splits into two through the appearance
of a second peak at the high frequency side. Finally, above 103 ◦C the in-
tensity decreases rapidly and eventually vanishes at 108 ◦C within a few
minutes. Apparently, shape alterations start already below Tg because
of surface melting and breaking the particle’s spherical symmetry. With
106
f / GHz
   Intensity /  a.u.   
 
       
4.3. Elastic Vibrations in Nanostructered Colloids
the onward and patchy destruction of the spherical shape, the resonance
modes of the unarmed PS particles finally disappear, and the destruc-
tion of the spheres is, of course, irreversible. SEM pictures show only
an undefined surface. Probing the shape of the PS mesospheres on the
average during the melting process is another interesting but demanding
task. In the context of the present study, it would require finite element
modelling of all eigenfrequencies supported by a temperature dependent
SEM or atomic force microscopy.
In summary, the morphological and thermo-mechanical response to
heating of PS-SiO2 core-shell particles at submicrometer length scales
is probed by BLS that delivers quasi in situ insight into the individ-
ual colloids. By monitoring the eigenmode spectra, it was shown that
silica shells of a few tens of nanometers protect a polymer core from
changing its spherical shape even at temperatures well above Tg, which
can be determined from the same experiment. The shape persistence of
these hybrid systems is accompanied by a remarkable enhanced rigidity
of the confined core above its Tg. The inevitable change of the amor-
phous packing inside the rigid nanocontainers will likely prove useful for
fundamental studies.
4.3.2. The Influence of the Components: SiO2-PMMA
Core-Shell Particles
Colloid science can create novel materials that possess spatial variation
of density and elastic constants at the nanoscale but their mechanical
characterization remains difficult. Similarly, the influence of the parti-
cle architecture (core-shell spheres or hollow capsules), size, and shape
on the eigenfrequencies, and the localization of elastic energy in spe-
cific regions of an individual particle is not known. In addition to the
characterization on the nanomechanical properties, tailored acoustic con-
finement will be important for precise phonon management by structural
engineering.
In this section, BLS is employed for the first measurement of the
resonant modes (the ‘music’) in sub-micron core-shell spheres (silica-
107
4. The Vibrations of Individual Colloids
poly(methylmethacrylate) − SiO2–PMMA) having constant core radius
and varying shell thickness and in the corresponding spherical PMMA
hollow nanoshells after dissolving the silica core. We observed up to
nine particle vibrational frequencies with increasing size of the core-shell
spheres and revealed the strong impact of the empty core of the hol-
low capsules on the size dependence of the resonance frequencies. The
observed vibration eigenfrequencies are identified by detailed and thor-
ough numerical calculations as the resonance eigenmodes of the individ-
ual spheres. The good overall agreement with the experiment allowed
to determine the core density and the two elastic constants of both con-
stituents in the hybrid particles. We found a significant deviation of these
material properties in the nanostructured hybrid spheres from their val-
ues in the macroscopic bulk systems, which underlines the importance of
such measurements at the nanoscale. The simulation of the displacement
fields of the different elastic modes allowed to visualize their localization
in different regions of the hybrid particles and provides a deeper insight
into their origin. These first findings illustrate qualitatively general fea-
tures of the localization of the elastic energy in nanostructured colloids
beyond core-shell particles.
The SiO2–PMMA particles with rigid silica cores and softer PMMA
shells were prepared by my coworker Dr. P. Spahn (DKI, Darmstad) in a
two-step process starting with the Stöber synthesis of the core followed by
emulsion polymerization of the shell.[67] The silica core, with a diameter
of 181± 3 nm, was coated with PMMA spherical shells of three different
thicknesses (in average 25, 57, and 112 nm), leading to core-shell particles
of final (outer) diameter ranging from d=232 nm to 405 nm. All samples
were characterized by scanning electron microscopy (SEM), using a 1530
Gemini SEM by LEO with acceleration voltages setup between 0.2 kV
and 1.0 kV. Fig. 4.17a shows exemplary details for the uncoated silica
cores and the three core-shell particles. Although there is some degener-
ation of the PMMA by the electron beam at 1 kV or some blurring using
lower voltage (200 V), the particles’ diameters d can be determined by
averaging the software-aided gauged diameters over about 100 spheres
for each sample. The size polydispersity is about 5%, which is confirmed
108
4.3. Elastic Vibrations in Nanostructered Colloids
a)                                                        b)
Figure 4.17.: SEM-images of a the bare silica (left, d = 181 nm) and silica–
PMMA core-shell particles (from right: d=405 nm, 294 nm, 232 nm), and b the
hollow PMMA capsules (bottom left: d=232 nm, top right: d=294 nm). For the
smaller particles in b, holes in the spheres are observed, whereas there are nearly
no defects in the thicker shells.
by the formation of crystalline films (Fig. 4.17a) after vertical lifting de-
position from the particle suspensions (cf. section 3.1).[114] Fig. 4.17b
shows SEM images of the spherical hollow capsules with d=232 nm and
d=294 nm. After dissolution of the silica core with aqueous hydrofluoric
acid, the diameters are found to be unchanged within the experimental
error. For the smaller hollow capsules (bottom left) a considerable frac-
tion (approximately 50%) of the particles possess holes, while the 294 nm
(top right) and the 405 nm (not shown) hollow shells are nearly defect
free due to their thicker shells. Notably these defects in the 232 nm
hollow capsules lead to a broadening of the spectral lines, but do not sig-
nificantly change their peak positions in the BLS spectra (see Fig. 4.20a
below). The sizes obtained from the SEM were confirmed by the hy-
drodynamic radii, Rh, of the core-shell particles in dilute suspension
measured by photon correlation spectroscopy (section 3.3.1). Within
3%, Rh amounts to 214 nm for bare silica and 262 nm, 352 nm and
504 nm, respectively, for the three core-shell particles. These values are
expectedly higher than the geometric radii R measured by SEM since
for homogeneous spheres the ratio R/Rh assumes the value of 0.78.
[115]
109
4. The Vibrations of Individual Colloids
a)
f)
Figure 4.18.: a BLS eigenmode spectra of bare silica (top) and the three core-shell
particle films. The representation of the BLS spectra with up to nine Lorentzians
(solid lines) is shown on the Stokes-side. (Note the different frequency scale for the
spectrum in the top.) b-e Enlarged Stokes-sides of the spectra from a (b: pure sil-
ica, c-e: core-shell). The experimental frequencies are accented by orange spheres,
the small vertical lines denote the corresponding calculated resonance frequencies,
each of them characterized by its angular momentum, l, shown at the top of the
lines. All experimental and theoretical (blue bars) values are summarized in f, with
the dotted lines connecting modes of the same angular momentum l; n=1 for all
observed modes.
Experimentally, R/Rh is found to be slightly higher, varying between
0.80 and 0.85. For the BLS experiment, films of all seven particles were
prepared on a thin glass substrate using the vertical lifting technique.[8]
In analogy to the experiments in the previous sections, Fig. 4.18a shows
the q-independent BLS spectra of the bare silica particles and the three
110
4.3. Elastic Vibrations in Nanostructered Colloids
core-shell particles. The spectra can be well represented by up to nine
Lorentzian line shapes as shown by the solid lines on the Stokes side of the
BLS spectra. The peak position of the spectral lines yields the resonance
frequencies of the eigenmodes indicated by solid circles in Fig. 4.18b-e
for the four particles. Advantageously, BLS can, in principle, record all
thermally excited modes within one measurement, which is not possible,
e.g., in the pump-probe technique. For the bare silica spheres, only two
resonance frequencies at about 13 GHz and 19 GHz can be resolved.
However, with increasing the PMMA-shell thickness of the core-shell
particles (d=232 nm, d=294 nm and d=405 nm), the BLS spectra become
richer as it was observed for pure polystyrene colloidal particles.[46] For
the particle with the thinnest shell a third weak peak is discernible in the
spectrum in the inset of Fig. 4.18c, whereas five modes are observed in
medium-thickness shell (Fig. 4.18d), and even nine modes are resolved
in the BLS spectrum of the thickest shell (Fig. 4.18e). The increased
number of the resolved modes in the BLS with increasing particle size
relates to the intensity of the resonance signals which depends on qd. In
the present case of strong multiple scattering q ≤ 2ki (the backscattering
vector), and the number of resolved modes increases with the 2kid as
discussed in section 4.2.1.[52]
The elastic parameters (longitudinal and transverse velocities, cl and
ct) of the two constituents (core- and shell-materials) are not a priori
known for such nanostructured systems. An access to these material
properties at these length scales and high frequencies is important since
they can considerably differ from their values in macroscopic systems.
The elastic constants are frequency dependent, and BLS specifically
yields their limiting high frequency values, which relate to local pack-
ing and interactions, as well as, the glass transition temperature.[49] The
detection of more than two particle elastic excitations in the experimen-
tal (BLS) spectra of Fig. 4.18 allows for an unambiguous determina-
tion of the elastic moduli, shear modulus G = ̺c2t and Young modulus
E = ̺c2l (1 + σ)(1 − 2σ)/(1 − σ), with σ = (c2l − 2c2t )/[2(c2 2l − ct )] be-
ing the Poisson ratio and ̺ being the mass density. The experimental
values of the resonance frequencies are compared with the resonance fre-
111
4. The Vibrations of Individual Colloids
quencies obtained from the calculated density of states (DOS) spectra
of a single constituent sphere of the experimental systems. The theoret-
ical computations were performed by Dr. R. Sainidou (Univ. le Havre)
using a formalism, appropriately developed for this case and presented
elsewhere.[76, 116] Each resonance mode appearing at frequency f(n, l) in
these DOS spectra is characterized by the angular momentum l, imposed
by the spherical symmetry of the particle, where n denotes the n-th or-
der solution for a given l. All the shell-localized modes reported in this
section have n=1. The materials elastic parameters (cl, ct) and densities
are used as adjustable parameters in order to achieve the least devia-
tion between theoretical and experimental eigenfrequencies. Obviously,
in the theoretical calculations the constituent spheres are considered as
homogeneous and isotropic, and their elastic coefficients are frequency-
independent.
First, the bare silica particles are considered (Fig. 4.18b). The two
sound velocities treated as adjustable parameters are obtained from rep-
resentation of the two experimental frequencies (solid circles) by the
calculated resonance frequencies (small vertical lines in Fig. 4.18b). The
obtained sound velocities for the bare-silica particles, c =4420 ms−1l and
ct=2780 ms
−1, are significantly lower (≈25%) than the values of dense
bulk amorphous silica (cl=5970 ms
−1, ct=3760 ms
−1, ̺=2200 kgm−3),
indicating the presence of porosity in these particles.[97] It also underlines
the necessity of the BLS experiment to determine these values avoiding
erroneous assumptions. The mass density does not sensitively affect the
DOS spectra, due to the huge impedance difference between silica and
air. Nevertheless, these porous silica spheres should be less dense than
bulk silica (see below).
Next, the PMMA coated silica particles consisting of same silica cores
are considered. For a first description of the DOS spectra, representing
the experimental frequencies seen in the BLS spectra of Figures 4.18c-e
(solid circles), the elastic constants measured for the bare silica parti-
cles were used and the values for the sound velocity and density of the
PMMA shell were fixed on the values for bulk PMMA (c =2800 ms−1l ,
ct=1400 ms
−1, ̺=1190 kgm−3).[33] However, this choice for the set of
112
4.3. Elastic Vibrations in Nanostructered Colloids
the elastic parameters of both the silica core and the PMMA does not
quantitatively represent the experimental resonances in the BLS spectra
of Figures 4.18c-e.
A systematic theoretical analysis based on the DOS spectra of the
three core-shell spheres has shown that changes in the elastic parameters
of both materials of the hybrid particles are required. Notably, it turned
out that for the silica core one must assume sound velocities, which are
about 3% higher than in the bare silica particle, i.e., cl,c=4540 ms
−1 and
c =2860 ms−1t,c . This hardening of the silica core is probably due to
a partial infiltration of methymethacrylate in the pores and subsequent
polymerization to PMMA during the formation of the shell. Due to
the reduced impedance contrast between the silica core and the PMMA
shell relative to the bare silica spheres vs. air, the core mass density ̺c
has now a substantial influence on the DOS calculations. For a given
density ̺c=1900 kgm
−3, the sound velocities in the PMMA shell are
cl,s=3080 ms
−1 and ct,s=1540 ms
−1, i.e., about 10% higher than in bulk
PMMA; the PMMA mass density was kept at the bulk value. Therewith,
nearly all measured signals can be identified as spherical eigenmodes with
angular momentum l and can be captured quantitatively by the theory
within about 3%. The calculated resonance frequencies are shown by
small vertical lines in Figures 4.18c-e and summarized in Fig. 4.18f along
with the corresponding experimental values. The Young modulus E and
the shear modulus G are directly accessible for both the core and the
shell components.
In order to obtain an insight into the nature of the experimentally ob-
served modes (Fig. 4.18f), the elastic field at the resonance frequencies in
the region of the sphere was calculated, assuming a longitudinal acoustic
plane wave of the same frequency, impinging on the sphere (cf. section
3.4.2). An example of elastic-field intensity plots is given in Fig. 4.19 for
the case of the thickest PMMA shell sample (d=405 nm). Their topology
is that of a field-intensity having 2l maxima on the internal interface of
the outer circumference of the core-shell particle: the two of them are
strong (global) maxima along the direction of the incident field, while the
rest 2(l − 1) are weaker and distributed equidistantly along the circum-
113
4. The Vibrations of Individual Colloids
Low
High
Low
0
High
Figure 4.19.: a-c Elastic-field distribution for the first three resonances (n=1) of
the 405 nm core-shell particle (Fig. 4.18e) in a cross section through the center of
the sphere. Plane wave incidence is assumed along the horizontal axis, from the
left. d An example of core-localized mode with l=1 (n=2) at 9.3 GHz, which is
not observed experimentally. Plots e and f give, respectively, the real part of the
radial and polar component of the elastic field for the case shown in c, with the
arrows visualizing the vibrational mode of the shell.
ference. The vibration of the observed modes is radial at the maxima
changing alternatively direction (outwards or inwards) from maximum
to maximum. In the regions of minima the field is directed tangentially.
Based on Fig. 4.19, the experimentally observed modes (Fig. 2f) are shell-
localized with l = 2− 4. Core-localized modes either do not exist within
the considered frequency range (films with d=232 nm and 294 nm), or
they are not observable in the experimental BLS spectra (thickest-shell
case, Fig. 4.18e), probably due to the very strong localization (virtually
delta-functions in the DOS spectra). It is worth mentioning, however,
114
4.3. Elastic Vibrations in Nanostructered Colloids
Figure 4.20.: a BLS eigenmode spectra of spherical hollow PMMA capsules with
d=232 nm and d=294 nm. b Comparison between calculated (solid lines, different
colors for the individual modes) and experimental (spheres) resonance frequencies
for the hollow capsules (n=1).
that the elastic parameters affect the frequencies of both core- and shell-
like eigenmodes.
Fig. 4.20a shows the BLS spectra of two hollow capsules (d=232 nm
and 294 nm), akin to those of double-shelled hollow carbon microspheres
published recently.[117] By comparison of the two spectra, it is appar-
ent that the thicker hollow capsules display significantly sharper peaks
than the thinner hollow capsules. For solid spheres, the line width of the
spectra is associated with the size polydispersity.[46] For the thin hollow
spheres (d=232 nm), however, an additional source for the line broaden-
ing is the significant fraction of particles with holes as seen in Fig. 4.17b.
These defects are neither uniform in size nor in shape and hence cause
a distribution of the elastic properties that further broadens the experi-
mental spectra (Fig. 4.20a). In order to identify the nature of the experi-
mental eigenmodes in the two hollow capsules, the resonance frequencies
of shell-localized modes were computed, adopting for the PMMA shell
the values of the elastic moduli obtained from the representation of the
eigenfrequencies of the core-shell particles (Fig. 4.18f). Fig. 4.20b shows
115
4. The Vibrations of Individual Colloids
the calculated (solid lines) eigenfrequencies for the modes characterized
by the angular momentum l=2-4 for hollow PMMA nanoshells with con-
stant inner diameter (181 nm) as a function of the (outer) particle di-
ameter d. The comparison with the experimental frequencies (orange
spheres) identifies the observed frequencies in the three hollow PMMA
spheres with the two lowest modes with l=2 and l=3. Interestingly, the
frequencies of these two modes vary very little with d due to two compet-
ing effects. At a constant particle diameter, the resonance frequencies of
hollow particles increase with shell thickness,[80] whereas the resonance
frequencies of filled spheres decrease with d since f(n, l) ∼ 1/d.[45, 46, 85]
Since in our case both shell thickness and total diameter increase simul-
taneously, the net effect is essentially the apparent insensitivity of the
eigenfrequencies to the d variation of the branches with l=2 and 3 seen
in Fig. 4.20b. Notably, the elastic parameters of the PMMA were not
affected by the core etching.
In summary, this section reports the first study of localized vibrational
excitations in silica-PMMA core-shell particles and PMMA hollow cap-
sules using the powerful optical technique of BLS, which is applicable for
turbid films. The BLS spectra show up to nine eigenfrequencies of the
core-shell particles, which sensitively depend on the particle architecture
and the mechanical moduli of the constituent parts. The values of the
Young moduli E and shear moduli G of the constituent components are
computed from the elastic parameters cl and ct and densities, obtained
for each component from the identification of the experimental modes
with the resonance modes appearing at f(n, l) for the n-th order of the
l-th harmonic in the calculated density of states (DOS) spectra. The
anticipated reduction of the moduli of the neat core compared to the
bulk material due to its porosity and subsequent subtle increase above
the bulk values upon grafting with PMMA chains are revealed. The
anchoring and confinement of the PMMA nanoscopic layer impact both
types of moduli which were found to exceed the bulk PMMA values.
The observed eigenfrequencies in the hollow capsules exhibit a peculiar
but apparent insensitivity to the variation of the diameter as a result of
antagonistic trends inferred by the DOS calculations, while the PMMA
116
4.4. Materials
elastic constants are not affected by removal of the silica core. In addition
to the fundamental understanding of localized elastic modes in hybrid
particles and the concurrent determination of the mechanical moduli, the
findings of this study will contribute to a rational design of nanostruc-
tured colloids with strong resonances that can be selectively excited.[27]
4.4. Materials
Homogeneous Polymer Colloids
Homogeneous polystyrene, poly(methyl methacrylate), and poly(methyl
methacrylate–co–n-butyl acrylate) colloids have been prepared by my co-
worker Dr. M. Retsch at MPIP by emulsifier-free emulsion polymeriza-
