electromagnetic levitation part ii: thermophysical property measurements in terrestrial conditions

Copyright c© 2008 Tech Science Press FDMP, vol.4, no.3, pp.163-184, 2008

Electromagnetic Levitation Part II: Thermophysical PropertyMeasurements in Terrestrial Conditions

Sayavur I. Bakhtiyarov1 and Dennis A. Siginer2

1 Introduction

Undercooling is a natural extension in a levitationenvironment due to the elimination of containerinduced nucleation giving access to a metastableregion of the phase diagram. It is important tosolidification processes, and therefore, the knowl-edge of thermophysical properties in this regionis crucial. The degree of undercooling determinesthe growth velocity, and is essential in selectingthe metastable solid phase. The use of hydro-gen gas reduces and removes heterogeneous nu-cleation catalysts (oxide and sulfide inclusions),and facilitates undercooling.

EML has the following advantages compared toother levitation methods:

• Contactless melting of the sample; elimina-tion of container walls is important in stud-ies of flow dynamics, glass formation, prepa-ration of pure substances, undercooling andsolidification.

• Molten sample can be protected from oxida-tion by a vacuum or an inert gas.

• It is feasible to solidify the sample in levi-tated position.

• Electromagnetic stirring occurs during themelting process.

• Equilibrium in gas-liquid metal systems isquickly achieved due to efficient stirring andthe relatively large surface exposed to the gasphase.

1 Department of Mechanical Engineering, New Mexico In-stitute of Mining and Technology, Socorro, NM 87801-4796 USA

2 Department of Mechanical Engineering, Wichita StateUniversity, Wichita, KS 67230-0133, USA

• Alloying additions can be made to the levi-tated molten sample.

However, broadly speaking the relatively lowelectrical heating efficiency of induction coils andthe limited amount of the specimen are discour-aging factors in the use of EML on a large com-mercial scale. Another issue is the introduc-tion of impurities which can occur through reac-tion with the gaseous environment during melt-ing. Therefore using very pure gas in the pro-cess is a strict requirement. The greatest difficultyin EML is keeping the suspended droplet stable.There are two kinds of instabilities, global and lo-cal to the surface. Global instabilities cause themetal to move as a whole. They can be avoidedby properly choosing the external current distri-bution that creates the supporting field. That canbe realized by arranging axisymmetric coils withcounter-windings at the top. However, Lorentzforce along the axis of symmetry is zero, and sur-face tension is the only force to balance the hy-drostatic pressure. As a result, the droplet has aconical shape with its apex pointing downwards.Surface instabilities do not allow levitating sam-ples of large masses. Another disadvantage re-sults from difficulties of temperature control andmeasurement. Only pyrometric methods can beused; either the emissivity of the material mustbe known or reliance has to be placed on mea-surements using multicolor pyrometers. In thepast due to difficulties in stabilizing and control-ling the liquid sample, a number of researchersconsidered the levitation melting industrially un-suitable as a production method and discontin-ued work on further developing it. However, thelevitation melting method has been recognizedas a useful small-scale laboratory melting tech-nique. Liquid droplets with a mass of up to 50g

164 Copyright c© 2008 Tech Science Press FDMP, vol.4, no.3, pp.163-184, 2008

have been levitated by axisymmetric coil design.Non-axisymmetric coil design allows levitatingmuch bigger samples, for example Sagardia andSegsworth (1977) levitated 1 kg of aluminum us-ing multi-frequency non-axisymmetric coils.

2 Thermophysical Property Measurements

Accurate and reliable values for the thermophys-ical properties of molten metals become increas-ingly significant as new casting techniques are de-veloped and progress is made in numerical mod-eling of these processes. At high temperatureschemical reactions can take place at the interfacebetween liquid sample and crucible leading to thecontamination of the test sample. EML offers thefollowing advantages in the study of the thermo-physical properties of liquid metals:

• Container effect is eliminated.

• High temperatures (up to 2400oC) can beachieved.

• Undercooled state can be maintained for anextended period of time.

• Non-contact diagnostic techniques (temper-ature measurements, imaging analysis, etc.)can be applied.

However, ground based EML also has some lim-itations in measuring the thermophysical proper-ties of molten metals:

• Electromagnetic fields with high flux densitydeform the shape of the specimen.

• High frequency electromagnetic fields in-duce turbulent currents inside the sample.

• Convective cooling by high-purity inertgases is required.

• Convection is present within the moltenmetal.

2.1 Specific Heat

Specific heat is traditionally measured by adia-batic calorimetry techniques. The calorimeter is

isolated from the environment and requires longrelaxation times. However, this method cannot beused for non-contact measurements at high tem-peratures. Betz and Frohberg (1980) proposed amethod to determine the enthalpy, the heat, andentropy of fusion for some refractory metals. Inthis method, the levitation fields are switched offwhen the desired temperature is reached. Then thesample is dropped into a copper block, and thetemperature rise ΔT of the copper block is mea-sured. Given the specific heat Cp of the copperblock, the enthalpy loss ΔH of the liquid speci-men and the specific heat of the sample can bedetermined,

ΔH = CpΔT, cp =∂H∂T

. (1)

Betz and Frohberg (1980) also measured the heatof mixing by alloying two liquid metals duringlevitation and detecting the related change in tem-perature. The enthalpy of molybdenum was mea-sured between 2282 and 3383 K by levitationcalorimetry. Special attention was paid to the ac-curacy of optical temperature measurements (±1.5%) and to the evaluation of heat losses fromthe sample during the fall. But, comparison of ex-perimental results with data from literature is notencouraging and shows rather large discrepancies.Ohsaka et al. (1992) measured the specific heatof undercooled liquid metals using the same tech-nique. The sample temperature is monitored byan optical pyrometer and is varied by controllingthe flow rate of the cooling gas. The measuredenthalpy is translated into the specific heat. Thespecific heats of Al and a Ti60Cr40 alloy are de-termined in this work. Maximum undercoolinglevels reached are less than those expected undermicrogravity conditions suggesting that measure-ments can be extended to a deeper undercoolinglevel if similar experiments were performed in amicrogravity environment.

The AC temperature technique requires shorterrelaxation times and it can be used for non-contactcalorimetry. Fecht and Johnson (1991) applied anon-contact AC pulse heating method for specificheat measurements in molten metals and alloys inan electromagnetic levitation device. The methodis a variant of non-contact modulation calorime-

Electromagnetic Levitation Part II 165

try. The power of the heater is modulated sinu-soidally,

Pω (t) = Pω0 cos2(ωt

2

). (2)

The power modulations result in a modulatedtemperature response (ΔTω) of the sample,

(ΔT)ω =Pωo

2ωcp

[1+(ωτ1)

−2 +(ωτ2)2]− 1

2, (3)

in which τ1 and τ2 are external and internal relax-ation times, respectively. The choice for the mod-ulation frequency ω should satisfy the inequality,

τ2 � 1ω

� τ1. (4)

Then, the specific heat cp is defined by the follow-ing expression,

cp =Pω0

2ω (ΔTω). (5)

The proposed method has been applied to esti-mate the specific heat of Zr-alloys. It is shownthat for droplet diameters of the order of 5-10 mm,the heat capacity data can be obtained with an ac-curacy of 1%. Near the melting point the valueof the specific heat for most pure metals is around30 J K−1 mol−1, close to the crystal specific heatvalue at the melting point,

cp = cv +cd +ce.p.d., (6)

where cv, cd, and ce.p.d. represent the vibrational,dilatational and the equilibrium point defects con-tributions, respectively. Above the melting pointspecific heat decreases with temperature, reachesits minimum value, and then increases up to 8-0.33 cv. Temperature dependence of alloys is sim-ilar to that of pure metals, and Neumann-Kopprule can be applied to estimate the additive val-ues, Herlach et al. (1993).

