Mean survival times of absorbing triply periodic minimal surfaces

Understanding the transport properties of a porous medium from a knowledge of its microstructure is a problem of great interest in the physical, chemical, and biological sciences. Using a first-passage time method, we compute the mean survival time $\tau$ of a Brownian particle among perfectly absorbing traps for a wide class of triply periodic porous media, including minimal surfaces. We find that the porous medium with an interface that is the Schwartz $P$ minimal surface maximizes the mean survival time among this class. This adds to the growing evidence of the multifunctional optimality of this bicontinuous porous medium. We conjecture that the mean survival time (like the fluid permeability) is maximized for triply periodic porous media with a simply connected pore space at porosity $\varphi =1/2$ by the structure that globally optimizes the specific surface. We also compute pore-size statistics of the model microstructures in order to ascertain the validity of a “universal curve” for the mean survival time for these porous media. This represents the first nontrivial statistical characterization of triply periodic minimal surfaces.

Figure 1

(Color online) Unit cells of three different minimal surfaces. (a) Schwartz $P$ surface, (b) Schwartz $D$ surface, (c) Schoen $G$ surface. Image is adapted from Ref. [16].

Figure 2

(Color online) Unit cells of two different pore-channel models. (a) Cubic pore-square channel and its cross section. (b) Spherical pore-circular channel and its cross section. Image is adapted from Ref. [16].

Figure 3

(Color online) Two-dimensional depiction of continuous first-passage time method applied to the spherical pore-circular channel model. The Brownian particle walks, by jumping to the surface of the largest possible first-passage sphere at each step, until it gets trapped at the two-phase interface (maroon boundary layer, which appears as dark gray if the image is being viewed in black and white).

Figure 4

(Color online) Pore-size density function data generated from the presented algorithm for the three minimal surfaces and the circular-channel model along with the best-fit fifth degree polynomial.

Figure 5

(Color online) The dimensionless mean survival time, $\tau /{\tau }_0$, versus the scaled mean pore size squared, ${⟨\delta ⟩}^2\left({\tau }_{0}D\right)$, for various two-phase media. The solid curve is the universal scaling relation (12) [28].