Structure of mass, momentum, and energy transfer equations under highly nonequilibrium conditions

The structure of mass, momentum, and energy transfer equations under highly non-equilibrium conditions is considered when the traditional assumption of nonequilibrium thermodynamics (the local equilibrium condition) is violated. The derived transfer equations based on particle mass, momentum, and the law of energy conservation are related to heterogeneous systems with arbitrary density, i.e., for three aggregate states and their interfaces. Fluxes of the mentioned properties are described at the atomic-molecular level by nonequilibrium discrete unary and binary distribution functions (in the lattice gas model) with regard to interparticle potential interactions of system components. It is found that the total set of local transfer equations consists of five modified mass, momentum, and energy transfer equations for each of the system sites, and of 15 new equations describing the correlated characteristics of the density, rate, and temperature for the sites of a pair. The relationship between the derived equations and previous theories is discussed.     

Calculation of the stability and of the phase equilibrium in the system polystyrene+cyclohexane+carbon dioxide based on equations of state

In order to calculate spinodals for polymer systems with an equation of state (EOS), we developed a stability theory using continuous thermodynamics. Here, the mixture considered consists of a polydisperse polymer and two monodisperse components as for example a solvent and a gas. We derived the spinodal equation on the base of the segment-molar Helmholtz energy of the mixture. As a result, a determinant similar to that of the monodisperse case is obtained, but the polydisperse polymer is identified by its weight average of the molecular weight. Furthermore, our paper provides the equations for the cloud-point curve derived with the aid of continuous thermodynamics. The final equations are applied to the system polystyrene+cyclohexane+carbon dioxide using the EOS of Sako, Wu and Prausnitz (SWP-EOS). For parameter fit and to prove the accuracy of the treatment, experimental data of the high-pressure equilibrium of the binary subsystems and of the ternary system were taken from literature.     

Principles of Considering the Effect of the Limited Volume of a System on Its Thermodynamic State

The features of a system with a finite volume that affect its thermodynamic state are considered in comparison to describing small bodies in macroscopic phases. Equations for unary and pair distribution functions are obtained using difference derivatives of a discrete statistical sum. The structure of the equation for the free energy of a system consisting of an ensemble of volume-limited regions with different sizes and a full set of equations describing a macroscopic polydisperse system are discussed. It is found that the equations can be applied to molecular adsorption on small faces of microcrystals, to bound and isolated pores of a polydisperse material, and to describe the spinodal decomposition of a fluid in brief periods of time and high supersaturations of the bulk phase when each local region functions the same on average. It is shown that as the size of a system diminishes, corrections must be introduced for the finiteness of the system volume and fluctuations of the unary and pair distribution functions.     

Nonequilibrium thermodynamics of transport processes on membrane surfaces

Nonequilibrium processes have been considered on interfaces between media, with one medium being a permeable membrane. The nonequilibrium processes have been described in terms of nonequilibrium thermodynamics combined with the Fokker?Planck kinetic equation. Phenomenological equations derived within the framework of the combined approach have been discussed. It has been shown that the formal use of linear equations of nonequilibrium thermodynamics may be, in some cases, reduced to nonlinear equations for heat and mass transfer through an interface. It has been noted that the asymmetry of transport characteristics of asymmetric membranes may be explained in terms of the derived relations.     

Small Systems and Limitations on the Use of Chemical Thermodynamics

Limitations on using chemical thermodynamics to describe small systems are formulated. These limitations follow from statistical mechanics for equilibrium and nonequilibrium processes and reflect (1) differences between characteristic relaxation times in momentum, energy, and mass transfer in different aggregate states of investigated systems; (2) achievements of statistical mechanics that allow us to determine criteria for the size of smallest region in which thermodynamics can be applied and the scale of the emergence of a new phase, along with criteria for the conditions of violating a local equilibrium. Based on this analysis, the main thermodynamic results are clarified: the phase rule for distorted interfaces, the sense and area of applicability of Gibbs’s concept of passive forces, and the artificiality of Kelvin’s equation as a result of limitations on the thermodynamic approach to considering small bodies. The wrongness of introducing molecular parameters into thermodynamic derivations, and the activity coefficient for an activated complex into the expression for a reaction rate constant, is demonstrated.     

