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2.2. Principle of nuclear magnetic relaxation in liquids 13the relaxation times can be correlated with η. Instead of using the Arrhenius expression(Eq. 2.8) for the effective correlation time τ c,eff , we can link the time to the rotationaldiffusion coefficient D R of the molecules [50]. This diffusion coefficient follows from aStokes-Einstein equation, so that [50, 51, 57]τ c,eff = 1 ≃ 4πηa36D R 3k B T , (2.11)where k B is the Boltzmann constant. Likewise, the diffusion coefficient D in Eq. 2.9 canbe related to the viscosity η byD = k BTCaη , (2.12)where C is a coefficient which is related to the molecular shape, amongst others [58].Hence, it follows from Eq. 2.10 that1∝ η T 1,2 T . (2.13)Zega et al. [59] showed that, for a series of alkanes, the values of 1/T 1 against η/T can beplotted approximately on a single curve.2.2.3 Binary mixturesIn a mixture of liquids A and B, there is an additional inter-molecular contribution dueto interaction between hydrogen nuclei of A and B. The individual relaxation times T1,2Aand T1,2 B in the mixture are defined as an extension to Eq. 2.10 by1T A 1,21T B 1,2==( µ04π( µ04π) 2γ 4 2 [c A τ c,eff,A +) 2γ 4 2 [c B τ c,eff,B +π5a A(N0AD AA+ N 0BD AB)], (2.14)π5a B(N0AD BA+ N 0BD BB)], (2.15)where c A and c B are constants related to the molecular structures, and D ij are the mutualdiffusion coefficients. Suppose the longitudinal magnetization of the mixture in equilibriumwith an external magnetic field is brought to zero by a 90 ◦ rf pulse. Subsequently,based on superposition of the magnetic moments, the total magnetization shows a biexponentialrelaxation behavior as described byM z (t) = M z (0)[1 − f exp(− tT A 1)(− (1 − f) exp − tT1B)], (2.16)where f is the proton density fraction of A. Thus, the proton fraction and the relaxationtimes are functions of composition. If T1 A is equal or almost equal to T1 B or in case f isclose to zero or close to one M z relaxes according to a single exponential as defined byEq. 2.3. Similarly, the transverse magnetization of the mixture relaxes back to zero afterapplication of the 90 ◦ pulse according to the bi-exponential functionM T (t) = M T (0)[f exp(− tT A 2)+ (1 − f) exp(− tT B 2)]. (2.17)Eq. 2.16 and Eq. 2.17 can be extended to describe the multi-exponential relaxation inmulti-component mixtures.
14 2. Coupled mass transfer and sol-gel reaction in a two-phase bulk system2.2.4 Correlation of relaxation times with viscosity in binary mixturesIn (simple) binary liquid mixtures the effective viscosity often depends on the composition,provided that the viscosities of the pure components, η A and η B , are not equal to eachother. Several mixture laws exist [60]. For example, the following mixture law wasintroduced by Grunberg and Nissan [61]:ln η mix = x ln η A + (1 − x) ln η B + Gx(1 − x), (2.18)where η mix is the effective viscosity of the mixture, x is the mole fraction of componentA, and G is an interaction parameter.The relaxation times will be a function of composition (through the proton fractionsbelonging to the components A and B) as long as T A 1,2 is higher or lower than T B 1,2. If, inaddition, η A is different from η B , the relaxation times can be correlated with the effectiveviscosity via the composition dependence. On a molecular level, one can expect, in caseη A ≠ η B , the diffusivities of the molecules to be a function of the composition. The higherthe viscosity, the lower are the diffusivities. This will affect the inter-molecular relaxation(see Eq. 2.9). As for intra-molecular relaxation, the effective correlation times of thespecies may depend on η mix as indicated by Eq. 2.11, but this is not always the case [62].2.3 Experimental section2.3.1 Chemicals and preparationTetra-methyl-ortho-silicate, Si(OCH 3 ) 4 , was obtained from Aldrich (>99% pure). Forthe oleic phase, n-hexadecane, or n-C 16 H 34 (obtained from Merck, >99% pure) was usedas a clean and well-defined hydrocarbon liquid which has a significant difference in T 1compared to water and TMOS. The viscosities of TMOS, n-hexadecane and mixtures ofTMOS and n-hexadecane were measured at 25 ◦ C with a rheometer (Contraves LS40).For the aqueous phase we used double-demineralized water.Several gel samples were prepared at ambient conditions with demineralized water andTMOS in different volume ratios. The initial volume fraction φ g T(hence, also the molefraction) of TMOS in the mixture was chosen to be a measure to characterize the final gelstate. It was found that for 2.5 vol% (or 0.31 mol%) of TMOS in water no gel was formed,even after several months. For 5.0 vol% and more (tested up to 50 vol%) a homogeneousgel formed. It was observed that after several weeks the high-concentration gels showed asmall degree of syneresis. Calibration solutions of methanol in demineralized water wereprepared as well.2.3.2 NMR apparatus and sequencesRelaxation times and T 1 -weighted images of the samples were recorded with a 4.7 TeslaNMR spectrometer operating at a frequency of 200 MHz for 1 H. The setup consists ofa super-conducting magnet (Oxford Instruments, Oxon, UK) with a vertical, narrowboreinsert (Doty, Columbia, USA) with an inner diameter of 40 mm. The insert has
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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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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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