tion,[118] adapting recipes from the literature.[119–122] A general recipe
for the used particles is given by the following:[118]
“The polymerization reactions were carried out either in a 300 ml
reactor with a thermostat jacket or a 500 ml three necked flask. At
first water was heated to the reaction temperature. Then all additives,
monomers, and comonomers were added; solid chemicals were dissolved
in water prior to addition. Strong magnet stirring was used and the
chemicals were allowed to equilibrate for 5 min. Finally, the initiator was
added as aqueous solution. The reactor was flushed with N2 or Ar during
charging with the chemicals and a slight inert gas flow was maintained
throughout the reaction. Reaction times were typically between 12 and
20 h. [...] Latex and silica nanoparticles were either purified by several
cycles of centrifugation and redispersion, by ultrafiltration, or dialysis.
In all cases the purification protocol was adjusted to the particles size and
composition by means of centrifugation speed and duration, or filtration
and dialysis membrane molecular weight cut-off.”
PS Styrene (>99%, Aldrich) was washed three times with a 10% KOH
solution and three times with MilliQ water and then distilled under re-
duced pressure. Acrylic acid (AA, 99%, Aldrich) was distilled with-
out washing with KOH. Sodium 4-vinylbenzenesulfonic acid (NaPSS,
117
4. The Vibrations of Individual Colloids
90%, Aldrich) and potassium persulfate (KPS, 99%, Acros) were used
as received. By playing around with monomer, comonomer (KPS, AA),
and initiator concentration, M. Retsch could prepare monodisperse PS
spheres with diameter between 180 nm and 1 µm. All reactions were
carried out with 0.1-0.2 g KPS, 0.005-0.06 g NaPSS, and 0-0.3 ml AA in
250 ml H O at 70 ◦C or 80 ◦2 C.
PMMA Methyl methacrylate (MMA, 99%, Acros) was washed three
times with a 10% KOH solution and three times with MilliQ water and
then distilled under reduced pressure. Sodium 4-vinylbenzenesulfonic
acid (NaPSS, 90%, Aldrich), potassium persulfate (KPS, 99%, Acros),
and 2,2’-azobis(2-methyl propionamidine) dihydrochloride (ABA, 97%,
Aldrich) were used as received.
Cationic initiator (ABA) as well as an anionic system (KPS, NaPSS)
have been used. The particle diameter is found to depend mainly on the
initial MMA concentration.
PMMA-nBA Copolymers Methyl methacrylate (MMA, 99%, Acros)
was washed three times with a 10% KOH solution and three times with
MilliQ water and then distilled under reduced pressure. n-Butyl acrylate
(nBA, >99%, Aldrich) was run over a basic alumina column. Sodium
4-vinylbenzenesulfonic acid (NaPSS, 90%, Aldrich) and potassium per-
sulfate (KPS, 99%, Acros) were used as received. All reactions were
carried out with 0.2 g KPS and 0.02 g NaPSS in 250 ml H2O at 780
◦C.
Homogeneous Silica Colloids
There are principally two synthetic strategies to obtain monodisperse
silica colloids: A one-step batch process, which is also termed ‘Stöber’
process,[123, 124] and a seeded growth process that can be performed step-
by-step or continuously.[125–127] While the Stöber process is the simplest
and fastets method, the monodispersity obtained by that is less good
then that achieved using the seeded growth processes.
118
4.4. Materials
All processes avail themselves of the hydrolysis of tetraethyl orthosil-
icate (TEOS) to orthosilicic acid and subsequent condensation of the
silicic acid to poly(silicon dioxide) (silica) under alcalic catalysis (ammo-
niac, NH3(aq)) and strong magnetic stirring. As solvent a water/ethanol
mixture is utilized due to the poor solulbility of TEOS in water. The
temperature is chosen between room temperature and 60 ◦C.
In order to control the particle size over a wide range and achieve
high monodispersity by the same time, the typical approach is to start
with a sample of monodisperse particles from the Stöber process (smaller
diameter than the wished size) and then TEOS in EtOH and NH3(aq)
either in steps or in a continuous flow over several hours to days. It
is found that the diameter increases monotonically with the amount of
added monomer following a cubic root dependency and that the polydis-
persity decreases with increasing amount of added monomer, i.e., with
the size. Both, step-by-step and continuous process, deliver high quality
particles with comparable monodispersity.[118] The silica particles used
in this thesis have been synthesized by my coworkers M. Retsch at MPIP
and D. Kiefer at DKI, Darmstadt.
PS-SiO2 Core-Shell Particles
PS-SiO2 core-shell particles have been provided by my coworkers M.
D’Acunzi and G. Schäfer at MPIP. Therefore they started with the
soap-free emulsion polymerization of the polymer core and used them
as templates for the synthesis of the silica shell employing the Stöber
method.[109, 128] As the details are clearly and completely given in the
mentioned literature, here I just want to introduce the main principles
of the synthesis.
The synthesis of the PS core by soap-free emulsion polymerization
was carried out using ammonium persulfate as initiator. Divinylbenzene
(DVB) and acrylic acid were facultatively added as crosslinking (DVB)
comonomers. Monomers and Comonomers were dropwise added to a
stirred aqueous solution of the initiator ammonium persulfate. The cores
were cleaned by centrifugation.
119
4. The Vibrations of Individual Colloids
To ease the coating with silica, the PS spheres were treated with
polyelectrolytes (poly(allylamine hydrochloride)) and then functional-
ized with poly(vinylpyrrolidone) (PVP). The functionalized PS beads
were transfered into ethanol. To form the silica shell, ammonia and
tetraetoxysilane (TES) were added under stirring. The completed core-
shell particles were washed three times in ethanol.
SiO2-PMMA Core-Shell Particles
The SiO2–PMMA particles with rigid silica cores and softer PMMA shells
were prepared by my coworker P. Spahn at DKI Darmstadt in a two-
step process starting with the Stöber synthesis of the core followed by
emulsion polymerization of the shell.[67] The silica core, with a diameter
of 181± 3 nm, was coated with PMMA spherical shells of three different
thicknesses (in average 25, 57, and 112 nm), leading to core-shell particles
of final (outer) diameter ranging from d=232 nm to 405 nm.
120
5. Phononic Behavior of Colloidal
Systems
5.1. Introduction
The first papers on photonic effects by Yablonovitch[129] and John[130]
in 1987 stimulated over the years much theoretical and experimental
work on the propagation of electromagnetic waves through appropri-
ately structured materials and subsequently led to the birth of the new
research field ‘photonic crystals’.[1] The tremendous interest in photonic
crystals with specially designed periodic variations in dielectric constant
largely originates from their display of propagation band gaps for light.
The appearance of band gaps makes an advanced control over light prop-
agation possible and permits as well a series of novel optical phenomena
such as slowing and localization of light or negative refraction.[1] Soon af-
ter the discovery of photonic crystals, it was found that in analogy to the
electromagnetic waves, band gaps also exist for the propagation of acous-
tic waves, and the so-called phononic crystals[4–6, 40, 90, 131, 132] are the
elastic analogue of photonic crystals replacing the role of the dielectric
constant by the elastic parameters and density. Such an analogy exists
as a consequence of the common origin of the band gaps in both cases,
i.e., the destructive interference of Bragg diffracted waves in periodic
structures,[10, 133] and hence these gaps are also termed as Bragg gaps.
However, the different nature of electromagnetic and acoustic waves also
guarantees the existence of some important differences between photonic
and phononic phenomena. Unlike electromagnetic radiation that is char-
acterized as transverse waves, acoustic waves in general are full vector
waves with both longitudinal and transverse polarizations, and their
121
5. Phononic Behavior of Colloidal Systems
propagation depends additionally on the material density. Even for a
homogeneous and isotropic medium, the acoustic wave propagation is
governed by three parameters, the two Lamé coefficients and density,
in contrast to the single parameter, the dielectric constant, that deter-
mines the propagation of light. Evidently, the phononic phenomena are
anticipated to be more complex and rich.
For phononic crystals, the band diagram depends on several param-
eters such as the elastic constants and density of the component ma-
terials, symmetry of the lattice, shape of the inclusions, and the filling
fraction. The width of the band gap generally increases with the con-
trast between the densities and sound phase velocities of the component
materials, and the center of the gap can be tuned by changing the lat-
tice parameter.[131, 132] The search for phononic structures started with
two theoretical works in 1993, which predicted the existence of phononic
band gaps in periodic two-dimensional (2D) elastic composites of parallel
cylinders embedded in a host matrix.[131, 132] The experimental verifica-
tion of phononic band gaps followed few years later, realized in metallic
macrostructures with gaps at sonic or ultrasonic frequencies.[4, 5, 90, 134]
Further explorations also revealed a number of peculiar phenomena with
potential applications associated with acoustic wave propagation includ-
ing tunneling effect,[135] negative refraction and focusing,[136] double re-
fraction, etc.[137]
Theoretical calculation of phononic band diagram requires no specifi-
cation of the lattice constant of the structure or the corresponding wave
frequencies as long as the crystal is defined by the same set of frequency-
independent elastic parameters. In other words, a fundamental length
scale does not exist for phononic phenomena, which is a direct conse-
quence of the invariance of the wave equation of elasticity under the
simultaneous transformation of space coordinates and frequency. How-
ever, acoustic waves of different frequencies do bear distinct character-
istics, particularly when their applications are concerned, and therefore
in practice the frequency range of the waves of interest constitutes an
important consideration. Recently, growing attention has been paid to
hypersonic (GHz) phononic crystals and the first experimental observa-
122
5.1. Introduction
tions of the hypersonic Bragg gaps have been lately reported in three-
dimensional (3D) colloidal crystals made by self-assembly[8] and in 2D
polymeric porous structures with hexagonal symmetry created by laser
interference lithography.[91] Hypersonic waves, owing to their high fre-
quencies, display certain unique features that are not possessed by ordi-
nary acoustic waves such as being thermally excited, acting as the main
heat carrier in dielectrics, and interacting with electrons and photons in
a rich manner. Consequently, hypersonic crystals with the potential to
mold the flow of hypersound may be utilized to achieve high-level control
over many important physical processes involving heat transport and
complex phonon-photon or phonon-electron couplings. A detailed un-
derstanding of phonon propagation in hypersonic crystals thus becomes
important.
The fabrication of hypersonic crystals, compared with their sonic and
ultrasonic counterparts, is much more demanding as the dimension typi-
fying the structure has to be scaled from macroscopic down to sub-micron
scale. On the other hand, advance in relevant nanofabrication, partially
driven by the desire of creating various photonic structures of similar
dimensions, has offered some available means to achieve such a purpose
including, for example, holographic interference lithography,[138] direct
laser-writing,[139] two-photon polymerization,[140] or self-assembly.[17, 87]
Colloidal superstructures, self-assembled from colloidal particles, repre-
sent a promising material class for phononic applications. The matura-
tion of colloidal science enables the preparation of colloidal particles with
well-defined size and shape for a great many of materials ranging from or-
ganic to inorganic, thus providing abundant building blocks with varied
elastic properties. The progress in colloidal particle self-assembly allows
easy and cheap fabrication of large area high quality single crystalline
2D and 3D crystals as well as non-crystalline structures. The appear-
ance of binary[56] and ternary[58] crystals further enriches the available
structure types. By filling the interstitials between the colloidal parti-
cles with different materials, additional freedom in elastic parameters of
the system is provided. Moreover, in most cases the colloidal particles
possess a spherical shape due to surface tension effects, thus represent
123
5. Phononic Behavior of Colloidal Systems
a strong scattering unit when the elastic contrast between the particle
and surrounding becomes large, which may cause additional gap forma-
tion with origin different from Bragg diffraction as will be encountered
later.[33]
The experimental exploration of hypersonic crystals, however, faces
additional challenge in monitoring the phonons in such small structures.
Apparently, the commonly used sonic and ultrasonic transmission tech-
niques for macroscopic sized structures cease to work. It has been demon-
strated that Brillouin light scattering (BLS), which takes advantage of
the inelastic scattering of photons by thermally excited high frequency
phonons, represents a powerful tool to record the phonon dispersion re-
lation in hypersonic crystals.
This chapter deals with the phononic band diagrams of transparent
colloidal systems that can be measured by BLS. A short introduction
into the nature of the bands in the dispersion relation is given as well
as into the properties of the effective medium. Experimental results
are discussed treating the effective medium velocity of infiltrated defect
doped colloidal crystals and the nature of acoustic band gaps in phononic
systems, including the first realization of a hybridization gap.
5.1.1. The Phononic Band Diagram
The information about the propagation of elastic waves in periodic me-
dia is described by the band diagram. It contains the dispersion relation
frequency as a function of the wave vector, ω(k). Therefore the vertices
of the irreducible Brillouin zone (BZ) are connected with a line for each
acoustic mode appearing. Fig. 5.1a shows for example the band diagram
for an fcc lattice (see also Fig. 5.5) of lead spheres in a beryllium ma-
trix (filling fraction 8.23%), calculated by Economou and Sigalas using
the plane wave method (cf. section 3.4.2).[74] The lowest two branches
(indeed, the lowest three as the two very lowest ones are degenerated)
are the so-called acoustic ones. At the central point of the BZ, Γ, their
frequency is zero, and around Γ the frequencies increase linearly (dis-
persionless) in all crystallographic directions, i.e., in these regions phase
124
5.1. Introduction
and group velocities of the corresponding acoustic waves are equal.
Away from this long wavelength (small k) limit, when the branches
approach the edge of the (first) BZ, the linear behavior vanishes and
a bending occurs. That bending can be best explained for the sim-
plest case of a periodic elastic system - an one dimensional lattice of
vibrating identical spheres with mass m connected with massless elastic
springs with spring constant D and equilibrium distance a as shown in
Fig. 5.2.[24, 141] Let’s assume a longitudinal wave, which creates displace-
ment of the spheres in the direction of the chain. When assuming (in
reasonable approximation) that the elastic forces that determine a vi-
bration have their origin in the relative displacement only of neighboring
particles, symbolized by the compressed and stretched springs in Fig. 5.2
leading to forces (black arrows) in the same direction, while the inertia
originates from the absolute displacement of the spheres in space. With
u(na) being the absolute displacement of the n-th sphere, the harmonic
Figure 5.1.: a (From Ref. [74]:) Calculated acoustic band diagram for lead spheres
in a beryllium matrix forming an fcc lattice with an occupancy of 8.32 vol%. the
lowest (transverse) branch is degenerated. The three flat branches around 2.8 ωa/c
are degenerated, too. There are two complete band gaps. b Dispersion of the
longitudinal waves in a one dimensional simple crystal (black line) over several
Brillouin zones. The dashed line denotes the dispersionless ‘free particle’ case.
125
5. Phononic Behavior of Colloidal Systems
Figure 5.2.: Illustration of a longitudinal mechanical wave in an 1D elastic lattice.
potential between all neighboring spheres becomes
1 ∑
Uharm = D (u(na)− u([n + 1]a))2. (5.1)
2
n
Since elastic forces and inertia must compensate each other, the equation
of motion becomes
∂2u(na) ∂Uharm
m = − = −D [2u(na)− u([n− 1]a)− u([n + 1]a)] .
∂t2 ∂u(na)
(5.2)
Assuming a plane longitudinal wave of the form
u(na, t) ∝ ei(kna−ωt) (5.3)
and inserting into Eq. 5.2 delivers
−mω2ei(kna−ωt) = −D[2− e−ika − eika]ei(kna−ωt). (5.4)
After algebraic transformation the dispersion relation is obtained:
√ √
∣
− ( )
∣
2D(1 cos(ka)) D ∣ ka ∣
ω(k) = = 2 ∣sin ∣ . (5.5)
m m ∣ 2 ∣
126
5.1. Introduction
The dispersion relation for this simple case is shown in Fig. 5.1b for
several BZ’s. The periodicity of the lattice is recovered also in the dis-
persion relation in reciprocal space. In the long wavelength limit, i.e.,
k ≪ πd , Eq. 5.5 can be simplified to
√
D
ω(k) ≈ kd, (5.6)
m
using the approximation cos(kd) ≈ 1 − (kd)2/2. This corresponds to
the mentioned linear behavior at small k’s. In Fig. 5.1b, dashed lines
are drawn for comparison that show purely linear slope. This behavior
corresponds to the ‘free particle’ case, i.e., a phonon in a continuum.
Of course, in this case the axis caption including the periodicity d is
meaningless and k is simply given by k = 2π/Λ with phonon wavelength
Λ. The slopes of the linear regions of the acoustic branches give the
effective longitudinal and transverse velocities cl and ct.
At the edge of the first BZ (k = π/d), Eq. 5.5 becomes
√
4D
ω = , (5.7)
m
√
with corresponding longitudinal phase velocity cl = D/m · 2d/m and
group velocity zero. Waves that fulfill that condition are reflected into
themselves (for normal incidence) and form a standing wave.
Above the acoustic branches, in Fig. 5.1a some flat bands appear
around 2.8 reduced wavelength (indeed, there are three of them). Since
these branches are apparently almost k-independent and because of their
degeneracy, they are identified as eigenwaves basically trapped within the
lead spheres with little leaking out.[74] These are resonance modes similar
to the eigenvibrations discussed in chapter 4, however, this time no free
boundary conditions exist, but there is an exchange of energy between
the spheres and the matrix. Only if the elastic mismatch (i.e., densities
and moduli) is large enough between spheres and matrix, such a trap-
ping can happen. With decreasing mismatch the amount of dissipation
127
5. Phononic Behavior of Colloidal Systems
increases and finally, for no elastic mismatch, the systems behaves like a
continuum, i.e., only acoustic modes appear.
At higher frequencies also mixed modes with transverse as well as lon-
gitudinal components appear. These modes may become quite numerous
even for relatively simple systems. For systems with liquid matrix that
do not support shear waves, the calculation might become a bit easier,
however, it is still more complex than the analogue investigations for
electromagnetic waves. In systems where one component supports shear
waves and the other component does not, one must not ignore the conver-
sion between waves of different nature into each other that is described
mathematically by the T -matrix (Eq. 3.28).
Regions in the band diagram where no band exists for all k’s are
called phononic band gaps. These are frequency regions where no me-
chanic wave can propagate. Since the existence of such gaps is one of
the striking features of the band diagram and promises also some techni-
cal importance, the search for and investigation or tailoring of phononic
band gaps is a demanding question in the young field of phononics.
In the following parts of the introduction, the properties of the effec-
tive medium as well as a short overview over the phononic band gaps is
given. The rest of this chapter is concerned with experimental studies
on colloidal phononic systems, including the investigation of the effective
medium velocities in defect doped opals, the first experimental realiza-
tion of a hybridization gap and the band diagram of nanostructured
systems.
5.1.2. The Effective Medium
