Int J Thermophys (2014) 35:1725–1748DOI 10.1007/s10765-014-1748-4

The Discharge Crucible Method for MakingMeasurements of the Physical Properties of Melts: AnOverview

T. Gancarz · W. Gąsior · H. Henein

Received: 31 October 2012 / Accepted: 15 September 2014 / Published online: 16 October 2014© Springer Science+Business Media New York 2014

Abstract The physicochemical properties, viscosity, density, and surface tension,were measured using the discharge crucible method (DC) on a wide range of pure meltsand alloys and for AZ91D in two gas atmospheres. The DC method was confirmedon pure metals, Sb, Sn, Zn, and compared with corresponding literature data. Resultsare also reported for Sn–Sb alloys containing (10, 20, 25, 50, and 75) at% of Sbat 550 K to 850 K, for Sn–Ag alloys containing (3.8, 32, 55, and 68) at% Ag, forcommercially pure Al, and for an AZ91D Mg alloy under an argon atmosphere. Theproperties for AZ91D were also measured under an atmosphere of air containing 2 %SF6. The results are compared with published data on all alloys. The experimentallymeasured surface-tension values are compared with the Butler model. Several modelsare compared and discussed for the viscosity measurements.

Keywords Al · AZ91D · Density · Sb · SbSn alloys · Sn · SnAg alloys · Surfacetension · Viscosity · Zn

Nomenclature

Cm Cumulative mass, kgD(T ) Diameter of lower container in CM method as a function of temperature

T , m

T. Gancarz · W. GąsiorInstitute of Metallurgy and Materials Science, Polish Academy of Science,Warsaw, Poland

H. Henein (B)Department of Chemical and Materials Engineering, University of Alberta,Edmonton, AB, Canadae-mail: hani.henein@ualberta.ca

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1726 Int J Thermophys (2014) 35:1725–1748

E Activation energy, kJ·mol−1M Molecular weight, kg·mol−1Qexp Experimental volumetric flow rate, m3·s−1R Universal gas constant, J·mol−1·K−1T Temperature, KV Volume of draining fluid, volume liquid, m3

a Polynomial constant describing slope of the discharge coefficient curve,dimensionless

b Polynomial constant describing the y-intercept of the discharge coeffi-cient curve, dimensionless

g Gravity acceleration, m·s−2h Planck’s constant, J·shexp Liquid head, mro Orifice radius, mt Time, su Velocity, m·s−1η Viscosity, mPa·sηo Pre-exponential, mPa·sηAZ91D Viscosity of AZ91D at temperature T , mPa·sρ Density, kg·m−3σ Surface tension, mN·m−1Bo Bond number

(ρgr0hexp

σ

), dimensionless

Cd Discharge coefficient, dimensionlessFr Froude number (Q2exp/(2πgr

3o)), dimensionless

Re Reynolds number(

2ρQexpπr0η

), dimensionless

1 Introduction

There is a wide range of methods that have been developed for measuring the physicalproperties of high-temperature liquids, namely, gas–liquid surface tension, density,and viscosity. These methods include the maximum bubble pressure (MBP) [1], thesessile drop (SD) [2], the electromagnetically levitated droplet (ELD) [3]; the modi-fied capillary (CM) [4], torsional oscillation (OM) [5], and pycnometric methods (PT)[6], the capillary rise (CR) [7], drop-weight (DW) [8], and dilatometric methods [9].Each of these methods presents advantages as well as disadvantages. For example,each of these methods when used terrestrially only provide two of the three propertiesoutlined above. The ELD is the only approach that is containerless. All of the methodspresent significant challenges eliminating oxygen from the atmosphere. Neverthe-less, by measuring the oxygen content, reported values can be compared. In addition,while this is a drawback for evaluating the properties of unoxidized melts, the valuesdetermined can be used and related to industrial conditions of processing these melts.These processing operations that are affected by these properties range widely. Metalsextraction [10], strip casting [11], atomization [12], spray forming [13], and soldering[14] are a few examples.

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Int J Thermophys (2014) 35:1725–1748 1727

Recently, Roach and Henein (DC) [15,16] developed a new method enabling thesimultaneous measurement of surface tension, viscosity, and density in one run forfluids. The method is based on a fluid draining from a crucible as a stream flowingunder gravity. The model developed can be used to calculate these physical propertiesfrom only one experiment at a desired temperature. The aim of this work is to presentan overview of the DC method and of the values for density, surface tension, andviscosity obtained using the DC method and comparing them with those obtainedusing other methods for pure metals Al, Sb, Sn, and Zn, and Sn–Ag, Al–Mg, andSn–Sb alloys.