2.2 Emissivity

The spectral normal emissivity ε as a function ofwavelength λ and temperature T is determinedfrom the following expression,

Ir (λ ,T ) = ε (λ ,T) Ib (λ ,T) , (7)

where Ir and Ib are the spectral radiance of thereal and blackbody, respectively. Hansen et al.(1989) combined the electromagnetic levitationtechnique with ellipsometry to measure the emis-sivity in liquid metals. The spectral emissivity iscalculated by Kirchhoff’s law,

r (λ ,T )+ε (λ ,T) = 1,

r (λ ,T ) =(n−1)2 +κ2

(n+1)2 +κ2

(8)

where r (λ ,T) is the spectral reflectivity, and nand κ are the real and imaginary parts of the com-plex spectral refractive index n = n− iκ . Usingthis method Hansen et al. (1989) measured theemissivity of liquid Co, Ag, Au, Ni, Pd, and Pt asa function of temperature and wavelength. Rhimand Paradis (2001) propose using a high-powerlaser beam to apply a torque to electromagneti-cally levitated samples in a vacuum. The torquecan be used to control the rotation rate of the sam-ple in measuring its properties. The magnitudeof the torque which depends on the power of thelaser beam, the fraction of incident photons ab-sorbed, and the size of the moment arm can be es-timated by applying the laser to a spherical sampleof mass m and radius R. Assuming that absorptionand emission of radiation by the sample is gov-erned by the Stefan-Boltzman law, the torque isdetermined as

τ =4πR2σεaT 4

c, (9)

where σ , a, c and T represent the Stefan-Boltzman constant, the moment arm, the speed oflight and the steady-state temperature of the sam-ple if the same laser beam is used to heat the sam-ple. The angular acceleration of the sample is es-timated by,

f =5σεaT4

mc. (10)

2.3 Thermal Expansion and Mass Density

Knowledge of the density of liquid metals is cru-cial in most theories and in particular for con-traction simulation during solidification. Thereare several methods for measuring the density


of the high-melting point liquid metals, bal-anced columns, pyknometer, immersed-sinker,maximum-bubble among others. However, theapplication of all these techniques is limited dueto the reaction between liquid metal sample andthe apparatus. Therefore, any EML technique isa good alternative for density measurements inmolten metals. Assuming a spherical shape forthe specimen to determine the volume V of thedroplet of mass m and repeating the density mea-surements at different temperatures the coefficientof thermal expansion β can be obtained

ρ =mV

, β =1V

∂V∂T

. (11)

The determination of thermal expansion coeffi-cient of these materials requires the measurementof the linear size of the specimen as a function ofthe temperature. There are two optical methodsof length measurements, geometrical and inter-ferometric. Some dilatometric techniques are de-scribed in review papers of Ruffino (1984, 1989,1992) and Kirby (1992). Geometrical dilatome-ters use light rays, photometry, or optical imagingsystems. The most precise geometrical dilatome-ter is the optical comparator, in which two mi-croscopes compare the lengths of the specimenin a thermo-stated bath and standard conditionsat room temperature. The method is well suitedfor thermal expansion coefficient measurementsat low temperatures, up to 100oC. At high tem-peratures there are problems, such as the distancebetween the hot specimen and the microscopelenses, precise definition of specimen tempera-ture, and marks on the specimen. The most ad-vanced geometrical dilatometer for measurementsat high temperatures, up to 1600oC, is describedby Rothrock and Kirby (1967). Interferometricdilatometers utilize the interference phenomenonin light waves. The application of laser lightsources in interferometric dilatometers increasesthe resolution and accuracy of the measurements.However, interferometric dilatometry requires thespecimen end faces to be flat and polished whichis difficult to maintain at high temperatures.

El-Mehairy and Ward (1963) developed a tech-nique to determine the density of liquid metalsfrom the profile of a levitated drop obtained by

emitted light photography and calibration. Thelevitating system consists of a stabilizing ringused in conjunction with a three-turn levitated coilenergized by a 10 kW Toccotron. The density ofliquid copper was determined in the temperaturerange 1370o - 2100oK. Temperature dependenceof the density is expressed by the following equa-tion:

ρ (T ) = 9.370−9.442×10−4T ±0.026, (12)

where ρ is the density in g cm−3 and T is the tem-perature in oK. Molar volumes and thermal ex-pansion coefficients were calculated from the dataobtained. It is found that upon melting copper ex-pands by 3.34%. A satisfactory agreement existsbetween the data and those previously obtainedby the maximum-bubble and the immersed-sinkermethods. Density measurements of liquid metalsby the EML technique were reported by Shiraishiand Ward (1964). The method used is based onthe determination of the volume of a liquid dropletof known weight (∼ 1.5g) levitated in the field ofa high-frequency coil. The method avoids the cor-rections for the temperature-dependent expansionof the apparatus. But, the deviation of the dropletgeometry from the spherical shape could increasemeasurement errors. The coil-ring levitating sys-tem was powered by a 10 kW Tocco generator.The temperature was measured with an accuracyof ±10oC by a two-color pyrometer focused tothe top of the droplet. A carbonyl nickel dropletsample of 99.9% purity was tested. During theexperiments the specimen lost 10-20 mg. In thismethod one camera monitors the sample from theside and another from the top, so that one cameracan always be used to determine the distance be-tween the moving sample and the second camera.The known diameter of the solid sample is used tocalibrate the camera image. Recorded frames areanalyzed with an image-processing system and acomputer to monitor the size of the spherical spec-imen. The density of the test sample is measuredover a 300oC range below the melting temperatureto determine that density varies linearly with tem-perature over the whole temperature range stud-ied. The following relationship summarizes the


findings,

ρ (T ) = 9.966−12.000×10−4T ±0.055, (13)

where ρ represents the density in g cm−3, and Tthe temperature in oK.

Racz and Egry (1995) describe the EML appara-tus, method of image acquisition, digital imageprocessing, and calculation methods to measurethe density and thermal expansion of undercooledliquid metals as a function of temperature. Theiranalysis shows that the main sources of error are:

• temperature measurement

• image calibration

• image processing steps to determine dropletsize

• Legendre polynomial fit.

Gorges et al. (1996) developed a method for mea-suring the density of levitated droplets using aCCD camera and image processing. Results forundercooled nickel samples are in good agree-ment with density measurements of previous re-searchers. An improved algorithm for edge detec-tion in digital image processing is also described.Lohöfer et al. (2003) conducted density measure-ments using the experimental setup shown in Fig-ure 1. A new optical measurement technique iscombined with EML. The sample is levitated andmelted in an argon atmosphere. As electromag-netic forces lift up the sample induced currentsstart heating it up. The sample is carefully cooledat controlled temperature by exposing it to flow-ing argon gas. The temperature is measured usingan infrared pyrometer. When measuring the vol-ume of the sample, it is important that the sam-ple is fully visible from the side. Lohöfer et al.designed a coil which assured that no part of theedge of the sample was hidden by the windings.To measure the sample volume an expanded, par-allel HeNe laser beam is used to illuminate thesample from behind, Figure 1. The laser light isthen focused on the small aperture of a pinhole,which removed all non-parallel parts of the beamcoming from interferences or the hot sample it-self. It is assumed that a band-pass filter addition-ally blocks all light not originating from the laser.