The slope at the long wavelength limit in the band diagram gives the lon-
gitudinal and transverse sound velocities, cl and ct, respectively. While
these velocities are well known for the most pure materials, their pre-
diction for composite materials of varying composition and geometric
structure is not straight forward and the topic of the effective medium
theory (EMT).
A very general approach to predict the effective longitudinal velocity
128
5.1. Introduction
of a two components system is given by Wood’s law as the harmonic
mean of the moduli:[105]
1 φ1 (1− φ1)
= + , (5.8)
Meff M1 M2
where M = c2l ̺ is the bulk longitudinal modulus of each component. In
principle this approach can be generalized for n components to
1 φ1 (φ2) (φn)
= + + . . . + , (5.9)
Meff M1 M2 Mn
∑
with n φn = 1. For equal mass densities (̺1 = ̺2), the effective lon-
gitudinal velocity would be given as the harmonic mean of the squared
sound velocities of the components (c ≡ [< 1/c2 >]−1/2l l ). For transverse
sound velocities, it should be sufficient to replace the bulk longitudinal
moduli in Eq. 5.8 by the share moduli.
Anyhow, the simple assumption of Wood’s law ignores for example
the interaction of longitudinal and transverse waves between different
components as it (in the longitudinal version) compares only the bulk
moduli. However, in certain cases, these interactions may lead to conver-
sion of energy between bulk and shear waves (T -matrix) and must not
be ignored anymore. In the theoretical calculations mentioned above,[74]
the authors compare the calculated effective velocities for several differ-
ent cases with the arithmetic mean and the harmonic mean of the sound
velocities as well as with the square root of the harmonic mean of the
squared velocities of the components - all weighted with the filling frac-
tion, but not with the mass densities. This deviation from Wood’s law
does not lead to a reproducible result for all cases, since in some cases
the one mean is nearer to the calculated result while in other cases an-
other mean is nearer to the theoretical findings. However, even applying
Wood’s law, which is not done in the paper, does not deliver perfect
agreements for all cases, although a relatively large deviation is achieved
only in the case of lead inclusions in a beryllium matrix with a filling frac-
tion of 8.23%. Here, using the parameters given by the authors, Wood’s
129
5. Phononic Behavior of Colloidal Systems
law delivers an effective longitudinal sound velocity of 9250 ms−1, while
the calculated velocities are about 10 % higher, between 10226 ms−1 and
10619 ms−1, depending on the crystallographic direction.
For the special case of spherical inclusions (and the practical appli-
cation for approximately spherical inclusions), a more sophisticated but
still easy-to-apply EMT method was developed mainly by Gaunaurd
and Wertman, reviewed by the same in Ref. [142]. In their calculations
they take the ratios of longitudinal and transverse sound velocities for
both components into account as well as the ratio of the mass densities.
Furthermore, the effective density is calculated differently for the two
general subcases of matrices that can only support insignificant amounts
of shear (‘fluid matrices’) and ‘elastic matrices’ supporting non-negligible
amounts of shear.
When calculating the effective longitudinal sound velocities for the dif-
ferent cases in Ref. [74] following this approach (utilizing a small C++
program written by myself), reasonable results are obtained for all sub-
cases (Au/Si, Pb/Si, Pb/Be, and Au/SiO2), although for some subcases
from the theoretical band diagram there are different effective velocities
in different crystallographic directions, which are described unequally
well by this EMT. E.g., for the Pb/Be subcase mentioned above, the
program gives an effective sound velocity of 10820 ms−1, which is only
about 2% more than predicted for the Γ–L direction.
In Tab. 5.1 three general cases of phononic systems with filling fraction
φ are compared for Wood’s law and the EMT of Gaunaurd and Wertman:
Polystyrene spheres in silicon oil (solid/liquid), alumina spheres in PS
(solid/solid), and air bubbles in silicon oil (gas/liquid).1 For all subcases
the values following the Gaunaurd and Wertman approach are given for
fluid (GWf ) and elastic matrix (GWe), although principally the latter
should be meaningful only in the alumina/PS subcase since silicon oil
that is matrix in the other two subcases does not support shear waves.
1Material parameters: PS: ct: 1200 ms
−1, cl: 2350 ms
−1, ̺: 1050 kg/m3; silicon oil:
: 0 ms−1ct , cl: 1400 ms
−1, ̺: 1000 kg/m3; Al2O3:
−1 −1
ct: 6345 ms , cl: 10850 ms ,
̺: 3970 kg/m3; air: ct: 0 ms
−1, cl: 343 ms
−1, ̺: 1.2 kg/m3
130
5.1. Introduction
Table 5.1.: Comparison between calculated effective longitudinal sound velocities
(in ms−1) using the EMT of Gaunaurd and Wertman (GW) and Wood, respectively,
for three cases and five filling fractions φ; GW calculations for fluid and elastic
matrix, GWf and GWe, respectively.
PS in silicon oil Al2O3 in PS air in silicon oil
φ GWf GWe Wood GWf GWe Wood GWf GWe Wood
0.74 1714 1714 1925 2854 2522 2590 42 27 27
0.50 1588 1587 1691 2479 2104 2136 34 24 24
0.20 1466 1466 1496 2362 2095 2103 35 30 30
0.05 1416 1415 1422 2351 2257 2258 57 54 54
0.001 1400 1400 1400 2350 2348 2348 364 364 364
For the first case of PS in silicon oil, it does not make any significant
difference which kind of matrix is taken into account. Both calculations
come to practically the same results, leading to sound velocities lower
than those predicted by Wood’s law. With increasing filling fraction the
difference increases up to 12% for the dense packing case φ=0.74.
In the subcase of a harder solid embedded in a softer one (alumina in
PS), GWe and Wood’s law deliver quite similar results, differing by less
than 3% at φ=0.74, while the theoretically meaningless GWf delivers
significantly higher values.
The last subcase of air bubbles in silicon oil shows that in this case,
where both components’ transverse sound velocities are set to zero, GWe
delivers exactly the same results as Wood’s law, while the GWf calcula-
tions give higher values. However, even for very low filling fractions all
theories give extremely low values. This means that even a low amount
of air bubbles should slow down the effective sound velocity in a liquid.
Another EMT method is described by Waterman and Truell.[143] Their
approach is in principle a full multiple scattering theory including nu-
merical calculation with n equations for n scatterers. For every scat-
terer the exact form of the multipole coefficient must be known, and so
this method is too complicated to be used for simple calculations of the
effective sound velocity. Taking the effective sound velocity from the
131
5. Phononic Behavior of Colloidal Systems
Figure 5.3.: Infiltration of a turbid colloidal crystal with an index matching liquid
resulting in a transparent ‘wet’ crystal (after [8]).
theoretical band diagrams calculated by the multiple scattering method
touched on in section 3.4.3 is very akin to this approach.
5.1.3. Phononic Band Gaps
To measure the phononic dispersion relation of colloidal crystals by BLS,
the strong multiple scattering that leads to the eigenmode spectra dis-
cussed in chapter 4 should be overcome. This is achieved by infiltra-
tion of the dry opals by a fluid with refractive index close to that of
the colloidal spheres (Fig. 5.3). For poly(methylmethacrylate) (PMMA,
n=1.49) or PS (n=1.59) colloids for example silicon oil (n=1.45) or liquid
poly(dimethylsiloxane) (PDMS, n=1.41) are suitable infiltration liquids.
Using such ‘wet’ colloidal crystals, the first experimental access to the
phononic band diagram along high symmetry directions in submicrome-
ter artificial crystals was recently reported,[8] leading to the observation
of phononic band gaps in the GHz-range. Till then, only sonic and
ultrasonic crystals had been investigated by other techniques, mostly
by sonic- and ultrasonic transmission experiments or by a special video
spectroscopy methods applied to 2D[144] and 3D[145] experiments. The
observation of a hypersonic band gap in a 2D hypersonic phononic crystal
132
5.1. Introduction
Figure 5.4.: The dispersion relations of a colloidal PS (d=307 nm) opal infiltrated
with silicon oil (left, see also Ref. [8]) and with PDMS (right, see also Fig. 5.11 and
Ref. [33]) in the Γ–M direction of the fcc crystals are compared. In both spectra
there is a clear BG (diagonal pattern). The dashed lines indicate the Γ–M distance
in the reciprocal space, hence the edge of the first BZ. The dotted lines denote the
frequencies that correspond to qBZ (cf. section 5.1.3). For the opal infiltrated with
PDMS a HG (vertical pattern) appears, too.
by BLS was subsequently reported.[146]
The nature of the observed band gaps is clearly that of a Bragg gap
(BG). A BG exists due to annihilation of the incident mechanical wave
and the waves scattered on the individual spheres in the crystal. To
create a band gap, the periodic structure length in the crystal must be
of the order of the relevant phonon wavelength. If the phonon wave-
length and the distance between the crystal planes in the direction of
the phonon propagation become equal, the scattered mechanical waves
interfere constructively and the energy of the incident wave is reflected
back opening a phononic gap.
Fcc crystals of sub-micron PS spheres are prepared by vertical lift-
ing deposition. After infiltration with PDMS, the dispersion relation,
f(q), was recorded by angle dependent BLS measurements along high–
133
5. Phononic Behavior of Colloidal Systems
symmetry directions in the reciprocal space. As shown in Fig. 5.4 at low
q’s, the dispersion relation is that of mechanical waves traveling through
a homogeneous medium with an effective sound velocity, i.e.,
ω = 2πf = ceffq. (5.10)
√
As q reaches the edge of the first BZ at qBZ = (3/2)
3/2π/( 2d) in the
probed Γ–M direction (cf. section 5.1.3), the fulfillment of the Bragg
condition leads to a gap opening. Accordingly, the frequency of the
observed gap is[8, 32]
f = qBZceff/2π. (5.11)
The position of the gap can be widely tuned by changing the lattice pa-
rameters of the hypersonic phononic crystal due to the linear dependence
of the gap frequency from the inverse diameter.
The width of the BG increases with elastic mismatch between the
spheres and the infiltrated liquid. In fact, the width of the gap in the
wet PS opals systematically increases when the infiltrated liquid changes
from glycerin (cl ≈2200 ms−1) to silicon oil (cl=1400 ms−1) and to
PDMS (cl=1050 ms
−1). According to the results of effective medium
theory[142, 147], the value of ceff lays in between the components’ indi-
vidual sound velocities. It is noteworthy that the calculation of effective
sound velocities following the approach of Gaunaurd and Wertman[142]
tends to result in moderately but certainly too small values compared
with the experiment, actually the deviations are smaller than 10%.
In section 5.3.1 the first experimental realization of an additional gap
besides the BG, the so-called hybridization gap (HG) is demonstrated.[33]
This HG originates from the interaction of the band of quadrupole par-
ticle eigenmodes with the band of the effective medium. This kind of
gap opens up from level repulsion when two bands of the same sym-
metry cross each other (in analogy to the linear combination of atomic
orbitals).[73] In this case, the involved bands are the acoustic field of the
extended states in the effective medium and the bands from the multi-
134
5.1. Introduction
pole modes of the interacting spherical particles, which can be treated
as local resonant elements.
The first hint on this gap was found in BLS experiments in colloidal
suspensions,[148, 149] but a clear demonstration, assignment and the con-
current observation along with the BG was just reported in the case of
PS and PMMA wet opals infiltrated with PDMS.
Normalization
In general, the condition that has to be fulfilled to create a Bragg gap
is that the distance between two scattering planes in the direction of
the incident wave matches the wave length of the wave, or, in reciprocal
space, that the wave vector q is half the diameter of the first Brillouin
zone in a certain crystallographic direction. By the method of vertical
lifting deposition (section 3.1) the colloidal fcc crystals show a preferred
orientation relative to the substrate. When inserting the sample in the
same orientation as prepared into the BLS setup and utilizing the special
transmission geometry, where the wave vector is parallel to the substrate,
the probed direction in the reciprocal lattice is usually the Γ−M direc-
tion. The first BZ of an fcc lattice is shown in Fig. 5.5. In the Γ −M
direction, the distance in reciprocal space between Γ and M is
3 3/2
|Γ− (M | = 2√) π (5.12)
2d
with sphere diameter d. Therefore, the wave vector qBZ in this crystallo-
graphic direction must have the same value to fulfill the Bragg conditions,
because |Γ−M | is half the diameter of the Brillouin zone DBZ . I.e.,
D (3 3/2BZ
q = = 2
) π
BZ √ . (5.13)
2 2d
135
5. Phononic Behavior of Colloidal Systems
Figure 5.5.: First Brillouin zone (BZ) of an fcc lattice. The black line denotes
the equator for the body standing on a hexagon, on which the point M is located.
If one assumes pure acoustic behavior with an effective sound velocity
ceff the frequency at qBZ would be
ceffqBZ ceffDBZ (
3
2)
3/2
√ cefffBZ = = = (5.14)
2π 4π 2 2d
with f = ω/2π = cq/(2π). Now one can normalize the measured values
for the frequency and the wave vector to the values at the edge of the
Brillouin zone:
√
f 2f 2d fd
fnorm = = ≈ 1.54 (5.15)
f (3)3/2BZ 2 ceff ceff
√
q q 2d
qnorm = = 3 ≈ 0.245qd (5.16)q ( )3/2BZ 2 π
136
5.2. Effective Medium Velocity in Defect Doped Opals
5.2. Effective Medium Velocity in Defect Doped
Opals
BLS can be used to measure the effective sound velocity in homogeneous
materials as well as in structured materials if the structure size is smaller
than the traveling phonon, or in other words, if 2π/q > d with structure
spacing d. E.g., for q=0.01 nm−1 (that corresponds to an angle of ≈50◦
in transmission geometry for the setup described in section 2.3.2) the
structure must be clearly smaller than 600 nm.
While in bulk, not too thin films of a single polymer of course the
polymer’s sound velocities (if it support shear waves) are found, in films
of thin multilayers of different components, e.g., for two different poly-
mers, the individual sound velocities of both components are found when
measuring parallel to the interfaces.[150] If the wave vector is chosen per-
pendicular to the layers, at least for thin (<30 nm) layers, only one
signal corresponding to the effective sound velocity is found. Following
the effective medium theories, ceff lays in between the velocities of the
individual components.[151]
Effective sound velocities are also found in nano- and mesoscopic 3-D
structures like the infiltrated colloidal crystals presented in section 5.1.3.
In Fig. 5.4 ceff is shown for PS colloidal crystals (φ = 0.74) infiltrated
with silicon oil (left) or PDMS (right). The effective sound velocities are
1990 ms−1 and 1670 ms−1, respectively. These are between those of PS
(c =2350 m/s, ̺=1050 kgm−3l ) and PDMS (cl=1050 m/s, ̺=965 kgm
−3)
or silicon oil (cl=1400 m/s, ̺=1000 kgm
−3).[33] The EMT of Gaunaurd
and Wertman[142] delivers effective velocities of 1714 m/s and 1488 m/s,
for silicon oil and PDMS as infiltration liquid, respectively, using the
numbers given above. Wood’s law, on the other hand, delivers 1925 m/s
and 1617 m/s, respectively. I.e., both EMT’s seem to underestimate the
real effective sound velocities, however for Wood’s simple law, the error
is significantly smaller.
A more systematic study of the influence of the filling fraction on the
effective sound velocity in PS colloidal crystals infiltrated with PDMS
137
5. Phononic Behavior of Colloidal Systems
Figure 5.6.: SEM pictures: a-d PS:SiO2 colloidal crystals with composition 2:1,
4:1, 8:1, and 12:1 (a to d); e-h The corresponding defect doped crystals after
etching away the silica beads. Analytics: i TGA measurement on the composite
PS:SiO2 colloidal crystals (decomposition of PS). The inset shows the loss of water
(legend like in j); j UV/VIS spectra of the PS:SiO2 colloidal crystals shown in a-d.
For comparison also the spectra of the pure PS colloids, the pure silica colloids and
of two mixtures with higher amount of silica are shown. (according to [118])
138
5.2. Effective Medium Velocity in Defect Doped Opals
and the comparison with EMT was performed within the framework of
this thesis. Therefore, a set of colloidal crystals consisting of PS colloids
(d=260 nm) and a certain amount of silica particles with practically the
same diameter (d=255 nm) were synthesized by my coworker Markus
Retsch (who also performed the characterization in Fig. 5.6). Since
the size difference is clearly below 5%,[152] the particles form random
co-crystals during vertical lifting deposition under stirring. Fig. 5.6a-d
shows the SEM pictures of four different co-crystals with composition of
the initial particle dispersion PS:SiO2 between 2:1 (a) and 12:1 (d); the
brighter spheres correspond to the silica particles. It can be seen that
the samples are nicely crystalline and nearly crack- and defect-free over
many µm, i.e., the filling fraction φ is in good approximation that of an
fcc lattice, φ=0.74.
It must be expected that the real composition in the composite crys-
tals differs from that in the initial dispersions due to the different mass
density of both types of particles and other possible surface effects that
can not be fully discussed in this context. In order to determine the real
ratios, TGA (Fig. 5.6i) and UV/VIS measurements (Fig. 5.6j) were per-
formed and the results were compared with the counting of the SEM pic-
tures. From the TGA measurements it is easy to calculate the PS:silica
ratio; at high temperatures the PS is decomposed and only the silica
spheres remain. By comparing the relative mass loss and taking into
account the different mass densities and the exact radius of the spheres
the relative amount of the polymer is obtained.[118] It should be noted
that it is meaningful to compare the change for the dry samples, i.e., to
set the starting point of the comparison at T>100 ◦C (inset in Fig. 5.6i).
The counting of the SEM pictures shows that systematically less silica
particles are built in the crystals than should be expected from the initial
dispersion concentrations. However, the SEM pictures show only the
distribution on the imaged surface and, by what reason ever, the ratio
inside the opal could be different. Anyhow, the comparison between the
data obtained by the TGA measurements and the SEM counting lead
to a systematic and consistent result.[118] The amount of silica is always
given by approximately 0.8 times the expected fraction, i.e., the real
139
5. Phononic Behavior of Colloidal Systems
ratio of the ‘12:1’ PS:SiO2 sample is ≈12:0.8 etc., however, for the rest
of this section the names originating from the initial ratio are kept to
avoid confusion.
The so found real compositions are verified by the UV/VIS measure-
ments shown in Fig. 5.6j. From the graph that contains also two addi-
tional samples with initial ratio 1:1 and 0.3:1 it can be seen that there
is a clear blue shift of the Bragg peak for increasing amount of silica,
resulting from the change in the effective refractive index (cf. section
3.3.6). The utilization of the modified Bragg equation (Eq. 3.10)
√
λ 2 2 2Bragg,theo = 1.63d 0.74χPSnPS + 0.74χSiO2nSiO2 + 0.26nair (5.17)
delivers excellent agreement between the theoretically expected Bragg
wavelengths λBragg,theo and the experimentally measured λBragg as far
as refractive indices a bit lower than for the bulk materials are assumed
(nPS=1.55, n
[118]
SiO2=1.37).
Table 5.2.: Characterization of the PS:SiO2 composite colloidal crystals. The
initial ratio is also the name of the samples in Figs. 5.6 and 5.7.
initial ratio experim. ratio mass lossa λ bBragg,theo λBragg
PS:SiO2 PS:SiO2 % nm nm
0:1 0:1 - 533.7 530.0 ±7.9
2:1 2:0.8 60.7 588.3 586.0 ±1.6
4:1 4:0.8 75.5 595.3 593.0 ±3.6
8:1 8:0.8 86.1 599.7 599.0 ±0.8
12:1 12:0.8 90.3 601.4 601.5 ±0.5
1:0 1:0 - 605.0 602.1 ±3.3
a: ̺PS=1.05 gcm
−3, ̺ −3SiO2=1.7 gcm , dPS=260 nm, dSiO2=255 nm
b: n =1.55 gcm−3PS , nSiO2=1.37 gcm
−3, d=260 nm
Since the number ratio of PS and SiO2 spheres in the composite is
clear now, the next step is to remove the silica spheres by etching with
hydrofluoric acid. Fig. 5.6e-h shows the SEM pictures of the composite
crystals after etching away the silica beads. The removed spheres appear
140
5.2. Effective Medium Velocity in Defect Doped Opals
 