2 Principle of DC Technique-Mathematical Model

A detailed description of the model used for a draining crucible has been presentedelsewhere [15–17], therefore, only a brief overview of the model will be presentedhere. For a vessel having an orifice in the bottom, if it is sufficiently large, a free jetwill form when the vessel is filled with a liquid. To predict the flow rate, the Bernoulliformulation is used. Refer to Fig. 1 for an illustration of unsteady flow through anorifice. The two reference locations are labeled as points 1 and 2 on Fig. 1. TheBernoulli formulation is applied on a small cylindrical element, illustrated in Fig. 2.Depicted are the forces acting relative to the direction of a streamline. Daugherty etal. [18] considered the case of unsteady flow (velocity as a function of both positionand time, i.e., a draining vessel) shown in Fig. 1 giving the relationship:

− dPdAc + ρgdAcdz = ρdAcdz(

u∂u

∂z+ ∂u

∂t

)(1)

Dividing Eq. 1 by −ρdAc, rearranging the terms, and integrating between the limitsof points 1 and 2, yields

Fig. 1 Schematic of drainingvessel system depicting flow rateof a fluid through an orificeplaced at the bottom

1

ro

h

z2

z1

Q

2

z

Patm

Patm rv

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1728 Int J Thermophys (2014) 35:1725–1748

Fig. 2 Control element onstreamline neglecting frictionallosses PdAc

(P+dP)dAc

dz gdAcdz z

Streamline

(P2 − P1)ρ

− g (z2 − z1) +(u22 − u21

)

2= − (z2 − z1)

(∂u

∂t

)(2)

Note that the formulation has thus far neglected viscosity effects. Equation 2 is there-fore considered to be valid for a draining vessel under inviscid flow with negligiblesurface forces. The term (z2 − z1) is the head of fluid just above the exit of the orifice,h, as indicated in Fig. 1. By continuity, the velocity at point 1 can be expressed withrespect to the velocity at point 2:

u1 = r2o

r2vu2 (3)

Since rv � ro, u1 ≈ 0. Rearranging Eq. 2 in terms of the outlet velocity, u2, yields

u2 =√

2g

(h − (P2 − P1)

ρg− h

g

(∂u

∂t

))(4)

The term hg(

∂u∂t

)represents the acceleration of the head within the vessel. This quantity

is neglected in the analysis since it is a factor equal to 3.5×10−3 less than the magnitudeof the first term (described by h) [19].

The pressure at reference points 1 and 2 is atmospheric in Fig. 1. However, consid-ering the principles of interfacial phenomena, there is a pressure induced at the outletof the orifice if the liquid surface tension is considered. The pressure induced from thesurface tension occurs at curved interfaces that separate liquid from the atmosphere.Such is the case of a cylindrical stream exiting an orifice. The Laplace equation relatesthe pressure difference across a curved interface to the radii of curvature:

�P = σ(

1

Ra+ 1

Rb

)(5)

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Int J Thermophys (2014) 35:1725–1748 1729

�P is the pressure difference across the liquid/gas interface, and Ra and Rb representthe radii of curvature that describe the shape of the surface. If there is a planar surface,R → ∞, and there is no pressure induced as Eq. 5 indicates. At reference point 1 (topof the liquid in the vessel), it is assumed that the pressure induced is negligible sincethe interface is considered to be planar (Ra and Rb approach infinity).

Directly below reference point 2 (orifice tip), the stream exiting the orifice is essen-tially cylindrical with the curvatures illustrated in Fig. 7. At a location just below theorifice (where the stream is first in contact with the atmosphere), Ra is taken to be theradius of the orifice, ro. The second radius of curvature, Rb is assumed to be infinitysince the stream is assumed to be a perfect cylinder at this location. From the Laplaceequation, the pressure induced due to the effects of surface tension directly below theorifice is

�P = σ(

1

ro

)(6)

Considering Eq. 4, and assuming quasi-steady state conditions, and the pressure atreference point 2 is included in the formulation that corresponds to a magnitude ofσ/ro.

u2 =√

2g

(h − (P2 − P1)

ρg

)=

√2g

(h − (σ/ro + Patm − Patm)

ρg

)

=√

2g

(h − σ

ρgro

)(7)

In terms of flow rate, Qexp under viscous conditions, Eq. 7 can be expressed as

Qexp = πr2o Cd√

2g

(h − σ

ρgro

)(8)

or in dimensionless form,

Fr + Bo−1 = 1 (9)

where Fr is the Froude number and is given by

Fr =(

Qexpπr2o Cd

)2

2gh(10)

and Bo is the Bond number given by

Bo = ρgrhσ

(11)

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1730 Int J Thermophys (2014) 35:1725–1748

Note that Cd is a function of Re and is given by

Reexp = I nertial ForcesV iscous Forces

= ρuexp2roη

= 2ρQexpπroη

(12)

where ρ and η are the density and viscosity of the fluid, respectively. Plotting Cd versusthe Reynolds number, Eq. 12, provides a measure of the frictional characteristics ofthe orifice. A polynomial fit can be derived from the Cd versus Re plot and substitutedinto Eq. 8.

The DC method is used in the following manner. A crucible with a well machinedorifice is calibrated with a fluid of known properties and a function of Cd versus Reis experimentally determined. An experiment is then carried out using a fluid withunknown properties. The flow rate as a function of the draining crucible height aremeasured during the experiment. The only unknowns in Eq. 1 are the surface tension,density, and viscosity of the fluid with unknown properties. A matrix solution is thenapplied to seek the values of these properties while minimizing the error. The detailsof the solution method is described elsewhere [15].

3 Materials, Apparatus, and Experimental Procedures

99.95 % purity aluminum used in these experiments was melted using induction heat-ing, and the liquid metal temperature was monitored using a thermocouple immersedinto the melt. It is essential to measure the temperature of the melt near the orifice[15,16]. 99.99 % purity of Ag, Sb, Sn, and Zn was used in the experiments for puremetals and for Sn–Ag and Sn–Sb alloys. The equipment for the Ag, Sb, Sn, and Zntests was placed inside a glove box to ensure inert conditions [20]. The experimentalprocedure used for the Al measurement was adopted for Sn–Ag alloys.

