## Polymers in Solvents

### Quick Start

Put a polymer molecule into a solvent and three things might happen to it.

• It might curl up on itself
• It might not really notice the solvent
• It might prefer the solvent to itself and stretch out.

It really matters which of these it is (the Flory-Huggins χ parameter), as it affects viscosity, rheology, solubility and how the polymer interacts with other components of a formulation. But it also matters how many monomer units, N, of length b and the angles, θ that they can swing around. The amount of polymer is φ2.

Have a play with the values then read what's going on.

### Polymers in Solvents

N of monomers
b Monomer size
θmin Bend angle
Flory-Huggins χ
Re End-to-End
Rg R. of Gyration
φ2 Concentration
φ* Critical

A polymer dissolved in a solvent can be classified in two different ways. The first is on its size if it were an isolated blob of polymer. The second is on how isolated, or not, those blobs are.

### The size of a polymer chain

Assuming that the polymer is made of N links of chains length b where those chain segments cannot bend by more than an angle θ, and that the "happiness" of the polymer in the solvent can be measured by χ where a value of 0.5 is "neutral" then the polymer has an average end to end distance, Re and a typical radius of gyration Rg given by

Re² = b²Nv(1+cos(θ))/(1-cos(θ))

Rg² = Re²/6

But what is the exponent v? For a theta solvent (χ=0.5), v is 0.5, for a very poor solvent (χ>0.75), such that the polymer curls up on itself to go from a coil to a globule,v approaches 0.33. In a very good solvent (χ=0, often called athermal) v rises to 0.59

These formulae tell us the obvious: more monomers, N, longer chain segments, b, less and less freedom to curl back on itself (lower θ down to 0°) and a greater happiness in the solvent means a larger blob of polymer in solution.

But we also can learn more. At low concentrations ("dilute") the polymer coils are not touching so the viscosity is almost unchanged from the pure solvent. At a critical volume fraction, φ*, the coils are touching each other so above this concentration ("semidilute") the viscosity is increasingly strongly affected by the polymer. Finally at φ** the solution is "concentrated" and viscosity rises very strongly. φ** is generally taken to be around 0.2-0.3

The critical volume fraction φ* is calculated as:

φ* = N1-3v

The dependence on N is obvious - the larger the polymer the more space is taken up so the smaller is φ*. Similarly, the larger v the smaller is φ* because the polymer expands more into the solvent.

Officially a larger b makes the blobs larger it doesn't affect φ* because φ* = Nb³/Rg³ = Nb³/(bNv)³ and the b's cancel out. For app purposes, a b effect is included.