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2.4. Results and discussion 211.51.0n-hexadecane1/T 1[s -1 ]0.5TMOS0.02.0x10 -6 4.0x10 -6 6.0x10 -6 8.0x10 -6 1.0x10 -5η mix/ T [Pa s K -1 ]Fig. 2.6: Reciprocal T 1 (mono-exponential) of the TMOS/n-hexadecane mixtures as a functionof the viscosity-temperature ratio, η mix /T . T is fixed at 25 ◦ C.Table 2.1: Fitting parameters for the relaxation time model of the pure liquids.n-hexadecane TMOSτ ′ [ps] 0.052 1.64E A [kJ mol −1 ] 15.2 2.46τ c,eff [ps] at 20 ◦ C 26.7 4.49used (not presented here), which seems acceptable given the value of the self-diffusioncoefficient for the pure liquids (i.e. 1.59×10 −9 m 2 s −1 for TMOS and 0.40×10 −9 m 2 s −1for n-hexadecane at 25 ◦ C).The intra-molecular part of the relaxation contains the effective correlation time τ c,effwhich is found by an optimization process as follows. For both species the correlationtime τ c,eff as defined in Eq. 2.8 is assumed to be independent of concentration, thereforethe term is derived from the experimental data of pure TMOS and pure n-hexadecane,respectively. The predicted inter-molecular term is subtracted from the experimental,total T 1 for each temperature. Then the remainder is fitted as a function of temperaturewith Eqs. 2.7 and 2.8. The summation term ∑ r −6ij is estimated by considering a rigidmolecular model for each species. The resulting fits for TMOS and n-hexadecane, yieldingthe unknown parameters E A and τ ′ (see Table 2.1), are shown in Figure 2.4 as the solidcurves. The excellent fit indicates that the model equations adequately describe the spinlatticerelaxation of the pure liquids.Finally, the total relaxation times in the mixture are calculated (defined by Eqs. 2.14and 2.15) using the obtained fitting parameters (see Table 2.1). The Saturation Recoverydecay is subsequently constructed as given by Eq. 2.16, and the result is fitted with
22 2. Coupled mass transfer and sol-gel reaction in a two-phase bulk system43.53experiment T = 20 o Cexperiment T = 40 o Cmodel T = 20 o Cmodel T = 40 o C2.5T 1[s]21.510.500 0.2 0.4 0.6 0.8 1φFig. 2.7: Relaxation time T 1 of TMOS/n-hexadecane mixtures as a function of concentration φafter interpolation of the experimental data for different temperatures. The predictedT 1 , based on Eqs. 2.14 - 2.16, is shown as well.the mono-exponential decay function in order to compare the calculated values to theexperimental results. The comparison is shown in Figure 2.7 for T = 20 ◦ C and T =40 ◦ C. An excellent match between the calculated and the experimental T 1 is found for0 < φ < 0.4. At higher concentrations the difference is larger but within 10% of theexperimental value.Relaxation in prepared gelsThe relaxation times of the prepared gel samples were measured after 14 days of preparationand at ambient temperature. In Table 2.2 the results are summarized togetherwith T 1 and T 2 for water, methanol, mixtures of water and methanol, and TMOS, whichare substances involved in the sol-gel reaction. The longitudinal relaxation in the gelsappeared to be mono-exponential. For the lowest initial concentration of TMOS (φ g T =0.05), T 1 is almost equal to that of water, but decreases with increasing concentration.Between φ g T = 0.30 and φg T = 0.50, T 1 is constant (about 1.70 s). The reduction of T 1of the wet gels with respect to T 1 of water is thus limited to about 50% for the range ofconcentrations considered.The CPMG data were analyzed using the inversion routine CONTIN. The T 2 spectraof water and TMOS appeared to be mono-exponential. Subsequent fitting of the datawith a single exponent yielded a T 2 of 1.66 s and 1.33 s for water, respectively, TMOS(with G = 100 mT/m and ∆τ E = 7.2 ms). The differences between T 2 and T 1 for waterand TMOS are explained by the effect of molecular diffusion (see Section 2.3.2).The gel T 2 spectra and the T 2 spectra of water/methanol showed a multi-exponentialbehavior with well-separated components within the range of the inversion. Additionally,
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148 Appendix B
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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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