Next: Effects of interfacial
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The influence of the ITZ on the elastic properties and shrinkage of mortars has recently begun to be addressed [20,21]. The quantative effect of this third phase is dependent upon its inherent properties and location within the microstructure. The location of the ITZ is handled well by the present model, but three unknowns for the interfacial region must still be quantified: two elastic moduli and the unrestrained shrinkage strain of the ITZ. The elastic moduli are considered in this section.
To explore the effect of the ITZ, models were created using monosize circles for sand grains, where the surface area of the sand grains, at a constant area fraction of 55%, was adjusted by using different diameters (110-910 micrometers), in order to compare with recent experimental results using similar systems . A 20 micrometer-thick interfacial zone surrounded each of the sand particles. The Young's moduli and Poisson's ratios from Section II were used for sand (Es) and bulk cement paste (Ep), while the Poisson's ratio of the interface was set to 0.3 (the same as the paste) and the Young's modulus of the interface (Ei) was varied from 20% to 500% of the paste (0.2 < Ei/Ep < 5). Due to the usually higher porosity of the ITZ paste in real mortars and concrete, the Young's modulus of the interfacial region is expected to be lower than the Young's modulus of the bulk. Helmuth and Turk  described their experimentally obtained moduli for a wide range of cement pastes and degrees of hydration by the expression:
|E = Egel(1 − Vc)3||(2)|
where E is the overall modulus of the paste, Egel is the average modulus of the hydration products, and Vc is the capillary porosity. However, in the model Ei/Ep was allowed to be greater than one, because we wish to see the full range of behavior, and because for lightweight aggregates and chemically treated aggregates, Ei/Ep > 1 could be physically realistic .
Figure 2 shows the results of these numerical experiments, with E*/Ep plotted against Ei/Ep, where E* is the overall composite modulus. Two distinct regions are apparent. As expected, at points where the interface moduli are less than the paste, the modulus of the composite material is decreased by the presence of the interfacial zone. Similarly, when the modulus of the interface is stiffer than the paste matrix, the modulus of the overall composite is increased. The magnitude of the increase or decrease varies significantly with the diameter of the aggregate particle, due to the varying amounts of interfacial zone phase present. As this diameter increases, there is less interfacial zone area per unit area of inclusion for isolated sand grains and sand grains at area concentration of 55%, as shown in Table 3. More importantly, as can also be seen in the last column of Table 3, the mortars with higher surface area sand have a larger area fraction of interfacial zone, and so the moduli of this phase have a correspondingly larger effect on the overall composite moduli. Also in Table 3, it can be seen that for particles larger than 210 micrometers, the area of the ITZ per particle area is only slightly smaller for the 55% sand systems than for isolated particles, implying that indeed the overlap area is small, even at this fairly high sand area fraction. However, for the 110 and 210 micrometer diameter sand grains, since h = 20 micrometers is a large fraction of the sand grain diameter, there is a fairly large amount of ITZ overlap, as can be seen in the differences between the third and fourth columns in Table 3.
Figure 2: Effect of the ratio of the interfacial transition zone cement paste Young's modulus, Ei, to the bulk cement paste Young's modulus, Ep, on the composite Young's modulus, E*, for different sand particle diameters at a constant area fraction of 55%.
|Diameter (micrometers)||Area of ITZ / Area of particle (at dilute sand content)||Area of ITZ / Areaof particle (at 55% sand content)||Area fraction of ITZ (at 55% sand content)|
Table 3: Particle and interfacial transition zone areas for different size sand particles (h = 20 micrometers), at dilute limit (single particle values) and at 55% sand content.
Separating the two regions of Figure 2 is a point where the interface moduli are equal to the bulk paste moduli. At this point, the interface has elastic properties equal to the bulk paste and "disappears", in effect acting as simply more bulk paste. The moduli of all particle size composites converge to this point, as was discussed in Section II.
