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A two-dimensional binary (pore/solid) image of porous Vycor glass was taken from Refs.  and . The image provided in the reference was digitized and thresholded, to obtain a binary image with a porosity fraction of 0.31, in agreement with the original measured value . The autocorrelation function measured for this image was then used to reconstruct 3-D porous Vycor microstructures, as was discussed earlier. Four different surface area (units of square meters of surface per gram of solid) systems were generated. A 70 m2/g system was generated to compare with the measured surface area of the TEM image, a 100 m2/g system was generated to compare with the actual measured nitrogen BET surface area of this same material, a 129 m2/g system was designed to compare with the material used by Amberg  for shrinkage measurements, and a 173 m2/g model was made to match with the material used by Yates for his low RH expansion measurements . Figure 4 shows a slice from each of these four systems, all taken at the same point in the material. The top left is the 70 m2/g model. The surface area then increases clockwise among the four pictures. It is interesting to note that the typical pore size appears to decrease as the surface area increases, with fine scale roughness showing up in the pores for the higher surface area material, reminiscent of what has been referred to as the "gel" porosity in porous Vycor glass .
Figure 4: Showing the same slice of different surface area reconstructed porous Vycor glass structures (clockwise from upper left: 70, 100, 129, and 173 m2/g). Solid is dark gray, white is pore.
These generated 3-D microstructures were then tested in several ways, in order to check how well their properties compared with real porous Vycor. These computations included: specific surface area, TEM generation, electrical conductivity, thermal conductivity, fluid permeability, and elastic moduli.
One should note that in computing the surface area of a digital image, there is a correction factor needed. Because the surface area is made up of pixel faces oriented in the x,y, and z directions, the actual surface area is overcounted by a factor of 3/2. Consider an isolated spherical pore with radius r. Project the pixel faces onto the ±x, ±y, and the ±z directions. One gets six circles, each with area πr2, for a total surface area of 6πr2, instead of the true surface area of 4πr2. Therefore the digital surface area, computed by counting pixel faces, was corrected downwards by a factor of 2/3.
The surface area of the reconstructed system per total volume was computed to be 70 m2/g in comparison to the value of 74 m2/g computed directly on the 2-D image . For this same material, a surface area of 100 m2/g was measured using BET analysis of nitrogen sorption , which probes the surface structure at a higher resolution than that of the transmission electron microscope image. The resolution of the TEM image was about 1 nm/pixel, so pores smaller than about 1 nm will be missing from the image, and pores up to a few nanometers in size will be distorted from their original shape, since they will be represented by only a few pixels . These pores will tend to keep water in them down to lower relative humidities. This will cause errors in the model computations of both water sorption and shrinkage , as will be discussed further below.
Having a 3-D digital model image available makes it simple to approximately simulate the appearance of a TEM micrograph. A slice of the appropriate thickness is taken from the model, and then mapped through the thickness into a 2-D image, where the gray level of each pixel in the mapped image depends on the solid density through the thickness of the microstructure at the location of the pixel of interest . Figure 5 shows a comparison between the real (a) and the simulated (b) TEM images. The simulated image has the same surface area as does the experimental TEM image. There is a definite qualitative resemblance between the two images, which, by itself, does not of course tell us very much about the accuracy of the reconstruction algorithm.
Figure 5(a): Real TEM image of porous Vycor glass microstructure (40 nm thick), adapted from Ref. . The scale bar is 1000 Angstroms = 100 nm.
Figure 5(b): Simulated TEM image of reconstructed 3-D porous Vycor glass
(40 nm thick, 500 nm by 500 nm).
Electrical, thermal, and fluid transport coefficients
Table 1 gives the results of various transport computations on the four systems. Data for a fifth system, listed in the next to the last row of the table, is included for comparison. This model's pore space was made up of three different sizes of non-overlapping spherical pores, such that the porosity (0.35) and the surface area (180 m2/g) was similar to the 173 m2/g porous Vycor glass models. The last row of data, for the cylinder model, will be discussed later.
