4666
Macromolecules 1991,24, 4666-4672
Translational Diffusion, Relaxation Times, and Quasi-Elastic Scattering of Flexible Chains with Excluded Volume and Fluctuating Hydrodynamic Interactions. A Brownian Dynamics Study Antonio Rey and Juan J. Freire' Departamento de Quimica Fisica, Facultad de Ciencias Quimicas, Universidad Complutense, 2.8040 Madrid, Spain
Jose Garcia de la Torre Departamento de Quimica Fisica, Facultad de Ciencias Quimicas y Matemhticas, Universidad de Murcia, 30071 Murcia, Spain Received October 17, 1990; Revised Manuscript Received March 4, 1991
ABSTRACT A polymer model with excluded volume interactions represented by a relatively soft potential is employed in Brownian dynamics simulations. The model and method have been shown in previous work to reproduce equilibrium properties and diffusion coefficients calculated with preaveraged hydrodynamic interactions. Trajectories obtained with fluctuating hydrodynamic interactions are now analyzed to obtain the diffusion coefficient, relaxation times of the Rouse coordinates, and the quasi-elastic scattering form factor, together with its first cumulant, of chains with different numbers of units. The main conclusions are as follows: an abnormal increase of the diffusion coefficient with respect to its preaveraged value for the shortestchains,tentatively attributed to effects in the relativelylarge surfaceof conformationswith overlapping units. These effects seem to be eliminated through extrapolations to the long-chain limit. Thus, the extrapolated ratio of the radius of gyration to the hydrodynamic radius is higher than the estimations for unperturbed Gaussian chains and is in good agreement with upper and lower bounds. The relaxation times are consistently higher than those obtained for Gaussian chains. The results for the cumulants are generally lower than those predicted by Benmouna and Akcasu, the differencesbeing attributed to the lack of consistency between the blob model and the chain models with long-range interactions employed in simulation work. Finally, the Pecora procedure to extract diffusion coefficients and relaxation times from the quasi-elastic form factor is favorably tested for the case of excluded volume chains. Introduction The dynamics of a flexible chain in a continuous solvent has been the object of great interest from the theoretical point of view.l Perhaps the major difficulty in describing this dynamic is to find an adequate mathematical treatment of the fluctuating hydrodynamic interactions (FHI) between polymer units. Recently, Brownian dynamics (BD) simulations have been performed in order to check the theoretical approximations that try to overcome this problem. For instance, BD has been applied to the correct prediction of diverse properties of linear chains such as fluorescence anisotropy,2 polarized314 and depolarized2 quasi-elastic light scattering, and hydrodynamic propertiesa6 An analysis of the Rouse internal motions of the chains, which have a major influence on many properties, have also been p e r f ~ r m e d . ~However, these studies considered chains without long-range intramolecular interactions, which constitutes good representations of real polymers only in the unperturbed or B conditions, because of the difficulties in introducing such interactions. Thus, BD simulations for chains with excluded volume (EV) have only been attempted to predict equilibrium properties, neglecting hydrodynamic interactiom6 The dynamics of chains under the EV or good solvent conditions, for which most universal features of polymers are manifested, have only been described through theories that contain approximate hydrodynamic treatments, such as the popular preaveraged hydrodynamic interactions (PHI)procedure.lJ Also, we should mention some numerical investigations of polymer dynamics based on the Monte Carlo (MC) method! though this method does not admit an easy introduction of hydrodynamic interactions. Consequently,only in the case of hydrodynamic properties are they able to provide quantitative estimations of the 0024-9297/91/2224-4666$02.50/0
