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Universit and L.C., E.P. Reviewers: and data; analyzed E.T. and R.G., E.P., research; uhrcnrbtos ..adET eindrsac;EP,RG,adET performed E.T. and R.G., E.P., research; designed E.T. and E.P. contributions: Author owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To h itneidpnec fblitctempoei force nanomanipulations. thermophoretic in important ballistic velocity. be of could group independence flexural a distance the permit The with to that moving given phonons accumulation is longitudinal mass which involves of some adsorbate, by momentum, the mechanism real anharmonic carry of they The regime which responsible. a are in disclosing regime, ballistic gradient, phonons, this flexural of soft The thus thermophoresis. and ballistic length of to proportional dent force phoretic submicrometer a for length show sheet Simulations doesn’t. it that—except lse nammrn-iesse uha rpeesheet graphene a temperatures as between such system suspended membrane-like a a physisorbed cause on small a cluster fluid Reasonably, body. a a on in force proportional gradients temperature thermophoresis, In Significance oee,frsse ie mle hno oprbewt the with comparable or than smaller sizes system for However, den- current heat the system, macroscopic a in with, begin To ∇T J spootoa otetmeauegain Fuirslaw) (Fourier’s gradient temperature the to proportional is K hudyedtesm etcretadtesm adsor- same the and current heat same the yield should > 0 λ o lodpnso dobt,sbtae and substrate, adsorbate, on depends also now i a xedt udeso aoees(3 or (13) nanometers of hundreds to extend can d PNAS osgi ainl el ieceIttt fcn dei Officina Ricerche–Istituto delle Nazionale Consiglio iCgir;adBG,IMResearch–Zurich. IBM B.G., and Cagliari; di a ` κ | and λ ulse nieAgs ,2017 3, August online Published J J a i fteha ares(hnn nour in (phonons carriers heat the of = = 1 K −κ∇T −K nai ttennsae Ballistic nanoscale. the at invalid hudntdpn on depend not should λ 1 κ: i ∇T nyatial ihsamples with attainable only agrthan larger ∆T . www.pnas.org/lookup/suppl/doi:10. , pr hudd just do should apart L u h aegradi- same the but ∆T L u indepen- but ar h heat. the carry | E7035–E7044 These L. 1 and [2] [1] 1

APPLIED PHYSICAL PNAS PLUS SCIENCES by replacing the constant κ with κ(x −x 0), with the resulting con- approach. The approximate temperature profile of suspended volution leading to the product of Fourier-transformed quanti- graphene clamped between two unlike temperatures is discussed, ties (19). We note that κ(x −x 0) will, however, in our case depend and some of its features related to the phonon flux as described on system size, a point to which we shall return later. by McKelvey–Shockley-type theory (25). Next, a gold cluster is The question which we address here is what will happen to adsorbed on graphene and initially shown to be freely diffus- thermophoresis in the small size and distance regime, in partic- ing in thermal equilibrium. After that, a left-to-right tempera- ular whether Eq. 2 would still be valid or not in that case. As ture difference ∆T = Thot − Tcold is turned on in the graphene we shall see, a ballistic regime emerges for thermophoresis too sheet, and phoretic motion of the cluster is readily observed in at short distances, where Eq. 2 breaks down, and a new under- the simulation. To measure accurately the phoretic force, the standing is necessary. This understanding will be mandatory for actual cluster motion is subsequently harnessed by a harmonic thermally induced transport of matter at the nanoscale. spring, acting as a dynamometer. The phoretic force so obtained As a specific test case, we consider here the thermophoretic and its dependence on L is examined and found independent of force felt by a gold cluster physisorbed on a graphene sheet L up to at least 150 nm, indicating ballistic thermophoresis. of length L suspended between two baths at temperatures To gauge the ballistic flux of ZA phonons, which appears ∆T apart. Graphene has an extremely large heat conductivity, as the driving agent of thermophoresis, the frequency-selected amongst the highest of any known material, with measured val- energy transmission spectrum of monochromatic flexural waves ues ranging from 2,600 to 5,300 Wm−1K −1 (17, 20). Thermal is examined. The ability of a ZA phonon mode to carry physical conductivity of suspended graphene is known to be dominated momentum is shown to occur as a result of the mass-carrying by acoustical lattice vibrations [even if the electron contribu- mechanism also associated with an anharmonically entangled tion to the total heat conductivity, estimated initially to be as LA mode, a mechanism best understood by viewing graphene low as 1% (13), might be underestimated especially in doped as a nearly inextensible membrane. Finally, the thermophoretic and in short samples (21)]. The acoustical lattice vibrations of force resulting from scattering on the adsorbed cluster of this free graphene are in-plane transverse (TA), in-plane longitudi- “ZA + LA” complex is demonstrated. The gradient-independent nal (LA), and out-of-plane flexural (ZA). The contribution of character of this nanoscale phoretic force invites a short final the latter, much lower in frequency and therefore much more discussion. populated, has been shown to dominate in suspended graphene (14, 22, 23). System and Methods The thermophoretic motion of gold nanoclusters in CNTs has Because the physical results to be reached in this work are entan- been associated to collective motions of the carbon atoms of gled with technical aspects of the simulation, we find it best to CNT by Schoen et al. (7), and the subnanometer motion of car- describe the system and methods first here, rather than letting gos adsorbed on suspended nanotubes has been observed in a the reader wonder about them until later chapters. thermal gradient (3). Very recently, it was shown via phonon Graphene is described by a C–C Tersoff potential, repara- wave packet molecular dynamics (MD) simulations in CNTs that metrized to better reproduce the experimental phonon spec- thermophoretic motion is driven by the scattering of LA phonons trum (26). A gold cluster (N = 459) with internal fcc struc- between the external and internal CNTs (12). In all of theoreti- ture and truncated-octahedral shape was described with Au–Au cal studies so far, the temperature gradient ∆T was nonetheless interactions of the embedded atom method (EAM) type, mod- considered as the relevant quantity which controls thermophore- ified through a smooth cutoff (27). The cluster 36-atom (111) sis as in Eq. 2. facet is physisorbed on graphene. Interaction between gold and Our present nonequilibrium MD (NEMD) study between graphene atoms is assumed to be of Lennard–Jones type with Thot = 475 K and Tcold = 325 K shows that in the nanoscale ε = 22 meV and σ = 2.74 A,˚ as parametrized by Lewis et al. regime below graphene sheet lengths ∼150 nm, the accurately (28), a choice meant to reproduce the gold–graphite corrugation, measured thermophoretic force acting on the gold clusters is not rather than the adhesion energy. at all proportional to the temperature gradient. Instead, the force We simulate a suspended graphene sheet of x length L and is found to depend basically only on the