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Further discussion of digital resolution

As is the usual case for random geometry models, the percolation properties of the cement hydration model do depend fairly strongly on digital resolution. They are robust, however, with respect to other factors like w/c ratio, and details of the model like cyclic vs. continuous operation, dissolution, etc. The question then arises, is there a "right" resolution?

This is not a trivial question. When solving differential equations using, say, a finite difference technique, the error gets smaller as the mesh size becomes smaller. In this case, full accuracy is only achieved in the limit of infinitely fine resolution, although usually there is a resolution at which the errors are "small enough." If the cement hydration model is like this case, then up to the limit of resolution tested, 0.125 µm per pixel, was not small enough, as it appeared that two out of three percolation thresholds were still changing (the capillary pore space percolation threshold was not computed for the 8003 case). However, the cement hydration model is designed to simulate a real material, cement paste, so we may turn to the real material to find a length scale that can give us clues as to the "right" resolution. It cannot be too strongly emphasized that a cellular automaton model is not the same mathematically as a finite difference model, and so will respond to digital resolution differently.

One note must be made before the discussion proceeds. If the notion of a dissolution length in the model is accepted, then the pixel representation of this length scale did not scale with resolution as did the number of pixels per particle. The reason for this is that going beyond the third nearest neighbor for dissolution brings in unphysical aspects of the dissolution. If dissolution can go beyond this point, then a pixel that is underneath solid pixels can "jump out" over its neighbor, dissolving before a solid pixel that is closer to the pore space. This does not happen if the model stays at a 3 x 3 x 3 dissolution box.

The first length scale of interest to the model comes from the resolution of the original cement image generated by scanning electrom microscopy that is used to generate the starting particles [8,22]. This resolution is only 0.5 µm per pixel, so in one way it doesn't make much sense to go beyond this resolution in hydration, as the original materials are only known to this resolution. This resolution would correspond to the 2003 models considered in this paper. However, this does not give an absolute limit, and could be improved in the future.

Second, the structure of the hydration model itself sets a lower limit to how many micrometers per pixel make sense. In this model, the C-S-H phase is represented as a featureless, uniform solid. Actually, there are nanometer-scale pores in this material, which are not resolved in this model. To begin to see these pores would require a resolution of about 50 nm/pixel, or 0.05 µm per pixel, which would correspond to a model size of about 20003, for a fixed unit cell length of 100 µm. So if the model is designed to not see the C-S-H pores, there is no reason to go larger than this size model. A size of 8003, corresponding to a resolution of 0.125 µm per pixel, is approaching this limit.

Another size scale is set by the dissolution kinetics. The first few hours of hydration time corresponds to the first few cycles of dissolution/hydration in the model [8]. It is known that small cement particles, 1 µm or less in diameter, dissolve and hydrate in the early hours of hydration. In the same way, 1 pixel particles dissolve very rapidly in the model, with increasing slowness as the number of pixels in the particle increases. Probably, in terms of matching this dissolution behavior, the lower limit on the length per pixel is 0.2 µm per pixel, which would correspond to a 5003 pixel model in this paper. Having more pixels in a 1 µm diameter particle would make it dissolve too slowly to match reality.

Another length scale in cement paste might be given by the thickness of the initial C-S-H layer that forms on the surfaces of the cement grains, which determines the initial percolation of the C-S-H phase. If this layer was on average, say, 0.25 µm thick, then it would make sense to have a pixel length scale of no less than 0.25 µm per pixel, or a 4003 size model. Then there would be no chance of forming an unphysically thin C-S-H layer of less than this thickness, as the one pixel coating that percolates would be the right thickness. The thickness of this layer might be able to be determined using a technique like atomic force microscopy (AFM), which has the capability of imaging the topography of a cement particle through water, so that the thickness of a thin surface layer of C-S-H might be able to be measured.

The set point can also be used to guess at an appropriate resolution for the model. Note that the set point of the larger models was not directly measured, since extra memory is required for this determination, which made the larger models too large for our study. However, the C-S-H percolation results can be used to show how the set point scales with resolution.

The set point in the portland cement model is determined by the percolation of the joint C-S-H/ettringite cement phase. Physically, the set point is defined as the point at which the growing C-S-H/ettringite first "glues" the cement particles together [31]. Cement particle flocculation is ignored in the model, as particles only count as being connected if a C-S-H or ettringite bridge connects them. To make a connected phase out of the joint cement/C-S-H/ettringite phase is easier than percolating the C-S-H phase separately. Clearly, the cement particles can be "glued" together into a percolated phase without the C-S-H part of the "glue" being itself percolated. Therefore, the degree of hydration at which the C-S-H percolation point occurs must be an upper bound for the degree of hydration at which the set point occurs. Since the set point is still dependent on C-S-H production, the set point must then scale similarly with model resolution as did the C-S-H percolation threshold, as seen in Fig. 12.

Set point results of the 1003 model have been found to be in fairly good agreement with experimental measurements. The differences tended to be that the predicted degree of hydration was a bit too low [31]. The higher resolution C-S-H percolation data in the present study would only further reduce the degree of hydration at set, implying that the model would agree less well with experiment. So certainly the model resolution cannot go too high above 1003, if at all, if agreement with set point experiments with this version of the model is to be preserved. More good measurements of the degree of hydration at set are needed to help calibrate the model's resolution. Of course, one must remember that the model predicts geometrical percolation, and the experiments predict set point via some kind of physical measurement. Geometrical percolation almost certainly takes place before the threshold for physical property measurement is reached, so the experimentally measured degree of hydration at set is most likely above that determined by the model. Therefore the lower values of degree of hydration at set that would be produced by a somewhat higher resolution model might still be in good agreement with reality.

One piece of evidence for a resolution near that of 1 µm per pixel is found in electrical conductivity measurements of frozen cement paste using impedance spectroscopy [32]. In this work, a low enough temperature was achieved (-40 ºC) so that the only conduction paths were considered to be through the C-S-H phase, and not the capillary pores. There was very good agreement between predictions of the model and experimental measurements, and the percolation threshold of the C-S-H appeared to be about 0.12 to 0.16. Since the experimental measurements were on portland cement paste, the model values of the C-S-H percolation threshold for this material should be used to compare. This value was found to be about 0.18, which argues for a resolution near to that used in Ref. [32], 1 µm per pixel.

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