Up: Microstructure of Mortar
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Concrete conducts electricity via the cement paste matrix, and the aggregates are assumed to be simply insulating obstacles to the flow of current. As was mentioned before, diffusivity is handled mathematically and physically the same way, so we specialize to conductivity language in the present discussion. Adapting the hard core/soft shell concrete model to study conductivity/diffusivity requires three input parameters: (1) the thickness of the interfacial layer, (2) the contrast in properties between this layer and the bulk cement paste, and (3) the concentration and size distribution of the aggregates. Following the previous discussion we will assume that all aggregates are spherical and that the interfacial transition zones are always spherical shells of constant thickness. As stated previously, while the thickness of the interfacial transition zone, h, may be as large as 50 µm, mercury intrusion measurements and modelling results  suggest that h = 20 µm is a more typical value, especially since this is the main region having higher porosity and therefore higher conductivity. We assume that the conductivity within the interfacial shell takes a constant value, σs, and that the conductivity of the cement paste, σp, is also constant. This is not, strictly speaking, quite true.
The actual situation is somewhat more complicated. Since the cement paste is mixed at a fixed w/c ratio, having extra porosity and thus extra water in the interfacial transition zone regions, which can occupy up to one third of the total matrix phase, means that the bulk paste must have a somewhat lower porosity and consequently a lower w/c ratio. Therefore, the conductivity of the matrix paste of a mortar or concrete cannot be simply taken as that of the cement paste from which the composite was made. Also, the interfacial transition zone region actually shows a gradient of properties , since it has a gradient in porosity . The above model is still good, though, as a first approximation.
Since there is therefore no experimentally established value for the ratio σs/σp , we allow this parameter to vary freely in our calculations. For a given aggregate volume fraction, we have studied the dependence of the composite conductivity on the value of σs/σp. We have also studied the conductivity as a function of aggregate volume fraction for several fixed values of σs/σp. To compute the conductivity of such a model, we have adopted a random walk algorithm, used extensively in studies of disordered porous media and composite materials [71,72,73].
However, it is useful to first consider the limit of dilute aggregate concentration. The composite conductivity in this regime contains important information about the conductivity and size of the interfacial transition zone. This is the case because exact analytical calculations can be made of the influence of a small volume fraction of aggregates (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 aggregates 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, can be analytically approximated by an equation of the form σ/σ3 = 1 + m c + O(c2), with m given by :
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. This 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. In a system with N different sand particle sizes, each with volume fraction ci, the average slope <m> is then defined by,
Using the four diameter model sand particle size distribution for a mortar mentioned previously , we can find the value of <m> averaged over the appropriately-weighted four values of bi . Figure 16 shows a graph of this average slope < m> as a function of σs/ σp.
Figure 16: The exact initial slope of the conductivity, in the limit of dilute sand
concentration, is shown as a function of
for the four size model mortar.
Note in the limit of σs/σp = 1, the slope < m > = - 3/2, 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, 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.
The only exact results available for general three dimensional composite materials, in addition to the dilute limits presented above, are variational bounds . However, effective medium theory (EMT) can often be employed to estimate composite properties at arbitrary volume fractions of the phases [77,78]. By EMT we mean only those theories that properly describe the dilute limit, and are then built up via some sort of averaging assumption into approximate analytical equations. The two examples that have been previously considered are the self-consistent  and the differential  methods.
In Figure 17 we present the random walk simulation results for the same four diameter aggregate mortar model together with the predictions of the differential (D-EMT) and self-consistent (SC-EMT) effective medium theories .
Figure 17: Composite conductivity for the four size model mortar is plotted vs. the
interfacial zone conductivity. [Both are normalized by bulk paste conductivity.] The solid dots
are the random walk data; also shown are the effective medium results (SC=self consistent and
The simulations were of how the effective conductivity, at a single aggregate volume fraction, 55%, and a single interfacial transition zone thickness, 20 µm, varied as a function of the conductivity of the interfacial transition zone phase, σs, expressed as the dimensionless ratio σs/σp. The overall shape of the curves is concave down. In this case, the effective conductivity could at most be linear in σs/ σp, for the following reason. If the two conducting phases, interfacial transition zone and bulk cement paste, were exactly in parallel geometrically, 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. Since the microstructure is such that these two cement paste phases are not in parallel, the curve must be sub-linear, or concave down. As σs/σp approaches infinity, the curve will eventually become straight as predicted , since the interfacial transition zone conductivity will dominate the effective conductivity. Note that by the point that σs/σp = 20, the effective conductivity is almost double the pure matrix conductivity, σ/σp 1.8.
The data in Fig. 17 indicate that 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. This is remarkably close to the dilute limit result σs/σp 8.26 found in Fig. 16. Because the dilute limit seems to define an essential feature of the overall curve, effective medium theories that are based on the dilute limit have a good possibility of successfully predicting the essential structure of the composite conductivity. The two EMT's displayed in Fig. 17 do show reasonably good agreement with the simulation data.
Figure 18 shows computed conductivity data for the random mortar model as a function of sand volume fraction, for σs/σp = 20,5, and 1,
Figure 18: Composite conductivities (calculated by random walk simulations) for the
model are shown as a function of sand concentration for three values of the interfacial zone
conductivity. Also shown are the predictions of the SC and D-EMT calculations, with the same
normalization as in the previous figure.
The curve for σs/σp = 20 is concave up, with an initial slope that is positive, since σs/σp = 20 > 8.26. Both EMT's predict that when the inital slope is positive, the value of the effective conductivity, σ/σp, will always be greater than 1. The simulation data do obey this prediction. The value of σs/ σp = 20 is not high enough in order to see clear evidence of the interfacial zone percolation threshold, which, for much higher values of σs/σp, would manifest itself as a sharply increasing conductivity near the percolation threshold. The σs/σp = 5 and 1 curves have negative initial slopes, and the efffective conductivities remain below one, as predicted also by EMT. 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 .
Figure 19 shows analytical results for <m> for a typical mortar  and a typical concrete .
Figure 19: The exact initial slope of the conductivity, in the limit of dilute sand
shown as a function of
for a typical mortar and a typical concrete aggregate
particle size distribution.
At a given value of σs/σp, the value of <m> for the mortar is always higher than that for the concrete. This comes back to the same point made earlier, that the mortar has a larger surface area at a given volume fraction, due to the smaller average particle size, and so the interfacial zone has a larger volume and plays a bigger role in its composite properties than it does in the concrete's composite properties.