- Email: [email protected]
The role of the torsional potential in relaxation dynamics: a molecular dynamics study of polyethylene
Computational
and Theoretical Polymer Science Vol. 8, No. l/2, pp. 93-98, 1998
0 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0189-3156/98/$19.00 + 0.00
PII: SlOSS-3156(98)00020-S
The role of the torsional potential in relaxation dynamics: a molecular dynamics study of polyethylene Richard H. Gee Department
of Chemistry Snow College, Ephraim, UT 84627,
USA
and Richard H. Boyd” Department
of Materials
Science
and Engineering
and Department
Engineering, University of Utah Salt Lake City UT 84112, (Received 2 October 1997; revised 9 January 1998)
of Chemical
and Fuels
USA
The role of the torsional potential in bulk polymer chain dynamics is investigated via molecular dynamics simulation using polyethylene as a model system. A number of three-fold barrier values, both greater and less than the standard one, were invoked. The one-fold potential that determines the gauche vs tran~ energy difference was also varied. For each of the selected torsional potentials, the MD volumetric glass transition temperature, Ts, was located. It was found that Ts is quite sensitive to the three-fold barrier magnitude, moving from below 100 K to nearly 400 K as the barrier goes from zero to twice the standard value. However Tg was found to be quite insensitive to the gauche trans energy difference. Details of the conformational dynamics were studied for the case of a zero torsional potential. This included the rate and location of conformational transitions, the decay of the torsional angle autocorrelation function (ACF) and the cooperativity of conformational transitions, all as a function of temperature. The temperature dependence of the conformational transition rate remains Arrhenius at all temperatures. The relaxation time characterizing the torsional angle ACF decay exhibits WLF temperature behavior. The conformational transitions are randomly distributed over the bonds at high temperature, but near Tg they become spatially heterogeneous and localized. The transitions show next-neighbor correlation as well as selfcorrelated forward-backward transitions. All of these features are similar to those found in previous simulations under the standard torsional potential. 0 1998 Elsevier Science Ltd. All rights reserved. (Keywords: glass transition temperature; torsional barrier; molecular dynamics)
INTRODUCTION One of the appealing features of molecular simulations is that structure versus property relationships can be explored by manipulating input parameters in the simulation. If, for example, one wished to know the relative effect of polar forces on a property, molecular dipole moments or partial charges could be varied in a simulation. One of the enduring structure property questions in polymer science is the relationship of the glass transition temperature (a,) to molecular features. An aspect that comes to mind is the ease and kind of conformational transitions in the polymer chain or, as put another way, what are the relative roles of the barrier to internal rotation and the inhibiting effect of the surrounding matrix of chains. In some *To whom correspondence should be addressed. Tel: 001 801 581 6865; Fax: 001 801 581 4816; e-mail: [email protected]
favorable cases at least, a volumetric glass transition can be easily located in molecular dynamics (MD) simulations’-‘. Polyethylene (PE) is one of these cases5-7 and it is invoked here as a model for exploring the role of the torsional barrier in determining the conformational dynamics and the attendant glass transition temperature. The plan is a simple one. The intrinsic torsional barrier, V(4), used consists of a three-fold term with parameter Vs that determines the barriers and a one-fold term with parameter Vi, that determines the gauche versus trans energy difference, as indicated by equation (I), IQ> = ;olWl
+ cos(34)] + $Wt[l
+ cos($)]
(1) The potential as written here also contains two scale factors, one (a) for the three-fold barrier and
COMP. AND THEOR. POLYMER SCIENCE
Volume 8 Number l/2
1998
93
perature
in runs of several
: 20 a E 15 % 2 10 s' 5 0 0
60
120
i80
240
300
360
otenkial function parameters are the same d there also ex~3pt for the scaling of Ihe
$ Figure 1 The torsional potentials selected for the study of their effect on Ts (a) The four potentials in which the three-fold barrier is modified by the scale factor shown (a in equation 1). The one-fold term is unmodified @= 1). (b) The potentials in which the one-fold term is modified by the scale factor shown (9 in equation I). The three-fold term is unmodified (IX= 1)
variation is be expected to be a ~t~~~t~~a~ one rather than a directly a dy~a~~~ one. The barriers but ween eonforrnations are not sensibly gauche tvam population ratio will ged chain extension or and hence the equilibriu tio will be also. In a 0 the in Figure 1, the value set 0 is included, that is, the case of a zero torsiona? potential. For each selected scale factor set, the s dete volumetric nation, etails of In additi e a, space exploration are studied for the case of zero torsional results are compared to formational transitions urnd tial ~cq/?= I>. Tabb
1
Potential
e iraertial term were the same as
es near the en
T g essns in the thr
ature results obtained for the arrier, _F&ureI(a): are results
for zero torsional
function?
