J. Phys. Chem. 1992, 96, 4100-4111
4100 2.0 1
15-
1.0 -
C , JaVk
0.5
I I
T’ = 1.5 ..*
/
-1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P*
Figure 19. Comparison of they and ny contributions with Cumfor T. = 1.5: C,- (solid line), Guam (dashed line), and Cu,nylcI(dotted line).
effects and critical point effects. Nevertheless, we have only been able to give a general and descriptive discussion because we cannot yet quantitatively predict how these factors combine to give the individual maxima, minima and inflection points. Nor do we know
what happens to the extrema very near the critical point. However, Figure 12 is consistent with the locus of maxima extrapolating to the critical point where there is an infinite heat capacity. If that is the case, it would be plausible to regard the locus of extrema as beiig a continuous curve starting with the infinity at the critical point, winding its way as a locus of maxima and then minima, and finally disappearing upon intersection with the liquid-vapor coexistence curve. These qualitative features of the locus of extrema are in agreement with those reported by Bearman et a1.,4 regardless of the validity of the analysis of experimental data by Theeuwes and Bearman.2 As mentioned in the Introduction, we are unaware of any previous simulation work which reports C, extrema of this nature. It is possible that liquid models that do not allow for large fluctuations of the internal energy in the repulsive core may well not manifest these C, extrema. Work is currently in progress to test that possibility . Acknowledgment. We are grateful to C. E. Woodward for suggesting that the extrema in C:, be discussed in terms of the structural information given by the radial distribution function. We are also grateful to B. Smit for providing an extract of his thesis. R.J.B. gratefully acknowledges helpful conversations with J. H. Levelt-Sengers, W. A. Steele, and G. Stell. Supplementary Material Available: Tables of the residual heat capacity, internal energy, and compressibility factor (9 pages). Ordering information is given on any current masthead page.
Local Knot Model of Entangled Polymer Chains. 1. Computer Simulations of Local Knots and Their Coilectlve Motion Kazuyoshi Iwata* and Mitsuya Tanaka Department of Applied Physics, Fukui University, Bunkyo 3-9-1, Fukui 910, Japan (Received: October 2, 1991)
The local knot (LK) theory recently proposed is confirmed by computer simulations of entangled ring polymer chains. The simple cubic lattice model chains (ring) of length L = 512 (volume fraction c = 0.5) is used. By tracing local maxima of = 2.3 Mu.t.; Gauss integral along polymer chains, many “true” LKs (lifetime ryUc= m) and “temporary” LKs (lifetime rtrmP u.t. = unit of time) are found. It is observed that the orders of true and temporary LKs along rings are conserved and that they perform a collective motion (reptation) as predicted by the theory. Various strange motions of true LKs, such as “merging effect”, “multipeak effect”, “ghost effect”, and “probe fluctuations”, are found. In this (part 1) and the following paper (part 2), we discuss in detail how to separate the true Markov motion of LKs and their collective motions from these non-Markov = motions. The average number of true and temporary LKs per ring (t= 512) are estimated to be &e = 3.44 and GmP 3.06. The average chain length per true LKs is Le= 149. The diffusion coefficient of single LK is estimated to be do = 0.0172 bond*/u.t. The mean-square displacement of LK coordinate [ along a ring, g(t) = ( ( [ ( t )- [(0))2),approaches this suggests that the temporary LKs join to the the Markov line computed for the diffusion coefficient do/(& + hemp); collective motion of LKs. The empirical entanglement spacing neof this system is estimated to be 230 or slightly less; this ne is much larger than ne = 120-133 estimated by Skolnick et al.
I. Introduction (A) Backgrod. The concept of reptation has been applied to the viscoelastic and transport phenomena of concentrated polymer solutions and melt with great success. In 1971, de Gennes’ proposed the reptation model of a polymer chain which is allowed to move in a rigid tube formed by surrounding chains. A major result of the theory is the prediction of sedimentation constant D, of a reptating polymer chain, D, W z .Doi and Edwards* later applied this model to the viscoelastic properties, and their results are summarized in their monograph.’ One of
-
~
~~~~
~
(1) de Genncs, P.-G. J. Chem. Phys. 1971, 55, 572. (2) h i , M.; Edwards, S. F. J. Chem. Soc., Faraday Trans. 1978.2, 1789, 1802.
0022-3654/92/2096-4100$03.00/0
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-
the difficulties of the reptation theory is the predicted viscosity which conflict the experimental law, M‘.4; a law 9 M3, similar discrepancy occurs in the terminal relaxation time re.’ A few modifications of the model have been proposed to improve the M dependence of 9 and re;Doi4 considers contour length fluctuation of the tube, and Klein,5 configurational change of the tube due to constraint release in the intermediate part of a polymer chain (tube renewal); the latter model was farther developed by Daoud and de Gennes6 and Grae~sley.~ These are partial (3) h i , M.;Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. (4) Doi, M.J. Polym. Sci. 1983, 21, 667. (5) Klein, J. Macromolecules 1978, 11, 852.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4101
Entangled Polymer Chains. 1
Figure 1. LK model.
Figure 2. Various topological states of a LK.
modifications of the reptation theory, but the concept of reptation itself is subjected to criticism, because it is based on the extremely simplified model of polymer motion. Formulation of reptation motion based on more fundamental physical foundation has been recently tried by Fixmaa8 Starting from kinetic equations of many chains in the full phase space, he derived diffusion equations in the curvilinear spaces parallel and orthogonal to the contours of the polymer chains and, by assuming a drag force due to the surrounding polymer matrix acting in parallel to the contour of a probe chain, obtained 0, hf2and Q @.5, which agree with experiments. Similar works are done by Curtiss and Birdg and by Hess.lo Curtiss and Birdg assumed that the diffusion coefficients in the parallel and orthogonal direction are different and show that the model goes from the Rouse to reptation-like as the orthogonal friction increases with increase of M and c. Hess’O obtained a similar result for a polymer model with the excluded volume interaction. Although these works give a new insight in the molecular interpretation of reptation, the origin of reptation is still not understood well. For many years, we have tried to construct an unified theory of entanglement based on the topology of We consider that polymers are topological in nature and entanglements must be concerned with much broader properties of polymers than their viscoelasticity. This theory includes not only the ordinary entanglement problems such as mechanical and dynamical properties of linear14-lsand network polymers13 but also dilute solution properties of ring polymers,’* which are usually out of the scope of entanglement. Since the reptation model can explain the most viscoelastic properties very well, it must at least be a good mathematical model. In the previous we have proposed a new model of reptation, a “local knot (LK) model”, whose phenomenological features are consistent with Doi-Edwards theory3 and which has the common framework with our unified topological theory. In this and the following paper,I6 referred as parts 1 and 2, the LK model is tested by computer simulations. We first review briefly the previous LK theories and explain what is to be done in parts 1 and 2. (B)LK Theory in Quasi-EquilibriumStates.I4 This model is based on the assumption that entanglement of polymer chains may be decomposed into twebody local entanglements, which are called ‘local knots (LKs)”. In Figure 1, each polymer chain, say A, is divided into many ‘local chains (LCs)”, ..., ai, ai+l,..., which are entangling with other LCs bj, Ckr ... to form LKs, (a,,bj),(ai+]&), .... In principle, more complex knots formed by three or more
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-
(6) Daoud, M.; de Gennes, P.-G. J . Polym. Sci., Polym. Phys. Ed. 1979, 17, 1971. (7) Graessley, W. W. Ado. Polym. Sci. 1982, 47, 61. (8) Fixman, M. J . Chem. Phys. 1988, 89, 3892, 3912. (9) Curtiss, C. F.; Bird, R. B. J. J . Chem. Phys. 1981, 74, 2016. (10) Hess, W. Macromolecules 1986, 19, 1395; 1987, 20, 2585. (11) Iwata, K. J . Phys. Soc. Jpn. 1974, 37, 1413, 1424, 1429. (12) Iwata, K.; Kimura, T. J . Chem. Phys. 1981, 74,2039. Iwata, K. Ibid. 1985, 78, 2778; Macromolecules 1989, 22, 3702. (13) Iwata, K. J. Chem. Phys. 1982,76,6363,6375; Ibid. 1985,83, 1969. (14) Iwata, K.; Edwards, S. F. Macromolecules 1988,21,2901; J . Chem. Phys. 1989, 90, 4567. (15) Iwata, K. Macromolecules 1991, 24, 1107. (16) Iwata, K. J . Phys. Chem., following paper in this issue.
Figure 3. Reptation in the LK model.
