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Computational experiments on the intramolecular energy flow in macromolecules
Chemical Physics North-Holland
160 ( 1992) 393-403
Computational experiments in macromolecules
on the intramolecular
energy flow
Bobby G. Sumpter and Donald W. Noid Chemrstry D~wsron, Oak Ridge Natlonal Laboratory, Oak Ridge. TN 378314182, USA and Department of Chemrstry, The Umverxty of Tennessee, Knoxvdle, TN 37996.1600. US‘4 Received
9 September
199 1
The microscopic details of the flow of energy in a single chain of polyethylene contaming 300 atoms is discussed. The intramolecular dynamics of the polyethylene molecule is studied as a function of CH stretch excitation, temperature, and pressure. The rate of energy flow from CH stretching modes is found to be very rapid and irreversible, occurring on a timescale of less than 0.5 ps at low temperatures, and increases with temperature. A general characteristic two-phase energy flow behavior is observed, where there is mitially a very rapid flow (due to the decay of the initial excitation) followed by a slower flow (due to energy redistribution throughout the system). The mechanism for the initial facile energy flow is shown to involve strong resonant pathways. In particular, a CH stretch/HCH bend Fermi ( 1.2) resonance is shown to dommate the short-time dynamics and facilitates the overall process of energy redistribution. The increase m the rate of energy flow as a function of the backbone temperature is found to be due to the increase in the density of the bath states for energy redistnbution which subsequently results m the formation of new low-order resonant interactions ( 1.1, etc). The long-time dynamics, associated to complete redistribution of the nutial CH stretch energy with all of the 894 available vibrational modes, occurs within a time of 2 ps. This timescale corresponds to the time for mtramolecular redistribution. A comparison of the mtramolecular redistribution time to that of mtermolecular redistribution (redistribution m the condensed or solid phase as opposed to a single chain) is also made. A preliminary study of energy flow m a crystal of polyethylene (system contaming 19 polyethylene chains) shows that the energy flow exhibits two very different time behaviors, The first is for the mtramolecular redistribution as m the single chain study and the second is for mtermolecular (chain-to-chain) redistribution. The timescale for intermolecular redistribution is found to be on the order of 0.2 ns at room temperature and pressure. about two orders of magmtude larger than the mtramolecular timescale.
1. Introduction A fundamental property of all matter is its ability to absorb and transfer energy. In terms of molecular systems, once a molecule has been energized (by laser, thermal radiation, collisions, etc. ) the future change (structural, physical, or chemical) will be determined by how the energy is redistributed on the timescale of an observation [ 11. This is particularly important in chemical reaction dynamics where a molecule that is sufficiently energized will ultimately undergo dissociation to form products. For these reasons, the determination of the pathways of energy flow within molecules has been the focus of intense research for a number of years. Unravelling the mechanisms that control internal energy redistribution is an essential step in the attempt to understand and predict how various feaElsevier Science Pubhshers
B.V.
tures of a chemical system control its dynamics [ 21. While internal energy flow is fundamental to both spectroscopy and reaction kinetics [ 11, a strong stimulus for much of the recent activity in this area results from interest in the possibilities of mode-specific chemistry. The question of whether pathways to internal energy flow leading to subsequent chemical reaction are dependent upon the initially prepared state has been a major outstanding problem in chemical physics for a number of years [ 21. In recognition of the scientific and practical importance of this question, much research has been devoted toward determining the feasibility of such a process. Most results show that by far most chemical systems exhibit statistical energy flow and thus follow the traditional ideas central to the RRKM theory [ 31. However, there are a few studies both theoretical and experimental (for example, deuterated water [4], alkane
394
B.G. Sumpter, D. Pf‘. Nold / Intramolecular energy flow an macromolecules
chain radicals [ 5 1. van der Waals molecules [ 61, some isomerization reactions such as allyisocyanide [ 7 1, and a number of silicon compounds [ 8 ] ) which have demonstrated that specific routes to unimolecular reactive energy flow do exist. The central question that needs to be answered is what is the nature of the energy flow in these systems. In order to probe the nature as well as the rate of internal energy flow one must examine the microscopic details of various complex processes which involve molecular vibrations, rotations, and collisions. Energy flow may occur on sub-picosecond timescales and thus it lends itself most acceptable to theoretical studies [ 9- 141. Indeed, there has been a substantial amount of computer simulation studies of internal energy flow in molecular systems. These studies have found that energy flow out of specifically excited vibrational modes generally follows a well defined path which is marked by resonant interactions. The most prominent resonance path among polyatomic molecules has been the stretch-bend Fermi resonance. This interaction leads to facile energy flow occurring on a timescale of a few hundreds of femtoseconds. Other nonlinear resonances are then tuned into existence and the energy is propagated away from the stretchbend interaction. In general, the initial energy is fairly well redistributed throughout all of the vibrational modes within about 5 ps. A prominent example of energy flow due to Fermi resonance is found for the CH overtone excitation of benzene [ 12-141. In this case the energy in a CH stretching vibration is transferred almost exclusively to the HCH bend on a timescale of approximately 100 fs. Detailed analysis carried out by Sibert and coworkers [ 12 ] has shown that the ultra-fast energy flow from the CH stretch to the HCH bend was due to a Fermi resonance between the two vibrations. The current literature on energy flow studies in small polyatomic systems clearly demonstrates the importance of the resonance mechanism in energy flow [ 141. Indeed this mechanism seems to be responsible for energy flow in the majority of reported X-H overtone decay studies. The characteristic nature of intramolecular energy flow, with nonlinear resonance such as the Fermi resonance at the root of the mechanism, appears to be fairly general for the decay of energy from excited bond modes in a variety of molecular systems. It has even been found to play
a dominant role in adsorption/desorption of atoms from surfaces and in the energy flow pathways of macromolecular crystals [ 15 1. With the advent of new state-of-the art experimental methods to study ultra-fast energy relaxation [ 21, there has been considerable interest in extending the present understanding of energy flow in small polyatomic systems to the study of energy flow in macromolecular systems. In recent years, the technique of time-resolved vibrational spectroscopy has been used to examine energy relaxation in condensed matter systems and in solids [ 16 1. Graener and Laubereau [ 161 were among the first to study the vibrational population decay of crystalline polyethylene. They excited the CH asymmetric vibrational modes using 10 mJ picosecond pulses from a Nd: YAG mode-locked laser. By doubling a small fraction of the fundamental laser pulse they were able to create a probe pulse which was then used in anti-Stokes Raman scattering. They found that the time delay between the absorption of the initial laser pulse and the population of the Raman active modes (symmetric stretch) occurred on a timescale of less than 5 ps. They hypothesize that all of the CH vibrational modes are involved in the energy flow process. A second and much slower time constant (on the order of 260 ps) was also measured and loosely attributed to energy redistribution throughout the crystal (thermalization time). They concluded that the energy randomization in polyethylene could not be described by a single decay constant and that the observed decay must be characterized by a yet unknown subensemble of vibrational modes including the CH stretching vibrations. The results on polyethylene have some striking similarities to the energy flow characteristics of small molecules. The observation of an initial rapid decay of the excited vibrational modes closely parallels that observed in X-H overtone decay. In particular, it occurs on a rapid timescale compared to other slower dynamical processes (such as reaction). Energy flow mechanisms for this type of decay are generally a result of nonlinear resonance. Indeed, Graener and Laubereau [ 16 ] suggested that Fermi resonance could provide a possible coupling for the randomization of energy. The second and orders of magnitude slower process is undoubtly due to intermolecular energy redistribution and demonstrates the radical difference
B.G. Sumpter, D. I+’ Nold /Intramolecular
between intramolecular and intermolecular processes. In order to bring together the experimental results on polyethylene with that of theoretical studies on XH overtone decay in small polyatomics, we have performed a series of very detailed computational experiments. In this paper, our study begins with the investigation of the energy flow behavior of CH stretches in a single chain of polyethylene and its sensitivity to details such as potential energy surface, CH stretch excitation energy, temperature, and pressure. We extend this study further by examining the microscopic details of the energy flow process in crystalline polyethylene. The methods used in the computational experiments are described in section 2. A discussion of the result is given in section 3 followed by the conclusions in section 4.
