Breaking in polymer chains. II. The Lennard-Jones

Breaking in polymer chains. II. The Lennard-Jones chain F. A. Oliveira, and P. L. Taylor

Citation: The Journal of Chemical Physics 101, 10118 (1994); doi: 10.1063/1.468000 View online: https://doi.org/10.1063/1.468000 View Table of Contents: http://aip.scitation.org/toc/jcp/101/11 Published by the American Institute of Physics

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Breaking in polymer chains. II. The Lennard-Jones chain F. A. Oliveira International Centre of Condensed Matter Physics, Universldade de Brasilia, 70910 Brasilia DF Brazil and Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106-7079

P. L. Taylor Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106-7079

(Received 24 May 1994; accepted 1 September 1994) Extensive simulations were performed in order to determine the conditions under which an anharmonic chain will break. The dynamics of a rectilinear chain of 100 monomers interacting via a Lennard-Jones potential were followed by solving a set of simultaneous Langevin equations. There are two principal results from this study. First, in order for irreversible breaking to occur in a stretched chain, a bond must be extended to a length considerably greater than the length at which the restoring force is maximized. Second, the breaking rate of a bond may be expressed in terms of the product of an attempt frequency and an Arrhenius factor. While. the Arrhenius factor may be satisfactorily described in terms of the height of an effective energy barrier, the attempt frequency is found to be several orders of magnitude smaller than the dominant phonon frequencies. © 1994 American Institute of Physics.

I. INTRODUCTION

The problem addressed in this series of papers is that of calculating the breaking strength of a polymer fibril. In the preceding paper! some of the most fundamental questions were answered by reference to an exactly calculable model, namely a classical harmonic chain. There it was found that the average time· to breaking, 7, of a chain consisting of masses connected by breakable Hookean springs was a very sensitive function of the extension of the chain. Conversely, it was possible to define a breaking strain as the strain necessary to produce a certain average time to breaking. This strain was very insensitive to the choice of breaking time, which could typically vary many orders of magnitude without causing a change of more than a few percent in the strain. The concept of a well defined breaking strain for a single chain thus has validity even though the approach to equilibrium of a stretched chain is a statistical process. In the present paper we move one step closer to a realistic model of a polymer chain. While we continue to confine our attention to a single chain, and do not yet introduce interchain interactions, we move from a harmonic to an anharmonic model. The principal reason for this change is· the need to introduce reversibility into the breaking process. That is, the breaking of a chemical bond in an actual polymer chain may immediately be followed by the reformation of that same bond. This phenomenon was absent in the treatment of the harmonic chain given in the preceding paper. This step towards realism comes at a considerable cost. No longer will it be possible to derive exact expressions for the average time to breaking of a chain; instead we must resort to computer experiments to determine this quantity and its dependence on strain and temperature. Before this work can begin, however, it is necessary to examine the definition of a breaking time for a chain in which neighboring atoms are linked by attractive potentials of long range. On a macroscopic scale, the fracture of a fiber occurs when a large number of chemical bonds are simultaneously 10118

J. Chern. Phys. 101 (11), 1 December 1994

ruptured and an avalanche process then redistributes the applied stress among the remaining bonds, which then rapidly fail? For a single molecular chain, on the other hand, the breaking process is less· well defined. 1\vo adjacent monomers may at some instant become separated by a distance much larger than the normal bond length, and one would then say that the bond had been broken. However, the same statistical fluctuation process that gave rise to this large separation could then reverse itself, and bring these two monomers close enough for the bond to be healed and the chain to become whole again. The approach that we shall take will be to follow the evolution of a stretched chain through the breaking process, while keeping track of the maximum separation between adjacent pairs of monomers. In this way it will be possible to identify the magnitude of the bond extension at which irreversible breaking of the chain becomes highly probable. We shall find the rather surprising result that a bond extension much larger than that necessary to break a small molecule is required when the collective dynamics of a long polymer chain are taken into consideration. However, we shall also find that some aspects of small-molecule behavior are retained in the long chain, and that some expressions cast in terms of single-particle dynamics do retain some predictive power for the case of a macromolecular system. II. LENNARD-JONES CHAINS

Since the pioneering work of Fermi et al. 3 the dynamics of nonlinear systems has been a very intensive field of study, with impact in many areas_of physics. 4 ,5 ill particular, the study of anharmonic chains has led to the development of a number of concepts that aid in our understanding of the transport of energy in macromolecules. Unfortunately, some of the models that have been most fruitful in shedding light on processes of solitary-wave int~raction have not involved breakable bonds. Thus, for example, large-amplitude compressive waves can be given exact expression in the Toda

0021-9606194/101 (11 )/1 011818/$6.00

© 1994 American Institute of Physics .

