The Discharge Crucible Method for Making Measurements of the Physical Properties of Melts: an Overview

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

The Discharge Crucible Method for Making Measurements of the Physical Properties of Melts: An Overview

T. Gancarz · W. Ga˛sior · 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 melts and alloys and for AZ91D in two gas atmospheres. The DC method was conﬁrmed on pure metals, Sb, Sn, Zn, and compared with corresponding literature data. Results are also reported for Sn–Sb alloys containing (10, 20, 25, 50, and 75) at% of Sb at 550 K to 850 K, for Sn–Ag alloys containing (3.8, 32, 55, and 68) at% Ag, for commercially pure Al, and for an AZ91D Mg alloy under an argon atmosphere. The properties for AZ91D were also measured under an atmosphere of air containing 2% SF6. The results are compared with published data on all alloys. The experimentally measured surface-tension values are compared with the Butler model. Several models are compared and discussed for the viscosity measurements.

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

Nomenclature

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

T. Gancarz · W. Ga˛sior Institute 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, Canada e-mail: [email protected] 123 1726 Int J Thermophys (2014) 35:1725–1748

E Activation energy, kJ·mol−1 M Molecular weight, kg·mol−1 3 −1 Qexp Experimental volumetric flow rate, m ·s R Universal gas constant, J·mol−1·K−1 T Temperature, K V 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 coefficient curve, dimensionless g Gravity acceleration, m·s−2 h Planck’s constant, J·s hexp Liquid head, m ro Orifice radius, m t Time, s u Velocity, m·s−1 η Viscosity, mPa·s ηo Pre-exponential, mPa·s ηAZ91D Viscosity of AZ91D at temperature T ,mPa·s ρ Density, kg·m−3 − σ Surface tension, mN·m1 ρgr0hexp Bo Bond number σ , dimensionless Cd Discharge coefficient, dimensionless ( 2 /( π 3)) Fr Froude number Qexp 2 gro , dimensionless ρ Q Re Reynolds number 2 exp , dimensionless πr0η

1 Introduction

There is a wide range of methods that have been developed for measuring the physical properties of high-temperature liquids, namely, gas–liquid surface tension, density, and viscosity. These methods include the maximum bubble pressure (MBP) [1], the sessile 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 properties outlined above. The ELD is the only approach that is containerless. All of the methods present 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 values determined can be used and related to industrial conditions of processing these melts. These processing operations that are affected by these properties range widely. Metals extraction [10], strip casting [11], atomization [12], spray forming [13], and soldering [14] are a few examples. 123 Int J Thermophys (2014) 35:1725–1748 1727

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

2 Principle of DC Technique-Mathematical Model

A detailed description of the model used for a draining crucible has been presented elsewhere [15–17], therefore, only a brief overview of the model will be presented here. For a vessel having an orifice in the bottom, if it is sufficiently large, a free jet will form when the vessel is filled with a liquid. To predict the flow rate, the Bernoulli formulation is used. Refer to Fig. 1 for an illustration of unsteady flow through an orifice. The two reference locations are labeled as points 1 and 2 on Fig. 1.The Bernoulli 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 et al. [18] considered the case of unsteady flow (velocity as a function of both position and time, i.e., a draining vessel) shown in Fig. 1 giving the relationship: ∂u ∂u − dPdA + ρgdA dz = ρdA dz u + (1) c c c ∂z ∂t

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

Fig. 1 Schematic of draining Patm vessel system depicting flow rate rv of a fluid through an orifice 1 placed at the bottom z1 z

