Effect of branch points on the dynamics of polymer chains

Eur. Polym. J. Vol. 32, No. 12, pp. 1421-1423,1996 Copyright 0 19% El.wicr Science Ltd Printed in Great Britain. All rights reserved PII: soo14-3057(9a)ooo89-4 0014-3057/%s15.00 + 0.00

Pergamon

EFFECT OF BRANCH POINTS ON THE DYNAMICS POLYMER CHAINS ZHANG

OF

LINXI* and XIA AGEN

Department of Physics, Hangzhou University, Hangzhou, 310028, People’s Republic of China (Received 20 June 1995; accepted in final form 2 November 1995) Akstraet-The dynamics of polymer chains, such as (-A-)., (-AR-). and A(-AR-A-).,, where R is the side group, are studied using the bond-fluctuation model in order to investigate the effects of branch points on the dynamics of polymer chains. It is found that the relaxation times TRof the end-to-end vector of linear polymer chains are less than those of chains with side groups, especially for the excluded-volume case. The ratio ‘&FRO (where TRois the relaxation time of the linear chain) is almost the same for various chain lengths, and the relaxation times Tr, of chains with side groups also obey the scaling law Tn - (n - 1)2 without excluded volume and TR- (n - l)2.5 with excluded volume for d = 2, where d represents a dimension. Copyright 0 1996 Wsevier Science Ltd

INTRODUCI’ION Computer

simulations

have

proven

to be a very

valuable tool for statistical physics for many years [l]. Since polymers are very complicated topological objects, computer simulations often are the best tool to obtain precise theoretical information on proposed models. Lattice models have long been extensively used to estimate equilibrium dimensions of polymer chains. Lattice models may also be used to simulate dynamic properties of polymer chains. Several lattice models have been adopted to investigate the dynamics of polymer chains [2-6]. However, these models cannot simulate the dynamics of polymer chains in d = 2 space (d represents a dimension) and they do not allow for the simulation of branched polymer chains exactly, only approximately [A. It is important to study the dynamics of comb-like polymer chains or polymer chains with side groups because they cover most types of polymer chains. Recently, a new effective algorithm, the bond-fluctuation model, was developed to investigate the dynamics of polymer chains [B] and the glass transition in polymer melts [9, lo]. In this paper, the dynamics of polymers with side groups, such as (-AR-). and A(-AR-A-).., where R is the side group, are studied using the bond-fluctuation model, and comparisons with the dynamics of linear polymers are made.

random. Then it tries to jump at random the distance of one lattice unit into one of the four lattice directions. If the move complies with the bond length restriction, it is accepted (for the random-tight chains) and the cycle is terminated. If not, a new bead is selected at random and so on. If excluded volume is present, the new bead position is checked. If the new position is occupied, the bead remains at its original position and the cycle is terminated. If it is unoccupied, the bead is moved and the cycle terminates. It is repeated again and again, the chain moves from one conformation to another, eventually taking on all the conformations of its equilibrium ensemble. Repeated sampling of the chain dimensions then gives estimates of equilibrium ensemble averages. To analyze the dynamics of the chain, we use the value of the end-to-end vector, R(t), to compute the autocorrelation function, pa(t), defined by pR(f) = < R(t)‘R(O) > / < R2 >

(1)

where < > denotes an equilibrium ensemble average. The elementary time unit is taken to be n-bead cycles. The ensemble average is computed as a time average over a trajectory begimring from a fully equilibrated chain conformation. The relaxation time, Ta, is estimated by fitting an unweighted least-squares line to the linear long-time region of a semilog plot of pa(t) vs time. The inverse of the relaxation time is the negative of the slope of the line WI.

MODEL The bond-fluctuation model is described in detail elsewhere [S]. In this paper, the single-site model is adopted. The chain is modeled as a random walk of (n - 1) steps (n beads) on the square lattice, Each of the steps is referred to as a bond, and the bond length is 1 or J2. To move the chain, a bead is selected at *To whom all correspondence

should be addressed.

RFSUL’IS AND DIBCUSSION

Simulation runs are performed on chains of length 11, 21, 31, 41 and 51 beads with various chain structures in the absence and presence of excluded volume. All computations were performed on a VAX8350 computer using FORTRAN source codes. At least 2000 runs were made for each case. The simulation requires approximately 1200 hr of CPU

1421

Zhang Linxi and Xia Agen

1422

I

0

I

I

I

I

400

800

1200

1600

t

Fig. 1. Semilogarithmic plots of the end-to-end vector autocorrelation function pa(r) vs f for various polymer chains with n = 31 beads in the absence of excluded volume: (a) (-AR-),; (b) (-A--)..