F. A. Oliveira and P. L. Taylor: Breaking in polymer chains. II

chain,6 but waves of expansion are less revealing, as the attractive force between neighbors tends to a constant at large extensions in this model. In the simulations to be described in this paper we make use of the Lennard-Jones potential

10119

-0.93 , - - - , - - , - - , - - -.....- , - - . - - - , - - ,

-0.94 ""'IiIiIIiWftfl~\M~.M~ -0.95

E(t)

-0.96 -0.97

(1)

-0.98 -0.99

Like the Morse potential used by Crist et al} it is breakable in the sense that the restoring force vanishes at large distances. OUf model consists of a chain of N monomers of mass M interacting through nearest-neighbor forces determined by the Lennard-Jones potential. The equilibrium length of the chain is Na, and monomers are initially located at positions X~O) = La. We now stretch the chain to a total length of N( a + S); when each bond is equally stretched the new positions of the monomers are O) = I (a + S). We define the displacements from these positions as XI == X l - X}O). We impose periodic boundary conditions, so that Xo =XN • A Langevin equation for the classical motion of this chain will relate the acceleration of any monomer to the total forces acting on it. These will have three components. The first will be the force due to its nearest neighbors, and will be of the form F(xl-xI-l +a+S) - F(xl+l -xz+a +S) where F(x) is -dUldx. The second component is a damping force introduced to represent the interchange of kinetic energy between a monomer and its neighbors on other chains; this is assumed to take the form of a viscous force, and is written as - M 'YXI with 'Y a viscosity parameter and representing the velocity. Finally, we include a Brownian-motion term h(t) to model the interaction with a thermal reservoir at temperature T. This force is generally assumed to have the property that its value at one position and time will be uncorre1ated with that at other positions and times. Thus

xl

-1.00 L-_..l..-_....L_-1-_---1._---''--_-'--_-'---_-'

o

200

400

600

800

t

1000

1200

1400

1600

FIG. 1. Energy as a function of time for a stretched and unstretched anharmonic chain in contact with a thermal reservoir. The chain is composed of 100 monomer units interacting via a Lennard-Jones potential. Eriergy is given in units of the binding energy E, distance in units of the lattice parameter a, and time in units of 'To = 27TWOI. Here T=O.Ol25 and r =O.25wo' We use this value of r in all the simulations. (a) Unstretched (S = 0), (b) stretched (S = 0.04).

III- RESULTS

with lUO = 12~(2EIMa2). The smallest period for phonon oscillations, 'To = 2 7f1 lUo, defines the time scale for the system and hence the maximum size of the time increment !:J. t that can be used in the numerical integration of Eq. (3); we chose !:J.t = O.OOS'To. The Brownian force f was then taken to be constant during each time increment and of magnitude 7J~6M ykBTI !:J.t. with 7J random numbers distributed uniformly in the interval - 1 < 7J < 1. The initial conditions for the numerical integration were that the distributions of velocities and displacements were uncorrelated Gaussians of

