Introduction

Surface tension is one of the most important physical properties of materials. For example, surface tension of liquid metal can determine the stability of nano-sized liquid metal and influence the stability of liquid phase and its phase transition temperatures. However, accurate measurement of surface tension at high temperatures is very challenging. In recent years, additive manufacturing (AM) technology has been widely employed to produce complex components and repair existing components.[1] AM process involves the melting of materials even above 3000 K followed by a rapid solidification. Among the various physico-chemical properties of materials, the surface tension emerges as a critical physical property, governing the wetting and spreading behavior of molten material on the solid substrate while also generating Marangoni stress due to temperature gradients, which induces circulatory motion in the melt pool.[2,3,4,5,6,7,8] These melt pool behaviors significantly impact the quality of fabricated parts, ultimately affecting their surface finish, microstructure, and mechanical properties. Having a comprehensive understanding of how surface tension of liquid metal varies with temperature and composition is crucial for modeling processes and designing new alloys.

In general, the surface tension of liquid metal \({\sigma }_{T}\) is a function of temperature known to decrease linearly with increasing temperature[9]:

$${\sigma }_{T}={\sigma }_{m}+\frac{d{\sigma }_{T}}{dT}(T-{T}_{m}),$$
(1)

where \({\sigma }_{m}\) is the surface tension of liquid metal at its melting temperature \({T}_{m}\), and \(d{\sigma }_{T}\)/\(d\text{T}\) is the temperature coefficient. Despite numerous studies determining both \({\sigma }_{m}\) and \(d{\sigma }_{T}\)/\(d\text{T}\) of liquid metals, the experimental values are often scattered and unreliable.[9] This is due to the difficulties in the measurement at high temperatures and the significant influence of impurities like oxygen and sulfur in liquid metals on the surface energy.

In light of these experimental challenges, the prediction of \({\sigma }_{m}\) and \(d{\sigma }_{T}\)/\(dT\) using theoretical approaches has been attempted by many researchers.[10,11,12,13,14] These approaches have ended up as semi-empirical or empirical equations. One of the most widely used semi-empirical equations for \({\sigma }_{m}\) was proposed by Iida et al.[15], who derived an equation for the surface tension of liquid metallic elements based on the work energy required to create a unit surface area. Iida et al. used a simple interatomic potential function between atomic pairs in liquid state. They calculated the work energy to separate an atomic pair by assuming a constant ratio between the separation distance and equilibrium distance of an atomic pair. Although the equation by Iida et al. can give a good prediction of \({\sigma }_{m}\) for many liquid metals, the \({\sigma }_{m}\) of transition metals and elements in groups 13 to 15 are predicted less accurately.

The purpose of this study is to present a new approach to predict the surface tension of pure liquid metal by modifying the existing model of Iida et al.[15] for \({\sigma }_{m}\). For this, the semi-empirical equation of Iida for \({\sigma }_{m}\) was revisited to derive more theoretical equation for \({\sigma }_{m}\) using available physical property data. Then, \(d{\sigma }_{T}\)/\(dT\) was also predicted from critical temperature (\({T}_{c}\)) by adopting Eötvös law.[16] In the end, the more accurate surface tension of liquid metallic elements can be calculated by using the results in the present study.

Iida’s Surface Tension Model for \({{{\sigma}}}_{{{m}}}\)

Iida et al.[15] assumed a simplified interatomic potential \({\emptyset }_{s}\) between a pair of atoms separated by a distance \(r\), as shown in Figure 1, to derive the semi-empirical equation for the \({\sigma }_{m}\) of liquid metals. The distance required to separate one atomic pair \({r}_{c}\) from its equilibrium distance \({r}_{e}\) is defined as \({r}_{c}=(1+\alpha ){r}_{e}\) where \({\emptyset }_{s}\) becomes nearly zero. Then, they derived an equation to calculate the \({\sigma }_{m}\) by determining the work needed to separate atoms to a distance \({r}_{c}\) using the \({\emptyset }_{s}\) based on Lindemann’s melting formula for atomic frequency.[17] In their original equation, \({\sigma }_{m}\) was expressed as a function of melting temperature \({T}_{m}\).

$${\sigma }_{m}=\frac{{\left(\pi c\right)}^{2}}{2{N}_{Av.}}\cdot {\left(\alpha \beta \right)}^{2}\frac{{T}_{m}}{{V}_{m}^{2/3}},$$
(2)

where \(c\) is a constant (2.8 ~ 3.1 \(\cdot \) 1012, roughly same for all metals); \(\alpha \), the parameter for \(\alpha ={r}_{c}/{r}_{e} -1\), as shown in Figure 1, \(\beta \), a correction factor of atomic frequency from the solid to the liquid state; \({N}_{Av.}\), Avogadro number; and \({V}_{m}\), the atomic volume of liquid metals at \({T}_{m}\).

