Eur. Phys. J. Special Topics 220, 101–110 (2013)© EDP Sciences, Springer-Verlag 2013DOI: 10.1140/epjst/e2013-01800-9

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Viscosity of liquid metal suspensions –experimental approaches and open issues

Dmitry Borina and Stefan Odenbach

Technische Universitat Dresden, Institute of Fluid Mechanics, 01062 Dresden, Germany

Received 17 December 2012 / Received in final form 5 February 2013Published online 26 March 2013

Abstract. Knowing the viscosity of metal melts with suspended par-ticles is necessary to interpret experimental results and to simulatefluid flow in such materials. At present, reliable viscosity data is onlyavailable for pure metals and alloys. In order to study the viscosity be-havior of metal melts with suspended solid particles in detail, sampleswith defined particle amounts are needed. Various methods of incor-porating particles into the metallic melts were evaluated, and viscositywas experimentally determined using an oscillating cup technique. Itwas shown that solid particles in suspension change the melts’ viscosi-ties dramatically, far beyond the effects expected from normal colloidalrheology.

1 Introduction

Liquid metal suspensions are essential to numerous applications, such as the pro-duction of metal foams and particle-reinforced metal composites [1–3]. Furthermore,metal in a mushy or semi-solid state includes both solid and liquid components andtherefore is identified as a suspension as well [4]. Thus, the thermophysical prop-erties, and particularly the viscosity, of metal melt suspensions is relevant to bothbasic research and industrial metallurgical tasks [5–7]. Realistic data are needed tointerpret experimental results and to simulate fluid flow in such materials. Previousfundamental assumptions about the rheology of the liquid metal systems suggeststhat they should be considered as Newtonian fluids, i.e. the viscosity of the liquidmetal is independent of shear deformation [8].However, data presented in the literature show a variation in the evaluated viscos-

ity of metals’ up to several hundred percent [6,8–12]. The Newtonian behavior is stillnot verified, and there is a lack of understanding of the liquid metals rheology [13].Moreover, theoretical studies using non-equilibrium molecular dynamics simulationspredict shear thinning behavior of the molten metal [14,15].Despite some discrepancies, investigations of the viscous behavior of pure melts are

quite comprehensive. However, information about melts containing solid particles israre. In the majority of cases, investigations are limited to the consideration of metalalloys in the semi-solid state [16–19] and significant deviation from the Newtonian hasbeen observed in their flow behavior. According to well-established colloidal rheology,

a e-mail: dmitry.borin@tu-dresden.de

102 The European Physical Journal Special Topics

a b c

Fig. 1. Schematic representation of the methods to incorporate the solid particles into metalmelts: a) mechanical stirring; b) electromagnetic stirring; c) laser injection.

the presence of solid particles in a liquid matrix leads to an increase in viscosity. Fur-thermore, even small changes in the particle concentration can dramatically influencethe rheology of the liquid [20]. Therefore, only suspensions with an accurately definedamount of solid should be used in rheological analysis.Incorporation of the powder material in the molten metal has been a technological

problem for many years, and numerous patents [21–26] and publications [1–3,27–29]are devoted to this topic. Nevertheless, there are still no satisfactory methods ofincorporating an exact amount of solid particles into metal melts [30].In this study, methods of particle incorporation into the liquid metal, including

mechanical and electromagnetic stirring, laser injection and metallurgical sinteringhave been considered and evaluated. Moreover, the viscosity of metal suspensionswas determined experimentally using an oscillating cup technique. Experimental ap-proaches, problems and results are discussed below.

2 Incorporation of the solid particles into metal melts

High melting points of liquid metals, as well as their chemical reactivity, restrictexperimental methods and make it difficult to incorporate a known concentration ofsolid particles. For the incorporation experiments, various combinations of metals andsolid particles were used. In addition to the problem of successful dispersion, the metalsuspension should be sufficiently stable. Therefore, particles with approximately thesame density as the matrix metal should be used, and so the choice of material forthe particles is limited.Pure metals including Sn, Ga and Pb, as well as low-melting alloys such as Wood’s

metal (eutectic alloy of Bi, Pb, Sn and Cd) and the eutectic of Ga, In and Sn wereused for the liquid phase. Both metallic and ceramic particles were used for the solidphase, inter alia oxides of the metals used for the liquid phase as well as Cu, ZrBand ZrO particles. We also considered different laboratory techniques for introducingsolid particles into metal melts.

