ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Regular Article
Surface Tension of Liquid Ag–Cu Binary Alloys
Jürgen BrilloGiorgio LaulettaLuca VaianellaElisabetta AratoDonatella GiurannoRada NovakovicEnrica Ricci
Author information

2014 Volume 54 Issue 9 Pages 2115-2119


Surface tension of liquid Ag–Cu binary alloys has been measured contactlessly using the technique of electromagnetic levitation in combination with the oscillating drop technique. The measurements were performed at temperatures above the melting point of alloys. The surface tension values were obtained from an analysis of the frequency spectra of droplet oscillations recorded with a CMOS-camera at 400 fps.

The alloy samples covered the entire composition range. The surface tension data can be described by linear functions of temperature with negative slopes. The new experimental data were compared to the corresponding theoretical values calculated combining the Butler equation with the ideal and subregular solution models, respectively. The agreement with the experimentally obtained data is excellent for the isothermal surface tension and quite reasonable for their temperature coefficients.

1. Introduction

Binary Ag–Cu liquid alloys serve as a model for the study of solidification and growth processes in eutectic materials.1,2) The phase diagram,3,4) exhibits an extended solid solubility range on both sides and an eutectic point at the composition of Ag60Cu40 with a transition temperature of 1053 K.

In the liquid state, the Ag–Cu displays ideal mixing behavior with respect to its density.5) This observation can be associated with a close similarity of the two components, Ag and Cu, which are both in group Ib in the periodic table and, hence, have a similar electronic configuration ([Ar]3d104s1 for Cu and [Kr]4d105s1 for Ag).6)

The Ag–Cu system is also interesting for a number of technical/industrial applications in joining processes. Such applications are brazing, soldering and welding. In particular, ternary Sn–Ag–Cu alloys (SAC) serve as replacement material for the lead containing Sn–Pb alloys conventionally used for soldering electronic compounds.7,8,9,10) Moreover, Ag–Cu alloys are also used in high temperature bonding applications where, at the interface, it comes to the formation of ternary Al–Ag–Cu alloys.11)

In these examples, nucleation, solidification and growth processes go hand-in-hand with key dynamic processes associated to surface and interfacial properties of a melt.12) In order to fully understand such processes, precise knowledge of the surface tension is indispensable.

Although Ag–Cu is chemically rather an inert system, there is still the risk of pollution due to chemical reactions if the surface tension is measured using a container-based technique. On the other hand, containerless methods, such as electromagnetic levitation,13) circumvent this problem. Moreover, and in contrast to container-based techniques, the free surface is the only interface in the system.14)

In this paper, we show the results of surface tension measurement performed on Ag–Cu alloy melts using electromagnetic levitation as a containerless technique. These were obtained and discussed as functions of temperature and alloy composition.

2. Butler Equation

In a number of previous studies, such as15,16,17,18,19,20,21,22,23,24,25,26,27) it was shown that the surface tension of liquid binary and ternary alloys can be successfully described by the Butler equation28) which treats the surface as an individual thermodynamic phase being in equilibrium with the bulk.

When i denotes one element of the alloy, (i = Cu, Ag), γi(T) denote their corresponding surface tensions at temperature T, Bxi, denotes the concentration of element i in the bulk phase, and Sxi the concentration in the surface phase, the Butler equation is given by the following expression for the present case:   

γ AgCu (T)= γ Ag + RT S Ag ln( S x Ag B x Ag ) + 1 S Ag { E G Ag S (T, S x Ag )- E G Ag B (T, B x Ag ) } = γ Cu + RT S Cu ln( 1- S x Ag 1- B x Ag ) + 1 S Cu { E G Cu S (T,1- S x Ag )- E G Cu B (T,1- B x Ag ) } (1)
Here, R is the universal gas constant, EGiB and EGiS (i = Ag, Cu) denote the respective partial excess free energies of component i in the bulk and in the surface layer, and Si is the partial molar surface area of pure liquid i which is estimated from the molar volume of the pure element, Vi, using a value of 1.091 for the geometrical factor.19,29)

In order to apply the Butler model another assumption becomes important, i.e. the approximation of the surface excess free energy, EGS(T, Sxi), by:30)   

E G S (T, S x i )ξ E G B (T, S x i ) (2)
The phenomenological factor ξ accounts in a crude way for the reduced coordination in the surface layer. A constant value of 0.83 was suggested by Tanaka and Iida as a default approximation for liquids with unknown structure.31) This value is also used in the present calculations.

