Swollen chain

With respect to the scattering behavior at > 1, the domains of Gaussian conformation and in the opposite case (Q , < 1) the domains of swollen chain conformation are probed. The crossover between both regimes is expected to occur at Q = l/ . 

The scaling exponent a can be related to the particle shape. One finds a = 2,0, 0.5, and 0.8 for a thin rod, solid sphere, ideal chain, and swollen chain, respectively. For most polymers K and a have been tabulated [23]. For a monodisperse sample Equation (36) can be used for a crude determination of the molar mass ... 

For determination of the value of a in equation (1) it is necessary to compare the experimental value of parameter A with theoretical theoretical dependences of parameter A on content of spin markers on macromolecule (P=m/P, here m - the number of units of macromolecule containing spin marker) at various values of a for Gauss and swollen chains are presented in works [9, 10], If we know values of a and v from ratio (1) we may easily calculate the mean-square distance between ends of polymer chain. 

The quality of solvent, reflected in the excluded volume v, enters only in the prefactor, but does not change the value of the scaling exponent u for any v > 0. The Flory approximation of the scaling exponent isu = 3/5 for a swollen linear polymer. For the ideal linear chain the exponent = 1/2. In the language of fractal objects, the fractal dimension of an ideal polymer is V — l/i/ = 2, while for a swollen chain it is lower T> — I/u = 5/3. More sophisticated theories lead to a more accurate estimate of the scaling exponent of the swollen linear chain in three dimensions ... 

Section 2.6.1 and is repeated for comparison with that of a swollen chain. The major difference between ideal and real chains is that in the latter there are excluded volume interactions between monomers that are far apart... 

On length scales larger than the thermal blob size in athermal and good solvents, the excluded volume repulsion energy is larger than the thermal energy kT and the polymer is a swollen chain of Njgj- thermal blobs (Fig. 3.14). The end-to-end distance of this chain is determined as a self-... 

Of course, the question becomes more complicated when one deals with partially swollen chains and, more precisely, when one studies real polymers. In... 

This salt concentration marks the transition from collapsed to Gaussian or swollen chains. 

The box-like cell model of a PE star can be considered as a generalization of a classical mean-field Flory approach, which was first suggested to describe the swelling of a polymer chain in a good solvent [90], The Flory approach estimates the equilibrium dimensions of a macromolecule, as a function of its parameters, by balancing the free energy of intramolecular (repulsive) interactions with the conformational entropy loss of a swollen chain. Within the box-like approximation, the star is characterized by the radius of its corona, R (end-to-end distanee of the arms), or by the average intramolecular concentration of its monomers ... 

Formally, this transition is very similar to the worming transition. Flory s approach to a real chain in a good solvent is the base of discnssion (see paragraph 3.6). Flory considered in his seminal paper the balance between attraction among the monomers, wanting to collapse the polymer chain into itself, and the entropy of the chain, wanting to expand the chain. Flory s approach revealed the end-to-end distance of the swollen chain in a good solvent... 

This corresponds to the classical Flory result for the dimensions of a swollen chain in a good solvent for the case P = 1 the chain remains swollen but the degree of swelling decreases as P increases, until the mushroom reaches the ideal, unswollen random walk dimensions which occurs when P =... 

In solvent welding, a solvent is applied which can temporarily dissolve/ swell the polymer at room temperature. When this occurs, the polymer chains have significantly more freedom to move and can entangle with other similarly dissolved/swollen chains in the other component. Given sufficient time, the solvent will permeate through the polymer and out into the environment, so that the chains lose their mobility. This leaves a solid mass of entangled polymer chains, which constitutes a solvent weld. 

This defines approximately the position of line L. Of course, L is not a sharp boundary it defines a region of crossover between ideal and swollen chains. 

Some readers may be surprised by the occurence of a quadratic law [eq. (V.35)] for gaussian chains which have a linear spring behavior. The reason is simple. When we extend our sample very much, its lateral dimensions decrease, and a (which is a force per unit area of cross-section) increases by one extra power of X. The really interesting feature is the difference between eq. (V.34) and eq. (V.35), which reflects the Fisher-Pincus scaling law for swollen chains in strong extension [eq. 

The assumption of ideal chain elasticity is incorrect in a good solvent. As discussed in eq. (I.4S), the spring constant of a swollen chain is much smaller than the spring constant of an ideal chain. The resulting corrections have been incorporated only recently into the theory. "... 

The swollen chain stmcture at length-scales 3> b is universal it is characterized by the unique size R = Rn in particular, the chain gyration radius Rg is proportional to R Rg R. 

A swollen coil (expanded by excluded-volume interactions) is a fractal. The size of any internal chain segment (blob) of g rmits (N < g < N) is g . The mean number of rmits in a sphere of radius r around any unit is Nr for b < r < R, that is, the chain fractal dimension is df= 1/v. The pair correlation function (cf. eqn [18]) of a swollen chain is... 

This result allows to obtain the partition function of closed (ring-like) swollen chain ... 

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