Thermalization and Thermal Transport in Molecules - The Journal of

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Thermalization and Thermal Transport in Molecules Hari Datt Pandey, and David M. Leitner J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b02539 • Publication Date (Web): 21 Nov 2016 Downloaded from http://pubs.acs.org on November 26, 2016

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The Journal of Physical Chemistry Letters

Thermalization and Thermal Transport in Molecules Hari Datt Pandey and David M. Leitner* Department of Chemistry and Chemical Physics Program, University of Nevada, Reno, NV 89557, USA * [email protected] Abstract The nature and rate of thermal transport through molecular junctions depend on the length over which thermalization occurs.

For junctions formed by alkane chains, in which

thermalization occurs only slowly, measurements reveal thermal resistance is controlled by bonding with the substrates, whereas fluorination can introduce thermal resistance within the molecules themselves, though the mechanism remains unclear.

Here we

present results of quantum mechanical calculations of elastic and inelastic scattering rates, the length over which thermalization occurs, and thermal conductance in alkane and perfluoroalkane junctions. The contribution to thermalization of quantum effects that give rise to many-body localization (MBL) in isolated molecules is examined. While MBL does not occur due to dephasing, thermalization is typically too slow to establish local temperature if the same molecule in isolation exhibits MBL. The results indicate limitations on the applicability of classical molecular simulations in modeling thermal transport in molecular junctions.

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Elucidation and control of thermal transport at the nanoscale remain major challenges in the design of nanoscale devices,1-2 small composite materials in which interfaces often mediate heat flow. Exploring thermal transport in molecular junctions has thus been the focus of considerable effort.1-12 On molecular scales of ~ 1 nm, the energy diffusion picture and Fourier’s heat law break down, as revealed by numerous experimental measurements probing vibrational energy and thermal transport in molecules,3-4, 9, 13-15 which indicate ballistic transport so that local temperature within the molecule is ill defined. For a molecule bridging two solid substrates (fig. 1) at two different temperatures thermalization occurs by quasiparticle dynamics, specifically inelastic scattering of the vibrational excitations by anharmonic interactions within the molecule. If the molecule were isolated it may not thermalize at all. Quantum mechanical interference effects can give rise to localization of a molecule’s vibrational states, an example of many-body-localization (MBL), where a closed quantum mechanical system does not thermalize under its own dynamics.16-18 Though molecular junctions are not isolated, effects that give rise to localization in the isolated molecule have a substantial impact on the time and length over which thermalization occurs when a molecule is coupled to two substrates at different temperature. Indeed, striking effects of quantum localization on chemical reaction kinetics involving large molecules in a variety of environments have been reported in the past.18-20 Here we point out and illustrate the influence of localization on thermalization and thermal transport in molecules. Thermal conductance across molecular interfaces can now be measured, e.g., by thermo-reflectance21 and scanning thermal microscopy22 techniques. Bonding between molecule and substrate has been found to control thermal transport at the interface.3, 12 In some studies that have controlled for bonding the length of the molecule has been varied22

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and in many cases conductance does not change, indicating that thermal transport through the molecule occurs essentially ballistically. Interfaces formed by alkane chains roughly a few nm in length exemplify these kinds of systems.3

In the absence of inelastic

scattering local temperature within the junction is not well defined. The length where thermalization begins to occur in alkanes is not yet known, only that it exceeds a few nm.7, 22-23

Molecular interfaces in which the origin of thermal resistance lies within the

molecules rather than bonding between molecules and substrate have also been identified. In a recent experimental study of carbon chains terminated by phosphates between a metal and sapphire, fluorination of the molecules in the junction yields thermal resistance that increases with the molecular size.9

The origin of scattering that gives rise to resistance

was suggested to be inelastic, but could also be elastic or both. Methods commonly used to calculate thermal boundary conductance highlight the distinction between elastic and inelastic quasiparticle dynamics in thermal conduction, where at one extreme classical molecular simulations can exaggerate thermalization, and in the other Landauer models neglect it.

