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We first present the results for the bcc periodic mortar model, in order to display overall trends and point out general behavior. Figure 6 shows, for the 54% sand volume bcc mortar model, the overall conductivity of the composite, σ, divided by the conductivity of the bulk cement paste matrix, σp. The x-axis is the ratio of the conductivity of the interfacial zone cement paste, σs, to σp (σs / σp). The interfacial zone cement paste is percolated in this model, and so plays a strong role in the overall conductivity, as can be seen from the graph. Several points are worthy of note.
The conductivity at σs / σp = 1 is that which would be obtained if the interfacial zone cement paste had the same porosity and therefore the same conductivity as the bulk cement paste. The presence of the insulating sand grains in this case reduces the overall normalized conductivity from 1 to 0.35. This is consistent with a 3/2 power law found in suspensions of spheres and many porous rocks, where the normalized conductivity goes like the 3/2 power of the conductive phase volume fraction. In this case, (0.46)3/2 = 0.31 [27,39,40].
Figure 6: Composite conductivity for the bcc model is plotted vs. the interfacial zone conductivity. [Both are normalized by bulk paste conductivity.] The solid dots are the numerical data, the solid line is the Pade approximant explained in the text.
The composite conductivity can be thought of as the result of a competition between the insulating sand grains, which tend to lower the overall conductivity, and the interfacial zone cement paste shells, which tend to raise the overall conductivity, when σs > σp. Figure 6 shows that when σs / σp = 6, the composite conductivity first achieves a value equal to the matrix conductivity, so for this microstructure, this value of σs / σp causes the greater interfacial zone conductivity to "cancel out" the effect of the zero conductivity sand grains. For values of σs / σp < 6, the overall conductivity is less than the bulk cement paste conductivity, while for σs / σp > 6, it is more. When σs / σp = 20, the highest value shown in the graph, the composite conductivity achieved is 2.6 times higher than the bulk cement paste conductivity.
The solid curve in Fig. 6 is a Pade approximant  that has been employed  to describe the conductivity of porous sandstone rocks where the sand grains have a thin clay coating that has a higher conductivity than the electrolytic pore fluid. This is a situation analogous to the mortar and concrete conductivity problem being studied here. The Pade approximant is a ratio of a quadratic polynomial to a linear polynomial,
where x = σs / σp . The parameters (a,b,c,d) are given by combinations of four well-defined parameters (F,Λ, f,λ), which are defined in terms of the following two limits of the curve shown in Fig. 6 .
The first limit, in which the interfacial zone is only slightly more (or slightly less) conductive than the bulk cement paste matrix, σs / σp 1 , is given exactly by perturbation theory as
where 1/F is the ratio of the conductivity of the mortar to the conductivity of the bulk paste when σs / σp = 1 , and Λ is a length parameter defined using the ratio of a volume integral to a surface integral of the electric fields obtained from the solution of the σs = σp problem . When σs / σp = 1 , σ / σp = 1 / F , and when σs / σp is close to 1, eq. (3) is linear in σs / σp with a slope given by the ratio 2h/(F Λ) .
The second limit is when the conductivity of the (assumed fully connected) interfacial zone is much larger than that of the bulk cement paste, σs / σp >> 1 . In this limit, the composite conductivity is given exactly by perturbation theory as 
where 1/f is the conductivity of the problem where σp = 0 and σs = 1, and λ is another length parameter that comes from a ratio of a surface integral and a volume integral of the electric fields found in a solution of this same problem . Fitting eq. (2) to eqs. (3) and (4) gives the four unknown coefficients (a,b,c,d).
Clearly, the Pade approximant provides an excellent fit to the computed data points in Fig. 6, so that this analytical curve could be used to accurately predict the composite conductivity at other values of σs / σp that were not numerically computed. The bcc periodic mortar model has served to illustrate some generic effects of the interfacial zone conductivity on the overall mortar conductivity. The random mortar model, which is expected to more faithfully describe the real material, is discussed next.
In Figure 7 we have the counterpart of Fig. 6 for the random mortar model, and we note that, as in Fig. 6, the overall shape of the curve is concave down. The curve could at most be straight. This would be the case if the two phases, interfacial zone and bulk cement paste, were exactly in parallel. Then the overall conductivity would be given by a simple linear combination of the two phase conductivities, and as σs increased, the overall conductivity would increase linearly in this parameter. Since the microstructure is such that the two cement paste phases are not exactly in parallel, then the curve must be sub-linear, or concave down. As σs / σp becomes very large, the curve will of course go to a straight line as in eq. (4).
To achieve an overall conductivity that is equal to the bulk cement paste conductivity, the value of σs / σp must be equal to approximately 8, as can be seen in Fig. 7. This is higher than in the bcc periodic model, due to the greater tortuosity of the interfacial zone cement paste phase in the random model and its smaller volume fraction compared to the bcc model. Increasing σs / σp has therefore a somewhat lesser effect on the overall conductivity. At σs / σp = 20 , the overall conductivity is about 1.8, which is also significantly less than in the bcc periodic mortar model, for the same reasons.
