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The well-known Sauer-Freise flux equation is derived analytically for the general case of a multiphase, multicomponent diffusion couple with variable molar volume. Discontinuities in concentration versus distance data are specifically treated. Fluxes with respect to the local center of volume are employed if the partial molar volumes are constants. Fluxes with respect to the local center of moles are employed for the more general case of variable partial molar volumes.

We modify a previous steady-state description developed by Genin [J. Appl. Phys. 77, 5130–5137 (1995)] for a grain boundary groove moving with a prescribed speed in a material subject to in-plane stress and a resultant grain boundary flux. The arbitrary assumption that the grain boundary flux is equally delivered to (or extracted from) the two adjacent free surfaces of the grains is replaced by a condition that requires continuity of surface chemical potentials at the grain boundary. Analytical results for the small-slope approximation as well as nonlinear results for large slopes are computed numerically for steady-state motion at a specified groove speed. We apply these results to a “partial loop” grain boundary geometry that moves by mean curvature induced by the groove conditions. In contrast to the ordinary effect that a grain boundary surface groove retards grain boundary motion, the presence of a compressive stress and resultant grain boundary flux toward the free surface can promote grain boundary motion.

We develop the irreversible thermodynamic basis of the phase field model, which is a mesoscopic diffuse interface model that eliminates interface tracking during phase transformations. The phase field is an auxiliary parameter that identifies the phase; it is continuous but makes a transition over a thin region, the diffuse interface, from its constant value in a growing phase to some other value in the nutrient phase. All phases are treated thermodynamically as viscous liquids, even crystalline solids. Phases are assumed to be isotropic for simplicity with reference to works that include anisotropy. The basis is an entropy functional which is an integral of an entropy density that includes non-classical gradient entropies. Equilibrium is investigated to identify a non-classical temperature and non-classical chemical potentials for a multicomponent system that are uniform at equilibrium in the absence of external forces. Coupled partial differential equations that govern the time evolution of the phase fi...

We present a rigorous irreversible thermodynamics treatment of creep deformation of solid materials with interfaces described as geometric surfaces capable of vacancy generation and absorption and moving under the influence of local thermodynamic forces. The free energy dissipation rate derived in this work permits clear identification of thermodynamic driving forces for all stages of the creep process and formulation of kinetic equations of creep deformation and microstructure evolution. The theory incorporates capillary effects and reveals the different roles played by the interface free energy and interface stress. To describe the interaction of grain boundaries with stresses, we classify grain boundaries into coherent, incoherent and semicoherent, depending on their mechanical response to the stress. To prepare for future applications, we specialize the general equations to a particular case of a linear-elastic solid with a small concentration of vacancies. The proposed theory creates a thermodynamic framework for addressing more complex cases, such as creep in multicomponent alloys and cross-effects among vacancy generation/absorption and grain boundary motion and sliding.

We present a numerical approach to modeling the deformation induced by the Kirkendall effect in binary alloys. The governing equations for isothermal binary diffusion are formulated with respect to inert markers and also with respect to the volume-averaged velocity. Relations necessary to convert between the two formulations are derived. Whereas the marker formulation is the natural one in which to pose constitutive laws, the volume formulation provides certain computational advantages. We therefore compute the diffusion and deformation with respect to the volume-centered velocity and then determine the corresponding fields with respect to the markers. Several problems involving one-dimensional (1-D) diffusion couples are solved for verification, and a problem involving two-dimensional (2-D) diffusion in a lap joint is solved to illustrate the power of the method in a more complex geometry.

We conduct a systematic study of the effect of various boundary conditions (bounce back and three versions of diffuse reflection) for the two-dimensional first-order upwind finite difference Lattice Boltzmann model. Simulation of Couette flow in a micro-channel using the diffuse reflection boundary condition reveals the existence of a slip velocity that depends on the Knudsen number @e=@l/L, where @l is the mean free path and L is the channel width. For walls moving in opposite directions with speeds +/-u"w, the slip velocity satisfies u"s"l"i"p=2@eu"w"a"l"l/(1+2@e). In the case of Poiseuille flow in a micro-channel, the slip velocity is found to depend on the lattice spacing @ds and Knudsen number @e to both first and second order. The best results are obtained for diffuse reflection boundary conditions that allow thermal mixing at a wall located at half lattice spacing outside the boundary nodes.

Diffusion equations are derived for an isothermal lattice Boltzmann model with two components. The first-order upwind finite difference scheme is used to solve the evolution equations for the distribution functions. When using this scheme, the numerical diffusivity, which is a spurious diffusivity in addition to the physical diffusivity, is proportional to the lattice spacing and significantly exceeds the physical value of the diffusivity if the number of lattice nodes per unit length is too small. Flux limiter schemes are introduced to overcome this problem. Empirical analysis of the results of flux limiter schemes shows that the numerical diffusivity is very small and depends quadratically on the lattice spacing.

