Surface tension is an essential thermophysical property of liquids, and the oscillating droplet method is particularly effective for investigations involving reactive molten alloys. The Rayleigh equation is commonly used to evaluate surface tension from measurements of the damping frequency response of an oscillating droplet with small deformation, but non-linear effects are expected to arise for larger deformation. This work describes an improved method for interpreting frequency analysis and validates previous numerical simulation and theoretical analyses which predict a decrease in observed frequency at moderate deformation amplitude. Experimental results from microgravity tests are used to determine a correction of the Rayleigh equation to eliminate the influence of finite deformation.

Surface tension is an important thermophysical property and essential in developing models for manufacturing processes where melts are subjected to conditions such as Marangoni effects, multi-phase flow, and agglomeration. Containerless processing techniques such as Electromagnetic Levitation (EML) can process reactive molten metals over a wide range of operating temperatures, and the oscillating droplet method^{1–6} is effective when applied to the measurement of key thermophysical properties. By conducting experiments in space on board the International Space Station (ISS) using the ISS-EML facility,^{7} the microgravity conditions allow for a reduction in the required electromagnetic field used to position and levitate the sample, resulting in significantly less static deformation of the sample. Splitting of the characteristic oscillation frequency^{8,9} due to the distorted equilibrium shape or influence from the supporting electromagnetic field is reduced, and the sample maintains a mean translational frequency of 1 Hz to 2 Hz, which introduces less than 0.4% deviation in Rayleigh frequency based on the correction formula of Cummings and Blackburn.^{8}

For a spherical droplet with oscillation at small deformation amplitude in mode *m *=* *2, defined in the spherical harmonic as the fundamental oscillation mode, the relation between the natural response frequency $f0$ and surface tension $\sigma $ can be described by the Rayleigh equation^{10} as presented in Eq. (1), where $m$ is the mass of the droplet

However, the Rayleigh equation may not be appropriate for application to oscillating droplets exhibiting non-linear behavior at increased deformation amplitude. Experimentally, Trinh and Wang^{11} and Becker *et al.*^{12} reported that the frequency decreases with the applied initial aspect ratio for the silicone oil/CCl_{4} mixture droplet immersed in distilled water and ethanol droplet discharged from a nozzle into a gaseous environment. Theoretically, Tsamopoulos and Brown^{13} calculated the decrease in oscillation frequency for an inviscid droplet, and Azuma and Yoshihara^{14} analysed the effect of deformation amplitude such that non-linear high-order terms were introduced into the kinetic equation to produce a negative frequency shift as compared to linear theory. Numerically, Basaran^{15} used the Galerkin finite element method, Watanabe^{16} used the level-set method, and Feng and Shi^{17} used arbitrary Lagrangian-Eulerian methods to predict the frequency shift behaviour for oscillating droplets. The current work introduces an enhanced method of frequency analysis, with experimental validation, to estimate correction factors that identify how frequency shifts can be correlated with observed deformation to improve surface tension accuracy and precision.

In this work, the microgravity experiments are conducted on the ISS using the ISS-EML SUPOS coil^{7} on a 1.2 g molten sample of LEK-94,^{18} a nickel-based superalloy developed recently by MTU, Munich Germany (liquidus temperature *T _{m}* = 1666 K). During a typical thermal cycle, a pyrometer is used to measure the sample temperature, and coil positioner and heater control voltages are varied to set the desired operating condition. At cycle initiation, the oscillating currents are maintained at a relatively high level to melt the sample quickly, and then, the heater reduced to near zero to let the molten sample cool until solidification. During the cooling process, one or more heater pulses at control settings between 5.0 and 9.0 V for 0.1 s are applied to deform the sample and excite free oscillations that are damped due to internal fluid flow, while the sample temperature decreases. For each pulse, the surface motion is captured by a 150 Hz top-view camera aligned along the coil center-line to record images along the polar axis, and a side-view camera is installed along the axial axis for solidification detection. An elliptical fit is applied to the projected area seen in each image to evaluate the radius change over time to correlated deformation and pyrometer temperature. It is worth noting that the temperature change of the molten sample during a single-pulse excited oscillation can be up to 100–200 K, depending on the cooling environment (vacuum or 450 mbar argon or helium), and it is necessary to analyse the oscillating frequency at a series of different temperatures in a narrow band of less than 10 K by segmenting the oscillation signal. The surface tension of the molten sample within specific and small temperature ranges can be calculated from the frequency of the surface oscillations in each temperature segment before surface motion damps out. The frequency resolution of FFT is limited to $fs/N$, where $fs$ is the sampling rate and $N$ is the number of data points in each segment, so time-domain fitting using the Levenberg-Marquardt method is used to determine the oscillation frequency, bypassing the resolution limitation of the FFT, resulting in an accuracy better than 0.1%.

