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.

Reciprocal relations in electroacoustics.

In a colloidal suspension, one can generate sound waves by the application of an alternating electric field (Electrokinetic Sonic Amplitude, i.e., ESA). Another phenomenon is electrophoresis (Electrophoretic Mobility, i.e., EM) where a colloidal particle moves relative to the solvent in an electric field. Vice versa one can generate electric fields or electric currents by sound waves (Colloid Vibration Potential/Current, i.e., CVP/CVI). In 1988 and 1990, O'Brien [J. Fluid Mech. 190, 71-86 (1988) and O'Brien, J. Fluid Mech. 212, 81-93 (1990)] derived a reciprocal relation between the proportionality coefficients of the EM and CVI phenomena. In this paper, we will generalize his proof by constructing the relevant entropy production from which the linear force-flux relations follow. General relations are derived for electrolyte solutions, of which colloidal suspensions are a particular case. The relations between CVI, CVP, EM, and ESA are discussed. O'Brien's reciprocal relation then follows as an Onsager relation. The relation is valid for any applied electric field frequency, particle surface charge and particle concentration (even in the presence of particle-particle interactions) provided the system is isotropic.

Curvature dependence of the interfacial heat and mass transfer coefficients.

Nucleation is often accompanied by heat transfer between the surroundings and a nucleus of a new phase. The interface between two phases gives an additional resistance to this transfer. For small nuclei the interfacial curvature is high, which affects not only equilibrium quantities such as surface tension, but also the transport properties. In particular, high curvature affects the interfacial resistance to heat and mass transfer. We develop a framework for determining the curvature dependence of the interfacial heat and mass transfer resistances. We determine the interfacial resistances as a function of a curvature. The analysis is performed for a bubble of a one-component fluid and may be extended to various nuclei of multicomponent systems. The curvature dependence of the interfacial resistances is important in modeling transport processes in multiphase 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.

Effect of compressibility in bubble formation in closed systems.

We analyze the stability of small bubbles in a closed system with fixed volume, temperature, and number of molecules. We show that there exists a minimum stable size of a bubble. Thus there exists a range of densities where no stable bubbles are allowed and the system has a homogeneous density which is lower than the coexistence density of the liquid. This becomes possible due to the finite liquid compressibility. Capillary analysis within the developed "modified bubble" model illustrates that the existence of the minimum bubble size is associated to the compressibility and it is not possible when the liquid is strictly incompressible. This finding is expected to have very important implications in cavitation and boiling.

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.

Resistances for heat and mass transfer through a liquid-vapor interface in a binary mixture.

In this paper we calculate the interfacial resistances to heat and mass transfer through a liquid-vapor interface in a binary mixture. We use two methods, the direct calculation from the actual nonequilibrium solution and integral relations, derived earlier. We verify, that integral relations, being a relatively faster and cheaper method, indeed gives the same results as the direct processing of a nonequilibrium solution. Furthermore we compare the absolute values of the interfacial resistances with the ones obtained from kinetic theory. Matching the diagonal resistances for the binary mixture we find that kinetic theory underestimates the cross coefficients. The heat of transfer is, as a consequence, correspondingly larger.

Transport of heat and mass in a two-phase mixture: From a continuous to a discontinuous description.

We present a theory that describes the transport properties of the interfacial region with respect to heat and mass transfer. Postulating the local Gibbs relation for a continuous description inside the interfacial region, we derive the description of the Gibbs surface in terms of excess densities and fluxes along the surface. We introduce overall interfacial resistances and conductances as the coefficients in the force-flux relations for the Gibbs surface. We derive relations between the local resistivities for the continuous description inside the interfacial region and the overall resistances of the surface for transport between the two phases for a mixture. It is shown that interfacial resistances depend among other things on the enthalpy profile across the interface. Since this variation is substantial, the coupling between heat and mass flow across the surface is also substantial. In particular, the surface puts up much more resistance to the heat and mass transfer than the homogeneous phases over a distance comparable to the thickness of the surface. This is the case not only for the pure heat conduction and diffusion but also for the cross effects such as thermal diffusion. For the excess fluxes along the surface and the corresponding thermodynamic forces, we derive expressions for excess conductances as integrals over the local conductivities along the surface. We also show that the curvature of the surface affects only the overall resistances for transport across the surface and not the excess conductivities along the surface.

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.

Numerical solution of the nonequilibrium square-gradient model and verification of local equilibrium for the Gibbs surface in a two-phase binary mixture.

In this paper we apply the general analysis described in our first paper to a binary mixture of cyclohexane and n -hexane. We use the square gradient model for the continuous description of a nonequilibrium surface and obtain numerical profiles of various thermodynamic quantities in various stationary state conditions. Details of the numerical procedure are given and discussed. In the second part of this paper we focus on the verification of local equilibrium of the surface as described with excess densities. We give a definition of the temperature and chemical potentials of the surface and verify that these quantities are independent of the choice of the dividing surface. We verify numerically that the surface in a stationary state of the mixture can be described in terms of Gibbs excess densities, which are found to be in good approximation equal to their equilibrium values at the stationary state temperature and chemical potentials of the surface.

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.

The dielectric response of a colloidal spheroid.

In this article, we present a theory for the dielectric behavior of a colloidal spheroid, based on an improved version of a previously published analytical theory [C. Chassagne, D. Bedeaux, G.J.M. Koper, Physica A 317 (2003) 321-344]. The theory gives the dipolar coefficient of a dielectric spheroid in an electrolyte solution subjected to an oscillating electric field. In the special case of the sphere, this theory is shown to agree rather satisfactorily with the numerical solutions obtained by a code based on DeLacey and White's [E.H.B. DeLacey, L.R. White, J. Chem. Soc. Faraday Trans. 2 77 (1981) 2007] for all zeta potentials, frequencies and kappa a1 where kappa is the inverse of the Debye length and a is the radius of the sphere. Using the form of the analytical solution for a sphere we were able to derive a formula for the dipolar coefficient of a spheroid for all zeta potentials, frequencies and kappa a1. The expression we find is simpler and has a more general validity than the analytical expression proposed by O'Brien and Ward [R.W. O'Brien, D.N. Ward, J. Colloid Interface Sci. 121 (1988) 402] which is valid for kappa a >> 1 and zero frequency.

Nonequilibrium properties of a two-dimensionally isotropic interface in a two-phase mixture as described by the square gradient model.

In earlier work a systematic extension of the van der Waals square gradient model to nonequilibrium one-component systems was given. In this work the focus was on heat and mass transfer through the liquid-vapor interface as caused by a temperature difference or an over- or underpressure. We will give an extension of this approach to multicomponent nonequilibrium systems in the systematic context of nonequilibrium thermodynamics. An explicit expression for the pressure tensor is derived valid also for curved surfaces. It is shown how the Gibbs relation should be modified in the interfacial region, in both equilibrium and nonequilibrium. The two-dimensional isotropy of a curved interface is discussed. Furthermore, we give numerically obtained profiles of the concentration, the mole fraction, and the temperature, which illustrate the solution for some special cases.

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.