Communication: Kirkwood-Buff integrals in the thermodynamic limit from small-sized molecular dynamics simulations

Concentration Fluctuation/Microheterogeneity Duality Illustrated with Aqueous 1,4-Dioxane Mixtures

The structural properties of aqueous 1-4 dioxane mixtures are studied by computer simulations of different water and dioxane force field models, from the perspective of illustrating the link between structural properties at the molecular level and measurable properties such as radiation scattering intensities and Kirkwood-Buff integrals (KBIs). A strategy to consistently correct the KBI obtained from simulations is proposed, which allows us to obtain the genuine KBI corresponding to a given pair of molecular species, in the entire concentration range, and without necessitating excessively large system sizes. The application of this method to the aqueous dioxane mixtures, with an all-atom CHARMM dioxane model and 2 water models, namely, SPC/E and TIP3P, allows one to understand the differences in the structure of the corresponding mixtures at the molecular level, particularly concerning the role of the water aggregates and its model dependence. This study allows us to characterize the dual role played by the concentration fluctuations and the domain segregation, particularly in what concerns the calculated X-ray spectra.

Breakdown of Langmuir Adsorption Isotherm in Small Closed Systems

For more than a century, monolayer adsorptions in which adsorbate molecules and adsorbing sites behave ideally have been successfully described by Langmuir's adsorption isotherm. For example, the amount of adsorbed material, as a function of concentration of the material which is not adsorbed, obeys Langmuir's equation. In this paper, we argue that this relation is valid only for macroscopic systems. However, when particle numbers of adsorbate molecules and/or adsorbing sites are small, Langmuir's model fails to describe the chemical equilibrium of the system. This is because the kinetics of forming, or the probability of observing, occupied sites arises from two-body interactions, and as such, ought to include cross-correlations between particle numbers of the adsorbate and adsorbing sites. The effect of these correlations, as reflected by deviations in predicting composition when correlations are ignored, increases with decreasing particle numbers and becomes substantial when only few adsorbate molecules, or adsorbing sites, are present in the system. In addition, any change that augments the fraction of occupied sites at equilibrium (e.g., smaller volume, lower temperature, or stronger adsorption energy) further increases the discrepancy between observed properties of small systems and those predicted by Langmuir's theory. In contrast, for large systems, these cross-correlations become negligible, and therefore when expressing properties involving two-body processes, it is possible to consider independently the concentration of each component. By applying statistical mechanics concepts, we derive a general expression of the equilibrium constant for adsorption. It is also demonstrated that in ensembles in which total numbers of particles are fixed, the magnitudes of fluctuations in particle numbers alone can predict the average chemical composition of the system. Moreover, an alternative adsorption equation, predicting the average fraction of occupied sites from the value of the equilibrium constant, is proposed. All derived relations were tested against results obtained by Monte Carlo simulations.

A streamlined molecular-dynamics workflow for computing solubilities of molecular and ionic crystals

Computing the solubility of crystals in a solvent using atomistic simulations is notoriously challenging due to the complexities and convergence issues associated with free-energy methods, as well as the slow equilibration in direct-coexistence simulations. This paper introduces a molecular-dynamics workflow that simplifies and robustly computes the solubility of molecular or ionic crystals. This method is considerably more straightforward than the state-of-the-art, as we have streamlined and optimised each step of the process. Specifically, we calculate the chemical potential of the crystal using the gas-phase molecule as a reference state, and employ the S0 method to determine the concentration dependence of the chemical potential of the solute. We use this workflow to predict the solubilities of sodium chloride in water, urea polymorphs in water, and paracetamol polymorphs in both water and ethanol. Our findings indicate that the predicted solubility is sensitive to the chosen potential energy surface. Furthermore, we note that the harmonic approximation often fails for both molecular crystals and gas molecules at or above room temperature, and that the assumption of an ideal solution becomes less valid for highly soluble substances.

