Phase-field investigation of the coarsening of porous structures by surface diffusion

Nano and microporous connected structures have attracted increasing attention in the past decades due to their high surface area, presenting interesting properties for a number of applications. These structures generally coarsen by surface diffusion, leading to an enlargement of the structure characteristic length scale. We propose to study this coarsening behavior using a phase-field model for surface diffusion. In addition to reproducing the expected scaling law, our simulations enable to investigate precisely the evolution of the topological and morphological characteristics along the coarsening process. In particular, we show that after a transient regime, the coarsening is self-similar as exhibited by the evolution of both morphological and topological features. In addition, the influence of surface anisotropy is discussed and comparisons with experimental tomographic observations are presented.

Figure 1

(a) Example of initial microstructure considered for volume fraction $\rho =50\%$. (b) Coarsened structure obtained at $tM/W^4=3600$. (c) Spherically averaged structure factors obtained from three different configurations along a simulation and the corresponding values of $k_0$ shown with dash lines (see text for explanations).

Figure 2

(a) Evolution of the characteristic length scale as function of simulation time obtained for various volume fractions. (b) Same graph with the $y$ axis elevated to the fourth power.

Figure 3

ISD obtained for the different volume fractions at the short times $t=240W^4/M$ (left column) and long simulation times $t=6000W^4/M$ (right column).

Figure 4

[(a) and (b)] Probability distribution of the average curvature $H$ rescaled with the characteristic curvature $\lambda$ at two different times and for volume fraction $\rho =50\%$ and $\rho =37.5\%$. (c) Temporal evolution of the standard deviation of the probability distribution depicted in (a) and (b). The symbols display the value average over ten independent simulations and the continuous lines a confidence interval.

Figure 5

(a) Time evolution of the connectivity obtained for $\rho =50\%$. (b) Rescale connectivity as a function of time for different volume fraction. The continuous lines represent confidence interval of one standard deviation.

Figure 6

[(a) and (b)] Structures obtained at $tM/W^4=1200$ without (a) and with (b) interface anisotropy. [(c) and (d)] Density of surface normal without (c) and with (d) interface anisotropy. (e) Time evolution of the characteristic length scale and rescaled connectivity for both isotropic (continuous lines) and anisotropic structures (dash lines).

Figure 7

(a) Tomographic imaging of a microporous FeCr sample with volume fraction $50\%$ obtained from liquid metal dealloying [51]. (b) Rescaled connectivity as a function of the size of the analyzed volume (see text for explanations). (c) Comparison of the average connectivity with numerical results.

Figure 8

(a) Slightly perturbed interface. (b) Exponential relaxation kinetics to equilibrium. (c) Growth rate as a function of wave vector of the perturbation, showing that dispersion relation ${\omega }_k=-M{k}^4$ is well verified.

Figure 9

Connectivity $(g-1)$ obtained from integration of the Gauss-Bonnet integral and from Euler formula for the gyroid structure as a function of the number of voxels in the system.