Modelling and Simulation in Materials Science and Engineering Extraction of effective permittivity and permeability of metallic powders in the microwave range

In this work, effective electric permittivity and magnetic permeability of metallic–dielectric mixtures are extracted from electromagnetic full 3D simulation data in the microwave range. The numerical method used here is the finite integration technique with periodic boundary conditions. Simulated mixtures have periodic extend in directions perpendicular to the direction of the plane wave. Thus, it is sufficient to analyze a unit element in order to extract the effective electric and magnetic properties. Using this procedure, the behavior of fine copper powders irradiated by microwaves at a frequency of 2.45 GHz is simulated. Then, the relation between particle size and the mixture’s effective properties is studied. By introducing a thin copper oxide or conductive layer it is possible to emulate the effective properties of metallic powder compacts in the early stage of sintering. Thus, this work contributes to improving the insight into the mechanisms of microwave absorption in powders of conductive materials in contrast to non-absorption in bulk metals. Microwave sintering of ceramic powders has been a well-established method in science and industry for the last few decades [1]. Microwave heating, in contrast to conventional heating methods, allows for volumetric heating of materials, which leads to time saving and reduced energy consumption. Additionally, high heating rates in metal carbide based materials used as microwave susceptors offer a combination of microwave and conventional heating to boost the processing of less absorbing materials such as most oxides and nitrides. Rapid and controllable heating and the use of fine powders lead to a smaller grain size and a more uniform grain size distribution which improves the mechanical properties of the sintered material. Recently microwave heating has emerged as a powerful tool for the processing of metallic powders. It was reported in 1999 by Roy et al [2] that porous metal powder compacts get heated up when subjected to microwave irradiation in either electric or magnetic field despite the well-known fact that microwaves do not penetrate bulk metals beyond skin depth and thus cannot deeply heat metals in a microwave furnace. Roy’s results show that porous metal 0965-0393/10/025015+13$30.00 © 2010 IOP Publishing Ltd Printed in the UK 1 Modelling Simul. Mater. Sci. Eng. 18 (2010) 025015 T Galek et al powder compacts are materials with both effective dielectric and effective magnetic losses, corresponding to the effective permittivity and the effective permeability of the porous metal compacts. There are numerous experimental studies of microwave heating of metallic powders. In the recent work of Ma et al [3] microwave heating of copper powder compacts in either a magnetic or an electric field in a single mode (TE102) cavity has been studied together with measurements of magnetic and electric properties of metal compacts. The dependence of the conductivity of the sample as a function of heating time is also measured giving insight into the mechanisms of pre-sintering stage with characteristic high heating rates. There are two important theories describing microwave absorption mechanisms in metallic powders based on experimental results. In the work by Luo et al [4] the heating rate of nickel– iron alloy powders is related to the theoretically derived formula of power absorbed by the compact. The paper of Rybakov et al [5] describes the absorption mechanism of microwaves in metallic powders with a thin oxide layer using the effective-medium approximation. In this work, we study electric and magnetic properties of metallic powders obtained from the finite integration technique (FIT) [6] simulations. Introducing a computer model of these materials together with an extraction of effective parameters of the mixtures gives us an opportunity to have an insight into the micrometer scale microwave absorption mechanisms in metallic powders. The computer simulations are performed with the cuttingedge electromagnetic simulation software, CST Microwave Studio® 2008 (CST MWS) [7]. Further studies of the microwave heating of the metal powder compacts are performed using the finite element method (FEM) multiphysics modeling software. For this purpose we have chosen the COMSOL Multiphysics® package [8], due to its flexibility and capability to deal with coupled problems. 1. Extraction of effective properties There are several methods for the extraction of effective material parameters for inhomogeneous mixtures. The most popular approach is the extraction from transmission and reflection characteristics of a sample. The method is based on free-space measurements of the complex permittivity and complex permeability [9, 10]. The same theory stands behind the extraction of effective properties from simulation data and is used with great success for example to describe metamaterials as homogeneous medium [11–16]. Figure 1 shows a slab of material of thickness d placed in free space. The slab is irradiated with a plane wave incident normally on the slab. Such a setup with microwave ports on the left and right sides, with a sample in between, can be considered as a two-port microwave network. The field amplitudes of the input and output waves at port 1 can be written as a1 and b1, respectively, and those of input and output waves at port 2 as a2 and b2, respectively. These parameters are either the complex field amplitudes of the electric field (E) or the magnetic field (H ). The scattering parameters are collected in the scattering matrix [S]. They are used to describe the relationship between input waves [a] and those of output waves [b]. In the case of a two-port network the output can be calculated as follows: [ b1 b2 ] = [ S11 S12 S21 S22 ] [ a1 a2 ]