Macromolecules 1993,26, 4196-4202
4196
Static and Dynamic Structure Factors for Star Polymers in 8 Conditions Marina Guenza' and Angelo Perico Centro di Studi Chimico-Fisici di Macromolecole Sintetiche e Naturali, CNR, Corso Europa 30, 16132 Genoua, Italy Received March 10,1993
ABSTRACT: The static and dynamic structure factors for star polymers in 8 solutions are derived for semiflexiblemodels and the partially stretched (PS)arm model. The PS model for the static structure factor is found to be in fairly good agreement with SANS experiments with exclusion of the high-k region, where only a qualitative agreement is obtained. The nonpreaveraged first cumulant for the PS model reproduces fairly well the peculiar effects found by neutron spin-echoexperiments even though those experimentswere carried out in good solvents. The effects of preaveraging and screening of the hydrodynamic interaction are considered. Results for the full dynamic structurefactor are presented only in the preaveraging approximation. The position of the ith bead at time t is Ri(t). The beads are conventionally ordered as i = 1for the star center and Our recent model for star polymers in 9 solutions was i = 2, ...,N + 1;N + 2, ...,2N+ 1;...;N f + 1for the sequence off arms. applied to the evaluation of static and dynamic properties such as the shrinking factor for the mean square radius We first introduce the dynamic structure factor exof gyration and the bond relaxation times in paper 1' and pression in the ORZLD approximation. Then we will paper 22 of this series. In the model the star polymer is consider the nonpreaveraged generalized Gaussian case. described as a semiflexible chain with partially stretched The dynamic structure factor is defined, taking into arms to take into account the segment concentration effect account the Gaussian nature of the random forces in the in the star core. The dynamics were derived in the ORZLD fluid, as (optimized Rouse-Zimm local dynamics) a p p r ~ a c h , ~ ~ ~ " which is of great utility for a general treatment of topological effects in a large hierarchy of polymer models. More recently, a reduced description was introduced which = exp(-k2Dt) S,(k,t) (2) uses the symmetry of the regular star to reduce the computation times required for the calculation of the with the second equation obtained after separation of the dynamic properties of long-arm stars. Results for the local center of mass motion, D the translational diffusion relaxation times were first presented.2 This reduced constant, and description is applied in this paper to the calculation of n the static and dynamic structure factors of the partially S,(k,t) = n-' exp{-(k2l2/6)di?(t)) (3) stretched model. ,J Extensive experimental resultsb9 obtained by SANS depending on the relative adimensional intramolecular and neutron spin-echo experiments were presented for dynamic distances dij2(t). The SIpart can be written for these structure factors, showing interesting peculiar effects a regular star in the computationally convenient form due to the star topology such as the emergence of a maximum in the Kratky plot for the static structure factor S,(k,t) = n-2{&(k,t) + f [ S , ( k , t )+ (f - l)Sc(k,t)l} and a related minimum in the normalized first cumulant (4) plot. These effects are only partially accounted by the with the self term classic Gaussian theory.'OJ1 Better agreement was achieved by RIS Monte Carlo simulations5taking into account both N+1 N+ 1 N N+l the detailed structure of the polymer and the structure of S , ( k , t ) =2 E I i ( k , t ) + Eii(k,t) + 2 Eij(k,t) the star center and by wormlike models with and without i=2 i=2 1=2 j=i+l specificstar centers.6 Evidence comes out of the relevance of the stretching in the arm interior part. In this paper, relative to the same arm and the cross term we will take into account both semiflexible arms and stretching to derive models for the static structure factor N+l N N+1 and the dynamic structure factor with its cumulant to explain the peculiar topological experimental effects. relative to different arms. The Ei, quantity is defined as Structure Factors for a Regular Star Eij(k,t)= expl-(k2Z2/6)dij2(t)] (7) Consider a regular star off arms, each with N bonds of The calculation of S, only requires a total of 2N + N(N length 1 and a total number of beads n - 1)/2dynamic distances on the same arm, and S, requires N(N + 1)/2 distances between two arms, thus optimizing n=Nff1 (1) 0024-929719312226-4196$04.00/0 0 1993 American Chemical Society Introduction
