## 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
*R*^{3}, and that the heat of mixing is proportional to the volume fraction of polymer segments in *R*^{3}. The overall system energy can be written as follows:

*E*_{mix} = <*N*_{pp}> *ε*_{pp} + <*N*_{ps}> *ε*_{ps} + <*N*_{ss}> *ε*_{ss}

where *ε _{ij}* is the interaction energy between two molecules, and <

*N*> is the average number of contacts between molecules

_{ij}*i*and

*j*. The probability

*P*

_{v}that a lattice site is occupied by a solvent molecule is

*P*_{v} = 1 - *φ*_{p} = 1 - *N *
*l*^{3}/*R*^{3}.

where *l* is the effective length of a monomer. Hence, the average number of solvent-polymer pairs is given by
<*N _{ps}*> =

*zN*(1 -

*φ*

_{p}), 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 (*ε*_{pp} + *ε*_{ss} - 2 *ε*_{ps}) / *kT*

This quantity is often called *Flory-Huggins* parameter
and was originally introduced by P. J. Flory^{1} and M. L. Huggins^{2}
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

*E _{mix}* =

*χ N*( 1 -

*φ*)

_{p}Let us assume that the polymer is
in a state where its end-to-end distance *R* is close to its most probable value,
*lN ^{α}*, 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

*R*

^{3}. 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[*-E _{mix}*(

*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:

*α*^{5} - *α*^{3} ≈ 4/3 ·**z**

where *α* =*R*^{1/2} /*N*^{1/2}*l*
is the swelling coefficient, **z** = (1 - 2*χ*)*N*^{1/2} =
*v* *N*^{1/2}*l*^{-3} and the
parameter
*v = l*^{3}(1 - 2*χ*) 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 (*α*^{3}) can be neglected if
*N* >> 1. It follows

*R* ∝ *N*^{3/5} (1 - 2*χ*)^{1/5} = *N*^{3/5} *v*^{1/5}

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 *α*^{3} ≈ 4/3 ·**z.** In this case we find

*R* ∝ *N*^{1/3} (-*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

- P. J. Flory,
*J. Chem. Phys.*9, 660 (1941); 10, 51 (1942); 17, 303 (1949) M. L. Huggins,

*J. Chem Phys.*9, 440 (1941);*J. Phys. Chem.*46, 151 (1942);*J. Am. Chem. Soc.*64, 1712 (1942)*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, l*^{3}*and v*_{exl}*are often assumed to be different quantities. Infact, v*_{exl}*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.*