The result of a typical simulation is shown in Fig. 1, in which the instantaneous average energy per monomer of the system is plotted as a function of time. As in all the results presented in this paper, energies are given in units of E, distances in units of a, temperatures T in units of ElkB' and times in units of 'To. The lower curve shows the fluctuating energy of an un stretched chain (S =0) of 100 monomers for the case where T = 0.0125. We use 'Y = 0.25lUQ and N = 100 unless otherwise stated for all the results presented in this paper. The upper curve shows the energy of a stretched chain (S=0.04) under the same conditions, and illustrates the reduction in energy that occurs on breaking as the broken ends recede to restore the chain fragments to their unstretched lengths. They do this in a time determined by the chain length, the speed of sound. and the viscosity coefficient. We verify that breaking occurs at a single bond by recording as a function of time the instantaneously largest value of YI = X 1- X 1- l ' This is shown as the upper curve in Fig. 2. The lower curve depicts the second-largest Y I, and shows that no other bond has experienced any significant extension during the breaking process. We now tum to the central problem in the analysis of breaking in the Lennard-Jones chain: What is the minimum value of Y I at which it can be said with a high degree of confidence that the chain has broken irreversibly? In Fig. 2 we see that at some point the length of one particular bond starts to arise from the noise and ascend inexorably to the vicinity of a saturation value given by the distance between the broken ends of the relaxed chain. But what is the threshold value of y at which a stretched bond becomes unlikely to be restored to its earlier size? To answer this question we have performed simulations on ensembles of identical chains and recorded the times at which Y max' the largest of the Y I, first exceeds a threshold value d. For brevity we refer to this event as a break. As long

width appropriate for the temperature under consideration.

as the breaks are infrequent, the time-in variance of the equa-

x

(2) with kB the Boltzmann constant. The equation of motion for the lth particle is then Mx/=F(x/-XI_l +a+S)-,F(XI+l-xl+a+S)

(3) In the absence of damping, the small-amplitude modes of oscillation of the unstretched chain are phonons of wavenumber q and angular frequency lU q =

lUosin(qaI2)

(4)

J. Chern. Phys., Vol. 101, No. 11, 1 December 1994

F. A. Oliveira and P. L. Taylor: Breaking in polymer chains. "

10120

3.0 0.30

2.5 2.0

0.20

YI(t)

1.5 1.0

0.10

0.5 0.00 L...I-_...I-_...L..._...L...-=::r::==*=_-L..._-L::::""....J 0.03 0.04: 0.05· 0.06 0.07 0.08, 0.09 0.10

0.0 0

100

200. 300

400

500

t

600

700

800

900

1000

S

FIG. 2. The evolution of a breaking process in a chain. The upper line shows as a function of time the instantaneously largest bond elongation YI = XI- XI-I' The parameters are T = 0.05 and S = 0.03. The lower curve depicts the second-largest YI' This shows that only one bond has shown significant extension during the breaking process.

tions of motion tells us that the probability per unit time that an unbroken chain will break will be constant. The number m(t) of unbroken chains in the ensemble will thus decay as met) = m(O)exp(-t/r), and we refer to r as the characteristic breaking time for the chain for this particular choice of d.

Before presenting the results of the simulations we first put the calculation in context by discussing the nature of the Lennard-Jones force. This force is long-range in nature, representing as it does the van der Waals interaction, and so exerts an attraction between monomers even after a break. It does, however, decay rapidlY after the maximum force, F max = 2.69, is reached at an inter-monomer separation of ( 13/7) 116 = 1.109. At zero temperature a chain would thus simultaneously fail at all bonds at an extension of S max == 0.109, and so the breaking threshold would then be d=O. This is illustrated in Fig. 3, in which curve (a) is the Lennard-Jones potential Uex). Curve Cd) represents U(a + Smax+ x) - xFmax' and is the effective potential in the critically stretched chain at zero temperature. A more sophisticated argument to estimate d might point out that the restoring force caused by the Lennard-Jones potential is still appreciable after the maximum value has been

FIG. 4. Parameters for the effective potential as functions of strain S. Here N = 100. (a) Position d p of the maximum of Ueff..x); (b) energy barrier

Eb •

passed at d = 0.109. A better cOJ;ldition for d might be found by arguing that irreversible breaking will occur only after this restoring force has been reduced to lie below the average tension F in the chain, which at zero temperature is -dU(a+S)/dS. The form of the effective potential U(a+S+x)-xF is shown as curve (c) in Fig 3. The maximum of this curve locates the predicted breaking extension d p ' and so we have as our condition dU(a+S) _ dU(a+S+d p) dS. dS

(5)

Finally, we note the modification that occurs in the effective potential when the chain is of finite length, and so we impose a condition of constant strain rather than constant stress. Then a chain of N particles in which one bond is stretched by an amount d while the other N - 1 bonds relax has a potential energy of Uerl..d)=U(a+S+d) +(N-l)U( a+S-

N~ 1)' (6)

The new condition for the breaking extension d p is then dU(a+S-dp/(N-l) _ dU(a+S+d p ) dS dS

(7)

This modification of Eq. (5) is significant in the model studied, as illustrated for the case of N = 100, in Fig. 3(b), which is perceptibly displaced from the infinite-chain result of Fig. 3(c). At smaller values of S the difference becomes significantly larger. The effective energy barrier to be overcome in order for the extension to reach d p will be Eb= Uerl..d p )

-

Uerl..O).

