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Thermal Boundary Conductance and Thermal Rectification in Molecules David M. Leitner* Department of Chemistry and Chemical Physics Program, University of Nevada, Reno, Nevada 89557, United States ABSTRACT: An approach is presented to calculate thermal boundary resistance in molecules, which occurs, for example, at the interfaces between moieties held at fixed temperatures and a molecular bridge that connects them. If the vibrational frequencies of each moiety lie outside of the band of heatcarrying modes of the bridge, anharmonic interactions mediate thermal conduction at the boundaries. We have expressed thermal boundary conductance in terms of the low-order anharmonic interactions between a moiety and a molecular bridge. Differences in the temperature-dependent boundary conductance at each end of the bridge can be exploited in the design of a molecular thermal diode. The approach is illustrated with the calculation of thermal boundary conductance and thermal rectification in azulene−(CH2)N−anthracene.

1. INTRODUCTION Interest in thermal conduction on the nanoscale1−7 has motivated many theoretical and experimental studies on thermal flow in molecules and molecular films8−19 in recent years, including alkane chains, 9,20−23 biological molecules,14,24−28 inorganic compounds,11 nanoporous silica materials,29,30 clathrate hydrates,31 nanoconfined oligomers,32 and self-assembled monolayers.33,34 Conditions for thermal rectification in molecules have been explored.34−40 Recently, attention has turned to thermal conduction across an interface between molecules and the solvent or substrate.33,41−46 This work includes computational studies on thermal boundary conductance between biomolecules and water41,42 and between molecular films and the substrate or solvent,33,43,45 as well as molecular simulation23 studies of interface resistance within a molecule itself. In the latter case, thermal boundary resistance has been observed between a moiety held at a fixed temperature and an alkane chain bridging it to a second moiety, for example, azulene and anthracene bridged by an alkane chain23 (Figure 1), resembling Kapitza resistance observed at the interface between many materials.47 In this article, we describe thermal boundary conductance between a moiety and alkane chain

using analogies to intramolecular vibrational redistribution (IVR).48−56 For two different moieties held at different temperatures on either side of the molecular chain, distinct values of the temperature-dependent thermal conductance across each of the moiety−alkane chain boundaries can be exploited for thermal rectification. Standard theoretical approaches to describe thermal boundary conductance, notably, the Acoustic Mismatch Model (AMM) and the Diffuse Mismatch Model (DMM), predict boundary conductance in terms of bulk properties of the materials that meet at the interface.47 Because the interactions between the materials at the boundary may, of course, contribute to thermal boundary conductance, there is currently effort in accounting for effects such as bonding and mass impurities.46,57−60 Anharmonic interactions at the boundary, which are neglected in the AMM and DMM, can affect thermal flow in several ways, for example, structural changes at the interface that occur with small changes in temperature15,16,61 and vibrational energy transfer via Fermi resonances.51 We expect the latter to be particularly important in facilitating thermal flow at the interface between a moiety and molecular bridge, including azulene and anthracene bridged by an alkane chain addressed in recent experimental22 and molecular simulation23 studies. Anharmonicity mediates thermal transport at a junction between an atom or diatomic molecule and thermal reservoir.18 In this paper, we develop an approach to calculate the thermal boundary conductance between different parts of a molecule facilitated by anharmonic interactions. Each moiety is held at a fixed temperature, which could be accomplished, for example, by attaching each to a

Figure 1. An alkane chain bridging two moieties, which serve as heat baths. Azulene−(CH2)N−anthracene is shown as an illustrative example, for which the thermal conductance at the boundaries, indicated by thick lines, is calculated.

