Effect of liquid bulk density on thermal resistance at a liquid-solid interface

Transcription

1 Journal of Mechanical Science and Technology 25 (1) (2011) 37~42 DOI /s Effect of liquid bulk density on thermal resistance at a liquid-solid interface Minsub Han * School of Mechanical Engineering and Robotics, University of Incheon, Incheon, , Korea (Manuscript Received October 5, 2010; Revised November 19, 2010; Accepted November 19, 2010) Abstract Thermal boundary resistance (TBR) at a liquid-solid interface that consists of simple Lennard-Jones atoms is studied by molecular dynamics simulation. A liquid layer is confined narrowly between flat solid surfaces in the simulation, and the mean density of the confined liquid is varied. A thermal gradient is imposed in the system by applying constant heat flux, and TBR is calculated using the temperatureprofile results. TBR is highly dependent on the bulk density of liquid when the liquid does not wet well with the solid. The bulk density gives a dominant influence over the liquid-side structure at the interface, which determines TBR in part. In the system of fluid on a high energy solid surface, the wetting system, even when the interfacial structure shows a marked dependence on the density, TBR depends less on the bulk fluid density. Keywords: Micro thermal system; Nanofluids; Thermal conductivity; Wetting Introduction When a thermal system gives the heat generated out to the environment by conduction, the transfer of heat is often more hindered at the interface between different materials in the system than in the bulk material. The greater resistance to the energy transport at the boundaries, so called thermal boundary resistance (TBR), results in a large temperature discrepancy between the neighboring materials. TBR for different solid materials are usually of a considerable size, which is primarily caused by the air gap of incomplete physical contacts. A better quality of contact between solid and liquid materials leads to a smaller resistance, which is not, however, negligible. TBR is therefore one of the basic elements to consider in design of thermal systems and has been under active investigation for more than half a century [1]. Large attention drawn recently on TBR is largely due to a rapidly growing field of the smallscale systems. For example, when we develop a thermal system of a small scale that consists of multi-phases, TBR at the liquid-solid boundary could give a dominant influence on the overall performance of the system. Such systems include nano-composite materials that contain a high density of interface, micro-devices that include nanoscale channels, and the surface scanning devices of a nanoscale resolution such as SThM [2-4]. This paper was presented at the MECT-10, Seoul, Korea, March This paper was recommended for publication in revised form by Guest Editors Seung Jin Song, S. Mochizuki * Corresponding author. Tel.: , Fax.: address: mhan@incheon.ac.kr Theoretical understanding on TBR is initially developed on the continuum ground [1]. One typical theory is acoustic mismatching model (AMM) in which phonons are treated as plane waves on a continuous material, and the interface is modeled as a plane that transmits or reflects the waves selectively. The macroscopic approach usually provides a phenomenological model that produces reliable predictions with enough empirical data, but the model hardly elucidates the fundamental relationship between TBR and the properties of constituent materials. For example, the nature of interaction among the phonon waves in the interfacial region can critically determine TBR, and the structure and dynamics in the region could directly be linked to the interaction. However, the details in the interfacial region are largely missing in the continuum picture. Molecular dynamics simulation (MD) has been increasingly used in overcoming the limitation. Classical MD can track down all the trajectories of the molecules in a material system and provide detailed information on the phonon dynamics on an atomic resolution with a minimum modeling effort [5]. Some progress has been made on characterizing the TBR at fluid-solid interface by MD studies. Maruyama et al. demonstrated that TBR clearly exists at the fluid-solid interface and is strongly dependent on the intermolecular energy [6]. Xue et al. further studied the dependence on interaction energy and revealed two distinctive regimes of heat transfer [7]. Ohara et al. closely examined the energy transfer mechanism between the molecules in the interfacial region according to their relative distance [12]. The interfacial region whose characteristics

2 38 M. Han / Journal of Mechanical Science and Technology 25 (1) (2011) 37~42 are critically related with TBR is of a small dimension and structurally stable because it is based on the balance among the strong intermolecular forces. Whenever circumstances require that the structural states of the interface depart from those of the normal one, TBR could also be affected significantly. The case that has gotten most attention thus far is when there are transfers of momentum as well as energy across the interface. The thin liquid film between solid surfaces under shearing motion was simulated for studying the effect of molecular structure and energetics on momentum and energy transfers. Simple Lennerd-Jones (LJ) fluid, polymeric liquid, and water are considered [8-10]. The main focus of this paper is to study the effect of various liquid states on the interfacial structure and the TBR. In the previous studies by MD on TBR at the liquid-solid interface, less attention is paid to the bulk state of the liquid. They considered the liquid in the state of normal coexistence with solids and paid chief attention on the effect of liquid-solid intermolecular interaction. However, for a number of nanoscale systems, the liquid being confined in a narrow space and departing from the normal saturated state is not unusual. Some of the examples are nanoscale lubricating film, liquids in nanoscale channel, and liquid bridge forming between surface probe microscopes [3, 4, 11]. When the narrowly confined liquids are pressurized, expanded or expanded sufficiently to induce bubble formation, this changes the bulk state and its thermal conductivity. This could also introduce the change of state at the interfacial region and, thus, TBR. The overall qualitative picture of the TBR variations based on the bulk density change and the analysis of their molecular origins are presented. 