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The pore volume to pore surface area ratio h is the first and simplest length scale of interest. In this paper, h is actually defined by twice the pore area to pore perimeter ratio, or 2Vp/S. Quantities such as h sometimes go by the name of hydraulic radius [1]. The use of h as a permeability length scale is motivated by the fact that the value of h in 3D for a circular cylindrical tube of radius R and length L is just 2 (πR2 L)/ (2 πRL) = R , which is obviously the length scale that controls the permeability of the tube. In 2D, h is equal to the tube width d. Physical assumptions that underlile the validity of using h to predict permeability include [1] that (1) the variation of pore size is small, and (2) the fluid-flow field is uniform throughout the pore space.
The second length scale that we consider is the Λ parameter [3], defined in the following way. If a material's pore space is filled by a conducting fluid with conductivity σo, and a potential drop is applied, then Λ is defined by the following ratio of integrals [3]:
where E(r) is the magnitude of the electrical field in the pore space, dVp is a volume element in the pore space, and dS is a surface element on the pore-solid interface. If the electric field were constant everywhere, like that in a circular cylindrical tube when the potential gradient is along the axis of the tube, then Λ = h. Since in a real porous medium, this would not be the case, Λ can be thought of as a dynamically weighted hydraulic radius [5], where regions of small electric field, which would probably have small fluid-flow velocities as well, contribute less than high-field regions. In particular, stagnant regions with little or no flow and fully isolated pores would contribute negligibly to Λ. How well pore space regions with low electric current flow rates actually match up with low fluid-flow rates will be studied in Sec. 6.
The third length scale we consider is denoted as dc, and is defined [4] by a non-wetting fluid injection experiment, typically using mercury, which is nonwetting for most porous materials of interest. Mercury is injected into an evacuated sample under hydrostatic pressure, and the additional amount of mercury intruded for each increase in pressure is monitored. The pressure at which the mercury first forms a continuous, percolating path through the sample is defined as the critical pressure Pc. The continuity of the mercury is monitored electrically [4,9]. If the solid phase of the porous material is an insulator, and the pore space is evacuated, then a non-zero bulk conductivity would only be achieved when the merucry percolates. The critical value of pressure Pc is converted into a critical pore size dc using the Washburn equation:
where γ is the surface tension of mercury, and θ is the contact angle of mercury on the solid phase of the porous material of interest. Equation (6) assumes a circular cylindrical local geometry for the pores [11].
Physically, the length scale dc can be thought of as the smallest member of a subset of the pore space containing the largest pores that form a continuous pathway through the pore space [4,9]. This is because, as can be seen in Eq. (6), the injected mercury fills the largest pores at the lowest pressures, and then fills progressively smaller pores as the injection pressure is increased.