time. In order to investigate the effect of branch points on the dynamics of polymer chains, we adopt

polymer chains with side groups, such as (-AR-),, A(-AR-A-),,, where R is the side group and n = 2n’ + 1 (n’ is the polymerization degree of polymer A(-AR-A-).,). The dynamics of linear polymer chains, such as (-A-),, are also investigated here. Typical semilog plots of the end-to-end vector autocorrelation function pR(f) for two types of polymer chain with n = 31 in the absence of excluded volume are shown in Fig. 1. Autocorrelation functions for all other cases behave similarly. The decay appears to be linear (on the semilog plot) for long times. This indicates that the decay is exponential for long times. The relaxation times, TR, calculated from the long-time slopes of the In pR(f) vs t plots, are collected in Table 1. Because of the two-dimensional polymer chains, chains move slowly, and long relaxation times are required, especially for long chains with many branch points. This is similar to polymer chains in the higher concentration [ 111.For linear polymer chains such as (-A-)., the scaling exponent, c(, obtained from a least-squares fit of logTRo vs log@ - 1), is also given in Table 1, see Fig. 2. This slope corresponds to a scaling exponent given by the relation TR - (n - 1)“.

(2)

In the absence of excluded volume, the value of tl is 2.01, which is very close to the Rouse value of 2.0 for d = 2. In the presence of excluded volume, the

I

-1.0

1.5

Log (n-l) Fig. 2. Plot of IogTRovs log@ - 1): (a) excluded volume; (b) no excluded volume.

value of c( is 2.53, which is also close to the scaling theory prediction of 2.50 for d = 2. The relaxation times of polymer chains with side groups are also given in Table 1. Here the side group R includes only one bead. To investigate the effects of branch points on the dynamics of the polymer chains, we adopt (-AR-). and A(-AR-A-)., to study the dynamics of polymer chains. In Table 1, the relaxation times TR increase with increasing number of branch points for constant n, especially in the presence of excluded volume. For example, the relaxation times TR of polymer chains of A(-AR-A-),. and (-AR-). of bead length n = 31 (n’ = 15) are 2030 and 2990, respectively. Although the bead length is the same (n = 31) the polymerization degree is different, and the number of branch points is 15 and 3 1, respectively. This means the polymer chains move slowly with the large number of branch points. We also calculate the ratio TRITRO, where TRo is the relaxation time of linear polymer chains, and the results are given in Fig. 3. In Fig. 3, it is found that the ratio TR/TRo is nearly the same for constant polymer model, especially in the absence of excluded volume; therefore, the

Table 1. Valuesof relaxation times, Ta, as a function of chain length n for various polymer structures with both nonexcluded volume and excluded volume cases (the time unit is n bead cycles [4-7, 1 I])

TR Nonexcluded Polvmer

(-A-).

n=ll

A(-AR-A-). (-AR-).

127 240 327

(-A-). A(-AR-A-)..

833 5950

n=Zl 513 1010 1350

n=31 1180 2030 2990

volume n=41

n=51

Scaling exoonent a

2190 3720 5840

3280 5900 9100

2.01

27100 185000

46500 295000

2.53

Excluded volume

Wheren=Zn’+l.

4720 24100

2.0

13300 84500

Dynamics of polymer chains

1423

same as the linear polymer chain. The result is in agreement with the approximation simulation [I. The dynamics of polymer chains with side group for d = 3 will be investigated using the bond-fluctuation model. Acknowledgement-This

project has been supported by the National Natural Science Foundation of China.

REFERENCFS

1.5 I 11

I 21

I 31

I 41

I 51

II

Fig. 3. Plot of the quantity TR/TROvs n for various polymers: ra(IAR--),,, no excluded volume; (b) A(-AR-A-).,, n no volume; and (c) excluded A(-ARLA-).,, (2n’ = n - 1), excluded volume.

relaxation

time of polymer chain with side group may

also be written in the form TR - (n -

1)“.

(3)

When the side group is considered, the relaxation time becomes larger, but the scaling exponent a is the

1. (a) K. Binder (ed.). Monte Carlo Methods in Statistical Physics, 2nd edn. Springer, Berlin (1986). (b) K. Binder (ed.). AppIications of the Monte Carlo Method in Statistical Physics. Springer, Berlin (1984). 2. P. E. Rouse. J. Chem. Phyr. 21, 1273 (1953). 3. F. Gerry and L. Monnerie. J. Polym. Sci. Polym. Phys. Edn 17, 131 (1979). 4. M. Dial, K. S. Crabb, C. C. Crabb and J. Kovac. Macromolecules 18,22 15 ( 1985). 5. J. P. Downey and J. Kovac. Macromolecules 20, 1357 (1987). 6. J. P. Downey, C. C. Crabb and J. Kovac. Macromolecules 19, 2202 (1986). I. Z. Linxi. Eur. Polym. J. 31, 947 (1995). 8. I. Carmesin and K. Kremer. Macromolecules 21. 2819

(1988). 9. J. Baschnage.1, K. Binder and H. P. Wittmamr. J. Phys. C: Condensed Matter. 5. 1597 (1993). Symp. 81, 63 10. J. Baschnagel and B.’ Lobe. iacrohol.

(1994). 11. C. C. Crabb, D. F. Hoffman, M. Dial and J. Kovac. Macromolecules 21, 2230 (1988).