Heat transfer at small scales
- Fernando Bresme (Imperial College London , United Kingdom)
- Miguel Rubi (University of Barcelona, Spain)
- Simone Wiegand (Forschungszentrum Jülich, Germany)
Understanding and controlling heat transfer is a problem of significant interest in practical applications concerned with heat management, as well as with conversion of heat in other forms of energy. Heat dissipation is a problem in current technologies, e.g., microelectronics, but heat dissipation can also be a source of clean energy, if this waste heat can be transformed using appropriate devices. There are several instances that show how dissipated heat can be converted in other forms of energy, e.g., thermoelectricity, or how the associated temperature gradients can be used to control the motion of colloidal particles or induce separation in multicomponent mixtures, hence, using thermal gradients to rectify the pervasive Brownian motion that becomes relevant a small scales.
Heat transport at small scales, from nano to micrometer lengths, is a generic problem that features in a wide range of disciplines; biology (molecular motors), physics, chemistry and engineering (chemical reactions at surfaces, microelectronic devices, condensation-evaporation processes) or in medical applications (thermal therapy treatments). Working at the micro/nano-scales provides interesting opportunities to develop materials that can exploit heat dissipation to make energy conversion devices, as it is possible to generate very large temperature gradients, e.g, 10^6 K/m, without destroying the material of interest. Such large gradients should produce strong coupling non-equilibrium effects. Hence it is desirable to develop materials that convert heat into other useful forms of energy at these small scales, as well as to control the Brownian motion, with a view to use extend the capabilities of current microfluidic applications.
Molecular machines, e.g, proteins that use light or chemical reactions (ATP) to perform work, also dissipate energy. Specifically, proteins have undergone a long evolution process and optimized energy transfer inside the structure and through the protein-aqueous interface. Similar optimization processes are desirable in the manufacture of manmade materials. At the small scales characteristic of nanomaterials and biomolecules the interfacial thermal conductance becomes a relevant variable in determining heat transport, and must therefore be quantified and understood at a microscopic level. In addition to heat transport through materials, which occurs through phonons or electron transport, thermophoretic effects can be employed to drive transport in solutions or in multicomponent mixtures. This approach is appealing as it provides a route to manipulate and transport small particles and biomolecules in small volume elements. Current challenges involve scaling down this approach to submicrometer scales. The combination of simulations and theory can play an important role to guide these efforts.
The discussion above underlines the need to develop approaches to quantify and understand heat transport at small scales. Computer simulations provide an ideal approach to tackle this problem from a microscopic perspective. The development of computational approaches to investigate heat transfer at small scales involves nonetheless many challenges: the variability in the nature of heat transport, ballistic vs. diffusive, the need to consider spatial and chemical heterogeneity, or the need to cover a wide range of length scales, from nano to micro. The latter is a particularly interesting problem, as it may require the development of coarse grained models that must correctly describe heat transport and the dynamic behavior of the material, if quantitative predictions of heat transport are required.
The aim of this Workshop Proposal is to bring together computer simulators, theoreticians and experimentalists to establish the state of the art on the investigation of the heat transport at small scales, as well as to delineate short-term objectives for the development of computational tools to assist in the design of materials for waste heat conversion, as well as to help in the interpretation of experiments and development of theoretical approaches. Recent computational, theoretical and experimental developments in the study of heat transfer at small scales make our proposal very timely. The Workshop should contribute to transfer simulation and theoretical approaches to the experimental community. With this purpose we have invited a number of experimental, theoretical and simulation groups with expertise in heat transport in a wide range of length scales.