The shadow image of the sample is then capturedby means of a digital CCD camera and fed into acomputer to be analyzed in real time by an edgedetection program. The optical setup is calibratedusing well defined ball-bearing spheres. This op-tical setup has the advantage to prevent apparentsize changes due to sample movements along theoptical axes, and to guarantee a constant contrastfor the edge detection algorithm independently ofthe brightness of the hot sample.

Wang et al. (2003) developed a containerlessand contamination-free measurement method us-ing EML techniques in order to determine accu-rately the thermal expansion coefficient and den-sity of metals in solid and liquid states. A com-puter model for predicting the levitating force, ab-sorption power and heating temperature is devel-oped as well, and the predictions of the numericalsimulations have been experimentally validated.Wang et al. also discuss a number of design is-sues with respect to a single levitation and heatingcoil. Several challenges related to this techniqueare discussed:

• Increasing the magnitude of the levitatingforce when the experiment is conducted interrestrial conditions.

• Controlling the position of the levitateddroplet and the temperature range by usinga single induction coil to maximize the effi-ciency of the optical techniques.

• Detecting accurately the edge of the dropletin the imaging process.

After getting a stable and good position (the sam-ple image is minimally blocked by the induc-tive coil) of the sample in the desirable temper-ature range, two "good" images, a side view anda top view, are then used to determine the vol-ume of the sample. A new edge detection methodhas been developed, which reduces the error byabout 30% as compared to a conventional edgedetection method often used the maximum inten-sity gradient method. Optical measurement er-rors are corrected, and thermal expansion coef-ficient and density data for representative solidand liquid metals and alloys are given. Dupac


Figure 1: Experimental set-up for density measurements; the shadow of the liquid, levitated sample insidethe levitation coil, illuminated from the back by an expanded HeNe laser beam, is recorded by a CCDcamera, Lohöfer et al. (2003).

et al. (2003) developed a new image processingtechnique to predict the volume of the levitatedaspherical droplet. The technique based on thecubic-spline convex-hull algorithm allows to re-store the missing (blocked by the coil) portion ofthe frontal image, and to recover the transversaland frontal surface shapes from image shading.The normalized volume of the levitated asphericaldroplet was determined by combining the recov-ered shapes. To validate the results, the algorithmperformance has been tested on the images of theductile iron specimens obtained experimentally.

2.4 Thermal Diffusivity

Thermal diffusivity is an important thermophysi-cal property of materials which controls the tem-perature distribution in the body of the specimenduring transient heat transfer processes as well asmass transfer through diffusion in heterogeneousboundary layers in many metallurgical processeslike melting, solidification, crystal growth, weld-ing and casting, among others. There are sev-eral methods to measure thermal diffusivity, suchas transient hot-wire, flash heating, dynamic andstepwise heating methods. In the popular flashheating technique, first introduced by Parker etal. (1961), a small disk-shaped sample receives ahigh-energy irradiation burst on one surface. Theincrease of temperature on the opposite surface isrecorded as a function of time. Comparison ofthe measured temperature with theoretical predic-

tions allows the determination of the thermal dif-fusivity of the specimen. The technique has sev-eral advantages:

• Small amount of energy needs to be added tothe specimen.

• The size of the specimen can be small

• The technique can be used by preheating orcooling the sample.

• Three thermal properties (thermal diffusiv-ity, thermal conductivity and heat capacity)can be determined for the same sample.

However, the presence of convective flows and thecontainer are the main drawbacks of this and othertechniques aimed at measuring thermal diffusiv-ity. To solve the problem of container contam-ination, a new measurement technique was pro-posed by Bayazitoglu et al. (1990). In this tech-nique two experiments with different droplet sizesand with Biot numbers less than 0.1 and greaterthan (or equal to) 0.1 are conducted to bypassdensity. An inverse conduction problem is for-mulated for the large-droplet case, and Laplacetransform methods are used to solve the problem.The method is demonstrated for three materials,Nickel, Niobium and Palladium. Bayazitoglu etal. use experimental surface temperature data in-stead of the inhomogeneous boundary condition,


and they determine thermal diffusivity by mini-mizing a function that complies with the heat bal-ance at the surface. The applicability of the pro-posed method is limited because of large spec-imen sizes required. Very large samples (∼ 20cm in diameter) must be levitated, which is prac-tically impossible from an experimental point ofview. The method proposed by Bayazitoglu et al.(1990) does not account for the two-dimensionaltemperature profile. To remedy this short comingMurphy and Bayazitoglu (1992) developed an-other two-step computational model of the tran-sient heat conduction in the spherical specimento account for the two-dimensional cooling pro-file. They consider a finite flash time which causesnon-uniform temperature fluctuations in the ab-sorption layer. Thermal diffusivity is estimated byminimizing the difference between the predictedand experimentally measured temperatures. Thetechnique is demonstrated for Nickel, Iron andCopper using analytical data. However, param-eters such as density, specific heat, incident heatflux, radiative emissivity must be known to esti-mate thermal diffusivity via this technique.

Shen and Khodadadi (1993) proposed an ex-tended single-step containerless flash techniquesuitable to measure thermal diffusivity of levi-tated spherical specimens. Radiation and con-vection heat losses from the surface were takeninto account to predict the temperature at anypoint analytically. Thermal diffusivity is deter-mined from simultaneous temperature measure-ments at least at two different points on the sur-face. The method presents some advantages be-cause it bypasses the need to know some parame-ters like incident energy flux intensity, absorptionlayer thickness and also heat loss. The diameterof the sample and the temperature rise history attwo different points on the surface are the onlytwo parameters needed to estimate thermal dif-fusivity. Shen et al. (1997) also proposed twoextended containerless flash techniques to levitatespherical specimens. The process is modeled asan axisymmetric transient conduction heat trans-fer problem within the sphere. To validate the ad-vantages of the techniques ground based experi-ments are conducted with high-temperature solid

samples of pure Nickel and Inconel 718 super-alloy near their melting temperatures. Compari-son of the experimentally determined thermal dif-fusivity values to data available in the literatureshow that the proposed techniques are potentiallypromising.

2.5 Electrical Resistivity

Information on the electrical resistivity of moltenmetals and alloys is especially important inmany metallurgical processes such as electroslugremelting, electromagnetic stirring in continuouscasting, electrolysis, and induction melting infoundries. Due to the disordered arrangement ofions in the liquid state, molten metals and alloysexhibit higher (∼ 1.5-2.3 times) electrical resis-tivity than those in solid state. However, rela-tively few studies have been reported on the elec-trical resistivity of molten metals and alloys, par-ticularly at elevated temperatures, since measure-ments are extremely difficult. Methods to mea-sure electrical resistivity can be categorized intothree groups:

• direct resistance measurements using contactprobes,

• contactless inductive measurements,

• non-contact containerless measurementtechniques.

The technique of choice for solid materials isthe direct resistance four-probe method based onOhm’s law. Although this direct method can beapplied to low melting point, non-reactive liq-uid materials, reactions between the probes andthe molten sample preclude using the four-probemethod with high-melting point materials. Volt-age drop along a sample in a capillary tube ofknown cross-section and length is measured atconstant current density. The probe cell has to becalibrated using a liquid metal (usually mercury)of known resistivity. Selection of proper materialsfor the capillary cell and the electrodes remainsthe primary difficulty with this method. The tech-nique is also limited to materials with very narrowfreezing ranges.


