Phase-field-crystal model for ordered crystals

Alster, Eli; Elder, K. R.; Hoyt, J. J.; Voorhees, Peter W.

doi:10.1103/physreve.95.022105

Cited by 40 publications

(35 citation statements)

References 56 publications

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“…Just dropping the ∇ · [n∇Ln] term, as done by Huang et al [24], means that the dynamics is not equivalent to a DDFT with mobility proportional to ρ. On the other hand, the alternative is to expand the logarithm up to O(n 4 ) in (51) in order to provide a nonlinear stabilizing term ( 1 12 n 4 ) in the free energy (57). However, in the original formulation, the logarithm comes from the ideal gas term in (10), and leads to a linear diffusive term in the dynamics.…”

Section: H Summarymentioning

confidence: 99%

“…Throughout we have made the simplest choices in the approximations, but other authors have made many other choices. For example, the original PFC paper [1], as well as later papers [22,24,[50][51][52], included a twocomponent (binary) version of the PFC model. Recently, Wang et al [53] took a much closer look at c (3) and c (4) , expressing these in terms of isotropic tensors and so allowing these functions to introduce bond angle dependence into the free energy.…”

Section: H Summarymentioning

confidence: 99%

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Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

et al. 2019

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Phase field crystal (PFC) theory is extensively used for modelling the phase behaviour, structure, thermodynamics and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇ · [n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are: (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L by a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to −∞ whilst being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalised exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallisation, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artefacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution log ρ(x) (rather than for ρ(x)), is surprisingly accurate. This approach gives a tantalising hint that accurate PFC-type theories may instead be derived as theories for the field log ρ(x), rather than for the density profile itself.

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Section: H Summarymentioning

confidence: 99%

Section: H Summarymentioning

confidence: 99%

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

et al. 2019

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“…al. [41] who effectively define σ 2 ∼ T /T o . This is choice is more consistent with the temperature dependence of the Debye-Waller factor observed in experiments.…”

Section: A a New Density Functional Approach: Expanding Around The Vmentioning

confidence: 99%

Thermodensity coupling in phase-field-crystal-type models for the study of rapid crystallization

Kocher

Provatas

2019

Phys. Rev. Materials

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We self-consistently derive a formalism that couples a Phase Field Crystal (PFC) density field to thermal transport. It yields a theory for non-uniform transient temperature and density evolution, and includes local latent heat release during atomic rearrangements of the PFC density field. The basic formalism is applied to the original PFC model, demonstrating its capacity to capture heat transfer and recalescence in solidification. With an aim towards consistently incorporating temperature and other thermodynamic variables into PFC modelling, a new classical density field theory for solid/liquid/vapor systems is then derived. It presents a different approach to those used in the PFC literature while retaining the major advantages that have become the hallmark of PFC modelling; the new model is also based entirely on physical density, temperature and pressure scales. We end the paper by applying the thermal-density coupling formalism to this new multi-phase density functional theory/PFC model.

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“…It has be shown that the method is derivable from the fundamental classical density functional theory (CDFT) [30,31], with certain approximations based on the Ramakrishnan and Yussouff [32] free energy formalism. Since its inception, the method has evolved to model structural phase transformations [33][34][35][36], alloy systems [30,37,38], multiferroic composite materials [39], order-disorder systems [40], nucleation and polymorphism [41,42], amorphous or glass transitions [43,44], quasicrystals [45] and crystal plasticity [46]. Through coarse graining procedures, the methodology may be used to generate models that operate on mesoscopic length scales.…”

Section: Introductionmentioning

confidence: 99%

Self-consistent modeling of anisotropic interfaces and missing orientations: Derivation from phase-field crystal

Ofori-Opoku

Warren

Voorhees

2018

Phys. Rev. Materials

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Highly anisotropic interfaces play an important role in the development of material microstructure. Using the diffusive atomistic phase-field crystal (PFC) formalism, we determine the capability of the model to quantitatively describe these interfaces. Specifically, we coarse grain the PFC model to attain both its complex amplitude formulation and its corresponding phase-field limit.Using this latter formulation, in one-dimensional calculations, we determine the surface energy and the properties of the Wulff shape. We find that the model can yield Wulff shapes with missing orientations, the transition to missing orientations, and facet formation. We show that the corresponding phase-field limit of the complex amplitude model yields a self-consistent description of highly anisotropic surface properties that are a function of the surface orientation with respect to the underlying crystal lattice. The phase-field model is also capable of describing missing orientations on equilibrium shapes of crystals and naturally includes a regularizing contribution. We demonstrate, in two dimensions, how the resultant model can be used to study growth of crystals with varying degrees of anisotropy in the phase-field limit.

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Phase-field-crystal model for ordered crystals

Cited by 40 publications

References 56 publications

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Thermodensity coupling in phase-field-crystal-type models for the study of rapid crystallization

Self-consistent modeling of anisotropic interfaces and missing orientations: Derivation from phase-field crystal

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