Institute of Metals Division - Electrical Resistivity of Dilute Binary Terminal Solid Solutions

THE classical work on the electrical conductivity of alloys was carried out by Matthiessen and his coworkers1 in the early 1860's. He attempted to correlate the electrical conductivity of alloys with their constitution diagrams, but the information regarding the latter was too meager for success. Guertler2 reworked Matthiessen's and other conductivity data in 1906 on the basis of volume composition (an application of Le Chatelier's principle with implications as to temperature and pressure effects), and obtained the following relationships between specific conductivity and phase diagrams (plotted as volume compositions) : 1—For two-phase regions, electrical conductivity can be considered as a linear function of volume composition, following the law of mixtures. 2—For solid solutions, except intermetallic compounds, the electrical conductivity is lowered by solute additions first very extensively and later more gradually, such that a minimum occurs in systems with complete solid solubility. This minimum forms from a catenary type of curve. Intermetallic compound formation with variable compound composition results in a maximum conductivity at the stoi-chiometric composition. Landauer" has recently considered the resistivity of binary metallic two-phase mixtures on the basis of randomly distributed spherical-shaped regions of two phases having different conductivities. His derivation predicts deviations from the law of mixtures which fit measurements on alloys of 6 systems out of 13 considered. Volency (Ionic Charge) Perhaps the first comprehensive discussion of the electrical resistivity of dilute solid-solution alloys was presented by Norbury' in 1921. He collected sufficient data to show that the change in resistance caused by 1 atomic pct binary solute additions is periodic* in character. The difference between the period and/or the group of the solvent and solute elements could be correlated with the increase in resistance. Linde5-7 determined the electrical resistivity (p) of solid solutions containing up to about 4 atomic pct of various solutes in copper, silver, and gold at several temperatures. He reported that the extrapolated"" increase in resistance per atomic percent addition is a function of the square of the difference in group number of the solute and solvent as follows: ?p= a + K(N-Ng)2 where a and K are empirical constants and N and Ng are group numbers of the constituents. This empirical relation was subsequently rationalized theoretically by Mott,8 who showed that the scattering of conduction electrons is proportional to the square of the scattering charge at lattice sites. Thus, the change in resistance of dilute alloys is propor-t,ional to the square of the difference between the ionic charge (or valence) of the solvent and solute when other factors are neglected. Mott's difficulty in evaluating the volume of the lattice near each atom site where the valency electrons tend to segre-gate: limited his calculations to proportionality relations. Recently, Robinson and Dorn" reconfirmed this relationship for dilute aluminum solid-solution alloys at 20°C, using an effective charge of 2.5 for aluminum. In terms of valence, Linde's equation becomes ?P= {K2 + K1 (Z8 -Za)2} A where K1 and K2 are coefficients, A is atomic percent solute, Z, is valence of solvent, and Zß, is valence of solute. Plots of these data for copper, silver, gold, and aluminum alloys are shown in Fig. 1. The values of K1 and K2 are constant for a given chemical period (P), but vary from period to period. The value of K, increases irregularly with increasing difference between the period of the solvent and solute element (AP), being zero when AP is zero. The value of K, appears to have no obvious periodic relationship. All factors other than valence that affect resistivity are gathered in these coefficients. Because of the nature of the coefficients, Eq. 1 is of limited use in estimating the effects of solute additions on resistivity unless a large amount of experimental data are already available on the systems involved. It is the purpose of the first part of this report to investigate the factors that may be included in the coefficients of Linde's equation. On this basis, it is hoped that the relative effects of solute additions on resistivity can be better estimated from basic data, leading to a more convenient alloy design procedure. It is well 10,11 that phenomena that decrease the perfection of the periodic field in an atomic lattice, such as the introduction of a solute atom or strain due to deformation, will also increase the electrical resistivity. Thus, in an effort to relate changes in electrical resistivity to alloy composition, it appears appropriate to consider the atomic characteristics related to solution and strain hardening