2.1 Methods

2.1.1 Mechanical stirring

Mechanical dispersion of particles in the melt can be achieved by a conventional over-head stirrer (Fig. 1(a)). According to the DIN 28131, a propeller mixer and diagonalflat blade agitator in axial arrangement are applicable. However, two crucial diffi-culties come with this procedure. First, mechanical stirring cannot be performed in

Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 103

a b

Fig. 2. Microscopic pictures of Wood’s metal with PbO (a) and Cu (b) particles as a resultof the mechanical stirring with an inorganic protection layer.

an oxygen-containing atmosphere, due to the strong oxidizing tendency of the metalmelt. Second, the stirrer is in direct contact with liquid metal, which has a high re-activity. Therefore, reaction of the melt with the stirrer should be prevented by aproper choice of stirrer material or with a suitable coating of its surface. A solutionto prevent oxidation is to use a vacuum oven filled with an inert gas. This is anappropriate technique for industrial applications. Otherwise, the free surface of themelt can be covered with a protective layer, a technique well-known in welding. Inour experiments, different flux melting agents including rosin, an organic and an in-organic agents were used to protect the melt from oxidation during stirring. The fluxcovers the surface of the metal before it has been melted to prevent the appearanceof a thick oxidation film. Particles are incorporated after the metal is in the liquidstate. The microstructure of solidified samples was studied using optical and scan-ning electron microscopy after the particles were incorporated into the melt. Samplesmade of a material with a low melting point were frozen in liquid nitrogen and frac-tured. Samples with a melting point higher than 200 ◦C were sawed and polished byconventional metallographic procedure prior to microscopy. With one of these twomethods, samples with various combinations of particles and metals were analyzed.Results showed that an inorganic layer can successfully be used to avoid oxidationof the liquid metal surface during the incorporation of the solid particles throughintensive mechanical stirring (Fig. 2). However, the migration or alloying of copperparticles into the liquid matrix was observed, which makes further rheological studyof this composite meaningless.

2.1.2 Electromagnetic stirring

The electromagnetic stirring of liquid metals has the advantage of being a contactlessprocess [31,32]. The melt is placed in a non-magnetic vessel under the influence of amoving magnetic field, which induces an electric current in the metal. The current andthe applied field generate a Lorentz force, which initiates flow in the liquid metal. Inour experiments, a combination of a rotating and traveling magnetic field was used toproduce an intense contact between the liquid melt and the solid particles (Fig. 1(b)).The microstructure of the solidified samples showed that all attempts to disperse

particles in the melts were unsuccessful. Flow induced by the Lorentz force in themetal is not strong enough to incorporate the particles.

2.1.3 Laser Melt Injection

Another possible method to disperse particles in the melt utilizes thermal Marangoniconvection [34]. In this process, the surface of the solid metal is locally melted by alaser beam and solid particles are injected into the melt pool through a special nozzle(Fig. 1(c)). Results of the laser melt injection are presented in Figure 3. Due to the

104 The European Physical Journal Special Topics

a b

Fig. 3. SEM pictures of Ga with suspended ZrB2 (a) and ZrO2 (b) particles as a result ofthe laser melt injection.

Fig. 4. Particles of SnO2 incorporated into the Sn as a result of metallurgical sintering.

high energy used in this method, it is necessary to use particles above a certain size.Particles may be divided or even destroyed by the laser if their size is underestimated.Furthermore, particles were dispersed in the metal inhomogeneously, and it was notpossible to sufficiently control the amount of solid that was incorporated.