2.5
composition before etching:
 260 nm PS
2.0  12:1 PS:Silica
   8:1 PS:Silica
   4:1 PS:Silica
1.5    2:1 PS:Silica
1.0
0.5
0.0
0.000 0.003 0.006 0.009
q [nm-1]
Figure 5.7.: Dispersion at low q’s and ceff for defect doped PS opals infiltrated
with PDMS.
now as randomly distributed hole defects in the colloidal crystal. From
the SEM pictures it is obvious that for the chosen compositions the
crystalline structure is not disturbed (e.g., by a collapse). Therefore, the
filling fraction of the whole crystal is now simply given by 0.74 times the
experimental ratio PS:(PS+SiO2) before the etching.
After infiltration with PDMS, the polarized Brillouin spectra are mea-
sured by BLS in the long wavelength limit. The results are displayed
in Fig. 5.7 All signals increase linearly with q with increasing slope for
increasing ratio PS:SiO2 before the etching, i.e., with increasing filling
fraction, which is expected due to the higher cl of bulk PS compared to
bulk PDMS. For all samples, ceff is calculated from the linear fits as
ceff = 2π · df/dq.
The experimental ceff , cexp, is compared to theoretical predictions
of the EMT in Tab. 5.3, utilizing the real filling fractions φ from the
experimental ratio in Tab. 5.2. First, the velocities are compared to the
141
f [GHz]
 
