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5.3. Results and discussion 693.53α T= 0.230 h -1k = 1.5 α Tn on wn t3.532.52.5n [mmol]21.5p [1] (10 -3 x m)21.5110.500 5 10 15 20 25 30time [h]0.5no bufferpH = 2.95pH = 9.4800 5 10 15 20 25 30time [h]Fig. 5.7: Predicted number of TMOS moleculesin the model system at pH =9.48 and T = 25 ◦ CFig. 5.8: First moment of the populationbalance. The thick lines indicatethe systems at 50 ◦ C, the thin linesindicate the systems at 25 ◦ C.In all systems p [1] increases monotonically towards a final value equivalent to the initialamount of TMOS n t (0), but the increase for the lower pH system is faster than for theintermediate and higher pH system in the initial stage.Several calculations were done varying the exponent ν and coefficient κ in the reactionrate kernel k p , in order to examine the cross-linking reaction in more detail. Two sets ofcalculations are discussed which form an illustrative example. In the first set κ = 10 h −1and in the second set κ = 20 h −1 . In both cases ν was varied, having the values 1/3, 2/3or 3/4. Though several parameters were calculated we here focus on the weight-averagemolecular weight ¯M w being equal to p [2] /p [1] . The results are demonstrated in Figure 5.9.Several aspects can be observed. First, in all cases ¯Mw is initially equal to 1 andincreases monotonically in time. Secondly, within the time interval considered, ¯Mw doesnot diverge when κ = 10 h −1 , irrespective of ν. However, when κ = 20 h −1 the parameter¯M w diverges for all pH systems when ν = 3/4 and also for the lower pH system andunbuffered system when ν = 2/3. The curve for the higher pH system with ν = 2/3 isclose to divergence and will do so after 30 hours. When ν = 1/3, the results of whichare not demonstrated here, none of the curves show critical behavior, even after extendedcalculation times. Other calculations were done as well with κ = 50, 100, 200, 500 and1000 h −1 . In general, the results show that increasing values of either ν or κ always leadto a faster monotonous increase of ¯Mw , and likewise, a shorter transition time at whichthe critical behavior occurs (in case ν = 2/3 or 3/4). Due to the slower mass transfer andhydrolysis in the higher pH system the silicic acid source is weaker and therefore, givena particular combination of ν and κ, the increase (and divergence) of ¯Mw is delayed withrespect to the lower pH system.