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Fig. 2 shows a simulated mercury intrusion into the Fig. 1 structure, using d = 7 pixels. This value of d corresponds to a pressure that is not high enough to cause breakthrough of the mercury, implying that dc is less than 7 pixels. Careful examination of Fig. 2 shows that for left to right, and top to bottom, a line of empty pores and solid particles separates the mercury into isolated parts. Fig. 3 shows the same structure as in Fig. 2, but with a higher intrusion pressure, corresponding to d = 5 pixels, so that breakthrough is achieved and the resulting mercury cluster is continuous across the sample. It is interesting to note in Fig. 3 the existence of pore space regions that are unintruded, but which obviously contain large pores. This happens because these regions are only accessible by pores that are still too narrow to be intruded at this pressure, a clear example of the well-known "ink-bottle" pore effect . When these regions are finally intruded at higher pressure, their area will be wrongly counted as small pore area. This result demonstrates that even a very simple structure will show ink-bottle pore effects, as long as there is randomness in its spatial configuration.
Figure 2: Showing the model from Fig. 1 after being intruded with mercury at a pressure corresponding to a pore diameter d = 7. Only surface intrusion has been occurred. Gray denotes intruded mercury.
Figure 3: Showing the model from Fig. 1 after being intruded with mercury at a pressure corresponding to a pore diameter d = 5. The mercury (gray) has established a continuous pathway, with some "ink-bottle" regions still unintruded (see text).
Katz and Thompson  found in their mercury intrusion experiments that the value of dc corresponded to the inflection point in the cumulative intrusion curve. This result was obtained by using an electrical conductivity measurement to confirm that the intruded mercury extended completely across the sample. Using the simple structural model described above, this result can be checked. Fig. 4 shows a complete intrusion curve for a model like that shown in Fig. 1, using 41 pixel diameter circles digitized onto a 1000 x 1000 pixel array for higher resolution. The porosity is 0.56. The solid points have been averaged over intrusions performed on ten independent configurations of the structure, where in each configuration a different set of random numbers was used to place the circles that define the microstructure. This was done to reduce noise that arose from the finite size of the structure. The solid line is the fitted cubic polynomial d(A), where the pore diameter is thought of as a function of the intrusion area A, rather than the reverse. The fit was done to extract the value of the inflection point, where the slope of the simulated intrusion curve changes sign. This would also correspond to the maximum on a differential intrusion area vs. pore diameter plot, since an inflection point, where the second derivative is zero, corresponds to a maximum in the first derivative. The inflection point is found to be at d = 17.8 ± 2.0 pixels, where the uncertainty comes from the uncertainty in the parameters of the fit. This value compares well with the exact value of dc = 16.3 ± 1.2 pixels, found by using the burning algorithm on the actual intruded mercury clusters. The uncertainty in this latter value comes from the differences between the ten different configurations used. Thus, the simulated results for dc in this simple two-dimensional model are considered to confirm the experimental result of Katz and Thompson obtained on real three- dimensional rock samples, that the value of dc corresponds to the inflection point in the cumulative intrusion curve.
Figure 4: Showing the cumulative intrusion curve for a model like in Fig. 1, but with 41-pixel diameter circles on a higher resolution 1000 x 1000 lattice. The porosity is 0.56. The results shown have been averaged over ten independent configurations. The value of dc was found to be 16.3 ± 1.2.
It is also instructive to study how dc depends on the microstructure of this simple structure. There are two obvious but interesting ways to vary this structure. The first is to change the porosity by varying the number of circles placed. In this manner, the porosity can be easily swept from 1.0 down to about 0.5. The absolute limit for circles in the random parking problem is a porosity of 0.45 . However, much computer time is required to build samples with porosities below 0.5, so that region is avoided in this paper.
Fig. 5 shows a plot of dc vs. porosity, computed using 21 pixel diameter circles on a 500 x 500 grid. The number of circles ranged from 180, at the high-porosity end, to 360, at the low-porosity end. The circular data points represent an average over 10 configurations, in the same sense as in Fig. 4. It is clearly seen that dc decreases non-linearly with porosity. The solid lines are simple theoretical predictions, based on computing the spacing between circles as if they were regularly spaced on a uniform lattice. The upper curve is for a triangular lattice, and the lower curve is for a square lattice. The dashed line is defined below. The calculation is carried out in the following way.
Figure 5: Showing dc vs. porosity for the circular particle model, where the porosity was varied by placing different numbers of particles. The solid and dashed lines are theoretical predictions explained in the text.
If N is the number of circles placed, then the equality
|N (a/b)2 = L2||(3)|
defines the lattice spacing, a, of a lattice on which a circle is placed at each lattice point, so that the density of circles in the L x L cell is the same as in the random case. The quantity (a/b)2 is the area of the lattice's primitive unit cell, where b=1 for a square lattice, and b = (4/3)1/4 for the triangular lattice. The primitive cell area is a2 for the square lattice, and (3/4)1/2 a2 for the triangular lattice, which define the values of b. In two dimensions, these are the only Bravais lattices that are the same in the x and y directions, so are the only lattices worth considering. The pore neck size, d, between circles is then just
|d = a - D||(4)|
where D is the diameter of the circles making up the microstructure. If a regular structure like the one defined here were to be intruded with mercury, there would be no intrusion at pressures corresponding to pore diameters greater than (a - D), but complete intrusion at (a - D). The rationale for this calculation is that if dc = a - D is the exact threshold diameter when the circles are arranged on a regular lattice structure, then this value might still be a reasonable approximation for dc when the circles are randomly placed. It can be seen in Fig. 5 that the two lattice predictions give fairly tight upper and lower bounds for dc. The exact equation plotted in Fig. 5 is
|dc = b (A/c)1/2 - D||(5)|
where A is the area of one circle of diameter D, c is the solid area fraction, is the porosity, and use has been made of the relation
A second way of modifying the microstructure is to fix the porosity, but vary the diameter of the circles that make up the structure. In effect, this is like changing the particle size of an agglomerate of particles. Fig. 6 shows the results of this calculation on a 1000 x 1000 lattice. Note that the size of the particles, even when the porosity is unchanging, controls the pore size, as the value of dc decreases with the particle size. The upper solid line is the triangular lattice prediction, and the lower solid line is the square lattice prediction from eq. (5). The dashed line is again the average of the triangular and square lattice lines. Again, the two theoretical predictions bracket the simulation data quite well, and the average of the two predictions again fits the simulation points quite well. It would be interesting to study a simple 3-d structure, say a monosize glass-bead pack , and see if these simple functional forms can describe the results of real MP experiments.
Figure 6: Showing dc vs. the circular particle diameter D used to generate the circular particle model microstructure. The porosity was held constant at 0.58. The solid and dashed lines are theoretical predictions explained in the text.
It is important to note that the above simulations can be repeated in models that employ real particle shapes to define the microstructure. Indeed, this algorithm can be applied to any digital image, thus broadening its usefulness. It can be thought of as part of a "tool kit" of algorithms that can be used to analyze digital images . The burning algorithm briefly mentioned above is also part of this "tool kit".