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Effective medium theory

Strictly speaking, the only exact results available for three dimensional composite materials with general properties are variational bounds [26] and the dilute limits presented above. However, effective medium theory (EMT) can often be employed to estimate composite properties at arbitrary volume fractions of the phases [27,28]. We consider only examples of EMT that describe properly the dilute limit, and then build up approximate analytical equations via some sort of averaging assumption. The two examples considered in this paper are the self-consistent (SC-EMT) [27] and the differential (D-EMT) [28] methods.

The SC-EMT has roots going back to Bruggeman [29] and Landauer [30]. Following the presentation of Hashin [27] we begin with an isotropic inhomogeneous system in which the applied electric field has magnitude E_o. In particular we are interested in a three-phase composite with conductivity σ_i and volume fraction c_i in the i'th phase. Then the effective conductivity, σ, of the entire composite can be defined by

Combining these two equations, we can then eliminate the average over the matrix, i=3, phase and write the effective conductivity as depending only on averages of electric fields in the non-matrix or inclusion phases:

where phase 1 is the sand grain and phase 2 is the shell. The above equation is exact. However, at arbitrary volume fractions, the above field averages cannot be evaluated analytically. [In principle, the finite element method discussed above could give these field averages numerically.] The SC-EMT is derived by making the assumption that the field averages can be approximated by inserting the values obtained from a modified dilute problem, in which the inclusions are embedded, not in phase 3, but in a matrix having the effective conductivity σ. This transforms Eq. (5) into an equation for σ, which can then be solved.

The four sizes of sand grains used in our model are simply treated as different phases. The shell volume fractions are estimated by assuming that the shells do not overlap. Numerical checks have shown that this is accurate to a few percent in even the concentrated sand case. This means that even though the shells percolate, the overlap volume is only a few percent of the total shell volume. For a single size sand grain embedded in a matrix of conductivity σ, the field averages for the sand grain and the shell are given by:

where

and the labels 1 and 2 denote the sand grain and shell respectively. Combining these with Eq. (5), and averaging over the different size sand grains, gives the final result. For the mortar problem, we take σ₁ = 1, σ₂ = σ>₃, and σ₃ = σ_p . The structure of the SC-EMT leads to a natural division of the possible behavior of the total conductivity for different values of σ_s / σ_p . When the value of σ_s / σ_p is such that the dilute limit slope, < m >, is zero, then the SC-EMT continues to give the matrix conductivity for any volume fraction of sand [i.e. σ = σ_p is a fixed point of the theory when < m > = 0 in Eq. (1)]. When σ_s / σ_p is greater than this value, σ σ_p, while for lower σ_s / σ_p values, σ σ_p, for all sand volume fractions.

In the D-EMT the dilute limit (1) is used in a different way to generate an approximate equation that can be solved for the effective conductivity [28,29]. Suppose that a volume fraction, c, of sand has been placed in the matrix. Treating this system as a homogeneous material with matrix volume fraction, φ, and conductivity σ, we then suppose that a differential volume element, dV, is removed and replaced by an equivalent volume of sand. The new conductivity, σ + dσ, is assumed to be given by the dilute limit

where V is the total volume and m(σ) is given by (1) with σ₃ → σ. When the volume element dV was removed, only a fraction φ of it was matrix material so that the change in the matrix volume fraction, dφ, is given by

Eq. (8) then reduces to dφ / φ = -dσ / m(σ) ,which can be integrated to yield

The function m(σ) is an average over the size distribution of sand grains when more than one size sand grain is used. In that case, the integral can be done numerically for chosen values of σ, with the sand volume fraction c = 1 − φ then treated as being a function of σ. For a single size sand grain, the final equation can be found analytically, and is given by:

where

As for the SC-EMT, the D-EMT gives a fixed point when the slope m(σ) equals zero. This is easy to see in the present (D-EMT) case, because the first step of adding inclusions to the original matrix will produce a transformed matrix with the same conductivity as the original bulk paste. Successive iterations, again, lead to no change in σ.

Both the SC-EMT and the D-EMT are correct in the dilute limit. Another figure of merit for an EMT is how well any percolation threshold is predicted [31]. In particular, the volume fraction of shells at which the shell phase first percolates is of interest, because of the possibility of having a large conductivity in the shells. In the EMT equations, the predicted percolation threshold may be found by allowing the matrix conductivity to go to zero, and determine at what value of the shell volume does the effective conductivity become non-zero. If the matrix conductivity is zero, the only way for the composite conductivity to be non-zero is if the interfacial zone regions percolate. The D-EMT gives a percolation threshold only at a sand volume of 1, which is clearly wrong (see Fig. 3). The SC-EMT for mono-size spherical sand grains of radius, b, and shell thickness, h, predicts a critical threshold for the shell phase at a shell volume fraction of (1/3) [1 − (b³/(b + h)³], or, alternatively, for a sand volume fraction of b³/[3 (b + h)³]. Figure 3 shows both these predictions plotted vs. b/a (a = b+h) along with the numerically determined percolation thresholds. There is indeed reasonable agreement. The region of interest is where b/a > 0.7, since that is the typical lower limit for b/(b+h) because the smallest b found in mortar is of the order of 50 µm, and h is usually of the order 20 µm. In this region, the SC-EMT prediction is qualitatively correct for the shell volume fraction, but does not give the correct curvature. The prediction for the sand volume fraction is increasingly off as h goes to zero, since the SC-EMT does not correctly predict the volume fraction at the random parking limit [14]. Still, this shows that since the dilute limit is correctly predicted, and the percolation threshold is reasonably well predicted, the SC-EMT might be expected to do fairly well in between these two limits. Because of this difference between the two methods, we expect that for σ_s / σ_p >> 1 , the SC-EMT will work better than the D-EMT. This is because the percolation threshold, at which the effective composite conductivity will eventually diverge, is not properly accounted for in the differential method.

If we consider the SC-EMT for the four sand grain radii model, we can numerically determine what is the predicted sand volume fraction at which the interfacial zones percolate. The SC-EMT prediction is c = 0.27, which is reasonably near the result of Fig. 3, c = 0.36.