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What is the use of these exact expressions? First consider the ITZ volume fraction formula. If we take the ITZ's as single thickness layers, then this formula gives the total volume occupied by ITZ's, taking into consideration the many, possibly multiple overlaps of this phase. This formula can also be used, as in Figs. 2 and 3, to give the matrix volume that is contained within a certain distance of an aggregate surface. For surface controlled phenomena, possibly including alkali-aggregate reaction, for example, this might be an important kinetic parameter. Freeze-thaw damage to the aggregate, for a porous aggregate, might also depend on the distance water has to move from the paste to the nearest aggregate surface. Also, the protected paste for a given air-void system, where the air-voids are the spherical particles, can also be investigated with this formula .
The exact dilute limit calculations can be applied in several ways. (1) They give a non-trivial, exact calculation of how ITZ properties can affect overall concrete properties. Real concretes have high volume fractions of aggregates, but the qualitative effect of the ITZ can be seen in the dilute case. Since the exact dilute limit is averaged over the volume distribution of the aggregates, differences can be seen, for instance, between mortars and concretes. (2) These formulas serve as useful checks on approximate analytical formulas that are either derived theoretically, in some effective medium approach, or experimentally, by fitting data or by using some other approximate approach. In the dilute limit, these kinds of expressions, if the assumption of a spherical aggregate is used, must reduce to the exact dilute limit formula . (3) It is often easier to think of the ITZ as a zone of some thickness that has a uniform property throughout, rather than as having a gradient of properties. Solving for the exact gradient, then mapping onto the exact solution for a single layer, can give the correct uniform property to be used for a given thickness . (4) Exact dilute limit calculations are in general very useful for checking the accuracy of various numerical schemes that attempt to handle the full volume fraction of aggregates . (5) Effective medium theories can be built out of the exact dilute limits in various ways. These effective medium theories can be quite accurate in analyzing the properties of concrete models .
Figures 5 and 6 show an example of application (1) above. In Fig. 5, the particle size distribution, in terms of the particle volumes, is shown for a C-109 mortar  and for a typical concrete . Notice that both go down to about 0.1 mm diameter particles, but the concrete also has particles in the 10 mm range, while the mortar stops at about one millimeter. If there were no ITZ, then the slopes for the mortar and concrete would be identical, as there is no size dependence for the initial slope for a spherical particle embedded in a matrix [21,22], and positive, as the aggregate particle being introduced into the cement paste is stiffer than the cement paste. This can be clearly seen at the point where the Young's modulus of the ITZ equals the Young's modulus of the bulk paste, effectively removing the ITZ. When the ITZ Young's modulus is less than that of the bulk paste, as would be expected because of its higher porosity, then it is clear that the mortar is more affected than the concrete, because of its smaller average diameter. One way of thinking of this is that the smaller the particle, the higher is the ratio of ITZ volume to particle volume. This ratio, for single spherical particles, is (1 + tITZ/r)3. Alternatively, for the same volume of aggregates, the mortar will have a higher surface area and therefore a higher volume fraction of ITZ than will the concrete. In Figure 6, the bulk and shear modulus of the paste were 20.8 and 11.3, respectively, and the bulk and shear modulus of the aggregate were 44 and 37, respectively, in units of GPa. These values were taken from a case in Zimmerman et al. . The ITZ paste had the same Poisson's ratio as the bulk paste but a smaller Young's modulus as shown in Figure 6.
Figure 5: Showing the two aggregate particle size distribution functions, in terms of volume, for a C-109 mortar and a typical concrete.
Figure 6: Showing the initial slopes for Young's modulus E and bulk modulus K for the mortar and concrete of Fig. 5.
In an example of application (3), Figure 7 shows a porosity gradient that might be found in a concrete or mortar. The "width" of the ITZ is about 10 µm. We then use the dilute limits to quantitatively map the true dilute limit, for this gradient and a given functional dependence of the properties on the porosity, onto the single uniform property shell case. This mapping is schematically shown in Fig. 8. We choose the following sets of properties: σp is the electrical conductivity of the bulk cement paste, σagg = 0 is the electrical conductivity of the aggregate, and the electrical conductivity of the cement paste in the ITZ scales as the square of the porosity, matching onto σp as the distance from the aggregate surface increases. In the elastic moduli case, we take K,G = 10,6 for the aggregate, and K,G = 3,1 for the bulk cement paste, both in arbitrary units. In calculating the bulk modulus slope, the units cancel out anyway. The ITZ cement paste has the same Poisson's ratio as the bulk cement paste, but its Young's modulus scales as the solid fraction cubed . It is important to remember that we are mapping an exact result, for the porosity gradient, onto another exact result, that for a single ITZ shell (N = 3).
Figure 7: Showing an artificial porosity gradient surrounding a spherical particle of diameter 400 µm. The gradient is reminscent of those found in real concretes.
Figure 8: Schematic picture showing how the dilute limit of an aggregate surrounded by a gradient in properties can be mapped into the case of an aggregate surrounded by a shell having a given thickness and uniform property.
Figure 9 shows the electrical conductivity results. It is clear that choosing a larger thickness for the ITZ results in a smaller value of the ITZ conductivity relative to the bulk paste conductivity. In this sense, one cannot really talk about "the ITZ conductivity" without specifying the value that is chosen for the thickness. Other papers have suggested that the median diameter of the cement grains should be chosen as the "best" value for the ITZ thickness . The point of Fig. 9 is that if the ITZ is modelled as a uniform property region of some conductivity, this conductivity depends on what thickness is chosen to represent the actual porosity gradient.
Figure 9: Showing the dependence of the ITZ conductivity, compared to the bulk paste conductivity, for various thicknesses of the ITZ.
Figure 10 shows the elastic results for a similar kind of problem. In the elastic case, the mapping can be done two different ways. The exact bulk modulus slope, for the porosity gradient, can be mapped onto the bulk modulus slope of the single ITZ shell case. This can also be done, with equal validity, using the shear modulus. Figure 10 assures us that, in least in this case, and when the ITZ moduli have the same Poisson's ratio as the bulk paste, essentially the same result is obtained using either the bulk or the shear modulus. As the thickness of the ITZ is increased, the ITZ moduli become larger and thereofore closer to the bulk paste moduli. This is because in the elastic case, the effect of increasing porosity is to make the moduli smaller, so that the ITZ moduli are less than the bulk values, and increase with distanceaway from the aggregate surface. This is the exact opposite of the case for the conductivity.
Figure 10: Showing the dependence of the ITZ moduli, compared to the bulk paste moduli, for difference thicknesses of the ITZ. The ITZ moduli are chosen so as to have the same Poisson's ratio as the bulk paste. The two curves are generated by matching the bulk modulus slope and the shear modulus slope independently to the true slope generated by the gradient of properties.