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Anomalous temperature dependence of the surface tension in liquid silver–lead alloys
Journal of Alloys and Compounds 458 (2008) 302–306
Anomalous temperature dependence of the surface tension in liquid silver–lead alloys P. Terzieff ∗ Institute of Inorganic Chemistry, Materials Chemistry, University of Vienna, W¨ahringerstrasse 42, A-1090 Wien, Austria Received 8 June 2006; accepted 18 January 2007 Available online 8 April 2007
Abstract The surface tension of liquid silver–lead alloys is discussed with special emphasis on the unusual temperature coefficients apparent at certain concentrations. In contrast to most other liquid alloys silver-rich mixtures of silver–lead adopt positive temperature coefficients. This anomalous behaviour is found to be well explained in terms of Butler’s model of surface tension. The phenomenological approach of Bhatia and March was applied in semi-empirical manner in order to achieve agreement with the experimental findings. In view of the high concentration fluctuations of the system – expressed in terms of Scc (0) – the anomalous temperature coefficients find a plausible interpretation. A short critical assessment of the model’s general applicability to systems with demixing properties is presented. Qualitatively, both treatments indicate that the unusual temperature dependence of the surface tension is the mere consequence of the thermodynamics of the system. © 2007 Elsevier B.V. All rights reserved. Keywords: Silver–lead alloys; Surface tension; Concentration fluctuations
1. Introduction Recent efforts to optimize the wetting behaviour of lead free solders have led to a renewed interest in the surface tension of liquid solder materials. In the course of numerous reinvestigations focused primarily on multicomponent alloys of copper and silver with tin, indium and other polyvalent metals it has become apparent that in some exceptional cases the surface tension increases with increasing temperature. In the past, most of the attempts to interpret the experimental findings have been based on Butler’s model which considers the surface layer as separate phase being in thermodynamic equilibrium with the bulk [1]. By applying the model to a series of liquid alloys Yeum et al. [2] illustrated that the model was a very successful approach to explain the variation of the surface tension with the composition of the alloys. From the theoretical point of view, less attention has been paid to the temperature dependence of the surface tension. There are recent indications that even the anomalous temperature dependences are well understood in terms of Butler’s model [3,4]. However, the first interesting conclusions are those of Met-
zger [5] who associated the anomalous temperature coefficient observed in silver–lead, silver–bismuth and copper–lead with the segregation tendency of these thermodynamically unstable liquid mixtures. Bathia and March [6] related the surface tension directly to the long-wavelength limit of the concentration–concentration structure factor Scc (0) which is a direct measure of the concentration fluctuations and thus of the thermodynamic instability of the mixture. It was suggested to apply this approach to systems with substantial size differences [6]. Although the model was rather concerned with the variation of the surface tension from one component to the other this paper will be primarily focused on its temperature dependence. The system silver–lead was considered to be the most suitable candidate since it is a typical demixing system void of any indications of compound formation, its anomalous temperature dependence is well established, the size effect is considerably large, and the thermodynamic quantities of mixing are well known. It was, furthermore, of particular interest to see how the two different concepts correlate. 2. Theory and input data
∗
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According to Butler’s model the surface phase is considered to be a monolayer which is in thermodynamic equilibrium with
P. Terzieff / Journal of Alloys and Compounds 458 (2008) 302–306
the bulk phase. Thus, the chemical potential of the ith component in the bulk is equal to its chemical potential in the surface layer, which includes the surface energy. Based on these assumptions, the surface tension of the alloy σ can be related to each component’s thermodynamic activity coefficient in the bulk γ i and in the surface monolayer γim , respectively: RT cm RT γm ln i + ln i σ = σi + Si ci Si γi
(1)
σ i denotes the surface tension of the ith component, ci and cim are the respective atomic fractions, and Si stands for the molar surface area of the pure component. Values of Si were derived from the molar volumes Vi of the pure liquids according to Ref. [7]: 1/3
2/3
Si = bNA Vi
(2)
