1 Introduction

Surface tension is a material property that helps describing the fluid behaviour and is defined as the force that is needed to increase the surface area against the tangentially acting bonding forces of the atoms/molecules or as the energy needed to increase the area of a surface. At a surface of a material towards another fluid or gas, the environment of an atom is different compared to the one in the bulk material. While an atom/molecule has typically twelve direct neighbours in the bulk material, an atom/molecule positioned in the surface layer sees only nine [1]. This difference results in forces parallel to the surface, which is experienced as the surface tension [2].

Surface tension plays an important role in many processes that involve liquids [3]. In thermal processing, the wetting behaviour of material on substrates (e.g. [4]) is defined by the surface tension as well as melt pool dynamics (e.g. [5]). In brazing, material wetting defines the process outcome and quality. When creating melt pools, the surface tension can play a major role to define the melt flow characteristics due to its temperature dependence. For most pure materials, the surface tension decreases at increasing temperature, which results in lower surface tension values in the hot areas of the melt pool compared to the areas with low temperature. This, in turn, creates a surface melt flow, called the Marangoni flow, from the high to the low temperature areas [6]. Certain additional elements in the alloy, e.g. sulphur, can change the flow field and alter the process outcome [7].

Surface tension influences many processes and processing outcomes. Therefore, there are many studies available that describe and explain that phenomenon. Basic understanding was gained about the relation of bonding forces between the atoms/molecules to surface tension for pure materials [8] as well as for material mixtures [9]. However, typically, materials were examined that are in liquid state at room temperature or are heated just above their melting temperatures. In advanced processing, temperatures above the melting temperatures of the materials can be reached. For steel processing, this means that temperatures above ~ 1808 K are reached.

1.1 Surface tension at high temperatures

In thermal processes of metals, such as welding, cutting or Additive Manufacturing techniques, typically melt pools are created that show temperatures above the melting temperatures of the metals. This applies to basically all heat sources used (e.g. furnace heating, arc sources) and in particular to beam process, using electron or laser beam sources, where the local high density energy input can create high temperatures and high thermal gradients. Such thermal cycles can show rapid heating and cooling.

For a better understanding of the process behaviour at such extreme conditions, a good understanding of the material properties is necessary, e.g. to develop reliable predictive process models. Therefore, efforts were made to derive the important property of surface tension. Figure 1 shows an excerpt as overview of surface tension measurements and simulations of pure iron by different authors. There are variations visible denoting variations of the measuring methods and probably possible impurities in the material or the ambient atmosphere.

Fig. 1
figure 1

Surface tension of pure iron and steel reported in literature [8, 10,11,12,13,14,15,16,17,18,19,20,21,22,23]

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The derived surface tension values show a general trend of decreasing values at increasing temperature. Typically, a linear decrease is assumed, and the surface tension values are calculated using the starting value at melting temperature and the experimentally determined surface tension gradient (e.g. [8, 9]). E.g. Morohoshi et al. [11] measured the surface tension of iron \(\sigma_{Fe}\) in highly pure Argon atmosphere and derived the temperature \(T\)-dependent linear equation

$$\sigma_{Fe} = \left( {1925 \pm 65} \right)\frac{N}{m} - \left( {0.455 \pm 0.034} \right)\frac{N}{m \cdot K} \cdot \left( {T - 1808K} \right)$$
(1)

However, Fig. 1 shows that the linear trend of the surface tension values was based on measurements just above the melting temperature, reaching maximum temperatures of only ~ 2500 K. It remains unclear if the linear trend assumed in Eq. 1 will continue when reaching higher temperatures.

The main hindering aspects to measure surface tension values at high temperatures are the limitations of the surface tension measurement methods and the challenges and effects due to the environment occurring at high temperatures. In addition, temperature sensors can often mechanically not withstand the high temperatures, are difficult to calibrate for such high temperatures or show large errors due to environmental effects (e.g. process radiation, surface dynamics) (e.g. [24]).

