Implications of Interfacial Bond Strength on the SpectralContributions to Thermal Boundary Conductance across Solid,Liquid, and Gas Interfaces: A Molecular Dynamics StudyAshutosh Giri, Jeffrey L. Braun, and Patrick E. Hopkins*

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, Virginia 22904, United States

ABSTRACT: The modal contributions to interfacial heat flowacross Lennard-Jones based solid/solid, solid/liquid, andsolid/gas interfaces are predicted via molecular dynamicssimulations. It is found that the spectral contributions to thetotal heat flux from the solid that comprises the interface arehighly dependent on the phase of the adjoining matter and theinterfacial bond driving the interaction between the solid andthe adjacent matter. For solid/solid interfaces, along with lowtemperatures, weak cross-species interaction strength canseverely limit the conductance owing to the inhibition ofinelastic channels that otherwise facilitate heat flow across the interface via anharmonic interactions. The increase in the cross-species interaction strength is shown to shift the modal contributions to higher frequencies, and most of the inelastic energyexchange is due to the longitudinal vibrational coupling across the interface. For solid/liquid interfaces, the increase in the cross-species interaction enhances the coupling of transverse vibrational frequencies in the interfacial solid region, which leads to anincrease in the total heat current across the interface. Our modal analysis suggests that very high frequency vibrations (withfrequencies greater than 80% of the maximum frequency in the bulk of the solid) have negligible contribution to heat flow acrosssolid/liquid interfaces, even for a strongly bonded interface. In the limit of weakly interacting solid/gas interfaces, the modescoupling in the solid to the gas have signatures of reduced dimensionality, as evident by the surface-like density-of-states in thesolid. Increasing the interfacial interaction shows similar trends to the solid/liquid case up to the limit in which gas atoms adsorbto the surface, enhancing the contribution of transverse phonons coupling at the solid interface. Our work elucidates generalsimilarities in the influence of interfacial bond strength to thermal boundary conductance across solid/solid, solid/liquid, andsolid/gas interfaces. In general, we find that the mode softening with a decrease in interfacial bond strength is more pronouncedin the longitudinal modes as compared to transverse modes, and we consistently observe a decrease in the transverse modecontribution from the solid across the interface as the interfacial bond strength is decreased, regardless of the phase of matter onthe other side.

■ INTRODUCTION

Heat flow in modern nanoscale devices is mostly limited by thehigh densities of interfaces rather than the materials comprisingthe device.1,2 These interfaces can provide resistance to energycarriers, which results in nanoscale temperature discontinuitiesbetween the materials comprising the interfaces. The relationbetween the temperature discontinuity, ΔT, and the impingingheat flux, Q, at the interface quantifies the efficacy of energyexchange across the interface, and is referred to as the thermalboundary conductance (hK = Q/ΔT).3 Over the past fewdecades, hK across interfaces comprising solid/solid materialshas attracted a considerable amount of interest bothexperimentally and theoretically because of the wide varietyof physical phenomena associated with interfacial thermaltransport and also because of the various device drivenapplications.1,2,4,5 Analytical expressions based on the phonongas model have been shown to reproduce the experimentallymeasured hK across various crystalline solid/solid interfacesreasonably well because of the periodicity of atoms in the solidlattices on either side of the interface.3,6−9 However, the lack of

periodicity and the inability to define group velocities indisordered phases such as in gaseous and liquid states renderthe direct application of these predictive models invalid, whichhas necessitated new predictive formalisms.10,11

The thermal interface coupling of heat carriers across solidsin contact with gases and liquids has recently garnered muchattention. For example, solid/liquid interfaces have beenstudied for their importance in understanding various biologicaland chemical processes,12,13 friction mitigation in hard disks,14

and evaporative cooling.15 Similarly, for solid/gas interfaces,heat transfer has been shown to play a critical role inunderstanding geothermal energy harvesting,16 heat pipeengineering,17,18 low-grade waste heat recovery, and seawaterenergy conversion.19−21 In this regard, prior works haveinvestigated the effects of interfacial structure and chemistryalong with effects of variations in temperature and pressure on

Received: August 11, 2016Revised: October 4, 2016Published: October 4, 2016

Article

pubs.acs.org/JPCC

© 2016 American Chemical Society 24847 DOI: 10.1021/acs.jpcc.6b08124J. Phys. Chem. C 2016, 120, 24847−24856

hK across solid/liquid10,22−27 and solid/gas interfaces.11,28−30

Most of these studies have alluded to the fact that the strengthof interaction at the interface between a solid and a liquid or gasis the dominant parameter that dictates hK across theseinterfaces. Despite these advances, a comprehensive under-standing of the spectral contribution to hK and the effect ofinterfacial bonding on the modal contributions across solid/liquid and solid/gas interfaces is still lacking in the literature;such a study would significantly enhance the fundamentalunderstanding of heat flow across these interfaces.More recently, molecular dynamics (MD) simulations have

