PHYSICAL REVIEW B VOLUME 46, NUMBER 4 15 JULY 1992-II Resistivity anomaly during the process of separation of phases of a binary alloy Ney Jose Luiggi and Oscar Febres {Received 20 September 1991; revised manuscript received 9 March 1992) It is well known that when one follows, isothermally, the process of separation of phases of certain binary alloys using electrical-resistivity measurements, an anomalous behavior is observed. The theoreti- cal explanation of this anomaly has been controversial. In this work, we discuss two aspects of that theory that have not received enough attention in the literature. The first is the real effect that an ex- ponential damping term produces on the resistivity anomaly when the mean free path is not a free pa- rameter but rather depends on the wave vector. This leads to an integral equation of the Volterra-type, the solution of which, by the iterative method of Newmann, exhibits rapid convergence when the time constant of the damping factor is associated with the internal mean free path of a Guinier-Preston zone. The second aspect concerns a possible reconciliation of the ideas of Rossiter and Hillel concerning a semiphenomenological model that reproduces well the clustering process. This model takes into account the effect of scattering by zones, separately, through the microstructure and through the boundaries, with a weight function that determines the centers by which the electron is scattered. The results ob- tained when this model is applied to the binary Al-Zn alloy are completely satisfactory. I. INTRODUCTION Consider a binary alloy A-B with atomic concentra- tion Co of solute atoms B. We assume that, after homo- genization at a temperature above the solubility limit, the sample is quenched so as to permit phase separation. Ini- tially all solute atoms will be — in unstable or metastable states — in solid solution, and a clustering process shall occur as the system stabilizes. At any time t )0, the sample will contain a system of scatterers associated with the solute atoms, as well as the usual scattering centers associated with phonons and lattice defects. Measure- ments of the residua1 electrical resistivity during this pro- cess will permit observation of a growth, a maximum, a decrease and a plateau of the resistivity as the alloy ages. This behavior, called the resistivity anomaly, attracted some attention in the specialized literature at the begin- ning of the last decade. ' Although at the present time the experimental electrical resistivity, because of the ease of measurement, continues to be a technique very useful for the characterization of binary alloys, the theoreti- cal aspects remain anchored to the ideas of Rossister ' and Hillel. ' The increasing use of the thermoelectric power as a technique to characterize alloys during phase separation and the close relation of this physical property with the electrical resistivity' ' have prompted us to reconsider the resistivity theory, since models that satisfactorily ex- plain the resistivity anomaly' do not do well in the analysis of the thermoelectric power' during clustering. We approach this theory of resistivity again, on the basis of the following two facts: First, as pointed out in Sec. II, the true meaning of the damping factor exp( — r, /A) in the isotropic model has not yet been fully considered. In previous calculations that have followed this scheme, the electronic mean free II. RESISTIVITY THEORY: INCLUSION OF THE EXPONENTIAL FACTOR A. Formulation The electrical resistivity, as derived from the Boltzmann equation and expressed within the free- electron model, may be written as P (l) ne r(k) where the relaxation time r(k) takes into account all of the dispersive centers that are present in the alloy. We are interested in the contribution of the solute atoms, which are either fully isolated or else in the form of Guinier-Preston zones. We will assume that this contri- bution and that of the phonons obey Mathiessen's rule. In the Born approximation, the relaxation time associ- ated with scattering by the solute atoms is written as r, (k)= Sk —k' k'8' — W, k 4~ h X ( l — cos81, z )d Ql, path that occurs in the damping factor has been taken to be independent of ~ q ~ = ~ k —k' ~, although in reality it has a ~q~ dependence. This would transform the relaxation- time equation into an integral equation solvable only by iterative techniques. Second, in Sec. III, we treat phenomenologically a model based on the evolution of the scattering power of different dispersive centers, particularly the Guinier- Preston (GP) clusters. This model was suggested in a previous work, ' in which the ideas of Hillel, Edwards, and Wilkes' and those of Rossiter tended to converge. 1992 1992 The American Physical Society