Robert F. Sekerka

On the thermodynamics of crystalline solids

A formalism for the analysis of thermodynamic equilibrium in an arbitrarily stressed multicomponent crystal is developed. We assume that the energy per unit cell is a function of the following per‐unit‐cell quantities: the entropy, the mole number of each chemical species, and six independent dot products of the three vectors that define the cell. Body forces specific to each species are included, but chemical reactions and capillarity are excluded. We use a Gibbsian variational method but refer all quantities to the unvaried state which may be one of large strain. We obtain the familiar equations of thermal and mechanical equilibrium and the uniformity of generalized potentials μi +φi for each species; the φi are associated with the body forces and the μi are identified as chemical potentials since we show them to be equal locally to the chemical potentials of components present in a fluid with which the crystal is in equilibrium. The equations of equilibrium are then rederived under the assumption that ...

Morphological stability of disc crystals

Abstract Several phenomenological models of the solidification and subsequent morphological instability of a discshaped crystal of ice growing edgewise from pure, slightly undercooled water (≈ 0.9 C° or less) are developed. We assume that growth along the c axis of ice is virtually prohibited by slow interfacial kinetics but growth in the basal plane is controlled principally by the flow of heat. Thermal properties of solid and liquid are taken to be equal for tractability and Laplaces equation is solved to account for the nearly steady-state heat flow which is assumed to accompany growth of a completely submerged disc. This leads to an unperturbed growth rate. d R/ d t = (2πk/Lh) ΔT/ In (16 eR/h) for a disc of radius R large compared to its thickness h ; here t is the time, k is the thermal conductivity of water, L is the latent heat of fusion per unit volume, and Δ T is the undercooling of the water bath below the melting point of ice. There is no strong size effect in agreement with experiment; but, agreement in the magnitude of the growth rate requires values of h several times smaller than observed. A model which accounts for additional losses of heat to ambient air via a boundary layer yields growth rates comparable to those observed provided that sufficiently large heat transfer coefficients can be justified. For a completely submerged disc, a critical radius above which the disc shape becomes unstable is calculated from perturbation theory.

Proof of the symmetry of the transport matrix for diffusion and heat flow in fluid systems

A phenomenological proof is given of the symmetry of the transport matrix (Bij) of coefficients in the constitutive laws that describe mass and energy transport in a multicomponent fluid for the following choices of frame, forces, and fluxes: the local center of mass frame; the forces ∇(μi/T), where μ0=−1, T is the temperature, and μi for i=1,...K is the chemical potential of the ith component; and the fluxes ji, where j0 is the nonconvective internal energy flux and ji for i=1,...K is the mass flux of the ith component. The proof is based on Onsager’s reciprocity theory and proceeds by using the equations of hydrodynamics and the constitutive equations, postulated to describe mass and energy transport in the fluid, to calculate the time development of a specially selected fluctuation, when that fluctuation is considered to be macrovariation. The resulting dynamical equations are cast into the Onsager form by use of the generalized forces obtained from a thermodynamic calculation of the entropy change ΔS ...

Irreversible thermodynamics of creep in crystalline solids

(Received 30 August 2013; revised manuscript received 19 October 2013; published 18 November 2013) We develop an irreversible thermodynamics framework for the description of creep deformation in crystalline solids by mechanisms that involve vacancy diffusion and lattice site generation and annihilation. The material undergoing the creep deformation is treated as a nonhydrostatically stressed multicomponent solid medium with nonconserved lattice sites and inhomogeneities handled by employing gradient thermodynamics. Phase fields describe microstructure evolution, which gives rise to redistribution of vacancy sinks and sources in the material during the creep process. We derive a general expression for the entropy production rate and use it to identify of the relevant fluxes and driving forces and to formulate phenomenological relations among them taking into account symmetry properties of the material. As a simple application, we analyze a one-dimensional model of a bicrystal in which the grain boundary acts as a sink and source of vacancies. The kinetic equations of the model describe a creep deformation process accompanied by grain boundary migration and relative rigid translations of the grains. They also demonstrate the effect of grain boundary migration induced by a vacancy concentration gradient across the boundary.

