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Water sorption and shrinkage results

Figure 7 provides a comparison between the water sorption curves determined using the spherical interface approximation on the 129 m²/g reconstructed model, and those measured experimentally. The surface area of the experimental system was not characterized, which may certainly lead to some disagreement between model and experimental results. Also, there is no temperature in the model algorithm, which could also lead to some disagreement with experiment, since the shape of the hysteresis loop in adsorption-desorption isotherms does depend on temperature [33] The hysteresis in the model results entirely from pore network percolation effects, and so is independent of temperature. Of course the Kelvin-Laplace depends on temperature somewhat, so the temperature will affect the isotherms slightly even in the model. For desorption, the agreement between model and experimental results appears to be reasonably good down to about 80% RH. After that point, the model appears to retain more water in it than does the experiment. However, there is no experimental data between 75% and 60% RH. The breakthrough point could be as far down as 70%, which would give a region where the model has less water than experiment (between 70 and 75% RH), followed by a region where the model retains more water than experiment (20 to 70% RH).

Figure 7: Simulated (129 m²/g) and experimental sorption curves for the porous Vycor glass (A=adsorption, D=desorption).

For adsorption, starting at low RH, the agreement with experiment is good up to about 20% RH, and for RH values larger than 80%. Below 20% RH, the amount of water retained is determined by the thickness of the adsorbed film and the surface area of the material. It appears that these match well between model and experiment. At intermediate humidities, the model picks up capillary water steadily, while the amount of water in the experimental curve stays low and then rises rapidly at about 60% RH. By differentiating the desorption isotherm, an estimated pore size distribution can be obtained. For the simulation and experimental data, a fairly broad peak with a peak value at a pore radius of about 3-5 nm was obtained. This value agrees fairly well with the values of 3.5 nm measured experimentally by Levitz et al. using nitrogen sorption [15] and 5.7 nm measured using nitrogen sorption by Scherer [8].

The model results in Fig. 7 were obtained using a 200³ digital image model. It is probable that using a higher resolution system would improve results, but how much we do not know, as the available computer power is not adequate to go much higher in resolution. In some respects, there is little point in going much higher in resolution, as we are limited by the experimental resolution of the original TEM image, about 1 nm / pixel. However, the adsorption-desorption algorithm uses digitized spheres in order to find the location of capillary-condensed water. The diameter of these spheres goes down to three pixels, since each pixel is a nanometer in length. A sphere that is three pixels in diameter does not look terribly spherical. Having more resolution available at these small diameters, or more pixels per nanometer, would allow better spherical shapes to be used to compute the location of capillary-condensed water at intermediate RH values. Also, the true shape of pores that are 1-5 nm in size would also be resolved better if there were more than one pixel per nanometer. Both these effects might improve the agreement with experiment in the intermediate RH regime. Since the surface area of the experimental sample was not measured, there could also be some differences between model and experiment due to different surface areas. However, at low RH, the model agrees quite well with experiment. In this regime, the water content is approximately just the surface area times the water layer thickness, so the experimental surface area was probably similar to that of the model, 129 m²/g, assuming of course that the layer thickness used was also accurate.

The known inaccuracy of the Kelvin-Laplace equation at small pore diameters (5-10 nm) would certainly seem to contribute to the differences between model and experiment in the intermediate RH regime. However, the hysteresis loop in the adsorption-desorption isotherms, which makes up the intermediate RH regime, may result mostly from pore network percolation effects [30,33]. And in fact, untangling single-pore adsorption-desorption effects from pore network effects is a difficult question [30]. So it is impossible, with any degree of certainty, to determine how much of the disagreement between model and experiment in the intermediate RH regime is due to the inaccuracy of the Kelvin-Laplace equation for small pores. It is also known that the percolation aspects of reconstructed microstructures can differ from the real microstructures [10]. This is because 2-D images have no 3-D percolation information in them, so that this information, in contrast to quantities like surface area and porosity, are put in entirely by the reconstruction process. Since hysteresis is sensitive to the pore space percolation, there will also be some error in the intermediate RH regime arising from this cause. For globally averaged quantities like electrical conductivity, fluid permeability, and elastic moduli, the model microstructure gave values close to experiment. A more local quantity like how much water is capillary condensed at a given relative humidity is clearly much more sensitive to the details of the pore space.

Figure 8 shows four partially saturated slices of the 129 m²/g model, all taken at the same position, but at different RH values, starting with RH = 0.494 in the upper left, then going clockwise to higher RH values (RH = 0.494, 0.656, 0.740, 0.791). The slices are taken from systems undergoing adsorption. White is capillary condensed water, gray is empty pore space, and black is solid material. It is clear that water is added first to the smaller pores, and then to progressively larger pores as the relative humidity is increased. In 3-D, using the adsorption algorithm described earlier, the menisci are roughly spherical, so their 2-D cross-sections are roughly circular.

Figure 8: Same slices of 129 m²/g model showing partially saturated conditions at different relative humidities (clockwise from upper left: RH = 0.494, 0.656, 0.740, and 0.791) during adsorption. Black = solid, light gray = empty pore, white = liquid water.

Figure 9 provides a plot of the computed shrinkage at general relative humidities, showing the experimental data of Amberg and McIntosh [9] for a 129 m²/g, non-heat-treated system. All results have been normalized to have zero shrinkage at 94% relative humidity, the maximum humidity for which experimental results were recorded [9]. At partial and full saturations, the surface energy was only put on the part of the solid backbone that was free from capillary-condensed water, since this term goes to zero for a thick capillary-condensed water layer. Very good agreement is demonstrated between model computation and experimental measurement from 94% RH down to about 80% RH. After that, the experiment stays higher than the model. Since the model, as can be seen in Fig. 7, retains more water than does the experimental desorption at intermediate relative humidities, it would seem that the model should go above the experiment, not below. We attribute this to the same phenomenon that gives a higher strain coefficient at low RH, that of water absorption into the solid backbone and thus swelling of the backbone. As water leaves the pore space it leaves areas of the backbone dry and able to lose water into the empty pore space. This will cause more shrinkage than due to capillary forces alone, and ultimately result in a higher, by about a factor of 2, strain coefficient at low RH. There could also be some differences between model and experiment due to different surface areas.

Figure 9: Experimental and model shrinkage for porous Vycor glass (129 m²/g) as a function of relative humidity.

The results of the model do tend to confirm a previously proposed explanation [9] for the observation of a maximum in the shrinkage vs. relative humidity curves for porous Vycor. Basically, capillary induced stresses are being replaced by stresses of a smaller magnitude caused by changes in the specific surface energy. Depending on the relative humidity at which the majority of the capillary pores empty, the new surface energy stresses may or may not exceed the lost capillary-induced stresses. As shown in Fig. 2, the surface energy contribution will be much larger if the pores empty at a low RH (less then 50%) then if they empty at a higher one (greater than 60% as in this study), since the specific surface energy is higher at lower RH values. Thus, the observation of a local maximum in the shrinkage curves will depend on the pore structure of the material as well as its surface energy characteristics. The model also clearly indicates how strongly capillary-induced shrinkage is dependent on the amount of capillary water in the microstructure, implying that sorption isotherms are critical measurements for characterizing the shrinkage behavior of porous materials.