been employed to resolve the mode level contributions to hKacross various solid/solid interfaces,31−41 thus providing anavenue to better understand and manipulate spectralcontributions to interfacial heat flow. Despite the advances instudying heat flow across solid/solid interfaces, relatively fewstudies have focused on the mode level details of hK acrosssolid/gas and solid/liquid interfaces. In this paper, we study theinfluence of interfacial bonding on the spectral contributions tohK across Lennard-Jones (LJ)-based solid/liquid, solid/gas, andsolid/solid interfaces. The spectral contributions across lattice-matched, mass-mismatched LJ solids are shown to be highlydependent on the temperature and the cross-species interactionstrength due to inelastic energy transfer mechanisms at theinterface. For LJ-based solid/liquid interfaces, the reduction incross-species interaction strength limits the coupling oftransverse vibrational frequencies and shifts the spectralcontributions at the solid to lower frequencies. In contrast tosolid/solid and solid/liquid interfaces, longitudinal vibrationalfrequencies contribute more to the heat current from the solidto the gas for a weakly bonded interface. However, the increasein cross-species interaction strength across the solid/gasinterface can eventually lead to hK with similar contributionsfrom transverse and longitudinal modes in the solid. Thedecrease in solid/gas interaction strength is also shown toreduce the maximum phonon frequencies in the interfacial solidthat can facilitate heat transfer (more so than in the case ofsolid/liquid interfaces).

■ METHODOLOGYThe thermal boundary conductances and their spectralcontributions across solid/liquid, solid/gas, and solid/solidinterfaces are calculated under the nonequilibrium moleculardynamics (NEMD) simulations framework by using theLAMMPS molecular dynamics package.42 The three types ofinterfaces are described by the 6−12 LJ potential, U(r) =

4ε[(σ/r)12 − (σ/r)6], where U is the interatomic potential, r isthe interatomic separation, and σ and ε are the LJ length andenergy parameters, respectively. For computational efficiencythe cutoff distance is set to 2.5σ for all the simulations and thetime step is set to 0.1 fs throughout the simulations. To gaugethe effect of cross-species interaction strength on the spectralcontribution to hK, we simulate structures with strong and weakinterfacial bonding (defined by ε for species across theinterface). We note that the use of the LJ potential in thiswork is to reinforce the generality of the results presented andnot to extract quantitative data for material specific properties.For the solid/solid systems, we considered two materials (A

and B) that are in contact with each other as shown by theschematic in the top panel of Figure 1a. The length and energyparameters are set to σAr = 3.405 Å and εAr = 10.3 meV,respectively (that is representative of solid Ar for both A andB). The materials are arranged in an fcc lattice with the samezero-temperature lattice constant of a0 = 1.55σ. This creates alattice matched interface between materials A and B. However,an acoustic mismatch is introduced by setting the mass ofmaterial B to be four times higher than that of material A (mB =4mA). Thus, material A (with mA = 40 g mol−1) represents solidargon and material B represents a fictitiously heavier solidargon. The systems consist of 10a0 × 10a0 × 60a0computational domains with 24 000 atoms each. Note,increasing the cross-sectional area or the length of thecomputational domain has negligible change in the observedresults. To investigate the effect of interaction strength betweenthe atoms across the interface, we vary εA−B in the range εAr toεAr/4. Periodic boundary conditions are used in the x- and y-directions, whereas fixed boundaries with four monolayers ofatoms at each end are placed in the z-direction. Initially thestructures are allowed to equilibrate at their prescribedtemperatures for a total of 2 ns under the Nose-Hooverthermostat43 (that is the NVT integration with the number ofatoms, volume, and temperature of the simulation heldconstant) followed by the NPT integration (which is theisothermal−isobaric ensemble with the number of particles,pressure, and temperature of the system held constant) for atotal of 2 ns at 0 bar pressure. After equilibration, a steady-statetemperature gradient is established under the NVE integration(with number of particles, volume, and energy held constant)by adding a fixed amount of energy per time step to a warmbath at one end and removing the equal amount of energy froma cool bath at the other end as shown in Figure 1a. The lengthof the baths are 10a0 in the z-direction and are shown in the

Figure 1. (Top panels) Atomistic illustration of the computational domain between LJ-based (left) solid/solid, (middle) solid/liquid, and (right)solid/gas systems. (Bottom panels) Temperature response of the computational domains for (a) solid/solid, (b) solid/liquid, and (c) solid/gassystems after a steady heat flux is imposed.

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schematic in Figure 1 as red (hot bath) and light blue (coldbath) atoms. A steady-state temperature profile is shown in thebottom panel of Figure 1a.For the solid/liquid systems, the masses of the solid and

liquid atoms are set at 40 g mol−1 and the length parameters areset to σ = 3.405 Å for both. The energy parameters defining theliquid−liquid and solid−solid interactions are set to εll = 10.3meV and εss = 103 meV, respectively as in earliersimulations.22,24,44,45 These values ensure that at the prescribedsimulation temperature, the liquid melts while the solid holdsits fcc crystallographic configuration. Moreover, as pointed outby Xue et al.,45 these parameters lead to an acoustic mismatchof ∼4, which captures the high impedance at the interface thatis characteristic to solid/liquid systems. Following the work inref 22, where two regimes of Kapitza resistance are observeddepending on the cross-species interaction strength (with εsl <εll distinct to “nonwetting” conditions), the solid−liquidinteraction strengths are varied from 5εll to 0.2εll, where εsl =5εll mimics a strongly bonded, “wetting” solid/liquid interface,while the εsl = 0.2εll mimics a weakly bonded, “nonwetting”interface.The atoms are initially placed in an fcc lattice in a

