Cooling rate dependence of specific heat in systems out of equilibrium

The anomaly of specific heat in systems out of equilibrium, especially the measurement procedure dependence of specific heat, is investigated by means of free energy landscape. Introducing measurement procedure which is based on experimental method, we propose a calculation method of specific heat in systems out of equilibrium and find an abrupt change in specific heat between annealed and quenched states. For longer observation time the change in specific heat occurs at lower temperature and becomes sharper. For slower cooling of a system the transition temperature becomes lower. This cooling rate dependence of the transition temperature is consistent with experiments and thus the abrupt change in specific heat can be regarded as the glass transition which is thermally identified.     

Two-Gaussian excitations model for the glass transition

We develop a modified "two-state" model with Gaussian widths for the site energies of both ground and excited states, consistent with expectations for a disordered system. The thermodynamic properties of the system are analyzed in configuration space and found to bridge the gap between simple two-state models ("logarithmic" model in configuration space) and the random energy model ("Gaussian" model in configuration space). The Kauzmann singularity given by the random energy model remains for very fragile liquids but is suppressed or eliminated for stronger liquids. The sharp form of constant-volume heat capacity found by recent simulations for binary mixed Lennard-Jones and soft-sphere systems is reproduced by the model, as is the excess entropy and heat capacity of a variety of laboratory systems, strong and fragile. The ideal glass in all cases has a narrow Gaussian, almost invariant among molecular and atomic glassformers, while the excited-state Gaussian depends on the system and its width plays a role in the thermodynamic fragility. The model predicts the possibility of first-order phase transitions for fragile liquids. The analysis of laboratory data for toluene and o-terphenyl indicates that fragile liquids resolve the Kauzmann paradox by a first-order transition from supercooled liquid to ideal-glass state at a temperature between T(g) and Kauzmann temperature extrapolated from experimental data. We stress the importance of the temperature dependence of the energy landscape, predicted by the fluctuation-dissipation theorem, in analyzing the liquid thermodynamics.     

Size effects during phase transformations in stratifying systems

Size effects during phase transitions in binary stratifying mixtures with small systems are simulated by means of equilibrium chemical thermodynamics. Conditions are described under which stable and metastable thermodynamically equilibrium heterogeneous states exist. It is established that degeneracy is split in nanosize and submicron systems and the inversion of equilibrium and metastable states is possible. It is shown that size effects include a change and shift of the heterogeneity region in the phase diagram. It is concluded that in the limiting case, reducing the size of a system can homogenize a stratifying mixture of any composition.     

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.     

Phase equilibria in phospholipid-water systems

This work is a critical account of phase studies of phospholipid-water mixtures. The Phase Rule and equations of thermodynamics of heterogeneous systems are applied in the analysis of calorimetric and dilatometric results for these systems. It is inferred that the approach in which a lipid-water mixture of low lipid content is regarded as a one-component system is misleading and unhelpful. A mixture containing water and a pure synthetic phospholipid is a binary system, and the Phase Rule can be applied to it in a straightforward manner. Non-isothermal transitions and invariant three-phase reactions of the eutectic, peritectic and polytectic type can be encountered in lipid-water systems, like in other binary mixtures. The concentration dependence of the enthalpy of the isothermal three-phase transition yields a well-known Tammann triangle. For the volume change at such transition an analog of the Tammann triangle can be drafted which can be useful in interpretation of dilatometric and densitometric data for heterogeneous binary systems.The literature thermodynamic data for the aqueous lipids are analyzed on the basis of the phase diagram of the DPPC-water system. The measured values of the transition enthalpy and volume change at the pre- and main transitions refer to invariant phase reactions. The measured heat capacities and thermal coefficients of expansion actually characterize the biphasic lipid-water mixtures and may reflect phenomena specific for heterogeneous binary systems only. A reinterpretation of the literature DSC data for the DPPC-water system at temperatures below zero is suggested.Experience gained in the study of equilibrium phase diagram, kinetic and colloidal aspects of phase behaviour of dioctadecyldimethylammonium chloride-water system indicates that the common procedures of the sample preparation of lipid-water mixtures may lead to the states which are non-equilibrium in two respects: the system contains a metastable phase, and the phase is dispersed to such an extent that an irreversible colloidal structure is formed. Definitive phase studies of the phase behaviour of phospholipid-water systems in the whole concentration range are urgently needed.     

