Amplitude expansion of the phase-field crystal model for complex crystal structures

Donno, Marcello De; Benoit–Maréchal, Lucas; Salvalaglio, Marco

doi:10.1103/physrevmaterials.7.033804

Cited by 3 publications

(7 citation statements)

References 74 publications

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“…We devised a coarse‐grained description of the HCP lattice by exploiting recent extensions of the amplitude PFC model. Importantly, the description of lattices with a basis introduced in the literature[26], namely the definition of

\mathcal {B}_n

, Equation (), is here applied beyond its original scope. Indeed it effectively encodes a local structure and differs from setting the relative position of atoms forming a basis per Bravais lattice site.…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, we use a formulation supporting the definition of Bravais lattices with basis, featuring modified amplitude functions

\widetilde{\eta }_n= \mathcal {B}_n\eta _n

, where

\mathcal {B}_n

are complex constants defined as [26]

\begin{equation} \mathcal {B}_n = \sum _{j=1}^{J} e^{-\mathrm{i} \mathbf {k}_n \cdot \mathbf {R}_j}, \end{equation}

with J the number of atoms in the unit cell, and

\mathbf {R}_j

their positions. The energy functional in this framework can be written as [26]

\begin{equation} \begin{split} F_{\widetilde{\eta }}=\int _{\Omega } & {\left[A \sum _{n=1}^N (\widetilde{\eta }_n \mathcal {M}_{n}\widetilde{\eta }_n^* +\widetilde{\eta }_n^* \mathcal {M}_{n}\widetilde{\eta }_n)+ B\zeta _2+C\zeta _3+D\zeta _4 + f_\mathrm{h} \right]} \rm d\mathbf {r}, \end{split} \end{equation}

where ζ 2, 3, 4 are polynomials in the amplitudes whose terms depend on the lattice symmetry, defined as

\begin{matrix} ζ_{} \end{matrix}

…”

Section: Modelmentioning

confidence: 99%

“…To model the HCP crystal structure, we need to consider a multi-mode APFC model. Moreover, we use a formulation supporting the definition of Bravais lattices with basis, featuring modified amplitude functions η𝑛 =  𝑛 𝜂 𝑛 , where  𝑛 are complex constants defined as [26]…”

Section: Apfc Modelmentioning

confidence: 99%

“…with 𝐽 the number of atoms in the unit cell, and 𝐑 𝑗 their positions. The energy functional in this framework can be written as [26]…”

Section: Apfc Modelmentioning

confidence: 99%

“…Here, we show how the APFC model can describe HCP lattices and provide numerical simulations of HCP crystalline systems hosting three-dimensional dislocation networks. To this goal, we consider extensions of the APFC model we recently introduced in the literature [26], which successfully described crystals with complex symmetry and of technological interest (e.g., the diamond lattice). The framework proposed therein allows for modeling lattices with a basis.…”

Section: Introductionmentioning

confidence: 99%

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Amplitude phase‐field crystal model for the hexagonal close‐packed lattice

De Donno,

Salvalaglio

2023

Proc Appl Math and Mech

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The phase‐field crystal model allows the study of materials on atomic length and diffusive time scales. It accounts for elastic and plastic deformation in crystal lattices, including several processes such as growth, dislocation dynamics, and microstructure evolution. The amplitude expansion of the phase‐field crystal model (APFC) describes the atomic density by a small set of Fourier modes with slowly‐varying amplitudes characterizing lattice deformations. This approach allows for tackling large, three‐dimensional systems. However, it has been used mostly for modeling basic lattice symmetries. In this work, we present a coarse‐grained description of the hexagonal closed‐packed (HCP) lattice that supports lattice deformation and defects. It builds on recent developments of the APFC model and introduces specific modeling aspects for this crystal structure. After illustrating the general modeling framework, we show that the proposed approach allows for simulating relatively large three‐dimensional HCP systems hosting complex defect networks.