LCs will appear, but they may be neglected because their occurrence probabilities are far smaller than that of the two-body knots. The topological state of L C pair (a,b) is represented by the Gauss integral (GI) T=
C C ekk’
kto k’tb
where ekk’ is GI in regard to bond k of L c a and bond k’of L c b. Let us consider how the entanglement state of (a,b),which is initially in state B of Figure 2, changes into A or C; clearly, the whole entanglements of A or B must be untied before changing the entanglement state of (a,b). LKs are defined in the so-called ’plateau time-region”, which is longer than re,the onset time of entanglement restriction, but sufficiently shorter than Td, the disengagement time of the tube model.3 It is assumed that, in this time region, GIs of the all LKs should be conserved as an average or, more precisely, they should fluctuate around integer number 0, f l , f 2 , ... according as they are in state B, A, C, .... When the local knots are conserved, they behave like cross-links and form a transient network all over the system. As the time passes beyond the plateau time region, reptation begins and renewal of LKs occurs initially at the terminal and then proceeds to the inner parts of polymer chains as shown in Figure 3. In the previous workI4 it is assumed that (a,b) forms a LK when 17‘l is larger than a certain positive number To(-0.5); Towas equated to 0.5, but it should be considered as a universal constant near 0.5. Now, let P( T,r) be a simultaneous probability density for GI be equal to T and for the distance between the centers of a and b be equal to r. By definition, the entanglement probability for (a,b) at distance r is given by Pent(r) =
J I ~ ~ T ~ P ( Td~ , ~ )
The average number of entanglement partner per LC, by fie.p.(v,C)
= (c/v) Jpent(r) dr
(2)
is given (3)
where c is the concentration of chain elements and v is the length of the LCs. The equilibrium length of LCs, ~ ( c )is, determined
a
4102 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992
_[cT
Iwata and Tanaka
‘*---‘e crankshaft motion
Jyq-
A nomal motion
*
j
B
cl
I
*---a
‘
c /
‘
t
&--4 c 2
c1
AFT
---*.---dc 3
three bond mot ion Figure 4. Elementary processes of SCLM. Normal motion A and crankshaft motion B are used in Kovac’s model.” Three bond motions, C,, C2 and C3, are later introduced by Kolinski et aLzo
by the requirement that they should have exactly one entanglement partner as an average, or fic.p.(iW,c)= 1
(4)
By definition, B corresponds to the entanglement spacing, or the average chain length between nearest-neighbor LKs, which is usually identified to empirical parameter ne = ckBT/GN,where GN is the plateau modulus. According to the previous work,14 ij is roughly equal to the two-thirds of ne:
= 0.65ne (5) (C) Dynamic LK Theory.15 In the dynamics, LC pairs are used as probes in tracing motion of LKs along polymer chains. Let at and br be LCs in polymer chain A and B, and [ and [’, the positions of the central bonds of at and b,. A modified G I for (a,,b,) is defined by si
T ( 4 f ‘ )=
kaA kPB
(6 1
uk(uk’$ekk’
where Ukt’s are weight factor for, say, bond k:
‘‘ [ =
sin [ n ( ( - k ) / v ] otherwise
fort 5 k 5
0
v is the length of the LCs. Position
5+
v
(7)
(&Q)of a LK is determined
by the local maximum condition for T aT/af = 0, a T / a Q = o
(8)
Although the modified G I is different numerically from the true GI, the local maximum position determined by eq 8 represents a rough position of a LK and describes sufficiently its long-time behaviors. The theory is presented for lattice models, say the simple cubic lattice model (SCLM), of which Brownian motion is generated by random Markov jumps J of their small part of chains as shown in Figure 4. By the topological restrictive condition that no polymer chain should go across each other during jump J, the variation of T by jump J , AjT, becomes much smaller than T itself and the motion of T becomes very slow. Displacement of coordinate [ due to jump J is given by = -(aAJT/a[)/(a2T/a[2)
(9)
which is far smaller than v, the length of LCs; thus the motion of [ is also very slow in this model. In the dynamics, it is important to use the modified GI instead of the true GI defined by eq 1; for the modified GI, AjT and Aj[ become very small so that T and F move slowly and smoothly; if the true GI is used, on the other hand, AjT becomes as large as T itself when jump J occurs in the terminal parts of LCs, motion of T and E are mixed with large high-frequency fluctuations generated by the terminal jumps, and thus formulation of their motion becomes difficult. It will however be shown in this work that even when the modified GI
I I
I
I
I
I
ts
tc
log t Figure 5. A schematic representation of At), the mean-square displacement of coordinate of a LK along a polymer chain. The logarithmic plot of At) has two Markovic parts of unit slopes, A and C; the former represents single LK motion and the latter, the collective motion of LKS; the horizontal separation between A and C gives log (&.), Nl,k,, the average number of LKs per chain joining in the collective motion.
is used, another non-Markov motion called “probe fluctuation” occurs and makes the analysis of the simulations difficult; the theoretical treatment of the “probe fluctuation” is presented in part 2.16 If Aj[ is assumed to be simple Markov, the diffusion coefficient of a single LK, do, is given by = y2c(PJ(AJ[)2) J
(10)
where p J is the occurrence probability of jump J per unit time. It is argued that (1) due to higher topological constraints which are not considered explicitly in the theory, the orders of LKs along a chain should be conserved and repulsive forces should act among adjacent LKs and (2) these forces lead to the collective motion of LKs, which is assigned to the reptation motion. These behaviors are schematically represented in the time evolution of the mean-square displacement of LKs, g ( t ) = ( [ [ ( t )- [(O)]*) shown in Figure 5 . The logarithmic plot of g(t) in Figure 5 has two Markov parts (of unit slope), A and C; line A, which appears in short-time region t I t,, corresponds to the Markov motion of single LK, and its diffusion coefficient dois given by eq 10; line C, which appears after t,, represents the Markov motion of the collective motion, of which diffusion coefficient is given by d,, = dO/Nl.k.,where LT,,~. is the average number of LKs joining in the collective motion; in Figure 5, the horizontal distance between line A and C is equal to log iVl,k,. The intermediate part B represents the slowing down of LK motions due to the repulsive interaction among neighboring LKs. The upper limit of the single Markov part in time and coordinate is (t,,%,), and the lower limit of part C is (tC&); is equated with the effective range of the repulsive force, re, between LKs; Qcmay be equated to the average distance between nearest-neighbor LKs, L1.k = L/fi,,k,, where L is the length of the polymer chain. It must be noted that the reptation (Le., the collective motion of LKs) considered here is different from the usual r e p t a t i ~ n ;the ~ former occurs in the index number space of bonds, while the latter occurs in a “tube” fixed to the lab space; the usual reptation motion is obtained by projecting the present reptation onto a time-averaged contour of the polymer chain (which corresponds to the contour of the “tube”). In this way, the disengagement time ?d and the sedimentation constant 0,are computed in the previous work.15 The main purpose of this (part 1) and the following work (part 216)is to confirm this theory, particularly the existence of LKs and their collective motion by computer simulations. As will be shown, the essential parts of the theory are well confirmed by the present simulations, but many new features, such as ‘probe
a,
Entangled Polymer Chains. 1 fluctuations", 'temporary LKs", and strong short-memory effects among AJt, are found. To compare the simulation results quantitatively with the theory, several minor modifications are necessary, but its essential part remains much as it is. The results of computer simulations are mainly given in this paper (part l), and theoretical works, in the following paper (part 216). 11. Simulation Method and Equilibrium Properties
(A) Method of Simulation. There are two typical techniques of computer simulation, Monte Carlo (MC) and the molecular dynamic (MD) methods, both of which are used in the studies of polymer chain motion. The technique most often used is MC,17-21in which polymer chains are assumed to be restricted on a lattice, say, simple cubic lattice, and their motions are generated by random jumps of elements such as shown in Figure 4. In MD,22polymer chains are composed of spheres connected by springs and time development of the system is followed by numerical integration of the Newtonian equations of motion.22 The MD method takes much more computational time than MC, but it is suitable for performing vectorized computations. To study entanglement, one must simulate motion of many long chains for a very long time. Such simulations are done by Kolinski et al. using the M C method on simple cubic19q21and tetrahedral lattices20 They found that the diffusion coefficient of the center of mass decreases rapidly with increases of chain length but found no clear evidence for reptation. Recently, an extensive study of MD simulations for entangled chains was done by Kremer and Grest,22who found the crossover from the nonentangled to the entangled regime and presented positive evidence for reptation. The results of these MC and MD simulations agree in the existence of the crossover but contradict in regard to reptation. The contradiction seems to be that the polymer chains used by Kolinski et al.19921are not long enough for the reptation motion to be observed clearly. The more fundamental reason, however, seems to be that the concept of "reptation" itself has not been understood well. In this and following paper,I6 we will study this problem in terms of the local knot (LK) t h e ~ r y . ' ~ ' ' ~ In MC methods, chain structures considered are more realistic but chain motions are simplified, while in MD, the motion of each element is treated exactly but the fine structure of chains is smoothed out. We cannot say at present which of the two methods is better, and we should study both of them. In this work, we use the MC method for the following reasons: (1) The previous LK theorylS has been presented for the MC model. Although the theory can be translated into the MD model, it is natural for us to study the MC model first for testing the LK theory. (2) A more practical reason is that we cannot a t present use supercomputers in our laboratory. MC simulations can be performed efficiently by small computers, since the algorithm of their elementary motions is simple and lattice chains move rapidly. By working on workstations (MIPS RS3330, RS3230, etc.) for several thousand hours, we can do work equivalent to that done by supercomputers using the MD method.22 In this work, we consider a system composed of ring chains entangling permanently with each other. If LKs really exist in this system, we will be able to observe their motion for a sufficiently long time under a constant topological condition, for neither annihilation nor creation of 'true" LKs occurs in the rings. Reptation (in our terminology, the collective motion of LKs) itself must be a cyclic motion along the rings. As the polymer model, we use the simple cubic lattice model (SCLM) with excluded volume; Le., the chains are restricted on a simple cubic lattice, no elements can occupy the same lattice site simultaneously, and their Brownian motion is generated by Markov random jumps (local jumps) of normal type A and crankshaft type B as shown (17) Kranbuehl, D.E.;Verdier, P. H. J . Chem. Phys. 1984, 17, 749. (18)Stockley, C.; Crabb, C. C.; Kovac, J. Macromolecules 1986, 19, 860. Naghizadeh, J.; Kovac, J. J . Chem. Phys. 1986,84, 3559. (19)Kolinski, A.; Skolnick, J.; Yaris, R. J . Chem. Phys. 1987, 86, 7164. (20)Kolinski, A,; Skolnick, J.; Airie-st, R. J . Phys. Chem. 1987,86, 1567. (21)Skolnick, J.; Yaris, R.; Kolinski, A. J . Chem. Phys. 1988, 88, 1407. (22)Kremer, K.;Grest, G. S. J . Chem. Phys. 1991, 92, 5057.