2.1.4 Metallurgical sintering

Pressure-induced deformation and sintering processes should force the particle toconnect with the metal [35]. In this procedure, metal and oxide powders in the desiredfractions are shear-mixed and pressed uniaxially. To avoid the appearance of the drossin the melt, an additional extrusion should be done after the sample has been pressed.With a technique described in [36,37] the sample is pressed through an angular dieso that the deformation is purely shear.This leads to the destruction of the oxide film on the surface of the powder’s

particles. As a result, the molten metal will be almost free of dross. Whatever of it isleft in the melt after extrusion should be removed by the first melting. Additional im-provement could be achieved if the powder processing is done in an inert atmosphere.The successful result of sintering the Sn02 particles in the Sn-matrix is shown inFigure 4. This method produces a concentrated composite which can be further di-luted. The pure metal is mixed with a previously prepared composite in the solidstate and melted afterwards. High intensity ultrasonic pulses are used in the nextstep to mix the suspension by cavitation and the acoustic streaming effect [33]. Met-allographic studies of the samples prepared in this way have shown its efficiency.However, the optically observed concentration of the solid particles is mostly incon-sistent with the intended value.

Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 105

2.2 Discussion

Productive incorporation of particles into liquid metal depends mainly on three mutu-ally correlated factors. The first of these is the tendency of the melt to wet the parti-cles’ surfaces. In particular, liquid metals do not wet ceramic particles well [26,30,39].In addition, the relatively high surface tension of the liquid metals opposes the incor-poration of the particles [8,40]. Furthermore, the size of the particles has an impact onthe dispersion. According to [26], the smaller the size of the particles, the greater thesignificance of surface effects compared to volume effects, hindering the incorporationprocess.Successful particle incorporation has been observed in the case of mechanical stir-

ring with an inorganic protection layer and through the laser melt injection process.An important disadvantage of the mechanical stirring procedure is the high toxic-ity and corrosive action of inorganic agents. Moreover, these agents contain halides,which can influence the rheology of the liquid metals [41–43]. In contrast to mechan-ical stirring, only an undefined amount of solid particles can be introduced throughlaser melt injection. Furthermore, particles injected by the laser are distributed in thevolume inhomogeneously. It may be presumed that electromagnetic stirring shouldresult in increased homogeneity of the particles’ distribution. However, whether theLorentz force can provide enough energy for the dispersion process is still an openquestion.The most appropriate method for controlled particle incorporation is metallurgical

sintering, combined with ultrasonic stirring for further dilution of the composite inthe liquid state. However, the results of this method are also not certain.

3 Viscosity of liquid metals with suspended particles

The experimental determination of thermophysical properties like the viscosity ofmelts has been a research focus for many years. Due to the low viscosity of liquidmetals, their chemical reactivity, and high melting point, only a few experimentalmethods have been proven suitable [6,8,44]. The most commonly used method is theoscillating cup technique [9,45–48]. In this method, the viscosity is determined fromthe decrement and time period of the motion of a vessel filled with the liquid which isput into oscillation around the vertical axis of the vessel. The shear rate applied to thefluid during the oscillating cup measurement is not constant with time and differs foreach oscillation. The maximal estimated magnitude of the shear rate in the oscillatingcrucible is usually ∼1 s−1 [49]. The applicability of the rotational technique for thisrange of slow deformations in liquid metals has not been proven [13,44]. However, tocalculate the zero-shear viscosity of suspensions, measurements of the minimal sheardeformation are necessary [20].We performed experiments with an oscillating cup viscometer, built according to

[8], and results for different metal suspensions are presented below.

3.1 Theoretical predictions

The presence of solid incorporations will change the velocity distribution in a flowingliquid and will lead to the increase in viscosity due to the extra energy dissipation.The zero-shear viscosity of diluted colloidal suspensions can be estimated with theclassical theoretical predictions given by Einstein [50,51] and by Batchelor [52].