dynamic properties (MC calculations have been devised to constitute lower or upper bounds of the real values of these properties with FHP). In a previous paperlo (paper 11,we proposed a new model of flexible chains represented by Gaussian units with longrange EV interactions. These interactions are treated as relatively soft potentials so that the useful time scales in which they act along the trajectories are not very different from those of the Rouse entropic potential joining neighboring units. We have shown in paper 1that this model reproduces remarkably well the equilibrium properties of an equivalent, more conventional, model with hard-sphere EV interactions, studied through a MC method. Furthermore, BD results were also reported there for the diffusion coefficient calculated with PHI. In the present work, we describe the results obtained with this model and FHI for the diffusion coefficient, the relaxation times of the Rouse internal motions, and the quasi-elastic light scattering (QELS) function and its first cumulant. All these results are compared with those obtained for simple Gaussian chains without EV. The effect of introducing FHI or employing PHI as an approximation is also discussed. Moreover, we have extracted values of the diffusion coefficient and the relaxation times from the QELS functions through the numerical methods that are commonly used to treat experimental data. These values are then compared with those obtained directly from the trajectories for the same property in order to check the performance of different treatments of experimental data. Some of the results are also compared with available theoretical predictions. 0 1991 American Chemical Society
Dynamics of Excluded Volume Chains 4667
Macromolecules, Vol. 24, No.16, 1991
Method We have computedBD trajectories for linear chains constituted of N statistical units so that, in the absence of long intramolecular interactions, the distance between neighboringunita would follow a Gaussian distributionwith a root-mean-squarevalue b. Two types of intramolecular forces are then considered those due to harmonic springs between first neighbors, described by the usual Rouse potential,' and repulsive forces between nonneighboring unita that mimic the EV interactions. The latter forces are introduced by means of a potentialof the form Ae*ij, with a cutoff distance r,. Adequate numerical values of the parameters were investigated in paper 1,a relatively wide range of these values giving a good description of the behavior expected for severalequilibriumproperties of chainsin the excludedvolume conditions, as different averages of internal distances and the distribution of the end-to-end distance, also calculated with a Monte Carlo procedure for an equivalent hard-spheres model. In this way, we selected the final values A = 75.0,@ = 4,and rc = 0.512, which are also used in the present work, in reduced unita (see below). As in earlier simu1ations:JO hydrodynamic interactionsare also introduced through the diffusion tensor proposed by Rotne and Prager" and Yamakawa.12 This tensor is configuration-dependent; i.e., it is able to describe FHI in a realistic way. Trajectories are generated with this scheme of forces and hydrodynamic interactions, using the fiist-orderErman and McCammon algorithm,2aJo so that the forces and hydrodynamic interactions are kept constant within a given time step. In order to evaluate the influence of the hydrodynamic interaction description on the dynamic properties of flexible chains,we have also performed BD simulationswith PHI,a usual approximation in the standard theories? For this purpose, we have employedthe preaveraged Oseen tensor,' whose components depend on mean reciprocal averages of distances between pairs of unite, (Rij-l),calculated in the previous MC simulation with the hard-spheres model. This way, we avoid the approximations included in analytical expressions for these averages,such as the ones provided by the blob modeP which, as discussed in paper 1, are not in exact agreement with the simulation results. We generate five different trajectories for each chain. The trajectories are sufficientlylong (each one includes 40 OOO time steps) so that an adequate signal-to-noiseratio is achieved, as can be verified from the resulta for different properties, which correspond to means and uncertainties calculated over these five independent samples. Further details on the model and the simulation techniques can be found in ref 3 and paper I, while in subsequent sections we will give the detailson the calculation of different properties. The results are always referred to in the following basic unite: b for length, the Boltzmann factor kBTfor energy, and the friction coefficient of a unit, (,for friction. This coefficient can be expressed as 4 = 611)ou,where ~0 is the solvent viscosity and u is the friction radius of the Gaussian unit, which we set as u = 0.256b. Then .