absolute temperature y width w ∼ 6.5 nm, with periodic boundary conditions along difference ∆T between heat source and sink, independently of y. The first C-atom row (x = 0) and the last one (x = L) are the sheet size L between them. That unmistakable evidence of frozen, thus clamping the graphene sheet. A left–right tempera- ballistic thermophoresis is, we further establish, associated with ture difference is introduced by coupling the first and the last 40 the known ballistic heat transport, caused chiefly by ZA flex- mobile atomic rows (∆x ' 4.5 nm) of the graphene sheet to two ural vibrations. The phoretic force arises due to scattering of Langevin thermostats at temperatures Thot and Tcold , respec- these phonons on the adsorbed cluster which is immersed in tively (Fig. 1). We typically use Thot ' 475 K, Tcold ' 325 K, and the phonon flow. Momentum is tangibly transferred from these a Langevin damping coefficient of 10 ps−1 strictly limited to the phonons to the cluster. That is surprising at first, since harmonic two thermostatted regions, leaving the largest middle part of the phonons are not supposed to carry real momentum; that can only graphene sheet (and the cluster when present) unthermostatted. be carried by a moving mass. All initial equilibrium and subsequent NEMD simulations are As it turns out, a specific anharmonic process of flexural conducted by using our home-developed code. Two main phonons in a flexible but nearly inextensible 2D membrane con- approximations are the neglect of quantum effects and of the centrates extra 2D-projected mass when it is corrugated, as the finite-size acoustical phonon gaps at k = 0. ZA phonons do. These phonons then carry real momentum as Quantum effects are absent in our entirely classical simula- they move with the ballistic ZA phonon phase velocity. Some tions. For that very reason (besides practical ones, including of that momentum is picked up by the adsorbed object, which experimental accessibility), we choose to work at T ∼ 400 K, a as a result is phoretically pushed from hot to cold regions. The temperature that compromises between three constraints. (i) It overall ballistic thermophoresis of clusters proposed here, yet to is well above the temperature where the quantum effects of ZA be verified experimentally, appears to bear some similarity with flexural phonons (the dominant thermophoresis agent) become that which enables fast diffusion of water clusters on graphene irrelevant (29). (ii) It is ∼1/5 of the LA (and TA) Debye tem- ripples (24). peratures (30), so that their specific quantum effect, although Our paper is organized as follows. First, we describe the model not irrelevant, are at least not dominant. LA and TA modes will system, gold clusters on a graphene sheet, and the simulation anyway turn out not to contribute to thermophoresis. (iii) It is

E7036 | www.pnas.org/cgi/doi/10.1073/pnas.1708098114 Panizon et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 n mohsed-tt pe utbet nacrt extraction accurate an well-defined to of but suitable speed small at steady-state artificially smooth (e.g., and an state obtain commensurate to poorly as a into lattice lock- orientation artificially cluster by obtained the simula- be ing can of long improvement estimate Some very case. the requires our complicating is, Such it times, stochastically. as tion diffuse occur interesting positionally which will behavior, events it erratic pinning then misaligned, and when and rela- lattice, misalign only rotations, graphene to the jumps, fluctuation to thermal cluster a tive needs of cluster sequence The events. a of result svsbei i.3 Different 3. γ Fig. in visible is that showed the velocity and mean flow the heat regime, the with steady-state hv both the condition reach once drift Alternatively, rest the adsorbate v). a Small from at from extracted tion cluster be the can to v of large force acceleration sufficiently the initial be to fluctuations, force dominate thermophoretic the Assuming velocity CM is average force the friction for coefficient thermophoretic friction average cluster–graphene viscous a an the assuming Given that (32). indicates viscous motion (CM) 3. Fig. in shown as cold, difference temperature con- T the positional equilibrium Once both (32). these diffusion, angular thermal and In undergoes equilibrium. cluster the adsorption thermal ditions, in (temperature-dependent) Ther- energy the to free of Contribution calculation Constrained Energy direct of Free Effects and Cluster, Alignment, mophoresis, on of in Motion Function described Angular a procedure as adiabatic Adsorption the following least sheet, at of time simulation a over 2. Fig. and ns; slice 15 the in theorem, atoms the equipartition over gradient the the along by nm obtained 0.5 of slices in direction sheet graphene temperature the local divide the of evaluation approximate irrelevant. therefore are gaps finite-size graphene for is our only to close came sidered, magnitude their if temperature problem working a be would hand, (31). size system phonon the some than shorter where anharmonically regimes get temperature MFPs higher the below still nthe on 1. Fig. aio tal. et Panizon (0) cold and i(t ete ehdi eypeie oee.Ps ok(32) work Past however. precise, very is method Neither center-of-mass cluster the of uniformity approximate The graphene the of center the near deposited then is cluster The an For first. simulated is sheet graphene adsorbate-free The at gaps phonon acoustical finite-size The F th 1.4 ' ∞) → stre n h lse sosre oditfo o to hot from drift to observed is cluster the on, turned is T h la ifrnebtentetopoei regimes phoretic two the between difference clear The . hv hot 0 ceai fa Au an of Schematic e,or meV, L goigapsil nua eedneo momen- of dependence angular possible a Ignoring i. esrn h hrohrtcFrefo Accelera- from Force Thermophoretic the (Measuring and x 0n,hwvr h Apoo gap phonon LA the however, nm, 30 = vrgn h taysaeaoi temperature— atomic steady-state the averaging , = γ T (T F cold ∼ th ) 16 ein fgraphene. of regions rtclwihas emt the permits also which protocol a S1, Fig. /γ 5 s ntedfuiergm,tetime-averaged the regime, diffusive the in is, × (for K a eetatd(,3 ,9 33). 9, 7, 3, (2, extracted be can M T 10 459 dv 0 .Ee o h mletsz con- size smallest the for Even K. 400 = − dt 3 lse eoie na60 a on deposited cluster (t θ v e or meV LA v ) eaiet h neliggraphene underlying the to relative θ is = ∼ auswudla oadifferent a to lead would values F 2k/) h Agap ZA the km/s); 22 th 60 − γ K At mK. γ h qaino motion of equation the , v T γ . n hsof thus and , = k (2/3)NE ,o h other the on 0, = T × T ∼ nm 7 γ ∆T θ (x ∼ 400 n large a and 30 = esub- we ), 2 v kin ∼ = LA rpeeset ooe ohglgttetemlgain.Temsasaeapidonly applied are Thermostats gradient. thermal the highlight to colored sheet, graphene F ,both K, Cluster th F /k T C/4L 2 is /2L ◦ th hot so ) B and [3] — in – 2 i.2. Fig. at width and 2 ) h eprtr a enaeae vrgahn lcso nm highlighted. 1 is profiles of temperature In slices two Methods ). graphene the and over between (System averaged direction gradient been the has along temperature The K). 