2. Methods In the past decade, science has seen a marked advancement in the understanding of the intramolecular dynamics of polyatomic molecules. Insight into the role of anharmonicity in molecular vibrations and the breakdown of statistical energy redistribution has been gained by detailed nonlinear dynamics studies of small molecules. This type of detailed knowledge is also needed in the description of the dynamics of macromolecules. In the present study we will utilize some techniques from nonlinear dynamics studies of small molecules to develop an equally detailed understanding of the dynamics of macromolecular systems. 2.1. Classical trajectories The classical trajectory method has been the principle means by which the knowledge of the nonlinear dynamics of small molecules was obtained [ 9- 141. By utilizing this method one gets the time dependence of the individual particles of a system. Imbedded within this simple set of data is a virtual wealth of information. The classical trajectory method is well known and there exist several recent reviews on the procedure as well as its merits [ 171. In short, it involves the numerical solution of Hamilton’s equations of motion
energy flow m macromolecules
395
or any other formulation of the classical equations of motion, starting with some initial positions and velocities of the atoms or particles of a system. Fundamental to the classical trajectory method is the necessity of describing the potential energy surface. For this purpose we have used the Hamiltonian: H= kinetic energy + Vbbonded + V,,nonbonded ,
(1)
where
Vbonded +
1
=
1
vCC
V”CC
+
+
1
1
vCC
1
vC”
+
1
vCCC
+
1
VHCH
VCCCC
(2)
and
Vnonbonded
=
+
1
VCH
+
2
(3)
v”H
The many-body expansion of the potential energy was chosen to give a relatively accurate description of the vibrational motion. The individual potential energy functions are:
V(r)=CD{1-exp[a,(r,,-re,)l}2,
(4)
where r= C-C or the C-H bond distances the ith andjth atom of the polymer chain; V(e)=CfKer,k(~~,rlk-~e/i)2 %k
=
@CC
? (&CC)
or
3
&CC
Vcccc( T) = I( - 4.4 COS
and g are
;
T,,k( +
Vnonbonded=~Q)/R~+Pexp(-~R,,)
(5)
6.4 COS37,k,) ; .
(6)
(7)
All of the potential parameters are given in table 1. The integration of Hamilton’s equations of motion was accomplished as described in ref. [ 18 ] _While we have studied three different ensembles (microcanonical, canonical or isothermal, and isobaric) the majority of the results discussed in section 3 are from the microcanonical ensemble. In these cases, the total molecular Hamiltonian was conserved to at least 6 digits. The other constants of motion, total linear and angular momentum were likewise conserved. 2.2. Normal mode analysis We have carried out a normal mode analysis in order to make comparisons with experimental values. First, it should be noted that our model consists of a single (initially all-trans) polyethylene chain of 100
B.G. Sumpter, D. l+: Nerd /Intramolecular enerDflow m macromolecules
396 Table I Potential
energy parameters
Intramolecular
‘) for eqs. (4)-(7)
bonds
DC_, bond = 80.0 kcal/mol D C_Hbond = 106.7 kcal/mol oc,c=1.94A-1 c&-n= 1.75 A-’ &=1.53A r”,,=l.O9A a) The parameters
were adapted
of the text Nonbonded
Angle bends
I&,=72.3 kcal rad-* mol-’ K ,cc=41.85 kcal rade2 mol-’ K ,,-=38.9 kcal rad-* mol-’ B’,cc=112” 6& = 108.9” B&c” = 109.3” from refs.
interactions
term
o (A6)
p (kcal/mol)
Y (A-‘)
H-H H-C C-C
35.967 289.75 534.61
2556.53 14313.4 149104.5
3.74 3.67 3.60
[ 37-391.
Table 2 Normal mode frequencies Mode
Calculated
CH(asym)
2922 2852 1465 729 1376 1137 503 111 1056
CH(sym) HCH(bend) HCH(rock) HCH (wag) CC(str) CCC(bend) CCCC(torsion) CHZ(twist)
al Calculations from the present work. b, Expenmental values taken from ref. [ 371. ‘1 Inferred from spectral analysis of paraffms
(cm-
’ ) ‘)
Expertmental 2919 2851 1463 721 1370 1133 z 533” 108 1050
(cm-
I ) b,
Symmetry J&u &u &u Bzu &g B,g B2 &g T&u
[ 401.
backbone carbon atoms with two hydrogen atoms per carbon, giving a total of 300 atoms. For an isolated system such as this, one expects to obtain 894 normal modes. A normal mode analysis is in principle simple to perform. There are several well-known algorithms available and the method is well documented in most text books on spectroscopy [ 191. Here we give only the details of the method that we have employed. Since we are ultimately interested in carrying out dynamics calculations, we always start a system in Cartesian coordinates where the total kinetic energy is exactly represented. The atoms of the polyethylene chain are positioned in Cartesian space to obtain the correct all-tram conformation (dihedral angles, bend angles, and bond distances). These initial positions are then oriented about the center-of-mass (centerof-mass reference frame ). Next the system is rotated such that the moment-of-inertia tensor is diagonal
(principle moment-of-inertia frame). This new coordinate frame satisfies the Eckart condition for separating the vibrational motion from the overall molecular rotational motion. In this reference frame the second derivatives of the potential with respect to the Cartesian coordinates are calculated to obtain the force field matrix. This is a real symmetric matrix of 3Nx3N (N is the number of atoms) of the second derivatives and is used to form the secular equation. In order to find solutions to the secular equation, the force matrix is diagonalized using a standard routine [ 201 and the resulting eigenvalues and eigenvectors correspond to the normal modes of the system. Symmetry species can be assigned to the normal modes following the Wilson GF method [ 211. A comparison of the calculated normal mode frequencies to those obtained from experiments is given in table 2. As can be seen, there is very good agreement. The largest deviations are those associated with
B.G. Sumpter, D. R: Noid/lntramolecular
energyjlow in macromolecules
391
the acoustic skeletal modes for which the frequencies of vibration are very small.
3. Results The details of the intramolecular dynamics of a single chain of 300 atoms ( 100 backbone carbons and 200 substituent hydrogens) has been examined in order to develop a better understanding of the energy flow pathways in macromolecular systems. The sensitivity to the initial level of excitation was characterized by examining the rate of energy flow out of specific CH overtones. The overtones were energized according to the equation E(CH)
=w,( v+O.5) -0,x,(
u+OS)~,
(8)
where o, corresponds to the mechanical frequency of the CH vibration and o,xe =0,2/4D,and v the quantum level of excitation (v- 1= the overtone). This excitation energy was added on top of an energy due to thermal motion of the backbone carbon atoms. As discussed in section 2, the total angular momentum was set to zero so that there was only vibrational energy present in the system. With the initial energy deposited into a single CH bond, the equations of motion are integrated. The energy in the CH stretch is calculated in time by determining the displacements of the CH bond and using eq. (4) to calculate the potential energy. The kinetic energy for the CH bond is calculated by projecting the total velocity of the carbon and hydrogen atoms onto the corresponding bond. The bond energy is then, simply, the sum of the potential plus the kinetic energy. This separation is necessary as well as the zero angular momentum restriction since we were only interested in the vibrational energy flow out of the CH bond and not to any components due to the bending, torsional, or overall molecular rotational motions. In a similar fashion, the mode energies for the various CH and CC stretches, HCC, HCH, CCC bends, and CCCC torsions can be obtained. The details of the transformation from Cartesian to curvilinear coordinates such as 8 and r can be found in ref. [ 22 1. Fig. 1 shows the results of the CH energy flow as a function of time for the excitation levels of v=2, 3, 4, 5, 6, 10, 12 for a backbone temperature of 20 K. These results were obtained from averaging an en-
b
1
Fig. 1. (a) The intramolecular energy flow out of a CH bond excited to Y= 2, 3,4,5 (correspondmg to curves in increasing order of imtial energy) at T= 20 K. Energy is m units of kcal/mol and time in units of picoseconds ( ps) (b ) Same as for (a) except for excitations ofv=6, 10, 12.
