## Statistical physics of many chains

In the previous section, we discussed the properties of an isolated chain, using the random walk model as a basis. In this section, we discuss the properties of many chains together; in other words, the properties of polymer solutions and polymer melts. Again, many of the fundamental ideas are due to Flory and his coworkers.

We start with Flory-Huggins theory, which is a model for the free energy of a polymer solution or melt. It is traditionally based on a lattice model, where the lattice spacing l is taken to be of the order of the Kuhn length. Let us suppose that the lattice volume is V, so the number of lattice sites is V/l3. The polymers have N Kuhn segments, and let the segment concentration be c. The number of polymers per unit volume is then p = c/N (the total number of polymers is Vc/N), and the fraction of sites occupied by polymers (the polymer volume fraction) is A = l3c.

With these definitions, various equivalent methods can be used to estimate the number of ways Vc/N polymers can be inscribed on the lattice and obtain the configurational entropy S. Without going into the details (see de Gennes (1979), for instance), the result is l3S A

In this, A and B are unimportant constants. We add to this a mean field estimate of the energy E. The approach is very similar to Bragg-Williams theory for alloys, and a host of other mean field models in statistical physics. We write

In the first term, for example, VA2/2l3 is an estimate of the number of polymer-polymer contacts and epp is the energy per contact. We combine this with the entropy estimate to arrive at the Flory-Huggins free energy: l3 F A

VkBT = N iog A + (1 - A) iog(1 - A) + xA(1 - A). (8)

The free energy in Equation 8 is a free energy of mixing, with the constants A and B in Equation 6 chosen such that F ^ 0 at A ^ 0 and A ^ 1. With this choice, the energy term only depends on the so-called Flory x-parameter, defined in Equation 9. The x-parameter is seen to be a measure of the chemical dissimilarity between the solvent and the polymer.

We can make a connection to the excluded volume problem as follows. Expand Equation 8 about small A to obtain l3F A, (1 - 2x)A2 (10)

VfcBT N or r 2

where A is an unimportant constant. In the excluded volume approach, we can write a similar virial expansion of the free energy,

VfcBT 2

The first term in this is for an ideal gas of polymers at a number density p = c/N, and the second term accounts for the second virial coefficient between polymer segments. Comparing this with the Flory-Huggins expansion, we see that v = (1 - 2x)13. (12)

This is a most important result. It shows, for example, that good solvent conditions correspond to x ~ 0, ^-solvent conditions to x = 1/2, and poor solvent conditions to X > 1/2. From Equation 9, we expect that x ~ 1/T; thus, x should increase (the solvent quality gets poorer) with decreasing temperature. It turns out this is true for many polymers in organic solvents, but is often untrue for aqueous systems (e.g., PEO in water).

If the polymer and solvent are chemically identical, all the energies in Equation 9 are the same and x = 0 (the so-called 'athermal' solvent case). Thus, a polymer dissolved in a solvent of its own monomers is expected to be fully swollen, which is a somewhat counterintuitive result.

A more counterintuitive result is the so-called 'Flory theorem', which states that a polymer dissolved in a solvent of equal polymers (in other words a polymer in a melt) is ideal, and not swollen at all. Various proofs of this can be constructed: via a spin-model mapping, via Edwards calculation of screening of the excluded volume interaction, or via a simple extension to Flory-Huggins theory. This last approach, although not rigorous, is quite interesting. We generalise Flory-Huggins theory to consider polymers of length N and volume fraction in a solvent of other polymers of length M and volume fraction 1 - The required generalisation is quite straightforward and is l"F = ^r l0S ^ log(1 - ¿)+ x^(1 - (13)

VkBT N^ M We now make the expansion about small 6 to get l3F 6 1 / 1

Comparing with Equation 11, we see that in this case v = (1/M - 2x)l3. Thus, v is reduced if the solvent is polymerised. If all the polymers are chemically the same, x = 0 and v = l3/M. To prove the Flory theorem from this, we use Equation 5 from the previous section. This shows that the excluded volume interaction is only a small perturbation to ideal chain statistics (E C kBT) if v/l3 C N-1/2. Applying this to the present mixture, excluded volume is unimportant if 1/M C N-1/2, in other words, if N C M2. If the polymers are all of the same length as well as chemically identical, then N = M and the condition N C M2 is trivially satisfied. Thus, polymers in a melt are ideal.