Structure Factors for Star Polymers 4197
Macromolecules, Vol. 26, No. 16, 1993 the computation time. The intramolecular dynamic distances dij2(t)are calculated in the ORLZD bead model as12 “-1
Equation 18 is evaluatedls by averaging first over the orientations and subsequently over scalar distances using a generalized Gaussian distribution to obtain Q(k)correct to second-order moments. The generalized result is Q(k)/U
= (k212/3n)rS(k,O)l-’(l+
c
(S;/n)
“-1
(1 - d,)(Z/R,)(3/4)H(x)~ (21)
ZJ
with with Xa and Q i a the eigenvalues and eigenvectors of the product matrix H A and l’a = ((QIT AQ)aa. The matrix H,defined as
Hij = dij + Sr(l/Rij)(l-dij)
(9)
is the matrix of the preaveraged hydrodynamic interactions, while A is the star structural matrix described elsewhere.’ The bond rate constant u is given as u = 3kBT/z2r
(10)
with {the bead friction coefficient and
s, = f/‘/6qIl
(11)
the reduced friction coefficientwith 70 the solvent viscosity. The important first cumulant relative to S(k,t)
using eqs 8 and 9 and the properties of X,and P’a takes the ORLZD form
H(x) = (xs
+ 2x-’) exp(-x2)Joxexp(t2)dt - x-2 x 2 = k2(Ri;)/6
(23) and (Rij2) = Z2di?(0) the static mean square distances in the chosen model. These equations for the nonpreaveraged first cumulant are not restricted to simple Gaussian chains but, using the proper second moments and (Z/Rij),describe stiffness as well as solvent effects. Inspection of eqs 13 and 21 shows that the preaveraging approximation amounts to substituting (3/4)H(x) with exp(-x2) independently of the assumed specificgeneralized Gaussian model. Following the same procedure of the preaveraging case, eq 21 can be rewritten to obtain an equation identical to (14) with the definition of Fij(k) (eq 17) changed toF’ij(k):
F’ij(k) = (3/4)H(x)(VRij) (17’) Finally, the ORLZD result for the translational diffusion constant D takes the Kirkwood f o r m 9
D = (kBT/nn[l+ (cr/n)
Q ( k ) / a= (k2l2/3n)[S(k,O)I-’X IJ
For a regular star, eq 13 can be written as in the case of S ( k , t ) in the computationally covenient form Q ( k ) / u= (k2Z2/3n)[S(k,0)l-’f1+ (S;/n)f[B,(k) + cf - UB,(k)l) (14)
with B, and B, the contributions from one arm and two arms, respectively:
and Fij(k) = Eij(k,O)(VRij) (17) It is noteworthy that in the case of the first cumulant a nonpreaveraged expression is available13J4 and may be expressed in the general form
Q(k) = (k,T/n
(l/Rij)]
(24)
121
(1 - djj)(l/Rij) exp[-(k2Z2/6)di;(O)1) (13)
(1 + NJn)
(22)
( (k*Hij*k)exp(ik.Rij))/[n2S(k,0)l 1J
(18)
The tensor Hij
+
Hij = ldij {(1 - 6ij)Tij (19) describes the nonpreaveraged hydrodynamic interaction with
+
Tij = (8rq,,Rij)-’(1 RijRij/Ri:) the Oseen tensor.
(20)
It remains to calculate the dynamic and static mean square distances together with the inverse static distances. As in the following we assume for (Z/Rij)the Gaussian value
( l / R i j )= ( 6 / 7 ~ ) ” ~ ( ( R ~ ~ ) ) - ” ~ (25) we are left with the computation of only the mean square distances. In this paper star models with semiflexible arms described as freely rotating chains (FRC) or partially stretched freely rotating chains (PS)are considered. Each arm has the first N‘ (IN)bonds characterized by a valence angle 0’ and the remaining N - N’ bonds by a valence angle 8, with
p’ = -cos B/ (26) p = -COS e (27) the stiffness of the core and the stiffness of the outer part of the star, respectively @’ Ip ) . Following paper 1, the correlation of the bond vectors attached to the star center is assumed as (l;.l:) = (Y (28) for any two different arms, with 11’ the first bond of arm i. In this paper a is assumed to take its minimum value -