For surface tension measurement, a variety of measurement methods were developed. According to Drelich et al. [25], measurement methods can be categorized in:

  1. (1)

    Direct measurement methods that include the Wilhelmy plate and Du Noüy ring method, which are contact measurement methods that measure the resistive force during removal of the measuring device from the liquid, while the maximum force is related to the surface tension.

  2. (2)

    Capillary pressure measurement methods that include the maximum bubble pressure and the growing drop methods, while the maximum inner pressure of the created bubbles can be related to surface tension values. The methods are fast, also liquids with surfactants can be measured and no contact of a measuring device to the liquid is necessary. However, the assumption of spherical drop creation must be fulfilled.

  3. (3)

    Capillary gravity forces that can be used for measurement of surface tension using the capillary rise or drop volume methods. At rise of a liquid in a tube, the height of the liquid is related to the surface tension. Drop detachment from a tube is related to the surface tension counteracting the gravity forces, which enables the correlation of the force at detachment to the surface tension of the liquid.

  4. (4)

    Gravity-distorted drops that indicate the surface tension using the pendant drop or the sessile drop methods.

  5. (5)

    Reinforce distortion of drops that uses initiated distortion e.g. by drop spinning, while the elongation is related to the surface tension.

The maximum bubble pressure method and sessile and pendant drop methods are often used. The laying drops in the sessile drop method contain the risk of measurement errors due to interaction with another material with often unknown interfacial tension values and the wetting phenomena that depend also on surface roughness and temperature effects that can alter the drop shape. For metals, non-contact methods are typically preferred due to the high temperatures needed to reach the liquid states.

It is also clear that dynamic surface behaviour can be related to surface tension values. The dynamic oscillation characteristics of liquid drops can be related to surface tension from the oscillation mode and the related frequencies of a drop with known mass [26]. Liquid drops can be observed when falling (Falling drop method) [27] or being levitated (e.g. [28, 29]). Falling drops can be only observed for a short time until they solidify, or their oscillation is damped. Levitation techniques were used to derive surface tension values for metals [30]. Different ways of levitation were applied also during parabolic flights for simulating no gravity [31], while it is challenging to maintain a certain temperature of the levitated drop. Acoustic, ultrasound or conductive levitation give the opportunity not only to levitate metal drops, but also initiate controlled drop movements. Surface tension can be either derived from observing the natural frequencies after initiating drop oscillation [26] or constantly applying a forced oscillation [3].

Another relation to surface tension can be given by surface waves [32]. Standing or travelling waves on liquids are physically described by forces acting on the wave surfaces. The Bond number \(Bo\) indicates the importance of surface tension \(\sigma\) versus gravity depending on the liquid density \(\rho\) to

$$Bo = \frac{\rho \cdot g}{{\sigma \cdot k^{2} }}$$
(2)

For so-called gravity waves (\(Bo \gg 1\)), the wave speed \(c\) is mainly defined by the gravitational acceleration \(g\) and the depth of the liquid film \(d\) to the ground of the liquid film to

$$c^{2} = \frac{g}{k} \cdot \tanh \left( {k \cdot d} \right)$$
(3)

with the wave vector \(k = \frac{2\pi }{\lambda }\) and the wavelength \(\lambda\). For shallow liquid films (\(k \cdot d \ll 1\)), all waves travel at the same speed \(c = \sqrt {g \cdot d}\), while for deep liquid films (\(k \cdot d \gg 1\)), the impact of the ground can be neglected, and the speed of the waves depends on the wavelength to \(c = \sqrt{\frac{g}{k}}\).

Capillary waves (\(Bo \ll 1\)) are defined as waves that are mainly driven by surface tension as restoring force with the travelling speed of

$$c^{2} = \frac{\sigma \cdot k}{\rho } \cdot \tanh \left( {k \cdot d} \right)$$
(4)

At shallow liquid films, the equation simplifies to \(c = \sqrt {\frac{\sigma \cdot d \cdot k}{\rho }}\), while for deep liquid films, the wave speed becomes \(c = \sqrt {\sigma \cdot k} \cdot \rho\).

