This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 10445–10454 10445

Cite this: Phys. Chem. Chem. Phys., 2012, 14, 10445–10454

Ballistic energy transport along PEG chains: distance dependence of the

transport efficiency

Zhiwei Lin, Nan Zhang, Janarthanan Jayawickramarajah and Igor V. Rubtsov*

Received 18th January 2012, Accepted 11th April 2012

DOI: 10.1039/c2cp40187h

Dual-frequency relaxation-assisted two-dimensional infrared (RA 2DIR) spectroscopy was used

to investigate energy transport in polyethylene glycol (PEG) oligomers of different length, having

0, 4, 8, and 12 repeating units and end-labeled with azido and succinimide ester moieties

(azPEGn). The energy transport initiated by excitation of the NRN stretching mode of the azido

group in azPEGn in CCl4 at ca. 2100 cm�1 was recorded by probing the CQO stretching modes

(reporters) of the succinimide ester moiety. Sensitive to the excess energy delivered to the reporter

modes, RA 2DIR permits observation of both the through-bond and through-solvent energy

transport contributions. The cross-peak data involving the reporter modes with different thermal

sensitivity and the data for mixtures of compounds permitted concluding that through-bond

energy transport is the dominant mechanism for most cross peaks in all four azPEGn compounds.

The through-bond energy transport time, evaluated as the waiting time at which the cross peak

maximum is reached, was found to be linearly dependent on the chain length of up to 60 A,

suggesting a ballistic energy transport regime. The through-bond energy transport speed

determined from the chain-length dependence of Tmax in CCl4 is found to be ca. 450 m s�1.

The cross-peak amplitude at the maximum decays exponentially with the chain length; a

characteristic decay distance is found to be 15.7 � 1 A. The cross-peak amplitude at zero waiting

time, determined by the end-to-end distance distribution, is found to decay with the chain length

(L) as BL�1.4, which is close to predictions of the free flight chain model. The match indicates

that the end-group interaction does not strongly perturb the end-to-end distribution, which is

close to the ideal random coil distribution with the Gaussian probability density.

1. Introduction

Essentially all chemical reactions involve some vibrational

energy transfer and for many reactions the vibrational energy

flowing into reactants and/or out of products plays an essen-

tial role. Understanding the energy transport properties of

molecules can lead to developing better models of chemical

reactivity1–5 and provide a thorough control of the outcome of

chemical reactions.6 Efficient energy transport and/or energy

dissipation is vital for devices of different dimensions, which

range from a molecular scale, such as molecular junctions and

molecular wires,7 to a macroscopic scale, such as thermo-

electric energy converters and optical limiters.

Energy transport in general can occur diffusively or ballis-

tically. The diffusive transport in macroscopic objects, referred

to as heat conduction, is governed by a Fourier law stating

that the heat flux is proportional to the temperature gradient.

In microscopic objects, such as molecules, the local tempera-

ture is ill defined; the diffusive energy transport results from an

intramolecular vibrational energy redistribution (IVR) process,

which involves energy hopping between various vibrational

states.4,8 The IVR process occurs between spatially over-

lapping modes and therefore results in the energy transport

in the molecule. The IVR steps in both forward and backward

directions in the molecule result in the diffusive energy trans-

port regime. In the ballistic regime the energy is transferred

by vibrational states delocalized over the whole transport

region. As a result the ballistic energy transport can be very

efficient. The ballistic transport regime has been observed

experimentally in macroscopic systems at low temperatures

(crystals)9,10 and in mesoscopic samples, such as carbon

nanotubes.11 Both acoustic and optical phonons can transfer

energy ballistically.3,12

In molecules both energy transport regimes have been

intensively studied theoretically3,7,13–18 and experimentally.

Recent development of time-resolved spectroscopic methods,

including infrared,19 two-dimensional infrared (2DIR),20,21

relaxation-assisted 2DIR (RA 2DIR),22–24 and combined

infrared excitation and Raman probing25,26 spectroscopies,

opened an avenue for investigating experimentally the energy

transport in molecules. Diffusive energy transport was observedDepartment of Chemistry, Tulane University, New Orleans,LA 70118, USA. E-mail: irubtsov@tulane.edu

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10446 Phys. Chem. Chem. Phys., 2012, 14, 10445–10454 This journal is c the Owner Societies 2012

in many molecular systems22,24,27–30 and for various amounts

of excess energy.31,32 Ballistic energy transport in molecules

following electronic excitation (B17 000 cm�1) was reported

by Troe and coworkers; the transport with nearly constant

velocity was observed through alkane chains of various

lengths containing up to 6 carbon atoms.14,33,34 A dynamic

transition in a peptide helix in response to a temperature

change was attributed to a switch between the diffusive and

ballistic energy transfer mechanisms.35 Dlott and coworkers

have found the energy transport in long-chain stretched

hydrocarbons exposed to ca. 800 K transient temperature

gradient to be ballistic in self-assembled monolayers of alkanes

at a gold surface.36

Ballistic energy transport in polyethylene glycol (PEG)

oligomers of various chain lengths in a chloroform solution

has recently been discovered.37 The transport was initiated by

excitation of the NRN stretching mode of the azido moiety

(2100 cm�1), attached at one end of the PEG oligomer (Fig. 1).

The vibrational energy transport was detected at the other end

of the oligomer by an infrared tag using the RA 2DIR method.

The energy transport has been detected at ca. 60 A distance

with the transport velocity of ca. 550 m s�1.37 Many intriguing

questions remain unanswered, for example: what is the energy

transport efficiency? How the transport speed and efficiency

are affected by the oligomer conformation and by the solvent?

In this paper the energy transport in PEG oligomers is

reported in a different solvent, CCl4, where the PEG mean

conformation is expected to be different than that in chloro-

form. The efficiency of the energy transport along the chain

was evaluated for the first time. In addition, several test

measurements are reported (Sections 3.3 and 3.6), which target

the nature of the transport. The energy transport data measured

in the two solvents are compared in Section 4.2.

