As has been mentioned earlier, computer modelling of the properties and performance of cement-based materials is complicated by the large range of relevant size scales. Processes occurring in the nanometer-sized pores ultimately affect the performance of these materials at larger length scales. At present, it is impossible to simultaneously handle all these length scales, from nanometers to millimeters and larger, using a single computer-based model. One approach to alleviating this complication is the development of a suite of models, consisting of individual digital-image-based or continuum structural models for the calcium silicate hydrate gel at the nanometer level, the hydrated cement paste at the micrometer level, and a mortar or concrete at the millimeter level. Computations performed at one level provide input properties to be used in simulations of performance at the next higher level. This is the ultimate goal of this research, and will be demonstrated for the property of ionic diffusivity in saturated concrete, using the models discussed in previous sections. Here the relative diffusivity of concrete is computed. The relative diffusivity is defined as the ratio of the diffusivity of ions in a composite medium to their diffusivity in bulk water. Based on the Nernst-Einstein relation, the relative diffusivity D/Do is equivalent to the relative conductivity Γ = σ/σo . In this case, saturated means that all the pore space is filled with the same pore fluid.
The Nernst-Einstein equation is used to determine the diffusivity by solving the equivalent electrical problem for the electrical conductivity of the composite material . As was discussed above, to solve for the properties of these models, the properties of each phase in a microstructure must be known or assumed. The multiscale approach we use is to utilize properties computed at one scale level of modelling as input into the computation procedures employed at a higher scale. For example, the conductivity/diffusivity computed for the C-S-H gel nanostructure model can be used as an input property into the cement paste microstructure model, so that the conductivity/diffusivity of cement paste can be computed as a function of w/c and α. Likewise, the diffusivity computed for bulk and interfacial transition zone cement paste can be used in the model of concrete at the level of millimeters, to compute the diffusivity of a concrete, the quantity of actual interest to a design or structural engineer.
The first step is the computation of the relative diffusivity of the C-S-H gel. Using the electrical analogy and the nanostructural model shown in the top portion of Fig. 20, a value of 1/300 is computed for the relative diffusivity of the gel. Here, we assume that diffusive transport occurs only in the cluster-level pores and not in the much smaller pores shown in the particle-level model in the top portion of Fig. 20. This is a reasonable assumption, since these smaller, nanometer size pores are on the order of the size of a water molecule and would be virtually inaccessible to many diffusing ions. A further assumption is that this relative diffusivity value is characteristic of all C-S-H gel regardless of cement composition or when during the hydration process the C-S-H is produced. When trying to match simulation results for the ionic diffusivity of cement paste to experimental results by adjusting the value of the relative diffusivity assigned to the C-S-H phase, we found a value of approximately 1/400 , so reasonably good agreement, albeit in an indirect way, has been obtained for the nanometer-scale model.
The second step is to use the value computed for the relative diffusivity of the C-S-H gel in a computation of the relative diffusivity of cement pastes of various w/c ratios and degrees of hydration. A diffusivity value of 1.0 is assigned to the capillary porosity, while the unhydrated cement and calcium hydroxide are assigned diffusivities of 0.0 since they contain no porosity. Again using the electrical analogy, computations performed for a variety of 1003 element microstructures have resulted in the development of an equation which relates relative diffusivity to the capillary porosity of the cement paste. Good agreement has been observed in comparing these computed model values to ones measured experimentally both for chloride ion diffusivity [29,34] and cement paste conductivity .
Finally, the relative diffusivity values computed for cement paste can be used as input into the structural model for mortar to determine the effect of aggregates and their surrounding interfacial zones on the diffusivity of the mortar, DM. At this level, one must select the thickness of the interfacial zone paste and the value of its diffusivity DITZ compared to that of the bulk paste DP. Techniques like those described in Refs. [61,66] can then be used to compute DM. Care must be taken to properly allow for the redistribution of cement between the interfacial transition zone regions and the bulk cement paste. For a given water:cement ratio cement paste surrounding the aggregate grains, higher porosity in the interfacial transition zone region means a lower porosity in the bulk cement paste regions. This redistribution of cement determines the correct ratio of diffusivities that is needed to be used in the models of Refs. [61,66]. A technique that approximately allows for this redistribution has been described [74,83,84].