Anisotropic Interface Kinetics Research Articles

The reversible inward growth of a particle in an alloy melt induced by anisotropic interface kinetics

The growth of particles in the alloy melt with anisotropic interface kinetics is studied in terms of the asymptotic method. The resulting asymptotic solution for the dynamics model of the particle shows that because of the anisotropic effect of interface kinetics the particle undergoes the reversible inward growth in the early stage of crystal growth along some growth directions. In the early stage of crystal growth, the particle first grows inward along some growth directions, while it grows outward along other growth directions. When the inward growth along these growth directions attains a certain depth, the particle starts to grow outward with the interface along other growth directions. The locally reversible inward growth induced by anisotropic interface kinetics results in the petal-like morphological pattern of the particle in the early stage of crystal growth. The petal-like morphology of the particle may favor grain refinement of microstructures when it is exerted to some perturbations during solidification of alloy melts.

The Effect of the Shear Flow on the Morphological Pattern of Particles in an Undercooled Melt

The effect of shear flow on the morphological pattern of particles in an undercooled melt is studied by using the asymptotic method. The mathematical model of the particle includes the anisotropic interface kinetic undercooling. The asymptotic solution for the mathematical model of the particle shows that shear flow in an undercooled melt intensifies the deformation and distortion of the particle in the initial stage of crystal growth. Due to the shear flow, the growth rate of the interface increases in the shear direction of the flow and strengthens the inward decay of the part of the interface induced by the anisotropic interface kinetics in the initial stage of crystal growth. As the shear rate of the flow increases, the interface of the particle is seriously deformed and distorted until it breaks into smaller particles. The analytical result provides the prediction of the formation of interface microstructures during solidification through the change of processing parameters.

Effect of anisotropic interface kinetics on morphological pattern of a particle in an undercooled melt

We study the morphological pattern of a particle in an undercooled melt with anisotropic interfacial atoms attachment kinetics by using the asymptotic method. The resulting asymptotic solution reveals the locally inward concave growth of the particle in the initial stage of crystal growth after nucleation and the variation of orientation selection. In the initial stage of crystal growth after nucleation, due to the effect of anisotropic interfacial atoms attachment kinetics, some parts of the interface first grow locally inward. Until the locally inward growth attains a certain distance, these parts start to grow outward with other parts. The locally inward concave growth and subsequent outward growth form the time window of inward growth of the particle, within which the particle has an inner radius less than the critical radius of nucleation. With the anisotropy parameters are systematically changed, the particle changes continuously its preferred growth direction from one preferred growth direction to another preferred growth direction. This orientation selection induced by anisotropic interfacial atoms attachment kinetics displays the rich morphological patterns of the particle with various deformed petal-shaped patterns.

Quantitative determination of tip undercooling of faceted sea ice with in situ experiments

Sea ice growth with lamellar microstructure containing brine channels has been extensively investigated. However, the quantitative growth information of sea ice remains lack due to the uncontrolled crystalline orientation in previous investigations. For the first time, we in situ observed the unidirectional growth of lamellar sea ice with well-manipulated ice crystal orientation and visualized tip undercooling of lamellar sea ice. With the real-time observation, interesting phenomena of doublon tip in cellular ice growth and growth direction shift of unidirectionally grown ice tip are discovered for the first time, which are attributed to the complex solutal diffusion and anisotropic interface kinetics in ice growth. The quantitative experiments provide a clear micro scenario of sea ice growth, and are suggested to promote relevant investigations of sea ice.

Combined effect of anisotropic surface tension and interface kinetics on pattern formation during the growth of two-dimensional crystals

Combined effect of anisotropic surface tension and interface kinetics on pattern formation during the growth of two-dimensional crystals

Effects of anisotropic interface kinetics and surface tension on deep cellular crystal growth in directional solidification

