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Shear Thinning: Determination of Zero-Shear Viscosities 
from Measurements in the Non-Newtonian Region
Bernhard A. Wolf
2. Modeling
Experimental information on the viscosities, η, of polymer solutions and 
of polymer melts as a function of shear rate is modeled by means of an The present treatment is based on some 
approach that describes the diminution of ln η as a function of shear stress, assumption/premises.
τ, in terms of an exponential decay. The approach uses the following three •	 In contrast to the usual procedure, 
adjustable parameters: the zero-shear viscosity of the system, a character- the changes in the viscosity η of 
istic shear stress, quantifying its susceptibility toward shear thinning, and polymer containing systems caused 
a dimensionless parameter stating the magnitude of the effect. This pro- by shear should not be modeled as 
cedure gives access to the Newtonian behavior also in cases where direct a function of the shear rate, γ
 , but 
rather as a function of shear stress, 
measurements are impractical or impossible; it discloses two phenomena τ  =  ηγ. The reason is that it is the 
not reported so far: a qualitative change in the efficacy of τ at a characteristic force per area that should be decisive 
concentration and indicates the occurrence of two different disentanglement for the fraction of entanglements 
mechanisms in thermodynamically unfavorable solvents. that is opened. This approach does 
not claim the existence of a threshold 
stress.
1. Introduction •	 Treating the disentanglement process by way of trial as a 
first order reaction, it appears expedient to study the rela-
tive changes in η as a function of τ. This assumption im-
The present study was initiated by problems with the experi- plies that ln η should decrease exponentially with rising τ.
mental determination of zero shear viscosities: Some polymer •	 Finally, for the modeling of the shear thinning of polymer solu-
solutions—like that of high molecular weight polysaccharides tions it appears recommendable to replace ln η by ln (η/η sol-
in water—are so susceptible against shear that standard rheom- vent) (i.e., by the relative viscosity ln η rel) because the solvent 
eters fail to reach the Newtonian flow regime. For that reason we is normally not directly involved in disentanglement processes.
wanted to find out whether there exist ways to obtain the desired 
information by reliable extrapolation methods. To the best of our The above considerations yield the following Ansatz
knowledge no such procedures were reported in the literature, 
despite the fact that the influence of shear rate on the viscosity of *lnητ = lnηo −τ /τ1rel rel − A1(1− e )  (1)
polymer solutions was very topical from the beginning of polymer 
science and still is, as demonstrated by a recent review article on where the superscripts τ and o indicate data referring to arbi-
that subject.1 This contribution describes an option to solve the trary shear stresses τ and to their limiting values for τ = 0. The 
problem using a “kinetic” model that quantifies the effects of the characteristic shear stress τ *1  measures the susceptibility of the 
disentanglement processes on the viscosity as a function of shear system toward shear thinning (resistance against disentangle-
stress. The present study uses exclusively published data; detailed ment) and the dimensionless parameter A1 = lnηorel − lnη∞rel 
experimental information can be found in the cited literature. states the maximum shear effect that would be hypothetically 
reached for infinite shear stress. Equation (1) describes the 
majority of experimental data reported in the literature, as will 
Prof. B. A. Wolf be demonstrated hereinafter.
Institute of Physical Chemistry For some of the systems under investigation a modifica-
Johannes Gutenberg-University Mainz
Welderweg 11, Mainz D-55099, Germany tion of Equation (1) turned out to be necessary. Equation (2) 
E-mail: Bernhard.wolf@uni-mainz.de accounts for two dissimilar disentanglement processes by intro-
The ORCID identification number(s) for the author(s) of this article ducing the additional parameters A2 and τ
*
2 .
can be found under https://doi.org/10.1002/macp.202000130.
© 2020 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, lnητ = lnηo − A1(1− e−τ /τ * *1rel rel ) − A2(1− e−τ /τ2 )  (2)
Weinheim. This is an open access article under the terms of the Creative 
Commons Attribution-NonCommercial License, which permits use, dis- The subsequent examination of the suitability of the above 
tribution and reproduction in any medium, provided the original work is 
properly cited and is not used for commercial purposes. relations for the modeling of experimental data is not confined to 
polymer solutions but also includes some examples of polymer 
DOI: 10.1002/macp.202000130 melts. In the latter cases ln ηrel is simply replaced by ln η.
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The following relation2 was used for the modeling of the 
composition dependence of the zero shear viscosities
c 2 + α c
lnηorel =  (3)
1 + β c + γ c 2 
c  =  c [η] is the reduced polymer concentration, where c is 
given in mass/volume, and α, β, and g are system specific 
parameters; α can be set zero in the majority of cases.