Inductive techniques for measuring electrical re-sistivity are contactless and thus prevent chem-ical reactions between molten samples and con-tacting probes as in the direct method. There aretwo different types of contactless methods. Therotating field method is based on the phenomenathat when a metal sample rotates in a magneticfield or when the magnetic field rotates arounda stationary sample circulating eddy currents areinduced in the sample generating an opposingtorque proportional to the electrical conductivityof the sample. In liquid metals and alloys appliedmagnetic field also causes significant rotation ofthe liquid in the crucible decreasing the angularvelocity between the field and the sample. Re-cently, Bakhtiyarov and Overfelt (1999a,b) devel-oped a rotational contactless inductive measure-ment technique to measure both the viscosity andthe electrical conductivity of liquid metals. Pre-liminary tests conducted with low melting pointmetals (lead and tin) and alloys (LMA-158 andPb/Sn binary systems) agree well with data fromthe literature. But, it must be pointed out that thismethod leads to difficulties caused by convection.

In both direct resistance and contactless induc-tive techniques chemically reactive componentscontained in the crucible may easily contaminatethe molten sample altering its electrical proper-ties. The container may provide heterogeneousnucleants that generate early solid-phase nucle-ation. Electromagnetic levitation is very usefultechnique for the containerless measurement ofelectrical conductivity of liquid metals. Electri-cally conducting samples represent an additional,inductively coupled electric circuit that changesthe impedance Z(σe) of the coil, Lohöfer et al.(1991). It increases the resistance and decreasesthe inductance depending on the electrical con-ductivity of the sample. Since the molten sam-ple in this method is isolated from container wallsthe sample will maintain its intrinsic propertiesallowing the study of the structures and proper-ties of the material in deeply undercooled states.Impedance affects the alternating current in thecoil causing changes in the power loss of the coilcurrent at fixed voltage across the coil, in thephase difference between coil current and volt-

age and the resonance frequency of the alternatingcoil current, Herlach et al. (1993). Nyberg andBurgess (1962) developed electrodless methodsfor determining the electrical conductivity, theHall mobility, and the magnetoresistance effectof semiconductors. Conductivity measurementswere made by placing the sample into the coreof a sinusoidally excited solenoid and by relatingthe impedance changes reflected into the solenoidto the conductivity of the sample. Tomlinson andLichter (1969) and Lee and Lichter (1972) ap-plied the inductive coupling method to measurethe change of the coil resistance after having in-serted a cylindrical sample into a solenoid. Thiselectrodless technique is used to measure the elec-trical resistivity of liquid Cd-Bi, Cd-Sn, Cd-Pb,In-Bi, and Sn-Bi alloys. Positive temperature co-efficients of electrical resistivity were obtained inall samples except Cd10Bi90 alloy. Zhuravlev etal. (1982) measured the inductance change us-ing an Owen bridge circuit. The electrical con-ductivity of liquid Fe, Co, and Ni is measuredby this method. The data agrees well with thoseobtained by the rotational magnetic field method.The uncertainty of the method is ± 4%. How-ever, the method has some shortcomings like thesmall quality factor of the measuring coil, signif-icant variation of the coil impedance itself withtemperature, thermal deformation of the coil, andrecrystallization of the winding wire among oth-ers.

The change in both the resistance and the in-ductance was simultaneously measured by Del-ley et al. (1980). In this work the electrical re-sistivity of the specimen is determined from theimpedance change of the coil when the specimenis inserted. The system automatically adjusts themeasuring frequency for a constant loss angle ofthe impedance change corresponding to a con-stant skin depth in the sample. The sample electri-cal resistivity is directly proportional to the mea-suring frequency, and the empty coil impedanceis automatically compensated for. Electrical re-sistivity measurements of liquid tin, copper, man-ganese, nickel and iron are reported. Data fromother sources agrees well with the results of thestudy.


Garnier and Moreau (1983) investigate the stabil-ity of the interface between a liquid metal and aninsulating atmosphere, in which an inductor gen-erates a uniform alternating magnetic field. Inparticular the influence of the electrical conduc-tivity of the liquid sample was studied. Linearstability analysis shows that the influence of thealternating magnetic field on the perturbed inter-face is neutral for wave-vectors perpendicular tothe magnetic field. The stabilizing effect is largestwhen the angle between the wave-vector and themagnetic field is zero. It increases with increas-ing wave-number, is maximum for an infinitelyconducting medium decreasing with the electri-cal conductivity. Kraftmakher (1991) proposeda method for contactless electrical conductivitymeasurements based on the determination of thephase angle of the effective magnetic susceptibil-ity of cylindrical samples in an axial AC mag-netic field. The output voltage of a differentialtransformer is compensated by a voltage with aphase shift which depends on the sample size,so that compensation becomes possible only atthe magnetic-field frequency numerically equal tothe electrical conductivity of the specimen. Lo-höfer (1994) analytically calculated the magneti-zation and the impedance of an electrically con-ducting sphere inductively coupled with an exter-nal, sinusoidally alternating current density distri-bution. The general results are applied to the spe-cial case, where the external current density distri-bution is concentrated on thin, toroidal rings sur-rounding the spherical specimen. It is shown thatimpedance measurements can be used for the con-tactless determination of the electrical conductiv-ity. When the skin depth is much less than thesample radius (δ �R), the changes in resistanceand inductance are determined as

ΔRωL0

∼ δ2R

,ΔLL0

∼(

δ2R

− 13

). (14)

L0 represents the inductance of the solenoid with-out the sample. These changes are related to thephase shift between current and voltage, whichcan be measured experimentally. The relation be-tween the phase shift and the skin depth allows anestimate of the electrical conductivity.

Enderby et al. (1997) describe the combinedapplication of aerodynamic levitation and elec-trodless conductivity techniques to determine thehigh-temperature electrical conductivities of liq-uid metals. The hybrid method allows the de-termination of the electrical conductivity at tem-peratures as high as 2500oC. Rhim and Ishikawa(1998) report a non-contact technique to measurethe electrical resistivity of liquid metals basedon a conducting drop that is levitated by a high-temperature electrostatic levitator in a high vac-uum. The relative changes in torque are mea-sured as a function of temperature when a rotat-ing magnetic field is applied to the sample. Thetechnique was tested for pure aluminum at solidand liquid states. Richardsen and Lohöfer (1999)describe a facility for non-destructive measure-ment of the electrical conductivity of liquid met-als above and below the melting temperature. Anew technique combines the containerless posi-tioning method of electromagnetic levitation withthe contactless method of inductive conductivitymeasurements. Earlier, this technique was appliedonly to low-melting point liquid metals containedin an ampoule, which gives the sample a well-defined cylindrical shape. In the modified methodadvocated by Richardsen and Lohöfer the sam-ple is freely suspended within the measuring field,and it does not have a predefined (nearly spheri-cal) shape. Lohöfer et al. (2003) conducted elec-trical resistivity measurements using the experi-mental set-up in Figure 2, which shows the ar-rangement of the measuring transformer betweenthe levitation coils. The alternating current in theprimary coil of the transformer generates a highfrequency magnetic field that induces a voltage inthe secondary coil, which depends on the electri-cal resistivity of the sample, its radius, and thedeviation of its shape from spherical symmetry.Through the measurement of the absolute valuesof the current in the primary coil and the voltagein the secondary coil as well as the phase differ-ence between both, the (complex) impedance isdetermined. Next electrical resistivity of the liq-uid droplet is calculated from the theoretical re-lationship between impedance and electrical re-sistivity, which is well known except for cali-bration constants that depend on the radius and


Figure 2: Electrical resistivity measurements; primary and secondary measurement coils are integrated withthe surrounding levitation coil in an UHV chamber, Lohöfer et al. (2003).

the shape factor. To determine these constants,measurements were performed at different cur-rent frequencies in the range between 10 kHz and1 MHz. Further, a containerless calibration ex-periment with a spherical sample of well-definedresistivity and radius was carried out. The sam-ple was positioned by the levitation field in thecenter of the measurement transformer. To pre-vent inductive interactions between the high fre-quency magnetic levitation field and the measur-ing coils, the measurement itself was performedonly in short time intervals of about 1 ms du-ration during which the levitation field is com-pletely switched off.

