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Effects of the percolation of the interfacial transition zones can be observed from mercury porosimetry data [59]. Cement paste by itself has a certain threshold pore diameter, corresponding to a certain mercury pressure. For mercury pressures below this value, the mercury only intrudes the surface of the sample. For mercury pressures above this value, the mercury percolates throughout the material. When aggregate is added, this threshold diameter does not increase at first, but remains at a value typical of cement paste. As the volume of aggregate increases to a critical value, the mercury breakthrough pressure or critical diameter abruptly changes from values typical of bulk cement paste to those typical of the interfacial transition region, pore sizes roughly ten times bigger. This is interpreted as the point at which there are enough aggregates present so that the associated interfacial zones can overlap and percolate [59].
To study the percolation or connectivity of the interfacial transition zones in concrete is computationally not simple, as the geometry of this phase is complex. Fortunately, in the percolation literature there is a model that is perfectly suited for this study, the hard core/soft shell (HC-SS) model [60]. This model starts with a random suspension of hard spherical particles, which are hard in the sense that they are packed without being allowed to overlap, as in a suspension. Then concentric spherical shells are placed around each particle, where the spherical shells all have the same thickness, and are allowed to freely overlap. The volume fraction of shells required to make the shells percolate, which is when a continuous soft shell pathway first becomes established, is then computed. The volume fraction of soft shells required for percolation is a function of how many hard core particles are present, and the thickness of the soft shell. Obviously, when more hard core particles are packed in a given volume, there is less space between them, so that thinner soft shells will be sufficient for percolation. When there are fewer hard core particles present, thicker shells will be required for percolation of the shell phase. For concrete, clearly the hard cores correspond to the aggregates, and the soft shells to the interfacial transition zones. In the percolation literature, usually the value of h varies for different radii particles so that the ratio of radius + h to the radius remains constant for different radii. In the case of concrete, the value of h remains constant for different particle radii.
Figure 11 shows a two dimensional slice through a simple HC-SS three-dimensional model of a mortar, where four different sizes of spherical sand grains were used, ranging between 0.5 and 3 millimeters in diameter [61]. Note that there are more than four diameters of particles apparently present in the picture, because the slice did not always go through sphere centers. In this two-dimensional slice, the interfacial zone regions do not appear to percolate at all. As a further example, Fig. 12 shows slices through two other similar systems, but with both having a wider range of aggregate sizes than in Fig. 11. The right hand system in Fig. 12 is built up of ellipsoidal particle shapes [62].
Figure 11: Showing a slice of a three dimensional model of mortar, with four
aggregate diameters ranging between 0.5 and 3 millimeters. The four colors
(dark blue, light blue, green, and red, indicate, from small to large, the four
aggregate sizes used. The interfacial transition zone is shown as yellow.
Figure 12:
Showing slices of two different
three dimensional concrete models, with spherical and ellipsoidal
shaped aggregate particles. Aggregates are white, bulk cement paste is gray,
and the interfacial transition zones are black.
To study the percolation in 3-D of the interfacial transition zones, it is good to first consider a simple, yet non-trivial example of these ideas: a HC-SS model for monosize spherical hard spheres [63,64]. We study the percolation of the soft shell for different thicknesses of the shell (h=a-b) compared to the radius of the hard cores (b). The total radius of the composite particle is a=b+h. The relevant variable in this case is the ratio b/a, where b/a = 0 is just the overlapping sphere problem mentioned earlier in the chapter, and b/a = 1 is the random sequential absorption or random parking problem for non-overlapping spheres in 3-D [65]. The larger the value of b/a (i.e. the thinner the shells), the greater the value of the volume fraction of hard cores, c, that will be required and the smaller the fraction of space that will be occupied by the percolating shells. Figure 13 shows the volume fraction of the shells at percolation vs. b/a.
Figure 13: Showing the volume fractions of the hard cores and the soft shells when the
soft shell
phase has just percolated, for monosize spherical hard core particless. The solid lines are from an
effective medium theory described later in the text.
