First-principles nonequilibrium deterministic equation of motion of a Brownian particle and microscopic viscous drag.

We present a first-principles thermodynamic approach to provide an alternative to the Langevin equation by identifying the deterministic (no stochastic component) microforce F_{k,BP} acting on a nonequilibrium Brownian particle (BP) in its kth microstate m_{k}. (The prefix "micro" refers to microstate quantities and carry a suffix k.) The deterministic new equation is easier to solve using basic calculus. Being oblivious to the second law, F_{k,BP} does not always oppose motion but viscous dissipation emerges upon ensemble averaging. The equipartition theorem is always satisfied. We reproduce well-known results of the BP in equilibrium. We explain how the microforce is obtained directly from the mutual potential energy of interaction beween the BP and the medium after we average it over the medium so we only have to consider the particles in the BP. Our approach goes beyond the phenomenological and equilibrium approach of Langevin and unifies nonequilibrium viscous dissipation from mesoscopic to macroscopic scales and provides new insight into Brownian motion beyond Langevin's and Einstein's formulation.

Nonequilibrium thermodynamics. II. Application to inhomogeneous systems.

We provide an extension of a recent approach to study nonequilibrium thermodynamics [Gujrati, Phys. Rev. E 81, 051130 (2010), to be denoted by I in this work] to inhomogeneous systems by considering the latter to be composed of quasi-independent subsystems. The system Σ along with the (macroscopically extremely large) medium Σ[over ̃] form an isolated system Σ0. The fields (temperature, pressure, etc.) of Σ and Σ[over ̃] differ unless at equilibrium. We show that the additivity of entropy requires quasi-independence of the subsystems, which results from the interaction energies between different subsystems being negligible so the energy also becomes additive. The thermodynamic potentials such as the Gibbs free energy that continuously decrease during approach to equilibrium are determined by the fields of the medium and exist no matter how far the subsystems are out of equilibrium, so their fields may not even exist. This and the requirement of quasi-independence make our approach differ from the conventional approach used by de Groot and others, as discussed in the text. We find it useful to introduce the time-dependent Gibbs statistical entropy for Σ0, from which we derive the Gibbs entropy of Σ; in equilibrium this entropy reduces to the equilibrium thermodynamic entropy. As the energy depends on the frame of reference, the thermodynamic potentials and the Gibbs fundamental relation, but not the entropy, depend on the frame of reference. The possibility of relative motion between subsystems described by their net linear and angular momenta gives rise to viscous dissipation. The concept of internal equilibrium introduced in I is developed further here and its important consequences are discussed for inhomogeneous systems. The concept of internal variables (various examples are given in the text) as variables that cannot be controlled by the observer for nonequilibrium evolution is also discussed. They are important because the concept of internal equilibrium in the presence of internal variables no longer holds if internal variables are not used. The Gibbs fundamental relation, thermodynamic potentials, and irreversible entropy generation are expressed in terms of observables and internal variables. We use these relations to eventually formulate the nonequilibrium thermodynamics of inhomogeneous systems. We also briefly discuss the case when bodies form an isolated system without any medium to obtain their irreversible contributions and show that this case does not differ from when bodies are in an extremely large medium.

Nonequilibrium thermodynamics. III. Generalization of Maxwell, Clausius-Clapeyron, and response-function relations, and the Prigogine-Defay ratio for systems in internal equilibrium.

We follow the consequences of internal equilibrium in nonequilibrium systems that has been introduced recently [Gujrati, Phys. Rev. E 81, 051130 (2010) and Gujrati, Phys. Rev. E 85, 041128 (2012).] to obtain the generalization of the Maxwell relation and the Clausius-Clapeyron relation that are normally given for equilibrium systems. The use of Jacobians allows for a more compact way to address the generalized Maxwell relations in the presence of internal variables. The Clausius-Clapeyron relation in the subspace of observables shows not only the nonequilibrium modification but also the modification due to internal variables that play a dominant role in glasses to which we apply the above relations. Real systems do not directly turn into glasses (GL) that are frozen structures from the supercooled liquid state L; there is an intermediate state (gL) where the internal variables are not frozen. A system possesses several kinds of glass transitions, some conventional (L→gL; gL→GL) in which the state changes continuously and the transition mimics a continuous or second-order transition, and some apparent (L→gL; L→GL) in which the free energies are discontinuous so that the transition appears as a zeroth-order transition, as discussed in the text. We evaluate the Prigogine-Defay ratio Π in the subspace of the observables at these transitions. We find that it is normally different from 1, except at the conventional transition L→gL, where Π=1.