|Model||Surface area (m2)||Porosity||Fp||Fs||Permeability (10-19 m2)||s||E/Es|
|Vycor||100||0.31||16.1||2.21||0.8 ± 0.2||0.1||0.283||0.218|
|Vycor||129||0.31||15.8||2.18||0.42 ± 0.06||0.1||0.293||0.206|
|Vycor||173||0.31||16.2||2.24||0.22 ± 0.02||0.1||0.302||0.197|
Table 1: Computed physical property data for various porous models
Electrical conductivity is examined first. Glass itself is an electrical insulator. If the pores of the porous Vycor were filled with a conductor, however, then the entire sample would be a conductor, since the pore space is percolated. The formation factor Fp is defined as  the ratio of the pure conducting phase conductivity to the bulk sample conductivity, so that Fp is always bigger than 1. This parameter can be calculated using a finite element scheme similar to that described for the elastic properties, or a finite difference scheme [44,50, 51]. Values of Fp ranging from 15.8 to 16.4 were found for the various surface area porous Vycor models using a finite difference method. There was very little difference between the x,y, and z directions. An experimental result has been previously obtained for porous Vycor filled with indium metal . A resistivity of 150 ohm-cm was measured at room temperature, while the room temperature resistivity of pure indium is 8.37 ohm-cm. That gives a formation factor of 17.9, so that the model value is in good agreement with the measured one. Other experimental results for Fp by the same authors give F = 20 , so there can be some variation from system to system. The experimental papers did not give a measured value of surface area, but as shown above, the value of Fs does not greatly depend on surface area and pore size. The three-sphere model has a disconnected pore space by construction, and so has an infinite value of Fp.
If the pore phase of porous Vycor glass is considered to be empty, then the thermal conductivity will be dominated by heat conduction through the solid phase. A formation factor Fs can be defined analogously for this case, being defined as the solid backbone thermal conductivity divided by the porous Vycor thermal conductivity. This can be solved for in an analogous way as the electrical conductivity, with resulting formation factors Fs ranging from 2.06 to 2.24, again with very little difference between the three principal directions. A measured value of the formation factor is Fs = 2.1 , so again the model prediction agrees well with experiment. As for the pore space conductivity, the value of the thermal conductivity Fs does not seem to greatly depend on the surface area. Supporting this point is the fact that the three-sphere model has a value of Fs for the solid of 2.12, very close to experiment and to the other model results.
As a check on how well the pore space is simulated, the electrical conductivity of the pores and the thermal conductivity of the backbone were not very sensitive to pore space details, as has been seen. The permeability, however, should be much more sensitive to pore size, because of the zero velocity boundary condition at all fluid:solid interfaces . The pore space of porous Vycor glass is continuous, so that if the pores are saturated with a fluid, and a pressure gradient is applied, Darcy flow will be observed in the bulk and the fluid permeability can be determined. Using previously developed numerical techniques [50,55], and neglecting the effect of molecular size, we can compute the fluid permeability, averaged over direction. In Table 1, the permeability can be seen to range from 1.8 · 10−19 m2 for the lowest surface area, and therefore highest hydraulic radius, down to 0.22 · 10−19 m2 for the highest surface area and therefore lowest hydraulic radius material. The standard deviations listed in the table are for the three different principal directions, so that these samples were rather anisotropic as far as permeability was concerned, which is a finite size effect. Experimental values in the literature, using various fluids and techniques, range from 0.4 to 1.0 · 10-19 m2 [56,57,58], so again in general there is good agreement between the range of model permeabilities and the range of experimental permeabilities. Some of the differences in experimental permeabilities were probably due to molecular size and viscosity differences in the fluids used, but some were on different systems. Unfortunately none of the references that we found for the permeability of porous Vycor glass also measured the surface area, so we cannot directly compare equivalent experiments and computations. The three-sphere model has zero permeability, because of its disconnected pore space.
For the porous Vycor glass models, clearly the surface area, and therefore the hydraulic radius, is setting the scale for the pore size, since we can think of the permeability as proportional to the inverse of the electrical formation factor times a typical pore size squared . Since the electrical formation factor is roughly constant for the different surface area materials, the pore size must be changing with the surface area. If this is true, then the square root of the permeability ratio should scale as the inverse of the surface area ratio. For the 70 m2/g and 173 m2/g models, the square root of the permeability ratio is 2.9, while the inverse of the surface area ratio is 2.5, so the above statement is approximately true. Comparing the other systems with each other leads to similar results.
The elastic properties of the porous Vycor model were computed using the finite element code described above. For a given assignment of bulk and shear moduli for the solid phase, the program computes the porous material moduli with reasonable accuracy. The main question is then--what moduli to use for the solid backbone? One can not just use the moduli of fused silica, which is what porous Vycor glass becomes after being fully sintered. Scherer  has shown that the moduli of porous Vycor changes as it is heat-treated, even without any changes in the porosity. Apparently this heat treatment drives off hydroxyl groups that tend to soften the solid backbone structure. The solid backbone of as-received porous Vycor glass therefore can have a much lower Young's modulus than that of fused silica . The effective Poisson's ratio for porous Vycor glass was found to range from 0.15 to 0.19, however, with little change after heat treatment.
Scherer's data can be used to determine the solid backbone moduli to be used in the model in the following way. The porous Vycor glass that Scherer used had a surface area of 149 m2/g, so the 173 m2/g model was compared to the experimental results. For heat treatment roughly equivalent to that of Yates , Scherer measured a Young's modulus E of 19.6 GPa. The model computation gave E/Es = 0.3, so the solid backbone modulus Es would be 65 GPa. (The elastic data for all the systems is in Table 1.)