computational domain size of 8a0 × 8a0 × 70a0. The solidregion is bounded by liquid regions on either side in the z-direction and we apply periodic boundary conditions in alldirections. Similar to the solid/solid case, the solid/liquidstructures are allowed to equilibrate under the NVT followedby the NPT integration for a total of 4 ns at a prescribedtemperature of 170 K and at a pressure of 0 bar. The top panelof Figure 1b shows a schematic of the equilibrated solid/liquidstructure. Note, only half of the computational domain isshown due to symmetry.Similarly, for the solid/gas systems, the solid is defined by the

same energy and length parameters as for the solid/liquid case.The cross-section for the solid−gas systems are 10a0. For thegas, we choose the potential and mass parameters tocorrespond to that of Ar. The solid/gas interaction strengthis varied from εsg = 5εgg (εsg/kBT = 2) to εsg = εgg (εsg/kBT =0.4) to ensure a gradual transition from a very high to anegligible gas adsorption coverage at the solid interface (similarto the parameters used in ref 29). The simulations on the solid/gas systems are performed at an average temperature of T =300 K, consistent with previous works.11,28−30

For both the solid/liquid and solid/gas systems, the length inthe z-direction is prescribed as shown in the figures. Periodicboundary conditions are applied in all three directions,therefore only half of the simulation cells in the z-directionare shown in the top panels of Figure 1b and Figure 1c due tosymmetry. Similar to the solid/liquid case, the atoms areinitially placed in an fcc crystal orientation for the solid−gasdomains and are then allowed to equilibrate at their prescribedtemperatures for a total of 2 ns under the Nose-Hooverthermostat43 (NVT integration with the number of atoms,volume, and temperature of the simulation held constant)followed by the NPT integration (which is the isothermal−isobaric ensemble with the number of particles, pressure, andtemperature of the system held constant) for a total of 2 ns. Forthe solid/gas systems, an additional Berendsen barostat isprescribed to control the pressure.46 The pressure for thesolid−liquid and solid−gas systems are set at 0 and 15 bar,respectively. For the solid/gas systems, the higher pressure and300 K temperature ensures that the Knudsen number (Kn = λg/L, where λg is the mean free path of the gas atoms; λg ∼ 5 nm

for Ar atoms47) is between 0.01 and 0.1, which belongs to thetemperature jump regime in gas dynamics problems.30,48 Arelatively higher pressure is prescribed for the solid/gas systemsso as to attain better statistics and reduce the finite simulationtime for NEMD calculations.29 For the solid/liquid systems, weconduct our simulations at zero bars pressure since we are onlyinterested in the effect of cross-species interaction strength oninterfacial heat flow. However, the effect of increasing pressureon interfacial heat flow has been shown to increase theefficiency of heat flow across solid/liquid interfaces, andinterested readers are referred to ref 27 and ref 44 for adetailed study of pressure dependence of interfacial heat flowacross solid/liquid interfaces. A temperature of 170 K is chosenfor the solid/liquid systems to ensure that the liquid melts andalso to be consistent with temperatures used in prior works.24,44

After equilibration, the reverse NEMD process (as detailed inref 49) is applied to create a steady heat flux across the systemsunder the NVE integration (with number of particles, volume,and energy of the system held constant). The heat flux is givenas q = ΔE/2AΔt, where A is the cross-sectional area, Δt is thetime step, and ΔE is the kinetic energy swapped between thehot and the cold slab at each time step. This procedure resultsin a steady-state temperature gradient as shown in the bottompanels of Figure 1b and Figure 1c. Usually hK across interfacesare determined by the relation, q = hKΔT, where ΔT isdetermined from the temperature profiles. To accuratelydetermine ΔT, a linear regression analysis to the temperatureprofiles of the constituent materials is calculated from the linearfit to the MD-data. The linear fits to the temperature profilesreduce the uncertainty associated with determining the exacttemperature at the boundaries.50−56 For the solid/liquidsystems, the steady-state temperature profiles are averaged for15 ns after equilibration. Similarly, for the solid/gas systems,temperature profiles are averaged for a total of 80 ns after asteady heat flux is applied.The NEMD procedure outlined above does not directly lend

insight into the modal contributions to hK at the interfaces.Therefore, to quantify the spectral contributions to hK, weapply a similar method to that detailed in ref 40, where thecorrelations between the force−velocity at the interfaces is usedto predict mode level details to understanding interfacial heatcurrent. Briefly, the heat flux is spectrally resolved by therelation,40

