International Journal of Heat and Mass Transfer

International Journal of Heat and Mass Transfer 61 (2013) 287 292 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Diffusive-ballistic heat transport in thin films using energy conserving dissipative particle dynamics Toru Yamada a,, Sina Hamian b, Bengt Sundén a, Keunhan Park b,, Mohammad Faghri b a Department of Energy Sciences, Division of Heat Transfer, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden b Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI 02881, USA article info abstract Article history: Received 27 December 2012 Received in revised form 29 January 2013 Accepted 1 February 2013 Available online 1 March 2013 Keywords: Nanoscale Thin film Heat transport Temperature jump Energy conserving dissipative particle dynamics Diffusive-ballistic heat transport in thin films was simulated using energy conserving dissipative particle dynamics (DPDe). The solution domain was considered to be two-dimensional with DPD particles distributed uniformly under constant temperature boundary conditions at the top and bottom walls and periodic boundaries at the side walls. The effects of phonon mean free path were incorporated by its relation to the cutoff radius of energy interaction. This cutoff radius was based on the Knudsen number using the existing phonon-boundary scattering models. The simulations for 0.1 < Kn < 10 were obtained with the different modifications of the cutoff radius. The results were presented in form of a nondimensional temperature profile across the thin film and were compared with the semi-analytical solution of the equation of phonon radiative transport (EPRT). When the phonon-boundary scattering is not considered, the DPDe simulation results have more discrepancies compared with the EPRT solution as Kn increases, indicating that the phonon-boundary scattering plays an important role when the heat transport across the film becomes more ballistic. The results demonstrate that the DPDe can simulate the diffusive-ballistic heat transport for a broad range of Kn, but phonon-boundary scattering should be considered for the accurate simulation of the ballistic heat transport. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Micro/nano structures, such as thin films, nanotubes and nanowires, have been widely utilized for a number of electronics, micro- and nano- electromechanical systems (MEMS and NEMS) and molecular-scale microscopic devices [1]. The size of these structures are in the order of less than a micrometer. In such small scales, the heat transport is quite different from the macro-scale continuum assumption (e.g., Fourier s law) due to the small characteristic length that is comparable to or smaller than the mean free path of the energy carriers. Therefore, it is imperative to understand the detailed phenomena of heat transfer at micro/nano scale for the further development of these devices. For dielectric materials and semiconductors, the heat transfer is dominated by lattice vibration. The quanta of the vibration energy are called phonons. Phonon-based heat transport is crucially important when the phonon mean free path becomes comparable to the characteristic length of the micro/nano structures. Joshi and Majumdar [2] derived the equation of phonon radiative transfer (EPRT) from the Boltzmann transport equation (BTE) to study the Corresponding authors. Tel.: +46 462228604; fax: +46 462224717 (T. Yamada); Tel.: +1 4018745184; fax: +1 4018742355 (K. Park). E-mail addresses: Toru.Yamada@energy.lth.se (T. Yamada), kpark@egr.uri.edu (K. Park). heat transport in diamond thin films of different film thicknesses. The EPRT was numerically solved using the explicit upstream differencing method. The results showed that the heat transport in thin films is not purely diffusive and becomes partially ballistic when the thickness becomes comparable to the phonon mean free path. Chen [3,4] modified the BTE with the gray relaxation time approximation and derived the ballistic-diffusive equation by dividing the phonon distribution function (or the phonon intensity) into ballistic and diffusive parts. This ballistic-diffusive approximation is much simpler than BTE. However, the results showed that this model is not accurate when the diffusive heat transport is dominant, i.e., the phonon mean free path is smaller than the film thickness. In addition to these studies, other analytical [1,5] and numerical investigations [6 11] showed similar trends in nanotubes and nanowires. However, most analyticial studies to date have been limited to simple one-dimensional nanostructures, such as thin films and wires, to be tractable to analytical and numerical solutions and physical interpretation. For most numerical studies, molecular dynamics (MD) has been employed since it is flexible in terms of the degree of freedom for computation domain and in dealing with geometrical complexities. However, MD is an extremely expensive numerical simulation and hence can hardly be applied for simulations at meso- or macroscales. For this reason, 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.02.011