Fig. 1
figure 1

Schematic diagram of simplified interatomic potential

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In the original paper by Iida et al.[15], they compared the experimental \({\sigma }_{m}\) and \({T}_{m}\)/\({V}_{m}^{2/3}\) for many pure metals, and derived

$${\sigma }_{m}=4\cdot {T}_{m}/{V}_{m}^{2/3}.$$
(3)

Then, \(\alpha \cdot \beta \) = 0.24 was obtained from Eqs. [2] and [3]. Based on the \(\beta \) = 0.7 reported by Egelstaff,[18] they derived \(\alpha \) = ~ 0.4. This generalized relationship in Eq. [3] can be used for the prediction of unknown \({\sigma }_{m}\) of liquid metals.

Later, Kasama et al. (Iida and his colleagues)[19] tried to get a more theoretical value of \(\alpha \). The isothermal work necessary to create a unit area of free surface is thought to decrease due to thermal expansion.[19] If this assumption holds over all temperatures, the surface tension becomes zero at \({r}_{c}\).[16] This relationship can be expressed as below, where \({V}_{c}\) represents the molar volume at the critical temperature \({T}_{c}\).

$$\alpha ={\left(\frac{{V}_{c}}{{V}_{m}}\right)}^{1/3}-1.$$
(4)

Kasama et al.[19] took the result of Ashcroft and Lekner[20] who used the Percus-Yevick equation[21] to fit the experimental structure factor with the effective hard-sphere diameter \({d}_{H}\), as shown in Eq. [5].

$$\eta =\frac{4}{3}\pi {\left(\frac{{d}_{H}}{2}\right)}^{3}\frac{{N}_{Av.}}{{V}_{m}},$$
(5)

where \(\eta \) is the packing density parameter which was found to be 0.45.[20] This \(\eta \) value was also supported by experimental data, ranging from 0.4 to 0.6, of Waseda.[22] In addition, \({d}_{H}\) has the relationship with \({V}_{c}\) as in Eq. [6], based on the van der Waals model by Young and Alder.[23]

$${V}_{c}=2.417\cdot {10}^{24} {{d}_{H}}^{3}.$$
(6)

Kasama et al.[19] proposed a generalized constant of \(\alpha \) = 0.51 for all liquid metals using Eqs. [4] through [6]. That is, instead of using their earlier value \(\alpha \) = ~ 0.4 in their original paper,[15] Kasama et al.[19] derived more theoretical value of \(\alpha \) = 0.51 and tried to fit the \(\beta \) using the experimental relationship \({\sigma }_{m}=4\cdot {T}_{m}/{V}_{m}^{2/3}\). The obtained \(\beta \) for various liquid metals was ranged in \(\beta \) = 0.41 ~ 0.84.

Equations [2] and [3] were reformulated by expressing the work needed to separate atoms in terms of the cohesive energy, also known as enthalpy of evaporation of liquid metals at \({T}_{m}\), \({\Delta H}_{m}^{ev.}\). Iida et al.[15] obtained the following relationship by plotting the experimental \({\sigma }_{m}\) data against \({\Delta H}_{{T}_{m}}^{ev.}\)/\({V}_{m}^{2/3}\) for 21 pure liquid metals.

$${\sigma }_{m}=1.7\cdot {10}^{-9} \frac{{\Delta H}_{m}^{ev.}}{{V}_{m}^{2/3}}.$$
(7)

Later, based on the proportional relationship between \({T}_{m}\) and \({\Delta H}_{m}^{ev.}\) (assumption of constant \({\Delta S}_{m}^{ev.}\)), Iida and Guthrie[24] plotted again the experimental \({\sigma }_{m}\) data against \({\Delta H}_{m}^{ev.}/{V}_{m}^{2/3}\) for available pure liquid metals (more than 40 elements) and obtained the following Iida’s equation:

$${\sigma }_{m}={c^{\prime}\cdot \left(\alpha \beta \right)}^{2}\frac{{\Delta H}_{m}^{ev.}}{{V}_{m}^{2/3}}=1.8\cdot {10}^{-9} \frac{{\Delta H}_{m}^{ev.}}{{V}_{m}^{2/3}},$$
(8)

where \({c}^{\prime}\) is a modified constant including the proportionality between \({T}_{m}\) and \({\Delta H}_{m}^{ev.}\). This Eq. [8] is commonly used to predict the \({\sigma }_{m}\) of liquid metals.

It should be noted that the slopes for the \({\sigma }_{m}\) expression in Eqs. [7] and [8] are slightly different. This can be highly dependent on the availability of experimental data and their experimental accuracy.

Eötvös Law for \({{d}}{{\upsigma}}\)/\({{d}}\mathbf{T}\)

Eötvös is recognized for his pioneering work on the temperature dependence of surface tension in liquid phase.[16] He provided insights into the temperature dependency of surface tension by incorporating two key concepts: van der Waals' equation of state, also known as the ideal gas law (\(PV=nRT\)), and the principle of corresponding states.