Temperature can always be defined as

proportional to the mean square atomic velocity in a classical system even if it is ill defined in a molecule. A recent experimental-computational study of thermal transport through a molecular interface between two metal leads highlights shortcomings of classical simulations in modeling thermal transport through molecular junctions.4 The measured interface conduction between two metal leads, at least one gold, bridged by selfassembled monolayer junctions decreases with increasing mismatch of the vibrational density of states of the metal leads, whereas the molecular simulations predict an increase over the same range of vibrational density mismatch. The authors suggest4 anharmonic scattering within the junction is likely exaggerated in the classical simulations, allowing


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vibrational energy entering from Au (Debye temperature 170 K) to up-convert and transfer to Pd (Debye temperature 275 K), channels apparently not available in the molecular interface probed experimentally.4 On such length scales a Landauer approach may fair better. The Landauer formalism, introduced to quantify electrical conductance in mesoscopic systems,24 can be applied to thermal transport through molecules between two solids at different temperature, but while elastic scattering of quasiparticles can be accommodated, inelastic scattering that ultimately yields thermalization is neglected.4, 21, 25 The scope of applicability of Landauer and classical simulation approaches can only be determined when the extent of thermalization is clarified. We address thermalization in two steps, first by processes within the molecule itself, then contributions of coupling to the environment.

The vibrational modes of the

molecule, described by the zero-order Hamiltonian, H0, are coupled by anharmonic interactions V, and the vibrational Hamiltonian of the isolated molecule is H = H0 + V. We take V to include coupling to third order in the anharmonicity; terms of higher order can be included but typically decrease exponentially in magnitude26 and are thus neglected. If for this molecule, H, the population of its vibrational modes is distributed as in thermal equilibrium then for a given mode α with frequency ωα the population is . (Though we consider for now a many-body system at fixed energy the temperature, T, is defined as 1/T = ∂S/∂U.) If the molecule were vibrationally excited by a laser and in the absence of anharmonic coupling the molecule could never relax to

at the new effective temperature T. However, the

anharmonic molecule may not relax either (see fig. 1), an example of many-body localization (MBL), which was predicted by Logan and Wolynes17 and has been observed in numerous vibrationally excited polyatomic molecules in the gas phase.18 The criterion


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for localization in the vibrational state space (fig. 1) of the anharmonic molecule is17

T (E) ≡

2π Vαβγ 3

2

ρ 2 (E) < 1, where ρ is the local density of states, and

Vαβγ

is the

average size of the matrix element coupling the triple of modes α, β and γ. This criterion emerges from a self-consistent analysis of the distribution of rates at which states on the energy shell (fig. 1) relax towards equilibrium.17 The most probable rate in a large molecule is always 0 when T < 1, despite finite coupling between vibrational states due to anharmonic interactions.

When the vibrational states of the molecule are localized

thermalization does not occur, but localization does not impede the transfer of energy by the delocalized heat-carrying modes of the molecule. Even when the molecule is not isolated the most probable relaxation rates remain in practice slow if T < 1. It is convenient to separate the interactions between vibrational states into two relaxation processes that can occur while conserving energy. In one process, decay, a quasiparticle in a mode of frequency ωα decays into two quasiparticles of frequency ω β and ωγ .

In the other process, collision, a quasiparticle in a mode of frequency ωα

combines with another of frequency ω β to create a quasiparticle of frequency ωγ . The matrix elements are of the form Vαβγ = Φαβγ process and Vαβγ = Φαβγ

nα ( nβ +1) ( nγ +1) for interactions in a decay

nα nβ ( nγ +1) for interactions in a collision.

