J . Phys. Chem. 1993,97, 3451
Reply to Comment on "Local Knot Model of Entangled Polymer Chains" Kazuyoshi Iwata' and Mitsuya Tanaka Department of Applied Physics, Fukui University, Bunkyo 3-9-1, Fukui 910, Japan Received: September 22, 1992 Our simulationI.2 is based on an assumption that reptation of linear chains is essentially the same as that of ring chains, gthey are in the topological equilibrium. The assumption seems to be soevident that even it has not been stated explicitly in our The comment of Kolinski and S k ~ l n i c k ,however, ~ seems to question it, so we first explain its meaning. It must be stressed that the purpose of our work is to study entanglement of linear chains and not of rings; rings were considered only for a technical reason, i.e., to remove the difficulty of terminal fluctuations in our computer simulations, in which chain length is limited to a rather low value. In linear chains, the average size and lifetime of local knots (LKs) in the terminal part must be different from those in the intermediate part of the chains, and particularly in short chains near n,, their reptation may be much disturbed by the terminal fluctuations. In rings, on the other hand, no such effect exists in principle even in chains as short as n n,. It is important that 'topological equilibrium" is a necessary condition in our simulations. In linear chains, the topological equilibrium is identical to the thermodynamic equilibrium, which is realized by Brownian motion either inhibiting crossing among chains (realchains) or permitting it (phantom chains). In rings, on the other hand, there are many different topological states, which are kept constant in real chains, while in the phantom chains, these states are mixed and equilibrated by Brownian motion. The topological equilibrium of rings is therefore defined as the thermodynamic equilibrium of the phantom chains. The definition is significant in calculating molecular parameters of entanglement, Llk, the average chain length per LK, and do, the diffusion coefficient of single LK. In the phantom chains, these parameters must have the same values in both linears and rings, if they are sufficiently long, because these parameters are determined only by conformation of 'local chains" (small parts of chains).',2-6Note that (1) in this argument do is assumed to be computed using a theoretical expression,* and (2) the equilibrium conformation of "local chains" is the same in linear and ring phantom chains, if they are sufficiently long. By definition, the real ring chains have the same Llk and do, when they are in the topological equilibrium. According to our theory,l.2-6if two systems (such as linear chains and ring chains in the topological equilibrium) have the same entanglement parameters, Llk and do, they are equivalent in entanglement phenomena. Of course, there are many differences between linear and ring chains (say, the centers of rings can move only about their average positions, stress of rings does not relax to zero, etc.), but they are not essential to the discussion on the origin of reptation, though important in viscoelastic problems. These features will be introduced after establishing the basic equation of motion of LKs. The assumption stated in the opening of this reply means primarily that we can estimate Llk and do in rings (in the topological equilibrium) and secondly that if we observe LKs and their collective motion in rings, so we will do in linear chains. The assumption seems to be self-evident to us, but if necessary, we can confirm it by simulations of long linear and ring chains. Now, Kolinski and Skolnick (KS) argued in their comment' that our ne = 230 should be compared with n, = 530 of their
-
0022-3654/93/2091-3451 $04.00/0
3451
noncatenated rings, rather than with ne = 120-1 33 of their linear chains. This is wrong, for the noncatenated rings are far from the topological equilibrium. In the real ring chains, there is many different topological states, each of which has a different thermodynamicstate. In our simulation, a topological equilibrium ensemble of rings is prepared by dynamics with crossing (phantom chain dynamics), and then one of its members is chosen as an initial configuration of the farther dynamics without crossing (real chain dynamics). Our system is therefore a typical member of the topological equilibrium ensemble (in fact, our R , averaged over time and chains agrees well with the theoretical value'), while the noncatenated rings made by KS are not typical in the topological equilibrium ensemble. (In fact, the scaling exponent of their R, deviates from its theoretical value, unity, as seen from Table I of KS.3) The noncatenated rings should be considered as a completely different system from those in the topological equilibrium. Although KS referred it as one reason for doubting our results that our system behaves so differently from their noncatenated system, the reason of the discrepancy is obvious. In conclusion, it is not their ne = 530 for the noncatenated rings but their n, = 120-133 for linear chains that is to be compared with our ne = 230. Kolinski and Skolnick (KS)3further give comment on what we mentioned' in comparing of our work with those of Kolinski, Yaris, and Skolnick (KYS)4 and of Kremer and Grest (KG).5 Their comment comes from misunderstanding our remark. We have not said that KYS's simulational results contradict those of KG; what we have really meant is that KYS's conclusion contradicts that of KG's (Le., KYS questioned the existence of reptation, while KG admitted it). We have only referred to the contradiction in their final conclusions and not to the details of their simulational results. In the comment to our work, KS state that the 11/4 regime in the single mer autocorrelation function, the n2 scaling of the self-diffusion coefficient, and a tubelike object found by KG in a series of superimposed snapshots of long linear chains do not necessarily mean the existence of reptation. We agree, and this is why we have argued none of these quantities in our simulations.'-2 In our works, we have observed a collective motion of LKs along chains and assigned it to the original reptation. The usual reptation is not equal to the collective motion of LKs but to its projection onto the lab space. The collective motion of LKs can be observed easily, but its projection onto the lab space is much indistinct particularly in short chains; this is why we observed 'reptation" while KS did not. Why KYS and KG have arrived at the opposite conclusions from similar results (according to the comment of KS') is not a matter to be discussed here, but let us point out that the controversy seems to come from confusion in the concept of reptation. Weconsider that the "tube" usually associated with reptation is not a physical existence but just a metaphor, and predictions based on the physical tube model are doubtful or need further justification. We believe that the collective motion of LKs is the true picture of reptation, and it is different in many important points from the tube model (cf. discussions in our original paper2). We cannot get the final answer to this problem without having a common understanding of reptation. We need much more work and discussions before doing so.
References and Notes ( I ) Iwata. K.; Tanaka, M. J. Phys. Chem. 1992, 96, 4100. (2) Iwata, K. J. Phys. Chem. 1992, 96, 4111. (3) Kolinski, A.; Skolnick, J . J. Phys. Chem.. preceding paper in this issue. (4) Kolinski, A.; Yaris. R.; Skolnick, J. J . Chem. Phys. 1988,88, 1407. ( 5 ) Kremer. K.; Grest, G.S. J. C h m . Phys. 1990, 92, 5057. (6) Iwata. K.; Edwards, S. F. J. Chem. Phys. 1989, 90,4564.
0 1993 American Chemical Society
3452
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993
ADDITIONS AND CORRECTIONS
1992, Volume 96 C. Chipot,
B. Maigret,' J.-L. R i d , * and H. A. Scheraga':
Modeling Amino Acid Side Chains. 1. Determination of Net Atomic Charges from a b Initio Self-Consistent-Field Molecular Electrostatic Properties Page 10278. Equation 16 should read
instead of
Equation 17 should read A'q
+ A = B'
ITq = qtotal instead of
Additions and Corrections
