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Results

Fig. 7 shows the 11 shapes studied, seven of which are shapes defined by collections of needles, and four of which are shapes defined by collections of pixels. Each shape was studied on a 1000 x 1000 (1000 x 999 for the needle shapes) lattice, and the longest dimension of each shape was either 20 bonds long for the needle shapes, or 20 or 21 pixels long for the pixel shapes. The values of n_I (suitably averaged over orientation) and n_c were computed for each shape. The quantity L_eff was defined via eq. (4), using the computed value of n_I.

Figure 7: Showing the various shapes that were used in this paper to study the relationship between n_c and n_I. The numbering scheme is the same as that used in Table III.

Table III shows the results for n_I, n_c, and n_c L_eff². With the exception of shapes no. 3 and 4, the quantity

is invariant to within 5% or less, for needle shapes and for solid pixel shapes. The corresponding values for shapes no. 3 and 4 are still within 30% of this number, however. Equation (12) should hold up under a true continuum analysis, as the lattice method would be expected to underestimate both n_I and n_c, as was discussed above, and so the ratio might well be unaffected.

We have also checked the invariance of x_c for combinations of shapes. Table IV shows the results for different proportions of two sizes of circles, and Table V shows similar results for two sizes of needles. In this case, the quantity L_eff² is used to normalize n_c, where L_eff² is averaged over the relative number concentrations of the two different size objects. The first column in both tables is the ratio of large to small diameter or needle length, and the next three columns contain the results for n_c L_eff² for 1:1, 2:1, and 3:1 ratios of the number of small objects to the number of large objects. In both cases, it is clear that n_c L_eff² is the same for these cases, within 5 or 10%. There is some effect of the ratio of unit cell to needle length in Table V, as the results for the 20:4 needle length ratio are somewhat different from those for the 40:8 needle length ratio. This could also be due to statistics, as there are fewer of the larger needles used, since the unit cell was always 1000 x 1000 lattice spacings.

We have obtained one final result, for a system where 41 pixel diameter circles and three times as many 40 lattice-spacing long needles were simultaneously mixed. The value for x_c was 5.6 ± 0.3, invariant within computational uncertainty, so that this invariant also holds for mixtures of different shapes, as well as for collections of the same shape.