Φ αβγ are the

coefficients of the cubic terms in the expansion of the interatomic potential in normal coordinates. To determine whether an isolated molecule can thermalize under its own dynamics we need to calculate T(E) by averaging over all states on the energy shell. For a large molecule most states are occupied very nearly according to


, so

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for nα we use nα . Averaging | Φαβγ | nα ( nβ +1) ( nγ +1) and | Φαβγ | nα nβ ( nγ +1) for resonant coupling we define T = Td + Tc , where

Td (ωα ) =

2 2π 2 | Φαβγ | nα ( nβ +1) ( nγ +1) ρres (ωα ) 3

(1a)

Tc (ωα ) =

2 2π 2 | Φαβγ | nα nβ ( nγ +1) ρres (ωα ) . 3

(1b)

Finally, we average over the modes at a given temperature (internal energy), ,

(2)

where Q is the partition function. The average quasiparticle relaxation rate is often estimated with the golden rule, which to third order in the anharmonicity has decay and collision contributions, Wd and Wc, respectively, which are given by eq. (S1). However, the golden rule is only valid when T is significantly larger than 1. Near the MBL transition the most probable rate is smaller than the average rate, meeting it only at large T.17 Correcting for the effects of a finite local density of states in the molecule the most probable relaxation rate becomes 1/2

Wdmp (ωα ) = Wd (ωα ) (1− T −1 (ωα )) , 1/2

Wcmp (ωα ) = Wc (ωα ) (1− T −1 (ωα )) ,

T ≥1

(3a)

T ≥1

(3b)

As with T, we take a thermal average of the rates, .

(4)

We use the transition rates given by eq. (3) – (4) to estimate the quasiparticle relaxation rate in molecules that form a thermal bridge between two substrates when T > 1.


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The molecules are coupled to the substrates, and we address energy transfer to and from the substrates in thermalization assuming steady state, where quasiparticles enter and leave the molecule via the substrates at the same rate. The molecules may also couple to each other, which we do not explicitly address though it can also be added to the environmental contribution to thermalization. To account for energy propagation in the molecules, we find in our simulations of dodecane (SI, fig. S2) a rate 1768 m s-1, comparable to the measured speed of sound in polyethylene, reported as low as 1250 m s-1 and has high as 2300 m s-1.23, 27

We find 641 m s-1 in perfluorododecane (SI, fig. S2).

MBL is lost with dephasing and we calculate the dephasing rate, η, using a ballistic transport time as an upper estimate; if there is elastic scattering the time for a quasiparticle that has entered the molecule to leave it will be longer and η smaller. We shall see that using the larger, ballistic estimate has very little practical effect on the rate of thermalization, which is generally small if T < 1. For the molecules we consider, η ranges from 2.4 ps-1 to 1.0 ps-1, respectively, for the alkanes that range in length from hexane to pentadecane, and from 1.0 ps-1 to 0.5 ps-1, respectively, for the fluorinated alkanes from perfluorohexane to perfluorododecane.

Calculation of the effects of dephasing on

thermalization within molecules is discussed in the SI and the rate of thermalization when T < 1, in practice small, is given by eq. (S3). Values of T defined by eq. (1) - (2) are plotted in fig. 2(a) and (b) for the alkanes and fluorinated alkanes with internal energies corresponding to 100 K, 200 K, 300 K and 400 K. At 100 K and 200 K, the threshold T = 1 has not been met by any of the alkanes, so that even for alkanes as large as pentadecane thermalization in the isolated molecule does not occur. At 300 K the threshold is exceeded only by dodecane and pentadecane and at 400 K all of the alkanes exceed the threshold of T = 1. In contrast, all perfluorinated


