Flory–Huggins solution theory

Flory–Huggins solution theory is a mathematical model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change Δ G m {displaystyle Delta G_{m}} for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments. Flory–Huggins solution theory is a mathematical model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change Δ G m {displaystyle Delta G_{m}} for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments. The thermodynamic equation for the Gibbs energy change accompanying mixing at constant temperature and (external) pressure is A change, denoted by Δ {displaystyle Delta } , is the value of a variable for a solution or mixture minus the values for the pure components considered separately. The objective is to find explicit formulas for Δ H m {displaystyle Delta H_{m}} and Δ S m {displaystyle Delta S_{m}} , the enthalpy and entropy increments associated with the mixing process. The result obtained by Flory and Huggins is The right-hand side is a function of the number of moles n 1 {displaystyle n_{1}} and volume fraction ϕ 1 {displaystyle phi _{1}} of solvent (component  1 {displaystyle 1} ), the number of moles n 2 {displaystyle n_{2}} and volume fraction ϕ 2 {displaystyle phi _{2}} of polymer (component  2 {displaystyle 2} ), with the introduction of a parameter χ {displaystyle chi } to take account of the energy of interdispersing polymer and solvent molecules. R {displaystyle R} is the gas constant and T {displaystyle T} is the absolute temperature. The volume fraction is analogous to the mole fraction, but is weighted to take account of the relative sizes of the molecules. For a small solute, the mole fractions would appear instead, and this modification is the innovation due to Flory and Huggins. In the most general case the mixing parameter, χ {displaystyle chi } , is a free energy parameter, thus including an entropic component. We first calculate the entropy of mixing, the increase in the uncertainty about the locations of the molecules when they are interspersed. In the pure condensed phases — solvent and polymer — everywhere we look we find a molecule. Of course, any notion of 'finding' a molecule in a given location is a thought experiment since we can't actually examine spatial locations the size of molecules. The expression for the entropy of mixing of small molecules in terms of mole fractions is no longer reasonable when the solute is a macromolecular chain. We take account of this dissymmetry in molecular sizes by assuming that individual polymer segments and individual solvent molecules occupy sites on a lattice. Each site is occupied by exactly one molecule of the solvent or by one monomer of the polymer chain, so the total number of sites is N 1 {displaystyle N_{1}} is the number of solvent molecules and N 2 {displaystyle N_{2}} is the number of polymer molecules, each of which has x {displaystyle x} segments. From statistical mechanics we can calculate the entropy change, the increase in spatial uncertainty, as a result of mixing solute and solvent. where k {displaystyle k} is Boltzmann's constant. Define the lattice volume fractions ϕ 1 {displaystyle phi _{1}} and ϕ 2 {displaystyle phi _{2}}