2 z2 ro

Patm 123 1728 Int J Thermophys (2014) 35:1725–1748

Fig. 2 Control element on streamline neglecting frictional losses PdAc

z gdAcdz dz

Streamline (P+dP)dAc

(P − P ) u2 − u2 ∂u 2 1 − g (z − z ) + 2 1 =−(z − z ) (2) ρ 2 1 2 2 1 ∂t

Note that the formulation has thus far neglected viscosity effects. Equation 2 is therefore considered to be valid for a draining vessel under inviscid flow with negligible surface 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 with respect to the velocity at point 2:

r 2 u = o u 1 2 2 (3) rv

Since rv ro, u1 ≈ 0. Rearranging Eq. 2 in terms of the outlet velocity, u2, yields (P − P ) h ∂u u = 2g h − 2 1 − (4) 2 ρg g ∂t h ∂u The term g ∂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 magnitude of the first term (described by h)[19]. The pressure at reference points 1 and 2 is atmospheric in Fig. 1. However, considering the principles of interfacial phenomena, there is a pressure induced at the outlet of the orifice if the liquid surface tension is considered. The pressure induced from the surface 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 relates the pressure difference across a curved interface to the radii of curvature: 1 1 P = σ + (5) Ra Rb 123 Int J Thermophys (2014) 35:1725–1748 1729

P is the pressure difference across the liquid/gas interface, and Ra and Rb represent the 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 (top of the liquid in the vessel), it is assumed that the pressure induced is negligible since the 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 the orifice (where the stream is first in contact with the atmosphere), Ra is taken to be the radius of the orifice, ro. The second radius of curvature, Rb is assumed to be infinity since the stream is assumed to be a perfect cylinder at this location. From the Laplace equation, the pressure induced due to the effects of surface tension directly below the orifice is 1 P = σ (6) ro

Considering Eq. 4, and assuming quasi-steady state conditions, and the pressure at reference point 2 is included in the formulation that corresponds to a magnitude of σ/ro. (P − P ) (σ/r + P − P ) u = 2g h − 2 1 = 2g h − o atm atm 2 ρg ρg σ = 2g h − (7) ρgro

In terms of ﬂow rate, Qexp under viscous conditions, Eq. 7 can be expressed as σ = π 2 − Qexp ro Cd 2g h (8) ρgro or in dimensionless form,

− Fr + Bo 1 = 1(9) where Fr is the Froude number and is given by 2 Qexp πr2C Fr = o d (10) 2gh and Bo is the Bond number given by

ρgrh Bo = σ (11) 123 1730 Int J Thermophys (2014) 35:1725–1748

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

Inertial Forces ρuexp2ro 2ρ Qexp Reexp = = = (12) Viscous Forces η πroη where ρ and η are the density and viscosity of the fluid, respectively. Plotting Cd versus the Reynolds number, Eq. 12, provides a measure of the frictional characteristics of the orifice. A polynomial fit can be derived from the Cd versus Re plot and substituted into Eq. 8. The DC method is used in the following manner. A crucible with a well machined orifice is calibrated with a fluid of known properties and a function of Cd versus Re is experimentally determined. An experiment is then carried out using a fluid with unknown properties. The flow rate as a function of the draining crucible height are measured 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 then applied to seek the values of these properties while minimizing the error. The details of 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 immersed into 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 pure metals and for Sn–Ag and Sn–Sb alloys. The equipment for the Ag, Sb, Sn, and Zn tests was placed inside a glove box to ensure inert conditions [20]. The experimental procedure 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 does not leak back into the unit. A continuous purge of argon is run through the unit until a nominal oxygen level of 20ppm is registered from the oxygen monitor. For the aluminum and magnesium alloy experiments, the charge is heated using a 20 kW induction furnace. When the temperature in the crucible (high density graphite for Al and steel for Mg alloy) reaches the desired value, the melt is maintained at a constant temperature for at least 15 min to 20 min. For the Ag, Sb, Sn, Zn, and Sn–Ag and Sn–Sb alloys, an electrical resistance furnace and graphite crucible are used following a similar procedure. Once the temperature and oxygen in the atmosphere are equilibrated, the melt is allowed to flow through an orifice placed in the bottom of the crucible. The melt is captured by a load cell which registers the cumulative mass of the melt draining from the crucible. For the case of Al and Mg alloy experiments, the relationship between the crucible volume and melt height was calibrated using a known fluid prior to the melt 123 Int J Thermophys (2014) 35:1725–1748 1731

