Effect of surface roughness and void fraction on thermal transportation of a solid: A molecular dynamics study Muhammad Rubayat Bin Shahadat, Ahmed Shafkat Masnoon, Shafkat Ahmed, and AKM M. Morshed
Citation: AIP Conference Proceedings 1919, 020037 (2017); View online: https://doi.org/10.1063/1.5018555 View Table of Contents: http://aip.scitation.org/toc/apc/1919/1 Published by the American Institute of Physics
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Effect of Surface Roughness and Void Fraction on Thermal Transportation of a Solid: A Molecular Dynamics Study Muhammad Rubayat Bin Shahadat a), Ahmed Shafkat Masnoon, Shafkat Ahmed, AKM M Morshed Department of Mechanical Engineering Bangladesh University of Engineering and Technology (BUET) Dhaka, Bangladesh a)
Corresponding author:
[email protected]
Abstract. Interstitial phenomena functioning as surface roughness, voids and irregularities in nano scale influence the heat transfer and these effects become very significant with the reduction of material size. Non-equilibrium Molecular Dynamics (NEMD) simulation was employed in this study to understand the effects of interfacial thermal resistance named kapitza resistance on solid. Argon like solid was considered in this study and LJ potential model was employed for the calculation of atomic interaction. Surface roughness as well as voids was created in the solid and the void radius was varied. From the simulation, it was observed that a large interfacial mismatch due to these irregularities in homogenous solid causes distortion of phonon frequency causing an increase in thermal resistance. The size of voids has a profound effect on thermal conductivity of solid. Voids positioned perpendicular to heat flow direction causes sharp reduction in thermal conductivity. The reduction of thermal conductivity due to large surface to volume ratio for the generation of surface roughness has also been observed in this study. Keywords: Non-equilibrium molecular dynamics, interstitial atom, effective thermal conductivity, kapitza conductance.
INTRODUCTION In the recent times, the application of micro and nano structured materials has been extended from micro fabrication to microelectronics or from nanotechnology to advanced materials metrology or from nano medicine to green nanotechnology [1]. Thermal properties of nano composites have very profound effects especially in their application to thermal interface materials, thermal insulation, and the third-generation solar cells [1]. Being thermal interface materials many nano structured composite materials have been used to provide better heat dissipation in electronic devices or different microchips [2]. Very high heat dissipation is possible by using low thermal resistance at interfaces. This is of particular concern to the development of microelectronic semiconductor devices as defined by the International Technology Roadmap for Semiconductors in 2004, where an 8-nm feature size device is projected to generate up to 100000 W/cm2 and would need efficient heat dissipation of an anticipated die level heat flux of 1000 W/cm2 which is an order of magnitude higher than current devices [3]. On the other hand, applications requiring good thermal isolation such as jet engine turbines would benefit from interfaces with high thermal resistance. This would also require material interfaces which are stable at very high temperature. Examples are metal-ceramic composites which are currently used for these applications. High thermal resistance can also be achieved with multilayer systems [3]. The subject of void structures is no more than two decades old [4]. Ciobanu et Proceedings of the 1st International Conference on Mechanical Engineering and Applied Science (ICMEAS 2017) AIP Conf. Proc. 1919, 020037-1–020037-8; https://doi.org/10.1063/1.5018555 Published by AIP Publishing. 978-0-7354-1611-6/$30.00
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al. presents a void structure that confine light in low-index region [4]. Thermoelectric (TE) materials can provide electricity when subjected to a temperature gradient, or provide cooling performance when electrical current through it. They have the advantages of lightweight, environmentally benign and without moving parts. A proposal to reduce thermal conductivity further obviously by introducing more surface scattering by introducing roughness and hole [5] Interfacial thermal resistance, also known as thermal boundary resistance or kapitza resistance is a measure of an interface's resistance to heat flow. Due to kapitza resistance when an energy carrier attempts to traverse the interface, it scatters at the interface. The probability of transmission after scattering depends on the available energy states. Understanding the thermal resistance at the interface between two materials is of primary significance in the study of its thermal properties. Interfaces often contribute significantly to the observed properties of the materials. This is