Effects of digital resolution on phase percolation

Increasing the fineness of the digital resolution of the hydration model, while keeping the physical size fixed, will increase the computational size of the models. Even having access to the computational facilities of NIST, when the hydration models become large, the computer time for hydrating a model becomes large as well. Having more continuous dissolution/ reaction also slows down the model, compared to the older versions. We have used grids as large as 800³ pixels in size for the C-S-H results. Since most of the percolation results for other phases required longer hydration times, only a maximum of 400³ size systems were able to be used for these results. Larger sizes are possible with the memory on machines available to us, but the slow turnaround time for systems above about 1000³ is prohibitive, even for the C-S-H results. Memory requirements, with the stripped-down version of the model used in this paper, which does not include the set point computation, is about 1.7 x n³ bytes, where the model is n pixels x n pixels x n pixels in size. Since monitoring the setting process takes much more memory (to store the original particle label in the appropriate pixel [8]), this was not carried out for the larger sizes, and so no digital resolution comparisons for the set point are available. However, insight into the effect of digital resolution on setting can be obtained from the results for the C-S-H phase percolation, and will be discussed later.

One aspect of the model must be addressed before the digital resolution effect can be discussed. A key part of the model is the dissolution of cement particles. This has been done in the past by considering all cement pixels that are touching water. For such a pixel, one of the six nearest neighbors of the cement pixel is selected randomly. If it is a water-filled pore pixel, then the cement pixel can dissolve into it. If it already occupied by a solid phase, then the cement pixel cannot dissolve.

When simulating systems with coarser particle size distributions, and therefore larger particles, it has been found that hydration using this dissolution rule stops too soon. To correct this deficiency, and allow the use of larger particles (on the order of 40 µm), the 2nd and 3rd nearest neighbor pixels were also allowed to be queried as possible dissolution sites [25,24]. There are a total of 26 neighbors, when 1st, 2nd, and 3rd nearest neighbors are considered (all the pixels in a 3 x 3 cube, with the central pixel being the cement pixel being investigated). This change allows further hydration to be carried out on larger particles, and provided excellent agreement between experiment and model data [25].

Physically, one can think of a "dissolution length", a length over which the material can dissolve. By going from 6 to 26 neighbors, we are effectively going from a dissolution length of 1 pixel to about 1.4 pixel. Of course, as the resolution increases, and the physical length per pixel decreases, the physical length of this pixel size decreases. This will be discussed further in Section 8.

The question then arises, what is the effect of digital resolution on the percolation properties of the cement paste model? In the results to be discussed next, the effect of having 6 or 26 neighbor dissolution will be considered simultaneously with the effect of digital resolution on the percolation properties of the model.

Section 5 showed that different w/c ratio pastes have percolation quantities ("Fraction connected") that collapse onto a single curve when plotted against the volume fraction of the phase being considered. Therefore, the following results will be for a single w/c ratio, 0.3, plotted only against the volume fraction of interest, and not the degree of hydration. This lower value of water:cement ratio was chosen, as it was computationally easier to drive it through its percolation thresholds, since less hydration was needed. Also, only C₃S cement particles were used, because they were simpler and quicker to hydrate, yet had the same qualitative behavior as did the portland cement systems.

Each system was created with the same number of randomly-placed particles, but with each particle appropriately larger, in pixels, according to the resolution. In continuum terms, the same size particles were used at each resolution, but with a different number of pixels per unit length. All systems were the same physical size, 100 µm x 100 µm x 100 µm. The resolutions were: 1 µm per pixel for the 100³ pixel system, 0.5 µm per pixel for the 200³ pixel system, 0.25 µm per pixel for the 400³ pixel system, and 0.125 µm per pixel for the 800³pixel system. For comparison, the 100³ and 200³ sizes were investigated using both 6 and 26 neighbor dissolution. At the 400³ and 800³ sizes, the 26 neighbor dissolution was necessary to be able to hydrate the cement to the point of capillary porosity de-percolation and CH percolation. Both the 6 and 26 neighbor dissolution schemes were able to hydrate to the point of C-S-H percolation, which comes fairly early in the hydration process (see Fig. 4).

Figure 8 shows a comparison, at the same w/c ratio and the same porosity, 0.32, between the 100³, the 200³, and the 400³ resolutions. Note that each system had the same physical size, 100 µm x 100 µm x 100 µm. The images shown in Fig. 8 are not full slices, but show an area of approximately 50 µm x 50 µm in size. The capillary pore space is in white, and all solid phases have been turned black, in order to focus on the capillary pore space.

Figure 8: Showing 2-D slices through the microstructure of an 0.3 w/c cement C₃S paste at three different resolutions: (top left) 100³, (top right) 200³, and (bottom) 400³ pixels, and the same capillary porosity = 0.32. Black is solid, white is the capillary pore space. The physical size of the slices is about 50 µm x 50 µm.