Since the death of Prof. Dr. Jan Czochralski nearly 50 years ago, crystals grown by the Czochralski method have increased remarkably in size and perfection, resulting today in the industrial production of silicon crystals about 30 cm in diameter and two meters in length. The Czochralski method is of great technological and economic importance for semiconductors and optical crystals. Over this same time period, there have been equally dramatic improvements in our theoretical understanding of crystal growth morphology. Today we can compute complex crystal growth shapes from robust models that reproduce most of the features and phenomena observed experimentally. We should care about this because it is likely to result in the development of powerful and economical design tools to enable future progress. Crystal growth morphology results from an interplay of crystallographic anisotropy and growth kinetics by means of interfacial processes and long-range transport. The equilibrium shape of a crystal results from minimizing its anisotropic surface free energy under the constraint of constant volume; it is given by the classical Wulff construction but can also be represented by an analytical formula based on the ξ-vector formalism of Hoffman and Cahn. We now have analytic criteria for missing orientations (sharp corners or edges) on the equilibrium shape, both in two (classical) and three (new) dimensions. Crystals that grow under the control of interfacial kinetic processes tend asymptotically toward a “kinetic Wulff shape”, the analogue of the Wulff shape, except it is based on the anisotropic interfacial kinetic coefficient. If it were not for long range transport, crystals would presumably nucleate with their equilibrium shape and then evolve toward their “kinetic Wulff shape”. Allowing for long range transport leads to morphological instabilities on the scale of the geometric mean of a transport length (typically a diffusivity divided by the growth speed) and a capillary length (of the order of atomic dimensions). Resulting crystal growth shapes can be cellular or dendritic, but can also exhibit corners and facets related to the underlying crystallographic anisotropy. Within the last decade, powerful phase field models, based on a diffuse interface, have been used to treat simultaneously all of the above phenomena. Computed morphologies can exhibit cells, dendrites and facets, and the geometry of isotherms and isoconcentrates can also be determined. Results of such computations are illustrated in both two and three dimensions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A model is used to describe the shape change of a binary diffusion couple when the diffusivities of the two species differ. The classical uniaxial Kirkendall shift is obtained only if the displacement is constrained to be in the diffusion direction. For traction-free conditions at the external surfaces of a diffusion couple, a more general displacement field is obtained that accounts for the lateral shape change data of Voigt and Ruth [Journal of Physics-Condensed Matter 7 (1995) 2655–2666]. The model employs an isotropic stress-free strain rate and equal and constant partial molar volumes. In this case the displacement field is shown to be independent of the various elastic/plastic moduli. Depending on the lateral dimension of the diffusion couple, the displacement in the diffusion direction can be reduced by up to a factor of three compared to the case of a pure uniaxial displacement.

The equilibrium shape of a crystal is the shape that minimizes its anisotropic interfacial free energy subject to the constraint of constant volume. This shape can be determined geometrically by using the Wulff construction; it can have missing orientations, sharp edges and corners, and its faces can be rounded or flat (facets). In two dimensions, when the surface free energy, γ , is a function of a single angle θ , the analytical criterion for the onset of missing orientations is that γ + γ θ θ changes sign from positive to negative. In three dimensions, for which γ depends on two angles, θ and ϕ , no such analytical criterion is known. A geometrical criterion for missing orientations can be based on the Herring sphere construction. By means of inversion through the origin, an equivalent criterion becomes whether any portion of a polar plot of the reciprocal of γ ( 1 / γ -plot) lies outside a tangent plane. The onset of missing orientations occurs when the 1 / γ -plot changes from convex to concave. An equivalent analytical criterion is obtained by showing that the normal to the 1 / γ -plot is proportional to the ξ vector of Hoffman and Cahn and then using an invariant representation of the Gaussian curvature of the 1 / γ -plot to test its convexity. Results are illustrated for cubic symmetry. As the magnitude of anisotropy is increased, the equilibrium shape loses orientations and tends to an octahedron (positive anisotropy) or a cube (negative anisotropy). A similar formalism can be used to find an analytical criterion for missing orientations on growth shapes of crystals that are growing under the control of anisotropic interface kinetics.

We examine Gibbs’ conditions for equilibrium of a non-hydrostatically stressed single component solid in equilibrium across one of its faces with a pure liquid at pressure p F . We show that the equilibrium melting temperature T N for the non-hydrostatically stressed solid in contact with a melt at pressure p F is below the equilibrium melting temperature T H of the hydrostatically stressed solid at p F . Furthermore, for small strain and linear isotropic elasticity, the deviation, T H − T N , is shown to be quadratic in the differences between the principal values of the stress tensor and − p F . The result depends on both the bulk modulus and the shear modulus of the solid. Even for stresses as large as a typical yield stress, T H − T N is equal to 1 K or less. Nevertheless, the liquid in equilibrium with this non-hydrostatically stressed solid is always unstable with respect to the formation of hydrostatic solid.

