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The computer model developed to simulate interfacial zone percolation in mortar and concrete is based on the hard core/soft shell problem outlined in the percolation literature [8,9,10,11]. The application of this problem class to concrete has been described in detail previously [12]. The aggregates represent impenetrable hard cores while the interfacial zones represent soft shells surrounding each hard core. While the hard core aggregates may not overlap, the soft shell interfacial zones may overlap one another and may even partially overlap other hard cores without penetrating them. Given an aggregate size distribution (such as the one in Table 1) and an interfacial zone thickness, the question to be answered using the model is: What quantity of aggregate is required for the interfacial zones to percolate across a concrete specimen? Since this problem cannot be solved analytically, a computer program has been written to simulate this system. A general overview of percolation theory can be found in Stauffer [13] while the application of percolation concepts to the connectivity of the capillary porosity and the other individual phases in cement paste has been examined previously using a cement paste microstructure model [14].
The model mortar or concrete specimen consists of a three-dimensional cube, typically 10 or 30 millimeters on a side. The smaller specimens are used to represent mortar while the larger specimens are used to represent concrete. It should be noted that these specimen sizes are the same or larger that those samples actually used for the mercury intrusion experiments described in this paper and in Ref. [1]. Aggregates are represented as spherical particles. Each aggregate is surrounded by a concentric interfacial zone as shown in two dimensions in Fig. 1. Since the model is continuum in nature, each aggregate is characterized by four numbers, a center location consisting of (x,y,z) coordinates, and a radius. This representation allows many aggregates to be included in the model since on the order of ten thousand aggregates are required to represent a typical mortar specimen and up to half a million may be needed to represent a 27,000 cubic millimeter concrete specimen.
The computer model procedure consisted of three major steps: particle placement, percolation assessment, and phase fraction estimation. During particle placement, the aggregates of a specific size distribution are placed in order of largest to smallest at random locations within the 3-D cube such that no two aggregate particles overlap. To eliminate artificial edge effects, periodic boundaries are used during particle placement. This means that any part of particle that extends beyond the cube boundaries is reflected into the cube on the opposite face. The particle locations are all stored in computer memory for access throughout the execution of the program.
Once the predetermined number of particles have been placed, establishing a volume fraction of aggregates, the percolation of the interfacial zones is assessed using a burning algorithm [13]. Beginning at one edge of the top face of the cube, the program finds all aggregate particles whose interfacial zones overlap the top face of the cube. For each aggregate particle found, the program checks to see if any other aggregate particles present in the system contain interfacial zones that overlap this original interfacial zone. This process is repeated in a iterative fashion until all aggregate particles connected to the original particle via overlapping interfacial zones are located. Finally, this list of particles is examined to see if any of the particles contact the bottom face of the cube, indicating the existence of a percolated pathway through the specimen. The sample shown in Fig. 1 does contain such a percolated pathway as indicated by the red interfacial zones. Thus, after percolation assessment all aggregates and their surrounding interfacial zones are classified either as being inaccessible from the top surface, accessible from the top surface but not part of percolated structure, or accessible from the top surface and part of a percolated structure.
Figure 1: Two-dimensional schematic of interfacial zone percolation. The red interfacial zones are percolated from top to bottom. Black = aggregate, light gray = unpercolated interfacial zones.
Finally, systematic point sampling [15] is used to estimate the volume fractions of the phases present in the 3-D cubic system. Using a 100 x 100 x 100 point grid, the program classifies each point as belonging to one of the following six phases: (1) bulk cement paste, (2) aggregate that is not part of a percolated pathway, (3) aggregate that is part of a percolated pathway, (4) inaccessible interfacial zone cement paste, (5) accessible interfacial zone cement paste that is not part of a percolated pathway, and (6) accessible interfacial zone cement paste that is part of a percolated pathway. Based on these point counts, the program calculates a variety of results including the aggregate volume fraction, the fraction of interfacial zone volume that is contained in percolated pathways, and the fraction of the total cement paste volume that is contained in interfacial zones. By executing the program for different numbers of aggregate particles, the volume fraction of aggregate required for interfacial zone percolation, for a given aggregate size distribution and interfacial zone thickness, can be determined and compared to experiment.
The computer program was checked by computing the critical volume fraction required for percolation for the case of completely overlapping (no hard core) monosize spherical particles. The computed critical volume fraction of 0.29 was same as previously published results [8,9]. When executing the computer model for mortars, the sand size distribution given in Table 1 was duplicated as closely as possible by using the sieve size analysis and distributing particles uniformly by volume between sieve sizes. This allows a direct comparison between experimental and computer model results.