Article pubs.acs.org/JPCB
Temperature Dependence of the Ballistic Energy Transport in Perfluoroalkanes Natalia I. Rubtsova, Arkady A. Kurnosov, Alexander L. Burin, and Igor V. Rubtsov* Department of Chemistry, Tulane University, New Orleans, Louisiana 70118, United States ABSTRACT: Temperature dependence of intramolecular energy transport in perfluoroalkane oligomers with a chain length of 3−11 carbon atoms terminated by a carboxylic acid moiety on one end and a −CF2H group on another end was studied in solution experimentally and theoretically. Experiments were performed using a dual-frequency relaxation-assisted two-dimensional infrared spectroscopy method. The energy transport was initiated by exciting the CO stretching mode of the acid and recorded by measuring a cross-peak amplitude between the CO stretching and the CH bending modes as a function of the waiting time between the excitation and probing. An efficient transport regime with a mean free path of 16.4 ± 2 Å is observed at 35 °C. The energy transport speed decreases at elevated temperatures, indicating a switch from the ballistic transport regime to diffusive. The modeling of the energy transport involving both ballistic and diffusive mechanisms is performed. It explains the temperature dependence of the energy transport speed and confirms a switch of the transport regime from ballistic at lower temperatures to diffusive at higher temperatures.
1. INTRODUCTION Understanding the routes, mechanisms, and dynamics of energy transport in molecules is one of the important objectives in the field of physical chemistry that has the potential of paving the way to a broad range of applications in nanotechnology, organic chemistry, and biochemistry. These include development of the effective cooling in microscopic and macroscopic molecular systems, such as nanowires1 and optical limiters, designing efficient energy transport schematics for energy signaling,2 as well as optimizing and even promoting chemical reactions by concentrating the excess energy at the reacting center.3,4 There are two limiting regimes associated with energy transport in materials, i.e., diffusive and ballistic transport mechanisms. Both regimes can be found in intramolecular energy transport. The diffusive process is expected to occur in molecules lacking periodic structure, and therefore lacking vibrational states delocalized over a large group of atoms. It originates from an intramolecular vibrational energy redistribution (IVR) process, which involves energy hopping between various vibrational states;5,6 as such, it is also referred to as a hopping mechanism. A single IVR event, known as a driving force of diffusive energy transport,7 is characterized by a change of two or more quantum numbers of the involved spatially overlapping vibrational modes, which requires the anharmonic coupling of these modes. On the other hand, in the ballistic regime, the energy is transferred via a free-propagating vibrational wavepacket, formed by vibrational states delocalized over the whole transport region; such transport can be very efficient.8 Both regimes may be important for certain applications and a switch between their contributions can be expected if the conditions, such as temperature or solvent, are changed. Ballistic transport efficiency is characterized by a mean free path distance of the vibrational wavepacket free propagation. Longer mean free path indicates a more efficient ballistic process, while the diffusive mechanism can be © 2014 American Chemical Society
associated with a small mean free path and low transport efficiency. Both regimes of energy transport in molecules have been intensively studied theoretically1,9−16 and experimentally. While the diffusive mechanism of energy transport is prevalent in the majority of investigated molecular systems,17−22 there are only a few studies that report on experimentally detected ballistic energy transport and the topic remains relatively new. Ballistic transport is successfully found in macroscopic systems such as carbon nanotubes23,24 and amorphous silicon nitride membranes at temperatures below 6 K.24 The study evidences an increase of a phonon mean free path with lowering the temperature. Ballistic transport is well-known as a main energy