Experimental Investigation and Modeling of the Influence of Microstructure on the Resistive Conductivity of a Cu-Ag-Nb in situ Composite
D. Raabe, F. Heringhaus
A Cu-8.2 wt% Ag-4 wt% Nb in situ metal matrix composite was manufactured by inductive melting, casting, swaging, and wire drawing. The final wire (h = ln(A0/A) = 10.5, A: wire cross section) had a strength of 1840 MPa and up to 46% of the conductivity of pure Cu. The electrical resistivity of the composite wires was investigated as a function of wire strain and temperature. The microstructure was examined by means of optical and electron microscopy. The observed decrease in conductivity with increasing wire strain is interpreted in terms of inelastic electron scattering at internal phase boundaries. The experimental data are in very good accord with the predictions of an analytical size-effect model which takes into account the development of the filament spacing as a function of wire strain. The experimentally obtained and calculated resistivity data are compared to those of the pure constituents.
Due to their combination of high strength and good electrical conductivity, in situ processed Cu based metal matrix composites (MMCs) are considered as candidate materials for the production of highly mechanically stressed electrical devices. Applications in the fields of steady state and long-pulse high-field resistive magnet design [1-7] and industrial robotics  are of particular interest in this context.
Most studies in the past concentrated on the investigation of binary Cu-Nb [9-20] and Cu-Ag composites [21-27]. Cu and Nb form a quasi monotectic system and have no mutual solubility by practical means [28,29]. During casting the Nb solidifies dendritically and forms into thin elongated curled filaments during wire drawing. Alloys consisting of Cu and 20 wt% Nb have an ultimate tensile strength (UTS) of up to 2.2 GPa (h = 12) [12,13,16,17].
Cu and Ag form a simple eutectic system with limited substitutional solubility . Alloys containing more than 6% Ag usually reveal two phases, a Cu-rich solid solution and a Cu-Ag eutectic. During deformation the eutectic and the Cu-rich matrix form into lamellar filaments. An intermediate heat treatment leads to Ag precipitations. These can be used to increase the matrix strain and to form additional fibers during further drawing. After large wire strains Cu-Ag MMCs reveal an UTS of up to s = 1.5 GPa (h = 10) [22,23].
In continuation of these studies on binary alloys recent work focussed on the development of a new generation of ternary in situ Cu-Ag-Nb composites with the aim to combine the hardening effects of both, Nb and Ag [31-35].
While the processing and the mechanical properties of these ternary compounds have been the subject of previous publications [31-35], their electromagnetic properties have not yet been studied. The resistive electrical conductivity of heavily deformed Cu or Ag based composites is typically smaller than expected from the linear rule of mixtures (ROM). This effect is commonly referred to as size-effect. It is due to inelastic scattering of the conduction electrons at the internal phase boundaries [9,20,27,36,37].
This study presents an experimental investigation of the evolution of the resistive electrical conductivity of a heavily wire drawn ternary Cu‑8.2 wt% Ag‑4 wt% Nb in situ processed metal matrix composite as a function of wire strain and temperature and its dependence on the microstructure. The experimental observations are quantitatively compared to predictions of an analytical size-effect model which takes into account the development of the filament spacing as a function of the wire strain.
2 Sample preparation
A Cu‑8.2 wt% Ag‑4 wt% Nb alloy was prepared by inductive melting using a frequency of 10 kHz and a power of 50 kW . All constituents had an initial purity of at least 99.995%. Ingots of 18 mm diameter were cast under an Argon atmosphere at a pressure of 0.6×105 Pa. A crucible and a mould of high purity graphite were used. The mould was preheated to about 600°C to ensure good fluidity and filling. From the cast cylindrical ingots wires were produced by rotary swaging and drawing through hard metal drawing bench dies. A maximum true wire strain above h = 10 (h = ln(A0/A), A: wire cross section) was attained without intermediate annealing. Further processing details are reported elsewhere [34,38].
3 Experimental procedure
3.1 Experimental investigation of the microstructure
Optical and scanning electron microscopy (SEM) were used to determine the morphology and topology both of the Cu matrix and of the Ag and Nb filaments. Due to insufficient contrast, an unambiguous optical identification of the various phases was sometimes not possible. The samples were thus additionally analyzed using energy-disperse X-ray spectrometry (EDX). The morphology of the isolated Nb fibers was also investigated by use of a selective etching technique, where the Cu and Ag were dissolved by dilute nitric acid. Details about the experimental procedure were reported previously .
