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### Including the ITZ in the dilute limit

In the concrete problem, as was mentioned in the Introduction, each aggregate is surrounded by a thin shell of different material, called the ITZ. Since any D-EMT must be built up from the exact dilute limit, the dilute limit for such a composite particle is now discussed.

Consider an idealized aggregate particle, like those shown in Fig. 1. Real aggregates generally have non-spherical shape, but for many kinds of aggregates, a spherical shape is a reasonable approximation. A spherical shape is used in the multi-scale model [4,15]. Consider spherical aggregate particles of conductivity σagg and radius b, each surrounded by a concentric shell of thickness h and conductivity σITZ, a = b + h, and all embedded in a matrix of conductivity σbulk. The left side of Fig. 2 shows these parameters pictorially. The volume fraction of aggregate grains, not counting the ITZ regions, which are only modified (more porous) matrix material, is still denoted by c. Eq. (1) is still valid, but now the slope m for the linear term in c is given exactly by [17,26]

 (8)

The parameter α is defined by the radius of the particle and the thickness of the ITZ:

 (9)

When σagg = 0, the usual case for concrete, then eq. (8) becomes

 (10)

This slope is negative when

 (11)

and is nonnegative otherwise. For most concrete cases, even though usually σITZ > σbulk, the slope m is negative when averaged over all particle sizes, as is discussed next. In all cases considered in this paper, the slope m was always negative, so there were no difficulty with zeroes in the denominator of eq. (5).

Concrete has a size distribution of aggregate grain radii {bj}, while the value of h is essentially fixed [4,15]. That implies that the slope mj for each kind of particle will be a function of bj, because the parameter αj = [(bj + h)/bj]3 will be different for each particle. The aggregate size distribution is usually given by a sieve analysis characterized by di, fj, i = 1, M+1, j = 1, M, where M is the number of sieves used, di < d i+1 are the endpoint diameters of the i-th sieve, and fj is the fraction of the total aggregate volume that is taken up in the j-th sieve ( j fj = 1). For now we assume that all the particles in the j-th interval have the same radius, bj (dj, dj+1). Later on, this assumption will be relaxed.

The dilute limit is then defined the same way, but the slope used, , is first averaged over the aggregate particle size distribution (sieve analysis) before being used in the dilute limit formula. The slope mj for the j-th size class is given by eq. (8), but with α going to .

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