The synthesis of novel materials with nanometer length scales has opened many opportunities to control heat dissipation and conversion of heat into other forms of energy. These experimental developments have produced a new range of materials that can provide solutions to current societal problems, such as the development of novel heat harvesting devices for energy conversion, or for the efficient dissipation of heat in microelectronic or fuel cell applications. With the development of these materials a number of scientific problems have been identified . One major issue is connected to the prevalent role of interfaces, and the thermal resistivity of these interfaces to heat transfer, which ultimately influences heat dissipation. For macroscopic systems, scientists and engineers use Non Equilibrium Thermodynamics (NET) and finite element approaches to analyse heat transport in terms of a few macroscopic properties. Recent extensions of non-equilibrium thermodynamics to heterogeneous systems have significantly expanded the capabilities of NET to investigate interfaces, which are prevalent in the small scales relevant to the present proposal . At these small scales fluctuations also become relevant, and further extensions of the theory through Mesoscopic NET  have enabled tackling heat transfer problems at small scales. We can mention in this context the motion of artificial motors induced by a temperature gradient and the growth of proteins stimulated by thermal inhomogeneities. The incorporation of the fluctuations needs a probabilistic description, which is provided by that theory that establishes the link between thermodynamics and randomness. Often these theories require previous knowledge of the transport coefficients that can be obtained by microscopic or fluctuating theories. These coefficients can be accessed in some instances from experiments, but when these are not available, computer simulations become a powerful tool to access such information.
It is fair to say that our understanding of the mechanisms controlling heat transport at small scales is still scarce, but there is evidence that the description of heat transport can significantly benefit from a molecular approach . In nanostructured materials featuring solid-solid interfaces, the wavelengths of the phonons taking part in heat transport range from 1-100 nm , i.e., in some cases they are of the same order as the material size (e.g., field effect transistors). This fact has exposed the limitations of macroscopic approaches in the description of heat transport at these length scales, which need to be understood and solved. One difficultly in developing new theories is the lack of data at the nanoscale. This problem is connected to the difficulties in measuring the trends of thermal transport with size and chemical composition in nanoscale materials. Computer simulations and numerical approaches represent a viable alternative to attain this knowledge. The Boltzmann Transport Equation (BTE) has been used with this purpose before.[7,8] Unlike BTE, computer simulations do not require a priori knowledge of phonon behaviour in the nanostructure. The only information needed is an accurate force-field to compute the interatomic interactions. Several non-equilibrium simulation approaches have been developed over the years, [9-17] which explicitly model thermal gradients. These approaches have been employed to investigate heat transfer in liquids and solids, as well as across interfaces [2,11,14,15] and in biomolecules . Solids introduce complications due to the finite size of the simulation cells and the truncation of the long wavelength phonons contributing to heat transfer. Corrections have been proposed and comparisons between the direct and Green Kubo (GK) methods have been reported . Recent investigations using the GK approach in combination with the BTE have shown that the thermal conductivity strongly depends on the surface structure of nanomaterials . The relevance of this finding to thermoelectricity has also been addressed . Non-equilibrium algorithms have also been employed to quantify the thermal conductance of organic and biological interfaces [15,21-23], showing some evidence for a dependence of the thermal conductance with curvature and substrate roughness. Equilibrium methods based on GK have also been used to estimate thermal conductance [24,25]. The agreement between GK and direct methods may depend on the convergence of the GK integrals, and differences between these two approaches have been noted before . Hence, further work is needed to establish a computationally robust approach. The development of such approach will benefit from discussions between simulators, theoreticians and experimentalists. In addition to these interfacial effects, recent experimental and simulation works have shown nanoscale rectification effects at interfaces, namely, the interfacial conductance depends on the direction of heat flow [23,27,28], opening many opportunities to design “thermal diodes” with potential uses in thermal management problems, e.g., microelectronics. Recently it has been shown that by coating a nanotube with a molecular layer that is thicker on one end than the other, it is possible to make a thermal rectifier that allows heat to flow easily along the tube in one direction, but not so easily in the opposite direction .