Thermodynamic Theory of Equilibrium Copolymerization

General equations are derived for the thermodynamics of equilibrium copolymerization based on the diad model of copolymer. Physical interactions between components in solution are taken into consideration. Several special cases are treated in detail. When these physical interactions are absent, the relations obtained from the theory agree completely with those derived from the kinetics. Equilibrium monomer concentrations are illustrated as functions of excess free energy due to formation of heterodiad, equilibrium constants of homopolymerization, and initial monomer concentrations. It is shown that when the physical interactions are present, their effects cannot be ignored even in dilute solution. In bulk copolymerization, the physical interactions have a considerable effect on equilibrium, especially when the equilibrium monomer volume fractions are relatively small.     

Molecular theory of irreversibility

A generalization of the Gibbs entropy postulate is proposed based on the Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy of equations as the nonequilibrium entropy for a system of N interacting particles. This entropy satisfies the basic principles of thermodynamics in the sense that it reaches its maximum at equilibrium and is coherent with the second law. By using a generalization of the Liouville equation describing the evolution of the distribution vector, it is demonstrated that the entropy production is a non-negative quantity. Moreover, following the procedure of nonequilibrium thermodynamics a transport matrix is introduced and a microscopic expression for this is derived. This framework allows one to perform the thermodynamic analysis of nonequilibrium steady states with smooth phase-space distribution functions which, as proven here, constitute the states of minimum entropy production when one considers small departures from stationarity.     

Molecular thermodynamics: Allowance for momentum and structural equilibrium

The possibility and the value of updating the theoretical base of classical thermodynamics were demonstrated. The energy of particles is divided between kinetic and potential energy. Universal coordinates are introduced to take account of the motion and the interaction of particles (with each other or with an external field). The theoretical potential and the application range of the formulated thermodynamics (called molecular thermodynamics) are substantially extended. The new criteria of structural and momentum equilibrium and their repercussions are introduced and illustrated, and statistical analogs are established. The mutual convexity and other unexpected characteristics of the thermodynamic functions are exposed. The applications of the theory are extended to classical and nonequilibrium systems.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 31, No. 6, pp. 339–350, November–December, 1995.     

Model hysteresis dimer molecule II: deductions from probability profile due to system coordinates

The hysteresis dimer reaction of the first sequel is applied to test the Gibbs density-in-phase hypothesis for a canonical distribution at equilibrium. The probability distribution of variously defined internal and external variables is probed using the algorithms described, in particular the novel probing of the energy states of a labeled particle where it is found that there is compliance with the Gibbs’ hypothesis for the stated equilibrium condition and where the probability data strongly suggests that an extended equipartition principle may be formulated for some specific molecular coordinates, whose equipartition temperature does not equal the mean system temperature and a conjecture concerning which coordinates may be suitable is provided. Evidence of violations to the mesoscopic nonequilibrium thermodynamics (MNET) assumptions used without clear qualifications for a canonical distribution for internal variables are described, and possible reasons outlined, where it is found that the free dimer and atom particle kinetic energy distributions agree fully with Maxwell–Boltzmann statistics but the distribution for the relative kinetic energy of bonded atoms does not. The principle of local equilibrium (PLE) commonly used in nonequilibrium theories to model irreversible systems is investigated through NEMD simulation at extreme conditions of bond formation and breakup at the reservoir ends in the presence of a temperature gradient, where for this study a simple and novel difference equation algorithm to test the divergence theorem for mass conservation is utilized, where mass is found to be conserved from the algorithm in the presence of flux currents, in contradiction to at least one aspect of PLE in the linear domain. It is concluded therefore that this principle can be a good approximation at best, corroborating previous purely theoretical results derived from the generalized Clausius Inequality, which proved that the PLE cannot be an exact principle for nonequilibrium systems.      