288 T. Yamada et al. / International Journal of Heat and Mass Transfer 61 (2013) 287 292 a new method needs to be developed which captures both the applicability to multi-scale domains and requires relatively reasonable computational cost. Dissipative particle dynamics (DPD) is a relatively new particlebased simulation method introduced by Hoogerbrugge and Koelman [12] and has great potential applications to mesoscopic simulations. It is a coarse-grained version of molecular dynamics in which each DPD particle represents a cluster of actual molecules moving in a Lagrangian fashion [13,14]. DPD was originally applied to isothermal systems. However, an energy conserving DPD (DPDe) method [15,16] was developed for non-isothermal problems by adding the energy conservation equation to the DPD simulation. The DPDe scheme has been utilized for the simulations of various heat conduction, forced convection and natural convection scenarios [17 22]. and the results were in good agreement with analytical and continuum-based numerical solutions. There are also some studies investigating the heat transfer in nano-composites and nanofluids using DPDe [23,24]. However, these models are based on the continuum-based assumption. There has been no previous study to date on heat transport in diffusive-ballistic regimes using DPDe. The present article demonstrates the application of DPDe for diffusive-ballistic heat transport in thin films. A methodology will be introduced to model the mean free path of phonon by its relation to the cutoff radius of energy interaction. The effect of phonon-boundary scattering is incorporated by employing the existing analytical models. Benchmark simulations are first performed to examine the effects of computational parameters on the numerical results and computational cost. In order to benchmark this method, steady heat transport in one-dimensional thin films is solved over a wide range of Knudsen numbers and the results are compared with the semi-analytical solutions of EPRT. 2. Methodology 2.1. DPD governing equations While the detailed description of the DPDe for conduction and convection heat transfer simulations can be found in Ref. [22], its formulation will be briefly reviewed in this section. In the DPD method, each DPD particle interacts with the surrounding particles through a set of distance- and velocity-dependent forces and energies within a certain cutoff radius. In the heat conduction problems, the motion of the DPD particles are not considered, and thus the DPD momentum equation is ignored. For the energy conservation in a DPD system, the following energy equation is used [15,16]: C v dt i dt ¼ q ij where T i is temperature. Cv is the heat capacity at constant volume of a DPD particle. It is suggested in Ref. [25] that the dimensionless heat capacity, Cv/k B, at meso-scale is Cv/k B 1.0. q ij is the volumetric inter-particle energy interaction rate between DPD particles and consists of three parts as follows: ð1þ q C ij ¼ X 1 j ij xðr ij Þ 1 T j i i T j q R ij ¼ X a ij xðr ij Þ 1 2 f e ij Dt 1 2 j i As shown in these equations, q C ij is proportional to D(1/T), instead of DT, which is related to the irreversible internal nergy interactions between particles in the absence of random heat fluxes [16]. The weight function x(r ij ) decreases monotonically with the particle particle separation distance and becomes zero beyond the cutoff radius. f e ij is a random number with zero mean and unit variance. Each pair has an antisymmetric value of f e ij and fe ij ¼ fe ji to ensure the energy of the interacting pair of particles is conserved. j ij and a ij are the strengths of the conductive and random heat fluxes. These coefficients are related and satisfy the following equation introduced by Ripoll [17]: j ij ¼ k o k B T 2 eq qffiffiffiffiffiffiffiffiffiffiffiffiffi a ij ¼ 2k B j ij i þ n j j 2k B T eq where = CvT and k o is a constant having positive value that determines the thermal conductivity of DPD particles. T eq is the equilibrium temperature of the system and n j is a constant which can be freely chosen. n j is selected to be 2 in this study as is common in the previous DPDe studies [20,21,23,24]. For all the simulations, the dimensionless heat capacity, Cv/k B, and k o were chosen to be 1.0 10 5 and 1.26 10 4, respectively. These are the same values used in the benchmark simulations of one-dimensional unsteady heat conduction problems performed in our previous study [22] which were compared with the analytical and other DPDe studies. 