At b/a = 0, we get a volume fraction of 0.29 [8], and as b/a approaches 1, the volume
fraction of the shells needed to percolate goes to zero [66]. Shown also are the value of the
limiting concentration c 
0.38 for mono-size random parking [65], and curves
based on an effective medium theory [66].
Note that there is a threshold value of b/a 0.9615 above which it does not seem
possible to form connected percolating shells based on the random parking algorithm for
monosize spheres.
To study interfacial transition zone percolation in concrete, we take a fixed shell thickness to represent the interfacial transition zone, and then randomly place spherical aggregate particles that are then each surrounded by these shells. The width of the interfacial transition zone should be independent of aggregate size, as long as the median aggregate size is at least 5-10 times the median cement particle size, and will depend only on the median cement particle size [47]. The size distribution of the hard core particles are taken from measured aggregate size distributions [59,67,68,69]. The fraction of the total shell volume that forms part of a connected cluster is then computed as a function of the volume fraction of aggregate present.
Results for a mortar [59,68,69] are shown in Fig. 14, in which each curve shows the connectivity of the interfacial transition zones for different choices of interfacial transition zone thickness.
Figure 14: Showing the fraction of the total interfacial zone volume that is a part of a
percolated (continuous) cluster as a function of aggregate volume fraction and for several
interfacial zone thicknesses.
When comparing against portland cement mortar mercury intrusion data [59,68,69], it was found that a choice of 20 µm for the interfacial transition zone thickness gave the best agreement with the mercury data. The mercury data gave an idea of what the percolation threshold of the interfacial transition zones was by showing a large increase in large pores intruded at the mercury breakthrough point [70] at a certain aggregate volume fraction. The width given in scanning electron microscopy studies of the interfacial transition zone, 30- 50 µm, is defined by measuring from the aggregate edge to where the measured porosity assumes its bulk value. This would not be the width seen by mercury porosimetry, because it is probable that the larger pores will be found in the larger porosity regions nearer to the aggregate, which will be seen first by the mercury. Also, as to the effect on transport properties, the inner region of the interfacial transition zone is of more importance, since its transport properties will be higher than the outer region because of the larger pore size and porosity. The width of 20 µm given by the hard core/soft shell model is then an effective width, where this width contains the larger pores that would be important for transport. Fig. 14 also shows that for an aggregate volume fraction of 40% or more and an interfacial transition zone thickness of at least 20 µm, the interfacial transition zones will be percolated at least partially, and will be fully percolated for aggregate volume fractions greater than 50%. Most concretes have aggregate volume fractions well above 50%, so that in general, we can conclude that the interfacial transition zones in usual portland cement concrete are percolated, and so will have an effect on transport properties.
The fraction of the cement paste that lies in interfacial transition zones can also be calculated using the hard core/soft shell model for a given aggregate volume fraction and particle size distribution. Results are given in Fig. 15 for a specific mortar and concrete [59].
Figure 15: Showing the fraction of the total cement paste volume fraction that lies within
a given distance from an aggregate surface, for a mortar with a sand volume fraction of 0.552,
and a concrete with an aggregate volume fraction of 0.646.
Figure 15 shows that, for the mortar, quite a large part of the cement paste matrix lies within an interfacial transition zone, with about 20% lying within 20 µm, and about 50% lying within 50 µm of an aggregate surface. For the concrete, the volume of cement paste matrix in the interfacial transition zone is smaller at each distance than for the mortar. This is related to the fact that the mortar, at an equal volume fraction of aggregate, has a much larger surface area because of the smaller size of the aggregates, and therefore more interfacial transition zone material. This difference will also be reflected in the transport properties, as will be seen further below.
The interfacial transition zone percolation problem has also been studied for ellipsoidal aggregate particles [62] as a first look at how the aggregate shape could affect interfacial transition zone percolation. The aspect ratio of the ellipsoidal aggregates was varied between one and four, since most aggregates found in real concrete are reasonably spherical. The interfacial transition zone percolation threshold in terms of the aggregate volume fraction was found to be dependent on the aspect ratio of the aggregate. However, the interfacial transition zone percolation threshold in terms of the volume fraction of interfacial zone (paste) was not dependent on the aggregate aspect ratio [62].