3. Results and Discussion
3.1. Polymer Solutions
All experiments evaluated in this section were performed at 25 °C. 
Solvents, polymers, and their molar masses as well as concentra-
tions are detailed in Table 1, which also collects the parameters Figure 1. Relative viscosities as a function of shear stress τ for the indicted 
obtained from the reported data by means of Equations (1) and (2). concentrations. The curves are modeled by means of Equation (1).
Kniewske3 with respect to the influences of polymer concentra-
3.1.1. Toluene (TL)/Polystyrene (PS)3 tions. The dependencies of the relative viscosities on the shear 
stress τ are shown in Figure 1. The graph reveals that the cur-
The shear thinning behavior of PS in the thermodynamically vature changes with rising concentration from positive to nega-
good solvent TL has been studied extensively by Kulicke and tive. Directed experiments would be required to find out how 
Table 1. Collection of the polymer solutions for which flow curves were modeled according to Equations (1) and (2) plus parameters obtained.
Solventa) Polymer 10−6 m wt% lnη 0 A1 τ * [Pa] A2 τ *rel 1 2 [Pa]
Toluene PS 23.60 0.50 5.30 2.56 2.32
23.60 1.00 8.10 4.80 7.39
23.60 2.00 11.70 8.29 23.30
23.60 3.00 13.30 10.22 55.50
23.60 4.00 14.10 −22.30 −402.60
Decalin PS 2.00 11.60 11.00 4.15 421.00 6.15 3068.00
Water PAAm 0.51 5.00 5.26 −0.14 −81.20
1.70 2.00 6.24 3.61 60.40
3.90 1.00 2.56 −0.09 −1.36
3.90 2.00 8.85 6.73 62.20
3.90 5.00 13.80 −17.00 −777.00
5.30 1.00 6.70 3.69 13.70
5.30 2.00 10.70 7.20 52.50
5.30 5.00 14.60 −13.74 −633.00
Glycerol PAAm 1.70 1.00 6.02 1.73 62.80
5.30 1.00 8.42 2.35 56.40 1.56 7.06
Formamid PAAm 1.70 1.00 4.48 102.00
5.30 1.00 5.61 2.89 8.77
Pentamer of PDMS 0.17 5.00 2.20 1.28 37.16
PDMS
0.17 10.00 3.57 −2.15 −115.50
0.17 20.00 5.65 −0.28 −41.80
0.17 40.00 8.35 −0.39 −185.00
a)PS: polystyrene 25 °C, PAAm: polyacrylamide 25 °C, PDSM: polydimethylsiloxane 60 °C.
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Figure 2. Concentration dependence of the parameters τ *1  and A1 for the 
dilute solutions. Both curves are modelled by an exponential decay.
Figure 4. As Figure  1 but for the system decalin/polystyrene. Only the 
the thus defined characteristic composition correlates with coil- data for the two lower concentrations can be modeled by Equation (1). For 
overlap and crossover concentrations. 11.6 wt% this relation fails (dotted line) and Equation (2) must be used.
Figure 2 shows how the parameters of Equation (1) vary 
with composition for the four lowest concentrations; sign and g  =  –0.0002. This outcome is typical for the present type of 
magnitude of the data for 4  wt% reveal (cf. Table  1) a funda- polymer solutions.
mentally different dependence beyond a critical composition 
of the system located between 3 and 4  wt%. Both parameters 
become zero in the limit of infinite dilution, consistent with 3.1.2. Decalin/Polystyrene5
theoretical considerations.
In order to check whether the zero shear viscosities obtained The shear thinning behavior of PS in the thermodynamically 
from the above evaluation can be modeled by means of the unfavorable solvent decalin has been studied by Kulicke and 
generally applicable three parameter approach2 formulated Porter.5 The evaluation of these results is shown in Figure 4.
in Equation (3), the relative viscosities are in Figure 3 plotted The shear stress up to which the measurements were per-
accordingly. formed for the highest concentration is uncommonly large. In 
The experimental errors in ηo were estimated to be 5% this case the results can no longer be modeled quantitatively by 
for all concentrations, except for lowest for which it is 15%. means of Equation (1). The reason why two different disentan-
Calculating the intrinsic viscosity [η] from the known molar glement processes are required for the description is presently 
mass of the polymer by means of a published Kuhn–Mark– unclear. It could simply lie in the large range of τ values under 
Houwink relation4 and setting the parameter α = 0 yields the investigation; on the other hand one could speculate that this 
following values for the two remaining parameters: β = 0.075, behavior is caused by the marginal solvent quality of decalin 
for polystyrene. Under such conditions it would not be unrea-
sonable to distinguish between two types of entanglements, the 
normal ones typical for good solvents and “dry” entanglement 
for which the polymer segments are in direct contact and no 
longer separated by solvent molecules. Such a situation should 
necessitate two sets of parameters. Analogous observation for 
the system glycerol/polyacrylamide (cf. Figure  7) favors the 
latter interpretation.