We now turn to another aspect of Flory-Huggins theory. This is the prediction that is made for the phase behaviour of polymer solutions. Whilst the entropic term in Equation 8 always favours mixing, we see that the energetic term favours demixing if x > 0. In fact, the free energy develops a double minimum if x becomes large enough, shown in Figure 2, and Flory-Huggins theory predicts liquid-liquid demixing, shown in Figure 3. Let us write f = l3F/VkBT as a dimensionless free-energy density. Then a

Figure 2. Flory-Huggins free energy for small (left) and large (right) x-parameters. On the right, the 'double tangent construction' is illustrated, the points of common tangency giving the densities of coexisting phases.

condition that the free energy has a double minimum is that there is a region where the second derivative d2f/d4>2 < 0. The boundary of this 'spinodal' region is the spinodal line d2f/d^2 = 0 as indicated in Figure 3. The minimum value of x on the spinodal curve is the point where additionally d3f/d^3 = 0. This point is a fluid-fluid demix-ing critical point (Ising universality class). Solving d2f/d^2 = d3f/d^3 = 0 for the Flory-Huggins free energy shows that the demixing critical point occurs at « N-1/2 and xc ~ 1/2 + N-1/2, where N > 1 is assumed. Thus, we see that increasing N shifts the critical point to small volume fractions and closer to ^-solvent conditions. The corresponding critical Edwards excluded volume parameter is vc = -l3N-1/2. Thus, a small negative virial coefficient between segments will result in phase separation (i.e., collapsed chains in a solution are dilute).

Flory-Huggins theory correctly indicates that increasing x, i.e., decreasing temperature since x ~ 1/T usually, or increasing N favours demixing, but being a mean-field theory there are obviously some things it does not get right, for example, the shape of the coexistence curve in the vicinity of the critical point. Less obviously, the prediction for the solubility in bad solvent conditions (i.e., the dilute-solution coexistence curve) is poor. This is because in bad solvent conditions, the mean-field estimate of the energy is inappropriate for polymers, which are essentially dense collapsed coils.

As another example, we can use the extended Flory-Huggins theory to predict mis-cibility in polymer blends. We leave as an exercise for the reader the proof that in the symmetric case (N = M), Equation 13 has a demixing critical point at &c = 1/2 and Xc = 2/N. Thus, a very small positive x-parameter (a small chemical incompatibility) results in phase separation in a polymer blend. In fact, even deuterated and non-deuterated polymers may phase separate!

The third aspect of Flory-Huggins theory that we shall consider is the prediction that is made for the osmotic pressure of a polymer solution. Recall that the osmotic pressure is the pressure difference required to maintain equilibrium across a semi-permeable membrane that seperates the solution from a reservoir of solvent. In these practically incompressible systems, it can be shown that the osmotic pressure n is (minus) the volume derivative of the free energy, n = —dF/dV. Osmotic pressure is here defined to be a thermodynamic quantity, but don't get hung up on this! It is capable of exerting real mechanical forces, as will be testified by any bacterium that has burst after being placed in distilled water.

Flory-Huggins theory, Equation 8, predicts that l3n/kBT = &/N — log(1 — &) — & — X&2, or on expanding for small &,

Let us focus on the good solvent case (x = 0). We notice that there is a crossover at &* - 1/N. If & < &*, we have n = &kBT/Nl3 = pkBT. In this regime, the van't Hoff law is obeyed (osmotic pressure equals kBT times the number density of objects in solution). The van't Hoff law can be used to determine the molecular weight of the polymer, but the fact that it only obtains for very low polymer volume fractions was a stumbling block in the early days of polymer science. For & > &*, Flory-Huggins predicts a regime where n — &2 is independent of N.

It turns out that these predictions of Flory-Huggins theory are nearly, but not quite, right. The modern approach to these problems originated with a mapping of polymer solution statistics onto another spin model by des Cloizeaux (1975). In this approach, &* is the overlap volume fraction for the coils, and the regime where &* C & C 1 is known as the semi-dilute solution regime. Briefly, &* can be computed as the point where R3 x (&*/Nl3) - 1. This results in &* - N1-3v, or &* - N-4/5 using the Flory value for the swelling exponent. In the semi-dilute regime, it is still true that the osmotic pressure does not depend on N (the chain ends do not count). If we suppose that power law scaling holds in the semi-dilute regime, n — &a, and demand continuity of the osmotic pressure with the van't Hoff law as one approaches &*, we can derive a = 3v/(3v — 1). With the Flory value for v, this predicts n — &9/4. Other aspects of the theory of semi-dilute solutions indicate how the chain shrinks from being fully swollen at & < &*, to being ideal as & ^ 1 (the Flory theorem). If we assume R — Nv(&/&*)-13, where ft is an exponent characterising the shrinkage, then an interesting exercise is to prove that ft = (2v — 1)/(6v — 2) (i.e. ft = 1/8 if v = 3/5).

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