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Interfacial zone volume: Step (1)

An analytical estimate for the total interfacial zone volume around a collection of spheres of various sizes can be obtained from the literature on the statistical geometry of composites [26]. In this work, a collection of spheres of various sizes are randomly placed in a volume according to equilibrium statistics. These involve treating the spheres as being dispersed in a liquid, where the effect of gravity is neglected, and allowing them to be "shook" sufficiently so as to achieve their desired positions. This process is actually similar to how a real concrete is mixed. In the case of our concrete model, however, the particles are placed according to non-equilibrium random parking statistics [27], as described above. Notably, the equilibrium and the random parking statistics are essentially identical for single size spheres up to about 20% volume fraction of spheres.

It would seem that this analytical estimate could not be applied to our model which has randomly parked aggregates. However, we were encouraged to do so by results we obtained for monosize spheres, randomly parked at 27% by volume [28]. Monosize spheres are the worst case, where the most differences can be seen between random parking and equilibrium distributions. However, the Lu and Torquato analysis worked very well even for this system, so we have applied it as well to these concrete systems. Why this should be is not known at present, but is the focus of further research.

There are many analytical results contained in the paper by Lu and Torquato [26] that are relevant to the concrete problem. In this paper we focus on the quantity eV(r), the "void exclusion probability" as denoted in Ref. [26] (note: in our case, "void" means outside the aggregates). As formulated by Lu and Torquato, if one adds a spherical shell of thickness r around each one of the spherical particles, then the volume fraction of material outside of both the particles and the shells is just
eV(r). The ITZ volume fraction, VITZ, is then just

where η is the volume fraction of aggregates [26]. The functional form of eV(r) is

where ρ is the total number of aggregates per unit volume, and the coefficients c, d, and g are given in terms of averages (<...>) over the particle size distribution of the aggregates in terms of number, not volume. These averages can be determined from the sieve analysis, using certain reasonable assumptions, as is shown in Appendix B. The coefficients c, d, and g are:

where ε2 = 2 π ρ < R2 >/3, and A is a parameter that can have different values (0,2, or 3) depending on the analytical approximation chosen in the theory [26]. The actual choice used may be fixed by experiment. In all the work on concrete, the actual value of A used did not make much difference, but A = 0 was always slightly better than A = 2 or 3, as judged by comparison to the exact (numerical) values listed in Table 2. Note that there is a misprint in Eq. (4.27) in Lu and Torquato [26], which is corrected in the present eqs. (4), (5), and (6).

System Vagg Vair PSD tITZ (mm) DITZ/Dbulk VITZ D/Dbulk
1 0.599 0 cfcc 0.03 2.24 0.0580 0.34
2 0.757 0 cfcc 0.01 4.94 0.0208 0.23
3 0.754 0 fffc 0.03 2.54 0.1647 0.28
4 0.601 0 cffc 0.03 4.22 0.0595 0.42
5 0.594 0 fffc 0.01 5.00 0.0431 0.42
6 0.757 0.0948 cfcc 0.03 4.48 0.1167 0.30
7 0.753 0 cffc 0.01 2.95 0.0242 0.20
8 0.752 0 ffcc 0.03 3.31 0.1577 0.34
9 0.601 0.0948 cffc 0.01 2.15 0.0422 0.25
10 0.602 0 ffcc 0.01 2.84 0.0458 0.36
11 0.752 0.0825 ffcc 0.01 2.34 0.0655 0.16
12 0.754 0.0775 fffc 0.01 1.18 0.0744 0.11
13 0.599 0.0948 cfcc 0.01 1.26 0.0421 0.22
14 0.594 0.0948 fffc 0.03 5.34 0.2008 0.69
15 0.753 0.0948 cffc 0.03 7.24 0.1214 0.47
16 0.602 0.0948 ffcc 0.03 8.34 0.2029 1.05
17 0.675 0.0454 medium 0.02 6.49 0.0927 0.46
18 0.524 0 cfcc 0.01 4.06 0.0164 0.42
19 0.824 0 cfcc 0.01 4.14 0.0231 0.16
20 0.675 0 cfcc 0.01 2.32 0.0190 0.26
21 0.675 0 cfcc 0.01 7.53 0.0190 0.33
22 0.675 0 cfcc 0.01 1.08 0.0190 0.23
23 0.675 0 cfcc 0.01 1.88 0.0190 0.24

Table 2: Parameters of model systems used, including point-counting result for VITZ and
myopic random walk results for D/Dbulk.

It should be noted that eqs. (2) and (3) can be used to map out the volume fraction of cement paste that is within a distance r of an aggregate surface. The accuracy of eq. (2), as will be demonstrated in Section 4, might very well allow it to be used to help theoretically analyze a phenomenon like alkali-silica reaction, where cement paste constituents, namely alkalis in the pore solution [29], must diffuse to an aggregate surface in order for the alkali-silica reaction to take place. Another possibility might be the reactivity of calcium hydroxide with fly ash, where calcium from the calcium hydroxide particle must diffuse to the silica-rich fly ash, because of the higher mobility of calcium in solution compared to silica. The distance of the cement paste to the nearest aggregate surface or of a calcium hydroxide crystal to the nearest fly ash particle then must obviously be important for the kinetics of these processes.

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