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Dynamics of Regular Star Polymers: The Intrinsic Viscosity

The dynamics of a regular star polymer is investigated under good-solvent and Θ conditions. A self-consistent approach to the equilibrium and dynamics is proposed within the Gaussian approximation in terms of the polymer normal modes, i.e., suitable statistically-independent linear combinations of the bond vectors. The equilibrium normal modes used to study the star expansion through self-consistent free-energy minimization (see Allegra, G. ; Colombo, E. ; Ganazzoli, F. Macromolecules 1993, 26, 330) form a first-order approximation to the dynamical normal modes under partial-draining conditions. The latter modes are obtained by diagonalization of a matrix that depends on the intramolecular elasticity and on the hydrodynamic interaction in the preaveraging approximation, the effect of chain expansion on both being easily accounted for. If the equilibrium normal modes are used, the relaxation times, from which the intrinsic viscosity is calculated, are somewhat in error only for the collective modes describing the concerted motion of the arms, unlike those related with the independent motion of the arms ; on the other hand, they yield the intrinsic viscosity to a very good approximation, thus avoiding the lengthy diagonalization. Numerical results for regular stars and linear chains are summarized by analytical equations giving the mean-square radius of gyration , the hydrodynamic radius R H and the intrinsic viscosity [η]. The influence of the topology is expressed through the ratios g Q = Q star /Q lin , Q being any of the above quantities. In a good solvent, the star is only slightly more expanded than the linear chain, and so the g ratios are very close to the theoretical phantom-chain value both for (S 2 ) and for R H . Conversely, [η] increases less in the star than in the linear chain because of the different rate of change with expansion of the intramolecular elasticity and of the hydrodynamic interaction, and so the corresponding g ratio is lower than the phantom-chain value. In the Θ state, the residual three-body interactions give a finite expansion to the star, unlike the linear chain. Therefore, we have g Q ph ≤ g Q * and R H , in agreement with experiment (the asterisk refers to the good-solvent conditions) ; on the other hand, we get g η * < g η ph < g η Θ , whereas the experimental finding is g η * < g η Θ < g η ph . The reason for this discrepancy is attributed to the preaveraging approximation, which becomes questionable for stars in the Θ state because of their large density near the branch point.