transfer mechanism of acoustic and optical phonons.9,25 In molecules, the ballistic energy transport with nearly constant velocity was found using azulene−anthracene featured compounds bridged by linear alkane chains of up to six carbon atoms.11,26,27 Dlott and co-workers have reported on the ballistic transport through self-assembled monolayers of alkanes anchored to a gold surface, exposed to ca. 800 K transient temperature gradient.28 The vibrational energy transport in a helical peptide was studied as a function of temperature, and a switch of the mechanism from diffusive to ballistic was suggested upon reaching temperatures below 270 K.29 A recent study of poly(ethylene glycol) (PEG) molecules2,30 showed an efficient, constant speed, long-range energy transport regime in oligomers of different size, from 0 to 12 PEG repeating units. Experiments were performed using the Special Issue: James L. Skinner Festschrift Received: February 27, 2014 Revised: March 31, 2014 Published: April 3, 2014 8381
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dual-frequency relaxation-assisted 2DIR (RA 2DIR) spectroscopy method. The energy transport was initiated by excitation with an IR pulse of the terminal azido group (2100 cm−1), serving as a tag, and detected by the frequency shift of the reporter mode, CO stretching, attached to another end of the molecule. The tag/reporter cross-peak amplitude was measured as a function of the waiting time between excitation and probing, and the energy transport time, Tmax, taken as the waiting time at which the maximal frequency shift of the reporter is observed, was found to be in a linear dependence with the chain length. The established transport speed was 550 m/s in chloroform2 and 450 m/s in CCl4.30 An exponential dependence of the amount of energy delivered to the reporter site on the chain length was found with a characteristic decay distance of 15.7 Å, and thus, a ballistic mechanism was suggested. A set of linear perfluoroalkane compounds of different length is very attractive for studying the ballistic energy transport. Adopting a rod-like conformation in solution helps to reduce the energy dissipation into the solvent, while their symmetry and structural simplicity promote delocalization, which spreads over the entire molecule.31 A linear dependence of the energy transport time with the chain length has been recently reported for a series of perfluoroalkane compounds, ranging from 3 to 11 carbon atoms per chain. The energy transport speed of 290 m/s was found based on the Tmax vs chain length measurements. In this work, we have applied experimental and theoretical approaches to describe the temperature dependence of the energy transport in perfluoroalkane oligomers (Cn) in solution. The CO stretching mode of the carboxylic acid terminal moiety served as a tag, while the CH bending mode of another CF2H terminal moiety served as a reporter for RA 2DIR measurements. It is found that the energy transport speed decreases at elevated temperatures. The effect is weaker in shorter oligomers. The energy transport efficiency was also measured, and a mean free path distance was evaluated. Maximum values of the cross-peak amplitude were found to be temperature and chain length dependent. In addition, the temperature dependence of the CO mode lifetime is reported. To investigate the change in the transport mechanism from ballistic to diffusive, we implemented a toy model of energy propagation using an optical phonon band of the polymer chain. Evolution of the density matrix is computed where decoherence is introduced via random fluctuations of vibrational site energies of the chain. The modeling shows that the transport is ballistic at chain lengths smaller than v/W, where v is the characteristic group velocity and W is the characteristic decoherence rate, while it is diffusive at longer lengths. The temperature dependence of energy transport is introduced via temperature dependence of the decoherence rate, W,32,33 which is small at low temperatures, resulting in the ballistic regime, and larger at higher temperatures, resulting in a transition to the diffusive regime. The latter is accompanied by a sharp increase in the transport time, which can be used to interpret experimental observations.
