Packing constraints model

The early Hartley model [2, 3] of a spherical micellar stmcture resulted, in later years, in some considerable debate. The self-consistency (inconsistency) of spherical symmetry witli molecular packing constraints was subsequently noted [4, 5 and 6]. There is now no serious question of tlie tenet tliat unswollen micelles may readily deviate from spherical geometry, and ellipsoidal geometries are now commonly reported. Many micelles are essentially spherical, however, as deduced from many light and neutron scattering studies. Even ellipsoidal objects will appear... 

Fig. 8.—Plot of optimal aggregation number M against (y/oo/c) for transition shapes of the ellipsoid of revolution model. M = Atriyiv is the largest possible aggregation number for spheres consistent with packing. Lower curve corresponds to spheres with no packing constraints.

In this section, we mention very briefly some recent theoretical developments, which go far beyond the simple Flory-Huggins theory. As was emphasized above, the Flory-Huggins theory suffers from two basic defects (i) Using a lattice model where polymers are represented as self-avoiding walks is a crude approximation, which neglects the disparity in size and shape of subunits of the two types of chain in a polymer blend, as well as packing constraints, specific interactions etc. (ii) Even within the realm of a lattice model, the statistical mechanics (involving approximations beyond the mean field approximation) is far too crude. 

On a broader level, the topology of the diagrams can be used to infer global characteristics of potential landscapes that control the dynamic relaxation of ensembles. For example, most speculations as to whether protein landscapes have funneling or glassy properties [49,50] have been based on computational studies performed on lattice models that attain simplicity at the expense of accuracy [51-53]. However, whereas these models may adequately account for important packing constraints... 

A simple example is one where the hydrophobic interactions result in < i for all > 1, but the packing constraints on the chains and heads result in a minimum energy for a finite value of N = M (i.e., m < for N Af). (Below, a curvature energy model is discussed this model can also be used to motivate, but not to calculate in detail, a study of micellar sizes and shapes.) If this minimum is deep n rises sharply compared to ksT around N = Af), the distribution of micelles will be nearly monodisperse. In this approximation, one can consider monomers and micelles of aggregation number Af only. At small values of 4>s (or equivalently, at small values of fi). Pi Pm (Af > 1) almost all the surfactant exists as monomers and the number of micelles is exponentially small. The requirement that all amphiphiles have the same chemical potential in equilibrium, Eq. (8.2) and the definition of the CMC (where Pi = Pi, Pm = Pmc ft = = c) allows us to calculate... 

Because of its occurence in diseased tissue, the mode of association of cholesterol esters with biomembranes is of interest. Possible modes of association could be droplets within the hydrophobic core of the membrane bilayer, binding to membrane protein or as part of membrane attached serum lipoproteins. A potentially useful model system for investigating this association is the membrane of the microorganism Mycoplasma capricolum. The Mycoplasma due to their simplicity have served as model membrane systems in many studies. As mentioned previously, cholesterol esters show complex behavior that is a function of thermal history, impurities and physical packing constraints. Using DSC on native membranes and extracted membrane material, it was possible to demonstrate that the majority of cholesterol esters associated with the membranes of M. capricolum exist as relatively large and pure liquid droplets (17). 

In essence, the packing parameter is a measure of the ratio between the effective areas occupied by the hydrophobic (5 ) and hydrophilic (a,) parts of the surfactant. This model is ideal when considering aqueous systems, but may also be applied to dispersions in oil. Depending on the surfactant structure, Pp assumes specific values and the packing constraints in the medium allow for the formation of a preferred aggregate shape configuration. This theoretical approach is also considered in microemulsion systems, as discussed below. 

It is important to observe that only the first term in equation (137) depends on chain architecture and the lattice coordination number, and that both it and the second term cancel out of the entropy of mixing, leaving only the third term equal to Q/Qq. Consequently the theory would apply alike to solutions of flexible and stiff-chain polymers (except that above a certain concentration the latter would respond to packing constraints and minimize the free energy by separation of an ordered phase). The factorization of the combinatorial factors in equation (137) is the fundamental reason why the lattice calculation works at all, despite the extreme artificiality of picturing the chain as fitting a sequence of regular lattice sites with a definite coordination number. These aspects of the model simply disappear in the final result. The independence of the intermolecular factor also implies that the chain conformation should be independent of dilution Rq should be the same in pure liquid polymer as in solution. Naturally this rationale would not hold for dilute solutions, for which the intermolecular factor in equation (137) is not valid. 

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