Macropores calculations

Also shown are the corresponding curves calculated for the same system assuming a diffusion model in place of the linear rate expression. For intracrystalline diffusion k = 15Dq/v, whereas for macropore diffusion k = 15e /R ) Cq/q ), in accordance with the Glueckauf approximation (21). 

Pore. size and surface area distribution. Pore sizes and pore volume distributions may be calculated from the relative pressures at which pores are filled (in the adsorption mode) or emptied (in the desorption mode). Fig. 3.45 shows the pore size distribution of a commercial y-alumina. The distribution is very broad both meso- and macropores are present. In practice this is usually a desired situation a texture consisting of a network of large pores (main roads) and small pores (side roads) is ideal. 

While for macroporous structures the inner surface can be calculated from the geometry, meso and micro PS layers require other methods of measurement First evidence that some PS structures do approach the microporous size regime was provided by gas absorption techniques (Brunauer-Emmet-Teller gas desorption method, BET). Nitrogen desorption isotherms showed the smallest pore diameters and the largest internal surface to be present in PS grown on low doped p-type substrates. Depending on formation conditions, pore diameters close to, or in, the microporous regime are reported, while the internal surface was found to... 

In contrast to the micro- and mesoporous regimes, for which only a few empirical laws for the growth rate and porosity are available, the detailed pore geometry for macropore arrays in n-type silicon can be pre-calculated by a set of equations. This is possible because every pore tip is in a steady-state condition characterized by = JPS [Le9]. This condition enables us to draw conclusions about the porosi-... 

The steady-state condition (/ap=Jps) at the pore tip determines not only the pore diameter but also the pore growth rate. The rate rp of macropore growth can be calculated if the local current density at the pore tip is divided by the dissolution valence nv (number of charge carriers per dissolved silicon atom), the elementary charge e (1.602 xlO-19 C) and the atomic density of silicon Nsi (5xl022 cm-3) ... 

It should be emphasized that n.. and JPS, and therefore c and T, refer to the condition at the pore tip. The dissolution valence and the temperature can be assumed to be independent of pore depth. This is not the case for the HF concentration c. Because convection is negligible in macropores, the mass transport in the pore occurs only by diffusion. A linear decrease in HF concentration with depth and a parabolic growth law for the pores according to Pick s first law is therefore expected, as shown in Fig. 9.18 a. The concentration at the pore tip can be calculated from the concentration in the bulk of the electrolyte c, the pore length l, the diffusion coefficient DHf (Section 1.4) and the flow of HF molecules FHf. which is proportional to the current density at the pore tip ... 

Fig. 9.17 Calculated growth rates of macropores for nv=2 and nv=4 (lines) and experimentally obtained initial macropore etch rates rp (squares) versus the HF concentration Chf of the electrolyte (J/JPS = 0.2S, 1015 crrT3 n-type, RT, square pattern). After [Le9],...

To calculate the optical transmittance of a single macropore, it can be modeled by two opaque screens with an aperture of diameter d at a distance l, as shown in the inset of Fig. 10.13. For such an arrangement the ratio nt of transmitted power to incident power of a wavelength X can be approximated by... 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

Still, this theory is over-simplified, and holds only for a limited part of the sorption isotherm, which is usually the case for relative pressures between 0.05-0.30, and the presence of point B (Fig. 1.14). Thus, isotherms of Types II (macroporous polymer supports) and IV (mesoporous polymer supports), but not Type I and III, are those amenable to BET analysis [21, 80]. Attention should also be paid to the constant C, which is exponentially related to the enthalpy of adsorption of the first layer. A negative or high value of C exceeding 200-300, is likely to indicate the presence of micropores and the calculated surface area should be questioned since the BFT theory would not be applicable [79, 80]. 

Calculation of Schematic Phase Diagrams for the Preparation of Solvent-Modified and Macroporous Thermosets via CIPS. . Guidehnes for the Preparation of Solvent-Modified and Macroporous Thermosets via CIPS. ... 

Calculation of Stress Distribution in Macroporous Epoxies Fracture Toughness of Solvent-Modified and Macroporous Epoxies Prepared via CIPS. 

