Exercise No. 4 Mercury Porosimetry

Program: MERCURY.EXE

In this exercise, the dependence of the critical pore diameter d _c, on microstructure, will be studied, using the 2-d mercury porosimetry algorithm. The user will examine models of porous materials, constructed in the following way. Equal-size circles are placed at random inside a square 200 x 200 cell with periodic boundary conditions, but are not allowed to overlap each other. This creates a model with a continuous pore space at any porosity, where the pore space is composed of the "capillary" regions consisting of the leftover space between solid circular particles.

At a fixed porosity, one might expect that the value of d_c would change as the solid particles that define the pore space are varied in size. For example, in one dimension, if a line segment of unit length is covered with two smaller segments of length 1/4, then the space between them is 1/4, and the covered length fraction is 1/2. This case is illustrated in the top of Fig. 4. If instead four segments, each of length 1/8 are used, the covered length fraction will still be 1/2, but now the empty spaces will only be of length 1/8, as shown in the bottom of Fig. 4. So the two line segments still have the same "porosity," but different "pore" sizes. Therefore one might expect, since average pore sizes will certainly change with the fineness of the solid particles that define the pore structure, the value of d_c would change as well.

Figure 4. Effect of line length on pore size. Gray = solid, white = pore.

In this exercise, the user will create pore spaces at a fixed porosity of about 70%, by randomly placing circles of diameters 7, 13, and 19 pixels. The areas in pixels occupied by these circles are 37, 137, and 293 pixels respectively. The mercury porosimetry program will intrude mercury from all sides, from top to bottom, and finally from left to right, in order to check for continuity in both directions. As you will recall, d_c is defined as that pore diameter corresponding to the injection pressure at which continuity is achieved. The value found should be averaged over the two different directions. Note that when percolation is first achieved for a given direction, intrusion in that direction will not be executed for any smaller meniscus diameters.

The program is called MERCURY.EXE. Switch to the appropriate directory and type MERCURY at the DOS prompt to begin execution. The program will ask you initially how many and what diameter circles to use to generate the microstructure. (Although not required for this exercise, the user may add multiple size circles as this option will appear iteratively until a value of 0 is entered for the number of circles to add. The user should always add multi-size circles in order of largest to smallest.) The diameter of the circles must be odd integer so that the circles may be centered on a pixel. The user will have to calculate the number of monosize circles needed for each diameter to achieve a porosity of about 70%. Next, the program will ask what range of menisci diameters to evaluate (3 to 17 for example). The program will also ask whether to run a complete intrusion or only check for percolation and the user may choose the latter by entering a 1 (to save time) or choose both by entering a 2 to produce a complete intrusion curve. When intruding from one side only, if the mercury reaches the opposite side from where it started, then the intrusion is stopped before completion, and the connectivity is recorded. During all intrusions, the previous intrusion image is left on the screen as the pressure is increased, so that the user may study how pore accessibility changes as pressure increases. Each pressure is color coded, so that the user may determine for any pixel in the pore space, the lowest pressure (highest meniscus diameter) at which this pore would be accessible from the exterior. When the program is finished executing for all the chosen intrusion diameters and directions, the user must press the ESCape key to remove the final image. At this point, if option 0 or 2 has been selected for execution, a graph of the area intruded vs. meniscus diameter will be produced on the screen. Finally, the user may press any key to exit the program.

The results of this exercise should be expressed in the form of a graph plotting d_c vs. the particle size used to generate the microstructure [18].

Tips. (1) Try to get at least three realizations at each circle size, and average d_c over them. (2) For a given direction, d_c is the average of two menisci sizes: the first being the largest value for which the mercury reaches the opposite side and the second being the smallest value for which the mercury doesn't reach the opposite side (e.g., if the mercury does not percolate for d=9 but does for d=7, then d_c=(9+7)/2=8).