New D-EMT

The standard D-EMT is a two-phase theory. In the present case, the ITZ causes conceptual problems, since it introduces a third phase. To use D-EMT in this case, should the ITZ be treated as part of the particle, or should it be considered as part of the matrix? If the ITZ conductivity is given independently of the matrix, then it should stay the same as the matrix is renormalized in the D-EMT calculation process. However, if it is given in terms of a ratio with the matrix conductivity, and if the ratio stays the same during the calculation process, then the absolute value of the ITZ conductivity will change [15]. The form of D-EMT previously used for the concrete problem [15] took a weighted average between these two extreme cases, with the weights determined by a fit to random walk computations. The agreement with computations was not spectacular ($\simeq 20$ %), and there was no guarantee that the fitted weights would be the same for all concrete systems studied.

An answer to the conceptual dilemma stated above would be to construct a version of D-EMT in which the ITZ regions were either unambiguously aggregate or matrix. This would eliminate the need for adjustable parameters. Since the ITZ regions, disregarding overlaps, are the same shape as the spherical aggregate particles, one is drawn to the option of making the ITZ regions part of the aggregates. This is accomplished using the following idea: Map each aggregate particle plus its accompanying ITZ region into a single effective particle, with an effective uniform conductivity, σ_p, which is embedded in the bulk matrix. This idea is illustrated in Fig. 2. The radius of this effective particle will then be a_j = b_j + h, rather than simply b_j. This procedure can be carried out by equating the exact result for m_j , eq. (8), to the exact result for m_j when the particle has uniform conductivity.

The dilute limit slope m_j for a spherical particle of conductivity σ_p, radius b_j + h, embedded in a matrix of conductivity σ_bulk, is given by

$$m_j = 3 \alpha_j \frac{ (\sigma_{p} - \sigma_{bulk})} {( 2 \sigma_{bulk} + \sigma_{p})}$$ (12)

where m_j is referred to c, which is the volume fraction of aggregates only (see eq. (1)). This dilute limit is referred to c, rather than c´, in order to be able to equate it to eq. (8) . When this dilute limit is equated to eq. (8), the value of σ_p turns out to be:

$$\sigma_{p} = \frac{[ 2 (\sigma_{agg} - \sigma_{ITZ}) + \alpha... ... - \sigma_{ITZ}) + \alpha_j (\sigma_{agg} + 2 \sigma_{ITZ})]}.$$ (13)

Therefore, the dilute limit for a particle of radius b_j + h, with conductivity σ_p (which is a function of j), referred to the volume fraction of aggregate c, is the same as for the real particle, of radius b_j and conductivity σ_agg, and accompanying ITZ of thickness h and conductivity σ_ITZ . Figure 3 shows this mapping between σ_p and the ITZ conductivity, for four different diameter (diameter = 2 b) aggregate particles, where σ_agg = 0, and h = 20 µm. The dependence on the value of α and thus the particle size can be seen clearly.

Figure 3: Showing the value of σ_p / σ_bulk, as a function of the value of σ_ITZ / σ_bulk, for four different diameter aggregate particles (2b), from eq. (13), for h = 20 µm.

This effective particle is then treated in usual differential EMT, as described above. When an aggregate size distribution is used, the function m ( σ ) is an average over this size distribution, as was stated above. The integral can be done numerically for chosen values of σ, with the aggregate volume fraction c = 1 − φ then treated as being a function of σ. There are a few differences, however, involving the effective aggregate volume fraction. Each particle is now of radius b_j + h, so that the volume fraction of "effective aggregate" now goes to c´, not c. The value of c´ must be known in order to perform the integral in eq. (5).

These differences can be worked out simply by considering the number of particles of a certain type. If V_i is the total volume of the i-th kind of particle, and N_i is the total number of this kind of particle, the

$$N_i {4 \pi \over 3} ( b_i )^3 = V_i$$ (14)

and therefore

$${N_i \over V} {4 \pi \over 3} ( b_i )^3 = {V_i \over V} = f_i \: c$$ (15)

$$n_i {4 \pi \over 3} ( b_i )^3 = f_i\: c$$ (16)

where V is the total volume of the system and n_i is the number of particles of type i per unit volume.

Now the new values of f_i and c, f _i´ and c´, are defined via rewriting the previous equation:

$$n_i {4 \pi \over 3} ( b_i + h )^3 = f_i^{\prime} \: c^{\prime}.$$ (17)

The values of f_i´ and c´ can also be defined directly by

$$c^{\prime} = \sum_{j=1}^M n_j {4 \pi \over 3} ( b_j + h )^3$$ (18)

$$f_i^{\prime} = \frac{n_i ( b_i + h )^3}{ \sum_{j=1}^M n_j ( b_j + h )^3}$$ (19)

By combining the above equations, one can then derive forms for f _i´ and c´ that involve only f _i, c, h, and α_i:

$$f_i^{\prime} = \frac{c_i \alpha_i}{ \sum_{j=1}^M f_j \alpha_j}$$ (20)

$$c^{\prime} = c \sum_{j=1}^M f_j \alpha_j$$ (21)

It should be noted that while the value of c was for non-overlapping aggregate particles, the value of c´ is for the volume occupied by each aggregate particle and its surrounding ITZ region, where the ITZ regions are assumed to not overlap. In a real concrete, these ITZ regions do overlap, causing percolation phenomena, as was mentioned earlier. This treatment of the ITZ volume fraction is another approximation of the D-EMT method.

In summary, a D-EMT calculation is carried out as follows. First the sieve analysis is used to compute c´ and f_i ´. Then the integral in eq. (5) is carried out numerically by Gaussian quadratures [27], where $\langle m \rangle$ is numerically averaged over the sieve analysis. Since the diameter range of each sieve is rather large, the assumption is made that within each sieve, the particles are uniformly distributed by volume, thus relaxing the assumption made earlier in this paper (see Sec. 2.2) that all particles in a certain sieve had the same radius. This enables an integral to be performed over each bin, and then a summation over all the sieves ( see Appendix 2 in Ref. [15]). This procedure is also used to compute c´ and f_i´ as well. The actual FORTRAN software (newDemt.f) used to calculate the D-EMT for an arbitrary sieve analysis is available on the Internet [28].