In this paper, we study the effects of anisotropic interface kinetics and surface tension on deep cellular crystal growth in directional solidification. The following assumptions are made: the process of solidification is viewed as a two-dimensional problem; the minor species in this binary mixture system is considered as an impurity; the solute diffusion in the solid phase is negligible; the thermodynamic properties other than the diffusivities are the same for both solid and liquid phases; there is no convection in the system; the anisotropic interface kinetics and the anisotropic surface tension are a four-fold symmetry function each; neither the preferred directions of the anisotropic interface kinetics nor the anisotropic surface tensions are necessarily the same as their counterparts for the solid and liquid phases respectively; the angle between the preferred directions of the two anisotropies is 0. By using the matched asymptotic expansion method and the multiple variable expansion method, we obtain the diagram of interface morphology for a deep cellular crystal in directional solidification. The results show that there exists a discrete set of the steady-state solutions subject to the quantization condition (35). The quantization condition yields the eigenvalue ???106801-20170033???* as a function of parameter and other parameters of the system, which determines the interface morphology of the cell. The results also show the variation of the minimum eigenvalue ???106801-20170033???*(0) with parameter . It is seen that when the preferred directions of the two anisotropies are the same, i.e., 0 = 0, the minimum eigenvalue ???106801-20170033???*(0) reduces with the increase of anisotropic surface-tension coefficient 4 , increases with the augment of parameter , and is unrelated to anisotropic interface kinetic coefficient 4 in the low order; when the angle 0 0 /4, as the 0 increases, the minimum eigenvalue ???106801-20170033???*(0) increases; when the angle /4 0 /2, as the 0 increases, the minimum eigenvalue ???106801-20170033???*(0) decreases. In addition, the results show the composite solution for the interface shape function B described on (X, Y) plane. It is seen that both of the anisotropy and the angle 0 have a significant effect on the total length and the root of deep cellular crystal, however, have little influence on the other solid-liquid interface, such as the top of deep cellular crystal. When the angle 0 is 0, as anisotropic coefficient increases, the total length of the finger increases, the curvature of the interface near the root increases or the curvature radius decreases. It is found that the influence of the anisotropic surface-tension coefficient on interface morphology is more remarkable than that of the anisotropic interface kinetics coefficient. when the angle 0 0 /4, as the 0 increases, the total length of the finger decreases, the curvature of the interface near the root decreases or the curvature radius increases; when the angle /4 0 /2, as 0 increases, the total length of the finger increases, the curvature of the interface near the root increases or the curvature radius decreases.

Effect of anisotropy on deep cellular crystal growth in directional solidification

The effect of anisotropic surface tension and anisotropic interface kinetics on deep cellular crystal growth is studied. An asymptotic solution of deep cellular crystal growth in directional solidification is obtained by using the matched asymptotic expansion method and the multiple variable expansion method. The results show that as the anisotropic parameters increase, the total length of deep cellular crystal increases and the root depth increases, whereas the curvature of the interface near the root increases or the curvature radius decreases.

Bridging the phase-field and phase-field crystal approaches for anisotropic material systems

In this paper the amplitude representation of the anisotropic phase-field crystal (APFC) model recently proposed as a generalized model for isotropic as well as anisotropic crystal lattice systems is developed. The relationship between the amplitude equations and the standard phase-field model for solidification of pure substances with elasticity effects is derived which provide an explicit connection between the phase-field and APFC models. On the one hand we present a computationally more efficient model and highlight its potential as a bridge between the PFC and phase-field models with anisotropic interface energies and kinetics on the other hand.

Velocity and cross-section shape of liquid cylindrical inclusions migrating normally to close-packed planes of a non-uniformly heated crystal under stationary thermal conditions

The mass transfer in a liquid cylindrical inclusion migrating normally to close-packed planes of a non-uniformly heated crystal under the stationary thermal conditions has been analyzed theoretically. The developed model of the mass transfer takes into account anisotropic interfacial energy density, anisotropic interface kinetics, and capillary effects. The dependences of both the inclusion velocity and the cross-section shape on the cross-sectional area, the temperature gradient, the interfacial energy density, and the interface kinetics parameters have been calculated and discussed. It has been shown that (1) the reason of deviation in shapes of migrating inclusions from equilibrium ones is the interface kinetics limiting the mass transfer in the liquid; (2) the interface kinetics generates the dependences of the shape distortion on the temperature gradient, the cross-sectional area, and the interfacial energy density; (3) the temperature-gradient dependences of the inclusion shapes are significantly different for the cases of the screw-dislocation and two-dimensional (2D) nucleation mechanisms of interface processes; (4) the mass transfer in the cylindrical inclusions is almost not limited by the growth kinetics at the singular interface, and, therefore, the cylindrical inclusions migrate faster than the liquid layers of the same thicknesses; (5) distortion in cross-section shapes of migrating inclusions is not an unambiguous function of their velocities.

Ice-templated porous alumina structures

The formation of regular patterns is a common feature of many solidification processes involving cast materials. We describe here how regular patterns can be obtained in porous alumina by controlling the freezing of ceramic slurries followed by subsequent ice sublimation and sintering, leading to multilayered porous alumina structures with homogeneous and well-defined architecture. We discuss the relationships between the experimental results, the physics of ice, and the interaction between inert particles and the solidification front during directional freezing. The anisotropic interface kinetics of ice leads to numerous specific morphological features in the structure. The structures obtained here could have numerous applications, including ceramic filters and biomaterials, and could be the basis for dense multilayered composites after infiltration with a selected second phase.

