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# 6 Physical Properties

## 6.1 Exact Results

[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 is defined by

 (68)

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

 (69)

[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 . [235.1.12] A fixed potential gradient was applied between two parallel faces of the cubic sample , and no-flow boundary condition were enforced on the four remaining faces of .

[235.2.1] The macroscopic effective transport properties are known to show strong sample to sample fluctuations. [235.2.2] Because calculation of requires a disorder average the four microsctructures were subdivided into eight octants of size . [235.2.3] For each octant three values of were obtained from the exact solution corresponding to application of the potential gradient in the -, - and -direction. [235.2.4] The values of 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 are indeed rather strong. [236.0.1] If the system is ergodic then one expects that 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 in the -direction, in the -direction, and in the -direction [65]. [236.0.3] All of these are seen to fall within one standard deviation of . [236.0.4] The numerical values have been confirmed independently by [47].

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

Table 3: Average and standard deviation for effective macroscopic transport property calculated from subsamples (octants) for and .
0.01880 0.01959 0.00234 0.00119
0.00852 0.00942 0.00230 0.00234

## 6.2 Mean Field Results

[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 that utilizes the partial geometric characterization contained in the quantities and . [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 as background phase

 (70)

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

 (71)

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

 (72)

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

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

 (73)

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

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

 (74)

where and 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 . [237.2.6] In the limit it also reduces to eq. (72) albeit with in eq. (72) replaced with . [237.2.7] In both limits the basic assumptions underlying all effective medium approaches become invalid. [237.2.8] For small 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 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 . [237.2.11] The question which 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 has been used which is defined through the condition

 (75)

assuming that it is unique. [238.0.2] The idea behind this definition is that at the inflection point the function changes most rapidly from its trivial value at small to its equally trivial value at large (assuming that the pore space percolates). The length is typically larger than the correlation length calculated from [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 and seems to give an improved estimate for the transport coefficient.

Table 4: Effective macroscopic transport property calculated from Clausius-Mossotti approximations ( ,), effective medium theory and local porosity theory compared with the exact result (for and ).
0.094606 0.0 0.0 0.025115 0.020297