Eliminating finite-size effects on the calculation of x-ray scattering from molecular dynamics simulations

Structural studies using x-ray scattering methods for investigating molecules in solution are shifting focus toward describing the role and effects of the surrounding solvent. However, forward models based on molecular dynamics (MD) simulations to simulate structure factors and x-ray scattering from interatomic distributions such as radial distribution functions (RDFs) face limitations imposed by simulations, particularly at low values of the scattering vector q. In this work, we show how the value of the structure factor at q = 0 calculated from RDFs sampled from finite MD simulations is effectively dependent on the size of the simulation cell. To eliminate this error, we derive a new scheme to renormalize the sampled RDFs based on a model of the excluded volume of the particle-pairs they were sampled from, to emulate sampling from an infinite system. We compare this new correction method to two previous RDF-correction methods, developed for Kirkwood-Buff theory applications. We present a quantitative test to assess the reliability of the simulated low-q scattering signal and show that our RDF-correction successfully recovers the correct q = 0 limit for neat water. We investigate the effect of MD-sampling time on the RDF-corrections, before advancing to a molecular example system, comprised of a transition metal complex solvated in a series of water cells with varying densities. We show that our correction recovers the correct q = 0 behavior for all densities. Furthermore, we employ a simple continuum scattering model to dissect the total scattering signal from the solvent-solvent structural correlations in a solute-solvent model system to find two distinct contributions: a non-local density-contribution from the finite, fixed cell size in NVT simulations, and a local contribution from the solvent shell. We show how the second contribution can be approximated without also including the finite-size contribution. Finally, we provide a "best-practices"-checklist for experimentalists planning to incorporate explicit solvation MD simulations in future work, offering guidance for improving the accuracy and reliability of structural studies using x-ray scattering methods in solution.

Smart Polymers for Soft Materials: From Solution Processing to Organic Solids

Polymeric materials are ubiquitous in our everyday life, where they find a broad range of uses-spanning across common household items to advanced materials for modern technologies. In the context of the latter, so called "smart polymers" have received a lot of attention. These systems are soluble in water below their lower critical solution temperature Tℓ and often exhibit counterintuitive solvation behavior in mixed solvents. A polymer is known as smart-responsive when a slight change in external stimuli can significantly change its structure, functionm and stability. The interplay of different interactions, especially hydrogen bonds, can also be used for the design of lightweight high-performance organic solids with tunable properties. Here, a general scheme for establishing a structure-property relationship is a challenge using the conventional simulation techniques and also in standard experiments. From the theoretical side, a broad range of all-atom, multiscale, generic, and analytical techniques have been developed linking monomer level interaction details with macroscopic material properties. In this review, we briefly summarize the recent developments in the field of smart polymers, together with complementary experiments. For this purpose, we will specifically discuss the following: (1) the solution processing of responsive polymers and (2) their use in organic solids, with a goal to provide a microscopic understanding that may be used as a guiding tool for future experiments and/or simulations regarding designing advanced functional materials.

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.

Computing chemical potentials of solutions from structure factors

The chemical potential of a component in a solution is defined as the free energy change as the amount of the component changes. Computing this fundamental thermodynamic property from atomistic simulations is notoriously difficult, because of the convergence issues in free energy methods and finite size effects. This paper presents the so-called S0 method, which can be used to obtain chemical potentials from static structure factors computed from equilibrium molecular dynamics simulations under the isothermal-isobaric ensemble. This new method is demonstrated on the systems of binary Lennard-Jones particles, urea--water mixtures, a NaCl aqueous solution, and a high-pressure carbon-hydrogen mixture.