The geometric factor b was assumed to be equal to that of typically close packed structures (b = 1.091, [8]). The volumes were taken from the compilation of Crawley [9]. The activity coefficients were deduced from the thermodynamic activities (ai = γ i ci ) which, in turn, were obtained in usual manner from the derivative of the free energy of mixing Gm with respect to the composition ci RT ln ai = Gm − (1 − ci )
∂Gm ∂ci
(3)
Assuming that the free energy of the alloy is always proportional to the number interactive contacts between neighbouring atoms, γ i and γim can be related to the respective coordination numbers in the surface layer and the bulk [10]: ln γim zm = ln γi z
(4)
In typical close-packed solid structures (bcc, hcp) the coordination number is 12 for the bulk and 9 for the surface layer. In the liquid bulk, the coordination numbers are in general smaller (z ≈ 11), but the ratio zm /z ≈ 0.75 is presumably not very different from that of the solid state. Eq. (1) holds for each constituent of the system, therefore the solution of the problem was to find those surface concentrations (c1m , c2m ) which via the activity coefficients (γim and γ2m ) satisfy both equations. Based on the phenomenological model of Bhatia and March [6] the surface tension of a binary mixture is in first approximation given by the ratio of the surface thickness and the isothermal compressibility κT . In presence of a finite volume difference between the constituents (δ = 0) the ratio /κT is weighted by an additional factor which takes account of the concentration fluctuations in the long wavelength limit Scc (0) via σ=α
1 2 κT 1 + βδ Scc (0)/RTκT
(5)
α and β are semi-empirical factors introduced in this paper so as to bring the calculated values of σ in scale with the experimental values. The quantitiy was identified with the cubic root of the
303
mean atomic volume. The difference in volume is defined by the size factor δ which is related to partial molar volumes of the constituents (v1 , v2 ): δ=
v2 − v1 c 1 v1 + c 2 v 2
(6)
The variations of the molar volumes with temperature and composition were those quoted by Sauerwald et al. [11]. Due to the lack of experimental results the adiabatic compressibility κS was assumed to vary ideally according to κS =
c1 v1 κS1 + c2 v2 κS2 c1 v 1 + c 2 v 2
(7)
The adiabatic compressibilities of the elements (κS1 , κS2 ) were deduced from the velocity of sound as given by Iida and Guthrie [12] by making use of experimental mass densities [9,11]. The conversion to the isothermal compressibility was effected as usual with the help of the coefficient of thermal expansion αp , the mass density ρ, and the heat capacity Cp of the alloy: α2p T κT =1+ Cp ρκS κS
(8)
The thermal expansion was estimated from the experimental data of Sauerwald et al. [11]. The heat capacities were those given in literature for the pure liquid elements [13] in combination with the excess heat capacities reported by Kawakami [14]. The concentration fluctuations Scc (0) were obtained from the second derivative of the free energy of mixing Gm with respect to the composition: Scc (0) =
RT (∂2 Gm /∂c2 )P,T,N
(9)
The free energies of mixing derived directly from the original emf-measurements of Hager and Wilkomirski [15] yielded results which were not very different from those assessed by Karakaya and Thompson [16]. Therefore, the application of Eqs. (3) and (9) was based on the polynomial presentation given in Ref. [16]. The reference temperature throughout the paper was 1273 K. 3. Discussion Butler’s equation has proven to be a very convenient tool for describing the surface tension of binary and multicomponent mixtures. It contains no adjustable parameters and the required input data are easily accessible. At the system boundaries (cAg = 0, cPb = 0) Eq. (1) matches automatically with the surface tensions of the pure elements. Their values at their melting points and their respective temperature coefficients for the conversion to the reference temperature were taken from Joud et al. [17]. The variation of the experimental surface tension with the composition is shown in Fig. 1, that of the temperature coefficient is illustrated in Fig. 2. As regards the surface tension itself the experimental results of Metzger [5] and those of Joud et al.
304
P. Terzieff / Journal of Alloys and Compounds 458 (2008) 302–306
Fig. 3. Variation of the surface concentrations and their temperature dependence with the nominal composition in liquid alloys of Ag–Pb calculated at 1273 K: m ; (b) surface concentration cm ; (c) temperature (a) surface concentration cAg Pb m . coefficient of cAg Fig. 1. Variation of the surface tension with the composition in liquid alloys of Ag–Pb at 1273 K: (, ) experimental [5,17]; (a and b) full calculation Eqs. (1) and (5); (c) ideal case Eq. (1); (d) simple case Eq. (5) (δ = 0).