1.2 Surface waves and surface tension

One promising way of deriving surface tension is the surface wave propagation observation. Surface wave propagation is directly influenced by the surface tension and offers information about the surface tension theoretically at any temperature. Already Einstein showed that surface gravity waves are based on the pressure components in the crest and trough along the surface streamline. The static pressure based on gravity and height of the fluid column and the dynamic pressure related to the flow speed (Bernoulli equation) act in opposite directions and form the physical description of surface gravity waves [33]. For capillary waves, the static component is defined by surface tension instead. This relation makes the capillary waves interesting for surface tension measurements.

The waves are driven by the inertia of the fluid, viscous forces, gravity and pressure gradients (e.g. Marangoni flow) and can result as moving or stationary waves [34]. Kinetic vapor recoil pressure from surface vaporization was shown to neither stabilize nor destabilize the melt surface in case it is homogeneously distributed [35]. The Marangoni flow can decrease the surface tension and lower the frequency of the surface waves [36]. In order to translate wave propagation characteristics into material surface tension values, those physical relations need to be clear and combined into an equation. The Kelvin relation (e.g. [37]) describes the surface tension based on the wave frequency, wavelength and material density [38] and is applicable for all temperatures of liquid metals [39]. The underlying assumptions, namely assuming small values of disturbance and the constancy of surface tension over the oscillating surface, allow the elimination of the nonlinear term from the Navier–Stokes equation to justify the use of the Kelvin equation [37]. The Kelvin equation is applicable since the wavelength of the capillary waves are small compared to the liquid film depth and the exact layer height is not mandatory to derive reliable surface tension data \(\sigma_{k}\) [37]

$$\sigma_{k} = \frac{\rho }{{k^{3} }} \cdot \left( {4\pi^{2} \cdot f^{2} - g \cdot k} \right)$$
(5)

\(f\) is the frequency related to the angular frequency \(\omega = 2\pi \cdot f\). The combined equation considering gravity and surface tension effects on the wave speed of a surface wave is

$$c_{surface} = \sqrt {\left( {\frac{g}{k} + \frac{\sigma \cdot k}{\rho }} \right) \cdot \tanh \left( {k \cdot d} \right)}$$
(6)

The dispersion relation can be made more accurate taking the viscosity in account, which can be relevant in particular at high frequencies due to its increasing impact on the wave attenuation and slight influence on the wavelength at increasing frequencies [40,41,42]. The adaption to the wave speed calculation considering viscosity is [43]

$$\begin{aligned}c_{visc} &= c_{surface} \\ &\quad\cdot \left[ {1 - \frac{1}{{\sqrt[4]{{\frac{{g \cdot \lambda^{3} }}{{2\pi^{3} \cdot \vartheta^{2} }}}} \cdot \sqrt[4]{{\tanh \left( {\frac{2\pi \cdot d}{\lambda }} \right)}} \cdot \sinh \left( {\frac{4\pi \cdot d}{\lambda }} \right)}}} \right]\end{aligned}$$
(7)

In deep liquid films, viscosity has little impact on the wave speed. The adaption of the surface tension results in [37]

$$\sigma_{visc} = \sigma_{k} + \sqrt {\frac{{16\pi \cdot \vartheta^{3} \cdot v}}{\rho }} - \frac{{4 \cdot k \cdot \vartheta^{2} }}{\rho }$$
(8)

The correction was shown to be in the dimension of 0.5% at 2500 Hz [37].

1.3 Surface wave measurements

Measuring surface waves can be challenging due to the high propagation speed or frequencies at small amplitudes. However, already in 1890, Lord Rayleigh [26] could show that surface tension values can be derived based on the fluid wave observations. Later, standing water waves were analysed (e.g. [44]). In order to resolve such small waves, light scattering was used to measure wave frequencies and propagation wavelengths to be used in the dispersion relations [39].