3.1 Description of Experimental Technique

The experiments were carried out in an inert chamber depicted schematically in Fig. 3.Argon is pumped into the unit until a small overpressure is obtained; thus, air doesnot leak back into the unit. A continuous purge of argon is run through the unit untila nominal oxygen level of 20 ppm is registered from the oxygen monitor. For thealuminum and magnesium alloy experiments, the charge is heated using a 20 kWinduction furnace. When the temperature in the crucible (high density graphite for Aland steel for Mg alloy) reaches the desired value, the melt is maintained at a constanttemperature for at least 15 min to 20 min. For the Ag, Sb, Sn, Zn, and Sn–Ag andSn–Sb alloys, an electrical resistance furnace and graphite crucible are used followinga similar procedure.

Once the temperature and oxygen in the atmosphere are equilibrated, the melt isallowed to flow through an orifice placed in the bottom of the crucible. The melt iscaptured by a load cell which registers the cumulative mass of the melt draining fromthe crucible. For the case of Al and Mg alloy experiments, the relationship between thecrucible volume and melt height was calibrated using a known fluid prior to the melt

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Int J Thermophys (2014) 35:1725–1748 1731

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0.04 0.06 0.08 0.1

Head, m

Aluminum 1073 KAverage

, N/m

1/Bo = 0.29 1/Bo = 0.13

Fig. 3 Surface tension of aluminum determined at 1073 K, with expected error in surface tension due toerror in head measurement represented by solid lines

880 920 960 1000 1040 1080

Temperature, K

6400

6440

6480

6520

6560

Den

sity

,kg.

m-3

SbDC, [20]PT, [22]

[23]MBP, [24]

Fig. 4 Density of antimony as a function of temperature compared with different measurement methods

experiments. Thus, the cumulative mass captured by the load cell could be convertedto the melt height in the crucible as part of the mathematical model to determine thephysical properties of the melt. For Ag, Sb, Sn, Zn, and Sn–Ag and Sn–Sb alloys, adilatometric technique is used to yield the metal volume in the crucible as a functionof time [20].

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1732 Int J Thermophys (2014) 35:1725–1748

900 1000 1100 1200

Temperature, K

330

340

350

360

370

380

390

Surf

ace

tens

ion,

mN

. m- 1

SbDC, [20]MBP, [25]MBP, [26]

MBP, [27]MBP, [28]

Fig. 5 Surface tension of antimony as a function of temperature compared with different measurementmethods

A data acquisition system records signals from the oxygen analyzer, thermocouple,and load cell. The mass flux is obtained from the cumulative mass, Cm, versus timecurve recorded by the load cell (see Fig. 5). A second-order polynomial curve is fit tothe data. A higher-order polynomial curve did not improve accuracy. For example, apolynomial fit on a set of data collected for aluminum through a 5 mm diameter orificeat 1073 K is expressed as

Cm = −8.356 × 10−4(t)2 + 6.491 × 10−2(t) + 2.212 × 10−2 (kg) (13)

Differentiating Eq. 13 with respect to time, and dividing by the cross-sectional areaof the orifice provides the flux,

Vexp = 1πr2o

dCmdt

= −8.52 × 101(t) + 6.491 × 10−2 (14)

The head, hexp, is obtained using load cell information as well. Knowing the geometryof the crucible, the head can be determined by monitoring the quantity of material thatpoured through the orifice as a function of time. The density of the liquid is requiredfor the calculation and was provided from the literature for molten aluminum [21].This value of density was used to initiate the iteration process and was subsequentlyupdated during the course of the solution as will be illustrated in the results section.Figure 6 illustrates the head as a function of time using this approach.

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Int J Thermophys (2014) 35:1725–1748 1733

400 800 1200 1600

Temperature, K

0.4

0.8

1.2

1.6

2

2.4

Vis

cosi

ty,m

Pa.

sSnDC, [17]DC, [20]CM, [25]OM, [40]

Fig. 6 Viscosity of antimony as a function of temperature compared with different measurement methods

3.2 Propagation of Error

The propagation of error analysis, applied to the surface tension, is written as follows:

δσ =√(

∂σ

∂hexp

)2 (δhexp

)2 +(

∂σ

∂Vexp

)2 (δQexp

)2 +(

∂σ

∂Cd

)2(δCd)2 (15)

The differentials in Eq. 15 are determined from Eq. 8 and are as follows:

∂σ

∂hexp= ρgro (16)

∂σ

∂ Qexp= − ρro

Cdπr2o

(Qexp

Cdπr2o

)(17)

∂σ

∂Cd= ρro

Cd

(Qexp

Cdπr2o

)2(18)

The equation for the error of density:

∂ρ =√(

∂ρ

∂h

)2(∂h)2 +

(∂ρ

∂ Qexp

)2 (∂ Qexp

)2 +(

∂ρ

∂Cd

)2(∂Cd)2 (19)

∂ρ

∂h= −ρ

2gr0σ

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1734 Int J Thermophys (2014) 35:1725–1748

∂ρ

∂ Qexp= ρ

2 QexpCdπ2r30σ

∂ρ

∂Cd= − ρ

2 Q2expC3dπ

2r30σ(20)

