Excluded volume forces chain repulsion

Expanded chains are found in dilute solution in good solvents. The effective interaction energy between two monomers is always repulsive here, and, as a consequence, chains become expanded. Expansion will come to an end at some finite value since it is associated with a decreasing conformational entropy. The reason for this decrease is easily seen by noting that the number of accessable rotational isomeric states decreases with increasing chain extension. The decrease produces a retracting force which balances, at equilibrium, the repulsive excluded volume forces. 

Although the rigorous solution was not presented until 1972, Flory offered arguments in support of the scaling law Eq. (2.83) much earlier. The point of concern is the equilibrium conformation of a chain, being the result of a balance between repulsive excluded volume forces and retracting forces ar-ing from the decreasing conformational entropy. Above, with Eq. (2.78), we have already introduced Flory s expression for the potential produced by the excluded volume forces... 

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

The early molecular theories of rubber elasticity were based on models of networks of long chains in molecules, each acting as an entropic spring. That is, because the configurational entropy of a chain increased as the distance between the atoms decreased, an external force was necessary to prevent its collapse. It was understood that collapse of the network to zero volume in the absence of an externally applied stress was prevented by repulsive excluded volume (EV) interactions. The term nonbonded interactions was applied to those between atom pairs that were not neighboring atoms along a chain and interacting via a covalent bond. 

On the other hand, the internal excluded volume results from the finite thickness of the macromolecular chain, and its influence is retained, even at infinite dilution. The internal excluded volume can be formally separated into two components. Repulsive forces lead to a positive excluded volume. Attractive forces, in contrast, tend to reduce the volume occupied by two contacting parts of the chain from the value obtained by the summation of their individual volumes the resulting excluded volume is negative. 

The addition of an alkyl group to every other carbon atom of one repeat unit of a copolymer should stiffen the copolymer chain. The group limits the number of positions the chain can adopt without crushing the added alkyl group into some other part of the copolymer. This is stiffening the polymer chain by a "steric" or "excluded volume" method. The steric (par-b-to-part, repulsive) forces in the molecule then keep the copolymer from occupying some of the volume (excluded volume) inside the polymer coil with the result that the alkyl-substituted molecule is larger in volume than an unsubstituted polymer with the same number of repeat units. 

The Kratky-Porod formula may not be applicable for long molecules with L Ip, for which excluded volume interactions should be taken into account. Intramolecular excluded volume effects result from repulsion between segments within the same molecule, which result in an increase of the end-to-end distance. These effects are particularly strong for 2D systems, which demonstrate an increased density of segments and do not permit chain crossings. Both methods require complete visualization of a statistical ensemble of single molecules in order to determine L, 0, and (R ). In addition, they assume the observation of molecules in their natural state, in which molecules are not constrained and freely fluctuate around their equilibrium conformation. The concurrent effects of adsorption, solvent evaporation, and capillary forces can, however, lead to kinetically trapped conformations. The question arises whether and under what conditions an equilibrium 2D conformation can be achieved. 

Consider a polymer chain adsorbed on a surface, all immersed in a liquid phase. In general, the polymer chains are partly dissolved, with those portions composed of loops and tails. These may have conformations similar to that of random coils. When two such adsorbed polymer chains approach one another, there will be an entropic repulsive force between them, arising from the excluded volume effect, as well as other forces (65,66). Sometimes the repulsive forces are called steric or overlap repulsion. 

Nanocomposites equate to a distribution of nanoparticles within a continuous phase and, consequently, share certain features with colloids, where a range of factors influence the structure of the system. These include excluded volume effects that are related to the direct interactions between hard particles where particle surfaces are covered with long chain molecules, additional steric repulsive forces will also... 

At the same time two polymer segment compete for space occupation. No segment allows the other segment of the same chain to be too close because of a repulsive force of polymer segments against each other. This repulsive force results in the excluded volume u, which is defined as... 

For a polymer system to be capable of responding strongly to slight changes in the external medium, a first-order phase transition accompanied by a sharp decrease in the specific volume of the macromolecule should occur. The theoretical foundation of such process was laid by Flory. One of the main conditions for the manifestation of critical phenomena in swollen polymer networks or linear macromolecules is the presence of a poor solvent. In such a solvent, the forces of attraction between the segments of the polymer chain may overcome the repulsive forces associated with the excluded volume, leading to the collapse of the polymer chain. 

This is an empirical equation which accounts for the fact that for ideal chains, i.e. vanishing excluded volume interactions, the coil size in thermal equilibrium equals Rq j3 is a dimensionless coefficient of order unity. The first term gives the repulsion experienced on squeezing a polymer chain, the second term represents the retracting force built up on a coil expansion. As only the second term appears relevant for the case under discussion we ignore the first term and write... 

---