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 [53]. 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 "shaken" sufficiently to achieve their desired positions. This process is actually similar to how a real concrete is mixed. In the case of the concrete model, however, the particles are placed according to non-equilibrium random parking statistics [68], as described above. However, this analytical formalism works quite well for the concrete model [51,54].

There are many analytical results contained in the paper by Lu and Torquato [53] that are relevant to the concrete problem. In this paper we focus on the quantity e_V (r), the "void exclusion probability" as denoted in Ref. [53] (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, the volume fraction of material outside of both the particles and the shells is just e_V (r). The ITZ volume fraction, V_ITZ , is then just

$$V_{ITZ} = 1 - e_V (t_{ITZ}) - \eta$$ (2)

where $\eta$ is the volume fraction of aggregates [53]. The functional form of e_V (r) is

$$e_V (r) = (1 - \eta) exp [ - \pi \rho (c r + d r^2 + g r^3)]$$ (3)

where $\rho$ 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 an aggregate sieve analysis, using certain reasonable assumptions, as is shown in Section 4.4. The coefficients c, d, and g are:

$$c = \frac{4 \langle R^2 \rangle}{1 - \eta}$$ (4)

$$d = \frac{4 \langle R \rangle}{1 - \eta} + \frac{12 \epsilon_2 \langle R^2 \rangle}{(1 - \eta)^2}$$ (5)

$$g = \frac{4}{3 (1 - \eta)} + \frac{8 \epsilon_2 \langle R \rangle}{(1 - \eta)^2} + \frac{16 A \epsilon_2^2 \langle R^2 \rangle}{3 (1 - \eta)^3}$$ (6)

where $\epsilon_2 = 2\pi \rho \langle R^2 \rangle / 3$, and A is a parameter that can have different values (0, 2, or 3) depending on the analytical approximation chosen in the theory [53]. The actual choice used may be fixed by experiment. In previous work on concrete models [51], the actual value of A used did not make much difference, but A=0 was always slightly better than A=2,3.