[17] are in excellent agreement. There is some divergency in the temperature coefficients, but the main feature – the anomalous temperature dependence in Ag-rich alloys – is found to be well reflected by both sets of data. The experimental accuracy of the temperature coefficient, dσ/dT, is presumably not much better than the scattering of the data points in Fig. 2. The theoretical results obtained from Butler’s model are compatible with the experimental findings. The differences in the surface tension are small only (curve a, Fig. 1) and even the remarkable variation of the temperature coefficient is found to
Fig. 2. Variation of the temperature coefficient of the surface tension with the composition in liquid alloys of Ag–Pb at 1273 K: (, ) experimental [5,17]; (a and b) full calculation Eqs. (1) and (5).
be reproduced in a rough manner (curve a, Fig. 2). The calculated positive coefficients are considerably smaller than experimentally observed, but the highly asymmetric shape of the curves turns out to be the same. Since no other input data are required to solve Eq. (1), the thermodynamics of the system is obviously responsible for the change in sign of dσ/dT. The drastic decrease of the surface tension of Ag due to the admixture of Pb is brought about by the pronounced migration of Pb-atoms into the surface layer. The extremely different concentrations of the constituents on the surface illustrate that throughout the system Pb is the surface active element (Fig. 3). It has been argued that the increase in surface tension with the temperature is due to the increased segregation of the less surface active element into the surface layer [3]. It is indeed informative to see that in this particular range of compositions the calculated surface concentration of Ag increases strongly with the temperature which enhances the surface tension of the alloy. It is apparent from Fig. 3 that the calculated temperature coefficient m passes through a pronounced maximum on the Ag-ride of cAg of the system. The application of the Bhathia–March formula involves a whole series of physical input parameter and not all of them are known with the desired precision. In particular the thermal expansion coefficient, the adiabatic compressibility, and thus the isothermal compressibility are rough estimates only. The variations of the crucial input parameters with the composition are shown in Fig. 4. In absence of a size difference (δ = 0) the surface tension of the alloys is expected to vary as α/κT . In the present paper the factor α was used to fit the calculated curve at cPb = 0 and cPb = 1 to the values of the pure elements (αAg = 7.519 × 10−6 , αPb = 6.432 × 10−6 ). This first approach (curve d, Fig. 1) ignores the effect of alloying and should thus not be very different from the ideal case of Butler’s model which emerges from Eq. (1) by setting all activity coefficients unity (curve c, Fig. 1). In fact, at least in their first approximations the two different models are compatible.
P. Terzieff / Journal of Alloys and Compounds 458 (2008) 302–306
305
Table 1 Surface tension and temperature coefficient in liquid alloys of Ag–Pb calculated at 1273 K Composition cPb
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Surface tension σ (N m−1 )
Temperature coefficient dσ/dT (10−3 N m−1 K−1 )
(a)
(b)
(a)
(b)
0.920 0.668 0.569 0.519 0.489 0.468 0.451 0.436 0.421 0.408 0.395
0.920 0.677 0.548 0.448 0.371 0.327 0.311 0.313 0.328 0.357 0.395
−0.150 0.079 0.066 0.034 0.002 −0.031 −0.056 −0.076 −0.092 −0.102 −0.110
−0.150 −0.038 0.020 0.099 0.173 0.193 0.155 0.093 0.026 −0.046 −0.110
(a) Calculated from Eq. (1). (b) Calculated from Eq. (5) with αAg = 7.519 × 10−6 , αPb = 6.432 × 10−6 and β = 0.1.
Fig. 4. Variation of the input quantities used for the calculation of the surface tension with the composition in liquid Ag–Pb at 1273 K: volume of mixing Vxs , size factor δ, isothermal compressibility κT , excess heat capacity Cp , excess free energy of mixing Gxs , concentration fluctuations Scc (0).