Function C-C bond stretch energy = l/2 kR (R - R,# Bond bending energy = J/2 k8 (8-B&CHT (PE) Torsional potentialb = c~/2 Vs(1+ cos 34) + /J/2 Vr(l i CQS#) AUA nonbonded potential, Lennard-Jones 6-12c-CHz-
Constants kR =663 ko=482
V,= 13.4 E = 0.585 a=3510
Re = I.54 0#)=!31.5
Vr =3.35 &in = 3.940 cd= 6.42
aEnergies in kJmol-‘, distances in A, angles in radians (shown in degrees). The potentials are from Ref. 5. bThe a and ,f?parameters are multiplication factors introduced to modify the torsional potential energy barrier. CFor the CHZ group the potential is of the ‘AUA’ (anisotropic united atom) type, the interaction center is offset form the C atom by the distance = d along the bisector of the C-C-C angle in the direction of the hydrogens; E is the well-depth; R,,,, the distance at the minimum; is the corresponding distance at the energy zero. The nonbonded potential is invoked between centers of 1-5 spacing and further (4 bonds) as the torsional potential explicitly represents the 14 center (3 bond) interaction via the V, term.
The role of the torsional potential in relaxation dynamics: R. H. Gee, R. H. Boyd
350
t
300 250 7
1.25
f
1.20 >
TE 200 150
/
1.15
100 r 50 t 01 0
0.5
1
2
1.5
a
Figure 4
Figure 2 Volume vs temperature curves (at 1 atm.) from MD simulation for the four torsional potentials of Figure 1 (a) in which the three-fold barrier is varied by the scale factors shown. In addition, the case of zero torsional potential is included. Vertical dashed lines mark breaks in slope indicative of the volumetric glass transition
potential (a,/?= 0) are included as well. The V-T curves for the one-fold variations, Figure I(b), are shown in Figure 3. Breaks in the slopes of the curves in both figures indicative of vitrification are found. These MD Tg values are indicated by dashed vertical lines from the temperature axis. Straight line segments indicating above and below Tg slopes are also drawn. It is apparent from Figure 2 that the volumetric Tg depends strongly on the three-fold potential. As expected, the Tg values increase with increasing three-fold barrier. This dependence in Figure 2 is plotted as Tg vs the barrier scaling factor in Figure 4. It is of interest to note that the simula-
1.35
1.25 _y 1.20 mg s 115 1.10 1.05 1.oo
0.90
” 0
”
” 100
”
” ‘I/“’ 200 T
” ” 300
”
400
”
” 500
W
Figure 3 Volume vs temperature curves (at 1 atm.) from MD simulation for the four torsional potentials of Figure 1 (b) in which the onefold barrier is varied by the scale factors shown. Vertical dashed lines mark breaks in slope indicative of the volumetric glass transition
The MD volumetric
Ts vs the three-fold
barrier
scale factor
tions show the presence of a finite Tg value even for zero torsional barrier. It is also to be noticed that in spite of the variation in T,, the slopes of the V-T curves are nearly the same above Tg. The relation between Tg and the barrier is roughly a linear one. It suggests, apart from the intercept effect at zero barrier, that Tg divided by barrier height might be a useful reduced variable for correlation of glass temperature with structure. In contrast to the three-fold barrier case, Tg varies very little as a function of the one-fold potential (Figure 3). This indicates that the rate effects on conformational transitions controlled by the threefold term are much more important in determining T, than the structural effect of variation in trans gauche ratio. CONFORMATIONAL
TRANSITIONS
A conformational transition is conveniently defined as a transition between a value of a torsional angle in the vicinity of one of the minima in the torsional potential to another value at one of the other minima. Transitions can occur between any of the three local states, tram (t) and gauche (gf , g-). In previous work on bulk PE7, it was found that the temperature dependence of the conformational transition rate was Arrhenius in character with an activation energy close to the barrier in the torsional potential. This circumstance persisted even below the volumetric glass transition. However, the autocorrelation function (ACF) associated with values of the torsional angles had contrasting behavior. At higher temperatures, well above the MD Tg, the torsional angle ACF behaved in a similar manner to the conformational transition rate with an activation energy close to the barrier value. As temperature approaches Tg, the torsional angle ACF vs l/T plot diverges from that of the transition rate, falling off in response in Vogel-Fulcher or WLF fashion.
