OSMOTIC PRESSURE OF POLYMER SYSTEMS

The osmotic pressure is the minimum pressure that has to be applied to a polymer solution to prevent a solvent to flow through a semipermeable membrane that separates pure solvent from polymer dissolved in the same solvent.

Osmometry is frequently used to determine the number-average molecular weight of polymers. A basic osmometer design is shown in the Figure below.

The osmotic pressure is also the thermodynamic driving force for mixing. It is always positive. However, for large molecules, this force is relative small. This follows directly from the osmotic pressure relationship:

Π = RT ln [a_s] / V_s

where V_s is the molar volume of the solvent and a_s is its activity. For polymer-solvent blends, the solvent activity can be calculated with the Flory-Huggins expression for activities:^1,2

ln [a_s] = Δμ_s/kT ≈  ln [1- φ_p] + φ_p + χ_psφ_p²

where φ_p is the polymer volume fraction. Substitution of the Flory-Huggins expression for solvent activity into the osmotic pressure relation gives

Π ≈ RT V_s^-1 · {ln [1- φ_p] + φ_p + χ_psφ_p²}

This expression can be simplified by expanding the logarithmic term in a Taylor series:

Π ≈ RT V_s^-1 · {φ_p / N + φ_p² · (0.5 - χ) + φ_p³ +...}

or

Π / c ≈ RT M_n^-1 · {1 + (M_nv_p²/V_s) · (0.5 - χ) c + (M_nv_p³/V_s) c²/3 + ...}

where v_p is the specific volume of the polymer, c the polymer concetration, c = φ_p / v_p, and χ the Flory-Huggins interaction parameter. The equation above can be rearranged to give the widely used virial expansion relation for the osmotic pressure³

Π / c ≈ RT · {M_n^-1 + A₂ c + A₃ c² + ...}

where A₂ and A₃ are the second and third virial coefficients:

A₂ = v_p²/V_s · (0.5 - χ₀)

A₃ = (v_p³/V_s) / 3

Values of the the second (A₂) and third osmotic virial coefficient (A₃), defined by the equation above, have been determined for many polymers, for example by light-scattering or osmotic pressure measurements. They are often tabulated as a function of polymer moelcular weight. Of great importance is the second osmotic virial coefficnent A₂, because the polymer-solvent interaction parameter χ₀ (at infinit dilution) can be calculated from it using the equation:

χ₀ = 0.5 - V_s A₂ / v_p² = 0.5 - V_s A₂ ρ_p²

For very dilute ideal polymer solutions, χ ≈ 0.5, φ → 0, the second and third term can be neglected. This gives the classical van't Hoff equation for the osmotic pressure of an ideal, dilute solution:

Π / c ≈ RT M_n^-1

According to this equation, the osmotic pressure of an ideal polymer solution rapidly decreases with increasing molecular weight, i.e., the driving force for polymer-polymer mixing is rather small.

In the Figure below, plots of the osmotic pressure for different concentrations of polystyrene (M_n = 404000 g/mol) in cyclohexane are shown. For low polymer concentrations in a good solvent (χ = 0.5), the osmotic pressure can be calculated with the vant Hoff law. However, for concentrations above 0.1 g/cc, the fit is poor and Flory's virial expansion relation gives much better results.