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In the approach to computing the diffusivity of concrete that has been previously discussed [58,59], the ratio of the concrete diffusivity to the bulk cement paste diffusivity as a function of the ITZ diffusivity and the aggregate volume fraction was computed. The value of diffusivity produced was relative to the value assigned to the bulk paste phase.
One might then naively think that an analogous experimental measurement would be to measure the concrete diffusivity as a function of the degree of hydration, and also measure the diffusivity of the cement paste from which the concrete was made (same w/c ratio). Normalizing the concrete diffusivity by the cement paste diffusivity at the same degree of hydration would presumably then give a number that could be compared to the model number [58,59]. As was pointed out in Ref. [58], this approach is not entirely correct. The main reasons are the way concrete is made, and the way in which the interfacial transition zone, as mentioned earlier, serves to redistribute the cement. There is, therefore, a higher than average w/c ratio in the ITZ regions and a lower than average w/c ratio in the bulk regions. The following example will serve to illustrate this point more precisely.
Suppose a concrete was made by mixing a 0.5 w/c cement paste with enough aggregates so that the final volume fraction of aggregates is 60%. We assume here that the aggregates neither absorb nor give off water. If the ITZ is 20 µm thick, all cement paste within this distance from an aggregate will, on average, have a higher porosity and therefore a w/c ratio higher than 0.5. This is because there is known to be less cement in the ITZ than in the bulk paste [48,63]. But the only way this can happen is for the cement paste outside the ITZ region to have a w/c ratio that is less than 0.5 and, therefore, a lower porosity, since the average w/c ratio has been specified to be 0.5 by the initial mixing conditions. If the ITZ volume fraction is small, this effect is also small, which would be the case for a small aggregate volume fraction. However, as the volume fraction of aggregate increases, the ITZ volume fraction also increases, and this effect can become quite appreciable. So for this hypothetical concrete, it would be incorrect to consider an 0.5 w/c ratio cement paste to be the matrix or bulk phase for the concrete. The actual bulk paste, where "bulk" means outside the ITZ region, would have a lower w/c ratio, possibly as low as 0.4. Therefore, concrete is not a simple two-phase composite, of cement paste plus aggregates, and not even a simple three-phase composite, of aggregates plus interfacial transition zone and bulk cement paste. Concrete is rather an interactive composite, where the amount of the aggregates influence the properties and amounts of the cement paste phases. The multiscale model approach described in this paper is designed to approximately take this effect into account, by combining models to actually compute this redistribution of cement and w/c ratio.
The multi-scale model works in three steps. It is not a particularly simple model, as it is addressing the transport properties of a complex interactive composite material.
(1) The cement particle size distribution (PSD) is used to establish the interfacial zone thickness, tITZ. The value of tITZ is taken to be equivalent to the median cement particle diameter, ignoring any effects of bleeding [13]. The first key step is then, using the millimeter-scale concrete microstructure model, to place aggregate particles, following the aggregate PSD of interest, into the concrete volume. Systematic point sampling is then used to determine the volume fractions of interfacial transition zone (V ITZ ) and bulk (Vbulk) paste for this particular choice of aggregate PSD and value of t ITZ [48].
(2) The second key step uses the micrometer-scale cement hydration model. The shape of the model volume is shown in Fig. 16, with aggregate, ITZ, and bulk paste regions defined. A double interface is used in order to allow easier use of periodic boundary conditions [6,57]. We want to determine the local microstructure near a single aggregate surface, in order to be able to compute the contrast between ITZ and bulk paste diffusivities. The dimensions of the model box are chosen to match the ratio of VITZ/Vbulk (determined in step 1). By matching the ratio VITZ/Vbulk as determined in the concrete volume, we approximate the microstructure that exists near a typical aggregate surface.
