Formation and Aggregation of Clusters

4 INORGANIC POLYMERS TO PREPARE MICROPOROUS MEMBRANES 7.4.1 Formation and Aggregation of Clusters 

A quite different approach from that of colloidal sols in the preparation of sol-gel derived membranes utilizes polymeric sols. In this category of sols the dispersed phase results from the hydrolysis and condensation of metal organic precursors in organic media. In most cases this process deals with the polymerization of metal alkoxides in alcohol according to the following reactions  

Silicon alkoxides exhibit very slow hydrolysis and condensation reactions compared with other alkoxides of aluminum, titanium or zirconium generally used for membrane preparation. Accordingly, acid or basic catalysts are used in the case of silicon alkoxides while methods for the control of hydrolysis are advisable with transition metal alkoxides [36,37]. 

In such reactive media branched clusters which do not contain fully condensed metal oxide cores are formed by kinetically limited growth processes. The structure of these clusters can be described using the fractal concept in which a mass fractal dimension D relates the cluster mass M to its radius according to 

A concept of mutual transparency or opacity based on the relative evolution of fractal dimension and radius of the clusters has been developed by Mandelbrot [38]. The tendency of fractal systems to interpenetrate is inversely related to the mean number of intersections 2 of two mass fractal objects of size and mass fractal dimensions Dj and D2 placed in the same region of space of dimension d  

---