Entropic model/theory

EXAMPLE 13.6 An Entropic Model for Steric Stabilization Due to Adsorbed Polymer Layers. Picture a flat surface to which rigid rods are attached by ball-and-socket-type joints. The free ends of the rods can lie anywhere on the surface of a hemisphere. The approach of a second surface blocks access to some of the sites on the cap of the hemisphere. Outline the qualitative argument that converts this physical picture to a theory for stabilization. What are some of the shortcomings of the model ... 

Figure 6.33 Crystallization according to the entropic barrier theory, (a) Representation of a lamellar crystal, showing stems (chain direction vertical) and a step in the growth face. The inset provides a description of the step in terms of units that are shorter than the length of the surface nucleation theory (one molecule making up a whole stem). The dotted lines indicate where the row of stems in (b) is imagined to occur, (b) The basic row of stems model, showing mers along the chains as cubes, chain direction vertical, as in (a).

Recently, we have introduced an entropic model for predicting the miscibility behavior of PS-NP/PS nanocomposites with very good agreement between theory and experiment [8]. Additionally, the theory has been employed for the prediction of the interaction parameter, the miscibility behavior, and the melting point depression of athermal poly(ethylene) (PE)-nanoparticle/linear-PE nanocomposites using chain dimensions data from Monte-Carlo (MC) simulations [9]. Our main findings indicate that dilution of contact, hard sphere-like, nanoparticle-nanoparticle interactions plays a key role in explaining the miscibility behavior of polymer-nanoparticles dispersed in a chemically identical linear-polymer matrfac [8, 9]. 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

Crosslinked polymers are rather peculiar materials in that they never melt and they exhibit entropic elasticity at elevated temperatures. The present review on the influence of crosslink density is structured around model polymers of uniform composition but with widely varying numbers of crosslinks. The degree of crosslinking in the polymers was verified by use of the theory of rubber elasticity. 

Although the basic concept of macromolecular networks and entropic elasticity [18] were expressed more then 50 years ago, work on the physics of rubber elasticity [8, 19, 20, 21] is still active. Moreover, the molecular theories of rubber elasticity are advancing to give increasingly realistic models for polymer networks [7, 22]. 

Prausnitz and coworkers [91,92] developed a model which accounts for nonideal entropic effects by deriving a partition function based on a lattice model with three categories of interaction sites hydrogen bond donors, hydrogen bond acceptors, and dispersion force contact sites. A different approach was taken by Marchetti et al. [93,94] and others [95-98], who developed a mean field theory... 

For the united atom models of realistic polymers the wall PRISM theory predicts interesting structure near the surface [95]. For example, the side chains are found preferentially in the immediate vicinity of the surface and shield the backbone from the surface. This behavior is expected from entropic considerations. Computer simulations of these systems would be of considerable interest. 

It may be asserted that the fundamental reason arises from the fact that, while parallel arrangements of anisotropic objects lead to a decrease in orientational entropy, there is an increase in positional entropy. Thus, in some cases, greater positional order will be entropically favorable. This theory therefore predicts that a solution of rod-shaped objects will undergo a phase transition at sufficient concentration into a nematic phase. Recently, this theory has been used to observe the phase transition between nematic and smectic-A at very high concentration (Hanif et al.). Although this model is conceptually helpful, its mathematical formulation makes several assumptions that limit its applicability to real systems. 

The solubility of a gas is an integral part for the prediction of the permeation properties. Various models for the prediction of the solubility of gases in elastomeric polymers have been evaluated (57). Only a few models have been found to be suitable for predictive calculations. For this reason, a new model has been developed. This model is based on the entropic free volume activity coefficient model in combination with Hildebrand solubility parameters, which is commonly used for the theory of regular solutions. It has been demonstrated that mostly good results are obtained. An exception... 

Birshtein and Luisi (17) made attempts to prove the theoretical validity of the above model by applying the theory of VOLKENSTEIN (137, 138, 139), Birshtein and Ptitsyn (16) on the conformation of isotactic macromolecules, to optically active poly-a-olefins. From their results it is possible to conclude that the presence of asymmetric carbon atoms in the lateral chains might cause the prevalence of main chain sections spiraled in one of the two possible screw senses, mainly for entropic reasons. 

This model is based on classic rubber theory, suggesting that elastin is made up of a network of random chains that are kinetically free and exist in a high entropic state. Stretching orders the chains and limits their conformational freedom, thus decreasing the overall entropy of the system (Hoeve and Flory, 1974). This provides the restoring force to the relaxed state. 

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. 

Regular solution theory, the solubility parameter, and the three-dimensional solubility parameters are commonly used in the paints and coatings industry to predict the miscibility of pigments and solvents in polymers. In some applications quantitative predictions have been obtained. Generally, however, the results are only qualitative since entropic effects are not considered, and it is clear that entropic effects are extremely important in polymer solutions. Because of their limited usefulness, a method using solubility parameters is not given in this Handbook. Nevertheless, this approach is still of some use since solubility parameters are reported for a number of groups that are not treated by the more sophisticated models. 

Another limitation of the regular solution theory is the assumption diat is negligible. While this assumption may be valid for solutions in which all components (solute and solvent) are of similar sizes, it breaks down when the molar voinmes of the components are significantly different, i.e., in the case of high molecular weight (polymeric) solvents and low molecular weight solutes. For such cases, more rigorous models that include entropic considerations, such as... 

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