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5.2. Theory 61Derivative parameters of momentsThe distribution of the aggregates can be characterized by the number-average molecularweight ¯Mn , equal to p [1] /p [0] , and by the weight-average molecular weight ¯Mw , equalto p [2] /p [1] [111]. The gel transition of the system can, from a mathematical point ofview, be linked to the divergence of the second and higher order moments, which occursat a transition time t c when ν is greater than a critical value ν c . Likewise, ¯Mw willdiverge at such a transition, which, for a batch system containing a fixed initial amountof aggregating monomers, is often represented by the power law [111](¯M w (t) = ¯M w (0) 1 − t ) −γ, (5.19)t cwhere the critical parameters are the gel time t c and the exponent γ.5.2.3 Gelation and percolationStructural evolution of aggregatesThe population balance model yields the number- and weight-average molecular weightas a function of time, but does not predict the structural evolution of the aggregates.The growth mechanisms are complex and depend strongly on the type of catalyst and thewater/TMOS ratio [33, 99]. Furthermore, after gelation the internal structure changesdue to dissolution and secondary cross-linking of the silica, which is referred to as aging[24]. A rigorous description of the mechanisms is out of the scope, and a phenomenologicalapproach is employed instead.In the early stage of the condensation process the monomers and small oligomerscombine into compact, spherical aggregates with a characteristic size R A and weight x A .The weight x A follows from( ) 3x A RA= , (5.20)x m awhere a is the size of a monomer. Subsequently, in the intermediate stage the nonfractalaggregates interlink to form fractal clusters with a fractal dimension d f [113, 114].Eventually, the clusters start to link into larger networks. On the scale of the networkthe fractal character is lost [115]. Gelation occurs when all clusters combine into a singlenetwork. Percolation, i.e. macroscopic gelation, occurs when the network spans the entirevolume of the system. The size R A ranges from several nanometers (at low pH) to severalmicrons (at high pH) [34], which is shown schematically in Figure 5.2. The growth of thelarge compact aggregates under base-catalyzed conditions is promoted since nucleationand growth is the dominant growth mechanism, which is facilitated by the enhanceddepolymerization rate [33] (see also Figure 5.3).In the intermediate stage the effective cluster radius R eff is related to the averagemolecular weight throughR eff ≈ R A( ¯Mwx A) 1/df. (5.21)
0+,62 5. Cross-linking of silica with a time-dependent monomer source - 4, 1 - 4+ ; + 1 +F 0 % 4 F 0 % 9 1 6 0 5 ) 6 5 2 4 - 5 - 62 ) 4 6 1 + - # ) * ! F 0 % 9 1 6 05 ) 6 5 ) * 5 - 6 6 0 4 - - , 1 - 5 1 ) / - - 6 9 4 5 5 5Fig. 5.2: Polymerization behavior of aqueous silica. In basic solution (B) particles grow in sizewith a decrease in number; in acid solution or in the presence of flocculating salts (A),particles aggregate into three-dimensional networks and form gels. After Iler [34]. / 4 ) 6 - / - 6 1 - 7 + / 9 6 04 1 2 - 1 / " $ & F 0Fig. 5.3: Schematic representation of the pH-dependence of hydrolysis (H), condensation (C),and dissolution (D) in an alkoxy-silane solution. In this case H 2 O:Si = 1.5. AfterBrinker [33].
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TWO-PHASE REACTIVE TRANSPORTOF AN O
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Dit proefschrift is goedgekeurd doo
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4. The effect of pH on mass transfe
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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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112 7. Permeability reduction in po
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114 8. Model of reactive transport
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132 9. Concluding remarks and outlo
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140
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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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Untitled - Technische Universiteit Eindhoven

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