Chapter 6: Osmometry

Measuring Properties of Polymers:

We have run into several key properties of polymers like molecular weight, end-to-end distance, etc., and there are a number of other key polymer behaviors that depend on some of these key properties like

• Non-cross linked rubber elasticity
• Shear thickening
• Elastic modulus of cross linked networks
• Electrical conductivity

The question then becomes if these polymer properties are so critical to behavior how to do we measure them. This question is quite difficult for example it will be important in terms of rubber elasticity to know the distance between cross-links or how do we measure χ? Well we will be talking about several different techniques to understand:

• Understand the origin of osmotic pressure in polymer solutions and relate this to molecular size and polymer-polymer interactions.

Membrane Osmometry

We now know that we can use intrinsic viscosity measurements to determine molecular weight and solvent conditions via our a scaling parameter. We can gain even more insight into a polymer solution using a measurement technique called membrane osmometry, which measures the the osmotic pressure of a polymer solution.

Osmotic pressure, π, is a colligative property of a solution, it depends only on the number of solute molecules in solution. It does not depend on the chemical composition of those solute molecules. For a physical interpretation, osmotic pressure is a thermodynamic force that arises due to the fact that when species mix the solution tends to maximize entropy via dilution with solvent. Osmotic pressure is typically measured using an osmometer which has two separate containers. One has the solvent and the other the solute. The containers are separated by a semi-permeable membrane which allows only the small solvent molecules to cross, the polymer solute cannot cross the membrane.

So you can imagine that when you put a polymer solute on one side the solvent will tend to flow from the pure solvent side to the polymer side until the chemical potential of each side are equal. Or in other words until thermodynamic equilibrium has been achieved. This driving force comes from the mixing terms we went over exhaustively from Flory-Huggins, in short the flow of solvent to the polymer side increases the entropy of the system. You can imagine this is the case physically because the number of microstates or configuration states of polymer and solvent increases as opposed to the pure polymer solution. Additionally this flow of solvent also increases the volume of the system which further increases the number of microstates. And the driving force is thermodynamic in nature and due to the osmotic pressure which pushes the solvent through the membrane into the increasingly dilute solution of polymer.

This flow of solvent will increase the volume as we just mentioned and on the polymer solution size you will see the solution to raise up the vertical tube which extends from the solution side of the chamber. This motion in the vertical direction is obviously opposed by gravity so we have a mechanical pressure that opposes the osmotic pressure. At equilibrium, i.e. when the solution stops rising, this mechanical pressure will be equal to the osmotic pressure. Once thermodynamic equilibrium is reached we can measure the osmotic pressure via the final height of the fluid in the solution part of the chamber and we can calculate the mechanical pressure and thus the osmotic pressure.

Once we have the osmotic pressure we can use this critical quantity to determine other properties of polymers. However, first we have to calculate the osmotic pressure by relating it to the condition of equilibrium which is that the chemical potential on both sides of the container must be equal

$$\mu _1^0 = \mu_1 +\int_{P_0}^{P_0 + \pi} \frac{\partial \mu_1}{\partial P} dP \nonumber$$

where is the chemical potential of the solvent alone (left side) which has to be equivalent, at equilibrium, to the chemical potential of the solution, µ₁, plus the change in the chemical potential due to the change in pressure, where the change in pressure is equal to the osmotic pressure π. To calculate the derivative we can rewrite the chemical potential in terms of its definition as the derivative of the Gibbs free energy:

$$\frac{\partial \mu_1}{\partial P} = \frac{\partial}{\partial P} \left ( \frac{\partial G}{\partial n_1} \right ) = \frac{\partial}{\partial n_1} \left ( \frac{\partial G}{\partial P} \right ) \nonumber$$

Here, we have exchanged the order of the partial derivatives so that we can relate the Gibbs free energy to a different and more convenient thermodynamic variable. Remember that G = U − TS − PV so again we have

$$dG = VdP - SdT + \mu dn \nonumber$$

$$\left(\frac{\partial G}{\partial P}\right)_{n, T} = V \nonumber$$

Now we can substitute back in and we get

$$\frac{\partial \mu_1}{\partial P} = \frac{\partial}{\partial n_1} \left ( \frac{\partial G}{\partial P} \right ) = \frac{\partial V} {\partial n_1} = \overline V_1 \nonumber$$

where V ₁ is the partial molar volume of pure solvent. This is a quantity that is constant with pressure. So we can finish by writing that

$$\mu _1^0 = \mu_1 + \int_{P_0}^{P_0 + \pi} \frac{\partial \mu_1}{\partial P} dP \nonumber$$

$$\mu _1^0= \mu_1 + \int_{P_0}^{P_0 + \pi} \overline{V}_1 dP \nonumber$$

$$\mu _1^0= \mu_1 + \pi \overline V_1 \nonumber$$

$$\mu_1 - \mu_1^0 = - \pi \overline V_1 \nonumber$$

So the change in chemical potential between the pure and the solution side is simply the osmotic pressure multiplied by the partial molar volume of the pure solvent. With this change in chemical potential we can also relate this equation back to Flory-Huggins in the limit of dilute solutions:

$$\mu _1 - \mu _1^0 = RT \left [ -\frac{\Phi_2}{x_2} +(\chi -\frac{1}{2})\Phi_2^2 \right ] \nonumber$$