In the microgravity environment, a steady-state liquid droplet is a spheroid with an aspect ratio of polar to equatorial lengths $RL/Rw$ less than 1.01, observed from side-view camera. $Rw$ can be determined as the mean value of the length of maximum and minimum semi-axis of the droplet's top-view projection area. An elliptical fit shows the background noise to be between 0.1% and 0.5%. During any given pulse-excited oscillation, the dimensionless deformation amplitude $\delta $ of the spheroidal droplet is characterized as the percent change of length $Rw$. Here, $43\pi RLRw2=43\pi R03$; therefore, $Rw=(1\xb1\delta )R0$ and $RL=(1\xb1\delta )\u22122R0$ under the assumption that the static shape is a sphere with radius $R0$ and that the deformed droplets are spheroids, and volume is conserved between the non-deformed and deformed droplets in each oscillation process. The radius difference due to the density change with temperature is neglected since over the entire test the change is less than 0.3%, while over an individual segmentation interval it is less than 0.03%.

Figure 1(a) presents a typical time-temperature profile for a molten sample during the damping of a single-pulse excited oscillation. The temperature steadily decreases, while the deformation amplitude oscillates from positive to negative values as it is internally damped. Three time-segment periods are shown at the beginning and two at the end—after which the background noise dominates deformation. Figure 1(b) shows the FFT while (c) shows how the top-view image is fit to an ellipse to overcome shading by the cage used to contain the sample within the coil.

The frequency analysis method prescribed is applied to each successive segment to determine the frequency at specific time and temperature ranges. In this example, the positioner power is maintained at 5.21 V, and a heater pulse at 6.46 V is applied for 0.1 s. The droplet is squeezed and starts to oscillate from an initial deformation of around 6%, while the temperature decreases from 1790 K at a cooling rate of approximately 20 K/s. Each overlapping segment is an interval of 0.5 s representing a temperature change of about 10 K. For each single-pulse excited oscillation, the oscillating frequency is represented as a function of beginning deformation amplitude $\delta $ and dimensionless temperature $\theta =(T\u2212Tm)/Tm$ for each time segment, as shown in Fig. 2.

The observed frequency decreases significantly for a deformation amplitude larger than 1.0% regardless of the temperature range and changes moderately for deformations of less than 1.0%, where the influence of sample temperature dominates. The surface tension correlated with the frequency usually shows linear dependency of temperature based on existing theories.^{19} A polynomial fit using a least-squares approach is applied to the frequency data from all experiments, yielding a relationship which is linear with dimensionless temperature $\theta $ and quadratic with deformation amplitude $\delta $ based on the work of Tsamopoulos and Brown^{13} and Azuma and Yoshihara,^{14} and displayed in the following equation with coefficients displayed in Table I,

The observed frequency $f$ approaches the Rayleigh frequency $f0$ in Eq. (1) at arbitrary given temperature when the deformation amplitude approaches zero. This value can be extrapolated from Eq. (2) at $\delta $ = 0, and these values are displayed as the bold line on the temperature-frequency plane and equivalent to $f0=f\theta =0(1+p\theta \theta )$, where $f\theta =0$ represents the Rayleigh frequency at *T _{m}*, and $p\theta $ is the slope. Thus, the frequency shift as a function of deformation can be readily evaluated for any temperature. The observed frequencies of molten LEK-94 sample droplets are plotted as a function of deformation amplitude up to 7% in Fig. 3 with segmentation ranges of ±5 K, corresponding to four selected temperature ranges centered at

*T*– 80 K,

_{m}*T*– 40 K,

_{m}*T*, and

_{m}*T*+ 40 K, respectively. Data are taken from multiple experiments that partially cover these selected temperature ranges. The extrapolated values at $\delta $ = 0 in Eq. (2) are shown as the filled makers on the ordinate axis; the frequency increases as the temperature is reduced. The fitting curves for $\delta $ > 0 at these temperatures are also superimposed on the data in the figure. At each temperature, the observed frequency increases quadratically with the deformation amplitude from up to 7.0% (as initially applied) to around 0.5% (where the amplitude dependence is lost in the noise of frequency measurement).

