Differential effective medium theory (Step 3)

Most "effective medium theories" for two or more phase composites are derived in two steps. In the first step, the dilute limit of the composite, where the inclusion phases are present in small amounts in the matrix material, is solved exactly. The second step takes this exact solution for the dilute limit and uses a statistical approximation of some kind to get an analytical form for the case of arbitrary amounts of the phases.

The type of effective medium theory (EMT) that has been found useful for concrete diffusivity problems uses the differential scheme to generate the EMT, and is hereafter referred to as D-EMT. The mathematical basics are found in McLaughlin [70], and have been previously discussed in the context of concrete [51,59]. The first step in generating the D-EMT equation is to solve the dilute limit properly. In this case, the dilute limit is given by a single spherical aggregate particle, with D_agg = 0, surrounded by a spherical shell of thickness t_ITZ , of diffusivity D_ITZ, embedded in a matrix with diffusivity D_bulk. Diffusivity or electrical conductivity can be used interchangeably for this problem, as the mathematical structure is identical. If there is a very low volume fraction of such particles, then the diffusivity of the composite is described by the expansion

$$\frac{D}{D_{bulk}} = 1 + \langle m \rangle_V c + O(c^2)$$ (7)

where $\langle m \rangle_V$ means that the slope is averaged over the volume distribution of the aggregates, since the expansion (7) is in terms of the aggregate volume fraction, not the number fraction. The slope m for a single size particle is given in Appendix A.

To generate the D-EMT equation, the dilute result above is used in the following way. Suppose that a volume fraction c' of aggregate has been added to the cement paste matrix, so that the total diffusivity is now D '. The current cement paste matrix volume fraction is ' = 1 - c'. The aggregates have been "smeared" out so that the concrete is a uniform material. Suppose that a differential volume element of volume fraction dV is now taken out and replaced with aggregate. The new actual volume fraction of aggregate is not just c' + dV, since some of the material that was removed was also aggregate, but is equal to c' + dV - c'dV. The change in aggregate volume fraction is then just dc' = dV( 1 - c' ). The dilute limit is used to get the new diffusivity, D ' + dD ',

$$D^{'} + dD^{'} = D^{'} + D^{'} \langle m \rangle dV$$ (8)

or, using the relation between c' and dV,

$$\frac{dD^{'}}{D^{'} \langle m \rangle_V} = \frac{dc^{'}}{1 - c^{'}}$$ (9)

This equation can be integrated on the left from D _bulk, the diffusivity when no aggregates are present, to D, the diffusivity when the aggregates have volume fraction c, and on the right from 0 to the desired aggregate volume fraction c, with the final result

$$- \int_{D_{bulk}}^D \frac{dD^{'}}{D^{'} \langle m \rangle_V} = \ln (1 - c)$$ (10)

In eq. (10) D ' replaces D_bulk in the expression for $\langle m \rangle_V$ (see Appendix A of Ref. [51]), so that <m> _V is a function of D '. To obtain the predicted value of D for any value, c, of aggregate volume fraction, one simply varies the value of D until the left hand integral equals the desired value of ln(1-c).

There is one complication of this three-phase model, compared to two phase models [70]. In two phase models, in the D-EMT process, the diffusivity of the inclusion would remain invariant, while only the matrix diffusivity would change. In this three-phase case, there is the question of whether the ITZ diffusivity, D_ITZ , is also renormalized in the D-EMT process, or whether it should stay invariant. The ITZ phase is outside the aggregate volume, and so must be considered part of the matrix phase. Intuitively and physically, there are two extremes that are worth considering. One is that the actual value of D_ITZ remains invariant. The other is that the value of D_ITZ /D ' remains constant at what the original value, D_ITZ/D_bulk , was chosen to be. If the concrete diffusivity is being reduced with the addition of aggregates, the first choice will clearly give a larger result. If the overall concrete diffusivity is going up with the addition of more aggregate, the second choice will clearly give the larger value. Theoretically, it is not possible to choose between these ways of carrying out the D-EMT process, as the value of D_ITZ is essentially a free parameter in the problem. Ref. [51] showed how the numerically exact random walker calculations of D/D_bulk can show how to make a choice between these two limits.

We note that, a priori, there is no reason to expect an EMT to work well for a particular problem. The dilute limit used in the development of an EMT is exact, but the approximation used to produce predictions for larger volume fractions of aggregate are essentially uncontrolled. Having numerically exact diffusivity calculations available makes it possible to quantitatively evaluate the D-EMT predictions and optimize them for practical use. The D-EMT results can now be used to replace the random walker numerical results, but the validity of this replacement could only have been determined by having the random walker techniques available in the first place.