1206 Macromolecules 1985, 18, 1206-1214 and n = 1, K = 106.6(in 0.01 M NaAc), respectively. Be- Acknowledgment. We are grateful to Mrs. A. R. So- cause the range for the extent of binding r is limited (0.7 chor for technical assistance, Dr. I. S. Ponticello and Mr. < r < 1.0) for this strong binding polymer, the agreement K. R. Hollister for the preparation of some of the polymers between the calculated curves and the data points in these used in this study, and Dr. H. Yu and Dr. J. L. Lippert plots does not support the validity of the independent-site for discussions during this study. model, however. One way to circumvent this difficulty was to increase References and Notes the ionic strength of the medium to 0.05 M NaCl and (1) Frank, H. S.; Evans, M. W. J. Chem. Phys. 1945, 13, 507. hence to weaken the binding so that the data points can (2) Kauzmann, W. Adu. Protein Chem. 1959, 14, 1. (3) Franks, F. “Water, A Comprehensive Treatise”; Franks, F., cover the whole range of r. These data were indeed ob- Ed.; Plenum Press: New York, 1975;Vol. 4, p 1. tained and analyzed in the same manner as those above (4) Klotz, I. M.; Royer, G. P.; Sloniewsky, A. R. Biochemistry for MPVI + MO and are summarized in Table IV. Direct 1969, 8, 4752. comparison of the parameters and w for the two poly- (5) Quadrifoglio, F.; Crescenzi, V. J. Colloid Interface Sci. 1971, KO 35, 447. mer listed in this table is not appropriate because the two (6) Reeves, R. L.; Harkaway, S. A. “Micellization, Solubilization, systems were not measured at the same ionic strength. and Microemulsions”:Mittal. ’ K. L..I, Ed.: Plenum Press: New Note that the cooperative model seems to describe well York, 1977; Vol. 2, p’819. both systems when the polymer sites are approximately (7) Takagishi, T.; Nakata, Y.; Kuroki, N. J. Polym. Sci. 1974,12, 807.__ . 70% occupied (Le., r = 0.6-0.7). (8) Ando, Y.; Komiyama, J.; Iijima, T. J. Chem. SOC.Jpn. 1981, 3, 432. Conclusions (9) Tan, J. S.; Handel, T. M., to be submitted for publication. 1. The 1:l stoichiometry for complexes of methyl orange (10) Orchard, B. J.; Tan. J. S.; Hopfinger, A. J. Macromolecules and the quaternized poly(N-vinylimidazole) homopolymers 1984, 17, 169. (11) Tan, J. S.;Sochor, A. R. Macromolecules 1981, 14, 1700. suggests that binding capacity is dictated primarily by (12) Tan, J. S.;Gasper, S. P. J. Polym. Sci. Polym. Phys. Ed. 1974, charge. 12, 1785. 2. The binding strength between methyl orange and the (13) Scholtan, W. Makromol. Chem. 1953, 11, 131. quaternized polymers is influenced by (a) Coulombic in- (14) Klotz, I. M. “The Proteins, Chemistry, Biological Activity, and Methods”; Neurath, H., Bailey, K., Eds.; Academic Press: New teraction between the anionic dye and polycations, (b) York, 1953; Vol. 1, p 727. nonionic interaction between the quaternizing side chain (15) Scatchard, G. Ann. N.Y. Acad. Sci. 1949, 51, 660. and the dye, and (c) nonionic bound dye-dye interaction. (16) Tanford, C. “Physical Chemistry of Macromolecules”;Wiley: 3. The binding behavior in the saturation range 0 < r New York, 1967. (17) von Hippel, P. H.; Schleich, T. Acc. Chem. Res. 1969,2,257. < 0.7 can be described by the McGhee-von Hippel ex- (18) Cantor, C. R.; Schimmel, P. R. “Biophysical Chemistry”; W. pression. The bound dye-dye interaction is associated with H. Freeman: San Francisco, 1980; Part 111. the aggregation tendency of the dye, which contributes an (19) McGhee, J. D.;von Hippel, P. H. J. Mol. Biol. 1974,86,469. additional force to the overall polymer-dye interaction. (20) Schwarz, G. Eur. J. Biochem. 1970, 12, 442. The cooperative binding cannot persist, however, at a (21) Crothers, D. M. Biopolymers 1968, 6,575. (22) Hill, A. V. J. Physiol. (London) 1910, 40, iv. higher saturation level, possibly because of the steric (23) Coates, E. J. SOC.Dyers Colour. 1969, 85, 355. hindrance of the two neighboring dye molecules. (24) Hewitt, M.;Handel, T. M.; Tan, J. S., unpublished data. Shape of Unperturbed Linear Polymers: Polypropylene Doros N. Theodorou and Ulrich W. Suter* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. Received July 30, 1984 ABSTRACT Large numbers of conformations of unperturbed polypropylene chains are generated in Monte Carlo experiments, based on a rotational isomeric state scheme, and the average instantaneous shape in the system of principal axes of gyration is evaluated. Several new shape measures are introduced to characterize the shape anisotropy, asphericity, and acylindricity. Significant differences are found between short- and medium-length chains of different tacticity, while for long chains all shape measures converge to a common limit. The detailed three-dimensional segment density distributions are examined, and they are found to be bimodal along the longest