Heat Conduction in Solids

1. Heat Conduction in Solids

2. Overview of Heat Conduction in Solids • Many MEMS devices are actuated thermally. Key issues involve : • The amount of heat required to invoke the desired action. • The time required for initiating and terminating the action. • The associated thermal stresses and strain induced in the device. • The possible damage to sensitive components of the device due to heating.

3. General Principle of Heat Conduction • Consider conduction in a slab of thickness d and temperatures Ta and Tb on the left- and right-hand side, respectively (Ta > Tb). • The amount of heat Q flowing in a time t is • k is the thermal conductivity of the solid expressed in units of W/m-°C in SI system.

4. Fourier Law of Heat Conduction • In terms of heat flux, q, defined as the heat flow per unit area and time • In a solid situated in space defined by r(x,y,z)

5. The Heat Conduction Equation • When a solid is subject to heat input source, or dissipates heat to the surrounding medium at a rate Q(r,t) per unit volume, the temperature field T(r, t) may be obtained using the heat conduction equation • The thermal diffusivity a is measured in units of m2/s in SI system.  is a measure of how fast the solid can conduct heat and is given by • c is the specific heat of the solid • A common heat source in MEMS is electric resistance, R, in which a current, i, generates a power P given by

6. Newton’s Cooling Law • Heat transmission in solids is via conduction, whereas in fluids convection is the mode of heat transfer. • For the fluid shown the heat flux q is expressed as • The constant h has several names one of which is “transfer coefficient”.

7. Important Relations in Fluids • h is normally embedded in “Nusselt number”, Nu, via • where L is a characteristic length • The following relations are normally used to determine Nu • the parameters a, b,g, and din the expressions for Nu, are experimentally determined parameters) • Reynold’s number, Re • Prandtl number, PR • Grashoff number, Gr • In the above relations cp is the specific heat of fluids under constant pressure, b is the volumetric coefficient of thermal expansion, and Dt is the duration. • For most convective heat transfer in fluids • where for a given fluid the parameter f is determined experimentally.

8. Solid/Fluid Interaction • Thermally actuated MEMS may involve heat transfer between solids and fluids. • Consider the scenario where heat is dissipated from solid to fluid or vice versa. • A boundary layer, which is a barrier for freer heat transfer, is built in the fluid immediately adjacent to the solid surface : the resistance of the layer is ~ 1/h • It can be shown that

9. The Boundary Conditions (BCs) : i) Prescribed Surface Temperature • The heat conduction equation is used to determine T(r,t). For design purposes it is necessary to determine locations in the solid with maximum temperature as well as thermal stress distribution • In the prescribed surface temperature BC • The function F(t) can be a constant in special cases.

10. ii) Prescribed Heat Flux at the Boundary • This is the BC in which the flux qin entering and qout leaving the solid are specified. These heat fluxes can be expressed by the Fourier law. • The signs in the equations are determined using the table.

11. iii) Convective Boundary Conditions • In this BC the solid boundary is in contact with fluid at temperature Tf • One notes that is equivalent to the BC in (i) , whereas the case of h = 0 leads to the insulated boundary condition

12. Heat Conduction in Multilayered Thin Films • MEMS structure involves stacks of several thin films. This necessitates heat conduction analysis through the films. • The temperature field in the ith film, Ti • Prescribed initial conditions in xi ≤ x ≤ xi+1 at t = 0, and prescribed BCs at x = 0 and x = xi+1 for t > 0. • The continuity conditions imply

13. Thermal Conductivity of Thin Films • A simple model based on molecular heat transfer gives k in thin films as • A more refined theory gives for the effective k, keff , in a thin film of thickness H (kb is the thermal conductivity of the bulk material of the film

14. Heat Conduction Equation for Thin Films • The lag time between heat flow and temperature gradient is insignificant in bulk materials, but it must be accounted for in thin films. • This results in a modified heat conduction equation in thin films • The last term represents the velocity of heat transmission in solids: it is in the form of wave propagation of T(r,t) and is called thermal wave propagation in the solid. This term is insignificant when H >> lo, where lo is the average mean free path. • t is the “relaxation time”, which is the average time that a carrier, e. g., a phonon, travels between collisions

15. Thermomechanics

16. Introduction • Many MEMS are fabricated at elevated temperature, such as in bonding or oxidation, or are operated at elevated temperatures. • Thermal effects are an important factor in the design and packaging in MEMS. • There are generally three serious effects on MEMS exposed to elevated temperatures • Thermal effects on mechanical strength of materials • Creep deformation • Thermal stresses

17. Variation of material properties with temperature

18. Temperature-Dependent Thermophysical Properties of Silicon

19. Creep deformation of materials at elevated temperature

20. Thermal Stresses • A bar with fixed ends • The temperature rise DT = T2 - T1 • The parameters sT and eT are temperature stress and strain, respectively. • The thermal expansivity of the material is a.

21. A bimaterial strip

22. Designation of stress components in a solid

23. Thermal stresses in thin plates (diaphragm) with DT through the thickness

24. Thermal stresses in beams with DT through the depth