Consistent multiphase-field theory for interface driven multidomain dynamics

We present a multiphase-field theory for describing pattern formation in multidomain and/or multicomponent systems. The construction of the free energy functional and the dynamic equations is based on criteria that ensure mathematical and physical consistency. We first analyze previous multiphase-field theories and identify their advantageous and disadvantageous features. On the basis of this analysis, we introduce a way of constructing the free energy surface and derive a generalized multiphase description for arbitrary number of phases (or domains). The presented approach retains the variational formalism, reduces (or extends) naturally to lower (or higher) number of fields on the level of both the free energy functional and the dynamic equations, enables the use of arbitrary pairwise equilibrium interfacial properties, penalizes multiple junctions increasingly with the number of phases, ensures non-negative entropy production and the convergence of the dynamic solutions to the equilibrium solutions, and avoids the appearance of spurious phases on binary interfaces. The approach is tested for multicomponent phase separation and grain coarsening.

Figure 1

Time evolution in an $N=5$ symmetric Cahn-Hilliard problem with starting conditions $u_1=u_2=u_3=u_4=u_5=0.2$. Different colors stand for different phases (where $u_i>0.99)$. Snapshots taken at $t=(1,3,10,30,100,300)\times 25000$ dimensionless time are displayed. (Time increases from left to right and from top to bottom.)

Figure 2

$N=5$ symmetric Cahn-Hilliard problem with starting conditions $u_1=u_2=0.5$ and $u_3=u_4=u_5=0$. (a),(b) Snapshots of the phase fields $u_1$ and $u_2$ at $t=2.5^5$ dimensionless time; (c) in the left half panel $|u_3|+|u_4|+|u_5|$ is shown, whereas the right half panel shows ${\sum }_{i=1}^5{u}_i-1$. Note the absence of spurious phases [criterion (ii)] and that the sum of the phase fields is 1 [criterion (i)]. (d) Multiphase-field map at the same instant.

Figure 3

$N=5$ symmetric Cahn-Hilliard problem with starting conditions $u_1=u_2=u_3=u_4=0.25$ and $u_5=0$. (a),(b) Snapshots of the phase fields $u_1,u_2,u_3$, and $u_4$ at $t={10}^6$ dimensionless time; (e) in the left half panel $u_5$ is shown, whereas the right half panel shows ${\sum }_{i=1}^5{u}_i-1$. Note the absence of spurious phases [criterion (ii)] and that the sum of the phase fields is 1 [criterion (i)]. (f) Multiphase-field map at the same instant.

Figure 4

$N=4$ asymmetric Cahn-Hilliard problem with initial condition $\mathbf{u}=\{1/3,1/3,1/3,0\}$. The results on the left were obtained with a Lagrangian mobility matrix, whereas those on the right were obtained using the Bollada-Jimack-Mullis-type mobility matrix. The top four rows [panels (a)–(h)] show the maps for $u_1,u_2,u_3$, and $u_4$, respectively, whereas in the bottom row [panels (i) and (j)] phase distribution maps are displayed. Results corresponding to $t={10}^6$ dimensionless time are presented. Note that using the Lagrangian mobility matrix spurious phase generation in the vicinity of the phase boundaries could not be avoided for $u_4$ [see panel (g)], whereas the Bollada-Jimack-Mullis mobility matrix suppresses the formation of spurious phases entirely [see panel (h)].

Figure 5

$N=4$ asymmetric Cahn-Hilliard problem with anisotropy. (a) Time evolution towards the equilibrium shapes of $u_1$ and $u_2$ embedded into $u_3$ at $t=3.125\times {10}^6$ dimensionless time. (b)–(e) Snapshots of the phase fields $u_1,u_2,u_3,$ and $u_4$, respectively, taken at $t=2\times {10}^5$ dimensionless time. Note the lack of third-phase generation at the phase boundaries and that $u_4$ remains absent even at the phase boundaries.

Figure 6

Snapshot of grain/phase map taken at dimensionless rime, $t={10}^4$, for a simulation relying on misorientation-dependent grain boundary energy obeying the Read-Shockley relationship [51]. The simulation was performed on a ${4096}^2$ rectangular grid. About 4000 grains can be distinguished at this stage.

Figure 7

Limiting grain size distributions (LGSDs): (a) for the isotropic simulation on a ${8192}^2$ grid. For comparison, experimental results [53] for metallic films (solid line, lognormal distribution fitted to experimental data [53]) and predictions by the theories of Mullins [55] and Hillert [56] (dashed and dotted lines, respectively) are also shown. (b) Comparison of the LGSD from XMPF (histogram) with predictions from previous MPFs by Kim et al. [27] (triangles) and Schaffnit et al. [52] (circles). The solid line indicates a lognormal fit to the experimental results [53]. (c) The same as panel (b), except that LGSD from an asymmetric XMPF model (the simulation shown in Fig. 6) is displayed (histogram), in which the grain boundary energy follows the Read-Shockley relationship [51]. Note that some improvement relative to the previous MPF models has been achieved, but the population of the small grains is larger than desirable.

Figure 8

$u\left(z\right)=[1+sin(z\left)\right]/2$ models. (a) Initial condition (top panel); (b)–(g) final states after ${10}^6$ time steps. From second to bottom row, respectively: the model of Nestler and Wheeler $p=1$ [17, 19, 20], the model of Steinbach and Pezzola [13], and the model of Steinbach and Pezzola with nonvariational dynamics [29]. (Left column, final states; right column, difference of final state and initial condition.)

Figure 9

$u\left(z\right)=[1+tanh(z/2\left)\right]/2$ models. (a) Initial condition (top panel); (b)–(g) final states after $2.5\times {10}^5$ time steps. From second to bottom row, respectively: the model of Steinbach et al.  [12] (coincides with the model of Nestler and Wheeler for $p=2$ [16, 17, 18, 19, 20]), the model of Steinbach et al. with nonvariational dynamics [12], and the XMPF model proposed in this work. (Left column, final states; right column, difference of final state and initial condition.)

Figure 10

Comparison of the time evolution of phase transition (a),(c) in the nonvariational model of Steinbach et al. [12] (left) and (b),(d) in the present model (right). Snapshots of one of the fields are displayed. The time dependence of the free energy is shown in panel (e): dashed line, model of Steinbach et al.; solid line, the XMPF model. The symbols correspond to snapshots shown in panels (a)–(d).