325 in ofrhrrdc h ro n ceeaeteaveraging, the angle accelerate ing and alignment error direc- cluster the gradient the reduce the further along To displacement tion. CM time, cluster simulation average the shorter much a to and term spring sheet, potential a a tie of artificially means X We by (8). CM others cluster by the done also as con- thermophoretic object, more the therefore directly is measure force It to immediate tricky. and thus venient and indirect, is trajectory product their sheet, of the independent from relatively cluster be the should by rate pickup tum h oiino h lse a ag uoorlto ie due times autocorrelation large has cluster the of position the of fluctuations the by ther- calculated the Cluster influence (see not force does mophoretic constraint angular the soft a by spring, place in kept this cluster, our Experimentally, of an spring. role 4. cantilever by the reproduced Fig. playing be har- tip, possibly in AFM might the shown geometry of harnessed are of trajectories kind CM Typical (unther- central system. cluster representative the nessed most of the part to mostatted) motion cluster the inlycekdb h utain ftefreaogtedirec- gradient, the thermal along the force to the perpendicular of tion fluctuations the by checked tionally nalcss oee,teetato of extraction the however, cases, all In h ttsia ro fetn u eut a eetmtdboth estimated be can results our affecting error statistical The the by permitted displacement cluster modest the to Owing l stecutrC oriaerltv otecne fthe of center the to relative coordinate CM cluster the is 22 m.Tefis ls)4n r hrotte at thermostatted are nm 4 (last) first The nm). 22–24 = θ 30 = F oa eprtr rfiei rpeelyro length of layer graphene a in profile temperature Local th and F th w rmteNM iuain yhresn h mobile the harnessing by simulation, NEMD the from ◦ k eoesseaial on ht ecekdthat checked we that, doing systematically Before . m ihadwtotteasre odcutr(located cluster gold adsorbed the without and with nm, 7 = is S2 Figs. s a eotie ihtedsrdacrc within accuracy desired the with obtained be can > 0 sasrn osat togeog orestrict to enough strong constant, spring a is and F PNAS fet fCntandAglrMto on Motion Angular Constrained of Effects th S3 itself, o oeinformation). more for | θ ulse nieAgs ,2017 3, August online Published a ekp xdb constrain- by fixed kept be can S X = F θ th sdesired. as hk = s (X k s F hX hX − S th Y (1/2)k h difference the Inset, where i, rmtecluster the from = i) k s 2 T hY s n addi- and i, 7 (T K 475 = X 2 L 2 Since i. where , 0nm 50 = hX | E7037 i F th is =

APPLIED PHYSICAL PNAS PLUS SCIENCES the sample depends on the scattering probability between the two opposite currents. Considering only phonons with a given energy , dI +(x) dI −(x) I Q = Q = − Q . [4] dx dx λ The temperatures of the two thermostats enter as boundary con- dition in the values of the two currents. With an assumed ideal nature of the contacts, the right-flowing current through the left contact is in equilibrium with the left thermostat at Thot , and similarly the left-flowing current through the right contact is in equilibrium at temperature Tcold , then M I + =  n(T ) [5] Q,0 h hot and M I − =  n(T ), [6] Q,0 h cold where M () is the distribution of modes of the thermal conduc- tor at energy , h is Planck’s constant, and n(T ) is the Bose– Einstein function. The coupled Eqs. 5 and 6 for the currents can be solved to yield a value for the thermal gradient ∇T which is not, in general, equal to ∆T /L = (Thot − Tcold )/L, thus imply- ing the two temperature jumps at the two thermostatted regions with ballistic transport variables. The ballistic thermal resistance Fig. 3. (Top) The evolution of the position along x of the center of mass of ball the Au cluster for a graphene length L = 110 nm with an applied ∆T = 100 K equals in fact R = IQ /2δT , where δT is the temperature between Thot = 450 K and Tcold = 350 K. (Inset) The velocity evolution for the jump between thermostatted–unthermostatted regions, assumed first 1.5 ns of the same simulations. (Middle and Bottom) Cluster transverse to be the same at left or right contacts since the thermostat effi- displacement along y (Middle) and angular velocity ω (Bottom) during the ciency is fixed (given by the Langevin damping rate and by the same NEMD simulation. extent of the thermostatted area). Note that this resistance is present even for ideal, nonreflective contacts. Summing up, the border temperature jumps represent the to the slow dynamics, a block analysis has been performed (see prime evidence for at least some ballistic heat transport. If all Fig. S3). Our overall error in this procedure is estimated to be of the heat current was transported ballistically, with zero dis- Ferr ' 0.5–1.0 pN, compared with Fth ' 2.0–10 pN. To enhance sipation from hot to cold, then both temperature jumps would the resolution on the calculated average force, we set the spring be (in a symmetric case) −∆T /2, and ∇T = 0 in between. If, 2 on the contrary, the heat flux was completely diffusive, then the constant ks = 0.001 meV/A˚ ' 16 µN/m, a value much smaller than the typical ∼1 N/m of AFM cantilevers. However, we note jumps would disappear, Fourier’s law would be obeyed through- that for a stiffer spring, the same resolution could be achieved by out, and ∇T = −∆T /L everywhere. The temperature profile just increasing the simulation time. and the jumps in our simulated graphene are intermediate (see Figs. S4 and S5). The jumps are sizable, indicating a large amount Temperature Profile and Heat Flux of ballistic transport, besides some back-reflection. The jumps Let us begin with the results of an NEMD simulation of the free graphene sheet with the temperature imbalance, but without the adsorbed cluster. Fig. 2 shows the typical average steady-state temperature profile. The thermal gradient that will become rel- evant later is the slope, obtained as a linear fit, in the central region. The steep temperature jumps near the two thermostatted regions are a common feature found in all NEMD simulations (8, 34, 35). A reasonable explanation generally offered for the jumps is the same as the Kapitza resistance jump (36). The con- strained left and right borders are mechanically different from the inside, and a traveling wave through that interface is partially reflected, causing a thermal resistance similar to that at the inter- face between two different materials (37). In reality, the temperature jumps and their magnitude con- tain a much more interesting underlying source. Recent work by Maassen et al. (25) has shown that near-border jumps in the effective temperature profile are present even for “ideal” contacts—where no reflection occurs—when at least some of the thermal current is ballistic, in the following sense. In standard Boltzmann theory of thermal transport, the net heat current is the difference between forward and backward + − Fig. 4. In plane trajectories for a spring-constrained gold cluster with (red) currents, IQ = I I . Following McKelvey (38) and Shock- Q − Q and without (blue), an imposed angular constraint subject to a temperature

ley (39), the scattering between forward and backward currents difference ∆T = 100 K between Thot = 450 K and Tcold = 350 K, on a graphene is controlled by a parameter, λ, roughly reflecting the phonon sheet of length L = 50 nm. The black spot corresponds to the rest position MFPs. In this respect, the evolution of the heat currents inside of the spring applied to the cluster CM.

