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Lu and Torquato  have recently derived an analytical formula for the statistical geometry of composites that is basically set up to predict the ITZ volume fraction. For a packing of spherical particles in a matrix, where the spheres can have any size distribution, they derived an approximate formula for a quantity they denoted as ev(r). Figure 1 shows a schematic view, in 2-D, of the geometry of the problem being considered. If a shell of thickness r is put on every sphere in the packing, the quantity ev(r) is defined as the volume fraction of matrix material that is outside all the spheres and all the shells. Clearly, ev(r) → 0 as r → ∞, since eventually the shells will overlap enough to fill up all the remaining matrix material. This function was designed to take into account the overlaps of these shells, so as to be accurate not just in the small r limit, where the volume of the shell phase is just the surface area of the particles times r, but for all values of r. In the concrete case, with spherical aggregates, if we take r = tITZ to be the ITZ thickness, then clearly the ITZ volume fraction is just
|VITZ = 1 - Vagg - ev(tITZ)||(1)|
where Vagg is the volume fraction of aggregates in the concrete.
Figure 1: A schematic view of the quantities in the Lu and Torquato formula . The thickness of every shell is r.
There are certain assumptions that go along with this formula. The aggregate size distribution must be known in terms of the number of particles with a certain size, not the volume of particles with that size. Using some simple assumptions, it is easy to generate this kind of distribution function from a typical sieve analysis. A recent publication gives details of how this can be done . Also, the spherical particles must be in an equilibrium arrangement, that is, arranged as they would be if they were suspended in a liquid and free to move.
Using the result that was derived for ev(r), at r = tITZ, the ITZ volume fraction is then
where ρ is the total number of particles per unit volume. Using the size distribution function, the quantities c, d, and g in the equation can be defined in terms of the number averages of the particle radius and the particle radius squared:
where z2 = 2 π ρ < R2 >/3 , A is a parameter equal to 0, 2, or 3, depending on the analytical approximation chosen in the theory , and indicates an average over the aggregate size distribution. In all our work on model spherical aggregates, A = 0 was always the best choice to use, as decided by comparison to numerically exact model data, although the value of A did not seem to make much difference. The term controlled by the value of A was a small contribution to the prediction of the ITZ volume fraction.
Figure 2 shows a comparison of the above formula to numerical model data for a typical concrete distribution, with aggregate particle sizes ranging from 0.1 mm to 10 mm, which is a range of a factor of 100 in size. Even for distances r that are larger than the usual width of the ITZ, the formula evidently correctly handles the large amount of overlapping of the ITZ's, for two different values of Vagg , 0.27 and 0.75. Figure 3 shows the same formula applied to a system with 27% by volume of monosize aggregates, where it does not work quite so well. This is because the formula was derived for spherical particles that were arranged like they would be if they were really suspended in a liquid, according to equilibrium statistics. Our models are made by randomly placing spherical particles in the cement paste, and not allowing them to move after placement. For monosize particles, it is well-known that after about 20% volume fraction of particles are present, the statistics of the two kinds of particle placing become gradually different. If our monosize particles were arranged according to equilibrium statistics, then the formula would agree with numerically exact model data to better than 1% accuracy . So in Figure 3, our system does not meet the conditions necessary for the analytical formula of eq. (2) to hold, although the agreement between theory and model is still reasonably good, especially for the usual ITZ thickness values (10 to 40 µm).
Figure 2: Showing the fraction of the matrix phase taken up by the ITZ's, of width r, as a function of r, for a typical concrete, at two different aggregate volume fractions, 0.27 and 0.75. The symbols are data points determined by pointcounting, the lines are the result from Lu and Torquato's formula .
Figure 3: Showing the fraction of the matrix phase taken up by the ITZ's, of width r, as a function of r, for monosize aggregates, at an aggregate volume fractions of 0.27. The symbols are data points determined by numerical point counting, the lines are the result from Lu and Torquato's formula .
In principle, the formula should not apply to our models for the reason given above. However, somewhat surprisingly, for a very wide particle size distribution, the statistics of the two kinds of particle placing are seemingly almost the same, as the formula fits the typical concrete distribution extremely well (see Fig. 2). This is an interesting result, and deserves closer study to determine under what conditions the statistics of the two different particle arrangements become nearly the same. Experimentally, the aggregates in a concrete are probably in a close-to-equilibrium arrangement, if gravitational settling is not too important, there is a wide size distribution of aggregates, and the aggregates are reasonably spherical. Therefore the formula should work well in real concrete, as long as these conditions are met.