Proceedings of IMECE2008 2008 ASME International Mechanical Engineering Congress and Exposition November 2-6, 2008, Boston, Massachusetts, USA Paper No. IMECE2008-69152 INTERFACIAL THERMAL RESISTANCE IN NANOSCALE HEAT TRANSFER Ganesh Balasubramanian Virginia Polytechnic Institute and State University Blacksburg, VA, USA Soumik Banerjee Virginia Polytechnic Institute and State University Blacksburg, VA, USA ABSTRACT We investigate nanoscale thermal transport across a solidfluid interface using molecular dynamics simulations. Cooler fluid argon (Ar) is placed between two heated iron (Fe) walls, thereby imposing a temperature gradient within the system. Fluid-fluid and solid-fluid interactions are modeled with Lennard-Jones potential parameters, while Embedded Atom Method (EAM) is used to describe the interactions between solid molecules. The Fe-Ar interaction causes ordering of fluid molecules into quasi-crystalline layers near the walls. This causes temperature discontinuity between these solid-like Ar molecules and the adjacent fluid. The time evolution of the interfacial (Kapitza) thermal resistance (Rk) and Kapitza length (Lk) are observed. The averaged Kapitza resistance (Rk,av) varies with the initial temperature difference between the wall and the fluid (∆Tw ) as Rk,av ∝ ∆Tw-0.82 . INTRODUCTION Thermal properties of nanostructures differ from the corresponding ones at the continuum since their length scales are comparable to those of the energy carriers at the nanoscale, viz. phonons[1, 2]. The large surface area to volume ratio exerts a strong influence on thermal transport at the molecular level[3, 4]. The structural ordering that occurs due to the attraction of liquid molecules to hydrophilic surfaces enhances the local thermal conductivity during heat transfer across nanoscale solid-liquid interface[5]. A finite temperature discontinuity is observed because of the local intermolecular interactions. Thus, the role of thermal resistance at nanoscale interfaces is much pronounced than at larger scales[6]. P.L. Kapitza[7] was the first to report the existence of this interfacial contact resistance (also known as Kapitza resistance[8], Rk) through his experiments on heat transfer between liquid helium and copper. It is expressed as the ratio of Ishwar K. Puri Virginia Polytechnic Institute and State University Blacksburg, VA, USA temperature drop (∆T) at the interface to the normal heat flux across it Rk = ∆T / q (1) This resistance to thermal transport is also often expressed in terms of a length parameter (Kapitza length, Lk)[9], which represents the width of the bulk medium over which there will be same temperature jump as at the interface Lk = ∆T / (dT/dx) (2) Previous studies have reported impact of Rk and Lk on heat transfer across interfaces at steady state[10, 11]. We attempt to understand the nature of temporal evolution of the resistance and length parameters using calculation of the heat flux from the first principles, instead of the more widely adopted approach using continuum approximation. Further, the dependence of Rk on initial temperature difference between the walls and the fluid (∆Tw) provides some more insight to the role of thermal resistance in nanoscale heat transfer. METHODOLOGY Molecular Dynamics (MD) is a fundamental tool applied to the study of thermal transport at the nanoscale. These atomistic simulations are applicable for systems both in equilibrium, where absence of temperature gradients allow free interactions between atoms, as well as non-equilibrium, such as interfacial heat transfer, where temperature gradients are imposed on the system[10]. An empirical potential function is used to represent atomic interactions in MD simulations. We model the interactions between solid-solid and solid-fluid atoms using Lennard-Jones (LJ) potential parameters (values listed in Table 1)[12], which is represented as, 1 Copyright © 2008 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use ⎡⎛ σ uij = 4ε ij ⎢⎜ ij ⎢⎜⎝ rij ⎣ 12 ⎞ ⎛ σ ij ⎟⎟ − ⎜⎜ ⎠ ⎝ rij ⎞ ⎟⎟ ⎠ 6 ⎤ ⎥ ⎥ ⎦ (3) where, ‘rij’ denotes the scalar distance between sites ‘i’ and ‘j’, ‘σij’ and ‘εij’ are the LJ interaction parameters. Interactions ε (eV) σ (Å) Ar-Ar Ar-Fe 0.0103 0.0516 3.4 3.7 Table 1: Lennard Jones interaction parameters for Fe and Ar The interaction amongst solid atoms has been previously modeled with pair-potential schemes like LJ as well as with harmonic potential functions. These, however, fail to incorporate the many-atom interactions characteristic of solids. Hence, we represent Fe-Fe interactions with a more realistic Embedded Atom Method (EAM) potential[13, 14], expressed as, ⎡ 1 E = ∑ ⎢ Fi ρi + 2 i ⎣ ( ) ⎤ Φij ( Rij ) ⎥ ∑ j ( ≠i ) ⎦ (4) Here, E is the total energy of a system of atoms, the sums are over the atoms ‘i’ and ‘j’, the embedding function ‘Fi’ is the energy to embed an atom of type ‘i’ into the background electron density at site ‘i’, ‘ρi.’ ‘Φij’ is a pair interaction between atoms ‘i’ and ‘j’ whose separation is given by ‘Rij’. The electron density of the solid is considered as a linear superposition of the densities (ρa) of the individual atoms. The simulation configuration as shown in Fig. 1 consists of a 3 nm × 3 nm × 58 nm cuboid that contains two solid Fe walls, each 3 nm × 3 nm × 1 nm in volume. The Fe blocks extending from 0-10 nm and 29-30 nm in the x-direction restrain 3402 atoms of liquid-vapor mixture of Ar (with an initial 33% vapor volume fraction) between them. The fluid is initialized in the form of face centered cubic lattice that equilibrates through simulations. We eliminate wall effects by applying periodic boundary conditions in all three dimensions. A cut-off distance of 10 Å is applied for both Ar-Ar and Fe-Ar interactions modeled by LJ potential parameters. Fig. 1: A 3-dimensional view of the MD simulation domain in which Ar (cyan) fills a space of 28 nm between two 1 nm thick blocks of solid Fe. Fluid atoms on both sides of the walls indicate periodicity. After initializing the system at a temperature of 100K, velocity-rescaling temperature control is applied over all the atoms for the first 300 picoseconds. An initial equilibrium state for a uniform temperature distribution over the entire system is achieved. For the next 700ps, temperature of the solid blocks is controlled and the fluid is allowed to equilibrate. Subsequently, at 1000ps, the Fe walls are provided with a step increase to a higher temperature, which is thereafter maintained with constant velocity rescaling for the remainder of the simulation. Linear momentum is conserved throughout the entire simulation. The simulations are performed with the massively parallel LAMMPS code[15] advanced through successive timesteps of 0.1 femtosecond. RESULTS In order to obtain the spacial distribution of temperature and number density of the Ar atoms when the system attains near steady-state configuration, we divide the fluid domain into several slabs, each 4 Å wide along the x-direction. The solid walls are maintained at a constant temperature of 125K and the density and temperature distribution for the fluid averaged over uniform time intervals is shown in Fig. 2. In accordance with previous literature [6, 10, 16], we observe the tendency of the fluid atoms to migrate towards the heated walls. This localized higher concentration of Ar atoms increases the interfacial pressure so that the fluid atoms behave like a quasi-crystalline solid. Atoms further away from the wall exhibit a vapor behavior. The migration of Ar atoms, few molecular diameters away from the solids, towards the Fe walls thereby creates local vacancies. Fluid atoms that are further removed from the walls are sparsely affected by the intermolecular interaction with the Fe atoms, and hence remain homogeneously distributed. The fluid being essentially a liquid-vapor mixture, phase segregation occurs and causes this inhomogeneous density distribution. 2 Copyright © 2008 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 60 128 Temperature Density Fe 50 124 Temperature(K) 40 30 120 Ar 20 Number of Molecules Fe 116 10 112 0 0 50 100 150 200 250 300 Distance between Iron Walls(Angstrom) Fig. 2: The temperature (dotted lines) and density (solid lines) distributions of the atoms across the x-wise direction after the system is near steady state. The solid Fe walls are located between 0-10 nm and 29-30 nm. temperature gradients (dT/dx) by central difference method on the temperatures computed over the successive fluid slabs where a drop is observed. The temporal variation of Rk and Lk for two different cases of wall temperature are presented in Fig. 3. Both the interfacial thermal resistance and the Kapitza length increase with time, though their natures of increment are different. As the system approaches steady state configuration, the rate of flow of thermal energy across the interface decreases, and as evident from the simulations, this decrease is faster than the corresponding rate of reduction of ∆T. Further, the rate of change in the temperature gradient is steeper at the interface than the corresponding the temperature drop, and hence the observed nature of Lk. However, from the presented results for the two cases, it seems that the rate of change of interfacial temperature gradient and temperature drop remain proportional till a critical time, after which this length shoots upwards from a moderately constant value. 