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alkanes lie above the threshold already at 200 K so that restrictions on thermalization due to MBL effects are smaller than in alkanes. The thermalization thresholds differ mainly because of the larger local density of states of the fluorinated alkanes. Vibrational mode frequencies of these molecules are plotted in fig. S1. Using the values of T plotted in fig. 2(a) and (b) we calculate the most probable quasiparticle relaxation rates, plotted in fig. 2(c) and (d) along with estimates for the quasiparticle relaxation rates in the limit of no MBL effects. We refer to the latter as the large molecule limit, which is the average rate calculated for the largest molecule in each class, recognizing that the relaxation rate may in fact be somewhat different for larger molecules, i.e., while the relaxation rate at some length becomes independent of the length of the molecule,28 we cannot be sure the largest molecules we consider have converged to that limit. At 300 K in the large molecule limit, quasiparticle relaxation times are about 1 ps, typical of relaxation times in large molecules, e.g., in peptides and proteins.28-31 However, MBL effects significantly lengthen thermalization times in alkanes; e.g., at 300 K, the relaxation time is about an order of magnitude longer for nonane than the 1 ps time in the large molecule limit. The differences between the most probable relaxation rate and the average quasiparticle relaxation rate in the large molecule limit are smaller for fluorinated alkanes but still significant at lower temperature. The thermalization length depends on the speed at which vibrational energy propagates, 1768 m s-1 and 641 m s-1 for dodecane and perfluorododecane, respectively. Dividing this speed by the quasiparticle relaxation rate gives the length over which inelastic scattering occurs. We plot the inelastic scattering length in fig. 3 for dodecane, perfluorododecane, and for alkanes and perfluoroalkanes in the large molecule limit. For alkanes in the large molecule limit the thermalization length is 2 nm at 300 K and > 10 nm ACS Paragon Plus Environment

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at 100 K. When the effects of MBL are included the length over which thermalization occurs in dodecane exceeds 5 nm at 300 K and over 300 nm at 100 K. Experiments on alkane chains to about 3 nm indicate little or no resistance to thermal transport at room temperature22-23 and to about 2 nm over temperatures reaching 400 K,9 consistent with the results of the calculations presented here. Resistance in a fluorinated molecule about 2 nm in length has been found at temperatures from 100 K to 400 K.9 The observed temperature dependence is much weaker than the temperature dependence of the inelastic scattering lengths plotted in fig. 3, so we also examine effects of elastic scattering, which occur due to the aperiodicity of the system. We found in an earlier simulation of energy diffusion in perfluorododecane in harmonic approximation a coefficient 1.77 x 10-7 m2 s-1.32 Since the speed of sound in that molecule is 641 m s-1, the elastic scattering length, twice the ratio of those values, is 0.55 nm, also plotted in fig. 3. The calculated inelastic scattering lengths indicate that local temperature cannot be defined in even ~ 1 nm fluorinated molecules, except perhaps at 400 K. We thus adopt a Landauer model to estimate the thermal boundary conductance

,

(5a)

τ (ω ) = τ 0 (ω ) / (1+ ( L / l ) τ 0 (ω )) ,

(5b)

τ 0 (ω ) = v2 ρ 2 (ω ) / ( v1ρ1 (ω ) + v2 ρ2 (ω )) .

(5c)

The transmission function, τ, accounts for detailed balance for transport between the two substrates, labeled 1 and 2, as well as elastic scattering through the molecule of length L with mean free path, l. v1 is the phonon velocity in substrate 1, ρ is the mode density per unit volume. The parameter A can account for surface contact. Its value has been ACS Paragon Plus Environment

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estimated in some studies4 but here it is the one fitting parameter for a given substrate pair; we find the values .82 and .18 to fit the data well for the Au-sapphire and Al-sapphire interface, respectively. The factor of 4 is the result of averaging over incident angles of the phonon as it strikes the interface. The Debye frequency, ωD, is the lower of the values for the two substrates. To compare with the results of Ref. [9], we use eq. (5) with a L = 2 nm perfluorinated alkane, the junction length in the experiments,9 and we use the mean free path l = 0.55 nm found in the elastic scattering calculation above. In fig. 4 we plot the thermal conductance from Au to Sapphire and from Al to Sapphire, both with and without a molecular layer between them, at 100 K, 200 K, 300 K and 400 K. The calculated elastic mean free path introduced into the Landauer formula describes well the effect of fluorination on thermal transport in the molecular layer. In summary, quantum effects that give rise to localization in the vibrational state space of isolated molecules can substantially reduce the rate of thermalization when the molecules form a junction even at room temperature, and by over an order of magnitude in alkanes at low temperature. The effect in fluorinated alkanes is smaller but temperature is nevertheless ill defined over lengths of order 1 nm. The observed length-dependent thermal conduction can be explained by elastic scattering in the molecule and a Landauer formalism provides a good description of the temperature-dependence of the thermal boundary conductance, as we observe here. When the length of the molecule is much greater than the thermalization length the Landauer picture will break down. In large molecules such as proteins, for instance, inelastic scattering via anharmonic interactions mediate thermal transport.28, 33 For alkane chains, lengths of many 10s or perhaps 100s of nm are needed, depending on temperature. For perfluorinated alkanes beyond roughly 5 ACS Paragon Plus Environment