If the actual size difference is taken into account (0.5 ≤ δ ≤ 0.7) the picture is drastically changed. On both sides of the system the admixture of the second component (case of infinite dilution) would lead to dramatic decrease in surface tension which is – at least on the Pb-side – in contrast to the experimental facts (curve e, Fig. 1). Presumably, the model in its original form (β = 1) overestimates the influence of the size effect, since even for the ideal mixture (i.e. Scc (0) = cAg cPb ) the predictions are unrealistic. The semi-empirical treatment of Eq. (5) in terms of the factor β suggests 0.05 < β < 0.15 to be in rough accordance with the experimental results. The full calculation shown in Fig. 1(curve b) refers to the arbitrary choice of β = 0.10. The numerical values are listed in Table 1 together with the calculations derived from Butler’s model. The accordance with the experimental results is not adequate, however, it was not possible to improve the agreement significantly by simply changing the value of β. Irrespective of the choice the model always turned out to overestimate the influence of Scc (0) on the Pb-rich side. Judged by Eq. (5) the temperature coefficient is primarily determined by the variation of the term Scc (0)/κT T with the temperature. The denominator, κT T, is expected to increase with the temperature whereas the variation of Scc (0) depends strongly on the type of interactions occurring in the system. Since mixtures of Ag–Pb tend to segregate, Scc (0) is found to decrease with the temperature. The resulting overall decrease of Scc (0)/κT T with the temperature, in turn, suggests dσ/dT to be positive over a wide range of compositions. This effect is found to be most pronounced around the equiatomic composition where Scc (0) passes through its maximum (Fig. 3). As a result, the calculated curve in Fig. 2(curve b) has a broad symmetric maximum around cPb ≈ 0.5 instead of the highly asymmetric peak at cPb ≈ 0.1,
which is reflected by the experimental points and the calculations from Butler’s model. Bhatia and March emphasized that their equation should not be applied in the neighbourhood of phase separation where the fluctuations tend towards infinity. The present case is far away from such a critical behaviour, and Scc (0) is not much higher than for the ideal mixture, nevertheless the model in original form (i.e. β = 1) seems not to be applicable to present system. Strictly, the actual input data do not justify the application of Eq. (5) since with β = 1 their condition (κT /V )(RT/Scc (0)) > β(1 − δ)δ
(10)
is not fulfilled. It is unclear whether the lack of better agreement is exclusively due to the violation of Eq. (10) or also to the theory itself. The approximative character of the input data can be ruled out as major source of disagreement, their influence is marginal only. The question is rather whether Eq. (5) is applicable to segregating mixtures at all. In view of the small compressibilities of typical metals (κT ≈ 10−11 m2 N−1 ), the usual concentration fluctuations observed in systems which tend to demixing (Scc (0)>0.25), and the large size differ˜ ences which favour the segregation (δ ≈ 0.5) this condition is hardly ever satisfied. The present results suggest that this model is rather applicable to compound-forming systems with small size effects, high compressibilities and negligible fluctuations. In conclusion, Butler’s model yields a fairly good description of the surface tension. The dependences of both composition and temperature are obviously well explained. The model of Bhatia and March seems to be less applicable to the present system. However, due to its direct connection with the thermodynamic instability of the system expressed in terms of Scc (0) it opens an interesting new aspect. Moreover, by virtue of Scc (0) it gives a first estimate of the sign of the temperature coefficient which affords a much more careful analysis if Butler’s model is used instead. It should be noted that – regardless the model – the thermodynamic properties of the system are responsible for the
306
P. Terzieff / Journal of Alloys and Compounds 458 (2008) 302–306
anomalous temperature dependence of the surface tension in Ag–Pb. References [1] J.A.V. Butler, Proc. Roy. Soc. A135 (1935) 348. [2] K.S. Yeum, R. Speiser, D.R. Poirier, Metall. Trans. B20 (1989) 693. [3] M. Kucharski, P. Fima, Monatsh. Chemie—Chemical Monthly 136 (2005) 1841. [4] J. Lee, W. Shimoda, T. Tanaka, Monatsh. Chemie—Chemical Monthly 136 (2005) 1829. [5] G. Metzger, Z. phys. Chem. 211 (1959) 1. [6] A.B. Bhatia, N.H. March, J. Chem. Phys. 68 (1978) 4651.