2.6 Thermal Conductivity

Accurate measurement of thermal conductivity ofliquid metals and alloys is usually more difficultthan the measurement of electrical conductivityand thermal diffusivity. The difficulties concernaccurate heat flow measurements. Thermal con-ductivity is directly related to the change in theatomic vibrational frequency. A number of non-metallic substances abide by

λ√M

= 2.4×10−3, (15)

where λ and M are the thermal conductivity andM the molecular weight, respectively. Sincefree electrons are responsible for the electricaland thermal conductivities of conductors in bothsolid and liquid states many researchers use the

Wiedemann-Franz-Lorenz law to relate thermalconductivity to electrical resistivity,

λ ρe

T=

πκ2

3e2 = L0, (16)

where κ and e represent the Boltzman constantand the charge of the electron, respectively. Theconstant

L0 =π2κ2

3e2 = 2.45×10−8W ΩK−2,

is the Lorenz number. The validity of this rela-tionship was confirmed experimentally with highaccuracy by many researchers.

Haller et al. (1977) developed a new techniqueto determine the Lorenz ratio directly for temper-atures up to 500oC. The Kohlrausch method isapplied to a spherical specimen with a constric-tion. The measurements on liquid tin are in agood agreement with the Wiedemann-Franz law.Fecht and Johnson (1991) applied the AC tem-perature technique to the problem of non-contactcalorimetry. The proposed method allows an ac-curate determination of the thermal conductivityof metastable undercooled liquid metal dropletsof spherical shape using an electromagnetic levi-tator. It is shown that the external (τ1) and internal(τ2) relaxation time constants are the most criticalparameters in achieving high accuracy in the mea-surements. An accuracy of better than 1% can beachieved if ωτ1 >10 and ωτ2 <0.1.


2.7 Surface Tension

Surface tension is determined by the microscopicstructure of the liquid near the surface. At aliquid-vapor interface density undergoes a steepchange from a high value in the liquid state to avery low value in the gas phase. Therefore, sur-face atoms experience an attraction toward the liq-uid phase, which is the cause of the surface ten-sion. Due to its energetic and entropic origin, it isnecessary to calculate the free energy of the sys-tem to determine the surface tension. Thus sur-face tension is determined as the additional freeenergy required to generate a unit surface areaseparating the liquid from its vapor phase.

There are many techniques for surface tensionmeasurements: sessile drop, pendant drop, max-imum bubble pressure, maximum pressure in adrop, detachment or maximum pull, capillary-rise, drop weight, and oscillating drop methods.The sessile-drop technique has been widely useddue to its many advantages. The method utilizesa molten drop resting on a horizontal ceramicsubstrate, and allows measurements over a widerange of temperatures. However, the surface ten-sion data may be affected by contaminants. Thesurface tension of a liquid droplet can be mea-sured by exciting surface oscillations. The fre-quency of the oscillations is related to the surfacetension. The radius of the viscous sphere under-goes oscillations of the form, Egry and Szekely(1991)

R ∼ R0 cos(2πνlt)e−Γl tPl (cosϑ ) , (17)

where l denotes the normal modes, Pl representsthe Legendre polynomials, and νl and Γl are thefrequency and damping of the oscillations. Theoscillating drop technique which uses electro-magnetic levitation prevents surface contamina-tion and provides deep undercooling of the moltenmetal. The sessile-drop and the oscillating droptechniques have been used for surface tensionmeasurements on superalloy 718 Overfelt et al.(2000). Due to chemical reactions between thespecimen and the substrate the temperature de-pendence of the surface tension obtained by thesessile-drop method is higher than expected. Asignificant sample “shaking” was observed during

the heat-up in sessile-drop experiments. Keene(1993) presents a literature survey of the exper-imentally determined values for the surface ten-sion of molten pure metals which shows signif-icant discrepancies between published values ofsurface tension and its temperature coefficient.The discrepancies are attributed to:

• experimental errors during measurementsdue to both the characteristics of the appa-ratus and the competence of the operator,

• incorrect density data used for surface ten-sion calculations,

• effect of impurities namely surfactants,

• surface tension data at various temperaturesmay not be enough to obtain reliable valuesfor the slope of the γ(T) curve.

Surface tension is the only restoring force forsurface oscillations, and consequently it deter-mines the oscillation frequency. In the EML tech-nique surface tension can be calculated from thefrequency of the surface oscillation of the non-rotating spherical sample utilizing the followingequation, Lamb (1945),

ν2 =l (l −1) (l +2)γ

3πm=

l (l−1)(l +2)γ4π2ρR3 , (18)

where γ , ρ , m, R, ν and l represent the surfacetension, the density, the sample mass, the un-perturbed radius of the droplet, the frequency ofthe surface oscillation, and an integer denotingthe normal modes, respectively. In (18) it is as-sumed that the amplitude of deviations from thespherical shape is small, the liquid viscosity islow, the damping effects on the natural oscilla-tions of the liquid droplet are negligible, and theliquid is incompressible. The lowest two modesare excluded by the requirements of conservationof mass (l = 0) and fixing the center of mass ofthe drop at the origin of the coordinate system (l= 1). The Rayleigh frequency is the frequency ofthe fundamental mode, which corresponds to l =2 in (18),

γR =38

πmν2R. (19)


For non-spherical rotating droplets the fundamen-tal mode is shifted and split into five peaks,whereas for non-rotating droplets only threepeaks exist, Busse (1984). Experimental observa-tions show that two pairs of oscillation frequen-cies are related, Natarajan and Brown (1986)

ν8 = ±2ν5 ν16 = ±2ν10. (20)

This means that quadratic nonlinearities in equa-tion (18) will create internal resonances betweenthese two pairs of modes. A resonance betweentwo modes causes secular terms to appear in(18) for higher-order corrections. Natarajan andBrown (1986) derive the interaction equations thatdescribe the quadratic internal resonance betweentwo oscillating modes of an inviscid droplet. Spe-cific oscillation modes are coupled by quadraticnonlinearities caused by inertia, capillarity, anddrop deformation. The equations describing theinteractions of these modes were derived fromthe variational principle for the appropriate La-grangian. It is shown that axisymmetric drop mo-tions due to the internal resonance are unstable tonon-axisymmetric perturbations.

The oscillations can be identified as changes inthe shape of the droplet. Frequency of the oscilla-tions can be defined from the Fourier transforma-tion of the time signals. The position of the peaksis related to the frequency. To obtain surface ten-sion values from measured frequency spectra, thecorrect labels (l,m) need to be assigned to eachpeak in the spectrum. As is clear from (18) themass of the sample is required to estimate the sur-face tension, but that can be easily measured withhigh accuracy. Consequently this technique hashigher accuracy than the methods which requireknowledge of the density. However, (18) is validonly for free linear oscillations of inviscid, non-gravitating spherical drops.

