Phase-field modeling of isothermal quasi-incompressible multicomponent liquids

In this paper general dynamic equations describing the time evolution of isothermal quasi-incompressible multicomponent liquids are derived in the framework of the classical Ginzburg-Landau theory of first order phase transformations. Based on the fundamental equations of continuum mechanics, a general convection-diffusion dynamics is set up first for compressible liquids. The constitutive relations for the diffusion fluxes and the capillary stress are determined in the framework of gradient theories. Next the general definition of incompressibility is given, which is taken into account in the derivation by using the Lagrange multiplier method. To validate the theory, the dynamic equations are solved numerically for the quaternary quasi-incompressible Cahn-Hilliard system. It is demonstrated that variable density (i) has no effect on equilibrium (in case of a suitably constructed free energy functional) and (ii) can influence nonequilibrium pattern formation significantly.

Figure 1

Equilibrium contact angles in a four-component Cahn-Hilliard liquid. The spatial distribution of the scalar variable $h(\widehat{\mathbf{r}},\widehat{t}):={\sum }_{i=1}^4{c}_i(\widehat{\mathbf{r}},\widehat{t})[(i-1/2)/4]$ is displayed at $\widehat{t}=0,\widehat{t}=7.5$, and $\widehat{t}=12500$ on panels (a)–(c), respectively. The fluid flow is indicated by vectors on panel (b). Panels (d) and (e) show the $c_i(\widehat{\mathbf{r}},\widehat{t})=1/2$ contours (solid lines) in the small vicinity of the trijunctions indicated by the small black squares in panel (c), while the dashed lines indicate the corresponding contact angles.

Figure 2

Equilibrium trijunction in a four-component Cahn-Hilliard liquid. Panels (a)–(d) show the individual density fields ${\widehat{\rho }}_i\left(\widehat{\mathbf{r}}\right)$ ($i=1,\cdots ,4$, respectively) at the $(1,3,4)$ trijunction shown in Fig. 1. Note the lack of spurious phases.

Figure 3

Phase separation in a four-component Cahn-Hilliard liquid. Snapshots of simulations taken at time points $\tilde{t}=375,\tilde{t}=1500$, and $\tilde{t}=6000$ are shown (from top to bottom, respectively) in case of constant density (on the left) and variable density (on the right). The scalar field defined in the caption of Fig. 1 with the same color bar is shown.

Figure 4

Time evolution of the total free energy in phase separating four-component systems shown by Fig. 3 for constant (solid line) and variable (dash-dotted line) density.