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Department of Physics and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115 

# Phase-field-crystal model for fcc ordering 

Kuo-An Wu ∗ , Ari Adland, and Alain Karma 

We develop and analyze a two-mode phase-field-crystal model to describe fcc ordering. The model is formulated by coupling two different sets of crystal density waves corresponding to 〈 111 〉 and 〈 200 〉 reciprocal lattice vectors, which are chosen to form triads so as to produce a simple free- energy landscape with coexistence of crystal and liquid phases. The feasibility of the approach is demonstrated with numerical examples of polycrystalline and (111) twin growth. We use a two-mode amplitude expansion to characterize analytically the free-energy landscape of the model, identifying parameter ranges where fcc is stable or metastable with respect to bcc. In addition, we derive analytical expressions for the elastic constants for both fcc and bcc. Those expressions show that a non-vanishing amplitude of [200] density waves is essential to obtain mechanically stable fcc crystals with a non-vanishing tetragonal shear modulus ( C$_{11}$ − C$_{12}$ ) / 2. We determine the model parameters for specific materials by fitting the peak liquid structure factor properties and solid density wave amplitudes following the approach developed for bcc [K.-A. Wu and A. Karma, Phys. Rev. B 76, 184107 (2007)]. This procedure yields reasonable predictions of elastic constants for both bcc Fe and fcc Ni using input parameters from molecular dynamics simulations. The application of the model to two-dimensional square lattices is also briefly examined. 

The phase-field-crystal (PFC) method has emerged as an attractive computational approach to simulate the evolution of crystalline patterns [1–6]. By resolving the crystal density field, it naturally incorporates defects and elastic interactions arising from localized and large scale distortions of this field, respectively. Moreover, this method can in principle be used to simulate microstruc- tural evolution on diffusive time scales that are much longer than typical time scales accessible by molecular dynamics (MD) simulations. 

Like classical density function theory (DFT), the PFC method is based on representing the free-energy of a ma- terial by a functional of its density [7–13]. However, clas- sical DFT and the PFC method use different functionals to achieve different goals. Classical DFT seeks a phys- ically realistic mean-field description of the crystal den- sity field n ( ⃗ r ) to reproduce quantitatively as accurately as possible the properties of a material. Since n ( ⃗ r ) is sharply peaked around mean atomic positions, this gen- erally requires a very large number of terms in the tra- ditional expansion of the number density as a sum of density waves 

where each ⃗ K$_{i}$ represents a different reciprocal lattice vector (RLV) in this unrestricted sum. In contrast, by us- ing a considerably simplified density functional, the PFC 

PACS numbers: 61.72.Mm,68.08.De,68.08.-p,81.16.Rf 

## I. INTRODUCTION AND SUMMARY 

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method essentially restricts this sum to a much smaller set of reciprocal lattice vectors in order to simulate effi- ciently the evolution of the crystal field on length and time scales as large as possible. Recent studies have shown that, despite this loss of realism, PFC models are able to reproduce quantitatively certain key proper- ties that influence microstructural evolution such as the crystal-melt interfacial free-energy [14, 15], the bulk mod- ulus [15], and grain-boundary energies [15], which have been computed for the test case of pure Fe. 

Despite this progress, the PFC method has only been developed for a small set of crystal structures. The orig- inal formulation of Elder et al. [1, 2] uses the same free- energy functional as the Swift-Hohenberg model of pat- tern formation [16, 17] of the form 

with the free-energy density 

where φ represents the crystal density field. This one- mode model essentially truncates the sum (1) to one set of RLVs with equal magnitude | ⃗ K$_{i}$ | = q$_{0}$ since higher K modes have much smaller amplitude. As a result, it favors crystal structures for which the principal RLVs can form “triads” (i.e. closed triangles), which include hexagonal and body-centered-cubic (bcc) ordering in two and three dimensions, respectively. Aside from favoring those structures, triad interactions are essential for solid- liquid coexistence. This is because in a weakly nonlinear expansion of the bulk free-energy density of the form, f = c$_{2}$u $^{2}$+ c$_{3}$u $^{3}$+ c$_{4}$u $^{4}$+ . . . (with u$_{i}$ = u for all principal RLVs), triads contribute a cubic term with a negative 

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