Competition of Several Energy-Transport Initiation Mechanisms Defines the Ballistic Transport Speed

The ballistic regime of vibrational energy transport in oligomeric molecular chains occurs with a constant, often high, transport speed and high efficiency. Such a transport regime can be initiated by exciting a chain end group with a mid-infrared (IR) photon. To better understand the wavepacket formation process, two chemically identical end groups, azido groups with normal, 14N3-, and isotopically substituted, 15N3-, nitrogen atoms, were tested for wavepacket initiation in compounds with alkyl chains of n = 5, 10, and 15 methylene units terminated with a carboxylic acid (-a) group, denoted as 14N3Cn-a and 15N3Cn-a. The transport was initiated by exciting the azido moiety stretching mode, the νN≡N tag, at 2100 cm–1 (14N3Cn-a) or 2031 cm–1 (15N3Cn-a). Opposite to the expectation, the ballistic transport speed was found to decrease upon 14N3 → 15N3 isotope editing. Three mechanisms of the transport initiation of a vibrational wavepacket are described and analyzed. The first mechanism involves the direct formation of a wavepacket via excitation with IR photons of several strong Fermi resonances of the tag mode with the νN=N + νN–C combination state while each of the combination state components is mixed with delocalized chain states. The second mechanism relies on the vibrational relaxation of an end-group-localized tag into a mostly localized end-group state that is strongly coupled to multiple delocalized states of a chain band. Harmonic mixing of νN=N of the azido group with CH2 wagging states of the chain permits a wavepacket formation within a portion of the wagging band, suggesting a fast transport speed. The third mechanism involves the vibrational relaxation of an end-group-localized mode into chain states. Two such pathways were found for the νN≡N initiation: The νN=N mode relaxes efficiently into the twisting band states and low-frequency acoustic modes, and the νN–C mode relaxes into the rocking band states and low-frequency acoustic modes. The contributions of the three initiation mechanisms in the ballistic energy transport initiated by νN≡N tag are quantitatively evaluated and related to the experiment. We conclude that the third mechanism dominates the transport in alkane chains of 5–15 methylene units initiated with the νN≡N tag and the wavepacket generated predominantly at the CH2 twisting band. The isotope effect of the transport speed is attributed to a larger contribution of the faster wavepackets for 14N3Cn-a or to the different breadth of the wavepacket within the twisting band. The study offers a systematic description of different transport initiation mechanisms and discusses the requirements and features of each mechanism. Such analysis will be useful for designing novel materials for energy management.


S2. Additional 2DIR data and analysis details
The  NN /  C=O cross peak spectra were measured for 15 N 3 Cn-a where n = 5, 10, 15 as a function of waiting time and the cross-peak amplitude kinetics were obtained. Figure S1 shows three waiting time dependences for each chain length, separately fitted with an asymmetric double sigmoidal function: (S1) Here is the offset, A is the amplitude, is the center of the peak, is the full width of half maximum, 0 1 is the variance of low energy side, and is the variance of high energy side. The data were fitted with an asymmetric double sigmoidal function (red lines), Eq. S1.

S3. Calculating mean group velocity.
The (N=N) mode frequency falls within two chain bands, corresponding to the CH 2 wagging and twisting motions. Wagging and twisting dispersion curves computed for an alkyl chain are shown in Figure  S2. A model of a linear chain of coupled states results in dispersion relations described by Eq. S2.