According to Eötvös,[16] the \({\sigma }_{T}\) of liquid approaches zero when temperature is close to \({T}_{c}\), indicating the disappearance of the interface between liquid and gas phases. This phenomenon occurs because the intermolecular forces in liquid and gas phases become identical to each other, resulting in the liquid phase and its saturated vapor phase having the same density \(\rho \) at \({T}_{c}\).

To express the relationship between the surface tension of a liquid–gas interface and temperature based on his deduction, Eötvös established the following empirical equation for liquids:

$${\sigma }_{T}{{V}_{T}}^{2/3}={k}_{y}\left({T}_{c}-T\right),$$
(9)

where \({\sigma }_{T}\) and \({V}_{T}\) are the surface tension and molar volume at given T, respectively, and \({k}_{y}\) is a constant that varies for each liquid. \({k}_{y}\) is approximately equal to 2.1 \(\cdot \) 10-7 J K-1 mol-2/3 for symmetric non-associated molecules of covalent or molecular liquids, while it tends to be 6.4 \(\cdot \) 10-8 J K-1 mol-2/3 for liquid metals.[25]

The Eötvös’ law has been reevaluated theoretically and shown to be a valid empirical model, particularly when the density of vapor is negligible compared to the density of liquid.[26,27,28,29]

Present Surface Tension Model for Pure Liquid Metal

Experimental Surface Tension Data and Other Physio-chemical Property Data

The surface tension data for many liquid metals have been obtained by various experimental methods, including a sessile drop, pendant drop, maximum bubble pressure, electromagnetic levitation, etc. Frequently, however, these data are quite scattered. For example, the data of pure iron[30,31,32,33,34,35,36] are plotted in Figure 2. As shown, the data of liquid Fe near its melting temperature are scattered within the range of 1.750 to 1.975 J/m2. Because most of the experimental methods for liquid metals involve a certain degree of dynamic state at high temperatures, error can be naturally involved. In addition, the impurity elements known as surface-active elements like O and S can significantly decrease the surface tension of liquid metal.

Fig. 2
figure 2

Experimental surface tension data of pure iron available in the literature[30,31,32,33,34,35,36]

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Mills and Su[9] collected these experimental data of liquid metals published until 2005 and proposed the adopted mean value for 25 metals. In their evaluations of the experimental data, the mean value was obtained when there was reasonable agreement among the surface tension results obtained in various investigations. Recently, Gheribi and Chartrand[37] re-examined the experimental data for 20 elements and proposed the most appropriate dataset of \({\sigma }_{m}\) to validate their prediction of \(d{\sigma }_{T}\)/\(dT\). Although the specific criteria for selecting the most reliable \({\sigma }_{m}\) experimental data were not outlined clearly in their paper, most of the selected data by Gheribi and Chartrand were from the experiments by non-contact measurement methods or the experimental results explicitly considering the influence of atmosphere and surface-active elements. In the present study, the most recent \({\sigma }_{m}\) data evaluated by Gheribi and Chartrand[37] were taken for those 20 elements. For the rest of elements (22 elements), the highest \({\sigma }_{m}\) value among the data collected by Mills and Su[9] was used for each metallic element because \({\sigma }_{m}\) tends to decrease with impurity surface-active elements. These experimental \({\sigma }_{m}\) data for 42 metallic elements used in the present study are summarized in Table I.

Table I Experimental Values of \(\eta \), \({\rho }_{m}\), \(d{\rho }_{T}\)/\(dT\), \({\sigma }_{m}\), and \(d{\sigma }_{T}\)/\(dT\) of 42 Elements from the literature
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The \({V}_{m}\) can be directly calculated from the density at melting temperature \({\rho }_{m}\) and molar mass of metallic elements. \({V}_{c}\) can also be calculated using packing density \(\eta \) (see Eqs. [5] and [6]). Table I shows the collected experimental data of \({\rho }_{m}\) and \(\eta \) for 42 elements. Thermodynamic properties like \({\Delta H}_{m}^{ev.}\) and \({T}_{m}\) of pure metals are taken from FactSage FACTPS thermodynamic database.[38,39]

The Predictivity of \({{{\sigma}}}_{{{m}}}\) by Iida’s Model

Iida proposed simple linear relations in Eqs. [3) and (8) for the prediction of the \({\sigma }_{m}\) of liquid metals using thermodynamic and molar volume data of metals. The \({\sigma }_{m}\) values were calculated using Eq. [8], and the results are plotted in Figure 3 in comparison with the selected experimental data in Table I. The predicted \({\sigma }_{m}\) values were calculated using \({\Delta H}_{m}^{ev.}\) from the FactSage thermodynamic database[38,39] and \({V}_{m}\) directly calculated from experimental \({\rho }_{m}\) (see Table I). Figure 3(a) compares \({\sigma }_{m}\) of 20 elements evaluated by Gheribi and Chartrand,[37] while Figure 3(b) compares \({\sigma }_{m}\) of 42 elements for which experimental \(\eta \) are available. In both comparisons, it is found that Iida's model shows a linear relationship with a slope of 1.7 \(\cdot \) 10-9 as described in the original paper (see Eq. [7]) by Iida et al.[15]. The corresponding R-squared value is 0.863 for Figure 3(a) and 0.895 for Figure 3(b).

