Swelling in good solvents

In a good-solvent that is not in the athermal limit (0 v b ), Eq. (5.48) must be used for the osmotic pressure  

Recall from Chapter 5 that the crossover concentration (p Ki jb [Eq. (5.36)] denotes the boundary between semidilute and concentrated solutions. For 0 0 chains are nearly ideal in concentrated solutions, whereas for 0 0 chains are swollen on intermediate scales. Network modulus and equilibrium swelling depend on the relative value of preparation and fully swollen concentrations (0o and l/Q) with respect to the crossover concentration 0. Since the swollen concentration is always lower than the preparation concentration (l/Q 0o) there are three 

In the intermediate regime (ii), the chain is ideal in the preparation state Rq but is swollen in the final state [Eq. (5.37)]. The modulus of the swollen gel is calculated from Eq. (7.71)  

Since u = 0.588, the modulus of a gel in the intermediate regime decreases as good solvent is added as G p) v - 0q 0 - . Increasing the excluded volume at constant 0, 0o and T, lowers the modulus because the larger excluded volume only increases 7 ref- 

Balancing this network elastic modulus in the intermediate regime [Eq. (7.88)] with the osmotic pressure [Eq. (7.86)] produces the expression for equilibrium swelling in the intermediate regime  

---

If an inert good solvent is used in solution polymerization, the gel thus obtained will have a supercoiled (expanded) structure (Gel B). Gel B swells in good solvents much more than Gel A which is synthesized in bulk. If the amount of the crosslinking divinyl monomer in the reaction mixture is increased while the amount of solvent remains constant, highly crosslinked networks are formed that cannot absorb all solvent molecules present in the reaction mixture and a heterogeneous structure results (Gel C). A part of the solvent separates from the gel phase during polymerization and the formed Gel C consists of two continuous phases, a gel and a solvent phase. If the amount of solvent is further increased, a... 

The relative swelling in good solvents can also be written as a function of the chain interaction parameter z [Eq. (3.21)]. 

In sttmmary, coils swell in good solvents and collapse in poor solvents. Somewhere in between these limits, tmder -conditions, coils behave ideally. These important resrrlts are strmmarized in Table 1. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

These simplified relationships offer a clearer insight into the dependence of the equilibrium swelling ratio qm on the quality of the solvent as expressed by Xh on the extent of cross-linking. Because of the nature of the approximations introduced to obtain Eqs. (40) and (40 ), their use as quantitative expressions must be limited to networks of very low degrees of cross-linking in good solvents. 

Highly cross-linked rubbers swell less in good solvents than do lightly cross-linked rubbers((188,189). For this reason, swelling in solvents is often used to determine quantitatively the degree of cross-linking. However, in... 

Further development of the Flory-Huggins method in direction of taking into account the effects of far interaction, swelling of polymeric ball in good solvents [4, 5], difference of free volumes of polymer and solvent [6, 7] leaded to complication of expression for virial coefficient A and to growth of number of parameters needed for its numerical estimation, but weakly reflected on the possibility of equation (1) to describe the osmotic pressure of polymeric solutions in a wide range of concentrations. 

Most of the carriers used in organic synthesis are lightly crosshnked gel-type resins. In contrast to macroreticular (macroporous) resins which are characterized by a permanent porosity, the gelatinous resins have to swell in appropriate solvents in order to build up the polymeric network and to make the reactive sites located on the polymeric strands accessible to the reactants. Good swelling properties are therefore essential for these gel-type resins. The functional groups are introduced either by copolymerization with functionalized monomers or by a posteriori functionalization of the polymeric component. Thus the reactive sites are statistically distributed on the polymer chains and more than 99% are positioned inside the polymeric beads and not on the surface. 

The general relation for the equilibrium swelling in any solvent has three branches that correspond to the good solvent case [Eq. (7.87)], the intermediate case [Eq. (7.89)], and the -solvent case [Eq. (7.76)]. 

Figure 4. Schematic illustration of the aggregation and swelling of HD AM in poor and in good solvents.

The structure and the behaviour of polymer chains in solutions have, since the 1930s, been the object of intensive experimental work. The main effort has been carried out on the isolated chain in good solvent, and the swelling caused by repulsive interaction between monomers. The effect had already been predicted by Kuhn, in an article published in 1934, describing the spatial occupancy of chains (Raumerfullung). The swelling of a rubber (cross-linked chains) in a solvent can be observed directly with the naked eye. In the case of linear chains, the observation of all aspects of this effect has, however, required the use of instrumentation which has grown heavier, year by year. 

In good solvents, the quantity zCS3/2 oc S2 is much greater than g CXi oc S3v. Hence, for d = 3, the theory which accounts for the chain swelling predicts for given p and S, a much lower pressure than the simple-tree approximation. 

The law (16.1.1) has been established by observing the swelling of the chain in good solvent and by interpreting the experiments in the framework of a one-parameter theory, this parameter being... 

Tf can be determined by measuring the temperature at which the two-body interaction b vanishes. We showed in Chapter 15 how values of b can be deduced from the swelling of a chain in good solvent. By letting the temperature T vary in good solvent, one obtains, in the zero concentration limit, a function b T) which is linear with respect to /T (see, for instance, Fig. 15.9). The extrapolation for b - 0 gives a value of Tr defined by ... 

Critical exponent for the swelling of a polymer chain in good solvent this exponent is the inverse of the Hausdorff dimension of a Kuhnian chain it is also the exponent v of the Land-au-Ginzburg model for n = 0... 

---