  3D bicontinuous two-phase materials are increasingly gaining interest because
of their unique multifunctional characteristics and advancements in techniques
to fabricate them. Due to their complex topological and structural properties,
it still has been nontrivial to develop explicit microstructure-dependent
formulas to predict accurately their physical properties. A primary goal of the
present paper is to ascertain various microstructural and transport
characteristics of five different models of triply periodic bicontinuous porous
materials at a porosity $\phi_1=1/2$: those in which the two-phase interfaces
are the Schwarz P, Schwarz D and Schoen G minimal surfaces as well as two
different pore-channel structures. We ascertain their spectral densities,
pore-size distribution functions, local volume-fraction variances, and
hyperuniformity order metrics and then use this information to estimate certain
effective transport properties via closed-form microstructure-property
formulas. Specifically, we estimate the recently introduced time-dependent
diffusion spreadability exactly from the spectral density. Moreover, we
accurately estimate the fluid permeability of such porous materials from the
second moment of the pore-size function and the formation factor, a measure of
the tortuosity of the pore space. We also rigorously bound the permeability
from above using the spectral density. For the five models with identical cubic
unit cells, we find that the permeability, inverse of the specific surface,
hyperuniformity order metric, pore-size second moment and long-time
spreadability behavior are all positively correlated and rank order the
structures in exactly the same way. We also conjecture what structures maximize
the fluid permeability for arbitrary porosities and show that this conjecture
must be true in the extreme porosity limits by identifying the corresponding
optimal structures.