Special Issue: Peter G. Wolynes Festschrift

© XXXX American Chemical Society

Received: February 26, 2013 Revised: April 25, 2013

A

dx.doi.org/10.1021/jp402012z | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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substrate, nanoparticle or nanotube held at fixed temperature. We assume that almost all of the vibrational frequencies of the moieties lie outside of the band of low-frequency heat-carrying modes of the bridge, as is reasonable for fairly small moieties such as azulene and anthracene bridged by a long alkane chain. The molecule is taken to be sufficiently large that energy flow among vibrational states via anharmonic interactions is unimpeded by effects of quantum localization.49,51,62−65 In section 2, we express thermal conductance across an interface using the DMM as a starting point, which we then adopt to predict the interface conductance for the boundary between a protein and water in section 3. That calculation is presented to indicate the range of values that the DMM yields for the thermal boundary conductance between molecules and to illustrate limitations of the theory when compared with results of molecular simulations. Moreover, the DMM expresses thermal boundary conductance in terms of bulk properties of the materials at the interface and therefore may not apply for calculating conductance across an interface within a molecule. It does not account for anharmonic interactions, which are particularly important when the vibrational frequencies of the object on one side of the interface are in a different range than those on the other. In section 2B, we introduce the contribution of anharmonic interactions to the calculation of thermal boundary conductance, and in section 2C, we show how a distinct temperature-dependent thermal boundary conductance on each end of a molecular bridge enables thermal rectification. In section 3B, we estimate thermal boundary conductance for a model azulene−(CH2)N−anthracene molecule, and in section 3C, we discuss thermal rectification in this system. We conclude in section 4.

hBd =

Q̇ AΔT

×

1 d 4 dT

(1)

∫ dω

ℏωv1(ω)ρ1̅ (ω)n(ω , T )α(ω)

(2)

where the transmission probability is α(ω). In a convenient approach to eliminating the transmission probability, α(ω), the DMM assumes that after crossing the boundary, a vibrational excitation has no memory of which side of the interface it emerged from. Detailed balance gives47 α(ω) =

v2(ω)ρ2̅ (ω)n(ω , T ) (v1(ω)ρ1̅ (ω)n(ω , T ) + v2(ω)ρ2̅ (ω)n(ω , T ))

1

1̅

2

2̅

e (e βℏω − 1)2

(4)

We shall illustrate the utility of eq 4 in estimating the interface thermal conductance between a protein and water in section 3A. As we shall see, while the values are comparable to results of molecular simulations on specific proteins, there are trends that cannot be captured by this approach. Missing from the theory is any information about interactions at the boundary itself. For example, contacts between water and hydrophobic and hydrophilic regions have been observed in molecular simulations to give rise to distinct boundary conductance.44 Curvature at the boundary can influence the boundary conductance,41 an effect that could be accounted for in the DMM, though in eq 4, a flat boundary is assumed. The DMM is a theory for boundary conductance in the harmonic approximation and neglects potentially important anharmonic effects in thermal flow. Vibrational energy transfer via Fermi resonances could also mediate thermal flow through a boundary between or within molecules. Finally, the DMM expresses the boundary conductance in terms of bulk properties of the subsystems, which may not apply to the calculation of boundary conductance within a molecule. Anharmonic effects become particularly important when the vibrational mode frequencies of the object on one side of the interface are in a different range than those on the other. This is largely the case for the moiety−bridge interface depicted in Figure 1. The vibrational modes of an alkane chain bridge carry phonons efficiently below 600 cm−1,9 with by far the highest density of such modes below 200 cm−1. Azulene has only one vibrational mode below 200 cm−1, and anthracene has only two.23 If these moieties are heated to several hundred Kelvin, moiety modes significantly higher in frequency than the relatively high-density heat-carrying modes of the bridge are excited. These modes will transfer energy to the bridge by anharmonic interactions. In the following section, we account for low-order anharmonic contributions to the thermal boundary conductance. 2B. Boundary Conductance in a Molecule. We calculate the boundary conductance as the rate of energy transfer from one portion of the molecule, a heat source, to a bridge (Figure 1). We take the heat source to have discrete vibrations, whereas the bridge modes form a quasicontinuum, and we address energy transfer via cubic anharmonic interactions, which in sizable organic molecules make the largest contribution to the rate of vibrational energy transfer.49 There are two processes by which cubic anharmonic interactions transfer energy. One is “decay” of a vibrational excitation into two others, such that energy is conserved, and the other is “collision” of a vibrational excitation with a second to produce a third such that energy is conserved. In a decay process in which some energy is transferred to the chain, a mode of the heat source moiety, which we refer to as mode α, can decay into two modes, β and γ, both of which are chain modes, or it can transfer some energy to a different mode, β, of the moiety and the rest to a chain mode, γ. In a collision process that transfers energy to the chain, a vibrational excitation in mode α can combine with an excitation in mode β of the moiety to add a vibrational excitation in mode γ of the chain, or it can combine with an excitation in mode β of the