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nm noticeable differences from the Landauer calculations would be found at room temperature or higher.

Computational Methods

The lowest energy structures, the Hessian matrix, normal

modes and anharmonic constants for all the alkanes and fluorinated alkanes were obtained using Density Functional Theory (DFT). B3LYP functionals were used together with the 6-31G** basis set.

All of the calculations were performed with the Gaussian-09

computational package.

The resulting normal mode frequencies for dodecane and

perfluorododecane are presented in figure S1. All parameters for the calculation of T and W using equations (1) – (4) were thereby obtained. The computational method to simulate vibrational energy propagation in the molecules is described in the SI.


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Figure 1. Vibrational states of one molecule at a molecular junction, shown schematically in the inset as a layer of alkane chains bonded to two substrates at different temperatures. Each vibrational state depends on the number of vibrational quanta that occupy the modes of the molecule, three of which are plotted explicitly. If a mode is excited to the state indicated by the red dot, then thermalization, if it occurs, yields an equilibrium distribution within the molecule, which is indicated by the yellow dot. However, if the anharmonic coupling or the local density of resonantly coupled states of the molecule is sufficiently small, relaxation occurs only very slowly, or not at all if the molecule were isolated, an example of many-body localization (MBL).

While the molecule is not isolated and

localization does not strictly occur due to dephasing, the rate of thermalization in the junction is nevertheless substantially reduced if the isolated molecule exhibits MBL, thereby minimizing effects of inelastic scattering on thermal transport through the molecular junction.


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1.6

10

✕

(a)

5

2

T

1

✕ ■

0.5

● ▲

● ▲

0.2

200

300

50

10 5

T

2

■

1 0.5 0.2

■

(b)

20

1.0

■

0.6 0.4

■

●

●

▲

0.0

400

■ ■ ✕ ▲ ●

100

■

1.6

●

1.4

✕ ▲ ■ ●

▲ ●

200

300

▲

■ 1.0 0.8

■

0.6

0.2

▲ 200

300

Temperature (K)

400

400

1.2

0.0

● ▲ ● ▲

●

0.4

0.1

● ▲

■

(d)

▲

●

100

✕

0.8

0.2

● ▲

0.1 100

✕

1.2

Rate (ps-1)

✕ ■

(c)

1.4

■

Rate (ps-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


■ ● ▲

100

▲ 200

300

400

Temperature (K)

Figure 2. (a) Value of the MBL transition parameter, T, as a function of the internal energy, converted to temperature, T (K), for alkanes with N = 6 (red), 9 (green), 12 (blue) and 15 (black) carbon atoms. The calculations were carried out at the temperatures indicated by points; lines connecting points are a guide to the eye. The thick black line indicates the MBL transition at T = 1. The lowest energy structure for dodecane appears in the inset. (b) same as (a) but for perfluoroalkanes, where the inset shows the lowest energy structure of perfluorododecane. (c) Rate of inelastic scattering in alkanes, where the colors are the same as in (a). The dashed curve is the rate calculated with the golden rule (eq. (S1)), where effects of MBL are neglected. (d) same as (c) for perfluoroalkanes.


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Figure 3. Mean free path due to inelastic scattering calculated for alkanes (black) and perfluoroalkanes (red). Results using a golden rule formula for the thermalization rate in the large molecule limit are plotted as dashed curves. Accounting for quantum coherence effects that give rise to many-body localization (X) the length over which thermalization occurs in alkanes increases dramatically; the increase is much smaller for perfluoroalkanes (squares).

