Finite-size integral equations in the theory of liquids and the thermodynamic limit in computer simulations

Chilling alcohol on the computer: isothermal compressibility and the formation of hydrogen-bond clusters in liquid propan-1-ol

Molecular dynamics simulations have been performed to compute the isothermal compressibility [Formula: see text] of liquid propan-1-ol in the temperature range [Formula: see text] K. A change in behaviour, from normal (high T) to anomalous (low T), has been identified for [Formula: see text] at [Formula: see text] K. The average number of hydrogen bonds (H-bond) per molecule turns to saturation in the same temperature interval, suggesting the formation of a relatively rigid network. Indeed, simulation results show a strong tendency to form H-bond clusters with distinct boundaries, with the average largest size and width of the size distribution growing upon decreasing temperature, in agreement with previous theoretical and experimental studies. These results also emphasise a connection between the behaviour of [Formula: see text] and the formation of nanometric structures.

Kirkwood-Buff integrals: From fluctuations in finite volumes to the thermodynamic limit

The Kirkwood-Buff theory is a cornerstone of the statistical mechanics of liquids and solutions. It relates volume integrals over the radial distribution function, so-called Kirkwood-Buff integrals (KBIs), to particle number fluctuations and thereby to various macroscopic thermodynamic quantities such as the isothermal compressibility and partial molar volumes. Recently, the field has seen a strong revival with breakthroughs in the numerical computation of KBIs and applications to complex systems such as bio-molecules. One of the main emergent results is the possibility to use the finite volume KBIs as a tool to access finite volume thermodynamic quantities. The purpose of this Perspective is to shed new light on the latest developments and discuss future avenues.

Effects of structural inhomogeneity on equilibration processes in Langevin dynamics

In recent decades, computer experiments have led to an accurate and fundamental understanding of atomic and molecular mechanisms in fluids, such as different kinds of relaxation processes toward steady physical states. In this paper, we investigate how exactly the configuration of initial states in a molecular-dynamics simulation can affect the rates of decay toward equilibrium for the widely known Langevin canonical ensemble. For this purpose, we derive an original expression relating the system relaxation time τ_{sys} and the radial distribution function g(r) in the near-zero and high-density limit. We found that, for an initial state which is slightly marginally inhomogeneous in the number density of atoms, the system relaxation time τ_{sys} is much longer than that for the homogeneous case and an increasing function of the Langevin coupling constant, γ. We also found, during structural equilibration, g(r) at large distances approaches 1 from above for the inhomogeneous case and from below for the macroscopically homogeneous one.

Solvation shell thermodynamics of extended hydrophobic solutes in mixed solvents

The ability of various cosolutes and cosolvents to enhance or quench solvent density fluctuations at solute–water interfaces has crucial implications on the conformational equilibrium of macromolecules such as polymers and proteins. Herein, we use an extended hydrophobic solute as a model system to study the effect of urea and methanol on the density fluctuations in the solute’s solvation shell and the resulting thermodynamics. On strengthening the solute–water/cosolute repulsive interaction, we observe distinct trends in the mutual affinities between various species in, and the thermodynamic properties of, the solvation shell. These trends strongly follow the respective trends in the preferential adsorption of urea and methanol: solute–water/cosolute repulsion strengthens, urea accumulation decreases, and methanol accumulation increases. Preferential accumulation of urea is found to quench the density fluctuations around the extended solute, leading to a decrease in the compressibility of the solvation shell. In contrast, methanol accumulation enhances the density fluctuations, leading to an increase in the compressibility. The mode of action of urea and methanol seems to be strongly coupled to their hydration behavior. The observations from this simple model is discussed in relation to urea driven swelling and methanol induced collapse of some well-known thermo-responsive polymers.

Finite-size scaling and thermodynamics of model supercooled liquids: long-range concentration fluctuations and the role of attractive interactions

We compute partial structure factors, Kirkwood-Buff integrals (KBIs) and chemical potentials of model supercooled liquids with and without attractive interactions. We aim at investigating whether relatively small differences in the tail of the radial distribution functions result in contrasting thermodynamic properties. Our results suggest that the attractive potential favours the nucleation of long-range structures. Indeed, upon decreasing temperature, Bathia-Thornton structure factors display anomalous behaviour in the k→0 limit. KBIs extrapolated to the thermodynamic limit confirm this picture, and excess coordination numbers identify the anomaly with long-range concentration fluctuations. By contrast, the purely repulsive system remains perfectly miscible for the same temperature interval and only reveals qualitatively similar concentration fluctuations in the crystalline state. Furthermore, differences in both isothermal compressibilities and chemical potentials show that thermodynamics is not entirely governed by the short-range repulsive part of the interaction potential, emphasising the nonperturbative role of attractive interactions. Finally, at higher density, where both systems display nearly identical dynamical properties and repulsive interactions become dominant, the anomaly disappears, and both systems also exhibit similar thermodynamic properties.

