Previous: Water sorption and shrinkage
A general pore space cannot accurately be thought of as pores with a certain shape that are linked up like tubes in a network [34,71]. The finite element techniques discussed in this paper for computing shrinkage, along with the algorithm for estimating the location of capillary condensed water at a given RH value , constitute a novel technique for modelling moisture-induced volume changes, including drying shrinkage, of porous materials with general pore shapes and pore network topology. The technique for locating capillary condensed water is only approximate, however, because the Kelvin-Laplace equation becomes inaccurate for very small pore sizes, and because the spherical interface approach has only one radius of curvature, whereas real meniscii have, in 3-D, two radii of curvature. Techniques were previously developed to produce a reconstructed three-dimensional representation of the porous medium from a 2-D micrograph. In this paper, this reconstruction technique has been applied to porous Vycor glass, giving a 3-D pore structure upon which the shrinkage algorithm has been demonstrated and the reconstruction algorithm checked. Computed properties like elastic moduli, fluid permeability, and electrical and thermal conductivity have been used to compare to experimental measurements, in order to test how close the reconstructed model comes to reality.
The thermal conductivity of the backbone agreed well with experiment. However, this quantity is not very sensitive to microstructure, because the three-sphere model gave results very close to that of the porous Vycor models, and these all had similar results, even though their surface areas were much different. The formation factor of the pore space also was nearly invariant with surface area, and agreed well with experiment. Even though these quantities are not very sensitive to microstructure, and can not be used by themselves as a check on a model, it does seem true that if a model cannot reproduce these quantities in agreement with experiment, then there is something lacking in the model's representation of the microstructure. Taking the cylinder model with a ratio of cylinder radius to unit cell size of x=0.355 gives a model with a porosity of 0.32. While the computed solid thermal formation factor is reasonable, 1.9, the pore space formation factor is only 6.3, three times less than the experimental measurements. The cylinder model is isotropic with respect to conductivity, as the conductivity is a second rank tensor, and cubic symmetry is enough to make a second order tensor isotropic. Based on this result alone, the microstructure of this model does not reproduce the porous Vycor microstructure as well as does the reconstructed models. However, the cylinder model has only a single pore size, compared to the distribution of pore sizes available in the reconstructed models. Because of this fact, one might not expect it to describe Vycor as well as the reconstructed models.
The fluid permeabilities were really a test of pore size, since the tortuosity of the pore space was reasonably given by the reciprocal of the pore space formation factors. Reasonable agreement with experimental results were obtained, although experimental and model results could not be precisely matched because the surface area of the samples used in the experiments was not measured. If the length scale of the cylinder model is chosen so that the cylinder radius is 3.33 nm at x=0.355, with L = 9.4 nm, the cylinder model then has 173 m2/g of surface area. The permeability can easily be computed using the same techniques, and is found to be 0.57 · 10-19 m2 along one of the principal directions, in reasonable agreement with the range of experimental and model values found. However, the square opening between the cylinders along one of the principal directions has width L-2a = 2.7 nm. This size opening seems small, compared to experimental pore diameters of about 7-11 nm [8,15]. This combination of a too small "pore width" and a too large value of 1/F for the pore space combine to give a reasonable permeability along one of the principal directions. Since the Darcy permeability tensor is second rank, the cylinder model will also have an isotropic permeability tensor. However, if we assume that the cylinder model represents only the main skeleton of Vycor, and not the very fine pores of the "gel" , a better value of surface area might be 100 m2/g, as about one third of the surface area of Vycor is in the very smallest pores . In this case, we get an opening of about 5 nm in diameter, which is in better agreement with experimental measurements of the principal pore size in Vycor [8,15].
The backbone elastic moduli of the reconstructed systems were chosen by fitting to either the measured elastic moduli, the low RH shrinkage, or the high RH shrinkage. The backbone moduli that came from these experiments agreed well with each other, giving additional validity to this process. The model therefore did a good job of predicting these three quantities using an average backbone Young's modulus. However, the actual value of s* for Vycor is probably about 0.2, while the model gave a value of about 0.3. This implies that there is a subtle difference in microstructure between real Vycor and the model that causes this disagreement. The reasons that a microstructure has a certain value of s* is a subject of further research .
At intermediate RH values, the amount of water retained by the model was more than in experiment, causing some error. But also, the phenomenon of water-swelling of the solid backbone may have caused most of the disagreement in shrinkage strain at intermediate humidities (see Fig. 9), since the additional water in the model would tend to make the model shrinkage greater than the experimental value, not less. This phenomenon certainly made the experimental low RH shrinkage higher than was predicted, as the argon results on the 173 m2/g system  were very well predicted, and there was no reason to believe that the 129 m2/g material used by Amberg  would not be equally well predicted.
The analysis of the model results indicate that some of the properties of porous Vycor glass are sensitive to the surface area, namely fluid permeability, low RH shrinkage, and water adsorption and desorption. Properties like the solid thermal conductivity, the pore space conductivity, and the high RH shrinkage strain are mainly functions of porosity only, and are not very sensitive to pore surface area.
One difficulty in comparing model results to experimental results was that in many cases, the surface area of the porous Vycor glass was not measured. The other difficulty was that water was used as the fluid for the desorption/adsorption measurements, which brought in the solid backbone water-swelling effect, which was beyond the scope of our model. In areas that were comparable, however, the model compared well to experimental results. In light of this good comparison with porous Vycor glass, we feel that the reconstruction method offers a reasonable method for obtaining accurate 3-D models of porous materials from 2-D images.
Finally, in the future we feel that it would be very interesting if experimental measurements of the kind described in this paper (both kinds of formation factor, fluid permeability, elastic moduli, adsorption/desorption, and shrinkage) were all made on a single kind of porous Vycor glass, with a measured surface area, and using a gas like argon for the adsorption/desorption measurements, so that the effect of swelling of the solid backbone was not present. Then a complete comparison could be made with model predictions. Based on the results in this paper, we feel that the reconstructed models would continue to do quite well in such a more rigorous comparison.