Fig. 3
figure 3

Comparison between predicted \({\sigma }_{m}\) by Iida et al.[24] and the selected representative surface tension data. (a) 20 elements evaluated by Gheribi and Chartrand,[37] and (b) 42 elements, taken from the data collected by Mills and Su,[9] for which experimental \(\eta \) are available

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Although the agreement is generally good, certainly there is still room to improve the prediction accuracy. For example, the \({\sigma }_{m}\) of transition metals like Fe and Co, and elements in groups 13 to 15, such as Si, Bi, Sn, Ga, Ge, and Sb, are less accurately predicted using Iida’s equation.

Present Model

Prediction of \({\alpha }\) value for individual liquid metal

To obtain a more accurate \(\alpha \) value for each metal, we attempted to estimate \({r}_{c}\) which is the distance required to separate a pair of atoms to a state where their interatomic force becomes negligible. As shown in Figure 1, the difference between \({r}_{c}\) and \({r}_{e}\) is denoted as \(\delta \) in the present study. Given that the pair potential in liquid metal is assumed to be the effective potential arising from long-range oscillatory interactions of valence electrons,[40,41,42] \(\delta \) can be a function of the atomic radius \({r}_{a}\) representing the distance of maximum charge density in the outermost electron shell of the atoms.

$$\delta ={r}_{c}-{r}_{e}=f({r}_{a})$$
(10)

It is important to note that there is no unique self-consistent way to calculate \({r}_{a}\) for each element. The comparison of the \({r}_{a}\) values from various studies[43,44,45,46,47,48] are presented in Figure 4. Bragg[43] determined the \({r}_{a}\) of 38 elements, ranging from lithium (Li) to bismuth (Bi), on the basis of a hard-sphere approximation model by examining the distance between neighboring atoms in crystals. Slater[44] suggested an analytical form for a part of an electron function that allows to calculate the theoretical \({r}_{a}\) values. Slater[45] reported values of \({r}_{a}\) for 85 elements, ranging from hydrogen (H) to Americium (Am), by assuming the sum of the radii of two atoms forming a bond in a crystal or molecule as an approximate internuclear distance. Waber and Cromer[46] calculated values of \({r}_{a}\) through relativistic Hartree-Fock wavefunctions. Nagle[47] used experimentally measurable atomic polarizability to drive \({r}_{a}\) value. Ghosh and Biswas[48] calculated \({r}_{a}\) corresponding to the principal maximum in the radial distribution function for the outermost orbital. As shown in Figure 4, the \({r}_{a}\) values obtained in the literatures[43,44,45,46,47,48] show the same periodic trends depending on atomic number of elements, but the absolute values are quite different each other.

Fig. 4
figure 4

Comparison between the proposed \({r}_{a}\) value in the previous studies[43,44,45,46,47,48]

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Theoretically, \(\delta \) values represented in Figure 1 may be calculated from \({r}_{e}\) at \({T}_{m}\) and \({r}_{c}\) at \({T}_{c}\). That is, \({r}_{e}\) can be calculated using Eq. [11] with experimental \({\rho }_{m}\), and \({r}_{c}\) can be calculated using Eq. [12] after obtaining \({V}_{c}\) using Eqs. (5) and (6) with experimental data of \({\rho }_{m}\) and \(\eta \). It should be noted that both \({r}_{e}\) and \({r}_{c}\) are derived from the occupied apparent atomic volume obtained based on \({V}_{m}\) and \({V}_{c}\), respectively, to calculate the distance between a pair of atoms. The experimental values of \({\rho }_{m}\) and \(\eta \) are listed in Table I.

$${{r}_{e}=\left(\frac{M/{\rho }_{m}}{{N}_{Av.}}\right)}^{1/3}{=\left(\frac{{V}_{m}}{{N}_{Av.}}\right)}^{1/3},$$
(11)
$${{r}_{c}=\left(\frac{{V}_{c}}{{N}_{Av.}}\right)}^{1/3}.$$
(12)

Figure 5 shows the plots between theoretical \(\delta \) values obtained using Eqs. (11) and (12) and several different sets of \({r}_{a}\) values from the literature[43,44,45,46,47,48] (see Figure 4). The linear regression equations between \(\delta \) and \({r}_{a}\) with R-squared values are also noted in the plots. Interestingly, a linear relationship between \(\delta \) values and \({r}_{a}\) values for 42 elements is found for all cases. Among all the datasets of \({r}_{a}\), the linear relationship between the \(\delta \) values and the set of \({r}_{a}\) by Slater[45] showed the highest R-squared value (0.890) and a slope of 0.922, which is close to 1.0. Considering the R-squared value, the \({r}_{a}\) dataset by Slater[45] was selected in the present study. That is, the \(\delta \) is calculated using the linear regression equation, and \({r}_{c}\) is calculated as follows.

