Asakura and Oosawa

The possibility of occurrence of instability of colloidal dispersions in the presence of free polymer was first predicted by Asakura and Oosawa (5), who have shown that the exclusion of the free polymer molecules from the interparticle space generates an attractive force between particles, DeHek and Vrij (1) have developed a model in which the particles and the polymer molecules are treated as hard spheres and rederived in a simple and illuminating way the interaction potential proposed by Asakura and Oosawa. Using this potential, they calculated the second virial coefficient for the particles as a function of the free polymer concentration and have shown that... 

In the basic model, put forward by Asakura and Oosawa (5), the hard spherical particles immersed in a solution of macromolecules are considered to be surrounded by depletion layers from which the polymer molecules are excluded. When two particles are far apart with no overlap of the depletion zones, the thermal force acting over the entire particle surface is uniform. However, when the particles come closer, such that their depletion zones begin to overlap, there is a region in which the polymer concentration is zero and the force exerted over the surfaces facing this region is smaller compared to that exerted over the rest of the surface. This gives rise to an attractive force between the two particles which is proportional to the osmotic pressure of the polymer solution. 

In one of the limiting cases, the free polymer is allowed to penetrate the adsorbed layer around the particles. One may note that when the free polymer and the adsorbed polymer are both present in the steric layer around the particle, the interactions between the two must be taken into account while evaluating the in-terparticle forces. However, in the absence of a detailed knowledge of the structure of the adsorbed layer, it is difficult to evaluate this contribution to the interaction potential. Then, the situation is similar to the one considered by Asakura and Oosawa (16), and the force of attraction between two bare particles of radius a in the presence of free polymer molecules of radius / can be expressed as the product of the osmotic pressure Pmm and the area of the intersection of the two overlapping volumes ... 

The force between two large hard spheres is a fluid of small hard spheres, obtained from (R), is plotted in Fig. 3. It is oscillatory and has the characteristics that are missing in the DLYO theory. The force in Fig. 3 is just the depletion force discussed by Asakura and Oosawa [40]. However the Asakura-Oosawa formula is valid only at low densities of the small hard spheres. 

The original geometrical analysis of Asakura and Oosawa (1954, 1958), generalized by Vrij (1976) and others, neglects the internal degrees of freedom of the polymer molecules to obtain simple, useful expressions for the interaction potential. The SCF theory reviewed here (Joanny et al., 1979) demonstrates the validity of the simpler approaches. 

Several theories have been put forward to account for the distributicm of polymer segments in the depletion zone. The theories of Feigin and Napper [48] and Scheutjens and Fleer [49] are qualitatively different from the theory of Asakura and Oosawa and de Cannes and coworkers [50,51] in that they predict not only depletion flocculation but also depletion stabilization. Depletion stabilization has not to date been verified experimentally although depletion fiocculation has been verified experimentally for several systems [52,53]. The effect of an adsorbed poljnner layer [54] and ordered solvent layers [55] on depletion flocculation is also under theoretical attack. The depletion stabilization interaction energy cannot simply be added to the other interaction energy terms to give the total interaction energy. 

The volume restriction effect as discussed in this paper was proposed several years ago by Asakura and Oosawa (12,13). Their theory accounted for the instability observed in mixtures of colloidal particles and free polymer molecules. Such mixed systems have been investigated experimentally for decades (14-16). However, the work of Asakura and Oosawa did not receive much attention until recently (17,18). A few years ago, Vrij (19) treated the volume restriction effect independently, and also observed phase separation in a microemulsion with added polymer. Recently, DeHek and Vrij (20) have reported phase separation in non-aqueous systems containing hydrophilic silica particles and polymer molecules. The results have been treated quite well in terms of a "hard-sphere-cavity" model. Sperry (21) has also used a hard-sphere approximation in a quantitative model for the volume restriction flocculation of latex by water-soluble polymers. 

A nonadsorbing polymer added to dispersions will often cause aggregation of particles. Asakura and Oosawa were the first to describe the cause of this instability as two particles approach the finite size of the polymer chains ensures their exclusion from the region between the two particles. The osmotic pressure is dependent on the concentration of macromolecule and hence it is diminished in this overlap region. The excess osmotic pressure in the main bulk fiuid causes the two particles to be pushed together. This is called the depletion potential and has been extensively studied.This is shown in Fig. 6. 

The second case involves non-adsorbing polymer chains in solution. It was realized by Asakura and Oosawa (AO)... 

The theory of Asakura a Asakura and Oosawa (1954) were the first to recognize that depletion effects could give rise to the flocculation of colloidal dispersions. They considered the specific example of two parallel flat plates immersed in solutions of rigid molecules, either spherical or rod shaped. When the distance between the plates is smaller than the diameter of the solute molecules, assumed for the purposes at hand to be spherical, none of these molecules can enter the domain between the plates. This region is then composed entirely of solvent. The solution outside the plates retains its bulk concentration of solute and so it exerts an inward force, arising from its osmotic pressure, on the plates. 