Because each particle is the same physical size, but is made up of more pixels at higher resolutions, the amount of hydration per dissolution cycle will be smaller as the resolution increases. The amount of hydration per cycle will also be affected by the number of neighbors allowed to be considered for dissolution. Figure 9 shows a plot of the degree of hydration vs. number of dissolution cycles for three different resolutions at w/c=0.3, with both the 6 and 26 neighbor dissolution. Note that decreasing resolution and increasing number of neighbors considered for dissolution both tended to increase the amount of hydration per cycle.

Figure 9: Showing the degree of hydration attained vs. the number of model cycles run, for three different resolutions and 6 vs. 26 neighbor dissolution, for 0.3 w/c C₃S cement paste.

Figure 10 shows the capillary pore space percolation for this paste, at three resolutions, 100³, 200³, and 400³. For the two lower resolutions, the 6 and 26 neighbor dissolution curves are identical in shape, and displaced slightly in percolation threshold. So there is only a small quantitative, and no qualitative, effect, of the different number of neighbors dissolution on capillary pore percolation. More interesting is the progressive lowering of the capillary pore percolation threshold as the resolution is increased, from 0.22at 100³ to 0.12 at 400³. Essentially this happens because smaller pores can be resolved at the greater resolutions, and a pathway that would seem to be closed off at low resolution is seen to be narrowly open at higher resolutions.

Figure 10: Showing the fraction of the capillary pore space that is part of a connected pathway as a function of the capillary porosity for 0.3 w/c, at different resolutions, for a C₃S cement paste.

This resolution effect is also seen in simpler systems, for example in 2-D percolation of digital circles [26]. When the circles are made up of only one pixel, and are randomly placed on a square digital lattice, this process is equivalent to site percolation, which has a percolation threshold of 0.59 for the circle phase. As the resolution is increased, the continuum circle limit of 0.68 area fraction of circles is achieved. In 3-D, the same process for spheres leads from site percolation (one pixel per sphere), where the percolation threshold is 0.249, to continuum sphere percolation, where the threshold is 0.29 [27].

The same story holds true for CH phase percolation. Figure 11 shows the percolation curves for CH. The difference between the 6 and 26neighbor dissolution now seems a little more pronounced than in Fig. 10, but still small compared to the effect of digital resolution. The movement of the CH threshold with resolution seems a bit different from the capillary percolation threshold case. The 800³ result seems close to an equilibrium point, with the infinite resolution value of the CH percolation threshold at about 0.1. Only a small part of this curve could be generated with the available computer capacity. CH phase percolation is topologically that of shapes growing at random locations, which gradually impinge enough so that a connected backbone is achieved. Certainly the shape of the crystals, which is a function of their growth habit, sensitively determines their percolation threshold. For the case of simply overlapping, equal size and shape ellipsoids, it was found that the percolation threshold was sensitively dependent of the aspect ratio of the ellipsoids of revolution [27]. It is certainly possible that the way the model grows CH does not agree well with real cement paste. For example, at times CH tends to form flat, hexagonal crystals, at least when there is room in the pore space for them to grow. The model grows them as roughly isotropic masses. CH morphology is also strongly dependent on temperature and silica fume content [28].

Figure 11: Showing the fraction of the CH phase that is part of a connected pathway as a function of the total CH volume fraction for 0.3 w/c, at different resolutions, for a C₃S cement paste.

The "Fraction connected" curves are shown in Fig. 12 for the C-S-H phase. Here it is seen that there are only very small differences between 6 and 26 neighbor dissolution. However, the C-S-H phase percolation threshold is probably the most sensitive, of the three phases considered, to the digital resolution. The value of this percolation threshold is seen to move regularly downward as resolution is increased.

Figure 12: Showing the fraction of the C-S-H phase that is part of a connected pathway as a function of the total C-S-H volume fraction for 0.3 w/c, at different resolutions, for a C₃S cement paste.

In fact, the asymptotic value of percolation threshold should scale like 1/L, where L is the size of the system in pixels, for the following reason. It is conjectured that the C-S-H phase becomes percolated when a thin sheet partially covers each cement particle and also touches a neighboring cement particle. This sheet can be made up of material that is dissolved and reacted from the first layer of the cement particles. If this is true, then the degree of hydration at C-S-H percolation should scale like 1/L, because the surface to volume ratio of the particles also scales this way, and essentially all that is happening is that the surface of the cement particles are dissolving and reacting to percolate the C-S-H layer. The volume fraction of the C-S-H should also scale the same way, since it goes like the total surface area of the cement divided by the volume of the system. Again, this ratio goes like 1/L. At the two highest values of L, 400 and 800, this scaling is approximately true, as the percolation threshold for the 800³ system is about one half that of the 400³ system. The comparison of this result to real cement paste is made in Section 8.