Fig. 5
figure 5

Evaluated \(\delta \) of 42 elements plotted with different sets of \({r}_{a}\) values from the literature: (a) Ref. [43], (b) Ref. [44], (c) Ref. [45], (d) Ref. [46], (e) Ref. [47], and (f) Ref. [48]

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$${r}_{c}= {r}_{e}+\delta ={r}_{e}+\left({0.922 r}_{a}+0.047\right).$$
(13)

The periodic variations of \({r}_{e}\), \({r}_{a}\), \({r}_{c}\), \(\alpha \), \({\Delta H}_{m}^{ev.},\) and \({\rho }_{m}\) of pure elements are plotted in Figure 6. The \({r}_{e}\) are calculated from Eq. [11] using experimental \({\rho }_{m}\),[49] \({r}_{a}\) are taken from Slater,[45] and \({r}_{c}\) are calculated in the present study using Eq. [13]. The \(\alpha \) values were calculated using the equation \(\left(1+\alpha \right){r}_{e}={r}_{c}\) from Iida et al.[15] as shown in Figure 1. The periodic trend of \({r}_{a}\) can be explained by considering the effective nuclear charge and the shielding effect. Moving across a period from left to right, the increasing number of protons in the nucleus results in a stronger effective nuclear charge, which attracts the outermost electrons more strongly, pulling them closer to the nucleus and thereby reducing \({r}_{a}\). Meanwhile, the shielding effect remains relatively constant due to the similar electron configuration in the same period. This effect counteracts the increasing nuclear charge to some extent, resulting in a relatively stable \({r}_{a}\) within the period. The variation of \({r}_{e}\) can be understood in terms of \({\Delta H}_{m}^{ev.}\) , as can be seen in Figure 6(b). The higher \({\Delta H}_{m}^{ev.}\) of liquid metals means the higher cohesive bond energy between atomic pair, resulting in a higher \({\rho }_{m}\) and the lower \({r}_{e}\). The \(\alpha \) values in the present study are calculated from \({r}_{a}\) from Slater[45] and \({r}_{e}\) from Eq. [11]:

Fig. 6
figure 6

Periodic variations of (a) \({r}_{a}\), \({r}_{e}\), \({r}_{c}\) (left y-axis), and \(\alpha \) (right y-axis), and (b) \({\Delta H}_{m}^{ev.}\) and \({\rho }_{m}\) of elements. P# indicates the #th horizontal row of the periodic table. The values of \({r}_{a}\) are from Slater,[45] and \({r}_{e}\), \({r}_{c},\) and \(\alpha \) are derived from Eqs. (11), (13), and (14), respectively. Dashed lines indicate the first p orbital elements in each period

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$$\alpha =\frac{{r}_{c}}{{r}_{e}}-1=\frac{\left({0.922r}_{a}+0.047\right)}{{r}_{e}}.$$
(14)

As seen in Figure 6, the \(\alpha \) values vary significantly with atomic number even within a period in the periodic table. In the periods 2 and 3, the \(\alpha \) values decrease from 0.50 to 0.45 with increasing atomic number. In periods 4 and 5, they show local maximums (\(\alpha \) = ~ 0.55) in the middle elements of the periods. In period 6, mostly for rare-earth elements (REEs), the \(\alpha \) values are about 0.53 for all REEs. In general, most elements exhibit an \(\alpha \) value between 0.5 and 0.6, and the elements in groups 13 to 15 with p orbitals, such as Si, Bi, Sn, Ga, Ge, and Sb, generally have lower \(\alpha \) values.

Prediction of \({{{\sigma}}}_{{{m}}}\)

The surface tension equation of Iida is presented in Eq. [8]. As it is hard to obtain the \(c{\prime}\) in the model equation, Iida et al.[15] found a correlation between the experimental \({\sigma }_{m}\) data and \({\Delta H}_{m}^{ev.}\)/\({V}_{m}^{2/3}\) to determine the pre-factor of Eq. [8], as shown in Figure 3(a). In the present study, we went back to the original relationship in Eq. [2] and attempted to find the relationship between \({\sigma }_{m}\) and \({\alpha }^{2}{\Delta H}_{m}^{ev.}\)/\({V}_{m}^{2/3}\) of all liquid metals.