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\mathcal {B}_n

Section: Discussionmentioning

confidence: 99%

“…Moreover, we use a formulation supporting the definition of Bravais lattices with basis, featuring modified amplitude functions

\widetilde{\eta }_n= \mathcal {B}_n\eta _n

, where

\mathcal {B}_n

are complex constants defined as [26]

\begin{equation} \mathcal {B}_n = \sum _{j=1}^{J} e^{-\mathrm{i} \mathbf {k}_n \cdot \mathbf {R}_j}, \end{equation}

with J the number of atoms in the unit cell, and

\mathbf {R}_j

their positions. The energy functional in this framework can be written as [26]

\begin{equation} \begin{split} F_{\widetilde{\eta }}=\int _{\Omega } & {\left[A \sum _{n=1}^N (\widetilde{\eta }_n \mathcal {M}_{n}\widetilde{\eta }_n^* +\widetilde{\eta }_n^* \mathcal {M}_{n}\widetilde{\eta }_n)+ B\zeta _2+C\zeta _3+D\zeta _4 + f_\mathrm{h} \right]} \rm d\mathbf {r}, \end{split} \end{equation}

where ζ 2, 3, 4 are polynomials in the amplitudes whose terms depend on the lattice symmetry, defined as

\begin{matrix} ζ_{} \end{matrix}

…”

Section: Modelmentioning

confidence: 99%

Section: Apfc Modelmentioning

confidence: 99%

“…with 𝐽 the number of atoms in the unit cell, and 𝐑 𝑗 their positions. The energy functional in this framework can be written as [26]…”

Section: Apfc Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Amplitude phase‐field crystal model for the hexagonal close‐packed lattice

De Donno,

Salvalaglio

2023

Proc Appl Math and Mech

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Gradient elasticity in Swift–Hohenberg and phase-field crystal models

Benoit-Maréchal,

Salvalaglio

2024

Modelling Simul. Mater. Sci. Eng.

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The Swift-Hohenberg (SH) and Phase-Field Crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that inherently includes elasticity effects. This study addresses how gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths, is incorporated into SH and PFC models. After presenting the fundamentals of these theories and models, we first calculate the characteristic lengths for various lattice symmetries in an approximated setting. We then discuss numerical simulations of stress fields at dislocations and comparisons with analytic solutions within first and second strain-gradient elasticity. Effective GE characteristic lengths for the elastic fields induced by dislocations are found to depend on the free-energy parameters in the same manner as the phase correlation length, thus unveiling how they change with the quenching depth. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation and crystalline systems using SH and PFC models and, importantly, complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.

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Phase field crystal modeling of grain boundary structures in diamond cubic systems

Blixt,

Hallberg

2024

Phys. Rev. Materials

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Phase field crystal (PFC) modeling has proved to be a versatile numerical tool in the analysis of crystalline microstructures. Most often, however, the focus is put on bulk crystal behavior, while crystal defects such as grain boundaries (GBs) are less explored. This is, in particular, the case for crystal structures beyond fcc and bcc. In this work, the possibilities and challenges in adopting PFC to diamond cubic (DC) crystal structures is investigated. Three different PFC models are considered for this purpose. One of them was published previously, and two are modifications proposed in the present work. The models are compared in terms of both DC phase stabilization and their ability to provide relevant GB structures. The models employ combinations of two- and three-point correlations, and the addition of a three-point correlation is found to be required for stabilization of the expected DC GB structures. It is concluded that although each of the models has limitations in terms of the GB structures which can be stabilized and performance in terms of phase stability, key PFC components for successful modeling of DC structures can be identified. Published by the American Physical Society 2024

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Amplitude expansion of the phase-field crystal model for complex crystal structures

Cited by 3 publications

References 74 publications

Amplitude phase‐field crystal model for the hexagonal close‐packed lattice

Amplitude phase‐field crystal model for the hexagonal close‐packed lattice

Gradient elasticity in Swift–Hohenberg and phase-field crystal models

Phase field crystal modeling of grain boundary structures in diamond cubic systems

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