2. Experimental details

2.1. Heterodyned dual-frequency 2DIR measurements

Details of the dual-frequency 2DIR setup with heterodyned

detection can be found in ref. 38 and 39. Two in-house-built

optical parametric amplifiers followed by different frequency

generation units were used to generate independently frequency-

tunable mid-IR pulses of ca. 120 fs pulse duration. One of the

beams was split into two equal parts, each of ca. 1.3 mJ energy,which served as excitation pulses interacting with the sample

(k1 and k2). A small portion (B4%) was split from the second

beam to serve as a local oscillator (LO) for heterodyned

detection, while the main part (B1.5 mJ) was used as a third

beam (k3) interacting with the sample. The spectra of the k1,

k2, k3, and LO pulses were tuned to ca. 2100 and 1780 cm�1,

respectively. A third-order signal generated by the sample was

picked at the phase matching direction (�k1 + k2 + k3),

mixed with the LO, delayed by the time delay t, and detected

by anMCT detector (Infrared Associates). The delays between

the first and the second and the second and the third pulses are

referred to as the dephasing time, t, and the waiting time, T,

respectively. Linear-motor translation stages (PI Inc.),

equipped with hollow retroreflectors, were used to control

the delays between the IR pulses. During the experiments the

positions of the translation stages were accurately measured

with an external interferometric system based on a continuous-

wave HeNe laser.29 2DIR spectra were obtained by a double

Fourier transformation of the M(t, t) data sets. The waiting

time dependencies for the relaxation-assisted 2DIR measure-

ments were measured by acquiring 2DIR F(t, T) data sets

while keeping the dephasing time (t) constant at 167 fs.22 The

F(t, T) data sets were then Fourier transformed along the t

direction and presented as a set of one-dimensional ot spectra

at various T values. The experimental conditions for acquiring

the F(t, T) data sets were selected so that only a single peak

along ot was excited. The cross-peak amplitude at each T

delay was determined by integrating the ot absolute-value

peak in the vicinity (at ca. 50% level) of its maximum and

subtracting the integrated and normalized background. The

resulting cross-peak amplitudes were plotted as a function of

the waiting time, T.

To suppress completely the CQO diagonal peaks at around

1740–1820 cm�1 which were generated due to a tail in the

spectrum of the k1 and k2 pulses, a high-frequency pass

filter was used in the k1 beam to cut-off the frequencies below

1950 cm�1. All experiments were performed at room temperature,

23.5 � 0.6 1C.

For the measurements targeting absolute values of the cross

peaks in different compounds the concentrations were made

the same within 5%. To minimize laser fluctuations, the

measurements for all four compounds were performed on

the same day one after another recording the cross peak

amplitudes at the waiting times close to zero, around the peak,

and at the plateau. Complete waiting time dependencies were

then measured for each compound.

2.2. Sample preparation

The azPEG4, azPEG8, and azPEG12 compounds were purchased

from Quanta BioDesign, Ltd. and were used as received

(Fig. 1). Succinimidyl-4-azidobutyrate abbreviated here as

azPEG0 was synthesized according to a literature procedure.40

Carbon tetrachloride (Fisher, 99.9%) was used as a solvent.

To speed up mixing of azPEG12 and CCl4, the mixture was

sonicated and the resulting solution was filtered through a

0.45 mm PTFE syringe filter to remove any particles. A sample

cell made of two CaF2 wafers and a 50 mm thick Teflon spacer

was used for all IR measurements.

2.3. DFT calculations

Normal mode harmonic calculations were performed for the

azPEG4 compound using the laboratory cluster and the

Gaussian 09 software package.41 The calculations were done

Fig. 1 Structures of the azPEG0, 4, 8, and 12 compounds.

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using the density functional theory (DFT) method with

B3LYP functional and a 6-311G(d,p) basis set under vacuum.

A tight optimization criterion and an ultrafine integration grid

were used.

3. Results

The four studied compounds (Fig. 1) feature azido and

succinimide ester end groups linked by a PEG oligomer with

various numbers of repeating PEG units (n) of 0, 4, 8, and 12

(azPEGn). Note that the azPEG0 compound has three bridging

carbon atoms, compared to two expected formally from the

molecular formula at n= 0 (Fig. 1). The RA 2DIR experiments

for all four compounds were performed at ca. 55 mM concen-

tration. At this concentration there are ca. 190 solvent molecules

per one solute molecule. This amount is expected to be sufficient

to solvate well azPEG0, 4, and 8 compounds, but might be a bit

low for azPEG12. In solution PEG oligomers are expected to

adopt a random coil conformation.42 Only a weak specific

attraction of the end groups is expected based on the chemical

nature of the groups, which was confirmed by the RA 2DIR

measurements (vide infra). Carbon tetrachloride is a poorer

solvent for PEG oligomers compared with chloroform; a more

compact coiled structure is expected in CCl4.

3.1. Linear absorption spectra of azPEGn in CCl4

The absorption peak of the N3 moiety at around 2106 cm�1

(Fig. 2) belongs to a N1RN2 stretching motion located

at the two outer nitrogen atoms (N1N2N3–R); the N2N3

stretching mode of the two inner nitrogen atoms is found at

ca. 1300 cm�1,43 indicating that the N2N3 bond order is about

one and a half. The NRN peak has very similar shapes for

the azPEG4, 8, and 12 compounds, but is substantially sharper

for the azPEG0 compound. The NRN transition dipole in

azPEG0 is ca. 1.09 fold larger than those in the other three

compounds.

The linear absorption spectrum of the succinimide ester

moiety has three characteristic peaks at 1748, 1789, and

1819 cm�1 (Fig. 2). The peak at 1748 cm�1 originates from an

asymmetric stretching motion of the two carbonyls of succinimide.

