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Disordered Models

To study a mortar with a more realistic random sand grain arrangement, we use the model illustrated in Fig. 1, based on the sand grain size distribution given in Table 1.

Figure 1: A slice through the four size random sphere parking model is shown. The total volume fraction of sand is 54% and the distribution of the four aggregate grain sizes is summarized in Table 1. The 500 µm diameter grains are shown as open circles, the 1000 µm diameter grains are shaded horizontally, the 1500 µm diameter grains are shaded at 45º, and the 3000 µm diameter grains are shaded at 270º The thickness of the interfacial zone (unshaded) is 20 µm, a value that guarantees percolation of the interfacial shells [see Fig. 2]. The bulk cement paste occupies the interstitial region outside of all the interfacial zones. Note, that there are several instances in which an interfacial shell intersects the plane of the image but the associated sand grain does not. These appear as isolated (rather than concentric) circles.

The models were generated by a three dimensional hard core parking algorithm [14]. [The sand grains were randomly placed, largest first and smallest last, such that no sand grains overlapped.] The maximum grain concentration studied was about c=0.55, based on a model with 6500 particles. Models with 5000 (c=0.42), 2000 (c=0.17), and 1000 (c=0.09) particles were also generated to study systems with lower sand concentrations. In Fig. 2 we show the fraction of the matrix occupied by the interfacial zone, as a function of the interfacial zone thickness. Results are shown for the several aggregate concentrations used. As with the periodic model, interfacial shells of thickness 20 µm were added to each grain, so that the interfacial zone comprised about 28% of the total matrix when c = 0.55.

Figure 2: Given the experimentally determined size distribution [10] of sand grains (assumed spherical), the fraction of the total cement paste volume occupied by the interfacial zone cement paste is shown as a function of the interfacial zone thickness. Results are given for four aggregate concentrations.

The connectivity of the interfacial zones was computed using a burning algorithm [10] and is displayed in Fig. 3 as a function of sand volume fraction. The interfacial layers first percolate at a sand concentration of about 36% and form a single spanning cluster at roughly c=0.51. The arrow marks the prediction of the self-consistent effective medium theory (SC-EMT) which will be discussed in Section 3.4. To illustrate the interplay between sand volume fraction and interfacial zone thickness, we also studied a simpler disordered model, where the sand grains were all the same size, but the interfacial shell thickness, h, was allowed to vary [15,16]. The controlling variable is then b/a, where b is the sand radius and a = b + h. For a given choice of b/a, suppose that we randomly park spherical grains until their concentration, c, is such that the shells form connected channels. The larger the value of b/a (i.e. the thinner the shells), the greater the value of c that will be required and the smaller the fraction of space that will be occupied by the percolating shells. This behavior is clearly shown in Fig. 4. Shown also are the value of the limiting concentration c=0.38 for mono-size random parking [14], and curves based on the SC-EMT [Section 3.4]. Note that there is a threshold value of about b/a = 0.9615 above which it is not possible to form connected percolating shells based on the random parking algorithm.

Figure 3: Percolation curve for the four size random sand grain model with a 20 µm thick interfacial zone. The x-axis is the sand volume fraction, and the y-axis is the fraction of the interfacial zone phase that is contained in the percolating cluster.

Figure 4: The percolation properties of interfacial zone in the mono-sized random parking model are illustrated by two curves. As functions of the ratio of the sand grain radius, b, to the interfacial zone radius, a, we plot the volume fractions occupied at the percolation threshold by (1) the interfacial shell and (2) the aggregate grains. Also shown are the volume fraction corresponding to the limit of dense random parking and the SC-EMT estimates of the shell and aggregate fractions at percolation. The arrow indicates the largest b/a value at which percolating shells can be achieved for dense parking.

While the model we pursue in the remainder of this paper is highly simplified, we emphasize that its essential features could be specified in detail if more experimental data on the structure of mortars were available. In particular, we feel that nuclear magnetic resonance (NMR) studies would be of great value in determining the model's parameters. NMR results would be particularly useful if the larger pores in the interfacial zone could be seen in relaxation studies as an independent contribution to the pore size distribution [17,18]. In principle, similar information is available directly from microscopy, but NMR has the advantage of being a non-destructive, non-invasive measurement.