S2
( ) = 0 +2 cos ( ) +2 cos (2 ) where is the site energy, is the nearest neighbor interaction, is the next nearest neighbor interaction, 0 is the wavevector and is the lattice period. Equation S2 can be used to approximate actual dispersion curves for linear alkane chains. Detailed description of chain band construction is given elsewhere. 1 The mean group velocity supported by the whole band was computed as: . If a small portion of a band at  end-gr. is involved in the wavepacket, covering a wavevector range from q 1 to q 2 , the group velocity can be computed as . If the portion of the band involved in the wavepacket is very narrow the mean group velocity of such wavepacket can be approximated as a point derivative to the dispersion curve at  end-gr. : .    As apparent from Figure S2 and Table S2, the point derivatives to both wagging and twisting ∂ ∂ band curves are significantly larger at the position of ( 15 N= 15 N) compared to those at ( 14 N= 14 N). Therefore, before performing the experiment, we expected that the red frequency shift of  N=N by isotopic substitution would lead to a higher transport speed in 15 N 3 Cn-a but opposite behavior was observed.
In an effort to explain experimentally observed speeds, molecule specific anharmonic effects on the wagging chain bands were considered. Wagging bands were constructed using DFT anharmonic calculations for the N 3 C5 compounds (Fig. S2B). Chain modes showing wagging motion type were grouped into a band and fitted with Eq. S2. A wavevector for each chain state was identified by the number of the nodes for the delocalized chain-state wavefunction. The wavevector for mode i was computed as = ( + 1) , where n is the number of methylene groups in the chain, a is the unit-cell length, computed at 1.53 ( +1) Å, and N i is the number of nodes. As expected, the wagging band is not affected by the end-group isotope editing (Fig. S2B). The bandwidth obtained with anharmonic state frequencies appeared to be wider than that for harmonic states by ca. 15%. As a result of the band changes, the transport speed obtained as < gr appears to be higher than that for harmonically determined band, at 43 and 41 Å/ps for 14 N 3 C5 >~∂ ∂ | end -gr. and 15 N 3 C5, respectively.

S4. Atomic displacements of the low-frequency modes involved in  N=N relaxation
Intramolecular vibrational energy redistribution (IVR) is an essential process for the thermalization of a molecule. This process is often described via anharmonic interactions of normal modes in the molecule. A large portion of relaxation pathways of  N=N (i) involve combination states of one of the twisting modes (j) and a low frequency mode (k). The low frequency modes (k) involved in such relaxation are shown in Table  S3.

S5. Fermi resonances computed of  NN
The effect of CNN angle variation and Fermi resonance on the  14NN transition has been studied elsewhere for 14 N 3 C3. 2 Here we performed DFT calculations of Fermi resonances for 15 N 3 C3 and found the presence of multiple Fermi resonances. Although Fermi resonances were found for a wide range of CNN angles for 15 N 3 C3 (Fig. S3B), their number is smaller than that for 14 N 3 C3 (Fig. S3A). It is conceivable that Fermi resonances contribute to the smaller T max values observed for 14 N 3 Cn-a compounds.

S6. Rocking chain band using anharmonic frequencies
Mixing of the  N-C mode with the rocking band states is negligible, so the rocking band does not change upon 14 N 3  15 N 3 end-group substitution. Figure S4 shows that the changes of the rocking band are not very large if anharmonic rocking states are considered.   Figure S4. CH 2 rocking band constructed from DFT harmonic and anharmonic calculations of 14 N 3 C5 and 15 N 3 C5. Harmonic and anharmonic  N-C frequencies for both isotopes are shown.

S7. Relaxation pathways of  N=N
Analysis of the IVR relaxation pathways for  N=N revealed that many bands receive energy with a significant rate contribution. All relaxation pathways of  N=N for N 3 C5 in DCM is shown below. Both isotopes prominently relax to the twisting band states, N-C stretch mode and an overtone of the NNN bending mode.  Figure S5. IVR relaxation pathways and rates for  N=N for both isotopes ( 14 N -blue and 15 N -green). The  N=N frequency is marked (red line).