COMP. AND THEOR. POLYMER
SCIENCE
Volume 8 Number ‘I /2 1998
95
ome spatiaUy
heterogeneous. Some bonds have far more transitions ess than given by the Poisson statistic23 with random di$~~i~~~i~~~ The few transition rake have bonds with hig their aver~~~~~~~~~~~nto he rehxation of the s
ere we have carried out a sin&u me case of zero
torsional
analysis potential.
for In
the eonventioPd of counting a transition as a $0, 180 or 300 degree torsional angle
ere f$Qt>is the value
expressed
of the torsimal angle at lue at time=0 and :’ e averages. It was co
as ~~a~~~~ions per bon e relaxation
3.0
time,t,
re shown in F&m2
as a
function
4 along with
2.5
2.0 2 ‘2 B !z .g v
t 1.30
1.5
0.30 -1.00
1.0 2 a ;=
2 0.5
F 7
-2.30 -3.30 -4.00 -5.00 -6.00
-ask 0
2
4
J
’
6
8
’
10
12
1000/-r(K) Figure 5 The temperature dependence of conformational transition rates (‘flip rate’ on ordinate). Open circles are for zero torsional potential. Filled triangies are for the unmodified (CI= 1, fi = 1) torsional potential (the latter date is from Ref. 7). Activation energies are also showi
-7.00 0
2
4
6 1000/T
8
10
!2
Figure 6 The temperature dependecce of the stretched exponential rekxation times, equation 3, for the torsional angle autocorrelation function. Open circles are for zero torsional potentia:. Filled triangies are for the unmodified (CX = 1, B = 1) torsional potential (the latter data is from Ref. 7)
The role of the torsional potential
T (K) Figure 7 The j3 stretching parameter, eq. 3, for the torsional autocorrelation function. Same symbols as in Figure 6. Lines are linear regressions
the zero torsional potential case. However, it is of great interest to note the temperature dependence is still WLF in nature. The /I stretching parameter vs temperature is shown in Figure 7. The values are somewhat greater for the zero barrier case. This could be expected on the basis of a more dynamically flexible chain. Spatial heterogeneity of transition rates
The 4000 transitions that were generated for the zero potential case at each temperature were cataloged according to the sequential bond number at which they occurred. The occurrence plots at
0
200
300 Bond?&nber 500
600
700
800
Figure 8 Spatial heterogeneity of conformational transition location for the zero torsional potential case. At each temperature there are 4000 total transitions. The number occurring at each bond, as numbered serially along the polymer chain, is shown. The standard deviations, cr, about the average number of transitions per bond (4000/ 765 = 5.23) are also shown
in relaxationdynamics: R. H. Gee, R. H. Boyd
100 K, 200 K and 300 K are shown in Figure 8. It is apparent qualitatively that at 100 K, a temperature close to the volumetric Tg, the distribution of transitions has become very uneven. More quantitatively, for a random occurrence of transitions over the bonds a Poisson distribution should prevail. For the latter, the standard deviation in distribution should be X112where X is the average number of transitions per bond (X=4000/ 765 = 5.23; X1i2=2.3). It may be seen in Figure 7 that the Poisson value is approached at 200 K and 300 K but at 100 K the deviation from randomness is considerable. The situation is very much like that found for the unaltered torsional potentia17. Thus the phenomenon of onset of dynamic heterogeneity as vitrification sets in appears to be a general one associated with chain dynamics in bulk systems. Cooperativity of transitions