The excess energy EGB, needed in order to derive the partial excess energies is usually written in the following Redlich-Kister form:32)   

E G B = B x Ag B x Cu ν=0 ( ν A+ ν BT) ( B x Ag - B x Cu ) ν (3)
In Eq. (3), the coefficients νA and νB do not depend on temperature and concentrations. Details of the solution of the Butler equation are also described in Refs. 15), 16), 17), 18), 19).

All parameters used to solve the Butler equation are listed in Table 2. The molar volumes are calculated from the densities which have been taken from Ref. 5). They are represented as linear functions of temperature with the two adjustable parameters, ρL being the density at TL and ρT which is the temperature coefficient. The parameters νA and νB needed in Eq. (3) for the calculation of the excess free energy are taken from Witusievicz33) and listed in Table 3.

Table 2. Parameters entering the Butler equation: Surface tension, γL, and density, ρL, of the pure components at liquidus, TL, and their respective temperature coefficients, γT and ρT. Density data were taken from Brillo.5)
TL [K]12341358
γL [N/m]0.8941.334
γT [10–4·N/m/K]–1.91–2.56
ρL [g/cm3]9.157.90
ρL [10–4·g/cm3/K]–7.4–7.6
Table 3. Parameters νA and νB entering the Redlich-Kister equation, Witusievic.33)
ννA [kJ/mol]νB [10–3 kJ/mol/K]

3. Experimental

Experiments were performed in a standard stainless steel high vacuum chamber filled with 800 mbar He, Ar or a mixture of both after prior evacuation. The purity of the gases was 99.9999 vol.%. The samples having typical diameters of 4 mm were positioned in the centre of a levitation coil. The high frequency inhomogeneous electro-magnetic field induces eddy currents inside the sample so that it can be positioned stably against gravity. Due to ohmic losses of the current the sample is heated and molten. In order to adjust a certain desired temperature, it is cooled in a laminar flow of the He gas. The temperature was measured by a pyrometer focussed at the sample from the side. As the emissivity is generally not known, it is determined from the pyrometer output signal at the known liquidus temperature TL under the valid assumption that within the operating wavelength range of the pyrometer the emissivity of the specimen material does not change with temperature. The details of this procedure are described in.5)

In the present work, values for TL were obtained from the phase diagram reported in,3) see Table 1.

Table 1. Parameters TL, γL, and γT of each of the investigated alloys. The table also shows the interpolated surface tension at 1423 K.
CompositionTL [K]γL [N/m]γT [10–4·N/m/K]γ(T = 1423 K) [N/m]

The frequencies of the droplet oscillations serve as a direct measure of the surface tension γ. In ground based electromagnetic levitation, the symmetry of the sample is reduced as compared to a droplet under micro gravity which is almost spherical. This leads to a split up of the l = 2 Rayleigh oscillation mode into a set of five frequencies ωm with m = –2, –1, 0, 1, 2.

These are measured by recording of 4096 images of the oscillating drop using a C-MOS camera with 400 frames per second (fps). An edge detection and analysis algorithm produces frequency spectra from the time signals, such as the one shown in Fig. 1. They are Fourier transforms of the difference and the sum signal of two perpendicular radii. Using symmetry rules13) the frequencies ωm, marked by arrows in Fig. 1, can be identified. γ is calculated from these frequencies using the formula of Cummings and Blackburn:34)   

γ= 3M 160π m=-2 2 ω m 2 -1.9 Ω 2 -0.3( g a ) 2 Ω -2 (4)
Here M is the mass of the sample, a its radius, and g is the gravitational acceleration. In Eq. (4), the parameter Ω corrects for the magnetic pressure. It is calculated from the three translational frequencies, ωX, ωY, ωZ corresponding to the horizontal and vertical movements of the center of gravity of the sample: Ω2 = 1/3(ω2X + ω2Y + ω2Z). From measurements published in17) the relative error Δγ/γ is estimated to be about 5%.
Fig. 1.

Frequency spectra of the surface oscillation of a Ag60Cu40 sample at 1440 K. The solid curve is the Fourier transform of the sum signal of two perpendicular radii and the dotted curve is the Fourier transform of the difference signal.

4. Results

The Ag–Cu samples can be levitated and molten. Due to its high vapour pressure, the maximum temperature was limited to TTL + 300 K. Mass loss due to evaporation did not exceed 1% of the original mass. On the other side, an undercooling was not observed.

Measured surface tension values for pure liquid Ag and Cu, as well as for the investigated binary alloys, Ag60Cu40, Ag40Cu60, Ag20gCu80, and Ag10Cu90 as function of temperature are shown in Fig. 2. For some of these compositions, such as Ag40Cu60 and Ag60Cu40, the measurements were repeated.

Fig. 2.

Measured surface tension γ of liquid Ag–Cu samples versus temperature.