The equation for the error of viscosity:

∂η =√(

∂η

∂h

)2(∂h)2 +

(∂η

∂ Qexp

)2 (∂ Qexp

)2 +(

∂η

∂Cd

)2(∂Cd)2 +

(∂η

∂a

)2(∂a)2 +

(∂η

∂b

)2(∂b)2

(21)

∂η

∂h= 2aρr0gQ

2exp

π2r20 [Cd − b]2[2gh

(1 − σ

ρgr0h

)]3/2 (22)

∂η

∂ Qexp= 2aρd

π r0(Cd − b)2∂η

∂a= 2ρr0 Qexp

π r2o [Cd − b]∂η

∂b= πr0η

2

2aρQexp(23)

where the errors determined for the experiment conducted on the various parameterswere as follows:

∂h =√∑

(hpoly − hexp)2n − 3 = 8.881 × 10

−5 m

∂ Qexp = 1/ρ ∂Cm∂t

= (in m3 ·s−1) where

∂Cm =√∑

(Cm,poly − Cm,exp)2n − 3 = 1.145 × 10

−3 kg

∂Cd =√∑

(Cd,poly − Cd,exp)2n − 4 = 0.018 (24)

given the error in the measured and varying values of hexp, Qexp, and Cd, The expectederror in s as a function of the change in these variables can be determined fromexperiments. This will be illustrated later for the effect of hexp on s for aluminum. Aswill be seen, the error in s due to hexp is very small. A similar analysis can be donefor the other variables Qexp and Cd as well as for r and m as a function of these samevariables.

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Int J Thermophys (2014) 35:1725–1748 1735

4 Results

4.1 Pure Metals

Experimental data obtained from different measurement methods for the pure metalsare presented in the figures: density (Figs. 4, 7, 10, 13, 16), surface tension (Figs. 5,8, 11, 14, 17, 19), and viscosity (Figs. 6, 9, 12, 15, 18, 20). The values of the densityfor Sb presented in Fig. 4 are different by about ∼10 kg·m−3 for DC [20], PT [22,23];only data from MBP [24] are higher by ∼40 kg·m−3 but this it still

1736 Int J Thermophys (2014) 35:1725–1748

500 600 700 800 900 1000

Temperature, K

440

480

520

560

600

Surf

ace

tens

ion,

mN

. m-1

DC, [17]DC, [20]MBP, [25]CR, [36]

CR, [37]CR, [38]BMP, [39]SD, [39]

Sn

Fig. 8 Surface tension of tin as a function of temperature compared with different measurement methods

400 800 1200 1600

Temperature, K

0.4

0.8

1.2

1.6

2

2.4

Vis

cosi

ty, m

Pa.

s

SnDC, [17]DC, [20]CM, [25]OM, [40]

Fig. 9 Viscosity of tin as a function of temperature compared with different measurement methods

methods. The results of the viscosity for pure Sn are shown in Fig. 9. We observe verygood agreement among the data obtained using DC [17,20], CM [25], and OM [40]methods.

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Int J Thermophys (2014) 35:1725–1748 1737

600 700 800 900 1000 1100

Temperature, K

6200

6300

6400

6500

6600

Den

sity

, kg.

m-3

ZnDM, [1]DC, [20]PM, [36]PM, [41]

Fig. 10 Density of zinc as a function of temperature compared with different measurement methods

700 800 900 1000 1100

Temperature, K

720

740

760

780

800

820

Surf

ace

tens

ion,

mN

. m-1

ZnMBP, [1]DC, [20]MBP, [36]MBP, [42]MBP, [43]MBP, [44]DW, [45]

Fig. 11 Surface tension of zinc as a function of temperature compared with different measurement methods

Zinc is the metal, which has a very high chemical affinity to oxidize and a high vaporpressure. Physicochemical properties of zinc as a function of temperature comparedamong different measurement methods are presented in Figs. 10, 11, and 12. In Fig. 10which shows the density of Zn, we can see very good agreement among the dataobtained from all methods of DC [20], DM [1], and PM [36,41]. The surface tensions

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1738 Int J Thermophys (2014) 35:1725–1748

700 800 900 1000 1100

Temperature, K

2

3

4

Vis

cosi

ty, m

Pa.

sZnDC, [20]CM, [29]OM, [46]OM, [47]OM, [48]

Fig. 12 Viscosity of zinc as a function of temperature compared with different measurement methods

400 600 800 1000 1200 1400

Temperature, K

6400

6800

7200

7600

8000

DC, SnAg3.5, [20]DC, SnAg32.5, [20]SD, SnAg30, [34]SD, SnAg3.5, [51]

SD, SnAg3.8, [52]SD, SnAg30, [52]DM, SnAg3.8, [53]DM, SnAg25, [53]

Sn-Ag

Den

sity

,kg.

m-3

Fig. 13 Density of Sn–Ag alloys as a function of temperature compared with different measurementmethods

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Int J Thermophys (2014) 35:1725–1748 1739

400 600 800 1000 1200 1400

Temperature, K

480

520

560

600

DC, SnAg3.5, [20]DC, SnAg32.5, [20]SD, SnAg30, [34]SD, SnAg3.5, [51]SD, SnAg3.8, [52]