Binding reactions at finite systems

A perpetual yearn exists among computational scientists to scale down the size of physical systems, a desire shared as well with experimentalists able to track single molecules. A question then arises whether averages observed at small systems are the same as those observed at large or macroscopic systems. Utilizing statistical-mechanics formulations in ensembles in which the total numbers of particles are fixed, we demonstrate that properties of binding reactions are not homogeneous functions. This means that averages of intensive parameters, such as the concentration of the bound-state, at finite systems are different than those at large systems. The discrepancy increases with decreasing temperature, volume, and to some extent, numbers of particles. As perplexing as it may sound, despite variations in average quantities, extracting the equilibrium constant from systems of different sizes does yield the same value. The reason is that correlations in reactants' concentrations ought to be accounted for in the expression of the equilibrium constant, being negligible at large-scale but significant at small-scale. Similar arguments pertain to the calculations of the reaction rate constants, more specifically, the bimolecular rate of the forward reaction is related to the average of the product (and not to the product of the averages) of the reactants' concentrations. Furthermore, we derive relations aiming to predict the composition only from the equilibrium constant and the system's size. All predictions are validated by Monte-Carlo and molecular dynamics simulations. An important consequence of these findings is that the expression of the equilibrium constant at finite systems is not dictated solely by the chemical equation of the reaction but requires knowledge of the elementary processes involved.

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.

Thermoresponsive Ionic Liquid/Water Mixtures: From Nanostructuring to Phase Separation

The thermodynamics, structures, and applications of thermoresponsive systems, consisting primarily of water solutions of organic salts, are reviewed. The focus is on organic salts of low melting temperatures, belonging to the ionic liquid (IL) family. The thermo-responsiveness is represented by a temperature driven transition between a homogeneous liquid state and a biphasic state, comprising an IL-rich phase and a solvent-rich phase, divided by a relatively sharp interface. Demixing occurs either with decreasing temperatures, developing from an upper critical solution temperature (UCST), or, less often, with increasing temperatures, arising from a lower critical solution temperature (LCST). In the former case, the enthalpy and entropy of mixing are both positive, and enthalpy prevails at low T. In the latter case, the enthalpy and entropy of mixing are both negative, and entropy drives the demixing with increasing T. Experiments and computer simulations highlight the contiguity of these phase separations with the nanoscale inhomogeneity (nanostructuring), displayed by several ILs and IL solutions. Current applications in extraction, separation, and catalysis are briefly reviewed. Moreover, future applications in forward osmosis desalination, low-enthalpy thermal storage, and water harvesting from the atmosphere are discussed in more detail.

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.

Solvation energetics of proteins and their aggregates analyzed by all-atom molecular dynamics simulations and the energy-representation theory of solvation

Solvation is a controlling factor for the structure and function of proteins. This article addresses the effects of solvation from an energetic perspective for the fluctuations and cosolvent-induced changes in protein structures and the equilibrium of aggregate formation for a peptide. A theoretical framework to analyze the solvation effects with an explicit solvent is introduced by adopting the energy-representation theory of solvation, and the connection of the solvation free energy to the protein structure and the aggregation tendency is quantitatively described in combination with all-atom molecular dynamics simulations. The interaction components that govern the solvation effects on the structural variations of proteins are further identified through correlation analysis, and a computational scheme to assess the shift of an aggregation equilibrium due to the addition of a cosolvent is provided.

How sensitive are physical properties of choline chloride-urea mixtures to composition changes: Molecular dynamics simulations and Kirkwood-Buff theory

Deep eutectic solvents (DESs) have emerged as a cheaper and greener alternative to conventional organic solvents. Choline chloride (ChCl) mixed with urea at a molar ratio of 1:2 is one of the most common DESs for a wide range of applications such as electrochemistry, material science, and biochemistry. In this study, molecular dynamics simulations are performed to study the effect of urea content on the thermodynamic and transport properties of ChCl and urea mixtures. With increased mole fraction of urea, the number of hydrogen bonds (HBs) between cation-anion and ion-urea decreases, while the number of HBs between urea-urea increases. Radial distribution functions (RDFs) for ChCl-urea and ChCl-ChCl pairs shows a significant decrease as the mole fraction of urea increases. Using the computed RDFs, Kirkwood-Buff Integrals (KBIs) are computed. KBIs show that interactions of urea-urea become stronger, while interactions of urea-ChCl and ChCl-ChCl pairs become slightly weaker with increasing mole fraction of urea. All thermodynamic factors are found larger than one, indicating a non-ideal mixture. Our results also show that self- and collective diffusivities increase, while viscosities decrease with increasing urea content. This is mainly due to the weaker interactions between ions and urea, resulting in enhanced mobilities. Ionic conductivities exhibit a non-monotonic behavior. Up to a mole fraction of 0.5, the ionic conductivities increase with increasing urea content and then reach a plateau.