Experimental \({\sigma }_{m}\) data and \({\alpha }^{2}{\Delta H}_{m}^{ev.}/{V}_{m}^{2/3}\) data for pure liquid metals are plotted in Figure 7. The \(\alpha \) value of each individual element was obtained using the method explained in the Section IV–C–1 (Eq. [14] with \({r}_{a}\) taken from Slater[45]). The results of 20 elements evaluated by Gheribi and Chartrand[37] are plotted in Figure 7(a), and the results for all 42 elements for which experimental \(\eta \) are available in the literature are plotted in Figure 7(b). The linear relationship is obtained with a slope of 6.3 \(\cdot \) 10-9 and 6.4 \(\cdot \) 10-9, from Figure 7(a) and (b), respectively.

Fig. 7
figure 7

Comparison between predicted \({\sigma }_{m}\) in this study and the selected representative surface tension data. (a) 20 elements evaluated by Gheribi and Chartrand,[37] and (b) 42 elements, taken from the data collected by Mills and Su,[9] for which experimental \(\eta \) are available

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$${\sigma }_{m}=\left(6.264\;or\;6.384\right)\times {10}^{-9}\cdot {\alpha }^{2}\frac{{\Delta H}_{m}^{ev.}}{{V}_{m}^{2/3}}.$$
(15)

The present approach yields the R-squared value of 0.938 and 0.929 for the data in Figure 7(a) and (b), respectively, which is significantly improved than the results of the original Iida’s model presented in Figure 3 (R-squared values of 0.863 and 0.895).

The improvement of the prediction of \({\sigma }_{m}\) in Figure 7 indicates that the original Iida’s relationship assuming \(\alpha \cdot \beta \) as a constant value could be less accurate for the \({\sigma }_{m}\) prediction. Compared to the results in Figure 3 for the original Iida’s model, the predictions of \({\sigma }_{m}\) of the groups 13 to 15 elements with p orbitals such as Si, Ga, Ge, Sn, Sb, and Bi are largely improved. The individual \(\alpha \) values of Si, Ga, Ge, Sn, Sb, and Bi estimated in this study are 0.404, 0.467, 0.431, 0.455, 0.441, and 0.468, respectively. That is, these elements with p orbitals have \(\alpha \) values largely different from constant \(\alpha \) = 0.51 used in the original Iida’s equation.[15] Therefore, the improvement in the present study can be understood by the usage of proper \(\alpha \) values in surface tension calculation rather than the constant values in the original Iida’s model.

Calculation of \({{d}}{{{\sigma}}}_{{{T}}}\)/\({{d}}{{T}}\)

If \({T}_{c}\) of an element is determined, \(d{\sigma }_{T}\)/\(dT\) can be calculated from the Eötvös law,[16] which is a well-known empirical relationship between \({\sigma }_{T}\) and \({T}_{c}\) as shown in Eq. [9]. By differentiating Eq. [9] with respect to \(T\), while considering \({V}_{T}\) as a function of \({\rho }_{T}\), and substituting \({k}_{y}\) using Eq. [9], the expression for the temperature coefficient of surface tension, denoted as \(d{\sigma }_{T}\)/\(dT\), can be derived as follows:

$$\frac{d{\sigma }_{T}}{dT}=\frac{{\sigma }_{T}}{\left({T}_{c}-T\right)}\left(\frac{2\left({T}_{c}-T\right)}{3{\rho }_{T}}\frac{d{\rho }_{T}}{dT}-1\right).$$
(16)

That is, if \({T}_{c}\) and \(d{\rho }_{T}\)/\(dT\) of an element are known, \(d{\sigma }_{T}\)/\(dT\) can be predicted using Eq. [16].

However, the reported \({T}_{c}\) data of element in the literature are very scattered depending on the estimation methods.[50,51,52,53,54,55,56,57,58] For example, the estimated \({T}_{c}\) values from different methods[50,51,52,53,54,55,56,57,58] are presented in Figure 8(a) as a range of values for each element. Direct experimental values of elements (14 elements are known) are plotted in the figure to compare the accuracy of the estimated values. It should be noted that the semi-empirical estimation methods in the literature are mostly developed based on these available real experimental data. However, some of experimental data are still not reproduced and some semi-empirical methods are obviously over- or under-estimating \({T}_{c}\) values.

