Instability of spatially periodic patterns due to a zero mode in the phase-field crystal equation

Ohnogi, Hiroshi; Shiwa, Yuh

doi:10.1016/j.physd.2008.06.011

Cited by 10 publications

(11 citation statements)

References 30 publications

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“…Equation (3) is sometimes called the derivative Swift-Hohenberg equation [37,38]; many papers use a different sign convention for the parameter r (e.g., [27,29,[39][40][41][42][43][44][45] [46] or f 35 = −b 3 φ 4 /4 + φ 6 /6 [19] have also been extensively studied, subject to the conditions b 2 > 27/38 (resp., b 3 > 0) required to guarantee the presence of an interval of coexistence between the homogeneous state φ = 0 and a spatially periodic state. Note that in the context of nonconserved dynamics [19,46] one generally selects a nonlinear term g nl directly, although this term is related to f nl through the relation g nl ≡ −df nl /dφ, i.e., g 23 or g 35 .…”

Section: A Equation and Its Variantsmentioning

confidence: 99%

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

et al. 2013

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The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.

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Section: A Equation and Its Variantsmentioning

confidence: 99%

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

et al. 2013

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“…where ε ∈ R is constant. When β = 0, equation (1.1) is known as the phase field crystal (PFC) equation: it has been employed to model and simulate the dynamics of crystalline materials, including crystal growth in a supercooled liquid, dendritic and eutectic solidification, epitaxial growth [6,7,11,12,31,33]. In the phase field approach, the number density of atoms is approximated by the phase function u.…”

Section: Introductionmentioning

confidence: 99%

Energy stable and convergent finite element schemes for the modified phase field crystal equation

Grasselli

Pierre

2016

ESAIM: M2AN

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We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and consists in a Galerkin approximation which allows low order (piecewise linear) finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn-Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. Using a Lojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations illustrate the theoretical results.

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“…The so-called phase-field crystal (PFC) equation has been recently employed to model and simulate the dynamics of crystalline materials, including crystal growth in a supercooled liquid, dendritic and eutectic solidification, epitaxial growth, and so on (see [7,8], cf. also [35,37]). In the phase-field crystal approach, the number density of atoms is approximated by using a phase function φ.…”

Section: Introductionmentioning

confidence: 88%

Well-posedness and long-time behavior for the modified phase-field crystal equation

Grasselli

2014

Math. Models Methods Appl. Sci.

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We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K.R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the global longtime behavior of the corresponding dynamical system. In particular, we establish the existence of an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space does converge to a unique equilibrium. This is done by means of a suitable version of the Lojasiewicz-Simon inequality. A convergence rate estimate is also given.

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Instability of spatially periodic patterns due to a zero mode in the phase-field crystal equation

Cited by 10 publications

References 30 publications

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

Energy stable and convergent finite element schemes for the modified phase field crystal equation

Well-posedness and long-time behavior for the modified phase-field crystal equation

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