A range of values for s, the Poisson's ratio of the solid frame, was chosen, and the effective Poisson's ratio, , computed. It was found, in agreement with previous work , that there was a critical value of s, denoted s*, such that for s < s*, > s, and for s > s*, < s. This behavior is ubiquitous in two-phase materials. It can be shown exactly in 2-D , and has been seen in 3-D [39,61], for different kinds of porous microstructures. The particular value of s* found of course depends on the microstructure. Figure 6 shows the numerical results for the Vycor 173 m2/g model, along with the results of two effective medium theories [62,63,64] at the same solid volume fraction of 0.69. The effective medium theory curves were added to show that this behavior is ubiquitous in porous materials . The actual vs. s curves are quite linear, and have a slope less than one. The two effective medium theories have an exact fixed point of s* = 0.2.,
Figure 6: Showing the vs. s curves for the 173 m2/g Vycor model and two effective medium theories [62,63]. The dashed line is the line of equality, = s.
The slope of the vs. curves will depend on porosity. It is easy to see why this is so. If the porosity were zero, then the vs. s curve would be exactly linear, passing through the origin with a slope of unity, since for any value of s, one must have = s. When the porosity becomes non-zero, consider the physically allowed limits for the Poisson's ratio of the solid frame, s = -1 and 0.5 . At the s = -1 endpoint, the only way that the porous solid Poisson's ratio can change upon introducing porosity is to increase, since cannot be less than −1. At the s = 0.5 endpoint, the porous solid Poisson's ratio must decrease when the porosity becomes non-zero, since must be less than 0.5. Therefore the vs. s curve goes above the = s line at negative values of s, and goes below the = s line at values of s near 0.5. Somewhere in between the physical limits, the vs. s curve must cross the = s line, and therefore the behavior seen here comes about. More analysis of this behavior is given in Ref. .
For porous solids made up of non-overlapping spherical pores, where the pores are monosize and the porosity is small, a few percent at most, one can use the exact dilute limit [44,45] for spherical pores and directly see how porosity affects the slope of the vs. s curve:
For the reconstructed Vycor models, s* was about 0.3, so that in order to achieve a value of between 0.15 and 0.19, the value s = 0.1 was chosen, which gave = 0.2. This value of s* is probably higher than in reality, for the following reason. The elastic moduli of fused quartz, which presumably is what the backbone of Vycor approaches as it is heat-treated , are Es = 72 GPa and s = 0.16 . Using s = 0.16 in the 173 m2/g model would give too high of a value for of the porous solid. If, however, the value of s* for the 173 m2/g model were lower, say about 0.2 or so, then a backbone value of s = 0.16 would give about the correct value of . Looking at Fig. 6, if the Vycor data points were each lowered by about 0.05, then they would agree closely with the Berryman EMT, and would give a value of s* of about 0.2. Also, looking at Table 1, one notices that the porous solid Poisson ratio decreased with increased surface area, or increasing fine structure (see Fig. 4). Since all of the models are built from the TEM picture, which only has 1 nm resolution, some of the finest Vycor structure is missing from even the 173 m2/g model. If this fine structure could be included in the model, the value of obtained would then likely be reduced, in analogy with Table 1, and so agree better with the experimental value.
Comparison with the measured Young's modulus of the non-heat treated porous Vycor glass gave a solid backbone Young's modulus of 49 GPa. To show the effect of microstructure, the next to the last row of Table 1 gives elastic data for the non-overlapping spherical pore system previously mentioned. The value of E/Es is significantly higher, 0.435, but the effective Poisson's ratio increased to 0.143 from the backbone value of 0.1, because the value of s* for this system is 0.22 .
For the heat treatment carried out in Yates' work , Scherer  estimated that the backbone Young's modulus would be about Es = 50 GPa. For Amberg's material, which was non-heat-treated, the backbone Young's modulus was estimated to be about Es = 38 GPa. The cylinder model mentioned previously was used to determine these values. However, this model is anisotropic, since it is inherently cubic. There are therefore three independent elastic moduli, C11, C12, and C44, not just a Young's modulus and Poisson's ratio, as reported by Scherer [68,69]. Scherer predicted the Young's modulus and Poisson's ratio in the three principal directions, which are not the same as those determined by carrying out an isotropic average of the elastic moduli tensor. Because this cubic model is stiffer in the three principal directions, along the cylinders, the Young's modulus of the porous Vycor glass is over-estimated by this model. Table 1 shows that E/Es = 0.44 for this model, in the principal directions and at a porosity of 0.32, in good agreement with the approximate analytical value predicted  but significantly stiffer than the reconstructed model. That is why the backbone modulus predictions of Scherer  are smaller than in the present work. Also, the Poisson's ratio in any of the three principal directions for the cylinder model is predicted to always be smaller than the solid backbone Poisson's ratio , (which checks with computed results). This is not the case for an isotropic single solid phase material , for the reasons given above. Rather, one would expect a critical value of s, as has been demonstrated for the reconstructed models and for other microstructures [39,60,61].