1/Bo = 0.29 1/Bo = 0.13

1 0.95 0.9 0.85 0.8 , Aluminum 1073 K 0.75 N/m Average 0.7 0.65 0.6 0.55 0.5 0.04 0.06 0.08 0.1 Head, m Fig. 3 Surface tension of aluminum determined at 1073 K, with expected error in surface tension due to error in head measurement represented by solid lines

6560

Sb DC, [20] 6520 PT, [22] [23] -3

m MBP, [24] . g ,k

y 6480 Densit

6440

6400 880 920 960 1000 1040 1080 Temperature, K 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 converted to the melt height in the crucible as part of the mathematical model to determine the physical properties of the melt. For Ag, Sb, Sn, Zn, and Sn–Ag and Sn–Sb alloys, a dilatometric technique is used to yield the metal volume in the crucible as a function of time [20]. 123 1732 Int J Thermophys (2014) 35:1725–1748

390 Sb DC, [20] MBP, [27] 380 MBP, [25] MBP, [28]

1 MBP, [26] - m . 370 N m

360 ce tension, a rf

u 350 S

340

330 900 1000 1100 1200 Temperature, K Fig. 5 Surface tension of antimony as a function of temperature compared with different measurement methods

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 time curve recorded by the load cell (see Fig. 5). A second-order polynomial curve is fit to the data. A higher-order polynomial curve did not improve accuracy. For example, a polynomial fit on a set of data collected for aluminum through a 5mm diameter orifice at 1073K is expressed as

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

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

1 dC − V = m =−8.52 × 101(t) + 6.491 × 10 2 (14) exp π 2 ro dt

The head, hexp, is obtained using load cell information as well. Knowing the geometry of the crucible, the head can be determined by monitoring the quantity of material that poured through the oriﬁce as a function of time. The density of the liquid is required for 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 subsequently updated 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. 123 Int J Thermophys (2014) 35:1725–1748 1733

2.4

2 Sn DC, [17] s .

a DC, [20] 1.6 CM, [25] mP

, OM, [40] y

1.2 Viscosit

0.8

0.4 400 800 1200 1600 Temperature, K 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:

∂σ 2 ∂σ 2 ∂σ 2 2 2 2 δσ = δhexp + δQexp + (δCd) (15) ∂hexp ∂Vexp ∂Cd

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

∂σ = ρgr (16) ∂h o exp ∂σ ρ Q =− ro exp 2 2 (17) ∂ Qexp Cdπr Cdπr o o 2 ∂σ ρr Qexp = o (18) ∂ π 2 Cd Cd Cd ro

The equation for the error of density:

∂ρ 2 ∂ρ 2 ∂ρ 2 2 2 2 ∂ρ = (∂h) + ∂ Qexp + (∂Cd) (19) ∂h ∂ Qexp ∂Cd ∂ρ ρ2 =− gr0 ∂h σ 123 1734 Int J Thermophys (2014) 35:1725–1748

∂ρ ρ2 Q = exp ∂ π 2 3σ Qexp Cd r0 ρ2 2 ∂ρ Qexp =− (20) ∂ 3π 2 3σ Cd Cd r0

The equation for the error of viscosity:

∂η 2 ∂η 2 ∂η 2 ∂η 2 ∂η 2 2 2 2 2 2 ∂η = (∂h) + ∂ Qexp + (∂Cd) + (∂a) + (∂b) ∂h ∂ Qexp ∂Cd ∂a ∂b (21)

∂η 2aρr gQ2 = 0 exp / (22) ∂h σ 3 2 π 2r 2 [C − b]2 2gh 1 − 0 d ρgr0h ∂η ρ = 2a d 2 ∂ Qexp π r0(Cd − b) ∂η 2ρr Q = 0 exp ∂ π 2[ − ] a ro Cd b ∂η πr η2 = 0 (23) ∂b 2aρ Qexp where the errors determined for the experiment conducted on the various parameters were as follows:

2 (hpoly − hexp) − ∂h = = 8.881 × 10 5 m n − 3 ∂C − ∂ Q = 1/ρ m = (in m3 ·s 1) where exp ∂t 2 (Cm,poly − Cm,exp) − ∂C = = 1.145 × 10 3 kg m n − 3 2 (Cd,poly − Cd,exp) ∂C = = 0.018 (24) d n − 4

given the error in the measured and varying values of hexp, Qexp, and Cd, The expected error in s as a function of the change in these variables can be determined from experiments. This will be illustrated later for the effect of hexp on s for aluminum. As will be seen, the error in s due to hexp is very small. A similar analysis can be done for the other variables Qexp and Cd as well as for r and m as a function of these same variables. 123 Int J Thermophys (2014) 35:1725–1748 1735

4 Results

4.1 Pure Metals

Experimental data obtained from different measurement methods for the pure metals are 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 density for 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 <1% different of the other measurement values. The surface tensions presented in Fig. 5 compare the obtained values with the MBP method, which is regarded as the best method for the determination of the surface tension of liquid metals. The obtained values of the surface tension for Sb from DC [20] and MBP methods [25–27] agree within 1% except for those in [28] which are significantly lower over the whole temperature range. The viscosity of Sb shown in Fig. 6 from DC [20], CM [29], and OM [30–32] show very good agreement. The values of the viscosity obtained from DC show a slightly larger temperature dependence than for the other methods. The obtained values of density from the DC method presented in Fig. 7 for two experimental conditions and equipment [17,20] are almost identical. Comparing the density from DC [17,20], DM [9], and MBP [33] for Sn, good agreement was observed; only values of the density from SD [34,35] deviate significantly from the rest. In the comparison of Fig. 8, the surface tension of tin are shown for the DC [17,20], BMP [25], CR [36–38], and SD [39] methods. As previously observed, the values of the surface tension for DC from [17,20] are almost identical. We can see very good agreement among the obtained values of surface tension for Sn for the various

7000

6900 3 - m

. 6800 g ,k y Sn DM, [9] 6700 Densit DC, [17] DC, [20] MBP, [33] 6600 SD, [34] SD, [35]

6500 400 600 800 1000 1200 Temperature, K Fig. 7 Density of tin as a function of temperature compared with different measurement methods 123 1736 Int J Thermophys (2014) 35:1725–1748

600

560 -1 m . mN

520 ce tension, a rf u S 480 DC, [17] CR, [37] DC, [20] CR, [38] MBP, [25] BMP, [39] CR, [36] SD, [39] 440

500 600 700 800 900 1000 Temperature, K Fig. 8 Surface tension of tin as a function of temperature compared with different measurement methods

2.4

2 Sn DC, [17] s .

a DC, [20] 1.6 CM, [25] , mP

y OM, [40]

1.2 Viscosit

0.8

0.4 400 800 1200 1600 Temperature, K 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 very good agreement among the data obtained using DC [17,20], CM [25], and OM [40] methods. 123 Int J Thermophys (2014) 35:1725–1748 1737

6600

Zn DM, [1] DC, [20] 6500 PM, [36] PM, [41] -3

m .

6400 Density, kg Density,

6300

6200 600 700 800 900 1000 1100 Temperature, K Fig. 10 Density of zinc as a function of temperature compared with different measurement methods

820 Zn MBP, [1] DC, [20] 800 MBP, [36]

-1 MBP, [42] m . MBP, [43] 780 MBP, [44] DW, [45]

760 Surface tension, mN

740

720 700 800 900 1000 1100 Temperature, K 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 vapor pressure. Physicochemical properties of zinc as a function of temperature compared among different measurement methods are presented in Figs. 10, 11, and 12.InFig.10 which shows the density of Zn, we can see very good agreement among the data obtained from all methods of DC [20], DM [1], and PM [36,41]. The surface tensions 123 1738 Int J Thermophys (2014) 35:1725–1748

4 Zn DC, [20] CM, [29] s . OM, [46] a OM, [47]

, mP OM, [48] y 3 Viscosit

700 800 900 1000 1100 Temperature, K Fig. 12 Viscosity of zinc as a function of temperature compared with different measurement methods

DC, SnAg3.5, [20] SD, SnAg3.8, [52] DC, SnAg32.5, [20] SD, SnAg30, [52] SD, SnAg30, [34] DM, SnAg3.8, [53] SD, SnAg3.5, [51] DM, SnAg25, [53] 8000

Sn-Ag

7600 -3 m .