The other two peaks (1789 and 1819 cm�1) are due to almost

equal-contribution combinations of the symmetric stretch at

succinimide and the CQO stretching motion of the ester.44

Orthogonality of the transition dipole of the asymmetric CQO

stretch of succinimide to the CQO transition moment of the

ester is the reason for a zero contribution of the ester CQO

motion to the mode at 1748 cm�1. The CQO region spectra

of all four compounds are also similar. A small peak at

ca. 1710 cm�1, which is most prominent in azPEG0, is due

to traces of water in the sample. We found that, even when the

sample is placed into the sample cell, this peak can grow

substantially with time if the sample cell is exposed to the

laboratory air that typically has ca. 55% relative humidity.

This process is even faster when the sample is dissolved in

more volatile chloroform, which slowly evaporates from the

‘‘sealed’’ sample cell and is replaced by wet air. Therefore, all

experiments were performed with fresh samples (not older

than 2 days); the sample cells with the samples were held in

a dry-air purged box. To target accurately the absolute

cross-peak amplitudes in RA 2DIR measurements the

concentrations of all four compounds were prepared to be

the same within 5%.

3.2. RA 2DIR measurements

The dual-frequency RA 2DIR measurements were performed

for the four azPEGn compounds focusing on the cross peaks

between the NRN stretching mode of the azido moiety and

the three CQO modes of the succinimide ester moiety. The

waiting time dependencies for the three cross peaks for each

compound are shown in Fig. 3. All twelve dependencies have

essentially the same shape type. All dependencies show a

substantial cross peak amplitude at the waiting times close

to zero (T B 0), which comes from a direct through-space

coupling between the initially excited NRN mode and the

reporter modes (CQO). Since the oligomer chains are flexible,

the end-to-end distance in some conformations is small.

Fig. 2 Linear absorption spectra of the four azPEGn samples in CCl4used in the RA 2DIR measurements. The concentrations differ by less

than 5%.

Fig. 3 Waiting-time dependencies of the NRN/1748 cm�1 (black),

NRN/1789 cm�1 (blue), and NRN/1819 cm�1 (green) absolute-

value cross peaks for the four compounds indicated. Best fits with an

asymmetric bell-shaped function, y= y0 + A*exp(�exp(�z)� z+1),

where z = (x � xc)/w, are shown with thin red lines.

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The through-space interaction of the excited and reporting

modes in such conformations is the source of the cross peak at

zero waiting time. In azPEG0 an additional mechanical

through-bond coupling may also contribute to the mode

interaction, although its contribution is expected to be small

due to a substantial through-bond distance of ca. 7.3 A.

Naturally the cross peak amplitude at T B 0 depends on the

coupling strength of the two IR labels. When two modes

interact, their frequencies change due to the interaction. In

addition, the combination band level of the two modes is

shifted as well. While it is difficult to measure experimentally the

shifts of the fundamental transitions, the shift of the combi-

nation band is easily accessible via 2DIR spectroscopy.45,46 One

can think of the shift of a combination band level in a simpler

way as a shift of the reporter mode frequency in response to

excitation of the initially excited mode.

With an increase in the waiting time the cross peak ampli-

tude increases, reaches maximum, and then decreases. This

dynamics is associated with a relaxation-assisted contribution

(vide infra).22,23 The NN lifetimes measured for azPEG0 and

azPEG4 in CCl4 using a pump–probe method44 were found to

be 1.2 � 0.2 ps and 1.1 � 0.2 ps, respectively, which are rapid

on the time scale of the dynamics in Fig. 3. The NN excited

mode relaxes to combinations of other modes in the molecule,

as a part of the IVR process. Several relaxation pathways,

often with comparable efficiencies, typically contribute to

relaxation of a particular mode.8,26,29,47 The energy relaxation

and the IVR process have a spatial component—with a time

increase the excess energy propagates further and further from

the group initially excited by a photon, exciting more and

more remote vibrational modes. It has been demonstrated

for numerous molecular systems that the through-bond

energy transport is very efficient at early time delays after

excitation.24,32,44 When the excess energy arrives at the region

where the reporter mode is located it excites various modes

there, including those strongly coupled to the reporter. Transition

dipole interaction is not required (although may contribute)

for this coupling as a substantial mechanical interaction

among the reporter mode and many other modes spatially

overlapping with the reporter is often found. Excitation of the

modes strongly coupled to the reporter modes via energy

transport results in a shift of the reporter-mode frequency,

which in turn causes enhancement of the cross peak.22,24 The

cross peak enhancement is delayed by the time needed for the

excess energy to propagate to the region in the molecule where

the reporter group is located. The delay time at which the cross

peak reaches maximum, Tmax, referred to as the energy trans-

port time, can be used to characterize the transport. Further

decrease in the cross peak amplitude is associated with the

energy dissipation to the solvent. Interestingly, the cross peak

amplitudes do not decay to zero but reach a plateau. The

plateau is associated with a complete thermalization of the

excess energy in the excitation region of the sample. This region

is macroscopic (a transient grating period is ca. 10 mm) and

further thermalization occurs on much longer time scales.37,44,48

Note that to have a plateau, the frequency of the reporter mode

must be sensitive to temperature—the frequency should change

if the sample temperature is changed.44 We found that the majority

of vibrational modes in molecules are sensitive to temperature,

so it is actually more difficult to find a reporter which is not

sensitive to temperature. The temperature increase in the

sample introduced through relaxation of the excited NRN

stretching mode was evaluated to be ca. 0.1 1C.44 Note that

since only a fraction of all NN modes in the sample is excited

by k1 and k2 pulses (ca. 20%), only this fraction of the CO

groups contributes to the cross peak at waiting times close to

zero. When thermalization is completed all CO modes will

respond to the temperature increase and contribute to the

cross peak. While this, ca. 5-fold, enhancement makes it easier

to measure the plateau, it makes harder to distinguish the

through-bond energy transport from the overall thermaliza-

tion involving a through-solvent transport. Notice also that

the pairwise correlation (in this case between NN and CO

modes) is lost when complete thermalization occurs as

several reporter modes (in average) respond to a single excited

NRN mode.