This concept was subsequently extended as follows by Asakura and Oosawa (1958) to embrace spheres. 

Flexible macromolecules. Calculations of the attractive potential energy according to equation (15.11) show that for spherical colloidal particles immersed in a dilute solution of rigid spheres, the attraction rarely exceeds k T. For articulated macromolecules, the configurational entropy of the chains is decreased in the neighbourhood of the interfaces and this provides a source of non-zero values for w(x,d). Asakura and Oosawa (1954) approximated this entropy decrement by analysing the problem in terms of the classical theory (Carslaw, 1921) for diffusion in a vessel with walls that absorb diffusing particles. The end result for parallel flat plates of area A is... 

Asakura and Oosawa (1958) showed that the force becomes stronger in solutions of chain molecules (or of molecules of dissymmetric shape) than in solutions of rigid spherical macromolecules (e.g. globular proteins) at the same net concentration. If the macromolecules are charged (i.e. polyelectrolytes), the force can be greatly intensified. 

The aggregation process. Asakura and Oosawa (1958) also presented formulae for calculating the number N(i) of small aggregates composed of i particles ... 

A further quantity calculated by Asakura and Oosawa (1954) is Ae mtical volume concentration 3 of the colloidal particles at which macroscopic aggregation takes place. They showed that for relatively small changes in the depth of the potential energy well into which the particles flocculate (e.g. an increase in well depth of ca 30%), the state of the particles changes from essentially total dispersion to complete aggregation. 

The polymer second virial coefficient A trivial extension of the theory of Asakura and Oosawa(1958) has been given by Sie aff (1959), who included the second virial coefficient (82) in the osmotic pressure (jt ) expression, i.e. 

The particle second virial coefficient Vrij (1976) and de Hek and Vrij (1981) have used the Asakura and Oosawa potential between colloidal particles suspended in solutions of rigid spheres to derive an expression for the second virial coefficient of the particles. They calculated the force on the particles from... 

The theories of Feigin and Napper (1979) and Scheutjens and Fleer (1982) are qualitatively different from those of Asakura and Oosawa (1954 1958) (and subsequent elaborations thereof) and de Gennes and coworkers (Joanny et ai, 1979 de Gennes, 1981 1982) in that they predict not only depletion flocculation but also depletion stabilization. For this reason, presentation of the former two omnibus theories will be delayed until their predictions with regard to both depletion stabilization and depletion flocculation are elaborated. The de Gennes approach, which does not predict the occurrence of depletion stabilization, will be presented at this juncture. 

In the first version of the theory expounded by Sato (1979), these solvation layers were assumed to be irreversibly adsorbed. This was subsequently modified (Sato and Sieglaff, 1980) to permit reversible adsorption. The latter allows the solvation layers to undergo mutual interpenetration on close approach of the particles. Interpenetration was postulated to produce desorbed solvent that dilutes the polymer in the bulk phase. This dilution is reminiscent of that first postulated by Asakura and Oosawa (1954) for quite different reasons. Dilution leads to an attraction between the particles. 

Finally, we note that Sperry (1982) has recently used the theory of Asakura and Oosawa (1954 1958) to explain the results obtained by Sperry et al. (1981) for the flocculation of polystyrene latex particles by hydroxyethyl cellulose. Although semi-quantitative agreement was claimed between theory and experiment, it should be noted that the theory contains too many adjustable parameters to allow a satisfactory comparison to be made with experiment. 

Asakura and Oosawa (5) first identified depletion as a mechanism for generating an attractive interparticle potential. Numerous elaborations of their simple model followed, including sophisticated lattice and self-consistent field theories. Recently, Evans (27) resolved some inconsistencies in the evaluation of the effective pair potentid between the original niave model and the subsequent detailed analyses and achieved quantitative consistency between predictions and the detailed experiments employing bilayer membranes in a micropipette device. 

Depletion interactions are interactions between two surfaces (particles) in the presence of firee, i.e., non-adsorbed, macromolecules, micelles, or very fine particles. Asakura and Oosawa (1954) first pointed out that if the distance between two surfaces h is smaller than the diameter of solute molecules Jm, this region will contain pure solvent depletion zone, cf. Fig. 3.14, left). Thus, an attractive force corresponding to the osmotic pressure of the bulk solution is acting on the two surfaces. Agglomeration caused by this effect is called depletion flocculation. In a second paper, the authors calculated the potential energy of this interaction for... 

We note that in the original paper of Asakura and Oosawa [54], where expression (1.21) was first derived, the polymers were regarded as pure hard spheres. Vrij [40, 56] arrived at the same result by describing the polymer chains as penetrable hard spheres, see Sect. 2.1. Inspection of (1.21) and (1.22) reveals that the range of the depletion attraction is determined by the size 2S of the... 

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