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model (3) and the present two-mode model (4). We use a brute force approach that consists of calculating to quadratic order the change of solid free-energy den- sity, modeled by a one- or two-mode approximation for bcc and fcc, respectively, due to small dilation or shear transformations of the unit cell. We have checked that we obtain identical expressions to those derived recently by Spatschek and Karma for general lattices using an am- plitude equation framework [22], which provides a non- trivial self-consistent test of our calculations. For the one-mode bcc model (3), the elastic constants are 

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where q$_{0}$ = | ⃗ K$_{110}$ | . For the two-mode fcc model (4), 

and 

where q$_{0}$ = | ⃗ K$_{111}$ | and R$_{1}$ = 0 for simplicity. Using values of C ′′ ( q$_{0}$ 

) and density wave amplitudes from molecular dynamics simulations for parameters of bcc Fe and fcc Ni, we find that the above expressions give reasonable estimates of elastic constants (e.g., C$_{11}$ ≈ 90 GPa for one-mode bcc PFC model compared to C$_{11}$ ≈ 128 GPa in MD Fe and C$_{11}$ ≈ 106 GPa for the two- mode fcc PFC model compared to C$_{11}$ ≈ 155 GPa in MD Ni). The predicted values generally tend to be lower than the constants computed from MD simulations, but such discrepancies are to be expected given the PFC models are based on one or two modes. 

The analytical predictions for the elastic constants al- low us to draw two important general conclusions per- taining to the development of PFC models for different crystal structures and to the method used to determine the parameters of those models. 

The first conclusion, which follows directly from Eqs. (6) and (7), is that the presence of the second mode, which corresponds to [200] density waves, is essential to obtain a physically meaningful set of elastic constants for fcc. Without this second mode ( u$_{200}$ = 0), Eqs. (6) and (7), predict that C$_{11}$ = C$_{12}$ = C$_{44}$ . This implies that the tetragonal shear modulus C ′ = ( C$_{12}$ − C$_{22}$ ) / 2 vanishes, and that the system is mechanically marginally stable. Of course, these analytical expressions for the elastic con- stants neglects the contributions of higher modes that are present in a full solution of the PFC equations. However, those higher modes are generally small for the small val- ues of ϵ corresponding to Fe and Ni parameters. There- fore, the contributions of those modes will generally be small and will not change qualitatively this picture. 

While it is in principle possible to select energetically different crystal structures in the PFC model with the addition of other nonlinearities in the free-energy density 

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(such as |∇ φ | 4 and φ $^{2}$| ∇ φ | $^{2}$) [23], this approach will be of limited applicability for crystal structures like fcc where one mode does not suffice to produce the correct elastic properties. This is also true for simple cubic lattices and two-dimensional square lattices. The latter are briefly examined in section VI by coupling 〈 10 〉 and 〈 11 〉 density waves. 

The second conclusion, which is general, is that the elastic constants are uniquely determined once the phase- field model parameters have been fitted to the peak liquid structure properties, which fixes C ′′ ( q$_{0}$ ), and the solid density wave amplitudes, as in the approach of Wu and Karma [14] summarized above. This also fixes the value of the elastic bulk modulus 

In general, the bulk modulus can also be defined from the thermodynamic relation 

where F is the total free-energy, V is the volume, and n = N/V is the number density. The second equality in the last equation can in principle be used to compute the bulk modulus directly from the PFC solid free-energy curve ( F/V versus n ), without computing the elastic con- stants. For a perfect crystal without vacancy, Eqs. (8) and (9) should in principle predict the same bulk modu- lus. However, the two definitions can give different pre- dictions for the PFC model because the number of atoms per peak of the crystal density field is not constrained to unity. While the average number of atoms per peak will also differ from unity in a real crystal with vacancies, thereby altering the open-system elastic constants [24], the vacancy concentration is generally very small even at melting. How to meaningfully relate the predictions of Eqs. (8) and (9) for the bulk modulus is unclear in the PFC approach that, by construct, does not use a realistic description of the crystal density field, and also does not model vacancy formation explicitly. 

Despite these limitations of the PFC approach, Jaati- nen et al. [15] have recently proposed a modified one- mode PFC model to remedy the fact that, for the stan- dard one-mode PFC model with the free-energy density (3), the bulk modulus predicted by Eq. (9) is several times smaller than the experimental value for parame- ters of bcc Fe. Their model yields a value of the bulk modulus computed through Eq. (9) that is in better agreement with experiment and also gives an improved prediction of the density difference between solid and liq- uid. It gives similar predictions of crystal-melt interfacial free-energies for bcc Fe as obtained previously by Wu and Karma using the standard one-mode model [14]. 

In the light of Eq. (5), it is apparent that any one- mode model that fits the correct peak structure factor properties and solid density wave amplitudes should pre- dict the same elastic constants. This is consistent with 

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