Selection of boundary conditions for modeling thermowave processes in multilayer objects

Abstract

Solution of many problems related to the heat transfer involves the calculation or determination of the influence of interfaces between two or more neigh boring regions. Possible applications of this vary from fundamental investigations in nanophysics to practi cally important calculations of heat transfer in high power electronics and construction engineering. A standard approach to solution of these problems consists in replacing the boundary layer of finite thick ness by a negligibly thin layer with finite thermal resis tance. Description of the interaction between a heat flux and the boundary layer then reduces to the intro duction of a temperature drop at the interface, which is proportional to the heat flux, which is considered continuous. This approach in many cases gives satis factory results and is quite well justified.Moreover, based on the electrothermal analogy, various combinations of layers can be treated as the corresponding combinations of thermal resistances [1]. By analogy with electrical engineering calcula tions, it is then possible to determine the total thermal resistance of several regions. However, in following this analogy, one should assume that any material pos sessing a heat capacity will influence the process of nonstationary heat transfer and the degree of this influence increase with the process rate. Various cal culations of nonstationary heat transfer have been described in classical monographs [2] and special guides on the calculation of heat transfer, e.g., in radio electronics [3] and building engineering [4,5]. How ever, in practical modeling of heat fluxes via interfaces, this aspect is usually ignored even in cases of purely nonstationary processes, e.g., in investigations of the scattering of thermal waves and in applications of ther mowave diagnostics [6][7][8][9], where only the variable component of temperature oscillations is considered.At present, thermal waves in these methods of diag nostics are excited using laser radiation in a regime with continuous or pulsed temporal modulation. With the aid of ultrashort laser pulses, it is possible, on one hand, to probe nanodimensional objects and, on the other hand, to study ultrafast energy transfer pro cesses. It appears that the total thermal impedance of interfaces involved in these nonstationary processes can be of great significance.The present study was aimed at (i) determining the limits of applicability of the well known model of heat transfer via interfaces with allowance of their thermal resistance and (ii) developing this model for, and justi fying its application to, nonstationary processes.Let us consider the problem of reflection and trans mission of plane thermal waves at interfaces of the intermediate layer of a planar three layer system com prising semi infinite external layers with tempera tures T 1 and T 2 , and the inner layer with tempera ture T i . For a plane thermal wave incident from the first external layer, the solution of the thermal conduc tivity equations for this three layer system can be writ ...

Laser Thermal Wave Diagnostics of the Thermal Resistance of Soldered and Bonded Joints in Semiconductor Structures

“…This situation can be illustrated by the example of the simplest structure shown in Figure 2 . Using the concept of incident and reflected thermal waves and following the approach developed in [ 57 , 58 ], the alternative temperature components for three regions can be presented in the form of plane waves where x is the coordinate shown in Figure 2 , t is the time, l is the thickness of the second layer, q n were obtained from the heat equations for each layer and q n = (−1 + i ) ( ω /2 κ n ) 1/2 , ω is the cyclic frequency of thermal waves, κ n is the thermal diffusivity of the n th layer, a is the amplitude of the incident thermal wave, and r , b , c , τ are coefficients to be determined from boundary conditions.…”

“…In [ 57 , 58 ], the coefficients at the exponents r , c , d , and τ as functions of the wave amplitude a were found in the analytical form by solving the heat equations for the structure shown in Figure 2 with common boundary conditions representing the continuity of temperature and heat flux at interfaces. In these publications, it was shown that in the case of a thin intermediate layer, the thermal problem for the structure in Figure 2 can be reduced to the problem of two regions between which there is the real TR r th and the reactive thermal conductance of the capacitive type z th where K 2 and ρ 2 C 2 are the thermal conductivity and the volumetric heat capacity of the intermediate layer, respectively, and i is the imaginary unit.…”

“…The TR r th depends only on the thermal conductivity of the intermediate layer, while the capacitive thermal conductance depends on its volumetric heat capacity ρC p . The estimates given in [ 57 ] show that neglecting the reactive thermal conductance z th is possible only in the case of η 2 2 << η 1 η 3 , where η = ( ρC p K ) 1/2 is the thermal effusivity. Otherwise, one can just ignore the TR r th .…”

Studies of the thermal conductivity and thermal resistance of solder layers with lead-free pastes by the photodeflection method

“…It was also demonstrated in [7] that a change in the solder composition induces considerable variations of its thermal conductivity and thermal capacity. The latter factor may be taken into account by introducing thermal impedance [15] into the boundary condition for heat fluxes. In the discussed case of lead-free solder pastes, it was also needed to introduce a similar parameter into the Ge substrate-solder layer boundary condition to characterize the PD signal accurately.…”