Heat flow across the interface can be expressed in terms of phonons, each with energy ℏω, that pass through it. The vibrational mode density per unit volume on side j is ρ̅j(ω), the mode occupation number is n(ω,T) = (exp(βℏω) − 1)−1, where β = 1/kBT, and the phonon speed on side j is vj(ω). Then, the thermal boundary conductance is47 hBd =

v (ω)v (ω)ρ (ω)ρ (ω)

∫ dω (βℏω)2 ℏω (v (ω1 )ρ (2ω) +1̅ v (ω)2̅ ρ (ω)) β ℏω

2. THEORETICAL METHODS 2A. Thermal Conductance at the Boundary between Molecules. The thermal boundary conductance, hBd, between two subsystems, 1 and 2, the inverse of the thermal boundary resistance or Kapitza resistance,47 is expressed in terms of the heat flow, Q̇ , driven by the difference in temperature, ΔT, between the two sides of the interface and the area of the interface, A hBd =

1 kB 4

(3)

Combining eqs 2 and 3 B



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chain to add a vibrational excitation to mode γ of the chain. These four distinct contributions to hBd are decay collision collision hBd = hβdecay = moiety + hβ = chain + hβ = moiety + hβ = chain

(ωα − ωβ )

× n(ωα)(n(ωβ ) + 1)(n(ωα − ωβ ) + 1)

hβdecay = moiety

1 d = A dT

1 d = A dT

1 d = A dT

∑ (ℏωα −

[ωα(n(ωα) + n(ωα − ωβ ) + 1)

ℏωβ )Γ decay α ; β = moiety

+ ωβ (n(ωβ ) − n(ωα − ωβ ))]

(6a)

α ,β

hβdecay = chain =

∑ ℏωα Γ decay α ; β = chain

(6b)

α

1 d A dT

hβcollision = moiety =

hβcollision = chain

∑ α ,β ωβ ωα − ωβ < ωc

n(ωα)

× [ωα(n(ωα) + n(ωα − ωβ ) + 1) + ωβ (n(ωβ ) − n(ωα − ωβ ))] hβdecay = chain =

(13a)

⎡ ⎛ ωα ⎞2 ⎛ ωα ⎞⎤ 2 6πkB3 2⎢ + S ∑ ⎜ ⎟ ∑ ⎜ ⎟⎥T Aℏ4 ⎢⎣ ω < ω ⎝ ωc ⎠ ω < ω < 2ω ⎝ ωc ⎠⎥⎦ α c c α c (13b)

hβcoll. = moiety =

(12a)

⎡ ⎛ ωα + ωβ ⎞⎤ 2 6πkB3 2⎢ S ⎜ ⎟⎥T ∑ Aℏ4 ⎢⎣ a , β ; ω + ω < ω ⎝ ωc ⎠⎥⎦ α

If both modes β and γ are chain modes, |Φ(ωα,ωβ,ωγ)| decreases as 1/N2, so that ρchain(ωβ)|Φ(ωα,ωβ,ωγ)|2ρchain(ωγ) is independent of N. We thus use in eqs 9b and 9d

hβcoll. = chain =

β

c

⎡ ⎤ ωα(ωc − ωα) ⎥ 2 6πkB3 2⎢ S T ∑ ⎥⎦ Aℏ4 ⎢⎣ a; ω < ω ωc 2 α c

(13c)

(13d)

2C. Thermal Rectification. For two different moieties, we may find rather different values of the temperature-dependent thermal boundary conductance at each end of the bridge. We can exploit the differences in thermal boundary conductance at each moiety−bridge interface and their dependence on temperature for thermal rectification. Consider two boundaries and three regions, as depicted in Figure 1, in which each moiety is held at a fixed temperature. Though it is straightforward to generalize to a bridge with a finite thermal gradient, we take the bridge to have just a single temperature for simplicity, that is, heat transport is ballistic and Fourier’s Law does not hold in the bridge. We note that this assumption appears to be well approximated by the azulene− (CH2)N−anthracene molecule studied numerically in ref 23, where in Figure 6, the temperature everywhere along the chain indeed appears to be nearly the same except very close to the boundary with the moieties. Moreover, experiments and other