[7] T.D. Turkdogan, Physical Chemistry of High Temperature Technology, Academic Press, New York, 1980. [8] T.P. Hoar, D.A. Melford, Trans. Faraday Soc. 53 (1957) 315. [9] A.F. Crawley, Int. Met. Rev. 19 (1974) 32. [10] T. Tanaka, T. Iida, Steel Res. 65 (1994) 29. [11] F. Sauerwald, W. Kepp, G. Metzger, Adv. Phys. 16 (1967) 545. [12] T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals, Claredon Press, Oxford, 1988. [13] A.T. Dinsdale, Calphad 15 (1991) 317. [14] Kawakami, Sci. Rep. Tohuku Imp. Univ. 19 (1930) 521. [15] J.P. Hager, I. Wilkomirski, Trans. Met. Soc. AIME 242 (1968) 183. [16] I. Karakaya, W.T. Thompson, Bull. Alloy Phase Diagrams 8 (1987) 326. [17] J.C. Joud, N. Eustahopoulos, A. Bricard, P. Desre, J. Chim. Phys. 70 (1973) 1290.
Anomalous temperature dependence of the surface tension in liquid silver–lead alloys P. Terzieff ∗ Institute of Inorganic Chemistry, Materials Chemistry, University of Vienna, W¨ahringerstrasse 42, A-1090 Wien, Austria Received 8 June 2006; accepted 18 January 2007 Available online 8 April 2007
Abstract The surface tension of liquid silver–lead alloys is discussed with special emphasis on the unusual temperature coefficients apparent at certain concentrations. In contrast to most other liquid alloys silver-rich mixtures of silver–lead adopt positive temperature coefficients. This anomalous behaviour is found to be well explained in terms of Butler’s model of surface tension. The phenomenological approach of Bhatia and March was applied in semi-empirical manner in order to achieve agreement with the experimental findings. In view of the high concentration fluctuations of the system – expressed in terms of Scc (0) – the anomalous temperature coefficients find a plausible interpretation. A short critical assessment of the model’s general applicability to systems with demixing properties is presented. Qualitatively, both treatments indicate that the unusual temperature dependence of the surface tension is the mere consequence of the thermodynamics of the system. © 2007 Elsevier B.V. All rights reserved. Keywords: Silver–lead alloys; Surface tension; Concentration fluctuations
1. Introduction Recent efforts to optimize the wetting behaviour of lead free solders have led to a renewed interest in the surface tension of liquid solder materials. In the course of numerous reinvestigations focused primarily on multicomponent alloys of copper and silver with tin, indium and other polyvalent metals it has become apparent that in some exceptional cases the surface tension increases with increasing temperature. In the past, most of the attempts to interpret the experimental findings have been based on Butler’s model which considers the surface layer as separate phase being in thermodynamic equilibrium with the bulk [1]. By applying the model to a series of liquid alloys Yeum et al. [2] illustrated that the model was a very successful approach to explain the variation of the surface tension with the composition of the alloys. From the theoretical point of view, less attention has been paid to the temperature dependence of the surface tension. There are recent indications that even the anomalous temperature dependences are well understood in terms of Butler’s model [3,4]. However, the first interesting conclusions are those of Met-
zger [5] who associated the anomalous temperature coefficient observed in silver–lead, silver–bismuth and copper–lead with the segregation tendency of these thermodynamically unstable liquid mixtures. Bathia and March [6] related the surface tension directly to the long-wavelength limit of the concentration–concentration structure factor Scc (0) which is a direct measure of the concentration fluctuations and thus of the thermodynamic instability of the mixture. It was suggested to apply this approach to systems with substantial size differences [6]. Although the model was rather concerned with the variation of the surface tension from one component to the other this paper will be primarily focused on its temperature dependence. The system silver–lead was considered to be the most suitable candidate since it is a typical demixing system void of any indications of compound formation, its anomalous temperature dependence is well established, the size effect is considerably large, and the thermodynamic quantities of mixing are well known. It was, furthermore, of particular interest to see how the two different concepts correlate. 2. Theory and input data
∗
Tel.: +43 1427752908; fax: +43 142779529. E-mail address: [email protected].