2. Simulation methods The non-equilibrium molecular dynamics simulation is conducted. The molecule in the present simulation consists of a single atom such as noble gases. The interaction with other molecules is modeled by the truncated and shifted pairwise LJ 12-6 potential. ULJ( r) ULJ( rc) if r < rc U = 0 else (1) 12 6 σ σ ULJ () r 4ε = r r (2) where the value of r c =2.5 is used in the simulation. ε is the interaction energy and σ is the inter-atomic distance. The simulation system is composed of two or three phases. Following the recommendations in [7], the coexisting state of two phases is set up by choosing the interaction energy between solid atoms, ε ss, as ten times as large as that between the fluid atoms, ε ff (i.e., ε ss =10 ε ff ). The interaction energy between fluid and solid atoms and liquid density are varied to set up various configurations. The equations of motions for the atoms are integrated Fig. 1. Schematic of simulation system. Fig. 2. Temperature profile for two-phase and three-phase systems. according to velocity Verlet algorithm [5]. The time step of [ε ff / (m σ ff 2 )] 1/2 is used in the integration. The simulation system is composed of five slabs of coexisting phases, two solids and three fluids (Fig. 1). The calculation of TBR is based on the data at the two interfaces of the liquid slab in the middle, as indicated as liquid 1 in Fig. 1, and the neighboring solids. At every time step, heat is put into the part of the liquid slab in lower boundary, liquid 3. The same amount of energy is subtracted from the atoms in upper liquid slab, liquid 2. As a result, a temperature gradient is maintained across the solid-liquid interface (Fig. 2). The heat in the form of kinetic energy is added to or subtracted from the atoms in a third of the region of each liquid slab by velocity-rescaling method. The rationale for using five slabs of phases rather than the fewer ones is similar to that in Ref. [12]. The source and sink regions using the velocity scaling method can act as an interface where the phonon waves are refracted as well as transmitted, and they must be separated well from the region of the major interest to minimize unintended artificial effects such as the size effect. For example, using three slabs and locating the sources in the solid slabs may cause some problem because the mean free path of the phonon in solid is much larger than that in liquid. One remedy may be the use of a sufficiently thick solid-region. Instead, additional liquid layers for the source regions are used in the present study. The total energy is conserved in the system, which therefore simulates the micro-canonical ensemble. The artificial drift of

3 39 M. Han / Journal of Mechanical Science and Technology 25 (1) (2011) 37~42 (a) (b) (c) Fig. 3. The snapshots for the three-phase systems εls/ εll = (a) 0.447, (b) 1.0, (c) The number of fluid atoms in the middle slab is the source or sink region can result in over-scaling of the kinetic energy, which is avoided by the method suggested in [13]. The total energy in the system is also controlled as constant by scaling the velocities of atoms in the heat source and sink region in every t. The solid slabs are fixed in space by ghost solid atoms. The atoms are fixed in space and interact with neighboring solid atoms by the same LJ potential. However, they do not interact with fluid atoms. Their primary function is to stabilize the simulation system and prevent the liquid 1 between the solid slabs from being compressed or expanded severely due to the states of heat source and sink regions. The periodic boundary condition is applied in the x and y directions. At the boundaries in the z direction, perfectly elastic walls are located to ensure a minimum interaction with the boundaries. In the simulation process, the system is equilibrated during the first t with a temperature of 0.73 (εff /k) and the data for analysis are produced in the next t. At every 20 t, the data are collected and averaged over atoms in the bins which is Ax Ay z in size. Here, Ai is the area of the simulation domain normal to the i direction. z~1.66 is chosen for most of the temperature calculations. This is indicated explicitly whenever another size of the bin is used. The thermal boundary conductivity, htb, can then be obtained by using the given heat transferred, q, and the temperature discontinuity, T, calculated at the interface. htb = q ( A t T ) Fig. 4. Density distribution (ρl=0.709, εfs/εff=3.16, Z=0.1). Fig. 5. Thermal Conductivity dependence on density and temperature. (3) where A is the area across which the heat is transferred. The thermal resistance length, RTB, is an alternative definition regarding TBR and equivalent to the length over which the same amount of temperature drop occurs as in the bulk. RTB can then be obtained as RTB = k htb (4) where k is the thermal conductivity for the bulk material. Vari- Fig. 6. Thermal Resistance Length for the two-phase and three-phase systems. The density at the midplane for the three-phase system is also plotted.