106 The European Physical Journal Special Topics

Einstein has calculated the shear viscosity η of a suspension with a volume con-centration φ in a carrier liquid with a viscosity η0 as follows:

η(φ) = η0(1 + 2.5φ). (1)

Einstein’s formula is obtained from the viscous dissipation produced by the flowaround a single hard sphere and is valid for very dilute systems (φ <3 vol.%) withoutany interaction between neighboring particles. Moreover, such assumptions as incom-pressibility of the fluid, creeping flow, neutral density, no slip between the particleand the fluid, no particle migration as well as locality of the velocity perturbation dueto a particle were used in the analysis. The numerical factor 2.5 in the equation (1) isvalid for spheres and represents the intrinsic viscosity, which is the dilute limit of theviscosity increment per unit volume fraction, divided by the carrier liquid viscosity.In Batchelor’s prediction, which is one of the extensions of Einstein’s results,

the influence of thermodynamic forces on particles is considered. Batchelor solvednumerically the differential equation representing the effects of the bulk deformingmotion and the Brownian motion on the probability density of the separation vectorof particle pairs in a dilute suspension. As a result the effective viscosity is calculatedas follows:

η(φ) = η0(1 + 2.5φ+ 6.2φ2). (2)

This prediction takes into account the effect of two-body interactions and is valid forφ <10 vol.%.Further predictions [20,53] evaluate the viscosity of concentrated suspensions and

are not related to the current topic. A detailed theory of viscosity of liquid metalswith suspended particles is actually missing and the experiments discussed below mayserve as a first step towards an understanding of viscosity of liquid metal suspensions.

3.2 Experimental results

3.2.1 Water-based colloid

To prove that the oscillating cup technique can provide realistic viscosity measure-ments for classical colloids, a suspension of hollow glass spheres in an iron(III)-chlorideaqueous solution was characterized. The concentration of the solid phase φ was variedwithin the range of 0 and ∼3 vol.%. The dependence of the viscosity η on the con-centration of the solid is shown in Figure 5. During the oscillating cup experiment,the temperature was not controlled and varied between 22.7 and 23.2 ◦C. The resultswere compared to the theoretical predictions considered above. The decrease in thesuspension’s shear viscosity due to the temperature increase during measurement andthe tolerance determined by statistical error calculations do not invalidate classicalcolloid rheology as a framework for analyzing the results. Thus, the oscillating cupmethod is applicable for measurements of the colloids’ shear viscosities.

3.2.2 Metal suspensions

The viscosity of suspensions of ZrB2 and ZrO2 particles in liquid Ga was measured atthe temperature of 45 ◦C. The solid particles were introduced by the laser melt injec-tion method, and their exact amount is undefined. Nevertheless, the concentration ofparticles φ did not exceed 3 vol.%. Measured viscosity values η are given in Table 1.The presence of particles does not influence the viscosity of the liquid Ga.

The experimental values cannot be compared to theoretical predictions because theconcentration of the powder is unknown. Calculation of the viscosity according to

Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 107

6

8

10

12

14

-0.5 0 0.5 1 1.5 2 2.5 3

Vis

cosi

ty, m

Pa*

s

Concentration, vol. %

experimentmodel by Einstein

model by Batchelor

Fig. 5. Viscosity of the water-based suspension as a function of the concentration of thesolid fraction measured with the oscillating cup technique. Solid lines show the theoreticalprediction for the viscosity based on the models by Einstein and Batchelor.

Table 1. The viscosity of gallium-based suspensions at 45 ◦C.

Ga Ga+ZrB2 Ga+ZrO2φ - < 3 vol.% < 3 vol.%η 2.59± 0.15mPa · s 2.58± 0.16mPa · s 2.69± 0.10mPa · s

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5

Vis

cosi

ty, m

Pa*

s

Concentration of SnO2, vol. %

experimentprediction by Einstein

Fig. 6. Measured viscosity of liquid Sn as a function of the concentration of suspended SnO2particles at the temperature of 350 ◦C. The solid line shows the theoretical prediction basedon the model of Einstein (Eq. (1)).

