FIG. 13: (Color online) Example of polycrystalline solidifica- tion for two-dimensional square lattices. The snapshots are at dimensionless times t = 10 , 100 , and 1000. The parameters are ϵ = 0 . 15, R$_{1}$ = 0, Q$_{1}$ = √ 2, and ¯ ψ = − 0 . 23. 14 TABLE III: Input parameters for the PFC model computed from MD simulations of pure Fe [26] using the EAM potential from MH(S A) 2 [29] and corresponding PFC parameters. MD input parameters Value PFC parameters Value q$_{0}$ = K$_{110}$ ( ˚ A − $^{1}$) 2 . 985 λ (eV ˚ A $^{7}$) 0 . 291 n$_{0}$ ( ˚ A − $^{3}$) 0 . 0765 g (eV ˚ A $^{9}$) 9 . 705 C $^{′′}$( q$_{0}$ ) ( ˚ A $^{2}$) − 10 . 40 ϵ 0 . 0923 S ( q$_{0}$ ) 3 . 012 u$_{110}$ 0 . 72 T$_{M}$ (K) 1771 u$_{110}$ 0 . 72 VI. TWO-DIMENSIONAL SQUARE LATTICE As an example of application of the two-mode model to other lattice structures, we briefly examine the exam- ple of t wo-dimensional square lattices. Those lattices are obtained by coupling 〈 10 〉 and 〈 11 〉 density waves with Q$_{1}$ = √$_{2, as demonstrated previously by Lifshitz and}$ Petrich [18] for a modifed Swift-Hohenberg model that corresp onds to the R$_{1}$ = 0 limit of the present two-mode model. The liquid free-energy density is given by and the solid free-energy density is obtained by substi- tuting the two-mode crystal density field into the free-energy functional defined by Eqs. (19) and (20) with Q$_{1}$ = √$_{2 , which yields}$ For R$_{1}$ = 0, we obtain ψ$_{c}$ = − 0 . 6782 numerically from a log-log plot of the mean equilibrium density versus ϵ similar to Fig. 6 which is determined from the common tangent construction. The feasibility of the two-model to model p olycrystalline solidification and grain boundaries is illust rated in Fig. 13. As for fcc, the second mode turns out to be essential to obtain physically meaningful elastic constan ts. Following the same procedure as for fcc in the last section (with deformations of the unit cell now constrained to the x − y plane) we obtain Δ f sq 3 = ˜ C$_{44}$ 2 ξ 2 = δ$_{sq}$ξ $^{2}$, (87) Δ f sq 2 = ( ˜ C$_{11}$ − ˜ C$_{12}$ ) ξ 2 = ( α$_{sq}$ + Λ$_{sq}$) ξ $^{2}$, (86) Sunday, November 22, 2009 Δ f sq 1 = $^{(}$˜ C$_{11}$ + ˜ C$_{12}$ $^{)}$ξ 2 = ( α$_{sq}$ + β$_{sq}$ ) ξ $^{2}$, (85) f sq s ( ¯ ψ$_{s}$ ) = 2( − ϵ + 3 ¯ ψ 2 $_{s}$) A$_{s}$ 2 + 2( − ϵ + 3 ¯ ψ 2 s + R$_{1}$ ) B$_{s}$ 2 +24 ¯ ψ$_{s}$A$_{s}$ $^{2}$B$_{s}$ + 36 A$_{s}$ $^{2}$B$_{s}$2 + 9 A$_{s}$ 4 + 9 B$_{s}$ 4 − ϵ 2 ¯ ψ 2 s + R$_{1}$ 2 ¯ ψ 2 s + 2 ¯ ψ 2 s + 1 4 ¯ ψ 4 $_{s}$. (84) ψ ( ⃗ r ) ≈ ¯ ψ + 2 A$_{s}$ (cos x + cos y ) + 4 B$_{s}$ (cos x cos y ) (83) f sq l ( ¯ ψ$_{l}$ ) = − ( ϵ − 4 − R$_{1}$ ) ¯ ψ 2 l 2 + ¯ ψ 4 l 4 , (82)