The elastic scattering length (.55 nm) calculated for perfluoroalkanes is

indicated by the blue line, and appears to control thermal conduction in these molecules to lengths of a few nm.


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Figure 4. Calculations of thermal conductance between Au and sapphire (red) and between Al and sapphire (black) with carbon-chain molecular interface that is fluorinated (solid) or not (dashed), for temperatures from 100 – 400 K. The Landauer model is used with calculated elastic scattering length, as discussed in the text. Experimental results from Ref. [9] are plotted (Au-sapphire red circles, Al-sapphire black squares, closed and open, respectively, for carbon-chain molecular interfaces that are fluorinated and not fluorinated) with reported error bars.

The results are consistent with incomplete

thermalization in these molecules to at least 2 nm.


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Acknowledgements Support from NSF grant CHE-1361776 is gratefully acknowledged.

Supporting Information Available The golden rule relaxation rate and effects of the dephasing rate on the most probable thermalization rate, the vibrational modes computed for the alkanes and fluorinated alkanes, and the calculation of vibrational energy propagation in terms of normal modes.

References 1. Cahill, D. G.; Ford, W. K.; Goodson, K. E.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R., Nanoscale Thermal Transport. J. Appl. Phys. 2003, 93, 793 - 818. 2. Cahill, D. G.; Ford, W. K.; Goodson, K. E.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R., Nanoscale Thermal Transport. Ii. 2003–2012. Appl. Phys. Rev. 2014, 1, 011305. 3. Losego, M. D.; Grady, M. E.; Sottow, N. R.; Cahill, D. G.; Braun, P. V., Effects of Chemical Bonding on Heat Transport across Interfaces. Nat. Mat. 2012, 11, 502 - 506. 4. Majumdar, S.; Sierra-Suarez, J. A.; Schiffres, S. N.; Ong, W.-L.; Higgs, C. F.; McGaughey, A. J. H.; Malen, J. A., Vibrational Mismatch of Metal Leads Controls Thermal Conductance of Self-Assembled Monolayer Junctions. Nano. Lett. 2015, 15, 2985 - 2991. 5. Galperin, M.; Nitzan, A.; Ratner, M. A., Heat Conduction in Molecular Transport Junctions. Phys. Rev. B 2007, 75, 155312 - 1 - 15. 6. Lee, W.; Kim, K.; Jeong, W.; Zotti, L. A.; Pauly, F.; Cuevas, J. C.; Reddy, P., Heat Dissipation in Atomic-Scale Junctions. Nature 2013, 498, 209 - 213. 7. Segal, D.; Nitzan, A.; Hänggi, P., Thermal Conductance through Molecular Wires. J. Chem. Phys. 2003, 119, 6840-6855. 8. Segal, D.; Agarwalla, B. K., Vibrational Heat Transport in Molecular Junctions. Ann. Rev. Phys. Chem. 2016, 67, 185 - 209. 9. Gaskins, J. T.; Bulusu, A.; Giordano, A. J.; Duda, J. C.; Graham, S.; Hopkins, P. E., Thermal Conductance across Phosphonic Acid Molecules and Interfaces: Ballistic Versus Diffusive Vibrational Transport in Molecular Monolayers. J. Phys. Chem. C 2015, 119, 20931 - 20939. 16 ACS Paragon Plus Environment