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4103 TABLE I: Equilibrium Properties of Ring SCLM Chains
64 128 256
3.05
3.27
0.33
4.48 6.42
4.62 6.53
1.12 2.40
0 0.019 0.11
0 0.001 0.002
0.33 1.14 2.51
"Computed for 6* = 1.98.
in Figure 4. Equilibrium and dynamic properties of this model have been studied well by many people.17-19.21323 Verdier17 found that when the normal jumps alone are considered, relaxation times increase as h!f even when no interchain entanglement exists. This contradictory result has been explained by Hilhorst and D e ~ t s c h ? ~ who showed that the model has a kind of local memory called 'extrema" that cannot be removed by the normal jumps alone; they also showed that the extrema can be removed by introducing the crankshaft motions. Later, Kovac showed that the self-diffusion coefficient of entangled SCLM in fact increases dramatically by adding a small amount (say, 3%) of the crankshaft motions. Kolinski et al. further introduced three-bond-jumps CI, C2, and C3, shown in Figure 4.19 By nature, the equilibrium properties of the two models must be the same. In this work, we adopt Kovac's model since (1) theoretical calculation of dobased on eq 10 is easier and (2) its algorithm is simpler and permits us to write more efficient programs. (B) Average Size of LKs. Before starting the dynamic simulations, we must first estimate the average length of LCs, I, using eqs 2-4, but these equations remain an ambiguity in regard to To (cf. section IIB). To estimate 3 in this system, LKs are modeled by links formed by small ring chains of length vring, since their entanglement states are defined with no ambiguity. Since both LCs in LKs and rings in linksare in the similar entanglement state, the average length I defined by eq 4 must be similar in both systems. In this section, we consider rings of length uring = 64, 128, and 256 as a model of LCs. Throughout this work, chains are restricted to a simple cubic lattice of size 32 X 32 X 32, the periodical boundary condition is imposed, and the concentration of chain elements, c, is fmed to 0.5. Equilibrium samples are made by the dynamic method: Initially, n rings of length vrinSare packed randomly in the lattice (in this stage, elements are overlapping mutually) and their Brownian motion is performed for a sufficiently long time, first permitting overlap and then reducing the overlap probability gradually to zero. Overlap among elements induces crossing across rings and thus leads the system to its topological equilibrium. This process is repeated, and the conformation of all chains are recorded at the end of each process (a set of data in one process is called a lot); 10, 20, and 70 lots are made for vring = 64, 128, and 256 rings. To check samples, R, @ computed and compared favorably with theoretical value vringbZ/12, where 6 = 1.98 is the effective bond length of this model reported by Kolinski et al.;19 the results are given in Table I. To see the distribution of the Gauss integral T, it is computed numerically for all pairs of rings in each lot and total number of pairs, Npir(T), which are in linking state T, is computed. The average number of linking partners per ring, &(T), which are in linking state T, is given by r ) = 2Npi,( T)/nNIot,where n is the number of rings per lot and Nlotis the number of lots made; iie,p,(T) and their sum over l7l 1 1, fie.+,, are given in Table I. As seen from the table, most pairs are in the lowest linking state 17l = 1 and only small amount of higher linking state l 7 l 1 2 appears; we may therefore neglect the difference among linking states and consider only their sum, as has been assumed in the previous w0rks.'~9~~ The average length vring of rings, which are in the entanglement state, is given by eq 4. Interpolation of i i , , on column 7 of Table I gives F,,,~ = 1 19. To estimate F of linear LCs roughly, the pseudoequilibrium ensembles of short linear chains of length Y = 64 and 128 are made from the configuration samples for rings of length L = 512 obtained in the dynamic simulation done in section IV as follows: (23)Hilhorst, H.J.; Deutsch, J. M. J . Chem. Phys. 1975, 63, 5153.
4104
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992
Iwata and Tanaka
TABL 11: Average Number of Entanglement Partner in Linear L C s Y
TO 0.3 0.4 0.5 0.6 0.7
64
128
0.53 0.33 0.21 0.14 0.09
1.74 1.22 0.89 0.64 0.47
18.8 108.8
TABLE III: Distribution of ne.+.in Rings of V~ ne 0 1 2 3
n
36.4 189.8 = 128 4
5
fraction 0.3023 0.3922 0.2281 0.0633 0.0117 0.0023 rings of length L = 5 12 are cut into eight LCs of Y = 64 or four LCs of Y = 128; since each lot contains 32 rings of length L = 512, it contains 256 LCs of Y = 64 or 128 LCs of Y = 128; 21 lots are taken, with a 1 Mu.t. interval, from the 20 Mu.t. dynamic simulation samples; the positions of cutting are different in each lot. T a r e computed for the all possible pair of LCs for these samples, and the average number of entanglement partner, is computed with use of eqs 2 and 3, changing To between 0.3 and 0.7; the results are given in Table 11. If To= 0.5 is assumed as in the previous works,14the average length of LCs determined by eq 4 is estimated roughly as I = 140, which is rather close to vring= 119. We cannot however determine fie,p, definitely for linear LCs, because it depends strongly on Toas seen from Table I1 (in fact, even a slight change of Toleads to a large difference in D). Since the entanglement state of rings are defined definitely, we had better use as the index of the average size of LKs and Put D = KIring (1 1) where K is a universal constant near unity. We consider next the distribution of ne,p,in an individual link of rings. The distribution function Pe,p,(ne,p,) is computed using T already determined, and the result for uring = 128 is given in Table 111 (the difference among T, 17‘l 2 1, is discarded). Since uring= 128 is close to pring = 119, the Pe,p.(ne.p.) given in Table 111 must roughly represent the distribution of ne.p,in LKs. Table 111 shows that the main peak occurs at h,p, = 1 as expected, but higher multiple links such as ne,p,= 4 have considerable probabilities (say, PJ4) = 0.012). At this point, the questions may rise that since chains are clouded in higher multiple links as shown in Figure 6A, might their motions be slowed down drastically and might they become a ratedetermining step of the collective motion, even if their occurrence probability is very small? This is however not the case in LKs which are made of linear LCs, for lengths of LCs are variable and higher multiple links stay most of the time in a loose conformation shown in Figure 6B;in such a loose conformation, LCs must move as freely as LCs in ordinary LKs. In the calculation of fie,p,, an LC having ne,p,entanglement partners is counted as ne,p.“imaginary” LCs moving independently; such LCs are multiply counted but they are balanced by LCs having no entanglement partner; the net number of entanglement partner per chain is thus given by fie.p. in Table I. In the previous theory,15 it is inexplicitly assumed that the ne.p,“imaginary” LCs have the same diffusion coefficient, do, as the ordinary LCs; this is equivalent to assuming that the diffusion coefficient of an LC in when it joins the collective the multiple link is reduced to do/ne.p., motion. The motion of such an L C must be slowed due to the multiple entanglements, and this effect is considered approximately by reducing its diffusion coefficient from doto do/nc,p,.Although this is a crude approximation, it would lead to no serious effect in the diffusion coefficient of the collective motion, for higher multiple links seldom appear. (C) True and Temporary LKs. The average number of LKs per polymer chain of length L is given by
Figure 6. (A, left) Multiple link made of rings. (B, right) Multiple LK made of linear LCs. In A, chains are clouded because chain lengths are fixed in the rings, while in B chains are in a loose conformation because lengths of linear LCs are variable.
temporary LK Figure 7. True LK and a temporary LK.
