Percolation theory deals with the connectivity of components in a system. First applied by Hammersley in the 1950s , an excellent introduction is provided by Stauffer and Aharony . The microstructure of cement-based materials provides numerous examples of percolation phenomena . Often in percolation studies, one is interested in the fraction of a phase (or phases) which is connected across the microstructure as a function of the total volume fraction of the phase. For example, as first noted by Powers based on observing a significant (sudden) reduction in permeability , the capillary porosity in cement paste exhibits a percolation transition (from connected to disconnected) at a volume fraction of about 20 % porosity, as further verified by computer simulations  and measurements of water imbibition during chemical shrinkage [29,30]. This transition is relatively independent of w/c ratio, but does depend somewhat on the particle size distribution of the cement . The setting of cement pastes, mortars, and concretes, illustrates the percolation of the total solids, as the individual cement particles become bonded one to another by hydration products, leading to the formation of a percolated backbone and the strength development of the material. Other phases in the cement paste such as the calcium hydroxide and C-S-H gel also exhibit percolation thresholds .
Once the capillary porosity depercolates, the permeability of the system is greatly reduced. In most cement pastes (w/c <=0.5), after 28 days or so, the permeability would be sufficiently low that any water vapor formed at elevated temperatures would have difficulty in escaping from the system. However, in mortars and concretes, another percolation phenomena comes into play, the percolation of the ITZ regions . Because the ITZ regions have a substantially larger w/c ratio than the bulk paste, they typically have a much greater (2 times to 3 times) capillary porosity. Thus, if the capillary porosity in the individual ITZs remains percolated and the ITZs themselves are percolated, a convenient escape route for the "steam" generated during fire exposure might exist. The percolation properties of these ITZ regions are conveniently studied using a hard core/soft shell (HCSS) percolation model , the implementation details of which will be discussed in detail in the next section of this paper. Briefly, each aggregate particle is viewed as an impenetrable hard core, surrounded by a concentric soft shell (ITZ), which may overlap other soft shells or portions of other hard core particles.
Figure 1 illustrates this HCSS model in two dimensions. In the upper left figure, the hard core aggregate particles are each surrounded by an ITZ region, but the ITZ regions do not percolate across the system. In the upper right figure, the thickness of each ITZ region has been increased, such that percolation from top to bottom is achieved. In the lower left figure, conversely, percolation has been achieved by adding more hard core/soft shell particles. Finally, in the lower right figure, percolation is achieved by the addition of just a few fibers to the system. From this simple illustration, one can clearly see the potential efficiency of fibers in percolating the ITZ regions in an originally non-percolated concrete. The percolation aspects of totally overlapping ellipsoids of revolution (a convenient geometrical representation of a fiber) have been simulated in detail by Garboczi et al. , who observed that for fibers with a 50:1 aspect ratio, approximately 1.5 % by volume would be required to form a percolated pathway across a 3-D microstructure. For aspect ratios of 100:1 and 200:1, this volume fraction is reduced to about 0.7 % and 0.3 %, respectively, suggesting that longer fibers should be more efficient in causing percolation of non-percolated systems. This is in agreement with the rapid chloride permeability measurements of Toutangi et al. who, for concretes containing equal volume fractions of fibers, measured greater permeabilities for the systems containing longer (19 mm vs. 12.5 mm) fibers .