5. Phononic Behavior of Colloidal Systems
Table 5.3.: Comparison between experimental effective longitudinal sound veloc-
ities and results from Wood’s law and the EMT of Gaunaurd and Wertman using
standard and fitted parameters (in ms−1).
PS:SiO2 φ cexp cWood ∆/% cGW,s ∆/% cGW,f ∆/%
2:1 0.529 1382 1362 -1.5 1304 -6.0 1395 1.0
4:1 0.617 1464 1451 0.9 1372 -6.7 1473 0.6
8:1 0.673 1522 1520 0.1 1421 -7.1 1530 0.5
12:1 0.694 1567 1548 -1.2 1441 -8.7 1553 -0.9
1:0 0.740 1627 1617 -0.6 1488 -9.3 1609 -1.1
constant: ̺(PS)=1050 kgm−3, ̺(PDMS)=965 kgm−3, ct(PDMS)=0 ms
−1;
c −1 −1 −1Wood, cGW,s: cl(PS)=2350 ms , ct(PS)=1200 ms , cl(PDMS)=1050 ms ;
c −1 −1 −1GW,f : cl(PS)=2470 ms , ct(PS)=1100 ms , cl(PDMS)=1109 ms
results of Wood’s simple law, using the standard elastic parameters for
PS and PDMS (given under the table). Interestingly, cWood seems to
capture the experimental values quite well within a non-systematic error
of 1.5% or less, which is in the range of the error of φ. Using the same
parameters in the algorithm of Gaunaurd and Wertman (cGW,s) leads
to a significant deviation from the experimental values between 6% and
more than 9%, while the deviation increases systematically for increasing
φ.
Anyhow, it is possible to bring the results of their EMT in better
accordance with the experimental results when taking cl(PS), ct(PS),
and cl(PDMS) as fit parameters in a mean square fit for all five filling
fractions. So the cGW,f are obtained, keeping the parameters in rela-
tively reasonable ranges. The PDMS is assumed to be about 8% softer
(1109 ms−1 instead of 1200 ms−1), for the PS the transverse sound veloc-
ity must be chosen smaller (1100 ms−1 instead of 1200 ms−1) and the lon-
gitudinal one must be chosen higher (2470 ms−1 instead of 2350 ms−1).
Doing so, the error stays within about 1%, however changing systemati-
cally from a little to high values for low φ to a little to low values for the
undoped crystal. Even better results can be achieved using really free fit
parameters without any fitting ranges. However, the best fit is achieved
142
5.3. Band Gaps in Polymer Opals and Disordered Systems
with unreasonable parameters, especially for PS (cl(PDMS)=1069 ms
−1,
cl(PS)=2622 ms
−1, and ct(PS)=1082 ms
−1).
In summary, it can be said that for different filling fractions the investi-
gated colloidal system shows the expected trend in the effective medium
sound velocity. The increase in ceff with increasing amount of the harder
component was predicted qualitatively by Wood’s law as well as by the
more elaborated EMT of Gaunaurd and Wertman, developed for the
special present case of spheres embedded in an elastic or fluid matrix.
However, while the predictions utilizing Wood’s law agree also quantita-
tively well with the experimental results, there is a systematic deviation
to the velocities calculated by the other EMT. The reason of the devi-
ation is unclear. It might originate in a kind of size or frequency effect
that leads to deviations for small spheres (possibly originating from elas-
tic confinement inside the spheres) or for high frequencies. Anyhow,
these assumptions are somehow speculative.
5.3. Band Gaps in Polymer Opals and Disordered
Systems
5.3.1. The Influence of the Order in Colloidal Systems
Structured materials with a periodic modulation in the density and elas-
tic coefficients, so-called phononic crystals,[10] can exhibit phonon band
gaps at Bragg frequencies or wavelengths commensurate to their lattice
constant. In addition to Bragg gaps (BG), theory predicts gaps evoked
by resonance modes of the constituent components interacting with the
extended acoustic branch of the composite structure.[153] These gaps pre-
vent elastic waves with certain frequencies to propagate through the crys-
tal at least in certain crystallographic directions. The width and the po-
sition of the BG in general depends on the contrast between the densities
(̺), longitudinal and transverse sound velocities (cl and ct) of the compo-
nent materials, and on the lattice parameter.[131, 132, 144] Yet, structures
with strong localized resonant elements can shift the gap well below the
143
5. Phononic Behavior of Colloidal Systems
Bragg frequency associated to the lattice constant.[90, 154] Soon after the
first experimental observation of ultrasonic band gaps,[4–6, 90, 134] pecu-
liar phenomena with potential applications dealing with the propagation
of elastic waves in periodic composite materials such as tunneling,[155]
negative refraction, focusing,[156, 157] and enhanced transmission through
one–dimensional gratings[136] have been discovered. Earlier realized sys-
tems with periodicity in the millimeter range deal with sonic and ul-
trasonic frequencies,[5, 6, 40, 90, 134, 154–157] while periodic patterns at the
submicrometer scale becomes necessary to shift the gap to hypersonic
frequencies.[91, 158]. The phononic dispersion relation of such a single
crystalline pattern created by vertical lifting deposition of polymer col-
loids and measured by BLS led to the first observation of a hypersonic
BG along high–symmetry crystallographic directions.[8] By varying the
particle size and the infiltrated fluid, the hypersonic frequency and the
width of the gap could be tuned. More recently, a hypersonic BG
was also observed in two–dimensional structures fabricated by optical
lithography.[146]
In soft opals, the spherical particles represent local resonant elements
and hence the bands, which originate from the multipole modes of these
interacting particles, can overlap with the acoustic field of the extended
states in the effective medium. As a result of this hybridization, a gap
opens up in the vicinity of the eigenfrequency of the quadrupole par-
ticle modes,[153] which is refered to as hybridization gap (HG). Albeit
there was a hint of this HG in colloidal suspensions,[95, 148, 149, 159] its
demonstration, assignment and the concurrent observation along with
the BG has been missing. In this section, the first realization of a
double phononic gap in fcc colloidal crystals formed by colloidal self–
assembly during vertical lifting deposition with subsequent fluid infiltra-
tion is presented.[33] The effect of the elastic contrast between the fluid
matrix and solid inclusions as well as the crystalline order of the struc-
ture on the two gaps is distinct. Furthermore, in amorphous colloidal
glasses, only the HG persists.
The fabrication of the colloidal films starts with the deposition of
the particles by vertically lifting a glass substrate from the aqueous
144
5.3. Band Gaps in Polymer Opals and Disordered Systems
Figure 5.8.: SEM top-view images of the PS-307 crystal (left) and the PS-hybrid
(right) - both “dry” before infiltration (scale bar is 1 µm). The insets display the
Fourier-transform images computed from the SEM pictures over an area of about
4 µm by 4 µm.
colloid suspension.[54] For different experiments, two kinds of monodis-
perse polystyrene (PS) spheres with diameters of d=307 nm and 360 nm
and poly(methylmethacrylate) (PMMA) particles with a diameter of
d=327 nm were used for the fabrication of the opals. Additionally, a non-
crystalline hybrid film consisting of PS sphere mixture with two different
diameters (300 nm and 360 nm) was prepared in the same way. After
complete drying, the films were infiltrated with liquid poly(dimethylsil-
oxane) (PDMS) and any liquid excess was blown off (Fig. 5.3). By this
method uniform wet colloidal films with a thickness of about 10 µm were
obtained. The phononic dispersion relation was measured by BLS for the
dry and liquid infiltrated state of these films. BLS probes the thermal
phonon propagation in the sample along a selected direction determined
by the transmission scattering geometry adumbrated by Fig. 5.9 (cf.
145
5. Phononic Behavior of Colloidal Systems
Table 5.4.: Sound velocities, particle sizes and lowest eigenfrequencies in colloid–
based phononics.
Material ct/ms
−1 c −1l/ms d/nm f(1, 2)/GHz c/ms
−1
PS 1200±30 2350±50 307 3.3a
360 2.9a
PDMS 0 1050±20a
PS/PDMS 1670±30a 307 2.2b 1490b
1570±30a 360 1.9b 1490b
Hybrid/PDMS 1510±20a 300/360 1400b,c
PMMA 1400±40 2800±50 327 4.0a
PMMA/PDMS 1720±30a 327 2.3b 1560b
[densities ̺ (kg/m3): PS: 1050, PMMA: 1190, PDMS: 965]
a: Measured by BLS. b: Computed for the wet opals.
c: for 65% filling fraction of spheres.
appendix).[8] For the given experimental conditions, the scattering wave
vector of the photon lies in the (111) plane of the fcc lattice and its am-
plitude q = (4π/λ) sin(θ/2) depends on the scattering angle θ and the
wavelength of the incident laser beam λ. At low q values for which the
system appears homogeneous, the wave vector of the phonon k and that
of the inelastically scattered light q are equal. Under these conditions,
the BLS spectrum at a given q consists of a doublet with a Doppler fre-
quency shift ω = ±ck, where c is the speed of sound with longitudinal (or
transverse) polarization in the effective medium (e.g. lowest spectrum in
Fig. 5.10, cf. section 2.3).
The precursor dry PS and PMMA opals have a highly ordered crys-
talline morphology as indicated by the SEM image in Fig. 5.8. The
single crystalline order extends over a few hundred µm. The dry films
exhibit strong multiple light scattering due to the large optical contrast
between the particles and the surrounding air and due to the particles
elastic form factor. The wave vector q is, therefore, ill–defined which pre-
cludes the measurement of the dispersion relation ω(q). Nonetheless, the
146
5.3. Band Gaps in Polymer Opals and Disordered Systems
Figure 5.9.: BLS parallel to the substrate in a colloidal crystal.
BLS spectrum of the dry opals reveals several localized (q–independent)
modes.[45, 85] As discussed in chapter 4, these are identified as vibration
eigenmodes of the particles and each mode can be specified by a pair
of indices (n, l) defining the l–th spherical harmonic of the n–th radial
mode. The frequencies of these sphere eigenmodes depend on the size
of the particle, the mass density and the speed of sound in air and in
the particle for both polarizations (i.e. compression and shear moduli).
Consequently, the mechanical properties of the samples can be reliably
determined from the theoretical fit of the experimental eigenfrequencies.
The obtained longitudinal and transverse sound velocities in the two
types of particle materials along with the lowest f(1, 2) eigenfrequency
are given in Table 5.4. For the particles embedded in the infiltrated
PDMS, the eigenfrequencies can be theoretically computed using the
elastic parameters of Table 5.4.[80, 95] Expectedly, f(1, 2) decreases when
the spheres are embedded in PDMS.
The infiltration of the thin dry opals by PDMS with a refractive index
(n=1.45) close to that of the PS particles (n=1.59) diminishes multiple
light scattering and hence q is well defined. To obtain the desired disper-
sion relation ω(q), the BLS spectra of the wet opals were recorded along
147
5. Phononic Behavior of Colloidal Systems
7
q=0.0163 nm-1
6
q=0.0152 nm-1
5
q=0.0135 nm-1
4
q=0.0118 nm-1
3
q=0.0107 nm-1
2
q=0.0100 nm-1
1
0 q=0.0076 nm
-1
-6 -4 -2 0 2 4 6
f (GHz)
Figure 5.10.: BLS spectra of the wet opal of PS spheres with diameter 307 nm in
a PDMS matrix at different wave vectors in the (111) plane of the fcc crystalline
colloidal film. The edge of the BZ along the probed direction corresponds to qBZ ≈
0.013 nm−1. The deconvolution in different spectral components is indicated for the
Stokes side of the spectrum. Each spectrum is normalized to the total integrated
intensity, for comparison of the relative contributions.[41]
the high–symmetry directions of the reciprocal space. In the present case
of the fcc lattices, the first Brillouin zone (BZ) is a truncated octahe-
dron (Fig. 5.5). The experimental q is confined in the hexagon formed
by the intersection of the (111) plane with the BZ.[8] The direction of q
is selected along Γ–M , where M is the edge center of the hexagon and
the evolution of the BLS spectra with q near the BZ for the PS wet
opal is shown in Fig. 5.10. The simple picture of single phonon prop-
148
Intensity (a.u.)
 
