Figure 7: Phase volume fraction as determined by the cement paste microsructure model as a function of distance from an aggregate surface.
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To investigate the intrinsic strain of the interfacial zone, the 3-D hydration model developed by Bentz, et. al. [8], was used to simulate the formation of microstructure in the presence of an aggregate. This model has been used for a variety of studies of the interfacial zone region [11,30,31,32]. For this particular study, a scale of 1 pixel = 2 micrometers was used.
Initially, a 20 mm wide aggregate was placed down the middle of a 512 x 512 x 512 micrometers cube. The width of the aggregate is irrelevant since only the edge-effect on particle packing and one-sided growth effect will have an impact on the development of the interfacial zone [30]in the model.
A particle-size distribution for cement was adapted from Coverdale's measurements [33]. The resulting distribution is shown in Table 5. Each particle was placed randomly in the simulation area to avoid overlapping each other or the aggregate.
| Diameter (micrometers) | Volume percent of distribution |
| 39 | 26.5 |
| 35 | 6.0 |
| 31 | 11.25 |
| 27 | 11.25 |
| 23 | 11.25 |
| 19 | 11.25 |
| 15 | 11.25 |
| 11 | 11.25 |
Table 5: 3-D cement particle size distribution for simulation of interfacial shrinkage.
In the basic hydration model [8], any hydrating tricalcium silicate (C3S) pixel produces 1.7 pixels of C-S-H and 0.61 pixels of CH. One pixel of the produced C-S-H replaces the hydrating C3S, and the additional 0.7 pixels of C-S-H and 0.61 pixels of CH are collected and placed randomly within the pore area of the simulation. The result is a more narrow interfacial zone than those recorded experimentally [30].
The simulation described in this paper allowed the additional C-S-H to form only within a cube of side length 11 pixels, centered on the original position of the dissolving pixel. The goal is to localize the formation of C-S-H and, therefore, widen the resulting interfacial zone, to match experiment. CH is still allowed to be placed at random within the pore space. This seems a reasonable assumption given that there is very little silicon in solution, suggesting the silicon ions probably do not travel far from the cement particle before reacting. However, there is a large amount of calcium in solution [34], requiring a much longer diffusion period before reaction occurs. Additionally, calcium ions (ionic radius = 0.099 nm) [35] will exist individually in the solution (with a small amount of CaOH+), while silicon will exist only in the form of much larger ionic species and, thus, will diffuse much more slowly than the individual ions [36].
The simulation was run to approximately 55% hydration. Phase fractions were measured by taking square slices parallel to the surface of the aggregate, every 2 pixels (4 micrometers) out to a distance of 50 pixels (100 micrometers) from the aggregate surface. Slices were then taken every 10 pixels (20 micrometers). The results are summarized in Fig. 7, showing an interfacial zone of about 40 micrometers in width, a value comparable to those noted by other authors [8,10,36]. The individual phase fractions indicate large gradients within this interfacial zone, as expected.
The porosity has a value of approximately 45% near the aggregate and drops off to a bulk value of 15%. These values are in line with Scrivener's values (35% at the interface, 7% in the bulk) [10], measured on a 28-day old specimen, while 55% hydration implies a younger and more porous specimen. If the simulated porosity values are inserted into equation 3, the interface/paste modulus ratio is 0.55. Data from a recent study by Bourdette [37] show that the ratio of the interfacial cement paste porosity to the bulk cement paste porosity changes with hydration, assuming a fixed width for the interfacial zone. Thus, the ratio of moduli between interface and bulk will also change with hydration.
The amount of restraining phases in the paste, unreacted C3S and CH, show a minimum at approximately 20 micrometers from the surface of the aggregate. This results from the way the CH concentration varies with distance from the aggregate; it starts at a value of 36% near the aggregate and drops to a value of 15% in the bulk, decreasing faster than the amount of unreacted cement increases. The CH concentration is much higher at the interface than anywhere else, due to the completely random placement of newly formed CH in the pore space, and the higher volume of pores at the interface than in the bulk paste. These results are supported by experiments which often show elevated CH levels in the interfacial zone [38,39]. The volume fraction of unhydrated cement exhibits an opposing trend, starting at very low values close to the aggregate and increasing to its bulk value further away from the aggregate. The C-S-H content, on the other hand, exhibits a maximum rather than a minimum at 20 micrometers, in agreement with recent experimental work by Breton, et. al. [36].
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