Over the last 50 years, there has been tremendous progress in the quantification of crystal growth morphology. In the 1950s, the dynamics of crystal growth from the melt was based on the sharp interface model (interface of zero thickness separating solid and liquid), often under the assumption of isotropy. Ivantsov had discovered analytical solutions to the Stefan problem for the special class of shapes known as quadric surfaces (ellipsoids, hyperboloids and paraboloids, including their special cases spheres, cylinders and planes). But in the 1960s, these solutions were shown to be morphologically unstable, resulting in cellular and dendritic growth forms that had long been known to exist from experimental work. Sharp interface models were used to model these growth forms, but it was necessary to include corrections of the interface temperature for capillarity and curvature (Gibbs–Thomson equation) in order to avoid instabilities at all wavelengths and to set the size scale of the resulting morphologies. Except for the case of total interface control, for which exact solutions even for facetted crystals had been provided by Frank using the method of characteristics, little could be done analytically to treat anisotropies. By the 1980s, our reliance on the sharp interface model began to change with the adaptation by Langer and others of diffuse interface models, of the Cahn–Hilliard type, to solve dynamical problems. This class of models, now known as phase field models, replaced the sharp interface model by the solution in the entire computational domain of coupled partial differential equations for thermal and compositional fields and for an auxiliary variable that keeps track of the phase. Moreover, the phase field equations incorporate automatically the Gibbs Thomson equation, anisotropy and even departures from local equilibrium (interface kinetics) asymptotically for a sufficiently thin diffuse interface. But it has been only in about the last decade that massive improvements in computing power have rendered the numerical solution of the phase field model tractable. By means of this model, complex morphologies and related phenomena over a vast range of length scales can now be studied.

We reexamine similarity solutions for composition in a very long binary diffusion couple for the case in which the diffusivity and the density are functions of composition. For such solutions, the composition depends for sufficiently short times only on a similarity variable x/ t where x is distance and t is time. The classical Boltzmann–Matano treatment holds for the case in which the diffusivity is a function of composition but the density is independent of composition. It results in the selection of a unique (Matano) interface as the origin of coordinates for x and a formula for the diffusivity that depends on integrals of the concentration profile measured with respect to that interface. For density dependent on composition, Sauer and Freise generalized this solution by introduction of two Matano interfaces, one of which is at rest with respect to the left end of the diffusion couple and the other which is at rest with respect to its right end. Wagner reworked this generalization in terms of a unique (Wagner–Matano) interface and went on to derive a formula for the diffusivity that was indepenent of the location of that interface. We reconcile these treatments by examination of a long but finite diffusion couple with careful identification of reference frames for the fundamental description of diffusion.

Facet formation during crystal growth is simulated by using the phase field model in two dimensions. Instead of moderate anisotropy of the often-used form 1 þ d cos 4y; several functions having strong anisotropy are explored. For simplicity, the interfacial energy is assumed to be isotropic, so only the anisotropy in the kinetic coefficient is considered. This results in the formation of a nearly flat face when the anisotropy function has a narrow minimum at a certain direction, for example 45 � for four-fold symmetry. Two types of functions are studied in this paper; Type 1: q1ðy Þ¼ 1 þ d � 2dð1 � cos4yÞ n =2 n ; and Type 2: q2ðy Þ¼ 1 � d þ 2dtanhðk=jtan2yjÞ: A ‘‘facet’’ is formed at the 45 � direction for each case. This ‘‘facet’’ is not completely flat for q1; but a real facet is obtained for q2: The crystal shapes depend on the parameters d; n and k in the anisotropy functions. A wider facet is formed for larger d for both q1 and q2; whereas, larger values of n in q1 and k in q2 lead to more pronounced facets. Results obtained by using the phase field model are in good agreement with Wulff shapes for the kinetic coefficient. Finally, corner formation is simulated by using similar anisotropy functions with maxima at 45 �

Two-dimensional finite difference lattice Boltzmann models for single-component fluids are discussed and the corresponding macroscopic equations for mass and momentum conservation are derived by performing a Chapman-Enskog expansion. In order to recover the correct mass equation, characteristic-based finite difference schemes should be associated with the forward Euler scheme for the time derivative, while the space centered and second-order upwind schemes should be associated to second-order schemes for the time derivative. In the incompressible limit, the characteristic based schemes lead to spurious numerical contributions to the apparent value of the kinematic viscosity in addition to the physical value that enters the Navier-Stokes equation. Formulae for these spurious numerical viscosities are in agreement with results of simulations for the decay of shear waves.

The equilibrium criterion of maximum entropy for an isolated system is used to derive the equivalent criterion of minimum internal energy at constant entropy. Alternative equilibrium criteria for chemically closed systems are derived for other conditions and thermodynamic potentials: minimum Helmholtz free energy for constant temperature and no external work; minimum enthalpy for constant pressure or minimum Gibbs free energy for constant temperature and pressure, both with no external work in excess of that against the external pressure. For an open system at constant temperature, constant chemical potentials, and no external work, the Kramers potential is a minimum at equilibrium. According to any of these criteria, the conditions for mutual equilibrium of heterogeneous systems are uniformity of temperature, pressure, and chemical potentials of each chemical component. We also derive the Gibbs phase rule that bounds the number of macroscopic degrees of freedom, depending on the number of phases in mutual equilibrium.

