Ideal Gaussian chain

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... 

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. 

Monte-Carlo simulations are applied to estimate the characteristic ratios and p parameters from the RIS models for PE, POM, polybutadiene, and polyisoprene. Here the p parameter is defined as the ratio of the radius of gyration to the hydrodynamic radius. The p parameters of these real chains in the unperturbed state show only a slight dependence on the microconformation in the limit of large molecular weights and are found close to 1.504, which is the value for an idealized Gaussian chain. The estimated p parameters of the real chains appear to be correlated to the chain stiffness and increase with the characteristic ratios. 

The reduced scattering vector is given by using the characteristic scale screening of Coulombic interaction by ideal Gaussian chains, r0, as follows... 

Note that. In a sense, the polyelectrolyte behaves now as If it were an "ideal" (Gaussian) chain with a relatively small number of Kuhn lengths. This coil size Is determined by the local stiffness, but not by long-range excluded volume. Should q Increase even furher (approaching L), then the wormlike chain would be better described by a slightly curved rod (as expressed by (5.2.21)) than by a random-flight chain. 

The freely jointed chain model describes ideal Gaussian chains and does not account for interactions between monomers that are not necessarily close neighbors along the backbone. Including these interactions will give a different scaling behavior for long polymer chains. The end-... 

We now consider an ideal Gaussian chain confined between two Garge) flat plates with area A at a plate separation h, see Fig. 2.9. 

A standard ideal Gaussian chain can be viewed as a sequence of point-like beads connected by bonds, l, = r(.,i-r,. The statistical distribution of the bond vector I is defined by the 1-dependence of the bond energy Ub = Lfb(l) by virtue of the Boltzmann statistics, the probability density for I is... 

Consider a branched polymer stmeture that does not involve any cycles (an example of such polymer tree strurture is shown in Figure 1(g)). Each strand (arm) of the polymer is an ideal Gaussian chain. For each pair i, J of units (heads) in this structure there is one and only one connerting linear strand hence (] ) -n] ) = b riip where n,j is the number of bonds in the connecting strand. Equation [20] then simplifies to... 

DDFT is able to capture the effect of electrostatic interactions through the Flory-Huggins-type interaction of the ideal Gaussian chains however, a more immediate solution comes from the ideas of Donnan (for a review, see Reference 300). Here, all ions are assumed to be in equilibrium, leading to conditions of perfect saeening and, therefore, the computationally convenient cancellation of all long-range forces. 

Here is the partition functional for the ideal Gaussian chains in the external field Uj, and is the contribution from the nonideal interac-... 

The aqueous solution of triblock copolymer Pluronic L64 EO PO qEOi [23] is modeled as a compressible system consisting of ideal Gaussian chain molecules in a mean field environment. The free energy is a functional of a set of particle concentrations p [11,12] ... 

Theta solvent A solvent in which a polymer molecule behaves as an ideal Gaussian chain. 

Theta temperature The temperature at which a polymer in solution behaves like an ideal Gaussian chain. 

We first consider numerical calculations for an ideal Gaussian chain at infinite dilution with the isotropic Forster transfer rate [10]. The three-body Fade is given by... 

The main results for the dynamics of dilute solutions reflect the importance of hydrodynamic interactions each moving monomer in the solvent creates a backflow field which decays very slowly with distance. In a semi-dilute solution, the interference between all these velocity fields induces a screening of the backflow field of a given monomer, which falls off exponentially after a characteristic distance A. This idea was originally proposed by Edwards and Freed we shall briefly summarize their theory for ideal Gaussian chains. 

In a thermodynamically poor solvent at the 0 temperature where A2 = 0, a linear polymer molecule can be regarded as an ideal Gaussian chain. However, in a thermodynamically good solvent, a polymer molecule is expanded by the excluded-volume effect and in a poor solvent the polymer molecule contracts into a more compart form in order to reduce unfavorable contact with the solvent. We write the molecular weight dependence of Rg in a power-law form ... 

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