4 40 M. Han / Journal of Mechanical Science and Technology 25 (1) (2011) 37~42 ous amount of the transferred heat is tested for the given simulation system. The value of ε ff is optimally chosen considering the data clarity and temperature range. It is about 2 per cent of the average energy of an atom in the system. 3. Results The simulation cases are categorized into two systems, twophase and three-phase. The three-phase system usually refers to that composed of three distinguishable phases, such as solid, liquid, and vapor. However, as is typical in the narrowly confined geometry, the stable fluid-state of a transitional nature, between liquid and vapor, is also observed. Fig. 3 shows various fluid states, where different values of interaction energy between fluid and solid are used. The vapor phase in Fig. 3(a) is not recognizable clearly, but the density at the mid-plane, z=0, is not of a constant value in the direction of the x coordinate like those in Fig. 3(b), (c). The three-phase system in the present study is therefore defined as that displaying an appreciable change in the density profile at the mid-plane, z=0. The fluid in the mid-plane is less influenced by the solid and prone to start the transition from one phase to the other earlier than the rest in the present geometry. In addition, the results of the three-phase system discussed below require special caution in their interpretation. In the three-phase system, as is shown in Fig. 3, the fluid properties vary in more than one direction. However, the variations of the properties are analyzed regarding only one dimension, that of the z, in this study. This is sufficient for the two-phase system. In the threephase system, the one-dimensional analysis would result in an averaged value of thermal boundary resistance. The analysis may then provide, for example, some global information on the thermal effects of a trapped bubble in a confined fluid. A multi-dimensional analysis on the three-phase system may reveal rich thermal phenomena, such as those involving the contact line, capillary force, disjoining pressure, and so on (See [14] for example.). However, these topics are beyond the scope of the present study. A typical density profile of the two-phase system is shown in Fig. 4. The bin height, z, used in the figure is about 0.1 to give a more refined view of the structure. The solid density distribution, dotted line, displays the periodic nature of a lattice. The fluid density next to the solid also has a periodic peaks and troughs, but the heights of the peaks decrease as the distance from the solid increases. Around the mid-plane at z=0, the structure of peak and trough still remains, but the density reaches a constant mean level. The confined nature of the geometry keeps the density from reaching the macroscopic bulk state of a constant density. The bulk density, ρ l =0.752, of this two-phase system implies the mean density at the midplane averaged with the grid size of the temperature measurement, z=1.66. The saturated density of the truncated and shifted L-J liquid with a cutoff length of 2.5 is about ρ l = at the temperature of This indicates that the fluid in Fig. 4 is under an expanded state. The state results directly from the geometry of confinement and the attraction by the solid in simulation. This kind of the expanded state by the long-range influence of the solid forces can usually be represented by the concept of disjoining pressure, and the scale below which the departure from the saturated state occurs can be estimated from the disjoining pressure data. On the other hand, other end in the results to be discussed, for example, ρ l =0.8901, stands for the state of a compressed fluid. The bulk density of the three-phase system calculated as in the two-phase system, however, also accounts for the vapor phase as well as liquid. The density, ρ l MP, for the three-phase system must be interpreted as the average density of a liquid-vapor mixture at the mid-plane. Better quantifications for the state of the threephase system may exist, such as volume fraction or neck width, but ρ l MP is used for the sake of convenience in comparison with the results of the two-phase system. To calculate the thermal boundary resistance, the data of the temperature discontinuity at the interface and the thermal conductivity at the bulk liquid are needed. The data are obtained from the temperature profiles, two of which are shown in Fig. 2. The profiles cover the whole range of the simulation domain in the z co-ordinate, and they include the layers of liquids where the heat comes in and out as well as the solids. For ρ l =0.709, the temperature slop in the liquid layer between solid layers is almost of a constant value, which indicates that the nature of transport is within the linear Fourier regime. The linear Fourier regime for the Lenneard-Jones liquid is suggested as about (ε ff /k σ ff ) in [15], and all the temperature gradient results are in the same order. The size of slops in the liquid is also far larger