Below a critical temperature, occupation of the ground state of a Bose gas becomes comparable to occupation of all excited states. This Bose condensation increases with decreasing temperature and affects thermodynamic functions. Only particles in excited states contribute to the pressure, internal energy, and entropy. Pressure remains equal to two-thirds of the energy density and becomes independent of molar volume. Heat capacity per particle is zero at zero temperature and rises to a sharp maximum at the critical temperature; with further increase of temperature it decreases to the constant value of a classical ideal gas. Its graph somewhat resembles the Greek letter lambda. A similar behavior occurs in helium with mass number four at its so-called lambda-point, although helium is not ideal because its atoms attract. We explore condensate regions that are bounded by an isentrope in the volume-temperature and volume-pressure planes.

The first law of thermodynamics is stated in terms of the existence of an extensive function of state called the internal energy. For a chemically closed system, the internal energy changes when energy is added by heat transfer or work is done by the system. Heat and work are not state variables because they depend on a process. Reversible quasistatic work can be done by a system by using pressure to change its volume very slowly. Heat capacities are defined as the amount of energy needed to cause temperature change at constant volume or pressure. Processes are illustrated for an ideal gas whose energy depends only on temperature. Sudden volume changes can result in irreversible work during which pressure is undefined. We define an auxiliary state function known as enthalpy to relate to processes at constant pressure. Phase transformations such as melting involve enthalpy changes that liberate latent heat.

Cooperative phenomena are introduced via the simple Ising model in which spins having two states occupy a lattice and interact with nearest neighbors and an applied magnetic field. We study this model in the mean field approximation. Correlations among spin states are neglected, so each spin interacts with a self-consistent mean field. With no applied magnetic field, the model predicts ordering of spins below some critical temperature for lattices of all dimensionalities, 1, 2, 3, …, and enables properties such as heat capacity and magnetic susceptibility to be calculated. Exact solutions for a one-dimensional lattice show no ordering transition; the mean field model fails badly in that case but otherwise shows reasonable trends. Exact solutions exist in two dimensions and show ordering. Better approximate solutions (Boethe cluster model) or numerical solutions can be obtained for lattices of all dimensionalities. We introduce Monte Carlo simulation for numerical solution of the Ising model as well as for models involving interacting classical particles.

The theory of morphological stability provides a dynamical analysis of the stability of the interface that separates phases during a phase transformation. We focus on crystallization from either a pure or alloy melt. One solves the governing equations for heat flow, including diffusion for alloys, and uses perturbation theory to analyze the stability of a base state, such as a planar or spherical interface. A linear stability analysis for small perturbations shows for a pure melt that thermal effects are destabilizing if more of the latent heat flows into supercooled liquid than into the solid, but capillary effects of interfacial free energy are stabilizing. Stability depends on competition of heat flow and capillarity effects. For alloys growing into a positive liquid temperature gradient, the effect of solute is destabilizing and competes with thermal and capillary effects to determine stability. This is the dynamical replacement for the principle of constitutional supercooling. Instability first occurs for perturbations of a given wavelength, but neighboring wavelengths become unstable for more severe conditions of instability. A nonlinear analysis shows that nonplanar periodic steady-state interfaces can sometimes be stable for a range of wavelengths. This can lead to cellular interfaces that can be computed numerically for even larger amplitudes. A number of other extensions of the Mullins-Sekerka analysis are discussed briefly.

Chemical reactions entail making or breaking of bonds, so energy is conserved for an isolated system. Reactions at constant volume or pressure exchange heat with the environment by change of internal energy or enthalpy, respectively. Reaction extent is measured by a progress variable; reactions progress until equilibrium is reached or some component is depleted. We define standard states of components and heats of formation of compounds. Affinity is defined as the decrease of Gibbs free energy per unit progress variable; its sign determines the direction of the reaction such that entropy is produced. Change of enthalpy per unit progress variable determines whether the reaction is endothermic or exothermic. At equilibrium the affinity is zero. Equilibrium conditions are expressed by equating a function of temperature and pressure called the “equilibrium constant” to a reaction product that depends on activities and fugacities of chemical components. Special cases include reaction products that can be approximated in terms of partial pressures of ideal gases.

Since the 1800s and the work of Clausius and Boltzmann, it was believed that the entropy function, which can only increase for an isolated system, was a measure of a state of greater probability, a more disordered state in which information is lacking. In 1948, Shannon developed a quantitative measure of information in the context of communication theory. Shannon’s measure is a function of an abstract set of probabilities and provides a quantitative measure of disorder. It is maximum when all probabilities are the same, in which case it becomes equal to Boltzmann’s formula for the entropy within a multiplicative constant. This provides us with a modern basis for the microcanonical ensemble in the next chapter. We give a demonstration of Boltzmann’s Eta theorem for an ideal gas based on a statistical analysis of elastic collisions of hard spheres. Boltzmann’s Eta function decreases as time increases. Its negative is the dynamical equivalent of Shannon’s measure of disorder.