2.2. DPD application for diffusive-ballistic heat transport For the DPDe simulation of the diffusive-ballistic heat transport in thin films, the solution domain is set to be a two-dimensional lattice with L and N cells in x- and y-directions, respectively, as seen in Fig. 1. A cell is a square having a unit length, a. DPD particles are distributed uniformly in this domain and enclosed within the walls in extra layers outside of the domain having the constant temperatures T H and T C at the top and bottom of the walls, respectively. The number of wall layers was adjusted based on the cutoff radius of particle energy interaction. A periodic boundary condition was imposed in the x-direction, such that the simulations were considered to be one-dimensional. The initial temperature of the DPD particles for the simulations was set to be 1.0, and T H and T C were chosen to be 1.1 and 0.9, respectively, in DPD units. The following section presents a detailed method that incorporates the effects of phonon mean free path into the DPD system by relating the length to the cutoff radius of energy interaction. ð3þ ð4þ ð5þ ð6þ q ij ¼ q V ij þ qc ij þ qr ij ð2þ The change in mechanical energy causes viscous heating and is indicated in the form of q V ij, which is ignored in the solid heat conduction because of no motion of the particles. The change in internal energy consists of two terms: one corresponds to the heat transport due to a temperature difference q C ij, and the other takes into account the random heat flux caused by thermal fluctuation q R ij. q C ij, and qr ij are expressed as [20,21] Fig. 1. Schematic of the boundary conditions of 1D heat conduction simulations.

T. Yamada et al. / International Journal of Heat and Mass Transfer 61 (2013) 287 292 289 2.2.1. Relating phonon mean free path to DPD cutoff radius Phonon mean free path is related to the DPD cutoff radius through Knudsen numbers, Kn, in both DPD and physical units. The Knudsen number is a dimensionless parameter defined as the ratio of the mean free path to the characteristic length, which is expressed as: Kn ¼ k=h where k is the phonon mean free path and H is the thickness of thin film. This value in physical units is equated with that in DPD units as described in the following equations: ðknþ R ¼ðKnÞ D ¼ k D H D where the respective subscripts R and D are the nomenclatures for Real and DPD, respectively. H D is calculated by the number of cells in y-direction, N, multiplied by the unit length of the cell, a, to yield H D ¼ N a Solving Eq. (8) for k D, k D ¼ðKnÞ R N a ð7þ ð8þ ð9þ ð10þ In order to reflect both ballistic and diffusive features in the DPDe simulation, the weight function, x(r ij ), and the cuttoff radius, r c, need to be carefully determined. Thus, we employ the following exponential weight function [26]: r ij xðr ij Þ¼ eb rc e b ð11þ 1 e b where b is an adjustable parameter. It should be noted that the weight function determines the probability of direct energy interactions between particles. Thus, in Fig. 2, the weight function has a plateau close to unity in the short r ij range, suggesting that most of the energy will be transferred directly, or ballistically, between DPD particle pairs. As r ij increases, however, the energy transport becomes more diffusive owing to the decrease of the weight function. The DPD mean free path, k D, is approximated as the interparticle distance at which the weight function becomes 0.99 to yield the following equation: k D ¼ r ij j x¼0:99 ¼ C 0:99 r c ð12þ where r c is the cutoff radius at which the weight function becomes zero, and C 0.99 is a constant (<1.0) depending on the value of b. Substituting Eq. (12) into Eq. (10), r c can be calculated using the following equation. Fig. 2. Exponential weighting function [26]. r c ¼ NðKnÞ R C 0:99 a ð13þ In this study, the cell unit length, a, was chosen to be unity for all the simulations. 