Macromolecules 1985,18, 211-216 (13) (14) (15) (16)
(17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27)
(28) (29) (30) (31) (32) (33)
(34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45)
Kholodenko, A.; Freed, K. F. J . Chem. Phys. 1983, 78, 7390. Kosmas, M.; Freed, K. F. J. Chem. Phys. 1978,69, 3647. Adler, R.; Freed, K. F. J. Chem. Phys. 1980, 72, 4186. (a) Douglas, J.; Freed, K. F. Macromolecules 1984, 17, 2354. (b) Douglas, J. F.; Freed, K. F. ’Polydispersity Corrections on Excluded Volume Dependence in Flexible Polymers”, to appear in J. Phys. Chem. Douglas, J.; Freed, K. F. Macromolecules 1983, 16, 1800. Oono, Y.; Freed, K. F. J . Chem. Phys. 1981, 75, 993. Ohta, T.; Oono, Y. Phys. Lett. A 1982, A89, 460. Ohta, T.; Oono, Y.; Freed, K. F. J.Chem. Phys. 1981, 74,6458. Oono, Y.; Freed, K. F. J.Phys. A 1982, 15, 1931. Oono, Y.; Ohta, T. Phys. Lett. A 1981,85,480. Oono, Y.; Kohmoto, M. J. Chem. Phys. 1983, 78, 520. Lipkin, M.; Oono, Y.; Freed, K. Macromolecules 1981,14,1270. Miyake, A.; Freed, K. F. Macromolecules 1983, 16, 1228. Miyake, A.; Freed, K. F. Macromolecules 1984, 17, 678. (a) Freed, K. F. J. Chem. Phys. 1983, 79,3121. (b) Nemirovsky, A. M.; Freed, K. F. “Excluded Volume Effects for Polymers in the Presence of Interacting Surfaces”, submitted manuscript. The motivation for choosing this rescaling is the minimization of the errors involved in the c-expansion procedure. Discussion of the rescaling is made in ref 11. Tanaka, G. Macromolecules 1980,13, 1513. Miyaki, Y.; Einaga, Y.; Fujita, H. Macromolecules 1978, 11, 1180. Stockmayer, W. H.; Albrecht, A. J. Polym. Sci. 1958, 32, 215. Weill, G.; des Cloizeaux, J. J . Phys. (Orsay,Fr.) 1979,40, 99. Akcasu, A.; Han, C. Macromolecules 1979,12, 276. We note that Mattice (preprint) has considered numerically the ratio aRz2/agz2 in the self-avoiding limit for long chains and found in his simulations p = 1.028,which is in much better agreement with the RG theory. Daoud, M. Thesis, Universit6 de Paris VI, Paris, France, 1977. Farnoux, B.; et al. J. Phys. (Orsay,Fr.) 1978, 39, 77. Cotton, J. P. J. Phys. Lett. 1980, 41, 231. Le Guillou, J.; Zinn-Justin, J. Phys. Rev. Lett. 1977, 39, 95. Lawrie, I. D. J . Phys. A 1976, 9, 435. Miyaki, Y.; Einaga, Y.; Hirosye, T.; Fujita, H. Macromolecules 1977, 10, 1356. Gobush, W.; Solc, K.; Stockmayer, W. H. J. Chem. Phys. 1974, 60, 12. Witten, T.; Shiifer, L. J. Chem. Phys. 1981, 74, 2582. (a) des Cloizeaux, J. J. Phys. (Orsay,Fr.) 1981,42,635. (b) des Cloizeaux, J.; Noda, T. Macromolecules 1982, 15, 1505. Kniewske, R.; Kulicke, W. M. Makromol. Chem. 1983, 184, 2173. Shimada, J.; Yamakawa, H. J . Polym. Sci. 1978, 16, 1927. Tanaka, G.; Imai, S.; Yamakawa, H. J. Chem. Phys. 1970,52, 2639.