even more critical for nano scale systems where interfaces could significantly affect the properties relative to bulk materials [7]. There are two primary models that are used to understand the thermal resistance of interfaces, the acoustic mismatch and diffuse mismatch models (AMM and DMM respectively). Both models are based only on phonon transport, ignoring electrical contributions. Thus, it should apply for interfaces where at least one of the materials is electrically insulating. For both models, the interface is assumed to behave exactly as the bulk on either side of the interface (e.g. bulk phonon dispersions, velocities, etc.). The thermal resistance then results from the transfer of phonons across the interface. Energy is transferred when higher energy phonons which exist in higher density in the hotter material propagate to the cooler materials, which in turn transmits lower energy phonons, creating a net energy flux [3]. While the experimental work becomes relatively difficult, several atomistic simulation techniques have been utilized to model photon transport in nano structured materials [6]. Two prevailing methods are Monte Carlo (MC) simulation and Molecular Dynamics (MD) simulation. The MC method has been used to solve the Boltzmann Transport Equation (BTE) for phonon transport under the relaxation time approximation [6]. The distribution function obtained from Boltzmann’s equation can be easily related to energy and therefore to temperature. The basic principle of the MC simulation is to track the phonon energy bundles as they drift and collide through the computational domain [6]. On the contrary Molecular Dynamics Simulation has been used to examine the thermal properties in nano structured materials where phonon-phonon scattering dominates heat transfer [9]. This method is now-a-days very popular method for calculating different transport properties. It is a very powerful toolbox in modern molecular modeling and enables us to follow and understand structure and dynamics with extreme detailliterally on scales where motion of individual atoms can be tracked [8]. This process is simple and can deal with complex geometries. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are calculated using inter atomic potentials or molecular mechanics force fields [5]. Molecular Dynamics Simulation can be of two approaches- Equilibrium Molecular Dynamics (EMD) Simulation and Non-Equilibrium Molecular Dynamics (NEMD) Simulation. EMD method calculates heat transfer by Green-Kubo formulation; whereas, NEMD is a direct approach to calculate thermal conductivity directly from heat flow. As argon-like solids with Lennard-Jones (LJ) potential are used as the model system where electrons are not involved in heat conduction, NEMD approach is preferable.
MOLECULAR MODELING The three-dimensional simulation cell of 80 X 5 X 5 nm3 was constructed of Argon molecules arranged in FCC lattice. In FCC lattice structure, each face has attained an atom in its center position. This atom is equally shared by the two-unit cells. Hence, in a sum each unit cell gets 4 atoms. Periodic boundary conditions were chosen to be imposed in all the directions so that if the interacting particles across the boundary lost from one end of the box can reappear through the other end. Hence, it ensures a constant number of particles in the simulation domain. Four layers of solid at both ends of the simulation domain along the length were assigned as hot region, and eight lattices in the middle were assigned as cold region. Temperature of this hot and cold region was maintained by velocity rescaling. 8000 atoms were created inside the solid. Three nano voids each having radius 0.5 nm were implanted perpendicular to heat flow direction with a view to observing the effects of voids on heat transfer. Then the radius of the voids was increased to 1 nm and 1.5 nm to observe the effect of void fraction. 4, 16 and 64 atoms were deleted respectively in this case. With a view to understanding the effect of irregularity some atoms were placed in the void.
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These atoms were bonded with the neighboring atoms from one side and there is a crescent shaped void in another side. To visualize the effect of surface roughness on thermal transport six hemi spherical holes were created on the surface of the solid by removing 64 atoms in total. Simplified Lennard-Jones (LJ) model was used for atom-atom interaction. Previous research confesses us those results from the simplified LJ model and atomically realistic models can be equally treated [9].
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(b)
(c)
(d)
(e) FIGURE 1. Simulation domain (a) Solid argon matrix with nano voids of 1 nm (b) Solid argon matrix with nano voids of 1.5 nm, (c) Solid argon matrix atoms in a crescent shaped void (d) Surface roughness on the top surface by creating 3 holes (e) Surface roughness on the bottom surface by creating 3 holes (OVITO Picture).