E7038 | www.pnas.org/cgi/doi/10.1073/pnas.1708098114 Panizon et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 n xrm lutain epeeti i.5afwsnapshots few from a starting 5 front, Fig. corrugation in thermal present visual a purely we where a illustration, as extreme First, and thermophoresis. difference simulate temperature we left–right graphene, the on turning By Force and Motion Thermophoretic main Cluster the of origin different. The be results. therefore must our force to thermophoretic compatible not is the see as to shall to construction contribution by pretense this scales Moreover, any force small. thermophoretic without too far Even actual is the below. this for accuracy, 15–40 described account is be to this to necessary out, value phoresis, turns the it than As smaller temperatures. of times detachment higher additional at small cluster the includes the that result a K/nm, 1 as culated this to Due of temperature. dependence increasing with magnitude in weaken to information energy (more free adsorption cluster free-standing in purely interact- provided is a fully to a system from removing gradually ing passing of interaction, method graphene–cluster standard the the by data, simulation extract intro- was We gradient temperature duced. spatial a once force, mophoretic In possible). T but (rare, should drop case, or either Unlike (commonest) upon may rise equilibrium. energy either free thermal heating interface (negative) full the in energies, cluster–graphene free temperature, bulk the of is function a gradients energy thermal free interface on turning before cluster depinned rotationally 42). the (32, translations where place, fast states take executes vibrations diffusive of only of alternation where an angles and this of specific diffu- therefore at In overall consists states mobile. cluster locked The the occur. more of to motion much sive diffusion is positional dramat- allowing cluster drops ically, barrier the translational the and state, “incommensurate” worse, adhe- the Alignment interlocking, is no of is sion there Function orientations, other a of majority in as reported is Adsorption discussion ter detailed zero-temperature more the (a energy of adhesion minima of to orientations corresponding interlock- angular generally strong some a ing, shows At interface 42). gold–graphene the 32, free- cluster, (28, the of cluster degrees the translation of cor- and strong dom a rotational found the graphite between or diffusion graphene relation on the clusters of gold early such adsorp- simulations of Following previous the gradient. (41), temperature studies consider a experimental zero to in at diffusion of thermal interest equilibrium, its full facet characterize of to and is close-packed thermodynamics tion it the motion, graphene mophoretic on octahedral adsorb truncated we Next, Adsorption Cluster small large at very regime at limit ballistic diffusive the the from to locality) (i.e., zero to infinity otc un u norcs ob once ihteballistic of the range with the connected flow: be heat to the case of hetero- our thermal fraction in a out in turns distribution contact temperature nonlinear the for behavior. κ(x this verify to possible be should it (40), niques for in vanish seen increasing be will for can consistently As decrease region. jumps middle border the in place taking than ∇ smaller clearly are aio tal. et Panizon T htdpnec ol nisl rvd oreo ther- of source a provide itself in would dependence that , eodfaueo h dobdcutrt eunderstood be to cluster adsorbed the of feature second A nve fteaoe h olclt fha conductivity heat of nonlocality the above, the of view In − snneo orsodn osm muto dissipation of amount some to corresponding nonzero, is x 0 eyaporaeyitoue yAln(9 oaccount to (19) Allen by introduced appropriately very ), F ad .The Thermophoresis). to Contribution Energy Free = L G −dG ihtenwnntemmti tech- nanothermometric new the With ∞. → ntmeaue hroyai oc scal- is force thermodynamic a temperature, on G G G (T /dx Au apnt eedsrnl nuhupon enough strongly depend to happen G (T ) −∆ (T 459 = esrn h lse deinas adhesion cluster the measuring ), rmtemdnmcitgainof integration thermodynamic from eaieqatt,i on here found is quantity, negative a ), T −dG lse.Bfr tdigisther- its studying Before cluster. ota h vrg gradient average the that so /2, L. /dT ∇ κ(x T ∼ and − .9p for pN 0.29 ∇T x 0 ) .A a At S1). Fig. ota they that so L, T hc swe as which , ayn from varying L the S5, Fig. ∆ 0 K, 700 = T ∇T across Clus- = L eie hsi tiighlmr fblitcbhvo.Ee at Even behavior. of ballistic size of sheet graphene hallmark largest striking our a gradient is Eq. This local from regime. the expected on be dependence would difference no temperature remarkably, overall the but, on dependence linear a 1. thermophoretic Table in spring-constrained reported forces the obtaining orientations, harnessed differences of size temperature series sheet ferent graphene systematic increasing a for clus- out simulations the cluster carry determine we to displacement, needed time ter the optimizes roughly is which force thermophoretic average as the sheet, extracted the of midpoint the at harness- displacement by mean spring the harmonic force dynamometer, a average by in motion the proceed cluster the directly we ing measure quanti- anticipated to for as ideal following and not the purposes, and accurate sizes and small for tative converge to hard rather the fitting v of Small By mates at Thermophoretic earlier. the Acceleration explained (Measuring from cluster Force as the lubricity, of acceleration improve Eq. initial to using the as by of evolution estimate, so the we with from time drift velocity, obtained of that cluster force on function thermophoretic Based the a ther- 3. 3, as the Fig. X-distance confirm in the Flux shown drift, Heat and cluster Profile mophoretic Temperature of tocol motion. thermophoretic flexural the ZA for the sit- responsible that sketchy suggestion are this initial vibrations as an provide Unrealistic does wave. it “tidal” is, a uation by along swept if as T of motion forward normal the plane. their equilibrium to of perfect corresponds track the atoms to carbon keeping relative the gold help displacement of the code to of color atom The red cluster. single in the one while colored indi- sheet, arrows is graphene black the cluster in The point visible. fixed are a phonons cate left flexural the right-moving where only simulation that a to with set only, total is purpose thermostat a pictorial to for (corresponding obtained snapshots been three These along. it 5. Fig. et htteZ eua irtosaeteatr responsible actors the are vibrations flexural ZA the that gests here. out borne not therefore depen- is The sizes on nanoscale profile. for force temperature the component graphene ballistic of the large dence a by on has indicated flux pre- nanoobject as heat our matches that absorbed this observation turn, on vious In ballistic. force largely remains thermophoretic graphene the work, R h hrohrtcforce thermophoretic The pro- standard the by out carried simulations realistic More h ls aallwt alsi ettasoto rpeesug- graphene of transport heat ballistic with parallel close The this in considered sizes the least at to up that conclude We F ,ht h lse rmtehtrgo,cryn talong it carrying region, hot the from cluster the hits K, 0 = th ssilinsignificant. still is hral xie eua orgto istecutr pushing cluster, the hits corrugation flexural excited thermally A F th F . N hsdrc prahis approach direct This pN. 6.7−9.3 range the in th = T R k 0 n h ih hrotti e to set is thermostat right the and K 700 = s hX ∆T PNAS ∇T fe hoigaconvenient a choosing After i. 5 n with and K 150 = eeal noe nltrtr even literature in invoked generally | F th ulse nieAgs ,2017 3, August online Published hc svldi h diffusive the in valid is which 2, L ∆ ipas ihntlrbeerrors, tolerable within displays, 5 m h xetddecline expected the nm, 150 = T and n w ifrn graphene different two and , hX ,w banesti- obtain we S6), Fig. i rmters position rest the from k s θ esrn,a na in as Measuring, . 30 = ∆t ◦ constrained 5p)have ps) 15 = w dif- two L, T k R s ∇T ,so K, 0 = | value E7039 ∆ T as ,