(8)

For an infinite chain this reduces to -0.4

0.0

0.2

x

0.4

0.6

FIG. 3. The effective potential Ueff..x). For simplicity the curves are drawn to coincide at their minima. (a) S = 0, N -+ 00 (unstretched chaill); (b) S = 0.035, N= 100; ec) S= 0.035, N -+ 00; Cd) S = 0.109, N ..... co (critical extension).

EbOO}=U(a +S+dp ) - U(a+ S).

(9)

Equations (5) and (7) suggest that the extension d that characterizes true breaking will be a function of the d~gree to which the chain is stretched. A plot of dpCS) for N = 100 along with a plot of the height Eb of the effective energy barrier is shown in Fig. 4. For an extension of S = 0.039, we find d p = 0.21. We shall see from the simulations, however,


10121

F. A. Oliveira and P. LTaylor: Breaking in polymer chains. II

900

a

700

5.5

b r

In(m(t))

.100 300

c

4.5

100

o

20

40

80

100

t

120

140

160

180

0.0

200

FIG. 5. Natural logarithm of the number mCt) of surviving chains as a function of time for an ensemble of 500 chains of 100 particles each. Here T "'0.05; S '" 0.039. The vertical lines indicate the limits of the steady-state region after thermalization and before too few chains remain for good statistics. The slope of the straight-line fit gives the characteristic time T. From left to right d = 0.42; d=O.69; d =0.96. This corresponds to T =41; 61; 69, respectively.

that this value also underestimates the actual breaking extension at non-zero temperature. The oscillatory nature of the forces acting within the anharmonic chain evidently allows rapid healing of the incipieht breaks that occur when d exceeds the values of breaking extension predicted from the static theory. Figure 5 illustrates the result of simulations performed on an ensemble of 500 chains of 100 monomers with S = 0.039. It shows the natural logarithm of the number of surviving unbroken chains at time t for three different threshold criteria. After an initial transient period the decay becomes exponential as expected. Less expected is the fact that the slopes of these .three lines are markedly different, even though the thresholds d chosen are several times d p • For d = 0.42 the inverse slope (i.e., the characteristic breaking time, r) is 41, while for d = 0.96 it is 69. We explore this phenomenon further by plotting in Fig. 6 the characteristic time r (found from the reciprocal of the slopes of plots similar to Fig. 5) as a function of the threshold extension d at various temperatures and for S = 0.035. To be meaning~l, the breaking time must be independent of the choice of d. Inspection of Fig. 6 shows that this requires

1.0

0.5

1.5

d

FIG. 7. The characteristic time T as a function of the distance d for fixed temperature T = 0.05 and various strains. (a) S= 0.03; (b) S = 0.0325; (c) S = 0.0345:

us to choose a threshold value d t of d sufficiently large that it lies on the plateau region on the right of the figure. The surprising aspect of this result is the very large value of d required for us to be able to say that irreversible breaking has occurred. For S = 0.035, the value of d t is comparatively insensitive to temperature changes, and is about 0.8, while d p = 0.23. At this extension of a bond, the restoring force. is less than a tenth of its maximum value, and the potential energy is only a little greater than 1120 of its maximum value. In Fig. 7 we show the variation of 7 with d at T = 0.05 for various values of the extension S. When S = 0.03, we must choose d t to be about 1.2, at which range the restoring force and potential energy are both only 1.7% of their maximum values! It is now clear that the collective oscillatory nature of the forces in the chain acts to stabilize the unbroken structure, and that the healing of incipient breaks is very efficient. Thisqealing effect is most pronounced at small applied strains. Now that the breaking process has been clearly defined in terms of the d-independent characteristic time 7, one can examine the dependence of 7 on the temperature and strain S. Figure 8 shows the variation oflne 7) with 13 (13 = 11k B T) for S = 0.035, and has the approximately linear form expected from elementary ideas of thermal activation over a barrier. We might then expect to find a result similar to the usual Arrhenius expression for a single particle in a single

