Gaussian polymer coils

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]. 

In this context, it is instructive to have a look at some previous studies on spin-cast thin polymer films [10-13, 15-18,47]. Being able to prepare smooth thin polymer films of variable thickness allowed to study questions related to the chain-like nature of macromolecules. In particular, it has been speculated that polymer properties like viscosity, chain orientation, interdiffusion rates, or mechanical properties and the glass transition temperature may change once the thickness of the film decreased below the diameter which Gaussian polymer coils have in bulk samples. 

Figure 3 Log-log plots of l q) vs. gfor the unified function for (a) Debye equation for Gaussian polymer coils (eqn [3]) using S= 2S//. Adapted from Beaucage, G. J. Appl. Cryst. 1996, 29,134-146, Figure 1. ...

A polymer coil does not only possess a structure on the atomistic scale of a few A, corresponding to the length of covalent bonds and interatomic distances characteristic of macromolecules are coils that more or less, obey Gaussian statistics and have a diameter of the order of hundreds of A (Fig. 1.2) [17]. Structures of intermediate length scales also occur e. g., characterized by the persistence length. For a simulation of a polymer melt, one should consider a box that contains many such chains that interpenetrate each other, i. e., a box with a linear dimension of several hundred A or more, in order to ensure that no artefacts occur attributable to the finite size of the simulation box or the periodic boundary conditions at the surfaces of the box. This ne-... 

Equations (18) and (16) define a temperature where Gaussian behavior is observed (the phase separation temperature) where % — 1/2 and thermal energy is just sufficient to break apart PP and SS interactions to form PS interactions. Equation (12) using (17) for Vc is called the Flory-Krigbaum equation. This expression indicates that only three states are possible for a polymer coil at thermal equilibrium ... 

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... 

Nevertheless, A.N. Semenov in his works [56-59] have used the concept of local quasi-knot for Gaussian coil state, because the end-to-end distance in this state ( N1/2) is still less than the chain length ( N). Within the framework of this approach, the authors of Refs. [56, 57] have shown that the topological constraints do not influence the dynamics of the whole polymer coil suspended in dilute 0-solutions but lead to the essential slowing-down of the internal coil excitation modes. In accordance with Refs. [56-59], the maximum relaxation time of a coil is limited by the diffusion of the local knots along the chain, i.e. the... 

The accuracy of the analysis presented in this paper is determined by the validity of two key approximations (1) the description of the energy transfer dynamics by the first order cumulant expansion method, (2) the use of a Gaussian chromophore pair distribution function. Although originally developed for, and successfully applied to, the problem of energy transfer in disordered infinite volume systems, the cumulant method can be modified to provide a highly accurate description of energy transfer in finite volume systems such as polymer coils (2 ). ... 

Quite naturally, a free single polymer coil is often called a Gaussian coil, after the distribution (6.16). 

Each polymer coil connecting two neighboring cross-links follows the Gaussian distribution regarding its end-to-end distances ... 

Fig. 7.10. Limiting slope from log-log plot of H - SALS profiles in a function of time when crystallizing EH064 at various crystallization temperatures. Note that the limiting slope for a Gaussian chain (random polymer coil) is 2...

For ideal polymer coils, the most significant and distinctive property is the Gaussian distribution of the end-to-end distances. By considering the different segments of the freely jointed chain which are statistically independent and can be represented by a Markov chain, one can derive the correction to the Gaussian distribution of the end-to end distances (see Reference 99). 

Similar to the ideal (Gaussian) coil, the swollen polymer coil is characterized by single characteristic length scale. This characteristic length scale is exemplified by the mean-square, end-to-end distance or the mean-square gyration radius, both of which scale as... 

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