Polymer concentration depletion

At even higher polymer concentrations, depletion flocculation is observed and. Anally, at the highest polymer concentrations depletion stabilization is observed. 

The adliesion and fiision mechanisms between bilayers have also been studied with the SEA [M, 100]. Kuhl et al [17] found that solutions of short-chained polymers (PEG) could produce a short-range depletion attraction between lipid bilayers, which clearly depends on the polymer concentration (fignre Bl.20.1 It. This depletion attraction was found to mduce a membrane fusion widiin 10 minutes that was observed, in real-time, using PECO fringes. There has been considerable progress in the preparation of fluid membranes to mimic natural conditions in the SEA [ ], which promises even more exciting discoveries in biologically relevant areas. 

A non-adsorbing polymer in solution can also destabilise a dispersion through a mechanism called depletion flocculation. When polymer molecules do not interact favourably with the particle surfaces from an enthal-pic perspective, they are repelled from the surface regions due to entropic reasons. A depletion zone around the particles is created which has a lower average polymer concentration than the bulk solution. The osmotic... 

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 2. Attraction energy per surface site due to depletion as a function of the plate separation, for two polymer concentrations. The distance is expressed in lattice units (step length 1). r = 1000, x = 0.5, hexagonal lattice.

Equation 8 was also applied by Sperry (12), although the underlying assumptions are different in his model. There is also a close analogy between Equation 8 and the pair potential used by De Hek and Vrij. Indeed, Equation 4 of Ref. 6 reduces to our Equation 8 for H = 0, provided that 2A is interpreted as the hard sphere diameter of the polymer molecule. Hence, in dilute solutions (where A a rg) the two approaches are very similar. However, in our model A is a function of the polymer concentration. Because most experimental depletion studies are carried out at values for that are comparable in magnitude to <)>, our model... 

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.

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. 

Here, A is the depletion layer thickness (assumed equal to the radius of gyration of the polymer, RG). H = r - 2a is the surface-to-surface particle separation, V ° is the molar volume of the solvent, and ji and ji are the solvent chemical potentials for the polymer solution and the pure solvent. It appears that the assumption A = RG is generally acceptable providing that the polymer solution is in the dilute concentration regime. At higher polymer concentrations, however, the value of A is reduced according to the relationship (Vincent, 1990) ... 

Vincent, B. (1990). The calculation of depletion layer thickness as a function of bulk polymer concentration. Colloids and Surfaces, 50, 241-249. 

At moderate to high polymer concentrations, the free polymer chains in the solution may begin to exercise an influence. One such effect is the so-called depletion flocculation caused by the exclusion of polymer chains in the region between two particles when the latter are very close to each other (i.e., at surface-to-surface distances less than or equal to approximately the radius of gyration of the polymer chains). The depletion effect is an osmotic effect and is discussed further in Section 13.6. 

At high polymer concentrations, one may also have what is known as depletion stabilization. The polymer-depleted regions between the particles can only be created by demixing the polymer chains and solvent. In good solvents the demixing process is thermodynamically unfavorable, and under such conditions one can have depletion stabilization. 

Because of restrictions on the number of possible configurations, non-adsorbing polymers tend to stay out of a region near the surfaces of the particles, known as the depletion layer. As two particles approach, the polymers in the solution are repelled from the gap between the surfaces of the particles. In effect the polymer concentration in the gap is decreased and is increased in the solution. As a result, an osmotic pressure difference is created which tends to push the particles together. The resulting attractive force is the reason for depletion flocculation. In contrast to this, depletion stabilisation has been mentioned above. 

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. 

Figure 17. Schematic cross sections of a photopolymer material in which a grating pattern is being written, (a) The image exposure forms polymer and depletes the monomer concentration, (b) Additional monomer diffuses into the exposed areas, (c) An overall development exposure completes the polymerization to give a polymer with a modulated density.

Most food systems are of a colloidal as well as a polymeric nature. The presence of a nonadsorbing polymer in a skim milk dispersion induces an effective attraction between the casein particles, called depletion interaction, resulting in phase separation at sufficiently high polymer concentration. Tuinier et al. (2003) discussed the influence of colloid-polymer size ratio, polymer concentration regime, size, poly-dispersity and charges in colloid/biopolymer mixtures, demonstrating the challenging complexity of the subject. 

Figure 10.18 is a schematic representation of depletion stabilization in which the polymer is prevented from the zone of close approach between two particles. As a result of this low polymer concentration between the particles due to size exclusion, there is a lower osmotic pressure, which results in (1) an attractive force for greater than theta solvents and (2) a repulsive force for less than theta solvents. Theta solvents will be discussed in the section on the thermodynamics of polymer solutions, but first a discussion of pol3naaer properties. 

A similar, and even more dramatic, viscosity enhancement was observed by Buscall et al. (1993) for dispersions of 157-nm acrylate particles in white spirit (a mixture of high-boiling hydrocarbons). These particles were stabilized by an adsorbed polymer layer, and then they were depletion-flocculated by addition of a nonadsorbing polyisobutylene polymer. Figure 7-9 shows curves of the relative viscosity versus shear stress for several concentrations of polymer at a particle volume fraction of 0 = 0.40. Note that a polymer concentration of 0.1 % by weight is too low to produce flocculation, and the viscosity is only modestly elevated from that of the solvent. For weight percentages of 0.4-1.0%, however, there is a 3-6 decade increase in the zero-shear viscosity ... 

Keep in mind that IKmin is negative.) The potential well depth was calculated using Eq. (7-8) for depletion flocculation. At the highest polymer concentrations, the dimensionless well depth reaches — 18. (At polymer concentrations high enough that... 

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