Prosperetti (1980) formulated the problem as aninitial-value problem, and determined that themotion consists of modulated damped oscillationswith the damping parameters and frequency ap-proaching only asymptotically the results of thenormal-mode analysis. An estimate of the orderof magnitude of the convective term in the mo-mentum equation shows that it is negligible com-

pared with the inertial term. Marston (1980) con-sidered the forced oscillation case. Using spher-ical harmonic expansions for the radial and tan-gential stresses droplet deformations opposed bysurface tension and driven by radiation stresses atthe interface were calculated. The decay time offree oscillations is also computed, where a newterm is found which is small but significant forliquid surrounded droplets. Reid (1960) introduc-ing the viscosity of the liquid droplet obtained therelationship between the oscillation frequenciesof inviscid (σl;0) and viscous (σl;ν ) liquid dropletsof radius R and kinematic viscosity ν ,

σl;0 =α2νR2 ,

σl;ν

σl;0=

q2

α2 . (21)

And calculated for the principal mode l = 2, α2

= 3.69, q2

α2 = 0.968. Myshkis et al. (1987) solvethe problem of free oscillations of liquid dropletsby introducing a self-gravity factor. Asymptoticformula for free oscillating and self-gravitatingliquid globe with low and high viscosity are de-rived. Pozrikidis (2001) developed a numericalmethod for simulating the surface tension inducedthree-dimensional oscillations of inviscid liquiddroplets. The numerical procedure is based ona generalized vortex formulation, which employsthe double-layer representation for the harmonicpotential. Results of simulations are presentedto illustrate the performance of the numericalmethod for axisymmetric and non-axisymmetricoscillations.

Temperature dependence of surface tension is re-lated to surface entropy and surface excesses.Thus, the changes in the structure of the liquidspecimen with temperature are reflected in thetemperature coefficients of the surface tension.The temperature coefficient of surface tension ofa liquid can be thermodynamically expressed bythe following equation, McLean (1957)

− dγdT

= Ss0 +∑

i

Γidμi

dT, (22)

where Ss0 is the surface entropy, Γi is the surface

excess concentration of the ith species in the liq-uid, and μi is the corresponding chemical poten-tial. The variation of surface tension with temper-


ature for most liquid metals can be expressed by alinear relationship,

γT = γm − dγdT

(T −Tm) , (23)

where γT and γm are the surface tension values attemperature T and at the melting point Tm, respec-tively.

The first technique to determine the surface ten-sion of molten metals through the measurementof the natural oscillation frequency of levitateddroplets (∼0.5g) was developed by Lu and hiscoworkers, Fraser et al. (1971), Murarka et al.(1971, 1975). A high-speed video camera hasbeen used to record the surface oscillations. Asystematic analysis of the effect of impurity level,drop size, viscosity, inertia, electromagnetic field,electrostatic charge, and amplitude of vibration onsurface tension measurements is conducted usingthe electromagnetic levitation method. The sur-face tension of pure liquid iron and nickel wasmeasured in a 6% H2-He gas mixture in the tem-perature ranges 1550-1780oC and 1475-1650oC,respectively. A linear relationship was found be-tween surface tension and temperature for bothtest samples. Surface tension of pure iron does notstrongly depend on temperature in the range fromits melting point to 1700oC. Supercooled iron at1480oC has a higher surface tension than its cor-responding liquid at temperatures above its melt-ing point. A good agreement is found betweenthe surface tension values obtained for iron andnickel at 1550oC using the electromagnetic levi-tation technique and the sessile drop method fa-vored by previous researchers. However, somesurface tension values were ∼10% higher thanthose measured by static methods, and the tem-perature dependence of surface tension was foundto be positive. The effect of trace amounts of oxy-gen on the surface tension of liquid iron is studiedby Murarka et al. (1975). The oscillating droptechnique was used to measure the surface ten-sion of liquid iron-oxygen alloys in the tempera-ture range 1560o-1645oC. It is shown that surfacetension does not change significantly with temper-ature, and the surface activity of oxygen in iron is2.1×106 dynes/cm.

To evaluate both the applicability of the levitation

technique and the validity of Rayleigh’s equationfor determining the surface tension of liquid met-als, Soda et al. (1977) developed a new meltingtechnique. The surface tension of liquid copper isevaluated in an atmosphere of hydrogen at 1000o-1330oC, including an 80oC incursion into the su-percooled temperature range. The surface tensionof copper is found to vary linearly with tempera-ture as,

γ = 1.390−0.00043(T −1356) , (24)

and the applicability of Rayleigh’s equation incalculating the surface tension of the levitatedmetallic droplets is demonstrated. It is shown thatoscillation frequency, and hence the simulatedsurface tension value strongly depends on oscil-lation amplitude, and that reliable surface tensionvalues can be achieved at low amplitude oscilla-tions. The surface tension of liquid iron and iron-oxygen alloys has been measured by Kasama etal. (1983) via EML. They found that the surfacetension of liquid iron varies linearly with temper-ature over the range 1781oK to 2015oK. These re-sults are 50 mN m−1 higher than those obtainedby the sessile drop method. The discrepancy isattributed to sample contamination in the sessiledrop technique. It is shown that the levitationtechnique eliminates container contamination andproduces an extraordinarily clean droplet surface.Surface tension data were used to analyze the oxy-gen adsorption layer.

Nogi et al. (1986) measured the surface tensionof liquid Fe, Co, Ni, Cu, Ag, Zn, Pb, Cd, andSn by the sessile droplet method and/or the levi-tation droplet method over wide range of temper-atures. The values of surface tension measured bythe levitated droplet method are higher than thoseobtained by the sessile droplet method. The dis-crepancy is explained by decreased droplet con-tamination in the containerless levitation method.Negative temperature coefficients of surface ten-sion were obtained for all test samples. The posi-tive values of the temperature coefficients of sur-face tension for liquid Zn and Cd found in litera-ture are attributed to impurity effects. Surface ten-sion measurements for Zn in a purified hydrogenatmosphere and in hydrogen saturated with water


vapor at 273oK lead to the conclusion that surfacetension decreases linearly with increasing temper-ature in the former and that it increases with in-creasing temperature near the melting point in thelatter. Schade et al. (1986) use the electromag-netic levitation technique to measure the surfacetension of undercooled pure liquid metals (Fe,Ni, Cr and Co) and binary alloys (Fe-Si and Co-Si). The natural frequency of oscillations is deter-mined using a Fourier wave analyzer, and surfacetension is then calculated from Rayleigh’s equa-tion. The technique allows reaching quite largeundercoolings for test samples (∼0.3Tm). Exper-imental observations show that surface tension ofall test specimens varies linearly with tempera-ture.

Two independent laboratories conducted mea-surements of the surface tension of some metalsby the levitating drop technique to establish con-fidence levels, Keene et al. (1986). Results agreereasonably well for pure iron, cobalt, gold andcopper. However, significant differences were ob-served in the data for stainless steel, which is at-tributed to the different hydrogen concentrationsin the respective environmental gases. These ex-periments and others similar in nature by Schadeet al. (1986) and Nogi et al. (1986) use photo-diodes to record the incoming light intensity asan integrated signal; therefore to define the actualtype of oscillations becomes an impossible task.

