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Chapter 5Cross-linking of silica with a time-dependentmonomer sourceIn this chapter the cross-linking of the hydrolyzed TMOS molecules is consideredand described using a simple population balance model. The mass transfer of themolecules from the oleic to the aqueous phase and the hydrolysis step were includedin the form of a time-dependent monomer source. Gel transition times were predictedbased on the critical behavior of the weight-average molecular weight of the species.Average structural properties of the silica aggregates in the gelling stage were derived.The results were used to predict the transverse relaxation time T 2 of the aqueousphase, which is typically measured in the NMR experiments.5.1 IntroductionIn sol-gel processing a solution of reactive monomers or colloids changes into a gel due toaggregation of the particles into larger clusters and networks. Macroscopically, the sol-geltransition occurs when the network percolates to span the entire volume of the system[100]. This transition is generally manifested by a sharp change in rheological properties,i.e. the bulk viscosity increases over several orders of magnitude [101, 102].The aggregation of particles into clusters and fractal structures has been studied anddescribed extensively [24]. Numerical models typically comprise the simulation of trajectoriesand aggregation of diffusing particles in two or three dimensions. In diffusion limitedcluster-cluster aggregation models (DLCCA) the particles or clusters combine directlyupon contact [103, 104]. Reaction limited cluster-cluster aggregation models (RLCCA)are similar, except that the particles only combine with a finite probability [103]. Thefluctuating bond aggregation (FBA) model, which is an extension to the DLCCA model,incorporates the flexibility of the formed clusters [105]. The flexibility depends on thetype of catalyst present in the sol-gel [106]. The numerical models mentioned typicallyyield the fractal dimension, the correlation lengths, percolation time, etc.Population balance models describe the evolution of a molecular-weight distributionthat characterizes a system of cross-linking particles. For a discrete distribution theevolution of the number of each species with a particular weight can be calculated usinga kinetic equation such as the Smoluchowski equation [107, 108]. Li and McCoy proposedan elaborate model for the cross-linking of reactive species which includes the degree ofcross-linking in addition to the molecular-weight balance [109].In this chapter the aim is to describe the aggregation of hydrolyzed TMOS monomersand the evolution of the aggregates and clusters in the aqueous phase as the precursor57
58 5. Cross-linking of silica with a time-dependent monomer sourcemonomers are transferred from the oleic to the aqueous phase. We develop an algebraicmodel based on a population balance of polymeric species [110, 111]. The mass transfer ofthe molecules from the oleic to the aqueous phase and the hydrolysis step are included inthe form of a time-dependent monomer source. A concise set of equations is obtained thatcan be solved analytically only for a limited set of parameters, but in many cases needs tobe solved numerically. The evolution in the population balance is used to predict certaincharacteristics of the gelling solution, like effective surface area, and cluster mobility. Inaddition, the gel transition times are obtained for a set of representative systems. Finally,the transverse relaxation time (T 2 ) of the water molecules present in the gel, which istypically measured in experiments involving NMR (see Chapter 4), is considered. A simplemodel is proposed to predict the average transverse relaxation time T 2 of the deuteriumnuclei of the water molecules due to their interaction with the gelling silica content. Theaverage T 2 decays in time due to the increasing silica content and increasing cluster size.The effect of pH and temperature on the reaction rates, and hence on the T 2 profiles, wasmodeled. The results agree qualitatively with previously obtained experimental results.5.2 Theory5.2.1 Mass transfer and hydrolysisWe consider again a volume consisting of two liquid phases which are in contact butseparated by an interface (see also Section 3.2.1). Phase 1 initially consists of (heavy)water, and phase 2 consists of a solution of TMOS in oil with an initial concentrationn t (0) = n 0 . The system is shown in Figure 3.1. As the phases are put together the TMOSpartitions into the aqueous phase in which it undergoes a heterogeneous sol-gel reaction.The overall mass transfer is governed by the hydrolysis rate in the aqueous phase. Eachphase is well-mixed and diffusion or convection processes do not limit the mass transfer.The total number of TMOS molecules n t in the system decreases in the course of timedue to the hydrolysis reaction. The change is described by the following kinetic equationdn tdt = d(n o + n w )dt= −kn w , (5.1)where n o and n w are the number of TMOS molecules in the oleic, respectively, the aqueousphase and k is the rate of hydrolysis (where we assume full hydrolysis of the TMOSmolecules). The methanol concentration n m increases in the course of time as given byn m (t) = 4∫ t0kn w (τ)dτ = 4 [n t (0) − n t (t)] . (5.2)Here we are only concerned with the typical profiles for n w and n m in the aqueous phase.The TMOS concentration in oil, n o , is modeled as a simple empirical function. Theexperiments showed (see Chapter 2) that the concentration n o (t) can be adequately approximatedby an exponential decay functionn o (t) = n t (0) exp(−α T t), (5.3)
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TWO-PHASE REACTIVE TRANSPORTOF AN O
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2 1. Introductionproducerbarrierlow
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114 8. Model of reactive transport
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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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