Microstructures and Percolation of ITZs

Three views of representative 3-D fiber-reinforced concrete microstructures are provided in Figs. 3, 4 and 5. Figure 3 provides a simple two-dimensional surface view for two concretes with the same volume fractions of aggregates and fibers, but one based on spherical aggregates and the other on ellipsoidal aggregates with an aspect ratio of 1.5:1:0.6667. In Figure 4, for simplicity, only those aggregates which are part of a percolated pathway for an ITZ thickness of 20 $\mu$m are shown in grey, with all of the fibers shown in white. Interestingly, in all three figures, one can observe that it is mainly the larger aggregates which are a part of the percolated pathway, and not the smaller aggregates which provide a major fraction of the aggregate surface area. This is illustrated quantitatively for the system with spherical aggregates in that 73 % of the aggregate volume is part of the percolated pathway, while only 32 % of the ITZ regions are part of this pathway. For the system with ellipsoidal aggregates, the corresponding volume fractions are 78 % and 39 %. Larger aggregates provide a large aggregate volume, but a proportionately smaller ITZ volume (surface area/volume ratio) than smaller ones. These results are supported by the simulations of Snyder [40], who showed that for air voids, the probability of a given air void having a neighbor within a fixed distance was larger for the larger air voids. Correspondingly in this study, the greater surface area of an individual large aggregate makes it more likely that one of the fibers or another aggregate will intersect its ITZ volume. This finding could have a significant impact on mixture proportions for HPC as will be outlined for the case of lightweight HPC in the mixture proportioning section to follow.

Figure 3: 2-D surface views of concrete microstructures based on spherical (left) and ellipsoidal (right) aggregates. Aggregates and fibers which are part of a percolated pathway for ITZ=20 $\mu$m (and 30 $\mu$m) are white, those which are part of a percolated pathway for ITZ=30 $\mu$m (but not 20 $\mu$m) are light grey, other aggregates and fibers are dark grey, and bulk cement paste is black. Aggregate volume fraction (cCcF distribution) is 0.6 and fiber content is 0.0012 (100 fibers of length 10 mm and diameter 0.25 mm). For ellipsoidal aggregates, ratio of major to minor axis is 1.5 to 0.6667. Each image is 30 mm in each direction. Periodic boundaries are employed such that an aggregate extending across one face of the 3-D system is completed penetrating into the opposite face.

Figure 4: View directly into one face of a concrete microstructure. Aggregates which are part of the percolated pathway are grey and all fibers are white. Bulk paste and non-percolated aggregates are transparent. Aggregate volume fraction (cCcF distribution) is 0.6 and fiber content is 0.0012 (100 fibers). Image is 30 mm in each direction.

Figure 5: 3-D view of a concrete microstructure with ellipsoidal aggregates. Yellow aggregates are part of a percolated pathway for an ITZ thickness of 20 (and 30) $\mu$m, blue for a value of 30 (but not 20) $\mu$m, and magenta are not part of a percolated pathway for either of these ITZ thicknesses. Aggregate volume fraction (cCcF distribution) is 0.6 and fiber content is 0.0012 (100 fibers). Image is 30 mm $\times$ 30 mm $\times$ 20 mm.

For each of the aggregate gradations examined in this study, the minimum ITZ thickness (5 $\mu$m, 10 $\mu$m, 20 $\mu$m, or 30 $\mu$m) necessary to create a percolated pathway was determined as a function of aggregate volume fraction. The results are summarized in Table I, in which the aggregate distributions are listed from coarsest (cCcF) to finest (fCfF). For the two finest distributions, results were not generated for V_agg = 0.75, since well more than one million aggregates would have been required and the V_agg = 0.70 systems were already percolated for an ITZ thickness of 5 $\mu$m. For the cCcF distribution, three separate simulations, each using a different random number seed, were executed at each volume fraction to provide some indication of the variability due to the random configuration of the 3-D microstructure. From Table I, one can clearly see that the greater the aggregate surface area (finer distributions), the lower the ITZ thickness needed to achieve percolation. In general, as would be expected, the greater the aggregate volume fraction, the lower the requisite ITZ thickness and volume fraction of ITZ paste. Since more of the concrete volume is filled with aggregates, less ITZ paste is needed to achieve percolation. For a volume fraction of aggregates of 75 %, all of the distributions are percolated for an ITZ thickness of 10 $\mu$m, and one would expect that these concretes would have less propensity for spalling during a fire. However, as the volume fraction of aggregates is reduced (as it typically is in HPCs [19,20] with their higher cement content), the systems based on the coarser particle size distributions remain unpercolated for an ITZ thickness of 10 $\mu$m, and could thus be susceptible to spalling.

