Anisotropic Surface Tension Research Articles

EFFECT OF ANISOTROPIC SURFACE TENSION ON THE MORPHOLOGICAL STABILITY OF DEEP CELLULAR CRYSTAL GROWTH IN DIRECTIONAL SOLIDIFICATION

This paper studies the effect of anisotropic surface tension on the morphological stability of deep cellular crystal in directional solidification by using the matched asymptotic expansion method and multiple variable expansion method. We find that the morphological stability of deep cellular crystal growth with anisotropic surface tension shows the same mechanism as that with isotropic surface tension. The deep cellular crystal growth contains two types of global instability mechanisms: the global oscillatory instability, whose neutral modes yield strong oscillatory dendritic structures, and the low-frequency instability, whose neutral modes yield weakly oscillatory cellular structures. Anisotropic surface tension has the significant effect on the two global instability mechanisms. As the anisotropic surface tension increases, the unstable domain of global oscillatory instability decreases, whereas the unstable domain of the global low-frequency instability increases.

The effect of anisotropic surface tension on interfacial evolution of a particle in the binary alloy melt

The dynamics of interfacial evolution of a particle in the binary alloy melt affected by anisotropic surface tension is studied by means of the multiple variable expansion method. The resulting analytical solution reveals the critical radius of nucleation in the binary alloy melt and the inward growth mechanism of interfacial evolution of the particle induced by the anisotropic effect of surface tension in the initial crystal growth immediately after nucleation. After nucleation, because of the anisotropic effect of surface tension, some parts of its interface grow outward, while some parts first grow inward in the initial crystal growth. Until the inward growth attains a certain distance (which is defined as ‘the melting depth’), these parts start to grow outward with other parts. During such a process the interface of the particle will form a remarkable concave and convex formation with the ear-like shape whose radius for some parts of the particle may be less than the critical radius of nucleation. The solute diffusion in the binary alloy melt changes the critical radius of nucleation and decreases the anisotropic effect of surface tension on interface deformation.

A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets

We develop a novel finite element method for a phase field model of nematic liquid crystal droplets. The continuous model considers a free energy comprised of three components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard energy representing the interfacial energy of the droplet, and a weak anchoring energy representing the interaction of the liquid crystal molecules with the surface tension on the interface (i.e. anisotropic surface tension). Applications of the model are for finding minimizers of the free energy and exploring gradient flow dynamics. We present a finite element method that utilizes a special discretization of the liquid crystal elastic energy, as well as mass-lumping to discretize the coupling terms for the anisotropic surface tension part. Next, we present a discrete gradient flow method and show that it is monotone energy decreasing. Furthermore, we show that global discrete energy minimizers $\Gamma$-converge to global minimizers of the continuous energy. We conclude with numerical experiments illustrating different gradient flow dynamics, including droplet coalescence and break-up.

Effect of the misorientation angle and anisotropy strength on the initial planar instability dynamics during solidification in a molten pool

The initial planar instability will appear with the solute accumulation ahead of the solid/liquid (S/L) interface during solidification in a molten pool. The instability process is dominated by the misorientation angle and the surface tension anisotropy strength, where the misorientation angle is the angle between the preferred crystalline orientation of base metal and the thermal gradient direction in front of the S/L interface. In this study, their effects on the initial planar instability during gas tungsten arc welding of an Al-alloy are investigated using a modified analytic model and a quantitative phase-field model, respectively. Specifically, we apply the uniform fluctuation spectrum assumption, Aω(0) = kBTM/{γ0[1 − 15γ4cos(4θ0)]ω2}, to represent the influence of thermal noise on S/L interface evolution. The incubation time, average wavelength and detailed interface morphology of the initial planar instability are investigated with varying surface tension anisotropies (determined by anisotropy strength γ4 and misorientation angle θ0). The results indicate that -γ4cos(4θ0) is a reasonable indicator for the effect of surface tension anisotropy on the initial planar instability. Moreover, rather than influencing solute diffusion, the surface tension anisotropy just affects the planar interface stability during the solidification. Finally, the experimental observations with the same welding parameters was carried out, which are in general agreement with the simulated results.

