## Polymer physics

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## Flory-Huggins Lattice Theory of Polymer Solutions, Part 1

The thermodynamics of (binary) regular polymer solutions1 were first investigated by Paul Flory2 and Maurice Huggins3 independently in the early 1940s. They assumed a rigid lattice frame, that is, the molecules in the pure liquids and in their solution / mixture are considered to be distributed over N0 lattice sites, as illustrated in the figure below. The total number of lattice sites, N0, is assumed to be equal to the number of solvent molecules, Ns, and polymer repeat units, Npr:

N0 = Ns + Npr

where Np is the number of polymer molecules each consisting of r repeat units.

### Two-dimensional Lattice

The model has been described in great detail by Flory in his famous book "Principles of Polymer Chemistry" (1953).4 Following the standard theory of mixing for small molecules of similar size and using Stirling's approximation lnM! = M lnM - M, Flory and Huggins obtained following expression for the entropy of mixing:

ΔSmix ≈ -k (Np ln φp + Ns ln φs )

Alternately, the entropy of mixing can be written as

Δsmix = ΔSmix / N0 ≈ -k {φp / (r·vr) · ln φp + φs / vs · ln φs}

where φs and φp are the volume fractions of the solvent and polymer,

φs = Nsvs / (Nsvs + Nprvr),   φp = Nprvr / (Nsvs + Nprvr)

and vs and vr are the volumes of a solvent molecule and of a polymer repeat unit, respectively. Obviously, the solvent needs not necessarily to be made of a single unit. The solvent may, in fact, consist of several repeat units or of another polymer.

The Gibbs free energy of mixing, ΔGmix, often includes an enthalpy part, that is, mixing can be an endothermic or an exothermic process

ΔGmix = ΔHmix - T ΔSmix

where ΔHmix is the heat of mixing. Flory and Huggins introduced a new parameter, the so called Flory-Huggins interaction parameter to describe the the polymer-solvent interaction:5

χps = ΔHmix / (kT Ns φp)

which combined with the entropy term leads to the free energy of mixing:

ΔGmix / kT = Np ln φp + Ns ln φs + χps Ns φp

Assuming equal lattice volumes for both repeat units, the free energy per lattice site can be written

Δgmix / kTφp / r · ln φp + φs ln φs + χps φs φp

A more general expression for the free energy of mixing is

Δg'mix / kT = φp / (r·vr) · ln φp + φs / vs · ln φs + χps φs φp / √(vrvs)

where Δg'mix is the free energy of mixing per unit volume. These equations are the starting point for many other important equations. For example, the partial molar free energy of mixing (chemical potential) can be obtained by differentiation of the expression above with respect to Ns. This gives

Δμs/RT =  ln [1- φp] + (1 - 1/r) φp + χps φp2

where μs is the chemical potential of the solvent per mole. Substitution of this expression into the osmotic pressure relation Π = - Δμs / Vs gives

ΠRT Vs-1 · {ln [1- φp] + φp + χpsφp2}

where Vs is the molar volume of the solvent.

##### References
1. The word regular implies that the molecules mix in a totally random manner. Thus, there is no segregation or preference of any molecule for another molecule.

2. P. J. Flory, J. Chem. Phys. 9, 660 (1941); 10, 51 (1942)
3. M. L. Huggins, J. Phys. Chem. 46, 151 (1942); J. Am. Chem. Soc. 64, 1712 (1942)
4. Paul L. Flory, Principles of Polymer Chemistry, Ithaca, New york, 1953
5. This equation can be easily deduced; The total number of contacts between a polymer and its neighbors is z-2 per chain. The probability that a cell is occupied by a solvent molecule is equal to its volume fraction φs. Then the total number of solvent-monomer contacts is simply (z-2) rNpφs. If εps is the change in energy for the formation of an unlike pair contact, Δεpsεps - ½ pp + εss), then the total enthalpy of mixing is simply ΔHmix = Δεps (z-2) rNpφs = Δεps z Nsφp. This is the well-known van Laar equation for mixing for 2-component mixtures, which can be recast to ΔHmix = kT χ Nsφp, with χ = z Δεps / kT.

• ## Summary

### Flory-Huggins

describes the thermodynamics of polymer solutions and polymer blends. It is a simple mean-field lattice model that can be used to understand the nonideal nature of polymer mixtures and solutions.

• The model assumes simple mixing rules:

1. The solvent and monomer repeat units in the pure liquids and in their mixture are considered to be distributed randomly over the lattice sites.

2. The volume of the mixture is equal to the volumes of the individual components, i.e. no volume change on mixing.

3. The total number of pair interactions in the mixture is equal to the sum of pure component pair interactions.

4. The average volume and energy for cross-interaction is equal to the geometric average of the pure components volume and interaction energy.

5. The interaction between two molecules is assumed to be equal to the interaction of a given molecule and an average field, caused by all the other molecules in the system.

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