Concentrated Colloidal Dispersions
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IN/AUGUST1991
(a)
(b)
SOLID FLUID
the number density n and the thermal energy kT.1 Thermodynamic properties dépend on certain features of the interparticle potential: The effective hard core diameter d, the softness of the repulsion K~\ and the magnitude m and range rm of the attraction (Figure 2). The equilibrium phase diagram (Figu r e 3a), expressed in d i m e n s i o n l e s s form as kT/$m versus d> = 7rd3«/6, reveals b o t h d i s o r d e r e d fluid and ordered solid phases with the relative softness of the repulsion 1/d* determ i n i n g the s t r u c t u r e of the ordered phase. For longer range a t t r a c t i o n s , d/rm < 3, a région of fluid -fluid coexistence appears as well. Other transitions are also of interest. For adhesive hard sphères, i.e., with an infinitely steep repulsion and a short range attraction, the percolation transition (Figure 3b) signais the divergence of the average size of transient clusters. Note that it terminâtes in the critical point of the spinodal curve, where the osmotic compressibility diverges. Until relatively recently, this equilibrium p h a s e b e h a v i o r was largely o b s c u r e d by n o n e q u i l i b r i u m s t a t e s preserved by the slow équilibration of many Systems. Indeed, the characteristic relaxation time t, = (37r/idVJtT)exp(A/JfcT),
(1)
w i t h fi the viscosity of the solvent and A the barrier hindering particles from sampling new configurations, extends to years with strong attractions or highly crowded particles. Hence, glassy phases, instead of ordered solids, and gels, rather than macroscopic f luid-
Figure 2. Dependence of (a) shear viscosity on shear rate and (b) shear moduius on frequency.
viscosity 170 and shear moduius G' decaying as G' ~ a>2 for a> —» 0, and solid responses, with Tjy—» Ty as y —» 0 and G' —» G'0 as Û) —» 0 (Figure 1). Second, it serves to demonstrate the known relations between the viscometric functions and the interparticle forces. Thermodynamic Preliminaries The standard approach for assessing phase behavior treats a dispersion of identical p a r t i c l e s as a p s e u d o onec o m p o n e n t System c h a r a c t e r i z e d by
Figure 2. Typical interparticle potential deriving from superposition of attractions and repulsions.4
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Concentrated Colloidal Dispersions
solid coexistence, may persist. The nature of the glass 6 and gelation transitions, as vvell as any relationship between the latter and the percolation transition, remain topics of c u r r e n t research. Interparticle Forces Each force has a characteristic magnitude and range (Table I); e.g., Brownian m o t i o n acts on the scale of t h e particle radius a with strength kT/a. The dispersion attraction dépends on dielectric properties t h r o u g h the H a m a k e r constant as Aell while the electrostatic repulsion varies in strength with the s u r f a c e p o t e n t i a l ijjs a n d in r a n g e through the Debye length K" 1 . For the others, either attraction or repulsion is possible, even between identical particles. Considérable levera
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