Gaussian polymers

The classical statistical theory of rubber elasticity1) for a Gaussian polymer network which took into account not only the change of conformational entropy of elastically active chains in the network but also the change of the conformation energy, led to the following equation of state for simple elongation or compression 19-2,1... 

Fig. 2. Heat and internal energy changes as a function of deformation at P, T = const, for a Gaussian polymer networl7> 8). A — heat B — internal energy (d In (r2>0/dT < 0) C — internal energy (d In < r2 >0/dT > 0). The arrows indicate inversion points (see text)...

Schwarz31) has modified the equation of state for the Gaussian polymer network by introducing a coefficient taking into account the influence of the packing of chains and their sizes on the mean square end-to-end distance of deformed chains. 

At equilibrium the osmotic pressure is equal in the two phases and the electrostatic osmotic pressure is balanced by the excluded volume osmotic pressure in the complex that for Gaussian polymers (at the 6 point) reads IIct=w2c3/2. The concentration in the complex then reads... 

Fully Swollen/Collapsed Gaussian Polymer Blocks.92... 

For small expansions of Gaussian polymer chains the expansion parameter, Oif), for the hydrodynamic chain dimensions is given by a closed expression of the form... 

The possibihty of being able to change easily and controllably the film thickness by varying the ccaicentration of the solution used for spin-coating opened up a whole field of research on questions related to the chain-like nature of these macromolecules. In particular, at that time it was unclear (and partially still is ) if polymer properties like viscosity, chain conformation (as expressed for example by the radius of gyration), chain orientation, and interdiffusion rate, or mechanical properties and Tg change once the thickness of the film decreases below the diameter of Gaussian polymer coils in bulk samples [1]. 

For small wave numbers the polymer acts like a point scatterer, while in the intermediate wave number regime the polymer behaves like a flexible, Gaussian polymer, and for the largest wave numbers the polymer can be viewed as a stiff rod. 

The universality of the Gaussian polymer is actually a consequence of the Central limit theorem. Suppose we construct a flexible polymer from bonds with independent probability distribution iP t) for a bond vector r. The end-to-end distance is given by R= Ei r, so that the probability density of R is... 

Before ending this section we would like to point out the relationship between the statistical properties of a polymer chain and a variety of other physical problems. The partition sum, as written in Eq. (5), is in the form of a path integral over all possible chain conformations. Since many physical problems can be formulated in terms of path integrals, we can map the problem of a Gaussian polymer in a random media to a wide... 

In this section we discuss the static properties of a Gaussian polymer chain without excluded volume interactions that is confined to a medium populated with quenched... 

For a linear Gaussian polymer Rg is easily calculated. From eqns (2.42)... 

Consider a system composed of a large number N of equivalent gaussian polymer chair s enclosed in a box of volume V = 5 . The individual polymer chains are taken to be ideal so all intramolecular long-range interactions can be neglected. The monomer density is... 

For a solution of a Gaussian polymer with polydispersity, eqns (5.29) and (5.32) are modified as ... 

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 ... 

Figure 1.3 Brownian motion of a colloidal particle results from molecular collisions, leading to a path that is a random walk. The statistical analysis for the conformation of a Gaussian polymer chain is the same (Section 2.3.2)...

The change of free energy per segment AG, which is estimated through Gaussian polymer chains on an incompressible (pj + (pg= ) lattice [79,80],... 

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