For excitation of the waves, different methods are possible using contact and contactless external drivers [45]. Using mechanical contact methods like a submerged body, typically only low frequency oscillations can be initiated and contaminations due to physical contact are possible, which limits their use to gravity or gravity-capillary waves [37]. Besides mechanical and acoustic waves, electrical current [46] was used to initiate waves. Contactless mechanical excitation is also possible by vertical vibration initiation of a cuvette [47, 48], which is limited to create standing waves at low frequencies.

Laser pulses that create ring waves can be used as well using laser power modulation [49]. Deformation initiation can be induced by using inhomogeneous electric fields for fluids with the related electrical properties [50] to create higher frequencies [51]. Pressure fluctuations induced locally at the interface provide the possibility to create capillary waves on any liquid independent of its electrical properties [37].

Due to the comparably low melt viscosities of metals at high temperatures, capillary waves can be easily induced by recoil pressure or thermocapillary forces [45]. Many studies reported observing surface waves propagation in arc welding (e.g. [52,53,54]), less in laser processing (e.g. [55]). Due to the typically small melt pool sizes in laser processes, oscillation frequencies are expected in the kHz dimension, which were shown in high-speed imaging observations [45].

Therefore, in this work, surface tension at high temperatures of steel was investigated in order to (1) measure the surface tension using surface wave analysis and (2) derive theoretical explanations for the surface tension trends at such extreme conditions.

2 Methodology

Surface tension values were derived in both experiments and simulation. An experimental method is suggested that enables measuring the characteristics of surface waves at high temperatures. A molecular dynamic model was developed and used to help explaining the physical phenomena and derive general conclusions.

2.1 Experimental procedure

An experimental setup was build enabling the laser illumination of a base steel plate (S235) using a fibre laser (IPG YLR-15000, wavelength 1070 nm). A laser pulse of 1 s length of 1.5 kW at a defocussed spot size on the material of ~ 2.1 mm was applied. Argon shielding gas was used to cover the processing area to avoid surface contamination and possible reactions of the metal with ambient gases. The gas flow rate was low (~ 10 L/min) to avoid deformation of the liquid surface and related possible influences on the surface tension. The metal surface was recorded using a high-speed camera (Photron Fastcam Mini UX100 type 800 K-M-16G) at 10,000 fps with a bandpass filter that only transmits the reflections from the processing zone of the used illumination laser (Cavitar, continuous wave, 808 nm wavelength, multimode, no preferred polarization, spot diameter on the material of ~ 10 mm). An RGB-camera (DFK 23GM021, TheImagingSource) was used at a recording rate of 30 fps, exposure time of 1/200,000 s and a gain of 0 dB for temperature measurement. An IR filter was installed to avoid damage of the sensor by the laser light process reflections.

2.2 Surface waves measurement

Surface waves were recorded at high frame rates (10,000 fps) in order to identify and track melt pool movements. Such waves were initiated during the process probably by degassing bubbles from the melt pool and melt pool dynamics. A total of nine videos were examined that included multiple events, while the events happening below the boiling temperature were examined in this work. Some videos contained several events at different temperatures. The events were documented, and the moving speeds of the waves \(v\) were extracted by measuring the distance between the wave fronts from one to the next frame divided by the time elapsed between the two frames (Fig. 2). In addition, the wavelengths of two consecutive wave tips were measured \(d_{wave}\).

Fig. 2
figure 2

a Wave propagation measurements in high-speed images and b 2D sketches of a cross-sectional view of the wave propagation

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With those measurements, the frequencies \(f\) were determined to

$$f = \frac{v}{{d_{wave} }}$$
(9)

Since the melt pool depth \(> 0.5 \cdot d_{wave}\), deep water waves were assumed. The liquid was assumed to be uniform in depth, incompressible and irrotational at small wave amplitudes. Small values of disturbance and a constant surface tension on the measuring surface were assumed with waves of lengths longer than their amplitude by three orders of magnitude [37]. This enables the simplification of the Navier–Stokes equation and Eq. 5 (Kelvin equation) was used to calculate the surface tension [56].

