Flory 0-solvent chain dimensions

Following Flory (1969), a 0 solvent is a thermodynamically poor solvent where the effect of the physically occupied volume of the chain is exactly compensated by mutual attractions of the chain segments. Consequently, the excluded volume effect becomes vanishingly small, and the chains should behave as predicted by mathematical models based on chains of zero volume. Chain dimensions under 0 conditions are referred to as unperturbed. The analogy between the temperature 0 and the Boyle temperature of a gas should be appreciated. 

Flory has recently summarized the experimental evidence pertaining to local correlation and their effects on chain dimensions (49). There is experimental support for local alignment from optical properties such as stress-optical coefficients in networks (both unswelled and swelled in solvents of varying asymmetry), and from the depolarization of scattered light in the undiluted state and at infinite dilution. The results for polymers however, turn out to be not greatly different from those for asymmetric small molecule liquids. The effect of... 

As pointed out in Chapter III, Section 1 some specific diluent effects, or even remnants of the excluded volume effect on chain dimensions, may be present in swollen networks. Flory and Hoeve (88, 89) have stated never to have found such effects, but especially Rijke s experiments on highly swollen poly(methyl methacrylates) do point in this direction. Fig. 15 shows the relation between q0 in a series of diluents (Rijke assumed A = 1) and the second virial coefficient of the uncrosslinked polymer in those solvents. Apparently a relation, which could be interpreted as pointing to an excluded volume effect in q0, exists. A criticism which could be raised against Rijke s work lies in the fact that he determined % in a separate osmotic experiment on the polymer solutions. This introduces an uncertainty because % in the network may be different. More fundamentally incorrect is the use of the Flory-Huggins free enthalpy expression because it implies constant segment density in the swollen network. We have seen that this means that the reference dimensions excluded volume effect. 

In summarizing the intrinsic viscosity relations presented in this section, it must be admitted that they represent nothing more than rather small semi-empirical refinements of the Flory excluded volume theory and the Flory-Fox viscosity theory. For a large fraction of the existing body of experimental data, they offer merely a slight improvement in curve-fitting. But for polymers in good solvents it is believed that a more transcendental result has been achieved. The new equations permit more reliable assessment of unperturbed chain dimensions in such cases, and in some instances (e. g., certain cellulose derivatives see Section III B) they offer possible explanations of heretofore paradoxical solution behavior. 

The unperturbed dimensions of various condensation polymers obtained by the present method are listed in Table 10. A polyelectrolyte chain, sodium polyphosphate, has been included because theta-solvent results are available. The freely-rotating chain dimension (Lzyof of poly(dimethylsiloxane) in the table is due to Flory and his coworkers (705), that for the polyphosphate chains is taken directly from the paper of Strauss and Wineman 241 ), while most of the others have been calculated in the standard manner with the convenient and only negligibly incorrect assumption that all the aliphatic bond angles are tetrahedral. The free-rotation values for the maleate and fumarate polyesters are based on parameters consistent with those of Table 6 for diene polymers. 

A polymer molecule moving in a dilute solution undergoes frictional interactions with solvent molecules due to its motion relative to the surrounding medium. The effects of these frictional interactions are related t6 the size and shape of the polymer molecule. Thus, the chain dimensions of polymer molecules can be evaluated from measurements of their frictional properties (Flory and Fox, 1950). 

Flory (1969) has argued that the occupied-volume exclusion (repulsion) for an isolated chain is exactly balanced in the bulk state by the external (repulsive) environment of similar chains, and that the exclusion factor can therefore be ignored in the solid state. Direct observation of single-chain dimensions in the bulk state by inelastic neutron scattering gives values fully consistent with unperturbed chain dimensions obtained for dilute solutions in theta solvents (Cotton et al., 1972), although intramolecular effects may distort the local randomness of chain conformation. 

As repeatedly noted thus far, the residual ternary cluster interaction energy is expected to play a role in poor solvents near 9. Thus, since the pioneering work of Orofino and Flory [13] many theoretical studies have been made to clarify its effects on chain dimensions and the second virial coefficient. However, they have turned out to reveal some serious problems, which still remain unsolved. 

With regard to differences in polymer behavior in solution versus the bulk state, several points must be made. Clearly, it is now well-established that the choice of theta solvent can affect chain dimensions to some extent [42-44, 46, 47]. Hence, only the chain in an amorphous melt of identical neighbors can be considered to be in the unperturbed state. Particularly striking are some of the differences noted in temperature coefficients measured by different techniques. Is it possible that the thermal expansion of a polymer molecule is fundamentally different in the bulk and in solution Can specific solvent effects exist and vary in a systematic way within a series of chemically similar theta solvents Does the different range of temperatures usually employed in bulk versus solution studies affect K Are chains in the bulk (during SANS and thermoelastic experiments) allowed adequate time to completely relax to equilibrium All of these issues need further attention. Other topics perhaps worthy of consideration include the study of the impact of deuterium labelling on chain conformation (H has lower vibrational energy than does H ) and the potential temperature dependence of the Flory hydrodynamic parameter . 