One interesting aspect of working at small scales is that it is possible to generate very large thermal gradients, 10^6 K/m using optical tweezers, or 10^8 K/m using metallic nanoparticles [30,31]. These gradients involve temperature differences of a few degrees in a nanometer of micrometer length scale, and are strong enough to induce a number of non-equilibrium coupling effects. The large heat flux associated to these gradients can induce local phase transitions, e.g., liquid-solid. In the case of flat interfaces such fluxes would promote the formation of insulating vapour films at interfaces, but the curvature inherent to nanoscale interfaces can inhibit the formation of such films, showing the relevance of the length scale in determining the non-equilibrium response of the material . These large temperature gradients can be used to transport particles a phenomenon known as thermophoresis or Soret effect. This technique offers an alternative over other approaches, e.g. electrophoresis, where electrochemical reactions might be unavoidable, or optical tweezers, which are limited to larger length scales, ~micro-meter. Thermophoresis is a physical phenomenon that has been known for many years , but its microscopic explanation is still a subject of debate. The Soret effect provides a physical basis for applications in thermoelectricity, as well as novel phenomena such as thermal orientation [34-38]. But some questions are still outstanding. While there is now some understanding for diluted solutions there are many open questions in the area such as the Soret effect at high concentrations, in crowded media and also the cross effects in multicomponent systems. Overall, there is a need to combine simulations, theory and experiments to advance in scaling down this approach to submicrometer scales , and in this way devise approaches for the thermal manipulation and assembly of small materials.
 D.G. Cahill, W.K. Ford, K.E. Goodson, G.D. Mahan, H.J. Maris, R. Merlin, S.R. Phillpot, Nanoscale thermal transport, Appl. Phys. Rev., 93, 793 (2003).
 S. Kjelstrup and D. Bedeaux, Non-Equilibrium Thermodynamics of Hetergeneous Systems, World Scientific (2008).
 D. Reguera, J.M. Rubi and J.M.G. Vilar, The mesoscopic dynamics of thermodynamics systems, J. Phys. Chem. B, 109, 21502 (2005).
 G. Chen, Nanoscale Energy transport and conversion, Oxford U Press (2005).
 Microscale and Nanoscale Heat Transfer (107 Topics in Applied Physics), Editor S. Volz, Springer Verlag Berlin Heidelberg 2007.
 D.G. Cahill, K.E. Goodson KE and A. Majumdar, J. Heat Transfer, 124, 223 (2002).
 A. Majumdar, J. Heat Transfer, 115, 7 (1993).
 D. Lemonnier, Chapter 5 in Microsale and Nanoscale Heat transfer, Topics Appl. Phys., 107 (Springer, Berlin, Heidelberg, 2007) pp. 77-106.
 D. Macgowan and D.J. Evans, Heat and Mass transport in binary liquid mixtures, Phys. Rev. A, 34, 2133 (1986).
 M. Tesch and K. Schulten, A simulated cooling process for proteins, Chem. Phys. Lett., 169, 97 (1990).
 T. Ikeshoji and B. Hafskjold, Non equilibrium molecular-dynamics calculations of heat conduction in liquid and through liquid-gas interface, Mol. Phys., 81, 251 (1994).
 F. Bresme, B. Hafskjold and I. Wold, Non-equilibirum molecular dynamics study if heat conduction in ionic systems, J. Phys. Chem., 100, 1879 (1996).
 D. Reith and F. Muller-Plathe, On the nature if thermal diffusion in binary Lennard-Jones liquids, J. Chem. Phys., 112, 2436 (2000).
 F. Bresme, Equilibrium and non-equilibrium molecular dynamics simulations of the central force model of water, J. Chem. Phys., 115, 7564 (2001).
 H. Patel, S. Garde and P. Keblinski, Thermal resistance of nanoscopic liquid-liquid interfaces: dependence on chemistry and molecular architecture, Nano Lett., 5, 2225 (2005).
 X. Yu and D.M. Leitner, Heat Flow in proteins: computation of thermal transport coefficients, J. Chem. Phys. , 122, 054902 (2005).
 C. Liu and Z. Li, Molecular Dynamics of composite nanochannels as nanopumps driven by symmetric temperature gradients, Phys. Rev. Lett., 105, 174501 (2010).
 P.K. Schelling, S.R. Philpot and P. Keblinksi, title, Phys. Rev. B, 65, 144306 (2002).
 D. Donadio and G. Galli, Atomistic simulations of heat transport in silicon nanowires, Phys. Rev. Lett., 102, 195901 (2009).
 G. Galli and D. Donadio, Thermoelectric materials: silicon stops heat in its tracks, Nature Nanotechnology, 5, 701 (2010).