For the azPEG0, 4, and 8 compounds, the through-bond

transport cross-peak contribution clearly dominates over the

thermal (through solvent) cross peak contribution, signified by

the plateau. For example, in azPEG8 the NN/1748 cross peak

at the plateau amounts to only ca. 25% of that at the

maximum. The plateau is much smaller for NN/1789 and

NN/1819 cross peaks. As apparent from the plateau values,

the 1748 cm�1 peak has the largest thermal sensitivity com-

pared to the peak at 1789 cm�1 and especially at 1819 cm�1.

Indeed, the sensitivity of the central frequency to temperature

evaluated for azPEG4 in chloroform at 23.5 1C was found at

+0.057� 0.004, +0.028� 0.003, and �0.003� 0.003 cm�1 K�1

for the low-, middle-, and high-frequency CQO peaks, respec-

tively.44,49 A comparison of the RA 2DIR data measured in

chloroform and CCl4 indicates that the thermal sensitivity for

the modes at 1748 and 1789 cm�1 is substantially smaller in

CCl4. Note that the heat capacity per unit volume for CCl4 is

slightly smaller than that for chloroform (1.35 vs. 1.42 J cm�3 K�1),

which results in a slightly larger temperature increase in CCl4upon NRN group excitation. The reduced thermal sensitivity

in CCl4 permits separating better the through-bond energy

transport from the through-solvent transport. Even in

azPEG12 the plateaus for the NN/1789 and NN/1819 cross

peaks are much smaller than the values at the maximum

amounting only at ca. 25% and 35%, respectively. Thus, the

cross-peak dynamics in all four compounds (with the excep-

tion of the NN/1748 peak in azPEG12 where the two contri-

butions are comparable), especially at the waiting times less

than ca. 20 ps, has a dominant contribution from the through-

bond energy transport.

3.3. Thermal cross-peak contribution

Nevertheless, the thermal contribution to the cross peak

also has its own T-dynamics affecting the shape of the overall

T-dependence. It is challenging to isolate such contribution

from the whole dynamics, especially if needed for the same

molecular system and under the same experimental conditions.

Several experiments were performed to get an estimate of the

thermal dynamics. First, the waiting time dependencies were

measured in azPEG8 at much higher concentrations. The

temperature increase in the sample under such conditions is

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much larger resulting in an overwhelming contribution of the

thermal cross peak (Fig. 4A). No peaks in the waiting time

dependencies were observed under such conditions: the cross

peaks gradually increase with a characteristic time of ca. 20 ps

(depending on the concentration) until plateaus are reached.

Second, the RA 2DIR measurements were performed with a

mixture of two non-aggregating compounds, one featuring an

azido moiety (N,N-dimethylnicotinamide), which was excited,

another featuring an amide (methyl 4-azidobutanoate); the

amide-I mode served as a reporter. Indeed, the waiting-time

dependence for the NN/amide-I cross peak (Fig. 4B) shows a

very small cross peak at T = 0 confirming the absence of

aggregation. The NN/amide-I cross peak was found to grow

exponentially with a characteristic time of ca. 18.7 ps at

80 mM concentration of both compounds in chloroform

(Fig. 4B). Again, no peak was observed in the waiting time

dependence. The modelling of the energy dissipation was

performed using a heat conduction equation (vide infra).

Assuming that the frequency shift of the reporter mode is

proportional to the temperature increase, the random distri-

bution of the reporters (Fig. 4B) leads to a mean frequency

shift of the reporters, which is independent of the time delay.

Introduction of the excluded volume, the space around the

heat source where no reporters are allowed, leads to a growth

of the mean frequency shift with time, which can be seen as a

growth of the cross peak amplitude. Thus the dynamics in

Fig. 4B likely characterizes the excluded volume, while that in

Fig. 4A is also influenced by the reporter tethered to the

excited group. However, since the mean end-to-end distance

for the tethered reporter is comparable to the radius of the

sphere the energy is dissipated into, the through-solvent

transport to the tethered reporters is expected to result in a

gradual growth of the cross-peak until the plateau is reached.

3.4. Dependence of Tmax on the chain length

To determine the Tmax values, the data in Fig. 3 were fitted in

the vicinity of the maximum with an asymmetric bell-shaped

function (see Fig. 3 caption). Fig. 5 shows the energy transport

time, Tmax, as a function of the chain length for three cross

peaks measured (Table 1). The chain length, L, for the

NN/1789 and N/1819 cross peaks was taken as a through-bond

distance between the N2 atom of the azido moiety and the

carbon atom of the ester. Since the 1748 cm�1 is located solely

at the succinimide moiety, but not at the ester, the lengths of

the C–O and O–N bonds in the succinimide ester were added

to obtain the chain length for the NN/1748 cross peak.

Interestingly, all three dependencies in Fig. 5 can be

approximated well by a linear function, which suggests that

the energy transport occurs with a constant speed. A fit with a

linear function made separately for cross peak data resulted in

(2.72 � 0.23 ps) + (0.198 � 0.008 ps A�1)L, (2.54 � 0.31 ps) +

(0.225 � 0.011 ps A�1)L, and (3.17 � 0.05 ps) + (0.219 �0.002 ps A�1)L for the cross peaks involving the reporting

modes at 1748, 1789, and 1819 cm�1, respectively.

The Tmax values for the same compound but different cross

peaks are expected to be similar. Indeed, the points for the

NN/1748 and NN/1819 cross peaks are essentially overlapping

for all four compounds (Fig. 5). However, the Tmax values for

the NN/1748 cross peak are somewhat smaller than those for

the other two cross peaks, especially for azPEG8 and 12,

which results in a smaller slope in the distance dependence of

Tmax for this cross peak.

The NN/1748 cross-peak data contain the largest thermal

contributions as the 1748 cm�1 mode is the most sensitive to

temperature. This is likely a reason for smaller Tmax values found

for the NN/1748 peak in azPEG8 and 12, where the plateau

Fig. 4 Waiting-time dependencies for the NRN/CQO cross peaks

in pure azPEG8 (A) and for the NRN/amide-I cross peak in the

N,N-dimethylnicotinamide and methyl 4-azidobutanoate mixture (B).