In the previous ~ o r k s , LKs ~ ~ Jare ~ assumed to be all true LKs, but as a matter of fact, only a half of them are true LKs and the remaining are temporary LKs which have definite lifetimes. This is explained in Figure 7, in which conformation A and B look like similar entanglement states, but A is the true entanglement state while B is not. The difference appears when they are twisted as in A’ and B’; by twisting, A remains in entanglement state A’ while B turns into nonentanglement state B’, which is still partly entangled but will be sooner or later untied. In the dynamic simulation done in the next section, numbers of such true and temporary LKs are in fac;t,,fo’und. Since conformations A and B appear with the same frequency, the average number of true LK, if,,,,, must be equal to a half of f i I . k , , or
By definition, the average GI of the true LKs, ,T,;I must be near unity, while that of the temporary LKs, Tmp, must be considerably smaller than unity, for the latters are usually in relaxed state B’ (of which GI is small) rather than in strained state B. This meam that the average number of the temporary LKs per ring, iitemp, is larger than that of the true LKs. It must be noted that the average chain length between nearest-neighbor true LKs, &, is equal not to their average length, D, but to double it:
L,,,, = 2s = 2KD,ing
(14)
111. Search, Tracing, and Classification of LKs
(A) Brownian Motion of the Polymer Chains. In the dynamic simulation, we consider again a system composed of ring chains.
Entangled Polymer Chains. 1 This system is particularly suitable for studying motions of LKs, since they are cyclic along the rings and we can observe them for a sufficiently long time under the same topological conditions. The system studied here is composed of 32 ring chains of L = 5 12 contained in a box of size 32 X 32 X 32; the periodical boundary condition is imposed, and the volume fraction is again fixed to c = 0.5. In Kovac's model,'* elements are in one of the following three states: (1) can make a normal motion, (2) can make a crankshaft motion, or (3) cannot move (see Figure 4). In the simulation, elements are chosen at random and are moved or not according to their states. The Occurrence probabilities of these states arefno,lIf~anksha~lIfnomotion = 0.3887:0.0298:0.58 15, which agree with the literature values.1g In the simulation, the unit of time (u.t.) is chosen such that the all elements try jumps once per u.t. as an average. The initial conformation of the system, which is nearly in the topological equilibrium state, is prepared as in section IIB. The Brownian motion is performed for 20 Mu.t. (mega unit time), and conformations are recorded per each 1 h . t . (kilo unit time) interval for the study of long-time behaviors; simulations for 1 or 4 Mu.t. with 0.2 ku.t. sampling interval are also done for the study of short-time behaviors. (B)Search and Tracing of Local Maxima of T. To find and trace motion of a LKs, modified Gauss integral T([,p),defined by eq 6 for a pair of LCs, (a&), is used as a probe. Its position ([,Q) is determined by the local maximum condition of T, eq 8. According to the uncertainty principle presented in part 2,16 neither position ([,p)nor length (v,v') of LKs is determined definitely, and to determine their true positions, many probes of different sizes must be used simultaneously. Considering the average length of LCs found in section 11, I = pring= 119, we use four LCs of length v1 = 64, v2 = 96, vj = 128, and v4 = 192 and seven probes composed of them, PI,, P22, P33,P44, PIj, P24,and P42,24 where suffix i represents length vi ( i = 1, ..., 4). In the initial conformation at t = 0, the all local maxima of I?l for probe Pit are found as follows: T([,Q)is computed numerically for all possible pairs of LCs of length vi and up, sweeping [ and I'and searching for its local maximum along all the rings in the central and its neighboring boxes; if a local maximum exceeds constant Tmi,,a lower limit of IZl in the dynamics, it is tentatively listed on the LK table. To find all LKs in the system, Tminmust be much smaller than To (=OS), the lower limit of IZl in the equilibrium problem; in this work, Tmin= 0.1 is assumed. A large number of local maxima (say, 597 for probe Pj3) are found at t = 0, but most of them are small fluctuations of T of short lifetimes. Using the conformation data obtained in the simulations, the local maximum positions at time t, ([(t),.$'(t)), are searched for in the vicinity of their previous position at t - 1 in the range ( [ ( t - 1) & Lt, p(t - 1) f I&,,& where I&,,t is a cutoff length, which must be sufficiently smaller than the average displacement of [ during the sampling interval. We use 4ut = 20 for the simulations with 1 ku.t. interval and I&,,, = 16 for simulations with 0.2 ku.t. interval. At each t , the average of T([,Q)over proceeding 10 samples at t - 9, t - 2, ..., t is computed, and if its absolute value goes below T,i, = 0.1, the corresponding local maximum is removed from the LK table. Continuing this process, the number of local maxima, NI.,.(t), decreases rapidly in the initial short period ( t < 0.12 Mu.t.). The average of I7l over the local maxima surviving at time t, Tl,,,(r), is also computed. The dissipation rate of N,,,,(t) increases drastically with decreasing probe size and, at 20 Mu.t., 36 LKs survived for PMr13 LKs for P3j,but only 2 LKs for small probe P22.For long-time simulations, therefore, large probes should be used, and data for P22 are omitted from the discussion below. Nl,,,(f)and TI,,,(t) for P33and PU are shown in Figures 8 and 9. (C)Classification of LKs. Since N1,,.(t)has no simple exponential form, we assume that it is composed of many simple-relaxation modes:
(A) prclbeP,
I
I
0
10
x) (Mu.t.)
t I
0
I
I
I
L
1
20 (Mu.t)
10
t Figure 8. Decomposition of Nl,m,(f), the number of local maxima of T survived at f , into simple exponential decay modes (Lt = 20, traced forwardly): (A) for (B) for PM. A represents Nl,m,(f); B, N I . ~ . ( ~ ) - N.2.exp(-f/TC.).
0 0
20 (Mu.t)
10
t
0.2
1 I
I
10
I
I 20 (Mu.t)
t
(15)
Figure 9. Logarithmic plot of Tl,m.(f),the average of T of the local maxima survived at t. The solid line represents TI,&) computed by cq 9 for the parameters given in Table IV: (A) for P33;(B)for Pu.
(24) Probe P3, is also used, but its data are omitted because an error occurred in the process of calculation.
We can separate two slow decay modes, M Iand M2, from Nl,,.(t) as shown in Figure 8; strength and lifetime T [ $ , of these
NI.m.(t) =
Ei W k , ex~(-t/7f!k.)
M,:,
Iwata and Tanaka
The Journal of Physical Chemistry, Vol. 94, No. IO, 1992
4106
0
TABLE I V Characteristics of True and Temporary LKs Drobe
mode 1
p3 3 p44
1
p3 3
2 2
p44
M?.
.TK?,,Mu.t.
TI!!.
54.7 55.1 42.