∫ ωπ

ω=∞

Qd

q2

( )0 (1)

where ω is the angular frequency and q(ω) is the spectral heatcurrent. In general, this heat current between an atom i and j isproportional to the correlation between the interatomic forceFij between the atoms and the velocities, qi→j(ω) ∝ ⟨Fij·(vi +vj)⟩, where the brackets denote steady-state nonequilibriumensemble average.57−59 Since we are interested in knowing thespectral contribution of the flux across a planar interfacebetween solid/solid, solid/liquid, and solid/gas systems, wecalculate the force exerted on atoms in each monolayer of solid(within the cutoff distance from the interface) on the left sideof the interface due to the different species of atoms at theother side of the interface. Using the identity ⟨Fij·vi⟩ = ⟨Fij·vj⟩,from atom i in material A at one side of the interface to an atomj in material B at the other side of the interface, the heat fluxcan be calculated as,44

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∑ ∑ω ω ω=Δ

⟨ · *⟩→∈ ∈

qAM t

F v( )2

( ) ( )j i

i iA Bs B A

B

(2)

where F and vi are Fourier transforms of the force and velocityvectors, A is the surface area, M is the number of samples, Δts isthe sampling interval and Fi

B = ∑j∈BFij. Therefore, for thepurposes of this work, atomic velocities and forces are collectedfor a total of 10 ns during steady-state nonequilibriumconditions at a sampling interval of 10 fs in order to determinethe spectral contribution to the total heat flux. As such, theoutputs of forces and velocities include components in thethree Cartesian coordinates. Thus, the dot-product of the forceand velocity vectors, ⟨Fij·vi⟩, which describes the modalcontributions to the heat current has separate contributionsin the x-, y-, and z-directions. The x- and y- componentsdescribe the in-plane or transverse mode contributions to thetotal heat current and the z-component describes the out-of-plane or longitudinal mode contributions to the total heatcurrent.Note, this method of calculating the total force on an atom

on the left side of the interface due to the collective forces fromall the atoms in the other side of the interface reduces thecomputational cost by a large extent. Furthermore, instead ofconsidering forces on every atom i at the left side of theinterface, we consider an average force exerted on a plane ofatoms (due to periodic boundary conditions on the x- and y-directions) since atoms in a monolayer parallel to the interfacewill experience the same force due to the collective atoms fromthe other side of the interface. In this context, our approachconsiderably reduces the computational time and cost for themodal analysis calculations compared to previous methods inwhich the computational cost of storing the velocities andforces between each atom pair interaction during the simulationand taking the Fourier transforms require a large amount ofstorage space.60,61 Moreover, because of the random motion ofthe gas and liquid atoms in the simulation cell, the atom-by-atom approach adds another complexity due to the require-ment of tracking the nearest-neighbor atoms of the solid atomsfor force and velocity calculations.

■ RESULTS AND DISCUSSIONSTo validate the modal decomposition method described above,we perform NEMD calculations on LJ argon at 50 K by placing

an imaginary interface with a cross section at the middle of thesimulation cell in the z-direction. For the calculations, thesimulation domain size is 10a0 × 10a0 × 60a0, which confirmsthat no size effects due to boundary scattering affects thecalculations. Figure 2a shows the modal contributions of thenormalized heat flux accumulation (q(ω)) due to a steady-statetemperature induced across the LJ argon computationaldomain. For comparison, the predictions from a Boltzmanntransport equation (BTE) in conjunction with anharmoniclattice dynamics (LD) calculations performed at 50 K (with thesame LJ parameters as detailed in ref 62) are also shown. In thismethod, the phonon properties predicted via an anharmonicLD calculation, which takes into consideration three- and four-phonon processes, are used as input parameters in the BTEequation. The modal decomposition method and the BTE-LDmethod predict very similar spectral contributions to thethermal conductivity of LJ argon. In particular, both methodspredict that the largest contribution to thermal conductivity isdue to phonons with 20% to 80% of the maximum frequency.This is intuitive due to the large population of phonons in thisfrequency range (see the density-of-states of LJ argonrepresented by Solid A in Figure 2b). The good agreementbetween the two approaches gives us confidence in the resultspresented in this section. We note that the BTE-LD approachand the approach used in this work are fundamentally differentas the former uses information such as the relaxation times asinput parameters in the BTE, while our approach relies on theoutputs of atomic velocities and forces directly from MDsimulations.Figure 2b shows the modal contributions to the normalized

heat flux accumulation from Solid A (m = mAr) to Solid B (m =4mAr) at 30 K for εA−B in the range εAr to εAr/4; the exact valuesof the interaction strengths are given in the legend of Figure 2c.Figure 2b also shows the predictions for the case with εA−B =εAr (which is the relatively strongly bonded interface) at 1 Kaverage temperature. For this case, the heat flux from Solid A toSolid B has negligible contributions from frequencies greaterthan the cutoff frequency of Solid B (even though the phononspectrum in Solid A extends to twice the maximum frequencyof Solid B). However, at 30 K, frequencies greater than thecutoff frequency of Solid B contribute to more than 50% of thetotal heat flux from Solid A to Solid B for the strongly bondedinterface. The temperature dependencies can be understood by

Figure 2. (a) Normalized thermal conductivity accumulation predicted from the NEMD calculations for a homogeneous LJ argon with an imaginaryinterface in the middle of the computational domain (at 50 K) as a function of ωAr/ωAr, max. For comparison, the result from ref 62 that is based onthe Boltzmann transport equation in conjunction with anharmonic lattice dynamics calculation performed at 50 K are also shown. (b) Normalizedthermal boundary conductance accumulation at 30 K for the range of εA−B studied in this work. Also included are the predictions at 1 K for thestrongest cross-species interaction strength across the interface (with εA−B = 10.3 meV). The DOS (a.u.) of the bulk solids are included to emphasizethe cutoff frequencies in the two solids. (c) NEMD-predicted thermal boundary conductances across the LJ-based solids differentiated by mass as afunction of temperature for the range of cross-species interactions.