2

S ωαωβ ωγ ωc 2

ωc

(n(ωβ ) + 1)(n(ωα − ωβ ) + 1)

2

ρchain (ωβ )|Φ(ωα , ωβ , ωγ )|2 ρchain (ωγ ) =

ωαωβ

(12b)

With the large N approximations, the thermal boundary conductance is independent of chain length. We furthermore assume for simplicity that T ≫ ℏωc/kB but otherwise make no assumption about T relative to moiety modes. We use this latter assumption for the mode occupation numbers in eqs 9b−9d but not in eq 9a, which we see below makes the largest contribution to hBd at the temperatures considered. Combining eqs 9−12, we obtain the following approximations to the thermal boundary conductance between a moiety with discrete vibrational modes and a chain with a continuous band of vibrations D



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Figure 3. Thermal boundary conductance computed for azulene−(CH2)N−anthracene, where N is large (at least 10), as a function of temperature. The thermal boundary conductance at the azulene−alkane chain interface is plotted as a dashed curve; the conductance at the anthracene−alkane chain interface is plotted as a solid curve. Estimates for the thermal boundary conductance at the anthracene−alkane chain interface (circle) and azulene−alkane chain interface (square) deduced from MD simulations reported in ref 23 are also plotted.

15 Å ps−1.74 For myoglobin, we have used 18 Å ps−1, which is a representative value of the frequency-dependent speed reported in ref 73. Neither this speed nor the frequency-dependent mode density should vary all that much from protein to protein,25 and we would thus not expect the interface conduction based on eq 4 to be all that different for the boundary conductance between water and different proteins. We observe a modest increase in boundary conductance over the plotted range of 230−320 K, with a value of 301 MW K−1 m−1 at 300 K. We compare the protein−water boundary conductance computed with eq 4 with values obtained by molecular simulations of solvated ATPase-1KJU, ATPase-1SU4, GFP, and myoglobin reported in ref 42, which were found to be, respectively, 210, 260, 270, and 100 MW K−1 m−1 at 300 K. The latter exhibit a sizable variation, about a factor of 3 for just these four proteins. Our estimate using eq 4 provides a rough approximation in all cases, but it does not predict such a large variation in boundary conductance. Variation in predictions of the theory for different proteins can arise from what are usually relatively small differences in the speed of sound and vibrational mode density at low frequency,25 differences that could perhaps account for a 20% variation in the boundary conductance from protein to protein, but not a factor of 3. The theoretical prediction also generally overestimates the boundary conductance. These results illustrate that the DMM, while predicting the right order of magnitude, does not describe all that well thermal boundary conductance at the interface between a large molecule and solvent, the kind of boundary for which it is appealing to use. We now leave the topic of thermal boundary conductance between two subsystems with bulk-like properties and turn to the thermal boundary conductance within a molecule, the focus of this article, specifically the interface between a subsystem with a relatively small number of discrete vibrations, such as a moiety, and another subsystem that may have a much larger vibrational density, such as a long molecular chain. For this latter kind of system, it is unclear if the DMM

simulations on alkane17,18,35 and some other molecular chains20,21,45,69 of comparable size also appear to reveal ballistic thermal transport. We then write the heat flux as ⎡ h(i)(T )h(j)(T ) ⎤ Ji → j = ⎢ (i)Bd 1 Bd(j) 2 ⎥(T1 − T2) ⎢⎣ hBd(T1) + hBd(T2) ⎥⎦

T1 > T2 (14)

where i and j label the two different moieties forming boundaries with the bridge. A rectification coefficient can be defined as70

R=

Ji → j Jj → i

(15)

R can be calculated in terms of the boundary conductance using eq 14. Equation 14 resembles the thermal rectification model developed by Peyrard70 and later demonstrated experimentally in a solid-state system,71 in which two subsystems exhibit distinct temperature-dependent thermal conductivities. In eq 14, thermal rectification is controlled instead by the boundaries between subsystems, a possibility discussed by Dames.72 Above, we provide explicit expressions for the thermal boundary conductance when the modes of the reservoirs lie outside of the band of heat-carrying modes of the bridge. In this case, the conductance at each boundary depends sensitively on temperature, as is apparent in eq 13 and in the illustrative example below, which facilitates control of rectification.

