A general equation for the surface tension of metals based on the law of corresponding states

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1 High emperatures ^ High Pressures, 2002, volume 34, pages 393 ^ 400 DOI: /htjr044 A general equation for the surface tension of metals based on the law of corresponding states Hamid Reza Rafiee, Ali Boushehri Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71454, Iran; fax: ; rafiee@chem.susc.ac.ir Received 4 May 2001; in revised form 6 November 2001 Abstract. he temperature dependence of the surface tension of noble metals has been examined by phenomenological scaling and related to the law of corresponding states. Reduced coordinates s ^, where s is reduced surface tension and is reduced reciprocal temperature, are introduced for the prediction of the surface tension of metals. In the correlation, the melting temperature, m, is applied as a corresponding temperature for different metals. A relationship in which is the reduced reciprocal temperature at a scaled distance from the boiling point is used for molten copper, silver, and gold over an extensive temperature range. It is shown that a linear relationship between s and in the form: s ˆ a b exists, where a and b are temperature-independent constants. Using this relationship we have shown that the predicted values of the surface tension of 33 elements at different temperatures are in good agreement with experimental data. 1 Introduction Surface tension, s, is a temperature-dependent property of liquids. When the temperature is raised gradually, the surface tension of a pure liquid in equilibrium with its vapour decreases gradually, and in a nearly linear fashion, until the critical temperature is reached, where the interface fades out and the surface tension vanishes. Since this quantity is important from both theoretical and practical points of view, it has been the subject of study for several researches. Many investigators have tried to connect the interfacial free energy changes to the related bulk properties, in order to allow the determination of surface tension as a function of temperature. At the same time, considerable efforts have been made to develop experimental methods for measuring the surface tension. For example, Egry (1991) showed that the levitated drop technique is an elegant method for accurate measurement of the surface tension of liquid metals. Brooks and Mills (1993) have used this method for determining the surface tension of cobalt, nickel, iron, and gold. By means of this technique, Egry et al (1994) determined the surface tension of copper, silver, and gold even in the undercooled regime. Because they have made appropriate corrections, their reported results are independent of sample mass. We are interested in a general relationship between the surface tension of molten metals and any of its bulk properties that would permit us to predict the surface tension over a wide range of temperatures. Since the boiling temperatures of elements are easier to determine accurately than the critical temperatures, our aim in this work was to introduce a simple and useful empirical correlation between the surface tension at different temperatures and the boiling point of a metal. hus, s and, the reduced surface tension and the reduced reciprocal temperature, respectively, are used to predict the surface tension of noble metals. It is shown that choosing boiling temperature (Ghatee and Boushehri 1994) instead of critical temperature (Chhabra 1991), increases the validity and the range of accuracy of correlation. he recently proposed set of coordinates, s ^, by Ghatee and Boushehri (1994) has also been applied for the calculation of the surface tension of molten alkali halides (Ghatee et al 1997), liquid alkanes (Sheikh and Boushehri 1999), and a number of transition metals (Sheikh and Boushehri 2000). Our purpose here is to apply this correlation for the prediction of

2 394 H R Rafiee, A Boushehri the surface tension of a large number of metals, for which Keene (1993) has compiled accurate data. It is shown that this method can be regarded as a general useful method for the evaluation of surface tension of molten metals. 2 Correlation with selected physical properties Surface tension can be correlated with boiling temperature, b, a quantity that characterises the difference in intermolecular forces of different substances. he phenomenological reduced coordinates proposed by Ghatee and Boushehri (1994) are defined as: b ˆ 1 b 1, (1) m s ˆ s sm, (2) m where s and s m are the surface tensions at temperature and melting temperature, m, respectively. It should be noted that s m is a corresponding energy and m denotes a corresponding temperature. he experimental data on surface tension are taken from the work by Egry et al (1994) providing correlation for three noble metals: copper, silver, and gold. he boiling temperatures at atmospheric pressure were taken from the Dictionary of Chemistry (Daintith 1988). Figure 1 represents the correlation obtained with equations (1) and (2). he solid line in figure 1 is the linear fit to the experimental data, which is obtained by linear regression as: s ˆ 0:6312 0:3691. (3) he linear correlation coefficient for equation (3) is Equation (3) has been employed for the calculation of the surface tension of copper, silver, and gold. he results over a wide range of temperatures ranging from the undercooling temperatures of copper and silver and from the melting temperature of gold up to the highest temperature limit reported by Egry et al (1994) are listed in table 1. Comparison of the calculated values of the surface tension with the reported data by Egry et al (1994) in table 1 shows that the correspondence between the predicted values and the experimental ones is quite good. Comparison of the calculated and experimental values of ds=d, further confirms the usefulness and success of equation (3). his is shown in table copper silver gold s Figure 1. Correlation results for the reduced surface tension, s, versus the reduced reciprocal temperature,.