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considering inelastic channels and anharmonic phononprocesses that are activated due to the increase in temperature.Where harmonic interactions limit the accumulation to reach100% by the maximum frequency in Solid B, ωB,max,anharmonic interactions open up channels for heat conduction,thereby allowing modes with different frequencies tointeract.9,37,40,63−68 Even though in homogeneous crystals,anharmonicity decreases thermal conductivity due to variousscattering mechanims, heat conduction across interfaces isaided by anharmonicity as is suggested by the increase in thespectrum of frequencies in Solid A that can carry heat acrossthe interface at higher temperatures.In comparison to the strongly bonded case (with εA−B = εAr),

the weaker strength of cross-species interactions lead to a shiftfrom midfrequency phonons dominating heat flow to lowerfrequencies contributing the most to the heat flux from Solid Ato Solid B, as shown by the gradual shift to lower frequencieswith decreasing interaction strengths across the interface. Also,in contrast to the strongly bonded case at 30 K, inelasticchannels are inhibited for the weakest interaction as is evidentfrom the negligible contribution of phonon frequencies higherthan the cutoff frequency of Solid B to the total heat flux. Thisis further quantified by the temperature dependencies of hKacross the Solid A/Solid B interface for the various εA−Binvestigated (see Figure 2c). The increase in hK withtemperature gradually becomes less pronounced as the strengthof interaction across the interface is reduced. Furthermore, thehK values converge at low temperatures for all the interactionstrengths. This suggests that anharmonic phonon scatteringprocesses that increase hK with temperature for the stronglybonded case are inhibited for the weakly bonded case and thisinhibition is less pronounced at lower temperatures whereinelastic interactions are limited, in line with the calculations ofmodal contributions shown in Figure 2b and consistent withour previous work.69

Figure 3 separates the contribution from longitudinal andtransverse modes on hK across the Solid A/Solid B interfaces.Even though the absolute contributions from both thetransverse and longitudinal modes decrease with decreasingcross-species interaction strength, about 60% of hK is due to thetransverse modes for all cases; note, the relative contribution ofthe transverse modes decreases very slightly as the cross-speciesinteraction strength is lowered. As is expected, the transversemodes have negligible influence on the total hK beyond ∼1.2THz for all interaction strengths, whereas, the entire spectrumof modes in Solid A for the longitudinal modes can contributeto the total heat flux for the strongly bonded case. This suggests

that most of the inelastic contributions for the flux impinginginto Solid B from Solid A (that are beyond the cutoff frequencyof Solid B) are due to longitudinal modes in Solid A. Takentogether, these analyses suggest that at solid/solid interfaces, astrong bond will increase the inelastic phonon scatteringchannels of the longitudinal phonons, which will not onlyincrease hK, but also increase its temperature dependence.Similar to the solid/solid interfaces, the effect of cross-species

interaction on the spectral contributions to hK across solid/liquid interfaces with εsl = 51.7 and 10.3 meV are shown inFigure 4 panels a and b, respectively. The inset of Figure 4b

shows the predicted thermal boundary conductances for therange of cross-species interactions studied for the solid/liquidsystems. The decreasing trend in the value of total hK withlower cross-species interaction strength is in line with priorworks that have considered the influence of solid/liquid bondstrength on hK.

22,24,25 It is interesting to note that for εsl > εll, hKis proportional to εsl, whereas, for εsl ≤ εll, hK decreasesexponentially. These results are in line with the conclusionsreported in ref 22, in which NEMD simulations revealed twodistinct regimes of thermal transport for “wetting” and“nonwetting” liquids. We note that we have performedadditional simulations at 300 K for interfaces with the extremecross-species interaction strengths (εsl = 51.7 and 2 meV),

Figure 3. Modal contributions at Solid A to the total thermal boundary conductance at 30 K temperature for (a) εA−B = 10.3 meV, (b) εA−B = 8.4meV, (c) εA−B = 4.5 meV and (d) εA−B = 2.6 meV. The contributions due to transverse modes are calculated to be ∼60% for the interface with thestrongest interaction strength and ∼58% for the interface with the weakest interaction strength.

Figure 4. Accumulation of thermal boundary conductance across thesolid/liquid interfaces for (a) εsl = 51.7 meV and (b) for εsl = 10.3meV. The increase in the interaction strength is shown to increase thecontribution from higher frequency phonons. (inset) Thermalboundary conductance as a function of solid−liquid interactionstrength.