Equation (8) in Part I of this series of papersBakhtiyarov and Siginer (2008) is based on anunconstrained droplet of spherical geometry, andsurface restoring forces depend only on surfacetension. In terrestrial conditions oscillation spec-tra may deviate (splitting and/or shifting) fromthose predicted by Rayleigh’s equation due to fac-tors such as droplet rotation, asphericity of thedroplet, effect of the electromagnetic field on thesurface restoring forces, stirring, etc. Thereforecorrections are needed to surface tension valuescalculated via Rayleigh’s equation. There are twoapproaches to make corrections:

• application of the Lagrange equations basedon kinetic and potential energy considera-tions

• derivation of the equations of motion fromconsideration of applied forces

Gagnoud and Garnier (1985) analyzed the interac-tion between the equilibrium shape of the moltendroplet and the magnetic field distribution. Twomodels are used and coupled to simulate the sur-face tension, frequency and inducting current in-tensity, the electromagnetic model and the freeboundary model. To ensure an electromagneticskin depth smaller than the size of the levitateddroplet a high frequency is assumed. Thereforethe magnetostatic problem was solved, and theinfluence of the internal fluid motion on the freesurface shape was ignored. It is shown that theintensity of inducting current significantly affectsthe shape and the position of the molten sample.However, surface tension does not significantlyaffect the equilibrium of a levitated charge.

The inclusion of asphericity in the analysis is avery inexpensive alternative to maintaining thesphericity of the droplet by means of complexinstrumentation, Suryanarayana and Bayazitoglu(1991a). The effect of the asphericity and addi-tional restoring forces introduced by the levitatingfield on the vibrations of a non-rotating inviscidliquid droplet using the linear perturbation theoryis studied by Cummings and Blackburn (1991).The following equation, which takes into accountboth magnetic and gravitational forces, is derived,

νR =15

2

∑m=−2

ν22,m−ν2

t

[1.9+1.2

(z0

R

)2],

z0 =g

8π2ν2t

,

ν2t =

13

1

∑m=−1

ν2l,m.

(25)

ν2,m and ν1,m are the frequencies of surface os-cillation for the l = 2 mode and the frequenciesof translational oscillation of the droplet’s cen-ter of mass, respectively, and R is the radius ofthe droplet. The simulations show that the ex-pected single frequency is split into three or fiveunequally spaced bands. Frequencies are higherthan those of a spherical droplet, and the surfacetension is higher than its normal value. A fre-quency sum rule to predict Rayleigh frequency


from the observed frequencies is derived. Un-fortunately, the parameters that describe the as-phericity of the sample depend on gravity and thesurface tension of the sample and are actually un-known.

Equation (25) has been used by Przyborowskiet al. (1995) to measure the surface tension ofmolten silicon by an oscillating drop method us-ing the EML technique over a wide range oftemperatures (1100o-1500oC) including the un-dercooling condition. Corrected surface tensionof the specimens is determined with an accu-racy of 3-4%. Similar measurements carried outin microgravity would lead to a significant im-provement in accuracy. Equation (25) derived byCummings and Blackburn (1991) is also used byEckler et al. (1991) to evaluate the oscillationspectra in their experimental studies. They per-form the experiments in an electromagnetic lev-itation facility, which allows containerless mea-surements of surface tension of electrically con-ducting bulk samples in a vacuum or inert gas at-mosphere. Nickel-iron alloys of various concen-trations were used as the test samples. The ap-proximate sample diameter and mass were 6 mmand 1 g, respectively. The experiments with purenickel demonstrate the linear temperature depen-dence of the surface tension of pure nickel withdecreasing temperature. Iron additives to nickelreduce the surface tension. The fundamental l =2 mode was split into three peaks due to the as-phericity of the droplet.

Another theory for the dynamics of asphericaldroplets subjected to external forces is developedby Suryanarayana and Bayazitoglu (1991a,b) whoshow that for an arbitrary shape deformation thefrequency spectrum splits into 2l −1 peaks for amode l oscillation. The splitting of the frequencyspectrum for mode 2, 3 and 4 oscillations is calcu-lated, and the frequency split is expressed in termsof external forces. It is shown that the effects ofasphericity sufficiently explain the splitting of thefrequency spectrum observed in the experiments.The results were applied to some previous experi-mental results related to surface tension measure-ments, and it is shown that the accuracy of surfacetension measurements can be improved if correc-

tions are made for the asphericity of the droplet.However, difficulties persist in their analysis in in-terpreting the electromagnetic spectra more accu-rately as compared to acoustic levitation. Later,Bayazitoglu et al. (1996) calculated the magneticpressure distribution on the surface of the dropletas a function of the parameters that govern the ex-ternal magnetic field. They assume that the pres-sure distribution on the surface of the sphericaldroplet can be used even when the droplet be-comes aspherical because the static deformationis smaller than the droplet radius. They determinethat for small droplet deformations the order ofthe peaks in the frequency spectrum is a functionof the position of the droplet in the magnetic field,and the variation of the magnetic pressure distri-bution is not important in determining the surfacetension.

A new method of oscillation detection based onthe inspection of certain geometrical parametersby digital image processing is presented by Sauer-land et al. (1992). The method has been used tomeasure the surface tension of pure nickel in thetemperature range 1300-1620oC and in He-H2 at-mosphere to reduce oxidation. Equation (22) isused to calculate the surface tension. It is shownthat the experimental data can be described bythe linear relationship (23). The data are slightlyhigher than those in the literature, but the temper-ature dependence is the same. This is attributed tothe elimination of systematic errors in the assign-ment of the modes to the frequency peaks in thespectra and by the lower oxygen content of theirsamples.

An improved method for measuring the surfacetension of molten metals has been proposed byEgry (1991). The spectrum of surface oscillationsof an electromagnetically levitated liquid dropletis evaluated by digital image processing. Prelim-inary results for FeNi samples are presented. Ina follow-up work Egry et al. (1992) conductedterrestrial measurements in a conventional elec-tromagnetic facility, where a generator operatingat 330 kHz was connected to a conical coil pro-viding levitation and heating. A two-color ratiopyrometer was used to measure temperature. Thefacility was capable of processing samples up to


1 g mass with oscillations in the 40 Hz range. Amodified standard video camera with 150 Hz sam-pling rate has been used to obtain images of theoscillating droplet. The frequency spectrum form= 0.71 g 90%Fe-10%Ni alloy sample at 1520oChad two low-lying peaks and triply split l=2 mode.The individual m-labels could not be clearly iden-tified, and therefore the surface tension was es-timated by Rayleigh’s equation (19). The com-puted surface tension value (1.76 ± 0.06 N m−1)is in the range reported by Keene (1988). Egryet al. (1994) also measured the surface tensionof noble metals (copper, silver and gold) by theEML technique. Noble metals were chosen dueto their high resistance to surface oxidation. It isshown that surface tension changes linearly withtemperature in the undercooled regime. Experi-mental data is in good agreement with the resultsobtained with conventional techniques leading tothe conclusion of negligible contamination of thenoble metal samples in ceramic crucibles and thatthe levitated-drop technique is capable of obtain-ing accurate surface-tension values for liquid met-als.