Based on existing experimental measurements , the E/Ep data from Figure 2 for Ei/Ep < 1 was replotted versus surface area of sand in Figure 3. An easily-derived relation between surface area and the particle diameter for d-dimensional spheres was used:
where k is a constant given by k=2d (d is the number of dimensions), SA is the surface area in md−1/kg, ρ is the density of the aggregate (taken to be 2.56 g/cc, the value for quartzite, as in ), and dmean is the mean diameter of the inclusion particles in meters. The three- dimensional value of k=6 was used in order to relate more closely to the experimental results in the work of Cohen et al. , but, most significantly, the surface area and mean diameter are inversely related via equation 4. The actual value of the constant of proportionality is not important in this work.
Figure 3: Effect of aggregate surface area on composite Young's modulus with changing interfacial/paste Young's modulus ratio for Ei/Ep < 1 for different sand particle diameters at a constant area fraction of 55%.
Figure 3 clearly shows the effects of increasing the area fraction of interfacial zone by increasing the surface area of the sand. Each curve has a different value of Ei/Ep. Figure 3 is an important graph because, in principle, it allows for the quantification of Ei by comparison with experimental data in one of two ways, which both require knowledge of the surface area of the aggregate and the moduli of the composite material.
The first method also requires knowledge of the moduli for the paste alone. A direct comparison can then be made by normalizing the measured composite moduli with the paste moduli, using the proper surface area, and then finding the best fitting value of the Ei/Ep ratio from the graph.
The second method would be to determine the experimental drop in Ei/Ep (percentage) as the surface area of the mortar aggregate is increased, assuming Ep is a constant. This can then be compared to the percentage drops predicted by the algorithm for the various Ei/Ep ratios. The point at which theory and experiment agree determines the correct ratio.
The results of Figure 3 are from a 2-D model, and so cannot be rigorously used in either of these methods to analyze 3-D experimental data. Also, as sand surface area increases , ITZ area fraction also increases. Because of the higher porosity of the ITZ, it will have a higher water:cement ratio. Since the overall paste has a fixed water:cement ratio, conservation of water between bulk and ITZ implies that the bulk cement paste will increase in density by decreasing in water:cement ratio as the ITZ area fraction increases. Thus, strictly speaking, the moduli of the paste from which the mortar is made are not exactly the moduli to be used for the bulk paste in the mortar microstructure. However, such a comparison does provide some qualitative insight.
The elastic moduli of mortars as a function of surface area of sand for nearly monosize sand grains have been measured by Cohen et al. . In their work, the dynamic elastic moduli of portland cement and silica fume mortars were measured as a function of surface area and of hydration time. In the present work, only their 28-day portland cement mortar data has been considered. Since the value of Ep for their data is unknown, the second method described above was used.
In the data of Cohen et al., a Young's modulus of 33 GPa was measured for a mortar containing sand of surface area of 2.5 square meters per kilogram, and a Young's modulus of 29 GPa was measured for a mortar containing a sand surface area of 10 square meters per kilogram, representing a drop of 11.6% in modulus as the sand surface area was increased by a factor of four. Using Figure 3, these values were bracketed between the curves of Ei/Ep=0.4 , which had a drop of 16.2%, and Ei/Ep=0.6 , which had a drop of 8.7%. It was found that Ei/Ep = 0.5 fit the experimental data reasonably well. This implies that, when averaging over an interfacial zone thickness of 20 micrometers, the effective Young's modulus of this region will be about 1/2 that of the bulk cement paste, at least for fairly large degrees of hydration. This is a reasonable value, considering the higher porosity of the interfacial zone . Also, this value can be seen to be reasonable through the use of equation (3). By taking an approximate value of 35% porosity at the interface and 7% in the bulk, as reported by Scrivener , for a 28-day-old concrete specimen of w/c=0.5 , averaging the porosity over the interfacial region, and using eq. (3) for both the interfacial region and the bulk, a modulus ratio of Ei/Ep = 0.6 is computed. This value is, of course, only a rough approximation, but it is in accord with the value fitted from the data of Cohen et al . Doing the same fitting procedure with a 3-D model would of course give a somewhat different answer.