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 non standard anisotropy functions which favor growth in -, - and -directions

Lattice Boltzmann simulations of 3D crystal growth: Numerical schemes for a phase-field model with anti-trapping current

A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK) collision rules is developed for a phase-field model of alloy solidification in order to simulate the growth of dendrites. The solidification of a binary alloy is considered, taking into account diffusive transport of heat and solute, as well as the anisotropy of the solid-liquid interfacial free energy. The anisotropic terms in the phase-field evolution equation, the phenomenological anti-trapping current (introduced in the solute evolution equation to avoid spurious solute trapping), and the variation of the solute diffusion coefficient between phases, make it necessary to modify the equilibrium distribution functions of the LB scheme with respect to the one used in the standard method for the solution of advection-diffusion equations. The effects of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of D3Q7. The method is validated by direct comparison of the simulation results with a numerical code that uses the finite-difference method. Simulations are also carried out for two different anisotropy functions in order to demonstrate the capability of the method to generate various crystal shapes

Hydrodynamics of domain growth in nematic liquid crystals

We study the growth of aligned domains in nematic liquid crystals. Results are obtained solving the Beris-Edwards equations of motion using the lattice Boltzmann approach. Spatial anisotropy in the domain growth is shown to be a consequence of the flow induced by the changing order parameter field (backflow). The generalization of the results to the growth of a cylindrical domain, which involves the dynamics of a defect ring, is discussed.Comment: 12 revtex-style pages, including 12 figures; small changes before publicatio

Influence of external flows on crystal growth: numerical investigation

We use a combined phase-field/lattice-Boltzmann scheme [D. Medvedev, K. Kassner, Phys. Rev. E {\bf 72}, 056703 (2005)] to simulate non-facetted crystal growth from an undercooled melt in external flows. Selected growth parameters are determined numerically. For growth patterns at moderate to high undercooling and relatively large anisotropy, the values of the tip radius and selection parameter plotted as a function of the Peclet number fall approximately on single curves. Hence, it may be argued that a parallel flow changes the selected tip radius and growth velocity solely by modifying (increasing) the Peclet number. This has interesting implications for the availability of current selection theories as predictors of growth characteristics under flow. At smaller anisotropy, a modification of the morphology diagram in the plane undercooling versus anisotropy is observed. The transition line from dendrites to doublons is shifted in favour of dendritic patterns, which become faster than doublons as the flow speed is increased, thus rendering the basin of attraction of dendritic structures larger. For small anisotropy and Prandtl number, we find oscillations of the tip velocity in the presence of flow. On increasing the fluid viscosity or decreasing the flow velocity, we observe a reduction in the amplitude of these oscillations.Comment: 10 pages, 7 figures, accepted for Physical Review E; size of some images had to be substantially reduced in comparison to original, resulting in low qualit

Computer simulation of liquid crystals

A review is presented of molecular and mesoscopic computer simulations of liquid crystalline systems. Molecular simulation approaches applied to such systems are described and the key findings for bulk phase behaviour are reported. Following this, recently developed lattice Boltzmann (LB) approaches to the mesoscale modelling of nemato-dynamics are reviewed. The article concludes with a discussion of possible areas for future development in this field.</p

Lattice Boltzmann Algorithm for three-dimensional liquid crystal hydrodynamics

We describe a lattice Boltzmann algorithm to simulate liquid crystal hydrodynamics in three dimensions. The equations of motion are written in terms of a tensor order parameter. This allows both the isotropic and the nematic phases to be considered. Backflow effects and the hydrodynamics of topological defects are naturally included in the simulations, as are viscoelastic effects such as shear-thinning and shear-banding. We describe the implementation of velocity boundary conditions and show that the algorithm can be used to describe optical bounce in twisted nematic devices and secondary flow in sheared nematics with an imposed twist.Comment: 12 pages, 3 figure

Lattice Boltzmann Simulations for Anisotropic Crystal Growth of a Binary Mixture

Phase field models are powerful tools to simulate the interfaces evolution in glass melt solidification mechanism, including crystallization phenomena. The purpose of this work is the numerical implementation of a phase field model for solidification of a dilute binary mixture by using the Lattice Boltzmann equations. The proposed Boltzmann method is based on the BGK approximation for kinetic equations relative to the phase field, supersaturation and temperature. In order to simulate the anisotropic term in the phase field equation, the equilibrium distribution function is a modification of the one used in the standard method for the advection diffusion equation. Simulations are carried out for anisotropic crystal growth for various thermal conditions, i.e., for several values of undercooling and different values of Lewis number

Modeling Dendritic Solidification using Lattice Boltzmann and Cellular Automaton Methods

This dissertation presents the development of numerical models based on lattice Boltzmann (LB) and cellular automaton (CA) methods for solving phase change and microstructural evolution problems. First, a new variation of the LB method is discussed for solving the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term is treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The results showed that the model can deal with phase change problems more accurately and efficiently than explicit LB models. Furthermore, a new numerical technique is introduced for simulating dendrite growth in three dimensions. The LB method is used to calculate the transport phenomena and the CA is employed to capture the solid/liquid interface. It is assumed that the dendritic growth is driven by the difference between the local actual and local equilibrium composition of the liquid in the interface. The evolution of a threedimensional (3D) dendrite is discussed. In addition, the effect of undercooling and degree of anisotropy on the kinetics of dendrite growth is studied. Moreover, effect of melt convection on dendritic solidification is investigated using 3D simulations. It is shown that convection can change the kinetics of growth by affecting the solute distribution around the dendrite. The growth features of twodimensional (2D) and 3D dendrites are compared. Furthermore, the change in growth kinetics and morphology of Al-Cu dendrites is studied by altering melt undercooling, alloy composition and inlet flow velocity. The local-type nature of LB and CA methods enables efficient scaling of the model in petaflops supercomputers, allowing the simulation of large domains in 3D. The model capabilities with large scale simulations of dendritic solidification are discussed and the parallel performance of the algorithm is assessed. Excellent strong scaling up to thousands of computing cores is obtained across the nodes of a computer cluster, along with near-perfect weak scaling. Considering the advantages offered by the presented model, it can be used as a new tool for simulating 3D dendritic solidification under convection