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.

Kirkwood-Buff-Derived Force Field for Peptides and Proteins: Philosophy and Development of KBFF20

A new classical nonpolarizable force field, KBFF20, for the simulation of peptides and proteins is presented. The force field relies heavily on the use of Kirkwood-Buff theory to provide a comparison of simulated and experimental Kirkwood-Buff integrals for solutes containing the functional groups common in proteins, thus ensuring intermolecular interactions that provide a good balance between the peptide-peptide, peptide-solvent, and solvent-solvent distributions observed in solution mixtures. In this way, it differs significantly from other biomolecular force fields. Further development and testing of the intermolecular potentials are presented here. Subsequently, rotational potentials for the ϕ/ψ and χ dihedral degrees of freedom are obtained by analysis of the Protein Data Bank, followed by small modifications to provide a reasonable balance between simulated and observed α and β percentages for small peptides. This, the first of two articles, describes in detail the philosophy and development behind KBFF20.

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.

Interactions of a DNA G-quadruplex with TMAO and urea: a molecular dynamics study on co-solute compensation mechanisms

We study the individual and combined influence of TMAO and urea on a basket-type DNA G-quadruplex by means of atomistic molecular dynamics (MD) simulations. In combination with the Kirkwood-Buff theory of solutions, we propose a simple mechanism to elucidate the impact of TMAO and urea on the G-quadruplex. Our results reveal the importance of the molecular accumulation around the DNA in terms of stabilizing or destabilizing effects. The results for mixtures show only a weak interaction between both co-solutes, which highlights the additivity of contributions. Despite the fact, that TMAO can to some extent compensate the adverse impact of urea on the G-quadruplex structure, the destabilizing influence is not completely eliminated. This observation opens the door for further research on selective stabilization of DNA G-quadruplexes by modulating the concentrations of TMAO and urea in solution.

The transition from salt-in-water to water-in-salt nanostructures in water solutions of organic ionic liquids relevant for biological applications

Computer simulations show how nano-structural motifs in organic salts/water solutions change with salt content increasing from dilute to highly concentrated.

Fluctuation adsorption theory: quantifying adsorbate-adsorbate interaction and interfacial phase transition from an isotherm

How adsorbate-adsorbate interaction determines the functional shape of an adsorption isotherm is an important and challenging question. Many models for the adsorption isotherm have been proposed to answer this question. However, a successful fitting of an isotherm on its own is insufficient to prove the correctness of the model assumptions. Instead, starting from the principles of statistical thermodynamics, we propose how adsorbate-adsorbate interactions can be quantified from an isotherm. This was made possible by extending the key tools of solution statistical thermodynamics to adsorbates at the interface, namely, the Kirkwood-Buff and McMillan-Mayer theories, as well as their relationship to the thermodynamic phase stability condition. When capillary condensation and interfacial phase transition are absent, adsorbate-adsorbate interactions can be quantified from an isotherm using the Kirkwood-Buff integrals, and virial coefficients can yield multiple-body interaction between adsorbates. Such quantities can be obtained directly from the fitting parameters for the well-known isotherm models (e.g., Langmuir, BET). The size of the adsorbate cluster involved in capillary condensation and interfacial phase transition can also be evaluated from the isotherm, which was demonstrated for the adsorption isotherm of water on activated carbons of varying pore sizes from the literature. Signatures of isotherm classifications by IUPAC have been characterized in terms of multiple-body interactions between adsorbates.