We give a unified treatment of ideal Fermi, Bose, and classical gases for temperatures sufficiently large that energy levels can be treated as a quasi-continuous. Sums can be converted to integrals over a density of quantum states to evaluate thermodynamic functions. Pressure is equal to two-thirds of the energy density for all three gases. Relevant integrals can be represented by series expansions if the absolute activity is less than unity, which is always the case for bosons. For fermions, larger values of the absolute activity can be handled by an asymptotic expansion. Virial expansions for the pressure of these ideal gases are power series in the ratio of the actual concentration to the quantum concentration. For absolute activity less than unity, the deviation from ideal gas behavior is practically linear in that ratio, less pressure for bosons, and more for fermions. Formulae for the heat capacity of these gases at constant volume are calculated in terms of several integrals.

Even at absolute zero, the Pauli exclusion principle forces fermions into high energy states, a degenerate gas. States fill to the Fermi energy, equivalent to about 50,000 K for a free electron gas. At laboratory temperatures, small excitation into higher energy states is calculated by using an asymptotic Sommerfeld expansion. Heat capacity is linear in temperature and typically 100 times smaller than for a classical gas. A magnetic field can split spin states, resulting in weak Pauli paramagnetism; its effect on orbits causes Landau diamagnetism. Heating enables electron escape by thermionic emission, also affected by electric fields and radiation. Semiconductors have densities of states with a forbidden energy band. Electrons in intrinsic semiconductors can be thermally excited to a conduction band above a band gap leaving empty states called holes in the valence band. This results in electrical conductivity that can be enhanced by dopants called donors and acceptors that provide states that are easier to excite.

The grand canonical ensemble applies to a system at constant temperature and chemical potential; its number of particles is not fixed. We derive it from the microcanonical ensemble by contact with heat and particle reservoirs to form an isolated system. The probability of a system having a specified number of particles and being in a given stationary quantum state is proportional to its Gibbs factor, the product of a Boltzmann factor and a factor exponential in the number of particles. Summing all Gibbs factors gives the grand partition function that relates to the Kramers potential. We calculate dispersion of particle number and energy. The grand partition function factors for independent subsystems, dilute sites, and ideal Fermi and Bose gases whose distribution functions are derived. We treat a classical ideal gas with internal nuclear and electronic structure and molecules that can rotate and vibrate. A pressure ensemble is derived and used to treat point defects in crystals.

The canonical ensemble applies to a system held at constant temperature. We present two derivations based on the microcanonical ensemble by putting a system of interest in contact with a heat reservoir to form an isolated system. A third derivation employs the most probable distribution of ensemble members. The probability of a system being in a given stationary quantum state is proportional to its Boltzmann factor. We calculate dispersion of energy relative to its average. The sum of Boltzmann factors gives a system partition function that relates to the Helmholtz free energy. For a system composed of independent but distinguishable subsystems with negligible interaction energies, the system partition function factors. For such subsystems of identical particles, we recover the simplified ensemble of the preceding chapter. We treat an ideal gas and explore its Maxwell-Boltzmann distribution of velocities. Paramagnetism is treated both classically and quantum mechanically and compared. The partition function is related to the density of states by a Laplace transform.

Phase equilibria for a monocomponent system require uniformity of temperature, pressure, and chemical potential. In the temperature-pressure plane, single-phase regions are separated from one another by two-phase coexistence curves that meet at the triple point where all three phases, crystalline solid, liquid, and vapor, are in mutual equilibrium. The Clausius-Clapeyron differential equation depends on the ratio of enthalpy change to volume change and describes the coexistence curves that can be approximated for ideal vapors. The solid-vapor coexistence curve ends at a critical point; at larger pressures or temperatures there is no distinction between these phases. The chemical potential is continuous at the coexistence curves but its slope versus temperature or pressure is discontinuous. We develop equations for the thermodynamic functions and sketch them versus temperature and pressure. Finally we discuss phase equilibria in the volume-pressure plane where two phases in equilibrium are separated by a miscibility gap in volume.

Classical many-particle systems are governed by continuous variables, the positions and momenta of all particles in multi-dimensional phase space. Total energy depends on these variables and is called the Hamiltonian. Hamilton’s equations govern dynamics. According to Liouville’s theorem, the time rate of change of the density of a given set of particles in phase space is independent of time. For a system in equilibrium, this will be true if the density depends only on the Hamiltonian. The classical microcanonical ensemble is obtained by assuming that this density is uniform in the volume of phase space available to the system for a narrow band of energies; it plays the same role as the assumption of equal probability of microstates for the quantum ensemble. The entropy is calculated within an additive constant by assuming it to be proportional to the logarithm of available phase space. We illustrate this ensemble for an ideal gas and three-dimensional harmonic oscillators.