Scheme 1. Structures of Cn Compounds
Inc., 99.96%). Specified compounds will be referred to here as C3, C5, C7, etc. The infrared spectroscopic measurements were performed in a sample cell made of two CaF2 wafers separated by a Teflon spacer. A 100 μm thick Teflon spacer was used for all samples, except for C11, in which case a 200 μm spacer was used, because of the low solubility of C11. The concentration of C3, C5, C7, and C9 samples was 0.25 ± 0.02 M, while the concentration of C11 was made 0.12 ± 0.02 M. A twice thicker spacer used for C11 allows lowering the concentration to a half, maintaining the same excitation probability to match all other samples. Additional heating was required to maintain all samples fully dissolved under the specified concentrations. Elevated temperatures of the sample cell were set by an electric heating element and measured continuously during the timeresolved measurements by a thermocouple placed in direct contact with the center of the sample cell window. To cool the sample, we used a liquid flow temperature controller (9706A11C, PolyScience) and an insulated sample cell unit. 2.2. Heterodyned Dual-Frequency 2DIR Measurements. Details of the dual-frequency 2DIR setup with heterodyned detection can be found in refs 34 and 35. Briefly, two in-house-built optical parametric amplifiers followed by two difference-frequency generation units were used to generate independently frequency-tunable mid-IR pulses of ca. 150 fs pulse duration. One of the mid-IR beams was split into two equal parts, each of ca. 1.2 μJ energy, which served as excitation pulses, k1 and k2, interacting with the sample. A small portion (∼4%) was split from the second beam to serve as a local oscillator (LO) for heterodyned detection, while the main part (∼1.1 μJ) was used as a third beam, k3, interacting with the sample. The spectra of the k1 and k3 pulses were tuned to ca. 1773 and 1380 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, which was delayed by the time delay t, and detected by an MCT 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, τ, 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. The positions of all translation stages during the experiments were accurately measured with an external interferometer based on a HeNe laser.21 2DIR spectra were obtained by a double Fourier transformation of the M(τ, t) data sets. The waiting time dependences for the relaxation-assisted 2DIR measurements were measured by acquiring 2DIR F(t, T) data sets while keeping the dephasing time (τ) constant at 167 fs. The F(t, T) data sets were then Fourier transformed along the t direction and presented as a set of one-dimensional ωt spectra at various T values. The experimental conditions for acquiring the F(t, T) data sets were selected so that a single peak along ωτ, that at ca. 1774 cm−1, dominated the 2DIR spectra. The cross-peak amplitude at each T delay was determined by integrating the ωt absolute-value peak in the
2. EXPERIMENTAL DETAILS 2.1. Sample Preparation. Perfluoroalkanes with chain lengths of 3, 5, 7, 9, and 11 carbon atoms, featuring an acetic acid moiety at one end of the chain and a CF2H group at another end of the chain (Scheme 1), were purchased from SynQuest Laboratories Inc., and their solutions in deuterated chloroform were studied (Cambridge Isotope Laboratories, 8382
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vicinity of its maximum (at ca. 50% level) and subtracting the integrated and normalized background. The resulting crosspeak amplitudes were plotted as a function of the waiting time, T. To suppress completely the C−H diagonal peak, which was generated due to a tail of the k1 and k2 pulse spectrum, a shortwave-pass filter was used in the k2 beam to cut off the frequencies below 1650 cm−1. 2.3. Waiting-Time Dependence Analysis. The waiting time dependences of the ν(CO)/δ(CH) cross-peak amplitude were measured for each compound at various temperatures. The carbonyl stretching mode, the tag, was excited with the first two IR pulses (k1, k2) and the CH bending mode, the reporter, was probed by the third pulse (k3). Examples of the waiting time dependences for the C5 and C7 compounds are shown in Figure 1. For each compound, the
Figure 2. Linear absorption spectrum of the C7 compound in CDCl3.
3.2. Energy Transport Efficiency and Speed. The distance dependence of the energy transport efficiency can help in identifying the dominant mechanism of the transport. It also defines the potential usefulness of the compounds for nanotechnology and chemistry applications. The fast propagating constant speed regime of energy transport has been previously observed for the perfluoroalkane system,31 suggesting the ballistic mechanism, although the energy transport efficiency of this system has not been measured. In order to determine the efficiency, the waiting time dependences were measured for each oligomer of the C3−C11 series, featuring the same concentration, temperature, and instrumental adjustments. The temperature of the sample was kept at 35 °C to ensure full solubility. Note that a twice lower concentration in the case of C11 is compensated with a doubled thickness of the sample; see section 2.1. The crosspeak amplitudes, Amax, were obtained at Tmax from the fitting of the waiting time dependences and averaged among three experimental sets for each oligomer. The resulting Amax amplitudes were then plotted as a function of the chain length (Figure 3). The dependence can be well approximated with a decaying exponential function, A = A0*e−L/L0, where L0 is a characteristic decay distance, found to be 16.4 ± 2 Å. Figure 3B shows the dependence of Tmax on the chain length, measured at a constant temperature of 35 °C (squares). The slope of 0.26 ± 0.03 ps/Å results in a transport speed of 385 m/s at 35 °C. For comparison with the previously reported data on Tmax vs chain length, where the temperature was not constant, the new data obtained under the same conditions as before are plotted (blue circles). The slope obtained (0.342 ± 0.05 ps/Å) is the same as that previously reported (0.344 ± 0.04 ps/Å). 3.3. Temperature Dependence of the Energy Transport Speed. Temperature influences many processes in molecules; for example, due to thermal population of low frequency vibrational modes, the IVR process becomes faster, as more relaxation channels open up.6,36,37 To study how temperature influences the speed of the energy transport, the C5, C7, and C9 compounds were selected. Notice that the temperature range for C9 is much smaller than that for C5 and C7 due to the lower solubility of C9, which results in its precipitation at temperatures below 35 °C. The C7 and C5 compounds showed signs of precipitation at 19 and 2 °C, respectively, which determines the accessible temperature ranges in the measurements. The waiting time dependences were measured no less than three times at each temperature of 5, 10, 22, 30, 45, and 60 °C for C5, 22, 45, and 60 °C for C7,
Figure 1. Waiting time dependences of the ν(CO)/δ(CH) crosspeak measured for the C5 and C7 compounds. A double exponential fit line is shown for each curve. The Tmax and maximum amplitude, Amax, values were determined from the fitting curve.