Nonequilibrium thermodynamics: structural relaxation, fictive temperature, and Tool-Narayanaswamy phenomenology in glasses.

Starting from the second law of thermodynamics applied to an isolated system consisting of the system surrounded by an extremely large medium, we formulate a general nonequilibrium thermodynamic description of the system when it is out of thermal and mechanical equilibrium with the medium. Our approach allows us to identify the correct form of the Gibbs free energy and enthalpy. We also obtain an extension of the classical nonequilibrium thermodynamics due to de Donder in which one normally assumes thermal and mechanical equilibrium with the medium; see text. We find that the temperature and pressure differences between the system and the medium act as thermodynamic forces, which are normally neglected in the classical nonequilibrium thermodynamics. The Prigogine-Defay ratio is found to be greater than 1 merely due to the lack of equilibrium with the medium, even though we do not consider any internal order parameters. This shows that these forces should play an important role in relaxation processes. We then apply our approach to study the general trend during structural relaxation in glasses and establish the phenomenology behind the concept of the fictive temperature and of the empirical Tool-Narayanaswamy equation on firmer theoretical foundation.

Molecular cooperativity in the dynamics of glass-forming systems: a new insight.

The mechanism behind the steep slowing down of molecular motions upon approaching the glass transition remains a great puzzle. Most of the theories relate this mechanism to the cooperativity in molecular motion. In this work, we estimate the length scale of molecular cooperativity xi for many glass-forming systems from the collective vibrations (the so-called boson peak). The obtained values agree well with the dynamic heterogeneity length scale estimated using four-dimensional NMR. We demonstrate that xi directly correlates to the dependence of the structural relaxation on volume. This dependence presents only one part of the mechanism of slowing down the structural relaxation. Our analysis reveals that another part, the purely thermal variation in the structural relaxation (at constant volume), does not have a direct correlation with molecular cooperativity. These results call for a conceptually new approach to the analysis of the mechanism of the glass transition and to the role of molecular cooperativity.

Comment on "Entropy of polydisperse chains: solution on the Bethe lattice" [J. Chem. Phys. 128, 184904 (2008)].

The results presented in the above-mentioned recent paper by Neto and Stilck [J. Chem. Phys.128, 184904 (2008)] represent special cases of a more general investigation by Gujrati on recursive lattices and have already appeared either in this journal or elsewhere. Even the methodology adopted by these authors is almost identical to that of Gujrati. We show that their Eq. (27) remains valid even when interactions are present.

Compressible or incompressible blend of interacting monodisperse star and linear polymers near a surface.

We consider a lattice model of a mixture of repulsive, attractive, or neutral monodisperse star (species A) and linear (species B) polymers with a third monomeric species C, which may represent free volume. The mixture is next to a hard, infinite plate whose interactions with A and C can be attractive, repulsive, or neutral. These two interactions are the only parameters necessary to specify the effect of the surface on all three components. We numerically study monomer density profiles using the method of Gujrati and Chhajer that has already been previously applied to study polydisperse and monodisperse linear-linear blends next to surfaces. The resulting density profiles always show an enrichment of linear polymers in the immediate vicinity of the surface due to entropic repulsion of the star core. However, the integrated surface excess of star monomers is sometimes positive, indicating an overall enrichment of stars. This excess increases with the number of star arms only up to a certain critical number and decreases thereafter. The critical arm number increases with compressibility (bulk concentration of C). The method of Gujrati and Chhajer is computationally ultrafast and can be carried out on a personal computer (PC), even in the incompressible case, when simulations are unfeasible. Calculations of density profiles usually take less than 20 min on PCs.

Compressible or incompressible blend of interacting monodisperse linear polymers near a surface.