S8. Lifetime measurements for  NN
Diagonal  NN peaks for 15 N 3 C5-a were measured as a function of the waiting time (Fig. S5A), integrated, and plotted as a function of the waiting time (Fig. S5B). The trace was fitted with a twoexponential function (red line in Fig. S5B) resulting in decay times of 0.84  0.05 ps (77% amplitude) and 7.0  1.1 ps (23% amplitude). The dominant fast decay component is associated with the  NN lifetime, while the second component is likely associated with cooling of the excess energy at the azido group. The  1/e value of 1.4 ps was obtained for 15 N 3 C5-a. For comparison, the  1/e value of 1.47 ps was obtained for 14 N 3 C5-a, indicating that the  NN lifetimes for both isotopes do not differ much. S9. Details of the classical modeling of ballistic transport. Ballistic energy transport in N 3 Cn-a was modeled by solving classical Newton's equations using DFT-determined harmonic force constants (Hessian). Note that harmonic approximation is expected to work reasonably well for the wavepacket transport. The initial excitation at ca. 1300 cm -1 was introduced by exciting a local  N=N mode. This local mode has been identified as the second highest vibrational frequency of the azido group, obtained by diagonalizing the reduced Hessian matrix (the part of the Hessian matrix associated with the three nitrogen atoms only). Initial displacements for nitrogen atoms have been set proportional to corresponding eigenvector components, while other displacements and all momenta have been set equal to zero at the initial time t = 0. Energy arrival to the reporter was monitored by summing all local energies of atoms belonging to the COOH end group. The local energies are defined for each atom i and each Cartesian coordinate x as where H stands for the Hessian matrix and p are the momenta. The analysis of the end-group energy time evolution was used to estimate the ballistic transport time and velocity as described below.
The time dependence of the COOH end-group energy was evaluated solving Newton's equations employing MATLAB software. The same Hessian matrix was used for normal and isotopically modified samples and the difference is originated only from the mass difference.
Representative time dependencies of the end-group energy are shown in Figure S7 for the normal and isotopically modified N 3 C5-s samples at delay times between 0 and 10 ps. It turns out that these dependences are periodic. Consequently, one can estimate the transport time assuming that the period represents the double travel time of the energy along the chain.
A B Figure S7. COOH end-group energy vs. time for the normal and isotopically edited N 3 C5-s compounds. Energy is expressed in arbitrarily units.
The period, t R , of the recurrency peaks was determined, as indicated in Figure S7 by double-sided arrows for normal and isotope samples. The energy transport velocity was determined independently for compounds with every chain length, n (n = 5, 10, 15) as where V(n) is the transport velocity, t R is the transport period, and (1.54 Å) and (1.48 Å) are the C-C and C-N bond lengths, respectively. The results for all samples are summarized in Table S4. Table S4. Transport velocities calculated with Eq. S3 for normal and isotopically substituted N 3 Cn-a compounds with n = 5, 10, 15. No energy damping was introduced. 14 N 3 Cn-a 15 N 3 Cn-a n Period, t R (ps) Velocity, V (Å/ps) Period, t R (ps) Velocity, V (Å/ps) 5 1.9 9.7 2.5 7.3 10 2.3 14.7 5.0 6.8 15 5.0 9.8 9.0 5.5 Alternatively, one can investigate the transport time considering the arrival time in the presence of energy damping, which represents relaxation and/or decoherence of the chain states. Following previous work, we set the damping time to be  = 1.5 ps and incorporated it into Newton's equations for momenta via -p/ terms. After solving the Newton's equations, one obtains that the COOH excess energy, E(t), decays rapidly with the time increase (Fig. 7C of the main text). To estimate the transport time, we used the average arrival time, t D weighted with the function E(t) as The estimates of transport velocity based on arrival times are shown below in Table S2. Table S5. Transport velocities calculated using Eq. S5 for normal and isotopic N 3 Cn-a compounds with n = 5, 10, 15 with the damping factor of 1.5 ps -1 . 14 N 3 Cn-a 15 N 3 Cn-a n t D (ps) Velocity (Å/ps) t D (ps) Velocity (Å/ps) 5 0.8 11.5 1.0 9.1 10 1. The normal mode analysis of the initial excitation suggests that CH 2 wagging modes in the region 1330 -1360 cm -1 are strongly involved in the energy transport; the resulting velocities are different for the two isotopes and slightly depends on the length of the molecule. These results are used in the main text to discuss the origin of the isotope effect in the ballistic transport.