Effect of thermal anisotropy on binary alloy dendrite growth

A numerical model to study the effect of thermal anisotropy on binary alloy dendrite growth is presented. The model is based on the volume averaged enthalpy method with explicit surface tension anisotropy for crystal orientation. Thermal anisotropy is incorporated using anisotropic thermal conductivity in the energy equation. This is done by splitting the anisotropic conductivity into two parts, equivalent isotropic conductivity and anisotropic departure source term, enabling the use of a conventional isotropic solver to model anisotropic heat transfer. The proposed model is applied to study the effect of thermal anisotropy ratio on tip velocity, aspect ratio and equivalent radius of an equiaxed grain growing in an undercooled binary alloy melt. It is found that the thermal energy stored in the grain during solidification plays an important role in interface evolution, and thus anisotropic conductivity in the solid affects the grain morphology. There is a consistent increase in the aspect ratio of grains with increase in thermal anisotropy ratio, although the grain volume remains almost invariant. Due to unequal growth rates of the perpendicular arms, severe distortion of the solid crystal is seen at higher thermal anisotropy ratios. The model is further extended to study the growth of multiple dendrites in order to simulate microstructure evolution with thermal anisotropy. It is observed that thermal anisotropy significantly affects the grain morphology at low grain density but has a smaller influence at high grain density as compared to other governing factors such as solute transport.

Stability and selection of lamellar eutectic growth with anisotropic surface tension

The morphological stability of lamellar eutectic growth with the anisotropic effect of surface tension is studied by means of the interfacial wave (IFW) theory developed by Xu in the 1990s. We solve the related linear eigenvalue problem for the case that the Peclet number is small and the segregation coefficient parameter is close to the unit. The stability criterion of lamellar eutectic growth with the anisotropic surface tension is obtained. The linear stability analysis reveals that the stability of lamellar eutectic growth depends on a stability critical number [Formula: see text]. Similar to the case of isotropic surface tension, the system involves two types of global instability mechanisms: the “exchange of stability” invoked by the non-oscillatory, unstable modes and the “global wave instabilities” invoked by four types of oscillatory unstable modes, namely antisymmetric–antisymmetric (AA-), symmetric–symmetric (SS-), antisymmetric–symmetric (AS-) and symmetric–antisymmetric (SA-) modes. The anisotropic surface tension, by decreasing the corresponding stability critical number [Formula: see text], stabilizes the “exchange of stability” mechanism and “global wave instability” mechanism invoked by AA-, SA- and SS-modes. However, by increasing the corresponding stability critical number [Formula: see text], the anisotropic surface tension destabilizes the “global wave instability” mechanism invoked by AS-mode.

Effect of Surface Tension Anisotropy and Welding Parameters on Initial Instability Dynamics During Solidification: A Phase-Field Study

The effects of surface tension anisotropy and welding parameters on initial instability dynamics during gas tungsten arc welding of an Al-alloy are investigated by a quantitative phase-field model. The results show that the surface tension anisotropy and welding parameters affect the initial instability dynamics in different ways during welding. The surface tension anisotropy does not influence the solute diffusion process but does affect the stability of the solid/liquid interface during solidification. The welding parameters affect the initial instability dynamics by varying the growth rate and thermal gradient. The incubation time decreases, and the initial wavelength remains stable as the welding speed increases. When welding power increases, the incubation time increases and the initial wavelength slightly increases. Experiments were performed for the same set of welding parameters used in modeling, and the results of the experiments and simulations were in good agreement.

A coupled Ericksen/Allen–Cahn model for liquid crystal droplets

We present a model and discretization that couples the Ericksen model of liquid crystals with variable degree of orientation to the Allen–Cahn equations with a mass constraint. The coupled system models liquid crystal droplets with anisotropic surface tension effects due to the liquid crystal molecular alignment. The total energy consists of the Ericksen energy, phase-field (Allen–Cahn) energy, and a weak anchoring energy that couples the liquid crystal to the diffuse interface. We describe our discretization of the total energy along with a method to compute minimizers via a discrete gradient flow algorithm which has a strictly monotone energy decreasing property. Numerical experiments are given in three dimensions that illustrate a wide variety of droplet shapes that result from their interaction with defects.

Effect of anisotropic surface tension on morphological stability of lamellar eutectic growth in directional solidification

