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4.2. Theoretical background 45where 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). Assuming quasi-static equilibrium conditions, the ratio between n o and n w ateach instant is determined by the solubility of TMOS in both phases through the followingpartitioning equationn wn w + m w=[n oexp − ε(n ]m), (4.2)n o + m o RTwhere m w is the total number of water, methanol and silicic acid molecules, m o is thenumber of hydrocarbon molecules, R is the gas constant, T is the temperature and ε isan interaction parameter which depends on the methanol concentration n m . The exactchoice of ε(n m ) is arbitrary, but the parameter should decrease with an increasing valueof n m and vice versa. As a first-order approximation, the parameter ε is chosen asε = ε 0[1 − α n mm w (0)], (4.3)where ε 0 is the interaction parameter in absence of methanol and α is an arbitrary scalefactor. The methanol concentration n m increases in the course of time as given byn m = 4∫ t0kn w (τ)dτ = 4 [n t (0) − n t (t)] . (4.4)The variables n w and m w in Eq. 4.2 are rewritten in terms of n o and m o . Then, byrearranging Eq. 4.1 the following differential equation is obtaineddn odt = −kEn o [N − (n o + m o )] × {(n o + m o ) + 4ε 0 αβEn o [N − (n o + m o )]}ENm o + (1 − E)(n o + m o ) 2 , (4.5)where E = exp(−ε/RT ), β = (RT ) −1 and N is the total number of particles, which isconstant. Equation 4.5 can be solved numerically and optimized to fit the experimentaldata, yielding the parameters ε 0 and k. The parameter α is not optimized to limit thenumber of parameters in the optimization. We choose α = 1, so that E – and thusthe solubility of TMOS in water – increases significantly during the reactive transportdepending on ε 0 and given the estimated final amount of methanol (see Figure 4.2 for anexample calculation of E).4.2.2 NMR relaxation of deuterium in sol-gelFor a rigorous introduction to nuclear magnetic relaxation the reader should refer toAbragam [41]. In the gelling solution the deuterium nuclei are most abundant in the formof D 2 O molecules. The main relaxation mechanism for deuterium nuclei (deuterons) inD 2 O is due to quadrupolar interaction between the deuterons and the internal electricfield gradient at the nuclei. This intramolecular relaxation is driven by molecular fluctuations,such as molecular rotation. The fluctuations are often characterized by a molecularcorrelation time τ c [41].In the gelling solution the water molecules diffuse through the system and with acertain probability they adsorb onto or interact with the silica clusters. The molecules