Many Aggregate Models

A real concrete is composed of more than 50% by volume of aggregates, and thousands of aggregates per cm³, implying the existence of many local ITZ regions, which comprise a material phase different from the bulk cement paste. The first aspect of this new complicated material phase to be studied is the connectivity or percolation of these ITZs in a mortar or concrete. The percolation of these ITZ regions, or specifically the capillary porosity contained in the ITZs, through a concrete was first indicated by mercury intrusion porosimetry data [44,45] and has more recently been inferred from SEM observations of specimens intruded with Woods' metal [46]. This percolation of the individual ITZs would be expected to influence transport properties, as the large pores present in the ITZs offer much less resistance to the diffusive and convective transport of aggressive species into the concrete. Any such effect of the ITZ regions is of course in competition with the presence of aggregates, which tend to reduce the volume of pathways and increase the path lengths for transport. This competition will be explored more fully in the transport section below (section 4).

The percolation of ITZs can be examined using a hard core/soft shell (HCSS) model [47], first adapted to concrete by Winslow et al. [48], and later duplicated by Bourdette et al. [49]. This is a continuum model. The model has also been employed by Johansen and Thaulow [50] to estimate the volume of reactive aggregate (sand) which can be present in concrete without causing deleterious expansion. In the HCSS model, the aggregates are represented by a collection of impenetrable spheres that follow a measured particle size distribution. Each aggregate is surrounded by a concentric spherical shell that represents the ITZ. In general, the thickness of the ITZ is not a function of aggregate particle size, as was discussed above, but is instead controlled by the median size of the cement particles (see Ref. [51] for further discussion of this question). Typical computational volumes are 1000 mm³ for mortar and 27,000 mm³ for a concrete, which requires on the order of 1,000,000 individual aggregate particles for the concrete systems.

For mortars, the model has been calibrated against experimental mercury intrusion porosimetry data [48] that indicated a large increase in the volume fraction of porosity intruded by mercury at low pressures when the sand content of the mortar was increased from 44.8% to 48.6%. This increase in "coarse" porosity could be indicative of a percolation transition for ITZ connectivity. Using the same sand particle size distribution as that employed experimentally, simulations were conducted for different sand contents and values of ITZ thicknesses ranging from 10 to 40 µm. Figure 11 provides a plot of the results, showing the fraction of the ITZ volume that is part of a percolated pathway through the 3-D microstructure as a function of sand content and ITZ thickness. From this plot, one can observe that an ITZ thickness of 15-20 µm best agrees with the experimental observations of a large increase in connectivity in the range of 45-50% sand. This value is smaller than the 40-50 µm typically estimated using SEM techniques [23,31,32], but this would be expected since the largest pores in the ITZ should be those nearest to the aggregate surface, where the porosity gradient is steepest (see Figure 3). Thus, the measured thickness of the ITZ will depend on the experimental technique being employed.

Figure 11: Percolated ITZ volume fraction vs. aggregate content for various ITZ thicknesses in µm (adapted from Ref. 48).

The second feature of the ITZ phase is its volume fraction. The HCSS model can be used to determine the fraction of cement paste within a given distance of an aggregate surface, thus giving the volume fraction of the ITZ regions as a function of their thickness. This is done using a numerical point-counting technique [48]. Using measured aggregate size distributions for mortar [48] and concrete [45], results shown in Figure 12 indicate that nearly all of the cement paste is within 100 µm of an aggregate, consistent with the SEM-based observations of Diamond et al. [52]. Also, 20-40% of the total cement paste is within the typical ITZ thickness of 20-30 µm. This volume fraction of ITZ paste is more than sufficient to create a percolated pathway through a microstructure, suggesting that most real concretes will contain percolated ITZ regions. It has been found recently that this prediction of the HCSS model can also be carried out analytically, with great accuracy, using a result of Lu and Torquato [51,53,54] (see section 4.2.1), when the aggregates are spherical.

Figure 12: Fraction of total cement paste volume that lies within a given distance of an aggregate surface for a mortar with a sand volume fraction of 0.552, and a concrete with an aggregate volume fraction of 0.646.

The HCSS model has been extended to the case of elliptical particles of various aspect ratios [4]. Figure 13 provides a plot of ITZ connected fraction vs. sand volume fraction for systems with five different choices of semiaxis length combinations a:b:c. One can clearly observe that smaller volume fractions of the elliptical particles are needed to achieve percolation of the ITZ regions. This is mainly due to the increased surface area per volume ratio of an ellipsoid relative to a sphere [55]. Figure 14 shows how these connectivity values for different aspect ratios collapse onto a single curve (for a fixed ITZ thickness) when plotted against the surface area of the aggregate, corrected slightly for the non-uniform thickness of the ITZ layer when ellipsoidal particles are employed [4]. ITZ thicknesses of 50, 22.5, and 10 µm were used in Fig. 14.

Figure 13: Percolated ITZ volume fraction vs. aggregate content. Three data sets are shown for the 1:1:1 aspect ratio to provide some indication of variability. The ITZ thickness is 22.5 µm.

Figure 14: Percolated ITZ volume fraction vs. aggregate surface area for ITZ thicknesses of 50, 22.5, and 10 µm from left to right.

Figure 15 compares the spherical and ellipsoidal particle modes by showing cross-sections from two 3-D mortar models of these types. This figure also serves to illustrate an important difference in percolation characteristics between 2-D and 3-D. In a 2-D slice, the ITZ regions do not appear to be percolated, while in fact they are percolated in 3-D. One cannot determine 3-D percolation quantities by looking at 2-D models or 2-D slices of 3-D materials.

Figure 15: Two cross sections from 3-D mortar models, using spherical aggregates-left, and ellipsoidal aggregates-right. Gray is bulk cement paste, white is aggregates, and black is ITZ regions.

Having used percolation ideas applied to the HCSS model, coupled with various numerical techniques, to establish some idea of the geometry and topology of the ITZ phase in a concrete, the focus now turns to quantifying the effect of the ITZ phase on concrete transport properties.