7200 Density, kg

6800

6400 400 600 800 1000 1200 1400 Temperature, K Fig. 13 Density of Sn–Ag alloys as a function of temperature compared with different measurement methods

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

DC, SnAg3.5, [20] SD, SnAg30, [52] DC, SnAg32.5, [20] MBP, SnAg3.8, [53] SD, SnAg30, [34] MBP, SnAg25, [53] SD, SnAg3.5, [51] SD, SnAg10, [54] SD, SnAg3.8, [52] SD, SnAg30, [54] 600

Sn-Ag -1 m . 560 mN

520 Surface tension,

480 400 600 800 1000 1200 1400 Temperature, K Fig. 14 Surface tension of Sn–Ag alloys as a function of temperature compared with different measurement methods

2.4 Sn-Ag DC, Sn, [17] DC, SnAg3.5, [17] 2 DC, SnAg32.5, [17] CM, Sn, [17] CM, SnAg30, [17] CM, Sn, [55] s . 1.6 OM, Sn, [56] OM, SnAg24.4, [56]

1.2 Viscosity, mPa Viscosity,

0.8

0.4 400 800 1200 1600 Temperature, K Fig. 15 Viscosity of Sn–Ag alloys as a function of temperature compared with different measurement methods 123 1740 Int J Thermophys (2014) 35:1725–1748

DC, SnSb10, [20] DM, SnSb10, [55] DC, SnSb20, [20] DM, SnSb20, [55] DC, SnSb25, [20] DM, SnSb40, [55] DC, SnSb50, [20] DM, SnSb60, [55] DC, SnSb75, [20] DM, SnSb80, [55]

6900

Sn-Sb 6800 -3 m

. 6700

6600 Density, kg

6500

6400 600 800 1000 Temperature, K Fig. 16 Density of Sn–Sb alloys as a function of temperature compared with different measurement methods of Zn, presented in Fig. 11, show good agreement among the data from DC [20] and MBP [42,43]. Also, good agreement is seen with data showing the same slope but a higher MBP [44]oralowerDW[45] value. In Fig. 12 comparisons of the viscosity of Zn are shown for the methods, DC [20], CM [29], and OM [46–48]. Based on the comparisons, the DC method yields accurate and reliable results for the properties of density, surface tension, and viscosity when compared with measurements made using other techniques.

4.1.1 Aluminum

In a review of the surface tension of metallic elements, Mills and Su compared the reported values of a number of researchers for aluminum including those obtained by the DC method [49]. He showed that the values reported by Roach and Henein were within 2% with other published values for the case of oxygen-saturated aluminum. The reported values of density were estimated to have <4 % error, while the viscosity was as much as an order of magnitude greater than those given in the literature by an order of magnitude [16]. This has recently been attributed to a high oxygen level of 20 ppm [50].

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

DC, SnSb10, [20] BMP, SnSb10, [55] SD, SnSb14.5, [58] DC, SnSb20, [20] BMP, SnSb20, [55] SD, SnSb20, [57] DC, SnSb25, [20] BMP, SnSb40, [55] SD, SnSb30, [58] DC, SnSb50, [20] BMP, SnSb60, [55] SD, SnSb47.3, [58] DC, SnSb75, [20] BMP, SnSb80, [55] SD, SnSb80, [58]

520 Sn-Sb

-1 480 m . mN

440

Surface tension, 400

360 500 600 700 800 900 1000 1100 Temperature, K Fig. 17 Surface tension of Sn–Sb alloys as a function of temperature compared with different measurement methods

