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, 13]. 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. As 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, nonspanning clusters can exist. For p greater than pc, there is always a spanning cluster, although some isolated, nonspanning 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, 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 pcpaint = 0.68.
The percolation properties of a digital-image-based model are easy to compute. The percolation of any phase of interest may be determined by the use of a "burning" algorithm [12]. 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.
Another percolation model that arises in the study of cement-based composites is the hard core/soft shell (concentric particle) model introduced by Torquato [14]. Like the overlapping circle (paint) percolation problem discussed above, objects are randomly placed in a continuum, and their connectivity is studied as a function of how many have been placed, or of what area or volume fraction is occupied. In the hard core/soft shell (hcss) model, the objects, for example circles in 2-D or spheres in 3-D, are made up of an inner core of radius b, and an outer shell of thickness a-b, so that the total radius is a. Hard cores may not overlap each other, but soft shells may overlap anything freely. When b is zero, in two dimensions, then we recover the overlapping circle problem mentioned earlier. When b = a, we have the problem of random close-packing of circles or spheres. For intermediate values of b/a, the percolation threshold, expressed as number of objects per unit area or volume, or the area (volume) fraction of soft shells, will depend on b/a.
The hcss model has been used to examine the percolation of interfacial zones in concrete by Winslow et al [15]. In this mode, the 3-D hard core spheres represent the aggregates while the soft shells represent the interfacial zones surrounding each aggregate. As shown in computer exercise no. 1, the interfacial zone regions are more porous and contain larger pores than the bulk cement paste. If enough aggregates are present in a mortar or concrete, these interfacial zones (soft shells) will interconnect and provide a path of lesser resistance (than the bulk paste) for the ingress of deleterious species such as chloride ions. This phenomena has been verified experimentally by a series of mercury intrusion porosimetry experiments [15].
Recently, the hcss model has been successfully applied to two other material systems. The first is a three-dimensional model of the microstructure of MDF (Macro-defect-free) cements [16], where the hard cores represent the unhydrated cement particles and the soft shells represent the interphase region (polymer + hydration products) surrounding each cement particle. Here, the model has been used to examine the percolation characteristics of both the interphase regions and the bulk polymer phase. The second is a two-dimensional model for the microstructure of chemical-vapor-infiltrated ceramic fibers. Here, the hard cores represent the original impenetrable fibers and the soft shells represent the deposition product, and thus increase in size during the course of the reaction. Researchers at NIST and Northwestern University are using the generated microstructures to compute thermal conductivities and permeabilities of these composite microstructures as a function of porosity. These physical properties will then be utilized in a macro-level processing model to provide a better understanding of the relationships between processing variables (temperature, pressure, etc.) and resultant structure.