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The random arrangement of multi-sized aggregate particles in a real mortar or concrete plays an important role in determining the effective properties of the composite. However, ordered periodic arrangements of aggregate grains are easier to handle computationally. Although they cannot be quantitatively accurate, results obtained from periodic models can still give qualitative insight, as aggregate-aggregate interactions and interfacial zone overlap do play an important role.
The periodic model that is investigated in this paper is a body-centered cubic [32] arrangement of spherical sand grains, chosen to have a diameter of 400 µm and an interfacial zone thickness of h = 20 µm. The value of 400 µm was chosen from the sand particle size distribution used in Ref. [15], as being a compromise between the number fraction-weighted and volume fraction-weighted average particle diameters. The size of the unit cell was chosen so that the interfacial zones would be percolated, and so that the sand volume fraction was 54%, with the interfacial zone occupying approximately 1/3 of the total paste volume fraction of 46%. This choice of parameters produces a system in rough agreement with the model studied in Ref. [15]. A slice through the body diagonal of the cubic unit cell is shown in Fig. 4, with the three phases shown in different gray levels. By convention σp = 1, while the value of σs / σp was allowed to vary freely.
Figure 4: Cross-sectional view (taken through the cubic unit cell body diagonal) of the bcc packing of equal size spherical sand grains. The bulk cement paste is shown in white, the sand grains in gray, and the interfacial zone in black.
To compute the overall conductivity of this periodic composite, a cubic unit cell, containing two sand grains, was digitized into a 3-D array of pixels, typically 1283. A macroscopic electric field was applied in one of the principal cubic directions. Each pixel is then treated as a tri-linear finite element, which results in a set of 1283 linear equations that are solved with a conjugate gradient algorithm [33,34]. A resolution of 64 x 64 x 64 was also used, with only very small changes in overall results, so that the 1283 resolution was judged to be adequate to represent both the sand grain, and more importantly, the thin interfacial zone volume.
To study a mortar with a more realistic random sand grain arrangement, we used the system studied in Ref. [15]. From the experimentally measured sand grain size distribution used there, we chose four representative diameters: 250, 500, 750, and 1500 µm. In effect, we reduced the total number of particles needed in the model by eliminating the largest and smallest grain sizes. [The ratio of largest diameter to smallest diameter determines how many particles are required to achieve a statistically representative volume.] For the maximum sand grain volume fraction studied, a total of 6500 particles was used. Table 1 shows the amounts of each kind of grain used, in terms of the fractional volume of the total sand amount.
| Diameter (µm) | Volume fraction of total sand content |
| 250 | 0.1895 |
| 500 | 0.2233 |
| 750 | 0.2317 |
| 1500 | 0.3555 |
Table 1: Sand size distribution used in random mortar model
The sand grains were randomly placed, largest first and smallest last, such that no sand grains overlapped. The center coordinates and radius of each sphere was recorded in a data list. Interfacial zones, of thickness 20 µm, were added to each grain. The interfacial zone volume was determined by point counting [15], the aggregate volume by just adding up the volumes of individual sand grains, and the bulk cement paste was the remaining volume.
The connectivity of the interfacial zone phase was computed using a burning algorithm [15]. Figure 5 shows the connectivity of the interfacial zone phase as a function of sand volume fraction. The y-axis, labelled "Fraction Connected", is just the fraction of the total interfacial zone volume, at a given sand volume fraction, that is part of a connected path across the sample. The interfacial zone phase first becomes partially connected at a sand volume fraction of about 36%, and essentially all of the interfacial zones are connected to each other by a sand volume fraction of 51%.
Figure 5: Percolation curve for random sand grain mortar model, for a 20 micrometer 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.
The conductivity of the random sand grain mortar model was computed quite differently from that of the periodic model, for the following reason. The largest unit cell we can presently use for electrical computations using the conjugate gradient/finite element method is actually around 1283, which is dictated by the limitations of the computers to which we had access. This is not enough resolution to adequately represent more than a few grains and their associated interfacial zones. To compute the conductivity of statistically representative volumes of material containing thousands of sand grains requires a different approach.
Here we adopt a random walk algorithm, used extensively in studies of disordered porous media [35] and composite materials [36]. For a material where only one phase has a non-zero conductivity, the algorithm is especially simple. Random walkers are started at various positions in the conductive phase, and allowed to take unit length steps in random directions at every time step. The mean-square distance travelled by each walker is computed as a function of the number of time steps. If a projected step would take the walker into a non-conductive phase, then that step is not allowed, but the clock is still advanced one time step. Eventually, the mean squared distance vs. number of time steps is a straight line, whose slope is the diffusion coefficient of the conductive phase. Multiplying by the volume fraction of the conducting phase then gives the overall conductivity of the composite, normalized by the conductivity of the pure conducting phase [35].
When two or more phases are conducting, with different phase conductivities, then the algorithm is somewhat more complicated, although this case has also been studied by Hong et. al. [37]. Basically, the step rate in a phase increases with conductivity of the phase. Additionally, the probability of stepping across a phase boundary, from phase A to phase B, is biased based on the relative conductivity of the two phases. Again the mean squared displacement vs. number of time steps gives a diffusivity from which is derived the overall conductivity [38].
The random-walk algorithm allows the sand grains and interfacial regions to be stored as geometrical objects rather than collections of pixels, so that the resolution is essentially that of the step size used. This step size can be small compared to the interfacial zone thickness h, without any increase in the computational storage required. This means that a large number of particles as well as the thin interfacial zones can be simultaneously resolved. Run times will increase as the step size decreases, however.
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