Open systems exchange particles with their environment in addition to work and heat. This exchange entails energy transfer. Internal energy becomes a function of entropy, volume, and moles of particles; its partial derivative with particle mole number is called chemical potential. This is extended to multicomponent systems. The chemical potential of an ideal gas depends on temperature and the logarithm of pressure, with fugacity replacing pressure for real gases. Maxwell relations result by equating mixed partial derivatives and relate measurable physical quantities. Euler’s theorem of homogeneous functions formalizes relationships of extensive and intensive variables, allows integration of fundamental differentials (Euler equation), and connects differentials of intensive variables (Gibbs-Duhem equation). Mole fractions define composition of multicomponent systems. Legendre transformations are developed and used to define new potentials such as Helmholtz and Gibbs free energies. Partial molar quantities are calculated by the method of intercepts. Entropy of a chemical reaction is introduced.

Surfaces or interfaces of discontinuity where phases meet are modeled by a Gibbs dividing surface of zero thickness. The differences between extensive variables of an actual system and one in which phases are uniform up to the dividing surface are defined to be surface excess quantities that depend on location of the dividing surface. The excess Kramers potential divided by surface area is independent of location and called the surface free energy or surface tension. The Gibbs adsorption equation governs segregation of surface components. The Cahn layer model is used to represent physically meaningful surface excess quantities by determinants. Curved interfaces can exert forces that cause pressure jumps between adjacent phases. We derive conditions for equilibrium at contact lines where three interfaces meet. Shapes of liquid surfaces under forces due to gravity and surface tension, including sessile drops and bubbles, are computed by solving differential equations.

According to the third law of thermodynamics, the entropy of a system in internal equilibrium approaches a constant independent of phase as the absolute temperature tends to zero. This constant value is taken to be zero for a non-degenerate ground state, in accord with statistical mechanics. Independence of phase is illustrated by extrapolation due to Fermi of the entropy of gray and white tin as the temperature is reduced to absolute zero. The third law is based on the postulate of Nernst to explain empirical rules for equilibrium of chemical reactions as absolute zero is approached. As a consequence of the third law, the following quantities vanish at absolute zero: heat capacity, coefficient of thermal expansion, and ratio of thermal expansion to isothermal compressibility.

The van der Waals model of a fluid exhibits a liquid-vapor phase transition. Isotherms in the volume-pressure plane depend on a parameter accounting for the finite size of molecules and another for molecular interactions. Below a critical temperature, the pressure of an isotherm is not monotonic. The locus of its maximum and minimum has an inverted U-shape and is called the spinodal curve. Volumes inside the spinodal curve represent unstable fluid. For volumes just outside the spinodal the fluid becomes metastable. For volumes beyond another inverted U-shaped curve there are two stable phases, a liquid and a vapor, separated by a miscibility gap. The Helmholtz free energy as a function of volume is investigated by the chord and common tangent constructions to calculate the miscibility gap. Isotherms of the Gibbs free energy as a function of pressure can be multiple-valued and display cusps. The miscibility gap obeys an equal-area construction due to Maxwell.

We derive a simplified version of the canonical ensemble developed in the next chapter. We treat a system of identical particles that can be distinguished, perhaps by position in a solid. We derive a statistical distribution of particles, each in a quantum state, by maximizing the number of ways they can be distributed among quantum states, subject to the constraint of constant total energy. This results in a most probable distribution. The probability of occupation of a given quantum state is proportional to its Boltzmann factor, the exponential of the negative of the energy of that state divided by a thermal energy. The thermal energy is the product of temperature and Boltzmann’s constant. The sum of all Boltzmann factors is called the partition function and is used to determine thermodynamic functions. Examples include two-state subsystems, harmonic oscillators, and rotations of a rigid diatomic molecule. Results are used to model heat capacities of solids and blackbody (cavity) radiation.

We use the method of the most probable distribution to show that the entropy for a general ensemble can be expressed by the maximum value of the disorder function of information theory, derived in Chapter 15 , subject to the set of constraints appropriate to the ensemble. We illustrate this in detail for a grand canonical ensemble with two kinds of particles. We treat a number of other ensembles practically by inspection, including an ensemble that relates to a Massieu function that is the Legendre transform of the entropy. By using a degeneracy factor to sum over energy levels, particle numbers, and volumes, we show that all ensembles can be related in a similar way to their associated thermodynamic functions, as observed by Hill.

