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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)