3.1.3. Water/Polyacrylamide6
The shear thinning behavior of polyacrylamide (PAAm) in 
water, glycerol, and in formamide has been studied by Klein 
und Kulicke6 for different molar masses; Figures 5 and 6 show 
two examples for water.
The results shown in Figures 5 and 6 document once more 
that the dependencies of lnητrel on τ change their curvature as 
Figure 3. Evaluation of the zero shear viscosities shown in Figure  1 the polymer concentration rises. In both cases the critical com-
according to Equation (3). position lies between 2 and 5 wt%.
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Figure 5. As Figure 1 but for the system water/polyacrylamide M = 5.3 × 106. Figure 7. As Figure 1 but for the system glycerol/polyacrylamide with the 
molar mass 5.3 kDa.
3.1.4. Glycerol/Polyacrylamide6 considerably lower than with all other PAAm solutions. For this 
reason it is not surprising that Equation (1) suffices for the mod-
Like decalin for polystyrene, glycerol and formamide should eling in all cases. The fact that the extrapolated zero shear viscosi-
be unfavorable solvents for polyacrylamide, as compared with ties for glycol and for formamide are much lower than for water 
water, according to the zero shear viscosities of Figure 7. The documents once more the inferior solvent quality of these liquids.
fact that Equation (2) is required for the modeling of the results 
for 5.3  wt% might be taken as a further indication that the 
reason for this behavior lies in the solvent quality. However, it 3.1.6. Solutions of Polydimethylsiloxane (PDMS) in Its Pentamer7
remains open why this is not the case for 1.7 wt%. One might 
argue that the “wet” entanglements are already opened at low The shear thinning behavior of PDMS solutions in its homolo-
τ values and that the higher values required for the opening of gous pentamer (DMS 5) has been studied by Ito and Shishido7 
“dry” entanglements are not reached. covering uncommonly large ranges of shear stresses for dif-
ferent concentrations. Only the results for low τ values are eval-
uated here. The complete dependence of ln η rel on shear stress 
3.1.5. Comparison of Solvents for Polyacrylamide6 exhibits a sigmoidal form. It is presently unclear whether this 
phenomenon is a particularity of PDMS or typical for many sys-
Figure 8 compares the shear thinning of 1  wt% solutions of tems. In either case it demonstrates that the present modeling 
PAAm in different solvents. The τ region under investigation is refers to the initial behavior only.
Figure 8. As Figure 1 but for 1 wt% solutions of polyacrylamide with the 
Figure 6. As Figure 1 but for the system water/polyacrylamide M = 3.9 × 106. molar mass of 1.7 kDa in different solvents.
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Figure 9. As Figure 1 but for solutions of polydimethylsiloxane (1.7 kDa) Figure 10. Viscosities of polyethylene melts as a function of shear stress 
in its pentamer. τ at the indicated temperatures.
The behavior shown in Figure 9 proofs once more that the the poor characterization of the sample; the author only states 
curvature changes with increasing polymer concentration. In that the sample is branched, but does not give a molar mass. 
the present case the critical composition lies between 5 and This course of action appears justified because it is the sole pur-
10 wt%, i.e., in the same range as with the other systems. pose of the present evaluation to check, whether Equation (1) can 
also model flow curves of polymer melts.
According to the data presented in Figure 12, the shear rate 
3.2. Polymer Melts dependence of the viscosities for the PE melts looks very sim-
ilar to those observed for the polymer solutions, however, with 
Unlike the situation with polymer solutions the measuring tem- the fundamental difference that the τ values required to yield 
peratures vary widely. Information on the molar masses of the a comparable reduction in viscosity are by approximately three 
samples, where available, and the parameters of Equations (1) orders of magnitude larger. This behavior has to be expected 
and (2) obtained from the reported data is collected in Table 2. due to the much higher viscosities of polymer melts as com-
pared with that of polymer solutions.