13.8 42.5 2.1 2.3
0.92 0.96 0.6, 0.59
45.
assignment true true temporary temporary
un
0
256
256
512
UI,
I
0
W
I
I
A
2.5 t
512
L
5
15
10
20 (M.u.t)
t
Figure 10. Trajectories of three true LCs survived for 20 Mu.t. in ring no. 25. The figure shows that the orders among true LKs are conserved and the LKs made a collective motion. Note that coordinate $, is cyclic along the ring.
modes estimated for P33and P44are given in Table IV. In the same manner, T,,m,(t)is decomposed into these modes as follows: Tl.m.(t) = C U%.Nf.%. ex~(-t/7~i~,)/Nl.,.(t)
(16)
I
where 7fi, is the average of I7l of local maxima belong to mode Mi. are determined so that eq 16 agrees best with the are given in the fifth simulation result; the best values of column of Table IV, and the curves computed for these parameters are shown in Figure 9. The slowest mode M Iis assigned to the true LKs and the second mode M2 to the temporary LKs as shown now. Mode M I is assigned to the true LK, because its n.2,( ~ 0 . 9 2 for P33,0.96 for P44)is near unity, and its apparent lifetime ~f.2, (=13.8 Mu.t. for P33and 42.5 Mu.t. for Pa) is quite long. If all true LKs in the system are included in this mode, M.2,should be the same in P33and Pd4and should be close to N,,, = 68.s computed from ping = 119 assuming K = 1 in section 11; A$:, = 54.7 found for Pg3and 55.1 found for PU satisfy these conditions. Although definite lifetimes are found in this mode, this is considered to be an artificial effect of using the probes, and their lifetime is essentially infinite as discussed later. for P33and P44are different, because their sizes are different. Typical trajectories of true LKs are shown in Figure 10, in which motions of three LKs survived a t 20 Mu.t. in ring no. 25 are shown as an example. In Figure 10, we see that the orders of LKs along the ring are conserved and LKs move as if a repulsive force was acting among them. It must be added that the trajectories shown in Figure 10 are time-smoothed for the sake of visualization; their raw trajectories contain much more rapid fluctuations (the “probe fluctuations” discussed later) and temporary crossing occurs sometimes among them, but their overall orders are rigidly conserved. It is also found that all 7% in mode M Istay almost in the vicinity of f l . All these features agree well with what is expected of the true LK. Since the data for Pa are more accurate than those of P33,we use hereafter the former data; i s . , N,,,, = 55.1 and = 3.44. Mode M 2 is assigned to the temporary LKs, because its intermediate magnitude of n.2,(=0.65 for P33and O S 9 for P-) and its fairly long lifetime 71.2.(=2.1 Mu.t. for Pj3and 2.3* Mu.t. for P-) agree with those expected of the temporary LKs (cf. discussion in section IIC). The number of temporary LKs (q.2,= 42.5 for
n:i,’s
n!A,
n,:!
I
5 (M.u.t)
Figure 11. Trajectories of true and temporary LKs. This is the initial 5 Mud. part of Figure 10 and the trajectories of the three true LKs are drown in bold lines;in this period, four temporary LKs appear, and their trajectories are shown by thin lines. Note that the orders among true and temporary LKs are roughly conserved.
I
I
0
n
P33 and 45 for P 4) is near that of the true LK as expected. We consider that T$,, represents the true lifetime of the temporary LKs, 7tcmp,since the artificial dissipation rate due to the probes is much slower than the decreasing rate of this mode. If so, ”f2, and ~f.2, should be independent of the probes, and this is in fact satisfied in this mode. Since the data of P44are more accurate, we adopt them for the temporary LKs; i.e., Ntemp = 45 and rtcmp = 2.3 Mu.t.; the average number of temporary LKs per ring is thus estimated to be fitem = 3.OS. An example of trajectories of true LKs (bold lines) and temporary LKs (thin lines) appearing in ring no. 25 are plotted simultaneously in Figure 11. We see from the figure that crossing between the trajectories of the true and temporary LKs occurs frequently, but their order along the ring is conserved as an average; this means that a repulsive force also acts between the true and temporary LKs. (D) Mech.nisnrs of Dissipation and the Lifetime of Trw LKs. The lifetime of true LKs, T ~ is the ~ most ~ ~important , quantity in the LK theory. Although definite lifetimes are found for the true LKs, this is an artificial effect of the method used. When tracing local maxima which look like true LKs, we observe sometimes strange motions, such as a jump of, say, 100 bonds in just 10 ku.t. or an abrupt decrease of T below 0.1, which usually results in their removal from the LK table. To see the reason for those abnormal motions, we repeated the same sampling of local maxima as done at t = 0 and also at t = 5 and 10 Mu.t. and traced their motion backward. Simulations starting a t t = 0, 5, and 10 Mu.t. are hereafter called runs 0, 5, and 10. In the forward tracing (run 0) using probe P-, 64 LKs surviyed a t 3 Mu.t. and 46 LKs at 9 Mu.t.; thus, 18 LKs.are’lost in 6 Mu.t. It is estimated using N,: and ~ f given i ,in Table IV that, among the 18 LKs lost, ca. 6.7 are true LKs and ca. 11.4 are temporary LKs. In the backward tracing starting at t = 10 Mu.t. (run 10) using P33,64 LKs suMved at t = 9 Mu.t., among which 17 LKs out of the 18 LKs lost in run 0 are found; it is also confirmed that the orders of the lost LKs along the rings are also kept exactly. Most of the LKs lost between 3 Must. It 5 9 Mu.t. in run 0 are thus recovered. Although we cannot say which of the 18 LKs lost are true, it is almost certain that an estimated 6.7 true LKs are all included in the recovered 17 LKs. Thus we conclude that the dissipation of true LKs deduced from the decay curve of N1,,,(t)are an artificial effect of the method used and true LKs exist even if our program loses them; Le., the lifetime of the true LKs must be in effect infmite. So,why are the temporary LKs also revived? The explanation is that, in ring chains, most temporary LKs are also trapped topologically as shown in Figure 12 and continue appearing and disappearing; it is confirmed that even the orders of the reappearing temporary LK are conserved. We call these temporary LKs “nonlocalized knots (NLKs)”. At present, it is
Entangled Polymer Chains. 1
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4107
0
UJI
256
F i i 12. Nonlocalized knot (NLK). In configuration A, a, and a, are temporary LKs and a2, a3,and a5 are true LKs. In configuration B, al is untied but it is still trapped topologically for both A and B are ring. Such nonlocalized knot repeats appearance and disappearance.
t
1
512 2.5
0 t
5 (l4.u.t)
Figure 14. Comparison of forward and reverse trajectories of two true LKs on ring no. 29. The forward trajectories are shown in solid lines. The reverse trajectories are shown in broken lines, but they are scarcely seen because they agree almost with the forward trajectories. When a jump between brothers occurs, however, a large deviation appears as shown in part a in the figure.
A
B
A
(B 1 Figure 13. Possible mechanism of merging between brothers. (A)
difficult to say whether such NLKs are particular in such short rings as considered here or whether the temporary LKs are really temporary and seldom revive in the ordinary systems composed of long linear chains. Now, let us consider how true LKs are lost in the process of tracing. The trajectories of LKs in runs 0,5, and 10 agree exactly with each other, but, in very exceptional periods, large deviations appear and their abnormal motions are detected. Comparing the trajectories of runs 0, 5 , and 10, we find the following three mechanisms of the unusual motions: (1) merging effect; (2) multipeak effect; (3) ghost effect. (1) Merging effect: LKs are generally classified into “brothers” and “half brothers”; brothers come from the same parents (rings), and half brothers are from the same ring but different partners. Most LKs are half brothers, because LCs of a ring are surrounded mainly by LCs of foreign Mgs. There is no problem in identifying half brothers in different runs, for they are the only LK of particular parents. Our program, however, gets into trouble with brothers, for they sometimes merge with each other or jump among them. In the initial states of runs, many maxima of the brother type appear and merge rapidly with each other; this corresponds to the process for local peaks of absorbing into their main peak. After finishing this process in the initial short period, surviving local maxima keep their T around f l ; at this stage, ca. onefourth of them are brothers, which merge sooner or later into one of their families. One of the possible mechanisms of the merging process is explained in Figure 13. In conformation A of Figure 13 (al,bl) is distinguished clearly from its brother (a3,b3),since they are separated by half brother (a2,c)and (b2,d);when a fluctuation occurs to change conformation A into B, our program cannot distinguish (al,bl)and (a3,b3)and judges that merging occurred. Although conformation B is more strained than A, it occurs several times in the 20 Mu.t. simulation; once it occurs, our program runs after one of the brothers and this looks like the sudden disappearance of a true LK. When conformation B reappears after losing one of the brothers, the surviving brother merges into the lost one; this process looks like jumping between the brothers. Several such motions are in fact found by comparing the forward and backward tracing of runs 0,5, and 10; they are observed as
abrupt jumps, say, of 100 bonds in 10 ku.t., (see Figure 14), which is much larger than the average displacement of LKs (a. 20 bonds per 10 ku.t.). Not all brothers necessarily take part in this process. The brothers which are likely to merge are called “intimate”, and those that are not are “independent”. The “independent” brothers are separated by many half brothers and behave like half brothers, since there is little chance for their merging. It must be noted that although many “intimate” brothers appear in the present system, this is a particular case of short rings. This comes from the following argument: Since the average number of true LKs per ring is estimated to be ,& , = 3.44, brothers are separated by less than one half brother as an average and they cannot but be “intimate” in such short rings. In longer chains usually used in viscoelastic experiments, many more half brothers appear per chain so that brothers are usually separated by many half brothers; thus, brothers will be more “independent” in longer chains. The merging effect is therefore not important in the ordinary systems. (2) Multipeak effect: In tracing local maxima, we sometimes observe that a seemingly true LK, which keeps its T near f l for quite a long time, suddenly loses its T and disappears. To see the reason for this abnormal motion, we searched for another local maximum around the position and time of disappearance and found that the disappeared LKs are double-peaked for a considerable time before their annihilation;the main peaks have switched from the original to the new ones shortly before the annihilation. The distance between the old and new peak is typically 40-80 bonds. Since it is larger than cutoff length 4,,,, our program cannot find the new peak and continues to run after the old one until it disappears. The double peaks are not brothers, since they are singlets most of the time and only for a short while they become a doublet (this is confirmed by searching other peaks around their positions in their ordinary periods). Since these LKs exist even after loosing in our program, the slow dissipation process of the half brothers must be an artificial effect, which comes from the inability of our program to follow unusually large fluctuations (or multipeaks). When small probes such as P2*are used, local peaks frequently appear and LKs are easily lost by the multipeak effect, but when larger probes such as P4., are used, small peaks are averaged out and this effect is less effective. This explains why P has a longer (apparent) lifetime (~f.2, = 42.5 Mu.t.) than P33 (TI,,,,4tf = 13.8 Mu.t.). ( 3 ) Ghost effect: This is an illusive motion of a LK merging into its ghost image appearing one round before or after it along a ring. It is observed as an abrupt motion of a LK by the length of the rings, L = 512, in only a few hundred ku.t.; during this motion, IZl becomes very small. This motion occurs when a ring has only a few LKs, and therefore it is particular in such short
Jwata and Tanaka
4108 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992
I h
a\/./
I
-
4-
P 0
G
d
C’
3-
t /
0
I
I
1
1
2
3
log t
4
(K.u.t)
Figure 15. Mean-square displacement of 5, along rings, g&, computed for the 20 Mu.t. tracing with Per (&, = 20). Points a, (gcc(t))LK; points b, (g44(t))ring Line A is the Markov line for single LK motion computed for do =: 0.0172; line C, the Markov@line of the collective motion of true LKs alone computed for dml= do/”, = 0.0050 and C‘, that of true temporary LKs computed for dml = do/(lVlwe + lVlcmp)= 0.00265; Nl,, = 3.44 and lVtemp= 3.06 are used. Note that the slope of log g,(t) increases again above 1 Mud. and it seems to tend to line C’.