Another important conclusion concerning conformational dynamics in bulk PE1A,7, and in common with results found in Brownian dynamics simulations12J3, was that cooperative transitions are common. That is, a concerted motion occurs where after one bond changes conformation, a second nearby bond follows by changing its conformation. Considerable correlation involving the next-neighbor bonds was found. These can be classified into types such as gauche migration, same sign gauche pair formation, kink formation and kink inversion7T13,14. In addition, self-correlated transitions were common7. These involve a bond that has already jumped then undergone a back transition. A convenient and simple way to demonstrate correlation is to keep a list in temporal order of the bond numbers where transitions have occurred. From this list, after a transition at bond, i, the next bond to jump, at, j, within a fixed window of &tN bonds can be found. Then, from the total number of transitions within the window, a probability, P(i*j), that following a jump at i, a jump within the window at &j can be constructed. Correlation is easily discerned in a plot of P(iij) vsj as a deviation from the uniform background of random i *j jumps. Such a plot is shown in Figure 9 for the zero torsional potential case. It can be seen that considerable + 2 next-neighbor correlation is present. The types of next-neighbor correlation are the same as found previously7 for the standard torsional barrier and mentioned earlier. Also present is j= 0 self-correlation. Unlike the f 2 correlation, the self-correlation is highly temperature dependent, becoming relatively much more prominent at low temperature. This was observed previously for the unaltered potential7 and appears to
COMP. AND THEOR. POLYMER
SCIENCE
Volume 8 Number l/2
1998
97
ence
of the
relaxa-
0.35
eminence
0.25
of 0,
0.10 V
1
0
12
I
3
Correlation
4
5
position
6
I
I
7
8
(i&j)
Figure 9 Cooperativity of transitions for the zero torsiona! potential case. The probability that a jump at bond i is followed next by a jump at *i wherei is restricted to window of f 8 bonds. Temperatures are indicated
be related to the onset of spatial ~ete~o~ene~ty in transition location (Figure 8). The transitions at the high rate locations tend to be of the forward and back type.
It is apparent from these results that t very sensitive to the glass transition temperature arrier. Thus, the the magnitude of the torsionaj urely kinetic rate effect set by the % bmportant one in determining
enius temperature dependence of the confo~~~ti~~a~ t~a~§iti~~ rate,
J., J. Chhem. Phys.; 1987, 1. J., 9. CYrem. Yhys., 1988, 2. 1989, 22, 2259. 3. Rigby, D. and Roe, R. J., M~cromoiec&s, 1990. 23. 5312. 4. Riabv. D. and Roe. R. J.. Macromolecules. 5. Woe, R. J., in Atom& hkodeling of Physical Pro&&s, Advances in Polymer Science, Vol. 116, eds L. Monnerie and U. Suier. Springer, 1994, I1 l-144. 6. Pant, P. V. K., Han, J., Smith, G. D. and 99: 597. Gee, R. H., Han, J. and Jin, Y., S. Chm. Whys., I. ;;s. ee, F.. H. and Boyd, R. 8. 9. 10. 11. 12. 13.