Statistical mechanics of time independent nondissipative nonequilibrium states

We examine the question of whether the formal expressions of equilibrium statistical mechanics can be applied to time independent nondissipative systems that are not in true thermodynamic equilibrium and are nonergodic. By assuming that the phase space may be divided into time independent, locally ergodic domains, we argue that within such domains the relative probabilities of microstates are given by the standard Boltzmann weights. In contrast to previous energy landscape treatments that have been developed specifically for the glass transition, we do not impose an a priori knowledge of the interdomain population distribution. Assuming that these domains are robust with respect to small changes in thermodynamic state variables we derive a variety of fluctuation formulas for these systems. We verify our theoretical results using molecular dynamics simulations on a model glass forming system. Nonequilibrium transient fluctuation relations are derived for the fluctuations resulting from a sudden finite change to the system's temperature or pressure and these are shown to be consistent with the simulation results. The necessary and sufficient conditions for these relations to be valid are that the domains are internally populated by Boltzmann statistics and that the domains are robust. The transient fluctuation relations thus provide an independent quantitative justification for the assumptions used in our statistical mechanical treatment of these systems.     

Heterogeneous equilibria and the kinetics of isotropic liquid-nematic phase transition in a multicomponent mixture 4-alkyl-4′-cyanobiphenyls

The isotropic liquid-nematic phase transition and the kinetics of growth of nematic phase drops during this phase transition were studied by polarization optical microscopy and IR spectroscopy for a four-component system during cooling. The statistical drop size distribution was described in terms of the equilibrium thermodynamics of irreversible processes. An analysis of the time dependences of the mean diameter of drops showed the presence of two kinetic stages of nematic phase growth and allowed them to be described by the universal law of cluster growth. In conformity with the Gibbs phase rule, nematic phases with different compositions can coexist at equilibrium.     

Reaction pathways of propene pyrolysis

The gas‐phase reaction pathways in preparing pyrolytic carbon with propene pyrolysis have been investigated in detail with a total number of 110 transition states and 50 intermediates. The structure of the species was determined with density functional theory at B3PW91/6‐311G(d,p) level. The transition states and their linked intermediates were confirmed with frequency and the intrinsic reaction coordinates analyses. The elementary reactions were explored in the pathways of both direct and the radical attacking decompositions. The energy barriers and the reaction energies were determined with accurate model chemistry method at G3(MP2) level after an examination of the nondynamic electronic correlations. The heat capacities and entropies were obtained with statistical thermodynamics. The Gibbs free energies at 298.15 K for all the reaction steps were reported. Those at any temperature can be developed with classical thermodynamics by using the fitted (as a function of temperature) heat capacities. It was found that the most favorable paths are mainly in the radical attacking chain reactions. The chain was proposed with 26 reaction steps including two steps of the initialization of the chain to produce H and CH₃ radicals. For a typical temperature (1200 K) adopted in the experiments, the highest energy barriers were found in the production of C₃ to be 203.4 and 193.7 kJ/mol. The highest energy barriers for the production of C₂ and C were found 174.1 and 181.4 kJ/mol, respectively. These results are comparable with the most recent experimental observation of the apparent activation energy 201.9 ± 0.6 or 137 ± 25 kJ/mol. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

On the definition of a Monte Carlo model for binary crystal growth

We show that consistency of the transition probabilities in a lattice Monte Carlo (MC) model for binary crystal growth with the thermodynamic properties of a system does not guarantee the MC simulations near equilibrium to be in agreement with the thermodynamic equilibrium phase diagram for that system. The deviations remain small for systems with small bond energies, but they can increase significantly for systems with large melting entropy, typical for molecular systems. These deviations are attributed to the surface kinetics, which is responsible for a metastable zone below the liquidus line where no growth occurs, even in the absence of a 2D nucleation barrier. Here we propose an extension of the MC model that introduces a freedom of choice in the transition probabilities while staying within the thermodynamic constraints. This freedom can be used to eliminate the discrepancy between the MC simulations and the thermodynamic equilibrium phase diagram. Agreement is achieved for that choice of the transition probabilities yielding the fastest decrease of the free energy (i.e., largest growth rate) of the system at a temperature slightly below the equilibrium temperature. An analytical model is developed, which reproduces quite well the MC results, enabling a straightforward determination of the optimal set of transition probabilities. Application of both the MC and analytical model to conditions well away from equilibrium, giving rise to kinetic phase diagrams, shows that the effect of kinetics on segregation is even stronger than that predicted by previous models.     