SD, SnAg30, [52]MBP, SnAg3.8, [53]MBP, SnAg25, [53]SD, SnAg10, [54]SD, SnAg30, [54]

Sn-Ag

Surf

ace

tens

ion,

mN

. m-1

Fig. 14 Surface tension of Sn–Ag alloys as a function of temperature compared with different measurementmethods

400 800 1200 1600

Temperature, K

0.4

0.8

1.2

1.6

2

2.4

DC, Sn, [17]DC, SnAg3.5, [17]DC, SnAg32.5, [17]CM, Sn, [17]CM, SnAg30, [17]CM, Sn, [55]OM, Sn, [56]OM, SnAg24.4, [56]

Sn-Ag

Vis

cosi

ty, m

Pa.

s

Fig. 15 Viscosity of Sn–Ag alloys as a function of temperature compared with different measurementmethods

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1740 Int J Thermophys (2014) 35:1725–1748

600 800 1000

Temperature, K

6400

6500

6600

6700

6800

6900

DC, SnSb10, [20]DC, SnSb20, [20]DC, SnSb25, [20]DC, SnSb50, [20]DC, SnSb75, [20]

DM, SnSb10, [55]DM, SnSb20, [55]DM, SnSb40, [55]DM, SnSb60, [55]DM, SnSb80, [55]

Sn-Sb

Den

sity

,kg.

m-3

Fig. 16 Density of Sn–Sb alloys as a function of temperature compared with different measurementmethods

of Zn, presented in Fig. 11, show good agreement among the data from DC [20] andMBP [42,43]. Also, good agreement is seen with data showing the same slope but ahigher MBP [44] or a lower DW [45] value. In Fig. 12 comparisons of the viscosityof Zn are shown for the methods, DC [20], CM [29], and OM [46–48]. Based on thecomparisons, the DC method yields accurate and reliable results for the properties ofdensity, surface tension, and viscosity when compared with measurements made usingother techniques.

4.1.1 Aluminum

In a review of the surface tension of metallic elements, Mills and Su compared thereported values of a number of researchers for aluminum including those obtained bythe DC method [49]. He showed that the values reported by Roach and Henein werewithin 2 % with other published values for the case of oxygen-saturated aluminum.The reported values of density were estimated to have

Int J Thermophys (2014) 35:1725–1748 1741

500 600 700 800 900 1000 1100

Temperature, K

360

400

440

480

520

DC, SnSb10, [20]DC, SnSb20, [20]DC, SnSb25, [20]DC, SnSb50, [20]DC, SnSb75, [20]

BMP, SnSb10, [55]BMP, SnSb20, [55]BMP, SnSb40, [55]BMP, SnSb60, [55]BMP, SnSb80, [55]

SD, SnSb14.5, [58]SD, SnSb20, [57]SD, SnSb30, [58]SD, SnSb47.3, [58]SD, SnSb80, [58]

Sn-Sb

Surf

ace

tens

ion,

mN

. m-1

Fig. 17 Surface tension of Sn–Sb alloys as a function of temperature compared with different measurementmethods

4.2 Sn–Ag Alloys

In Figs. 13, 14, and 15 comparisons of the density, surface tension, and viscos-ity of Sn–Ag alloys, respectively, are presented. Results for the density of Sn–Agusing the DC [20], SD [34,51,52], and DM [53] methods are presented in Fig. 13.For the SnAg3.5 alloy, we can see excellent agreement between DC [20] and SD[51] experimental tests over the complete temperature range. We can also see thatthe density of the eutectic alloy SnAg3.8 has a lower density than the hypoeutec-tic SnAg3.5 alloy. For the alloys richer in Ag, the obtained data show greater dif-ferences but still no larger than 1 % from the SD method. The values of densityincrease with Ag content in the alloy; the data from the DC method for SnAg32.5are a good fit in the trends of the density dependence of Sn–Ag alloys. The surfacetensions of Sn–Ag alloys presented in Fig. 14 show comparisons among the differ-ent methods, DC [20], MBP [53], and SD [34,51,52,54]. The differences betweenthe values from DC [20] and SD [51] for SnAg3.5 are relatively large, ∼9 %. Theobtained data of the surface tension from the SD method show the large differ-ences in the slopes over a large range of data. It is therefore important during theliquid-drop experiments to take into account problems arising from the oxides orconcentration fluctuations which are very important and may distort the measure-ments for the viscosity of Sn–Ag alloys presented in Fig. 15; we can see that the

123

1742 Int J Thermophys (2014) 35:1725–1748

400 600 800 1000 1200

Temperature, K

0.8

1.2

1.6

2

2.4

Vis

cosi

ty, m

Pa.

s

DC, Sn, [20]

DC, Sb, [20]

OM, Sn, [23]

OM, Sn, [23]

OM, SnSb9.9, [23]

OM, SnSb19.8, [23]OM, SnSb39.7, [23]

OM, SnSb49.7, [23]OM, SnSb79.8, [23]

Sn-Sb

DC, SnSb10, [20]

DC, SnSb20, [20]

DC, SnSb25, [20]

DC, SnSb50, [20]

DC, SnSb75, [20]

Fig. 18 Viscosity of Sn–Sb alloys as a function of temperature compared with different measurementmethods

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

850 950 1050 1150 1250

, N ·

m-1

Temperature, K

Magnesium

This study

Alloy liquidus temperature

Fig. 19 Surface tension of AZ91D under argon from this work compared to literature values (Roach andHenein [5])

viscosity increases with Ag content in the alloy. Comparing the DC [20], OM [56],and CM [17,52] techniques, the viscosities of Sn–Ag alloys show very good agree-ment.