cross-peak amplitude grows with the waiting time due to the energy transport from the tag mode (CO) toward the reporter (CH), reaches its maximum, and then decays due to the overall cooling of the compound. The energy transport time, taken as the waiting time at which the maximum is reached (Tmax), increases with an increase of the chain length, as it takes longer for energy to reach the reporter mode across a longer chain; notice that the maximum for the C7 compound appears at larger waiting times compared to that for C5 (Figure 1). Linear dependence of the Tmax time on the chain length was previously reported.31 Each waiting time dependence was fitted with a double exponential function, and the Tmax and maximum amplitude values were determined from the fitting curve.
3. RESULTS 3.1. Linear Absorption Spectra. Linear absorption spectra of all Cn samples demonstrate a strong peak with a maximum at ca. 1775 cm−1 originating from the carbonyl stretching motion (ν(CO)) and a significantly smaller peak of the CH bending mode of the terminal CF2H group (δ(CH)) appearing at 1401 cm−1 (Figure 2). The extinction coefficient of the CH bending mode in the C7 compound is found to be ca. 37 M−1 cm−1; the value is similar in other Cn compounds. A broader discussion on the spectral features, observed in perfluorinated Cn homologues, can be found in our earlier work.31 8383
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Figure 5. Temperature dependence of Tmax measured for the C5 (squares), C7 (rhombs), and C9 (circles) oligomers. A fit with a linear function (red) resulted in slopes of 0.011 ± 0.002, 0.034 ± 0.002, and 0.020 ± 0.003 ps/K for C5, C7, and C9, respectively.
A linear fit of the dependence results in a slope of 0.011 ± 0.002 ps/K, which is about 3-fold smaller compared to that for C7. Slower transport at higher temperatures is again in agreement with the ballistic energy transport. An intermediate slope of 0.020 ± 0.003 ps/K is found for C9, suggesting a larger contribution from the diffusive mechanism (see section 4). 3.4. Temperature Dependence of the CO Mode Lifetime. The lifetime of the excited CO mode can affect the waiting time dependence if the energy release from CO is temperature dependent. A set of pump−probe experiments was performed to measure the CO-mode lifetimes for the C7 and C9 samples at temperatures of 22, 35, 45, 55, and 69 °C and 35, 42.5, 50, 58, and 65 °C, respectively. The kinetics were measured at 1730 cm−1, which is close to the peak of the 1 → 2 transition of the CO mode. The lifetime of the CO mode for C7 is found to be essentially temperature independent: the lifetimes of 0.90 ± 0.08 and 0.91 ± 0.05 ps were measured for 22.5 and 69 °C, respectively (the lifetimes at other temperatures were also 0.90 ± 0.06 ps). The CO mode lifetime for C9 also appeared to be temperature independent with a lifetime of 0.95 ± 0.05 ps. 3.5. Temperature Dependence of the Cross-Peak Amplitude. The amplitude of the ν(CO)/δ(CH) crosspeak is also found to be temperature dependent. Figure 6 shows the temperature dependence of the cross-peak amplitude measured at Tmax waiting time for the C5−C9 compounds.
