Ballistic thermophoresis of adsorbates on PNAS PLUS free-standing graphene Emanuele Panizona, Roberto Guerraa,b, and Erio Tosattia,c,d,1 aInternational School for Advanced Studies, 34136 Trieste, Italy; bCenter for Complexity and Biosystems, Department of Physics, University of Milan, 20133 Milan, Italy; cThe Abdus Salam International Centre for Theoretical Physics, 34151 Trieste, Italy; and dConsiglio Nazionale delle Ricerche–Istituto Officina dei Materiali, Democritos National Laboratory, 34136 Trieste, Italy Contributed by Erio Tosatti, July 6, 2017 (sent for review May 16, 2017; reviewed by Luciano Colombo and Bernd Gotsmann) The textbook thermophoretic force which acts on a body in a To begin with, in a macroscopic system, the heat current den- fluid is proportional to the local temperature gradient. The same sity J is proportional to the temperature gradient (Fourier’s law) is expected to hold for the macroscopic drift behavior of a dif- through the thermal conductivity κ: fusive cluster or molecule physisorbed on a solid surface. The J = −κrT : [1] question we explore here is whether that is still valid on a 2D membrane such as graphene at short sheet length. By means of Existing discussions of adsorbate thermophoresis similarly a nonequilibrium molecular dynamics study of a test system—a assume that the adsorbate mass current will similarly be propor- gold nanocluster adsorbed on free-standing graphene clamped tional to the temperature gradient between two temperatures ∆T apart—we find a phoretic force which for submicron sheet lengths is parallel to, but basically Ja = −K rT ; [2] independent of, the local gradient magnitude. This identifies a thermophoretic regime that is ballistic rather than diffusive, per- where K > 0 now also depends on adsorbate, substrate, and sisting up to and beyond a 100-nanometer sheet length. Analy- temperature—but not on its gradient. According to Eqs. 1 and sis shows that the phoretic force is due to the flexural phonons, 2, systems with different substrate lengths L but the same gradi- ent rT should yield the same heat current and the same adsor- whose flow is known to be ballistic and distance-independent up SCIENCES to relatively long mean-free paths. However, ordinary harmonic bate current—that is, κ and K should not depend on L. These phonons should only carry crystal momentum and, while imping- macroscopic expectations will of course be necessarily borne out APPLIED PHYSICAL ing on the cluster, should not be able to impress real momentum. for sufficiently large systems where transport is diffusive. We show that graphene and other membrane-like monolay- However, for system sizes smaller than or comparable with the ers support a specific anharmonic connection between the flex- mean-free paths (MFP) λi of the heat carriers (phonons in our ural corrugation and longitudinal phonons whose fast escape case), the heat transport may instead turn from diffusive to bal- leaves behind a 2D-projected mass density increase endowing listic, where phonons with MFP λi larger than L carry the heat. the flexural phonons, as they move with their group velocity, In graphene, λi can extend to hundreds of nanometers (13) or with real momentum, part of which is transmitted to the adsor- even more (14), making Eq. 1 invalid at the nanoscale. Ballistic bate through scattering. The resulting distance-independent bal- heat transport along with ballistic-diffusive crossover have been listic thermophoretic force is not unlikely to possess practical discussed experimentally (15) and theoretically (16, 17). Collec- applications. tive phonon excitations have also been proposed (16) with MFPs possibly larger than those of single phonons, which might render the pure diffusive regime of Eq. 1 only attainable with samples thermophoresis j graphene j ballistic j flexural phonons j heat transport of size 0.1–1 mm (18). To describe the small size regime, one can phenomenologically introduce a nonlocal generalization of Eq. 1 hermophoresis is the phenomenon by which a body im- Tmersed in a fluid endowed with a temperature gradient expe- Significance riences a force and, independent of convection, drifts from hot to cold (1). We address here the less common case of ther- In thermophoresis, temperature gradients in a fluid cause a mophoresis of a physisorbed nanoobject caused by an in-plane proportional force on a body. Reasonably, a small physisorbed temperature imbalance in the underlying solid substrate surface. cluster on a membrane-like system such as a graphene sheet Recent years have seen a surge of interest for methods to control suspended between temperatures ∆T apart should do just nanoscale transport and manipulation, also in view of potential that—except it doesn’t. Simulations show for submicrometer applications in nanodevices. The possibility to drive directional sheet length a phoretic force proportional to ∆T but indepen- motion of adsorbates by means of thermal gradients is interest- dent of length and thus of gradient, disclosing a regime of ing and has been explored both theoretically (2) and experimen- ballistic thermophoresis. The soft flexural phonons, ballistic in tally (3). By a similar principle, the controlled directional motion this regime, are responsible. The anharmonic mechanism by on graphene was also explored by means of strain or wettabil- which they carry real momentum, some of which is given to ity gradients (4–6). Carbon systems such as graphene and car- the adsorbate, involves longitudinal phonons that permit a bon nanotubes (CNTs) are prime candidate substrates (2, 3, 7– mass accumulation moving with the flexural group velocity. 11) for these phoretic phenomena, owing to their remarkable The distance independence of ballistic thermophoretic force mechanical strength and thermal conductivity. Computational could be important in nanomanipulations. studies have highlighted the possibility to drive thermally gold nanoparticles, water clusters, graphene nanoflakes, C60 clusters, Author contributions: E.P. and E.T. designed research; E.P., R.G., and E.T. performed and small CNTs over graphene layers or inside CNTs. Despite research; E.P., R.G., and E.T. analyzed data; and E.P. and E.T. wrote the paper. that, there appears to be so far insufficient intimate understand- Reviewers: L.C., Universita` di Cagliari; and B.G., IBM Research–Zurich. ing of that phenomenon, besides the obvious consensus that The authors declare no conflict of interest. the driving force stems from the spatial nonuniformity of the 1To whom correspondence should be addressed. Email: [email protected]. phonon population (3, 8, 12). Our scope will be to deepen that This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. understanding. 1073/pnas.1708098114/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1708098114 PNAS j Published online August 3, 2017 j E7035–E7044 Downloaded by guest on October 1, 2021 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).