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 $\overline{C}$ is defined by

$\left\langle\boldsymbol{j}({\boldsymbol{r}})\right\rangle=-\overline{C}\left\langle\boldsymbol{\nabla}u({\boldsymbol{r}})\right\rangle$

(68)

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

$C_{\mathbb{P}}=1,\qquad C_{\mathbb{M}}=0.$

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

[235.2.1] The macroscopic effective transport properties are known to show strong sample to sample fluctuations. [235.2.2] Because calculation of $\overline{C}$ requires a disorder average the four microsctructures were subdivided into eight octants of size $128\times 128\times 128$ . [235.2.3] For each octant three values of $\overline{C}$ 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 $\overline{C}$ are indeed rather strong. [236.0.1] If the system is ergodic then one expects that $\overline{C}$ 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 $\overline{C}_{x}=0.02046$ in the $x$ -direction, $\overline{C}_{y}=0.02193$ in the $y$ -direction, and $\overline{C}_{z}=0.01850$ in the $z$ -direction [65]. [236.0.3] All of these are seen to fall within one standard deviation of $\overline{C}$ . [236.0.4] The numerical values have been confirmed independently by [47].

[236.1.1] Finally it is interesting to observe that $\overline{C}$ seems to correlate strongly with $p_{3}(L)$ shown in Figure 11. [236.1.2] This result emphasizes the importance of non-Hadwiger functionals because by construction there is no relationship between $\overline{C}$ and porosity, specific surface and correlation functions.

Table 3: Average and standard deviation $\sigma$ for effective macroscopic transport property $\overline{C}$ calculated from subsamples (octants) for $C_{\mathbb{P}}=1$ and $C_{\mathbb{M}}=0$ .

	$\mathbb{S}_{{\sf EX}}$	$\mathbb{S}_{{\sf DM}}$	$\mathbb{S}_{{\sf GF}}$	$\mathbb{S}_{{\sf SA}}$
$\overline{C}$	0.01880	0.01959	0.00234	0.00119
$\sigma$	$\pm$ 0.00852	$\pm$ 0.00942	$\pm$ 0.00230	$\pm$ 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 $\overline{C}$ that utilizes the partial geometric characterization contained in the quantities $\mu$ and $\lambda$ . [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 $\mathbb{P}$ as background phase

$\overline{C}_{c}(\overline{\phi})=C_{\mathbb{P}}\left(1-\frac{1-\overline{\phi}}{(1-C_{\mathbb{M}}/C_{\mathbb{P}})^{{-1}}-\overline{\phi}/3}\right)=C_{\mathbb{P}}\left(\frac{3C_{\mathbb{M}}+2\overline{\phi}(C_{\mathbb{P}}-C_{\mathbb{M}})}{3C_{\mathbb{P}}-\overline{\phi}(C_{\mathbb{P}}-C_{\mathbb{M}})}\right),$

(70)

[page 237, §0] the Clausius-Mossotti approximation with $\mathbb{M}$ as background phase

$\overline{C}_{b}(\overline{\phi})=C_{\mathbb{M}}\left(1-\frac{\overline{\phi}}{(1-C_{\mathbb{P}}/C_{\mathbb{M}})^{{-1}}-(1-\overline{\phi})/3}\right)=C_{\mathbb{M}}\left(\frac{2C_{\mathbb{M}}+C_{\mathbb{P}}+2\overline{\phi}(C_{\mathbb{P}}-C_{\mathbb{M}})}{2C_{\mathbb{M}}+C_{\mathbb{P}}-\overline{\phi}(C_{\mathbb{P}}-C_{\mathbb{M}})}\right),$

(71)

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

$\overline{\phi}\frac{C_{\mathbb{P}}-\overline{C}}{C_{\mathbb{P}}+2\overline{C}}+(1-\overline{\phi})\frac{C_{\mathbb{M}}-\overline{C}}{C_{\mathbb{M}}+2\overline{C}}=0$

(72)

which leads to a quadratic equation for $\overline{C}$ . [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 $\overline{\phi}$ is the only geometric observable representing the influence of the microstructure. [237.0.3] Thus two microstructures having the same porosity $\overline{\phi}$ are predicted to have the same transport parameter $\overline{C}$ . [237.0.4] Conversely, measurement of $\overline{C}$ combined with the knowledge of $C_{\mathbb{M}},C_{\mathbb{P}}$ allows to deduce the porosity from such formulae.

[237.1.1] If the microstructure is known to be homogeneous and isotropic with bulk porosity $\overline{\phi}$ , and if $C_{\mathbb{P}}>C_{\mathbb{M}}$ , then the rigorous bounds [24, 8, 62]

$\overline{C}_{b}(\overline{\phi})\leq\overline{C}\leq\overline{C}_{c}(\overline{\phi})$

(73)

hold, where the upper and the lower bound are given by the Clausius-Mossotti formulae, eqs. (71) and (70). [237.1.2] For $C_{\mathbb{P}}<C_{\mathbb{M}}$ the bounds are reversed.

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

$\int _{0}^{1}\!\!\frac{\overline{C}_{c}(\phi)-\overline{C}}{\overline{C}_{c}(\phi)+2\overline{C}}\lambda _{3}(\phi,L)\mu(\phi,L)d\phi+\int _{0}^{1}\!\!\frac{\overline{C}_{b}(\phi)-\overline{C}}{\overline{C}_{b}(\phi)+2\overline{C}}(1-\lambda _{3}(\phi,L))\mu(\phi,L)d\phi=0$

(74)

where $\overline{C}_{b}(\phi)$ and $\overline{C}_{c}(\phi)$ 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 $L\to 0$ . [237.2.6] In the limit $L\to\infty$ it also reduces to eq. (72) albeit with $\overline{\phi}$ in eq. (72) replaced with $\lambda _{3}(\overline{\phi})$ . [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 $L_{p}$ has been used which is defined through the condition

$\left.\frac{d^{2}p_{3}}{dL^{2}}\right|_{{L=L_{p}}}=0$

(75)

assuming that it is unique. [238.0.2] The idea behind this definition is that at the inflection point the function $p_{3}(L)$ changes most rapidly from its trivial value $p_{3}(0)=\overline{\phi}$ at small $L$ to its equally trivial value $p_{3}(\infty)=1$ at large $L$ (assuming that the pore space percolates). The length $L_{p}$ 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 $\mu$ and $\lambda$ seems to give an improved estimate for the transport coefficient.

Table 4: Effective macroscopic transport property $\overline{C}$ calculated from Clausius-Mossotti approximations ( $\overline{C}_{c}$ , $\overline{C}_{b}$ ), effective medium theory $\overline{C}_{{\mbox{EMA}}}$ and local porosity theory $\overline{C}_{{\mbox{LPT}}}$ compared with the exact result $\overline{C}_{{\mbox{exact}}}$ (for $C_{\mathbb{P}}=1$ and $C_{\mathbb{M}}=0$ ).

$\overline{C}_{c}$	$\overline{C}_{b}$	$\overline{C}_{{\mbox{EMA}}}$	$\overline{C}_{{\mbox{LPT}}}$	$\overline{C}_{{\mbox{exact}}}$
0.094606	0.0	0.0	0.025115	0.020297

Author:	R. Hilfer
originally published in:	Räumliche Statistik und Statistische Physik, Lecture Notes in Physics Vol. 554, Springer, Berlin, page 203-241 (2000), ISBN - -
originally published:	2000