Effect of thermal losses on the microscopic two-step heat conduction model

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Abstract

The effects of radiative and convective thermal losses on the thermal behavior of thin metal films, as described by the microscopic two-step heat conduction model, are investigated. It is found that radiative losses from the electron gas are significant in thin films having (L/Te3)<10−22, while radiative losses from the solid lattice are significant when (L/Te3)(Te4/Tl4)<10−22. Also, it is found that convective losses from the thin metal film are insignificant in most practical operating conditions.

Introduction

High-rate heating of thin metal films is a rapidly emerging area in heat transfer [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. When a thin film is exposed to a very rapid heating process, such that induced by a short-pulse laser, the typical response time for the film is an order of picoseconds which is comparable to the phonon–electron thermal relaxation time. Under these situations, thermal equilibrium between solid lattice and electron gas cannot be assumed and heat transfer in the electron gas and the metal lattice needs to be considered separately. Models describing the non-equilibrium thermal behavior in such cases are called the microscopic two-step models. Two microscopic heat conduction models are available in the literature. The first one is the parabolic two-step model [1], [2], [3], [4], [5], [8], [9], [10] and the second one is the hyperbolic two-step model [1], [3], [7], [11].

Ultrafast heating of metals consists of two major steps of energy transfer which occur simultaneously. In the first step, electrons absorb most of the incident radiation energy and the excited electron gas transmits its energy to the lattice through inelastic electron–phonon scattering process [1], [3]. In the second step, the incident radiation absorbed by the metal film diffuses spatially within the film mainly by the electron gas. For typical metals, depending on the degree of electron–phonon coupling, it takes about 0.1–1 ps for electrons and lattice to reach thermal equilibrium. When the ultrafast heating pulse duration is comparable with or less than this thermalization time, electrons and lattice are not in thermal equilibrium. As a result, the thermal behavior of the thin film under the effect of the microscopic parabolic heat conduction model is described by [1], [3]Cl(Tl)Tlt=G(Te−Tl),Ce(Te)Tet=∇·(ke∇Te)−G(Te−Tl)+Qe.Using different solving techniques, the microscopic parabolic heat conduction model has been used numerously [4], [5], [8], [9], [10], [13] to describe the thermal behavior of thin metal films under different applications, operating conditions, geometrical parameters and metal properties. Most of the available investigations have assumed that heat losses are negligible [1], [2], [3], [8], [9], [10], [12]. This is due to two reasons.The first reason is the fact that the duration of the heating process is very short and as a result, the metal film does not have enough time to lose energy to the surrounding. The second reason is the fact that hot electrons exchange energy with cold lattice through a phonon–electron coupling factor G having an order of magnitude of 1016(W/m3K) which is much larger than any other thermal exchange coefficient. As a result, hot electron gas prefers to transmit most of its energy to the cold solid lattice instead of the surrounding. However, there are applications in which the intensity of the laser heating source is very high and the metal film is very thin. Under these applications, electron gas attains very high temperatures during the early stages of the heating process. As a result, radiative losses from the film through its very large surface area-to-volume ratio becomes comparable with the rate of energy exchange between electron gas and solid lattice.

Up to the authors knowledge, no qualitative or quantitative descriptions for the conditions under which thermal losses from the film may be neglected exist. The aim of the present work is to investigate the effect of thermal losses on the thermal performance of thin metal under the effect of the parabolic two-step heat conduction model. Also, the effects of operating conditions, geometrical parameters and metal properties on the thermal losses are investigated.

Section snippets

Analysis

Consider a very short laser pulse on a pure metal film of thickness L. In the following analysis, the interest is focused on the effect of thermal losses from the thin film. As a result, thermal diffusion may be neglected and the film is considered as a lumped system. This assumption is reasonable since the metal film is very thin and has a very high thermal conductivity. Neglecting the temperature dependence of the thermal properties and assuming the incident radiation to be totally absorbed

Quantitative analysis

To present a quantitative analysis for the importance of the radiative losses from the film, a modified version of , has to be solved numerically. The modified version of , assumes that the laser heating source evolves its energy instantaneously at η=0 and this energy is absorbed immediately by electron gas. This assumption is justified when the duration of the heating source is much less than the duration of the thermalization process. In the literature, heating process of very short

Concluding remarks

The effects of thermal losses on the microscopic two-step heat conduction model is investigated. It is found that the radiative losses from the electron gas may have significant effects on the thermal behavior of the thin metal film when (L/Te3)<10−22, while radiative losses from the solid lattice are significant when (LTe/Tl4)<10−22. Also, it is found that convective losses from the thin metal film are insignificant in most practical operating conditions. However, both kinds of thermal losses

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