Fig. 8
figure 8

Estimated \({T}_{c}\) values (a) from the literature[50,51,52,53,54,55,56,57,58] and (b) in the present study, compared to experimental \({T}_{c}\) data (circles). Experimental data are Li,[59] Na,[60] K,[61] V,[62] Fe,[63] Co,[64] Zn,[65] Rb,[66] In,[67] Cs,[66] Pt,[68] Au,[69] Hg,[70] and Pb[71]

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In the present study, \({T}_{c}\) was obtained by assuming a linear decrease in density with temperature:

$$d{\rho }_{T}/dT=\left({\rho }_{c}-{\rho }_{m}\right)/{(T}_{c}-{T}_{m}),$$
(17)

where \({\rho }_{c}\) and \({\rho }_{m}\) are the density of liquid metal at the \({T}_{c}\) and \({T}_{m}\), respectively, and \(d{\rho }_{T}\)/\(dT\) is the temperature dependence coefficient of density which is assumed to be constant. As \({\rho }_{c}\) = \(M/{V}_{c}\) where M and \({V}_{c}\) are molar mass of metal and molar volume of liquid metal at \({T}_{c}\), respectively, Eq. [17] can be modified using Eq. [12], as follows:

$${T}_{c}=\left(\frac{M}{{N}_{Av.}\cdot {{r}_{c}}^{3}}-{\rho }_{m}\right)/(d{\rho }_{T}/dT)+{T}_{m}.$$
(18)

Equation [18] is a new method to calculate \({T}_{c}\) value proposed in the present study. While most of the previous methods[50,51,52,53,54,55,56,57,58] obtained their model parameters to reproduce the available experimental \({T}_{c}\) data in the literature, a new method in Eq. [18] is only based on \({r}_{c}\) (see Eq. [13]) and \(d{\rho }_{T}\)/\(dT\) data (available experimental data in literature) instead of using experimental \({T}_{c}\) data directly. Therefore, this new approach of Eq. [18] can be considered to be a more predictive method than the previous studies.

The calculated \({T}_{c}\) results from Eq. [18] are plotted in Figure 8(b). Because \(d{\rho }_{T}\)/\(dT\) (see Table I) of each liquid metal reported in the literature has a certain range of values, the resultant \({T}_{c}\) in this study is also calculated with a certain range of values. In general, the calculated \({T}_{c}\) in this study tends to be higher than those estimated values by the previous methods[50,51,52,53,54,55,56,57,58] (see Figure 8(a)). When compared with experimental \({T}_{c}\) values of 14 elements,[59,60,61,62,63,64,65,66,67,68,69,70,71] except the elements V, Zn, Cs, and Hg, the rest of them (10 elements) are well predicted within 25 pct.

By using the predicted \({\sigma }_{m}\) (Figure 7(b)) and \({T}_{c}\) (Eq. [18]) in the present study, the \(d{\sigma }_{T}\)/\(dT\) for 20 elements at \({T}_{m}\) are calculated in this study. In the calculations, \(d{\rho }_{T}\)/\(dT\) values highlighted in Table I were used. The selection of the \(d{\rho }_{T}\)/\(dT\) value was based on the average of estimated \({T}_{c}\) from the literature, as well as the experimentally measured \(d{\sigma }_{T}\)/\(dT\) values and the estimated \(d{\sigma }_{T}\)/\(dT\) values from Gheribi and Chartrand.[37] Most of the elements had a \(d{\rho }_{T}\)/\(dT\) value that satisfied all the criteria. However, Al and Ti, the \(d{\rho }_{T}\)/\(dT\) values were selected to give a value closer to the experimentally measured and the estimated \(d{\sigma }_{T}\)/\(dT\). The calculated \(d{\sigma }_{T}\)/\(dT\) are also plotted in Figure 9 in comparison to the experimental \(d{\sigma }_{T}\)/\(dT\) values for the elements and the predicted values by Gheribi and Chartrand.[37] It is interesting to see that the comparison of the present results and the results by Gheribi and Chartrand[37] shows much more consistent results. Gheribi and Chartrand[37] used a thermodynamic self-consistent formulation based on Maxwell relations to predict \(d{\sigma }_{T}\)/\(dT\) without any empirical parameters. Given the large scatters (over 100 pct) among experimental \(d{\sigma }_{T}\)/\(dT\) data,[9] the proposed method in this study can be considered as a reasonable and adequate method for estimating the value of \(d{\sigma }_{T}\)/\(dT\) for liquid elements.

Fig. 9
figure 9

Comparison between the predicted \(d{\sigma }_{T}\)/\(dT\) of the present study with (a) the experimental data and (b) predicted values by Gheribi and Chartrand 37

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Predicted Surface Tension of Pure Liquid Metals

Using the proposed model in this study, the surface tensions of 42 liquid metals were calculated across the temperature range from their melting to evaporation temperatures at 1 atm, and plotted in Figure 10(a). The \({\sigma }_{m}\) were determined based on the linear relationship with a slope of 6.384 \(\cdot \) 10-9 as shown in Figure 7(b). Subsequently, the \({T}_{c}\) were calculated using the calculated \({r}_{c}\) and experimental \(d{\rho }_{T}\)/\(dT\) values (highlighted in Table I) as per Eq. [18]. For the additional 22 elements shown in Figure 10, the selection of the \(d{\rho }_{T}\)/\(dT\) value was based solely on the average of estimated \({T}_{c}\) from the literature. Then, the \(d{\sigma }_{T}\)/\(dT\) were obtained by the differentiation equation of the Eötvös law in Eq. [16]. The adopted surface tensions equations for liquid metals, published by Mills and Su[9] based on the collected experimental data, are plotted in Figure 10(b) for comparison with the predicted values in this study.

