Gaussian chains

The conformation of ideal polymer chains corresponds to that of a random walk. The mutual distance between segments within the chain obey Gaussian statistics [1]. The size of polymer chains is often given in terms of the end-to-end distance, Ro, or the radius of gyration, Rg. The radius of gyration is experimentally accessible from scattering experiments. 

Gaussian chains are characterized by the distribution function of the two segments separated by n segments (see e.g.. Ref. [1])  

Inserting this into the expression for the correlation function in the scattering function, we get the form factor of a Gaussian polymer chain equal the Debye function, g  

Thermodynamics of Polymer Blends and Solutions. Flory-Huggins Theory 

The phase behavior of polymer blends and solutions is, like any other mixtures, governed by enthalpic interactions between the different units and entropic effects, as described in thermcxlynamics [1]. The stable phase is determined from the minimum in free energy. 

We now consider the intensity of independent scattering from a random coil polymer molecule, consisting of (N + 1) beads connected by N bonds and obeying the Gaussian approximation (5.12). We assume that the volume of a bead is uu and the volume of the chain is v = (N + l)uu. Each bead thus contributes po u to the 

P(r) must obviously be related to the end-to-end distance distribution w(N,r) given by Equation (5.12). The exact relationship between the two is obtained as follows. For P(r) we have to include not only the end bead pair separated by N bonds, but all other bead pairs separated by one bond, two bonds, three bonds, etc., and also those separated by zero bonds (i.e., the beads by themselves). In a chain of (N + 1) beads 

Substituting (5.28) in (5.27) and performing the Fourier transform [see Equation (B.17)] give 

Although the summation in (5.29) can be evaluated exactly, we will approximate the summation by an integration. This is valid when N is large, which is the same assumption already incorporated in the Gaussian approximation. On using the variable u = K/N and letting N + 1 = N, (5.29) becomes 

In addition to the above result for the size exponent, several quantities such as the probability distribution function for finding a particular end-to-end distance can be derived for the Kuhn model. In particular, the results become simple if the end-to-end distance is smaller than the chain contour length Nl.ln this limit, for example, the probability distribution function for the end-to-end distance is a Gaussian function. In view of this, a Kuhn model chain with large enough N is called a Gaussian chain. The major properties of a Gaussian chain are now summarized. 

Since a Gaussian chain is a freely jointed chain of large number of Kuhn steps without excluded volume interactions, it does not describe any chain statistics 

This form factor for a Gaussian chain is known as the Debye structure factor. The asymptotic limits of this equation are as in Equation 2.11 with v = 1/2. 

The first term on the right-hand side is a constant independent of R. The second term shows that the chain entropy decreases quadratically with the end-to-end distance due to the reduction in the number of conformations. Therefore, coil-like conformations are entropically favorable, and it would cost energy to expand the chain to rod-like conformations. 

Free energy for a fixed end-to-end distance Substituting the expression for the chain entropy from the above equation, the free energy of a Gaussian chain with end-to-end-distance at R follows from F = E — TS as 

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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. 

Considering the chain dimensions, for very long gaussian chain, one predicts ... 

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... 

For Gaussian chains the spatial structure of the eigenmodes is given by the Rouse form... 

The other limiting case concerns locally smooth chains. For the case of a Gaussian chain in a network of rods, Edwards found d (L/V) 1/2 [66] which agrees with the above discussed binary contact model. Finally, considering worm-like chain bridges the differences in the power laws between the two limiting cases and exponents between — 1 and — 1/2 may be obtained. 

For small deviations from equilibrium, we average r — rm 1 by the statistical segment distribution. For a Gaussian chain which is realized under... 

Fig. 48. Reduced relaxation rates r 1,2/Q3 for stars with Gaussian chain conformation. The insert represents the rates directly. Note that T1 approaches a constant at lower Q. The broken lines give the effective reduced relaxation rates for various contrast conditions. Dashed line shell contrast dashed-dotted line core contrast dotted line average contrast. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)...

Rg)cai2 is the calculated radius of gyration with the parameters for PIP, taken from P.J. Flory (Statistical Mechanics of Chain Molecules, Wiley Intersciences, New York 1969) and the assumption of Gaussian chain statistics. 

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

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... 

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