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4.3 Sn–Sb Alloys

In Figs. 16, 17, and 18 are presented the density, surface tension, and viscosity ofSn–Sb alloys, respectively. The data for the density of Sn–Sb alloys from the DC [20]and DM [55] methods show excellent agreement. In Fig. 17 we show comparisons ofthe surface-tension data for Sn–Sb alloys obtained by three methods, DC [20], BMP[55], and SD [57,58]. Here we can see that the slope of the surface tension of Sn–Sb and from SD [58] is lower than that from DC and BMP. The obtained values ofthe surface tension for SnSb20 from DC and SD [57] show excellent agreement, butvalues for SnSb30 from SD [58] appear to be too high or maybe there is an error inthe preparation of the alloy. There is this same trend in all the compared methods,and good agreement is generally observed. In Fig. 18 comparisons of the viscosity ofSn–Sb alloys obtained by the two measurement methods, for the DC [20] and OM[23], are presented. The values of the viscosity from OM for the lower concentrationsof Sb in Sn–Sb alloys and lower temperatures to 800 K are lower than the values ofviscosity from the DC method. For SnSb50 and higher concentrations, we observegood agreement between the obtained values of the viscosity of Sn–Sb alloys fromOM and DC.

4.4 Effect of Gas Atmosphere (AZ91D Results) [59]

The effect of air with SF6 compared to argon for the surface tension is shown in Fig. 13,while the change in viscosity as a function of temperature is presented in Fig. 19. Thedata for the density of AZ91D agree with an ideal solution model for calculating thedensity of the alloy. Figure 20 shows that the presence of air with 2 % SF6 reduces thesurface tension of the alloy. While the viscosity is significantly lower than is typicallyreported in the literature for molten metals, the variation of viscosity as a function oftemperature follows an Arrhenius equation of the form;

μAZ91D = μo exp ERT

(25)

Fig. 20 Viscosity of AZ91D magensium alloy under argon as a function of temperature. Solid line is theArrhenius curve fit through all points and dashed line is the Arrhenius fit without the data point at thehighest temperature (Roach and Henein [59])

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Table 1 Viscosity of AZ91D as a function of temperature

Alloy Best fit based on data from μo(Pa·s) E(kJ·mol−1) R2

AZ91D (Ar) 921 K to 1169 K 5.325 × 10−6 35.91 0.518921 K to 1067 K 9.058 × 10−8 68.03 0.7119921 K to 967 K 1.181 × 10−13 173.5 0.9889

Mg Ref. [59] 2.45 × 10−5 30.5

Fig. 21 Surface tension of tinand Sn–Ag alloys as a functionof temperature compared withthe Butler model: solidline—Butler model,circle—MBP [25], square—DC[17]

where μAZ91D is the viscosity of the alloy at temperature T, μo is the pre-exponential,E is the activation energy, and R is the universal gas constant. Fitted values are shownin Table 1 and compared with the literature for Mg [60].

4.5 Butler Model

The Butler model [61] was used to model the surface tension as described in an earlierwork by Gancarz et al. [17]. The thermodynamic optimized parameters were takenfor Sn–Ag from Ohnuma et al. [62] and for Sn–Sb from Jonsson and Agren [63].The Butler model then yields the temperature dependencies of the surface tension forexperimental compositions that are plotted in Fig. 21 for Sn–Ag and in Fig. 22 forSn–Sb in comparison with experimental data obtained by other measurement methods.

Figure 21 illustrates a comparison between the results of calculations using theButler model [61]—line and the experimental data obtained by different methods:MBP—circles [25] and DC—squares [17] for respective compositions. The surfacetension calculated from the model decreases with increasing temperature and increaseswith increasing Sn concentration, which is in accordance with experimental results.Figure 22 presents the results of calculations using the Butler model [61] with data

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0 0.2 0.4 0.6 0.8 1

XSb

360

400

440

480

520

SD, [58]MBP, [25]SD, [57]DC, [20]Butler model

Sn Sb

923 KSn-Sb

Surf

ace

tens

ion,

mN

. m-1

Fig. 22 Surface-tension isotherms of liquid Sn–Sb alloys obtained from the Butler’s model (continuouslines) against the background of experimental values, with literate data (symbols) [5,6,30]. Dashed lineshows the linear changes of surface tension between pure Sb and Sn

received from MBP, SD, and DC methods. Obtained values of the surface tension fromthese three methods are almost identical and have slightly lower values of calculateddata from the Butler model.This may be due to the surface tension of pure Sb obtainedfrom the DC method that has been used in the modeling is slightly higher than thatobtained with MBP. Good agreement is observed between calculated data from theButler model and all experimental data.

