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.

A procedure to find thermodynamic equilibrium constants for CO2 and CH4 adsorption on activated carbon.

Thermodynamic equilibrium for adsorption means that the chemical potential of gas and adsorbed phase are equal. A precise knowledge of the chemical potential is, however, often lacking, because the activity coefficient of the adsorbate is not known. Adsorption isotherms are therefore commonly fitted to ideal models such as the Langmuir, Sips or Henry models. We propose here a new procedure to find the activity coefficient and the equilibrium constant for adsorption which uses the thermodynamic factor. Instead of fitting the data to a model, we calculate the thermodynamic factor and use this to find first the activity coefficient. We show, using published molecular simulation data, how this procedure gives the thermodynamic equilibrium constant and enthalpies of adsorption for CO2(g) on graphite. We also use published experimental data to find similar thermodynamic properties of CO2(g) and of CH4(g) adsorbed on activated carbon. The procedure gives a higher accuracy in the determination of enthalpies of adsorption than ideal models do.

Calculation of the chemical potential and the activity coefficient of two layers of CO2 adsorbed on a graphite surface.

We study the adsorption of carbon dioxide at a graphite surface using the new Small System Method, and find that for the temperature range between 300 K and 550 K most relevant for CO2 separation; adsorption takes place in two distinct thermodynamic layers defined according to Gibbs. We calculate the chemical potential and the activity coefficient of both layers directly from the simulations. Based on thermodynamic relations, the entropy and enthalpy of the CO2 adsorbed layers are also obtained. Their values indicate that there is a trade-off between entropy and enthalpy when a molecule chooses for one of the two layers. The first layer is a densely packed monolayer of relatively constant excess density with relatively large repulsive interactions and negative enthalpy. The second layer has an excess density varying with the temperature, an activity coefficient, which also indicates repulsion, but to a much smaller degree than in the first layer. Results for activity coefficients, entropies and enthalpies can be used to model transport through and along graphitic membranes for carbon dioxide separation purposes.

Compressibility, thermal expansion coefficient and heat capacity of CH4 and CO2 hydrate mixtures using molecular dynamics simulations.

Understanding the thermal and mechanical properties of CH4 and CO2 hydrates is essential for the replacement of CH4 with CO2 in natural hydrate deposits as well as for CO2 sequestration and storage. In this work, we present isothermal compressibility, isobaric thermal expansion coefficient and specific heat capacity of fully occupied single-crystal sI-CH4 hydrates, CO2 hydrates and hydrates of their mixture using molecular dynamics simulations. Eight rigid/nonpolarisable water interaction models and three CH4 and CO2 interaction potentials were selected to examine the atomic interactions in the sI hydrate structure. The TIP4P/2005 water model combined with the DACNIS united-atom CH4 potential and TraPPE CO2 rigid potential were found to be suitable molecular interaction models. Using these molecular models, the results indicate that both the lattice parameters and the compressibility of the sI hydrates agree with those from experimental measurements. The calculated bulk modulus for any mixture ratio of CH4 and CO2 hydrates varies between 8.5 GPa and 10.4 GPa at 271.15 K between 10 and 100 MPa. The calculated thermal expansion and specific heat capacities of CH4 hydrates are also comparable with experimental values above approximately 260 K. The compressibility and expansion coefficient of guest gas mixture hydrates increase with an increasing ratio of CO2-to-CH4, while the bulk modulus and specific heat capacity exhibit the opposite trend. The presented results for the specific heat capacities of 2220-2699.0 J kg(-1) K(-1) for any mixture ratio of CH4 and CO2 hydrates are the first reported so far. These computational results provide a useful database for practical natural gas recovery from CH4 hydrates in deep oceans where CO2 is considered to replace CH4, as well as for phase equilibrium and mechanical stability of gas hydrate-bearing sediments. The computational schemes also provide an appropriate balance between computational accuracy and cost for predicting mechanical and thermal properties of gas hydrates in the high temperature range (≥260 K), and the schemes may be useful for the study of other complex hydrate systems.

Mesoscopic non-equilibrium thermodynamic analysis of molecular motors.

We show that the kinetics of a molecular motor fueled by ATP and operating between a deactivated and an activated state can be derived from the principles of non-equilibrium thermodynamics applied to the mesoscopic domain. The activation by ATP, the possible slip of the motor, as well as the forward stepping carrying a load are viewed as slow diffusion along a reaction coordinate. Local equilibrium is assumed in the reaction coordinate spaces, making it possible to derive the non-equilibrium thermodynamic description. Using this scheme, we find expressions for the velocity of the motor, in terms of the driving force along the spacial coordinate, and for the chemical reaction that brings about activation, in terms of the chemical potentials of the reactants and products which maintain the cycle. The second law efficiency is defined, and the velocity corresponding to maximum power is obtained for myosin movement on actin. Experimental results fitting with the description are reviewed, giving a maximum efficiency of 0.45 at a myosin headgroup velocity of 5 × 10(-7) m s(-1). The formalism allows the introduction and test of meso-level models, which may be needed to explain experiments.