3 Surface tension of metals 395 able 1. Surface tension calculated from equation (3), s calc, compared with experimental values, s exp, of Egry et al (1994). Metal =K s exp =Nm 1 s calc =Nm 1 Deviation=% Copper Gold Silver able 2. Comparison of calculated values of ds=d with experimental ones. Melt Copper Silver Gold ds ds d exp d calc 3 Correlation at a reduced scaled temperature Correlations based on a scaling law have been used earlier, for example for liquids (Leilmezs and Merriman 1986) and liquid alkali metals (Chhabra 1991). Ghatee et al (1997) showed that by multiplying equations (1) and (2) by (s=s m ) n the following new set of scaled coordinates is obtained: n 1 s m s sc ˆ, (4) s m

4 396 H R Rafiee, A Boushehri and sc ˆ n b m s, (5) b m s m where n is a real number, s m is a corresponding energy, and m is a corresponding temperature. We examined this correlation for six transition elements including copper, silver, and gold (Egry et al 1994); and iron, nickel, and cobalt (Brooks and Mills 1993). If s sc is plotted against sc, a family of linear curves is produced for different values of the reduced scaled temperature, index, where this quantity, introduced by Ghatee et al (1997), is defined as: index ˆ b b m. (6) In figure 2 this behaviour is shown for n ˆ 4. he related data at different reduced temperatures are listed in table 3, and it is clear that the linear correlation coefficients are very close to unity. It should be noted that separate study for the exploration of a theoretical basis for this is required s sc sc Figure 2. he variation of s sc with sc for n ˆ 4 and values of index from 0.1 to 0.9. able 3. Coefficients in the linear relationship fit of s sc versus sc (with n ˆ 4) for different values of index for the six melts: s sc ˆ a b sc. LCC denotes the linear correlation coefficients. index a b LCC

5 Surface tension of metals Prediction of the surface tension of other elements In order to test the applicability of the correlation equation (3) for other metals, the surface tension of cobalt, nickel, and iron was calculated over a wide range of temperatures and able 4. he calculated surface tension, s calc, at different temperatures from equation (3) of this work, compared with the experimental values of Brooks and Mills (1993) and previous calculations by Sheikh and Boushehri (2000) (SB). Metal =K s exp =Nm 1 s calc =Nm 1 Deviation=% this work SB Cobalt Iron : Nickel

6 398 H R Rafiee, A Boushehri compared with the experimental values reported by Brooks and Mills (1993). able 4 shows that the agreement is very good. For comparison, deviations of the calculated values from the experimental ones for the surface tension of these elements at the same temperatures, reported by Sheikh and Boushehri (2000), have also been listed in table 4. A similar calculation and comparison have also been made for other elements of the Periodic able for which the experimental surface tension data have been reported. Analysis of the results listed in table 5 shows that we have devised a simple method for able 5. he calculated surface tension, s calc, at different temperatures from equation (3) of this work, compared with the experimental values taken from Keene (1993), and comparison between the deviations of this work and correlation equation of Sheikh and Boushehri (2000) (SB). Element =K s exp =Nm 1 s calc =Nm 1 Deviation=% this work SB Antimony Bismuth # # Cadmium Cesium # # Chromium # # # Erbium # Germanium # # # Indium # # # # Lanthanum # Lead # # # Magnesium # # # # Manganese Mercury # # # # # # #

7 Surface tension of metals 399 predicting the surface tension of elements that involves only three parameters: m, b, and s m, often available with acceptable accuracy. Although for some of the metals the lack of experimental surface tension data over a considerable range of temperatures prevents us from examining this correlation further, our present scheme covers most of the elements of the Periodic able. It can be seen that at least one element from each group (except halogens and noble gases) is present in table 5. Data for s exp in table 5 were taken from Keene (1993). able 5 (continued). Element =K s exp =Nm 1 s calc =Nm 1 Deviation=% this work SB Niobium # Palladium Platinum Rhodium # Rubidium Samarium # Scandium Selenium # # # Silicon ellurium # erbium # Uranium Vanadium # # Ytterbium # Yttrium # Zinc Zirconium # not reported by Sheikh and Boushehri (2000).

8 400 H R Rafiee, A Boushehri 5 Conclusion We are reporting here a simple correlation between s and based on the law of corresponding states. Using this relationship for 33 elements, we have predicted the surface tension at various temperatures and compared the results with experimental values. Considering the range of experimental error ( 7:5%) mentioned by Chhabra (1991), the agreement between our predicted values with experimental data, reported in tables 1 ^ 5 confirms the applicability and reliability of this treatment. his correlation has two important advantages: simplicity due to not containing the critical parameters, and implicit accuracy. hat is, its generality and its simplicity do not limit its accuracy. Acknowledgement. he support of this work by the Shiraz University Research Council is gratefully acknowledged. his work is supported in part by the Ministry of Sciences of I.R. Iran under grant 80-SC-7-RM. References Brooks R F, Mills K C, 1993 High emp. ^ High Press ^ 664 Chhabra R P, 1991 High emp. ^ High Press ^ 573 Daintith J (Ed.), 1988 Dictionary of Chemistry (Oxford: Oxford University Press) Egry I, 1991 J. Mater. Sci ^ 3003 Egry I, Sauerland S, Jacobs G, 1994 High emp. ^ High Press ^ 223 Ghatee M H, Boushehri A, 1994 High emp. ^ High Press ^ 514 Ghatee M H, Mousazadeh M H, Boushehri A, 1997 High emp. ^ High Press ^722 Keene B J, 1993 Int. Mater. Rev ^ 192 Lielmezs J, Merriman L H, 1986 hermochim. Acta ^ 389 Sheikh S, Boushehri A, 1999 High emp. ^ High Press ^ 202 Sheikh S, Boushehri A, 2000 High emp. ^ High Press ^238 ß 2002 a Pion publication