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.01.186
According to Butler’s model the surface phase is considered to be a monolayer which is in thermodynamic equilibrium with
P. Terzieff / Journal of Alloys and Compounds 458 (2008) 302–306
the bulk phase. Thus, the chemical potential of the ith component in the bulk is equal to its chemical potential in the surface layer, which includes the surface energy. Based on these assumptions, the surface tension of the alloy σ can be related to each component’s thermodynamic activity coefficient in the bulk γ i and in the surface monolayer γim , respectively: RT cm RT γm ln i + ln i σ = σi + Si ci Si γi
(1)
σ i denotes the surface tension of the ith component, ci and cim are the respective atomic fractions, and Si stands for the molar surface area of the pure component. Values of Si were derived from the molar volumes Vi of the pure liquids according to Ref. [7]: 1/3
2/3
Si = bNA Vi
(2)
The geometric factor b was assumed to be equal to that of typically close packed structures (b = 1.091, [8]). The volumes were taken from the compilation of Crawley [9]. The activity coefficients were deduced from the thermodynamic activities (ai = γ i ci ) which, in turn, were obtained in usual manner from the derivative of the free energy of mixing Gm with respect to the composition ci RT ln ai = Gm − (1 − ci )
∂Gm ∂ci
(3)
Assuming that the free energy of the alloy is always proportional to the number interactive contacts between neighbouring atoms, γ i and γim can be related to the respective coordination numbers in the surface layer and the bulk [10]: ln γim zm = ln γi z
(4)
In typical close-packed solid structures (bcc, hcp) the coordination number is 12 for the bulk and 9 for the surface layer. In the liquid bulk, the coordination numbers are in general smaller (z ≈ 11), but the ratio zm /z ≈ 0.75 is presumably not very different from that of the solid state. Eq. (1) holds for each constituent of the system, therefore the solution of the problem was to find those surface concentrations (c1m , c2m ) which via the activity coefficients (γim and γ2m ) satisfy both equations. Based on the phenomenological model of Bhatia and March [6] the surface tension of a binary mixture is in first approximation given by the ratio of the surface thickness and the isothermal compressibility κT . In presence of a finite volume difference between the constituents (δ = 0) the ratio /κT is weighted by an additional factor which takes account of the concentration fluctuations in the long wavelength limit Scc (0) via σ=α
1 2 κT 1 + βδ Scc (0)/RTκT
(5)
α and β are semi-empirical factors introduced in this paper so as to bring the calculated values of σ in scale with the experimental values. The quantitiy was identified with the cubic root of the
303
mean atomic volume. The difference in volume is defined by the size factor δ which is related to partial molar volumes of the constituents (v1 , v2 ): δ=
v2 − v1 c 1 v1 + c 2 v 2
(6)
The variations of the molar volumes with temperature and composition were those quoted by Sauerwald et al. [11]. Due to the lack of experimental results the adiabatic compressibility κS was assumed to vary ideally according to κS =
c1 v1 κS1 + c2 v2 κS2 c1 v 1 + c 2 v 2
(7)
The adiabatic compressibilities of the elements (κS1 , κS2 ) were deduced from the velocity of sound as given by Iida and Guthrie [12] by making use of experimental mass densities [9,11]. The conversion to the isothermal compressibility was effected as usual with the help of the coefficient of thermal expansion αp , the mass density ρ, and the heat capacity Cp of the alloy: α2p T κT =1+ Cp ρκS κS
(8)
The thermal expansion was estimated from the experimental data of Sauerwald et al. [11]. The heat capacities were those given in literature for the pure liquid elements [13] in combination with the excess heat capacities reported by Kawakami [14]. The concentration fluctuations Scc (0) were obtained from the second derivative of the free energy of mixing Gm with respect to the composition: Scc (0) =
RT (∂2 Gm /∂c2 )P,T,N
(9)
The free energies of mixing derived directly from the original emf-measurements of Hager and Wilkomirski [15] yielded results which were not very different from those assessed by Karakaya and Thompson [16]. Therefore, the application of Eqs. (3) and (9) was based on the polynomial presentation given in Ref. [16]. The reference temperature throughout the paper was 1273 K. 3. Discussion Butler’s equation has proven to be a very convenient tool for describing the surface tension of binary and multicomponent mixtures. It contains no adjustable parameters and the required input data are easily accessible. At the system boundaries (cAg = 0, cPb = 0) Eq. (1) matches automatically with the surface tensions of the pure elements. Their values at their melting points and their respective temperature coefficients for the conversion to the reference temperature were taken from Joud et al. [17]. The variation of the experimental surface tension with the composition is shown in Fig. 1, that of the temperature coefficient is illustrated in Fig. 2. As regards the surface tension itself the experimental results of Metzger [5] and those of Joud et al.
