Excluded volume effects scaling theory

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

The theoretical description of excluded volume effects on the adsorption from good solvents is still unsatisfactory. The scaling theory for polymer adsorption has not yet been subject to experimental tests. 

The behavior of PE chains can be analyzed in more detail by the scaling approach based on the concept of thermal and electrostatic blobs. The blob theory assumes that, on small length scales shorter than the correlation length (called also the blob size ), the energy of random thermal motion counterbalances the excluded volume effect of segments, and short parts of the chain behave as ideal chains. Therefore, it holds that where gn is the number of segments per... 

The development of adsorption theory provides the explanation of the macromolecule behavior in the adsorption layer and provides the basis of arguments on the experimental results. A few theoretical models, describing the adsorbed macromolecule, are widely used now. Self-consistent field theory or mean field approach is used to calculate the respective distribution of trains, loops, and tails of flexible macromolecule in the adsorption layer [22-26]. It allows one to find the segment density distribution in the adsorption layer and to calculate the adsorption isotherms and average thickness of the adsorption layer. Scaling theory [27-29] is used to explain the influence of the macromolecule concentration in the adsorption layer on the segment density profile and its thickness. Renormalization group theory [30-33] is used to describe the excluded volume effects in polymer chains terminally attached to the surface. The Monte Carlo method has been used for the calculation of the density profile in the adsorption layer [33-35]. 

In recent years, studies of solutions of polymer blends and of copolymers have aroused a substantial theoretical and experimental interest. This is motivated by both numerous applications and more fundamental issues concerning the usefulness of the scaling and universality concepts to describe the thermodynamic properties and the phase transitions in these systems. In this lecture, chain interactions in dilute and semidilute solutions are reviewed and it is discussed how and when the interactions between chemically different monomers lead to a macroscopic phase separation in the case of ternary polymer A-polymer B- solvent systems and to a mesophase formation in diblock-copolymer solutions. The important conclusion is that due to both the overall monomer concentration fluctuations (excluded volume effects) and the composition fluctuations, the classical Flory theory often fails. This requires the use of the renormalization method and of scaling concepts to give a correct description of the phase diagrams and the critical phenomena observed in these complex systems. We give only here a brief outline, a complete review has been published elsewhere, ... 

At the interface between coexisting polymer solutions a central layer enriched in solvent is expected, because the two incompatible polymers avoid mutual contact. Using mean field theory extended to include excluded volume effects to describe the mixing of two polymers in a common solvent, the structure of the interface can be estimated [21]. In this approach, the polymer solution is considered a melt of ideal chains of blobs. Blobs are sequences of monomers that are correlated by excluded volume effects. Beyond a blob, no such correlations exist and the chain of blobs is ideal. The interaction between blobs of the two types of polymers is responsible for phase separation. This interaction is dependent on concentration because the blob size is dependent on concentration. The calculation [21] starts from the density of the free energy of mixing, rescaled on the relevant distance scale, that is, the blob size... 

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