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# Appendix A: Discussion of dependence of ITZ thickness on aggregate size

In Ref. [9], it was found that the ITZ thickness decreased with aggregate radius, in contradiction to our assumption that the ITZ thickness is independent of aggregate size, at least for aggregates larger than a few times the median cement particle size. This conclusion in Ref. [9] was reached by making electrical conductivity measurements, equating the data to an equation for the composite conductivity that had as parameters the aggregate volume fraction, the thickness and conductivity of the interfacial zone, the bulk paste conductivity, and the average radius of the aggregates, taken to be spheres. The value of the interfacial zone thickness, normalized to some arbitrary value, was then found from this matching process. The trouble with this procedure is, as will be shown below, that the equation used was only an approximate one. Its dependence on the interfacial zone thickness, for example, is not correct and so will not give correct results in the above procedure.

A simple test is in the dilute limit, where the diffusivity is known exactly. In this limit, as was discussed in the text, the diffusivity, normalized by the bulk diffusivity, is

where c is the volume fraction of aggregates placed. The exact result for m, for a spherical aggregate of radius r, with zero aggregate diffusivity, is

where the parameter β = [ (r + tITZ) / r]3 contains all of the dependence on the particle and interfacial zone dimensions [6]. (There is a typographical error in Ref. [6], in the equation equivalent to eq. (13). In the numerator, the term that reads (b+h)/h should really be (b+h)/b, in accordance with the definition of the parameter β).

The equivalent value for m, derived from the equation in Ref. [8], with h = DITZ/Dbulk, is

where r is the aggregate radius. All the aggregate must be of the same size for this equation to apply to a concrete. In (14), if we treat tITZ / r as a small parameter, and expand the exact result (13) to second order in this parameter, then the following equation results:

Comparison of eq. (15) to eq. (14) shows that only the −3/2 term agrees. So eq. (14) is not correct, in the dilute limit, in either the first or the second order term in tITZ / r .

If we set eq. (14) equal to eq. (13), simulating the matching of (14) to an "experimental" result, then we can solve for the "effective" value of tITZ / r needed in order to match up eq. (14) to the correct value, (13). This equation is

where m is the exact slope from eq. (13). In this way, we can generate a graph of the perceived value of tITZ measured as a function of the particle radius, in a case where the actual value of tITZ is fixed. We take tITZ = 20 µm, and the ratio h of the ITZ diffusivity to the bulk diffusivity to be 4 and 10. Figure 4 shows the results of this process. Clearly, using the equation from Ref. [9], the ITZ thickness is perceived to be sharply decreasing with particle radius, when in fact it is staying constant, as indicated by the horizontal line. Even though the measurements in Ref. [9] were made at aggregate volume fractions well beyond the dilute limit, the same result will almost certainly hold. Therefore the conclusion of Ref. [9] that the ITZ thickness decreases with aggregate radius was based on a formula that was not correct in accounting for the effect of the ITZ, and so when used to "back out" a value for DITZ/Dbulk, gave incorrect results. This would be the case even if eq. (14) gave reasonable results for m for a concrete, since the specific dependence on DITZ/Dbulk is not correct. The error occurs when trying to extract a small quantity out of an experimental measurement using an inexact equation that does not have the correct dependence on the small quantity, in this case the ITZ thickness.

Figure 4: Showing the "effective" value of tITZ / r, as determined from matching the approximate formula of eq. (14) to the exact formula of eq. (13) (see eq. (16)), plotted against the radius of the aggregate particle r, for two different values of h = DITZ/Dbulk (solid line h = 4, dashed line h = 10).

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