1600 1400

7.5

~-----r---'~I---'

1200 1000 T

800

b

6.5

600

In(r)

400 C

200 5,5

0.5

1.5

1.0

2.0

2.5

d

FIG. 6. The characteristic time T as a function of the distance d for fixed strain S = 0.035 and various temperatures found from an ensemble of 800

FlG. 8. Natural logarithm of the breaking time as a function of the inverse

chains of 100 particles. (a) T= 0.025; (b) T = 0.030; (c) T=

temperature {3 for S == 0.035.

oms.


10122

F. A. Oliveira and P. L. Taylor: Breaking in polymer chains. II ,------,----,--'--'---,-----,----,-,-"

-0.6 r----,r---r---r--,----,--..,--,..--,-----,

0.07

-0.8 -1.0 -1.2

(a)

0.05

ECd)

.In(v)

0.03

-1.4 -1.6

~tiI------~..__---------i

-1.8 -2.0 -2.2 -2.4 -2.6 L---'_-'-_--'-_--"-_-L_-'-_-"--_....1..----l 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

(a)

0.01 ~~~

__

~

0.03

0.01

__

~

____ --L-____

0.07

0.05

~~

0.09

d

d

0.16 -1

0.14 0.12 0.10

,. \.

-3

ECd:) 0.08

(b)

In(v)

(b)

0.06

·5

0.04 0.02

·7

0.2

0.4

0.6

0.8

. 1.0

1.2 0.0

d

FIG. 9. The effective potential barrier E(d) as a function of the distance d for S = 0.035. The solid line shows data obtained from several experiments like those in Fig. 8. (a) Small d. The dotted line shows the theoretical construct UeI!..d). (b) Medium and large d. The smooth lirie is a polynomial fit to the data.

well, namely 'T- 1 "'" voe- PE , (here vo=wol2'Tr), which for N independent wells would become 'T - [ "'" N voe - PE. Instead we find Fig. 8 to be closer to the form 1

-=Nve- f3E (d)

(10)

'T

with v "'" 10- 3 Vo and E(d) .... 0.12. The attempt frequency v thus appears to be only a small fraction of the typical vibrational frequency vo, and the apparent barrier E(d) is only a tenth of the actual depth (unity) of the Lennard-Jones potential well of a single unstretched bond. To examine this manifestation further we look at the transition from harmonic to anharmonic behavior by returning to the cases where d is chosen to be below the threshold value d t at which 'T becomes independent of d. Because "breaking" is said to occur when some bond length Yl exceeds d we expect 'T to be close to the result found previously[ for the undamped harmonic chain, namely I

Vo

-=N r,:;exp(-tj3K(S)d 2 ), 'T y2 ~

(11)

where K(S) is the strain-dependent spring constant of the chain, which in our case is 32.9 at S = 0.035. The attempt frequency is then predicted to be the root-mean-square phonon frequency and the effective barrier height to be 16.5 d 2 •

The results of our simulations are indicated in Figs. 9 and 10 for three regions of d, the intervening regions being omitted for economy in computation. Because the characteristic breaking time retains its Arrhenius behavior we can plot the apparent potential barrier ECd) and the logarithm of the

0.2

0.4

0.6

0.8

1.0

d

FIG. 10. (a) Natural logarithm of the attempt frequency 11 as a function of the distance d for small d. The lower horizontal line is the average value in the range 0.01 < d < 0.02, the upper line is the prediction of the harmonic approximation, and the middle line is from reaction rate-theory. (b) Natural logarithm of the attempt frequency as a function of the distance d for large d, showing that 11 does not become independent of d until d>0.6. The dotted line is a polynomial fit to the data.