4.6 Viscosity Models

For the modeling of viscosity the following models were used: Moelwyn-Hughes[64], Sichen et al. [65], Seetharaman and Sichen [66], Kozlov et al. [67], Kaptay [68],and Morita et al. [69]. A detailed description of these models has been provided inearlier work [17]. A comparison of the experimental data of the viscosity of Sn–Agalloys received from DC and CM methods with each of these models is presented inFig. 23. The lack of a universal model and the difficulty in modeling the viscosity areapparent. The best agreement between measured and calculated viscosities is given bythe Seetharaman and Sichen model [66]. The data obtained from the Moelwyn-Hughes[64] model yield much higher values. One can notice that in the Moelwyn-Hughes[64] model, only the enthalpy of mixing is used in the calculation of the viscosity. Insuch a case for alloys with a negative value of the enthalpy of mixing, the calculatedviscosity will always be higher than that for the ideal solution, and for alloys witha positive value of the enthalpy of mixing, the viscosity will be lower than that for

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0 0.2 0.4 0.6 0.8 1XSn

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

CM, [17]DC, [17][64][65][66][67][68]

T=823 KSn-Ag

Vis

cosi

ty, m

Pa.

s

Fig. 23 Comparison of viscosity of Sn–Ag alloys with models: 1—Sheetharaman and Sichen [66], 2—Kozlov et al. [67], 3—Sichen et al. [65], 4—Moelwyn-Hughes [64], 5—Kaptay [68], this work: diamonds—MC method, inverted triangle—RH method

0 0.2 0.4 0.6 0.8 1XSn

0.4

0.8

1.2

1.6923 KSn-Sb

DC, [20]OM, [23][64][65][66][67][68][69]V

isco

sity

, mP

a.s

Fig. 24 Comparison of viscosity of Sb–Sn alloys with models: Moelwyn-Hughes [64], Sichen et al. [65],Seetharaman and Sichen [66], Kozlov et al. [67], Kaptay [68], Morita et al. [69] and this work and literaturedata of Sato and Munakata [23] for 923 K

ideal solutions. The obtained values of the viscosity from the DC and OM methodsare compared with the modeling data in Fig. 24. As can be seen, the model of Sichenet al. [65] gives the best fit to the experimental data.

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5 Conclusions

The presented DC method provides comparable data for the density compared withPM, MBP, DM, and SD methods, for the surface tension with BMP, CR, and SD, andviscosity with CM and OM. The advantage of the DC method over the other techniquesis that it provides three properties in a single test. Looking at the comparison of resultswith those obtained from other techniques, the values agree clearly within the estimateduncertainties.

Acknowledgments This work was supported in part by the Natural Science and Engineering ResearchCouncil of Canada. The authors from IMIM PAS wish to thank the Ministry of Science and Higher Educationfor financing the projects.

References

1. J. Pstrus, Physico-Chemical Properties of New Solder Alloys, An Example of Sn–Zn–In System, Ph.D.Thesis, Krakow, 2008

2. D.N. Staicopolus, J. Colloid Sci. 17, 439 (1962)3. R.W. Hyers, Meas. Sci. Technol. 16, 394 (2005)4. R.F. Brooks, A.T. Dinsdale, P.N. Quested, Meas. Sci. Technol. 16, 354 (2005)5. T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals (Clarendon Press, Oxford, 1988),

p. 776. A. Crawley, D. Kiff, Met. Trans. 3, 157 (1972)7. T.R. Hogness, J. Am. Chem. Soc. 43, 1621 (1921)8. Y. Matuyama, Sci. Tohoku Univ. 16, 555 (1927) in E. Pelzel. Z. Metallkde. 93, 248 (1952)9. J. Pstruś, Z. Moser, W. Gąsior, A. Dębski, Arch. Metall. Mater. 51, 335 (2006)

10. S.B. Martin Jr, H.E. Allen, Chemtech 26, 23 (1996)11. K. Shibuya, M. Ozawa, ISIJ Int. 31, 661 (1991)12. A.J. Yule, J.J. Dunkley, Atomization of Melts (Clarendon Press, Oxford, 1994), p. 1713. M. Hirai, Inst. Jpn. Int. 33, 251 (1993)14. P. Fima, T. Gancarz, J. Pstrus, K. Bukat, J. Sitek, Solder. Surf. Mount Technol. 24, 71 (2012)15. S.J. Roach, H. Henein, Can. Metall. Q. 42, 175 (2003)16. S.J. Roach, H. Henein, Metall. Mater. Trans. B 36, 667 (2005)17. T. Gancarz, Z. Moser, W. Gąsior, J. Pstruś, H. Henein, Int. J. Thermophys. 32, 1210 (2011)18. R.L. Daugherty, J.B. Franzini, E.J. Finnemore, Fluid Mechanics with Engineering Applications, 8th