Finite-size corrections for the static structure factor of a liquid slab with open boundaries

The presence of a confining boundary can modify the local structure of a liquid markedly. In addition, small samples of finite size are known to exhibit systematic deviations of thermodynamic quantities relative to their bulk values. Here, we consider the static structure factor of a liquid sample in slab geometry with open boundaries at the surfaces, which can be thought of as virtually cutting out the sample from a macroscopically large, homogeneous fluid. This situation is a relevant limit for the interpretation of grazing-incidence diffraction experiments at liquid interfaces and films. We derive an exact, closed expression for the slab structure factor, with the bulk structure factor as the only input. This shows that such free boundary conditions cause significant differences between the two structure factors, in particular, at small wavenumbers. An asymptotic analysis of this result yields the scaling exponent and an accurate, useful approximation of these finite-size corrections. Furthermore, the open boundaries permit the interpretation of the slab as an open system, supporting particle exchange with a reservoir. We relate the slab structure factor to the particle number fluctuations and discuss conditions under which the subvolume of the slab represents a grand canonical ensemble with chemical potential μ and temperature T. Thus, the open slab serves as a test-bed for the small-system thermodynamics in a μT reservoir. We provide a microscopically justified and exact result for the size dependence of the isothermal compressibility. Our findings are corroborated by simulation data for Lennard-Jones liquids at two representative temperatures.

Cavity Particle in Aqueous Solution with a Hydrophobic Solute: Structure, Energetics, and Functionals

Endpoints density functional theory (DFT) provides a framework for calculating the excess chemical potential of a solute in solution using solvent distribution functions obtained from both physical endpoints of a hypothetical charging process which transforms the solvent density from that of the pure liquid to the solution state. In this work, the endpoints DFT equations are formulated in terms of the indirect (solvent-mediated) contribution ω(x) to the solute-solvent potential of mean force, and their connections are established with the conventional DFT expressions which are based on the use of direct correlation functions. ω actually corresponds to the free-energy cost to move a cavity particle (a tagged solvent molecule which interacts with the other solvent molecules but not the solute) from the bulk to the configuration x of a solvent molecule relative to the solute and is a suitable variable to describe the solvent effects on the solute-solvent interactions. HNC and PY type approximations are then used to integrate the DFT charging integral involved in the exact expression for the excess chemical potential. With these approximations, molecular simulations are to be performed at the two endpoints of solute insertion: pure solvent without the solute and the solution system with the fully coupled solute-solvent interaction. An endpoints method thus utilizes the ensembles of intermolecular configurations of physical interest, which are often readily accessible with MD simulations given the present computational power. To illustrate properties of the formulation, we perform simulations of model systems consisting of a cavity particle in an aqueous solution containing a spherical hydrophobic solute of three different sizes from which ω(x) and the solute chemical potential can be calculated using endpoints DFT expressions. These are compared with corresponding results obtained using the approximations needed in order to evaluate the endpoints DFT charging integral when cavity particle simulation data is not available. We analyze a new approximation (two-points quadratic HNC) to the DFT charging integral which captures the correct behavior of the cavity distributions at both endpoints of the solute insertion. The behavior of the cavity particle in simple and complex liquids plays an important role in various theoretical treatments of the solute chemical potential. For pure Lennard-Jones fluids, the free energy to bring a cavity particle from the bulk to the center of a fluid particle is negative. However, for solutes of varying size, this is not generally true for Lennard-Jones fluids or the systems studied in this work. We carry out energetic and structural analyses of the cavity particle in aqueous solution with hydrophobic solutes of varying size and discuss the results in the context of the hydrophobic effect.

Kirkwood-Buff Integrals Using Molecular Simulation: Estimation of Surface Effects

Kirkwood-Buff (KB) integrals provide a connection between microscopic properties and thermodynamic properties of multicomponent fluids. The estimation of KB integrals using molecular simulations of finite systems requires accounting for finite size effects. In the small system method, properties of finite subvolumes with different sizes embedded in a larger volume can be used to extrapolate to macroscopic thermodynamic properties. KB integrals computed from small subvolumes scale with the inverse size of the system. This scaling was used to find KB integrals in the thermodynamic limit. To reduce numerical inaccuracies that arise from this extrapolation, alternative approaches were considered in this work. Three methods for computing KB integrals in the thermodynamic limit from information of radial distribution functions (RDFs) of finite systems were compared. These methods allowed for the computation of surface effects. KB integrals and surface terms in the thermodynamic limit were computed for Lennard–Jones (LJ) and Weeks–Chandler–Andersen (WCA) fluids. It was found that all three methods converge to the same value. The main differentiating factor was the speed of convergence with system size L. The method that required the smallest size was the one which exploited the scaling of the finite volume KB integral multiplied by L. The relationship between KB integrals and surface effects was studied for a range of densities.