than that of the solid layer because of the smaller thermal conductivity. The temperature discontinuities at the liquid-solid interfaces are of an appreciable size and, therefore, so are the thermal boundary resistance. The discontinuity in temperature at the interface exists mainly because there is the acoustic mismatch between the neighboring bulk phases. It is also turned out that the character of interaction among neighboring molecules in the interfacial region can also be critical in transmitting the phonon waves [7]. A temperature profile in the three-phase system is also shown in Fig. 2. In the case of ρ l MP =0.412, the vapor phase exists, and the liquid layer is turned into a bridge between the solid surfaces. The temperature is calculated in the same way as the calculation of density such that it is averaged over the planes parallel to the mid-plane, z=0. The non-linearity in the slop can largely be explained by the variation in the cross-sectional area of the liquid bridge in the z co-ordinate. By using Eq. (4), the thermal resistance length is calculated. Strictly speaking, the temperature discontinuity and the bulk thermal conductivity must be obtained at the same temperature. As is shown in Fig. 2, the variation of the temperature slop in the liquid layer is of a negligible amount. This indicates that, in the two-phase system, the thermal conductivity does not change appreciably within the liquid layer notwithstanding the variations in proximity to the solid and the tem-

5 M. Han / Journal of Mechanical Science and Technology 25 (1) (2011) 37~42 Fig. 7. Thermal resistance length vs. liquid density. Fig. 8. ρl/ρl= (ρ2-ρ1)/ρ1. For εfs/εff = 3.16, ρ1=0.709, ρ2=0.870; for εfs/εff = 0.44, ρ1=0.800, ρ2= perature. This argument agrees with the results in Fig. 5. That is, the thermal conductivity is well correlated with the density, but not with the temperature in the range of the present simulation results. Therefore, the thermal conductivity based on the temperature gradient at the mid-plane is used in the calculation. Fig. 6 shows the thermal resistance length (RTB) versus the relative interaction energy, εsf /εff. The thermal boundary conductivity partly accounts for the size of the thermal conductivity in the bulk. However, the energy transfer at the interface can differ with different interfacial structures of the same bulk thermal conductivity. In this aspect, TBR is represented better with thermal resistance length. As εsf /εff varies, so does the temperature at the interface because only the total heat flux and the temperature in the mid-plane is fixed. The range of the temperature variation is about Regarding the insensitivity of thermal conductivity, and the thermal resistance results, it is assumed that the temperature discontinuity does not vary appreciably in the range. The overall trends are in agreement with a previous study [7]. RTB results show that the energy transfer is quite different for wetting and nonwetting systems. For the systems of a small interaction energy, εsf /εff <1, the resistance is of a large amount and highly varying. On the other hand, those of a larger interaction energy, εsf /εff >1, results in the resistance that is of a small amount and 41 slowly varying. This implies that the interfacial region highly structured by strong liquid-solid interaction is more effective in transporting the energy across the interface. The RTB results for the three-phase system is also obtained from the upper interface between solid and liquid, in which the part of the solid surface in contact with the bubble is still covered with a liquid layer of a few molecular-diameter thickness in all results. RTB for three-phase system has the similar trend with respect to the interaction energy, but differ in value from that for the two-phase system of a weaker interaction. The bulk density is also plotted. Larger interaction energy induces the increase in the local density near the solid surface and, because of the confined geometry, the bulk density decreases. The density decrease induces the bubble formation in the three-phase system. Fig. 7 shows, given the same configuration of materials, the dependence of thermal resistance length on the bulk density. For the cases of strong interaction between solid and liquid, the change in TBR is of a negligible amount. For those of weak interaction, however, TBR varies significantly in the given range of the bulk density. This is congruous to the results in Fig. 2. That is, the energy transfer at the interface in a non-wetting system is more sensitive to the system setup, such as the bulk density and the interaction energy between the materials, than that in wetting system (See also the assessment of the characteristics in the interfacial region in Ref. [12, 16]). This trend is still observed in the three-phase system. Regarding the effect of a bubble inclusion, it is noted that TBR in the three-phase system is close to that in the two-phase when the results for the wetting system is considered. This suggests that