edn. (McGraw-Hill, New York, 1985), p. 45019. H. Henein, Can. Metall. Q. 44, 261 (2004)20. T. Gancarz, W. Gasior, H. Henein, Int. J. Thermophys. under review21. D. Lide (ed.), CRC Handbook of Chemistry and Physics, 88th edn. (CRC Press, Boca Raton, FL, 2008)22. A. Crawley, D. Kiff, Met. Trans. 3, 157 (1972)23. T. Sato, S. Munakata, Bull. Res. Inst. Min. Dress. Metall. 11, 183 (1955)24. H.J. Fisher, A. Phillips, JOM-J. Min. Met. Mat. S, 1060 (1954)25. W. Gasior, Z. Moser, J. Pstrus, J. Phase Equilib. 24, 504 (2003)26. I. Lauerman, F. Sauerwald, Z. Metallkde. 55, 605 (1964)27. V. Somol, M. Beranek, Anorg. Chem. Technol. B30, 199 (1984)28. V.B. Lazarev, Russ. J. Phys. Chem. 38, 325 (1964)29. T. Iida, Z. Morita, S. Takeuchi, J. Jpn. Inst. Met. 39, 1169 (1975)30. E. Gebhartd, K. Kostlin, Z. Metallkde. 48, 636 (1957)31. H. Nakajima, Trans. JIM. 17, 403 (1976)32. F. Herwing, M. Wobst, Z. Metallkde. 83, 35 (1992)33. M. Kucharski, Arch. Hut. 22, 181 (1977)34. M. Kucharski, P. Fima, Monatsh. Chem. 136, 1841 (2005)35. P. Fima, R. Nowak, N. Sobczak, J. Mater. Sci. 45, 2009 (2010)36. T. Hogness, J. Am. Chem. Soc. 43, 1621 (1921)

123

1748 Int J Thermophys (2014) 35:1725–1748

37. D. Melford, T. Haor, J. Inst. Met. 85, 197 (1956–57)38. G. Lang, J. Inst. Met. 101, 300 (1973)39. G. Lang, P. Laty, J. Joud, P. Desre, Z. Metallkde. 68, 133 (1977)40. E. Gebhardtd, M. Becker, E. Tragner, Z. Metallkde. 44, 379 (1953)41. K. Okajima, H. Sakao, Trans. Jpn. Inst. Metal. 23, 111 (1982)42. E. Pelzel, Berg. Huttenmann. Monatsh. 93, 247 (1948)43. W. Krause, F. Sauerwald, M. Micalke, Z. Anorg. Chem. 181, 353 (1929)44. L.L. Bircumshaw, Philos. Mag. 3, 1286 (1927)45. Y. Matuyama, Sci. Rep. Tohoku Imp. Univ. 16, 555 (1927)46. D. Ofte, L.J. Wittenberg, Trans. Met. Soc. AIME 227, 706 (1963)47. A. Sinha, E. Miller, Met. Trans. 1, 1356 (1970)48. H. Neumann, Ch. Dong, Yu. Plevachuk, Z. Metallkde. 91, 933 (2000)49. K.C. Mills, Y.C. Su, Int. Mater. Rev. 51, 329 (2006)50. T. Gancarz, J. Jourdain, H. Henein, unpublished research, University of Alberta, 201451. I. Kaban, S. Mhiaoui, W. Hoyer, J.-G. Gasser, J. Phys. Condens. Matter. 17, 7867 (2005)52. P. Fima, Appl. Surf. Sci. 257, 3265 (2011)53. Z. Moser, W. Gasior, J. Pstrus, J. Phase Equilib. 22, 254 (2001)54. J. Lee, W. Shimoda, T. Tanaka, Mater. Trans. 45, 2864 (2004)55. W. Gasior, Z. Moser, J. Pstrus, M. Kucharski, Arch. Metall. 46, 23 (2001)56. E. Gebhardt, M. Becker, E. Tragner, Z. Metallkde. 46, 379 (1955)57. Yu. Plevachuk, W. Hoyer, I. Kaban, M. Kohler, R. Novakovic, J. Mater. Sci. 45, 2051 (2010)58. R. Novakovic, D. Giuranno, E. Ricci, S. Delsante, D. Li, G. Borzone, Surf. Sci. 605, 248 (2011)59. S.J. Roach, H. Henein, Int. J. Thermophys. 33, 484 (2012)60. C.J. Smithells, Metals Reference Book, 7th edn. (1998), p. 1461. J.A.V. Butler, Proc. R. Soc. Lond., Ser. A CXXXV, 348 (1932)62. I. Ohnuma, X.J. Liu, K. Ishida, J. Electron. Mater. 28, 1164 (1998)63. A. Jönsson, J. Ågren, Mater. Sci. Technol. 2, 913 (1986)64. E.A. Moelwyn-Hughes, Phys. Chem. 434 (1970)65. A. Sichen, J. Bygdén, S. Seetharaman, Metall. Mater. Trans. B 25, 519 (1994)66. S. Seetharaman, D. Sichen, Metall. Mater. Trans. B 25, 589 (1994)67. L.Y. Kozlov, L.M. Romanov, N.N. Petrov, Izv. Vuzov. Chernaya Metall. 3, 7 (1983)68. G. Kaptay, Proceedings of MicroCAD 2003, International Conference Section: Metallurgy (University

of Miskolc, Hungary, 2003), p. 2369. Z. Morita, T. Iida, M. Ued, Liquid Metals, Institute of Physics Conference Series No. 30 (Bristol, 1977),

p. 600

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The Discharge Crucible Method for Making Measurements of the Physical Properties of Melts: An OverviewAbstract1 Introduction2 Principle of DC Technique-Mathematical Model3 Materials, Apparatus, and Experimental Procedures3.1 Description of Experimental Technique3.2 Propagation of Error

4 Results4.1 Pure Metals4.1.1 Aluminum

4.2 Sn--Ag Alloys4.3 Sn--Sb Alloys4.4 Effect of Gas Atmosphere (AZ91D Results) [59]4.5 Butler Model4.6 Viscosity Models

5 ConclusionsAcknowledgmentsReferences