Perturbation theory polymer chains

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

We should not close this section without touching upon some delicate technical point. A priori even for a discrete chain the critical chemical potential g s, defined as chemical potential per segment of an infinitely long chain, does not exist outside the -expansion. We already encountered this problem in Sect. 7.2, where we found that rC proportional to p, for d < 4 in naive perturbation theory suffers from infrared singularities. That problem has been considered in the field theoretic framework. The results, expressed in polymer language, show that the u-expansion is not invalidated, as long as we consider quantities which do not involve explicitly. Almost all the quantities of interest to us are of this type. Only in the equation of state relating gp(n) and cp(n) does an explicit contribution g n occur. But even... 

Flory-type free energy calculations show that the root mean square end-to-end distance of a polyelectrolyte increases linearly with the chain length at infinite dilution and without added salt [40]. Using the above perturbation theory, scaling relations at finite densities are obtained. The influence of the interaction with other polymer chains, counterions, and added salt is captured in the Debye screening length xT1. 

Experimentally the overall size of the polymer chain can be studied by light scattering and neutron scattering. A great deal of theoretical work is present in the literature which tries to predict the properties of mixtures in terms of their components. The analytical model by Rouse-Zimm [85,86] is one of the earliest works to derive fundamental properties of polymer solutions. Advances were made subsequently in dilute and concentrated solutions using perturbation theory [87], self-consistent field theory [88], and scaling theory [89],... 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

In the case of flexible polymer chains, the dominant problem for theory has been the understanding of the statistical properties of self-avoiding random walks in 2, 3 and 4 dimensions. The case of D>3 arises of course because the self-avoiding interaction only becomes perturbative in 4 dimensions and... 

The SAFT equation of state was proposed by Radosz, Gubbins, Jackson, and Chapman and is a model derived based on the perturbation theory of Weirtheim. SAFT is a noncubic equation with separate terms for the various effects (dispersion, polar, chain, hydrogen bonding). SAFT has already found extensive application in both polymer and oil industry, where different capabilities of the model have been exploited. In the oil industry, it is used for describing the complex multiphase equilibria of hydrogen bonding multicomponent systems, e.g., water-oil-alcohols (glycols). Several recent reviews of the SAFT equation of state are available, all of which present results for polymer solutions. 

The final results of many perturbation theories (often called two-parameter theories) contain the combinations Nb and N" /3 as the only sample dependent quantities. Here N is the number of segments, each of length b, in an equivalent freely jointed polymer chain, related by (r )o = Nb" (see p. 64), and /3, the so-called excluded volume integral, is the volume excluded by one chain segment to another. 

The second virial coefficient of polyelectrolytes is treated theoretically either by applying the theory for charged spherical colloids with a correction for the chain character of the polyion or as an extension of the theory of the second virial coefficient for nonionic linear polymers. An example of an extension of the theory of the second virial coefficient for nonionic linear polymers to polyelectrolytes is the above-mentioned Yamakawa approach of using perturbation theory of excluded volume to calculate A2 [42],... 

For the calculations, different EoS have been used the lattice fluid (LF) model developed by Sanchez and Lacombet , as well as two recently developed equations of state - the statistical-associating-fluid theory (SAFT)f l and the perturbed-hard-spheres-chain (PHSC) theoryt ° . Such models have been considered due to their solid physical background and to their ability to represent the equilibrium properties of pure substances and fluid mixfures. As will be shown, fhey are also able to describe, if not to predict completely, the solubility isotherms of gases and vapors in polymeric phases, by using their original equilibrium version for rubbery mixtures, and their respective extensions to non-equilibrium phases (NELF, NE-SAFT, NE-PHSC) for glassy polymers. 

Flocculation has also been observed when the diluent of the dispersion is a good solvent for the chains constituting the steric barrier. This is observed when free polymer, even when it is identical to that constituting the steric barrier, is dissolved in the diluent of the dispersion [3.77]. The magnitude of flocculation is a function of both the molecular mass and the concentration of the added polymer. The depletion flocculation theory suggests that this phenomenon is related primarily to the perturbation of the free polymer chains in solution rather than those in the solvated sheath [3.78]. 

HON Hong, S.-U. and Pretel, E.J., Predicting the solubility of l,l-difluoroethane in polystyrene rrsing the perturbed soft chain theory, Polym. Int., 45, 55, 1998. 

Today, there are two principal ways to develop an equation of state for polymer solutions first, to start with an expression for the canonical partition function utilizing concepts similar to those used by van der Waals (e.g., Prigogine, Flory et al., Patterson, Simha and Somcynsky, Sanchez and Lacombe, Dee and Walsh,Donohue and Prausnitz, Chien et al. ), and second, which is more sophisticated, to use statistical thermodynamics perturbation theory for freely-jointed tangent-sphere chain-like fluids (e.g., Hall and coworkers,Chapman et al., Song et al. ). A comprehensive review about equations of state for molten polymers and polymer solutions was given by Lambert et al. Here, only some resulting equations will be summarized under the aspect of calculating solvent activities in polymer solutions. 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

Although the above-mentioned perturbation theories account for the formation of chains in the repulsive contribution, the dispersion is still considered as resulting from the attraction of unbonded chain segments. This assumption is especially not justified in the case of polymer molecules where the segments do not interact independently but are influenced by the neighboring segments of the same molecule. Several attempts have been made to overcome this deficiency. Various models were suggested which use the square-well sphere (see, e.g.,... 

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