The Free Energy of a Globule

Where did the term U s a) come from It has to do with the poorer choice of shapes that a straightened polymer can take. Fewer possibilities means lower entropy, and a lower probability of the elongated state (see Section 7.5). Meanwhile, we have already said that the other term, U a), is the energy of the monomer interactions. Thus, when we write the free energy in the form (8.11), we automatically distinguish its entropy and energy parts. Notice that we never use the fact that a 1. It applies just as well if the molecule shrinks (a 1 ) instead of swells (a 1 ). We still have the free energy in the same form  

Here t/eff(a) is determined by the entropy of the final state of the coil, when it is either swollen or shrunk by the factor a. (Although in the case of shrinking, a is less than one, we would like to keep its previous name. 

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Volume approximation (when the surface contribution to the free energy of a globule is neglected) works the better the farther the system is from the point of the coil-to-globule transition. In the framework of this approximation, it coincides with the -point, whereas under the theoretical consideration where the surface layer is taken into account, a gap appears separating these two points. The less is the length of polymer chain l, the more pronounced is this gap. Hence, the condition, imposed on the thermodynamic and stoichiometric parameters of the system by the equation of the -point,... 

The same approximation gives the difference between the free energies of a globule and an unperturbed coil without excluded-volume effects... 

To describe the coil-globule transition in the vicinity of the Flory temperature, in 1968 Lifshitz8 proposed an original method which amounts to expressing the free energy of a polymer in terms of the local monomer concentration. This method was subsequently reexamined and developed in various articles.2 We shall describe it here but, for reasons of convenience, we shall use a slightly different formalism. 

Single phase microemulsions are treated in the next section. Two general thermodynamic equations are derived from the condition that the free energy of the system should be a minimum with respect to both the radius r of the globules as well as the volume fraction of the dispersed phase. The first equation can be employed to calculate the radius while the second, a generalized Laplace equation, can be used to explain the instability of the spherical shape of the globules. The two and three phase systems are examined in Sections III and IV of the paper. 

The transition between both globular structures and the coil state turns out to be the first order phase transition. The relation between the free energies of structures A and B depends on the temperature and on the characteristic dimensionless parameter /7a. The latter dependence is manifested already at the temperatures which are much lower than the coil-globule transition temperature, i.e. formally at A > l13. In this region, the following simple result can be obtained ... 

Let us assume that the microenulsion contains spherical globules of a single size. For given numbers of molecules of each species, temperature and external pressure, we consider an ensemble of systems in which the radii and volume fractions of the globules can take arbitrary values. Because the equilibrium state of the system is completely determined by the number of molecules of each species, the temperature and external pressure, the actual values of the radius and volume fraction will result from the condition that the free energy of the system be a mlnlnum. 

On the contrary, this set of experimental results would provide some ground for a theoretical and thermodynamical explanation of the evolution swollen micelle-microemulsion. Indeed each type of structure seems to reflect a domination of one or other component of the free energy of these nonionics at room temperature. Although a calculation and a discussion of these energy effects are well beyond the scope of the present paper, we can point out the importance of the forces specific to the hydrocarbon chain and to the oil beside the pure hydration forces. Van der Waals forces would favour the formation of a water layer, while entropic effects seem very important as far as the transitions hank-lamella and lamella-globule are concerned. These effects due to the solvent concentration (but also to the nature of the oil (2,5) are quite evident from the fine evolution of the phase diagrams, especially for water/surfactant ratios in the range 0.5-1.2. 

A tUflerent approach to the phase separation in polyelectrolyte solutions was proposed by Dobiynin and Rubinstein. They used a two-state model to describe counterion condensation inside beads of the necklace-like globules in dilute solutions. Within this approximation, the free energy of the dilute solution of necklaces is... 

For the B chains, we have taken as a reference-state isolated chains in a good solvent for the A chains the molten state. The dominant contribution to the free energy of collapsed chain is the surface energy between the solvent and the molten A globule being the relevant surface tension). 

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