How the characteristic τ *1  values obtained from the evalua-
3.2.1. Polyethylene (PE)8 tion of the data shown in Figure 10 depend on temperature is 
illustrated in Figure 11. Owing to the fact that τ *1  represents a 
Meissner8 has studied the flow behavior of the PE melts over derived quantity, the scattering of the data is considerable; the 
wide ranges of temperatures and shear stresses. The reported observed diminution with rising T reflects the reduction in the 
data were evaluated according to the Equations (1) and (2) despite viscosity of the melt.
Zero shear viscosities and (hypothetical) limiting viscosi-
ties for infinitely high shear stress obtained from the above 
Table 2. Collection of polymer melts for which flow curves were mod-
eled according to Equation (1) and parameters obtained.
Polymera) 10−6 m T [K] lnη 0 A1rel τ *1 [Pa]
PE Not specified 388.00 12.90 7.55 107.50
403.00 12.20 7.36 93.00
423.00 11.25 7.14 99.00
443.00 10.50 7.39 97.40
463.00 9.80 6.43 81.50
513.00 8.20 6.15 78.00
PDMS T [°C]
LG2 0.80 23.00 12.90 4.38 22.30
BG 0.45 23.00 10.70 6.63 49.80
a)PE: polyethylene, PDSM: polydimethylsiloxane; LG2 and BG are the abbreviations Figure 11. Temperature dependence of the characteristic shear stress for 
used by the authors for the PDMS samples. polyethylene.
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shear viscosities from measurements outside the Newtonian 
flow regime. This statement applies to polymer solutions as 
well as to polymer melts.
Moreover the approach yields two helpful system specific 
parameters: A characteristic shear stress τ *1, which quanti-
fies the sensitivity of the viscosity of polymer containing liq-
uids toward shear stress, and the dimensionless parameter 
A1, which measures the magnitude of the effect. The latter 
information must, however, be treated with caution because 
it normally characterizes the behavior only in a limited range 
of shear stress and—in the case of solutions—up to moderate 
polymer concentrations only.
The present method of data evaluation offers additional 
insight in two ways:
Figure 12. Arrhenius plots for zero shear viscosities and for lnη∞  (hypo- i. It reveals the existence of a critical polymer concentration for rel
thetical values for τ → ∞). solutions. Below this value the disentanglement processes 
slow down with rising shear stress τ; above it they accelerate 
with rising τ. It is presently unclear how this critical value 
evaluation are in Figure 12 evaluated in terms of Arrhenius 
≠ correlates with independently determined coil-overlap and plots. The activation energy E  for the fully entangled melt crossover concentrations. Directed experiments are required 
results expectedly much larger than for the disentangled case. to clarify this item.
ii. The current results yield indications for the existence of two 
9 separate disentanglement processes in thermodynamically 3.2.2. Polydimethylsiloxane unfavorable solvents. This observation is tentatively attrib-
9 uted to two dissimilar types of entanglements: Such that are El Kissi  and co-workers have studied a series of nine silicone surrounded by a high number of solvent molecules (good 
fluids (PDMS). Their results—as far as relevant for the present solvents) and others that are surrounded by polymer clusters 
considerations—are evaluated in Figure 13 using the nomen- (marginal solvents). This hypothesis requires directed experi-
clature of the authors. The non-Newtonian behavior of PDMS ments again.
resembles that of PE.
4. Conclusions Conflict of Interest
The author declares no conflict of interest.
The most important outcome of the present study lies in the 
aptness of the communicated approach to acquire reliable zero 
Keywords
modeling shear thinning, polymer melts, polymer solutions, shear 
thinning, zero shear viscosity
Received: April 14, 2020
Revised: June 5, 2020
Published online: June 30, 2020
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[2] B. A. Wolf, Ind. Eng. Chem. Res. 2015, 54, 4672.
[3] W. M. Kulicke, R. Kniewske, Rheol. Acta 1984, 23, 75.
[4] J. W. Breitenbach, H. Gabler, O. F. Olaj, Makromol. Chem. 1964, 81, 
32.
[5] W. M. Kulicke, R. S. Porter, J. Polym. Sci., Part B: Polym. Phys. 1981, 
19, 1173.
[6] J. Klein, W. M. Kulicke, Rheol. Acta 1976, 15, 558.
[7] Y. Ito, S. Shishido, J. Polym. Sci., Part B: Polym. Phys. 1974, 12, 617.
[8] J. Meissner, Kunststoffe 1971, 61, 576.
[9] N.  El Kissi, J. M.  Piau, P.  Attane, G.  Turrel, Rheol. Acta 1993, 32, 
Figure 13. Like Figure 10 but for polydimethylsiloxane. 293.
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