principal axis of gyration. The loci of highest segment density are always two clearly separated domains, not containing the center of gyration, and they lie on the major principal avis separated by ca. 1.3(S2)01/2-2.0(S2)01~2.The core of the segment distribution of an unperturbed chain is therefore dumbbell-like in shape. Introduction and spherical distributions of approximately Gaussian Flexible-chain molecules can assume a large number of shape are ~btained.l-~As early as 1934, however, Kuhn conformations and shapes due to the many internal degrees had realized5 that the instantaneous shape of polymer of freedom. The average shape of these molecules is of coils, observed without orientational averaging, is far from importance for a variety of phenomena, especially for spherical. Average shapes obtained without orientational dilute-solution hydrodynamics. Commonly the segment averaging are therefore necessarily aspherical, and the distribution for long chains around their center of mass accurate characterization of these shapes is relevant for is examined as an average over all possible orientations in the interpretation of phenomena with characteristic time space, i.e., integrating over the external degrees of freedom, scale smaller than the largest relaxation time of the 0024-9297/85/2218-1206$01.50/0 0 1985 American Chemical Society Macromolecules, Vol. 18, No. 6, 1985 Shape of Unperturbed Linear Polymers 1207 Table I Eigenvalues of the Average Instantaneous Radius of Gyration Tensor for Simple Linear Chain Models tvDe__ of chain random walk on a simple cubic lattice n = 50 0.1270 0.02906 0.01075 11.8 2.7 8, 9 n = 100 0.1256 0.02916 0.01077 11.7 2.7 8, 9 approximate analytical solution for freely jointed chain n-a 0.125 0.0298 0.0118 10.6 2.5 10 Monte Carlo estimation with the rotational isomeric state model of n = 1000 12.0 2.7 6 polyethylene molecule.6 Even in situations of long-time scale, however, tures general to all chains as well as structure-dependent the shape of the individual chains can play an important variations. role; in viscous flow, Zimm26 showed, orientational preaveraging of chain conformations may lead to signifi- Measures of Shape cant error, and consideration of the chains as an ensemble Consider a chain with n skeletal bonds, numbered from of rigidly translating and rotating bodies is a much better 1 to n, and n + 1 skeletal atoms, numbered from 0 to n, approximation. Considerable work has been done in ex- in a particular conformation. The center of gravity of the ploring size and shape characteristics; an excellpt review chain is defined so that Cy=osi= 0, where si = col (xi,yi,zJ of _the literature prior to 1977 was written by Solc.' is the position vector of skeletal atom i in a frame of Solc and St~ckmayer~,~were the first to explicitly ad- reference with origin at the center of gravity. We form the dress the shape of flexible polymer chains; they introduced dyadic sis? and consider its average value over all skeletal as shape__ measure the eigenvalues of the radius of gyration atoms, the radius of gyration tensor tensor, X2,y2, and z,and presented on elegant analytical method for the evaluation of the symmetric sums of the average moments of the form c ( X2"T2UF2") all permutations of u, u, w where the overbar denotes averaging over all backbone where the angle brackets denote an average over all con- atoms in the chain. Transformation to a principal axis formations. The individual principal components could system diagonalizes S, and we choose that principal axis only be estimated by numerical techniques, however. An system in which --- approximate analytical method was introduced by Koya- S = diag (X2,yZ,z2) ma,lo yielding essentially the same values for (%),(F), __ and (z)as the random-walk results of solc and Stock- so that the eigenvalues of S, X2, YZ, and z,are in de- mayer.8~~Yoon and FloryG were the first to estimate these scending order, Le., % 1 1 5. The first invariant1' quantities for a realistic chain model, the rotational isom- of S is the squared radius of gyration eric state model of polyethylene, and they obtained values s2 = = tr (s) = + F + 22 (3) in excellent agreement with the previously reported results x2 for long linear chains. Results for the eigenvalues of the a measure of the average size of the particular conforma- average instantaneous radius of gyration tensor, obtained tion. Chains consisting of segments of equal mass have by these different authors, are shown in Table I. More by definition18 a moment of inertia tensor (with respect recently Mattice has obtained values for (3), (F) , and to the center of gravity), I, which is diagonal in the same (z)by Monte Carlo calculations with rotational isomeric frame of reference as S (see Appendix).