Why Do Elastin-Like Polypeptides Possibly Have Different Solvation Behaviors in Water-Ethanol and Water-Urea Mixtures?

The solvent quality determines the collapsed or the expanded state of a polymer. For example, a polymer dissolved in a poor solvent collapses, whereas in a good solvent it opens up. While this standard understanding is generally valid, there are examples when a polymer collapses even in a mixture of two good solvents. This phenomenon, commonly known as co-non-solvency, is usually associated with a wide range of synthetic (smart) polymers. Moreover, recent experiments have shown that some biopolymers, such as elastin-like polypeptides (ELPs) that exhibit lower critical solution behavior T l in pure water, show co-non-solvency behavior in aqueous ethanol mixtures. In this study, we investigate the phase behavior of elastin-like polypeptides (ELPs) in aqueous binary mixtures using molecular dynamics simulations of all-atom and complementary explicit solvent generic models. The model is parameterized by mapping the solvation free energy obtained from the all-atom simulations onto the generic interaction parameters. For this purpose, we derive segment-based (monomer level) generic parameters for four different peptides, namely proline (P), valine (V), glycine (G), and alanine (A), where the first three constitute the basic building blocks of ELPs. Here, we compare the conformational behavior of two ELP sequences, namely -(VPGGG)- and -(VPGVG)-, in aqueous ethanol and -urea mixtures. Consistent with recent experiments, we find that ELPs show co-non-solvency in aqueous ethanol mixtures. Ethanol molecules have preferential binding with all ELP residues, with an interaction contrast of 6-8 k B T, and thus driving the coil-to-globule transition. On the contrary, ELP conformations show a weak variation in aqueous urea mixtures. Our simulations suggest that the glycine residues dictate the overall behavior of ELPs in aqueous urea, where urea molecules have a rather weak preferential binding with glycine as observed from the all atom simulations, i.e., less than k B T. This weak interaction dilutes the overall effect of other neighboring residues and thus ELPs exhibit a different conformational behavior in aqueous urea in comparison to aqueous ethanol mixtures. While the validation of the latter findings will require a more detailed experimental investigation, the results presented here may provide a new twist to the present understanding of cosolvent interactions with peptides and proteins.

Structure Factor of EuCl3 Aqueous Solutions via Coupled Molecular Dynamics Simulations and Integral Equations

Identifying the structure of an aqueous solution is essential to rationalize various phenomena such as crystallization in solution, chemical reactivity, extraction of rare earth elements, and so forth. Despite this, the efforts to describe the structure of an aqueous solution have been hindered by the difficulty to retrieve structural data both from experiments and simulations. To overcome this, first, undersaturated EuCl₃ aqueous solutions of concentrations varying from 0.15 to 1.8 mol/kg were studied using X-ray scattering. Second, for the first time, the theoretical X-ray signal of 1.8 mol/kg EuCl₃ aqueous solution was simulated, with precise details for the complete range of scattering vectors using coupled molecular dynamics and hypernetted chain integral equations, and satisfactorily compared with the 1.8 mol/kg experimental X-ray scattering signal. The theoretical calculations demonstrate that the experimental structure factor is dominated by Eu³⁺-Eu³⁺ correlations.

Theory of applying heat flow from thermostatted boundary walls: Dissipative and local-equilibrium responses and fluctuation theorems

We construct a microscopic theory of applying a heat flow from thermostatted boundary walls in the film geometry. We treat a classical one-component fluid, but our method is applicable to any fluids and solids. We express linear response of any variable B in terms of the time-correlation functions between B and the heat flows J_K from the thermostats to the particles. Furthermore, the surface variables J_K can be written in the form of space integrals of bulk quantities from the equations of motion. Owing to this surface-to-bulk relation, the steady-state response functions consist of dissipative and local-equilibrium parts, where the former gives rise to Fourier's law with Green's expression for the thermal conductivity. In the nonlinear regime, we derive the steady-state distribution in the phase space in the McLennan-Zubarev form from the first principles. Some fluctuation theorems are also presented.

