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5.3. Results and discussion 71Prediction of gel timesThe mass transfer rate and hydrolysis coefficient given in the previous section were usedagain. The exponent ν was set to 3/4, in agreement with Li and McCoy [111]. Thefractal dimension d f was always 2.2 [117]. The size of the colloids in the initial stageof the condensation process, R A , was varied as a function of pH and temperature (seeTable 5.2). The condensation coefficients κ were varied in order to reflect the difference ingelation rates, as observed in the experiments. The critical transition time t c was obtainedby fitting the profiles of ¯Mw (not shown) with Eq. 5.19 and fixing the exponent γ to 2.7[111]. The resulting values for t c are shown in the left-hand graph of Figure 5.10 for eachsystem as a function of κ. It can be observed that t c decreases with increasing κ. Thecalculated points in Figure 5.10 were fitted with the following power law:t c = cκ −β , (5.32)where c and β are constants. For each set the fit is adequate in the range of κ between20 and 1000 h −1 .In Table 5.2 the values for κ are listed that resulted in a satisfactory match between theexperimentally obtained gelation time t expg (from the NMR experiments) and the criticaltransition time t c . It is noted that the base-catalyzed systems in the experiments did notgel macroscopically. For these systems κ was chosen relatively high, in order to reflectthe fast condensation rate at a pH around 9 (see Figure 5.3). Next, given the specifiedvalues for κ, the initial concentration of TMOS, n 0 , was varied and the critical times t cwere extracted. The results are shown in the right-hand graph of Figure 5.10. Again, thepoints were adequately fitted with the power law in Eq. 5.32. It is noted that for eachsystem α T and k were fixed and independent of n 0 , which is a simplification. It followsthat an increase in the initial TMOS concentration by a factor 2 results in a reductionof the gel time by a factor between 1.3 and 2.1. This is consistent with the gel timesobtained in the experiments (see Chapter 4).Besides the critical behavior of ¯Mw the critical growth of φ was considered. Thetime, t g , at which φ ≈ 1 was extracted from the calculated data. Since φ increasesmonotonically with ¯M w it diverges like ¯M w close to t c . The obtained times t g are slightlylower than t c for each system (see Table 5.2). The magnitudes of R A had a negligibleeffect on the transition times t g . Despite the lack of macroscopic gelation (percolation) inthe experimental base-catalyzed systems, the transition (φ > 1) did occur in the modelsystems, since φ always diverged.Prediction of T 2T 2 was calculated for all model systems given the mass transfer rates, hydrolysis rates andcondensation coefficients in Table 5.2. The volume fraction of the total hydrolyzed silicacontent φ m was readily calculated using Eqs. 5.23 and 5.24. We used the upper-limitapproximation of the internal surface area A according to Eq. 5.26. The monomeric sizea is based on the density and molar weight of silica, yielding a ≈ 2.3 Å. The fractionf s then follows from Eq. 5.31 together with the choice λ = 1.5 Å. In these calculationsf s grows from zero to about 0.07. The final internal surface area would correspond to
72 5. Cross-linking of silica with a time-dependent monomer source0.70.6no bufferpH = 2.95pH = 9.480.5T 2[s]0.40.30.20.100 5 10 15 20 25 30time [h]Fig. 5.11: Calculated profiles of T 2 for the model systems. The correlation time τ s equals100×D −1R with a maximum of 2 ns. The thick lines indicate the systems at 50 ◦ C,the thin lines indicate the systems at 25 ◦ C.a value of about 6700 m 2 g −1 , which is rather high. Typical experimental values of thesurface area of aero-gels, with which the dilute wet gels are compared, are on the orderof 1000 m 2 g −1 [99]. Therefore, A(t) is divided by a factor 6.7 to yield a more realisticsurface area. The fraction f s scales accordingly. For all systems f s increases from zeroto 0.0104. With the assumption that the modulation of the nuclei can be described bya single effective correlation time τ s the average relaxation time T 2 was calculated usingEqs. 5.28, 5.29 and 5.30. The resonance frequency ω 0 was set to 3.90×10 −7 rad s −1 . Thebulk value T 2,b follows from the calibration measurements of the methanol-D 2 O mixtures,and the calculated methanol fraction.A qualitative agreement between the model calculation and the experiments is obtainedwith the approximation that τ s is inversely proportional to D R . Because D R is a functionof ¯Mw it will drop to zero once ¯M w diverges, which occurs in the higher pH system att ≈ 5 hours. This would result in τ s going to infinity. Instead a maximum value of 2ns is imposed on the correlation time so that T 2 does not become shorter than 0.050 s.For the T = 50 ◦ C case this is 54 ms, due to a difference in T 2,b . The profiles, calculatedwith the choice τ s = 100 × D −1R , for all systems are shown in Figure 5.11. At T = 50 ◦ Cthe decay of T 2 is always faster for a given pH than at T = 25 ◦ C, which agrees with theexperimental observations. The discontinuities in the graphs are due to the truncation ofτ s at 2 ns. A smoother truncation is required near the point where τ s diverges, to obtaina better physical representation of T 2 . The differences in the plateau values, observed inthe experiments, could either be modeled by scaling the fraction f s or by changing thetruncation value for τ s .
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TWO-PHASE REACTIVE TRANSPORTOF AN O
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Samenvatting . . . . . . . . . . .
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2 1. Introductionproducerbarrierlow
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4 1. IntroductionTwo inter-European
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Part ITwo-phase bulk systems7
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10 2. Coupled mass transfer and sol
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132 9. Concluding remarks and outlo
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-142 Appendix A, - C D *, - D
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144 Appendix Awhere S 0 is a consta
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146 Appendix AThe 0.95 Tesla scanne
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148 Appendix B
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150 Appendix CwhereΩ A (θ) = −2
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152 Appendix C
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154 Appendix D
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156 Appendix Eand equal to zero in
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158 Appendix EAs the saturations, d
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160 Bibliography[12] B. Vinot, R. S
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162 Bibliography[40] M. D. Mantle a
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164 Bibliography[70] F. J. M. van d
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166 Bibliography[97] K. Yoshida, A.
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168 Bibliography[127] G. W. Scherer
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170 Bibliography
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172 Summaryand gelation rates. The
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174 Samenvattingwaterfase. De T 1 v
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176 List of publications
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178 Dankwoordimmer opgewekte stemmi
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Prior to the speech, Harald Cramér
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