Toward a possibility to exchange CO2 and CH4 in sI clathrate hydrates.

We study the thermodynamic conditions for exchange of CH(4) with CO(2) in sI clathrate hydrates using Grand Canonical Monte Carlo simulations. From the variations in the Helmholtz energy, we suggest a thermodynamic path for exchange of CH(4) by CO(2). The results can be understood from single-component occupancy isotherms. Simulations of pure component systems show that all methane sites are essentially equivalent, whereas carbon dioxide distinguishes between two types of sites, large or small. Mixture of CO(2) and CH(4) in the clathrate can be regarded as ideal, as long as only the large sites are occupied. A strong preference in selectivity is demonstrated for methane when the smaller sites become filled. The Helmholtz energies of the hydrate with a CO(2)-CH(4) gas mixture for temperatures between 278 and 328 K and pressures between 10(4) and 10(9) Pa indicate that there exists a region of stability of a mixed hydrate referred to single-component hydrates.

Concentration fluctuations in non-isothermal reaction-diffusion systems. II. The nonlinear case.

In this paper, we consider a simple reaction-diffusion system, namely, a binary fluid mixture with an association-dissociation reaction between two species. We study fluctuations at hydrodynamic spatiotemporal scales when this mixture is driven out of equilibrium by the presence of a temperature gradient, while still being far away from any chemical instability. This study extends the analysis in our first paper on the subject [J. M. Ortiz de Zárate, J. V. Sengers, D. Bedeaux, and S. Kjelstrup, J. Chem. Phys. 127, 034501 (2007)], where we considered fluctuations in a non-isothermal reaction-diffusion system but still close to equilibrium. The present extension is based on mesoscopic non-equilibrium thermodynamics that we recently developed [D. Bedeaux, I. Pagonabarraga, J. M. Ortiz de Zárate, J. V. Sengers, and S. Kjelstrup, Phys. Chem. Chem. Phys. 12, 12780 (2010)] to derive the law of mass action and fluctuation-dissipation theorems for the random contributions to the dissipative fluxes in the nonlinear macroscopic description. Just as for non-equilibrium fluctuations close to equilibrium, we again find an enhancement of the intensity of the concentration fluctuations in the presence of a temperature gradient. The non-equilibrium concentration fluctuations are in both cases spatially long ranged, with an intensity depending on the wave number q. The intensity exhibits a crossover from a ∝q(-4) to a ∝q(-2) behavior depending on whether the corresponding wavelength is smaller or larger than the penetration depth of the reacting mixture. This opens a possibility to distinguish between diffusion- or activation-controlled regimes of the reaction experimentally. The important conclusion overall is that non-equilibrium fluctuations in non-isothermal reaction-diffusion systems are always long ranged.

The role of temperature in nucleation processes.

Heat and mass transfers are coupled processes, also in nucleation. In principle, a nucleating cluster would have a different temperature compared to the surrounding supersaturated old phase because of the heat release involved with attaching molecules to the cluster. In turn a difference in temperature across the cluster surface is a driving force for the mass transfer to and from the cluster. This coupling of forces in nonisothermal nucleation is described using mesoscopic nonequilibrium thermodynamics, emphasizing measurable heat effects. An expression was obtained for the nonisothermal nucleation rate in a one-component system, in the case where a temperature difference exists between a cluster distribution and the condensed phase. The temperature is chosen as a function of the cluster size only, while the temperature of the condensed phase is held constant by a bath. The generally accepted expression for isothermal stationary nucleation is contained as a limiting case of the nonisothermal stationary nucleation rate. The equations for heat and mass transport were solved for stationary nucleation with the given cluster distribution and with the temperature controlled at the boundaries. A factor was defined for these conditions, determined by the heat conductivity of the surrounding phase and the phase transition enthalpy, to predict the deviation between isothermal and nonisothermal nucleation. For the stationary state described, the factor appears to give small deviations, even for primary nucleation of droplets in vapor, making the nonisothermal rate smaller than the isothermal one. The set of equations may lead to larger and different thermal effects under different boundary conditions, however.

Mesoscopic non-equilibrium thermodynamics of non-isothermal reaction-diffusion.

