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One of the properties contributing to the service life of concrete structures is the resistance the concrete provides to the diffusive ingress of deleterious species such as chloride and sulfate ions [1,2]. Prediction of the diffusivity of a concrete based on its mixture proportions and expected curing is needed to help predict its service life in its expected service environment. The availability of such a prediction capability in terms of the initial mixture propertions and expected curing of a concrete will allow the rational development of durability-based, in addition to the current strength-based, design codes. In fact, durability specifications are already being issued, even though there has not been quantitative theory developed that can accurately predict the chloride ion diffusivity of a given concrete . The model described in this paper can help to overcome this barrier, even though other types of transport mechanisms, like sorption, are also important for chloride ingress .
Because of the wide range of feature sizes in concrete, from nanometer-sized pores to millimeter-sized aggregates, it is impossible, because of memory limitations on current computers, to simultaneously represent all of these structural features in a single microstructure model. For example, if in a 3-D digital image model we let the pixel size be one nanometer, which would be barely adequate for the resolution of the C-S-H phase, then the minimum size model that would incorporate enough aggregates to be statistically valid would be at least 30mm on a side, and contain 2.7 · 1022 pixels! Uniting multiple length scales in a single model in a more sophisticated way then obviously becomes necessary.
Multi-scale modelling techniques offer a promising solution to this restriction [5,6]. In this approach, properties computed at one scale, micrometers for instance, are input into a model which is constructed at a higher scale, such as millimeters. The present 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 has been demonstrated previously for mortars . 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.
In Ref. , 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 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  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 concrete. Therefore the results of Ref.  are incomplete, and in general, by themselves,should not be used to analyze an experiment.
In Ref. , 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 has to be based on large-scale computer simulations, using supercomputing-type computing power. Because of this approach, 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 this paper, we show how two out of the three key steps in the multi-scale model can be accurately replaced with analytical expressions. This will make the results of this model much more acessible. Previous numerical results are used to check the accuracy of this analytical replacement.
The multi-scale model has some limitations. The model applies to conventional (0.25 < w/c < 0.75 ), saturated concretes. Diffusion/sorption in partially saturated concrete is important in many field exposures but is not addressed in this study. However, we do allow for the presence of entrained air bubbles. In addition, we are only considering ionic diffusion under steady-state conditions, and so ignore the short time effect of chloride binding.