the change in the interfacial structure due to the nearby contact line, thus, the vapor phase, does not lead to some significant change in TBR. In the cases of the limiting values of the relative liquid-solid interaction energy of the two-phase system, εsf /εff =0.447 and 3.162, the interfacial density structures change appreciably as the bulk density changes from ρl=0.800 to and ρl=0.709 to 0.870, respectively (Fig. 8). The amount of the change is relatively larger for the higher-energy case. Considering the little variation in TBR for εsf /εff =3.162, this implies that a higher structural change in the interfacial region does not necessarily lead to a larger change in TBR. We can therefore postulate that the improvement margin in the phonon transfer by the solid-like structure of the neighboring liquid is more saturated in the wetting system. 4. Conclusions Non-equilibrium molecular dynamics simulation is conducted for a simple Lennard-Jones system to investigate the influence of the bulk density change on TBR. Some of the findings are summarized below: First, in the non-wetting system, whether it consists of two or three phases, the change in the bulk density can lead to a significant degree of change in TBR. While the bulk thermal

6 42 M. Han / Journal of Mechanical Science and Technology 25 (1) (2011) 37~42 conductivity is proportional to the density, the density dependence of TBR is primarily caused by the structural change in the liquid-side interfacial region. This is congruous with the previous finding that TBR in the non-wetting system is more sensitive to the size of the fluid-solid interaction energy than that in the wetting. In the wetting system, or the system of the lower fluid-solid interaction energy, the variation in TBR is not of an appreciable amount with respect to that in the density. It is so even when the density variation results in a significant degree of structural change in the interface. Finally, these conclusions made on the results of the two-phase system are also valid for the three-phase system. When the fluid is considered as a mixture of liquid and vapor and analyzed in average, TBR shows trends similar to those of the two-phase system. Acknowledgment This work was supported by the University of Incheon Research Grant in References [1] E. T. Swartz and R. O. Pohl, Thermal boundary resistance, Rev. Mod. Phys., 61 (3) [2] J. A. Eastman, S. R. Phillpot, S. U. S. Choi and P. Keblinski, Thermal Transport in Nanofluids, Ann. Rev. Mat. Res., 34 (2004) [3] R. B. Schoch, J. Han and R. Philippe, Transport Phenomena in Nanofluidics, Rev. Mod. Phys., 80 (2008) [4] B. Cretin, S. Gomes, N. Trannoy and P. Vairac, Scanning Thermal Microscopy, in Microscale and Nanoscale Heat Transfer, Topics Appl. Physics, Volz, S. ed., 107 (2007) [5] M. P. Allen and D. J. Tildesley, 1987, Computer Simulation of Liquids, Oxford, 494. [6] S. Maruyama and T. Kimura, A Study on Thermal Resistance over a Solid-Liquid Interface by the Molecular Dynamics Method, Therm. Sci. Eng., 7 (1) (1999) [7] L. Xue, P. Keblinski, S. R. Phillpot, S. U. S. Choi and J. A. Eastman, Two regimes of thermal resistance at a liquid-solid interface, J. Chem. Phys., 118 (1) (2003) [8] T. Ohara and D. Torii, Molecular dynamics study fo thermal phenomena in an ultrathin liquid film sheared between solid surfaces: The influence of the crystal plane on energy and momentum transfer at solid-liquid interfaces, J. Chem. Phys., 122 (2005) [9] D. Torri and T. Ohara, Molecular dynamics study on ultrathin liquid water film sheared between platinum solid walls: Liquid structure and energy and momentum transfer, J. Chem. Phys., 126 (2007) [10] S. Kalpakjian and S. R. Schmid, Manufacturing Processes for Engineering Materials, Second Ed., Addison-Wesley Publishing Company, New York, USA (1992). [11] R. Khare, P. Keblinski and A. Yethiraj, Molecular dynamics simulations of heat and momentum transfer at a solidfluid interface: Relationship between thermal and velocity slip, Int. J. Heat and Mass Transfer, 49 (2006) [12] S. M. Hsu, Nano-lubrication: concept and design, Tribology Int., 37 (7) (2004). [13] T. Ohara and D. Suzuki, Intermolecular Energy Transfer at a Solid-Liquid Interface, Microscale Therm. Eng., 4 (2000) [14] P. Jund and R. Jullien, Molecular-Dynamics calculation of the thermal conductivity of vitreous silica, Phys. Rev. B, 59 (21) [15] M. Han, J. S. Lee, S. Park and Y. K. Choi, Molecular dynamics study of thin film instability and nanostructure formation, Int. J. Heat Mass Transfer, 49 (2006) [16] P. K. Schelling, S. R. Phillpot and P. Keblinski, Comparison of atomic-level simulation methods for computing thermal conductivity, Phys. Rev. B, 65, [17] J.-G. Weng, S. Park, J. R. Lukes and C.-L. Tien, Molecular dynamics investigation of thickness effect on liquid films, J. Chem. Phys., 113 (14) (2000) Minsub Han received PhD in Mechanical Engineering from Northwestern University in 1996, is now working as Assistant Professor in the School of Mechanical Engineering and Robotics, University of Incheon.