Fig. 10
figure 10

Surface tensions of 42 pure liquid metals over the temperature ranges of their liquid state at 1 atm (a) predicted values in this study and (b) experimental values from Refs. 9 and 37

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In general, \({\sigma }_{m}\) and \(-d{\sigma }_{T}\)/\(dT\) of alkali and alkaline earth elements tend to be lower than those of other elements, while those of transition metals like Fe, Co, Ni, Cr, and V are showing typically higher \({\sigma }_{m}\) and \(-d{\sigma }_{T}\)/\(dT\) values. This is mainly attributed to \({\Delta H}_{m}^{ev.}\) and \({V}_{m}\) of liquid elements. In theory, \(\sigma \) is influenced by the cohesive energy within the liquid metal, which can be correlated with both \({\Delta H}_{m}^{ev.}\) and \({V}_{m}\). The higher \({\Delta H}_{m}^{ev.}\) implies the stronger interatomic interactions, resulting in the higher cohesive forces and, consequently, the higher \(\sigma \). Conversely, while the lower \({V}_{m}\) indicates the stronger interatomic interactions, it also suggests the smaller surface area affected, leading to the higher surface energy per surface area, consequently, resulting in the higher \(\sigma \). It should be also noted that \(d{\sigma }_{T}\)/\(dT\) of Zr and Ti are much less steep than those of other transition metals. This is because their predicted \({T}_{c}\) (16176 K for Zr and 14501 K for Ti) are significantly higher than those of neighboring transition metals.

Prediction of \({{\beta}}\) Parameter

In the original Iida’s model shown in Eqs. (2) and (8), both \(\alpha \) and \(\beta \) parameters are used to calculate surface tension. We focused our study to find a new method to calculate \(\alpha \) of each metallic element in this study, but not attempted to calculate \(\beta \). \(\beta \) is a correction factor of atomic frequency from the solid to the liquid state.

Iida et al.[15] considered a homogeneous liquid metal at its melting point, and assumed that liquid atoms of mass m show harmonic vibrations of the same frequency:

$${v}_{l}=\frac{1}{2\pi }{\left(\frac{{K}_{f}}{m}\right)}^{1/2},$$
(19)

where \({K}_{f}\) is the force constant. To represent \({K}_{f}\) in terms of well-known physical parameters, they employed Lindemann’s melting formula[17] for atomic frequency (\({v}_{s}\)) at solid metals and tried to obtain \({v}_{l}\) from \({v}_{s}\):

$${v}_{l}=\beta {v}_{s}=\beta \left(c\frac{R {T}_{m}}{M {V}_{m}^{2/3}}\right),$$
(20)

where \(\beta \) represents the ratio of the atomic frequency in the liquid state to that in the solid state. \(c\) is a constant (2.8 ~ 3.1 \(\cdot \) 1012, roughly same for all metals); \(R\), the gas constant; \({T}_{m}\), the melting temperature; \(M\), the molar mass; and \({V}_{m}\), is the molar volume at the \({T}_{m}\).

Of course, \(\beta \) can vary with metals. Although it was not attempted in this study. \({v}_{l}\) and \({v}_{s}\) can be calculated using Molecular Dynamics (MD) simulation. Therefore, MD simulation can offer a robust method for further optimization of the parameter, \(\beta \). This should be further investigated in future.

Conclusion

In the present study, a well-known semi-empirical equation for surface tension \({\sigma }_{m}\) of liquid metal at its melting temperature by Iida has been modified by assigning \(\alpha \) value for each liquid metallic element. \(\alpha \) value represents the ratio of the distance required to separate one atomic pair from its equilibrium distance, which was obtained by analysis of the equilibrium interatomic distance (\({r}_{e}\)) and the atomic radius (\({r}_{a}\)) for 42 elements. The predictability of \({\sigma }_{m}\) for liquid metals was improved significantly using the present model. In addition, a new method to predict \({T}_{c}\) was proposed in the present study. Then, the temperature coefficient of surface tension (\(d{\sigma }_{T}\)/\(dT\)) of liquid metal was calculated based on the Eötvös law with predicted \({T}_{c}\). By using new \({\sigma }_{m}\) and \(d\sigma \)/\(dT\) values derived in the present study, the surface tensions (\(\sigma \)) of 42 pure liquid metals were predicted over the entire temperature ranges of their liquid states. We believe that the present study provides the upmost accurate method to predict \(\sigma \) for pure liquid metal among all available methods in literature.