Toxvaerd, S., 1. C&m. Phys., 1 Nose, S., 1. Gem. Phys., 1984, Takeuchi, H. and Ok&&i, K., J. Chem. P&s., 1990,92, 5643. Weber. T. A. and Helfand. E.. _C Chew. Phvs., 1979. 71. 4760. Helfa&, E., Wasserman, a. k and Web& ?., Mocromokcules, 1980, 13, 526. 14. Welfand, E., J. Cham. Phys., 1971, 54, 4651.
and Theoretical Polymer Science Vol. 8, No. l/2, pp. 93-98, 1998
0 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0189-3156/98/$19.00 + 0.00
PII: SlOSS-3156(98)00020-S
The role of the torsional potential in relaxation dynamics: a molecular dynamics study of polyethylene Richard H. Gee Department
of Chemistry Snow College, Ephraim, UT 84627,
USA
and Richard H. Boyd” Department
of Materials
Science
and Engineering
and Department
Engineering, University of Utah Salt Lake City UT 84112, (Received 2 October 1997; revised 9 January 1998)
of Chemical
and Fuels
USA
The role of the torsional potential in bulk polymer chain dynamics is investigated via molecular dynamics simulation using polyethylene as a model system. A number of three-fold barrier values, both greater and less than the standard one, were invoked. The one-fold potential that determines the gauche vs tran~ energy difference was also varied. For each of the selected torsional potentials, the MD volumetric glass transition temperature, Ts, was located. It was found that Ts is quite sensitive to the three-fold barrier magnitude, moving from below 100 K to nearly 400 K as the barrier goes from zero to twice the standard value. However Tg was found to be quite insensitive to the gauche trans energy difference. Details of the conformational dynamics were studied for the case of a zero torsional potential. This included the rate and location of conformational transitions, the decay of the torsional angle autocorrelation function (ACF) and the cooperativity of conformational transitions, all as a function of temperature. The temperature dependence of the conformational transition rate remains Arrhenius at all temperatures. The relaxation time characterizing the torsional angle ACF decay exhibits WLF temperature behavior. The conformational transitions are randomly distributed over the bonds at high temperature, but near Tg they become spatially heterogeneous and localized. The transitions show next-neighbor correlation as well as selfcorrelated forward-backward transitions. All of these features are similar to those found in previous simulations under the standard torsional potential. 0 1998 Elsevier Science Ltd. All rights reserved. (Keywords: glass transition temperature; torsional barrier; molecular dynamics)
INTRODUCTION One of the appealing features of molecular simulations is that structure versus property relationships can be explored by manipulating input parameters in the simulation. If, for example, one wished to know the relative effect of polar forces on a property, molecular dipole moments or partial charges could be varied in a simulation. One of the enduring structure property questions in polymer science is the relationship of the glass transition temperature (a,) to molecular features. An aspect that comes to mind is the ease and kind of conformational transitions in the polymer chain or, as put another way, what are the relative roles of the barrier to internal rotation and the inhibiting effect of the surrounding matrix of chains. In some *To whom correspondence should be addressed. Tel: 001 801 581 6865; Fax: 001 801 581 4816; e-mail: [email protected]
favorable cases at least, a volumetric glass transition can be easily located in molecular dynamics (MD) simulations’-‘. Polyethylene (PE) is one of these cases5-7 and it is invoked here as a model for exploring the role of the torsional barrier in determining the conformational dynamics and the attendant glass transition temperature. The plan is a simple one. The intrinsic torsional barrier, V(4), used consists of a three-fold term with parameter Vs that determines the barriers and a one-fold term with parameter Vi, that determines the gauche versus trans energy difference, as indicated by equation (I), IQ> = ;olWl
+ cos(34)] + $Wt[l
+ cos($)]
(1) The potential as written here also contains two scale factors, one (a) for the three-fold barrier and
COMP. AND THEOR. POLYMER SCIENCE
Volume 8 Number l/2
1998
93
perature
in runs of several
: 20 a E 15 % 2 10 s' 5 0 0
60
120
i80
240
300
360
otenkial function parameters are the same d there also ex~3pt for the scaling of Ihe
$ Figure 1 The torsional potentials selected for the study of their effect on Ts (a) The four potentials in which the three-fold barrier is modified by the scale factor shown (a in equation 1). The one-fold term is unmodified @= 1). (b) The potentials in which the one-fold term is modified by the scale factor shown (9 in equation I). The three-fold term is unmodified (IX= 1)
variation is be expected to be a ~t~~~t~~a~ one rather than a directly a dy~a~~~ one. The barriers but ween eonforrnations are not sensibly gauche tvam population ratio will ged chain extension or and hence the equilibriu tio will be also. In a 0 the in Figure 1, the value set 0 is included, that is, the case of a zero torsiona? potential. For each selected scale factor set, the s dete volumetric nation, etails of In additi e a, space exploration are studied for the case of zero torsional results are compared to formational transitions urnd tial ~cq/?= I>. Tabb
1
Potential
e iraertial term were the same as
es near the en
T g essns in the thr
ature results obtained for the arrier, _F&ureI(a): are results
for zero torsional
function?