Heat capacity, enthalpy fluctuations, and configurational entropy in broken ergodic systems

A common assumption in the glass science community is that the entropy of a glass can be calculated by integration of measured heat capacity curves through the glass transition. Such integration assumes that glass is an equilibrium material and that the glass transition is a reversible process. However, as a nonequilibrium and nonergodic material, the equations from equilibrium thermodynamics are not directly applicable to the glassy state. Here we investigate the connection between heat capacity and configurational entropy in broken ergodic systems such as glass. We show that it is not possible, in general, to calculate the entropy of a glass from heat capacity curves alone, since additional information must be known related to the details of microscopic fluctuations. Our analysis demonstrates that a time-average formalism is essential to account correctly for the experimentally observed dependence of thermodynamic properties on observation time, e.g., in specific heat spectroscopy. This result serves as experimental and theoretical proof for the nonexistence of residual glass entropy at absolute zero temperature. Example measurements are shown for Corning code 7059 glass.     

A diffuse‐interface modeling for liquid solution–solid gel phase transition of physical hydrogel with nonlinear deformation

A diffuse‐interface model is presented in this paper for simulation of the evolution of phase transition between the liquid solution and solid gel states for physical hydrogel with nonlinear deformation. The present domain covers the gel and solution states as well as a diffuse interface between them. They are indicated by the crosslink density in such a way that the solution phase is identified as the state when the crosslink density is small, while the gel as the state if the crosslink density becomes large. In this work, a novel order parameter is thus defined as the crosslink density, which is homogeneous in each distinct phase and smoothly varies over the interface from one phase to another. In this model, the constitutive equations, imposed on the two distinct phases and the interface, are formulated by the second law of thermodynamics, which are in the same form as those derived by a different approach. The present constitutive equations include a novel Ginzburg–Landau type of free energy with a double‐well profile, which accounts for the effect of crosslink density. The present governing equations include the equilibrium of forces, the conservations of mass and energy, and an additional kinetic equation imposed for phase transition, in which nonlinear deformation is considered. The equilibrium state is investigated numerically, where two stable phases are observed in the free energy profile. As case studies, a spherically symmetrical solution‐gel phase transition is simulated numerically for analysis of the phase transition of physical hydrogel.     

The microcanonical thermodynamics of finite systems: the microscopic origin of condensation and phase separations, and the conditions for heat flow from lower to higher temperatures

Microcanonical thermodynamics [D. H. E. Gross, Microcanonical Thermodynamics, Phase Transitions in "Small" Systems (World Scientific, Singapore, 2001)] allows the application of statistical mechanics both to finite and even small systems and also to the largest, self-gravitating ones. However, one must reconsider the fundamental principles of statistical mechanics especially its key quantity, entropy. Whereas in conventional thermostatistics, the homogeneity and extensivity of the system and the concavity of its entropy are central conditions, these fail for the systems considered here. For example, at phase separation, the entropy S(E) is necessarily convex to make e(S(E)-E/T) bimodal in E. Particularly, as inhomogeneities and surface effects cannot be scaled away, one must be careful with the standard arguments of splitting a system into two subsystems, or bringing two systems into thermal contact with energy or particle exchange. Not only the volume part of the entropy must be considered; the addition of any other external constraint [A. Wehrl, Rev. Mod. Phys. 50, 221 (1978)], such as a dividing surface, or the enforcement of gradients of the energy or particle profile, reduce the entropy. As will be shown here, when removing such constraints in regions of a negative heat capacity, the system may even relax under a flow of heat (energy) against a temperature slope. Thus the Clausius formulation of the second law: "Heat always flows from hot to cold," can be violated. Temperature is not a necessary or fundamental control parameter of thermostatistics. However, the second law is still satisfied and the total Boltzmann entropy increases. In the final sections of this paper, the general microscopic mechanism leading to condensation and to the convexity of the microcanonical entropy at phase separation is sketched. Also the microscopic conditions for the existence (or nonexistence) of a critical end point of the phase separation are discussed. This is explained for the liquid-gas and the solid-liquid transition.     