+
rings as considered here; this effect will be negligible in larger rings and absent entirely in linear chains. The ghost effect increases with an increase in the size of the probes. From these considerations, we conclude that the true LKs are lost gradually due to the merging, multipeak, and ghost effects, even if their lifetime is infinite. Although these effects come from the insufficiency of our programs, we consider that the uncertainty in determining their positions is an intrinsic nature of LKs. In part 2, we will discuss this subject more in detail and propose an uncertainty principle in analogy with the quantum mechanics. Since true LKs are not true topological invariants, their lifetime is not necessary infinite but, if it is sufficiently longer than the disengagement time of the tube model, we may consider it essentially infinite. IV. Diffusion Coefficient of LKs and Their Collective Motion (A) Time Correlation Functions of Let a LK be on ring A and B and let ti and tj be its coordinate along A determined by probe Pii,and Pjl; then the correlation function between ti and is defined by
[.
gii(t)
+
([ti(? to)
- &(to)] [tj(t + t o ) - € / ( t o ) ] ) t o (17)
where ( ), represents the average over to. Its simple average over LKs, (gij(t)),+,,and its average over rings (gii(f.))ring
= C ( g i i ( t ) ) R / [ n o of . rings1 R
-
-
n
v
e
system, the effective range re of the repulsive force and the average distance d between acjacent LKs are roughly estimated to be re pring/2 = 60 and d L/i& = 149 or - L / ( & + = 79. Although the slope is considerably smaller than unity, the lower linear part in Figure 5 is assigned to the single LK motion, since it appears below r: = 3600. The upper part of (g&) seems to tend to line C’ of unit slope above ca. lo5, which is far larger than d2 22000 or 6400;it is thus assigned to the collective motion of LKs. More detailed discussions are given now. (B) Diffusion Coefticient of Sile LK. We first note that the trajectories shown in Figures 10-12 seem to contain rapid oscillation modes. We consider that these oscillations come from uncertainties induced by the probes used in tracing LKs. In part 2, it will be argued that when the size of a probe is fixed, the position of an LK becomes uncertain and, when the position is fixed, size of the probe becomes uncertain (the uncertainty principle). Hereafter, we call these oscillations “probe fluctuations”. The main source of the probe fluctuations is the multipeak effect discussed in section I11 (of course, the merging and ghost effects in brothers are another source of the probe fluctuations, but we omit brothers in this discussion, for they are not important in ordinary systems). When a new peak reappears far from the original peak, our program cannot find it and the LK is lost (or in terms of the uncertainty principle, its position becomes completely uncertain), and when it reappears near the original peak, it oscillates between the peaks. Further discussions on the probe fluctuations are given in part 2. To compute the diffusion coefficient of a single LK, it is necessary to separate the Markov mode of a single LK from the probe fluctuations in g&). It is shown in part 2 that the probe fluctuations can be removed by using an infinite number of probes simultaneously, and we take appropriate linear combination qo of the coordinates determined by them, Le.
(18)
are computed, where ( g i i ( t ) ) Ris the average of gii(t) over LKs found in ring R. In the 20 Mu.t. tracing with probe P-, 36 LKs suMved but one LK, which shows abnormal behaviorZSis omitted from the calculation; in Figure 15, (ged(t))l,k, (points a ) and (g44(t))ring (points b) computed for the 35 LKs (70 LCs) are shown. We have also computed ( g 2 2 ( r ) and ) ( g 3 3 ( t ) )but , they ) they are contain many more statistical errors than ( g e d ( f ) and omitted from the discussion on the long-time behaviors. According to the previous theory,Is the logarithmic plot of ( g i i ( t ) )should have two Markov parts (of unit slope.) A and C shown in Figure 5. In Figure 15, we see two linear parts correspond to A and C in Figure 5, but their slopes are considerably smaller than unity (compare lines A, C, and C’of unit slope in Figure 15). In this ~
(25) Among 36 LKs survived in the 20 Mu.t. tracing with PU, one LK shows strange behavior; Le., despite that its T always stays near zero, it survived for 20 M.u.t. A possible explanation for this LK is that it is a Boolian link. It is omitted from calculation of (gU(r)).
70 = CuoiEi i
(19)
where ti is the coordinate (along ring A) of a LK determined by probe Piitand uoi is the coefficient of the linear combination. By this transformation, correlation function gij(t) is changed into
f d t ) = CCuoiuojgij(t) i
J
(20)
When an infinite number of probes is use, qo represents the pure Markov mode and the logarithmic plot offo(t) should become a straight line of unit slope. Even when a limited number of probes is used as in the present (and in most future) work, we can observe the Markov part off,(t) in fortunate cases, if coefficients uoi are chosen properly. More exactly, in the latter case, the logarithmic plot off&) has an extended S shape; i.e., its slope is considerably smaller than unity in the beginning (t = l), increases with increasing t , becomes unity in fortunate cases, and then decreases again due to the repulsive interaction among LKs (the slope will increase again to unity as the collective motion of LKs is established, but this is irrelevant‘ to this discussion). According to the theory of part 2, the best linear combination is that which minimizesfo(t) at t = 1, where t should be measured by the sampling interval. To determine doby this method, we have performed simulations with a 0.2 ku.t. interval and traced LKs using the seven probes P I i ,P22, P33, PM,Pi3,Pz4, and P42. Among 36 LKs surviving in the 20 Mu.t. tracing with P33,11 half brothers, which show no irregular motion such as switching between brothers or merging into their ghost images, are chosen as samples. Tracing of these LKs is done under the following conditions: (1) for 1 Mu.t. using all seven probes and (2) for 4 Mu.t. using three probes, P22, P33, and P44rboth with sampling interval 0.2 ku.t. and kU, = 16. It is found that the smallest probe, P I , ,moves slowly but often makes abrupt motions of, say, 30 bonds in 0.2 ku.t., and it is easily lost in the process. By inspecting their trajectories in detail, we found that these unusual motions are jumps among small local peaks of T, which appear frequently when small probes such as PI1 are used; for larger probes, these peaks are smeared out and such motions occur less frequently. During the 1 Mu.t. simulation,
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4109
Entangled Polymer Chains. 1 TABLE V Results of 1 and 4 Mu.t. SiIfIUhtiOM"
Mu.t. simulation probes
LK 1
6, 4* 10,s. 13,2* 22,7* 29, 26*
2
3 4
5
4 Mu.t. simulation
1
ring paiP p 2 2 , p337
p44,p13*
p24, p42
pi13 P22r p339 p44, p 1 3 , p24, p42 p 2 2 , p339 p44, P24* p42
pI1,
p22, p339 p44, p24, p42 P22r
p33, P44r p13,
P24r p42
d0
probes
d0
0.00195
P22r p333 p 4 4
0.00170 0.00166 0.00 168 0.00148
p229
0.0175 0.0170 0.0165 0.0175 0.0160
p33, p 4 4
p229 p33, P 4 4 p 2 2 , p 3 3 , p44 p221
p33, p 4 4
a&,,l = 16, sampling intervals, 0.2 ku.t. bThe number represents sequential number of rings. The LC for which do is computed locate on the ring attached by an asterisk.