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All the interactions between molecules were calculated by LJ potential: ࣌ ࣌ ࣐ሺ࢘ሻ ൌ ࢿሾቀ ቁ െ ቀ ቁ ሿ ࢘ ࢘ Interaction potential’s length and energy parameters were collected from Ref. [14]: x Lengths are expressed in terms of σ =0.3405 nm, x The energy units are specified by ࢿ ൌ ࡷ Where ε= 120×1.3806×10-16 erg/atom x Given the mass of an argon atom m= 39.95×1.6747×10 -24g x The MD time unit corresponds to 2.161×10--12s. Typical time step size is Δt = 0.001ns. The Lenard–Jones potential function was truncated at 2.5σ. MD simulation can trace the probable trajectory of each particle. MD also requires the value of potentials, f (r) that is responsible for the interaction between atoms. Then it solves Newton’s equations of motion for every atom in the system numerically. In MD simulation, the force between a pair of atoms is calculated using ࢌ ൌ െ
ࣔሺ࢘ሻ ࢄ ࣔሺ࢘ሻ ൌ െሺ ሻሺ ሻ ࢘ ࣔ࢞ ࣔ࢘
Where xi stands for x, y, z components. Newton’s equations of motion ࢊ ࢘ ࢌ ൌ ࢇ ൌ ࢊ࢚ Can be then solved with a view to obtaining the new positions of the particles. This solve is basically followed by the force calculation. There must be a technique for solving the equations of motions and among all the algorithms feasible to solve it, the Varlet algorithm is the simplest and the best. The Varlet method is a direct solution of the second order differential equations. The velocities are eliminated by comparison of two expansions about the position at time t. The Taylor series expansion about +dt and -dt are ࢘ሺ࢚ ࢊ࢚ሻ ൌ ࢘ሺ࢚ሻ ࢊ࢚࢜ሺ࢚ሻ ൬ ൰ ࢊ࢚ ࢇሺ࢚ሻǤ ǤǤ ࢘ሺ࢚ െ ࢊ࢚ሻ ൌ ࢘ሺ࢚ሻ െ ࢊ࢚࢜ሺ࢚ሻ ൬ ൰ ࢊ࢚ ࢇሺ࢚ሻǤ ǤǤ Summing these two equations ࢘ሺ࢚ ࢊ࢚ሻ ൎ ࢘ሺ࢚ሻ െ ࢘ሺ࢚ െ ࢊ࢚ሻ ࢊ࢚
ࢌሺ࢚ሻ
The probable result of the new position incorporates an error term of order dt4, where dt is the time step in MD scheme. Once the trajectory is found, the velocities can be derived using: ࢜ሺ࢚ሻ ൎ
࢘ሺ࢚ ࢊ࢚ሻ െ ࢘ሺ࢚ െ ࢊ࢚ሻ ࢊ࢚
This expression is only accurate to order dt2[6]. The simulation was started from its initial configuration with a time step of 2.1 fs. The simulation ran with NPT ensemble followed by NVE ensemble. Heat flux required to establish the temperature gradient was measured. Efflux was set 2.78 units/timestep. Thus, heat flow started between the cold region and the hot region. The simulation ran for 10 ns to reach a steady temperature gradient.
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Once the system reaches a steady temperature, the thermal conductivity of the overall system was calculated using Fourier's Law: οࡽ ࡷ࢜ࢋ࢘ࢇ ൌ ൌ ࡾ࢚ ࣎ ࢚ࣔ ࣔ࢞ Where, డ௧ ∆Q is the total heat flux, τ is the simulation time, is the linearized temperature gradient of solid argon. డ௫
All simulations in this study were performed in LAMMPS [11] and visualizations were done by using OVITO.
RESULTS AND DISCUSSION MD simulation has an advantage in thermal transport modeling compared to the other theoretical approaches such as AMM and DMM [15]. The inputs provided in the MD simulation are the atomic structure and inter-atomic potentials. MD simulations have been used largely to calculate thermal resistance across any solid-solid interface [15]. The simulation starts from its initial configuration and temperature was scaled to achieve the equilibrium temperature of 20K. The temperature profile of the simulation domain was checked continuously to understand whether the simulation domain was in equilibrium state or not. Once the simulation domain was in equilibrium state, Nose-hoover thermostat was applied to establish the temperature gradient. Variation of temperature with time steps for the simulation of solid argon is shown in Fig 2.
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(b)
FIGURE 2. (a) Variation of temperature with time at initial stage (b) Variation of temperature with time at final stage.
