Flory-Huggins Solution Theory

Flory-Huggins Solution Theory jls164 Wed, 02/26/2014 - 12:24

Flory-Huggins solution theory attempts to describe the thermodynamics of polymer solutions in a more accurate way than ideal solution theory. We noted the two important flaws with ideal solution theory, namely the fact that polymers are large in comparison to solvent and that there are intermolecular interactions to account for. Let’s tackle the size issue first. How can we account for the fact that polymers are very large in comparison to the size of the solvent? We are going to imagine that a polymer has a bunch of segments, where each segment is the size of the solute. So in this picture (Figure 10.3) for example, each blue dot is a segment of a polymer, and all those blue dots are connected to make up one polymer molecule.

Figure 10.3: Simple model of a polymer molecule in a solvent

Source: Lauren Zarzar

Flory’s result for the combinatorial entropy of mixing is then:

$Δ S_{m} = - R [n_{1} \ln ϕ_{1} + n_{2} \ln ϕ_{2}]$

where $ϕ_{1}$ and $ϕ_{2}$ are the volume fractions (not mole fractions) of the two components (i.e., solvent and polymer), respectively.

EXAMPLE

In the picture above (Figure 10.3), what is the mole fraction (or number fraction) of polymer?

$\to$ There is 1 molecule of polymer and 75 molecules of solvent, thus the total number of molecules is 76. $Χ_{p o l y m e r} = 1 / 76$ .

What is the volume fraction of polymer in the picture above?

→ The volume of the polymer is 25 times larger than the volume of the solvent. The total volume is 100. Thus the volume fraction of polymer is $ϕ_{p o l y m e r} = 25 / 100$ .

What is $Δ S_{m}$ for the above picture using ideal solution theory?

$Δ S_{m} = - k [n_{1} \ln X_{1} + n_{2} \ln X_{2}]$

$Δ S_{m} = - k [1 * \ln (1/76) + 75 * \ln (75/76)]$

What is $Δ S_{m}$ for the above picture using Flory-Huggins theory?

$Δ S_{m} = - k [n_{1} \ln ϕ_{1} + n_{2} \ln ϕ_{2}]$

$Δ S_{m} = - k [1 * \ln (0.25) + 75 * \ln (0.75)]$

PROBLEM

Determine the entropy change that takes place when mixing 10 g of toluene with 10 g of a polystyrene sample with M_n = 100 000 g/mol. Assume the volume of a monomer is approximately the same as a solvent molecule. Molar mass of toluene = 92 g/mol, molar mass of styrene = 104 g/mol. R = 8.314 J/(K mol)

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ANSWER

First, let’s solve for the number of moles of solvent and polymer:

$n_{1}_{(s o l v e n t)} = \frac{10 g}{92 \frac{g}{m o l}} = 0.109 m o l$

$n_{2}_{(p o l y m e r)} = \frac{10 g}{100, 000 \frac{g}{m o l}} = 0.0001 m o l$

We need to find the degree of polymerization of the polystyrene so we can figure out the “volume” of the polymer in relation to the solvent:

$x_{n} = \frac{\frac{100, 000 g}{m o l}}{104 \frac{g}{m o l}} = 962$

$(1 m o l e c u l e o f p o l y m e r i s 962 t i m e s b i g g e r i n v o l u m e t h a n t h e s o l v e n t)$

$ϕ_{1 (s o l v e n t)} = \frac{0.109}{0.0001 m o l * 962 + 0.109 m o l} = 0.531$

$ϕ_{2 (p o l y m e r)} = \frac{(0.0001 m o l) * 962}{0.0001 m o l * 962 + 0.109 m o l} = 0.469$

$Δ S_{m} = - 8.314 J / (K m o l) [0.109 * \ln (0.531) + 0.0001 m o l * (0.469)]$

$Δ S_{m} = 0.574 J$

Next, we will tackle the intermolecular interactions and their contribution to mixing. We are going to first make a “nearest neighbor” assumption, which is to say that we are going to neglect long range interactions and only consider the interactions between molecules that are closest to each other. In the pure solvent before mixing, we have solvent-solvent interactions, and let’s call the free energy of this interaction g₁₁. In the pure polymer before mixing, we have polymer-polymer interactions, which we will call g₂₂. When we mix the solvent and polymer, we must break the solvent-solvent interaction, and break the polymer-polymer interaction, before forming a new solvent-polymer interaction, g₁₂. The change in energy per each new solvent-polymer pair formed is thus:

$Δ g_{12} = g_{12} - \frac{1}{2} (g_{11} + g_{22})$

Flory defined the following interaction parameter, which he made dimensionless by dividing by $k T$ , and it represents polymer-solvent interactions using this value of ∆g₁₂ and “ $z$ ”, which is the number of surrounding “sites” or “cells” around each position in our matrix:

$F l o r y - H u g g i n s i n t e r a c t i o n p a r a m e t e r = χ = \frac{(z - 2) Δ g_{12}}{k T}$

We use this interaction parameter as a contribution to the free energy of mixing. It is a way of trying to account for the change in enthalpy that can occur when the polymer and solvent mix:

$Δ G_{m} = R T [n_{1} \ln ϕ_{1} + n_{2} \ln ϕ_{2} + n_{1} ϕ_{2} χ]$

Now even though this model is still relatively simplistic – after all, we are still considering that the molecules are organized in this matrix which is of course unrealistic – it is still much better than ideal solution theory.

PROBLEM 2

Which is more favorable for mixing, a high or low Flory-Huggins parameter?

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ANSWER 2

Low. For favorable, spontaneous mixing, we need ∆G_m to be negative. We thus want χ to be small.