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Pore Space Percolation

Percolation theory was developed to mathematically deal with disordered media, in which the disorder is defined by a random variation in degree of connectivity [12]. The main concept of percolation theory is the existence of a percolation threshold, defined in the following way. Suppose p is a parameter that defines the average degree of connectivity between various sub-units of some arbitrary system. When p = 0, all sub-units are totally isolated from every other sub-unit. When p = 1, all sub-units are connected to some maximum number of neighboring sub-units. At this point, the system is connected from one side to the other, since there are paths that go completely across the system, linking one sub-unit to the next along the spanning cluster.

Now suppose, starting at p = 1, connections are randomly broken, so that p, the measure of average connectivity, decreases. The percolation threshold is that value of p, usually denoted p_c, at which there is no longer an unbroken path from one side of the system to the other. Alternately, if we start out at p = 0, and randomly create connections, so that p increases, p_c is defined as the point at which a spanning cluster first appears. For p less than p_c, only isolated, non-spanning clusters can exist. For p greater than p_c, there is always a spanning cluster, although some isolated, non-spanning clusters can still be present. For a network, p could be defined as the average number of connections per node. For a continuum material such as cement paste, the appropriate parameter is volume fraction, which for the capillary pore space is defined as φ, the capillary porosity.

The percolation properties of the cement paste microstructural model are easy to compute, since the digital-image-based model has an underlying lattice structure. This means that any algorithm designed for simple percolation problems can be applied to this more complex digital-image lattice. In particular, the percolation of the pore space, or any other phase of interest, may be determined by the use of a "burning" algorithm [13]. This algorithm is a simple way of identifying all pixels that are part of a spanning cluster, if such a cluster exists, and works as follows. Conceptually, all the pixels belonging to the phase of interest are classified to be "combustible". A "fire" is started on one side of the model's unit cell, and allowed to propagate only between pairs of combustible pixels. If any pixels on the opposite side of the model cell are found to have been "burned", then a spanning cluster of the phase of interest must exist. The number of "burned" pixels are counted to determine the fraction of the phase of interest that is a part of the spanning cluster. This connectivity can be computed at any degree of hydration.

Previous work using the model and this algorithm [10] has identified the following percolation thresholds for the cement hydration model: a) the capillary pore space becomes discontinuous at a critical porosity of φ_c 18% , b) the C-S-H gel phase becomes continuous at a critical volume fraction of about 17-20%, and c) the CH phase becomes continuous at a critical volume fraction of about 12-15%. The critical volume fraction for CH (pore product, in general) is particularly important in this study. It implies that pastes originally containing a volume fraction of CH that is greater than 15% and some finite amount of capillary pore space will, after complete leaching, contain connected (percolated) pathways of capillary pore space that have been left behind by the leaching process.