Two types of averaging occur in quantum statistical mechanics, the first for pure quantum mechanical states and the second for a statistical ensemble of pure states. We define and exhibit the properties of density operators and their density matrix representation for both pure and statistical states. For equilibrium states, a statistical density operator depends only on stationary quantum states. We exhibit it in the energy representation for the microcanonical, canonical, and grand canonical ensembles; its use is illustrated for an ideal gas and the harmonic oscillator. Density matrices for spin 1/2 are expressed in terms of a polarization vector and Pauli spin matrices and related to vectors called spinors. Symmetric wave functions for bosons and antisymmetric wave functions for fermions are constructed from single-particle quantum states in terms of occupation numbers by using permutation operators, or Slater determinants for fermions. Weighting factors for states are contrasted for bosons, fermions, and distinguishable classical particles.

An ensemble is a collection of microstates that are compatible with a specified macrostate of a thermodynamic system. The microcanonical ensemble represents an isolated system having fixed energy. For that ensemble, the fundamental assumption of statistical mechanics is that every compatible stationary quantum microstate is equally probable. Properties of a system in a macrostate are calculated by averaging its values over the ensemble microstates. The entropy is assumed to be proportional to the logarithm of the number of compatible microstates as proposed by Boltzmann and in agreement with the disorder function of information theory. The proportionality constant is known as Boltzmann’s constant. Temperature, pressure, and chemical potential are calculated from partial derivatives of the entropy. The ensemble is illustrated for two-state subsystems, harmonic oscillators, an ideal gas with the Gibbs correction factor, and a multicomponent ideal gas. The entropy of mixing of ideal gases is calculated.

We derive equilibrium criteria in the presence of conservative external forces. For a chemically closed isothermal system with constant volume, equilibrium requires virtual variations of the Helmholtz free energy plus the external potential to be positive. For a uniform gravitational field, use of the calculus of variations shows that the gravitational chemical potential, which is the chemical potential per unit mass plus the product of the gravitational acceleration and height, is constant for each component. Pressure increases with height and the composition changes with height, so such systems are not homogeneous. For a mixture of ideal gases and binary liquids, the segregation of chemical components with height is small for samples of laboratory size. For the non-uniform gravitational field in the atmosphere of the Earth, there can be larger segregation. Rotating systems are treated by equivalence to gravitational forces; a fast centrifuge causes significant segregation. For applied electric fields, the electrochemical potential of ions is constant.

The second law of thermodynamics is stated as the existence of an extensive function of state called the entropy that can only increase for an isolated system. Equilibrium is reached at maximum entropy. Reciprocal absolute temperature is defined as entropy change with energy. Entropy is additive for a composite system. Heat added to a chemically closed system increases entropy by an amount greater than the ratio of the heat to the absolute temperature for an irreversible process; entropy equals that ratio for a reversible process. We relate entropy to its historical roots including other postulates and the Carnot cycle for an ideal gas. The second law plus the first law establish a fundamental equation to calculate entropy changes as a function of state. Reversible and irreversible expansion of an ideal gas are illustrated. Enthalpy and entropy changes are calculated for an isobaric melting of ice. Entropy is related to quantum microstates of a system via probability of a macrostate.

We investigate whether a homogeneous system is stable with respect to breakup into a composite system of two or more homogeneous subsystems. Criteria to avoid breakup lead to requirements for the dependence of the entropy and thermodynamic potentials on their natural variables. For stability, the entropy must be a concave function of its natural variables (all extensive) and the internal energy must be a convex function of its natural variables (all extensive). The thermodynamic potentials (Helmholtz, enthalpy, Gibbs, Kramers) must be convex functions of their extensive variables and concave functions of their intensive variables. Properties of Legendre transformations are used to derive the stability requirements for intensive variables. Local stability criteria depend on the signs of second order partial derivatives. When these stability criteria are violated, there can be locally unstable regions and metastable regions that are locally stable but globally unstable. Then transformations can occur. Principles of Le Chatlier and Le Chatlier-Braun elucidate the approach to equilibrium.

The classical canonical ensemble employs a probability density function in phase space in which the energy in the Boltzmann factor for a quantum system is replaced by the classical Hamiltonian. The classical partition function is the integral of that Boltzmann factor over phase space. One can artificially divide the classical partition function by a factor containing powers of Planck’s constant to get results that agree with quantum mechanics at high temperatures. We illustrate this for an ideal gas and compute effusion from a small hole. The law of Dulong and Petit is derived for a harmonic potential. We compute classical averages of canonical coordinates and momenta. We derive the virial theorem for time averages and use it to treat a nonideal gas with particle interactions calculated by using a pair distribution function. We discuss the use of canonical transformations in calculating partition functions and calculate the partition function for a rotating polyatomic molecule by using Jacobians.

