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The phenomenon of drying shrinkage in porous materials is relevant to a variety of problems of technological interest such as the shrinkage [1] and cracking [2] of concrete and the drying of gels and aerogels [3]. In these porous materials, pore diameters are often such that capillary condensation will occur over some range of partial pressure of the gas phase of the adsorbent considered. The saturation, or fraction of the pore space filled with liquid phase, of these materials is therefore a strong function of the external partial pressure of the gas phase. The capillary tension induced by the condensed fluid will generate stresses at all fluid- solid interfaces, resulting in some deformation of the solid phase and hence an overall shrinkage in the composite. In addition, changes in specific surface free energy [4,5] in adsorbed fluid layers can also contribute to the overall shrinkage, especially at lower partial pressures. In this paper, we focus on the fluid being water, although the techniques described are general. The partial pressure of water vapor, divided by the saturated water vapor pressure at the same temperature, is usually called the relative humidity (RH).
While many measurements of shrinkage have been made, a direct linkage between microstructure and shrinkage is still lacking. Mackenzie [6] has presented a general analytical method for estimating the dilatation of porous materials based on their elastic properties and the amount of water-filled porosity at a given relative humidity. This approach is exact for a saturated porous material, but only approximate for partially saturated materials. In addition, the general elastic dynamics of unsaturated and fluid-saturated porous materials have been extensively discussed by Biot [7].
In this paper, a computer modelling approach is presented in which the shrinkage behavior of a three-dimensional microstructure is directly computed. Beginning with a discrete three- dimensional representation of microstructure, an algorithm based on the Kelvin-Laplace equation is used to estimate the equilibrium location of all capillary-condensed water at some fixed relative humidity. In addition, the surface energy and thickness of the water layer adsorbed on all exposed (no capillary condensation) solid surfaces is estimated. Finite element analysis is then employed to compute the shrinkage of the three-dimensional composite, which is caused by the capillary tension present in the condensed water and the surface energy changes in the adsorbed water. It should be noted that the techniques developed in this paper do not deal with the dynamics of the shrinkage process or flow of the water during shrinkage, but only estimate the final equilibrium value of shrinkage. Thus, for example, it is implicitly assumed that the shrinkage does not alter the microstructure enough to significantly change the locations of the capillary-condensed water at the resolution being considered. For the magnitude of the shrinkage strains considered, 0.001 or less, this is a reasonable assumption. Also, we presume that the interior of the material is always connected to the exterior, so that all parts of the pore space are ultimately accessible.
Other physical properties besides shrinkage depend on microstructure, too, of course. One way to test the accuracy of the reconstructed microstructures is to compute their physical properties and compare to experimental measurements on equivalent systems. By comparing to experimental data for real materials, where possible, we hope to indicate precisely the advantages and limitations of the 3-D media that are reconstructed for 2-D images using our algorithm.
There are several reasons why porous Vycor has been chosen as a material upon which to test these techniques. First, the total porosity and pore surface area of porous Vycor glass are similar to those of hardened cement paste, which is our ultimate research interest [1], and the shrinkage strains found in this material, of the order of 0.001, match the approximate tensile fracture strain [2] of cement-based materials. More important, however, is that many of the properties of porous Vycor glass are readily available [8], as are experimental measurements of porous Vycor shrinkage vs. external relative humidity [9]. Also, the drying shrinkage of porous Vycor, in both low [5,8] and high [9] partial pressure of absorbent regimes, has been shown to be well-described by linear elastic equations [5,8,9], so that the finite element analysis can also be simplified to be linear elastic only. This is unfortunately not the case in hardened cement paste [1], which is strongly viscoelastic. Because of all of the above reasons, porous Vycor glass has been chosen to test our reconstruction algorithm [10] and demonstrate our new shrinkage algorithm.