$b2/kBT is the basic unit time. All the reduced quantities are denoted by asterisks. Diffusion Coefficient The diffusion coefficient, Dt, of chains with different numbers of units, EV and FHI, has been obtained through the quadratic displacement along the trajectories of the center of masses, characterized by its position vector, R,:
([R,(t)- R,(t
+ T ) ] ~ =) 6Dt7
(1)
where the average extends to all the different values of the time t along the trajectory. This method was favorably tested in paper 1for the same model through trajectories generated with PHI. It was then verified that the BD results for Dt were essentially the same as those obtained from the preaveraged Kirkwood-Riseman formula in terms of the equilibrium averages (Rij-1) (eq 3 of paper 1). The BD results obtained with FHI can be found in Table I. We also include in Table I the preaveraged results obtained from the Kirkwood-Riseman formula with distance averages evaluated from the trajectories analyzed in paper
Table I Translational Diffusion Coefficients (in Reduced Units) Obtained by Different Methods for an Excluded Volume Chain of Varying Numbers of Units. 6 8 11
15 20 25
0.357 0.302 0.252 0.211 0.180 0.160
0.398 0.004 0.335 0.002 0.262* 0.002 0.231 0.002 0.186* 0.002 0.166 0.002
*
0.355 0.300 0.250 0.209 0.178 0.158
0.352 0.297 0.247 0.207 0.176 0.155
0.340 0.286 0.236 0.196 0.166 0.145
0 DO,resulta from the Kirkwood formula, eq 2 . BD, resulta from eq 1 calculated from the trajectories. P, results from the KirkwoodRiseman (preaveraged)formula. Z, Monte Carlo values from the Zi" (lowerbound) method. D1 is definedin the text. Error DO. According to Fixman: D1 can be independently calculated from the trajectories as Dl = ( 1 / 3 M $ O m C A (d~ ~)
(3)
+T))
(4)
where CA(T) (A(t).A(t
and (5)
where H = D/(kBn, D being the diffusion tensor and Fj the intramolecular force on unit j . We have performed the calculation for D1 from the trajectories with FHI and PHI. The ratio of the values of CA(t) obtained through these two different hydrodynamic treatments is shown in Figure 1 for different chain lengths. It can be observed that this ratio is always greater than 1,in accordance with the prediction of a preaveraged lower bound for DI. Moreover, the ratios closely follow the trends observed by Fixman for Gaussian chains: with a peak at t = 0 and approaching the value 1 relatively rapidly, which constitutes ita asymptotic long-time limit. Furthermore, since the decay of these correlation functions is fast, we have been able to estimate the integrated value of D1, which in all cases represents only a few percent of DO(it is even smaller than the D1 contribution for Gaussian chains calculated previously by Fixmans). The values of DO- D1 obtained with PHI are practically identical with the Kirkwood-Riseman results, while the values obtained with FHI are included in Table I for comparison with the other estimation for Dt. According to the preceding paragraphs, they are always slightly smaller than DOand the preaveraged results, and therefore, they are in contradiction with the numerical values of Dt obtained from eq 1. Therefore, we have found a disagreement between the results for Dt obtained from eq 1 and those calculated as DO- D1. Since the values of DOand D1 seem t o be
Macromolecules, Vol. 24, No. 16, 1991
4668 Reyet al.
+
N = 6
o
N = 8
Table I1 Values of p Obtained from the Results for Calculated with the Different Methods Included in Table 1. P
N = l l
0
0
N = 15
* N = 2 0
DO
N 6 8 11 15 20 25
1.45 1.48 1.53 1.51 1.53 1.51
BD 1.63 0.03 1.65f 0.02 1.59*0.02 1.65f0.02 1.58*0.02 1.59* 0.02
mC
1.59f0.03
1.54*0.04
P 1.45 1.47 1.52 1.50 1.51 1.49 1.56
* 0.03
DQ-DI 1.44 1.45 1.48 1.50 1.48 1.48
Zb 1.41 1.42 1.41 1.41 1.41 1.42
1.54 0.02 1.41
0 Thevalues of (9) are those obtainedfrom the trajectories,except for the Zimm (Z) method, where the Monte Carlo samplingestimates are considered. Error
