From finite-size thermodynamics to grand canonical molecular dynamics
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Abstract
Molecular Dynamics (MD) simulations play an increasing role in the studies of soft matter and statistical mechanics. The limitations of MD are unavoidable when comparing results to both,
experiments and theory as it is practically impossible to reach, computationally, large time and size scales. Nevertheless, the effects of these limitations can be understood and described in a
statistical mechanics compatible manner, therefore corrected. These effects, known as finite-size effects, come due to finite integration domains, differences in the ensembles and the contributions
attributed to periodic boundary conditions (PBCs) used to mimic an infinite system. Overcoming these limitations require either larger simulation boxes, implying the need of more resources, or
new techniques and methods to extract information and to extrapolate it to the thermodynamic limit (TL). In this thesis we focused in the latter. We divided this work into different branches
keeping as a ground basis, the connection between thermodynamics and computer simulations. First, we took the case of the Kirkwood-Buff integrals (KBI), as they provide a clean link
between thermodynamic properties and microscopic quantities, where the last ones are accessible directly by MD simulations. We identify and corrected a finite-size version of the KBI, bridging the
density fluctuations with integrals of the radial distribution functions, identifying and correcting independently all sources of finite-size effects during the process. Along these lines, we extended
the analysis to the computation of the finite-size two-body excess-entropy s_2 , where despite the fact of having no ensemble dependent finite-size effect it does scale with the inverse of the simulation
box, and more importantly, we found that the empirical Dzugutov scaling relation also exhibits a finite-size dependence, where our values in the thermodynamic limit, agree with those of the
literature.
Second, going to applications to the computation of thermodynamics, we moved to the understanding of structural properties of liquids by using KBIs, structure factors and chemical potentials.
On the one hand, we studied the effect of small differences in the tail of the radial distribution functions (RDFs) on the structure of prototypical supercooled liquids, Kob-Andersen mixtures
with and without attractive interactions. Our results suggest that the nucleation of long-range domains is induced by the attractive potentials when decreasing temperature. This was verified
using the limit k → 0 of the Bathia-Thornton structure factors and the bulk isothermal compressibility. Moreover, after performing similar studies were performed at higher density we found the
two systems indistinguishable regarding structure. Whereas on the other hand, we found similar behaviour of the creation of domains, for propan-1-ol simulations for a range of temperature
200 < T < 300 K. Where around ∼ 220 K a change in the isothermal compressibility points out to a transition from a normal to an anomalous liquid. The mechanism for the creation of domains
in this case is rather different, as we found them to nucleate from the creation of well defined networks of molecules connected via Hydrogen bonds (H-bonds). This results were verified by the
computation of the average number of H-bonds inside the simulation box, the size of the larges cluster, and more importantly NMR experiments.
Lastly, the development of a Density-functional-theory (DFT) approach to the Hamiltonian Adaptive Resolution Simulation (H-AdResS) allowed us to show that the external potential needed
to induce a constant density profile coincides with the system’s excess chemical potential. Hence, given the one-to-one correspondence between equilibrium densities and external potentials in DFT,
it is possible to obtain the excess of chemical potential by imposing a constant density profile. Which brings us to the most important point of this work, and that is to the study of the statisticsof the atomistic region in the (H-AdResS) method. We considered a slab geometry where the atomistic region was taken to be the system and the hybrid + Ideal gas the thermal bath. This
implies that the contact system-bath is done purely in one coordination whereas the other two keep their PBCs. The consequences of these considerations, is that it is necessary to have a
description of a grand canonical simulation where size effects are still present, more specifically, PBCs contributions. This description was proposed and validated as part of this thesis, were we
identified and corrected all sources of size effects in the Kirkwood-Buff integrals and therefore, in the density fluctuations. We computed the local density fluctuations inside the atomistic region,
and compare those with the results coming from our method to compute the finite-size Kirkwood-Buff integrals finding that in order to ensure the correct stable grand canonical statistics having
a H-AdResS set up is not enough. Nevertheless, a correct description of the grand canonical ensemble in the thermal baths, meaning the ideal gas, actually leads to an excellent agreement
between the H-AdResS and the finite-size integral equation. The way to improve the description of the bath, is by introducing a particle insertion/deletion algorithm with the correct statistics
which is possible due to the fact that the insertion probabilities for an ideal gas in the grand canonical ensemble can be computed analytically. Summarizing, the Adaptive Resolution with
Particle insertion/deletion Steps (AdResS+ PI) can be used as a grand canonical simulation method.