Low relative humidity shrinkage
At very low relative humidities, where the material has no capillary condensed water, length change (shrinkage on desorption and expansion on adsorption) depends mainly on changes in the specific surface energy of the adsorbed water layer. Comparison to experiment in this regime then does not depend on details of the absorption/desorption algorithm, but only on the elastic properties and surface area of the model.
Yates  has published data on the expansion due to the adsorption of various gases on the pore surfaces in porous Vycor. Based on Scherer's analysis  of this data, for argon, below capillary condensation, the expansion strain could be expressed as 0.00545 (m2/J) Δ γ. In other words, a change of 1 J/m2 in the specific surface energy would produce a strain of 0.00545, taking totally dry Vycor as the baseline of zero strain. Amberg  has done the same measurement for the adsorption of water vapor. The strain coefficient was found to be about 0.01 (m2/J) . Hiller, for water absorption in a similar material, also found a strain coefficient of about 0.01 (m2/J) . The porous Vycor glass used by Amberg had a lower surface area than Yates, 129 vs. 173 m2/g, which would tend to decrease the strain slope. However, the glass used by Amberg was non-heat-treated, so it would have a lower backbone modulus. These two effects roughly cancel out, so that both glasses should have had approximately the same strain slope . This relative increase for water vapor vs. argon has then been attributed by Scherer  to the absorption of water into the glass backbone, due to the many hydroxyl groups present in the glass before heat treatment at 500C, which causes swelling of the solid backbone, and hence swelling of the porous composite. Since this mechanism is not present in our computations, we will compare to Yates' value of 0.00545 (m2/J), Δ γ. or a strain coefficient of 0.00545. The measured nitrogen BET surface area of Yates material was 173 m2/g, so this reconstructed model was used for computing the surface energy-driven expansion/shrinkage. For the 173 m2/g model, a backbone value of Es = 59.1 GPa, with s = 0.1, gives a strain coefficient of -0.00545, in exact agreement with experiment and with Scherer's prediction. The surface energy-driven shrinkage of the cylinder model also agrees well with experiment, using the lower values for the backbone modulus . Scherer's prediction, with his values of the backbone modulus, also agrees well with the model prediction, when Es = 44.9 GPa, for the strain coefficient of the 129m2/g material: -0.0054 (Scherer) vs. -0.0052 (model).
High relative humidity shrinkage
Another limit that can be critically examined is the high relative humidity limit. In this limit, as the RH value decreases from 1, the sample stays essentially fully saturated, but the water is placed under a negative pressure (hydrostatic tension). Therefore Mackenzie's formula, eq. (10), applies, and is exact. Amberg has data for 94% and 82% RH, at which, according to the model and experiment (see Fig. 7), the pores are still filled. The experimental shrinkage strain increase between these two RH values is 0.000530. Using Mackenzie's formula applied to the model, with Es = 40.8 GPa and s = 0.1, makes the model agree perfectly with the experimental value.
Best values of backbone Young's modulus
Using these three experimental data points, the measured elastic moduli and the low and high RH shrinkage measurements, one can obtain a value for the backbone Young's modulus that makes the model agree well with these experiments. For the 129m2/g, non-heat-treated model, the elastic moduli imply Es = 49 GPa , while the high relative humidity shrinkage measurement implies that Es = 40.8 GPa. We then take the average of them to use, 44.9 GPa, which gives an accuracy of -8.4 % in the modulus and 10% with respect to the high relative humidity data point. For the 173 m2/g model, a value of Es = 65 GPa is necessary for agreement with the elastic modulus, while a value of Es = 59.1 GPa is implied by the low RH strain slope. Therefore a value of 62 GPa is used, the average of these two values, which gives an error of -4.6% in modulus and 4.9% in the strain coefficient. The Young's modulus ratio for porous Vycor glass, between the non-heat-treated and heat-treated 173 m2/g models, according to these choices of backbone moduli, is then 1.42, in good agreement with the measured ratio of moduli, 19.6/14.7 = 1.33 for a 149 m2/g system of similar porosity .
To infer solid backbone moduli from measured porous material moduli requires, in most cases, a model of how the pores affect the measured moduli. The above choices are the best for our reconstructed models. There is no reason to prefer the values of backbone moduli inferred in Ref. , since the cylinder model used was not isotropic. Its success in predicting the low RH strain coefficient was probably due to the fact that the backbone moduli were picked so that the composite, effective moduli were the same as experiment. Also, the shrinkage strain tensor is second order, and so is isotropic for a cubic system like the cylinder model.