APPLIED PHYSICAL PNAS PLUS SCIENCES Table 1. Average thermophoretic force Fth evaluated for a duced two symmetric flexural wave packets, a left-traveling one 2 Au459 cluster deposited on a L × 7 nm graphene substrate with which enters the left thermostat and gets dissipated, and another average temperature hTi = (Thot + Tcold )/2 = 400 K, and right-traveling across the graphene sheet, finally absorbed by the 0 nominal temperature difference ∆T = Thot − Tcold of 40 and right thermostat. By measuring the heat W absorbed by the right 150 K at the edges thermostat, alternatively without and with the adsorbed cluster, we calculate the variation of transmitted energy ∆W (ω) caused L (nm) ∇T (K/nm) ∆T (K) ∆T0 (K) F (pN) th by the cluster. For each flexural frequency ω the heat transmis- 30 0.42 38 40 2.1 ± 0.5 sion coefficient T = Wcluster /Wclean is extracted in this manner. 40 0.39 37 40 1.7 ± 0.6 To allow for a computationally viable and sufficiently precise cal- 50 0.33 37 40 1.9 ± 0.7 culation we study running waves of just a few (nc = 3–5) oscil- 60 0.29 37 40 3.6 ± 0.6 lations, so as to avoid interference with the backscattered wave, 70 0.28 37 40 2.9 ± 0.8 identifying the transmitted and reflected parts, finally absorbed 30 1.88 148 150 9.7 ± 0.5 by the opposite thermostats. Although ideally one could in the 40 1.47 147 150 9.2 ± 0.9 same way measure the flux of momentum besides energy, that is 50 1.26 147 150 8.9 ± 0.5 in practice substantially more difficult, and not really necessary. 60 1.10 147 150 10.4 ± 0.6 First, by injecting a small number of oscillations, the resulting 70 1.04 147 150 10.1 ± 0.6 force on the cluster is often insufficient to overcome the static 150 0.54 147 150 8.8 ± 0.6 friction, and the cluster just oscillates around its average posi- 30 1.95* 147 150 9.0 ± 0.4 tion. The magnitude of the oscillation is related to the momen- 40 1.71* 147 150 9.8 ± 0.6 tum transfer between the wave packet and the cluster, but the 50 1.27* 147 150 9.3 ± 0.5 net momentum transfer Pavg = M hvCM i is zero, absorbed in this 60 1.15* 147 150 8.7 ± 0.4 regime by the lattice underneath. Secondly, the flux of energy 70 1.03* 147 150 7.9 ± 0.5 and its variation are well defined, even in the absence of adsor- bates, and can always be measured with good precision, unlike The effective temperature difference is instead calculated as ∆T = the total momentum flux, which, as we will show in the follow- Tx=0 − Tx=L. The local thermal gradient at the cluster site ∇T is also given (linear fitting of the central region; Fig. 2). ing, has multiple components and cannot be measured with the *Thermal gradient applied in the zig-zag direction, while all of the other same precision. We will therefore stick to the energy transmis- simulations have thermal gradients applied in the armchair direction. sion, whose analysis is nonetheless quite informative. The phonon-resolved energy transmission of graphene sheets of transverse size w = 9 nm and lengths in the range L = 53– for the thermophoretic force. To check that, we repeat simula- 88 nm is shown in Fig. 6. At frequencies larger than ∼500 GHz, tions by artificially freezing the out-of-plane graphene motion, the transmittance increases smoothly from ∼0.80 to ∼0.97. The only allowing in-plane LA and TA vibrations. In that case we loss of transmission can be attributed to all processes that pro- obtain, for L = 50 nm and ∆T 0 = 150 K, a thermophoretic force duce backscattering. of 0.5 ± 0.7 pN, negligible in comparison with the force found In the low-frequency region, two transmittance dips are visible, with unconstrained graphene, and more comparable to the error. approximately at ω1 = 49 GHz and ω2 = 175 GHz. These two We conclude that the flexural ZA vibrations of graphene are features are connected with resonances between cluster-induced the main source of thermophoretic force on the adsorbed cluster. local vibrational modes and the incoming wave. The first mode, This conclusion now invites a more detailed analysis. ω1, coincides with the natural frequency of out-of-plane vibra- tion of the cluster as a whole on the graphene substrate. This fre- Scattering of Monochromatic Flexural Wave Packets. We need to quency depends on the Au–C interaction and on the cluster size. qualify the role of phonons in the observed ballistic ther- p Assuming ∼ω1 ∼ K /M , a cluster mass M and effective spring mophoresis. The connection between the thermophoretic force constant K scaling proportional to N and N 2/3, respectively, one and the atomic vibrations which transport the heat is direct at −1/6 the nanoscale sheet size, where phonons, whose MFP is larger, obtains ω1 ∝ N . Using the value obtained for N = 459, we remain sufficiently well defined. The temperature difference gives rise to a net imbalance of phonon population between the two sides, nhot and ncold . In coaxial nanotubes, Prasad et al. (12) showed how a phonon wave packet traveling in the outer (longer) tube scatters at the edges of the inner (shorter) tube, exchang- ing energy and momentum. This “push” from a single wave is felt identically whether it comes from the hot reservoir or from the cold reservoir. However, since the phonons that constitute the wave packets from the two opposite regions have different populations a net phoretic force arises. In the ballistic transport regime, where Eq. 1 is violated and δn dominates, the latter is a function only of ∆T , and so is the thermophoretic force result- ing from this unbalanced population. In our case, the ballistic propagation of flexural phonons is identified as the source of the thermophoretic force. Harmonic phonons, however, should carry crystal momentum but no physical momentum: How then can the flexural phonons cause a net force? We address this issue by simulating the time evolution of nearly monochromatic flexural phonon wave packets. A phonon is injected in a free graphene sheet by applying an external ver- tical force on the left side of the nonthermostatted region, shak- Fig. 6. The flexural phonon energy transmission coefficient T (ω) as a func- ing along z a single column of carbon atoms with a force Fex = tion of the phonon frequency. The low frequency resonances are discussed F0 sin(ωt)ˆz for a certain number of cycles nc . The shaking pro- in the text.