304
P. Terzieff / Journal of Alloys and Compounds 458 (2008) 302–306
Fig. 3. Variation of the surface concentrations and their temperature dependence with the nominal composition in liquid alloys of Ag–Pb calculated at 1273 K: m ; (b) surface concentration cm ; (c) temperature (a) surface concentration cAg Pb m . coefficient of cAg Fig. 1. Variation of the surface tension with the composition in liquid alloys of Ag–Pb at 1273 K: (, ) experimental [5,17]; (a and b) full calculation Eqs. (1) and (5); (c) ideal case Eq. (1); (d) simple case Eq. (5) (δ = 0).
[17] are in excellent agreement. There is some divergency in the temperature coefficients, but the main feature – the anomalous temperature dependence in Ag-rich alloys – is found to be well reflected by both sets of data. The experimental accuracy of the temperature coefficient, dσ/dT, is presumably not much better than the scattering of the data points in Fig. 2. The theoretical results obtained from Butler’s model are compatible with the experimental findings. The differences in the surface tension are small only (curve a, Fig. 1) and even the remarkable variation of the temperature coefficient is found to
Fig. 2. Variation of the temperature coefficient of the surface tension with the composition in liquid alloys of Ag–Pb at 1273 K: (, ) experimental [5,17]; (a and b) full calculation Eqs. (1) and (5).
be reproduced in a rough manner (curve a, Fig. 2). The calculated positive coefficients are considerably smaller than experimentally observed, but the highly asymmetric shape of the curves turns out to be the same. Since no other input data are required to solve Eq. (1), the thermodynamics of the system is obviously responsible for the change in sign of dσ/dT. The drastic decrease of the surface tension of Ag due to the admixture of Pb is brought about by the pronounced migration of Pb-atoms into the surface layer. The extremely different concentrations of the constituents on the surface illustrate that throughout the system Pb is the surface active element (Fig. 3). It has been argued that the increase in surface tension with the temperature is due to the increased segregation of the less surface active element into the surface layer [3]. It is indeed informative to see that in this particular range of compositions the calculated surface concentration of Ag increases strongly with the temperature which enhances the surface tension of the alloy. It is apparent from Fig. 3 that the calculated temperature coefficient m passes through a pronounced maximum on the Ag-ride of cAg of the system. The application of the Bhathia–March formula involves a whole series of physical input parameter and not all of them are known with the desired precision. In particular the thermal expansion coefficient, the adiabatic compressibility, and thus the isothermal compressibility are rough estimates only. The variations of the crucial input parameters with the composition are shown in Fig. 4. In absence of a size difference (δ = 0) the surface tension of the alloys is expected to vary as α/κT . In the present paper the factor α was used to fit the calculated curve at cPb = 0 and cPb = 1 to the values of the pure elements (αAg = 7.519 × 10−6 , αPb = 6.432 × 10−6 ). This first approach (curve d, Fig. 1) ignores the effect of alloying and should thus not be very different from the ideal case of Butler’s model which emerges from Eq. (1) by setting all activity coefficients unity (curve c, Fig. 1). In fact, at least in their first approximations the two different models are compatible.
P. Terzieff / Journal of Alloys and Compounds 458 (2008) 302–306
305
Table 1 Surface tension and temperature coefficient in liquid alloys of Ag–Pb calculated at 1273 K Composition cPb
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Surface tension σ (N m−1 )
Temperature coefficient dσ/dT (10−3 N m−1 K−1 )
(a)
(b)
(a)
(b)
0.920 0.668 0.569 0.519 0.489 0.468 0.451 0.436 0.421 0.408 0.395
0.920 0.677 0.548 0.448 0.371 0.327 0.311 0.313 0.328 0.357 0.395
−0.150 0.079 0.066 0.034 0.002 −0.031 −0.056 −0.076 −0.092 −0.102 −0.110
−0.150 −0.038 0.020 0.099 0.173 0.193 0.155 0.093 0.026 −0.046 −0.110
(a) Calculated from Eq. (1). (b) Calculated from Eq. (5) with αAg = 7.519 × 10−6 , αPb = 6.432 × 10−6 and β = 0.1.