Figure 7: Composite conductivity for the random mortar model is plotted vs. the interfacial zone conductivity. [Both are normalized by bulk paste conductivity.] The solid dots are the numerical data, the solid line is the Pade approximant explained in the text.
The solid line in Fig. 7 is a Pade approximant similar to that in Fig. 6. The fit is less good than in the bcc case, again an indication of the greater complexity of the random mortar model, and of real mortars. However, the asymptotic slope of the Pade approximant for large values of σs / σp should be accurate, so that this curve can be safely extrapolated to predict the effect on σ / σp of much higher values of σs / σp.
As measurements of the conductivity of mortars (and the cement paste they are made from) become available, curves like those shown in Figure 7 will serve to identify the conductivity value for the interfacial zone that provides the best match to the composite conductivity. A second important experimental data set will be obtained by measuring the conductivity of a mortar as a function of sand volume fraction. One important aspect of such a data set is the dilute limit, where the sand volume fraction is low. The composite conductivity in this regime contains important information about the conductivity and size of the interfacial zone. This is the case because exact analytical calculations can be made of the influence of a few sand grains surrounded by an interfacial zone shell placed in a matrix. For the composite to be considered to be in the dilute limit, the volume fraction of spherical inclusions must be small enough (less than 5%) so that particles can be considered individually and do not affect each other.
Consider mono-size spherical particles of conductivity σ1 and radius b, each surrounded by a concentric shell of thickness h and conductivity σ2, and all embedded in a matrix of conductivity σ3. The volume fraction of sand grains is c. Then σ / σ3 , the overall composite conductivity normalized by the matrix conductivity, is given by the expansion:
Equation (5) has been derived by exactly solving for the local electric fields and currents when the above physical system is placed in an (initially) uniform electric field. Appropriate current and field averages are then calculated to define the effective conductivity [43,44]. To make the connection to our mortar problem, let σ1 = 0, h = the interfacial zone thickness, σ2 = σs (interfacial zone conductivity), and σ3 = σp (bulk cement paste conductivity).
For the random mortar model, or indeed for a real mortar, there is a size distribution of sand grain radii bi, while the value of h is fixed. That implies that the slope mi for each kind of particle will be a function of bi, because the parameter [(bi+h)/bi]3 will be different for each particle. Eq. (6), for n different sand particle sizes, each with volume fraction ci ( c1 + c2 + ... = c) becomes
Using the sand particle size distribution given in Table 1 by volume, we can find the value of m for the random mortar model averaged over the appropriately-weighted four values of bi as in eq. (8).
Figure 8: The exact initial slope of the conductivity, in the limit of dilute sand concentration, is shown as a function of σs / σp for the sand size distribution (see Table 1) of the random mortar model.
Figure 8 shows a graph of this average slope <m> as a function of σs / σp. Note in the limit of σs / σp = 1 , the slope <m> = −1.5, which is the known exact result for insulating spherical inclusions of any size distribution . The marked point on the graph is at σs / σp = 8.26 , which is the point at which the slope <m> = 0. At this value, adding a few sand grains would, to leading order in the sand volume fraction, have no effect on the overall conductivity, keeping its value at the bulk cement paste value. In Fig. 7, it was found that a value of σs / σp 8 was required to make the composite conductivity equal to the bulk cement paste conductivity at a sand volume fraction of 55%. This implies that there is information about the shape of the conductivity vs. sand volume fraction curve for this particular sand size distribution and interfacial zone width obtainable without further computation. For σs / σp 8.26 , such a curve must start out with negative slope, and σ / σp will always lie under the bulk cement paste conductivity. For σs / σp > 8.26 , the curve will start out with positive slope and always remain above the bulk cement paste conductivity. This cutoff value for the dilute limit slope will of course vary with the sand size distribution. Having larger particles will, keeping the interfacial zone width the same, tend to reduce the dilute limit slope at the same value of σs / σp.
Figure 9 shows computed conductivity data for the random mortar model as a function of sand volume fraction, for σs / σp = 20, 5, and 1. The sand size distribution was preserved at every volume fraction. The curve for σs / σp = 20 is roughly concave up, as the initial slope is greater than one, and the interfacial zone conductivity is great enough so that the addition of more sand makes this slope even greater. [In principle, the percolation threshold at which the interfacial zone phase becomes continuous might be visible as a fairly sharp break in such a curve. Presumably, the required values of σs / σp are considerably larger than those shown here.] The σs / σp = 5 curve has a negative initial slope, and remains below one, as would be expected from the previous predictions. The σs / σp = 1 curve roughly follows a 3/2 power law in the total cement paste volume fraction, as would be expected since there is no difference between interfacial zone and bulk cement paste in this case.
Using the results of eq. (5) for the bcc model implies that the initial slope is zero for σs / σp = 5.5, nearly equal to the value needed to make σ / σp = 1 at 54% sand, so that a fairly similar picture will hold for the ordered model as well.
Figure 9: Composite conductivities (calculated by random walk simulations)
for the random
mortar model are shown as a function of sand concentration for several values
of the interfacial
[Normalization is as in Figs. 6 and 7.]