JOURNAL OF COLLOID AND INTERFACE SCIENCE 190, 294–301 (1997) ARTICLE NO. CS974838 Random-Walk Aggregation Phenomena in Solid Bimodal Liquid Dispersions: Transition to Nondeterminism from Si 3 N 4 to Si 3 N 4 / Al 2 O 3 Aqueous Systems Stefano A. Mezzasalma 1 and Rada Novakovic Materials Science and Engineering Laboratory of the Engineering and Materials Science Institute, University of Genoa, C.so Perrone 24A, 16161 Genoa, Italy Received September 27, 1996; accepted February 19, 1997 equation from which the unknown physical quantities can This paper, which is based on another recent work, (Mezza- be derived. Accordingly the (minimum) number of solid salma, S. A., Phys. Rev. E 55(4), (1997)) deals with experiments particles was obtained for different values of the suspension and theory concerning an aqueous dispersed system formed from pH and of the solid mass concentration. silicon nitride (Si 3 N 4 ), alumina (Al 2 O 3 ), and mixed silicon ni- In principle the developed method can be extended to tride / alumina (Si 3 N 4 / Al 2 O 3 ) solid agglomerates. From titra- more complex systems formed by two or more sets of solid tion data applied to a thermodynamic equilibrium condition, the particles dispersed in aqueous and / or other kinds of solution. minimum number of each agglomerate species and their maxi- However, this extension, which has been proposed in another mal average dimensions have been derived as functions of the aqueous solution pH. These parameters are of the order of, re- recent study (1), must overcome a certain number of diffi- spectively, (1–2) mm for Si 3 N 4 and Al 2 O 3 agglomerates and ( 20 – culties. 50) mm for the mixed agglomerates. The numbers of solid parti- The titration method gives the total number of H / and cles of all species are poorly correlated with changes in pH of the OH 0 ions adsorbed onto the solid surfaces no matter which liquid phase. This behavior has been interpreted as intrinsically kind of solid particles contribute to the dispersion. How to related to the complexity of the system which, due to the many obtain the numbers of ions adsorbed onto each solid surface interactions among the different species, probably becomes non- species is a first problem. In this paper an experimental deterministic. In order to describe such behavior a probabilistic method which leads to the partition of the adsorbed ions on approach has been developed. The probability of finding a given a single kind of solid surface will be presented. solid agglomerate number within a scatter band varies with the suspension pH. Furthermore, the scatter band amplitude be- To obtain the number of solid agglomerates vs the suspen- comes negligible near the isoelectric point. Accordingly, only sion pH from the adsorption data, the total Gibbs free energy the numbers of aggregates derived in the neighborhood of the of the complex system must be known. This is not an easy isoelectric point are predictable.  1997 Academic Press task, because it is not possible, for solid polyphasic systems, Key Words: aggregation; titration; bimodal suspensions; silicon to neglect a priori other antagonistic interaction terms be- nitride; alumina; random walk; thermodynamics; nondeter- sides the usual coulombic repulsion terms (4–14) which, minism. when solid monophasic systems were involved, were able to characterize globally the ionic adsorption of the agglomer- ated surfaces from liquid solution (2). On the other hand, INTRODUCTION for describing aggregation processes among mixed solid spe- cies, London–van der Waals effects must be considered. Recently ( 2, 3 ) a titration method, based on MOS / ISFET Moreover, the competition between these interaction terms pH meter measurements, has been developed for Si 3 N 4 pow- can make, in principle, the number of solid aggregates vs ders dispersed in aqueous solutions. To derive from these the liquid phase pH a very unstable system. measurements the number and the mean dimension of clus- It will be shown also that instability regions exist for the ters as a function of the pH dispersion, an expression for minimum number of agglomerates vs the pH, where it is the total Gibbs free energy of the suspension is required, not possible to predict exactly the agglomeration number so that the equilibrium condition can be written, giving an and size of the solid biphasic dispersed system. To overcome this difficulty the instability regions have 1 To whom correspondence should be addressed. been described by means of a random-walk stochastic pro- 294 0021-9797/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.