semble of 20 trajectories, each beginning with a different set of initial conditions. In the present study, the initial conditions were chosen so that the backbone carbon atoms started with different random momenta and coordinates and the phase averaged momenta and coordinates for the excited CH bond were chosen from a random distribution appropriate to a non-rotating Morse potential [ 17 1. From examination of fig. 1, it can be seen that the higher levels of excitation (v = 4,5,6, 10, 12) exhibit a rather rapid decay of energy followed by a much slower decay. For the lower levels of excitation (v= 2,3) the energy flow out of the CH stretch is very slow. In fact, the energy appears to be localized on the timescale of 2 ps. At
398
B.G. Sumpter. D. I+: Nord /Intramolecular
first thought, the instability of the higher levels of excitation versus the relatively stable lower levels could be attributed to the simple fact of increasing anharmonicity as a function of energy. This of course is the fundamental reason behind this observation but closer examination reveals a more detailed mechanism. In fig. 1, the higher levels of CH stretch excitation lead to a two-phase characteristic energy flow. The initial dynamics is very rapid and results in an almost exponential decay of the energy on a timescale of less than 0.5 ps. The dynamics following this advent(s) (referred to as long-time dynamics) is much slower and consumes as much as 1.5 ps to reach a stable energy. The two-phase behavior has been characterized as a prominent feature in studies of small polyatomic molecules. In fact, the decay of CH overtones in systems such as benzene [ 12-l 41, methyl isocyanide [ 231, models for alkanes such as butane [24], dimethylnitramine [ 25 ] and other systems [ 14 1, have very similar qualitative energy decay behavior. In these systems the mechanism for the energy flow was determined to be primarily due to resonant interactions. Therefore, we strongly expect that similarly there may be a low-order resonance between the CH stretch and the HCH bending vibrations. A first-order approximation to the vibrational frequency of an anharmonic vibration such as the CH stretch can be obtained from the frequency dependence on the bond energy. The equation is:
energy jlow rn macromolecules
versible decay (around 0.3 ps). Similar results can be found for the v levels 6, 10, and 12. The resonant pathway of energy flow from the CH stretch into the HCH bend can be better seen by examining the energy as a function of time in the HCH bend. In fig. 2a, the energy flow out of the excited CH stretch for v=6 is plotted as a solid line and the energy flow into the HCH bend is plotted as a dashed line. Initially there is no significant participation between the modes (the first 0.1 ps). An instability appears to occur around 0.15 ps and the result is an incredible flow of energy from the CH stretch into the HCH bend. The HCH bend energy then slowly decays and this energy is redistributed to the other vi-
,
. 02
04
06
08
1
12
14
16
I8
12
14
16
18
-2
Tlm(pr)
(9) where v” is the fundamental frequency for the CH stretch, ECH is the energy of the CH bond, and DCHis the dissociation energy of the CH bond. Substituting the correct parameters into this equation for the energy level corresponding to that where significant flow occurred for v = 5 yields a frequency of vCH= 2460 cm- ‘. Using the resonance relation nw, - mw2 = 0, we obtain n = 5 and m = 3 which indicates that there is a possible 3: 5 resonance between the CH stretch and the HCH bending vibrations. The existence of such a resonance will act to facilitate energy flow out of the CH stretch into the HCH bend. As the energy is removed from the CH stretch, the vibrational frequency increases and finally reaches and energy where it is in a 1: 2 (Fermi) resonance with the HCH bend. This is the point where the energy flow exhibits irre-
0
02
"4
06
08
2
Fig. 2. (a) Energy flow out of the initially excited CH stretch (u= 6) mto the HCH bend. The solid lme IS the energy in the CH bond and the dashed line is the energy of the HCH bend. (b) Same as (a) except the energy in the CH bond adjacent to the excited CH stretch and the energy in the terminal CH bond are also plotted.
B.G. Sumpter. D. W. Nold /Intramolecular
brational modes as shown in fig. 2b. As can be seen, after about 2 ps, the various vibrational modes have obtained approximately equivalent amounts of the initial energy. This timescale corresponds to the intramolecular redistribution of the initial energy. The redistribution is facilitated by the Fermi resonance between the CH stretch and HCH bend. As the energy in the initial excited CH stretch decays into the HCH bend, the CH stretch tunes into a 1: 1 resonance with one of the alternate CH stretches. Energy flows into this CH stretch which then tunes into a loworder resonance with the HCH bend. This process continues and results in rapid redistribution of the energy throughout the system. It should be pointed out that these characteristics are dependent on the model for the polymer chain. For example, if the torsional modes (gauche trans isomerization potential) is not included, that is, the polyethylene chain is constrained to be planar, the rate for the energy decay and redistribution are on completely different timescales. Although the resonant interaction mechanism is still in effect due to the incredible density of states for macromolecules, there is a significant difference in the timescales for the observed processes (the timescale is slower by about 1 ps). This demonstrates the importance of the torsional modes (gauche trans isomerizations) in the internal dynamics leading to energy flow. Similar conclusions have been discussed in a number of studies on small polyatomics [ lo]. The phenomenon of Fermi resonance is a well studied example of spectral intensity sharing between vibrational modes [ 26 1. In polyethylene, the existence of Fermi resonance has been inferred for many years [ 271 and recently verified in theoretical calculations [ 15,281. In the present study we have shown that energy flow from an excited CH stretch occurs through resonant pathways marked by low-order resonances such as the Fermi resonance. In order to obtain a better visual picture of the dynamics, a time dependent local configuration plot analysis [ 29 ] is carried out for the v = 6 CH stretch excitation. This analysis simply involves plotting sections of a trajectory at times where the dynamics appears to be interesting. From examination of fig. 2a it can be seen that the initial energy flow begins at about 0.15 ps and is irreversible at about 0.2 ps. If we plot the bond distance of the CH as a function of the bond distance of the adjacent CH stretch and also as a function of the
energyflow rn macromolecules
399
HCH bend, we can see if the characteristic shapes of resonances in configuration space are present [ 301. In general, a 1: 2 resonance leads to a C-shaped ligure. 1: 3 to a S-shaped figure. 2 : 3 to a gamma-shaped figure, 3 : 4 to a pretzel shaped figure [ 3 11, etc. In fig. 3a, a plot of the CH bond distance as a function of the adjacent CH bond distance is given. The plot is for the initial 0.1 ps of dynamics and demonstrates that on this timescale the energy is very localized in the initial excited CH stretch. This type of behavior is classified as local mode motion. As time increases, 0.15-0.4 ps, the local mode behavior is rapidly destroyed. In fig. 3b the CH bond distance is plotted as a function of the HCH bending angle. As the energy flows from the CH stretch into the HCH bend, the bending angle begins to have larger amplitude displacements from its equilibrium angle. After several hundred vibrations, a C-shaped figure appears in the center of this pot. This C-shape corresponds to the time of 0.2 ps and remains stable for about 0.1 ps. A better view of the C-shape is illustrated in fig. 3c. Clearly, the dynamics has resulted in a situation where the CH stretching motion is at a rate of 2 times faster than the HCH bending motion. This frequency degeneracy is coupled to a large set of “bath” states (the remaining 892 vibrational modes) and thus causes rapid and irreversible energy flow. Now that it has been determined why the energy flow exhibits the two-phase decay behavior we turn to the lower CH stretch excitations. As was shown in fig. la, for ~=2 and 3, the initial energy exhibited very little decay on a timescale of 2 ps. This is because there are no strong couplings to facilitate the energy flow. However, this does not mean that the vibrational states v=2 and 3 are stable. If we examine the energy flow properties for a slightly longer timescale (5 ps) it happens that both states become unstable and rapid energy flow occurs. This is shown in fig. 4 for the energy flow out of the v = 2 level of the CH stretch. After about 3.5 ps, the initially stable CH stretch becomes unstable and the energy flow exhibits the two-phase behavior as in the higher excitation cases. This type of behavior is very similar to that attributed to bottlenecks in phase [ 321 (in this case the bottleneck is the one referred to as the golden mean [ 331: a frequency ratio between the CH stretch and HCH bend of approximately 1.6 18). In short, this means that energy flow is localized due to a confining
B.G. Sumpter, D. W. Nord /Intramolecular
energy flow m macromolecules
a
Fig. 4. The intramolecular tov=2at T=20K.
Fig. 3. (a) A plot of the excited CH bond length versus the adjacent CH bond length. The plot is for a ttme period of 0.1 ps and the bond lengths are in units of angstrom. (b) A plot of the excited CH bond length (in angstrom) versus the HCH bend angle (m degrees). The time period is from 0. I5 to 0.4 ps. (c) Same as for (b) except for a shorter time period (0.2-0.3 ps). The labels q. and q, stand for CH stretching and HCH bendmg vibrations. respectively.