TABLE VI: Table of sij =
( ( t i- tj)2)for LK, and LK2 (Unit:
Bond2)
V. Considering these results, we adopt as the simulation value of do their average: do = 0.0172 bond2/u.t.
P24
87 105 131 83 111
P42
153
p22
P33 PM
PI3
8 30 64 20 39 65
35 22 20 81
19 14 59 25 94
9 7 22 30 92
18 15 8 17
3 8 16 12 19
LKI"
83
LK: "The upper right part represents uij of LKI, which is in a quite season. bThe lower left part represents uij of LK2, which is in a stormy season. PI survived in only 6 LKs among the 1 1 LKs studied. Among them, 7 LKs, in which more than 5 probes suMved in the 1 Mu.t. simulation, are chosen for further treatment; a list of the probes surviving in these LKs is given in Table V. To see the magnitude of the probe fluctuation, the mean-square deviations among coordinates, ab= ((ti- t,)2), are computed, and two typical examples for LKI and LK2 are given in Table VI. By inspecting the trajectories of these LKs, we found that they spend alternatively a quiet and active season which continue for a few Mu.t. In Table VI, aV of LKI represents an example of the quiet season (in which aijis no more than a few tens) and aijof LK2 represents an example of the relatively active season (in which aij becomes as large as a few hundreds). Parts A and C of Figure 16 show gii(t)of LKI and LK2 for the 1 Mu.t. simulation, and Figure 16B shows that of LKI for the 4 Mu.t. simulation. When a LK is in the quiet season, the deviations among gij(t)are small as shown in Figure 16A, and in the active season,the deviations are large as seen from Figure 16B. The correlation between the magnitude of aij and the deviations among gij(t)is evident ip these figures. The optimized linear combinations of g&) andfo(t) are determined using the method mentioned above for the 14 LCs, which constitute the 7 LKs chosen, for both the 1 and 4 Mu.t. simulations. The extended S shape appears in the allfo(t), but the Markov part (of unit slope) is observed in only five LCs in LKI,LK2, ..., LK,. The diffusion coefficient do is given by half of the intersection at t = 1 u.t. of the Markov line, and the results are given in Table V. Although there is a small deviation among do's obtained, it is within the statistical errors and they are essentially the same. It is important that we get essentially the same do irrespective of the condition of LKs-(i.e., either they are in the quiet or active season). Generally,fo(t) of an LK depends on its environcent (i.e., how LKs are clouded etc.). This is seen by comparingfo(t) of LKI in Figure 16A (1 Mu.t. simulation) and that in Figure 16B (4 Mu.t. simulation); the difference appears in the length of the Markov part of fo(t), which occurs between 2 and 5 ku.t. in Figure 16A and which continues up to the upper limit of the observation time (24 ku.t.) in Figure 16B. The Markov part ends where the interactions among neighboring LKs start, and thus its upper limit is determined by the distance among them; in Figure 16A, the interaction begins earlier (around 5 ku.t.) in LK, since LKs are crowded around it in this period (1 Mu.t.) observed while, in Figure 16B, it begins later than 24 ku.t. since LKI is relatively isolated in the period (4 Mu.t.) observed. In the LKs studied, their Markov parts appear at various time regions and with various lengths, but they give essentially the same do as seen from Table
(21)
Comparison of do with theoretical value computed from eq 10 is made in part 2. (C) Collective Motion of LKs (Reptation). In Figure 15, line A represents the Markov line of unit slope computed for do = 0.0172; since (gM(t))is determined using probe P4 alone, its single LK motion part comes naturally above line A. Let us first assume that the collective motions occur only among true LKs. According to the previous theory,I5 the diffusion coefficient of the collective motion on ring R , which has q,,(R) true LKs, is given by d,,(R) = do/ntrue(R). In the short-time region A, where LKs move independently, their gii(t)are different, but as their motions become correlated with each other due to the interactions among them, their gii(t) tend to the same Markov line of diffusion coefficient d,,(R). Since (g44(t))l.k.is the average of g44(t) over the all true LKs in the system, the diffusion coefficient determined from (g44(f))I,k, is an average of dCol(R) over the all rings with weight factor ntrue(R);i.e. R
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which gives = 0.0050. Line C in Figure 15 represents the Markov line computed for (dco,)true = 0.0050. Clearly, (gM(t))true (points a) does not tend to line C, and the assumption that the collective motion occurs among true LKs alone is rejected. Next, we assume that both true and temporary LKs join equivalently in the collective motion. For simplicity, we further assume that the diffusion coefficients of the true and temporary LKs are both equal to do = 0.0172 and each ring R has the same number of LKs (true or temporary), nl,k,(R)= he + (i.e,, distribution of q k , ( R ) among rings is neglected); in this approximation, therefore, the collective motions on all rings have the same diffusion coefficient, d,,(R) = d0/(fitme + iitemp).Since (g44(t))ring is the simple average of (g44(t))R over the rings, the diffusion coefficient determined from it corresponds to do/(filrue+ hemp) = 0.00265, which is equal to the diffusion coefficient of the collective motion of all (i.e., true + temporary) LKs; i.e., = 0.00265. The Markov line computed for (d,])l,k, 0.00265 is shown in Figure 15 by line C', to which (gM(t))k (points b) seems to approach. Although (g44(t))ring still does not reach Markov line C' in Figure 15, this result seems to suggest that both the true and temporary LKs join almost equivalently to the collective motion. This seems natural, since the lifetime of the temporary LKs, T , , , ~ = 2.3 Mu.t., is comparable to the beginning time of the collective motion, T , 1 Mu.t., and the orders among true and temporary LKs along rings are almost conserved as shown in Figure 11. For more accurate discussions, the simulation should be continued until the slope of log (g44(t)) becomes unity, but it will take much more computational time and be more difficult to perform with our workstations. Moreover, the present system is not suitable for accurate quantitative studies, because various inconvenient effects appear such as the wide distribution of q k ( R ) among rings, the merging between brothers, or the ghost effect, which are particular in such short rings as L = 512. For these reasons, we stopped this simulation at this stage. Although the present simulations are insufficient in many respects, we can conclude that the temporary LKs join at least partly to the col-
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in Figure 12, they will give some contribution to the stress, although it may be much smaller than that of the true LKs. The real role of temporary LKs in the entanglement phenomena is still open to question. From E,rue = 3.44 and ii1.k. = 6.5, the average chain length between true LKs and that between true + temporary LKs are estimated to be &, = 149 and = 79. Introducing €, = 149 and pring = 119 into eq 14, we can determine the universal constant, K = ( J / F , ~ , , J = 0.63. Once K is determined, we can calculate &, (and hence ne and n, as shown below) for other polymers using their $ring, which is computed from the equilibrium distributions without performing elaborate dynamic simulations. A more important meaning of eq 14 is that it enables us to relate viscoelastic quantities such as ne and GN to other equilibrium topological quantities. For example, the right-hand side of eq 3, which defines fie.,,, is identical with that defining the topological second virial coefficient in dilute solutions of ring polymers, A;, which turns out to be essentially the same quantity as GNas shown in the previous work.14 If the stress of the system in the plateau region comes purely from true LKs, J should be replaced by &me on the right-hand side of eq 5; i.e., LtNei= 0.65ne, which gives ne 230 for the present system. If we assume the well-known empirical relationship26between ne and the crossover chain length n,, n, = 2ne, we find n, 460. If the temporary LKs contribute partly to the stress, ne and n, are somewhat smaller than these estimations. It is interesting that n, of this model (SCLM with excluded volume, c = 0.5) is within the range 270 < n, < 750 of most vinyl polymers in bulk (n, is measured by a main-chain bond). (D) commentson the simuletiolrs of Kolimki, Skoloick, et d.19*21 The present systems have already been studied by Kolinski, Skolnick, et al.,19.21who found a negative conclusion on the existence of reptation in contrast to our results. Their conclusion however came mainly from simulations done for chains shorter than our n, 460, in which the collective motion of LKs (reptation) perhaps does not occur. They have also done some simulation for longer chains of length L = 800, in which reptation perhaps occurs, but it is still doubtful for “tubes” to be formed so clearly as observed by their method of visualizing chain contour motions, since L = 800 is not sufficiently larger than our n, 460. A similar method is used by Kremer and