Estimating the temperature dependence of peptide folding entropies and free enthalpies from total energies in molecular dynamics simulations

The temperature dependence of thermodynamic quantities, such as heat capacity, entropy and free enthalpy, may be obtained by using well-known equations that relate these quantities to the enthalpy of the molecular system of interest at a range of temperatures. In turn, the enthalpy of a molecular system can be estimated from molecular dynamics simulations of an appropriate model. To demonstrate this, we have investigated the temperature dependence of the enthalpy, heat capacity, entropy and free enthalpy of a system that consists of a beta-heptapeptide in methanol and have used the statistical mechanics relationships to describe the thermodynamics of the folding/unfolding equilibrium of the peptide. The results illustrate the power of current molecular simulation force fields and techniques in establishing the link between thermodynamic quantities and conformational distributions.     

硝酸盐型卤水体系的综合热力学模型与多温相图预测

硝酸盐型卤水是盐湖卤水、 硝酸盐工业、 废水处理中普遍遇到的电解质溶液体系. 硝酸盐具有极高的溶解度, 实现硝酸盐型复杂电解质体系物性和相平衡的精准热力学表达依然具有挑战性. 以煤化工废水的典型体系Na⁺//NO₃^- , Cl^-, SO₄^{2 -} -H₂O为对象, 以改进的eNRTL模型为基础, 由活度系数模型、 溶液物性模型、 物种热力学模型和固液相平衡模型构成了电解质体系的综合热力学模型. 利用二元体系的冰点、 饱和蒸汽压、 等压摩尔热容、 活度系数和渗透压系数等物性数据和三元体系等温相平衡数据, 采用多目标优化方法, 获得了表达研究体系的多温特性的12组液相特征参数和7个固相物种的热力学参数. 据此完成了3个二元体系、 3个三元体系等温相平衡的准确计算和三元、 四元完整相图的预测, 适用温度达到实验所及的全部温度范围(254.65~543.15 K); 适用浓度达到饱和程度, 其中NaNO₃的浓度高达226.88 mol/kg. 三元、 四元体系的多温相图预测结果与实验数据相吻合, 并给出了9个三元、 5个四元体系零变点的完整信息.     

Thermal treatment and analysis

Heating and/or cooling of substances is one of the oldest and basic methods for preparing materials with defined properties. This always leaves a definitive fingerprint of the thermal history. Beside knowing the structure we need to specify such materials by their thermodynamic behaviour, i.e., stability/metastability, phase relations and transitions, particularly establishing corresponding characteristic points. All this can be based on ordinary thermodynamics but its validity must be approved for non-equilibrium conditions of temperature changes where equilibrium and kinetic effects overlap. The slower the phase transition proceeds the greater is the deviation of the system state (kinetic curve) from its equilibrium state (equilibrium background). This makes possible to locate the actual phase boundary between two states investigated, resulting in the so-called kinetic phase diagrams. Most of modern technologies are intentionally based on non-equilibrium phenomena in order to create metastable/nonstable phases of specific properties. In this sense thermal analysis is understand as the method for determining the sample state on the basis of the sample interactions with the surroundings whose intensive parameters are controlled. Temperature is here considered as a basic parameter that connects all thermophysical measurements and thermal treatments. ICTA-TA Award lecture     

支化高分子溶胶-凝胶相变的热力学

以典型的Aa-Bb型缩聚反应为例, 应用统计力学和热力学的基本原理对反应体系的一些平衡特征进行研究. 基于从两种不同角度所构造的正则配分函数, 导出反应体系的平衡自由能以及质量作用定律的解析形式, 同时指出获得数量分布函数的新方法, 并通过计算反应体系的等温压缩系数从而得到反应体系的凝胶化条件. 进一步利用数量分布函数的不变性, 给出临界点后溶胶相和凝胶相的平衡自由能, 探讨了溶胶-凝胶相变的相关问题.     

A novel method for evaluating the critical nucleus and the surface tension in systems with first order phase transition

We introduce a novel method for calculating the size of the critical nucleus and the value of the surface tension in systems with first order phase transition. The method is based on classical nucleation theory, and it consists in studying the thermodynamics of a sphere of given radius embedded in a frozen metastable surrounding. The frozen configuration creates a pinning field on the surface of the free sphere. The pinning field forces the sphere to stay in the metastable phase as long as its size is smaller than the critical nucleus. We test our method in two first order systems, both on a two-dimensional lattice: a system where the parameter tuning the transition is the magnetic field, and a second system where the tuning parameter is the temperature. In both cases the results are satisfying. Unlike previous techniques, our method does not require an infinite volume limit to compute the surface tension, and it therefore gives reliable estimates even by using relatively small systems. However, our method cannot be used at, or close to, the critical point, i.e., at coexistence, where the critical nucleus becomes infinitely large.