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Cement particles are placed into this computational volume,
following the cement PSD, to achieve the specified total w/c ratio. Of
course, the w/c ratios in
the two regions will be different
because of the wall effect of the aggregate surface. The actual numerical
difference is a prediction
of the micrometer-scale microstructure model, and is not specified by the user. The cement
particles are then hydrated, using the model, up to a desired degree of hydration. After hydration, the porosity 
is measured as a function of distance from the
aggregate surface. The relative diffusivity, D/Do
, of cement paste as a function of
distance from the aggregate surface, x, can be
estimated using a previously established relationship [64]:
where relative diffusivity is defined as the ratio of the diffusivity D of ions in the material of
interest relative to their value in bulk water, Do,
(x) is the capillary porosity volume
fraction at a distance x from an aggregate surface, and H is the
Heaviside function having a value
of 1 when
> 0.18 and a value of 0 otherwise. This equation comes from fitting the results
of several different w/c cement pastes at many different degrees of hydration, where a value of
diffusivity for the
C-S-H phase was used which agrees with nanometer scale simulations of
C-S-H nanostructure and properties [56]. The constant term in eq. (1) comes from the limiting value
of diffusion through
C-S-H gel pores when the capillary porosity is zero.
The H term represents diffusion through percolated capillary porosity, and
turns off this transport mechanism when the capillary porosity percolation
threshold is reached. The second term in eq. (1) is
a fitting term that
connects the two limiting behaviors [64]. Equation (1) is
not exact, of course, but should
give results accurate to at least a factor of two for the absolute diffusivity, and better than that for
ratios of ITZ to bulk diffusivity, for the usual range of capillary porosity encountered (10-40%).
Refs. [64,65] give experimental validation of
this model and the associated eq. (1).
Using eq. (1), the diffusivity as a function of distance from the aggregate surface is computed. These diffusivity values are averaged in two subsets, those lying within a distance tITZ of the aggregate and those in the "bulk" paste, to give two values, DITZ and Dbulk. Averaging in this way assumes that the diffusive flux in the two phases is locally parallel to the aggregate surface, and so each layer can be simply summed up. By averaging over tITZ, the simplifying approximation is made that the interfacial transition zone has a fixed width with fixed properties. The average over the simulated local microstructure of the interfacial transition region is done to help make this approximation more accurate. Choosing a different value of tITZ would of course give a different value of ITZ diffusivity. Other ways of averaging over the local microstructure are possible, such as matching to an exact solution of a single specific size aggregate with a gradient of properties around it [66]. These methods do not, however, seem to make a significant difference in the final results [66].
After the first two steps, we now have a microstructure model of the concrete, using spherical aggregates that follow the correct PSD, and we have values of diffusivity for each of the three phases in the concrete: aggregate (Dagg = 0), bulk cement paste (Dbulk), and ITZ cement paste (DITZ). It is crucial to remember that the values of Dbulk and DITZ were determined interactively, since the amount and size of the aggregates was used to determine the value of VITZ/Vbulk that was used in the cement paste model, which in turn helped determine the value of DITZ/D bulk. The ratio of DITZ/D bulk is a function of the aggregate PSD and volume fraction and the degree of hydration of the cement.
(3) The third key step is to finally use the ratio of the bulk and ITZ cement paste diffusivities, DITZ/D bulk, as an input back into the original diffusivity concrete model [58,59]. Random walk numerical techniques are then employed to compute the diffusivity of the overall concrete system. The random walker techniques employed for this calculation have been previously described [58,59]. These techniques are quite computer-time intensive, but do not use huge amounts of memory. The relative diffusivity of the concrete, Dconc/D bulk, is calculated by this algorithm [57,58,59]. This value can then be converted into an absolute chloride ion diffusivity for the concrete, Dconc, by multiplying it by Dbulk/D o determined from the cement-level microstructural model [eq. (1)] and by Do, the diffusion coefficient of chloride ions in bulk water at room temperature (25ºC), given as 2.0 · 10-9 m2/s [67]. By changing the value of Do to correspond to that measured for the specific ion of interest, the above techniques can be generalized to other ionic species of relevance in cement-based materials. This model does not address effects such as chemical binding and chemical reactions of the diffusant.