E7040 | www.pnas.org/cgi/doi/10.1073/pnas.1708098114 Panizon et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 ie neapeo aorsnne(3.Tetasitnea a at transmittance pro- at The (43). resonance actually resonance Fano problem Fano linearly of coupled therefore example The is an vides and continuum. graphene, that of to modes coupled flexural ZA the cluster as This simulation. in to up yields clusters it where octahedra gold-truncated for verified rto l iil ntetmeauepol hw nFg 2: Fig. in shown profile temperature the is in cluster visible the all Waves. of of Traveling nature first momentum-reflecting) Flexural (and of reflecting Momentum Physical next. describe the trans- we momentum which to the cluster, to the contribute occurs to fer same ones, The flux. low-frequency heat transmitted longest, the especially energy. transmitted the of of 1.5% reduction on- contribute cluster-induced only two total resonances the the that find integrating we By parts, resonance dominates. which conse- spectrum, fre- whole a quency the the over as with contribution compared and continuum minor, integrated resistance, remaining is force, thermal thermophoretic the the contribution to to quence the resonances that shows these however, of analysis, clus- overall quantitative as Interesting is, the the resistance. it to thermal and cluster-induced contributing the phonon process, of increase scattering incoming the the enhances ter between coupling onant enough (24). drops large the wavelengths “contain” have the to by which dominated ripples is that graphene thermal on observation graphene drops recent water the of This to diffusion minimum. fast connected the be transmittance may a process in of kind scat- reflected increased is the and maximum phonon oscillation, flexural tering into the it set facet, effectively contact wavelength more cluster the can the when of Thus, size cluster. the the matches under dimple this of size with at at [when peak infinity symmetric from ranging parameter, asymmetry an obtain hits phonon ZA begins. the damping when thermostat reflection the no where basically boundary is the vibrations there flexural that ZA at Note of visible. cluster) propagation is (no and sheet excitation The graphene frequency. position clean 80-GHz the longitudinal shakes of force function nal a as mapped aio tal. et Panizon fre- dip the that find We (44). face is quency contacting graphene cluster on the cluster under the of adsorption at accompanied that recall we onance, where 7. Fig. ncnlso,alo h hral xie eua wavelengths, flexural excited thermally the of all conclusion, In res- the therefore, spectra, transmission the in dips both At  ] h Apoo rnmtac near transmittance phonon ZA The 0]. = q  ω The ∼ ω = 1 2 = . and 0.0 (ω orsod oawvlnt hc saottiethe twice about is which wavelength a to corresponds x poetdCCbn length bond C–C -projected 139N − ω ω T 1 0 i 3Gz ls ote3-H au obtained value 31-GHz the to close GHz, 33 = −1/6 )/Γ,  Γ yalcl“ipe ftegahn sheet graphene the of “dimple” local a by 0 = ω T ]t eo[when zero to 0] = = () = ∼ Γ H.Tevldt fti siaei well is estimate this of validity The GHz. ω H.T nesadtescn res- second the understand To GHz. 5 i en h it ftersnneand resonance the of width the being so h eea form general the of is 1+ (1 z -mode ( q 2 + ( + )(1 l ω x q vrgdoe the over averaged , ) 1 2 T a h aesymmetry same the has  () x 2 ) n ie steexter- the as time, and , a ymti dip symmetric a has ω 1 X h energy- The a efitted be can ' N T 0n with nm 20 y () 5,635, = direction, a a has [7] q rvln eua akt,woeoclaigprsaerespectively are parts oscillating whose packets, flexural traveling X F momentum. physical positive one a only carries the adsorbate, phonon, the flexural scatters that the the and However, separate, theory. packets Gruneisen wave in two as negative ZA– is the much pair of phonon momentum and LA total wavelength, combined The longer velocity. a frequency, group with larger double phonon ZA of requirement the phonon of Gruneisen LA association the anharmonic first, does the at This of momen- seem adsorbate. because would physical the it to positive as communicated violate, net is not a which phonons of reality flexural part inter- in of tum, quite packet conveys out, wave turns graphene localized it a in as that is, find puzzle We this esting. simu- of in solution observed force The momentum, thermophoretic lation. positive physical the negative with a odds carry at phonons flexural that gest (50). established coef- well is expansion graphene thermal of negative ficient the and parameter negative the bound- Gruneisen between connection the tem- This where expansion. pulling materials causes mem- therefore perature bulk clamped normal tension, of a contrary of inwards—the of state aries length a total causing other the brane, any increase or excited phonons Thermally underlying graphene, transparent. flexural and reason of simple actually physical parameter is the membrane, Gruneisen (49), negative literature the the in found (48), be negative is it puzzling. Alas, is first large. at indeed which is parameter Gruneisen where parameter, p Gruneisen monic tum momentum physical a to ated ther- the in involved be must anharmonicity force. This and mophoretic force. case, pres- thermophoretic the in ballistic not even zero is cluster, and the scattering, with therefore of exchange should ence momentum there net level, zero harmonic be the At physical not (45). but momentum momentum, crystal carry phonons picks setting, cluster monic and The packets, profile. phonon temperature incoming way. the the that in for in momentum up scatterer resulting K a resistance, 2 as thermal of acts localized drop a minute as a acts cluster The h Amd,ti Amd nege iil eeto u othermostat to due reflection boundary. visible the undergoes unlike at mode that onset damping LA Note this figure. mode, preceding ZA the the of visible, mode is mode flexural longitudinal the LA fast accompanying the of propagation and excitation monic at cluster) (no sheet graphene position of function a as i.8. Fig. q z oetbihta,w pl etcl iedpnetforce time-dependent vertical, a apply we that, establish To sug- thus would theory standard case, our to blindly applied If can that derivations full the in contained course of While ned codn osadr hoy phonon a theory, standard to according Indeed, har- completely a In problem. a face we anticipated, as Here, = = ' q Γ~q u oahroiiy If anharmonicity. to due ) F 0n.Ti oaie oc rae w opposite-moving two creates force localized This nm. 20 h oa – odlength bond C–C total The 0 sin(ω 4,4) o h eua hnn fgahn,the graphene, of phonons flexural the For 47). (46, t ) oterwo rpeecro tm tposition at atoms carbon graphene of row the to x PNAS n ie steetra oc hksteclean the shakes force external the as time, and X ' | 0n ih8 H rqec.Teanhar- The frequency. GHz 80 with nm 20 l p vrgdoe the over averaged , ulse nieAgs ,2017 3, August online Published q Γ dfeetfo rsa momen- crystal from (different = −d V ln d ω/ stevlm,then volume, the is ln y ieto,mapped direction, V ω steanhar- the is q sassoci- is | E7041