Mass and thermal diffusivity measurements conducted on Earth are prone to contamination by uncontrollable convective contributions to the overall transport. Previous studies of mass and thermal diffusivities conducted on spacecraft have demonstration the gain in precision, and lower absolute values, resulting from the reduced convective transport possible in a low-gravity environment. We have developed and extensively tested real-time techniques for diffusivity measurements, where several measurements may be obtained on a single sample. This is particularly advantageous for low gravity research were there is limited experiment time. The mass diffusivity methodology uses a cylindrical sample geometry. A radiotracer, initially located at one end of the host is used as the diffusant. The sample is positioned in a concentric isothermal radiation shield with collimation bores located at defined positions along its axis. The intensity of the radiation emitted through the collimators is measured versus time with solid-state detectors and associated energy discrimination electronics. For the mathematical algorithm that we use, only a single pair of collimation bores and detectors are necessary for single temperature measurements. However, by employing a second, offset, pair of collimation holes and radiation detectors, diffusivities can be determined at several temperatures per sample. For thermal diffusivity measurements a disk geometry is used. A heat pulse is applied in the center of the sample and the temperature response of the sample is measured at several locations. Thus, several values of the diffusivity are measured versus time. The exact analytic solution to a heat pulse in the disk geometry leads to a unique heated area and measurement locations. Knowledge of the starting time and duration of he heating pulse is not used in the data evaluation. Thus, this methodology represents an experimentally simpler and more robust scheme.

We report some computational results obtained by using the phase field model for binary alloy solidification. We study directional solidification and concentrate on the transition between a planar interface and steady shallow cells near the onset of morphological instability at low growth speeds. The model is formulated on the principle of local positive entropy production. This gives rise to a set of partial differential equations for the temperature, composition and the phase fields. We make a frozen temperature approximation and compute steady-state nearly periodic cellular crystal–melt interfaces in two dimensions. The cell shapes can be characterized by a small number of Fourier coefficients and become nearly sinusoidal as stability is approached. Wavelength selection by tip splitting or coarsening is also observed.

Two Bhatnagar–Gross–Krook (BGK) models for isothermal binary fluid systems—the classical single relaxation time model and a split collision term model—are discussed in detail, with emphasis on the diffusion process in perfectly miscible ideal gases. Fluid equations, as well as the constitutive equation for diffusion, are derived from the Boltzmann equation using the method of moments and the values of the transport coefficients (viscosity and diffusivity) are calculated. The Schmidt number is found to be equal to one for both models. The split collision term model allows the two fluid components to have different values of the viscosity, while the single relaxation time model does not have this characteristic. The value of the viscosity does not depend on the density in the split collision term model, as expected from the classical kinetic theory developed by Maxwell. Possible extension of BGK models to non-ideal gases and ideal solutions (where the Schmidt number is larger than 1) is also investigated.

In previous work, approximate solutions were found for paraboloids having perturbations with four-fold axial symmetry in order to model dendritic growth in cubic materials. These solutions provide self-consistent corrections through second order in a shape parameter e to the Peclet number vs supercooling relation of the Ivantsov solution. The parameter e is proportional to the amplitude of the four-fold correction to the dendrite shape, as measured from the Ivantsov paraboloid of revolution. The equilibrium shape for anisotropic surface free energy to second order in the anisotropy is calculated. The value of e is determined by comparing the dendrite tip shape with the portion of the equilibrium shape near the growth direction, [001], for anisotropic surface free energy of the form γ = γ 0 [1+4 e 4 ( n 1 4 + n 2 4 + n 3 4 )], where the n i are components of the unit normal of the crystal surface. This comparison results in e =−2 e 4 −24 e 4 2 + O ( e 4 3 ), independent of the Peclet number. From the experimental value of e 4 , it is found that e ≈−0.012±0.004, in good agreement with the measured value e ≈−0.008 of LaCombe et al . ( Phys. Rev. E , 1995, 52 , 2778)

We develop two analytical solutions for thermodynamic fluctuations that are present in the phase-field model of solidification. One solution deals with fluctuations in an isothermal single phase system. The other deals with fluctuations in a two-phase isothermal system having a planar diffuse interface. Explicit formulae are obtained in one, two and three dimensions. In the case of two-phase system, fluctuations of the phase field are somewhat suppressed in the interface region. These solutions provide targets for testing numerical solutions.

The Ivantsov solution for an isothermal paraboloid of revolution growing into a pure, supercooled melt provides a relation between the bulk supercooling and a dimensionless product (the Peclet number P ) of the growth velocity and tip radius of a dendrite. Horvay and Cahn generalized this axisymmetric analytical solution to a paraboloid with elliptical cross-section. They found that as the deviation of the dendrite cross-section from a circle increases, the two-fold symmetry of the interface shape causes a systematic deviation from the supercooling/Peclet number relation of the Ivantsov solution. To model dendritic growth in cubic materials, we find approximate solutions for paraboloids having perturbations with four-fold axial asymmetry. These solutions are valid through second order in the perturbation amplitude, and provide self-consistent corrections through this order to the supercooling/Peclet number relation of the Ivantsov solution. Glicksman and colleagues have measured the shape and the supercooling/Peclet number relation for growth of succinonitrile dendrites in microgravity. For a Peclet number of P ≈0.004 and the experimentally observed shape, we calculate a correction corresponding to a 9% increase in the supercooling, in general agreement with the experimental results.