energy flow out of a CH bond excited
boundary that results from the breakup of other stable motions. Escape channels through this barrier can be achieved by way of resonant interactions [ 29 1. However, in general the energy flow is localized for extended periods of time. The localization means that there is a lack of coupling to the bath states. In the present case, the energy is low causing the anharmonicity to be low and thus the density of states. Energy decay only occurs after a slow adiabatic leaking of the energy to the bend has occured. At this time, the CH stretch is “tuned” into a 1: 1 resonance (determined by a local frequency analysis [ 291) with a subset of the CH stretches in the system and energy flow occurs. Thus the bottleneck is broken on a relatively short timescale. The existence of phase space bottlenecks such as this are a common example of nonstatistical behavior [ 321. Needless to say, the observation that nonlinear dynamical features such as resonances and phase space bottlenecks also play an important role in the internal dynamics of macromolecular systems is of great interest and demonstrates the overall generic applicability of these phenomena. The sensitivity of the above results to pressure and temperature was also studied. In order to examine the dynamics under different external conditions, we have used the method of Nose dynamics [ 341. Briefly, Nose dynamics is a modified version of Newtonian dynamics so as to reproduce both the canonical and isothermal-isobaric probability densities in the phase space of an N-body system. It provides for a rela-
B.G. Sumpter, D. U: Nold /Intramolecular
tively simple way to induce external pressure and constant temperature. It was found that increasing the external pressure tends to only slightly affect the energy flow decay rates (the rates tending to be slower) while temperature can cause a significantly enhanced rate of energy flow. For a temperature change of 300 K, the energy flow decay time decreases by about 1 ps. This result is also fairly independent of the excitation level of the CH stretch. Again, the temperature is playing the role of increasing the total energy and thus increasing the total density of vibrational states available for coupling. As expected, this results in a dynamical behavior which is at faster rates. Finally, it is interesting to compare the results of our single polyethylene chain study to those of crystalline polyethylene. To accomplish this task, we have carried out a number of preliminary calculations on a model for orthorhombic polyethylene. The model consisted of 19, 100 CH2 group polymer chains. An initial excitation energy was deposited in the center of the crystal and the energy flow was followed as discussed above for the single chain. The results indicate that the intramolecular energy redistribution is enhanced somewhat, occurring on a timescale of about 0.1 ps. The total energy redistribution however involves the migration of the energy throughout the large crystal. This process is an intermolecular process, involving chain-to-chain energy flow. The intermolecular energy flow was found to occur on a timescale of about 0.2 nanoseconds, about two orders of magnitude slower than that for the intramolecular process of the single chain. The intermolecular process is also found to be dependent on the temperature and pressure of the system. These results are reported in more detail in ref. [ 35 1. Cross-correlation functions were calculated to examine the intermolecular energy flow pathways though the crystal. From this analysis, it was found that the intermolecular energy flow from the center of the crystal occurs by radial diffusion, crystal layerby-layer, until it reaches the outside edge. The diffusion of the energy was found to be reasonably described using the Langevin equation [ 361. In this regard, we have performed as series of calculations [ 351 (Langevin dynamics using different friction coefficients) to verify this fact.
energy flow in macromolecules
401
4. Conclusions In this letter we have discussed the results of a detailed study of the intramolecular energy flow in a single chain model of polyethylene. The intramolecular dynamics of the polyethylene molecule was studied as a function of the potential energy surface, CH stretch excitation, temperature, and pressure. The rate of energy flow from CH stretching modes ( v = 5- 12 ) was observed to be very rapid and irreversible, occurring on a timescale of less than 0.5 ps at low temperatures, and increases with overall increasing temperature. A characteristic two-phase energy flow behavior was observed, where the initial part consisted of a very rapid flow (due to the decay of the initial excitation) followed by a slower flow (due to energy redistribution throughout the system). The mechanism for the initial facile energy flow involves strong resonant pathways. In particular, a CH stretch/HCH bend Fermi ( 1: 2) resonance was shown to dominate the short-time dynamics and facilitates the overall process of energy redistribution. The long-time dynamics, associated to complete redistribution of the initial CH stretch energy with all of the 894 available vibrational modes, occurs within a time of 2 ps. This intramolecular redistribution timescale is compared to that of intermolecular redistribution (redistribution in the solid phase as opposed to a single chain). A preliminary study of energy flow in a crystal of polyethylene (system containing 19 polyethylene chains) revealed that the timescales for intramolecular and intermolecular energy redistributions are quite different. Intramolecular redistribution in the polymer crystal was similar to that in the single chain study but occurred on a slightly shorter timescale (0.1 ps). Intermolecular (chain-to-chain) redistribution, however, was two orders of magnitude larger than the intramolecular redistribution timescale. Thus the energy redistribution behavior in the solid phase of polyethylene demonstrates two distinctly different decay processes; one corresponding to the intramolecular and other corresponding to the intermolecular energy redistribution. The results of this study are in very good accord with the experimentally measured results of Graener and Laubereau [ 161. Our computer study shows for the first time a definite resonance enhanced intramolecular redistribution mechanism that occurs on a
402
B.G Sumpter, D. W. Nord /Intramolecular
timescale of about 2 ps and a slower intermolecular process on a nanosecond timescale (0.2 ns).
Acknowledgement This work was supported by the Office of Basic Energy Sciences, US Department of Energy, under Contract No. DE-AC05-840R2 1400 with Martin Marietta Energy Systems, Inc., and by the Polymer Program of the National Science Foundation, present Grant No. DMR-88 184 12. BGS acknowledges the support provided by the Direction General de Investigation Cientilica y Ttcnica of the Ministry of Education and Science (MEC) of Spain for his stay at the Universidad Complutense de Madrid during the summer of 199 1. The computations were performed on the IBM 308 1 and 3090 at the University of Tennessee, the CRAY-YMP/48 at the National Center for Supercomputing Applications at the University of Illinois (grant CHE890025N), and the CRAY YMP at the NSF Pittsburgh Supercomputing Center (grant CHE9 1OOOP Cray sponsored research). We would like to thank E.L. Sibert III and S.K. Gray for useful suggestions on the use of cross-correlation functions and Langevin dynamics. We also thank Coral Getino for many discussions on energy transfer in polyatomic molecules.
References I. Steinfold. J.S. Franctsco and W.L. Hase. Chemtcal Kmetics and Dynamics (Prentice Hall. Englewood Chffs. NJ. 1988). F.F. Cnm, Science 249 ( 1990) 1387; A. Zewail and R.B. Bernstein, Chem. Eng. News 66 ( 1988) 24. P.J. Robinson and K.A. Holbrook, Ummolecular Reacttons (Wiley. New York, 1972). R.L. Vander Wal, J.L. Scott and F.F. Cnm, J. Chem. Phys. 92 (1990) 803, R.L Vander Wal and F.F. Cnm. J. Phys. Chem. 93 ( 1989) 5331, A. Smhaim, M.C. Hstao and F.F. Grim, J. Chem. Phys. 92 (1990) 6333. [5] J Los. J.H.M. Berlersbergen, S. Kdrmg and P.G. Istemaker, J. Phys. Chem. 95 ( 199 1) 2 143. [6] N. Halberstadt and K.C. Janda, eds.. Dynamics of Polyatomic van der Waals Complexes, NATO AS1 Series B, Vol. 227 (Plenum Press, New York, 1990).
energy flow m macromolecules
[ 71 K.V Reddy and M.J. Berry, Chem. Phys. Letters 56 ( 1979) 223: J. Segall and R.N. Zare, J. Chem. Phys. 89 ( 1988) 5704. [ 81 H W. Schrantz, L.M. Raff and D.L. Thompson. J. Chem. Phys. 95 ( 199 1) 106, and references therein. [9] J.S. Hutchmson. J.T. Hynes and W.P. Reinhart, J. Chem. Phys. 79 (1983) 4247; Chem. Phys Letters 108 (1984) 353; T.A. Holme and J.S. Hutchmson, J. Chem. Phys. 83 (1985) 2860; 84 (1986) 5455: T. Uzer. J.T. Hynes and W.P. Reinhardt. Chem. Phys. Letters 117 (1985) 600: J. Chem. Phys. 85 (1986) 5791; T. Uzer, B.D. MacDonald, Y. Guan and D.L. Thompson, Chem. Phys. Letters 152 ( 1988) 405. [ 10lB.G. Sumpter and D.L. Thompson, J. Chem. Phys. 82 (1985) 4557, 86 (1987) 2805; 88 (1988) 6889: Chem. Phys. Letters 153 (1988) 243: B.G. Sumpter. C.C. Martens and G.S. Ezra, J Phys. Chem. 92 (1988) 7193. [I l] W.L. Hase and B.J. Orr. J. Chem. Phys. 86 ( 1987) 3380; W.L. Hase, R.J. Duchovic, K.N. Swamy and R.C Wold, J. Chem. Phys. 80 ( 1984) 7 14; D.-H. Lu and W.L. Hase, J. Phys. Chem 92 ( 1988) 3217; H. Gal, D.L. Thompson and G.A Fisk, J. Chem. Phys. 90 (1989) 7055. [ 121 E.L. Sibert III, J.T. Hynes and W.P. Reinhardt. J. Chem. Phys. 81 (1984) 1135. E.L. Sibert III and W.P. Remhardt. J. Chem. Phys. 77 (1984) 3583. E.L. Sibert III, W.P. Remhardt and J.T. Hynes. J. Chem. Phys.81 (1984) 1115; E.L. Sibert III. J.S. Hutchinson. J.T. Hynes and W.P. Remhardt, m: Ultrafast Phenomena IV, eds. D.H. Auston and K.B. Eisenthal (Spnnger. Berlin 1984) p. 336; E.L. Sibert 111. W.P. Remhardt and J.T. Hynes, Chem. Phys. Letters 92 ( 1982) 455. [ 131 K.L Bintz, D.L. Thompson and J.W. Brady, J. Chem. Phys. 85 (1986) 1848; Chem Phys. Letters 131 (1986) 398; J. Chem. Phys. 86 (1987) 441 I: P.J. Nagy and W.L. Hase. Chem. Phys. Letters 54 (1978) 73. ll4 T. Uzer and W.H. Miller. Phys. Rept. 199 ( 199 1) 73. ]15 Results of the present paper on polyethylene. Chem. Phys Letters 133 116 H. Graener and A. Laubereau, (1987) 378; See articles m Chem. Rev. 90 ( 1990) and m Chem. Phys. 146 (1990). !17 L.M. Raff and D.L. Thompson, in: Theory of Chemical Reaction Dynamics, ed. M. Baer (Chemical Rubber, Boca Ration, FL, 1985); M.L. Klein, Annu. Rev. Phys. Chem. 36 (1985) 525. ]18 D.W. Noid, B.G. Sumpter, B. Wunderhch and G.A. Pfeffer, J. Comput. Chem. 11 ( 1990) 236. ]19 G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules. (Van Nostrand Reinhold, New York. 1945). S.A. Teukolsky and W.T ]20 W.H. Press, B.P. Flannery, Vetterlmg, Numerical Recipes (Cambridge Univ. Press, Cambridge, 1986)
B.G. Sumpter. D. W: Nold /Intramolecular [Zl ] E.B. Wtlson. J.C. Deems and P.C. Cross, Molecular Vibrations (reprinted by: Dover. New York, 1955). [22] C. Getino. B.C. Sumpter and J. Santamaria, Chem. Phys. 145 (199O)l. [23 I B.C. Sumpter and D.L. Thompson, J. Chem. Phys. 87 ( 1987) 5809. [24 J.S. Hutchmson. J.T. Hynes and W P. Reinhardt. J. Phys. Chem. 90 (1986) 3528. J. Chem. Phys. 86 ]25 B.G. Sumpter and D.L. Thompson, (1987) 3301. [26 D.W. Noid and R.A. Marcus, J. Chem. Phys. 67 (1977) 559: J. Chem. Phys. 62 (1975) 2119. [27] R.G. Synder and J.R. Scherer. J. Chem. Phys. 71 (1979) 3221, L Rtcard, S. Abbate and G. Zerbi, J. Phys. Chem. 89 ( 1985 ) 4793. and references therein. [28] D.W. Notd, B.C. Sumpter and F. Cnmm, Macromolecules, to be submitted. [29] CC. Martens, G.S. Ezra and M.J. Davis, Chem. Phys. Letters 142 (1987) 519. [30] D.W. Notd and R.A. Marcus, J. Chem. Phys. 85 ( 1986) 3305. [ 311 C.C. Martens and G.S. Ezra, J. Chem. Phys. 86 ( 1987) 279.