APPLIED PHYSICAL PNAS PLUS SCIENCES of the two modulations produced in a cycle of the vertical force. The ratio, for the simulation discussed above, can be cal- culated as PLA/PZA ∼ (2λLAvLAδρLA)/(λZAvZAδρZA) ∼ −12. This result indicates that the total momentum is indeed nega- tive as Gruneisen’s theory indicates (the factor 2 in the formula above comes from the double frequency). Simulations thus confirm that the longitudinal modulation, although anharmonically created together with the flexural mode, moves away and separates very quickly. The remaining flexurally corrugated graphene has an increased mass density per unit projected area and its associated physical momentum is pos- itive. This mechanism bears a resemblance with the picture by Bassett et al. (51), who investigated the physical momentum of

Fig. 9. x-projected C–C graphene bond length lx averaged over the y direc- localized running waves for 1D systems; to the best of our knowl- tion, mapped as a function of the longitudinal x position and of time for a edge, no further investigation of this mechanism was done fol- graphene sheet with a physisorbed cluster located at X = 65 nm, excited by lowing that work. a z oscillation applied at X = 20 nm. The arrow indicates the point in time Consider now the interaction of these artificially excited and space where the first flexural corrugation reaches the adsorbed cluster. phonons with the gold cluster adsorbed in the middle of the graphene sheet (Figs. 9 and 10). The cluster produces intense + − scattering of the flexural wave packets, as visible from the pro- of the form z = A sin(qZAx − ωt) and z = A sin(qZAx + ωt) jected bond lengths (Fig. 9). In this way, it picks up positive for x > 0 and x < 0. Consider now the forward-traveling phonon physical momentum, which explains the thermophoretic force. (the backward one behaves in exactly the same manner). Anhar- At the same time, there is no visible cluster-related scattering of monically associated with this harmonic ZA phonon, there is the fast-moving longitudinal phonons. The LA cross-section on an increase of the C–C bond lengths along x, since the ZA the cluster is very small, as can be seen, comparing the evolu- corrugation extends the effective length of clamped graphene by tion of the true bond length in simulations without and with the 2 2 2 δL/L ∼ (qZAA /2) cos (ωt) (assuming qZAA  1). adsorbed cluster (Figs. 8 and 10): Indeed, the presence of the This forced bond length modulation gives rise to a LA phonon cluster has little effect until the ZA wave packet hits the clus- wave packet of double frequency ω˜ = 2ω, which travels away ter and additional anharmonic effects take place. As a result, the from X with velocity vLA and a small wave vector qLA, such negative LA momentum is not transferred to the cluster. that vLAqLA = 2ω. LA phonons are orders of magnitude faster than ZA phonons, and in extremely short times (t ∼ 1 ps) the LA Phoretic Force by a Flexural Phonon. We are now in a position to phonon carries away all longitudinal perturbations, leaving the present the resulting formulation for the phoretic force caused remaining flexurally corrugated graphene with equilibrium C–C by a single injected flexural phonon in graphene, valid also for bond lengths. The combination of corrugation and equilibrium a more general membrane-like 2D material. While (predomi- bond length produces a 2D projected density excess in the x–y nantly) propagating from hot to cold, each flexural phonon en plane δρ, and therefore a positive physical momentum density route impinges on the adsorbed cluster, which picks up momen- associated the slow flexural phonon p = δρvZA. tum by partly reflecting it backward. This can be described as Figs. 7 and 8 show as an example the case of ω = 80 GHz and a follows. z-shaking force F0 = 30 pN applied at X ' 20 nm of the sus- The excess density embedded in a traveling ZA wave moves pended graphene sheet (no adsorbed cluster for now). Upon with its phase velocity. This excess projected density δρC is increasing the intensity F0, it is verified that the ZA corruga- tion A scales linearly with F0, accompanied by an LA excitation δρC /ρC = (leq /lx − 1), [8] amplitude, which scales like A2, as expected for its anharmonic where lx is the projection along x of the average carbon– origin. In the first picoseconds after switching on of the force, carbon bonds length whose equilibrium length is leq . The physical we monitor the projected bond length, defined as the distance momentum density p per unit area is therefore: between carbon atoms projected on the x–y plane, and the true C–C bond length. In Fig. 7 the in-plane projected C –C bond lengths are reported as a function of position and time. The propagat- ing flexural ZA phonon involves a projected bond length which is shorter than the static equilibrium value, indicating that graphene is slightly “overdense” under the slowly moving mode of velocity v ∼ 0.9 km/s, close to the theoretical value dωZA/dqZA = 1.06 km/s (30). The reason for that higher den- sity is clarified in Fig. 8, where the true bond lengths are shown. At very short times there are compensating “underdense” modulations (bonds longer than the equilibrium value) which move away from the excitation point. Their apparent speed is v ∼ 22 km/s, close to the experimental speed of sound for LA phonons vLA ∼ 21 km/s (26). Note the scale difference of the two pictures: While the true bond length modulation is of 0.03%, the projected length shrinks by as much as 2%; and this for all Fig. 10. The total C–C bond length l, averaged over the y direction, ω mapped as a function of the longitudinal x position and of time, as the ZA frequencies z . The explanation for this mismatch lies in external force shakes the graphene sheet at X ' 20 nm with a physisorbed the fact that the modulations in the ZA and LA phonons are cluster located at X = 65 nm. The arrow indicates the point in time and space “diluted” over the relative wavelengths, which are very differ- where the first flexural corrugation reaches the adsorbed cluster. Note how ent: Indeed, λLA ≈ 138nm and λZA ≈ 7.9 nm. To better com- up to that time the presence of the cluster has little effect on the bond pare the two contributions, we can estimate the momentum lengths.