3. RESULTS AND DISCUSSION 3A. Protein−Water Interface. As an illustration of the magnitude of thermal boundary conductance between molecules that the DMM predicts and limitations of the theory, we have calculated the boundary conductance, hBd, for myoglobin and water with eq 4 and plotted the result in Figure 2. We have used the vibrational mode density of myoglobin reported in ref 73 and that for water in ref 74, both calculated in the harmonic approximation. For the speed of sound, we have used for water E



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Figure 4. Thermal rectification, R, defined by eq 15, for azulene−(CH2)N−anthracene as a function of difference in temperature, ΔT, between the hot and cold moiety. The cold moiety is held at 200 (solid curve), 300 (dashed), and 400 K (dotted).

interface and hBd ≈ 970 MW K−1 m−1 at the anthracene−bridge interface. On the basis of our calculations, we would typically expect hBd at the anthracene−bridge interface to be larger than that at the azulene−bridge interface, though not for the large difference in temperature between the moieties in the simulations. At 750 K, our illustrative calculation gives 780 MW K−1 m−1 for hBd at the azulene−bridge interface, close to the value obtained in the simulations reported in ref 23. However, our calculation gives 192 MW K−1 m−1 for hBd at 340 K at the anthracene−bridge interface, much smaller than the result obtained from the MD simulations. It is possible that the classical MD simulations overestimate hBd at lower temperatures, such as 340 K, at which only very few of the vibrational modes of either moiety are thermally accessible. A purely anharmonic model for energy transport across the interface of the larger moiety and bridge at the lower temperature could also underestimate the boundary conductance. Moreover, we have used the temperature of the moieties to represent the temperature at the boundaries when comparing the results of eq 13 for the model system with the results of the simulations in ref 23, a rough approximation given the large change in temperature across the interfaces. It is difficult to ascribe bulk properties to either of the moieties, so that we cannot apply eq 4 to, say, an azulene− alkane chain interface, but it is nevertheless interesting to compare predictions of eq 4 for the thermal boundary conductance between two large organic molecules. We have carried out such a calculation for the interface between two proteins (both myoglobin) and found results quite similar to those plotted in Figure 2 for myoglobin and water, that is, about 240 MW K−1 m−1 at 200 K, reaching 301 MW K−1 m−1 at 320 K and then increasing gradually to 367 MW K−1 m−1 by 1000 K. Returning to azulene−(CH2)N−anthracene, there are very few moiety vibrations with frequency corresponding to 200 cm−1 or lower, the range where the density of heat-carrying vibrations in the bridge is relatively large. For example, azulene has only one such mode, and anthracene has only two,23 so that virtually all vibrational modes of the moieties lie outside of the high-density band of heat-carrying modes of the bridge.

applies because of ambiguities in ascribing bulk properties to at least the moiety. We adopt instead the approach described in section 2B for the thermal boundary conduction within a molecule. 3B. Thermal Boundary Conduction in a Molecule: An Illustrative Example. We have used eq 13 to calculate hBd for the interface between azulene and an alkane chain bridge and anthracene and an alkane chain. The vibrational mode frequencies of each moiety are provided in Appendix A of ref 23. For the area of the interface, A, we assume a circle of radius 2.0 Å corresponding approximately to the van der Waals radius of CH2.75 The results are plotted in Figure 3 from 200 to 1000 K. The main anharmonic contribution to hBd at the moiety− bridge interface is decay, specifically the decay term in eq 13a. The collision terms make no contribution to the thermal boundary conductance at the azulene−bridge interface and an order of 0.1% contribution to thermal conductance at the boundary between anthracene and the alkane chain bridge. The collision terms can only make a significant contribution if the moiety has a sizable number of modes with frequency below ωc, which azulene and anthracene do not. The decay term given by eq 13b makes about a 1 and 6% contribution to the thermal boundary conductance at the azulene−bridge and anthracene− bridge interfaces, respectively. We observe the thermal boundary conductance to vary significantly with temperature, approaching T2 dependence at high temperatures, which should hold at high temperature as long as a perturbation approach remains valid. We can compare the results plotted in Figure 3 with results of the molecular dynamics (MD) simulations on azulene− (CH2)N−anthracene molecules reported in ref 23. In that study, azulene was held at 750 K and anthracene at 340 K. The temperature of the bridge appears in Figure 6 of ref 23 to be very close to 500 K everywhere except near the moieties, so that the drop in temperature is about 250 K at the interface with azulene and about 180 K at the interface with anthracene. The flux shown in Figure 7 of ref 23 is ∼22 nW; with our estimate for the interface surface area given above, this corresponds to hBd ≈ 700 MW K−1 m−1 at the azulene−bridge F



