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5.2. Theory 63As the clusters grow in time their volume fraction in the system, φ, increases relative to thevolume fraction of the monomeric units, according to [115] (provided the size distributionis narrow)( ) (3−df ) ( ) (3−df )/d f Reff¯Mwφ(t) = φ m (t)= φ m (t), (5.22)R A x Awhere φ m is the volume fraction of the monomeric units, given byφ m (t) ≈ 4 3 πa3 n s (t)V (t) −1 . (5.23)The volume V in Eq. 5.23 increases due to the transfer of TMOS, and is given byV (t) = V 0 + [n t (0) − n o (t)]¯v T , (5.24)where V 0 is the initial volume of the aqueous phase and ¯v T is the molecular volume ofTMOS. Formally, when the volume fraction φ becomes greater than one the clusters startto overlap and will aggregate to form a percolating network that spans the entire volume.Internal surface of silica clustersIn case the monomers would aggregate to form solid spheroids (d f = 3) the total surfacearea A per unit volume [m −1 ] isA ≈φ m. (5.25)3R effThe situation becomes more complicated when the aggregates or clusters are fractal. Themagnitude of accessible surface area is also dependent on the scale at which the fractalstructure is considered. An extreme case, at low concentrations, is where the networkconsists of weakly cross-linked chains of a monomer thick. The surface area is thenapproximated byA ≈ 3φ ma . (5.26)Equation 5.25 yields the lowest possible surface area for a given φ m , whereas Eq. 5.26gives the highest possible magnitude.Mobility of aggregatesFor small aggregates and clusters the mobility can be related to their sizes. With thecondition that the shapes of the aggregates are spherical the rotational diffusion coefficientD R is approximated byD R =k BT. (5.27)4πηReff3For larger clusters and in case of percolation the rotational mobilities will obviously beclose to zero due to steric effects.