Figure 3. (A) Dependence of the ν(CO)/δ(CH) cross-peak amplitude, Amax, on the chain length, measured at 35 °C and its fit with an exponential function. (B) Dependence of Tmax on the chain length, measured at 35 °C (squares) and its linear fit (red); points measured at the same temperatures as in ref 31 (blue circles) and its linear fit (blue line) are shown for comparison.
and 35, 50, and 66 °C for C9 samples, and the Tmax values obtained for each measurement were averaged. Figure 4 shows three fitting curves that correspond to experiments with the C7 compound at 20, 45, and 60 °C. A
Figure 4. Normalized waiting time dependences of the ν(CO)/ δ(CH) cross-peak measured for the C7 compound at 20, 45, and 60 °C.
clear shift of Tmax toward higher values can be seen. The Tmax values determined from these results and plotted as a function of temperature show a growth that can be approximated by a linear function with a slope of 0.034 ± 0.002 ps/K (Figure 5). Within the ballistic transport mechanism, a slower transport is expected at higher temperatures as the wavepacket dephasing and relaxation becomes faster. The energy transport governed by the diffusive mechanism can either become faster, if IVR is the dominant mechanism, or slower, if the dynamic disorder slows the hopping between the states being in close resonance (see modeling). A similar dependence measured for C5 oligomer also demonstrates a growth of Tmax with temperature (Figure 5).
Figure 6. Temperature dependence of the ν(CO)/δ(CH) crosspeak amplitude at Tmax waiting time, obtained for C5, C7, and C9. A fit with a linear function (red) resulted in slopes of 0.011 ± 0.002, −0.014 ± 0.001, and −0.015 ± 0.001 K−1 for C5, C7, and C9, respectively. 8384
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Interestingly, the cross-peak amplitude increases with the temperature increase for C5 but decreases for C7 and C9. The amplitude decrease for C7 and C9 is in agreement with the expectation that the wavepacket dephases and relaxes faster at elevated temperatures, which results in reduction of the energy transport efficiency and reduction of the cross-peak amplitude at Tmax. The distance between the tag and reporter modes in C5 is small enough (7.3 Å) that one could expect additional energy transfer channels, for instance, those involving the tag and reporter directly, which could be slow but efficient in delivering energy to the reporter site, possibly requiring only a single relaxation step. These channels may become more efficient at elevated temperatures, resulting is a small amplitude growth for C5. Such additional channels would be prohibitively slow for C7 and C9 because of a larger tag−reporter distance. Theoretical modeling was performed to describe ballistic and diffusional contributions to intramolecular energy transport in perfluoroalkane chains, and the predictions were compared with the experimental data.
ρij (0) = δi1δj1
This is the simplest possible initial condition for the density matrix, though in reality the initial excitation can be more extended in space.9 However, it should be qualitatively valid for long chains. Note that no thermal excitations are introduced in the model. The time dependence of the density matrix elements can be described by the equation32,33 dρij dt
N
∑ ωibi+bi + i=1
Δ 2
=i
Δ [(1 − δi1)ρi − 1, j + (1 − δiN )ρi + 1, j ℏ
− (1 − δj1)ρi , j − 1 − (1 − δjN )ρi , j + 1] − W (1 − δij)ρij − kρij
(4)
We investigated the time dependence of the probability of the excitation to reach the opposite end of the polymer which can be expressed by the time dependent matrix element ρNN(t). The cross-peak time, tc, taken at the maximum of this probability, is the main computed parameter, which is compared to the Tmax value measured experimentally. Several representative time dependences of ρNN(t), calculated for different decoherence rates W at a fixed coupling strength of Δ = 9.1 cm−1 and an excitation decay time of 11.5 ps, selected to obtain the energy transport times comparable to the experimental observations, are shown in Figure 7. Figure 8 shows the dependence of the cross-peak time on the decoherence rate W for the same parameters as in Figure 7 for the polymer chains with N = 5, 7, and 9.