3.2 Experimental investigation of the resistive electrical conductivity
Systematic measurements of the resistive conductivity at 298 K and 77 K as a function of the wire strain were carried out for Cu‑8.2 wt% Ag‑4 wt% Nb by means of the direct current (DC) four-probe technique using a sample current of 100 mA. For some Cu‑8.2 wt% Ag‑4 wt% Nb, pure Cu, pure Ag, and pure Nb wires of identical total wire strain the influence of the temperature on the resistive conductivity was studied within the temperature range 3 K to 350 K. The data were taken continuously during cooling. The cooling rate was controlled using a small heater, attached close to the samples, and by addition of a He exchange gas. For the measurement of the temperature a carbon-glass resistance sensor was applied.
4 Experimental Results
4.1 Microstructure evolution as a function of wire strain
Fig. 1 shows the development of the diameters of the Cu matrix phase (dCu), the Nb filaments (dNb), and the Ag filaments (dAg) in the composite as a function of the true wire strain h. At low strains the Ag filaments were thicker and shorter than the Nb filaments. With increasing strain their average thickness became more similar to that of the Nb filaments. At a wire strain of h = 3.6 the average Ag filament diameter amounted to dAg » 676 nm and at h = 6 to dAg » 260 nm. At a wire strain of h = 2.6 the average Nb filament diameter amounted to dNb » 529 nm and at h = 9.5 to dNb » 66 nm.
For including the evolution of the filament morphology in an analytical size-effect model the average phase diameters were exponentially fitted from the metallographic data according to dCu = 31767 nm ´ exp(‑0.6415 h), dAg = 2630 nm ´ exp(‑0.3861 h), and dNb = 1386.6 nm ´ exp(‑0.4143 h).
Fig.1: Development of the diameters of the Nb filaments, the Ag filaments, and the Cu matrix phase in Cu - 4 wt.% Nb - 8.2 wt.% Ag as investigated by quantitative SEM.
Fig.2: Electrical resistivity of Cu-8.2wt%Ag-4wt%Nb as a function of logarithmic deformation at 298K and 77K
4.2 Resistive conductivity as a function of wire strain and temperature
Fig. 2 shows the dependence of the electrical resistivity of Cu‑8.2 wt% Ag‑4 wt% Nb at 298 K and 77 K on the strain. At true wire strains above h = 8.5 the conductivity decreases drastically. The increase in the resistivity with increasing strain is more pronounced at 77 K (from ~10 nWm at h = 3.5 to ~20 nWm at h » 10) than at 298 K (from ~27 nWm at h = 3.5 to ~38 nWm at h » 10). Consequently, the resitivity ratio of Cu‑8.2 wt% Ag‑4 wt% Nb, r(273 K)/r(77 K), drops as a function of the true wire strain (Fig. 3).
At temperatures below the transition temperature of pure Nb the wire drawn Cu‑8.2 wt% Ag‑4 wt% Nb reveals superconducting properties. Fig. 4 shows for the composite the transition to the superconducting state as a function of strain. The data reveal that an increasing wire strain leads to a shift of the transition temperature to lower values. A detailed analysis of the superconducting properties of the ternary composite will be given in a subseqeunt paper .
The electrical resistivity of pure Cu wires and pure Ag wires was practically independent of the degree of deformation and was always lower than for the composite (Fig. 5). The effect of deformation on the resistivity temperature coefficient of pure Cu, pure, Ag, pure Nb, and Cu‑8.2 wt% Ag‑4 wt% Nb is given in Fig. 6. While the temperature coefficient of the pure wires is practically independent on wire strain, that of the composite drops with increasing wire strain.
Fig. 3. Electrical resistivity ratio r.298 K.=r.77 K. of the ternary Cu±8.2 wt% Ag±4 wt% Nb composite as a function of the true (logarithmic) wire strain h (h = ln.A0=A., A: wire cross section).
Fig.4: Transition of the electrical resistivity of the ternary Cu±8.2 wt% Ag±4 wt% Nb composite from the normal resistive state to the superconducting state as a function of temperature for two different degrees of the true (logarithmic) wire strain (h = 8, h = 5:96) (h = ln.A0=A., A: wire cross section).