Equilibrium free energy estimates based on nonequilibrium work relations and extended dynamics

Jarzynski's relation and the fluctuation theorem have established important connections between nonequilibrium statistical mechanics and equilibrium thermodynamics. In particular, an exact relationship between the equilibrium free energy and the nonequilibrium work is useful for computer simulations. In this paper, we exploit the fact that the free energy is a state function, independent of the pathway taken to change the equilibrium ensemble. We show that a generalized expression is advantageous for computer simulations of free energy differences. Several methods based on this idea are proposed. The accuracy and efficiency of the proposed methods are evaluated with a model problem.     

Surface Tension: Mechanics,Thermodynamics, and Relaxation Times

A microscopic analysis is presented of the existing definitions of equilibrium surface tension, which can be divided into two types: mechanical and thermodynamic. Each type of definition can be studied from the presentation below according to thermodynamic hypotheses or molecular calculations. An analysis of the planar interface is given and its generalization for curved (spherical) interfaces is considered. The distinction between approaches describing the surface tension of metastable and equilibrium droplets is discussed. Based on nonequilibrium thermodynamics, it is shown that the introduction of metastable droplets is due to a violation of the relationship between the times of impulse and chemical potential relaxation in condensed phases. Problems of calculating the surface tension in nonequilibrium situations are created.     

A differential fluctuation theorem

We derive a nonequilibrium thermodynamics identity (the "differential fluctuation theorem") that connects forward and reverse joint probabilities of nonequilibrium work and of arbitrary generalized coordinates corresponding to states of interest. This identity allows us to estimate the free energy difference between domains of these states. Our results follow from a general symmetry relation between averages over nonequilibrium forward and backward path functions derived by Crooks [Crooks, G. E. Phys. Rev. E 2000, 61, 2361-2366]. We show how several existing nonequilibrium thermodynamic identities can be obtained directly from the differential fluctuation theorem. We devise an approach for measuring conformational free energy differences, and we demonstrate its applicability to the analysis of molecular dynamics simulations by estimating the free energy difference between two conformers of the alanine dipeptide model system. We anticipate that these developments can be applied to the analysis of laboratory experiments.     

Equations for the evolution of a growing or evaporating free microdroplet under nonstationary conditions of diffusion and heat transfer in a multicomponent vapor–gas medium

A set of equations has been derived for the nonstationary composition, size, and temperature of a growing or evaporating multicomponent microdroplet of a nonideal solution under arbitrary initial conditions. Equations for local nonstationary diffusion molecular and heat fluxes in a mixture of a multicomponent vapor with a noncondensable carrier gas have been obtained within the framework of nonequilibrium thermodynamics with allowance for hydrodynamic flow of the medium. The derived closed set of equations takes into account the nonstationarity of the diffusion and heat transfer, effect of thermodiffusion and other cross effects in the multicomponent vapor–gas medium, the Stefan flow, and droplet boundary motion, as well as the nonideality of the solution in the droplet. The general approach has been illustrated by the consideration of the multicomponent medium at low concentrations of vapors taking into account its thermal expansion due to the Stefan flow in the case of a nonstationary diffusion regime of the nonisothermal condensation growth of a one-component droplet.     

Manifestation of nonequilibrium initial conditions in molecular rotation: the generalized J-diffusion model

In order to adequately describe molecular rotation far from equilibrium, we have generalized the J-diffusion model by allowing the rotational relaxation rate to be angular momentum dependent. The calculated nonequilibrium rotational correlation functions (CFs) are shown to decay much slower than their equilibrium counterparts, and orientational CFs of hot molecules exhibit coherent behavior, which persists for several rotational periods. As distinct from the results of standard theories, rotational and orientational CFs are found to dependent strongly on the nonequilibrium preparation of the molecular ensemble. We predict the Arrhenius energy dependence of rotational relaxation times and violation of the Hubbard relations for orientational relaxation times. The standard and generalized J-diffusion models are shown to be almost indistinguishable under equilibrium conditions. Far from equilibrium, their predictions may differ dramatically.     

On the calculation of pressure tensor in wedge-shaped cavity

A new simplified expression for the components of a pressure tensor in a wedge-shaped cavity of a solid with dispersion forces is derived. All four (three diagonal and one off-diagonal) components of the pressure tensor are calculated as functions of their positions inside the cavity. It is verified that these functions satisfy the conditions for mechanical equilibrium.