According to [34], the short wavelength in the current observations indicate that the waves are dominated by surface tension (capillary waves). However, to consider impacts of gravity, in this work, capillary-gravity waves were assumed (Eq. 8). Corrections due to viscosity were considered due to the high wave speeds and compared to calculations without. However, assuming a dynamic viscosity of the steel of 0.004 Pa*s (at melting temperature) [57], it was found that the deviation is small (in average 0.1%), which indicates that viscosity effects on the surface tension measurement can be neglected in this work. The observed surface waves moved fast and much faster than other melt movement impacts. Therefore, the impact of Marangoni flow on the surface wave movement can be neglected.

2.3 Temperature measurement

Temperature measurement is a difficult task, in particular at high temperatures due to fast moving surfaces that show extensive vaporization. The RGB-camera offers the possibility to extract the intensities of each colour picture separately. In this work, the red ® and blue (B) pictures were evaluated. Further details can be found in [24]. With the black body intensity

$$B_{v} = \frac{{2h\lambda^{3} }}{{c^{2} }} \cdot \frac{1}{{e^{{h\lambda /k_{B} T}} - 1}}$$
(10)

including the Planck coefficient \(h\), the vacuum light speed \(c\), the Boltzmann constant \(k_{B}\), the wavelength \(\lambda\) and the temperature \(T\), the intensities were calculated for each colour spectrum \(i\) in the measuring wavelength range between 400 and 800 nm to

$$I_{i} = \int\limits_{{400{\kern 1pt} {\text{nm}}}}^{{800{\kern 1pt} {\text{nm}}}} {B_{v} \cdot Q_{i} \cdot d\lambda }$$
(11)

with the spectral sensitivities of the camera sensors (\(Q_{i}\)) provided by the camera producer.

Since the measured body was of course not a perfect black body, the temperature values were corrected using the known melting temperature at the melting line of the melt pools as reference offset. The pixel intensity ratios \(I_{R} /I_{B}\) in each camera pixel were related to the actual temperature within the calibration range (Fig. 3).

Fig. 3
figure 3

Exemplary temperature field of a melt pool measured with the RGB camera

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When local saturation occurred e.g. due to reflections of local surface curvatures, some pixels needed to be neglected for the evaluation and are marked white in Fig. 3.

The temperature fields were extracted, and temperature values were evaluated at the time and location of the surface wave events using MATLAB. Since the temperatures can locally vary, the positions of surface wave events were identified in the high-speed video frames and identified in the related temperature frame to relate the measured surface tension values to the correct temperature.

2.4 Simulation procedure

A simplified molecular dynamic model was implemented to derive the forces between the Fe atoms of a system with a surface. For that, a simple ordering of 10 × 10 × 10 Fe atoms in a cube at constant initial distance \(a_{cube}\) was implemented in MATLAB (2019). No contamination was assumed. In addition, as boundary condition, two rows of fixed atoms were added to each side and to the bottom of the bulk to enable the interaction calculation of the atoms at the edges of the movable atom areas (Fig. 4).

Fig. 4
figure 4

Initial atom arrangement

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A simple Lennard–Jones pair potential was used to describe the atom interactions [58]. It is known that such simple pair potentials are not completely accurate to simulate metals in some cases due to the complex metallic bonds and Coulomb interactions and more complex Embedded Atom models can be used [59]. However, for the purpose of this work, it is assumed that the pair correlation can simulate the general trends of surface tension.

The strength of interaction \(\varepsilon_{strength}\) between the atoms (in eV) is calculated as temperature dependent to

$$\varepsilon_{strength} = \varepsilon_{LJ} - \varepsilon_{LJ} \cdot \frac{T}{{T_{b} }}$$
(12)

with the Lennard–Jones parameter \(\varepsilon_{LJ}\) and the boiling temperature \(T_{b}\). The thermal expansion is considered to determine the actual distance between two atoms as initial positioning of the atoms to

$$a_{i,x,y,z} = a_{cube} + a_{cube} \cdot \alpha \cdot \left( {T - T_{m} } \right)$$
(13)

with the initial atom distance \(a_{cube}\) (at melting temperature), the melting temperature \(T_{m}\) and the thermal expansion coefficient \(\alpha\).