Phonon wave packets are formed from linear combinations of vibration eigenstates of the perfect crystal. This wave packet is then allowed to propagate towards an interface or a scatterer where it scatters into transmitted and reflected waves [6]. To investigate the effect of voids on the interfacial scattering, voids of different radius are set in the perpendicular to the heat flow direction. The radiuses of the voids are varied 0.5 nm, 1 nm and 1.5 nm. When a void is placed, at the position of void the particles do not find any other particle to transmit the energy and therefore there is a decrease of thermal conductivity. As the radius of void increases, there will be more decrease of transmission of energy and hence more decrease of thermal conductivity. When the radius of the void is 0.5 nm only 4 nano particles are removed from the simulation cell. The thermal conductivity in this case is 1.58 W/mk which is 96.3% of solid argon structure. For understanding the size effect of void 16 atoms are removed from the cell and the radius has become 1 nm. Thermal conductivity reduces to 1.54 W/mk. That means 93.9 % of solid argon. When the radius of the void is increased further to 1.5 nm by removing 64 atoms there is a more decrease of transmission of energy and therefore effective thermal conductivity reduces more to 1.49 W/mk as 90.8% that of solid argon.
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ࡷ࢘ࢇࢊ࢛࢙Ǥ Ǥ ૢ ൌ ൌ Ǥ ૢૡ ࡷ࢙ࢊࢇ࢘ࢍ Ǥ ࡷ࢘ࢇࢊ࢛࢙ Ǥ ൌ ൌ Ǥ ૢૢ ࡷ࢙ࢊࢇ࢘ࢍ Ǥ ࡷ࢘ࢇࢊ࢛࢙Ǥ Ǥ ૡ ൌ ൌ Ǥ ૢ ࡷ࢙ࢊࢇ࢘ࢍ Ǥ
FIGURE 3. (a) Temperature profile of voids in solid matrix of different radius (b) Comparison of thermal conductivity among the void structures
With a view to observing the effect of irregularity some atoms were placed in the void. These atoms were bonded with the neighboring atoms from one side and there is a crescent shaped void in another side. In this situation, there is a very little place where the particles will not be able to transmit energy. Hence there will be a less distortion of phonon frequency and less thermal resistance. Therefore, aberration from the pristine argon matrix in this case is less than the previous case. Effective thermal conductivity is 1.59 W/mk. ࡷࢇ࢚࢜ࢊ Ǥ ૢ ൌ ൌ Ǥ ૠ ࡷ࢘ࢇࢊ࢛࢙Ǥ Ǥ ૢ
FIGURE 4. (a) Temperature profile of voids of 1.5 nm and crescent shape
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Surface roughness often shortened to roughness, is a component of surface texture. It is quantified by the deviations in the direction of the normal vector of a real surface from its ideal form. In order to understand the effect of surface roughness on thermal transport six hemi spherical holes were created on the surface of the solid by removing 64 atoms in total. The effect of thermal transport in this case is basically dominated by surface to volume ratio. Any void or hole on the surface increases the surface area and decreases the volume. Hence the value of surface to volume ratio increases a lot. This enhanced surface to volume ratio reduces the percentage of delocalized modes causing a large reduction of thermal conductivity [5]. The effective thermal conductivity for on-plane holes is less than the inner hole structure. As for inner void surface area does not change but volume is reduced. But for on plane holes or surface roughness surface area increases and volume decreases. So, surface to volume ratio increases more than the previous case. The effective thermal conductivity for surface roughness is 1.43 W/mk which is 87.1% of conductivity of solid argon matrix.
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(b)
FIGURE 5. (a) Temperature profile of surface roughness vs. solid argon (b) Overall comparison of all temperature profiles
CONCLUSIONS NEMD simulation was engaged to investigate the effect of irregularities due to void and surface roughness on thermal transport in a solid argon matrix. Nano voids of different radius were placed in the perpendicular to heat flow direction. Spherical as well as crescent voids were studied. Surface roughness were created by removing atoms from surface and all outcomes were compared with pristine argon matrix. From the simulation results, the following conclusions can be drawn: x Nano voids have a profound influence on the overall thermal conductivity of the solid matrix. Presence of voids inside the solid matrix causes interfacial mismatch faced by the phonon wave causing significant drop in the thermal conductivity.
x The size of the voids has a great influence on overall thermal conductivity. As the radius of the void is increased the thermal conductivity tends to decrease according to the power law-
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ࡷ ൌ Ǥ ࢋିǤ࢘ Where K is the effective thermal conductivity and r is the radius of the void x
For crescent shaped void thermal conductivity drops less than the spherical one due to less distortion of phonon wave frequency.
x
The thermal conductivity becomes lowest for the surface roughness due to a large surface to volume ratio as well as for the reduction of percentage of delocalized mode.
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