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(46) (a) Norisuye, T.; Kawahara, K.; Teramoto, A.; Fujita, H. J. Chem. Phys. 1968,49,4330. (b) Kawahara, K.; Norisuye, T.; Fujita, H. J. Chem. Phys. 1968,49, 4339. (47) We use the value of @o = 2.51 X loz3calculated by Zimm (Macromolecules 1980, 13, 592) which is obtained in a numerical simulation where the usual preaveraging approximation is not employed. Experimental results for a0which are close to Zimm’s values are frequently reported in the literature. The values of @eand Pet however, are not universal (see ref 1) so that it is best to regard these as phenomenologicalconstants which are ‘reasonably” estimated by the Gaussian theory. There is still some controversy regarding his simulation so that it must be regarded as a tentative result (see ref 15). (48) Tsitsilianis, C.; Pierri, E.; Dondos, A. Polym. Lett. 1983, 21, 685. (49) Sarizin, D.; Francois, J. Polymer 1978, 19, 699. (50) Suzuki, H. Br. Polym. J. Dec 1982, 137. (51) Nishio, I.; Swislow, G.; Sun, S.; Tanaka, T. Nature (London) 1982, 300, 243. (52) Tanaka, G. Macromolecules 1982, 15, 1028. (53) Sun,S.; Nishio, I.; Swislo, G.; Tanaka, T. J. Chem. Phys. 1980, 73, 5971. (54) Bauer, D.; Ullman, R. Macromolecules 1980, 13, 392. (55) Perzynski, R.; Adam, M.; Delsanti, M. J. Phys. (Orsay, Fr.) 1982, 43, 129. (56) Berry, G. C. J. Chem. Phys. 1966,44,4550. (57) Abdel-Azim, A.; Huglin, M. Polymer 1982, 23, 1859. (58) Shultz, A.; Flory, P. J. Polym. Sci. 1955, 15, 231. (59) Noel, R.; Patterson, D.; Somcynsky, T. J . Polym. Sci. 1960,42, 561. (60) Simionescu, C. I.; Simionescu, B. C. Makromol. Chem. 1983, 184, 829. (61) Matsumoto, T.; Nishioka, N.; Fujita, H. J . Polym. Sci. 1972, 10, 23. (62) Akita, S.; Einaga, Y.; Miyaki, Y.; Fujita, H. Macromolecules 1976, 9, 774. (63) Slagowski, E. Ph.D. Thesis, University of Akron, Akron, OH, 1973
(64) Fakida, M.; Fukutomi, M.; Kato, Y.; Hashimoto, T. J.Polym. Sci., Polym. Phys. Ed. 1974, 12, 871. (65) Fox, T., Jr.; Flory, P. J . Phys. Colloid Chem. 1949, 53, 197. (66) Berry, G. C. J. Chem. Phys. 1967, 46, 1338. (67) Norisuye, T.; Kawahara, K.; Fujita, H. Polym. Lett. 1968, 6, 849. (68) Noda, I.; Mizutani, K.; Kato, T. Macromolecules 1977,10,618. (69) Ooms, G.; Mijnlieff, P.; Beckers, H. J. Chem. Phys. 1970,53, 4123. (70) Noda, I.; et al. Macromolecules 1970, 6, 787. (71) Roovers, J.; Toporowski, P. M. J . Polym. Sci., Phys. Ed. 1980, 18, 1907. (72) Barrett, A. J. Phys. A 1976, 9, L-33.
Shape Distributions for Gaussian Moleculest B. E. Eichinger Department of Chemistry, BG-IO, University of Washington, Seattle, Washington 98195. Received June 29, 1984 ABSTRACT The general solution for the distribution function of the gyration tensor for Gaussian molecules in k-dimensional space is given in terms of zonal polynomials. The distribution for two-dimensional rings is reviewed, and that for three-dimensional ellipsoids of revolution is formulated so as to reduce the calculation to a sum of one-dimensional integrals.
Introduction The distributions of &apes of linear chains in three dimensions and of both linear and-circular chains in two dimensions have been studied by Solc, Stockmayer, and Gobush in a series of original paper~.l-~ Since its inception, the theory of these distributions has seemed to be for?Dedicated to Professor walkr H, Swkmayer on the occasion of his 70th birthday. 0024-9297/85/2218-0211$01.50/0
midable, but it has nonetheless attracted some attention and because of potential applications to rubber elasti~ity4.~ thermodynamics*697 to The gyration tensor S for a system of particles with masses mi, n, is defined as S = M’XMX’ (1) If the particles are imbedded in a k-dimensional space, X is a k X n matrix of coordinates in an arbitrarily oriented frame with origin at the center of mass, X’is the transpose 0 1985 American Chemical Society
212 Eichinger
Macromolecules, Vol. 18, No. 2, 1985
of X, M = diag (m1,m2,...,m,) and M = tr (M) = total mass. For present purposes, all particles will have the same mass, and eq 1 reduces to