Fig. 5 Energy transport time, Tmax, as a function of the chain length

is shown for the three cross peaks indicated in the inset. The linear fits

were made for cross peak data separately (thin lines of the matching

color). To avoid congestion, the chain lengths for the NN/1789 cm�1

and NN/1819 cm�1 cross peak points were shifted by +0.4 A and

�0.4 A, respectively. The linear fit of the Tmax data for the NN/1819

cross peak for azPEGn in chloroform (CDCl3) is shown with a dashed

line.37

Table 1 Energy transport times, Tmax (in ps), determined for threecross peaks for azPEGn in CCl4 and the chain lengths (in A)

azPEG0 azPEG4 azPEG8 azPEG12

NN/1748 4.7 � 0.4 8.1 � 0.4 11.5 � 0.5 14.4 � 1.1NN/1789 4.4 � 0.5 7.8 � 0.4 12.6 � 1.0 15.6 � 0.9NN/1819 4.7 � 0.6 8.2 � 0.7 12.1 � 1.0 15.8 � 1.3Chain lengtha 10.4 26.4 44.0 61.5

a The distances between the N2 atom of the azido moiety and the

N atom of succinimide, which were used for the NN/1778 cm�1 cross

peak; 3.06 A smaller distances were used for the NN/1789 and

NN/1819 cm�1 cross peaks.

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is substantial (Fig. 3). A very small distortion of the through-

bond transport dynamics is expected for the NN/1789 and

NN/1819 cross peaks in all four compounds as the plateaus

here are small; these peaks report most selectively on the

through-bond energy transport process. Therefore, these two

sets of data were fit together with a linear function resulting in a

slope of 0.223 � 0.006 ps A�1 and an intercept of 2.8 � 0.2 ps.

Notice that the through-space coupling contributes to the cross

peaks at small waiting times. The through-space NRN/CQO

coupling decays with the lifetime of the NN mode (B1.2 ps in

CCl4). Only ca. 3% of the NRN modes remain excited at the

waiting time of 4.5 ps, at which the maximum is reached for

azPEG0. However, other IR-active modes at the azido moiety

populated via vibrational relaxation of the NRN mode can stay

excited for longer time, also contributing to the through-space

coupling cross peak that does not require energy transport. Note

however that the transition moment of the NRN stretching mode

is by far the largest among all modes of the azido moiety and of the

PEG chain so a steep decrease of the through-space contribution

with time is expected. Thus it is unlikely that the through-space

dynamics contribute significantly to the overall dynamics at

the delay times larger than B3 ps. The T-dependencies for the

NN/1748 cross peak are affected most by the through-space

coupling due to a largest transition dipole of the 1748 cm�1 mode,

but no significant deviations are found for this cross peak for

azPEG0 compared to the other cross peaks for azPEG0 (Fig. 5).

3.5. Cross-peak amplitude at Tmax

The amplitude of the cross peak at Tmax is an important charac-

teristic of the energy transport process as it reports on the amount

of excess energy delivered from the initially excited mode to the

reporter mode. Careful measurements targeting absolute cross

peak amplitudes in different compounds were performed. To

minimize the changes under the experimental conditions for

different samples, all measurements were performed during the

same day and on samples of the same (within 5%) concentrations.

Fig. 6 shows the chain-length dependencies of the cross-

peak amplitude measured at the respective Tmax delays for the

three cross peaks in the four compounds. The cross-peak

amplitudes depend steeply on the chain length, L, and can

be approximated well by an exponentially decaying function

(y(L) = y0exp(�L/L0), Fig. 6). The best fit results in L0 of

13.2� 0.5, 15.6� 0.7, and 16.2� 1.6 A for the 1748, 1789, and

1819 cm�1 reporter modes, respectively. Again the decay

dynamics are similar for the NN/1789 and NN/1819 cross

peaks and slightly different for the NN/1748 cross peak. The

influence of the thermal contribution to the amplitudes for

azPEG8 and 12 is likely the reason for a different slope for the

NN/1748 cross peak (Fig. 6). The weighted average of the two

L0 values obtained for the 1789 and 1819 cm�1 reporting

modes gives hL0i of 15.7 � 0.6 A.

The cross peak amplitude at T = 0 depends on a pairwise

distribution of the vibrational modes in question. Assuming a

normal end-to-end distance distribution (ideal chain) with the

probability density given by a Gaussian function a L�3/2

power dependence is expected for the cross peaks determined

by a through-space coupling.50 The results suggest that the

NRN/CQO cross-peak decay is described as L�1.4.

The closeness of the experimental dependence to that predicted

by the free-chain flight model confirms that the end group

interaction is small. Very different distance dependence is

expected if the ends of the chain are bound specifically.50

3.6. Overtone excitation

PEG chains have a reasonably strong absorption at around

1080–1170 cm�1 due to C–O–C stretching modes; the absorp-

tion increases with an increase in the chain length. As a result,

the overtones of these transitions grow as well with an increase

in the chain length. Note that the optical density of the

overtones remains very small and is hardly observed even in

PEG12. To test that the cross-peak data in the compounds

with long chains (azPEG8 and azPEG12) are not associated with

the direct excitation of these overtones we performed measure-

ments with the PEG8 and PEG12 compounds where the azido

group was replaced by a methyl group. No cross peaks, except a

small nonresonant contribution decaying to zero by ca. 0.4 ps

delay, were found in these measurements. These experiments

proved that the observed cross peaks are not associated with

excitation of the overtone transitions of the PEG chains.

