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# Percolation Theory

Percolation theory was developed to mathematically deal with disordered media, in which the disorder is defined by a random variation in the degree of connectivity [9]. 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 pc, 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, pc is defined as the point at which a spanning cluster first appears. For p less than pc, only isolated, non-spanning clusters can exist. For p greater than pc, there is always a spanning cluster, although some isolated, non-spanning clusters can still be present.

A typical lattice example of a percolation problem is that of site percolation on a simple two- dimensional square lattice. The lattice starts off empty, with all sites unoccupied. The sites of the lattice are then randomly occupied, one at a time. If two occupied sites are nearest neighbors, a connection is made between them. When a critical fraction pc = 0.593 of the sites are present [9], a spanning cluster will come into existence, and the occupied sites will percolate.

A more complex continuum percolation problem is defined by the following. Take a white piece of paper, and randomly throw down equal-sized circular blobs of paint, studying the critical paint fraction needed to have a continuous path of paint from one side of the paper to the other. In this case, the critical threshold is expressed as a paint area fraction, with the value pc = 0.68 [10] .

Scher and Zallen [11] found that the critical thresholds for many lattice percolation problems, when expressed as area or volume fractions (in 2-d or 3-d), were approximately the same. In particular, for percolation problems where a structure was being randomly built-up of non-overlapping particles, they discovered that the critical volume fraction for percolation for 3- D systems was approximately 0.16, and was 0.45 for 2-D systems. The critical volume or area fraction is the phase fraction where a spanning cluster first appears. These approximate invariants were first defined for simple, model lattice problems [11], but have since turned up in experiments on real materials and in more complex continuum computer simulations [12,13,14], leading Scher and Zallen to hypothesize that these thresholds have more general or "universal" validity [9]. However, one case in which the Scher and Zallen hypothesis does not apply is when the particles that are being used to randomly build up the spanning cluster are allowed to overlap. For example, in the 2-D overlapping paint example described above, the circles only percolated when their area fraction was 0.68, not the 2-d "universal" value of 0.45.

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 [15]. 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 along these 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 assessed at any degree of hydration.

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