Depletion thickness

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

Figure 1. Illustration of depletion effects for two plates in a solution of nonadsorbing polymer of volume fraction < >. A is the depletion thickness.

Figure 3. Depletion thickness A for four chain lengths as a function of polymer concentration. The arrows indicate the solution concentration where the polymer coils begin to overlap, x = 0.5, hexagonal lattice.

Figure A. Overlap of depletion zones. The hatched region corresponds to the volume of solvent that is transferred when two particles of radius a and depletion thickness A come into close contact.

Our model predicts destabilization of colloidal dispersions at low polymer concentration and restabilisation in (very) concentrated polymer solutions. This restabilisation is not a kinetic effect, but is governed by equilibrium thermodynamics, the dispersed phase being the situation of lowest free energy at high polymer concentration. Restabilisation is a consequence of the fact that the depletion thickness is, in concentrated polymer solutions, (much) lower than the radius of gyration, leading to a weaker attraction. 

Afj free energy difference per particle, due to depletion, between the floe and the dispersion Ass configurational entropy difference per particle between the floe and the dispersion A depletion thickness... 

It is instructive to compare how the ion-interface interactions (image force, van der Waals interactions, ion hydration) affect the depletion thickness at various electrolyte concentrations. To estimate the magnitudes of these effects, it will be assumed that the interactions are non ion-specific (which implies to use Eq. (6a) for the potential of the image force, and B1=B2=B in Eq. (8) for the van der Waals interactions). Consequently, the interface is neutral and the surface adsorption can be calculated from the simple expression ... 

As that generated by ion hydration for c/, =d2, the depletion thickness generated by the van der Waals interactions, when Bx =B2, does not depend on concentration. When only the van der Waals interactions are accounted for, Eqs. (8) and (18) lead... 

There is no reason to assume that B, = B2 or that dl=d2. However, when the surface potential is low (hence in the absence of a surface charge and at sufficiently low electrolyte concentrations), Eq. (18) can be used to calculate separately the depletion thicknesses for each kind of ions. 

For cE- 0, the surface potential generated by the asymmetry of the ion concentrations vanishes, the anions are uniformly distributed from d, to infinity, and the total depletion thickness is given by the sum of the depletion thicknesses of each kind of ions, calculated independently. 

To illustrate the above discussions, the surface potential and the surface tension are plotted vs. electrolyte concentration in Fig. 6a and b, respectively, for dx = d2 = dE=5 A and various values for W,. The Poisson-Boltzmann Eq. (30b) was integrated numerically and its solution was employed in Eq. (15) to calculate the surface tension. If the potential well for anions is shallow, the total depletion thickness is positive at any electrolyte concentration and the surface tension always... 

M. Manciu, E. Ruckenstein / Advances in Colloid and Interface Science 105 (2003) 63-101 interface are taken into account, the depletion thickness is given by ... 

From the experimental values of the zeta potentials for air bubbles [32], one could infer that the surface potential should be approximately 0.060 V for 0.001 M NaCl at pH 7. However, this value is much too large to be in agreement with the experiments of Jones and Ray [33]. The depletion thickness vanishes at the minimum of the surface tension, and at this surface potential, the adsorption thickness, due to the Poisson-Boltzmann effect, would be over 200 A and could not be compensated by the other interactions discussed here. The value which had to be selected for N... 

In both the descriptions by De Hek and Vrij and by Gast, Russel and Hall, the depletion thickness S was assumed to be equal to the radius of gyration. This assumption was rationalized by Eisenriegler [118]. He calculated the density profile of ideal chains near a flat surface and from this density profile it follows that S/Rg = 2fy/nRi 1.13 [119], see Sect. 2.2. 

Experimental work on the determination of the depletion layer thickness commenced in this period. The depletion thickness S of polystyrene at a nonadsorbing glass plate was measured using an evanescent wave technique by Aflain et al. [120]. The value found for S was indeed close to the radius of gyration of the polymer. Ausserre et al. [121] measured the depletion thickness of xanthan (a polysaccharide) at a quartz waU below and above the polymer overlap concentration. In dilute solutions, below overlap, S was close to the radius of gyration of... 

Fig. 1.33 Sketch of the overlap zones between flat surfaces (upper) and between roughened surfaces lower. Depletion thickness is indicated by the arrows. Left drawings intermediate overlap of depletion layers. Right drawings large overlap of depletion layers...

Starting from (2.59) we can obtain an analytical expression for the depletion thickness around a sphere 6s, which is now defined by... 

Note that in the limit q 0, (2.62) yields, as expected, the flat plate result 6s/Rg = 2/y/n. The result in (2.62) holds for Gaussian ideal chains, implying the segment size b is smaller than all other length scales, Rg and R. For freely-jointed ideal chains the depletion thickness also depends on the size ratio b/R for R<50b [38]. 

Fig. 2.16 Depletion thickness of ideal chains at a sphere d = 3) as a function of the size ratio q = Rg/a. For comparison the flat wall case (d = 1) is given as straight solid line...

This brings us to the conclusion that as far as the depletion interaction is concerned ideal polymer chains to a good approximation can be replaced by penetrable hard spheres with a diameter a = 2, where the depletion thickness bs now depends on the size ratio q = Rg/R. In dilute polymer solutions the ideal chain description suffices to describe depletion effects. In Chap. 4 we shall see that for polymers with excluded volume the depletion thickness not only depends on the size ration q but also on the polymer concentration, see also [36, 39-41]. Also the (osmotic) pressure is no longer given by the ideal (Van t Hoff) expression. Both features significantly affect depletion effects. 

In Chap. 3 we introduced the phase behaviour of hard spheres mixed with penetrable hard spheres (phs). This provides a starting point for describing the phase behaviour of colloid-polymer mixtures. In Sect. 4.1 we show that the phs-description using penetrable hard spheres is adequate for mixtures in the colloid-limit small q with polymer chains smaller than the particle radius. In Sect. 4.2 we treat the modifications for the case that the polymers are treated as ideal chains. More advanced treatments accounting for non-ideal behaviour of depletion thickness and osmotic pressure for interacting polymer chains enable to also describe intermediate and large q situations. This is the topic of Sect. 4.3. In Sect. 4.4 we qualitatively consider work available on the effects of polydispersity on... 

The first step in taking into account more appropriate polymer physics compared to the simple description of penetrable hard spheres is by considering the polymers as ideal chains. Then one needs to ineorporate the correct depletion thickness of non-adsorbing ideal chains near a eoUoidal hard sphere into free volume theory. 

In Chap. 2 we saw that an analytical expression (2.62) can be derived for the depletion thickness around a sphere due to ideal polymer chains ... 

Fig. 4.5 Depletion thickness of ideal polymer chains around a sphere (solid curve) as a function of the polymer-to-sphere size ratio q. Solid curve (4.2), dashed line is the classical penetrable hard-sphere approach = Rg and the dotted curve follows the approximation 5/R = 0.938 see (4.24)...

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