Monte Carlo study of titration of linear polyelectrolytes

Abstract

An off‐lattice Metropolis Monte Carlo algorithm with reptation is used to find the average fractional ionization ᾱ as a function of pH for a generic ionizable linear polyelectrolyte in a salt solution. The polyelectrolyte is treated as a threefold rotational isomeric state model polymer; each unit can bear a negative charge or not with intrinsic ionization constant pK_a. Debye–Hückel screening is assumed between the charges. For computational convenience, the dielectric constant of the polymer is taken to be that of the solvent. The number of units N was either 50 or 100. Monte Carlo results were collected for various Debye screening lengths at six combinations of number of chain units N, bond angle θ, and Manning parameter when fully charged, ξ₀. For four of the combinations, ξ₀ was 1 to take partial account of counterion condensation. These runs had N and θ of 50 and 1°, 50 and 70°, 100 and 1°, and 100 and 70°. The fifth combination had N=50, θ=70°, and ξ₀=2.85. The sixth had N=50, θ=27.34°, and ξ₀=0.72, for comparison with data for hyaluronate. The Monte Carlo results are compared to third nearest‐neighbor linear Ising type calculations and to simple mean field theories in α. Mean field theory in α worked very well in the (nearly rodlike) θ=1° cases using the known distance between units. Mean field theory in α using an estimate for the distance between units based on the ideas of electrostatic persistence length and excluded volume worked equally well for the θ=1° cases and moderately well for the θ=70° cases. The free energy and entropy per simulated chain were calculated by thermodynamic integration of the Monte Carlo results for ᾱ as a function of pH.