Structure and thermodynamics of aqueous urea solutions from ambient to kilobar pressures: From thermodynamic modeling, experiments, and first principles simulations to an accurate force field description

Molecular simulations based on classical force fields are a powerful method for shedding light on the complex behavior of biomolecules in solution. When cosolutes are present in addition to water and biomolecules, subtle balances of weak intermolecular forces have to be accounted for. This imposes high demands on the quality of the underlying force fields, and therefore force field development for small cosolutes is still an active field. Here, we present the development of a new urea force field from studies of urea solutions at ambient and elevated hydrostatic pressures based on a combination of experimental and theoretical approaches. Experimental densities and solvation shell properties from ab initio molecular dynamics simulations at ambient conditions served as the target properties for the force field optimization. Since urea is present in many marine life forms, elevated hydrostatic pressure was rigorously addressed: densities at high pressure were measured by vibrating tube densitometry up to 500 bar and by X-ray absorption up to 5 kbar. Densities were determined by the perturbed-chain statistical associating fluid theory equation of state. Solvation properties were determined by embedded cluster integral equation theory and ab initio molecular dynamics. Our new force field is able to capture the properties of urea solutions at high pressures without further high-pressure adaption, unlike trimethylamine-N-oxide, for which a high-pressure adaption is necessary.

Density variations of TMAO solutions in the kilobar range: Experiments, PC-SAFT predictions, and molecular dynamics simulations

We present measurements, molecular dynamics (MD) simulations, and predictions using Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) of the density of aqueous solutions in a pressure range from 1 bar to 5000 bar, a pressure regime that is highly relevant for both biochemical applications and the fundamental understanding of solvation. The accurate determination of density data of pressurized solutions remains challenging. We determined relative density changes from the variations in X-ray absorption through the sample and developed a new water parameter set for PC-SAFT modeling that is appropriate for high pressure conditions in the kilobar regime. As a showcase, we studied trimethylamine N-oxide (TMAO) solutions and demonstrated that their compressibility decreases with the TMAO content. This result is linked to the stabilizing effect of TMAO on the local H-bond network of water. Experiments and calculations, which represent two independent methods, are in very good agreement and are in accordance with results of force field molecular dynamics simulations of the same systems.

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.

Size and shape dependence of finite-volume Kirkwood-Buff integrals

Analytic relations are derived for finite-volume integrals over the pair correlation function of a fluid, the so-called Kirkwood-Buff integrals. Closed-form expressions are obtained for cubes and cuboids, the system shapes commonly employed in molecular simulations. When finite-volume Kirkwood-Buff integrals are expanded over an inverse system size, the leading term depends on shape only through the surface area-to-volume ratio. This conjecture is proved for arbitrary shapes and a general expression for the leading term is derived. From this, an extrapolation to the infinite-volume limit is proposed, which converges much faster with system size than previous approximations and thus significantly simplifies the numerical computations.

From classical to quantum and back: Hamiltonian adaptive resolution path integral, ring polymer, and centroid molecular dynamics

Path integral-based methodologies play a crucial role for the investigation of nuclear quantum effects by means of computer simulations. However, these techniques are significantly more demanding than corresponding classical simulations. To reduce this numerical effort, we recently proposed a method, based on a rigorous Hamiltonian formulation, which restricts the quantum modeling to a small but relevant spatial region within a larger reservoir where particles are treated classically. In this work, we extend this idea and show how it can be implemented along with state-of-the-art path integral simulation techniques, including path-integral molecular dynamics, which allows for the calculation of quantum statistical properties, and ring-polymer and centroid molecular dynamics, which allow the calculation of approximate quantum dynamical properties. To this end, we derive a new integration algorithm that also makes use of multiple time-stepping. The scheme is validated via adaptive classical-path-integral simulations of liquid water. Potential applications of the proposed multiresolution method are diverse and include efficient quantum simulations of interfaces as well as complex biomolecular systems such as membranes and proteins.

