Hard-sphere polymers

Substituting (2.91) in (2.92) after some algebra yields (2.84). Note that in all cases considered so far (penetrable hard spheres, polymers) the quantity [r(h) — r(oo)] was always positive (or zero) for all values for h. Here we see that due to accumulation effects in the concentration profiles [r(/i) — r(oo)] is negative for a certain range of h values. This leads to a positive interaction energy as is clear from (2.92). 

In this chapter we have presented the free volume theory for hard spheres plus depletants and focused on the simplest possible case of hard spheres + penetrable hard spheres. In the next chapters we will extend the free volume theory to more realistic situations (Chap. 4 hard spheres + polymers. Chap. 5 hard spheres -I- small colloidal particles. Chap. 6 hard rods -I- polymers) and compare the results with experiments and simulations. 

Forsman, J. Woodward, C. E. (2003). An improved density functional theory of hard sphere polymer fluids at low density. /, Chem. Phys., Vol. 119,1889-1892 Forsman, J. Woodward, C. E. (2005). Prewetting and layering in athermal polymer solutions. Phys. Rev. Letts., Vol. 94,118301... 

Campbell, A. I. Bartlett, P. Fluorescent hard-sphere polymer colloids for confocal microscopy. J. Colloid Interface Sci. 2002, 256, 325-330. 

Anotlier model system consists of polymetliylmetliacrylate (PMMA) latex, stabilized in organic solvents by a comb polymer, consisting of a PMMA backbone witli poly-12-hydroxystearic acid (PHSA) chains attached to it [10]. The PHSA chains fonn a steric stabilization layer at tire surface (see section C2.6.4). Such particles can approach tire hard-sphere model very well [111. 

Altematively, tire polymer layers may overlap, which increases tire local polymer segment density, also resulting in a repulsive interaction. Particularly on close approach, r < d + L, a steep repulsion is predicted to occur. Wlren a relatively low molecular weight polymer is used, tire repulsive interactions are ratlier short-ranged (compared to tire particle size) and the particles display near hard-sphere behaviour (e.g., [11]). 

Experimentally, tire hard-sphere phase transition was observed using non-aqueous polymer lattices [79, 80]. Samples are prepared, brought into the fluid state by tumbling and tlien left to stand. Depending on particle size and concentration, colloidal crystals tlien fonn on a time scale from minutes to days. Experimentally, tliere is always some uncertainty in the actual volume fraction. Often tire concentrations are tlierefore rescaled so freezing occurs at ( )p = 0.49. The widtli of tire coexistence region agrees well witli simulations [Jd, 80]. 

FIG. 4 Sterically stabilized colloidal particles are coated with short polymer brushes. A hard sphere-like interaction arises. 

A plot of A versus r, the calibration curve of OTHdC, is shown in Fig. 22.2. The value of constant C depends on whether the solvent/polymer is free draining (totally permeable), a solid sphere (totally nonpermeable), or in between. In the free-draining model by DiMarzio and Guttman (DG model) (3,4), C has a value of approximately 2.7, whereas in the impermeable hard sphere model by Brenner and Gaydos (BG model) (8), its value is approximately 4.89. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Hyperbranched poly(ethyl methacrylate)s prepared by the photo-initiated radical polymerization of the inimer 13 were characterized by GPC with a lightscattering detector [51]. The hydrodynamic volume and radius of gyration (i g) of the resulting hyperbranched polymers were determined by DLS and SAXS, respectively. The ratios of Rg/R are in the range of 0.75-0.84, which are comparable to the value of hard spheres (0.775) and significantly lower than that of the linear unperturbed polymer coils (1.25-1.37). The compact nature of the hyperbranched poly(ethyl methacrylate)s is demonstrated by solution properties which are different from those of the linear analogs. 

Before concluding this discussion of the excluded volume, it is desirable to introduce the concept of an equivalent impenetrable sphere having a size chosen to give an excluded volume equal to that of the actual polymer molecule. Two such hard spheres can be brought no closer together than the distance at which their centers are separated by the sphere diameter de. At all greater distances the interaction is considered to be zero. Hence / = for a dey and fa = 0 for a[Pg.529]

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

A preferable system is poly(p-fluorostyrene) doped into poly(styrene). Since rotations about the 1,4 phenyl axis do not alter the position of the fluorine, the F spin may be regarded as being at the end of a long "bond" to the backbone carbon. In standard RIS theory, this polymer would be treated with dyad statistical weights to automatically take into account conformations of the vinyl monomer unit which are excluded on steric grounds. We have found it more convenient to retain the monad statistical weight structure employed for the poly(methylene) calculations. The calculations reproduce the experimental unperturbed dimensions quite well when a reasonable set of hard sphere exclusion distances is employed. 

Figure 5 is an ORTEP computer plot of the first 50 backbone carbons in a typical chain. Only the fluorine atoms of the sidechains are shown. The various hard sphere exclusions conspire dramatically to keep the fluorines well separated and the chain highly extended even without introducing any external perturbations. The characteristic ratio from the computer calculations is about 11.6 from data for poly(p-chlorostyrene), CR = I l.l, poly(p-bromostyrene), CR = 12.3, and poly(styrene), CR = 10.3 (all in toluene at 30°C), we expect the experimental value for the fluoro-polymer to be in the range of 10 to 12. 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

The quantity of primary interest in the study of nonuniform fluids is the density profile of the fluid at a surface. Dickman and Hall [28] reported the density profiles of freely jointed hard-sphere chains at hard walls. Their focus was on the equation of state of melts of hard-chain polymers, and they performed simulations of polymers at hard walls because the bulk pressure, P, can be calculated from the value of the density profile at the surface using the wall sum rule ... 

Interactions of such glassy polymeric particles should resemble the collisions of hard spheres. Phase diagrams of the type shown in Fig. 36 have been obtained for various polymer-organic solvent mixtures [85,94,345-353]. 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

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). 

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... 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

An important conclusion of this discussion is the fact that at very high <)> thermodynamic stability is re-established. Restabilisation is not a kinetic effect, as suggested by Feigin and Napper (10, 11), but is a consequence of lower free energy of the dispersion as compared to the floe. This conclusion is supported by experimental evidence for soft spheres (3, 5, 23). We should add, however, that for hard spheres is so high that experimental verification is difficult for most polymer-solvent systems due to the high viscosity of the solution. 

The only experimental data available to date for a system of hard spheres in a polymer solution are those of De Hek and Vrij (6). 

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