We consider a lattice model of a mixture of repulsive, attractive, or neutral monodisperse linear polymers of two species, A and B, with a third monomeric species C, which may be taken to represent free volume. The mixture is confined between two hard, parallel plates of variable separation whose interactions with A and C may be attractive, repulsive, or neutral, and may be different from each other. The interactions with A and C are all that are required to completely specify the effect of each surface on all three components. We numerically study various density profiles as we move away from the surface, by using the recursive method of Gujrati and Chhajer [J. Chem. Phys. 106, 5599 (1997)] that has already been previously applied to study polydisperse solutions and blends next to surfaces. The resulting density profiles show the oscillations that are seen in Monte Carlo simulations and the enrichment of the smaller species at a neutral surface. The method is computationally ultrafast and can be carried out on a personal computer (PC), even in the incompressible case, when Monte Carlo simulations are not feasible. The calculations of density profiles usually take less than 20 min on a PC.

Percolation of particles on recursive lattices using a nanoscale approach. I. Theoretical foundation at the atomic level.

Powdered materials of sizes ranging from nanometers to micrometers are widely used in materials science and are carefully selected to enhance the performance of a matrix. Fillers have been used in order to improve properties, such as mechanical, rheological, electrical, magnetic, thermal, etc., of the host material. Changes in the shape and size of the filler particles are known to affect and, in some cases, magnify such enhancement. This effect is usually associated with an increased probability of formation of a percolating cluster of filler particles in the matrix. In this series of papers, we will consider lattice models. Previous model calculations of percolation in polymeric systems generally did not take into account the possible difference between the size and shape of monomers and filler particles and usually neglected interactions or accounted for them in a crude fashion. In our approach, the original lattice is replaced by a recursive structure on which calculations are done exactly and interactions as well as size and shape disparities can be easily taken into account. Here, we introduce the recursive approach, describe how to derive the percolation threshold as a function of the various parameters of the problem, and apply the approach to the analysis of the effect of correlations among monodisperse particles on the percolation threshold of a system. In the second paper of the series, we tackle the issue of the effect of size and shape disparities of the particles on their percolation properties. In the last paper, we describe the effects due to the presence of a polymer matrix.

Percolation of particles on recursive lattices using a nanoscale approach. II. The effect of size and shape disparities.

The preparation of many composites requires the intermixing of several macromolecular fluids along with the addition of rigid filler particles. These fillers are usually polydisperse and there is extensive experimental evidence that their size and shape profoundly affect the properties of the resulting material. In particular, it is generally found that the percolation threshold decreases as the size disparity between the different particles present in a system increases, and that the threshold decreases with increasing aspect ratio of the particles. Here, a recursive approach that we have recently introduced is applied to the study of the percolation of particles of different sizes and shapes, without the presence of a polymer matrix, on a lattice in various phases including metastable states. In our approach, the original lattice is replaced by a recursive structure on which calculations are done exactly and interactions as well as size and shape disparities are easily taken into account. In the previous paper of this series, we introduced the recursive approach and showed how correlations among particles of the same size can affect percolation. Before considering the complete system made of particles of various sizes and shapes embedded in a polymer matrix, in the third paper of the series, we describe here the properties of systems made of particles without any matrix. The approach appears to be extremely successful since it is able to capture most of the important features observed in experiments.

Percolation of particles on recursive lattices using a nanoscale approach. III. Percolation of polydisperse particles in the presence of a polymer matrix.

We use a recently developed lattice model of polymers to study the percolation of particles of different sizes and shapes in the presence of a polymer matrix. The polymer is modeled as an infinitely long chain to simplify the calculation but we make it more realistic by considering it semiflexible. We study the effects of the stiffness of the polymer, the size disparity of the filler particles, their aspect ratio, and the interactions between fillers and polymer on the percolation properties of the system. The lattice model is solved exactly on a recursive square Husimi lattice, which is an approximation for a square lattice. The solution represents an approximate solution for a square lattice. Our results are able to reproduce most of the experimental findings that have been observed in the literature. In particular, we observe how an increase in the size disparity of the filler particles dispersed in the matrix as well as an increase of their aspect ratio decreases the percolation threshold.