Function C-C bond stretch energy = l/2 kR (R - R,# Bond bending energy = J/2 k8 (8-B&CHT (PE) Torsional potentialb = c~/2 Vs(1+ cos 34) + /J/2 Vr(l i CQS#) AUA nonbonded potential, Lennard-Jones 6-12c-CHz-
Constants kR =663 ko=482
V,= 13.4 E = 0.585 a=3510
Re = I.54 0#)=!31.5
Vr =3.35 &in = 3.940 cd= 6.42
aEnergies in kJmol-‘, distances in A, angles in radians (shown in degrees). The potentials are from Ref. 5. bThe a and ,f?parameters are multiplication factors introduced to modify the torsional potential energy barrier. CFor the CHZ group the potential is of the ‘AUA’ (anisotropic united atom) type, the interaction center is offset form the C atom by the distance = d along the bisector of the C-C-C angle in the direction of the hydrogens; E is the well-depth; R,,,, the distance at the minimum; is the corresponding distance at the energy zero. The nonbonded potential is invoked between centers of 1-5 spacing and further (4 bonds) as the torsional potential explicitly represents the 14 center (3 bond) interaction via the V, term.
The role of the torsional potential in relaxation dynamics: R. H. Gee, R. H. Boyd
350
t
300 250 7
1.25
f
1.20 >
TE 200 150
/
1.15
100 r 50 t 01 0
0.5
1
2
1.5
a
Figure 4
Figure 2 Volume vs temperature curves (at 1 atm.) from MD simulation for the four torsional potentials of Figure 1 (a) in which the three-fold barrier is varied by the scale factors shown. In addition, the case of zero torsional potential is included. Vertical dashed lines mark breaks in slope indicative of the volumetric glass transition
potential (a,/?= 0) are included as well. The V-T curves for the one-fold variations, Figure I(b), are shown in Figure 3. Breaks in the slopes of the curves in both figures indicative of vitrification are found. These MD Tg values are indicated by dashed vertical lines from the temperature axis. Straight line segments indicating above and below Tg slopes are also drawn. It is apparent from Figure 2 that the volumetric Tg depends strongly on the three-fold potential. As expected, the Tg values increase with increasing three-fold barrier. This dependence in Figure 2 is plotted as Tg vs the barrier scaling factor in Figure 4. It is of interest to note that the simula-
1.35
1.25 _y 1.20 mg s 115 1.10 1.05 1.oo
0.90
” 0
”
” 100
”
” ‘I/“’ 200 T
” ” 300
”
400
”
” 500
W
Figure 3 Volume vs temperature curves (at 1 atm.) from MD simulation for the four torsional potentials of Figure 1 (b) in which the onefold barrier is varied by the scale factors shown. Vertical dashed lines mark breaks in slope indicative of the volumetric glass transition
The MD volumetric
Ts vs the three-fold
barrier
scale factor
tions show the presence of a finite Tg value even for zero torsional barrier. It is also to be noticed that in spite of the variation in T,, the slopes of the V-T curves are nearly the same above Tg. The relation between Tg and the barrier is roughly a linear one. It suggests, apart from the intercept effect at zero barrier, that Tg divided by barrier height might be a useful reduced variable for correlation of glass temperature with structure. In contrast to the three-fold barrier case, Tg varies very little as a function of the one-fold potential (Figure 3). This indicates that the rate effects on conformational transitions controlled by the threefold term are much more important in determining T, than the structural effect of variation in trans gauche ratio. CONFORMATIONAL
TRANSITIONS
A conformational transition is conveniently defined as a transition between a value of a torsional angle in the vicinity of one of the minima in the torsional potential to another value at one of the other minima. Transitions can occur between any of the three local states, tram (t) and gauche (gf , g-). In previous work on bulk PE7, it was found that the temperature dependence of the conformational transition rate was Arrhenius in character with an activation energy close to the barrier in the torsional potential. This circumstance persisted even below the volumetric glass transition. However, the autocorrelation function (ACF) associated with values of the torsional angles had contrasting behavior. At higher temperatures, well above the MD Tg, the torsional angle ACF behaved in a similar manner to the conformational transition rate with an activation energy close to the barrier value. As temperature approaches Tg, the torsional angle ACF vs l/T plot diverges from that of the transition rate, falling off in response in Vogel-Fulcher or WLF fashion.