enerajlow
m macromolecules
403
[ 321 M.J. Davis and SK. Gray, J. Chem. Phys. 84 (1986) 5389; S.K. Gray, S.A. Rice and M.J Davis, J. Phys. Chem. 90 (1986) 3470; M.J. Davis. J. Chem. Phys. 83 (1985) 1016. [33] A.J. Ltchtenberg and M.A. Lieberman, Regular and Stochastic Motion (Sprmger, Berlin, 1983) [34] Wm. G. Hoover. m: Lecture Notes In Physics 258, Molecular Dynamics (Springer, Berlm, 1986). [35] B.C. Sumpter and D.W. Noid, J. Chem. Phys.. to be submitted. [ 361 J.T. Hynes. m: Theory ofChemical Reaction Dynamtcs. The Theory of Chemical Reactions in Solution, Vol 4.. ed. M. Baer, (CRC Press, Boca Raton, Fl, 1985); R.W. Pastor, m: The Molecular Dynamics of Liqtnd Crystals, Techniques and Applications of Langevm Dynamics Simulations, ed. G.R. Luckhurst, to be published. [ 371 J. Barnes and B. Francom, J. Phys. Ref. Data 7 ( 1978) 1309. [ 381 R.A. Sorensen W.B. Loam and R.H. Boyd, Macromolecules 21 (1988) 194. [ 391 S.L. Mayo, B.D. Olafson and W.A. Goddard III, J. Phys. Chem. 94 ( 1990) 8897; N. Karasawa. S. Dasgupta and W.A. Goddard III, J. Phys. Chem. 95 (1991) 2260. [40] R.G. Snyder, J. Chem. Phys. 47 (1967) 1316.
160 ( 1992) 393-403
Computational experiments in macromolecules
on the intramolecular
energy flow
Bobby G. Sumpter and Donald W. Noid Chemrstry D~wsron, Oak Ridge Natlonal Laboratory, Oak Ridge. TN 378314182, USA and Department of Chemrstry, The Umverxty of Tennessee, Knoxvdle, TN 37996.1600. US‘4 Received
9 September
199 1
The microscopic details of the flow of energy in a single chain of polyethylene contaming 300 atoms is discussed. The intramolecular dynamics of the polyethylene molecule is studied as a function of CH stretch excitation, temperature, and pressure. The rate of energy flow from CH stretching modes is found to be very rapid and irreversible, occurring on a timescale of less than 0.5 ps at low temperatures, and increases with temperature. A general characteristic two-phase energy flow behavior is observed, where there is mitially a very rapid flow (due to the decay of the initial excitation) followed by a slower flow (due to energy redistribution throughout the system). The mechanism for the initial facile energy flow is shown to involve strong resonant pathways. In particular, a CH stretch/HCH bend Fermi ( 1.2) resonance is shown to dommate the short-time dynamics and facilitates the overall process of energy redistribution. The increase m the rate of energy flow as a function of the backbone temperature is found to be due to the increase in the density of the bath states for energy redistnbution which subsequently results m the formation of new low-order resonant interactions ( 1.1, etc). The long-time dynamics, associated to complete redistribution of the nutial CH stretch energy with all of the 894 available vibrational modes, occurs within a time of 2 ps. This timescale corresponds to the time for mtramolecular redistribution. A comparison of the mtramolecular redistribution time to that of mtermolecular redistribution (redistribution m the condensed or solid phase as opposed to a single chain) is also made. A preliminary study of energy flow m a crystal of polyethylene (system contaming 19 polyethylene chains) shows that the energy flow exhibits two very different time behaviors, The first is for the mtramolecular redistribution as m the single chain study and the second is for mtermolecular (chain-to-chain) redistribution. The timescale for intermolecular redistribution is found to be on the order of 0.2 ns at room temperature and pressure. about two orders of magmtude larger than the mtramolecular timescale.
1. Introduction A fundamental property of all matter is its ability to absorb and transfer energy. In terms of molecular systems, once a molecule has been energized (by laser, thermal radiation, collisions, etc. ) the future change (structural, physical, or chemical) will be determined by how the energy is redistributed on the timescale of an observation [ 11. This is particularly important in chemical reaction dynamics where a molecule that is sufficiently energized will ultimately undergo dissociation to form products. For these reasons, the determination of the pathways of energy flow within molecules has been the focus of intense research for a number of years. Unravelling the mechanisms that control internal energy redistribution is an essential step in the attempt to understand and predict how various feaElsevier Science Pubhshers
B.V.
