6 Physical Properties

[235.1.3] One of the main goals in studying the microstructure of porous media is to identify geometric observables that correlate strongly with macroscopic physical transport properties. [235.1.4] To achieve this it is not only necessary to evaluate the geometric observables. [235.1.5] One also needs to calculate the effective transport properties exactly, in order to be able to correlate them with geometrical and structural properties. [235.1.6] Exact solutions are now becoming available and this section reviews exact results obtained recently in cooperation with J. Widjajakusuma [10, 65, 67]. [235.1.7] For the disordered potential problem, specified above in equations (2) through (7), the effective macroscopic transport parameter C¯ is defined by

where the brackets denote an ensemble average over the disorder defined in (25). [235.1.8] The value of C¯ can be computed numerically [66, 33]. [235.1.9] For the following results the material parameters were chosen as

[235.1.10] Thus in the usual language of transport problems the pore space is conducting while the matrix space is chosen as nonconducting. [235.1.11] Equations (2) through (7) need to be supplemented with boundary conditions on the surface of S. [235.1.12] A fixed potential gradient was applied between two parallel faces of the cubic sample S, and no-flow boundary condition were enforced on the four remaining faces of S.

[235.2.1] The macroscopic effective transport properties are known to show strong sample to sample fluctuations. [235.2.2] Because calculation of C¯ requires a disorder average the four microsctructures were subdivided into eight octants of size 128×128×128. [235.2.3] For each octant three values of C¯ were obtained from the exact solution corresponding to application of the potential gradient in the x-, y- and z-direction. [235.2.4] The values of C obtained from dividing the measured current by the applied potential gradient were then averaged. [235.2.5] Table 3 collects the mean and the standard deviation from these exact calculations. [235.2.6] The standard

[page 236, §0]    deviations in Table 3 show that the fluctuations in C¯ are indeed rather strong. [236.0.1] If the system is ergodic then one expects that C¯ can also be calculated from the exact solution for the full sample. [236.0.2] For sample  EX  the exact transport coefficient for the full sample is C¯x=0.02046 in the x-direction, C¯y=0.02193 in the y-direction, and C¯z=0.01850 in the z-direction [65]. [236.0.3] All of these are seen to fall within one standard deviation of C¯. [236.0.4] The numerical values have been confirmed independently by [47].

[236.1.1] Finally it is interesting to observe that C¯ seems to correlate strongly with p3⁢L shown in Figure 11. [236.1.2] This result emphasizes the importance of non-Hadwiger functionals because by construction there is no relationship between C¯ and porosity, specific surface and correlation functions.

[236.1.3] According to the general criteria discussed above in Section 3.1 a geometrical characterization of random media should be usable in approximate calculations of transport properties. [236.1.4] In practice the full threedimensional microstructure is usually not available in detail, and only approximate calculations can be made that are based on partial geometric knowledge.

[236.2.1] Local porosity theory [27, 28] was developed as a generalized effective medium approximation for C¯ that utilizes the partial geometric characterization contained in the quantities μ and λ. [236.2.2] It is therefore useful to compare the predictions from local porosity theory with those from simpler mean field approximations. [236.2.3] The latter will be the Clausius-Mossotti approximation with P as background phase

[page 237, §0]    the Clausius-Mossotti approximation with M as background phase

and the self-consistent effective medium approximation [37, 35]

which leads to a quadratic equation for C¯. [237.0.1] The subscripts b and c in (71) and (70) stand for "blocking" and "conducting". [237.0.2] In all of these mean field approximations the porosity ϕ¯ is the only geometric observable representing the influence of the microstructure. [237.0.3] Thus two microstructures having the same porosity ϕ¯ are predicted to have the same transport parameter C¯. [237.0.4] Conversely, measurement of C¯ combined with the knowledge of CM,CP allows to deduce the porosity from such formulae.

[237.1.1] If the microstructure is known to be homogeneous and isotropic with bulk porosity ϕ¯, and if CP>CM, then the rigorous bounds [24, 8, 62]

hold, where the upper and the lower bound are given by the Clausius-Mossotti formulae, eqs. (71) and (70). [237.1.2] For CP<CM the bounds are reversed.

[237.2.1] The proposed selfconsistent approximations for the effective transport coefficient of local porosity theory reads [27]

where C¯b⁢ϕ and C¯c⁢ϕ are given in eqs. (71) and (70). [237.2.2] Note that (74) is still preliminary, and a generalization is in preparation. [237.2.3] A final form requires generalization to tensorial percolation probabilities and transport parameters. [237.2.4] Equation (74) is a generalization of the effective medium approximation. [237.2.5] In fact, it reduces to eq. (72) in the limit L→0. [237.2.6] In the limit L→∞ it also reduces to eq. (72) albeit with ϕ¯ in eq. (72) replaced with λ3⁢ϕ¯. [237.2.7] In both limits the basic assumptions underlying all effective medium approaches become invalid. [237.2.8] For small L the local geometries become strongly correlated, and this is at variance with the basic assumption of weak or no correlations. [237.2.9] For large L on the other hand the assumption that the local geometry is sufficiently simple becomes invalid [27]. [237.2.10] Hence one expects that formula (74) will yield good results only for intermediate L. [237.2.11] The question which L to choose has been discussed in the literature

[page 238, §0]    [12, 3, 10, 66, 33]. [238.0.1] For the results in Table 4 the so called percolation length Lp has been used which is defined through the condition

assuming that it is unique. [238.0.2] The idea behind this definition is that at the inflection point the function p3⁢L changes most rapidly from its trivial value p3⁢0=ϕ¯ at small L to its equally trivial value p3⁢∞=1 at large L (assuming that the pore space percolates). The length Lp is typically larger than the correlation length calculated from G⁢r[10, 11].

[238.1.1] The results obtained by the various mean field approximations are collected in Table 4 [65, 67]. [238.1.2] The exact result is obtained by averaging the three values for the full sample  EX  given in the previous section. [238.1.3] The additional geometric information contained in μ and λ seems to give an improved estimate for the transport coefficient.