COMP. AND THEOR. POLYMER
SCIENCE
Volume 8 Number ‘I /2 1998
95
ome spatiaUy
heterogeneous. Some bonds have far more transitions ess than given by the Poisson statistic23 with random di$~~i~~~i~~~ The few transition rake have bonds with hig their aver~~~~~~~~~~~nto he rehxation of the s
ere we have carried out a sin&u me case of zero
torsional
analysis potential.
for In
the eonventioPd of counting a transition as a $0, 180 or 300 degree torsional angle
ere f$Qt>is the value
expressed
of the torsimal angle at lue at time=0 and :’ e averages. It was co
as ~~a~~~~ions per bon e relaxation
3.0
time,t,
re shown in F&m2
as a
function
4 along with
2.5
2.0 2 ‘2 B !z .g v
t 1.30
1.5
0.30 -1.00
1.0 2 a ;=
2 0.5
F 7
-2.30 -3.30 -4.00 -5.00 -6.00
-ask 0
2
4
J
’
6
8
’
10
12
1000/-r(K) Figure 5 The temperature dependence of conformational transition rates (‘flip rate’ on ordinate). Open circles are for zero torsional potential. Filled triangies are for the unmodified (CI= 1, fi = 1) torsional potential (the latter date is from Ref. 7). Activation energies are also showi
-7.00 0
2
4
6 1000/T
8
10
!2
Figure 6 The temperature dependecce of the stretched exponential rekxation times, equation 3, for the torsional angle autocorrelation function. Open circles are for zero torsional potentia:. Filled triangies are for the unmodified (CX = 1, B = 1) torsional potential (the latter data is from Ref. 7)
The role of the torsional potential
T (K) Figure 7 The j3 stretching parameter, eq. 3, for the torsional autocorrelation function. Same symbols as in Figure 6. Lines are linear regressions
the zero torsional potential case. However, it is of great interest to note the temperature dependence is still WLF in nature. The /I stretching parameter vs temperature is shown in Figure 7. The values are somewhat greater for the zero barrier case. This could be expected on the basis of a more dynamically flexible chain. Spatial heterogeneity of transition rates
The 4000 transitions that were generated for the zero potential case at each temperature were cataloged according to the sequential bond number at which they occurred. The occurrence plots at
0
200
300 Bond?&nber 500
600
700
800
Figure 8 Spatial heterogeneity of conformational transition location for the zero torsional potential case. At each temperature there are 4000 total transitions. The number occurring at each bond, as numbered serially along the polymer chain, is shown. The standard deviations, cr, about the average number of transitions per bond (4000/ 765 = 5.23) are also shown
in relaxationdynamics: R. H. Gee, R. H. Boyd
100 K, 200 K and 300 K are shown in Figure 8. It is apparent qualitatively that at 100 K, a temperature close to the volumetric Tg, the distribution of transitions has become very uneven. More quantitatively, for a random occurrence of transitions over the bonds a Poisson distribution should prevail. For the latter, the standard deviation in distribution should be X112where X is the average number of transitions per bond (X=4000/ 765 = 5.23; X1i2=2.3). It may be seen in Figure 7 that the Poisson value is approached at 200 K and 300 K but at 100 K the deviation from randomness is considerable. The situation is very much like that found for the unaltered torsional potentia17. Thus the phenomenon of onset of dynamic heterogeneity as vitrification sets in appears to be a general one associated with chain dynamics in bulk systems. Cooperativity of transitions