In Fig. 2, the new surface tension data measured values are represented as a linear function of temperature T by straight lines having negative slopes for each composition. Thus, the following linear law can be fitted to the data:   

γ= γ L + γ T (T- T L ) (5)
where γL is the surface tension at liquidus temperature TL and γT is the temperature coefficient ∂γ/∂T. The results of these fits are shown in Fig. 2 as solid straight lines. For each alloy sample, the fitted parameters, Eq. (5), γL and γT, are listed in Table 1.

In order to study the concentration dependence of the surface tension, γ was interpolated by Eq. (5) to T = 1423 K. This temperature was chosen, because it lies in the middle of the temperature ranges of all investigated samples. The results of these interpolations, γ(T = 1423 K), are listed in Table 1 as well. They are plotted in Fig. 3 versus the silver bulk mole fraction BxAg. As can be seen from this figure, the largest surface tension is obtained for pure copper. At 1423 K, γ equals 1.317 N/m. When the bulk silver mole fraction BxAg is increased to 0.1 the surface tension γ decays quickly down to approximately 1.1 N/m. This decay continues upon further increase of BxAg but with diminished speed. For BxAg > 0.5, γ < 1.0 N/m until finally the value of pure silver, γ = 0.858 N/m, is reached (see also Table 1).

Fig. 3.

Isothermal surface tension γ versus silver bulk mole fraction BxAg at T = 1423 K (triangles) at T = 1623 K (circles). The lines correspond to calculations of the Butler equation, Eq. (1), for the ideal (dashed) and the subregular solution model (solid)33) at 1423 K and for the subregular solution model33) at 1623 K (dash-dotted).

In Fig. 4, the temperature coefficient γT is plotted versus BxAg. The scatter of these data is obviously larger than that in Fig. 3. This has to do with the fact that, generally, γT is influenced by different sources of error, such as evaporation or strong sample rotations which directly or indirectly depend on temperature. Therefore an absolute uncertainty of γT can be estimated as ± 50%.

Fig. 4.

Temperature coefficient γT versus silver bulk mole fraction BxAg. The solid line was calculated by the Butler equation, Eq. (1), as described in the text and by using the subregular solution model.33)

Starting on the left hand side (Fig. 4), γT strongly increases from a value of –2.56·10–4 N/m/K, corresponding to pure liquid copper, up to approximately –1.0·10–4 N/m/K when BxAg is increased to only 0.1. Upon further increase of BxAg, γT scatters around values of roughly 5·10–5 N/m/K. A weak and broad maximum (or even a plateau) may be identified around BxAg = 0.5 before γT relaxes back to a value of –1.9·10–4 N/m/K for pure liquid Ag.

5. Discussion

The measured values of γL and γT for the pure elements Ag and Cu, respectively together with corresponding results from various literature sources15,17,35,36,37,38,39,40,41,42,43,44,45) are listed in Tables 4 and 5. Concerning pure silver γL varies around 0.91 (± 0.01) N/m (Table 4) if the freak value, γL = 0.966 N/m reported by Hibiya35) is ignored. In the present work γL is obtained for pure silver as 0.894 N/m. This value is lower than all other parameters, but still within the margins. The same holds for γT which is determined as –1.91·10–4 N/m/K. The average value is γT = –2.1 (± 0.1) 10–4 N/m/K, given in Table 4.

Table 4. Compilation of surface tension values, γL and γT, for liquid silver. (Method keys: EML = electromagnetic levitation, SD = variant of the sessile drop method, R = review).
γL (N·m–1)γT (10–4 N·m–1K–1)Author, ReferenceMethod
0.925–2.1Mills37, Keene38)R
0.894–1.9this workEML
Table 5. Compilation of surface tension values, γL and γT, for liquid copper. (Method keys: EML = electromagnetic levitation, SD = variant of the sessile drop method, R = review).
γL (N·m–1)γT (10–4 N·m–1K–1)Author, ReferenceMethod
1.334–2.56present workEML

In Table 5, the listed values γL of pure copper scatters around 1.31 (± 0.02) N/m. This fits well to the value of 1.334 N/m for γL obtained in the present work. The same was found for the temperature coefficient, γT, which scatters around a value of –2.4 10–4 N/m/K, given in Table 5. γT = – 2.56·10–4 N/m/K is within the margin of approximately ± 0.25·10–4 N/m/K.

The experimental data are compared with the corresponding values obtained by the Butler equation for the ideal and the sub-regular solution models at 1423 K (Fig. 4). In the ideal solution model the Butler equation is solved for EG = 0. As shown in Fig. 4 the course of the experimental data is qualitatively correctly predicted, i.e. that the calculation reproduces that the data collapse on a convex curve. However, the ideal solution model systematically overestimates the experimental data. Depending on composition, the deviation reaches up to 10%.

