Electrical Resistivity of Metal Powder Aggregates
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I.
INTRODUCTION
ANY aspect related to the modeling of real powder systems constitutes a complex task. One of the reasons for this complexity is that powder particles of any material have, in general, very different shapes and sizes. The first theoretical approaches simplified the problem by considering monosize particles preferentially with spherical shapes. Such models used to describe the represented phenomena only qualitatively. Other approaches considered the powdered material as a continuum material with a phase constituted by pores. This approach, although reasonable in the very low porosities range, clearly fails when extended to the high porosity range. Related with these systems, modeling of the mechanical problem compaction, despite its complexity, has received the attention of much research. Conversely, the electrical behavior of lowly compacted powder masses, which is a much more complex problem, is the objective of very few studies. Most of these studies are performed from an experimental point of view,[1,2] although the comprehension of this problem is a crucial task to model any of the multiple modalities of the electrical resistance sintering process. The cause of this additional complexity lies in the fact that metallic powder particles are coated by very thin oxide films, irrelevant for mechanical problems, which drastically alter the electrical properties of the powder mass. Other properties, including powder sinterability, are also affected by this oxide coating. In this work, the electrical resistivity of a powder mass, subjected to different compaction degrees (in the porosity range nearest the tap porosity, i.e., the porosity J.M. MONTES, F.G. CUEVAS, and J. CINTAS, Senior Lecturers, are with the Departamento de Ingenierı´ a Meca´nica y de los Materiales, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, E-41092, Sevilla, Spain. Contact e-mail: fgcuevas @us.es Manuscript submitted April 12, 2007. Article published online October 30, 2007. METALLURGICAL AND MATERIALS TRANSACTIONS B
of the powder mass after vibration), is modeled. This is done by means of a new theoretical tool: the equivalent simple cubic system of deforming spheres.[3] There are only a few bibliographic references tackling this or a similar problem. Most of which are localized in the context of the percolation theories.[4–6] The expression proposed here is also consistent with these theories.
II.
MODELING
A. Powder Systems Consisting of Bare Particles In previous studies,[3,7] the authors reported the following equation to calculate the effective electrical conductivity of a porous sintered compact: rE ¼ r0 ð1 HR Þt
½1
where the exponent t, which usually ranges between 1 and 2, is given by 4
t ¼ 1 þ ð1 HM Þ5
½2
The parameter rO represents the electrical conductivity of the bulk (completely solid) material and HR is the relative porosity. This latter parameter is defined as the ratio of the sample porosity, H, to the maximum porosity of the system, HM, which can be assimilated to
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