Lohöfer et al. (2003) conducted surface ten-sion measurements using the experimental setupshown in Figure 3. The levitated Cu-Ni samplemelts in a quartz-glass tube filled with a mixtureof helium and hydrogen. The magnetic levita-tion coils are located outside the glass tube. Thetemperature is measured by a pyrometer and con-trolled by the gas flow. The turbulent fluid flow ofthe Cu-Ni melt, excited by the strong rf magneticlevitation field, caused oscillations of the dropletsurface recorded by a camera. As a containerlesstechnique EML eliminates perturbations and dis-tortions from outside sources and allows the ob-servation of the unperturbed oscillations of theliquid droplet. Surface oscillation frequencies ofthe sample around its equilibrium shape can becomputed by the ‘oscillating drop method’ viaFourier analysis using the recorded image.

2.8 Viscosity

Viscosity is one of the most important transportproperties of molten metals. It is related to the in-ternal friction within the liquid and provides some

information about the structure of the material.Viscosity which relates the shear stress τ to theshear rate γ

τ = ηγ , (26)

enters the definition of some of the most crucialhydrodynamic criteria such as the Reynolds num-ber, the Rayleigh number, the Hartmann number,and the Marangoni number. The viscosity of thematerial in the glass forming process changes by14 orders of magnitude in a temperature range ofapproximately 500 ◦K between melting and glasstemperature, Egry and Szekely (1991). An earlyreview of experimental methods for liquid metalviscometry was done by Beyer and Ring (1972).There are a number of theories, such as Arrhe-nius, Vogel-Fulcher, power law, Vogel-Fulcher-Tammann for the temperature dependence of theviscosity. However, the temperature dependenceof the viscosity of undercooled liquid metals is notwell understood. The relationship (27) betweenviscosity and surface tension which is based onthe analogy with the Wiedemann-Franz-Lorenzlaw and the Stokes-Einstein relation is widelyused. κB and m are the Boltzmann constant andthe atomic mass, respectively.

γη

=1516

√κBTm

(27)

Since viscosity measurements are not affected byconvection as diffusion measurements, the diffu-sivity D may be estimated from the viscosity datausing Stokes-Einstein theory,

D =κBT

6πRη. (28)

Viscosity can be measured by exciting and detect-ing surface oscillations of a levitated specimen us-ing EML techniques. The damping of the oscil-lations (Γl) is related to the viscosity (η) of theliquid specimen as given by Lamb (1945),

Γl =4π (l−1)(2l +1)ηR

3m. (29)

The frequency does not change with viscosity inthe weak damping limit according to Reid (1960).


Figure 3: Experimental set-up for surface tension measurements; the sample is levitated in a quartz tube.The temperature and surface oscillations of the droplet are alternatively observed from the top over a mirrorby a pyrometer or a video camera, Lohöfer et al. (2003)

Under 1 g conditions the electromagnetically in-duced flow in the droplet may be turbulent, andthe prevailing magnetic field can generate an ad-ditional damping, which can lead to systematicerrors, Herlach et al. (1993). Damping of the os-cillations can be determined from the half widthof the peaks in the Fourier transformation of thetime signals. The damping effects are closely re-lated to a decay factor (τ) defined as the time thatmust elapse for the amplitude of the vibration todecay to e−1 of its original value,

τ =R2ρ

η (l−1)(2l +1). (30)

Chandrasekhar (1961) derived the following tran-scendental equation for the viscous damping of afree oscillating droplet,[

lγR3ρ

(l +2) (l−1)+ f 2]−⎧⎨

⎩ 2(l2 −1

)R2

c2 − 2RcQ(l, R

c )+

2l (l −1)R2

c2

[1− (l +1)Q

(l, R

c

)R2c −Q

(l, R

c

)]⎫⎬⎭

· f 2 = 0, (31)

Q

(l,

Rc

)=

Jl+ 32

(Rc

)Jl+ 1

2

(Rc

) ,

c =(

μf ρ

) 12

,

f = iω +Γ,

where Ji represent the Bessel functions of order i.Simulations were performed for low and for highReynolds numbers. The intermediate regimeswere investigated by Suryanarayana and Bayaz-itoglu (1991b). They developed a method to de-termine both surface tension and viscosity froma single experiment in which the damping rateand frequency of oscillations are measured. Forhigh Reynolds numbers equation (25) flows outequation (31). Bratz and Egry (1995) investigatedthe influence of gravity and Lorentz force on thedamping of surface waves on electromagneticallylevitated liquid-metal droplets. The case of highReynolds numbers and a linear magnetic field wassimulated. They found a correction of the damp-ing due to the static deformation. The correctiondisappears for fixed l-values if the average overdifferent m-values is considered. However, the in-fluence of external forces on the frequencies doesnot disappear on average.

2.9 Determination of Gas Content in Metals

Mechanical properties of metals, such as tensilestrength, yield stress, elongation, age hardening,


and drawing properties are strongly dependenton gas content. The gas content in metals isconventionally determined by the inert gas car-rier fusion/thermoconductometric method whichis based on melting the specimen in a graphitecrucible in an inert gas medium, and detectingthe released gas (hydrogen, nitrogen, carbon, oxy-gen, etc.) quantitatively with a thermal conductiv-ity detector. The melted specimen reacts with thecrucible causing the latter to release contaminantsnot easy to eliminate from the crucible. The EMLtechnique sidesteps this difficulty as metals meltwithout crucibles.

A new technique for quantitative determinationof hydrogen in steels using electromagnetic lev-itation was developed by Nishifuji et al. (1996).Melting is induced in steel specimens levitated ina nitrogen gas flow. Hydrogen is completely ex-tracted within 1 min after melting. The releasedhydrogen gas is detected and measured quanti-tatively with a thermal conductivity detector. Acylindrical sample of 6 mm diameter, 6 mm lengthand 1.5 g mass was melted using an electromag-netic levitator with the levitation coil of 1.5 kWpower and 200 kHz frequency. The calibrationof the technique showed an excellent linearity inthe range of 0.4-6.7 μg/g, with a relative standarddeviation of 13% at the 1 μg/g level. A levitationmelting/thermoconductometric technique also hasbeen applied to measure the nitrogen gas level insteel, Nishifuji et al. (1998). About 5 min was re-quired to extract nitrogen from the steel specimen.The calibration curve for nitrogen was linear upto 156 μm/g, with a relative standard deviation of6.5% at the 20 μg/g level. The technique was ap-plied to determine nitrogen content in a certifiedreference material. The result (17.5 ± 1.1 μg/g)is in good agreement with the certified value (17.0μg/g) obtained by alkalimetric method after dis-tillation. They conclude that the levitation melt-ing/thermoconductometric technique could be ap-plied to determine other gaseous elements, suchas oxygen and carbon in all kinds of conductivemetal.

3 Conclusions

The advantages of the EML as a technique inthe study of the thermophysical properties of liq-uid metals are many, elimination of container ef-fect, attainment of high temperatures, possibilityto maintain undercooled state for an extended pe-riod of time and application of non-contact diag-nostic techniques among others. However, theEML as a technique to measure thermophysicalproperties of molten metals in terrestrial condi-tions is plagued with substantial problems suchas instabilities (global and local to the surface)of the suspended droplet, shape deformations ofthe specimen, requirement of high purity inert gasfor convective cooling, and convection inside thesample to name some. Researchers try to avoidthese difficulties by properly arranging the coilswith counter-windings and by choosing the exter-nal current distribution that creates the supportingfield. Using the EML technique, a certain suc-cess was achieved in measuring some of the prop-erties of molten metals such as surface tension,viscosity, thermal diffusivity, density and thermalexpansion.

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electromagnetic levitation part ii: thermophysical property measurements in terrestrial conditions

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