Table 1: Minimum ITZ thickness (and ITZ volume fraction) for ITZ percolation in concrete

Agg. dist.	Vagg=0.6	Vagg=0.65	Vagg=0.7	Vagg=0.75
cCcF	30 $\mu$m (0.058)	20 $\mu$m (0.042)	10 $\mu$m (0.022)	10 $\mu$m (0.024)
cCcF	30 $\mu$m (0.058)	20 $\mu$m (0.042)	10 $\mu$m (0.022)	5 $\mu$m (0.012)
cCcF	20 $\mu$m (0.038)	20 $\mu$m (0.042)	10 $\mu$m (0.022)	10 $\mu$m (0.023)
fCcF	20 $\mu$m (0.038)	20 $\mu$m (0.042)	20 $\mu$m (0.045)	5 $\mu$m (0.011)
mCmF	20 $\mu$m (0.065)	10 $\mu$m (0.032)	10 $\mu$m (0.036)	5 $\mu$m (0.018)
cCfF	10 $\mu$m (0.043)	10 $\mu$m (0.047)	5 $\mu$m (0.024)
fCfF	10 $\mu$m (0.042)	10 $\mu$m (0.046)	5 $\mu$m (0.024)

For those systems which are unpercolated, the addition of fibers can be extremely efficient in creating a percolated pathway. This can be observed in Figure 6 which contrasts the efficiency of adding fibers vs. adding more aggregates to a system initially containing 60 % aggregates (cCcF distribution), for ITZ thicknesses of 20 $\mu$m and 30 $\mu$m. One can clearly observe that the fibers are nearly five times more efficient on a volume basis (0.004 addition of fibers is more or less equivalent to 0.02 addition of aggregates). This clearly illustrates the ability of the fibers to enhance the connectivity of the ITZs present in a concrete. Further illustration of the efficiency of fibers is provided in Tables II and III which list results for the addition of fibers to two specific concrete microstructures, one based on the medium coarse-medium fine (mCmF) distribution with 60 % by volume of aggregates and the other based on the coarse coarse- coarse fine (cCcF) distribution with V_agg = 0.75. For systems whose ITZ regions are unpercolated or weakly percolated, the addition of fibers is seen to significantly increase the fraction of both the aggregates (V_AGGp) and ITZ regions (V_ITZp) which are a part of the percolated network spanning the specimen. It should be noted that these volume fractions of fibers (0.1 % to 0.4 % by volume) correspond closely to those currently employed in structural concretes to prevent spalling [4].

Figure 6: Comparison of percolation efficiency of 10 mm (circles) and 20 mm (triangles) length fibers vs. aggregates (squares). Aggregate gradation (cCcF) is composed from coarse coarse and coarse fine aggregate distributions. Base system to which fibers are added contained 60 % aggregates by volume.

Table 2: Fiber effects on percolation properties for mCmF, V_agg = 0.60 concrete.

		ITZ=	10 $\mu$m		15 $\mu$m		20 $\mu$m
Number of	Fiber	Fiber	VAGGp1	VITZp2	VAGGp	VITZp	VAGGp	VITZp
fibers	length	vol.(%)
	(mm)
0			0.000	0.000	0.636	0.231	0.904	0.591
100	10	0.12	0.000	0.000	0.748	0.300	0.912	0.615
50	20	0.12	0.000	0.000	0.755	0.315	0.913	0.620
200	10	0.24	0.000	0.000	0.780	0.346	0.921	0.643
100	20	0.24	0.000	0.000	0.787	0.364	0.923	0.649
300	10	0.37	0.000	0.000	0.795	0.380	0.929	0.670
150	20	0.37	0.164	0.039	0.812	0.413	0.928	0.666

¹ volume fraction (0-1) of total aggregates which are part of a percolated pathway

² volume fraction of all ITZ regions which are part of a percolated pathway

Table 3: Fiber effects on percolation properties for cCcF, V_agg = 0.75 concrete.

