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Example: Random Tube Network

The mercury porosimetry-generated mapping can be tested using a simple lattice model with uniform connectivity. Consider a face-centered cubic (fcc) lattice, a regular lattice with 12 nearest-neighbor bonds per node [16]. Other regular three-dimensional lattices could be also be used, like simple cubic and body-centered cubic [16], with qualitatively similar results. Each of the 12 nearest neighbour bonds is taken to be an elliptical cylindrical tube, with semi- major axis a and semi-minor axis b. The value of a is taken to be a constant for the lattice, while b ( 0 < b < a ) is taken to be a random variable, independent and different for each tube, with probability density f(b) = 1/a.

The quantity p(b) is defined as the fraction of tubes with semi-minor axes greater than or equal to a given value b, so that p(0) = 1, and p(a) = 0. The quantity b_c is defined by the relation p(b_c) = p_c = 0.119, the threshold for bond percolation on the fcc lattice [17]. This means that all bonds with b > bc just form a connected pathway, since they make up the required fraction p_c of the lattice necessary for a continuous pathway to exist. With the above functional form for f(b), it is evident that p(b) = 1 − b/a, so that b_c = a(1 − p_c) = 0.881a. Using the definition of the effective diameter d_e in eq. (5), the value of d_c for this model is then d_c = 4ab_c / (a+b_c) = 1.873a.

With the use of effective medium theory, the macroscopic quantities k and σ can each be calculated analytically, via averages over f(b), with an accuracy of a few percent, thus avoiding the need for large numerical lattice simulations [18,19]. I will first calculate them using K_t and Σ_t, and then compare the result for the k/σ ratio with the same calculation done using K_e and Σ_e for the individual conductances. This will test how much the "exact" solution is affected by using the effective tube conductances instead of the true tube conductances. The effective medium equation is [18]

where the superscript indicates how the calculation was carried out (EMT = effective medium theory), and the subscript indicates what expression was used for the local tube conductances ("t" means eqs. (4) and (8), "e" means eqs. (7) and (9)). Thus using the effective local conductances instead of the true local conductances resulted in only a 12% difference in the k/σ ratio. This reflects a 1% difference in the K^EMT calculation and a 11.6% difference in that for Σ^EMT. The error is greater for the electrical conductance, as might have been expected from Fig. 1.

The next step is to evaluate the effect of using (K_e, Σ_e) vs. (K_t, Σ_t) in a Katz- Thompson calculation of k/σ. The main point is to determine what difference the effective mercury porosimetry network approximation has on the Katz-Thompson calculation, but the deviation from the "exact" effective medium theory result (k/σ)_t^EMT will also be discussed.

The Katz-Thompson calculation uses the quantity p(b) to construct lower bound functions for k and σ. Being lower bounds, these functions are then maximized to give the best results, and their ratio computed. The permeability and conductivity functions are given by [8]

where W is a constant involving lattice size and geomery parameters that relates the conductance (hydraulic or electrical) of individual tubes to the overall permeability or conductivity, t = 1.9 is the critical exponent for the percolation conduction problem in three dimensions [18,19], and K(b) = K_t or K_e, Σ(b)= Σ_t or Σ_e as defined in eqs. (3), (6), (7), and (8). The effective tubes have all been chosen to have the same length l_o [8].

The next step is to maximize k(b) and σ (b) independently, first using the true local conductances and then repeating the calculation with the effective quantities. When the ratios of k_max and σ_max are formed, the unknown constants W and l_o drop out, leaving an expression proportional to d_c². The values of b that maximize the two functions will not be the same, in general. It should be noted that, since p(b) is known exactly in this model, the maximization of k(b) and σ(b) can also be carried out exactly. The algebra is simple, and will not be given here.

Using p_c = 0.119 [17], the results are

where the superscript "K-T" indicates that the calculation was done with the Katz-Thompson method [3,4]. Eq. (13) shows that, whether one does the Katz-Thompson calculation on the real lattice or on the effective mercury porosimetry-generated network, there is only a 3% difference in the result for the k/σ ratio. One should notice that (k/σ)_e^K-T differs from (k/σ)_t^EMT by only 28%, which is in good agreement considering the typical 50% or more uncertainty in experimental measurements of k/σ [3,4]. Also, the numerical prefactor, 0.0138, for (k/σ)_e^K-T agrees within 17% with the prefactor, 0.0114, calculated for a general pore structure by Banavar and Johnson [8] by carrying out the Katz-Thompson calculation approximately, without using any knowledge of the (generally unknown) explicit form of p(b).