## Flory-Huggins Parameter and Excluded Volume of Polymers

It is well known that the size of a polymer depends on the type and temperature of the liquid in which it is placed. In a good solvent, a polymer coil will expand (swell) due to the affinity with the solvent molecules, whereas in a poor solvent the polymer coil will shrink, i.e., it will avoid mixing with solvent molecules.

A good starting point is a simple linear chain. We assume that the polymer segments and the solvent molecules are randomly distributed in the coil volume R3, and that the heat of mixing is proportional to the volume fraction of polymer segments in R3. The overall system energy can be written as follows:

Emix = <Npp> εpp + <Nps> εps + <Nss> εss

where εij is the interaction energy between two molecules, and <Nij> is the average number of contacts between molecules i and j. The probability Pv that a lattice site is occupied by a solvent molecule is

Pv = 1 - φp = 1 - N l3/R3.

where l  is the effective length of a monomer. Hence, the average number of solvent-polymer pairs is given by <Nps> = zN(1 - φp), where N is the number of repeat units (segments) in the polymer chain and z is the total number of contacts per repeat unit, also called lattice coordination number. Flory defined a dimensionless quantity which characterizes the interaction energy per solvent molecule divided by kT:

χ = -z/2 (εpp + εss - 2 εps) / kT

This quantity is often called Flory-Huggins parameter and was originally introduced by P. J. Flory1 and M. L. Huggins2 as an exchange interaction parameter in their lattice model of polymer solutions. zkTχ is simply the difference in energy between a solvent molecule immersed in pure polymer and in pure solvent. If we introduce this parameter, the energetic term can be written as

Emix = χ N ( 1 - φp)

Let us assume that the polymer is in a state where its end-to-end distance R is close to its most probable value, lNα, where l is the (average) length of a repeat unit and N is the number of repeat units in the polymer chain. Let us further assume that the majority of the segments can be found in a spherical volume of same size and that the solvent molecules are randomly distributed in this volume R3. These are crude assumption; nevertheless, the formulas that can be derived from it, describe, at least qualitatively, the approximate behavior of polymer chains immersed in solvent.

The probability W(R) of a chain having size R is proportional to the product of three factors:

W(R) ∝ P(R) · Q(R) · S(R)

or

The first factor, P(R), is the probability distribution of the end-to-end distance of an ideal chain, the second factor, Q(R) = exp[-Emix(R)/kT], describes the solvent-polymer interaction and the third, S(R), is the probability that the configurations described by the first factor are also allowable under excluded volume conditions.

To estimate the average or statistical size of a polymer chain we have to calculate the value R that makes W a maximum. If the volume fraction of polymer segments in the sphere of radius R is small, then the approximation 1 - x ≈ e-x is valid and we obtain

The maximum of W can be found by differentiating the logarithm of the equation above. This gives

The first term describes the elastic deformation of the chain, and the second term describes the swelling of the coil due to excluded volume interaction and mixing with solvent molecules. The expression above can be rewritten in the form:

α5 - α3 ≈ 4/3 ·z

where α =R1/2 /N1/2l is the swelling coefficient, z = (1 - 2χ)N1/2 = v N1/2l-3 and the parameter v = l3(1 - 2χ) is the so-called excluded volume parameter. The result is identical with that of the first order perturbation theory. In a good solvent, α > 1, the second term on the left-hand side (α3) can be neglected if N >> 1. It follows

RN3/5 (1 - 2χ)1/5 = N3/5 v1/5

The value of the exponent, ν = 3/5, is not the exact value but very close to it.3 Calculation based on renormalization group methods give a more accurate value of ν = 0.588. In a good solvent, χ is small or negative, indicating strong (polar) attractions between the polymer and solvent molecules. Under these conditions, a polymer coil will expand and dissolve in a solvent due to the affinity with the solvent molecules.

In a poor solvent, α << 1 and α3 ≈ 4/3 ·z. In this case we find

RN1/3 (-v)-1/3

In a poor solvent χ is large and polymer-solvent contacts are unfavorable. Under these condtions a polymer will avoid contacts with solvent molecules and the coil will be more compact (collapsed; and at a certain point it will be immiscible with the solvent (globule state).

### Theta-Temperature And Coil-Globule Transition

With increasing temperature, v will change sign from positive to negative. The temperarue where the excluded volume parameter v equals zero is called the θ temperature. At this temperature, the repulsive excluded-volume forces balance the attractive forces between the segments and the polymer coil behaves like an ideal polymer or a Gaussian chain with unperturbed dimensions, which are the same as those of a polymer in its melt state. Below the θ temperature, the size of the polymer is much smaller than an ideal chain and at a certain point, the polymer coil will collapse, i..e expel all solvent molecules. This change is called coil-globule transition.

In terms of the phase equilibrium of a poor-solvent-polymer mixture, the θ-temperature is equal to the critical miscibility temperature in the limit of infinite molecular weight.

At the θ-temperature, the intrinsic viscosity η is proportional to the unperturbed dimension or mean square end-to-end distance.

##### References
1. P. J. Flory, J. Chem. Phys. 9, 660 (1941); 10, 51 (1942); 17, 303 (1949)
2. M. L. Huggins, J. Chem Phys. 9, 440 (1941); J. Phys. Chem. 46, 151 (1942);  J. Am. Chem. Soc. 64, 1712 (1942)

3. In the derivation above, we made the assumption that the size of a statistical segment is identical with the excluded volume under athermal conditions. However, l3 and vexl are often assumed to be different quantities. Infact, vexl is the volume of one lattice element, the so-called reference volume of the lattice, which can be more or less arbitarily chosen, and l is the length of a statistical segment.

• ## Summary

• ### Excluded Volume PArameter

The size of a polymer coil in a solvent depends on the excluded volume parameter. The excluded volume is the volume that is inaccessible to other polymer molecules.

• In a good solvent, χ is small or negative. Under these conditions, a polymer will expand and dissolve in a solvent.

• In a poor solvent a polymer will avoid contacts with solvent molecules and the coil will be much more compact, and at a certain point it will be immiscible with the solvent (globule state).

• With increasing temperature, the excluded volume parameter v will change signs from positive to negative. The temperature where the excluded volume parameter v equals zero is called the theta temperature.

• At the theta temperature, a polymer behaves like a Gaussian chain with unperturbed dimensions (melt state).

• The excluded volume parameter of a polymer in its melt state is zero. This means the dimensions are the same as those of an ideal polymer chain.

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