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10. Leitner, D. M., Thermal Boundary Conductance and Rectification in Molecules. J. Phys. Chem. B 2013, 117, 12820 - 8. 11. Craven, G. T.; Nitzan, A., Electron Transfer across a Thermal Gradient. Proc. Natl. Acad. Sci. (USA) 2016, 113, 9421 - 9429. 12. O'Brien, P. J.; Shenogin, S.; Liu, J.; Chow, P. K.; Laurencin, D.; Mutin, P. H.; Yamaguchi, M.; Keblinski, P.; Ramanath, G., Bonding-Induced Thermal Conductance Enhancement at Inorganic Heterointerfaces Using Nanomolecular Monolayers. Nat. Mater. 2012, 12, 118 - 122. 13. Rubtsova, N. I.; Qasim, L. N.; Kurnosov, A. A.; Burin, A. L.; Rubtsov, I. V., Ballistic Energy Transport in Oligomers. Accounts of Chem. Res. 2015, 48, 2547 - 2555. 14. Lin, Z.; Rubtsov, I. V., Constant-Speed Vibrational Signaling Along Polyethyleneglycol Chain up to 60 Å Distance. Proc. Natl. Acad. Sci. (USA) 2012, 109, 1414-1418. 15. Rubtsova, N. I.; Rubtsov, I. V., Vibrational Energy Transport in Molecules Studied by Relaxation-Assisted Two-Dimensional Infrared Spectroscopy. Ann. Rev. Phys. Chem. 2015, 66, 717 - 738. 16. Nandkishore, R.; Huse, D. A., Many-Body Localization and Thermalization in Quantum Statistical Mechanics. Annu. Rev. Condens. Matter Phys. 2015, 6, 15 - 38. 17. Logan, D. E.; Wolynes, P. G., Quantum Localization and Energy Flow in ManyDimensional Fermi Resonant Systems. J. Chem. Phys. 1990, 93, 4994 - 5012. 18. Leitner, D. M., Quantum Ergodicity and Energy Flow in Molecules. Adv. Phys. 2015, 64, 445 - 517. 19. Keshavamurthy, S., Eigenstates of Thiophosgene near the Dissociation Threshold: Deviations from Ergodicity. J. Phys. Chem. A 2013, 117, 8729 - 8736. 20. Leitner, D. M.; Levine, B.; Quenneville, J.; Martínez, T. J.; Wolynes, P. G., Quantum Energy Flow and Trans-Stilbene Photoisomerization: An Example of a NonRrkm Reaction. J. Phys. Chem. A 2003, 10706-10716 21. Hopkins, P. E., Thermal Transport across Solid Interfaces with Nanoscale Imperfections: Effects of Roughness, Disorder, Dislocations and Bonding on Thermal Boundary Conductance (Review Article). ISRN Mechanical Engineering 2013, 2013, art. no. 682586. 22. Meier, T.; Menges, F.; Nirmalraj, P.; Hölscher, H.; Riel, H.; Gotsmann, B., Length-Dependent Thermal Transport Along Molecular Chains. Phys. Rev. Lett. 2014, 113, 060801. 23. Wang, Z.; Carter, J. A.; Lagutchev, A.; Koh, Y. K.; Seong, N.-H.; Cahill, D. G.; Dlott, D. D., Ultrafast Flash Thermal Conductance of Molecular Chains. Science 2007, 317, 787 - 790. ACS Paragon Plus Environment

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24. Datta, S., Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, 1995. 25. Li, Q.; Duchemin, I.; Xiong, S.; Solomon, G. C.; Donadio, D., Mechanical Tuning of Thermal Transport in a Molecular Junction. J. Phys. Chem. C 2015, 119, 24636 - 642. 26. Gruebele, M.; Bigwood, R., Molecular Vibrational Energy Flow: Beyond the Golden Rule. Int. Rev. Phys. Chem. 1998, 17, 91 - 145. 27. Mergenthaler, D. B.; Pietralla, M., Heat Conduction in Highly Oriented Polyehtylene. Z. Phys. B 1994, 94, 461 - 468. 28. Yu, X.; Leitner, D. M., Vibrational Energy Transfer and Heat Conduction in a Protein. J. Phys. Chem. B 2003, 107, 1698 - 1707. 29. Hamm, P.; Lim, M.; Hochstrasser, R. M., Structure of the Amide I Band of Peptides Measured by Fs Nonlinear-Infrared Spectroscopy. J. Phys. Chem. B 1998, 102, 6123-6138. 30. 259.