A much better agreement between the experimental data and the calculated values is obtained if EG is calculated from the regular solution model, i.e. from Eq. (3) with EG taken from Witusievic.33) In Fig. 4, the calculation reproduces the experimental data within their scatter and the agreement is excellent.

In order to compare the Butler equation with the measured temperature coefficients, γT, Eq. (1) is calculated for two different temperatures, T1 = 1423 K and T2 = 1523 K. Then γT is estimated from these runs as γT ≈ (γ(T1)–γ(T2)) / (T1T2). The result is shown in Fig. 4. Qualitatively, the course of the experimental data with their steep increase on the copper rich side and the plateau in the middle is reproduced. The overall agreement between the experimental data and the calculation can be judged as reasonable, although the calculation slightly underestimates the experiment for 0.4 ≤ BxAg ≤ 0.6. Relative to the surface tension, this deviation is in the order of less than <10–5. An uncertainty of γT being in the same order as γT itself should have a negligible effect on the prediction of γ. This is demonstrated in Fig. 3 where the Butler equation is compared to the experimentally obtained surface tensions at 1623 K. As visible, the agreement between calculation and experiment is very good at this temperature as well.

In addition to the surface tension and its temperature coefficient, Eq. (1) also provides information on the composition in the surface layer, see Novakovic.40) The surface mole fraction of silver, SxAg vs the bulk mole fraction, BxAg for T = 1423 K is plotted in Fig. 5. Obviously, silver is enriched in the surface layer as SxAg > BxAg. SxAg strongly increases to SxAg ≈ 0.8 when the corresponding bulk mole fraction BxAg is increased to 0.2. Upon further increase of BxAg the slope of SxAg has drastically become smaller and SxAg finally reaches unity for BxAg = 1.

Fig. 5.

Calculated surface silver mole fraction, SxAg, versus the bulk mole fraction BxAg at 1423 K (solid line) and at 2200 K (dash-dotted line). The symbols correspond to experimental data obtained by AES analysis.46) The dashed line marks the equality of BxAg and SxAg and was drawn in order to aid the eye.

It is justified to ask whether or not the calculated surface compositions are reliable. In general, experimental data for a validation are difficult to obtain. However, in the case of Ag–Cu liquid alloys, Williams and Norris46) have performed Auger electron spectroscopy (AES) measurements on surfaces of Ag–Cu melts. In their experiments, they also determined SxAg. Unfortunately, nothing was reported about the temperature at which these experiments were carried out. This circumstance makes it difficult to compare their results with our calculation. Nevertheless, their results are plotted in Fig. 5 as well.

As visible, the AES data exhibit qualitatively the same trend as our calculation. Namely, they confirm that silver is segregating at the surface. However, the calculation overestimates the experimental data of32) by nearly 100%. Hence, the quantitative agreement between the Butler equation and the AES data is not significant. Of course, the results obtained from the Butler equation can strongly depend on the used input data, for instance on the thermodynamic assessment used, or the fact, that the temperature, for which the calculation was carried out, 1423 K, is most likely not the same as the one at which the AES experiments have been performed. In order to discuss the effect of the different temperatures, T was adjusted in the Butler equation, Eq. (1), such that the AES data could be reproduced, see Fig. 5. However, this was possible only for T = 2200 K which is rather unrealistic. If their temperature was lower than 1423 K segregation would have become even more pronounced in their results.

One can also identify the following as potential reason for the discrepancy in Fig. 5: It is a fact that, although AES is regarded as surface sensitive,47) it will always average over the top few monolayers, depending on the penetration depth of the secondary electrons with their element specific kinetic energies. This effect will certainly decrease the experimental results obtained for SxAg, as segregation is less pronounced in sub surface layers.

6. Summary

Surface tension data of liquid Ag–Cu binary alloys have been measured in a containerless way by the oscillating drop technique. The measurements were performed on the alloy samples covering the entire composition range and for temperatures at and up to 300 K above the melting point. It was found that the surface tensions can be described by linear functions of temperature with negative slopes. The new experimental results of pure silver and copper were in good agreement with corresponding literature data.

Calculations based on the Butler equation using the ideal- and subregular solution approximation exhibited excellent agreement for the isothermal surface tension and quite reasonable agreement for their temperature coefficients. Comparison of the predicted surface composition with published results obtained from Auger analysis was successful only from a qualitative point of view.

Further research will be focussed on the influence of oxygen on the surface tension in this system.


The support by the ERASMUS foundation as well as the Italian Space Agency ASI (contract 2013-027-R.0) is gratefully acknowledged.

© 2014 by The Iron and Steel Institute of Japan