Macromolecules zyxwvuts 1985, 18, 2225-2231 2225 (25) zyxwvut Isono, Y.; Fujimoto, T.; Takeno, N.; Kajiura, H.; Nagasawa, M. Macromolecules 1978, zyxwv 11, 888. (26) Brochard, F.; de Gennes, P.-G. Macromolecules 1977,10,1158. (27) Brochard, F. J. Phys. (Les Ulis, Fr.) 1983,44, 39. (28) Hecht, A.-M.; Bohidar, H. B.; Geissler, E. J. Phys., Lett. 1984, (29) Matsushita, Y.; Noda, I.; Nagasawa, M.; Lodge, T. P.; Amis, E. J.; Han, C. C. Macromolecules 1984, zyxw 17, 1785. (30) Fukuda, M.; Fukutomi, M.; Kato, Y.; Hashimoto, T. J. Polym. Sci., Polym. Phys. Ed. 1974, 12, 871. (19) Endo, H.; Nagasawa, M. J. Polym. Sci., Part A-2 1970,8, 371. (20) Kajiura, H.; Endo, H.; Nagasawa, M. zyxwvuts J. Polym. Sci., Polym. (21) Endo, H.; Fujimoto, T.; Nagasawa, M. J. Polym. Sci., Part zyxwvuts A-2 Phys. Ed. 1973, 11, 2371. 1971, 9, 375. (22) Kurata, M. Macromolecules 1984,17, 895. 45, 121. (23) Fujimoto, T.; Kajiura, H.; Hirose, M.; Nagasawa, M. Polym. J. (Tokyo) 1972, 3,181. (24) Graessley, W. W.; Roovers, J. E. L. Macromolecules 1979, 12, 959. Calculation of the End-to-End Vector Distribution Function for Short Poly(dimethylsiloxane), Poly(oxyethylene), and Poly(methylphenylsi1oxane) Chains A. M. Rubio Departamento de Quimica General y Macromoldculas, Facultad de Ciencias, Universidad Nacional de Educacidn a Distancia, 28040 Madrid, Spain J. J. Freire* Departamento de Quimica Fisica, Facultad de Ciencias Quimicas, Universidad Complutense, 28040 Madrid, Spain. Received January 8, 1985 ABSTRACT The end-bend vector distribution function of short poly(dimethylsiloxane), poly(oxyethylene), and zyxwvuts poly(methy1phenylsiloxane) chains has been calculated through inference from generalized moments obtained by means of certain iterative equations previously derived. The numerical results obtained this way are tested with Monte Carlo values. A fair agreement is reached in most cases. For the PDMS chain the results are also compared with those calculated with the Hermite series procedure, whose performance is considerably poorer. The main features of the distribution function are analyzed for the different chains. Also, Monte Carlo results showing orientational preferences in the region of small end-to-enddistances are reported and discussed. Introduction An incisive description of the spatial distribution of a flexible polymer with a finite number of bonds, N, can be accomplished in terms of the end-to-end vector, R, and its density distribution function, F(R), where R is expressed in a reference frame embedded in the first bonds in the chain.’ Of course, a high number of bonds provides a Gaussian distribution with spherical symmetry. However, F(R) does not have these properties for short chains, which require detailed numerical calculations based on realistic polymer models. Notwithstanding, the calculations are not routine due to their computational difficulty. In recent years, the realistic rotational isomeric state model has been adopted to obtain F(R) for a few types of short chains. Thus, results for poly(methylene)2 (PM), poly(dimethylsi10xane)~ (PDMS) and polypeptides* have been obtained by Flory et al. by means of the Hermite series expansion procedure developed by Flory and Yoon.’ Nevertheless, these results do not agree in general with those generated by Monte Carlo calculations for the shortest chains, due to the poor convergence of the Her- mite series and limitations caused by the lack of efficiency in the evaluation of moments by the usual transfer matrix algorithm. Fixman et al.”’ have developed a more pow- erful method, based on the use of a spherical harmonic representation of rotational operators, to evaluate high moments of R, from which the distribution is inferred through a least-squares algorithm. This method was or- iginally5” aimed at the inference of the end-bend distance density distribution function F(R), i.e., the radial-de- pendent part of F(R), and was applicable only to simple “homogeneous” chains such as PM chains. It was subse- quently generalized to the calculation of F(R)’p8 by ex- pansion in a spherical harmonic series of the angular co- ordinates of R. The results obtained this way for PM chains have been satisfactorily compared with values calculated by simulation method^,^,^ being in good agree- ment with those values. Moreover, we have performed calculations of moments and radial distribution functions for “heterogeneous” chains, Le., chains with several dif- ferent bond lengths, bond angles, or sets of statistical weights for isomers, such as PDMS and poly(oxyethy1ene) (POE),l0and also for chains with asymmetric sets of iso- mers,” such as poly(methylphenylsi1oxane) (PMPS). These calculations used conveniently modified versions of the original procedure. The results obtained for all these chains are also in good agreement with Monte Carlo val- ues.loJ1 In this work, we complete the numerical investigation of the generalized inference method by obtaining F(R) for PDMS, POE, and PMPS chains. This study allows us to verify the numerical validity of our quasi-analytical (i.e., nonsimulation) procedure for significantly different types of chain molecules. Moreover, we analyze the variations in symmetry and other properties of these functions for each type of chain. The PDMS results are compared with the values previously obtained by Flory and Chang3using the Hermite expansion (we do not know of previously reported attempts to calculate F(R) for the other chains studied here). Numerical values of the components of the persistence vector (R) and the second-moment tensor zy ( RRT) are also calculated and analyzed (they are directly related to the “generalized” higher moments needed to infer F(R) as it will be explicitly described in the next 0024-9297/85/2218-2225$01.50/0 0 1985 American Chemical Society