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Dilute limit

Consider next the limit of dilute sand concentrations. 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 (each of which is surrounded by an interfacial 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 so that particles can be treated 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 . If the volume fraction of sand grains is denoted by c, then the composite conductivity, σ, will satisfy an equation of the form σ / σ₃ = 1 + m c + O(c² ), where [6]

To make the connection to our mortar problem, let σ₁ = 1, h = the interfacial zone thickness, σ₂ = σ_s (interfacial zone conductivity), and σ₃ = σ_p (bulk cement paste conductivity). For the random mortar model, or indeed for a real mortar, there is a size distribution of sand grain radii {b_i}, while the value of h is fixed. That implies that the slope m_i for each kind of particle will be a function of b_i, because the parameter [(b_i + h)/b_i]³ will be different for each particle. In a system with N different sand particle sizes, each with volume fraction c_i (Σ_i c_i = c) , we have

Using the sand particle size distribution given in Table 1 (second column) we can find the value of for the random mortar model averaged over the appropriately-weighted four values of b_i.

Figure 6 shows a graph of this average slope as a function of σ_s / σ_p . Note in the limit of σ_s / σ_p = 1 , the slope = −1.5, which is the known exact result for insulating spherical inclusions of any size distribution [24,25]. The marked point on the graph is at σ_s / σ_p ≈ 8.26, which is the point at which the slope = 0. At this value, to linear order in c, adding a few sand grains would have no effect on the overall conductivity. Here one has a perfect balance between the negative influence of the insulating sand grains, and the positive effect of the enhanced conductivity in the interfacial shells.

Figure 6: The exact initial slope of the conductivity, in the limit of dilute sand concentration, is shown as a function of σ_s / σ_p for the sand size distribution (see Table 1) of the random model.