Macromolecules zyxwvu 1983,16, 1701-1707 1701 Kirkwood-Riseman Calculations for Hydrodynamic Properties of Flexible Branched Polymers zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG R. Prats, J. Pla, and J. J. Freire* Departamento de Quimica Fisica, Facultad de Ciencias Quimicas, Universidad Complutense, Madrid-3, Spain. Received December 30, 1982 ABSTRACT: Numerical results for hydrodynamic properties of uniform flexible branched polymers (tri- functional comblike polymers and star polymers of different functionalities) have been calculated by means of the Kirkwood-Riseman theory. The properties have been expressed in terms of the coefficients g, zyxw h, and zy g', which correspond to the ratios of the mean radius of gyration, the translational diffusion coefficient, and the intrinsic viscosity of branched chains to those of linear chains with the same molecular weight. Two standard treatments of the hydrodynamic interactions have been used, leading to significantly different results. The numerical values obtained with the most rigorous treatment clarify the ranges of validity of the approximate relations between the ratios suggested by Zimm and Kilb, Stockmayer and Fixman, and Kurata and Fukatsu. The qualitative behavior of experimental data is, in general,reproduced. Moreover, theoretical and experimental values of h are in close agreement for several types of stars. Introduction The hydrodynamic properties of linear flexible polymers have been extensively studied from both the theoretical and experimental points of view.l These studies have made it possible to relate experimental data with micro- scopic characteristics through simple limit expressions. Thus, for high molecular weights, the translational friction coefficient, f, is obtained as f zyxwvutsrqponm = Kfr0(S2)'/2 (1) where zyxwvutsrqpo Kf is a numerical constant, fo is the solvent Viscosity, and ( S2) is the mean quadratic radius of gyration of the chain. For the intrinsic viscosity, [VI, Flory's relation is applicable (S2)3/2 [f] = @ ' - (2) M (a' is also a constant and M is the polymer molecular weight). However, the effect of branching on these properties has been described mainly on a qualitative bash2 Thus, the intrinsic viscosity of branched chains has been investigated through adequate hydrodynamic treatments only in a few particular case^.^-^ In one of the most detailed studies performed so far, Zimm and Kilb3 generalized the Rouse-Zimm theory for linear flexible molecules in their unperturbed state to two different types of star molecules. They analyzed the numerical results for the parameter y, defined by (3) where the subscripts b and 1 refer to a branched chain and a linear chain, both with the same molecular weight, and g corresponds to the ratio of the respective radii of gyration g' = [flb/[f11 = gr g = (s2)b/(s2)I (4) For both types of star chains, Zimm and Kilb obtained zyxwvu y N 0.5. On the basis of this coincidence they proposed that eq 3 with y = 0.5 might hold for all kinds of branched molecules. This result is, however, in conflict with simpler arguments. Thus, if one considers a chain with a very long backbone and many much shorter branches (for instance, a uniform comblike polymer with many branching units) so that its cross section is very small compared to the backbone's contour length, the chain's hydrodynamic be- havior must be defined by the backbone. Then eq 2 can be applied and, in consequence, the value y = 3/2 is pre- dicted. 0024-9297/83/2216-1701$01.50/0 0 Experimental data2i6-9 obtained for different flexible branched chains yield a broad distribution of values of y within the interval 0.5 zyxw 5 y 5 1.5. In fact, the experimental variation of y should be strongly dependent on the to- pology of the chain; i.e., it should be related to the degree and type of branching. Nevertheless, this dependence is not fully understood, in part due to the lack of a rigorous theoretical analysis. Other approximate relations between different ratios have been proposed. Stockmayer and Fixmanlo suggested g'= h3 (5) = fb/fl (6) which can be simply obtained by applying the approximate Kirkwood formula to the calculation of fb and fl. Some experimental data" show, however, certain deviation from the theoretical values of h obtained this way. We should also consider the inequality predicted by Kurata and Fu- katsu12 (7) The aim of this work is to obtain numerical results for the hydrodynamic properties of detailed theoretical models corresponding to different types of branched chains in their unperturbed state. These results can be useful to clarify the role of the chain topological form or, at least, to es- tablish the ranges of validity for the different approximate theoretical relations, eq 3,5, and 7. We study the influence on the results of the total number of hydrodynamic units, N (proportional to the number of real repeating units), for star polymers of different functionalities (denoted by the value of the variable F). Also, we study comblike polymers with F = 3 and with different values of m (number of branching points along the backbone). In both cases we consider a uniform distribution of units among the p subchains (p = F for stars and p = 2m + 1 for our tri- functional comblike structures). Our calculations are based on the general Kirkwood- Riseman theory, which accounts for hydrodynamic inter- actions between units.' In this theory, the form of the general expressions for the hydrodynamic properties is not affected by the type of connectivity between units. This constitutes an important practical advantage with respect to the RouseZimm theory as extended by Zimm and Kilb, which includes calculations through the coordinate matrix A, built in a different way for every type of branching. 1983 American Chemical Society where h is the ratio of friction coefficients 1 I h/g1I2 I 1.39