Standard D-EMT

In the usual D-EMT [19], when a particle with conductivity σ_p is embedded in a matrix with conductivity σ_bulk , the dilute limit is used to generate an approximate equation that can be solved for the effective conductivity. In the dilute limit, the value of c, the volume fraction of aggregates, is small enough so that the aggregate grains do not influence each other. The effective conductivity, σ, is then given exactly by [5, 17]:

σ = σ _bulk + σ_bulkmc +  O(c² ) (1)

where m is a dimensionless coefficient often called the dilute limit slope or intrinsic conductivity [25] that is a function of the shape of the particle, and the ratio ${\sigma_{p} \over \sigma_{bulk}}$. The higher order terms in the c expansion come from interactions between aggregate particles, and so are negligible in the dilute limit.

The dilute limit is now used to generate a differential equation for the conductivity when an arbitrary amount of aggregates is placed in the matrix. Suppose that a non-dilute volume fraction c of aggregates (of conductivity $\sigma_{p}$ _p ) have been placed in the matrix. The effective conductivity of the entire composite system is now σ. This system of matrix (volume fraction = φ = 1 − c) plus aggregates (volume fraction = c) is treated as being a homogeneous material. Suppose then that additional aggregates are added by removing a differential volume element, dV, from the homogeneous material, and replacing it by an equivalent volume of aggregates. The new conductivity, σ + dσ, is assumed to be given by the dilute limit

$$\sigma + d \sigma = \sigma + \sigma m(\sigma) {dV \over V}$$ (2)

where V is the total volume and m(σ) is the same as that in eq. (1), but with the replacement σ_bulk → σ. This is the key approximation that is made in order to generate the D-EMT. When the volume element dV was removed, only a fraction j was matrix material so that the actual change in the matrix volume fraction, dφ, is given by

$$d\phi = - \phi {dV \over V}.$$ (3)

Eq. (2) then reduces to

$$d\phi / \phi = - d\sigma / (\sigma m(\sigma)),$$ (4)

which can be integrated to yield

$$- \int_{\sigma_{bulk}}^{\sigma} \: {d \sigma^{\prime} \over m... ...} \: {d \phi^{\prime} \over \phi^{\prime}} = \ln(\phi) \: \: .$$ (5)

For spherical aggregates of only one size, with conductivity σ_p, and embedded in a matrix of conductivity σ [5],

$$m(\sigma) = 3 \frac{(\sigma_{p}-\sigma)}{(2 \sigma + \sigma_{p})}.$$ (6)

The integral in eq. (5) can be done exactly, using eq. (6), with the result

$$\frac{(\sigma - \sigma_{p})}{(\sigma_{bulk} - \sigma_{p})} \left (\frac{\sigma}{\sigma_{bulk}} \right )^{-1/3} = ( 1 - c ).$$ (7)