tures of a chemical system control its dynamics [ 21. While internal energy flow is fundamental to both spectroscopy and reaction kinetics [ 11, a strong stimulus for much of the recent activity in this area results from interest in the possibilities of mode-specific chemistry. The question of whether pathways to internal energy flow leading to subsequent chemical reaction are dependent upon the initially prepared state has been a major outstanding problem in chemical physics for a number of years [ 21. In recognition of the scientific and practical importance of this question, much research has been devoted toward determining the feasibility of such a process. Most results show that by far most chemical systems exhibit statistical energy flow and thus follow the traditional ideas central to the RRKM theory [ 31. However, there are a few studies both theoretical and experimental (for example, deuterated water [4], alkane
394
B.G. Sumpter, D. Pf‘. Nold / Intramolecular energy flow an macromolecules
chain radicals [ 5 1. van der Waals molecules [ 61, some isomerization reactions such as allyisocyanide [ 7 1, and a number of silicon compounds [ 8 ] ) which have demonstrated that specific routes to unimolecular reactive energy flow do exist. The central question that needs to be answered is what is the nature of the energy flow in these systems. In order to probe the nature as well as the rate of internal energy flow one must examine the microscopic details of various complex processes which involve molecular vibrations, rotations, and collisions. Energy flow may occur on sub-picosecond timescales and thus it lends itself most acceptable to theoretical studies [ 9- 141. Indeed, there has been a substantial amount of computer simulation studies of internal energy flow in molecular systems. These studies have found that energy flow out of specifically excited vibrational modes generally follows a well defined path which is marked by resonant interactions. The most prominent resonance path among polyatomic molecules has been the stretch-bend Fermi resonance. This interaction leads to facile energy flow occurring on a timescale of a few hundreds of femtoseconds. Other nonlinear resonances are then tuned into existence and the energy is propagated away from the stretchbend interaction. In general, the initial energy is fairly well redistributed throughout all of the vibrational modes within about 5 ps. A prominent example of energy flow due to Fermi resonance is found for the CH overtone excitation of benzene [ 12-141. In this case the energy in a CH stretching vibration is transferred almost exclusively to the HCH bend on a timescale of approximately 100 fs. Detailed analysis carried out by Sibert and coworkers [ 12 ] has shown that the ultra-fast energy flow from the CH stretch to the HCH bend was due to a Fermi resonance between the two vibrations. The current literature on energy flow studies in small polyatomic systems clearly demonstrates the importance of the resonance mechanism in energy flow [ 141. Indeed this mechanism seems to be responsible for energy flow in the majority of reported X-H overtone decay studies. The characteristic nature of intramolecular energy flow, with nonlinear resonance such as the Fermi resonance at the root of the mechanism, appears to be fairly general for the decay of energy from excited bond modes in a variety of molecular systems. It has even been found to play
a dominant role in adsorption/desorption of atoms from surfaces and in the energy flow pathways of macromolecular crystals [ 15 1. With the advent of new state-of-the art experimental methods to study ultra-fast energy relaxation [ 21, there has been considerable interest in extending the present understanding of energy flow in small polyatomic systems to the study of energy flow in macromolecular systems. In recent years, the technique of time-resolved vibrational spectroscopy has been used to examine energy relaxation in condensed matter systems and in solids [ 16 1. Graener and Laubereau [ 161 were among the first to study the vibrational population decay of crystalline polyethylene. They excited the CH asymmetric vibrational modes using 10 mJ picosecond pulses from a Nd: YAG mode-locked laser. By doubling a small fraction of the fundamental laser pulse they were able to create a probe pulse which was then used in anti-Stokes Raman scattering. They found that the time delay between the absorption of the initial laser pulse and the population of the Raman active modes (symmetric stretch) occurred on a timescale of less than 5 ps. They hypothesize that all of the CH vibrational modes are involved in the energy flow process. A second and much slower time constant (on the order of 260 ps) was also measured and loosely attributed to energy redistribution throughout the crystal (thermalization time). They concluded that the energy randomization in polyethylene could not be described by a single decay constant and that the observed decay must be characterized by a yet unknown subensemble of vibrational modes including the CH stretching vibrations. The results on polyethylene have some striking similarities to the energy flow characteristics of small molecules. The observation of an initial rapid decay of the excited vibrational modes closely parallels that observed in X-H overtone decay. In particular, it occurs on a rapid timescale compared to other slower dynamical processes (such as reaction). Energy flow mechanisms for this type of decay are generally a result of nonlinear resonance. Indeed, Graener and Laubereau [ 16 ] suggested that Fermi resonance could provide a possible coupling for the randomization of energy. The second and orders of magnitude slower process is undoubtly due to intermolecular energy redistribution and demonstrates the radical difference
B.G. Sumpter, D. I+’ Nold /Intramolecular
between intramolecular and intermolecular processes. In order to bring together the experimental results on polyethylene with that of theoretical studies on XH overtone decay in small polyatomics, we have performed a series of very detailed computational experiments. In this paper, our study begins with the investigation of the energy flow behavior of CH stretches in a single chain of polyethylene and its sensitivity to details such as potential energy surface, CH stretch excitation energy, temperature, and pressure. We extend this study further by examining the microscopic details of the energy flow process in crystalline polyethylene. The methods used in the computational experiments are described in section 2. A discussion of the result is given in section 3 followed by the conclusions in section 4.
2. Methods In the past decade, science has seen a marked advancement in the understanding of the intramolecular dynamics of polyatomic molecules. Insight into the role of anharmonicity in molecular vibrations and the breakdown of statistical energy redistribution has been gained by detailed nonlinear dynamics studies of small molecules. This type of detailed knowledge is also needed in the description of the dynamics of macromolecules. In the present study we will utilize some techniques from nonlinear dynamics studies of small molecules to develop an equally detailed understanding of the dynamics of macromolecular systems. 2.1. Classical trajectories The classical trajectory method has been the principle means by which the knowledge of the nonlinear dynamics of small molecules was obtained [ 9- 141. By utilizing this method one gets the time dependence of the individual particles of a system. Imbedded within this simple set of data is a virtual wealth of information. The classical trajectory method is well known and there exist several recent reviews on the procedure as well as its merits [ 171. In short, it involves the numerical solution of Hamilton’s equations of motion
energy flow m macromolecules
395
or any other formulation of the classical equations of motion, starting with some initial positions and velocities of the atoms or particles of a system. Fundamental to the classical trajectory method is the necessity of describing the potential energy surface. For this purpose we have used the Hamiltonian: H= kinetic energy + Vbbonded + V,,nonbonded ,
(1)
where
Vbonded +
1
=
1
vCC
V”CC
+
+
1
1
vCC
1
vC”
+
1
vCCC
+
1
VHCH
VCCCC
(2)
and
Vnonbonded
=
+
1
VCH
+
2
(3)
v”H
The many-body expansion of the potential energy was chosen to give a relatively accurate description of the vibrational motion. The individual potential energy functions are:
V(r)=CD{1-exp[a,(r,,-re,)l}2,
(4)
where r= C-C or the C-H bond distances the ith andjth atom of the polymer chain; V(e)=CfKer,k(~~,rlk-~e/i)2 %k
=
@CC
? (&CC)
or
3
&CC
Vcccc( T) = I( - 4.4 COS
and g are
;
T,,k( +
Vnonbonded=~Q)/R~+Pexp(-~R,,)
(5)
6.4 COS37,k,) ; .
(6)
(7)
All of the potential parameters are given in table 1. The integration of Hamilton’s equations of motion was accomplished as described in ref. [ 18 ] _While we have studied three different ensembles (microcanonical, canonical or isothermal, and isobaric) the majority of the results discussed in section 3 are from the microcanonical ensemble. In these cases, the total molecular Hamiltonian was conserved to at least 6 digits. The other constants of motion, total linear and angular momentum were likewise conserved. 2.2. Normal mode analysis We have carried out a normal mode analysis in order to make comparisons with experimental values. First, it should be noted that our model consists of a single (initially all-trans) polyethylene chain of 100
B.G. Sumpter, D. l+: Nerd /Intramolecular enerDflow m macromolecules
396 Table I Potential
energy parameters
Intramolecular
‘) for eqs. (4)-(7)
bonds
DC_, bond = 80.0 kcal/mol D C_Hbond = 106.7 kcal/mol oc,c=1.94A-1 c&-n= 1.75 A-’ &=1.53A r”,,=l.O9A a) The parameters
were adapted
of the text Nonbonded
Angle bends
I&,=72.3 kcal rad-* mol-’ K ,cc=41.85 kcal rade2 mol-’ K ,,-=38.9 kcal rad-* mol-’ B’,cc=112” 6& = 108.9” B&c” = 109.3” from refs.
interactions
term
o (A6)
p (kcal/mol)
Y (A-‘)
H-H H-C C-C
35.967 289.75 534.61
2556.53 14313.4 149104.5
3.74 3.67 3.60
[ 37-391.
Table 2 Normal mode frequencies Mode
Calculated
CH(asym)
2922 2852 1465 729 1376 1137 503 111 1056
CH(sym) HCH(bend) HCH(rock) HCH (wag) CC(str) CCC(bend) CCCC(torsion) CHZ(twist)
al Calculations from the present work. b, Expenmental values taken from ref. [ 371. ‘1 Inferred from spectral analysis of paraffms
(cm-
’ ) ‘)
Expertmental 2919 2851 1463 721 1370 1133 z 533” 108 1050
(cm-
I ) b,
Symmetry J&u &u &u Bzu &g B,g B2 &g T&u
[ 401.
backbone carbon atoms with two hydrogen atoms per carbon, giving a total of 300 atoms. For an isolated system such as this, one expects to obtain 894 normal modes. A normal mode analysis is in principle simple to perform. There are several well-known algorithms available and the method is well documented in most text books on spectroscopy [ 191. Here we give only the details of the method that we have employed. Since we are ultimately interested in carrying out dynamics calculations, we always start a system in Cartesian coordinates where the total kinetic energy is exactly represented. The atoms of the polyethylene chain are positioned in Cartesian space to obtain the correct all-tram conformation (dihedral angles, bend angles, and bond distances). These initial positions are then oriented about the center-of-mass (centerof-mass reference frame ). Next the system is rotated such that the moment-of-inertia tensor is diagonal
(principle moment-of-inertia frame). This new coordinate frame satisfies the Eckart condition for separating the vibrational motion from the overall molecular rotational motion. In this reference frame the second derivatives of the potential with respect to the Cartesian coordinates are calculated to obtain the force field matrix. This is a real symmetric matrix of 3Nx3N (N is the number of atoms) of the second derivatives and is used to form the secular equation. In order to find solutions to the secular equation, the force matrix is diagonalized using a standard routine [ 201 and the resulting eigenvalues and eigenvectors correspond to the normal modes of the system. Symmetry species can be assigned to the normal modes following the Wilson GF method [ 211. A comparison of the calculated normal mode frequencies to those obtained from experiments is given in table 2. As can be seen, there is very good agreement. The largest deviations are those associated with
B.G. Sumpter, D. R: Noid/lntramolecular
energyjlow in macromolecules
391
the acoustic skeletal modes for which the frequencies of vibration are very small.