Configurational entropy and its crisis in metastable states: ideal glass transition in a dimer model as a paragidm of a molecular glass.

We discuss the need for discretization to evaluate the configurational entropy in a general model. We also discuss the prescription using restricted partition function formalism to study the stationary limit of metastable states where a more stable equilibrium state exists. We introduce a lattice model of dimers as a paradigm of molecular fluid and study stationary metastability in it to investigate the root cause of glassy behavior. We demonstrate the existence of entropy crisis in metastable states, from which it follows that the entropy crisis is the root cause underlying the ideal glass transition in systems with particles of all sizes. The orientational interactions in the model control the nature of the liquid-liquid transition observed in recent years in molecular glasses.

Entropy crisis, ideal glass transition, and polymer melting: exact solution on a Husimi cactus.

We investigate an extension of the lattice model of melting of semiflexible polymers originally proposed by Flory. Along with a bending penalty epsilon, present in the original model and involving three sites of the lattice, we introduce an interaction energy epsilon (p), corresponding to the presence of a pair of parallel bonds and an interaction energy epsilon (h), associated with a hairpin turn. Both these new terms represent four-site interactions. The model is solved exactly on a Husimi cactus, which approximates a square lattice. We study the phase diagram of the system as a function of the energies. For a proper choice of the interaction energies, the model exhibits a first-order melting transition between a liquid and a crystalline phase at a temperature T(M). The continuation of the liquid phase below T(M) gives rise to a supercooled liquid, which turns continuously into a new low-temperature phase, called metastable liquid, at T(MC)

Significance of the free volume for metastability, spinodals, and the glassy state: an exact calculation in polymers.

A lattice model of semiflexible linear chains (with equilibrium polydispersity) containing free volume is solved exactly on a Husimi cactus. A metastable liquid (ML) is discovered to exist only at low temperatures and is distinct (and may be disjoint) from the supercooled liquid (SCL) that exists only at high temperatures. The free volume plays a significant role in that the spinodals of the ML and SCL merge and then disappear as the free volume is reduced. The Kauzmann temperature T(K) occurs in the ML without any singularity. At T(MC)>T(K), the ML specific heat has a peak. For infinitely long polymers, the peak height diverges and the free volume vanishes at T(MC), resulting in a continuous liquid-liquid transition. Contrary to the conventional wisdom, both T(K) and T(MC) occur in the ML and not in the SCL.

Importance of interactions for free-volume and end-group effects in polymers: an equilibrium lattice investigation.

We consider a lattice model of a polymer system in which we distinguish between the end (E) and the middle (M) groups. The free volume is represented as a "hole" or "void" (0), which constitutes a separate species in addition to the two "species" M and E. There are three different exchange interaction energies, and correspondingly three Boltzmann weights w(ij), i not equal j=0,E,M between different species. We define the free volume associated with the species j=M or E, as the average number of voids next to j. Using a recently developed equilibrium lattice theory, we calculate the free volume v(E) and v(M) associated with an end group and a middle group, respectively, and investigate the effects of interactions among them. Our calculations show that v(E) and v(M) are intricate functions of w(ij), the pressure and the molecular weight, and that their difference can change sign under certain conditions. These conditions are elucidated. We demonstrate that when the end group is chemically dissimilar from the middle group, the middle group may have more free volume than the end group. We find that the conditions that favor a middle group having more free volume over an end group are w(E0)<1, w(M0)<1, and w(ME)>1. The effect of pressure and molecular weight can be of either type and appears to be dependent on the interactions.

Entropy-driven phase separation and configurational correlations on a lattice: some rigorous results.

We prove that if there is a phase separation in a fully packed (FP) athermal system, it must be between pure components only. We then rigorously demonstrate that no phase separation in an athermal FP state of hard particle mixtures on a lattice is possible merely due to size disparity or nonadditivity, if the configurations are weakly correlated, i.e., are quasirandom. We consider a mixture of linear polymers at all packing fractions and argue that no phase separation is possible in an athermal state. The last result also applies to a mixture of flexible particles and hard dimers. Our results contradict many recent numerical results.