Flory-Krigbaum

We shall devote a considerable portion of this chapter to discussing the thermodynamics of mixing according to the Flory-Huggins theory. Other important concepts we discuss in less detail include the cohesive energy density, the Flory-Krigbaum theory, and a brief look at charged polymers. 

To apply these ideas to solution nonideality, we consider a theory developed by Flory and Krigbaum. This is only one of several approaches to the problem, but it is one which can be readily outlined in terms of material we have already developed. We shall only sketch the highlights of the Flory-Krigbaum theory, since the details are complicated and might actually obscure the principal ideas. 

The objective of the Flory-Krigbaum theory is to find a quantitative expression for the placement probability n(d) of the two coils as a function of their separation d. There are three stages to the derivation ... 

The full Flory-Krigbaum theory results in the following expression for the excluded volume ... 

The complicated form of the final result makes it clear why we have skipped over the details of the Flory-Krigbaum derivation ... 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

Intrinsic viscosity measurements revealed a conformational transition upon heating from 26 to 40 °C, while the UV absorbance of the solution was insensitive to the change. The entropy parameters for PA were also discussed in light of the Flory-Krigbaum correlation between the second virial coefficient and theta temper-... 

Equations (18) and (16) define a temperature where Gaussian behavior is observed (the phase separation temperature) where % — 1/2 and thermal energy is just sufficient to break apart PP and SS interactions to form PS interactions. Equation (12) using (17) for Vc is called the Flory-Krigbaum equation. This expression indicates that only three states are possible for a polymer coil at thermal equilibrium ... 

Theory of Steric Stabilization. A detailed description of the competing theories can be obtained from other publications (1-3) and only an outline will be given here. Almost all the acceptable theoretical descriptions have their origins in the Flory-Krigbaum theory (10) for a dilute polymer solution which has been adapted to the case of sterically stabilized particles. 

Replacing [(1/2) — x] by [1 — 0/71 offers an opportunity to test the ideas of the Flory-Krigbaum theory against experimental observations of the CFT. It is apparent from the material presented here that the CFT and the 0 temperature correspond to the same condition. Table 13.3 lists CFT values and independently determined 0 temperatures for several systems. The agreement between the two is quite satisfactory for these systems. Incidentally, electrolytes are added to the aqueous media in Table 13.3 to suppress the ion atmosphere mechanism for stabilization. 

There is another type of free energy change that can be considered within the overlap volume in addition to the concentration effect considered by the Flory-Krigbaum theory. This additional contribution to AGlens is likely to be more important for d < 5RS and should be considered when the outcome of the encounter is not determined by the initial approach of the colliding particles. This contribution arises from an elastic response by the adsorbed polymer, effectively pushing the approaching particles apart. 

Fig. 29. The function Afroj- ) related to the osmotic second virial coefficient. Full curve, modified Flory- Krigbaum-Orofino theory, Eq. (107) chain curve, Casassa-Markovitz theory, Eq. (108)...

Using the Flory-Krigbaum theory [10], it is possible to calculate the free energy of mixing, due to this unfavourable overlap that is. 

Cm is a parameter depending on the properties of the polymer-solvent system and ocg is an expansion parameter. Both the terms are defined in the following section on Excluded Volume Theories. At T = 9, X becomes zero, F X) becomes equal to 1, and the Flory-Krigbaum theory reduces to the Flory-Huggins theory. 

Problem 3.9 For polystyrene of molecular weight 100,000, calculate approximately the solution concentration below which the Flory-Krigbaum theory should be used. Use the following relationship between the RMS radius of gyration and the polymer molecular weight M 2 x cm. 

For polystyrene of molecular weight 10 the overlap therefore occurs above about 2% concentration. Below this concentration, one should therefore use the Flory-Krigbaum theory. 

C. Dilute Polymer Solutions (Flory-Krigbaum Theory)... 

C. DILUTE POLYMER SOLUTIONS (FLORY-KRIGBAUM THEORY)... 

Osmotic pressure, as indicated earlier, is one of the quantities that can be obtained experimentally from the Flory-Huggins and Flory-Krigbaum theories. Before we illustrate how thermodynamic parameters characteristic of polymers can be derived from osmotic pressure measurements, let us first explain very briefly the basis of these measurements. ... 

A further development of Equation (3.41) was to use a Flory-Krigbaum model [122] to evaluate the second virial coefficient giving [47]... 

Quantitative predictions of surfactant phase behavior can be made by constructing a thermodynamic model. The classical expression for the free energy of a microemulsion is a function of the interfacial tension, bending moment, and micelle-micelle interactions [47]. Two quantitative models have been developed to describe supercritical microemulsions based on this concept. Here, the key challenge is to find accurate expressions for the oil-surfactant tail interactions and the tail-tail interactions. To do this, the first model uses a modified Flory-Krigbaum theory [43,44], and the second a lattice fluid self-consistent field (SCF) theory [25]. 

The above mixing interaction can be calculated from a consideration of the free energy of mixing two polymer solutions, e.g. by using the Flory-Krigbaum theory (3), as follows ... 

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