4. Discussion

Energy transport pathways between the end groups in a

molecule include those through the backbone and through the

solvent (Fig. 7). Several experimental evidences indicate that the

through-bond transport (Fig. 7A) dominates for the most

measurements in all four azPEGn compounds. First, the peak

in the T-dependence was found for all three reporting modes,

including the one that is not sensitive to temperature. Fast local

thermalization is expected when the energy is transported via the

solvent as the dissipation occurs with small vibrational quanta.7

Fig. 6 Chain-length dependencies of the cross peak amplitude at Tmax

evaluated for the three cross peaks indicated in the inset. The best fits (solid

lines) with an exponentially decaying function (y(L) = y0exp(�L/L0))

results in L0 of 13.6� 0.7, 15.6� 0.7, and 16.2� 1.6 A for the cross peaks

involving 1748, 1789, and 1819 cm�1 reporter modes, respectively. The

cross-peak amplitude chain length dependence predicted by a purely

through-solvent energy transport is shown with black diamonds.

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Rapid thermalization at the reporter site and insensitivity of

the reporter to temperature result in the absence of the cross

peak involving this reporter.

The coiled structure of the PEG oligomers results in a distribu-

tion of the end-to-end distances for azPEGn compounds. Because

the mean end-to-end distance depends on the chain length, L12,

and the energy transport time via solvent is proportional to the

square of the distance, a linear dependence is expected. Amodelling

of the energy dissipation through solvent was performed using

the heat conduction equation. In spherical coordinates with

central symmetry, the equation is reduced to

1

r2@

@rr2@yðr; tÞ@r

� �¼ 1

a@yðr; tÞ@t

; ð1Þ

where y is the temperature and a is the heat diffusivity. The

problem is formulated as a thermalization process in a sphere

of a volume equal to the volume into which a single excited N3

label dissipates its energy. This volume depends on concen-

tration and excitation probability. The boundary conditions

are y(0, t) = finite and @yðr0;tÞ@r ¼ 0, where the latter signifies that

the excess heat does not leave the sphere. The initial condition

is y(r, 0) = (Ti � T0) exp(�r2/s2), where Ti � T0 is the initial

excess temperature at r=0 versus that at r0 and s is the width of

the initial temperature distribution taken as Gaussian (s { r0).

Note that the temperatures were selected so that the final

temperature, Tf, is equal to zero. The solution is given by51

yðr; tÞ ¼X1k¼1

ak expð�al2krÞsinðlkrÞ

r; ð2Þ

where lk are the roots of the equation

tan(lkr0) = lkr0, k = 1, 2, � � � (3)

and ak are given by

ak ¼R r00 yðr; 0ÞrsinðlkrÞdrR r0

0 sin2ðlkrÞdr: ð4Þ

The coefficients lk and ak were evaluated numerically and the

overall solution (eqn (2)) was obtained. A purely thermal dynamics

was evaluated for azPEGn oligomers of different lengths. It was

assumed that the cross-peak signal is proportional to the

temperature increase at the reporter. Two types of reporters

were considered: one tethered to the origin (heat source) and

another randomly distributed. The distance distribution for the

tethered reporter is described by the random flight chain model:

R2ee ¼ a20Nl2

1þ cosf1� cosf

� 2a20l2 cosf

1� cosNf

ð1� cosfÞ2ð5Þ

where N is the number of bonds in the oligomer, l is the mean

bond length, (1801 � f) is the bond angle taken as 109.51, and

a0 is the oligomer expansion factor, taken as 1.15 for chloro-

form and 1.1 for CCl4.42 The resulting Ree distances in CCl4

are 4.3, 8.6, 11.7, and 14.1 A in azPEG0, 4, 8, and 12,

respectively. The excitation probability determines the number

of randomly distributed reporters around a single excited N3

group; four such reporters were included into the sphere when

the excitation probability is 20%. In addition, a small excluded

volume is introduced at around r = 0; that is the area where

no randomly distributed reporters can be found. The excluded

volume was taken to be equal to the molecular volume of the

respective oligomer. Note that the excluded volume is small

and does not affect the modelling significantly.

The modelling predicts some growth in the cross-peak

waiting time dependencies, but the value of the thermal peak

is expected to be significantly smaller than that observed

experimentally. For example, for the conditions corresponding

to azPEG8 at 55 mM concentration, the temperature raise at

the peak in the waiting time dependence is expected to be less

than two fold of that at the plateau. The peaks in the

experimental waiting-time dependencies are 4.1, 5.7, and

>16 fold larger than the values at the plateaus, for the peaks

at 1748, 1789, and 1819 cm�1, respectively. Although some

contribution of the through solvent transport is expected,

especially for the mode at 1748 cm�1, the expected peaks are

much smaller than the experimentally observed peaks in the

waiting time dependencies.

The amplitudes of the purely thermal contributions at their

respective Tmax values were computed for the four compounds

and plotted in Fig. 6 with diamonds. The computed ampli-

tudes were scaled so that the value for the 1789 cm�1 cross

peak for azPEG0 is equal to the experimental amplitude. The

deviations from the experimentally observed dependence are

apparent, confirming that the through-solvent contribution

alone cannot explain the observed waiting time dependencies.

Additional support for the data interpretation comes from

the shape of the T-dependence for the cross peak with the

1746 cm�1 mode. The T-dependence has a characteristic dip at

TB 30–40 ps (Fig. 3), which has been previously assigned to a

switch of the sign of the effective anharmonicity.44 Because the

mode at 1746 cm�1 is shifted to higher frequencies by the thermal

effect, the effective anharmonicity is positive at the plateau.

The dip indicates that at smaller waiting times the frequency is

shifted to lower values. Indeed excitation of the modes higher

than 200 cm�1 (kBT) around the reporter will cause a

frequency shift of the reporter to lower frequencies. Such a

change of the sign is not expected for the thermal effect, as the

thermalized intramolecular vibrations and thermalized solvent

both shift the 1746 cm�1 mode frequency to higher values. In

contrast, a nonthermalized intramolecular coupling contri-

bution is expected to shift the frequency to lower values,

indicated by the predominantly negative anharmonicities of

the mode at 1746 cm�1 with the intramolecular modes with

frequencies higher than 100 cm�1; the mean anharmonicity for

all modes between 100 and 1600 cm�1 is�0.42 cm�1. Thus, thepresence of the dip in the T-dependence suggests that

the dominant contribution to the cross peak comes from the

excited intramolecular modes at frequencies above 200 cm�1,

Fig. 7 Various contributions to energy transport. (A) Through-bond

transport. (B) Transport to the solvent and via solvent. (C) Through-

bond transport with losses into the solvent along the chain. L1 and L2

denote the initially excited and reporting labels (modes).

