Flory- Krigbaum theory

Cm is a parameter depending on the properties of the polymer-solvent system and as is an expansion parameter. Both the terms are de ned in later seetions. AlT = 6.X becomes zero, F X) becomes equal to 1, and the Flory-Krigbaum theory reduces to the Flory-Huggins theory. 

Problem 3.8 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 5 2)F2 and the polymer molecular weight M cm. 

The radius of a sphere enclosing the segments of a polymer molecule is taken approximately as twice the value of that is, R Ax cm. The volume, V, of the solution containing n polymer 

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

The second virial coefficient for homogeneous polymers can now be expressed theoretically  

In relation to the dimensions of the molecule over those of its random-flight counterpart, we have, as we may recall from Chapter 5, 

If polymer molecules are not homogeneous in size, we need to modify the second virial coefficient  

The terms w, and wj are weight fractions of polymer species i and j and are related to Xij, 

The characteristic features of Hory s theory and Flory and Krigbaum s theory are summarized as follows  

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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 ... 

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. 

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. 

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

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. ... 

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 ... 

According to (4.44) in the Flory-Krigbaum theory for dilute solutions, when the solvent is the polymer (x 00), the molar volume of the solvent vy 00 too, then A2 —> 0. This implies that the bulk phase is the theta solvent of polymers. With this approach, Rory had already recognized the unperturbed chain conformation in concentrated polymer solutions (Flory 1953). 

Flory-Krigbaum theory n. A theory of dilute polymer solutions which describes the... 

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