Flory-Huggins theory dilute polymer solutions

In dilute polymer solutions, the polymer molecules are isolated from each other by regions of pure solvent, i.e., the polymer segments are not uniformly distributed in the lattice. In view of this, the Flory-Huggins theory is least satisfactory for dilute polymer solutions and only applies to concentrated solutions or mixtures. Furthermore, the interaction param-... 

Comparison of these two equations shows that for dilute polymer solutions, Flory-Huggins theory predicts the excess contribution to be... 

If a polymer solution undergoes phase separation into two liquid phases then the solution cannot be too far removed from the 0-point. It follows that the Flory-Huggins theory, which was derived for moderately concentrated solutions, can be applied to both of the polymer solution phases, one of which is dilute and the other concentrated. The chemical potentials of both the polymer and the solvent must be equal in the two co-existing phases ... 

The lattice theory of entropy of mixing, derived independently and contemporaneously by Huggins and Flory and the "Huggins constant K ", relating the concentration dependence of the viscosity of dilute polymer solutions are listed among Huggins major contributions to polymer science. He developed new procedures for more quantitative productions of solubiHty. 

Gibbs principle of multiple phase equihbria is applied to model polymer solutions to explore the possible types of heterophase coexistence and phase transitions. The fundamental properties of dilute polymer solutions and hquid-hquid phase separation driven by van der Waals-type interaction is reviewed within the framework of Flory-Huggins theory. No specific molecular interactions are assumed. Refinement of the polymer-solvent contact energy beyond Flory-Huggins description is attempted to study the glass transition of polymer solutions at low temperatures. The scaling description of semiconcentrated polymer solutions is summarized. 

Most solutions of rodlike polymers are not athermal. Heat-of-mixing terms can be added to the chemical potentials, just as in the ordinary Flory-Huggins theory of solutions. When the interaction parameter % is large enough, the solution will separate into a dilute phase and a concentrated phase, just as for any polymer solution. There will still be a miscibility gap between dilute isotropic solutions and ordered concentrated solutions. A typical phase diagram for a solution of rods with x = 100 is shown in Figure 9.3. The concentrated phase is now very concentrated and ordered. 

The Flory-Huggins lattice consideration of the polyelectrolyte solutions presented above incorrectly describes dilute polyelertrolyte solutions. In the Flory-Huggins approach, the monomers are uniformly distributed over the whole volrrme of the system, leading to underestimation of the effect of the short-range monomer-monomer interactions and of the intra-chain electrostatic interactions. A similar problem appears in the Flory-Huggins theory of phase separation of polymer solutions (see for discussion References 32 and 33). This leads to the incorrect expression for the low polymer density branch of the phase diagram. 

For sufficiently dilute polymer solutions, the only difference between the new approach and the original Flory-Huggins theory is in the second term. According to theoretical considerations and in accord with experimental findings, becomes zero under theta conditions (where the coils assume their unperturbed dimensions) and the conformational relaxation no longer contributes to... 

We shall in section 4.2 deal with regular solutions of small-molecule substances. The construction of phase diagrams from the derived equations is demonstrated. The Flory—Huggins mean-field theory derived for mixtures of polymers and small-molecule solvents is dealt with in section 4.3. It turns out that the simple Flory—Huggins theory is inadequate in many cases. The scaling laws for dilute and semi-dilute solutions are briefly presented. The inadequacy of the Flory-Huggins approach has led to the development of the equation-of-state theories which is the fourth topic (section 4.6) Polymer-polymer mixtures are particularly complex and they are dealt with in section 4.7. 

It is convenient to partition polymer solutions into three different cases according to their concentration. Dilute solutions involve only a minimum of interaction (overlap) between different polymer molecules. The Flory-Huggins theory does not represent this situation at all well due to its mean-field assumption. The semi-dilute case involves overlapping polymer molecules but still with a considerable separation of the segments of different molecules. 

Solution activity data obtained by osmometry on dilute solutions showed that the second virial coefficient is dependent on molar mass, contradicting the Flory-Huggins theory. These problems arise from the mean-field assumption used to place the segments in the lattice. In dilute solutions, the polymer molecules are well separated and the concentration of segments is highly non-uniform. Several scaling laws were therefore developed for dilute (c < c is the polymer concentration in the solution, c is the threshold concentration for molecular overlap) and semi-dilute (c > c ) solutions. In a good solvent the threshold concentration is related to molar mass as follows ... 

The theory presented above for dilute polymer solutions is based upon the Flory-Huggins Equation (3.22) which strictly is not valid for such solutions because of the mean-field approximation. Nevertheless, whilst Equations (3.38) and (3.45) do not accurately predict fix -/ ), they are of the correct functional form, i.e. the relationships... 

In dilute polymer solutions each polymer molecule excludes all others from its volume. Thus mean-field theories, such as Flory-Huggins theory, are inappropriate and more exact theories of dilute solutions calculate AG from the volume excluded by one polymer molecule to another, which in turn is calculated using to account for intermolecular segment-segment volume exclusion. These theories show that (jui... 

A typical plot of experimental data is shown in Fig. 3.10. Thus M can be evaluated from the intercept, and from the slope it is possible to calculate an estimate of x using Equation (3.97). However, since Flory-Huggins theory strictly is not valid for dilute polymer solutions. Equation (3.97) usually is written in the form of a virial equation such as... 

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