		ITZ=	5 $\mu$m		10 $\mu$m
Number of	Fiber	Fiber	VAGGp	VITZp	VAGGp	VITZp
fibers	length	volume(%)
	(mm)
0			0.000	0.000	0.895	0.627
100	10	0.12	0.000	0.000	0.908	0.662
50	20	0.12	0.404	0.147	0.911	0.670
150	10	0.18	0.327	0.111	0.914	0.680
75	20	0.18	0.228	0.085	0.916	0.695
200	10	0.24	0.589	0.208	0.918	0.692
100	20	0.24	0.617	0.245	0.924	0.706

In Figure 6, by comparing the results for the two different length fibers, it can also be observed that the 20 mm (triangles in Figure 6) length fibers are slightly superior to the 10 mm (circles in Figure 6) ones in creating a percolated network. These results are also confirmed by those presented in Tables II and III where in certain cases, at equivalent volume fractions, the 20 mm length fibers are able to percolate a system that remained unpercolated with the addition of 10 mm fibers. In general, for the PSD distributions investigated in this study, the 20 mm fibers were slightly more efficient both in percolating an unpercolated system and in increasing the connectivity of a partially percolated one. As stated earlier, this is consistent with the percolation characteristics of totally overlapping ellipsoids [33]. For the few systems examined with the fiber diameter reduced to 0.1 mm and the length maintained at either 10 mm or 20 mm, the percolated volume fractions of aggregates and ITZ regions were just slightly reduced relative to those observed for the 0.25 mm fibers. This suggests that it is the length and number of fibers that is critical in percolating the microstructure. Fibers of 0.1 mm diameter occupy only 16 % of the volume of 0.25 mm diameter ones, so that the needed volume fraction (fiber content) could be significantly reduced. This, however, must be contrasted against the structural rigidity of the fiber and the fact that its diameter needs to be significantly larger than that of the cement particles in order to promote ITZ formation at the fiber-cement paste interface (and provide a ``fiber channel'' of sufficient diameter to facilitate the escape of water vapor).

For systems containing fibers only, the following percolation thresholds were observed using the computer simulation. For 10 mm length, 0.25 mm diameter fibers, volume fractions on the order of 6 % and 4.5 % were required for ITZ thicknesses of 20 $\mu$m and 30 $\mu$m, respectively. For the equivalent diameter 20 mm fibers, by contrast, the percolation thresholds were observed to be about 3 % and 2.5 %. For fibers of reduced diameter (0.1 mm, length = 10 mm) percolation thresholds were substantially reduced to 1.1 % and 0.8 % for ITZ thicknesses of 20 $\mu$m and 30 $\mu$m, respectively. For the 20 mm length, 0.1 mm diameter fibers, the equivalent percolation values were once again reduced by a factor of five to values of approximately 0.6 % and 0.5 %, respectively. All of these values are somewhat higher than those previously observed for totally overlapping ellipsoids of equivalent aspect ratios [33], due to the restriction in the present study that the hard core fibers can not overlap one another, although their soft shell ITZ regions are free to overlap one another. For the 0.1 mm diameter fibers, however, the values are definitely approaching those reported earlier for the totally overlapping ellipsoid case (0.7 % for an aspect ratio of 100:1 and 0.3 % for 200:1). As the fibers become thinner, their ITZ regions occupy a larger proportion of their total volume and they would be expected to approach the case of totally overlapping particles, particularly for the 30 $\mu$m ITZ systems. In all cases, the longer (higher aspect ratio) fibers are seen to be the more efficient shape for creating a percolated ITZ network through a 3-D microstructure.