Flory s viscosity theory also furnishes confirmation of the w temperature as that in which a=V.2, and it permits the determination of the unperturbed dimensions of the Polymer chain. Even if a Q solvent is not available, several extrapolation techniques can be used for the estimating the unperturbed dimensions from viscosity data in good solvents. The simplest of these techniques seems to be that of Stockmayer. 

We have omitted a great deal of detail in this discussion of polymer viscosity. The interested reader will find some of the missing information supplied in Flory (1953). In particular, we have omitted all numerical coefficients, which limits us to ratios as far as computational capability is concerned. Numerical coefficients are available for Equation (92), for example, and this allows coil dimensions to be evaluated from viscosity measurements. A general conclusion that unifies all of this section is that any factor that causes a polymer chain to be more extended in space —whether by coil unfolding or swelling by solvent —tends to increase [77]. This is exactly what we expect in terms of the purely qualitative picture provided by Figure 4.8. Example 4.6 illustrates this for some actual polymers. 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

In a sufficiently poor solvent at a given temperature, the condition where a = 1 can be achieved, and the chain attains its unperturbed dimensions. This turns out to be the 6 temperature of Flory and Krigbaum previously described in Section 5.5 of this chapter. 

Ceescenzi, V., and P. J. Flory Configuration of the poly(dimethyl siloxane) chain. II. Unperturbed dimensions and specific solvent efiects. J. Am. Chem. Soc. 86. 141 (1964). 

According to Flory s definition, a chain has its unperturbed dimensions in a 0 solvent where the osmotic second virial coefficient is zero (55). 

A. Sariban and K. Binder (1988) Phase-Separation of polymer mixtures in the presence of solvent. Macromolecules 21, pp. 711-726 ibid. (1991) Spinodal decomposition of polymer mixtures - a Monte-Carlo simulation. 24, pp. 578-592 ibid. (1987) Critical properties of the Flory-Huggins lattice model of polymer mixtures. J. Chem. Phys. 86, pp. 5859-5873 ibid. (1988) Interaction effects on linear dimensions of polymer-chains in polymer mixtures. Makromol. Chem. 189, pp. 2357-2365... 

The quality of solvent, reflected in the excluded volume v, enters only in the prefactor, but does not change the value of the scaling exponent u for any v > 0. The Flory approximation of the scaling exponent isu = 3/5 for a swollen linear polymer. For the ideal linear chain the exponent = 1/2. In the language of fractal objects, the fractal dimension of an ideal polymer is V — l/i/ = 2, while for a swollen chain it is lower T> — I/u = 5/3. More sophisticated theories lead to a more accurate estimate of the scaling exponent of the swollen linear chain in three dimensions ... 

Flory (1953) has presented a celebrated theory of the excluded volume effect that relates the expansion factor to the thermodynamic properties of the polymer-solvent system. Basically what Flory calculated was the free energy of mixing of polymer segments with solvent that is associated with the expansion of the coil dimensions in a good solvent. Such expansion is opposed, however, by the loss of configurational entropy of the chain. The latter corresponds, of course, to an elastic contractive force. Expansion proceeds until the two opposing effects are in equilibrium. Flory s result was... 

Another important statement of French authors from their neutron scattering experiments was that the size of polystyrene strands in dry model networks is equal to the size of the precursor chains dissolved under -conditions (see Table 1.5). At first sight, this result confirms the idea [136, 137], expressed by Flory [138] in the middle of the last century and later picked up by De Gennes [139], that in bulk polymer the chain retains its unperturbed dimension characteristic of its solution in a -solvent. Flory believed to have experimentally confirmed his idea [140]. By analyzing elastic neutron scattering from a mixture of linear protonated and deuterated polyisobutylene with a molecular weight of 48,000+ 3,000 Da, he found that the gyration radii of the macromolecules... 

It seems appropriate to discuss here the probabUity of interpenetration of polystyrene coils in the model networks. As already mentioned, according to the theoretical considerations of Flory [138] and De Gennes [139], polymeric coils in an amorphous solid state retain unperturbed dimensions. Since the volume fraction of the polymer in an unperturbed coil under -conditions is weU known to be very smaU, only about 2%, the transition from swoUen coils to solid state has to be accompanied by the replacement of aU solvent molecules with fragments of other polymeric molecules. In other words, theoretical notions predict extremely high mutual interpenetration of the polymeric chains in bulk state. Indeed, in order to maintain the coil dimension that is characteristic for a -solution, the coil must accommodate, on removing the solvent, a 50- to 100-fold amount of alien polymeric matter. In the 1970s this problem was discussed in fiiU [149-165], The authors of the tailor-made networks also took part in the discussion. 

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