Analytical criteria for missing orientations on three-dimensional equilibrium shapes

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.

Applications of morphological stability theory

Recent applications of morphological stability theory are reviewed. For growth of a binary alloy from the melt, the temperature-dependence of the solute diffusivity can have a significant effect on the critical wavelength at the onset of instability. The response of the interface velocity to an electrical current pulse has been calculated by a linear perturbation analysis. The effect of a parallel shear flow and anisotropic interface kinetics on the onset of instability during growth from a supersaturated solution has been analyzed. The kinetic anisotropy arises from a model of step motion in which the morphological instability corresponds to step bunching. Kinetic anisotropy causes traveling waves along the crystal-solution interface and an enhancement of morphological stability. A shear flow in the direction of the step motion promotes morphological instability, while flow in the opposite direction is stabilizing. Oscillatory (in time) shear flows can be studied by Floquet theory.

The effect of oscillatory shear flow on step bunching

During crystal growth, an imposed shear flow at the crystal–fluid interface can alter the conditions for the onset of morphological instability. In previous work, we studied the effect of time-independent shear flows and anisotropic interface kinetics on the morphological stability of a crystal growing from supersaturated solution. The model assumes that growth is by the motion of elementary steps, which is treated by a macroscopic anisotropic kinetic law; morphological instability corresponds to the bunching of elementary steps. Predictions from linear stability theory indicate that a solution flowing above a vicinal face of a crystal can either enhance or prevent the development of step bunches, depending on the direction of the steady shear flow in relation to the direction of step motion; this is also observed in experiments. Here we extend the linear stability analysis to include the effect of an oscillatory shear flow on the morphological stability of a crystal growing from solution and present results for a model system for a range of oscillatory shear rate amplitudes and frequencies both with and without a steady shear component. In the presence of a steady shear flow, modulation can either stabilize or destabilize the system, depending on the modulation amplitude and frequency. Numerical solutions of the linearized Navier–Stokes and diffusion equations and an approximate analytical treatment show that the flow oscillations weakens both the stabilization and destabilization induced by steady-state flow. This weakening comes from mixing of solution above the perturbed interface and a modification to the phase shift between the interface perturbation wave and the corresponding concentration and flow waves. Optimal values of modulation frequency and amplitude are found when the steady flow is destabilizing.

Competition between kinetic and surface tension anisotropy in dendritic growth

The combined effect of an anisotropic surface tension and interface kinetics in dendritic crystal growth is studied numerically by a fully dynamical front-tracking method in two dimensions. It is shown how kinetic effects can be incorporated into the algorithm without causing numerical instabilities. The results are compared to the theory of E.A. Brener and V.I. Mel'nikov (Adv. Phys. 40, 53 (1991)). A particularly interesting case arises when the directions of minimum surface tension and minimum kinetic effect are different. In this case, when the deviation from local equilibrium is increased, the predicted transition from dendrites growing into the direction of the minimum surface stiffness to the direction of minimum kinetic effect is confirmed. Dendrites near this transition show strong oscillations and correlated side-branching. The transition where the oscillating dendrites change direction shows hysteresis.

Facetting during directional growth of oxides from the melt: coupling between thermal fields, kinetics and melt/crystal interface shapes

Implicit in most large-scale numerical analyses of crystal growth from the melt is the assumption that the melt/crystal interface shape and position are determined by transport phenomena. Although reasonable for many materials under a variety of growth conditions, this assumption is incorrect for a number of practical systems under realistic growth conditions. Specifically, the behavior of systems (e.g. certain oxides) which tend to develop facets along the melt/crystal interface is often affected both by transport phenomena and by interfacial attachment kinetics. We present a new modeling approach which accounts for interfacial kinetic effects during melt growth of large single crystals. The isotherm condition, typically employed at the melt/crystal interface, is replaced by an equation accounting for undercooling due to interface kinetics. A finite-element algorithm, designed to accommodate its numerical mesh to the appearance of facetted interfaces, is applied to this problem. Results are presented for the simulated directional growth of oxide slabs. The interplay between evolving thermal fields and anisotropic interface kinetics is investigated. In particular, the evolution of facets and the dependence of their size on growth conditions is explored. Trends reported here are in qualitative agreement with those appearing in the literature. Discrepancies between quantitative predictions of facet sizes using a theory (see Ann. Rev. Mater. Sci. 3 (1973) 397 and references within) and those calculated in this paper can, in a number of cases, be attributed to the simplifications on which this theory is based.

