The Mark-Houwink equation: intrinsic viscosity vs. molar mass

Mark (1938) and Houwink (1940) generalized Staudinger’s linear rule to \([\eta]=KM^a\) (mark_houwink_intrinsic_viscosity()), the relation still used to turn a viscosity measurement into a molar mass. Via Flory-Fox, \([\eta]\propto R^3/M\) with \(R\propto M^\nu\) gives \(a=3\nu-1\) (mark_houwink_exponent_from_flory()): \(a=0.5\) in a theta solvent, \(a=0.8\) in a good solvent, and Staudinger’s \(a=1\) is recovered only for stiffer chains. Here synthetic “measurements” with 3% seeded noise are fit back to recover \(K\) and \(a\) (fit_mark_houwink()).

theta solvent (nu=1/2): true a = 0.500, fitted a = 0.496; true K = 0.08, fitted K = 0.0835
good solvent (nu=3/5): true a = 0.800, fitted a = 0.811; true K = 0.01, fitted K = 0.00872

import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.polymer.systems.solution_viscosity import (
    fit_mark_houwink,
    mark_houwink_exponent_from_flory,
    mark_houwink_intrinsic_viscosity,
)

rng = np.random.default_rng(0)
M = np.logspace(4, 6.5, 12)
cases = {
    "theta solvent (nu=1/2)": (0.08, mark_houwink_exponent_from_flory(0.5)),
    "good solvent (nu=3/5)": (0.01, mark_houwink_exponent_from_flory(0.6)),
}

fig, ax = plt.subplots(figsize=(6.5, 4.5))
for label, (K, a) in cases.items():
    eta = mark_houwink_intrinsic_viscosity(M, K, a) * (1 + 0.03 * rng.standard_normal(M.size))
    K_fit, a_fit = fit_mark_houwink(M, eta)
    print(f"{label}: true a = {a:.3f}, fitted a = {a_fit:.3f}; true K = {K:.3g}, fitted K = {K_fit:.3g}")
    ax.loglog(M, eta, "o", label=f"{label}: a = {a_fit:.2f}")
    ax.loglog(M, mark_houwink_intrinsic_viscosity(M, K_fit, a_fit), "k-", lw=0.8)

ax.set_xlabel("molar mass M (g/mol)")
ax.set_ylabel(r"$[\eta]$ (mL/g)")
ax.set_title(r"Mark-Houwink: $[\eta] = K M^a$")
ax.legend()
plt.tight_layout()
plt.show()