We show how the law of mass action can be derived from a thermodynamic basis, in the presence of temperature gradients, chemical potential gradients and hydrodynamic flow. The solution gives the law of mass action for the forward and the reverse contributions to the net chemical reaction. In addition we derive the fluctuation-dissipation theorem for the fluctuating contributions to the reaction rate, heat flux and mass fluxes. All these results arise without any other assumptions than those which are common in mesoscopic non-equilibrium thermodynamics; namely quasi-stationary transport across a high activation energy barrier, and local equilibrium along the reaction coordinate. Arrhenius-type behaviour of the kinetic coefficients is recovered. The thermal conductivity, Soret coefficient and diffusivity are significantly influenced by the presence of a chemical reaction. We thus demonstrate how chemical reactions can be fully reconciled with non-equilibrium thermodynamics.

Thermal diffusion and partial molar enthalpy variations of n-butane in silicalite-1.

We report for the first time the heat of transfer and the Soret coefficient for n-butane in silicalite-1. The heat of transfer was typically 10 kJ/mol. The Soret coefficient was typically 0.006 K(-1) at 360 K. Both varied with the temperature and the concentration. The thermal conductivity of the crystal with butane adsorbed was 1.46 +/- 0.07 W/m K. Literature values of the isosteric enthalpy of adsorption, the concentration at saturation, and the diffusion coefficients were reproduced. Nonequilibrium molecular dynamics simulations were used to find these results, and a modified heat-exchange algorithm, Soft-HEX, was developed for the purpose. Enthalpies of butane were also determined. We use these results to give numerical proof for a recently proposed relation, that the heat of transfer plus the partial molar enthalpy of butane is constant at a given temperature. The proof is offered for a regime where the partial molar enthalpy can be approximated by the molar internal energy. This result may add to the understanding of the sign of the Soret coefficient. The technical importance of the heat of transfer is discussed.

Transfer coefficients for evaporation of a system with a Lennard-Jones long-range spline potential.

Surface transfer coefficients are determined by nonequilibrium molecular dynamics simulations for a Lennard-Jones fluid with a long-range spline potential. In earlier work [A. Røsjorde, J. Colloid Interface Sci. 240, 355 (2001); J. Xu, ibid. 299, 452 (2006)], using a short-range Lennard-Jones spline potential, it was found that the resistivity coefficients to heat and mass transfer agreed rather well with the values predicted by kinetic theory. For the long-range Lennard-Jones spline potential considered in this paper we find significant discrepancies from the values predicted by kinetic theory. In particular the coupling coefficient, and as a consequence the heat of transfer on the vapor side of the surface are much larger. Thermodynamic data for the liquid-vapor equilibrium confirmed the law of corresponding states for the surface, when it is described as an autonomous system. The importance of these findings for modelling phase transitions is discussed.

Unifying thermodynamic and kinetic descriptions of single-molecule processes: RNA unfolding under tension.

We use mesoscopic nonequilibrium thermodynamics theory to describe RNA unfolding under tension. The theory introduces reaction coordinates, characterizing a continuum of states for each bond in the molecule. The unfolding considered is so slow that one can assume local equilibrium in the space of the reaction coordinates. In the quasi-stationary limit of high sequential barriers, our theory yields the master equation of a recently proposed sequential-step model. Nonlinear switching kinetics is found between open and closed states. Our theory unifies the thermodynamic and kinetic descriptions and offers a systematic procedure to characterize the dynamics of the unfolding process.

Thermal effects during adsorption of n-butane on a silicalite-1 membrane: a non-equilibrium molecular dynamics study.

Non-equilibrium molecular dynamic (NEMD) simulations have been used to study the kinetics of adsorption of n-butane molecules in a silicalite membrane. We have chosen this simple well-known process to demonstrate that the process is characterized by two stages, both non-isothermal. In the first stage the large chemical driving force leads to a rapid uptake of n-butane in all the membrane and a simultaneous increase in the membrane temperature, explained by the large enthalpy of adsorption, DeltaH=-61.6kJ/mol butane. A diffusion coefficient for transport across the external surface layer is calculated from the relaxation time; a value of 3.4x10(-9)m(2)/s is found. During the adsorption, a significant thermal driving force develops across the external surface of the membrane, which leads to an energy flux out of the membrane during the second stage. In this stage a thermal conductivity of 3.4x10(-4)W/Km is calculated from the corresponding relaxation time for the surface, confirming that the thermal conduction is the rate-limiting step. The aim of this paper is to demonstrate that a thermal driving force must be taken into account in addition to a chemical driving force in the description of transport in nano-porous materials.

Interface film resistivities for heat and mass transfers-integral relations verified by non-equilibrium molecular dynamics.