attempt frequency vas functions of d. At the smallest values of d [Fig. 9(a)] harmonic behavior is displayed in the interval 0.01 < d < 0.02. For the intervals, 0.01 < d < 0.02, and 0.05 < d < 0.10, the experiments are consistent with E(d) = Ueft{d) as defined by Eq: (6). As d increases beyond 0.02, this quadratic dependence of E(d) on d then changes as shown on Fig. 9(b), until it saturates when d > d t , with d t "'" 0.7. The attempt frequency v displays more interesting behavior. Although the harmonic theory suggests we should find v to be independent of d, the simulations show an extremely rapid decrease with increasing d when d>O:05. For small d, we find that, apart from some fluctuations, this value tends to a constant, v = 0.23 ±, 0.03, as shown in the lower line of Fig. lO(a). The top line is the value predicted by harmonic theory, 1 namely v = .JK(S)/(2K(0». The factor .JK(S)/ K(O), is introduced here to take into consideration the change in frequency with stretching, and is absent in the harmonic chain. Finally the middle line is the result of the reaction-rate theory to be discussed later in Sec. IV. The full range of d is shown in Fig. lOeb). The magnitude of v is seen to drop by nearly 3 orders of magnitude as d increases. In the range 0.05 < d < 0.40 the decay appears exponential in d; it reaches its saturation value for d > 0.8. Finally we indicate in Fig. 11 the results of a series of simulations at fixed temperature (T = 0.05) but with varying chain extension S. As expected, the characteristic breaking time 'T changes rapidly with S. For example it decreases by


F. A. Oliveira and P. L. Taylor: Breaking in polymer chains. II

10123

maximum. The average rate at which a particle irreversibly crosses the barrier (i.e., the average rate for a bond to break) is given, according to Kramers' theory13,14 by

7.5 7.0 6.5

In(r)

6.0

(12)

5.5

Here the Kramers attempt frequency

5.0

VK

is given by

4.5

(13)

4.0 0.030

0.032

0.034

0.036

0.038

S

with

FIG. 11. Natural logarithm of the breaking time r as a function of the strain S for T = 0.05. The line is a quadratic fit to the data.

an order of magnitude (more precisely from about 950 to 70) as S is increased from 0.030 to 0.039.

IV. DISCUSSION No analytical theory capable of accounting for all the richness of this problem is available at present, and so we try to develop a simple picture of the phenomena as a stepping stone to a more complete theoretical explanation. The existence of a well-defined activation energy for small d suggests that a simple effective one-particle picture may be able to describe the main characteristics of the system. Indeed such a potential was already suggested by Eq. (6), in which the breaking configuration of minimum energy was identified as that in which one bond length increases while all the other bonds relax equally. For S = 0.035 and N = 100, this potential was plotted as Fig. 3(c). The activation energy Eb=0.14 of this potential agreed closely with that found by the simulations in Fig. 9(b), and thus leads us to ask the extent to which an effective potential of this type can be useful in predicting results in a strongly interacting manybody problem. We consider a single particle of effective mass f.L moving in a potential like Eq. (6), and in contact with a thermal bath. For our purposes f1- may be identified with M 12, since in the chain the mass M is acted on by its two neighbors. It is possible to make some estimate for the rate of transition from a metastable position to a position d t outside the barrier, where d t ~ d p • This hundred year old problemS is still a very active area of study in reaction-rate theory,9-15 and incorporates a large class of phenomena including tunneling in Josephson junctions lO and some biological processes. ll ,12 The first systematic approach to the problem is due to Kramers,13 who proposed that the main characteristics of a potential with a barrier could be described in a harmonic approximation at the minimum and the maximum of the potential, namely

)l.b=

~w~+ !I:- h.

(14)

The factor lIwo in Eq. (13) was introduced in order for v to be given in units of vo. For S = 0.035, Wa = 0.478wo, wb= 0.211wo, and vK = 0.273 for the effective potential. If we compare these values with the results of our simulations for large d (d;:.d t ) we see that while ECd) ... E b , the frequency v may be 350 times smaller than VK' The discrepancy in predicted and observed values of v indicates that an important ingredient is missing from our attempt to describe breaking in terms of an equivalent one-particle problem. This ingredient must necessarily involve the correlations between motions of adjacent particles, such as those discussed by Florencio and Lee 16 for a harmonic chain. If we are to include the effects of the collective motion in the equivalent one-particle model, then it will be necessary to include the time scale of the vibrational modes. To achieve this we generalize the frictional constant y to be non-local in time, and hence give it a memory of previous motion by writing it as yCt - t'). In terms of its Laplace transform 5'(z), the quantity )l.b in Eq. (14) will become a function of z, and we find the self-consistent-equation )l.b = )I.()l.b), where (15)

