Staudinger’s macromolecules: solution viscosity grows with chain length

Staudinger’s case that rubber and cellulose are genuine covalent macromolecules rested heavily on dilute-solution viscosity: he found the specific viscosity per unit concentration rising in proportion to the chain length, \(\eta_\text{sp}/c=K_mM\) (staudinger_specific_viscosity()). If the chains were colloidal aggregates held together by weak association, the “particle size” would fall apart on dilution or change of solvent; covalent chains keep their length, so a polymer-analogous reaction (e.g. hydrogenating rubber) leaves the degree of polymerization – and hence \(\eta_\text{sp}/c\) – unchanged. This example contrasts the two pictures.

import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.polymer.systems.solution_viscosity import staudinger_specific_viscosity

Km = 1.0e-4  # Staudinger constant (per molar-mass unit, per concentration unit)
M = np.array([1e4, 3e4, 1e5, 3e5])
c = np.linspace(0.001, 0.01, 10)

for Mi in M:
    ratio = staudinger_specific_viscosity(c, Mi, Km) / c
    print(f"M = {Mi:8.0f}:  eta_sp/c = {ratio[0]:.2f} at every concentration (constant: {np.allclose(ratio, ratio[0])})")

M =    10000:  eta_sp/c = 1.00 at every concentration (constant: True)
M =    30000:  eta_sp/c = 3.00 at every concentration (constant: True)
M =   100000:  eta_sp/c = 10.00 at every concentration (constant: True)
M =   300000:  eta_sp/c = 30.00 at every concentration (constant: True)

Colloidal-aggregate picture: particles of 1e5 units dissociate on dilution (a simple association equilibrium, apparent size ~ sqrt(c)), so eta_sp/c would fall as the solution is diluted, unlike a real macromolecule.

M_apparent = 1e5 * np.sqrt(c / c[-1])
eta_colloid = staudinger_specific_viscosity(c, M_apparent, Km) / c
eta_macro = staudinger_specific_viscosity(c, 1e5, Km) / c

fig, axes = plt.subplots(1, 2, figsize=(11, 4.2))
M_scan = np.logspace(3.5, 6, 50)
axes[0].loglog(M_scan, staudinger_specific_viscosity(1.0, M_scan, Km), "k-")
axes[0].loglog(M, staudinger_specific_viscosity(1.0, M, Km), "o")
axes[0].set_xlabel("molar mass M")
axes[0].set_ylabel(r"$\eta_{sp}/c$")
axes[0].set_title(r"Staudinger's rule: $\eta_{sp}/c = K_m M$")

axes[1].plot(c, eta_macro, label="covalent macromolecule (Staudinger)")
axes[1].plot(c, eta_colloid, "--", label="dissociating colloidal aggregate")
axes[1].set_xlabel("concentration c")
axes[1].set_ylabel(r"$\eta_{sp}/c$")
axes[1].set_title("Dilution test: real molecules keep their size")
axes[1].legend()
plt.tight_layout()
plt.show()