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which resulted in statistically insignificant changes in thepredicted thermal boundary conductance and the spectral heatflux.Unlike the solid/solid systems in which the relative

contributions from the transverse and longitudinal modes didnot significantly change with interfacial bonding, for the solid/liquid systems, the transition from a “wetting” to a “non-wetting” liquid leads to a marked decrease in the contributionfrom the transverse modes (as shown in Figure 4 for εsl = 5εlland εsl = εll that represent a “wetting” and a “nonwetting”liquid). To highlight the relative spectral contributions from thetransverse and longitudinal modes, in Figure 5, we plot the

normalized accumulations of thermal boundary conductancesfor the various interfacial interaction strengths considered.Overall, in line with the results for the solid/solid case, theincrease in the cross-species interaction strength for the solid/liquid interface shifts the mode contributions to higherfrequencies. Moreover, most of the contribution to the hKaccumulation across all the solid/liquid interfaces is facilitatedby modes ≤4 THz. Even for the case of the strongest bondacross the interface, more than ∼80% of the heat flux is carriedby frequencies below 4 THz, suggesting that in general, thevery high frequency phonons from the solid have negligibleinfluence on the heat transfer across solid/liquid interfaces.

Figure 5 also shows the total contribution of longitudinal andtransverse modes to the normalized heat flux accumulationacross solid/liquid interfaces with the varying cross-speciesinteractions. The gradual decrease in the interaction strengthfrom εsl = 51.7 to 20 meV leads to a slight decrease in therelative contribution of transverse modes while the contributionfrom the longitudinal modes to the total heat current increasesvery slightly. However, for εsl ≤ 10.3 meV (or εsl ≤ εll), a drasticreduction in the relative contributions from the transversemodes are observed. This is in line with the observations in ref22, where it has been shown that for solid−liquid interactionstrengths of εsl < εll mimic a “nonwetting” liquid. As our εsl getscloser to this regime, the contribution from transverse modes inthe solid decreases due to the decrease in the shear rate of theliquid near the interface, which consequently leads to lower hK.This is in line with the results of MD simulations by Torii etal.,70 where they demonstrate that the increase in slip lengthdue to stronger solid−liquid interaction leads to an increase inhK as a consequence of the liquid motion parallel to theinterface (i.e., shear states in the liquid molecules near the solidinterface).The results shown in Figure 5 are consistent with the results

presented in ref 44 (through rigorous calculations of spectralcurrent across solid/liquid interfaces with varying εsl) where ithas been shown that transverse modes are better coupled to theliquid by increasing the interaction strength between the solidand the liquid. Furthermore, in line with our results, the modaldecompositions of solid/liquid interfaces with varying inter-action strengths presented in ref 44, are also shown to shift theheat carrying vibrations toward the lower frequency spectrumin the interfacial solid owing to the reduced bonding betweenthe solid and liquid atoms. Apart from these observations, therelative contributions of transverse and longitudinal modes tothe total heat flow across the solid/liquid interfaces aspresented in Figure 5 demonstrate a clear transition from a“wetting” to a “nonwetting” liquid.For the LJ-based solid/gas systems, the results for the

accumulation of thermal boundary conductance across solid/gas interfaces with variable interfacial bonding is shown inFigure 6. The insets in Figure 6 show the number of adsorbedgas atoms on the solid interface (in bins of 3 Å perpendicular tothe solid interface) due to the varying cross-species interactionstrengths. For the case of εsg = 51.7 and 40 meV (shown inFigure 6a and Figure 6b), the coverage of adsorbed gas atoms isrelatively high and therefore the conductances are higher

Figure 5. Normalized accumulation of thermal boundary conductanceacross solid/liquid interfaces. The gradual increase in the interactionstrength is shown to increase the contribution from higher frequencyphonons. For the case with the “nonwetting” liquids (εsl ≤ εll), therelative contribution from the transverse modes are drastically reduced.

Figure 6. Accumulation of thermal boundary conductance across solid/liquid interfaces for (a) εsg = 51.7 meV, (b) for εsg = 40 eV, (c) for εsg = 30meV, and (d) for εsg = 10.3 meV. (insets) Number of atoms in bins of 3 Å perpendicular to the surface of the solid. (Insets a,b) The stronginteraction strength between the solid and the gas atoms causes adsorption on the solid surface, which is evident from the large number of atomsadjacent to the solid surface.