The movable atoms (Fig. 4) are repositioned depending on the acting forces of their neighbouring atoms. The interaction with each direct neighbour of each atom [60] was calculated and the resulting force was determined. First, for each interaction, the pair potential energy \(E\) (in J) was calculated [61] to

$$E_{i,x,y,z} = 24 \cdot \varepsilon_{strength} \cdot \left( {\left( {\frac{\varphi }{a}} \right)^{13} + \left( {\frac{\varphi }{a}} \right)^{7} } \right)$$
(14)

with the temperature-dependent distance parameter

$$\varphi = r_{avg} + r_{avg} \cdot \alpha \cdot \left( {T - T_{m} } \right)$$
(15)

Based on the resulting distances to the neighbour atoms, for each atom \(i\) the resulting forces \(F_{i,x,y,z}\) (in N) in all directions were determined

$$F_{i,x,y,z} = \frac{{E_{i,x,y,z} }}{{a_{i,x,y,z} }}$$
(16)

The resulting movements of the single atoms were defined based on the resulting force \(\vec{F}\) and the position coordinate \(pos\) of the atoms was adapted to the previous position \(pos_{0}\) based on atom movement within one time step for a certain force according to

$$pos = pos_{0} + 0.01 \cdot erf\left( {\frac{\sqrt \pi }{2} \cdot \vec{F}^{0.8} } \right)$$
(17)

For the calculations, the values shown in Table 1 were taken.

Table 1 Simulation parameters
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The initial temperature of the whole system was set to a constant temperature. The repositioning deviation between the calculation steps reaches nearly constant values already after three calculation steps, which indicates that the atom positioning reaches a quasi-static state for all tested temperatures (Fig. 5). The further evaluations were done using the results at the seventh calculation step.

Fig. 5
figure 5

Average position deviation at different calculation steps

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After seven iterations, the surface tension was calculated. Surface tension values were derived along a line of ten atom distances \(a_{cube}\) length in both x- and y-direction since the surface tension can be understood as a tension along a line [1] (Fig. 6).

Fig. 6
figure 6

Sketch of the top view of the calculated bulk including the calculation lines for surface tension determination in x- and y-direction

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The respective forces in the areas on the one and the other side of the evaluation line were averaged, and the resulting force difference was used to calculate the surface tension values. Since it is known that the liquid surface is not infinitely sharp (e.g. [64]), both the surface layer and the layer underneath were evaluated. In the simulation, the bulk temperature was varied to get a better understanding of the surface tension at different atom arrangements.

3 Results

3.1 Experimental determination of surface tension

Surface tension values were extracted from surface wave events and related to the temperature in the respective area at the time that they occurred. Figure 7 shows the derived values.

Fig. 7
figure 7

Measured surface tension values

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The horizontal error bars indicate the minimum and maximum temperature values in the area where the surface wave event occurred, while the vertical error bars denote the standard deviation of the surface tension values.

Just above melting temperature, the measured surface tension values are close to the linear curve assumed from literature (Eq. 1), which indicates that the surface tension measurement method shows valid results. However, when increasing the temperature, the surface tension values decrease faster than the linear extrapolation would suggest. Around boiling temperature, the surface tension minimum seems to be reached, while above boiling temperature, the surface tension values remain at the same low level.

3.2 Simulated surface tension

The simulated surface tension values were derived from averaging the forces on the atoms on each side of the evaluation line. Figure 8 shows the evaluations of surface tension forces in different directions and layers including an exponential fit. The regression analysis shows an R2 value of 73%, which seems acceptable for the suggested simple model approach.

Fig. 8
figure 8

Calculation of averaged forces along the surface in x- and y-directions in the topmost layer (surface layer) and the layer underneath (2nd layer)

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Figure 9 shows the calculated surface tension values. A similar trend was observed compared to the experimental surface tension measurements (Fig. 7). The values above melting are comparable to the ones known from literature but decrease rapidly at increasing temperature.