Lamellar eutectic solidification is very important in the development of new materials in which the periodic multiphase structures each may have a remarkable or enhanced functionality. The morphological instability during solidification may lead to various eutectic microstructures and greatly affect the physical and mechanical properties of final solidification products. In this paper, the morphological stability of lamellar eutectic growth with the anisotropic surface tension is studied by using the matched asymptotic expansion method and multiple variable expansion method. We assume that the process of solidification is viewed as a two-dimensional problem, The anisotropic surface tension is a four-fold symmetry function. The solute diffusion in the solid phase is negligible, and there is no convection in the system. On the basis of the basic state solution for the lamellar eutectic in directional solidification, the asymptotic solution for the perturbed interface shape of lamellar eutectic growth under the anisotropic surface tension is obtained in the case where the Peclet number is small, and then the quantization conditions of interfacial morphology for lamellar eutectic crystal is obtained. A dispersion relation between the wave number and the perturbation amplification rate, and the stability criterion of lamellar eutectic growth under the anisotropic surface tension are also obtained. The result shows that the anisotropic surface tension has a significant effect on lamellar eutectic growth in directional solidification. It shows that comparing the directional solidification system of isotropic surface tension, the interface morphological stability of anisotropic surface tension also involves two types of global instability mechanisms: the exchange of stability induced by the non-oscillatory, unstable modes and the global wave instability caused by four types of oscillatory unstable modes, namely antisymmetric antisymmetric (AA)-, symmetric antisymmetric (SA)-, antisymmetric symmetric (AS)-, and symmetric symmetric (SS)- modes. The linear stability analysis reveals that the stability of lamellar eutectic growth depends on stability critical number *. When *, the eutectic growth system is unstable; When q *, the eutectic growth system is stable. The anisotropic surface tension, by reducing the corresponding stability critical number *, stabilizes both the exchange of stability mechanism and the global instability mechanism for the AA-, SA- and SS-modes. It implies that the anisotropic surface tension parameter tends to reduce the stability zone. However, by increasing the corresponding stability critical number *, the anisotropic surface tension destabilizes the global instability mechanism for the AS-mode. It implies that the anisotropic surface tension parameter tends to increase the stability zone for AS-mode.

The dynamics of nucleation and growth of a particle in the ternary alloy melt with anisotropic surface tension.

The dynamics of nucleation and growth of a particle affected by anisotropic surface tension in the ternary alloy melt is studied. The uniformly valid asymptotic solution for temperature field, concentration field, and interface evolution of nucleation and particle growth is obtained by means of the multiple variable expansion method. The asymptotic solution reveals the critical radius of nucleation in the ternary alloy melt and an inward melting mechanism of the particle induced by the anisotropic effect of surface tension. The critical radius of nucleation is dependent on isotropic surface tension, temperature undercooling, and constitutional undercooling in the ternary alloy melt, and the solute diffusion melt decreases the critical radius of nucleation. Immediately after a nucleus forms in the initial stage of solidification, the anisotropic effect of surface tension makes some parts of its interface grow inward while some parts grow outward. Until the inward melting attains a certain distance (which is defined as "the melting depth"), these parts of interface start to grow outward with other parts. The interface of the particle evolves into an ear-like deformation, whose inner diameter may be less than two times the critical radius of nucleation within a short time in the initial stage of solidification. The solute diffusion in the ternary alloy melt decreases the effect of anisotropic surface tension on the interface deformation.

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

A phase field model for snow crystal growth in three dimensions

Snowflake growth provides a fascinating example of spontaneous pattern formation in nature. Attempts to understand this phenomenon have led to important insights in non-equilibrium dynamics observed in various active scientific fields, ranging from pattern formation in physical and chemical systems, to self-assembly problems in biology. Yet, very few models currently succeed in reproducing the diversity of snowflake forms in three dimensions, and the link between model parameters and thermodynamic quantities is not established. Here, we report a modified phase field model that describes the subtlety of the ice vapour phase transition, through anisotropic water molecules attachment and condensation, surface diffusion, and strong anisotropic surface tension, that guarantee the anisotropy, faceting and dendritic growth of snowflakes. We demonstrate that this model reproduces the growth dynamics of the most challenging morphologies of snowflakes from the Nakaya diagram. We find that the growth dynamics of snow crystals matches the selection theory, consistently with previous experimental observations.

Kernels with prescribed surface tension & mobility for threshold dynamics schemes

We show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces, including networks of them, in both two and three dimensions. Moreover, we discuss necessary and sufficient conditions for the positivity of the kernel which, in the case of two-phase flow, ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow. In particular, we show, in a barrier-type statement, that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions, and give a complete characterization. Among other results is a threshold dynamics scheme that is guaranteed to dissipate a non-local approximation to the interfacial energy in the fully anisotropic, multiphase setting, using the new kernel construction.

Periodic assembly of nanoparticle arrays in disclinations of cholesteric liquid crystals

An important goal of the modern soft matter science is to discover new self-assembly modalities to precisely control the placement of small particles in space. Spatial inhomogeneity of liquid crystals offers the capability to organize colloids in certain regions such as the cores of the topological defects. Here we report two self-assembly modes of nanoparticles in linear defects-disclinations in a lyotropic colloidal cholesteric liquid crystal: a continuous helicoidal thread and a periodic array of discrete beads. The beads form one-dimensional arrays with a periodicity that matches half a pitch of the cholesteric phase. The periodic assembly is governed by the anisotropic surface tension and elasticity at the interface of beads with the liquid crystal. This mode of self-assembly of nanoparticles in disclinations expands our ability to use topological defects in liquid crystals as templates for the organization of nanocolloids.