Another important conclusion concerning conformational dynamics in bulk PE1A,7, and in common with results found in Brownian dynamics simulations12J3, was that cooperative transitions are common. That is, a concerted motion occurs where after one bond changes conformation, a second nearby bond follows by changing its conformation. Considerable correlation involving the next-neighbor bonds was found. These can be classified into types such as gauche migration, same sign gauche pair formation, kink formation and kink inversion7T13,14. In addition, self-correlated transitions were common7. These involve a bond that has already jumped then undergone a back transition. A convenient and simple way to demonstrate correlation is to keep a list in temporal order of the bond numbers where transitions have occurred. From this list, after a transition at bond, i, the next bond to jump, at, j, within a fixed window of &tN bonds can be found. Then, from the total number of transitions within the window, a probability, P(i*j), that following a jump at i, a jump within the window at &j can be constructed. Correlation is easily discerned in a plot of P(iij) vsj as a deviation from the uniform background of random i *j jumps. Such a plot is shown in Figure 9 for the zero torsional potential case. It can be seen that considerable + 2 next-neighbor correlation is present. The types of next-neighbor correlation are the same as found previously7 for the standard torsional barrier and mentioned earlier. Also present is j= 0 self-correlation. Unlike the f 2 correlation, the self-correlation is highly temperature dependent, becoming relatively much more prominent at low temperature. This was observed previously for the unaltered potential7 and appears to
COMP. AND THEOR. POLYMER
SCIENCE
Volume 8 Number l/2
1998
97
ence
of the
relaxa-
0.35
eminence
0.25
of 0,
0.10 V
1
0
12
I
3
Correlation
4
5
position
6
I
I
7
8
(i&j)
Figure 9 Cooperativity of transitions for the zero torsiona! potential case. The probability that a jump at bond i is followed next by a jump at *i wherei is restricted to window of f 8 bonds. Temperatures are indicated
be related to the onset of spatial ~ete~o~ene~ty in transition location (Figure 8). The transitions at the high rate locations tend to be of the forward and back type.
It is apparent from these results that t very sensitive to the glass transition temperature arrier. Thus, the the magnitude of the torsionaj urely kinetic rate effect set by the % bmportant one in determining
enius temperature dependence of the confo~~~ti~~a~ t~a~§iti~~ rate,
J., J. Chhem. Phys.; 1987, 1. J., 9. CYrem. Yhys., 1988, 2. 1989, 22, 2259. 3. Rigby, D. and Roe, R. J., M~cromoiec&s, 1990. 23. 5312. 4. Riabv. D. and Roe. R. J.. Macromolecules. 5. Woe, R. J., in Atom& hkodeling of Physical Pro&&s, Advances in Polymer Science, Vol. 116, eds L. Monnerie and U. Suier. Springer, 1994, I1 l-144. 6. Pant, P. V. K., Han, J., Smith, G. D. and 99: 597. Gee, R. H., Han, J. and Jin, Y., S. Chm. Whys., I. ;;s. ee, F.. H. and Boyd, R. 8. 9. 10. 11. 12. 13.
Toxvaerd, S., 1. C&m. Phys., 1 Nose, S., 1. Gem. Phys., 1984, Takeuchi, H. and Ok&&i, K., J. Chem. P&s., 1990,92, 5643. Weber. T. A. and Helfand. E.. _C Chew. Phvs., 1979. 71. 4760. Helfa&, E., Wasserman, a. k and Web& ?., Mocromokcules, 1980, 13, 526. 14. Welfand, E., J. Cham. Phys., 1971, 54, 4651.