3. Results The details of the intramolecular dynamics of a single chain of 300 atoms ( 100 backbone carbons and 200 substituent hydrogens) has been examined in order to develop a better understanding of the energy flow pathways in macromolecular systems. The sensitivity to the initial level of excitation was characterized by examining the rate of energy flow out of specific CH overtones. The overtones were energized according to the equation E(CH)
=w,( v+O.5) -0,x,(
u+OS)~,
(8)
where o, corresponds to the mechanical frequency of the CH vibration and o,xe =0,2/4D,and v the quantum level of excitation (v- 1= the overtone). This excitation energy was added on top of an energy due to thermal motion of the backbone carbon atoms. As discussed in section 2, the total angular momentum was set to zero so that there was only vibrational energy present in the system. With the initial energy deposited into a single CH bond, the equations of motion are integrated. The energy in the CH stretch is calculated in time by determining the displacements of the CH bond and using eq. (4) to calculate the potential energy. The kinetic energy for the CH bond is calculated by projecting the total velocity of the carbon and hydrogen atoms onto the corresponding bond. The bond energy is then, simply, the sum of the potential plus the kinetic energy. This separation is necessary as well as the zero angular momentum restriction since we were only interested in the vibrational energy flow out of the CH bond and not to any components due to the bending, torsional, or overall molecular rotational motions. In a similar fashion, the mode energies for the various CH and CC stretches, HCC, HCH, CCC bends, and CCCC torsions can be obtained. The details of the transformation from Cartesian to curvilinear coordinates such as 8 and r can be found in ref. [ 22 1. Fig. 1 shows the results of the CH energy flow as a function of time for the excitation levels of v=2, 3, 4, 5, 6, 10, 12 for a backbone temperature of 20 K. These results were obtained from averaging an en-
b
1
Fig. 1. (a) The intramolecular energy flow out of a CH bond excited to Y= 2, 3,4,5 (correspondmg to curves in increasing order of imtial energy) at T= 20 K. Energy is m units of kcal/mol and time in units of picoseconds ( ps) (b ) Same as for (a) except for excitations ofv=6, 10, 12.
semble of 20 trajectories, each beginning with a different set of initial conditions. In the present study, the initial conditions were chosen so that the backbone carbon atoms started with different random momenta and coordinates and the phase averaged momenta and coordinates for the excited CH bond were chosen from a random distribution appropriate to a non-rotating Morse potential [ 17 1. From examination of fig. 1, it can be seen that the higher levels of excitation (v = 4,5,6, 10, 12) exhibit a rather rapid decay of energy followed by a much slower decay. For the lower levels of excitation (v= 2,3) the energy flow out of the CH stretch is very slow. In fact, the energy appears to be localized on the timescale of 2 ps. At
398
B.G. Sumpter. D. I+: Nord /Intramolecular
first thought, the instability of the higher levels of excitation versus the relatively stable lower levels could be attributed to the simple fact of increasing anharmonicity as a function of energy. This of course is the fundamental reason behind this observation but closer examination reveals a more detailed mechanism. In fig. 1, the higher levels of CH stretch excitation lead to a two-phase characteristic energy flow. The initial dynamics is very rapid and results in an almost exponential decay of the energy on a timescale of less than 0.5 ps. The dynamics following this advent(s) (referred to as long-time dynamics) is much slower and consumes as much as 1.5 ps to reach a stable energy. The two-phase behavior has been characterized as a prominent feature in studies of small polyatomic molecules. In fact, the decay of CH overtones in systems such as benzene [ 12-l 41, methyl isocyanide [ 231, models for alkanes such as butane [24], dimethylnitramine [ 25 ] and other systems [ 14 1, have very similar qualitative energy decay behavior. In these systems the mechanism for the energy flow was determined to be primarily due to resonant interactions. Therefore, we strongly expect that similarly there may be a low-order resonance between the CH stretch and the HCH bending vibrations. A first-order approximation to the vibrational frequency of an anharmonic vibration such as the CH stretch can be obtained from the frequency dependence on the bond energy. The equation is:
energy jlow rn macromolecules
versible decay (around 0.3 ps). Similar results can be found for the v levels 6, 10, and 12. The resonant pathway of energy flow from the CH stretch into the HCH bend can be better seen by examining the energy as a function of time in the HCH bend. In fig. 2a, the energy flow out of the excited CH stretch for v=6 is plotted as a solid line and the energy flow into the HCH bend is plotted as a dashed line. Initially there is no significant participation between the modes (the first 0.1 ps). An instability appears to occur around 0.15 ps and the result is an incredible flow of energy from the CH stretch into the HCH bend. The HCH bend energy then slowly decays and this energy is redistributed to the other vi-
,
. 02
04
06
08
1
12
14
16
I8
12
14
16
18
-2
Tlm(pr)
(9) where v” is the fundamental frequency for the CH stretch, ECH is the energy of the CH bond, and DCHis the dissociation energy of the CH bond. Substituting the correct parameters into this equation for the energy level corresponding to that where significant flow occurred for v = 5 yields a frequency of vCH= 2460 cm- ‘. Using the resonance relation nw, - mw2 = 0, we obtain n = 5 and m = 3 which indicates that there is a possible 3: 5 resonance between the CH stretch and the HCH bending vibrations. The existence of such a resonance will act to facilitate energy flow out of the CH stretch into the HCH bend. As the energy is removed from the CH stretch, the vibrational frequency increases and finally reaches and energy where it is in a 1: 2 (Fermi) resonance with the HCH bend. This is the point where the energy flow exhibits irre-
0
02
"4
06
08
2
Fig. 2. (a) Energy flow out of the initially excited CH stretch (u= 6) mto the HCH bend. The solid lme IS the energy in the CH bond and the dashed line is the energy of the HCH bend. (b) Same as (a) except the energy in the CH bond adjacent to the excited CH stretch and the energy in the terminal CH bond are also plotted.
B.G. Sumpter. D. W. Nold /Intramolecular
brational modes as shown in fig. 2b. As can be seen, after about 2 ps, the various vibrational modes have obtained approximately equivalent amounts of the initial energy. This timescale corresponds to the intramolecular redistribution of the initial energy. The redistribution is facilitated by the Fermi resonance between the CH stretch and HCH bend. As the energy in the initial excited CH stretch decays into the HCH bend, the CH stretch tunes into a 1: 1 resonance with one of the alternate CH stretches. Energy flows into this CH stretch which then tunes into a loworder resonance with the HCH bend. This process continues and results in rapid redistribution of the energy throughout the system. It should be pointed out that these characteristics are dependent on the model for the polymer chain. For example, if the torsional modes (gauche trans isomerization potential) is not included, that is, the polyethylene chain is constrained to be planar, the rate for the energy decay and redistribution are on completely different timescales. Although the resonant interaction mechanism is still in effect due to the incredible density of states for macromolecules, there is a significant difference in the timescales for the observed processes (the timescale is slower by about 1 ps). This demonstrates the importance of the torsional modes (gauche trans isomerizations) in the internal dynamics leading to energy flow. Similar conclusions have been discussed in a number of studies on small polyatomics [ lo]. The phenomenon of Fermi resonance is a well studied example of spectral intensity sharing between vibrational modes [ 26 1. In polyethylene, the existence of Fermi resonance has been inferred for many years [ 271 and recently verified in theoretical calculations [ 15,281. In the present study we have shown that energy flow from an excited CH stretch occurs through resonant pathways marked by low-order resonances such as the Fermi resonance. In order to obtain a better visual picture of the dynamics, a time dependent local configuration plot analysis [ 29 ] is carried out for the v = 6 CH stretch excitation. This analysis simply involves plotting sections of a trajectory at times where the dynamics appears to be interesting. From examination of fig. 2a it can be seen that the initial energy flow begins at about 0.15 ps and is irreversible at about 0.2 ps. If we plot the bond distance of the CH as a function of the bond distance of the adjacent CH stretch and also as a function of the
energyflow rn macromolecules
399
HCH bend, we can see if the characteristic shapes of resonances in configuration space are present [ 301. In general, a 1: 2 resonance leads to a C-shaped ligure. 1: 3 to a S-shaped figure. 2 : 3 to a gamma-shaped figure, 3 : 4 to a pretzel shaped figure [ 3 11, etc. In fig. 3a, a plot of the CH bond distance as a function of the adjacent CH bond distance is given. The plot is for the initial 0.1 ps of dynamics and demonstrates that on this timescale the energy is very localized in the initial excited CH stretch. This type of behavior is classified as local mode motion. As time increases, 0.15-0.4 ps, the local mode behavior is rapidly destroyed. In fig. 3b the CH bond distance is plotted as a function of the HCH bending angle. As the energy flows from the CH stretch into the HCH bend, the bending angle begins to have larger amplitude displacements from its equilibrium angle. After several hundred vibrations, a C-shaped figure appears in the center of this pot. This C-shape corresponds to the time of 0.2 ps and remains stable for about 0.1 ps. A better view of the C-shape is illustrated in fig. 3c. Clearly, the dynamics has resulted in a situation where the CH stretching motion is at a rate of 2 times faster than the HCH bending motion. This frequency degeneracy is coupled to a large set of “bath” states (the remaining 892 vibrational modes) and thus causes rapid and irreversible energy flow. Now that it has been determined why the energy flow exhibits the two-phase decay behavior we turn to the lower CH stretch excitations. As was shown in fig. la, for ~=2 and 3, the initial energy exhibited very little decay on a timescale of 2 ps. This is because there are no strong couplings to facilitate the energy flow. However, this does not mean that the vibrational states v=2 and 3 are stable. If we examine the energy flow properties for a slightly longer timescale (5 ps) it happens that both states become unstable and rapid energy flow occurs. This is shown in fig. 4 for the energy flow out of the v = 2 level of the CH stretch. After about 3.5 ps, the initially stable CH stretch becomes unstable and the energy flow exhibits the two-phase behavior as in the higher excitation cases. This type of behavior is very similar to that attributed to bottlenecks in phase [ 321 (in this case the bottleneck is the one referred to as the golden mean [ 331: a frequency ratio between the CH stretch and HCH bend of approximately 1.6 18). In short, this means that energy flow is localized due to a confining
B.G. Sumpter, D. W. Nord /Intramolecular
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a
Fig. 4. The intramolecular tov=2at T=20K.