(Note: Remember when we are dealing with molar quantities the prefactor of kT is replaced by RT and the variables n₁ and n₂ now refer to the number of moles, not molecules; however, practically speaking this only matters in terms of having the correct units.) This expression is only valid in the limit of dilute solutions - that is, φ₂ << 1. We have previously stated above that our experimental set up assumes a dilute solution in the osmometer, so we can again rearrange and get

$$\mu _1 - \mu _1^0 = -\pi \overline{V}_1 \nonumber$$

$$\mu _1 - \mu _1^0 = RT \left [ -\frac{\Phi_2}{x_2} +(\chi -\frac{1}{2})\Phi_2^2 \right ] \nonumber$$

$$\pi = RT \left [ \frac{\Phi _2}{\overline{V}_1x_2} +(\frac{1}{2} - \chi)(\frac{\Phi_2^2}{\overline{V}_1}) \right ] \nonumber$$

In the limit of very dilute solutions, Φ₁ >> Φ₂, so $\Phi_2 \approx \frac{n_2x_2}{n_1}$. You can also approximate the total volume of the solution as the volume of the solvent, or V = n₁V₁. This leads to V ₁ ≈ V₁. Again physical meaning ... the increase in volume of the solution due to the addition of solvent is approximately the same as the volume of a single mole of solvent molecules. When we substitute these approximations into the equation above

$$\pi = RT \left [ \frac{n _2}{V} + \left ( \frac{1}{2} - \chi \right ) \left ( \frac{n_2}{V} \right )^2 V_1x_2^2 \right ] \nonumber$$

Let’s take a look at the first term and what happens when χ = 1/2, we get the ideal gas law - πV = RTn₂. This first term is the contribution if we had an ideal solution.

We can simplify the expression a bit more by defining variables in terms of the concentration of polymer c₂ instead of the volume fraction φ₂, knowing that

$$c_2 = \frac{\text{Total mass of Polymer}}{\text{Volume}} = \frac{n_2 \overline{M}_n}{V} \nonumber$$

remember the total mass is related to the number averaged molecular weight and now we can use this to substitute into the equation above and get:

$$\frac{\pi}{c_2} = RT \left [ \frac{1}{\overline{M}_n} + \left ( \frac{1}{2} -\chi \right ) \frac{V_1x_2^2}{\overline{M}_n^2}c_2 \right ] \nonumber$$

Now we can see how important osmotic pressure is as we can now relate this to the number averaged molecular weight and χ. You can also see that this equation has the form of a virial expansion (move this to the first instance)

$$P = RT(A_1c + A_2c^2 +A_3c^3 + ...) \nonumber$$

We can match terms with our equation and get that the first virial coefficient $A_1 = \frac{1}{\overline{M}_N}$ and the second virial coefficient $A_2 = \left ( \frac{1}{2} -\chi \right )\frac{V_1x_2^2}{\overline{M}_n^2}$

Experimental Membrane Osmometry Technique

Now how do we utilize this equation. Well now we can measure the M_n and χ from experiments by plotting the reduced osmotic pressure $\frac{\pi}{c_2}$ vs. c₂.

This will once again give an y-intercept as

$$\frac{RT}{\overline{M}_n} \nonumber$$

and a slope as

$$\left ( \frac{1}{2} -\chi \right ) \frac{V_1x_2^2}{\overline{M}_n^2}c_2 \nonumber$$

So from a single series of measurements we can get both parameters! You can also see that for theta conditions χ = 0.5 and the slope disappears! (write problem with this)

Key Assumptions in Membrane Osmometry Derivation

For membrane osmometry, we make two key assumptions:

1. We assume mean-field conditions - local environments are similar everywhere throughout the polymer solution, we did this in the Flory-Huggins Theory.
2. We assume a dilute solution - Φ₂ << 1. We use this assumption to simplify the expression for the chemical potential.

These conditions are somewhat contradictory as mean field conditions break down in dilute solutions so we force experimental conditions to be

1. The system is thermodynamically concentrated - $c_2 > c_2^*$
2. The system is mathematically dilute - Φ₂ << 1

Our experimental conditions are very specific and only valid in the semi-dilute condition. The parameter is the overlap concentration, where polymer coils just begin to overlap in solution. Around this concentration, polymer coils are slightly interpenetrating such that the mean-field assumption of equivalent local environments is accurate; this allows us to apply Flory-Huggins theory. However, there is still enough solvent right at the overlap concentration that the assumption of a dilute solution is accurate, and we can make the assumptions of the limit of φ₂ << 1. We can simply approximate the overlap concentration as

$$c_2^* = \frac{\text{mass of polymer molecule}}{\text{volume of polymer coil}} = \frac{M/N_{av}}{\langle r^2 \rangle ^\frac{3}{2}} \nonumber$$

We know that M ∝ N¹ and $\langle r^2 \rangle ^\frac{3}{2} \propto N^{3/2}$, so the entire expression scales

$$c_2^* \propto N^{-1/2} \nonumber$$

This scaling law is interesting because it says that at large values of N, the overlap concentration actually gets quite small which should make sense. Additionally, this implies that the conditions under which our derivation of membrane osmometry apply can be achievable under a broad range of experimental conditions.

Summary: Membrane Osmometry

We have discussed membrane osmometry, where we place a dilute polymer solution in a container adjoined to a different container filled with only solvent. The two containers are separated by a semi-permeable membrane that permits passage of solvent but not polymer. Due to the free energy gain from the entropy of mixing (characterized as a thermodynamic driving force called the osmotic pressure), solvent will move in the direction of the polymer solution, leading to an increase in competing pressure that opposes the osmotic pressure. When the flow stops, we know the osmotic pressure is directly offset by the pressure from the increased volume of the polymer solution, allowing us to measure the osmotic pressure experimentally. We saw that measuring this quantity led to an estimate of χ and number-average molecular weight.