Small size effects in open and closed systems: What can we learn from ideal gases about systems with interacting particles?

Small systems have higher surface area-to-volume ratios than macroscopic systems. The thermodynamics of small systems therefore deviates from the description of classical thermodynamics. One consequence of this is that properties of small systems can be dependent on the system's ensemble. By comparing the properties in grand canonical (open) and canonical (closed) systems, we investigate how a small number of particles can induce an ensemble dependence. Emphasis is placed on the insight that can be gained by investigating ideal gases. The ensemble equivalence of small ideal gas systems is investigated by deriving the properties analytically, while the ensemble equivalence of small systems with particles interacting via the Lennard-Jones or the Weeks-Chandler-Andersen potential is investigated through Monte Carlo simulations. For all the investigated small systems, we find clear differences between the properties in open and closed systems. For systems with interacting particles, the difference between the pressure contribution to the internal energy, and the difference between the chemical potential contribution to the internal energy, are both increasing with the number density. The difference in chemical potential is, with the exception of the density dependence, qualitatively described by the analytic formula derived for an ideal gas system. The difference in pressure, however, is not captured by the ideal gas model. For the difference between the properties in the open and closed systems, the response of increasing the particles' excluded volume is similar to the response of increasing the repulsive forces on the system walls. This indicates that the magnitude of the difference between the properties in open and closed systems is related to the restricted movement of the particles in the system. The work presented in this paper gives insight into the mechanisms behind ensemble in-equivalence in small systems, and illustrates how a simple statistical mechanical model, such as the ideal gas, can be a useful tool in these investigations.

Conservative Potentials for a Lattice-Mapped Coarse-Grained Scheme

The conservative potential, arising from a coarse-grain (CG) mapping scheme for nonbonded atomistic particles, is studied. This is a bottom-up approach from first-principles that maps atomistic particles to fluid element-like subcells whose centers lie on a regular, cubic lattice. Unlike standard CG mapping schemes, the current one uses dynamic labeling which on-the-fly changes the CG labels of the particles. The subcells can also be different sizes and shapes, in principle. Equilibrium atomistic molecular dynamics trajectories for different Lennard-Jones fluids are calculated and converted to CG ones, from which CG probability distribution functions are calculated. Correlation studies show position and mass CG variables are uncoupled in a given subcell, as are different vector components of position. Furthermore, the strongest coupling occurs with neighboring cells in specific directions, and the resulting distribution is well described by a multivariate Gaussian. This implies the CG potential has a generalized quadratic form, whose derivative can be determined analytically. A microscopic rationalization is provided for the signs and relative magnitudes of different correlation coefficients, and in some cases, a connection is made with bulk properties of the fluid. We argue the generalized quadratic form should be robust to changes in the particulars of the CG scheme, as well as the nature of the atomistic intermolecular potential. Only a few potential parameters need to be calculated from the underlying atomistic system. This is significant because it indicates the transferability of this form to other, more complex systems. This transferability will be tested in future work, where mapping schemes with fuzzy boundaries will be considered.

Density-functional-theory approach to the Hamiltonian adaptive resolution simulation method

In the Hamiltonian adaptive resolution simulation method (H-AdResS) it is possible to simulate coexisting atomistic (AT) and ideal gas representations of a physical system that belong to different subdomains within the simulation box. The Hamiltonian includes a field that bridges both models by smoothly switching on (off) the intermolecular potential as particles enter (leave) the AT region. In practice, external one-body forces are calculated and applied to enforce a reference density throughout the simulation box, and the resulting external potential adds up to the Hamiltonian. This procedure suggests an apparent dependence of the final Hamiltonian on the system's thermodynamic state that challenges the method's statistical mechanics consistency. In this paper, we explicitly include an external potential that depends on the switching function. Hence, we build a grand canonical potential for this inhomogeneous system to find the equivalence between H-AdResS and density functional theory (DFT). We thus verify that the external potential inducing a constant density profile is equal to the system's excess chemical potential. Given DFT's one-to-one correspondence between external potential and equilibrium density, we find that a Hamiltonian description of the system is compatible with the numerical implementation based on enforcing the reference density across the simulation box. In the second part of the manuscript, we focus on assessing our approach's convergence and computing efficiency concerning various model parameters, including sample size and solute concentrations. To this aim, we compute the excess chemical potential of water, aqueous urea solutions and Lennard-Jones (LJ) mixtures. The results' convergence and accuracy are convincing in all cases, thus emphasising the method's robustness and capabilities.