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compared to the lower interfacial interaction strengths.However, as the gas atoms do not adsorb to the solid surfacefor εsg = 30 and 10.3 meV, the conductances associated withthese interfaces are relatively lower compared to the solid/gasinterfaces with high number of gas atoms adsorbed to thesurface. This is in line with previous NEMD simulations on Pt/Ar systems in ref 28 and Au/Ar systems in ref 29, where thethermal accommodation coefficients are shown to increase withincreasing interfacial bond strengths.As the solid−gas interaction strength decreases (see Figure

6), the relative contributions from the longitudinal modesgradually increase and ultimately leads to higher relativecontributions for cases with no adsorption of gas atoms onthe solid surface. To emphasize the relative contributions fromthe different modes, the normalized accumulation of the heatfluxes (for the different cases with variable solid−gas interactionstrengths) are plotted in Figure 7. The interfaces with the

relatively strong cross-species interactions (εsg = 51.7 and 40meV), both transverse and longitudinal modes contributealmost equally to the total hK. This can be attributed in part dueto the adsorbed layer of gas atoms on the surface of the solid(as shown in the insets of Figure 6), which effectively couplestransverse mode frequencies from the solid. In contrast, thereduction in the strength of bonding between the solid and gasatoms leads to a reduction in the contribution of transverse

modes. It is also interesting to note that the decrease ininterfacial bond strength and the decrease in adsorbed gassurface coverage leads to a decrease in the spectrum offrequencies responsible for the interfacial heat flow to lowerfrequencies. Moreover, only ∼20% of the total hK is due to thecontribution from frequencies in the higher half of the phononspectrum (>4 THz) in the solid for all the solid/gas interactionstrengths. This suggests that as the solid/gas interaction islowered, the maximum frequency that can non-negligibly affecthK and, therefore the thermal accommodation coefficient(TAC) at solid/gas interfaces, decreases considerably. Thisresult is consistent with recent predictions from a diffusemismatch model-based analytical expression for hK (and TAC)across solid/gas interfaces, which shows that the model fails toreplicate MD-predicted data for weakly bonded interfaces(εsgkBT).

11 In other words, the model cannot account for thereduction in the maximum frequency available for heat transferdue to the weak nature of the cross-species interactionstrength,11 whereas, the model is applicable to solid/gasinterfaces with strong cross-species interactions (εsg ≥ kBT, asdescribed in ref 11) due to the non-negligible contributionfrom relatively high frequency phonons in the solid, which areeffectively coupled to the adsorbed layer on the stronglybonded solid/gas interface.Prior works have shown that the energy transfer across solid/

gas interfaces is highly dependent on the pressure of thegas.11,47,71 Therefore, we conduct additional simulations on theLJ-based solid/gas systems at a higher pressure of 30 bar. Wefind that hK increases by a factor of 2 for the case of weakestsolid−gas interaction as the pressure in the MD simulation isincreased. Similar to the effect of the increase in the cross-species interaction strength, the effect of increase in pressureshifts the modal contribution to higher frequencies. This can beattributed to an increase in the collision frequency, Iϵ, of the gasatoms on the solid surface due to the increase in pressure (Iϵ ∝P).47,72 Therefore, with the increase in the pressure of the gas,higher frequencies from the solid can non-negligibly increasethe energy transfer rate to the gas. It should be noted thatincreasing the pressure did not increase the number ofadsorbed atoms on the solid interface in these simulations.In general, the weak bonding across interfaces shifts the local

density of states (DOS) at the interfacial solid to lowerfrequencies. In this context, it has been previously shown thatfor a weakly bonded solid/solid interface, the reduction in thestrength of cross-species interaction leads to a softening of thelocal DOS near the interface.69 To gauge the relative influenceof cross-species interactions to this ”softening” effect in solid/liquid and solid/gas systems, we calculate the DOS of solid

Figure 7. Normalized accumulation of thermal boundary conductanceacross solid/gas interfaces with variable solid−gas interactionstrengths. The weak bonding at the interface is shown to reduce themaximum frequency of phonons participating in thermal transport atthe solid interface. Unlike that in solid/solid systems, the weakbonding leads to an increase in the relative contribution of transversemodes, similar to the “nonwetting” solid/liquid case.

Figure 8. Local phonon density of states calculated for (a) bulk solid, solid monolayer adjacent to the solid/liquid, and (b) solid/gas interfaces withvarying cross-species interaction strengths. Density of states for the transverse modes in (c) solid/liquid and (d) solid/gas systems. Likewise, thedensity of states for the longitudinal modes in (e) solid/liquid and (f) solid/gas systems.

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monolayers in contact with liquid and gas atoms for cross-species interaction strengths in the range of 51.7 to 10.3 meV.Compared to the bulk solid DOS, for the solid monolayer incontact with liquid and gas atoms, the spectrum in the DOSshifts to lower frequencies (as shown in Figure 8a,b).Progressing from a solid/solid interface to a solid/liquid andsolid/gas interfaces, the gradual decrease in the amount ofcross-species interactions leads to further softening towardlower frequencies for the DOS of the interfacial solidmonolayers. Ultimately, for a weakly bonded solid/gas system,the interfacial DOS in the solid mimics that of a free surface.This is demonstrated by the strong peak in the lower frequencyregion ∼2 THz in Figure 8a and Figure 8b, which becomesmore prominent progressing from a liquid to a gas environmentnear the interface. Interestingly, for the solid/gas systems, theincrease in cross-species interaction strength has a negligibleinfluence on the sharp peak at ∼2 THz until the interactionstrength increase causes the gas atoms to adsorb at the solidsurface. This is depicted by the sudden shift to higherfrequencies in the local DOS for εsg ≥ 40 meV, which is aconsequence of adsorption of gas atoms at the solid surface. Incontrast, the softening in DOS for the solid/liquid system isgradual as the cross-species interaction strength decreases. Thesoftening of the vibrational DOS at the solid monolayer atsolid/liquid and solid/gas is consistent with our modaldecomposition analysis.We also plot the interfacial DOS for transverse and