Grest22in their MD simulations, in which they observed reptation-like motion, but in chains much longer than n,. The negative view of Kolinski and Skolnick on reptation comes from their underestimation of ne;in their early work, they seem to consider that ne is near the blob size (a. 18),19 but in the later work2’ they changed it to n, = 125-133, which is still much smaller than our ne 230. Their recent estimation2’ is based on their “blob contact model”, in which they assume that entanglement comes from long-lived “contacts” among blobs, where “contact” means that centers of blobs are within a certain distance, say, 5. In some meaning, their model resembles ours: Their blobs correspond to our LCs and entanglement among blobs ismdetectedby “contact”, while that in our LCs, by GI. Problems in their model are that (1) entanglements cannot be detected sufficiently by a nontopological probe such as ”contacting blobs”, (2) their probe is fixed to polymer chains and does not slide along chains (this motion is essential to our LK theory), and (3) their probe ( L = 18) is too small to represent sufficiently the entanglements in this system (compare with our probes of L = 64-192). To some extent, the entanglements (in our terminology, LKs) may be detected by their probes. They estimated the average chain length between long-lived “contacts” in linear chains of L = 216 to be equal to 133, which may be equated to the average chain length between true + temporary LKs, L1.k. = 79, which is enlarged to 133 due to end effects (in such short chains as L = 216, the end effects must be large). The lifetime of the long-lived “contacts” (-50 ku.t.) found in their simulations is however far shorter than that of our temporary LKs, T , ~ , , ,= ~ 2.3 Mu.t. Perhaps they observed a part of the entan-
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.Figure 16. Logarithmic plots of time-correlation functions, g&), between f i (determined by probe Pii,) and tj (determined by probe P j f ) . In the figures, only some of data are shown, for there are so many data that the all points cannot be plotted on single sheet. Symbols: X, PI,;. 0, P22; A, 0,Pu. Filled circles (0)representf,(t), the optimized linear combination of gii(r) and the straight line represents the Markovic part of unit slope. (A) LKI for 1 Mu.t. simulation; points for P I , , P24,and P42 are omitted. LK, is in a quiet season. (B) LK2 for 1 Mu.t. simulation; and P42 are omitted. LK2 is in a stormy season. (C) points for P I , , PZ4, LKI for 4 Mu.t. simulation;points for Pu are omitted. LKI is in a quiet season.
lective motion. Since 7,empis independent of chain length L and it is much shorter than the disengagement time of the ordinary linear chains, say, L = 5000, the “true” temporary LKs will not contribute to the stress and viscosity in the entanglement region. However, if temporary LKs are trapped topologically as shown
(26) Ferry, J. D.Viscoelastic Properties of Polymers, 3rd 4.;Wiley: New York, 1980.
J. Phys. Chem. 1992, 96. 41 11-41 18
of a repulsive interaction among neighboring LKs. It is found that true LKs are lost gradually in tracing their motion using probes (pairs of LCs). Three mechanisms of dissipation of true LKs are found, the merging, the multipeak, and the ghost effect. It is also found that probes make large and rapid fluctuations, which are called “probe fluctuations”. In the ordinary systems composed of long linear chains, the multipeak effect and the probe fluctuations are important. From the decay curve of the number of local maxima of T,the average numbers of true and temporary LKs per chain are estimated to be fitruc = 3.44 and rite,,+, = 3.06 in ring of L = 512. Seven kinds of probes are used simultaneously to separate the Markov motion of a single LK from the probe fluctuations. The diffusion coefficient of a true LK is estimated to be do = 0.0172 boud2/u.t. The mean-square displacement of F determined by probe PM, ( g M ( t ) )tends , not to the Markov line for but to that for do/(fitruc+ fitcmp); this means that the temporary LKs join the collective motion. The average chain length between true LKs is estimated to be L,,,, = 149, and the empirical spacing between entanglement, ne 230, which is much larger than the ne = 120-133 estimated by Skolnick et al.z’ The universal constant K is estimated to be K = 0.63, which is rather close to unity as expected. From these results, we concluded that the LK theory is well confirmed by the present simulations.
glement motion, but only through a very narrow slit. As Seen from the present simulations, the probe fluctuations increase with decrease of probe size, and in such a small probe ( L = 18) as used by Skolnick et al. the probe fluctuations must be very strong; it is therefore quite possible that the “rapid” dissipation process of the “long-lived contacts” observed in their simulations comes from the probe fluctuations which are irrelevant to the entanglement motion. They gave another estimation, ne = 120,*’based on a theoretical expression of the self-diffusion constant derived by themselves assuming Rouse-like behaviors of the entanglement motion; the assumption however contradicts not only our results but the usual t h e o r i e ~ . For ~ these reasons, the ne estimated by Skolnick et al. is not acceptable. Although our ne remains uncertain in regard to the temporary LKs, we believe our estimation of ne is based on a more firm physical foundation than that of Skolnick et al.
V. Summary The LK theories presented beforel4.l5were tested by computer simulations of entangled ring chains, in which cyclic reptation occurs. The simple cubic lattice model (SCLM) chains were used; chain lengths were L = 64, 128, and 256 in the equilibrium simulations and L = 512 for dynamic simulations. The following is found in this work: From the equilibrium distribution of Tin ring and linear chains, the average length of local chains (LCs), which form LKs, is estimated to be i j , = Kijring,where K is a universal constant near unity and ping = 119 is the average length of ring chains, which have one entanglement partner as an average. By dynamic simulations, the existence of true and temporary LKs is confirmed. The lifetime of the true LKs is essentially infinite, while that of the temporary LKs is considerably long ( T ~ = 2.3 Mu.t.) but definite. The order of true and temporary LKs along the polymer chain is conserved, which suggests the existence
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Acknowledgment. The basic idea of this work comes from early work” done by one of us (K.I.) in the research group of professor Marshall Fixman in 1972-1973. It is a great pleasure for K.I. to express his thanks to Dr. Fixman for many discussions and good friendship on his sixtieth birthday. This work is supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture, Japan.
Local Knot Model of Entangled Polymer Chains. 2. Theory of Probe Fluctuatlon and Diffusion Coefficient of a Single Local Knot Kazuyoshi Iwata Department of Applied Physics, Fukui University, Bunkyo 3-9-1, Fukui 910, Japan (Received: October 25, 1991)
The local knot (LK) theory is tested by computer simulations in parts 1 and 2. Here, theoretical problems of the simulations are mainly discussed. The probe fluctuation found in part 1 is studied theoretically, and a method for separating the Markov motion of a LK from its probe fluctuation is proposed. A detailed discussions on the mechanism of the probe fluctuation and the uncertainty principle are given. A modified expression of the diffusion coefficient of a LK is derived that cancels the interference of the probe fluctuations, and its numerical calculation is performed. A correction for short memory effects of LK motion is also done. The theoretical value of do thus computed is 0.0393 bond2/u.t. (u.t. = unit time) which is comparable to its simulation value 0.0172 bond2/u.t. obtained in part 1. Finally, it is concluded that the LK theory is proved by the results of parts 1 and 2 and, by this, a true molecular theory of entanglement has been first established.
I. Introduction In viscoelasticity, the reptation model due to de Gennes’ has been widely accepted. In its original form, this model predicts sedimentation constant D, M-2 and viscosity 7 M3,which agree roughly with expe.riments1v2 but needs various modifications to lead to more quantitative agreement.24 Although the reptation
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(1) de Gennes, P.-G. J . Chem. Phys. 1971,55, 572. (2) hi,M.; Edwards, S. F. The Theory ofPolymer Dynamics; Clarendon: Oxford, 1986.
model has succeeded to explain many viscoelastic phenomena of entangled polymers,2 its molecular origin has not yet been understood well. Recently, extensive computer simulations of entangled polymers have been performed by Kolinski, Skolnik, et al.5,6and Kremer and Grest’ to test the reptation and tube model, (3) Klein, J. Macromolecules 1978, J I , 852. (4) Graessley, W. W. Ado. Polym. Sci. 1982, 47, 67. (5) Kolinski, A.; Skolnick, J.; Yaris, R. J . Chem. Phys. 1987, 86, 7164. (6) Skolnick, J.; Yaris, R.; Kolinski, A. J . Chem. Phys. 1988, 88, 1407.
0022-3654/92/2096-4111%03.00/0 0 1992 American Chemical Society