Fig. 9
figure 9

Simulated surface tension values at different temperatures

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4 Discussion

The main aim of this work is to increase the understanding of surface tension of steel at elevated temperatures. Therefore, experimental and simulated surface tension values were determined and compared. Experimentally derived surface tension values correlate well with results from the molecular dynamic calculations, which indicates that surface tension values also at elevated temperatures are valid and can be used in high-temperature simulations in the future.

In the simulation results, the calculation of surface tension considering only the topmost atomic layer gives scattered and partly unphysically high surface tension values, even when averaging along the two 5 × 10 atom areas. It seems that the atoms of the second atomic layer support the surface structure and need to be considered to describe the final surface tension. The layers underneath show already similar forces in all directions indicating that those can be considered bulk material. This observation fits to previous findings that surfaces of liquids are not infinitely sharp but are undefined in around two atomic distances (e.g. [64]). It was also found that surface layering occurs, denoting slight density variations on the interfaces (e.g. [65, 66]). Therefore, it can be assumed that surface tension is not a single-atom-layer effect but an interplay of at least two atomic surface layers.

While just above melting temperature, the experimentally derived values in this work are close to the values given in literature. The values here are slightly lower, which can originate from impurities in the steel. However, at increasing temperatures, the surface tension decreases much steeper than expected from linear extrapolations suggested in literature (Eq. 1). Around boiling temperature, the surface tension values seem to stop decreasing and do not reach values close to zero.

The simulation model can support to explain such behaviour. The steep decrease of surface tension values was also seen in the simulation results, which indicates that the model assumptions can describe the observed trend. The decrease of forces between the surface layer atoms originates from the temperature-dependent calculations of both the interaction strength (Eq. 12) and the atomic distances (Eq. 13). Both the interaction strength and thermal expansion reduce the tangential forces in the surface layers and result in the steep surface tension decrease. Therefore, it is assumed that the increased reduction of bonding forces at higher temperatures is an important effect to reduce surface tension. Further effects can occur due to the increasing evaporation of the surface when approaching boiling temperature, which will be the focus of future studies.

A linear decrease of surface tension values of metals assumed in literature (Eq. 1) is usually based on measurements of values close to the respective material melting temperatures. However, such a linear decrease is not visible in the actual measurements and simulations when reaching higher temperatures. A linear approximation seems not sufficient to explain the steep decrease. The derived force values from the simulations show rather an exponential trend (Fig. 9).

An exponential description also accounts for the observation that the surface tension does not reach values around zero even at high temperatures around boiling (Fig. 7) and for the increased impact of reducing bonding forces at higher temperatures. Using the averaged forces in x- and y-direction for the two surface atomic layers and accounting for thermal expansion, the calculation of surface tension with \(\sigma = F_{(T)} /a_{(T)}\) was approximated to the measurements (Fig. 10).

Fig. 10
figure 10

Approximation (approx) of surface tension and surface tension values from simulation (sim) and experiment (exp)

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The exponential empiric model is able to describe the observed trend of surface tension values up to the material boiling temperature. The model underpredicts the surface tension value slightly with an R2 of 57.1%.

5 Conclusions

Based on surface wave propagation measurements and molecular dynamic simulations, surface tension values at high temperatures of steel could be determined. Those data can help process simulations and lead to a better understanding of surface tension at high temperatures:

  • Surface tension values are lower than expected compared to literature assumptions when reaching temperatures further above melting temperature. Simulation results indicate that the reduction of the strength of interaction between the atoms has a larger decreasing impact on surface tension at increasing temperatures.

  • Since the calculation of the surface tension only based on the direct surface layer gives scattered and unphysical values, it was assumed that the second atomic layer contributes to the surface tension. Using this assumption, the experimentally determined values can be sufficiently reproduced. Therefore, it can be concluded that at least two atomic layers contribute to the actual surface tension at elevated temperatures of steel.

  • In contrary to the assumptions of a linear decrease of surface tension with increasing temperature, it was shown that surface tension even at boiling temperature does not disappear. An exponential decrease is suggested as an empirical description of surface tension up to boiling temperature.