longitudinal modes in the solid/liquid and solid/gas systemsin Figure 8. For the solid/liquid case (see Figure 8c,e), theDOSs for the transverse and longitudinal modes are graduallyshifted toward lower frequencies with decreasing solid−liquidbond strength. This is consistent with the modal predictions inFigure 5, where the contributions to the total heat flow shift tolower frequencies for both transverse and longitudinal modes asinteraction strength is decreased. In contrast, the DOS of thetransverse modes in the solid/gas systems (Figure 8d) does notshow as much of a difference with varying interaction strengthsas compared to the solid/liquid case. However, the DOS for thelongitudinal modes in the solid/gas varies markedly withvarying interaction strengths across the interface (Figure 8f).This is in line with the observations in Figure 7 where the shiftto lower frequencies in the transverse modes is negligiblecompared to the shift in longitudinal modes with decreasingbond strength. Therefore, for the solid/gas systems, thelongitudinal modes can significantly alter the spectralcontributions to interfacial heat flow as evident from Figures7 and 8f.In the Debye approximation, the DOS for a free surface

increases linearly with frequency (DOS ∝ ω), whereas, theDOS for atoms in the bulk increases quadratically withfrequency (DOS ∝ ω2).73 Compared to the DOS for thesolid/gas systems, the DOS of the monolayer of solid atoms atthe interface for the solid/liquid systems demonstrates featuresthat are closer to the bulk solid DOS. This is intuitive since thesolid monolayer in contact with gas atoms (without adsorption)mimics the DOS of a free surface, whereas the solid surface inthe solid/liquid systems is confined by liquid atoms. Thisobservation lends insight into the similar modal contributionsof transverse modes across solid/solid and solid/liquidinterfaces. More specifically, by increasing the cross-speciesinteraction strength across the solid/liquid interface, thecontribution of transverse vibrational frequencies can beincreased to ∼60%, similar to that in the solid/solid case.

Whereas, even though the interaction strength across the solid/gas interface is strengthened, the modal contributions fromtransverse mode coupling has a modest influence on hK (seeFigure 7), until the strength reaches the threshold to facilitateadsorption on the surface, which facilitates additional degrees offreedom and effective coupling of the transverse modes in thesolid to the adsorbed layer. The difference in the DOS of theinterfacial solid due to the different bonding environmentsbetween solid/liquid and solid/gas systems helps shed morelight on the difference in their respective modal contributions,as discussed above.

■ CONCLUSIONS

For solid/solid interfaces, the strong cross-species interactionstrength enhances the longitudinal mode coupling due tocontributions from higher frequency phonons with frequenciesthat are greater than the cutoff frequency of the softer solid,thus facilitating inelastic energy exchanges across the interface.Furthermore, our results show that the majority of the inelasticscattering processes at solid/solid interfaces are dominated bythe longitudinal modes, with little evidence of this anharmonicmode conversion from the transverse modes. For the solid/liquid case, the enhancement in interfacial bond strength allowsbetter coupling of the transverse modes in the solid due to theincrease in the shear rate of the liquid near the interface.However, even for strongly bonded solid/liquid interfaces, veryhigh frequency modes do not contribute significantly to heatflow across the interface. For the weakly bonded solid/gasinterfaces, the DOS of the interfacial solid monolayer mimicsthat of a free surface with the spectrum shifted to lowerfrequencies. Increasing the interfacial bond strength across thesolid/gas interface leads to adsorption at the interface and ashift in the DOS to higher frequencies. This leads to an increasein the modal contributions from higher frequencies in thelongitudinal and transverse vibrational frequencies to the heatflow from the solid to the gas.Our work elucidates general similarities in the influence of

interfacial bond strength to thermal boundary conductanceacross solid/solid, solid/liquid, and solid/gas interfaces. Forexample, an increase in the bonding shifts the phonons in thesolid contributing to the interfacial heat flux to higherfrequencies (stated differently, decreasing the bond strengthsoftens the modes contributing to the heat flux). In the limit ofa weakly interacting solid/gas interface, the modes coupling thesolid to the gas have signatures of reduced dimensionality, asevident by the surface-like DOS. In general, we find that themode softening with a decrease in the interfacial bond is morepronounced in the longitudinal modes as compared to thetransverse modes; as the bond strength is increased to the limitof a strong solid/solid bond, these longitudinal modes are theprimary mechanisms driving inelastic scattering. Furthermore,we consistently observe a decrease in the transverse modecontribution from the solid across the interface as the interfacialbond is decreased, regardless of the phase of matter on theother side.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: phopkins@virginia.edu.

NotesThe authors declare no competing financial interest.

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■ ACKNOWLEDGMENTS

We would like to thank the Office of Naval research for support(N00014-15-12769).

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