4. THEORETICAL MODELING The theoretical model implemented is based on quantum treatment of vibrational energy transport via an optical phonon band in a one-dimensional polymer chain formed by coupled harmonic oscillators with fluctuating site frequencies ωi(t) and coupling strength Δ, which can be described by the Hamiltonian Ĥ =
(3)
N−1
∑ (bi+bi+ 1 + bi++ 1bi) i=1
(1)
where the index i enumerates sites of the oligomer. A single vibrational mode, such as the C−F stretching or F−F scissoring, is considered for each site. The mean frequency at each site i is set to be equal to ω0, while the fluctuations at different sites are assumed to be delta-correlated32,33 ⟨ωi⟩ = ω0 ⟨δωi(t1)δωj(t 2)⟩ =
W δijδ(t1 − t 2) 2
(2)
Figure 7. Time dependence of the probability of excitation of the end site for the chain length N = 7.
The first assumption is justified by the very high ordering of perfluoroalkane chains.31 The second assumption is an approximation, which treats the site frequencies as uncorrelated, while introducing the decoherence rate of W/2 for each site. However, we expect that the analysis should be qualitatively valid.32,33 The advantage of this approach is that it permits us to describe decoherence by introducing the decay rates for all off-diagonal density matrix elements, which leads to a straightforward analytical and numerical solution of the problem. In addition to decoherence, we introduced a decay of all density matrix elements with a rate constant k, associated with the finite lifetime of the excited states. Note that interaction of the vibrations with the environment includes anharmonic interactions within the chain as well as interactions with the solvent.6 Time evolution of the density matrix ρij(t) (i = 1, ..., N, j = 1, ..., N) describes excitation transport in the chain, where indices i and j denote the first excited vibrational state of each monomer site. The initial vibrational excitation is prepared at site 1, which corresponds to the initial conditions in the form
The transition between ballistic and diffusive transport regimes is clearly seen. In the ballistic regime, the cross-peak time is almost independent of the decoherence rate (W < 15 cm−1 for N = 5, W < 10 cm−1 for N = 7, and W < 8 cm−1 for N
Figure 8. Dependence of the cross-peak time on the decoherence rate (Δ = 9.1 cm−1, k = 0.46 cm−1, τ = 11.5 ps). 8385
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attributing it to a fully ballistic transport regime. The cross-peak amplitude increase, found experimentally for C5, cannot be explained by the presented model. Existence of relaxation channels of the CO mode that would populate directly the modes in the vicinity of the reporter, or even reporter itself, is suggested for C5 as a reason for the cross-peak amplitude increase at higher temperatures. It is conceivable that such channel(s) can be efficient in transporting energy, but slow, as the distance between the tag and reporter is large. The dependences of Tmax vs temperature, shown in Figure 5, have different slopes for compounds with different chain lengths, indicating that the dependence of Tmax vs chain length is not linear for all temperatures. The latter dependence is well described as linear at 35 °C (Figure 3B) but becomes curved at higher temperatures with a higher apparent speed for longer chains. This observation indicates the crossover between the regimes where the transport is dominated by the ballistic mechanism and where the diffusive mechanism is dominant. Note that at high disorder levels (large decoherence rate W) essentially site-localized states are obtained. Transitions between such states results in diffusive energy transport via hopping. The current model does not consider the IVR process, where excitation of one mode is used to excite the other two (or three) modes, as has been described previously for transport in disordered chains.6,38 Note, however, that perfluoroalkanes have a small number of degrees of freedom per site, and high intersite couplings, which cause the majority of the normal modes to be fully delocalized when the disorder is small, thus leaving less room for a traditional IVR process involving more than two modes. Inclusion of only a single exciton band is another assumption of the model. Variations in the coupling strength (Δ) and disorder level (W) for different bands can result in different mean free path distances for them, making some of the bands act as ballistic transporters while others as diffusive transporters. Inclusion of the bands with different transport properties permits describing a more complex competition of the two transport mechanisms. A more realistic model should also include an influence of the end groups on the states in the chain, which was observed both experimentally and computationally via normal-mode analysis. Note that the end groups perturb stronger the chain bands of shorter oligomers, which can result in larger ballistic transport contribution in longer chains. Such theoretical treatment, however, is much more involved and will be performed separately.