5.3. Band Gaps in Polymer Opals and Disordered Systems
agation in the effective medium at low q values becomes complex as q
increases towards the BZ boundary. Up to four Lorentzian curves are
required to represent the experimental BLS spectrum. In contrast to the
PS/silicon oil opal with a single peak splitting across the BZ boundary,[8]
the same opal infiltrated in PDMS providing higher elastic contrast dis-
plays richer spectral features and exhibits a second splitting at lower q
values within the first BZ. The experimental dispersion relation is de-
picted in Fig. 5.11. In the hypersonic PS/PDMS crystal (d=307 nm)
only one longitudinal phonon branch is observed at low q values with
c =1670 ms−1l , intermediate between the longitudinal sound velocities
in the pure component materials. Transverse phonons are not observed
experimentally, probably PDMS cannot support shear waves and the me-
chanical contact between the particles is weakened. The most striking
feature of the dispersion diagram is the simultaneous presence of two
band gaps at about 3 GHz and 4 GHz, respectively. The latter is clearly
a BG since it occurs at the edge of the BZ, q = qBZ ≈ 0√.0133 nm
−1, that
matches the distance Γ–M i.e. (3/2)3/2π/a, where a = 2d is the lattice
constant of the given fcc crystal. An analogue behavior is observed in
the second wet opal of the larger PS spheres (d = 360 nm) for which the
gap positions shift to lower wave vectors and frequencies (open circles in
the upper left diagram of Fig. 5.11).
In order to elucidate further the nature of the two gaps, the effect
of the crystalline order on the experimental band diagram was exam-
ined. The formation of a hybrid colloidal film consisting of a mixture
of an equal number of two PS spheres (d=300 nm and d=360 nm) ar-
tificially broadens the size distribution beyond the polydispersity limit
of about 5% necessary for crystallization.[152] Indeed, the crystallization
is prohibited in this hybrid colloidal film, as indicated by the lack of a
long–range order in the right-hand panel of Fig. 5.8, leading to an amor-
phous colloidal glass. The BLS spectra of the infiltrated monodisperse
opal and hybrid films are shown in Fig. 5.12 for two wave vectors. The
deletion of one peak in the spectrum of the hybrid is due to the disap-
pearance of the BG in the disordered hybrid, as it is clearly shown in the
dispersion plot in the upper left diagram of Fig. 5.11 (solid squares). Ap-
149
5. Phononic Behavior of Colloidal Systems
  
6
 PS-360 d=300 nm
5  PS-hybrid
4
3 d=360 nm
2
1
  
0
 
5  PS-307
 
l=5
4 l=1l=4
3 l=3
2 l=2
1
0
0.000 0.005 0.010 0.015 0.020  
q (nm-1) DOS (a.u.)
Figure 5.11.: Phononic band diagrams for the different PS wet opals under consid-
eration, along the Γ-M direction (see text). The hatched bands mark the phononic
gaps and the vertical arrows indicate the q values in the corresponding spectra of
Fig. 5.10 and Fig. 5.12 below. The vertical lines denote the first BZ limit (for
d=307 nm and d=360 nm respectively). The change in the DOS induced by the
corresponding single PS particle in PDMS is shown in the right–hand panel.
parently, the crystalline order is a prerequisite for the BG but not for the
newly observed lower frequency HG, which is omnidirectional in the col-
loidal glass. An identification of the latter as a theoretically anticipated
HG,[153] through density–of–states (DOS) calculations,[80] is examined in
Fig. 5.11. The right-hand panel of Fig. 5.11 displays several eigenmodes
of the individual PS spheres embedded in the fluid PDMS. The lowest
150
f (GHz)
  
  
 
  
5.3. Band Gaps in Polymer Opals and Disordered Systems
f(1, 2) appears to compare well with the frequency at the crossing with
the acoustic branch and the opening of the HG. Moreover, the lowest fre-
quency points at the experimental dispersion at high q’s compare very
well with the computed f(1, 2). This and the two higher flat bands of
localized modes in Fig. 5.11 cause sufficiently strong inelastic light scat-
tering at high q’s[95] and compare well with the particle resonances of
higher l. Their presence in the BLS spectra obscures the resolution of
the phonons in the second BZ. Interestingly, the bands originating from
particle resonant modes appear to be considerably narrower and occur at
higher frequency than theoretically predicted.[153] This can be ascribed
to viscous losses in the liquid matrix that were not taken into account in
the theoretical calculations, and which weaken the interparticle interac-
tions as a result of the reduced overlap between the corresponding wave
fields. Qualitatively, the opening of the two gaps is also observed in a
third wet opal of PMMA with d = 327 nm in PDMS (Fig. 5.13). The HG
occurs at 2.5 GHz, very close to the f(1, 2) = 2.35 GHz of this particle in
PDMS, whereas the wet opal exhibits a sound velocity of 1720±30 ms−1
in the long–wavelength limit (Table 5.4).
It is worth noting that the various effective-medium theories (EMT)
yield sound velocities about 10% smaller than the experiment,[142, 147]
even if the viscosity in the liquid matrix is taken into account. On the
other hand, if one considers the colloidal film as a polymer (solid) matrix
with fluid inclusions,[147] EMT strongly overestimates the sound veloc-
ity. The above, in view also of the fact that measured sound velocities
agree generally well with the results of EMT in non close-packed colloidal
crystals,[57] suggests the existence of consolidation, at least to some de-
gree that may be different for different samples, in the colloidal films.[160]
This also explains why the measured effective sound velocity is different
in the two PS/PDMS crystals (see Table 5.4).
In conclusion, this work presents, for the first time, the discovery of two
phononic band gaps of different nature coexisting at hypersonic frequen-
cies in the same physical system and elucidated the underlying physical
mechanisms. Induced disorder did not destroy the newly demonstrated
HG. This study has been possible by taking advantage of the opportuni-
151
5. Phononic Behavior of Colloidal Systems
q=0.0100  nm-1 q=0.0124  nm-1
PS-hybrid
 
PS-360
2 4 f  (GHz) 2 4
Figure 5.12.: Exemplary BLS spectra of the PS (d = 360 nm)/PDMS wet opal
(bottom) and the wet PS-hybrid colloidal film (top) at two q values indicated by
the arrows in the corresponding diagram of Fig. 5.11. The deconvolution into
different spectral components is shown below each spectrum (note the absence of
the BG-induced spectral features in the hybrid).
ties offered by the colloidal science to tailor the phononic band diagram of
nanostructured materials, measured directly by BLS. Manipulating the
flow of phonons may allow heat management, e.g. in thermoelectrics. Fi-
nally, it was pointed out the need of a detailed quantitative evaluation of
the dispersion diagrams of colloid-based phononic structures, by means
of full elastodynamic calculations that take into account consolidation
and soft matter properties (e.g. structural dynamics of the component
materials or interfacial effects), which still remains an open challenging
theoretical problem.
152
 Intensity (a .u.)
  
  
5.3. Band Gaps in Polymer Opals and Disordered Systems
  
7
PMMA-327 nm
6 (infiltrated with PDMS)
5 l=1
4 l=4
3 l=3
2 l=2
1
0
0.000 0.005 0.010 0.015 0.020 1 2 3 4
q (nm-1) DOS (a.u.)
Figure 5.13.: Phononic band diagrams for PMMA wet opals in PDMS with
d = 327 nm, along the Γ-M direction. The change in the DOS induced by the
corresponding single PS particle in PDMS is shown in the right–hand panel.
5.3.2. The Influence of the Composition in Disordered
Colloidal Systems
In section 5.3.1 the realization of a hybridization gap (HG) was presented
for monodisperse colloidal crystals as well as for disordered ‘hybrids’, e.g.,
for the 1:1 number ratio 300:360 nm PS hybrid in Fig. 5.11. However,
when discussing the HG at that point, the mixing of two different sizes
of colloids was only mentioned in order to introduce structural disorder,
i.e., to destroy the colloidal crystal, but the influence of the composition
on the position of the HG was not elucidated. Indeed, in the top panel
of Fig. 5.11 it seems that the HG for the pure 360 nm opal and the
1:1 hybrid come more or less at the same position. In this section, an
153
f  (GHz)
 
 
5. Phononic Behavior of Colloidal Systems
experiment is presented in which the number ratio of two kinds of PS
colloids with clearly different size (180 nm and 360 nm) is varied in a wide
range. The HG for these systems infiltrated with PDMS is observed and
the changes in width and position are discussed; of particular interest is
the question if it is possible to tailor this band gap by simply controlling
the size composition of the particles.
The set of samples has been realized by M. Retsch mixing distinct
amounts of 180 nm PS beads and 360 nm PS beads in aqueous dispersion
and preparing the hybrids by vertical lifting deposition under continuous
stirring in order to guarantee the homogeneous distribution of both par-
ticles in the whole sample. Doing so, hybrids with composition between
1:1 and 40:1 180:360 nm particles number ratio have been prepared, i.e.,
the volume ratio varied between 1:8 and 5:1 for the smaller particles
compared to the larger ones. After infiltration with PDMS, BLS was
performed in transmission geometry and the dispersion relation was ob-
tained fitting the spectra with Lorentzians in the q-range between about
0.006 nm−1 and 0.020 nm−1. The result is shown in Fig. 5.14.
Fig. 5.14 contains the experimental band diagrams for the mentioned
systems as well as for bare 180 nm PS particles. The position of the HG
is marked by colored rectangles. Note that for all systems only one HG
between ≈4-5.5 GHz was found in the experimentally accessible range,
only for the 1:1 hybrid, a second HG was observed between ≈2-2.5 GHz.
By comparison with the HG from the bare 180 nm PS particles and
taking into account the scaling of the HG with 1/d, it is obvious that
this gap originates (mostly) from the 360 nm particles. The 1:1 hybrid
is the only one whose volume fraction of the larger particles exceeds that
of the smaller spheres. Anyhow, in the rest of this section, I want to
focus on the higher frequency HG above 4 GHz.
Indeed, when comparing the HGs of the bare 180 nm particles and of
the hybrids, one finds that the HG for every hybrid is shifted to lower
frequencies in a mostly systematic way. The higher frequency HG of the
1:1 hybrid is distinctly lower and smaller than that of the opal. With
increasing amount of 180 nm particles in the mixture, the gap widens up
and is shifted to higher frequencies. This is true at least for the 9:1 and
154
5.3. Band Gaps in Polymer Opals and Disordered Systems
 
6
5 PS 180 nm
20:1
40:1 9:1
4 1:1
3
2
 PS 180 nm
PS Hybrids 180 nm:360 nm:
 1:1
1  9:1
 20:1 
 40:1
0
0.000 0.005 0.010 0.015 0.020
q / nm-1
Figure 5.14.: Development of the HG in binary PS hybrids infiltrated with PDMS
as a function of the composition. The colored rectangles mark the HGs for the
different systems. The empty black rectangle corresponds to the HG from the
180 nm PS opals after correction for effective sound velocity. The light red rectangle
around 2.5 GHz for the 1:1 hybrid belongs to the HG originating from the larger
spheres.
the 20:1 hybrid, the 40:1 hybrid, however, seems to contradict the trend,
although its HG is still broader and at clearly higher frequency than that
of the 1:1 hybrid. On the other hand, when ignoring the lowest point in
the upper branch of the 40:1 sample, also this sample would follow the
trend.
155
f / GHz
 
5. Phononic Behavior of Colloidal Systems
When comparing the opal with the hybrids, one should note that for
an fcc colloidal crystal of course the filling fraction (φ=0.74) is different
than for a disordered system. Approximating a uniform filling fraction of
φ=0.65 for all hybrids,[161] which is admittedly a bit arguable especially
for the extreme ratios, one can estimate that for the infiltrated opal the
effective longitudinal sound velocity ceff should be higher by about 6.3%
or 8.5%, applying the EMT of Gaunaurd and Wertman[142] or Wood’s
law,[105] respectively (cf. section 5.2) - in reasonable agreement with the
experimental ceff . Since the position of the band gaps scales with ceff ,
it is possible to normalize the frequencies by that. Doing so, also the HG
of the 180 nm opals is shifted towards somehow lower frequencies, still
following the trends mentioned above. In Fig. 5.14 the normalized (rela-
tive to the amorphous systems) HG of the PS 180 nm opal is symbolized
by the open black rectangle.
To rationalize the findings in this experiment one must go back on
the nature of the HG (section 5.3.1). The HG originates from the level
repulsion between the bands of the linear acoustic mode and those from
localized modes, i.e., the eigenvibrations inside the spheres. The genera-
tion of a band originating from these eigenvibrations can be described in
analogy to the tight coupling method for the electronic states in solids,
in which the overlap of atomic orbitals is assumed to be sufficient to
require corrections of the picture of the isolated atom, but not so much
to render the atomic description completely irrelevant.[141] This theory
predicts that in a sample of N atoms each electronic state consists of an
N -fold degenerated level. If the atoms come closer together, i.e., if their
wave functions overlap significantly, the levels broaden up into bands. In
the analogy of the phononic system, the individual spheres play the role
of the atoms, and instead of electronic wave functions one deals with the
acoustic resonance modes of the spheres (as ‘acoustic wave functions’),
which are both described by the same mathematics (cf. section 2.3.3).
This means that in order to obtain a strong band, the spheres must
be in close contact to each other and the acoustic wave function must
overlap with that of the next neighbor. Since the spheres are close-packed
(or randomly close-packed) in all cases, bands can be assumed. The
156
5.3. Band Gaps in Polymer Opals and Disordered Systems
question is how do modes of the same type but from different particles
sizes interact with each other. From symmetry considerations, there
is no reason for them not to interact, i.e., to hybridize. However, if
their energy levels are too different, they should be not able to interact
anymore. Anyhow, in the present experiment the frequency of the gaps
is shifted, which is a hint on a shift of the position of the resonance band
(in this case of the (1,2)-mode), too. The most reasonable explanation
is that, although there is a significant difference in energy for 180 nm
and 360 nm spheres, the (1,2) modes of the 180 nm spheres couple with
those of the 360 nm spheres, leading to shift to lower frequencies with
increasing fraction of the 360 nm spheres wave function.
On the other hand, the coupling seems to be weak enough to allow
two HGs for the 1:1 hybrid. Since the higher frequency HG is clearly
shifted towards lower frequency, it is obvious that there can’t be two
fully decoupled bands, at least there must be some influence of the bigger
spheres on the localized mode of the smaller spheres. For lower number
ratios of the larger spheres the low frequency HG completely disappears,
which is another hint for the rightness of the conclusion that this gap
must originate from the 360 nm particles, which are too separated in
the other hybrids to interact effectively. The smaller width especially for
the 1:1 hybrid can be rationalized by the larger average sphere to sphere
distance of the small spheres in the sample.
In summary, one can conclude that the experiment can be principally
captured qualitatively by simple hybridization considerations. However,
there are still open questions that could only be solved with additional
experiments and - even more needed - strong theoretical support. From
this section, I would like to retain the message that the distribution of
colloids of different size has an influence on the hybridization gap. Under
certain circumstances, the mixing of particles of different sizes may be a
way on which future studies to tailor the band gap could proceed.
157
5. Phononic Behavior of Colloidal Systems
5.4. Band Gaps in SiO2 Colloidal Systems
In the previous sections of this chapter it was shown that the elastic
contrast between matrix and scatterers is crucial for phononic phenom-
ena. Another important role is played by the density contrast, which is,
however, usually quite weak for polymer/liquid systems. In this section,
experiments on silica colloids in liquid or elastic matrix are shown, i.e.,
the elastic contast is relatively high as well as the density contrast is.
In such systems, additional effects can be found that can not be fully
captured by theory (and the explanations in the previous chapters), yet.
5.4.1. Phononic Behavior of Silica Suspensions
In section 4.2.4 a system consisting of dense packed silica spheres in an
index-matching liquid ethoxy-ethoxyethyl acrylate (SR256) matrix was
introduced to show the q-dependence of the light scattering intensity
of the particles eigenmodes. Since the eigenmodes appear at high f
compared to the longitudinal acoustic branch, there was no need to go
into the development of the acoustic behavior on that point. However,
since, in first approximation, the acoustic branch in Fig. 4.12 appears to
be linear, a more detailed investigation of the dispersion relation of the
acoustic mode shows some surprising details, which will be presented in
this section.
The investigated materials are the sample presented already in section
4.2.4 (d=375 nm) as well as an analogue sample with d=219 nm silica
spheres. The samples have been synthesized by D. Kiefer at DKI with
an initial concentration (filling fraction) of φ=0.34. A dense packing
(φ=0.74) was achieved by ultra-centrifugation on the bottom of an NMR-
tube (see also Fig. 1.3b).
Fig. 5.15 shows ten exemplary spectra for the dense packed sample
of d=375 nm spheres in the range 30◦≤ θ ≤75◦, i.e., 0.009 nm−1≤
q ≤0.020 nm−1, recorded at low free spectral range. In this q-range,
the general shape of the observed signal changes dramatically. Since at
the lowest angles the signal is clearly a single peak, the signal broad-
158
5.4. Band Gaps in SiO2 Colloidal Systems
Figure 5.15.: a Development of the acoustic signal in the BLS spectrum of
d=375 nm silica spheres in SR256 (φ=0.74) for 30◦≤ θ ≤75◦. b Exemplary spec-
tra at θ=40◦ for low (φ=0.34) and high (φ=0.74) filling fraction. c Experimental
dispersion relations for dense packed silica spheres with d=219 nm or 275 nm in
SR256. For comparison also pure SR256 as well as silica with d=375 nm and
φ=0.34 are shown.
ens rapidly with increasing angle. Furthermore, between 30◦ and ≈ 50◦
the frequency of the signal is virtually unshifted (dashed line), and only
the linewidth Γ increases. At higher angles, f(q) is not longer constant
and the frequency increases again with q, although the signals remains
extremely broad.
The corresponding dispersion relation is shown as open triangles in
159
5. Phononic Behavior of Colloidal Systems
Fig. 5.15c. For the lowest q’s, the acoustic signal behaves as an effective
medium with ceff ≈ cSR256. For comparison the pure SR256 is shown as
gray squares (with linear fit). From about q=0.008 nm−1 the dispersion
relation shows an abrupt bending, the effect that is seen in panel a. At
q≈0.015 f(q) increases again. Although the absolute values for f(q) at
high q’s, due to the large Γ, depend on the chosen limits of the fitting
range, the trend is clear and looks for an unchanged fitting range as
presented in Fig. 5.15c: f(q) increases monotonically and linearly again,
and the slope ∂f/∂q is essentially the same as that of the longitudinal
phonon in the matrix (second linear fit).
An akin dispersion relation is obtained for the smaller silica spheres
with d=219 nm. However, the bending starts at higher q’s and the second
linear range must be expected at q’s beyond the range measurable by
BLS. In fact, the position of the bending scales with d−1, and it appears
about π/d.
The appearance of the bending requires high filling fraction, since the
suspension with φ=0.34 (black circles) shows only the effective medium
acoustic phonon, which appears at slightly higher frequencies than for
the pure matrix. This is expected by effective medium theory (cf. section
5.1.2) since the density of the silica inclusions is much higher than that of
the matrix. The high density contrast leads to a strong scattering of the
phonons on the hard spheres. I.e., the phonons travel mostly through
the liquid matrix, and hence, ceff is nearly the same in both cases.
Correspondingly, the linewidth at a given q is smaller for lower filling
fraction as is shown for one exemplary angle (θ=40◦) in Fig. 5.15b. Γ
is inversely proportional to the phonon lifetime. If the phonon path is
disturbed by many scatterers, the phonon lifetime decreases drastically.
In fact, if Γ ≈ f , as it is found at higher q’s for φ=0.74, the free phonon
pathlength is akin to the phonon wavelength. I.e., the phonon cannot
travel anymore within the medium (The group velocity is zero.), the
system is overdamped. However, at higher q’s the group velocity comes
back to its value before the damping.
If one wants to summarize the experimental findings, one should start
with the fact that there is no band gap. At a certain q-range the group
160
5.4. Band Gaps in SiO2 Colloidal Systems
velocity becomes zero, however, there is no forbidden f -range. Since
the centrifugated sample is polycrystalline, the lack of a Bragg gap is
not surprising. On the other hand, it is noteworthy that the bending
occurs at π/d, which means q equals the shortest intersphere distance of
touching spheres. Anyhow, higher q’s within the bending correspond to
even shorter distances, while other crystallographic directions correspond
to q < π/d.
The lowest particle eigenmodes are found at clearly higher frequencies
(≈5 GHz for the d=375 nm particles, section 4.2.4). Therefore, it can be
excluded that the bending originates from a localization of the phonon
in the individual spheres, similar to HG (section 5.3.1). On the other
hand, a localization in the liquid ‘voids’ between the spheres is unlikely,
since the liquid forms a continuous matrix, hence, there are no closed
cavities to apply standing-wave conditions. Also the interpretation of
the the cavities as open Helmholtz resonators (similar to those in musical
instruments and subwoofers) seems to be not adjuvant.[153, 162]
Another possible explanation for strong localization that does not re-
quire a periodic structure, but, moreover, has ‘large’ disorder as a pre-
condition, is the Anderson localization of sound.[163, 164] Only recently,
John Page and coworkers could show the realization of the localization
of ultrasound in a three-dimensional elastic network consisting of disor-
dered, sintered, monodisperse aluminum beads.[165] However, the theo-
retical treatment of Anderson localization is extremely demanding. Fur-
thermore, it must be challenged that there is enough disorder in the
polycrystalline samples. In addition, in Page’s experiment a band gap
for certain frequencies was found, which is not the case in the experiment
presented here.
The bending must correspond to the appearance of a flat band in the
theoretical band diagram due to the folding of bands in an fcc lattice.[79]
An underlying symmetry can, theoretically, lead to the degeneracy of
bands in the theoretical band diagram. The perturbation introduced
by actual scatterers in the theoretical lattice may remove this degener-
acy, leading to non-degenerated flat bands, which can interact with the
acoustic band. In fact, layer multiple scattering calculations performed
161
5. Phononic Behavior of Colloidal Systems
 