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.

On anisotropy function in crystal growth simulations using Lattice Boltzmann equation

In this paper, we present the ability of the Lattice Boltzmann (LB) equation, usually applied to simulate fluid flows, to simulate various shapes of crystals. Crystal growth is modeled with a phase-field model for a pure substance, numerically solved with a LB method in 2D and 3D. This study focuses on the anisotropy function that is responsible for the anisotropic surface tension between the solid phase and the liquid phase. The anisotropy function involves the unit normal vectors of the interface, defined by gradients of phase-field. Those gradients have to be consistent with the underlying lattice of the LB method in order to avoid unwanted effects of numerical anisotropy. Isotropy of the solution is obtained when the directional derivatives method, specific for each lattice, is applied for computing the gradient terms. With the central finite differences method, the phase-field does not match with its rotation and the solution is not any more isotropic. Next, the method is applied to simulate simultaneous growth of several crystals, each of them being defined by its own anisotropy function. Finally, various shapes of 3D crystals are simulated with standard and nonstandard anisotropy functions which favor growth in 〈100〉-, 〈110〉- and 〈111〉-directions.

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.

Degenerate seaweed to tilted dendrite transition and their growth dynamics in directional solidification of non-axially oriented crystals: a phase-field study.

We report the results of a phase-field study of degenerate seaweed to tilted dendrite transition and their growth dynamics during directional solidification of a binary alloy. Morphological selection maps in the planes of (G, Vp) and (ε4, Vp) show that lower pulling velocity, weaker anisotropic strength and higher thermal gradient can enhance the formation of the degenerate seaweed. The tip undercooling shows oscillations in seaweed growth, but it keeps at a constant value in dendritic growth. The M-S instability on the tips and the surface tension anisotropy of the solid-liquid interface are responsible for the formation of the degenerate seaweed. It is evidenced that the place where the interfacial instability occurs determines the morphological transition. The transient transition from degenerate seaweed to tilted dendrite shows that dendrites are dynamically preferred over seaweed. For the tilted dendritic arrays with a large tilted angle, primary spacing is investigated by comparing predicted results with the classical scaling power law, and the growth direction is found to be less sensitive to the pulling velocity and the primary spacing. Furthermore, the effect of the initial interface wavelength on the morphological transition is investigated to perform the history dependence of morphological selection.

On the Origin of Grid Anisotropy in the Simulation of Dendrite Growth by a VFT Model

A virtual front tracking model, based on solute and heat diffusion in two dimensions, is chosen to capture the full microstructural behavior of dendritic solidification in a binary alloy. We use a simple method of calculation, easy to perform, with relatively high stable time step, to simulate the dendrite growth in an Al-8 wt pct Mg alloy for which no numerical simulation has been carried out in the past. Local equilibrium at the liquid solid interface and the buildup of solute ahead of the interface are solved, and the dendrite growth process is simulated in isothermal solidification conditions. We show that the artificial grid anisotropy originates from the four cell neighborhood method adopted for capturing the moving front. By a correct neighborhood configuration, a grid independent set of results and expected phenomena are reproduced for a free dendrite growing either aligned or inclined with the grid. The dendrite morphology and orientation, and the growth velocity are explored via physical simulation parameters such as undercooling and surface tension anisotropy.

Faceting control by the stoichiometry influence on the surface free energy of low-index bcc-In2O3 surfaces

The faceting of In2O3(0 0 1), (0 1 1), and (1 1 1) grown by plasma-assisted molecular beam epitaxy on yttria-stabilised zirconia was investigated under different growth conditions—conventionally used oxygen-rich growth conditions with and without heavy Sn-doping, and indium-rich growth conditions—by in situ reflection high-energy electron diffraction, scanning electron microscopy, x-ray diffraction, and atomic force microscopy. In a simple thermodynamic model that considers surface free energy only, the observed faceting is compared to recent theoretical predictions of the surface tension (also termed surface free energy) anisotropy and the related equilibrium crystal shape derived from a Wulff construction. These predictions and our comparison include the variation with growth-condition-dependent oxygen chemical potential. Our results demonstrate how the experimentally changed oxygen chemical potential controls the faceting or island shape of In2O3 by changing the surface tension anisotropy. While the experimental results largely agree with the theoretically derived surface tension anisotropy, they strongly suggest a lower relative surface tension of the (0 0 1) surface at lower oxygen chemical potential (In-rich growth conditions) than theoretically predicted or a significant surface entropy contribution.