Macromolecules zyxwvu 1989, zyxwvut 22, 333-336 333 zyxwvutsrqpo Moments and Distribution of the End-to-End Vector for Short Poly(methylphenylsi1oxane) Chains of Different Tacticities Ana M. Rubio and Juan J. Freire* Departamento de Quimica Fisica, Facultad de Ciencias Quimicas, Universidad Complutense, 28040 Madrid, Spain. Received February zyxwv 15, 1988; Revised Manuscript Received May 27, 1988 ABSTRACT: Moments of the end-to-endvector and the distribution function of the end-to-end distance have been obtained for short zyxwvut poly(methylphenylsi1oxane) chains with different lengths and tacticities (in- termediate between the purely syndiotacticand isotacticchains),using a slightly modified version of a previously established quasi-analyticalscheme. Sharp changes in the conformational Characteristics (values of the momenta and the cyclization probability,shape of the distribution function, asymmetry,etc.) are observed as the isotactic chain limit is approached. Introduction Most interesting conformational characteristics of a flexible polymer are directly related to the distribution function of the end-to-end vector, F(R), with R expressed in a reference frame embedded in the first bonds of the chain.' In fact, very useful information is provided by the simpler function F(R), the distribution function of the end-to-end distance. Thus, from F(R) one can obtain different averages of R that intervene in the calculation of equilibrium of hydrodynamic properties. Moreover, a detailed knowledge of F(R) is required to describe non- Gaussian effects in networks at high elongations (strain- stress Equilibrium charge transfer and cycliza- tion processes are also determined by F(R) or F(R).5 For long chains, F(R) is spherically symmetric and F(R) is Gaussian. However, the evaluation of these functions for realistic models of short chains, as the rotational isomeric model (RIS), is a difficult task. In recent years, calculations for F(R) and F(R) have been performed for different short polymer chains by two dif- ferent methods: (a) direct Monte Carlo simulations on conformational sample^;^^^ (b) inference from statistical moments of R that have been previously obtained ac- cording to exact iterative (quasi-analytical, i.e., non-Monte Carlo methods). Though method (b) is not exact (its accuracy depends on the number of moments employed to infer the distribution function), the results obtained so far have been shown to be sufficiently accurate to avoid costly simulations. In this way we have been able to obtain results with this inference method for poly- meth~lene,~!~ poly(oxyethylene),2J0 poly(dimethy1- siloxane),2J0and poly(methylphenylsi1oxane) (PMPS) chains.lOJ1 The latter chains were studied considering molecules with regular stereochemical configurations; isotactic, i.e., zyxwvutsrq ... ddd ... or ... zyxwvu 111 ... sterochemical sequences, or syndiotactic, Le., ... dldldl ... sequences (d refers arbitrarily to a Si atom with a phenyl group in front the plane of the chain backbone and 1 corresponds to the phenyl group behind that plane). However, the study of F(R) for asymmetric chains with intermediate, nonregular stereochemical structures has not been yet performed because the computational effort re- quired for such a study is considerably greater. Thus in these cases, the results should be averaged over samples of Monte Carlo generated chains with different stereo- chemical sequences but with the same replication proba- bility, zyxwvutsrqpo p, (p, is defined as the probability that a repeat unit being added to the growing chain will have the same configuration, d or 1, as its predecessor). In this paper we present results for PMPS chains with different values of p,. Our goal is to show that even in these cases, in which a further Monte Carlo average is 0024-929718912222-0333$01.50/0 necessary, the inference method can be usefully applied. Moreover, we discuss some interesting features of the re- sults concerning the very different conformational char- acteristics exhibited by PMPS chains with different ste- reochemical structures. Methods For a given stereochemical sequence, the calculation of different moments of R and its distribution function can be accomplished by means of the procedures explained in ref 10 and 11. However, we have performed some slight modifications in ow scheme to evaluate moments from real and imaginary terms associated to each bond in the iter- ative equations. These modifications are introduced in order to avoid the use of redundant statistical weight matrices and are inspired in those previously employed by Flory et d.'J2J3 within their scheme of Cartesian coordinate transformation matrices. (A scheme that is not efficient to obtain the high moments required for an adequate in- ference of F(R).699) This way, matrices ud, V,, QI, Udl, U,, and U,, defined in ref 11, that contain statistical weights associated to the different bond pairs are substituted by u = QUd = UlQ um = QUII = UddQ ur = udl = QuldQ with zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG . = ( A ; ;) (1) so that U, and ud, are identical, while U and U, are obtained from matrices U, and VIl by interchanging rows 2 and 3. U is now associated to bond pairs 0-Si-0 bracketing an Si atom of either d or 1 stereochemical configuration. U, is employed for Si-0-Si bond pairs containing a meso (dd or 11) sequence of asymmetric Si atoms, while U, is employed for Si-0-Si bond pairs con- taining a racemic (dl or Id) sterochemical sequence. Using U , U,, and U, implies changes both in the di- rection of the bond-based coordinate systems and in the sign of rotational angles associated to the bond pairs. It can be shown that a left-side multiplication by matrix Q (i.e., an interchange of rows 2 and 3) on a weight matrix associated to the N bond rotation is equivalent to reverse the YN-1 axis direction (change a). On the other hand, a right-side multiplication by Q on such a matrix (inter- change of columns 2 and 3) is equivalent to reverse the YN axis direction. Also, in this case, the different cpjN-' rota- tional angles should reverse their signs (change b). The net effect of both types of changes is equivalent as can be 0 1989 American Chemical Society