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The tricalcium silicate (C3S) cement hydration model used in this multi-scale model has been described in detail elsewhere [15,16]. The cement powder to be modelled is represented by non-overlapping digitized spheres following the particle size distribution (PSD) measured on actual cement samples. Various PSD's can be used , although we have only examined PSD's with median particle diameters of 10, 20, and 30 µm . Based on previous modelling studies , we take the ITZ widths equal to the median cement particle size. To minimize finite size effects, periodic boundaries are used during particle placement, such that a particle which extends outside of one face of the 3d computational volume is completed into the opposite side of the system. For the micrometer-scale model, each pixel element represents 1 cubic micrometer in volume. The system sizes needed ranged from 20-30 million pixels. The cement particles are placed in order of size from largest to smallest at random locations in the 3-D microstructure such that they do not overlap one another or the aggregate particle.
After initial particle placement, a simple cellular automaton model is used to model the hydration reactions between tricalcium silicate and water . In this model, cement pixels in contact with water dissolve at random, diffuse within the pore space, and react to form calcium hydroxide crystals in the pore space and calcium silicate hydrate gel (CSH) on the surfaces of the original cement particles and previously deposited CSH (C=CaO, S=SiO2, H=H2O). For these studies, the aggregate is considered inert and does not participate in the hydration reactions. At any degree of hydration, the porosity can be determined as a function of distance from the aggregate surface. Initially, after particle placement, the interfacial transition zone region contains a higher w/c ratio (more porosity) than the bulk paste due to the inefficient packing of the cement particles, the so-called wall effect [19,20]. During hydration, the porosity is reduced throughout the cement paste, but still remains higher in the ITZ regions. Thus, these regions will typically have a higher diffusivity than the bulk paste regions. Once porosity has been quantified, the relative diffusivity, D/Do , as a function of distance from the aggregate surface, x, can be estimated using a previously established relationship :
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 . 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 the second term in eq. (1) is a fitting term that connects the two limiting behaviors . 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. [21,22] give experimental validation of this model and the associated eq. (1).