Equation (1) for φ = 3vol.% gives a value of η = 2.78mPa·s, which is higher thanthe experimentally obtained results. Probably, the amount of effectively suspendedparticles is not sufficient for a pronounced effect.Using a material with a higher melting point, the solid was incorporated into the

metal through a sintering procedure performed in our own facilities. Solid particlesof Sn02 were introduced into the Sn-matrix with a concentration of 3 vol.%. Thesintered composite was diluted by the addition of pure Sn with further ultrasonicstirring for homogenization. The dependence of the viscosity on the concentration ofsolid particles for the tin-based suspension at T = 350 ◦C is shown in the Figure 6.

108 The European Physical Journal Special Topics

2

3

4

5

6

600 620 640 660 680 700

Vis

cosi

ty, m

Pa*

s

T,K

pure Pb (experiment)pure Pb (literature)

Pb with 0.9 vol. % of Pb203Pb with 6.6 vol. % of Pb203

prediction by Einsteinprediction by Batchelor

Fig. 7. Measured viscosity of liquid Pb with different concentrations of suspended Pb2O3particles [54,55]. Literature values have been taken from [8]. The dashed and solid lines showthe theoretical prediction for the viscosity based on the models by Einstein (for 0.9 vol.%,solid line) and Batchelor (for 6.6 vol.%, dashed line) respectively.

In contrast to the gallium-based suspension, the viscosity of the liquid tin significantlyincreases if solid particles are introduced. Einstein’s prediction (Eq. (1)) fails even forthe smallest concentration (∼0.7 vol.%), even considering the tolerance in the amountof particles. Thus, colloidal rheology cannot be applied to such metal suspensions.The influence of the solid particles on the viscosity of liquid lead were studied in

[54,55]. The solid particles have been incorporated into the metal through a sinteringprocedure. Figure 7 shows a dramatic increase in the viscosity of the melt with in-creasing concentration of solid particles. The relative change in the melt’s viscosity fora volume concentration of 0.9 vol.% of Pb2O3 particles is about 35%, and, for a con-centration of 6.6 vol.%, the increase is more than 100%. Moreover, the dependence ofviscosity on temperature is considerably weaker for the higher volume fraction. Theseresults cannot be explained by the classical theoretical predictions for the viscosityof colloidal suspensions (Eqs. (1) and (2)). The reason for the tremendous changes isunclear at this point. The influence of the dross, which may be present in the volumeof the melted lead, on the data cannot be disregarded.

4 Conclusion

With oscillating cup viscosimetry, it could be shown that solid particles suspended ina liquid metal can change the melt’s viscosity far beyond the effects expected fromnormal colloidal rheology, and the effect of the solid particles is dependent on thecomposition of the suspension. Particles of Sn and Pb oxides, when suspended inliquid Sn and Pb, dramatically change the viscosity of the melts, while the effect hasnot been confirmed for liquid gallium with incorporated solid particles. The variationin the behavior of these systems can be attributed to material-dependent properties.We have evaluated several methods of incorporating solids into the metallic melts,

but none of them were found to give consistent results. In most cases, the amount ofthe incorporated particles could not be controlled. Metallurgical sintering is the mostsuitable technique investigated. It can be combined with a stirring method for furtherhomogenization, as it was done for the dilution of the Sn-based suspension. However,it should be examined whether the sintering process itself can has an influence on thepurity and viscosity of the metal suspension through the presence of the dross.

Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 109

Furthermore, experiments on diverse metal-particle systems should be done toclarify the basis of the influence of solid particles on the rheology of metallic meltsand to provide realistic data for metallurgical applications and theoretical research.Moreover, possible influence of the shear deformation, known from the rheology ofcolloids, such as shear thinning or thickening, should be addressed in further studies.Therefore, the implementation of the rotational rheometry is the next challenge inthe rheology of the metal suspensions.

Financial support by the Deutsche Forschungsgemeinschaft through the CollaboratedResearch Center Electromagnetic flow control in metallurgy, crystal growth and electro-chemistry (SFB-609, project A11) is gratefully acknowledged.We acknowledge the support of project A2 during the measurements with electro-

magnetic stirring, as well as contributions to this study made by Christoph Jakobitz, LisaSprenger, Martin Queck and Thomas Richter.

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