_{m}The frequency shift ratio is defined as $(f\u2212f0)/f0$, and the values from the ISS EML experiments are compared with theories and numerical simulations from the literature (Fig. 4) with good agreement. Note that each author used different metrics for the definition of droplet deformation and the form of the definition has been standardized to $\delta $ which equals percent change $Rw$ for the plot. Tsamopoulos and Brown^{13} and Basaran^{15} used the prolate aspect ratio $RL/Rw$ which is equivalent to $(1\u2009\u2212\u2009\delta )\u22123$, and Azuma and Yoshihara,^{14} Watanabe,^{16} and Feng *et al.*^{17} used normalized amplitude of polar length $RL$ of the prolate droplet which is equivalent to $(1\u2009\u2212\u2009\delta )\u22122$.

The region with small deformations around 0.5% has a lower frequency shift ratio; however, the analysis is less accurate in this range since background noise is on the same order of magnitude as the signal. The opposite is observed for the region with deformations well above 1.0%, such that investigation of both regions is necessary for complete analysis. An empirical formula is derived by quadratic fitting through the origin showing frequency shift ratio versus deformation amplitude as defined in Eq. (3) which shares the same coefficients in Eq. (2)

The Rayleigh equation can also be modified as defined in Eq. (4) to eliminate the influence of moderate deformation with changes of up to 8% in equatorial length or a change of up to 18% in polar length for free oscillating spheroidal droplets. Corrected values of surface tension can now be calculated from the observed frequency and deformation amplitude of the droplet during the oscillating process

In Eq. (4), the value of $p1$ and $p2$ has been defined for LEK-94 in Table I and will be validated in a future publication for other alloy systems during EML processing. Additional future work involves generalizing the relationship for all sample sizes.

f _{θ = 0} | 34.69 ± 0.07 (Hz) |

p _{θ} | −0.1250 ± 0.004 |

p _{1} | −0.08138 ± 0.018 |

p _{2} | −2.032 ± 0.3 |

R-squared | 0.9653 |

f _{θ = 0} | 34.69 ± 0.07 (Hz) |

p _{θ} | −0.1250 ± 0.004 |

p _{1} | −0.08138 ± 0.018 |

p _{2} | −2.032 ± 0.3 |

R-squared | 0.9653 |

$\sigma Tm$ | 1.718 ± 0.01 (N m^{−1}) |

$d\sigma dT$ | −0.0002818 ± 0.000012 (N m^{−1} K^{−1}) |

R-squared | 0.9039 |

$\sigma Tm$ | 1.718 ± 0.01 (N m^{−1}) |

$d\sigma dT$ | −0.0002818 ± 0.000012 (N m^{−1} K^{−1}) |

R-squared | 0.9039 |

The surface tension of the molten LEK-94 alloy in the range of *T _{m}* – 100 K to

*T*+ 200 K is correlated with a linear fit in Eq. (5) based on corrections from Eq. (4)with coefficients displayed in Table II

_{m}This fit, along with raw data and corrected data, is compared to previous literature values^{18,20} and displayed in Fig. 5. For each colored symbol with different marker types, representing a segmented time slice from a given test, there exists a corrected value shown as an open red circle, and the solid line represents the linear fit. The significant deviation and data scatter are much reduced by removing the influence of systematic error induced by non-linear effects from larger deformation. Both accuracy and precision are improved substantially.

This work was funded by NASA under Grants NNX16AB59G and NNX17AH41G at Tufts University and NNX16AB40G at the University of Massachusetts, Amherst. Generous financial support by ESA Grant No. AO-19999-022 and DLR ThermoLab WM1170 is gratefully acknowledged. The authors wish to thank the Microgravity User Support Center (MUSC) team for successful operations using the ISS-EML facility on the International Space Station.