Theory of Crystal Growth Morphology

Publisher Summary Crystal growth morphology is derived from an interplay of the crystallographic anisotropy and growth kinetics. The growth kinetics consist of interfacial processes as well as long-range transport. The equilibrium shape results from minimizing the anisotropic surface free energy of a crystal under the constraint of constant volume and serves as the baseline for crystal morphology. The phenomenon of morphological stability is related to the spontaneous change (instability) of shape (morphology) of a surface or interface. The morphological instability concept is related to a well-defined base state, in which the crystal-melt interface evolves in time according to a growth law. This growth law comes from a solution to the appropriate free-boundary problem in such a way that all the transport equations and boundary conditions are satisfied. Then this well-defined base state is tested for stability by introducing a shape perturbation, solving the resulting perturbed problem, and deducing from the solution whether the perturbation will grow or decay in time. If the perturbation is very small, the problem can be linearized in its amplitude, leading to the linear stability theory.

On the validity of the Onsager reciprocal relations

This article examines the conditions under which the Onsager reciprocal relations hold for the matrix of transport coefficients in the constitutive laws for heat and mass transport in continuous systems. For a K component system and a suitable choice of reference frame, those laws may be expressed in the form vector J/sub i/ = summation from j = 0 to K (H/sub ij/vector F/sub j/) summation from j = 0 to K (H/sub ij/grad h/sub i/ where vector J/sub 0/ is the energy flux, vector J/sub i/ for i = 1,. . .,K is the mass flux of the i/sup th/ species, and the vector F/sub j/ = grad h/sub j/ are thermodynamic driving forces in which the h/sub j/ are potentials that are functions of the local thermodynamic state; and H/sub ij/, also functions of the local state, are the transport coefficients whose symmetry is under consideration. All terms in Equation 1 will depend in general on position.

Irreversible thermodynamic basis of phase field models

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 field and accompanying fields (such as temperature and composition) are formulated on the basis of local positive entropy production subject to suitable constraints on energy and chemical species. Fluxes of energy and chemical species, Korteweg stresses due to inhomogeneities and an equation for phase field evolution are obtained from the rate of entropy production by postulating linear constitutive relations. The full phase field equations are discussed and illustrated for the simple case of solidification of a pure material from its melt assuming uniform density.

Growth of an ice disk: dependence of critical thickness for disk instability on supercooling of water.

The appearance of an asymmetrical pattern that occurs when a disk crystal of ice grows from supercooled water was studied by using an analysis of growth rates for radius and thickness. The growth of the radius is controlled by transport of latent heat and is calculated by solving the diffusion equation for the temperature field surrounding the disk. The growth of the thickness is governed by the generation and lateral motion of steps and is expressed as a power function of the supercooling at the center of a basal face. Symmetry breaking with respect to the basal face of an ice disk crystal is observed when the thickness reaches a critical value; then one basal face becomes larger than the other and the disk loses its cylindrical shape. Subsequently, morphological instability occurs at the edge of the larger basal face of the asymmetrical shape (Shimada, W.; Furukawa, Y. J. Phys. Chem. 1997, B101, 6171-6173). We show that the critical thickness is related to the critical condition for the stable growth of a basal face. A difference of growth rates between two basal faces is a possible mechanism for the appearance of the asymmetrical shape.

Phase Field Modeling of Crystal Growth Morphology

Crystal growth morphology results from an interplay of crystallographic anisotropy and growth kinetics, the latter consisting of interfacial processes as well as long‐range transport. Mathematical modeling of crystal growth shapes is important to our understanding of fundamental crystal growth phenomena as well as to improvement and optimization of practical processes for crystal growth. Such modeling results in a difficult free boundary problem because one must piece together solutions of partial differential equations, via boundary conditions, on a crystal surface whose location and shape are yet to be determined. Moreover, this problem is complicated because the nature of long‐range transport leads to natural instabilities of shape, so‐called morphological instabilities, on the scale of the geometric mean of a transport length and a capillary length. The resulting shapes can be cellular or dendritic but can also exhibit corners and facets related to the underlying crystallographic anisotropy. Growth su...

Mean-field Density Functional Theory of a Three-Phase Contact Line

A three-phase contact line in a three-phase fluid system is modeled by a mean-field density functional theory. We use a variational approach to find the Euler-Lagrange equations. Analytic solutions are obtained in the two-phase regions at large distances from the contact line. We employ a triangular grid and use a successive overrelaxation method to find numerical solutions in the entire domain for the special case of equal interfacial tensions for the two-phase interfaces. We use the Kerins-Boiteux formula to obtain a line tension associated with the contact line. This line tension turns out to be negative. We associate line adsorption with the change of line tension as the governing potentials change.