the fact that Eq. (5) predicts a shear modulus C$_{44}$ ≈ 45 GPa for the standard one-mode model of bcc Fe, which is reasonably close to the value C$_{44}$ ≈ 53 GPa estimated by Jaatinen et al. [15] from numerical shearing experiments in their model for similar input parameters. Since elastic constants are a major determinant of grain boundary energies and long-range interactions be- tween crystal defects, reproducing those constants, and hence the bulk modulus predicted by Eq. (8), appears essential for modeling microstructural evolution. Also re- quiring that Eq. (9) predicts the correct bulk modulus using the solid free-energy curve may appear desirable. However, the motivation for doing so in the context of simple PFC models is somewhat less clear given the lack of realism of the crystal density field and the fact that Eqs. (5)-(7) predict reasonable values of the elastic con- stants. In fact, any one- or two-mode model with the same peak liquid structure factor properties and density wave amplitudes will predict essentially the same elastic constants associated with the free-energy cost of lattice distortions, and also the same interfacial energies as can be inferred from amplitude equations [14]. Since those elastic constants and interfacial energies are the quanti- ties that matter most for modeling microstructural evo- lution in a PFC context, we have not found it necessary to formulate the two-mode PFC model in such a way that the bulk modulus is also correctly predicted from the solid free-energy curve using Eq. (9). Accordingly, we follow essentially the same approach outlined in Ref. [14] for determining the PFC model parameters. where we set Q$_{1}$ = | ⃗ K$_{200}$ | / | ⃗ K$_{111}$ | = $^{√}$4 / 3 for fcc (i.e. Q$_{1}$ equal to the ratio of the magnitudes of the 〈 200 〉 and 〈 111 〉 RLVs). We also define the dimensionless variables Substituting the above definitions into Eqs. (2) and (4) yields the dimensionless form with the free-energy functional and free-energy density where we have dropped the prime symbol on the dimen- sionless spatial coordinate vector ⃗ r ′ for brevity. Even though most of the paper focuses on the two-mode model, we also compute in section V the elastic constants for the standard one-mode PFC model. For this model, we use the same scaling as in Ref. [14] with the parameter The PFC equations have the standard form for con- served dynamics where F is the free-energy functional defined by Eq. (2) with the free-energy densities given by Eqs. (3) and (4) for the one- and two-mode models, respectively. To min- imize the number of parameters, it is useful to rewrite the equations in dimensionless form. For the two-mode model, we define the dimensionless parameters where ⃗ r ′ is defined by Eq. (14). Substituting the above forms into Eqs. (2) and (3) yields (after dropping the prime symbol on ⃗ r ′ ) the dimensionless form of the one- mode PFC equations (18) and (19) with and dimensionless variables 4 ⃗ r ′ = q$_{0}$ ⃗ r, (14) ψ = √ g λq 8 0 φ, (15) t = Γ λq 7 $_{0}$τ, (16) F = g λ $^{2}$q 13 0 F . (17) ∂ψ ∂t = ∇ $_{2}$δF δψ , (18) F = ∫ d⃗ rf ( ψ ) , (19) f = ψ 2 $^{[}$− ϵ + ( ∇ 2 + 1) $^{2}$(( ∇ 2 + Q$_{1}$ $^{2}$) 2 + R$_{1}$ ) ] ψ + ψ 4 4 , (20) ∂φ ∂τ = Γ ∇ $_{2}$δ F δφ , (10) ϵ = − a λq 8 0 , (11) R$_{1}$ = r$_{1}$ q 4 0 , (12) Q$_{1}$ = q$_{1}$ q$_{0}$ , (13) f = ψ 2 $^{[}$− ϵ + ( ∇ 2 + 1) 2 ] ψ + ψ 4 4 . (25) F = g λ $^{2}$q 5 0 F , (24) t = Γ λq 3 $_{0}$τ, (23) ψ = √ g λq 4 0 φ, (22) ϵ = − a λq 4 0 , (21) II. PHASE-FIELD CRYSTAL MODEL A. Basic equations and scalings