Calculation of Schematic Phase Diagrams for the Preparation of Solvent-Modified and Macroporous Thermosets via CIPS... 

Interestingly, the values for volume fraction and IPD calculated via the classical method or via the density values differ significantly. These data are compared in Table 2 for the macroporous epoxies prepared with 6 and 7.5 wt % hexane displaying a narrow size distribution. A reasonable agreement for the density values exists, but the tendency for the IPD values is quite different. These two single values are, however, not sufficient to draw any conclusion concerning the error introduced by the simplification that t=d. 

The results of image analysis of macroporous epoxies showing a narrow and bimodal pore size distribution are summarized in Table 3. The volume fraction, ( ), is always calculated from density measurements. The validity of the data obtained with digital image analysis is of utmost importance in order to draw correct conclusions concerning the structure-property relationships. 

Very interestingly, the IPD is found to show practically no or very little dependence on the volume fraction and only depends on the reaction rate (Fig. 35). Here the IPD is calculated from the mean diameter taken from image analysis and the volume fraction, which has been determined directly from the porosity of these macroporous thermosets. Both values could be determined experimentally with high accuracy and no simplifications are needed for the calculations. 

Fond et al. [84] developed a numerical procedure to simulate a random distribution of voids in a definite volume. These simulations are limited with respect to a minimum distance between the pores equal to their radius. The detailed mathematical procedure to realize this simulation and to calculate the stress distribution by superposition of mechanical fields is described in [173] for rubber toughened systems and in [84] for macroporous epoxies. A typical result for the simulation of a three-dimensional void distribution is shown in Fig. 40, where a cube is subjected to uniaxial tension. The presence of voids induces stress concentrations which interact and it becomes possible to calculate the appearance of plasticity based on a von Mises stress criterion. 

As for the samples prepared without catalyst, the ability for energy absorption after crack propagation decreases strongly as the solvent is removed. This is reflected in Fig. 49, where the load displacement curves of solvent-modifled, semi-porous, and macroporous epoxies prepared with initially 22 wt % cyclohexane (( )=18.5%) are shown. The crack length is the same in all three cases. Therefore the decrease in maximum load is directly proportional to the decrease in Kj. It is also clearly seen that the fracture behavior changes drastically and that the surface under the load-displacement curve, which is used to calculate the fracture energy is signiflcantly lowered. 

The quantitative results of fracture energy, which are calculated from the total area under load-displacement curves, are presented in Fig. 50. It becomes obvious, that a brittle-tough transition exists at a volume fraction of around 10%. This brittle-tough transition is observed for solvent-modifled as well as semi-porous epoxies. A brittle behavior is observed in macroporous epoxies after complete solvent removal, thus giving low fracture energies similar to the neat epoxy for each porosity. 

Fig. 52. Critical stress intensity factor, Kj, and critical stress energy release rate, of solvent-modified, semi-porous, and macroporous epoxies prepared via kinetically controlled Cl PS with 1 wt % catalyst calculated from SENB tests...

The dielectric constant of the pure cyanurate network under dry nitrogen atmosphere at 20 °C is 3.0 (at 1 MHz). For the macroporous cyanurate networks, the dielectric constant decreases with the porosity as shown in Fig. 57, where the solid and dotted lines represent experimental dielectric results together with the prediction of the dielectric constant from Maxwell-Garnett theory (MGT) [189]. The small discrepancies between experimental results and MGT might be due to the error in estimated porosities, which are calculated from the density of the matrix material and cyclohexane assuming that the entire amount of cyclohexane is involved in the phase separation. It is supposed that a small level of miscibility after phase separation would result in closer agreement of dielectric constants measured and predicted. Dielectric constant values as low as 2.5 are measured for macroporous cyanurates prepared with 20 wt % cyclohexane. 

Different theoretical models applied to this pore size distribution can give relatively large variations of the calculated effective diffusivity value (DelT). The most commonly used approximations are (i) random-pore model (Wakao and Smith, 1962) using two characteristic transport pores (micropores ji and macropores M)... 

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