6
4
2
[111] directio n
0
6  
4
2
[001] direction
0
0 2 4 6 8 10
qd
Figure 5.16.: Normalized theoretical band diagrams for fcc opals of SiO2 spheres
in SR256 matrix along two crystallographic directions. The experimental dispersion
relations in Fig. 5.15 are shown as circles and squares.
by N. Stefanou and G. Gantzounis at Univ. of Athens,[76] can capture
the experimental band diagram quite well. Fig. 5.16 shows the calcu-
lated phononic band structure for an fcc crystal of the given materials
along two distinct crystallographic directions together wirth the normal-
ized experimental points. The theoretical band diagram describes nicely
the long wavelength limit, but there is also a flat band expected, which
agrees with the flat band observed by BLS. However, there should be
clear band gaps, which are not found in the experiment, which is most
162
d/c
l  
  
5.4. Band Gaps in SiO2 Colloidal Systems
probabely caused by a certain directionality in the band gaps, which is
not fulfilled in the polycrystalline sample.
Although the band diagram can be quantitaively captured by theoret-
ical calculations, the interesting question remains still unsolved: What
is the origin of the flat band? As mentioned above, the explanation
given for soft scatterers (section 5.3.1) taht the flat band corresponds
to a tight-binding-like process from localized single particle modes, i.e.,
eigenmodes, is not suitable for this case, neither is the localization in
single liquid cavities. Rather, the flat band must be understood as a
multiple-scattering effect. Its details, however, are still unclear and must
be developed by theoreticians in the future towards a concrete, descrip-
tive explanation.
5.4.2. Silica–Poly(ethyl acrylate) Films (PhoXonics)
Another realization of silica based phononic materials are melt com-
pressed films with silica inclusions.[17, 67, 166] The technique of melt com-
pression is presented in section 3.2. When choosing silica-poly(ethyl
acrylate) (PEA) core-shell particles as the starting materials, films can
be pressed with different filling fraction of silica particles in a rubbery
PEA matrix, i.e., with different spacing of the silica spheres in an fcc
lattice.
In this section, the results of BLS measurements on several silica-PEA
films is presented, with dSiO2=216 nm or 253 nm. The total diameter
of the initial core-shell particles was chosen between dCSP=308 nm and
498 nm, leading to filling fractions of φ = (d 3SiO2/dCSP ) between φ=0.08
and 0.36 in the pressed films. The crystallographic orientation of the
silica spheres in the pressed film is known to be an fcc lattice with
orientation of lines of next neighbors in the radial direction from the
center of the film.[68] Due to the melt compression, in the film the silica
cores come nearer then in the initial opal, as is schematically shown
in Fig. 5.17b. The shortest intercore distance becomes (0.74)1/3dCSP
along the direction marked by the white arrow, which is also the main
orientation of q in the following experiments. The corresponding lattice
163
5. Phononic Behavior of Colloidal Systems
√
constant for an fcc-lattice is a= 2(0.74)1/3dCSP .
Fig. 5.17a shows five exemplary spectra for such a silica-PEA film
(dSiO2 / dCSP=253 / 387 nm) at different q’s along the direction marked
by the white arrow in panel b. The spectra are represented by a fit of
one (lowest q) or two Lorentzian signals on the Anti-Stokes side. Around
q≈0.008 nm−1 a band gap opens up. The corresponding band diagram is
shown in Fig. 5.17c, normalized by the effective medium sound velocity
ceff , measured in the long wavelength limit, and by the intercore distance
in the investigated crystallographic direction, which depends only on
the initial total diameter of the core-shell particles (see above). The
diagram contains also the normalized dispersion relations for two films
with dSiO2=217 nm and dCSP=308 nm and 329 nm. The dashed lines
mark the qBZ , the wavevector whose magnitude is half the diameter
of the first Brillouin zone (BZ) along the observed direction, and the
corresponding frequency. Obviously, a band gap is observed for all three
films at the same normalized q and f , at the edge of the first BZ, hence
it is a Bragg gap (BG) as introduced in section 5.1.3.
The films compared in Fig. 5.17 have φ ≥ 0.28. In fact, also two films
with significantly lower φ have been investigated, but neither for φ=0.08
nor for φ=0.16 a BG was found. These films show only a linear acoustic
branch in the band diagram. These results demonstrate that a certain
filling fraction is required to realize a BG, however, for sufficient elastic
contrast, even with filling fractions much lower then in the crystalline
case (φ=0.74) or the dense amorphous case (φ=0.65) (cf. section 5.3.1),
a relatively large BG can be realized. The comparison of the three band
diagrams in Fig. 5.17c shows that the width of the gap increases system-
atically with increasing amount of silica in the continuous PEA matrix.
Indeed, the ratio w of width of the band gap to its frequency is even
higher for the melt flow pressed films (w=0.25 for the 217/308 nm sys-
tem with φ=0.36) than for the PS/PDMS opal presented in Fig. 5.11
(w=0.18).
So far, a large BG at low filling fractions was presented in a handy,
all solid material, but these results are basically not astonishing. How-
ever, additional, more elaborate experiments lead to results that elude
164
5.4. Band Gaps in SiO2 Colloidal Systems
Figure 5.17.: a Experimental spectra for a melt compressed film (dSiO2 /
dCSP=253 / 387 nm) at different q’s, represented by Lorentzian lines. b Scheme of
the result of melt compression (section 3.2); the white arrow indicates the orienta-
tion of q. c Normalized dispersion relations for three different films. d Insensitivity
of a melt compressed film (217/329 nm) to stretching e and the orientation of q.
165
5. Phononic Behavior of Colloidal Systems
the straightforward explanation of BG. Fig. 5.17d shows the dispersion
relation of the 217/329 nm sample for four different strains. The results
were measured by BLS utilizing a stretching machine constructed by my
colleague Nikos Gomopoulos. Surprisingly, it turned out that the disper-
sion relation is virtually unchanged from the unstretched (strain=100%)
film up to strains of 250%, i.e., films stretched to the two and a half
fold of their initial length (strain=250%), while the stretching is in the
direction of q.
Theoretically, since the band gap is assumed to be a BG, one must
expect that the gap scales inversely with the spacing between the spheres,
as is also observed in Fig. 5.17c. However, when stretching the film,
the distance of the spheres should also change accordingly. From light
diffraction experiments on similar samples, also synthesized by DKI, it
is experimentally proven that stretching clearly changes the intersphere
distances.[68] In the direction of the strain, the distance between the
spheres increases, while perpendicular to that the spheres come nearer
together. Accordingly, also the BG should be shifted to lower frequencies,
which is not the case within the experimental error, or, with other words,
the gap is robust against structural variation. At high strains, it is
even likely that the crystallographic order vanishes, since the spheres
rearrange in a random way.
How can such robustness be explained? As was shown in section 5.3.1,
a hybridization gap (HG) originates from the individual scatterers prop-
erties. But the present band gap is for sure not a HG, since the position
of the HG would scale with d−1 −1SiO2 and not as observed with dCSP . Fur-
thermore, the lowest eigenmodes of silica embedded in the PEA rubber
should be not so different from those found in the liquid SR256 matrix in
section 4.2.4, which appear at clearly higher frequencies than the band
gap does in the films. Additional to that, the evidence of the gap being
a BG is striking. Not only the scaling with d−1CSP , but also the correct
prediction of the gap position by theoretical consideration in the un-
stretched case, are clear proofs. Moreover, there is no reason why there
should be no BG.
So, the open question remains why the gap does not react on the
166
5.5. Materials
stretching. So far, there can be only speculation on this matter. A
possible explanation might be that BLS is probing a kind of ‘effective
distance’ between spheres, which is a somehow averaged value. There-
fore, one must assume that the probed phonons are not only on a well
defined path like in Fig. 5.17b, which might be explained by strong scat-
tering by the silica spheres. Even at higher strain, the average distance
between each two neighboring spheres does not change to much (in fact,
it is a function of Poisson’s ratio). On the other hand, this explanation
seems not to be very satisfactory. If there would be very strong scatter-
ing, the phonon lifetime would decrease and the linewidth would increase
drastically. Furthermore, an analogue band gap would have to appear
also in the dense silica suspensions of section 5.4.1, which is not the case.
Therefore, instead of presuming to absurd theories, I would like to let
the explanation of this effect as an open question. Additional experi-
ments to determine the crystallographic behavior at high strains (e.g.,
AFM, TEM) and further theoretical support will be needed to solve this
demanding problem in the future.
In summary, both investigated systems with hard silica spheres in soft
liquid or elastic matrix show unprecedented phononic effects, which will
challenge experimentalists and theoreticians. A dense packing of silica
spheres in an index matching liquid shows an unseen dispersion relation
with an intermediate flat band only for certain q’s, which is probably a
result of the folding of bands. In melt flow pressed films containing about
30 vol% of silica cores in a crystalline order, a band gap is presented that
can be assigned as a BG. However, stretching of the film results in an
unexpected robustness of the gap position, which can not be explained
satisfactory so far.
5.5. Materials
The synthesis of the polymer and silica colloids is already described in
Section 4.4. Opals and disordered opals (hybrids) are obtained by verti-
cal lifting deposition described in Section 3.1.
167
5. Phononic Behavior of Colloidal Systems
For the inflitration of opals shown in Fig. 5.3 silicon oil and PDMS
made in house or purchased from Sigma Aldrich were used with no fur-
ther purification. 2(2-ethoxyethoxy) ethyl acrylate (SR256) used in the
suspensions was purchased from Sartomer with no further purification.
Melt Compressed Silica–Poly(ethyl acrylate) Films
The preparation of the compression molded films is explained in Section
3.2. The synthesis of the initial core-shell particles was performed by
my coworker Diana Kiefer at DKI, Darmstadt and is briefly described
here:[17, 67, 167]
The synthesis starts with the preparation of the silica core following
the Stöber process and subsequent step-by-step growing as described in
Section 4.4. The silica core’s surface was functionalized by acryl silanes
on which a thin (∼5 nm) PMMA interlayer was crafted by emulsion
polymerisation of methy methacrylate (MMA) together with some allyl
methacrylate (ALMA) monomer, using sodium dodecylsulfate (SDS) as
an emulsifier and ammonium peroxodisulfate and sodium dithionite as a
redox initiator system. Finally the PEA shell was crafted onto the allylic
double bonds of the ALMA, utilizing again SDS as emulsifier.
168
6. Smaller than Colloids:
Characterization of Stable
Organic Glass
6.1. Introduction
A wide range of packing structures are available to glasses, with more
efficient packing leading to higher moduli materials.[30] Aging a glass al-
lows for better packing and a higher modulus, but even long aging times
increase the modulus by only a few percent;[168, 169] preparing high mod-
ulus materials in this manner is impractical. In this chapter it is shown
that physical vapor deposition can be used to circumvent this kinetic lim-
itation and produce glasses whose moduli exceed those of the ordinary
glass by up to 19%. These high modulus glasses resist thermal treatment
and take at least 104 times longer than the structural relaxation time to
transform to the supercooled liquid. The ability to easily produce high
modulus glasses will prove to be useful for fundamental investigations
and coating technologies.
Unlike their crystalline counterparts, glasses have a nearly limitless
array of packing arrangements. As a supercooled liquid is cooled, mole-
cular motions eventually slow to such an extent that equilibrium cannot
be maintained. Below this transition temperature Tg, a mechanically
stable, non-equilibrium glass is formed. Glasses slowly evolve towards
equilibrium (i.e., aging) in a process that optimizes packing and creates
higher moduli materials. The structural relaxation time τα dictates the
rate at which this process takes place, and due to the steep temperature
dependence near Tg, τα is on the order of days only a few degrees below
169
6. Characterization of Stable Organic Glass
the glass transition temperature Tg. The aging process thus changes
the moduli so slowly that in practice changes of only a few percent are
possible.[168, 169] If high modulus amorphous materials are to be utilized
for science and technology, new preparation techniques are needed which
circumvent these kinetic restrictions and allow for more optimized amor-
phous packing.[170]
6.2. BLS experiments on IMC
This section shows that high modulus glass materials can be made effi-
ciently with physical vapor deposition. Using this preparation technique,
one can avoid the kinetic limitations of aging and prepare high modu-
lus glasses in a matter of hours. Enhanced dynamics at the surface of
amorphous materials allows for rapid configurational sampling in the
top few nanometers.[110, 111, 171–173] Vapor deposition can build an effi-
ciently packed amorphous material in a layer-by-layer fashion by taking
advantage of the enhanced surface dynamics and thus is not limited by
the slow relaxation dynamics of the bulk.[31] The mechanical properties
of vapor-deposited films are determined using Brillouin light scattering
spectroscopy (BLS). Because of the non-destructive nature of BLS, the
moduli of the as-deposited glass, the supercooled liquid, and ordinary
glass (created by cooling the liquid) can be determined from a single
sample.
Physical vapor depositions were performed by my coworker Kenneth
Kearns in Prof. Mark Ediger’s group at University of Wisconsin, Madi-
son, separately on two organic glass-forming materials: indomethacin
(IMC, Tg = 315 K) and tris-naphthylbenzene (TNB, Tg = 348 K). These
two molecules are well-known glass-formers and their glasses have been
previously prepared with physical vapor deposition.[31, 174, 175] During
deposition, the temperature of the substrate Tsubstrate and the deposition
rate are the important control parameters. For this work, Tsubstrate was
held near 0.85 Tg, i.e., 265 K for IMC and room temperature (≈295 K)
for TNB. The rate of the deposition in all cases was 0.2±0.03 nm/s. It has
170
6.2. BLS experiments on IMC
Figure 6.1.: a Schematic representation of the physical vapor deposition cham-
ber with temperature controlled substrate. b and c Chemical structures of TNB
and IMC, respectively. d Representative BLS spectra for a stable vapor-deposited
IMC glass (lower panel) and supercooled liquid (upper panel) at the wave vector
q=0.0136 nm−1. (Inset: transmission scattering geometry with scattering angle θ
of 70◦.) The Rayleigh line region (shaded area) was removed for clarity. Both the
longitudinal (L, black) and transverse (T, red) spectra are shown. Longitudinal and
transverse phonon scattering for the IMC glass (LIMC and TIMC , respectively) and
amorphous SiO2 substrate (Lsubstrate and Tsubstrate, respectively) are indicated in
the lower panel. The vertical lines drawn near the Stokes scattering peaks illustrate
the difference between the supercooled liquid and as-deposited glass.
previously been shown that these deposition conditions produce glasses
with low enthalpy, high density, and high kinetic stability.[31, 174–177]
Tsubstrate was controlled by attaching the SiO2 substrates to a copper
temperature stage (Fig. 6.1a). The deposition rate was controlled by
adjusting the temperature of the crucible. Further details are given in
the Methods section.
In the BLS experiment, the incident laser polarization was chosen
to be perpendicular to the scattering plane. Scattered light polarized
171
6. Characterization of Stable Organic Glass
a b
Figure 6.2.: a Temperature dependence of phase velocity c for IMC. The lon-
gitudinal (squares, black) and transverse (circles, red) c are shown for the stable
as-deposited glass (SG), the supercooled liquid (SCL), and the ordinary glass (OG).
Arrows indicate the progression of the heating and cooling cycles. Arrow 2 indicates
the isothermal transformation of the SG to the SCL at Tg+10 K. b Temperature
dependent longitudinal and transverse c of TNB for the SG, OG and SCL. The
thermal cycle is similar to the one described for panel a.
perpendicular and parallel to the incident polarization was measured in
separate experiments, providing access to scattering from longitudinal
and transverse phonons, respectively. The scattering from these two
polarizations is shown in Fig. 6.1d for vapor-deposited IMC glass at
298 K (lower panel) and supercooled liquid IMC at 336 K (upper panel).
Stokes and anti-Stokes shifts by the longitudinal (L) and transverse (T)
phonons are observed to the left and right of the Rayleigh line region
(shaded area), respectively. The vertical lines drawn on the Stokes side
of the spectrum illustrate the temperature-independent scattering of the
SiO2 substrate and the temperature-dependent scattering of the IMC.
The absence of longitudinal phonons in the transverse spectrum indicates
no birefringence in the vapor-deposited film within the sensitivity.
The peaks in the BLS spectra provide access to the phase velocities
172
6.2. BLS experiments on IMC
cl,t for the L and T polarizations and through this route the moduli can
be obtained. Spectra similar to those found in Fig. 6.1d were obtained
at multiple temperatures for both IMC and TNB. Each peak was fitted
with a Lorentzian lineshape yielding the linewidth ΓL,T and the peak fre-
quency fl,t. The corresponding phase velocities cl,t for the L and T polar-
izations were calculated using cl,t = 2πfl,t/q, and are plotted in Fig. 6.2.
Four experiments are shown in Fig. 6.2; in each case, the as-deposited
stable glass was heated to a temperature above Tg, held isothermally
until equilibrium was attained, then further heated, and finally cooled.
During the initial heating of the stable glass (SG), cl and ct have high
values as compared to the ordinary glass (OG), indicating high moduli.
During isothermal annealing above Tg (arrow 2), significant decreases in
the phase velocities are observed. This change, discussed in detail be-
low, signifies the transformation of the stable glass into the supercooled
liquid (SCL). Once the transformation is complete, the sample, now a
supercooled liquid, is heated still higher and then cooled to 295 K (ar-
rows 3 and 4). A characteristic kink in the temperature-dependent phase
velocity indicates Tg and the value obtained is in reasonable agreement
with the value obtained from differential scanning calorimetry.[174, 175]
From the phase velocities cl,t in Fig. 6.2, the moduli of the stable glass
and the ordinary glass can be calculated. The longitudinal bulk modu-
lus M=̺c2l , the shear modulus G=̺c
2
t and the commonly used Young’s
modulus E=9MG/(3M +G) can all be determined from c. For the sta-
ble as-deposited glass of IMC, M , G, and E moduli were determined
to be 8.3, 1.7, and 4.9 GPa at Tg, respectively. The moduli calculated
for the ordinary glass of IMC are smaller than those obtained for the
stable glass samples. At 312 K, M for the stable as-deposited IMC glass
is 14% greater than the ordinary glass while G and E are 19% greater.
A similar situation holds for TNB where M , G, and E are 10%, 15%,
and 14% greater, respectively, for the stable glass samples as compared
to the ordinary glass. To obtain these values, the published densities of
1.31 g/cm3 for IMC[178] and 1.16 g/cm3 for TNB[179] were used. The
stable as-deposited glass was assumed to be 1.5% more dense than the
ordinary glass in accordance with previous measurements on TNB vapor-
173
6. Characterization of Stable Organic Glass
deposited under similar conditions.[31, 177] Notably, the Poisson ratio σ
has the same value (0.36±0.01) in both stable and ordinary glass samples
and increases for the supercooled liquid as expected.[180]
In contrast to the results presented here, aging experiments on glass-
forming systems show only small improvements in mechanical properties
during aging. BLS experiments were performed on glycerol by quench-
ing from above Tg to Tg-4.2 K; in this experiment, a 0.6% increase
in modulus was achieved over 20 days.[169] For silicone oil, tempera-
ture down-jumps of 1.5 K near Tg resulted in a 2.7% increase in the
modulus during the 104 s equilibration time.[168] Based on calorimetry
measurements,[31, 175] it was previously estimated that IMC and TNB
samples vapor-deposited under the conditions utilized here have prop-
erties similar to those expected after aging an ordinary glass for more
than 300 years. Thus it is not surprising that aging for hours or days
does not produce the large modulus changes reported here. Pressur-
ization experiments have been shown to change moduli by up to 20%,
but this effect is reversible and upon pressure release, the system reverts
back to the ordinary glass value rendering it of little use for potential
applications.[181]
Notably, the high modulus IMC and TNB glasses produced by vapor-
deposition exhibit remarkable thermal stability. Fig. 6.3 (panels a and b)
shows the change in cl and ct for vapor-deposited IMC during isothermal
annealing above Tg. In these experiments, the temperature of the stable
as-deposited glass was increased to 325 K in about 15 min and then held
for several hours. Both phase velocities (and moduli) remain relatively
unchanged for the first 120 min of the experiment. After this induction
time, the phase velocities begin to decrease from the high values of the
stable glass to the supercooled liquid values. Approximately 200 min
is needed to complete the transformation to the supercooled liquid. To
put this value into context, the structural relaxation time τα of the IMC
supercooled liquid at 325 K is about 1 s based on dielectric spectroscopy
studies;[182] thus the isothermal transformation of stable vapor-deposited
IMC into the supercooled liquid requires about 104τα. Similar results are
obtained for TNB where the stable as-deposited glasses required 104.8τα
174
6.2. BLS experiments on IMC
to transform into the supercooled liquid at 358.2 K.[183]
Conceptually similar experiments have been performed on ordinary
glasses that have stabilized by aging. For example, Kovacs aged poly-
(vinyl acetate) for 2 months below Tg and then performed dilatometry
during isothermal annealing above Tg. He observed the transformation
into the supercooled liquid in a period of 35τ .[184]α Thus, in comparison to
glasses prepared by cooling a liquid and then aging, the vapor-deposited
IMC and TNB samples maintain their extraordinarily high modulus for
a remarkably long time.
Changes observed in the BLS spectral linewidth during isothermal an-
nealing provide insight into the mechanism by which the stable glass
transforms into the supercooled liquid. Fig. 6.3a shows that a maxi-
mum in Γ is reached near the midpoint of the transformation (Γ is read
from the right axis). A similar feature is observed in Fig. 6.3b although
the smaller signal of the transverse scattering makes this less apparent.
The observation of a change in Γ during the transformation of a glass
to a liquid is unprecedented. The increase in Γ indicates that the sam-
ple is heterogeneous on the 500 nm (∼ 2π/q) length scale during the
transformation. Fig. 6.3c shows the BLS spectral lineshapes at three
times during isothermal annealing at 325 K. Before the transition begins
(30 min) and after the transition ends (270 min), narrow lines are ob-
served. In contrast, near the midpoint of the transformation (180 min),
a broad line is observed. The shape of the line suggests that the sample
is a mixture of the supercooled liquid and the stable glass at this stage.
This point is illustrated by the dotted red curve in Fig. 6.3c which is a
linear combination of narrow lineshapes from the 30 min (stable glass)
and 270 min (supercooled liquid) data. This conclusion is consistent with
a recent quasi-isothermal differential scanning calorimetry study of these
materials.[176] While that work suggested that stable glasses transform
to the supercooled liquid via two-phase intermediate state, the spatial
extent of these regions (at least 500 nm) has not been known prior to
this BLS study.
Significantly increasing the modulus of amorphous solid materials us-
ing conventional aging methods is a time consuming and ineffective pro-
175
6. Characterization of Stable Organic Glass
a c
b
Figure 6.3.: BLS experiments on vapor-deposited IMC glasses during isothermal
annealing above Tg (325 K). a Longitudinal, cl, and b transverse, ct, phase velocity
changes as a function of time. Sigmoidal fits are given as guides to the eye. Two
different samples (black squares and red circles) are shown in panel a to demonstrate
the reproducibility of the transformation. In panels a and b, the full width at
half height linewidth, Γ, is shown as crosses. c Spectral line shapes (black) and
Lorentzian fits (solid red lines) at three different times during isothermal annealing.
cess. The work presented in this chapter shows that physical vapor de-
position can be used to prepare glasses with moduli that are as much as
19% greater than those made by cooling the liquid. These glasses resist
thermal treatment and retain their high modulus values for at least 104
times longer than the structural relaxation time of the glass at tempera-