 A. Lervik, F. Bresme, S. Kjesltrup, D. Bedeaux and J.M. Rubi, Heat transfer in protein-water interfaces, Phys. Chem. Chem. Phys., 12, 1610 (2010).; A. Lervik, F. Bresme and S. Kjelstrup, Heat transfer in soft nanoscale interfaces: the influence of interface curvature, Soft Matter, 5, 2407 (2009).
 M. Hu, D. Poulikakos, C.P. Grigoropoulos and H. Pan, Recrystallizaion of picosecond laser-melted ZnO nanoparticles in a liquid: a molecular dynamics study, J. Chem. Phys., 132, 164504 (2010).
 H. Acharya, N.J. Mozdzierz, P. Keblinksi and S. Garde, How Chemistry, Nanoscale Roughness, and the Direction of Heat Flow Affect Thermal Conductance of solid-water interfaces, Industrial & Engineering Chemistry Research, 51, 1767 (2012).
 J.L. Barrat and F. Chiaruttini, Kapitza Resistance at the liquid-solute interface, Mol. Phys., 101, 1605 (2003).
 G. Domingues, S. Volz, K. Joulian and J.J. Greffet, Heat transfer between two nanoparticles through near field interaction, Phys. Rev. Lett., 94, 085901 (2005).
 K. Termentzidis, S. Merabia, P. Chantrenne and P. Keblinkski, Cross-place thermal conductivity of superlattices with rough interfaces using equilibrium and non-equilibrium molecular dynamics, Int. J. Heat and Mass transfer, 54, 2014 (2011).
C.W. Chang, D. Okawa, A. Majumdar and A. Zettl, Solid State Thermal Rectifier, Science, 314, 1121 (2006).
 M. Hu, J. Goicoechea, B. Michel and D. Poulikakos, Thermal Rectification at water/functionalized silica interfaces, Appl. Phys. Lett., 95, 151903 (2009).
 G. Casati, Device physics: The heat is on — and off , Nature Nanotechnology 2, 23 (2007)
 H.R. Jiang, H. Wada, N. Yoshinaga and M. Sano, Manipulation of colloids by a nonequilibrium depletion force in a temperature gradient, Phys Rev. Lett., 102, 208301 (2009) ; S. Duhr and D. Braun, Thermophoretic depletion follows Boltzmann distribution, Phys. Rev. Lett., 96, 168301 (2006).
 A.O. Govorov, W. Zhang, T. Skeini, H. Richardson and J. Lee, Nanoscale Research Letters, 1, 84 (2006).
S. Merabia, S. Shenogin, L. Joly, P. Keblinksi and K.L. Barrat, Heat transfer from nanoparticles: a corresponding state analysis, PNAS, 106, 15113 (2009).
 W. Köhler and S. Wiegand, in Thermal Nonequilibrium Phenomena in Fluid Mixtures, edited by W. Koehler and S. Wiegand, Lecture Notes in Physics Vol. 584 (Springer, Berlin/Heidelberg, 2002).
 B. Hafskjold, in Thermal Nonequilibrium Phenomena in Fluid Mixtures, edited by W. Koehler and S. Wiegand, Lecture Notes in Physics, Vol. 584 (Springer, Berlin/ Heidelberg, 2002), pp. 3–23.
 M. Braibanti, D. Vigolo and R. Piazza, Does thermophoretic mobility depend on particle size?, Phys. Rev. Lett., 100, 108303 (2008).
 F.M. Weinert, C.B. Mast and D. Braun, Optical fluid and biomolecule transport with thermal fields, Phys. Chem. Chem. Phys., 13, 9918 (2011).
 A. Wuerger, Transport in Charged Colloids Driven by Thermoelectricity, Phys. Rev. Lett., 101, 108302 (2008)
 F. Bresme, A. Lervik, D. Bedeaux, S. Kjesltrup, Water Polarization under Thermal gradients, Phys. Rev. Lett., 101, 020602 (2008); F. Roemer, F. Bresme, J. Muscatello, D. Bedeaux and J.M. Rubi, Thermomolecular orientation of non polar fluids, Phys. Rev. Lett., 108, 105901 (2012)