Phase-field modeling of two-dimensional crystal growth with anisotropic diffusion

Meca, Esteban; Shenoy, Vivek B.; Lowengrub, John

doi:10.1103/physreve.88.052409

Cited by 29 publications

(30 citation statements)

References 49 publications

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“…This very good agreement and the quadratic convergence seem to indicate, in accordance with similar results in other systems, e.g., Ref. [24], that the smallest order of the finite ε corrections to equation (3.76g) will be O ε 2 .…”

Section: Comparisons To Phase-field Simulationssupporting

confidence: 91%

Sharp-interface formation during lithium intercalation into silicon

2017

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In this study, we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as amorphous silicon (a-Si). The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in lithium ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface intersects the free boundary of the Si layer. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

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Section: Comparisons To Phase-field Simulationssupporting

confidence: 91%

Sharp-interface formation during lithium intercalation into silicon

2017

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“…In 2D, the anisotropy of conformational entropy is usually given by

κ = 1 + ϵ_{4} \cos (m Θ),

where m is the number of folds of anisotropy, ϵ 4 is a parameter for the anisotropy strength, and Θ defined as

\tan (Θ (ϕ)) = \frac{\partial_{y} ϕ}{\partial_{x} ϕ}

in 2D (cf. ). One can easily derive when m = 4, the fourfold symmetric model‐type anisotropy is given by the following: in 2D (cf.…”

Section: The Model Systemmentioning

confidence: 97%

“…We first briefly describe the phase field model for dendritic crystal growth . In a confined domain Ω ∈ R d ,( d = 2,3), a phase variable (or a labeling function) ϕ ( x , t )( x ∈ Ω , t is the time) is introduced to label the liquid and solid phase, respectively:

ϕ (bold-italicx, t) = ({, \begin{matrix} - 1 & fluid, \\ 1 & solid, \end{matrix})

with a smooth but steep transitional layer controlled by parameter ϵ .…”

Section: The Model Systemmentioning

confidence: 99%

“…To understand the process of dendritic crystal growth, many theoretical models have been developed and numerical simulations based on these models were carried out (cf. and the references therein) to study transport of heat and solute in the vicinity of the dendrite tip and the factors that control the stability of the shape of the dendrite tip, and to determine the tip radius and initial secondary spacing. In a pioneering work , the diffuse interface approach, also known as the phase field method, was first applied in the solidification problem that accounts for the Gibbs–Thomson equation at the solid–liquid interface for solidification problems.…”

Section: Introductionmentioning

confidence: 99%

“…Subsequently, the phase field dendritic crystal growth model (PF–DCG) has been improved quite a bit in the past two decades. In , based on the time dependent Ginzburg–Landau mesoscopic simulation method, the authors developed various phase‐field models that exhibited great versatility to study dynamics of atomic‐scale dendritic crystal growth dependent on temperature and isotropic/anisotropic diffusive time scales. The essential idea of the PF–DCG model is that an order parameter (phase field variable) is adopted to define the physical state (liquid or solid) of the system at each point, and a free‐energy functional is devised by incorporating a specific form of the conformational entropy with anisotropic spatial gradients.…”

Section: Introductionmentioning

confidence: 99%

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Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

Zhao

Wang

Yang

2016

Numerical Meth Engineering

176

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Summary We present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model, which is derived from the variation of a free‐energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient‐dependent anisotropic coefficient. We introduce a novel Invariant Energy Quadratization approach to transform the free‐energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations. Based on the reformulated equivalent governing system, we develop a first and a second order semi‐discretized scheme in time for the system, in which all nonlinear terms are treated semi‐explicitly. The resulting semi‐discretized equations consist of a linear elliptic equation system at each time step, where the coefficient matrix operator is positive definite and thus, the semi‐discrete system can be solved efficiently. We further prove that the proposed schemes are unconditionally energy stable. Convergence test together with 2D and 3D numerical simulations for dendritic crystal growth are presented after the semi‐discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes. Copyright © 2016 John Wiley & Sons, Ltd.

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Achievements and Challenges of Graphene Chemical Vapor Deposition Growth

Liu

et al. 2022

Adv Funct Materials

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Graphene, since the first successful exfoliation of graphite, has continuously attracted attention due to its remarkable properties and applications. Recently, the research focus on graphene synthesis has been directed to the controllable synthesis of large‐area and high‐quality graphene. In the last decade, there has been great progress in the chemical vapor deposition (CVD) growth of graphene. Theoretical investigations have led to an enhanced understanding of puzzles on hydrocarbon species stability, key reaction pathways, the role of hydrogen gas, the morphology of graphene islands, and the alignment of graphene on substrates. Experimentally, high‐quality graphene is epitaxially grown on both insulating and metal substrates. Progress has also been reported on low‐temperature graphene growth and on controlling the thickness and stacking of graphene layers. In this review, the authors summarize the previous theoretical and experimental studies on graphene CVD growth and discuss the future challenges on the growth of graphene i) on insulating substrates, ii) at low temperature, iii) with controllable thickness, and iv) with selected stacking twist angles. The authors assert that the key to the continuous advancement of graphene growth is the synergy of experimental and theoretical investigations.

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Phase-field modeling of two-dimensional crystal growth with anisotropic diffusion

Cited by 29 publications

References 49 publications

Sharp-interface formation during lithium intercalation into silicon

Sharp-interface formation during lithium intercalation into silicon

Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

Achievements and Challenges of Graphene Chemical Vapor Deposition Growth

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