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5.3. Results and discussion 67Table 5.2: Fitted mass transfer coefficient α T for the various bulk experiments. The hydrolysisreaction rate k is equal to 1.5 × α T . The condensation rate κ and the colloidal sizeR A which are used in the model calculations are listed as well. t c is the obtainedcritical transition time of ¯Mw , and t g is the gel transition time at which φ = 1. t expgis the experimental gelation time.T [ ◦ C] pH α T [h −1 ] k [h −1 ] κ [h −1 ] R A [nm] t c [h] t g [h] t expg [h]25 no buffer 0.52 0.78 100 10 7.5 7.4 8 ± 125 2.95 0.86 1.29 15 1 27.5 27.4 30 ± 325 9.48 0.23 0.35 500 1000 5.3 5.2 n.a.50 no buffer 0.67 1.01 700 20 2.6 2.5 2.0 ± 0.550 2.95 2.22 3.33 100 2 4.8 4.7 5 ± 150 9.48 1.06 1.60 1000 2000 1.7 1.6 n.a.which implies that the condensation rate is higher at T = 50 ◦ C.The faster decay of T 2 in the first 10 hours for the higher pH system compared to thelower pH system is indicative of a faster condensation rate, since the extent of the TMOStransfer between the two phases is actually greater for the lower pH system and the factthat a fast mass transfer must be linked to a high hydrolysis rate. For example, at t =3 hours about half of the TMOS in the pH = 9.48 system has transferred from the oleicto the aqueous phase, whereas the transfer is almost complete in the pH = 2.95 system.However, at t = 3 hours the T 2 for the latter system is 300 ms and for the higher pHsystem T 2 is 150 ms.5.3.2 Model calculationsPopulation balance evolutionModel calculations were performed using the simple mass transfer and population balancemodel described in Section 5.2. The ODEs were solved numerically with a straightforwardexplicit scheme with a sufficiently small time step. With regard to the mass transfer andhydrolysis rates six sets of parameters were chosen to reflect the different pH in the sixexperimental systems. Since the mass transfer is promoted by a higher rate of hydrolysis,the hydrolysis rate k is linked directly to the mass transfer rate α T . We arbitrarily choosethe linear relation k = 1.5 × α T . The hydrolysis reaction rate constants k are given inTable 5.2 for each system.The number of TMOS molecules in both phases n o , respectively, n w as functions oftime are calculated using Eqs. 5.3 and 5.4. The profiles for the base-catalyzed system areshown in Figure 5.7. Like the amount of TMOS in oil, n o , the total amount of TMOS, n t ,decreases monotonically with time and becomes almost zero after about 20 hours. Theprofile of the amount of TMOS in water, n w , exhibits a distorted bell shape. At t = 0,n w is zero and reaches a maximum after about 4 hours, after which it decreases graduallyto zero. The total amount of hydrolyzed TMOS, in the form of silicic acid monomers oraggregates, is equal to the first moment p [1] , the profiles of which are shown in Figure 5.8.