In our case, f.LW~ is the second derivative of Eq. (6) with

From Eq. (13) we thus see that VK also becomes a function 'of z; we refer to this new value as Vb' The value of )I.(z) given by Eq. (15) may differ substantially from the static one of Eq. (14), but even this adjustment may not be enough to account for the discrepancy observed in the values of v. A further factor that may bring theory and simulation closer lies in the fact that the reaction rate obtained by the Kramers theory is really only an upper bound. ll ,15,17 Recently Pollack et ai.,ls have presented a lower value of this upper bound for a particle moving in a potential with a barrier, and with a memory yet - t') of its classical trajectory. They went beyond the Kramers harmonic approximation of the particle potential [which in our case is given by Eq. (6)] and kept the anharmonicity up to the fourth order. The upper bound they obtained shows a strong coupling between the anharmonicity in the potential and the memory of the particle. The main result of their work is the introduction in the reaction rate of a corrective factor, P, which is always smaller than 1. The function P is strongly dependent on the factor {3E b , and on the variation of 5'(z) with z. It may

respect to x at the local minimum, and Wb is evaluated at the

range from unity down to a number some orders of magni-

tf.L. w~x2, for x'" 0

Uerix) = { Eb -

t!.LW~(x-dp)2, for x=dt •


10124

F. A. Oliveira and P. L Taylor: Breaking in polymer chains. II

tude lower. In our specific case it does not seem to be small enough to reduce the predicted value of v to lie inside the range found in our numerical simulations. Without a better understanding of y(z), and of the necessary corrective factor for our potential (which includes anharmonic terms of higher order than the fourth), it becomes unprofitable to pursue these arguments in the search for the explanation for the low value of v found in the numerical experiments. However, the ideas presented here suggest that it may not be impossible to find some predictive power in an effective one-particle theory for the problem. Finally, we look at the application of our results to single molecular chains of two commercially important polymers, polyethylene and the aramid poly(p-phenylene terephthalamide), or PPTA. The energy parameter € of the LennardJones potential was chosen to be 360 kJ mol- [ as being roughly equal to the dissociation energy of the C-C and C-N bonds in similar molecules. 18 Here we face the problem that in order to describe the bonds in a real chain we need at least three parameters, namelY the equilibrium bond length, the breaking energy,. and the second derivative of the potential near equilibrium. The Lennard-JOIies potential, on the other hand, contains only two parameters, € and a. We solve this problem by ceasing to assume that the length parameter a of the Lennard-Jones potential is "the monomer length (or lattice spacing) of the polymer. We thus now consider a to be a length parameter describing the elastic modulus of the chain at small extensions, and distinct from the lattice spacing ao. That is, the spring constant K of the Lennard-Jones bond is found from the second derivative of the potential in Eq. (1), and is equal to 72€/a 2 . The Cerius 19 software package was used to deterinine the spong constants of PE (considered as a chain of CH2 units of actual spacing ao = 0.13 nm) and PPTA (considered as a chain of C6H4 NHCO units of spacing ao = 0.63 nm) as 280 N m -1 and 84 N m - 1 respec~ tively. The dynamics of the PE chain of spacing ao are thus equivalent to those of a Lennard-Jones chain of spacing a when 72EI a 2 = 280 N m -1, or a = 0.39 nm, a number different from its actual spacing, ao. For PPTA a siIpilar calculation gives a = 0.72 nm. (While the Lennard-Jones PO-" tential is not an ideal choice to model a planar zigzag chain like polyethylene, as, the potential of a stretched chain is a combination of bond~angle and bond-length distortion terms, it is an adequate fit for the order-of-magnitude estimate that we are seeking.) We are now in a position to relate the results of the simulations to the cases of PE and PPTA, although with one severe limitation. A temperature of 300 K corresponds to a thermal energy of only about 7 X 1O~3"in units of E (i.e., /3= 150) and so lies outside the range of the computer simulations that we have performed, which were confined to {3