Polymers Gaussian chains

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

Maurits, N.M., Altevogt, P., Evers, O.A., Fraaije, J.G.E.M. Simple numerical quadrature rules for Gaussian Chain polymer density functional calculations in 3D and implementation on parallel platforms. Comput. Theor. Polymer Sci. 6 (1996) 1-8. 

The normal modes (Rouse modes) that characterize the internal dynamics of the polymer can be computed exactly for a Gaussian chain and are given by... 

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

A basic theoretical model for flexible polymers is the Gaussian chain which assumes N ideal beads with intramolecular distance between them following a Gaussian distribution, so that the mean quadratic distance between two beads separated by n-1 ideal and not correlated bonds is given by [ 15,20]... 

In particular it has been conjectured that the terminal relaxation of star polymers might be the most sensitive test of the dilution exponent P in Go theta solvents suggest a mean value of nearer 2.3 [32]. A physically reasonable scahng assumption for the density of topological entanglements in a melt of Gaussian chains leads to a value of 7/3 [31]. 

The dynamics of a generic linear, ideal Gaussian chain - as described in the Rouse model [38] - is the starting point and standard description for the Brownian dynamics in polymer melts. In this model the conformational entropy of a chain acts as a resource for restoring forces for chain conformations deviating from thermal equilibrium. First, we attempt to exemphfy the mathematical treatment of chain dynamics problems. Therefore, we have detailed the description such that it may be followed in all steps. In the discussion of further models we have given references to the relevant literature. 

The Rouse model starts from such a Gaussian chain representing a coarsegrained polymer model, where springs represent the entropic forces between hypothetic beads [6] (Fig. 3.1). 

The above expressions provide a universal description of the dynamics of a Gaussian chain and are valid for real linear polymer chains on intermediate length scales. The specific (chemical) properties of a polymer enter only in terms of two parameters N =Rl and The friction parameter is gov-... 

Models which also describe the molecular weight between crosslinks for neutral polymer networks but use a non-Gaussian chain distribution have also been derived. These models would be useful in cases of highly crosslinked polymer networks. Examples of these types of models include those of Peppas and Lucht [7], Kovac [8], and Galli and Brummage [9]. 

It must be noted that Eqs. (35) and (36) are for the case in which the crosslinks in the polymer network were introduced in solution as with the Peppas-Merrill equation for neutral hydrogels and also that a Gaussian chain distribution is assumed. The complete equilibrium expressions accounting for the mixing, elastic-retractive, and ionic contributions to the chemical potential for anionic networks in the two cases described above are then... 

Thurston,G.B., Morrison.J.D. Eigenvalues and the intrinsic viscosity of short gaussian chains. Polymer (London) 10,421-438 (1969). 

Smith, K. J., A. Ciferri, and J. J. Hermans Anisotropic elasticity of composite molecular networks formed from non-Gaussian chains. J. Polymer Sci., Pt. A, 2, 1025 (1964). 

All excluded volume theories for branched chains suffer, however, from a principal deficiency since the assumption is tacitly made that all monomeric units in the molecule may have, in principle, the chance to interact with each other. This is, however, a too extensive assumption since for sterical reasons two remote segments can never form a contact. Hence, the excluded volume effect is highly overestimated for densely branched chains. In fact, highly branched polymers show the phenomenon of swelling but no detectable distortion of Gaussian chain behavior117,137,138,179. ... 

Fig. 4.27 SAXS intensity as a function of wavevector for a PS-P1 diblock (Mw = 60 kg mol-1, 17wt% PS) (points) in dibutyl phthalate with a polymer volume fraction

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. 

This assumption is equivalent to considering the polymer molecule as a Gaussian chain. For a Gaussian chain the probability of the two ends colliding in three-dimensional space is proportional to its length to the power -3/2. For the Kuhn (or freely-jointed chain) model the same assumption maybe taken for sufficiently long chains [60]. For linear polymers in good solvents, no similar simple assumption can be adopted. To study cyclization one has to resort to more sophisticated mathematical treatments (see, e.g. [61]). 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

Another prediction of the model is the decrease in the size of the polymer as the salt concentration is increased [48]. If counterion adsorption is dominant, the interactions between the ion pairs are attractive in nature, and leads to the radius of gyration being less than that of a Gaussian chain [48],... 

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