Fig. 3. (a) A plot of the excited CH bond length versus the adjacent CH bond length. The plot is for a ttme period of 0.1 ps and the bond lengths are in units of angstrom. (b) A plot of the excited CH bond length (in angstrom) versus the HCH bend angle (m degrees). The time period is from 0. I5 to 0.4 ps. (c) Same as for (b) except for a shorter time period (0.2-0.3 ps). The labels q. and q, stand for CH stretching and HCH bendmg vibrations. respectively.
energy flow out of a CH bond excited
boundary that results from the breakup of other stable motions. Escape channels through this barrier can be achieved by way of resonant interactions [ 29 1. However, in general the energy flow is localized for extended periods of time. The localization means that there is a lack of coupling to the bath states. In the present case, the energy is low causing the anharmonicity to be low and thus the density of states. Energy decay only occurs after a slow adiabatic leaking of the energy to the bend has occured. At this time, the CH stretch is “tuned” into a 1: 1 resonance (determined by a local frequency analysis [ 291) with a subset of the CH stretches in the system and energy flow occurs. Thus the bottleneck is broken on a relatively short timescale. The existence of phase space bottlenecks such as this are a common example of nonstatistical behavior [ 321. Needless to say, the observation that nonlinear dynamical features such as resonances and phase space bottlenecks also play an important role in the internal dynamics of macromolecular systems is of great interest and demonstrates the overall generic applicability of these phenomena. The sensitivity of the above results to pressure and temperature was also studied. In order to examine the dynamics under different external conditions, we have used the method of Nose dynamics [ 341. Briefly, Nose dynamics is a modified version of Newtonian dynamics so as to reproduce both the canonical and isothermal-isobaric probability densities in the phase space of an N-body system. It provides for a rela-
B.G. Sumpter, D. U: Nold /Intramolecular
tively simple way to induce external pressure and constant temperature. It was found that increasing the external pressure tends to only slightly affect the energy flow decay rates (the rates tending to be slower) while temperature can cause a significantly enhanced rate of energy flow. For a temperature change of 300 K, the energy flow decay time decreases by about 1 ps. This result is also fairly independent of the excitation level of the CH stretch. Again, the temperature is playing the role of increasing the total energy and thus increasing the total density of vibrational states available for coupling. As expected, this results in a dynamical behavior which is at faster rates. Finally, it is interesting to compare the results of our single polyethylene chain study to those of crystalline polyethylene. To accomplish this task, we have carried out a number of preliminary calculations on a model for orthorhombic polyethylene. The model consisted of 19, 100 CH2 group polymer chains. An initial excitation energy was deposited in the center of the crystal and the energy flow was followed as discussed above for the single chain. The results indicate that the intramolecular energy redistribution is enhanced somewhat, occurring on a timescale of about 0.1 ps. The total energy redistribution however involves the migration of the energy throughout the large crystal. This process is an intermolecular process, involving chain-to-chain energy flow. The intermolecular energy flow was found to occur on a timescale of about 0.2 nanoseconds, about two orders of magnitude slower than that for the intramolecular process of the single chain. The intermolecular process is also found to be dependent on the temperature and pressure of the system. These results are reported in more detail in ref. [ 35 1. Cross-correlation functions were calculated to examine the intermolecular energy flow pathways though the crystal. From this analysis, it was found that the intermolecular energy flow from the center of the crystal occurs by radial diffusion, crystal layerby-layer, until it reaches the outside edge. The diffusion of the energy was found to be reasonably described using the Langevin equation [ 361. In this regard, we have performed as series of calculations [ 351 (Langevin dynamics using different friction coefficients) to verify this fact.
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4. Conclusions In this letter we have discussed the results of a detailed study of the intramolecular energy flow in a single chain model of polyethylene. The intramolecular dynamics of the polyethylene molecule was studied as a function of the potential energy surface, CH stretch excitation, temperature, and pressure. The rate of energy flow from CH stretching modes ( v = 5- 12 ) was observed to be very rapid and irreversible, occurring on a timescale of less than 0.5 ps at low temperatures, and increases with overall increasing temperature. A characteristic two-phase energy flow behavior was observed, where the initial part consisted of a very rapid flow (due to the decay of the initial excitation) followed by a slower flow (due to energy redistribution throughout the system). The mechanism for the initial facile energy flow involves strong resonant pathways. In particular, a CH stretch/HCH bend Fermi ( 1: 2) resonance was shown to dominate the short-time dynamics and facilitates the overall process of energy redistribution. The long-time dynamics, associated to complete redistribution of the initial CH stretch energy with all of the 894 available vibrational modes, occurs within a time of 2 ps. This intramolecular redistribution timescale is compared to that of intermolecular redistribution (redistribution in the solid phase as opposed to a single chain). A preliminary study of energy flow in a crystal of polyethylene (system containing 19 polyethylene chains) revealed that the timescales for intramolecular and intermolecular energy redistributions are quite different. Intramolecular redistribution in the polymer crystal was similar to that in the single chain study but occurred on a slightly shorter timescale (0.1 ps). Intermolecular (chain-to-chain) redistribution, however, was two orders of magnitude larger than the intramolecular redistribution timescale. Thus the energy redistribution behavior in the solid phase of polyethylene demonstrates two distinctly different decay processes; one corresponding to the intramolecular and other corresponding to the intermolecular energy redistribution. The results of this study are in very good accord with the experimentally measured results of Graener and Laubereau [ 161. Our computer study shows for the first time a definite resonance enhanced intramolecular redistribution mechanism that occurs on a
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timescale of about 2 ps and a slower intermolecular process on a nanosecond timescale (0.2 ns).
Acknowledgement This work was supported by the Office of Basic Energy Sciences, US Department of Energy, under Contract No. DE-AC05-840R2 1400 with Martin Marietta Energy Systems, Inc., and by the Polymer Program of the National Science Foundation, present Grant No. DMR-88 184 12. BGS acknowledges the support provided by the Direction General de Investigation Cientilica y Ttcnica of the Ministry of Education and Science (MEC) of Spain for his stay at the Universidad Complutense de Madrid during the summer of 199 1. The computations were performed on the IBM 308 1 and 3090 at the University of Tennessee, the CRAY-YMP/48 at the National Center for Supercomputing Applications at the University of Illinois (grant CHE890025N), and the CRAY YMP at the NSF Pittsburgh Supercomputing Center (grant CHE9 1OOOP Cray sponsored research). We would like to thank E.L. Sibert III and S.K. Gray for useful suggestions on the use of cross-correlation functions and Langevin dynamics. We also thank Coral Getino for many discussions on energy transfer in polyatomic molecules.
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