2.2.2. Phonon-boundary scattering effect The heat transport in dielectric and semiconductor materials is governed by different phonon-related scattering mechanisms, such as phonon phonon, phonon-electron, phonon-impurity, and phonon-boundary scattering [1]. When the characteristic length (film thickness) is larger than the phonon mean free path, the intrinsic phonon scatterings (i.e., phonon phonon, phonon-electron, and phonon-impurity scatterings) are dominant and implicitly considered in the shape of the weight function in Eq. (11), which shows a more diffusive feature as the inter-particle distance increases. However, when the film thickness is comparable to or smaller than the phonon mean free path, the phonon-boundary scattering becomes important [1] and should be integrated in the DPDe simulation accordingly. Several analytical models that describe the effect of phononboundary scattering on the phonon mean free path have been developed [27 30] and well summarized in Ref. [1]. The first model uses the same concept for the thin-film electrical conductivity introduced by Fuchs and Wills [27], which is expressed as follows: k eff k b ¼ k 1 eff Kn ¼ 1 þ ð14þ k b lnðknþþ1 where the subscriptions, eff and b, refer to effective and bulk, respectively. This model does not take into account the direction of phonon transport. Filk and Tuen [28] included a weighted average of the mean free path components to capture the anisotropy of the mean free path. According to their method, the effective phonon mean free path for the normal direction to thin films is described as: k 1 eff Kn ¼ 1 þ ð15þ k b 2 Kn 1 The above two equations are applicable for Kn > 5. For Kn < 1, the following equation can be used [28,29]: k eff ¼ 1 þ Kn 1 ð16þ k b m where m = 3 is used for thin films. While the above equations are derived using Matthiessen s rule, a different effective phonon mean free path model can be derived from the BTE under the relaxation time approximation [30]. Since the model is originally for the electrical conductivity of thin film metals, it was modified for the thermal conductivity of thin films [1], which is expressed as follows: k eff k b 3ð1 pþkn ¼ 1 2 Z 1 1 1 t 1 1 e t=kn 3 t 5 dt ð17þ 1 pe t=kn where p represents the probability ratio of diffusive scattering to elastic scattering of phonon at the boundary. Typically, p is assumed to be zero since the wall interfaces were considered to be diffusive (blackbody) in the previous analytical study on EPRT [2]. In Ref [1], it is written that p depends on the surface roughness and there is an equation to obtain its value. p was set to be zero because the analytical study in Ref. [2] employed p =0. Since the phonon mean free path in DPD units is proportional to the cutoff radius as shown in Eq. (12), the effective cutoff radius due to the phonon-boundary scattering can be calculated using the relation of r c,eff /r c,b = k eff /k b. DPDe simulations using different effective cutoff radii were performed, and the results are shown in the next section.

290 T. Yamada et al. / International Journal of Heat and Mass Transfer 61 (2013) 287 292 3. Results and discussion The steady heat conduction between two walls with the constant temperatures of T H and T C was simulated to benchmark the above methodology. The results of the DPD simulations are shown in form of the temperature profiles along the thin film. The aforementioned models for the phonon-boundary scattering effect are considered, and the results are compared with the semi-analytical solution of the one-dimensional EPRT, which is solved under the gray relaxation time approximation and the diffusive boundaries at the top and bottom walls [2]. 3.1. Preliminary simulations The effects of number density and the number of cells in the y- direction on the simulation results were studied because these values could significantly increase the computational time. The number density, n d, is defined as n d = N m /N c, where N m and N c are the total numbers of DPD particles and cells in the solution domain, respectively. The simulations for different n d were obtained at first and the results were compared. Fig. 3 shows the dimensionless temperature profile, i.e., (T T C )/(T H T C ), for different number densities, n d = 4 and 9, when r c = 3.0 and N = 10 (or Kn = 0.16). As can be seen in this figure, the local temperatures for different n d are in good agreement. Fig. 4 depicts the dimensionless temperature profiles for N = 10 and 30, when n d = 4. For both simulations, the Knudsen number were fixed at Kn = 0.21, which changes the cutoff radius as shown in Eq. (13). This figure demonstrates that the