S = n-lXX‘ (2) The shape distribution of the mechanical system is determined by the probability distribution of the principal components (eigenvalues or latent roots) of S when some potential acts between the particles. The Gaussian model is a useful starting point for this problem; the corresponding potential is harmonic with zero mean and is additive over bonded pairs of mass elements. These interactions are the equivalent bonds in the spring-bead model. In this approximation, the effective potential of mean force pV(X) for a molecule of arbitrary connectivity is pV(X) = y tr (XKX’) = tr (XK,X’) (3) where /3 = l/kBT, y = k / 2 ( 12)o, and K, = yK is a Kirchhoff matrix. (The notation used here follows that of a previous review of the literature on the distribution of the radius of gyration.8) The highest order probability distribution that is sought is for k unequal principal components of S, and this can be written as
P ( S )dS = ( d S / a l
...l
etr (-XK,X’)S(XJ’) dX/dS
where Qo is k X ( n - l),as obtained from Q by deletion of the coordinates corresponding to the center of mass motion. The 6 function is expressed as the Fourier integral 6(S - n-’QoQo’) = ( Z ~ ) - ~ ( ~ + ~ ) / ~(iYS { e t r- in-lYQoQo’) dY (7)
where dY =
n,,,dye,, and
Y kk
- -
Let q = (q11,q21,...,qn-11,q12,q22 ,...,Q,-,~) be the row form of Qo. It is easy to show that if Qo AQ,B, with A a k X k and B an ( n - 1) X ( n - 1)matrix, q q(A’ @ B). It is apparent also that dQ, = dq. Insert eq 7 into eq 6, collect terms, and integrate over dq to get
P ( S ) dS = d S l e t r (iYS)Il + in-lY
ni,lnna=t
@
hy-11-1/2 dY (9)
where use has been made of
z= , k ( n - U / 2 l ~ , l - k / 2
(4)
where 2 is the configuration integral, etr (.I = exp[tr (*)I, dX = dxia, and d S = dS,,. The 6 function, with J = (l,l,...,l),fixes the origin of coordinates a t the center of mass. The integral in eq 4 occurs in multivariate statistic^,^ as has been communicated to the author by Richards.lo This theory is used here to solve, as far as is generally possible, eq 4 and its lower dimensional analogues. The theory to be presented differs somewhat from that formulated by S O ~ C It. ~is, more ~ direct than his method, but it gives equivalent results. The problems considered under each heading are as follows: (I) the general k-dimensional problem, i.e., solution of eq 4; (11) the formulation of eq 4 in Eckart coordinates; (111) the asymptotic distribution for the general case; (IV) two-dimensional rings; and (V) two equal components of S in three dimensions, i.e., ellipsoids of revolution.
i
(10)
Since Y is symmetric, Y = hyh’, h E SO@) and y = diag (yl,...,yL). The volume element in polar (y,h) variables is computed with the following procedure. The metric di2 = dx G dx’, with G an m X m matrix and x a 1 X m vector, has an associated volume element d V = IG11/2nadx,. Consider di2 = tr (dY dY’) = tr (dY dY)
This metric has a natural volume element d V = 2-k(k-1)/4na5S dy,,. In terms of polar coordinates, dY = dh yh’+ h dy h’+ hy dh’= h(dy + 6hy -y6h)h’, with 6h = h’dh = -dh’h = -6h‘since h‘h = 1. Substitute this expression for dY into eq 11 to find dE2 = tr [dy2 + (6hy - y6h)’I k
I. The General Problem The probability distribution of S, P ( S ) dS, may be formulated as an integral over the configuration space in two different ways. The first, to be considered in this section, is had by restricting the configuration integral to yield zero when S # n-‘XX’. The other method, described in the next section, makes use of Eckart coordinates. The constraint is written as P ( S ) dS = (dS/Z)!d(S - n-’XX’)G(XJ’) etr (-XK,X’) dX ( 5 ) To solve eq 5, convert to normal coordinatess Q = XT’, t SO(n), chosen such that TK,T’ = A = diag (0, Y A ~ , . . . , ~ & - J = diag (O,h,). [The orthogonal group of n x n matrices is denoted by O(n);S O ( n ) is the subgroup of O h ) consisting of matrices with determinant +1.] The Jacobian of the transformation is +1. Since K, has a single zero eigenvalue if the molecule is connected, and since T contains a constant row, proportional to J, we have P(S)dS = ( d s / z ) S W - n-’QoQo’) etr (-QoAtQd) dQ, (6)
T
The volume element is thus
where dh is the (unnormalized) Haar measure on S O ( I Z ) . ~ [On the last point: tr (dh dh? = tr (h’ dh dh’ h) = tr (6h6h9, with volume element dh. This is both left and right invariant, since for any fixed hl,h2 t SO(k):h-hlhh2 we have tr (h, dh h2hd dh’h,’) = tr (dh dh?.] In eq 13, a factor of 2k-1has been divided out because of symmetry (see ref 9, p 104). Equation 9 now becomes dS =
22k-lTk(k+1j/2 dS
in-’y
@
l e t r (ihyhS)Il +
II Jy, - yplIIdya dh a
a