Leitner, D. M., Energy Flow in Proteins. Ann. Rev. Phys. Chem. 2008, 59, 233 -

31. Stock, G., Classical Simulation of Quantum Energy Flow in Biomolecules. Phys. Rev. Lett. 2009, 102, art. no. 118301. 32. Pandey, H. D.; Leitner, D. M., Vibrational Energy Transport in Molecules and the Statistical Properties of Vibrational Modes. Chem. Phys. 2016, (doi:10.1016/j.chemphys.2016.07.008). 33. Foley, B. M.; Gorham, C. S.; Duda, J. C.; Cheaito, R.; Szwejkowski, C. J.; Constantin, C.; Kaehr, B.; Hopkins, P. E., Protein Thermal Conductivity Measured in the Solid State Reveals Anharmonic Interactions of Vibrations in a Fractal Structure. J. Phys. Chem. Lett. 2014, 5, 1077 - 1082.


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Figure 1. Vibrational states of one molecule at a molecular junction, shown schematically in the inset as a layer of alkane chains bonded to two substrates at different temperatures. Each vibrational state depends on the number of vibrational quanta that occupy the modes of the molecule, three of which are plotted explicitly. If a mode is excited to the state indicated by the red dot, then thermalization, if it occurs, yields an equilibrium distribution within the molecule, which is indicated by the yellow dot. However, if the anharmonic coupling or the local density of resonantly coupled states of the molecule is sufficiently small, relaxation occurs only very slowly, or not at all if the molecule were isolated, an example of many-body localization (MBL). While the molecule is not isolated and localization does not strictly occur due to dephasing, the rate of thermalization in the junction is nevertheless substantially reduced if the isolated molecule exhibits MBL, thereby minimizing effects of inelastic scattering on thermal transport through the molecular junction. 157x188mm (72 x 72 DPI)



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Figure 2. (a) Value of the MBL transition parameter, T, as a function of the internal energy, converted to temperature, T (K), for alkanes with N = 6 (red), 9 (green), 12 (blue) and 15 (black) carbon atoms. The calculations were carried out at the temperatures indicated by points; lines connecting points are a guide to the eye. The thick black line indicates the MBL transition at T = 1. The lowest energy structure for dodecane appears in the inset. (b) same as (a) but for perfluoroalkanes, where the inset shows the lowest energy structure of perfluorododecane. (c) Rate of inelastic scattering in alkanes, where the colors are the same as in (a). The dashed curve is the rate calculated with the golden rule (eq. (S1)), where effects of MBL are neglected. (d) same as (c) for perfluoroalkanes. 241x177mm (72 x 72 DPI)


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Figure 3. Mean free path due to inelastic scattering calculated for alkanes (black) and perfluoroalkanes (red). Results using a golden rule formula for the thermalization rate in the large molecule limit are plotted as dashed curves. Accounting for quantum coherence effects that give rise to many-body localization (X) the length over which thermalization occurs in alkanes increases dramatically; the increase is much smaller for perfluoroalkanes (squares). The elastic scattering length (.55 nm) calculated for perfluoroalkanes is indicated by the blue line, and appears to control thermal conduction in these molecules to lengths of a few nm. 227x156mm (72 x 72 DPI)



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Figure 4. Calculations of thermal conductance between Au and sapphire (red) and between Al and sapphire (black) with carbon-chain molecular interface that is fluorinated (solid) or not (dashed), for temperatures from 100 – 400 K. The Landauer model is used with calculated elastic scattering length, as discussed in the text. Experimental results from Ref. [9] are plotted (Au-sapphire red circles, Al-sapphire black squares, closed and open, respectively, for carbon-chain molecular interfaces that are fluorinated and not fluorinated) with reported error bars. The results are consistent with incomplete thermalization in these molecules to at least 2 nm. 233x203mm (72 x 72 DPI)


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TOC Graphic 162x172mm (72 x 72 DPI)


Thermalization and Thermal Transport in Molecules - The Journal of

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