tures significantly above Tg. During this slow transformation, a mixture
of supercooled liquid and glass is observed on length scales greater than
176
6.3. Materials and Methods
500 nm. High modulus vapor-deposited glasses may prove useful for
applications because of their remarkable mechanical properties but also
their extraordinary thermal stability. Given the known correlation be-
tween amorphous packing and modulus, new insights into this packing
could be realized through fundamental studies of these materials.
6.3. Materials and Methods
Stable Glass
IMC and TNB stable glass films were prepared by my coworker Kenneth
Kearns at University of Wisconsin in Madison, WI:
IMC was purchased from Sigma Aldrich and was used without further
purification. TNB was synthesized by McMahon and co-workers using
published methods.[185] SiO2 substrates (5 mm diameter and 0.15 mm
thick, Fisher Scientific coverglass) were attached to a copper temperature
stage using double-sided conductive carbon black tape (SPI Supplies).
The temperature of the substrate during deposition was controlled us-
ing a Lakeshore 340 temperature controller. To deposit the IMC and
TNB glass films, a quartz crucible containing the crystalline material
was heated inside a vacuum chamber (∼10−8 torr) such that the desired
rate of 0.2 nm/s was achieved. The rate of deposition was controlled
by maintaining the temperature of the quartz crucible, and the rate
was monitored with a quartz crystal microbalance (Sycon instruments).
Deposition continued until a 10-15 µm film was deposited. After deposi-
tion, the substrates were removed from the deposition chamber, placed
in desiccant and stored in dry ice prior to analysis.
BLS Measurements
BLS was performed in a transmission geometry to obtain the phase ve-
locity and thus modulus values. The transmission, as opposed to the
commonly used reflective geometry, was used to remove the influence of
refractive index on the measurements, and the details of this geometry
177
6. Characterization of Stable Organic Glass
are described in detail elsewhere.[32, 49, 95] For all experiments, the sam-
ple holder was initially flushed with argon gas and desiccant was placed
in the holder to eliminate any possible plasticization by atmospheric
water.[186, 187] The sample temperature was monitored with a platinum
RTD and controlled to within 0.5 K with a temperature controller which
was built in-house. For each data point in Fig. 6.2, the temperature was
allowed to stabilize for 15 min and then the BLS spectra were acquired for
5 min. The heating/cooling rate between each temperature measurement
was approximately 2-3 K/min. The temperature history establishes an
effective heating/cooling rate of approximately 0.1 K/min. For isother-
mal experiments, the temperature was changed from room temperature
to the annealing temperature in 15-20 min.
178
7. Concluding Remarks
7.1. Conclusions
Colloidal self–assembly by vertical lifting is a potential and versatile
method to create and design artificial sub-micron periodic structures
of hard (e.g., SiO2) and soft (polymers) colloidal particles. The fabrica-
tion of high quality crystals as well as the creation of artificial glasses by
the introduction of structural disorder is possible. Such structures have
spatial dimensions of their components in the order of the wavelength
of visible light, hence they act as artificial opals. In addition, they show
interesting phononic properties in the hypersonic (GHz) range.
Brillouin light scattering (BLS) is found to be a powerful technique to
measure the elastic properties of nanostructured materials at hypersonic
frequencies. In case of dry, non-transparent, multiple scattering colloidal
samples, BLS measures q–independent localized modes of the individual
spheres. In particular, for soft spheres these spectra are found to be
richer with increasing diameter. This finding is explained by theoretical
and geometrical considerations. Nearly all predicted modes (especially
those that appear at frequencies significantly below that of the acous-
tic phonon in the back–scattering case) are found experimentally and
assigned to their corresponding spheroidal eigenmodes. The analysis of
the eigenmode spectra yields the longitudinal cl and transverse sound
velocity ct, allowing the calculation of the Young and shear elastic mod-
uli at the nanoscale. Additionally, the analysis of the line shape can
provide information on the size polydispersity and on interactions with
surrounding medium.[32, 46]
Based on the studies of several binary colloidal mixtures, the eigen-
frequencies do not depend on the crystalline order or on the kind and
179
7. Concluding Remarks
number of the neighboring spheres (section 4.2.2). Only in the extreme
case of dilute large spheres in a sea of small spheres a small blue shift
was observed.[32] On the open discussion with regard the eigenmode
BLS activity,[52, 93] density of states and finite element calculations have
shown that all eigenmodes (n,l) can contribute to the light scattering
spectrum of spheres with diameter in the order of magnitude of the
probing laser wavelength. The limitation on even ‘quantum numbers’ l,
as claimed by others, is disputed (section 4.2.1). Further, the utilization
of densely packed silica spheres in an index matching liquid led to the
first verification of the theoretical predictions for the q-dependent ampli-
tude of the eigenmode spectrum; the frequency of these localized modes
is expectedly q-independent.
After these fundamental questions concerning the spectrum of the re-
solved modes in homogeneous colloidal spheres have been convincingly
addressed, the effect of the elastic constants on the particle vibrational
modes was best addressed in the case of random copolymer (PMMA-
PnBA) spherical particles, where the elastic constants vary with the
composition (section 4.2.3). The randomness of the copolymer and the
amorphous state of the particle was verified by the presence of single
Tg by DSC, further confirmed by the single a-relaxation obtained by
dielectric spectroscopy, i.e., both kind of segments feel the same energy
landscape. The systematic red shift of the eigenfrequencies with increas-
ing composition of the softer (PnBA) component as well as the effective
medium velocities measured in bulk copolymer films were theoretically
captured when a higher ‘effective glassy sound velocity’ was assumed for
the PnBA in the copolymer than in its bulk state.
Two additional elaborate BLS experiments on nanostructured core-
shell hybrid systems were presented herein: (i) Hard silica colloids sur-
rounded by a soft polymer shell show eigenmode spectra that are domi-
nated by the scattering from the soft polymer shell (larger compressibil-
ity). Nevertheless, due to the boundary conditions and displacements in
the composite material, it becomes feasible the extraction of the elastic
constants in parts, core and shell at the nano- and mesoscale. The spa-
tial confinement causes a hardening of the thin polymer chain (section
180
7.1. Conclusions
4.3.2).[12] (ii) In the inverse particle architecture, where the polymeric
core is embraced by a thin silica shell, the former dominates the eigen-
mode spectrum. A temperature dependent study has revealed a striking
hardening of the polymeric core as compared to the bare polymer spheres
implying a modified glassy state under confinement. The glass transi-
tion temperature Tg of the core, however, remained virtually constant
whereas the thin silica shell assured a robust shape persistent polymeric
core at temperatures well above its Tg (‘nano armor’, section 4.3.1).
[100]
Since the hypersonic phonons are in the order of magnitude of few
hundreds nanometers, they do not only probe the effective medium
structured in a molecular scale (like in the copolymers), but also in the
mesoscale. By infiltration of the dry samples with a liquid close to op-
tical matching, it is possible to overcome the multiple light scattering
and, by that, to get access to the collective acoustic behavior of the col-
loidal systems. Pure effective medium behavior is observed by BLS for
such systems in the long wavelength limit, i.e., for small q’s, where the
phonon wavelength exceeds the structure length of the colloidal system.
Section 5.1.2 presents a short overview over the most important estab-
lished effective medium theories, while in section 5.2 their applicability
on a system of defect doped liquid infiltrated opals is presented.
At higher q’s, BLS can be used to record directly the dispersion rela-
tions that led to the first demonstration of a distinct hypersonic Bragg
gap at the edge of the first Brillouin zone of a colloidal crystal by Cheng
Wei et al.[8] In this thesis, the resonant character of the particles was real-
ized to demonstrate, for the first time, the presence of an additional band
gap - the hybridization gap, theoretically predicted six years earlier. This
HG originates from the interaction of the acoustic band of the effective
medium and bands from the multipole modes of the individual particles
(section 5.3.1), i.e., the eigenmodes. Hence, its realization demonstrates
the strong correlations between the individual colloid’s elastic properties
(the ‘music’) and the phononic characteristics of the ensemble (the ‘con-
cert’). For polymer colloids infiltrated with an index matching liquid like
PDMS or silicon oil, the HG is found to open up at frequencies below
that of Bragg gap.
181
7. Concluding Remarks
Increase of the mechanical mismatch by going from soft to hard opals
based on silica spheres significantly changes the propagation characteris-
tics (section 5.4). This includes an unseen bending in the band diagram
of polycrystalline silica colloids in a liquid matrix as well as a Bragg
gap, which is peculiarly robust against structural deformation, in silica
/ rubber phoXonic films.
7.2. Outlook
Phononic materials might have interesting potential applications starting
with the obvious use as acoustic shields or vibration isolators. However,
for most practical applications, gap frequencies in the range of sonic or
ultrasonic seem to be of higher interest than those in the hypersonic
range. On the other hand, if the blocked phonon’s wavelength is com-
parable to the structure dimensions, i.e., λ ∼ d, then the structure’s
components for ultrasonic devices must be near to the mm–range and
even larger to stop sound between kHz and Hz. The ultimate goal would
be to create materials, which can realize gaps at wavelengths much larger
then the inherent length scale of the gap material (λ ≫ d). Ping Sheng
and coworkers demonstrated an approach going in this direction.[90] They
prepared locally resonant sonic materials by combining a spherical lead
core and a silicone rubber coating in the millimeter size range and build-
ing up a crystal out of these. The resulted superstructure exhibits a
band gap around 400 Hz, which corresponds to a wavelength two orders
of magnitude bigger than the diameter of the spheres. Like HG the ex-
planation is due to localized modes mostly in the soft shells. Advances in
colloidal science can be utilized to provide materials with new functions.
For example core–shell particles with varying composition, hybrid ma-
terials, or materials with even hierarchical order can allow tailoring the
phononic properties of small but smart materials to frequencies, where
a broad range of applications is conceivable.
When we turn our attention back to the manipulation of the phonon
flow in the GHz range, hypersonic phononic materials hold promise to
182
7.2. Outlook
control heat flow in insulators and semiconductors. The typical phonon
wavelength responsible for the heat transport is even smaller (≈10 nm)
than the structures investigated in this thesis. If we will control the flow
of elastic energy by building phonon guiding devices in full analogy to the
lossless light guidance in photonics or by building devices with a strong
anisotropy of phonon transmission, we can also attain new knowledge in
advanced heat management. In this direction, colloidal science could of-
fer new materials, e.g., multilayer structures with different particle sizes
in each structure that could split and direct the elastic energy. Other
analogies to photonics, e.g., acoustic superlenses already exist,[156] but
still impose challenges for high frequency acoustics. It should be men-
tioned that hypersonic phononics can simultaneously act as photonics for
wavelengths in the visible spectrum allowing the realization on photonic
and phononic band gaps at the same time (PhoXonics).[89] The coupling
of optical and mechanical degrees of freedom can be applied to influence
the optomechanical dynamics of small systems, utilizing the radiation
pressure of light.[34] The upper frequency limit for such systems is, so
far, in the MHz-range, however, colloidal systems like those presented in
this thesis may expand the investigation of such effects to the GHz-range,
giving raise to novel fascinating phenomena.
In parallel to the development of materials with new phononic func-
tions, there is also a need for new experimental techniques to measure
the band diagrams of non-transparent structures exhibiting also strong
phononic gaps. BLS is a powerful technique for transparent samples
with serious shortcomings in opaque systems such as dry opals and in
measuring hypersonic transmission spectra. The main problem is the
availability of a selective and continuous generation of phonons in the
GHz–range, which will be hopefully overcome in the near future.
183
Acknowledgments
‘Begegnet uns jemand, der uns Dank schuldig ist, gleich fällt
es uns ein. Wie oft können wir jemandem begegnen, dem wir
Dank schuldig sind, ohne daran zu denken!’
J. W. v. Goethe, Maximen und Reflexionen
The work presented in this thesis would be nothing without the help
of so many people, who advised me, who provided samples, who eased
my daily life, or helped me in many other ways. Having in mind the
quotation I cited above, I hope not to forget anyone of these important
people, who helped me to make not only a happy but also a successful
time out of the last two and a half years.
First of all, my thanks go to Prof. XXXXXXXX, who gave me the
opportunity to do my PhD in his research group at Max Planck Institute
for Polymer Research. I enjoyed very much the freedom of science, which
is not given everywhere to a PhD student.
Prof. XXXXX XXXXX (J. Gutenberg-Univ. Mainz) and Prof. XXXXXX
XXXXXX (Univ. Utrecht) kindly agreed to be the second and third ref-
eree of my thesis.
The deepest gratitude I owe to Prof. XXXX XXXXX, who was a
great supervisor, giving me all the freedoms I needed to develop my own
ideas, but on the other hand pushing me in the best possible meaning
of the word to go on with them in a very consequent and productive
way. Although, as an external member of the Max Planck Society, he
was not physically present in Mainz too often, he was always available to
discuss and to share his enthusiasm on science. When he was around, our
discussions were very intensive and instructive, leading so many times to
new perceptions and ideas on both sides.
185
Acknowledgments
I’d like to thank Prof. XXXXXX XXXXXX for many scientific and
non-scientific discussions, technical support, proof-reading a part of this
thesis, and for hiring me.
When I started my PhD in Mainz, there were two fellows that sup-
ported me a lot by introducing me into BLS and some of the topics that
accompanied me for the next years, Dr. XXXX XXXX and Dr. XXXX
XXXXX. Especially XXXX XXXX was always available to help me to
solve my smaller and bigger problems with the setup (‘Touch it as if you
would touch a woman’) and to discuss my results. Beyond that, I en-
joyed his special interpretation of far eastern philosophy (‘You can even
ride on a donkey!’).
When reading this thesis, it becomes clear that nearly all presented
results are based on diverse cooperations, since the success of my studies
was up to the continuous supply with all kinds of high quality samples.
The most fruitful and longest lasting of these cooperations was with Dr.
XXXX XXXXX and Dr. XXXX XXXX. I really enjoyed our work to-
gether, since both of them came up with a great deal of improvements
and completely new ideas. XXXX was the best conceivable ‘colloid cook’,
who always delivered requested samples or realized just discussed ideas
incredibly shortly and with impressive quality. Beyond science, he be-
came a good friend.
I also acknowledge the fruitful cooperation with XXXXXX XXXXX,
XXXXX XXXXXX and Dr. XXXXX XXXXX (all MPIP), leading to
the investigation of PS-silica core-shell particles.
I thank Dr. XXX XXX and XXX XXXX from Dr. XXXX XXX’s
group at DKI Darmstadt for the ongoing cooperation on Silica-PMMA
core-shell particles, silica suspensions, and the phoXonic films.
XXX XXXXX and Prof. XXX XXX from Univ. Wisconsin must
be named thankfully for the cooperation on the stable organic glasses
project, which will be continued with Dr. XXXX XXXX.
For their theoretical support I’d like to thank Dr. XXX XXXX (Univ.
du Havre), Prof. XXXX XXXX, and XXXX XXXX (both Univ. of
Athens). Especially XXXXXX was involved in many projects, and I
want to thank her not only for her calculations, but also for her patience
186
in explaining me her results.
I thank my office mates XXXXX XXXX and XXXXX XXXXXXX for
many helpful discussions, as well as XXXXX for building the stretching
machine.
I benefited from the great infrastructure at the Max Planck Insti-
tute for Polymer Research that might be virtually unique in the world.
Therefore, I thank the people in the different service groups, the mechan-
ical shop, the electronics shop, and besides that the technical support
by XXXXX XXXX, XXXXX XXXXX and XXXXX XXXXX (SEM),
XXXX XXXXX and XXXXX XXXXXX (DSC), Dr. XXXXX XXXXX
(dielectric), and XXX XXXXXX (AFM).
Anyhow, also a good working atmosphere is needed to find full gratifi-
cation in what one is doing. This good atmosphere was greatly enhanced
by many non-scientific activities like barbecues, our daily lunch at 11.30
(or usually some minutes later, since there were always one or two people
we had to wait for), the coffee corner philosophy (with special thanks to
XXXXX XXXXX for his outstanding contributions), the weekly soccer
training, the table soccer matches, and so on. For all this I want to thank
all involved members of XXXXX, but also, in the same manner, the peo-
ple from the material research group (formerly known as XXXXXXX),
who treated me almost as one of themselves.
This thesis was proof-read by XXXXX XXXXX (MPI Golm), XXXXX
XXXXXX (Univ. Marburg), XXX XXXXX (LMU Munich), and XXX
XXXX (MPI Martinsried). Thank you all.
Coming to the end of these acknowledgments, I want to thank the
supervisor of my diploma work, Prof. XXXXXXXXXXXX XXXXXX
from Philipps-Universität Marburg for introducing me into the field of
inelastic scattering. Furthermore, I thank many colleagues from all over
the world that I met on several conferences and who gave new impulses
to me during many interesting discussions.
I want to thank my parents for all the support they gave me and all
chances they facilitated in my life.
And finally, I want to express my deepest thankfulness to the woman
I love, who gave me so much joy and happiness in the last years. Danke!
187
Abbreviations
2D twodimensional
3D threedimensional
α angle incident laser/sample
αη Mark-Houwink parameter
β(T,S) (isothermal, adiabatic) compressibility
χ
ǫ dielectric constant
ǫ extinction coefficient
φ glancing angle
ϕ,ψ, χ scalar functions
Γ central point of fcc BZ
η viscosity
λ wave length
λ Lamé coefficient
Λ phonon periodicity
µ Lamé coefficient
ν velocity of light in vacuum
θ scattering angle
̺ mass density
σ Poisson’s ratio
σ scattering cross section
σ̂ normalized scattering cross section
σik stress tensor
τ correlation time
ζ tandem FP angle
ω angular frequency
A free energy
189
Abbreviations
A area under the curve
A absorbance
Alml′m′ structure constants
BLS Brillouin light scattering
nBA n-butyl acrylate
BG Bragg gap
cF coefficient of finesse
cl/t/eff (longitudinal/transverse/effective) sound velocity
Cp/T specific heat (at constant p/T)
Cik components of the stiffness matrix
CCD charge-coupled device
d diameter
d1/2 FP mirror distances
D diffusion coefficient
DOS density of states
DSC differential scanning calorimetry
DTA differential thermo analysis
DVB divenyl benzene
E Young’s modulus
E electric field
E0 field amplitude
EMT effective medium theory
f frequency
F finesse
F force
flm(r, θ, φ) solution of the scalar Helmholtz equation
fcc face centered cubic
FP Fabry-Pérot interferometer
FSR free spectral range
G Green’s function
G reciprocal space vector
G(q, τ) time-correlation function
g2(q, τ) second order autocorrelation
190
GPC gel permeation chromatography
h, ~ Planck quantum (/2π)
hcp hexagonal close packing
HG hybridization gap
I (scattering) intensity
I unit tensor
IMC indomethacin
kB Boltzmann’s constant
KPS Potassium persulfate
K Gordon-Taylor parameter
K bulk modulus
Kη Mark-Houwink parameter
k(i/sc) (incident/scattered) wave vector
l longitudinal
L distinct point in reciprocal fcc lattice
L shear modulus
l,m,n three independent vectors
LMS layer multiple scatering method
M distinct point in reciprocal fcc lattice
MMA methyl methacrylate
MPIP Max Planck Institute for Polymer Research
MS multiple scattering method
m mass
me electron mass
Mn number averaged molecular weight
Mw weight averaged molecular weight
n refractive index
ni unit vector
NaPSS sodium sulfonated polystyrene
Nd:YAG Neodym doped yttrium aluminium garnet (laser)
OG ordinary glass
p pressure
p momentum
191
Abbreviations
PnBA poly (n-butyl acrylate)
PCS photon correlation spectroscopy
PDI polydispersity
PS polystyrene
PW plane wave method
PMMA poly (methyl methacrylate)
q scattering wave vector
q absolute value of q
qpara q parallel to the sammple plane
qperp q perpendicular to the sammple plane
Q heat
r radius
r space vector
R reflectivity
R rigidity
R position vector
R absolute value of R
Rh hydrodynamic radius
Rl(k, r) spherical Bessel functions
Rn Bloch’s vector
RS Raman scattering
S entropy
SCC supercooled liquid
SEM scanning electron microscope
SG stable glass
T T-Matrix
t time
t transverse
T temperature
T transmittance
TEM transmission electron microscope
TNB ααβ-1,3,5-tris-naphtylbenzene
u displacement vector
192
uik strain tensor
UV/VIS ultraviolett / visible light (spectroscopy)
V (scattering) volume
v velocity
w work
w relative width of a band gap
wc degree of crystallization
X distinct point in reciprocal fcc lattice
x(n, l) eigenmode constant
Ylm spherical harmonics
193
A. scattering geometry
A.1. transmission case
incident laser beam
n0=1 a
n gb
ks
ki
2
q
1 qpara
n ga
n0=1 aq scattered beam
Figure A.1.: Transmission geometry
Since BLS is a quasi-elastic scattering technique, incident and scattered
wave vector can be regarded as of equal length:
| 2πnki| = |ks| = . (A.1)
λ
With Snell’s law:
( )
n sinβ = sinα ⇒ β = sin−1 1 sinα , (A.2)
n
195
A. scattering geometry
( )
− ⇒ −1 1n sin γ = sin θ α γ = sin sin θ − α . (A.3)
n
The length of the scattering wave vector q is the lenght of the third
side of the isosceles triangle spanned by ki and ks. Dividing it into two
right-angled triangles allows to apply simple trigonometric rules:
| | · 2πn · β + γq = q = 2 sin (A.4)
λ 2
And with Eqs. A.2 and A.3
[ ( ( ) ( ))]
4πn 1
q = sin sin−1
1
sin (θ − 1α) + sin−1 sinα . (A.5)
λ 2 n n
Under the distinct condition that the incident angle α is half the scat-
tering angle, q becomes equal to its component parallel to the surface,
and Eq. A.5 can be further simplified for this special case:
q = qpara = |qpara| for γ = β ⇒ θ = 2α (A.6)
2πn
qpara = (sin β + sin γ) (A.7)
λ
2π 4π θ
qpara = (sinα+ sin (θ − α)) = sin . (A.8)
λ λ 2
In the last equation, valid only in this special case, the refractive index
disappears.
196
A.2. reflection case
A.2. reflection case
reflected beam
n0=1 a a incident laser beam
q
n d e
k bh s
r
qperp g
q b
h ki
n
a
n0=1
Figure A.2.: Reflection geometry
Starting from the same point as in the transmission geometry,
|ki|
2πn
= |ks| = , (A.9)
λ
applying Snell’s law and the internal angle of a triangle, we can define a
couple of angles shown in Fig. A.2:
( )
β = sin−1
1
sinα (A.10)
n
γ = 180◦ − α (A.11)
ρ = 180◦ − (90◦ − β)− ǫ = 90◦ + β − ǫ (A.12)
ǫ = 90◦ − δ (A.13)
n sin δ = sin 180◦ − θ − α (A.14)
197
A. scattering geometry
Now it is possible to express the angle γ in the sample between ki and
ks as a function of α and θ
( )
γ =180◦ − 90◦ − 1sin−1 sinα +
n
( ( ))
90◦ − 1sin−1 sin (180◦ − θ − α)
n
( ) ( )
1 1
=180◦ − sin−1 sinα − sin−1 sin (θ + α) , (A.15)
n n
and q turns out to become:
[ ( ( ) )]
4πn 1 1 1
q = sin 180◦ − sin−1 sinα − sin−1 ( sin (θ + α))
λ 2 n n
[ ( ( ) )]
4πn 1 1 1
= cos sin−1 sinα + sin−1 ( sin (θ + α)) (A.16)
λ 2 n n
The angle η between q and ki or ks is
180◦ − γ
η =
2
[ ( ) ( )]
1
= sin−1
1 1
sinα + sin−1 sin (180◦ − θ − α) . (A.17)
2 n n
Using this equation, the relation between q and its component perpen-
dicular to the surface, qperp, can be expressed again as a function of the
two angles θ and α,
qperp
=cos (β − η)
q
[ ( ) { ( )
1 1
=cos sin−1 sinα − sin−1 1 sinα
n 2 n
( )}]
1
+ sin−1 sin (θ + α)
n
(A.18)
198
A.2. reflection case
[ { ( ) ( )}]
1 1 1
= cos sin−1 sinα − sin−1 sin (θ + α)
2 n n
and qperp becomes
qperp =q · cos (β − η)
[ { ( ) ( )}]
4πn 1 1 1
= sin 180◦ − sin−1 sinα − sin−1 sin (θ + α)
λ 2 n n
[ { ( ) ( )}]
· 1 1 1cos sin−1 sinα − sin−1 sin (θ + α) .
2 n n
(A.19)
By comparison with Eq. A.16 it is clear that q = qperp if the last term
on the right side becomes one. Due to the periodicities of the inverse
sine function, this is the case for
θ − 180◦
α = . (A.20)
2
I.e., when chosing this special geometry in the BLS experiment, the
wave vector lies fully perpendicular to the substrate’s surface. However,
the careful comparison with Fig. 2.6 shows that the fulfillment of this
geometry would lead to a reflection of the laser directly into the Fabry-
Pérot. Therefore, only geometries very near to that can be chosen in
practice, in order not to destroy the detector, and the q’s probed in
reflection geometry are only approximately equal to the qperp’s.
199
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