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confirming the assignment. We believe that these arguments

provide a sufficient proof of the through-bond character of

the main peaks in the waiting time dependencies for all four

compounds.

4.1. Energy transport mechanism

A constant speed regime of the through-bond energy transfer

suggests a ballistic transport mechanism, where the energy is

transferred as a vibrational wavepacket. A wavepacket is described

as a linear combination of several states closely spaced in

frequency and delocalized over a substantial part of the molecule

(sample). A vibrational wavepacket in PEG oligomers can be

formed using exciton states in the chain formed by coupling

local vibrational states of the same type residing at the repeating

units of the oligomer. For example, there are two C–O bonds per

ethylene oxide monomer in the chain; the C–O stretching band

in PEG4 therefore consists of 8 excitonic states. Depending on

the disorder level in the chain, these states can be delocalized

over the whole chain or over a part of the chain. The normal

mode analysis for azPEG4 in the extended conformation performed

under vacuum or in the chloroform solvent, described by a

continuum polarized model, predicts almost fully delocalized

exciton states for several types of local motions involved.

The level of delocalization decreases in a coiled conformation

due to a disorder induced by a structural inhomogeneity and a

chain-to-chain interaction. Interactions with the solvent are also

expected to reduce the level of delocalization. To evaluate how

chain-to-chain interactions affect the delocalization level we

performed DFT normal-mode analysis for azPEG4 in a

randomly selected but strongly coiled structure (Fig. 8A).

Note that the end-to-end distance for the selected structure

(B5.5 A) is smaller than the mean end-to-end distance com-

puted for azPEG4 (7.8 A). A single parameter, the degree of

delocalization, was calculated for each mode according to the

following procedure. The atoms of the PEG chain in the

molecule were separated into five groups. Each of the PEG

repeating unit (–CH2CH2O–) formed a group; the two remaining

CH2 groups at the succinimide ester side and the carbonyl group

of the ester (CH2CH2CQO) formed the fifth group. The

distribution of atomic displacements for a mode among these

five groups was used to determine the degree of delocalization

of the mode. First the displacements (xi, yi, zi) of all atoms (i)

of the group j were summed: dj =P

i (x2i + y2i + z2i ). The end-

group modes, those localized mostly outside the five groups of

the chain, were identified by the criterion dS o 0.5, where

dS ¼P5

j¼1 dj; a zero delocalization factor was assigned to such

modes. For the modes with a large presence at the bridge

(dS > 0.5), the group displacements (dj) were normalized by

dS. A mode fully delocalized over the five groups has dj = 0.2

for all the groups. A mode localized on one group has one of

the dj values equal to unity and zero for the rest of them. The

number of sites, g, a mode is delocalized over was calculated

by summing the participation factors for each group. If the djvalue for a group is larger than 0.2, the participation factor for

this group is assigned to unity; the value of dj/0.2 is assigned

otherwise. The delocalization factor obtained by this method

reflects the number of groups or sites in the PEG chain

involved in the mode delocalization. Fig. 8B shows the

delocalization factor for all modes below 1850 cm�1 in

azPEG4 (Fig. 8A). The modes fully delocalized over the

PEG chain have g B 5. The localized modes, such as for

example the CO mode at 1789 cm�1, have g close to unity

(Fig. 8B). Note that the other two CO modes fell into the end-

groups mode category with g = 0.

Several types of vibrational motions result in delocalized

modes. Four such high-frequency groups are colored differently in

Fig. 8B. These groups involve CH2 scissoring (1503–1553 cm�1,

bandwidth 50 cm�1), CH2 wagging (1366–1456 cm�1, bandwidth

90 cm�1), CH2 twisting (1246–1340 cm�1, bandwidth 94 cm�1),

and C–O, C–C stretching with CH2 rocking (781–1167 cm�1,

bandwidth 386 cm�1) groups. The average delocalization factor

for each group is determined to be 1.9, 2.6, 2.5, and 2.8,

respectively. The deformation modes in the frequency region

below 580 cm�1 are found to be most delocalized with the

mean delocalization factor of 3.2.

The bandwidth of the exciton band for an ideal chain

consisting of an infinite number of units having the same

nearest-neighbour interaction (b) is equal to 4b; for a finite

number of units and/or introduced disorder for different sites

the bandwidth of the exciton band will be smaller.52 The

exciton bandwidth computed for the characteristic motions in

PEG indicate the coupling strength of the local modes involved.

The large widths of the bands involving CH2 wagging, CH2

twisting, and especially C–O, C–C stretching with CH2 rocking

motions suggest the large site interaction strength for them,

which makes them less susceptible to the disorder and results in

Fig. 8 (A) The structure of azPEG4 used for the mode delocalization

analysis. (B) The delocalization factor, g, is plotted as a function of

mode frequency (harmonic) for all modes below 2000 cm�1. Different

groups of modes, including CH2 scissoring (sc), CH2 wagging (w),

CH2 twisting (tw), CH2 rocking (r), and C–C, C–O stretching (n),are denoted with different symbols and colors.

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large delocalization factors. The vibrational states in these bands

are expected to be the most resistant to localization driven by the

solvent induced disorder. Notice also that in solution the inter-

chain interactions, unbalanced in calculations performed under

vacuum, could be balanced, at least partially, by the solvent-to-

chain interactions, which may even help delocalization.

