First Cumulant for Chains with Constraints
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FIRST CUMULANT FOR CHAINS WITH CONSTRAINTS
AKAUa
dU~
B HAc
A.Z AKCASUa, B HAMMOUDA , W.H STOCKMAYER AND G. TANAKA (a)Dept. of Nuclear Engin., U. of Michigan, Ann Arbor, MI 48109 (b)Research Reactor and Dept. of Physics, U. of Missouri, Columbia, MO 65211 (c)Dept. of Chemistry, Dartmouth College, Hanover, NH 03755 (d)The Clayton Foundation Labs., The Salk Inst., La Jolla, CA 92037
ABSTRACT The Fixman-Kovac formulation of chain dynamics with constraints is used to calculate the first cumulant 2(q) of the dynamic scattering function. This general formalism is applied to the case of freely jointed chains. It is shown that the large q limit (q being the scattering wavenumber) of Q(q) for a chain of N bonds in the absence of hydrodynamic interaction is proportional to the ratio (2N+3)/3(N+l) representing the fraction of unconstrained degrees of freedom of the chain. The inclusion of hydrodynamic interaction seems to enhance the apparent segmental diffusion. The use of constrained chain dynamics has no appreciable effects, however, on the behavior of Q(q) in the small and intermediate q regions for long enough chains. This formalism can be used to interpret neutron (spin echo) scattering experiments from semiflexible polymers in solution. INTRODUCTION The first cumulant of the dynamic expressed as[l]: 2(q)
scattering function can be
- /
(1)
where represents an equilibrium average, p(q)
- I exp(iq.V) VV
is the density of the N+l beads of a polymer chain in the Fourier space, and L is the Kirkwood-Riseman diffusion operator in the full 3N configuration space. Q(q) has been extensively studied for flexible (Gaussian) chains. 2 For instance, the high q limit yields the segmental diffusion q D regardless of chain length. When applied to freely joinfed chains, special care must be taken. Constrained degrees of freedom (bond lengths in this case) must be excluded before performing the equilibrium average. This was pointed out by Stockmayer and
Mat. Res. Soc. Symp. Proc. Vo4 79.
1987 Matertals Research Society
188
Burchard[2J
who considered the example of a Fraenkel dumbbell.
A generalized dynamical operator L is obtained using the Fixman-Kovac[3] formalism of chain dynamics with constraints and then used in Eq. (1) to derive the first cumulant for freely jointed chains. Other approaches such as the Titulaer-Deutsch[4] formalism could also be followed. DIFFUSION EQUATION WITH CONSTRAINTS Following Fixman and Kovac[31, a set of constraining forces 0 (j!=l .... N) are introduced in the Langevin equation of mttion of each bead: X+
dWAV/dt = HVX
CXjpj•
+ IX]
(2)
In this notation, Greek indices run over beads while Latin ones run over bond lengths, H,, is the diffusion matrix (case of preaveraged hydrodynamic interaction), r are the random Brownian forces acting on the beads, • are "soft" coupling forces between different bonds and
Cxj= 6, j- 6 X, j-1 is a constant matrix introduced for notation convenience. A better suited coordinate systeai corresponds to the center of friction 1 and bond lengths S.. The cons
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