number of cells can be reduced with almost no effect on the accuracy and stability of the solution, resulting in a significant p reduction in the computational cost. We defined N ffiffiffiffiffi n d as the resolution of the temperature distribution for the computation domain, representing the number of measured points available in the y-direction of the solution domain: for example, it is capable of calculating local temperatures at 20 positions in y-direction for N = 10 and n d = 4. The preliminary simulations show that the resolution of 20 is accurate enough, and thus it might be possible to obtain the same results for less cells and higher particle density. As shown in Figs. 3 and 4, the temperature profiles do not agree with the Fourier s law and obvious temperature jumps are observed near the walls, which is discussed in the next section. 3.2. Steady heat transport in thin films The results for different Knudsen numbers are shown in this section. In each simulation, the number of cells in x-direction, L, was chosen to be at least twice of the r c, while that in y-direction, Fig. 4. Dimensionless temperature profiles across the solution domain for the same Knudsen number and different number of the cells in y direction, N (n d = 4). N, fixed to be 10 based on the preliminary simulations. The number density, n d, was set as 4. The simulations were conducted for Kn = 0.1, 0.6, 1.0, 3.0 and 10 and the cutoff radius was modified using Eqs. (14) (17), which is shown in Table 1. It is clear from the table that the phonon-boundary scattering significantly reduces the cutoff radius for large Knudsen numbers, indicating that as the heat conduction becomes more ballistic the phonon-boundary scattering becomes more dominant to reduce the effective phonon mean free path. Fig. 5 shows the comparison of the dimensionless temperature profiles with different phonon-boundary scattering models for Kn = (a) 0.1, (b) 0.6 and (c) 1.0. For Kn = 0.1, the results for all simulations agree well with the EPRT result. k eff /k b was calculated to be 0.97 and 0.96 using Eqs. (16) and (17), respectively, and thus the mean free path is not significantly influenced by the phononboundary scattering. However, as the Knudsen number increases, more discrepancies are observed between the DPDe and EPRT results. Since the EPRT depicts the diffusive-ballistic heat conduction semi-analytically under the gray relaxation time approximation [2], Fig. 5(b) and (c) indicate that if boundary scattering is not considered, the DPDe over-predicts the ballistic feature of heat conduction. The boundary scattering of phonons needs to be considered to reduce the effective mean free path: the ratios of k are 0.75 and 0.68, respectively, when calculated from Eqs. (16) and (17) for Kn = 1.0. As compared in Table 1 and Fig. 5, Eq. (17) predicts a smaller effective mean free path and cutoff radius than Eq. (16), suggesting that Eq. (17) predicts a larger phonon-boundary scattering and provides a better agreement with the EPRT solution when integrated with the DPDe simulation for the Knudsen number up to Kn <1. Fig. 6 (a) and (b) show the comparison of the dimensionless temperature with the different k eff and EPRT for Kn = 3.0 and 10, respectively. For the simulations of the Knudsen numbers larger than 1.0, the k eff (or r c,eff ) was estimated with Eqs. (14), (15) and (17). For the simulations for r c,eff > 30, the energy interactions in x-direction were reduced due to large computation time. Some Table 1 The effective cutoff radius corresponding to each Kn. Kn r c,eff Eq. (13) Eq. (14) Eq. (15) Eq. (16) Eq. (17) Fig. 3. Dimensionless temperature profiles across the solution domain for different number density of DPD particle, n d (r c = 3.0, N = 10). 0.1 1.85 1.79 1.78 0.6 11.11 9.26 8.75 1.0 18.52 13.89 12.66 3.0 55.56 22.87 19.84 23.81 10 185.2 45.98 29.57 38.73