The wavepacket propagation is characterized by the momentum

conservation. The mean distance a wavepacket passes before it

is scattered or relaxed, the mean-free-path length, depends on

the disorder in the system. At the limit of very high disorder

the vibrational states are local, which corresponds to the

wavepacket with the mean-free-path length comparable to a

bond length. Diffusive energy transport is expected in this case

as the energy transfer events forward and backward the chain

will have similar probabilities. With moderate disorder an

intermediate case occurs where the mean-free-path length is

comparable to the overall chain length. Although scattering

events are important in this case, the wavepacket description of

the transport is appropriate. The characteristic decay distance

evaluated in this work (15.7 � 1 A) suggests that the inter-

mediate energy transport regime occurs in the PEG oligomers.

It is conceivable that because of the presence of several

delocalized bands in the azPEGn chain, the wavepacket

scattering not only can relax to the local modes7 but can also

scatter inelastically into other wavepackets.

A variety of conformations is available in the ensemble of

PEG molecules described as a random coil. More experiments

are needed to understand the energy transport properties in

different conformations.

4.2. Comparison of the energy transport in chloroform and CCl4

The coiling level of PEG oligomers can be characterized by a

Mark–Houwink exponent; a larger exponent report on a more

extended structure.53 Unfortunately, the literature data for the

PEG oligomers in CCl4 and chloroform are too scattered. For

example, the values summarized in the Polymer Handbook for

small PEG oligomers are 0.5054 and 0.6455 in chloroform and

CCl4, respectively. Notice that the original manuscripts cited

are different for the two solvents; indeed no comparison of the

two solvents is available from any single manuscript. At the

same time, the Mark–Houwink exponent for PEG oligomers

in chloroform reported more recently in a very extended essay

is ca. 0.68 (depending on the oligomer size),42 which is larger

than the 0.61 value reported for CCl4.55 The solubility data

also provide an estimate on how collapsed the structure is.42,53

Based on a smaller solubility of azPEGn oligomers in CCl4, we

anticipate a more collapsed azPEGn structure in it, but the

difference between the two solvents might be small.

The energy transport times measured in both solvents are

similar: the linear fit of the Tmax data for the NN/1819 cross

peak for azPEGn in chloroform is shown in Fig. 5 with a

dashed line. The slopes determined are somewhat different,

550 m s�1 in chloroform37 vs. 450 m s�1 in CCl4. Although the

dashed line in Fig. 5 lies within the error bars for the points

measured for the NN/1819 cross peak in CCl4, the difference

in the slopes is apparent.

Notice, however, that the parameter selected here as a

measure of the energy propagation speed (dTmax(L)/dL),

while convenient and reliably measurable, is not ideal for reporting

the energy transport speed as it is affected by the energy dissipation

to the solvent; smaller Tmax values are expected for larger cooling

rates from the chain. The energy dissipation could vary not only in

different solvents, but also for samples with different chain lengths,

as the excess energy in the chain differs for different chain lengths.

As clearly seen from Fig. 3, the signals start to grow much earlier

than Tmax; the T-dependencies for azPEG4, 8, and 12 do show an

S-like shape at small waiting times (Fig. 3). The delay at which the

growth starts, T0, represents better the energy arrival time.36 There

is a larger uncertainty, however, in evaluating T0 from the data

due to the presence of the through-space interactions, which is

apparent from a substantial cross peak amplitude at T = 0.

Although the NN/CO through-space coupling decays rapidly

with the delay time, other modes populated via NN relaxation

contribute to the through-space interaction cross peak. As a

result, the waiting time dynamics for the through-space cross-

peak contribution is difficult to predict and take into account,

especially at small waiting times.

Nevertheless, the waiting time at which the cross-peak

grows to a half of its maximum (T12) can be evaluated much

more accurately; it is more representative of the energy arrival

time than Tmax, as it differs little from T0 (by less than 3 ps).

The chain-length dependencies of T12are also found to be linear

in both CCl4 and chloroform; the slopes, however, are about

two-fold larger than those for the Tmax dependencies at

850 � 90 m s�1 and 990 � 70 m s�1 for CCl4 and chloroform,

respectively. The current error bars do not permit discrimina-

tion of these two slopes as different; certainly the transport

properties in these two solvents are not dramatically different.

Also, although the T12parameter represents better the energy

arrival time, such data are not completely free from the influence

of the chain cooling. Interestingly, the speed found is ca. 1.7-fold

smaller than the speed of sound in the PEG oligomers.56

Future applications of the ballistic energy transport in

molecules can be envisioned in different directions, including

developing novel signaling schemes for molecular electronics

where vibrational energy is used as a reporter and new ways of

delivering energy to chemical reactions.

5. Conclusions

The NRN/CQO cross peak enhancements observed for the

azPEG0, 4, 8, and 12 compounds as a function of the waiting

time are attributed to a dominating contribution from the

through-bond energy transport, rather than to a through-

solvent energy transport. A weaker thermal contribution to

these cross peaks in CCl4 compared to chloroform permits

more accurate Tmax measurements for the cross peaks involving

all three CQO reporters. The thermal contribution to the cross

peaks was measured in several molecular systems, including

azPEG8 of high concentration and mixtures of randomly

distributed compounds. The cross-peak contribution associated

with excitation of the overtones of C–O–C modes of the PEG

chain is found to be negligible. The through-bond transport in

azPEGn in CCl4 is found to be ballistic with the transport speed

of ca. 450 m s�1. The cross-peak amplitude, measured at the

Tmax delay, decays exponentially with the chain length with a

characteristic distance of 15.7 � 1 A.

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Acknowledgements

Support by the National Science Foundation (CHE-0750415

and CHE-1112091), the Air Force Office of Scientific Research

(FA9550-10-1-0007), and the US Army Medical Research

and Materiel Command (W81XWH-10-1-0377) is gratefully

acknowledged. Z.L. is thankful for the fellowship from the

IBM Corporation.

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50 Z. Lin, N. Zhang, J. Jayawickramarajah, A. L. Burin andI. V. Rubtsov, in preparation.

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