# CRI-08

**Large-Scale Simulation of Coupled Phonon-Gas Thermal Transport in Nanoscale Geometries**

**Graduate Students: Xiaohui Guo (AAE), Dhruv Singh (ME)**

**PI: Alina Alexeenko (AAE) Co-PI: Jayathi Murthy (ME)**

**A** physically rigorous computational algorithm and solver are developed and applied to calculate sub-continuum thermal transport in structures containing semiconductor-gas interfaces. The solution is based on a finite volume discretization of the Boltzmann equation for gas molecules (gas phase) and phonons (solid semiconductor phase). The gas-solid interface coupling is treated through conservation of energy. The computational method allows us to study a range of transport regimes varying from ballistic phonon transport and free molecular flow to continuum heat transfer in both gas and solid. In particular, the method is attractive in the mesoscopic regime where a solution of the Boltzmann equation is necessary and must incorporate simultaneously the transport properties of the solid and gas heat carriers.

Temperature jump in the gas and the formation of a Knudsen layer at a solid boundary is a primary mechanism that limits thermal transport in NEMS and particulate media. We investigate the behavior of temperature jump as an interfacial thermal resistance that may be used for reduced order modeling of these systems. Calculated values of interfacial thermal resistance for various accommodation coefficients and phonon transmission are plotted below [1].

**Figure.2** Temperature profiles for 1D heat conduction at gas-solid interface.

**Figure.1** Temperature profiles for 1D heat conduction in solid.

Subcontinuum thermal transport problems such as the contact thermal resistance of semiconductor-gas interface plays an important role in micro/nano-scale devices. Verification has been conducted by comparison with asymptotic analytical solutions as well as previously reported numerical results and experimental data. Thermal transpiration in nano-sized channels and thermal resistance of nano-sized constrictions has been studied using the new solver [2]. Thermal transpiration is the main mechanism for temperature-driven pumps using submicron-sized membranes. In order to maintain a high temperature ratio between the two ends of the gas path (capillaries), the thermal properties of membrane material are important and the gas-solid interaction must be understood to achieve high compression efficiency. It is shown from the calculations that the temperature gradient, capillary geometry, gas and solid Knudsen numbers, and the gas/solid thermal conductivity ratio are responsible for the over-all compression efficiency. As shown in figure 3, thermal transpiration is very sensitive to thermal conditions along the channel. The coupling effects must be taken into account if the ratio of gas to solid thermal conductivities is larger than 0.1

**Figure.3** Thermal transpiraiton in a closed system.**Top**: uncoupled; **Bottom**: coupled.

**Figure.4** Contact thermal resistance of a nano-sized constriction. **Left: **schematic;

**Right: **heat flux.

Heat transfer across mesoscopic constrictions poses a similar gas-phonon coupling problem. Simulations show that the analytical solution applies to limiting cases where the gas gap can be taken as a thin-film insulator. However, the heat flux through the gas thermal path may become a significant contributor at lower gas phase Knudsen numbers. As shown in figure 4, the heat flux through the gas phase is about 90% of the total at gas-phase Kn=8.

*Supported by Computing Research Institute, Special Initiative Research Grant.*

**<Papers>**

1. D. Singh, X. Guo, A. Alexeenko and J. Murthy, Modeling of subcontinuum thermal transport across semiconductor-gas interfaces, Proceedings of Summer Heat Transfer Conf., Jacksonville, FL, USA, August 10-14, 2008.

2. X. Guo, D. Singh, J. Murthy and A. Alexeenko, Gas-phonon interactions model for subcontinuum thermal transport simulations, Proceedings of 26th Rarefied Gas Dynamics Symposium, Kyoto, Japan, July 21-25, 2008.