T. Yamada et al. / International Journal of Heat and Mass Transfer 61 (2013) 287 292 291 Fig. 5. Dimensionless temperature profiles across thin films for Kn 6 1.0. (a) Kn = 0.1, (b) Kn = 0.6 and (c) Kn = 1.0. simulations were performed for r c,eff < 30 with the energy interaction limited within 2a in x-direction. The results showed the maximum deviation of 7% compared to those without this reduction. In Fig. 6(a), the dotted lines show the ±7% of the DPD results with no boundary scattering modification to depict the possible error by reducing the energy interaction in x-direction. As can be seen in this figure, the trend of the results are very similar to those for lower Knudsen numbers. It is also observed that the phonon-boundary scattering effect is larger than that of lower Kn, while the possible computational error has to be considered due to the reduced energy interaction in x-direction. In Fig. 6(b), the dotted lines show the ±7% of the DPD results including the boundary scattering effect using Eq. (17). Since all DPDe simulations in Fig. 6(b) were performed with the reduced r c in the x-direction as mentioned above, they cannot directly compare with the EPRT result due to the inherent existence of the ±7% error. However, all the DPDe simulation results with different phonon-boundary scattering models reside within 7% boundaries around the EPRT solution. Better agreement is expected by taking all the energy interaction in x-direction, which requires substantial computational costs. In Fig. 6, the temperature profiles without the modification do not significantly change as Kn changes from 3 to 10. This indicates that the heat transport becomes almost purely ballistic above a certain r c, i.e., when r c is larger than 55.56 above Kn = 3. However, the phonon-boundary scattering effect reduces the effective mean free path of phonon, and therefore, the heat transport becomes more diffusive. For Kn = 10, the phonon-boundary scattering does not make the temperature profile more diffusive because the effective mean free path is in the ballistic regime (close to 55.56) even after the modification. As can be seen the above DPD results for the wide range of Kn, the simulations using k eff calculated by Eq. (17) agrees well with the semi-analytical EPRT solutions better than the results using other models. This is because the equation was obtained based on the BTE for bulk materials with the same assumption as the derivation of EPRT, i.e. the gray relaxation time approximation, whereas the other equations were derived using the different approach as shown in the previous section. This model also can take into account the different specularity of the boundary scattering, while it is assumed that the boundary scattering is diffusive (inelastic) for both EPRT and the simulations (p = 0 in Eq. (17)). We strongly believe that our DPDe model integrated with this boundary scattering modification is able to simulate the diffusive-ballistic heat transport for two- and three-dimensional structures with different specularity of boundaries, which will be our future research. 3.3. Applicability of the DPD model One of the advantages of the present DPDe model is its applicability to complex geometries. In the previous analytical studies [2 4], the solution domain was considered to be one-dimensional thin films. This is due to the complexity of analytically solving two- and three-dimensional geometries. On the other hand, the present DPD model can be easily extended to complex geometries, such as the cantilever tips of AFM and nanostructures, which is analogous to

292 T. Yamada et al. / International Journal of Heat and Mass Transfer 61 (2013) 287 292 Acknowledgments This work is supported by Swedish Energy Agency and the National Science Foundation Grant (NSF-OISE-0530203 and NSF CBET-1067441). The authors also thank Professor Yutaka Asako at Tokyo Metropolitan University for his insightful advices and encouragements. References Fig. 6. Dimensionless temperature profiles across thin films for Kn > 1.0. (a) Kn = 3.0 and (b) Kn = 10. MD. Moreover, DPD is suitable for the mesoscopic simulations, which results in less computational cost than MD. In terms of the formulation of DPD, it is based on the Maxwell Boltzmann (MB) distribution whereas the phonon transportation follows the Bose Einstein (BE) distribution. For this reason, the present model may be applicable when the temperature is low. Deriving the corresponding relation between j ij and a ij may extend the applicability of DPD model based on the kinetic equation for the BE distribution function given by Kaniadakis and Quarati [31]. 4. Conclusions Energy conserving dissipative particle dynamics was utilized to model the diffusive-ballistic heat transport in thin films. The mean free path of phonon was modeled in the DPDe simulation by its relation to the cutoff radius of energy interaction. The existing phonon-boundary scattering models were implemented to obtain the effective phonon mean free path (cutoff radius). Steady heat conduction in one-dimensional thin films was solved for 0.1 < Kn < 10 and the results were compared with the semi-analytical solutions of EPRT. 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