= 9). In the diffusive regime, the cross-peak time increases with increasing decoherence rate approximately as W1/2, in agreement with the earlier predictions.38 There is a sharp crossover between the two regimes (Figure 8). This crossover can be understood considering the dramatic change in the shape of the waiting-time dependence between the two regimes (Figure 7). It is interesting that in the ballistic regime with W = 2 there are several peaks due to multiple reflections of the wavepacket from the chain ends, while they are not observed near the crossover. The increase of the cross-peak time by a factor of 1.5 for the C7 oligomer at higher temperatures can be explained by a sharp crossover between ballistic and diffusive regimes (Figure 8). Indeed, a minor change in the decoherence rate, W, due to the relatively small temperature raise from 20 to 60 °C (from 293 to 333 K, i.e., by a factor of 1.14) can lead to a substantial increase in the cross-peak time, tc. For the same changes in W, the cross-peak time for C5 does not change much, which is consistent with the experiment. Much weaker temperature dependence of the cross-peak time than that for C7 is also expected for longer polymer chains, such as C9 and C11, because there we expect tc ∼ W1/2 and the dependence of the decoherence rate on temperature is not expected to be very strong (e.g., W ∼ Ta with a ∼ 1.3−1.4 in heteropolymers at T < 200 K39). Indeed, the slope for C9 is found to be smaller than that for C7 by a factor of 1.7 (Figure 5). The sharp rise in the cross-peak time near the crossover can be understood as follows. In the ballistic regime, the cross-peak maximum occurs when the wavepacket reaches the end of the chain, while its decrease is associated with the wavepacket reflection. Therefore, the width of the peak is quite narrow and the peak position is determined by the ratio of the chain length and the wavepacket group velocity. In the case of diffusion, there is no reflection and the amplitude increases until the energy dissipation to the environment (k) becomes significant. Therefore, the cross-peak is expected to be wider in the diffusive regime (Figure 7) and this qualitative change at the crossover determines the sharp rise in the cross-peak time (Figure 8). Preliminary analysis of the cross-peak amplitude shows a faster decrease with the chain length compared to the observations shown in Figure 3, which is likely due to oversimplification of the model that involves only a single exciton band in the chain. While incorporation of a more detailed oligomer structure, resulting in introduction of multiple exciton bands, can improve the agreement between the theory and experiment, it will likely make it difficult to track the key factors affecting the transport.
6. CONCLUSIONS The energy transport in linear perfluoroalkane oligomers was investigated using the relaxation-assisted 2DIR spectroscopy method. The efficient energy transport regime with a mean free path of 16.4 Å and an effective transport speed of 385 m/s is observed in perfluoroalkane solutions at 35 °C. The transport speed is found to be temperature dependent, slowing down at higher temperatures. The temperature dependences of the cross-peak amplitude and the lifetime of the CO mode are also reported. The energy transport modeling describes well a switch between ballistic and diffusive mechanisms in the perfluoroalkane oligomers. The obtained results provide a credible support for the ballistic mechanism of the energy transport to be dominant for perfluoroalkane chains in solution at lower temperatures.
5. DISCUSSION The obtained results suggest that the mechanism of energy transport through perfluoroalkane chains is dominated by the ballistic regime at smaller temperatures. The presented modeling suggests that the transition between the ballistic and diffusive regimes occurs at higher temperatures. The transport in C7 shows a strong dependence on temperature, which can be explained as a transition from the ballistic regime at lower temperatures, to a diffusive regime at higher temperatures. Both the transport time and the cross-peak amplitude (amount of energy delivered) observed for C7 are consistent with the model. The energy transport time for C5, also showing an increase at higher temperatures but much smaller than that for C7, is explained within the model by 8386
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Support by the Army Research Office (vv911NF-13-1-0186) and by the National Science Foundation (CHE-1012371 and EPSCoR LA-SIGMA, EPS-1003897) is gratefully acknowledged.
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