[ ¯ 1 ¯ 12] [1 ¯ 10] [111] 6 FIG. 4: Simulation of an equilibrium coherent (111) twin boundary for R = 0  ϵ = 0 . 00823  and ¯ ψ = − 0 . 06269. FIG. 3: Simulation of polycrystalline solidification starting from three seeded fcc crystals in a supercooled liquid. The system is fully periodic  and the snapshots are taken at dimen- sionless times t = 5 × 10 $^{2}$  3 × 10 $^{3}$  and 10 $^{5}$. The parameters are R = 0  ϵ = . 00823  and ¯ ψ = − 0 . 06. The first example in Fig. 3 shows the growth of small fcc crystallites of different orientations for a value of ¯ ψ = − 0 . 06 > ¯ ψ$_{s}$ that is well inside the stable fcc-solid region of the phase diagram. The crystallites grow as ex- pected until they collide to form grain boundaries. The second example in Fig. 4 shows a (111) twin crystal for a value of ¯ ψ = − 0 . 06269 at coexistence and for a system size chosen such that a twin crystal with two stacking faults fits perfectly the periodic boundary conditions in all directions without any liquid present. A computa- tion of the excess free-energy of this twin boundary given in the appendix to this paper yields a value of approx- imately 30 mJ/m 2 that falls within the range of values typically reported in the literature for fcc metals. Fig. 5 then shows the growth of the same twin crystal in a super- cooled liquid for a much larger system with ¯ ψ = − 0 . 06. We now demonstrate the feasibility of the model with some numerical examples of fcc polycrystalline growth and (111) twin growth. The PFC conserved dynamics governed by Eq. (18) with the free-energy defined by Eqs. (19) and (20) was solved using the semi-implicit pseudo-spectral scheme given by Eq. (A2) in Appendix A of Ref. [25]. We used the parameters R = 0 and ϵ = 0 . 00823 obtained from our fit of pure Ni presented later in section IV  together with the grid spacing Δ x = Δ y = Δ z = 2 $_{π}$√$_{3}$$_{/}$$_{16  which determines the number of Fourier}$ modes  and the time step Δ t = 0 . 5. For this value of R and ϵ   the computations presented in the next section show that the size of the solid-liquid coexistence region is extremely small  i.e. ¯ ψ$_{s}$ − ¯ ψ$_{l}$ is two orders of magnitude smaller than ( ¯ ψ$_{s}$ + ¯ ψ$_{l}$ ) / 2 as can already be seen from the phase diagram in Fig. 2  and ¯ ψ$_{s}$ ≈ ¯ ψ$_{l}$ ≈ − 0 . 0627. C. Numerical examples