Steps, kinetic anisotropy, and long-wavelength instabilities in directional solidification.

We consider the effect of anisotropic interface kinetics on long-wavelength instabilities during the directional solidification of a binary alloy having a vicinal interface. Linear theory predicts that a planar solidification front is stabilized under the effect of anisotropy as long as the segregation coefficient is small enough, whereas a novel instability appears at high rates of solidification. Furthermore, the neutral stability curve, indicating the values of the principal control parameter (here the morphological number) for which the growth rate of a sinusoidal perturbation of a given wavelength changes its sign, is shown to have up to three branches, two of them combining to form an isola for certain values of the control parameters. We identify conditions for which linear stability theory predicts the instability of the planar interface to long-wavelength traveling waves. A number of distinguished limits provide evolution equations that describe the resulting dynamical behavior of the crystal-melt interface and generalize previous work by Sivashinsky, Brattkus, and Davis and Riley and Davis. Bifurcation analysis and numerical computations for the derived evolution equations show that the anisotropy is able to promote the tendency to supercritical bifurcation, and also leads to the development of strongly preferred interface orientations for finite-amplitude deformations.

Step bunching: generalized kinetics

Our previous model of the effect of a parallel shear flow and anisotropic interface kinetics on the onset of (linear) instability during growth from a supersaturated solution is extended to allow for a kinetic coefficient which is a nonlinear function of the supersaturation and crystallographic orientation. The presence of impurities in the binary solution is also treated, although for typical impurity levels there is no explicit effect of impurities on morphological stability. As previously, a shear flow counter to the step motion enhances stability and a shear flow in the direction of the step motion promotes instability. Instability is enhanced when the kinetic coefficient increases rapidly with supersaturation. Specific calculations are carried out for parameters appropriate to the growth of ADP and KDP from solution.

The effect of a shear flow on the morphological stability of a vicinal face: Growth from a supersaturated solution

The effect of a parallel shear flow and anisotropic interface kinetics on the onset of instability during growth from a supersaturated solution is analyzed. The model used for anisotropy is based on the microscopic picture of step motion. A shear flow (linear Couette flow or asymptotic suction profile) parallel to the crystal-solution interface in the same direction as the step motion decreases interface stability. A shear flow counter to the step motion enhances stability and for sufficiently large shear rates the interface is absolutely morphologically stable. For large wave numbers, the perturbed flow field can be neglected and a simple analytic approximation for the stability-instability demarcation is found.

Nonequilibrium effect of anisotropic interface kinetics on the directional growth of ice crystals

Directional growth experiments were carried out on ice crystals in a water-KC1 solution system. It is well known that the effect of anisotropic interface kinetics causes tilting of the cell axes. The nonequilibrium effect of anisotropic interface kinetics, namely the kinetic supercooling ΔT k, was estimated by observing the tilt angle of the cell axes, based on the theoretical prediction of linear stability analysis for small perturbations on a planar interface during directional growth. It was found that ΔT k of an ice crystal during directional growth is minimum in the 〈112̄0〉 direction, i.e., the dendrite tip growth direction, and maximum in the 〈101̄0〉 direction. Therefore, it is shown that the preferred growth direction determined by anisotropic interface kinetics coincides with the direction by anisotropic interface free energy.

Step bunching on a vicinal face of a crystal growing in a flowing solution

The effect of a parallel shear flow and anisotropic interface kinetics on the onset of (linear) instability during growth from a supersaturated solution is analyzed including perturbations in the flow velocity. The model used for anisotropy is based on the microscopic picture of step motion. A shear flow (linear Couette flow or asymptotic suction profile) parallel to the crystal-solution interface in the same direction as the step motion (negative shear) decreases interface stability. For large wavenumbers k x , the perturbed flow field can be neglected and a simple analytic approximation for the stability-instability demarcation is found. A shear flow counter to the step motion (positive shear) enhances stability and for sufficiently large shear rates (on the order of 1 s −1) the interface is morphologically stable. Alternatively, the approximate analysis predicts that the system is unstable if the solution flow velocity in the direction of the step motion at a distance(2 k x) −1 from the interface exceeds the propagation rate v x of step bunches induced by the interface perturbations. The approximate results are applied to the growth of ADP and lysozyme. For sufficiently low supersaturations, the interface is stable for positive shear and unstable for negative shear. More generally, there is a critical negative shear rate for which the interface becomes unstable as the magnitude of the shear rate increases. For a range of growth conditions for ADP, the magnitude of this critical shear rate is2 k xv x . Even shear rates due to natural convection may be sufficient to affect stability for typical growth conditions.