Excluded volume of a segment

Note 1 This type of interaction is closely related to the excluded volume of a segment, the latter quantity reflecting interactions involving segments and solvent molecules. 

Note The excluded volume of a segment depends on the Gibbs and Helmholtz energies of mixing of solvent and polymer, i.e., on the thermodynamic quality of the solvent, and is not a measure of the geometrical volume of that segment. 

The equation addresses the molar volume of a segment of the polymer v, the subscript m indicating a segmental molar quantity. The parameters of the equation are segmental parameters thus is the excluded volume of a segment q = bJ(Av is the density of the fluid expressed in a reduced dimensionless form, also called the packing fraction and is the attractive-pressure parameter of the segment. The eccentricity parameter of the rotators is e, equal to 1.078 for the model rotator. Results of data reduction indicate the attractive parameter to vary weakly and linearly with temperature. 

The excluded volume of a segment is given by what is called the cluster integral, whereby x) should be very much smaller than... 

The quantity is a so-called excluded volume of a segment, i.e. that which is excluded for all the other segments due to their repulsion. Then, the mean-square end-loend distance is expre.s.sed as (see Equation 87)... 

Provided that they arc normalized, G and Gh produce the distribution function of h, i.e. t hey must be proportional to each other. Fhe factor of proportionality Z is chosem so that some remaining principal parts in the e expansion arc absorbed. I hcreforc, Z. Zi, and the relation v vs vq lead to micro-macro relationships, which absorb all the principal parts of the bare perturbation series, if there is a macroscopic distribution function with two macroscopically-controllablc parameters, viz., the chain length and the excluded volume of a. segment. I hus,... 

The ideal gas free energy functional is defined exactly from statistical mechanics, dropping the temperature-dependent terms that do not affect the fluid structure. Free energy functional contribution due to the excluded volume of the segments is calculated from Rosenfeld s (1989) DFT for a mixture of hard spheres. The functional derivatives of these free energy functional contributions, which are actually required to solve the set of Euler-Lagrange equations, are straightforward. 

For most polymer-solvent combinations i > 0 because of a net segment-segment attraction. This implies that the energy effect opposes the entroplcally driven dissolution of the polymer in the solvent. Such solutions are still thermodynamically stable unless x becomes too high see sec. 5.2e. If > 0 the excluded volume is smaller than the real volume (= ] of a segment. FloryH, Edwards ), De Gennes ) and others have shown that the excluded volume may be written as v . where the dimensionless excluded volume parameter v is defined as... 

Let us begin with the estimation of the polymer volume fraction inside the coil formed by one long semiflexible macromolecule. It is well known that this estimation depends essentially on the strength of the excluded volume effect, i.e. on the value of the parameter z = vN /a, where v is the excluded volume of a monomer and a is the spatial distance between two neighbouring monomers. To be definite let us adopt for a moment the model shown in Fig. 1 b. Then, if we choose one segment as an elementary monomer, v 6. (see Eq. (2.4)) and a i.e. z p Consequently, the excluded volume effect is pronounced at N p and negligible at N [Pg.77]

A potential function (r) between two segments r apart must be assumed before the excluded volume of a coiled molecule can be calculated. The segments in question can belong to the same or to different molecules. Since every molecule possesses many segments, the expressions involved are very complex, and the derivations can only be sketched here. 

Overlap volume between depletion layers (Sect. 1.2) Ensemble-averaged free volume for the depletants (Sect. 3.3) Undistorted ensemble-averaged free volume (Sect. 3.3) Excluded volume of a polymer segment (Sect. 4.3)... 

Proceeding in a manner paralleling the derivation of the excluded volume for a pair of molecules, we consider the polymer molecule to consist of a swarm of segments distributed on the average about the molecular center of gravity in accordance with the Gaussian formula (see Eq. XII-51). This spatial distribution in the unperturbed molecule, as it would exist on the average in the total absence of inter-... 

We recall that we are interested in universal features, i.e. properties that are independent of the microstructure. Our model simple as it is - still has a definite microstructure n = no Gaussian segments of length = (h interacting via an excluded volume of strength q. Taking universality... 

The bond fluctuation model [72] is used to simulate the motion of the polymer chains on the lattice. In this model, each segment occupies eight lattice sites of a unit cell, and each site can be a part of only one segment (self-avoiding walk condition). This condition is necessary to account for the excluded volume of the polymer chains. For a given chain, the bond length between two successive seg-... 

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