To assess the set point, which we define rigorously as the point where percolation of solids occurs, we use a 3D model based on monophase C3S particles, thus avoiding the problem of starting with realistic 3D cement phase distributions. This simplification is justified by our assumption that the C-S-H surface product causes set and by the fact that this simple model generates almost the same quantity of surface products as those produced in real cements (Bentz and Garboczi 1991b).
In the model, to assess percolation, a 'burning algorithm' (Stauffer 1985) is utilized. This algorithm searches for a path across the 3D microstructure composed exclusively of the unreacted cement and C-S-H phases, proceeding iteratively from each such pixel to its nearest neighbours. We uniquely label each cement particle and allow the 'burning' to spread within any given cement particle and from any cement particle to C-S-H , and vice versa. Since the burning does not proceed directly across unreacted cement grains, this computational algorithm eliminates the problem of initial particle−to−particle contacts in a digital−image−based model which would result in very low W/C ratio pastes being 'set' before any hydration occurs since the particles themselves will form a connected backbone. The burning starts at the top of the microstructure and continues until no more 'fuel' remains, at which point we check to see if the bottom has been reached and if so, what fraction of the total volume of pixels is in the cluster that connects the top of the cube to the bottom (i.e. the fraction of solids connected). A plot of the fraction of solids connected versus degree of hydration is shown in Figure 15 for a W/C ratio of 0.35 and a particle size distribution as measured on a real cement (Coverdale 1992). As expected, the fraction connected begins at zero and approaches a value of unity as the hydration progresses.
The amount of hydration needed to achieve set would be expected to vary with W/C, as there are different initial solids fractions. Results for various values of initial W/C are presented as data points in Figure 16 which shows the fraction of the total volume that is solid (C3S + C-S-H) at set point versus W/C ratio. The line in the figure represents the initial solid fraction (only C3S) for a given initial W/C ratio. Higher W/C ratio values require more hydration to achieve set since the cement particles are initially spaced farther apart in the 3D microstructure. This is reflected in Figure 16 by the fact that points as high W/C ratio tend to be further above the initial C3S line than points at low W/C ratio. Although C3S is consumed during hydration, there is a net increase in volume of C3S + C-S-H.
The results shown are in quantitative agreement with the recent experimental results of Chen and Odler (1992) who measured the porosity of cements pastes of various W/C ratios at the beginning and end of set (American Society for Testing and Materials 1990). The actual solids percolation point probably occurs before the beginning of set as defined in the ASTM standard (American Society for Testing and Materials 1990). The results shown in Figure 16 are for several particle size distributions ranging from monosize cement particles to the particle size distribution corresponding to that of a real cement (Coverdale 1992).
From our viewpoint that it is the skeleton of cement particles surrounded by C-S-H gel which is critical to the realization of set, we can also study the effect of the initial dispersion of cement particles on the amount of hydration required for set. The amount of dispersion can also be modified experimentally by the use of superplasticizers or other additives (Legrand and Wirquin 1992). To achieve dispersion in the model, we modify our cement particle placement conditions to ensure that every pair of particles is separated by at least a one pixel (1 µm) distance, thus simulating the effect of a dispersive agent. Figure 17 shows the results obtained for a variety of systems of differing W/C ratios and particle size distributions. In every case, we find that the initial dispersion of the cement particles results in more hydration being required (i.e. a larger volume of C3S + C-S-H) to bridge the gaps between particles and to achieve set. These results are in agreement with recently reported experiments (Legrand and Wirquin 1992).