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Prospects et L, graphene: Lindsay of 14. conductivity thermal high Extremely (2008) al. et S, Ghosh 13. h igepoo oc aclto ok,teei oreason no would force is thermophoretic there work. integrated also overall works, the calculation that doubt force to Since phonon undertake. not single do we the which effort however, one massive That, a imbalance, force. demand would thermophoretic left–right total its the with approximate will integrating distribution and thermal modes ZA the all over for procedure the repeating by the ple, on acting 25 force of actual the cluster with agreement good reasonably in on, 1 earlier extracted coefficient transmission energy free coefficient transmission the average of state—an acceleration force initial z-shaking the phoretic by the with measured prediction cluster. above as the compare force can we Here, F of at sheet cluster adsorbed graphene the suspended the of simulation scat- the simplicity). of for assumed reflection behavior backward (1D for phonon accounts tered 2 factor the where coefficient contact of vector wave adsorbate the by and tion wavevector different very of phonon LA ZA an velocity. the and of ZA square the of the to proportional amplitude phonon effect, along anharmonic only the acts an generates that is which force physical force a external this phonon, the flexural that by produced again directly stressed not be should It aio tal. et Panizon .Zmrn A ate H omusksP blaiiI 20)Thermophoretic (2008) IF Sbalzarini P, Koumoutsakos JH, Walther HA, motion. nanoflake Zambrano drive to 9. graphene on gradients Thermal (2014) X Wang M, Becton ther- 8. assisted Phonon (2007) P Koumoutsakos D, Poulikakos gra- JH, wettability Walther surface PA, via Schoen graphene on 7. droplets by water graphene Actuating on (2015) B mass Xu molecular Q, of Liu transport Directional 6. (2014) T fields. Li gradient strain S, by Zhu Y, driven Motion Huang (2015) S 5. Chen C, gradients (2006) Wang thermal by 4. driven P cargoes of motion Koumoutsakos Subnanometer (2008) al. D, et A, Barreiro Poulikakos 3. S, Arcidiacono JH, Walther PA, gradient. Schoen temperature a 2. along move molecules Why (2006) D Braun S, Duhr 1. 0 T − rma needn,cutrfe iuainwt h same the with simulation cluster-free independent, an From specific a out carry we case, specific a in result this check To force phoretic The oini abnnntb oscillators. nanotube carbon in motion bons. gradient. temperature a by driven nanotubes. carbon inside confined 71. nanodroplets water of motion Comput Theor Chem J nanotubes. carbon inside nanoparticles 90:253116. gold of motion mophoretic dients. straining. nanotubes. carbon along carbon through transport mass nanotubes. 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(l 20:055708. 1). eq h /l fawv aktof packet wave a of n transmission and A, x z − h momentum The . x c Rep Sci L 1)v saigforce z-shaking oetmis momentum 5n with nm 85 = F ω α q aoLett Nano 2 th plPy Lett Phys Appl 0GHz. 80 = plPy Lett Phys Appl q , 5:13675. .8= 0.28 = ∼ 0pN, 20 rcNatl Proc [10] 9:66– [9] 6 ida ,Bod 21)OtmzdTrofadBenreprclpotential metals. fcc empirical for model nearest-neighbor Brenner Analytic (1988) and R Johnson Tersoff 27. Optimized (2010) D Broido L, Lindsay with 26. Ballistic-to-diffusive transport: nanodroplets heat water Steady-state (2015) of M diffusion few-layer Lundstrom Fast J, supported Maassen (2016) G and 25. Aeppli suspended A, Michaelides in G, in Tocci transport M, transport Thermal Ma thermal (2010) 24. and al. phonons et Flexural Z, (2010) Wang N 23. Mingo D, graphene. Broido of L, conductivity thermal Lindsay electronic The 22. carbon (2016) N nanostructured Marzari and CH, Park graphene TY, Kim of 21. properties Thermal (2011) arXiv:1612.01173v2. AA conductivity. thermal Balandin phonon Non-local 20. (2016) single-layer PB suspended Allen in conductivity 19. thermal Length-dependent (2014) al. et X, Xu monolayer in 18. conductivity thermal Intrinsic (2015) L Collective Colombo graphite: C, and Melis graphene G, of Barbarino conductivity 17. Thermal (2014) al. ribbons. et graphene in G, flow Fugallo heat of 16. crossover diffusive to Ballistic (2013) al. et MH, Bae 15. 9 oiiN agJ azr 21)Aosi hnnlftmsadtemltransport on thermal and nanoclusters lifetimes gold phonon Acoustic of (2012) Diffusion N Marzari (2000) J, JL Garg N, Barrat Bonini N, 29. Combe P, Jensen LJ, Lewis 28. nuh rsoe hc u iuain ontytdtc at detect yet not do simulations our L length which sheet crossover the a once enough, diffusive phonons to flexural ballistic as disappear from eventually evolve will and submicrom- sizes to sheet (52) specific eter al. is here et described Passian thermophoresis ballistic by D and described Gotsmann tips, by observed on force and Knudsen particles the gas include by phoresis exerted ballistic of onto Previously examples scattering transported. by known ballistically momentum is which physical cluster, its hot adsorbed of the from some flowing cedes flux cold phonon to flexural from The away region. is decrease scattering density that the compensating the turn, longitudinal takes a that In to phonon coupling anharmonic density. special their the projected with to heat, associated of owing Besides momentum, increase known. physical abil- well membrane-like carry and are also MFP ballistically phonons long heat flexural whose transport phonons, to flexural ity the are effect this ∇T to length ther- proportional theoretical sheet is the cold. force sizes, to sheet mophoretic hot nanoscale from submicrometer cluster to predicted For adsorbed is an sheet thermophoretically graphene suspended push vacuum a of extremes difference temperature imposed externally An Conclusions otdb uoenCoeaini cec n ehooyAto Grant Action Technology sup- and partly Science also in was Cooperation MP1303. and European MODPHYSFRICT by 320796 ported Grant Advanced Council ballistic– ACKNOWLEDGMENTS. expected the characterize able to be and should crossover. diffusive effect 60) (59, this microbalances detect crystal to quartz or Sensi- microscopes 58) force regime. atomic (57, ballistic “pendulum” inter- so-called the as molecular from such tools of benefit tive also evaluation nanoscale could or and (56)] (55) microscale actions at separation effect particle thermophoretic hot [i.e., current a the Other as molecules. of such or uses source clusters adsorbed heat long- on moving allow (54) a cantilever could by graphene action on noncontact ballis- range, thermophoresis the example, of an character As tic applications. not practical and find remarkable, to is unlikely force ballistic distance-independent a ize xiain n enfe paths. free mean and excitations Commun Nat aaeesfrltiednmc n hnntemltasoti abnnanotubes carbon in transport thermal graphene. phonon and and dynamics lattice for parameters law. Fourier’s graphene. on graphene. graphene. Lett Nano materials. graphene. simulations. atomistic by estimation direct A B Rev limited: Phys upper ultimately is graphene nfe-tnigadsrie graphene. strained and free-standing in graphite. Matter ∼ e vdneo alsi hrss h anaet of agents main The phoresis. ballistic of evidence key a , 150nm 37:3924–3931. hsRvB Rev Phys a Mater Nat 16:2439–2443. aoLett Nano B Rev Phys Commun Nat 91:035416. plPhys Appl J 4:1734. a Mater Nat eo htcosvrsz,tepsiiiyt real- to possibility the size, crossover that Below . L hsRvB Rev Phys n hsidpneto h hra gradient thermal the of independent thus and 61:16084–16090. 10:569–581. 11:113–118. 82:115427. eerhwscnutdudrErpa Research European under conducted was Research 5:3689. 15:66–71. 117:035104. 81:205441. PNAS aoLett Nano | ulse nieAgs ,2017 3, August online Published aoLett Nano rg(3.Tephonon-induced The (53). urig ¨ 14:6109–6114. ∆ 12:2673–2678. 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