1.0E-06 40 q& sl = ( E& ls − E& sl ) / 2 (5) The total thermal energy transferred from a high temperature wall to the low temperature fluid adjacent to it is q& = ∑s ∑l q& sl Kapitza resistance(m2.K/W) 35 Lk for 125K Lk for 120K Kapitza length (nm) Local equilibrium approximation is applied to obtain the averaged temperature distribution of the fluid domain, also shown in Fig. 2. The densely packed Ar atoms adjacent to the walls have a temperature nearly same as that of the constant temperature solid. The strong attraction of these fluid atoms (extending for about four atomic layers from the solid) to the Fe walls renders them practically motionless, thereby increasing their effective thermal conductivity. A distinct temperature discontinuity is observed at the interface between these quasi-solid atoms and the neighboring vacancies. This can be attributed to the impediment to energy transport across the thinly scattered region of Ar vapor atoms. The typically adopted approach to calculate the heat flux across nanoscale interfaces is by a quasi-continuum approximation using the material bulk thermal conductivity. We have instead determined this flux through a more fundamental first principles method by considering the energy transfer rate[17, 18] from a solid molecule s to a fluid molecule l as Rk for 125K Rk for 120K 30 25 20 1.0E-07 15 10 5 0 1.0E-08 0.5 1 1.5 2 2.5 3 3.5 Time (nanoseconds) Fig. 3: The temporal evolution of the interfacial thermal resistances and Kapitza lengths for two different cases with wall temperatures of 120 and 125 K. The interfacial temperature drop at any instant is calculated as a time average over 100 ps. The dependence of Rk on the wall temperature is also examined. We calculate an averaged Kapitza resistance, Rk.av, using averaged values for ∆T and heat flux over a period of 2000-4000 ps for each simulated case. q& av = (1 / 2000)∑i = 2000 ps q& i 4000 ps (6) (7) This enables us to determine Rk (Equation 1) explicitly from the simulation. To compute Lk, we obtain the spacial 3 Copyright © 2008 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use The results presented in Fig. 4 show the variation of Rk.av with increasing initial temperature difference between the wall and the fluid (∆Tw). Kapitza resistance, m2K/W 6.0E-07 4.0E-07 2.0E-07 Rk,av = (3x10-6)∆Tw-0.82 0.0E+00 5 15 25 35 45 Initial temperature difference, K Fig. 4: Change in the Kapitza resistance with respect to varying initial temperature difference between fluid and wall. The fluxes and temperature drops are averaged over 2000-4000 ps. With increasing wall temperatures, the pressure within the fluid and hence on the ordered layers of Ar atoms increase. This causes more pronounced intermolecular interactions. This leads to enhanced thermal transport across these layers as a result of which resistance to heat transfer at the interfaces reduces. A dependence of the form Rk.av∝ ∆Tw-0.82 is obtained. CONCLUSION In summary, we find that the inhomogeneous density distribution and the non-uniform temperature profile across the fluid domain occur due to phase segregation. The temperature discontinuity between the layers of quasi-crystalline Ar atoms and the adjacent rarified region is caused by hindrance to energy transfer through the sparse population of fluid atoms. Heat flux and temperature gradient decrease at a faster rate than the temperature drop. So, Rk and Lk exhibit an increasing trend with time. A denser packing of the solid-like fluid atoms with increasing wall temperatures enhances heat transfer and the averaged interfacial thermal resistance decreases with ∆Tw obeying a power law relation. ACKNOWLEDGMENTS We thank the Virginia Tech Advanced Research Computing Facility for use of the terascale System X. REFERENCES [1]Poulikakos, D., Arcidiacono, S. and Maruyama, S., 2003, "Molecular Dynamics Simulation in Nanoscale Heat Transfer: A Review", Microscale Thermophysical Engineering, 7(3), pp. 181-206 [2]Cahill, D. G., Ford, W. K., Goodson, K. E., Mahan, G. D., Majumdar, A., Maris, H. J., Merlin, R. and Phillpot, S. R., 2003, "Nanoscale Thermal Transport", Journal of Applied Physics, 93(2), pp. 793-818 [3]Eastman, J. A., Phillpot, S. R., Choi, S. U. S. and Keblinski, P., 2004, "Thermal Transport in Nanofluids", Annual Review of Materials Research, 34, pp. 219-246 [4]Swartz, E. T. and Pohl, R. 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S., Shiomi, J. and Maruyama, S., 2007, "A Study on the Thermal Resistance over Solid-LiquidVapor Interfaces in a Finite-Space by a Molecular Dynamics 4 Copyright © 2008 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Method", International Journal of Thermal Sciences, 46(12), pp. 1203-1210 [17]Ohara, T., 1999, "Contribution of Intermolecular Energy Transfer to Heat Conduction in a Simple Liquid", Journal of Chemical Physics, 111(21), pp. 9667-9672 [18]Ohara, T. and Suzuki, D., 2000, "Intermolecular Energy Transfer at a Solid-Liquid Interface", Microscale Thermophysical Engineering, 4(3), pp. 189-196 5 Copyright © 2008 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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