Next: Dilute limit Up: Computational Methods Previous: Periodic model: finite
The conductivity of the random sand grain mortar model was computed quite differently from that of the periodic model. The largest unit cell presently employed in our conjugate gradient-finite element bcc calculations is 1283. Such calculations typically require 300 megabytes of memory, which scales as the third power of the system length. 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 disordered systems containing thousands of sand grains requires a different approach.
Here we adopt a random walk algorithm, used extensively in studies of disordered porous media [20,21,22] and composite materials. For a system in which 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 steps of fixed length, ε, in random directions at every time step. The mean-square distance traveled by each walker is computed as a function of the number of time steps. If a projected step would take the walker into the insulating 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 [20]. More details of the method are available in the Appendix.
When two or more phases are conducting, with different conductivities assigned to each phase, then the algorithm is somewhat more complicated. Within the framework of lattice random walks this problem has been studied by Hong et al. [22]. In the Appendix we discuss the extension of the methods developed in Ref. [22] to the study of continuum systems. In the present framework, the principle advantage of the random-walk approach is that the aggregate and interfacial structure can be stored as geometrical objects rather than collections of pixels, so that the resolution is essentially that of the step size used. The step size must be small compared to the interfacial zone thickness h, but this can be accomplished without any increase in the computational storage required. Of course, run times will increase as the step size decreases, because the walkers must cover several aggregate grain diameters in order to properly estimate the overall conductivity. It should be emphasized that the number of time steps required in these calculations can be very large compared to the corresponding number for calculations of bulk (i.e. single phase) conductivity. Thus, in Fig. 5 we compare the behavior of the effective diffusion coefficient, D(t), for two bulk calculations with the limiting case of purely interfacial conduction. The longer times required in the latter calculation reflect the more tortuous paths connecting the overlapping interfacial shells. At short times the behavior of D(t) is controlled by the surface area to pore volume ratio, S/Vp, of the conducting phase [23]. This ratio is largest for the case in which the interfacial shells alone comprise the conducting channels, as is seen quite clearly in the insert to Fig. 5.
Figure 5: The effective diffusion coefficient is plotted as a function of time for three systems based on the four size, c = 0.54, packing shown in Fig. 1 . Results, from top to bottom, are shown for σs / σp = 1, σs / σp = 0, (i.e., insulating interfacial zone), and σs / σp = ∞ (i.e., insulating bulk paste). The x-axis is (Do t) 1/2 / ε, and the y-axis is D(t)/Do , where Do is the diffusivity of the matrix, and ε is the step size.
Next: Dilute limit Up: Computational Methods Previous: Periodic model: finite