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with 

Again we look in the limit R$_{1}$ ≈ 0. This yields the di- mensionless elastic constants 

These relations show that the one-mode crystal density field consisting only of a superposition of 〈 10 〉 density waves ( B$_{s}$ = 0) yield vanishing shear moduli, which be- come finite with the inclusion of the second mode. 

In summary, we have presented a two-mode PFC model with a phase-diagram that includes different tem- perature ranges for bcc-liquid and fcc-liquid coexistence. The relative sizes of these ranges can be changed by vary- ing one model parameter that controls the relative mag- nitudes of the amplitudes of the two modes, correspond- ing to [111] and [200] density waves, respectively. We have shown that the free-energy landscape for fcc-liquid coexistence has a double-well structure with a finite free- energy barrier between solid and liquid in the plane of the amplitudes of the two modes. We have demonstrated the feasibility of the model with some numerical examples of fcc polycrystalline growth and twin growth, as well as for two-dimensional square lattices. 

At a more quantitative level, we have determined the model parameters by fitting the peak liquid structure factor properties ( S ( q$_{0}$ ) and C ′′ ( q$_{0}$ )) and solid-density wave amplitudes as an extension of our previous study of bcc Fe [14]. Furthermore, we have derived analytical expressions for the elastic constants. With input values for those parameters from MD simulations of pure Ni, we have found that the PFC model elastic constants are in reasonable agreement with MD results given the sim- plicity of the model, which neglects the contributions of many other modes that are present in a realistic descrip- tion of the crystal density field. Those expressions also stress the necessity of having at least two distinct modes to obtain physically meaningful values of the elastic con- stants for fcc in the physically relevant small ϵ -limit of the PFC model, which is also true for square lattices. 

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## VII. CONCLUDING REMARKS 

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We have found that the standard one-mode PFC model also predicts reasonable values of the elastic constants for pure bcc Fe, and we have argued that any one- or two-mode model will predict similar elastic constants for bcc and fcc with the same peak liquid structure factor properties and solid density wave amplitudes 

Finally, while the numerical examples focused on crys- tal growth, it might also be possible to use the two-mode PFC model to study the bcc/fcc martensitic transforma- tion, which has been modeled by other phase-field ap- proaches that make use of structural order parameters (see Refs. [32, 33] and references therein). The abil- ity to vary the relative stability of fcc and bcc crystal structures, which was demonstrated here, should prove particularly useful for this application. 

This work was supported by DOE grant DE-FG02- 07ER46400 and the DOE sponsored Computational Ma- terials Science Network program. We thank Mark Asta and Jeff Hoyt for valuable exchanges and Stephen Foiles for providing values of the elastic constants computed from molecular dynamics simulations. 

We computed the coherent (111) twin boundary en- ergy using the method put forth in Ref. [25], which ex- ploits the dependence of the free-energy on system size. We performed simulations for four different lengths L$_{z}$ along the axis perpendicular to the boundary. By plot- ting the bulk free energy density f against the inverse of this length (Fig. 14), we then extracted the boundary energy from the slope of this plot using the relation 

where f$_{s}$ ( ¯ ψ ) is the free energy density of a perfect crystal. This method turns out to be more accurate than com- puting directly the excess free-energy of the boundary for a fixed system size [25]. This calculation gives twice the boundary energy since there are two boundaries in our periodic system. We convert the result to dimensional units through, γ$_{twin}$ = ( λ $^{2}$q15 0 /g )˜ γ$_{twin}$ , which yields the reasonable value of 29.9 mJ/m 2 for the same Ni param- eters used to compute the elastic constants. 

## Appendix A: Twin boundary energy 

## Acknowledgments 

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