Fig. 4. Variation of the input quantities used for the calculation of the surface tension with the composition in liquid Ag–Pb at 1273 K: volume of mixing Vxs , size factor δ, isothermal compressibility κT , excess heat capacity Cp , excess free energy of mixing Gxs , concentration fluctuations Scc (0).
If the actual size difference is taken into account (0.5 ≤ δ ≤ 0.7) the picture is drastically changed. On both sides of the system the admixture of the second component (case of infinite dilution) would lead to dramatic decrease in surface tension which is – at least on the Pb-side – in contrast to the experimental facts (curve e, Fig. 1). Presumably, the model in its original form (β = 1) overestimates the influence of the size effect, since even for the ideal mixture (i.e. Scc (0) = cAg cPb ) the predictions are unrealistic. The semi-empirical treatment of Eq. (5) in terms of the factor β suggests 0.05 < β < 0.15 to be in rough accordance with the experimental results. The full calculation shown in Fig. 1(curve b) refers to the arbitrary choice of β = 0.10. The numerical values are listed in Table 1 together with the calculations derived from Butler’s model. The accordance with the experimental results is not adequate, however, it was not possible to improve the agreement significantly by simply changing the value of β. Irrespective of the choice the model always turned out to overestimate the influence of Scc (0) on the Pb-rich side. Judged by Eq. (5) the temperature coefficient is primarily determined by the variation of the term Scc (0)/κT T with the temperature. The denominator, κT T, is expected to increase with the temperature whereas the variation of Scc (0) depends strongly on the type of interactions occurring in the system. Since mixtures of Ag–Pb tend to segregate, Scc (0) is found to decrease with the temperature. The resulting overall decrease of Scc (0)/κT T with the temperature, in turn, suggests dσ/dT to be positive over a wide range of compositions. This effect is found to be most pronounced around the equiatomic composition where Scc (0) passes through its maximum (Fig. 3). As a result, the calculated curve in Fig. 2(curve b) has a broad symmetric maximum around cPb ≈ 0.5 instead of the highly asymmetric peak at cPb ≈ 0.1,
which is reflected by the experimental points and the calculations from Butler’s model. Bhatia and March emphasized that their equation should not be applied in the neighbourhood of phase separation where the fluctuations tend towards infinity. The present case is far away from such a critical behaviour, and Scc (0) is not much higher than for the ideal mixture, nevertheless the model in original form (i.e. β = 1) seems not to be applicable to present system. Strictly, the actual input data do not justify the application of Eq. (5) since with β = 1 their condition (κT /V )(RT/Scc (0)) > β(1 − δ)δ
(10)
is not fulfilled. It is unclear whether the lack of better agreement is exclusively due to the violation of Eq. (10) or also to the theory itself. The approximative character of the input data can be ruled out as major source of disagreement, their influence is marginal only. The question is rather whether Eq. (5) is applicable to segregating mixtures at all. In view of the small compressibilities of typical metals (κT ≈ 10−11 m2 N−1 ), the usual concentration fluctuations observed in systems which tend to demixing (Scc (0)>0.25), and the large size differ˜ ences which favour the segregation (δ ≈ 0.5) this condition is hardly ever satisfied. The present results suggest that this model is rather applicable to compound-forming systems with small size effects, high compressibilities and negligible fluctuations. In conclusion, Butler’s model yields a fairly good description of the surface tension. The dependences of both composition and temperature are obviously well explained. The model of Bhatia and March seems to be less applicable to the present system. However, due to its direct connection with the thermodynamic instability of the system expressed in terms of Scc (0) it opens an interesting new aspect. Moreover, by virtue of Scc (0) it gives a first estimate of the sign of the temperature coefficient which affords a much more careful analysis if Butler’s model is used instead. It should be noted that – regardless the model – the thermodynamic properties of the system are responsible for the
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