Because of the wide range of feature sizes in concrete, from nanometer-sized pores to millimeter-sized aggregates, memory limitations on current computers make it impossible to simultaneously represent all of these structural features in a single microstructure model. For example, suppose one wanted to represent all phases, from C-S-H to aggregate, in a 3-D digital image model of concrete. The pixel size in this case should be one nanometer, which would be barely adequate for the resolution of the C-S-H phase. The minimum size model that would incorporate enough aggregates to be statistically valid would be at least 30mm on a side, assuming a maximum aggregate size of about 10mm. Such a model would contain (30 · 106)3 = 2.7 · 1022 pixels! Considering the limits of even modern-day computers, uniting multiple length scales in a single model in a more sophisticated (and less computer-memory intensive!) way is obviously necessary.
Multi-scale modelling techniques offer a promising solution to this restriction [56,57]. In this approach, properties computed at one scale, say micrometers, are fed into a model which is constructed at a higher scale, such as millimeters. The present multi-scale model combines microstructure models for the cement paste surrounding a single aggregate (micrometers), and for a representative volume of concrete (millimeters). These two microstructure models are used, in association with computational techniques for computing the diffusivity of a three-dimensional microstructure, to compute the diffusivity of a representative volume of concrete. This procedure was first demonstrated for mortars [57], and has since been extended to normal concretes [6,51]. The equation used for diffusivity at the micrometer scale also indirectly incorporates information from the nanometer scale, so that 6-7 orders of magnitude of length scale are actually contained in the model.
Computation of the transport properties of a concrete is made difficult by the fact that two length scales must be treated directly: the micrometer scale ITZ regions, and the millimeter scale aggregates. In Refs. [58] and [59], it was shown how to compute the overall diffusivity of a concrete model, where the difference between bulk and ITZ cement paste could be quantitatively taken into account. In that paper, it was mentioned that the redistribution of cement between bulk and ITZ regions, because of the different w/c ratios in these regions, could very well be important, and would have to be considered in future models. The main reason for this particular multi-scale approach being developed [6] was to approximately handle this redistribution of cement between ITZ and bulk cement paste. This effect does indeed turn out to play an important role in predicting concrete diffusivities, because it affects what diffusivity values need to be assigned to the bulk and ITZ phases in a composite analysis of concrete properties. Therefore, the results of Refs. [58] and [59] are incomplete, and need a multi-scale analysis in order to properly assign values to the diffusivity of various phases.
In Ref. [6], this multi-scale modelling approach was extensively discussed, showing how different length scales could be quantitatively linked to predict the diffusivity of a concrete material. Because of the complexity of the problem, several key steps used in generating the model results had to originally be based on large-scale computer simulations, using supercomputing-type computing power. It was hard to change parameters, because of the long run times involved. Also, the computing power necessary to solve the model would be unavailable for most concrete technologists. In Ref. [51], it was shown how two out of the three key steps in the multi-scale model could be accurately replaced with analytical expressions. This section will briefly present the fundamental aspects of the multi-scale diffusivity model, leaving details to the references.
The multi-scale model, as developed, has some limitations. The model applies to conventional (0.25 < w/c < 0.75), fully-saturated concretes. Diffusion/sorption in partially saturated concrete is important in many field exposures but is not addressed in this study. In addition, only ionic diffusion under steady-state conditions is considered, with the short time effect of chloride binding ignored. Mathematically, this is equivalent to electrical and thermal conductivity, so this analysis applies to these areas as well [60]. Also, only concrete without mineral admixtures can be presently handled.
However, the presence of entrained air bubbles is allowed. Air voids are considered to be equivalent to aggregate particles in terms of their effects on ionic diffusivity. They are assigned a diffusivity of 0 and an associated ITZ region like the aggregates [61]. It is assumed that the air voids are not filled with water. Concrete exposed to water for a long period of time may actually have the air voids filled with water. A fixed air void size distribution is used for all of the simulations based on a logarithmic probability density function [62]. Air voids smaller than 100 micrometers in diameter are not included in the model, however, as they are similar in size to the cement particles. The whole concept of the ITZ breaks down when the cement grains are similar in size to the aggregates, so that this small size air bubble cannot be handled with this methodolgy. Avoiding this size range air bubble should not significantly affect results, however.