5 Modeling of the electrical resistivity of the Cu-Ag-Nb composite
5.1 Fundamentals of the model
Any analytical calculation of absolute values of the electrical resistivity of metal matrix composites requires very detailed data both on the impurity content and distribution and on the density and distribution of the lattice defects (e.g. dislocation density, grain size, etc.). Since these data are usually not known with sufficient reliablility, analytical models of the resistivity of composites are commonly exclusively based on the size effect, which typically provides by far the most dominant contribution to inelastic internal scattering of the conduction electrons in such materials [9,20,27,36,37]. The predictions of such models can then be compared with the relative changes observed experimentally.
Analytical predictions of the electrical resistivity of composites on the basis of the size effect require the volume fractions and the topologies of each phase in the composite and some intrinsic constants.
The model starts with the calculation of the electrical resistivity of each phase according to the phenomenological expression for surface and phase boundary scattering given by Sondheimer .
where r(d) is the resistivity as a function of the filament thickness, r0 the resistivity for a sample without scattering at phase boundaries (infinite sample or phase size), p the scattering factor, l0 the mean free path of the conduction electrons in that particular phase, d the thickness of the filament and P its perimeter. According to Dingle  the scattering factor p represents the probability of elastic scattering and (1-p) that of inelastic scattering at the phase boundary.
The mean free paths of the conduction electrons were determined from the resistivities of the bulk phases using
where s is the conductivity and the electron density. The right hand side of eq.(2) was calculated using a value of =5.57×1029 m-3 under the assumption of mono-valence for Ag and Cu and a value of =5.325×1029 m-3 for Nb.
Finally, the individual resistivities ri of the phases were topologically combined to give the overall resistivity of the composite rMMC. The model treats all phases as resistors that are arranged parallel.
where the vi are the volume fractions of the phases i=1...n, and ri the electrical resistivities of the phases i=1...n.
Fig.6: Temperature coefficient of the electrical resistivity of Cu, Ag, Nb and Cu-8.2wt%Ag-4wt%Nb between 273K and 373K as a function of the logarithmic deformation h
5.2 Application of the model
The prediction of the electrical resistivity of the Cu‑8.2 wt% Ag‑4 wt% Nb composite on the basis of eqs.(1)–(3) requires some topological considerations about the incorporation of the topology of the Ag filaments. In the as–cast sample the Ag had a lamellar shape and formed a Ag-Cu eutectic whilst the Nb was precipitated in the form of isolated Wulff-polyhedra and dendrites [33,34]. Therefore, the composite was treated as a material consisting of two primary phases, namely, 95.82 vol.% Cu–Ag and 4.18 vol.% Nb, in which the former consists of two sub-phases, namely, 91.92 vol.% Cu and 8.08 vol.% Ag. The weight fractions of the primary phases and of the sub-phases were calculated from the weight composition of Cu‑8.2 wt% Ag‑4 wt% Nb as (90.66 wt.% Cu‑9.34 wt.% Ag)‑4 wt.% Nb and the volume fractions correspondingly as (91.92 vol.% Cu‑8.08 vol.% Ag)‑4.18 vol.% Nb. Eq. (3) was first applied to the two sub-phases Cu and Ag which then in turn were combined parallel with Nb to form the overall composite.
Values for the interface scattering factors p associated with the Cu‑Ag interfaces were taken from previous investigations and simulations on this binary system . These were for Cu‑Ag: p=0.81 at T=298 K and p=0.84 at T=77 K. The size effect approach as outlined in eq. (1) can be derived form the Boltzmann transport equation for cases where d>l0. This condition holds for Cu the (dCu/l0 Cu » 3), the Ag (dAg/l0 Ag » 3), and the Nb phase (dNb/l0 Nb » 20) corresponding to the current data for h=10 and T=293 K.
5.3 Predictions of the model
The results for the size effect of the composite, Cu‑8.2 wt% Ag‑4 wt% Nb, and all phases, Cu, Ag, CuAg, and Nb, are given in Fig. 10 and Fig. 11 as a function of the wire deformation for T=298K and T=77K. A strong size effect can be found in the Cu and Ag phases and thus in CuAg, whereas Nb reveals only small changes with deformation. Agreement with the experimental data is only found in the regimes of low and very high deformation, while intermediate deformation leads to lower values in the simulation than experimentally.