Integral relations that predict interface film transfer coefficients for evaporation and condensation have recently been derived. According to these relations, all coefficients can be calculated for one-component systems, using the thermal resistivity and the enthalpy profile through the interface. The integral relations were tested in this work using nonequilibrium molecular dynamics simulations for argon-like particles and n-octane molecules. The simulations confirm the integral relations within the accuracy of the calculation for both systems. Evidence is presented for the existence of an excess thermal resistivity on the gas side of the surface, and the fact that this property is decisive for interface heat and mass transfer coefficients. The integral relations were used to predict the mass transfer coefficient for n- octane as a function of surface tension. The findings are important for modeling of one-component phase transitions.

Thermodynamics for single-molecule stretching experiments.

We show how to construct nonequilibrium thermodynamics for systems too small to be considered thermodynamically in a traditional sense. Through the use of a nonequilibrium ensemble of many replicas of the system which can be viewed as a large thermodynamic system, we discuss the validity of nonequilibrium thermodynamics relations and analyze the nature of dissipation in small systems through the entropy production rate. We show in particular that the Gibbs equation, when formulated in terms of average values of the extensive quantities, is still valid, whereas the Gibbs-Duhem equation differs from the equation obtained for large systems due to the lack of the thermodynamic limit. Single-molecule stretching experiments are interpreted under the prism of this theory. The potentials of mean force and mean position, now introduced in these experiments in substitution of the thermodynamic potentials, correspond respectively to our Helmholtz and Gibbs energies. These results show that a thermodynamic formalism can indeed be applied at the single-molecule level.

Verification of Onsager's reciprocal relations for evaporation and condensation using non-equilibrium molecular dynamics.

Non-equilibrium molecular dynamic (NEMD) simulations have been used to study heat and mass transfer across a vapor-liquid interface for a one-component system using a Lennard-Jones spline potential. It was confirmed that the relation between the surface tension and the surface temperature in the non-equilibrium system was the same as in equilibrium (local equilibrium). Interfacial transfer coefficients were evaluated for the surface, which expressed the heat and mass fluxes in temperature and chemical potential differences across the interfacial region (film). In this analysis it was assumed that the Onsager reciprocal relations were valid. In this paper we extend the number of simulations such that we can calculate all four interface film transfer coefficients along the whole liquid-vapor coexistence curve. We do this analysis both for the case where we use the measurable heat flux on the vapor side and for the case where we use the measurable heat flux on the liquid side. The most important result we found is that the coupling coefficients within the accuracy of the calculation are equal. This is the first verification of the validity of the Onsager relations for transport through a surface using molecular dynamics. The interfacial film transfer coefficients are found to be a function of the surface temperature alone. New expressions are given for the kinetic theory values of these coefficients which only depend on the surface temperature. The NEMD values were found to be in good agreement with these expressions.

Active transport: a kinetic description based on thermodynamic grounds.

We show that active transport processes in biological systems can be understood through a local equilibrium description formulated at the mesoscale, the scale to describe stochastic process. This new approach uses the method established by nonequilibrium thermodynamics to account for the irreversible processes occurring at this scale and provides nonlinear kinetic equations for the rates in terms of the driving forces. The results show that the application domain of nonequilibrium thermodynamics method to biological systems goes beyond the linear domain. A model for transport of Ca2+ by the Ca2+-ATPases, nonlinear way. Our results unify thermodynamic and kinetic descriptions, thereby opening new perspectives in the study of different transport phenomena in biological systems.

Nonequilibrium Molecular Dynamics Simulations of Steady-State Heat and Mass Transport in Condensation. II. Transfer Coefficients.

We present coefficients for transfer of heat and mass across the liquid-vapor interface of a one-component fluid. The coefficients are defined for the Gibbs surface from nonequilibrium thermodynamics and determined by nonequilibrium molecular dynamics simulations. The main conductivity coefficients are found to become large near the critical point, consistent with the disappearance of the surface in this limit. The resistivities of transfer found by molecular dynamics simulations are compared to the values predicted by kinetic theory. The main resistivity to heat transfer is found to agree from the triple point to about halfway to the critical point. The resistivity to mass transfer was used to determine the condensation coefficient, which was found to be practically constant with a value of about 0.82. The resistivity coupling coefficient predicted by simulations also agrees with values predicted by kinetic theory from the triple point until about halfway to the critical point. Copyright 2001 Academic Press.

Nonequilibrium Molecular Dynamics Simulations of Steady-State Heat and Mass Transport in Condensation.

We present evidence for the hypothesis of local equilibrium for a liquid-vapor interface in a one-component fluid, using molecular dynamics simulations. Lennard-Jones/spline particles are studied in a two-phase system that is out of global equilibrium. Equilibrium simulations are first used to establish the equation of state for the vapor and interface. A procedure is developed to define the boundaries of the interface. Finally it is shown that the equation of state for the interface applies also when there is heat and mass transport through the interface. The temperature gradient in the vapor was 10(8) K/m in these studies. Copyright 2000 Academic Press.