THE UMBRELLA SHAPE OF STAR POLYMERS IN THE THETA-STATE

The chain conformation in the theta state is calculated by self-consistent free-energy minimization assuming that the overall long-range two- and three-body interactions mutually compensate. However, the repulsive screened interactions arising from the intrinsic chain thickness do survive, imparting an asymptotically finite expansion to the chain. Because of the medium-range character of these interactions, the proportionality between the unperturbed mean-square radius of gyration < S2 > O and the molecular weight is asymptotically maintained. Compared with the phantom chain, devoid of both intrinsic thickness (i.e. screened interactions) and long-range interactions, the arms experience an increasing expansion upon increasing the number of arms, f. At the same time, the fluctuations of the branch point around the centre of mass do increase as well, since the average angle between the arms decreases, and the macromolecule tends towards an umbrella-like shape. As a consequence, the macromolecular density becomes more uniform than for the phantom chain. This is reflected in the calculated structure factor S(Q) by a sharpening of the peak in the plot of Q2S(Q) versus Q (shere Q = 4n sin-(theta/2)/lambda, in good agreement with the experimental results by Huber et al. Through cancellation of contrasting effects, the topological index g = < S2 > O/< S2 > O, linear is close to the value of the phantom chain for any f, whereas h = R(HO)/R(HO), linear, R(H) being the hydrodynamic radius, and p = < S2 > 0(1/2)/R(HO) are somewhat larger. The negligible influence of the residual three-body interactions in the theta state compared to the screened interactions is finally commented upon considering also recent Monte Carlo simulations.