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Anharmonic interactions open up pathways for energy flow and give rise to the relatively high boundary thermal conductance predicted in Figure 3 and observed in molecular simulations.23 Nevertheless, for a large moiety, perhaps even as small as anthracene, results of a harmonic model based on bulk properties such as eq 4 might be added to the anharmonic contributions that we address here. 3C. Thermal Rectification in a Molecule: An Illustrative Example. We have calculated the thermal rectification, R, defined by eq 15, as a function of difference in temperature, ΔT, between the hot and cold moieties of our model azulene− (CH2)N−anthracene molecules. We take the cold moiety to be held at 200, 300, and 400 K. The results are plotted in Figure 4. For a large difference between the temperature of the hot and cold moieties, R goes over to the ratio R ≈ h(anthrace) (Tcold)/ Bd (Tcold), where Tcold is the temperature of the colder h(azulene) Bd moiety, because of the large increase in hBd with increasing T at both boundaries. As a result, the thermal rectification reaches 2.50 for large ΔT when the temperature of the colder moiety is 200 K because at that temperature, the ratio of the thermal boundary conductance for the anthracene−bridge interface is about 2.5 times as large as that for the azulene−bridge interface. If the cold moiety is held at higher temperature, this ratio becomes smaller. For example, R is 2.00 for large ΔT when the temperature of the colder moiety is 300 K and 1.79 for large ΔT when the temperature of the colder moiety is 400 K. For smaller ΔT, we obtain smaller R. For example, for ΔT = 100 K, we find that R is 1.82, 1.39, and 1.24 when the temperature of the colder moiety is 200, 300, and 400 K, respectively. The fact that R is nevertheless significantly greater than 1 reflects the sizable difference between hBd for the two boundaries and the significant dependence of hBd with temperature, both of which are a result of the finite sizes of the moieties, each with vibrations mainly outside of the band of heat-carrying modes of the bridge. While we expect the general trends that appear in Figure 4 to hold for azulene−(CH2)N−anthracene, the values of R obtained in our illustrative model calculation may differ from those of the molecule itself. We have, for example, assumed that cubic anharmonic interactions account for the thermal boundary conductance because almost all modes of the moieties are higher in frequency than the bridge modes. However, a small number of modes of the moiety lie within the band of heatcarrying modes of the bridge, providing additional channels for thermal flow, which also affect R.

boundary conductance at the moiety−molecular bridge interfaces in azulene−(CH2)N−anthracene. Differences in the temperature-dependent boundary conductance at each end of the bridge can be exploited in the design of a molecular thermal diode. The heat flux through the molecule depends on which moiety is held at the lower temperature. The boundary between the smaller moiety, with the lower density of low-frequency vibrational modes, and the bridge serves as a bottleneck to heat flow when the former is held at low temperature. Heat will flow optimally along the molecule when the temperature of the moiety with fewer lowfrequency modes is higher.

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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 from NSF Grant CHE-0910669 is gratefully acknowledged.

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REFERENCES

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4. CONCLUDING REMARKS We have presented an approach to calculate thermal boundary conductance in molecules where different parts of the molecule or moieties held at fixed temperatures are connected by a molecular bridge. If the vibrational frequencies of each moiety lie outside of the band of heat-carrying modes of the bridge, anharmonic interactions mediate thermal conduction at the boundaries between the moieties and the bridge. In this case, we cannot apply conventional approaches based on bulk properties of the materials that meet at the interface, such as the diffuse mismatch model, which, as illustrated above, does not even estimate all that well the thermal boundary conductance between water and organic macromolecules such as proteins. We have expressed the thermal boundary conductance in terms of the cubic anharmonic interactions between a moiety and a bridge. As an illustrative example of the anharmonic theory presented here, we calculated the thermal G



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