Chemical Potential Differences in the Macroscopic Limit from Fluctuations in Small Systems

We present a new method for computing chemical potential differences of macroscopic systems by sampling fluctuations in small systems. The small system method, presented by Schnell et al. [Schnell et al., J. Phys. Chem. B, 2011, 115, 10911], is used to create small embedded systems from molecular dynamics simulations, in which fluctuations of the number of particles are sampled. The sampled fluctuations represent the Boltzmann distributed probability of the number of particles. The overlapping region of two such distributions, sampled from two different systems, is used to compute their chemical potential difference. Since the thermodynamics of small systems is known to deviate from the classical thermodynamic description, the particle distributions will deviate from the macroscopic behavior as well. We show how this can be utilized to calculate the size dependence of chemical potential differences and eventually extract the chemical potential difference in the thermodynamic limit. The macroscopic chemical potential difference is determined with a relative error of 3% in systems containing particles that interact through the truncated and shifted Lennard-Jones potential. In addition to computing chemical potential differences in the macroscopic limit directly from molecular dynamics simulation, the new method provides insights into the size dependency that is introduced to intensive properties in small systems.

SJ, van der Vegt NFA. Characterizing Polymer Hydration Shell Compressibilities with the Small-System Method

The small-system method (SSM) exploits the unique feature of finite-sized open systems, whose thermodynamic quantities scale with the inverse system size. This scaling enables the calculation of properties in the thermodynamic limit of macroscopic systems based on computer simulations of finite-sized systems. We herein extend the SSM to characterize the hydration shell compressibility of a generic hydrophobic polymer in water. By systematically increasing the strength of polymer-water repulsion, we find that the excess inverse thermodynamic correction factor (Δ1/Γs∞) and compressibility (Δχs) of the first hydration shell change sign from negative to positive. This occurs with a concurrent decrease in water hydrogen bonding and local tetrahedral order of the hydration shell water. The crossover lengthscale corresponds to an effective polymer bead diameter of 0.7 nm and is consistent with previous works on hydration of small and large hydrophobic solutes. The crossover lengthscale in polymer hydration shell compressibility, herein identified with the SSM approach, relates to hydrophobic interactions and macromolecular conformational equilibria in aqueous solution. The SSM approach may further be applied to study thermodynamic properties of polymer solvation shells in mixed solvents.

Thermodynamic Analysis of n-Hexane-Ethanol Binary Mixtures Using the Kirkwood-Buff Theory

A complete thermodynamic analysis of mixtures consisting of molecules with complex chemical constitution can be rather demanding. The Kirkwood-Buff theory of solutions allows the estimation of thermodynamic properties, which cannot be directly extracted from atomistic simulations, such as the Gibbs energy of mixing (Δ_mix G). In this work, we perform molecular dynamics simulations of n-hexane-ethanol binary mixtures in the liquid state under two temperature-pressure conditions and at various mole fractions. On the basis of the recently published methodology of Galata [ Fluid Phase Equilib. 2018 , 470 , 25 - 37 ] , we first calculate the Kirkwood-Buff integrals in the isothermal-isobaric ( NpT) ensemble, identifying how system size affects their estimation. We then extract the activity coefficients, excess Gibbs energy, excess enthalpy, and excess entropy for the n-hexane-ethanol binary mixtures we simulate. We employ two approaches for quantifying composition fluctuations: one based on counting molecular centers of mass and a second one based on counting molecular segments. Results from the two approaches are practically indistinguishable. We compare our results against predictions of vapor-liquid equilibria obtained in a previous simulation work using the same force field, as well as with experimental data, and find very good agreement. In addition, we develop a simple methodology to identify the hydrogen bonds between ethanol molecules and analyze their effects on mixing properties.