Fluctuations, Finite-Size Effects and the Thermodynamic Limit in Computer Simulations: Revisiting the Spatial Block Analysis Method

The spatial block analysis (SBA) method has been introduced to efficiently extrapolate thermodynamic quantities from finite-size computer simulations of a large variety of physical systems. In the particular case of simple liquids and liquid mixtures, by subdividing the simulation box into blocks of increasing size and calculating volume-dependent fluctuations of the number of particles, it is possible to extrapolate the bulk isothermal compressibility and Kirkwood–Buff integrals in the thermodynamic limit. Only by explicitly including finite-size effects, ubiquitous in computer simulations, into the SBA method, the extrapolation to the thermodynamic limit can be achieved. In this review, we discuss two of these finite-size effects in the context of the SBA method due to (i) the statistical ensemble and (ii) the finite integration domains used in computer simulations. To illustrate the method, we consider prototypical liquids and liquid mixtures described by truncated and shifted Lennard–Jones (TSLJ) potentials. Furthermore, we show some of the most recent developments of the SBA method, in particular its use to calculate chemical potentials of liquids in a wide range of density/concentration conditions.

Size and shape effects on the thermodynamic properties of nanoscale volumes of water

Small systems are known to deviate from the classical thermodynamic description, among other things due to their large surface area to volume ratio compared to corresponding big systems. As a consequence, extensive thermodynamic properties are no longer proportional to the volume, but are instead higher order functions of size and shape. We investigate such functions for second moments of probability distributions of fluctuating properties in the grand-canonical ensemble, focusing specifically on the volume and surface terms of Hadwiger's theorem, explained in Klain, Mathematika, 1995, 42, 329-339. We resolve the shape dependence of the surface term and show, using Hill's nanothermodynamics [Hill, J. Chem. Phys., 1962, 36, 3182], that the surface satisfies the thermodynamics of a flat surface as described by Gibbs [Gibbs, The Scientific Papers of J. Willard Gibbs, Volume 1, Thermodynamics, Ox Bow Press, Woodbridge, Connecticut, 1993]. The Small System Method (SSM), first derived by Schnell et al. [Schnell et al., J. Phys. Chem. B, 2011, 115, 10911], is extended and used to analyze simulation data on small systems of water. We simulate water as an example to illustrate the method, using TIP4P/2005 and other models, and compute the isothermal compressibility and thermodynamic factor. We are able to retrieve the experimental value of the bulk phase compressibility within 2%, and show that the compressibility of nanosized volumes increases by up to a factor of two as the number of molecules in the volume decreases. The value for a tetrahedron, cube, sphere, polygon, etc. can be predicted from the same scaling law, as long as second order effects (nook and corner effects) are negligible. Lastly, we propose a general formula for finite reservoir correction to fluctuations in subvolumes.

Statistical thermodynamic foundation for mesoscale aggregation in ternary mixtures

The origin of persistent mesoscale aggregation around the plait point has been clarified from statistical thermodynamics and differential geometry.

Aqueous ionic liquids and their effects on protein structures: an overview on recent theoretical and experimental results

Ionic liquids (ILs) are used in a variety of technological and biological applications. Recent experimental and simulation results reveal the influence of aqueous ionic liquids on the stability of protein and enzyme structures. Depending on different parameters like the concentration and the ion composition, one can observe distinct stabilization or denaturation mechanisms for various ILs. In this review, we summarize the main findings and discuss the implications with regard to molecular theories of solutions and specific ion effects. A preferential binding model is introduced in order to discuss protein-IL effects from a statistical mechanics perspective. The value of the preferential binding coefficient determines the strength of the ion influence and indicates a shift of the chemical equilibrium either to the native or the denatured state of the protein. We highlight the role of water in order to explain the self-association behavior of the IL species and discuss recent experimental and simulation results in the light of the observed binding effects.