6.1 Resistive conductivity
The resistivity of Cu‑8.2 wt% Ag‑4 wt% Nb increases considerably with the degree of deformation. The strong dependence may be chiefly attributed to the scattering of conduction electrons at the various phase boundaries. This effect becomes particularly pronounced when the average filament spacing is, after heavy deformation, of the same order of magnitude as the mean free path of the conduction electrons in the Cu and the Ag phase (Fig. 2). Since the mean free electron path linearly decreases in the regime above the Debye temperature, due to the decrease of phonon scattering, the contribution of interface scattering to the overall resistivity is more pronounced at low than at elevated temperatures (Fig. 3). Since the Cu-Ag interfaces have a low, and the Cu-Nb and Ag-Nb interfaces a very high portion of inelastic scattering, the latter are assumed to be primarily responsible for the observed increase in resistivity.
The density of deformation induced lattice dislocations is of minor importance for the dependence of the resistivity on wire strain since only their cores add to the electrical resistivity, but contribute only a very small resistivity change per unit length of a dislocation. The here applied DC four-probe technique is not accurate enough for exactly accounting for such a change.
Fig. 7 Modeled and experimentally observed relative changes of the electrical resistivity of Cu, Nb, Ag, and the ternary Cu±8.2 wt% Ag±4 wt% Nb composite at T . 298 and 77 K as a function of the true (logarithmic) wire strain η (η . ln.A0=A., A: wire cross section).
(a) T . 77 K, size e€ect model for the various phases and for the compo-site.
(b) T . 77 K, comparison between size-e€ect model and experimental data.
(c) T . 298 K, size-e€ect model for the various phases and for the composite.
(d) T . 298 K, comparison between size-e€ect model and experimental data.
The composite (Cu-8.08vol.%Ag)-4.18vol.%Nb can be described as consisting of two first-order phases, 95.82vol.% CuAg and 4.18vol.% Nb, the former of which again consists of two second-order phases, 91.8vol.% Cu and 8.08vol.% Ag (cf. Fig. 7). With this approach, a calculation and prediction of the “size effect” due to interface scattering is possible. A determination of the absolute values of the electrical resistivity is not feasible due to effects of impurities and mutual solution. The latter, although of particular importance in the CuAg phase, cannot be incorporated owing to the unavailability of appropriate experimental data. For the same reasons of uncertainty, the calculations regarding the “size effect” may not reveal a good agreement with the experimental results (Fig. 10 and Fig. 11). The variance found at intermediate degrees of deformation may be due to deformation-induced changes in the solubility or the state of precipitation. These may either not be as dominant anymore at high degrees of deformation and strong interface scattering or be reverted owing to the presence of a high density of internal interfaces.
Regarding the effect of the microstructure of the individual phases, it may be concluded that the “size effect” in the Cu phase and the Ag phase are responsible for the observed increase in the electrical resistivity of the CuAg phase and ultimately the composite. In comparison to the Cu phase and the CuAg phase, the size effect in Ag, particularly at high strains, is much stronger at T=298K than at T=77K. This is due to the stronger temperature-dependence of the development of the mean free path in Cu.
A ternary in situ Cu‑8.2 wt% Ag‑4 wt% Nb and MMCs was manufactured by melting, casting, and wire drawing. The microstructure was investigated using electron microscopy and EDX. The resistive conductiving properties were examined using four probe DC tests at various temperatures. The main results are:
The Cu-Ag-Nb material was very ductile. A maximum wire strain of h = 10.5 was reached without intermediate annealing.
Cu-Ag-Nb wires of maximum strain (hmax = 10.5) had an UTS of 1840 MPa and 46% of the conductivity of pure Cu (IACS).
The electrical conductivity of the ternary composite at large strains decreased with increasing strain. This was attributed to the size effect, i.e. to the inelastic scattering of the conduction electrons at the internal interfaces.
The authors are indebted to G. Gottstein and H.-J. Scheider-Muntau for helpful discussions. One of the authors (D.R.) gratefully acknowledges the financial support by the Deutsche Forschungsgemeinschaft through the Heisenberg program and the by the National High Magnetic Field Laboratory in Tallahassee, Florida.
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