4.2 Sn–Ag Alloys

In Figs. 13, 14, and 15 comparisons of the density, surface tension, and viscosity of Sn–Ag alloys, respectively, are presented. Results for the density of Sn–Ag using 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 that the 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 differences but still no larger than 1% from the SD method. The values of density increase with Ag content in the alloy; the data from the DC method for SnAg32.5 are a good ﬁt in the trends of the density dependence of Sn–Ag alloys. The surface tensions of Sn–Ag alloys presented in Fig. 14 show comparisons among the different methods, DC [20], MBP [53], and SD [34,51,52,54]. The differences between the values from DC [20] and SD [51] for SnAg3.5 are relatively large, ∼9%. The obtained data of the surface tension from the SD method show the large differences in the slopes over a large range of data. It is therefore important during the liquid-drop experiments to take into account problems arising from the oxides or concentration ﬂuctuations which are very important and may distort the measurements 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

DC, Sn, [20] OM, Sn, [23] DC, SnSb10, [20] OM, SnSb9.9, [23] DC, SnSb20, [20] OM, SnSb19.8, [23] DC, SnSb25, [20] OM, SnSb39.7, [23] DC, SnSb50, [20] OM, SnSb49.7, [23] DC, SnSb75, [20] OM, SnSb79.8, [23] DC, Sb, [20] OM, Sn, [23]

2.4

Sn-Sb

2 s . a , mP

y 1.6 Viscosit

1.2

0.8 400 600 800 1000 1200 Temperature, K Fig. 18 Viscosity of Sn–Sb alloys as a function of temperature compared with different measurement methods

0.8 Alloy liquidus temperature 0.75 0.7 0.65 -1 Magnesium 0.6

, N · m 0.55 This study 0.5 0.45 0.4 850 950 1050 1150 1250 Temperature, K Fig. 19 Surface tension of AZ91D under argon from this work compared to literature values (Roach and Henein [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 agreement. 123 Int J Thermophys (2014) 35:1725–1748 1743

4.3 Sn–Sb Alloys

In Figs. 16, 17, and 18 are presented the density, surface tension, and viscosity of Sn–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 of the 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 of the surface tension for SnSb20 from DC and SD [57] show excellent agreement, but values for SnSb30 from SD [58] appear to be too high or maybe there is an error in the 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 of Sn–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 concentrations of Sb in Sn–Sb alloys and lower temperatures to 800 K are lower than the values of viscosity from the DC method. For SnSb50 and higher concentrations, we observe good agreement between the obtained values of the viscosity of Sn–Sb alloys from OM 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.The data for the density of AZ91D agree with an ideal solution model for calculating the density of the alloy. Figure 20 shows that the presence of air with 2% SF6 reduces the surface tension of the alloy. While the viscosity is signiﬁcantly lower than is typically reported in the literature for molten metals, the variation of viscosity as a function of temperature follows an Arrhenius equation of the form;

E μ = μ exp (25) AZ91D o RT

Fig. 20 Viscosity of AZ91D magensium alloy under argon as a function of temperature. Solid line is the Arrhenius curve ﬁt through all points and dashed line is the Arrhenius ﬁt without the data point at the highest temperature (Roach and Henein [59]) 123 1744 Int J Thermophys (2014) 35:1725–1748

Table 1 Viscosity of AZ91D as a function of temperature

−1 2 Alloy Best ﬁt based on data from μo(Pa·s) E(kJ·mol ) R

− AZ91D (Ar) 921 K to 1169 K 5.325 × 10 6 35.91 0.518 − 921K to 1067K 9.058 × 10 8 68.03 0.7119 − 921K to 967K 1.181 × 10 13 173.50.9889 − Mg Ref. [59]2.45 × 10 5 30.5

Fig. 21 Surface tension of tin and Sn–Ag alloys as a function of temperature compared with the Butler model: solid line—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 shown in 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 earlier work by Gancarz et al. [17]. The thermodynamic optimized parameters were taken for 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 for experimental compositions that are plotted in Fig. 21 for Sn–Ag and in Fig. 22 for Sn–Sb in comparison with experimental data obtained by other measurement methods. Figure 21 illustrates a comparison between the results of calculations using the Butler model [61]—line and the experimental data obtained by different methods: MBP—circles [25] and DC—squares [17] for respective compositions. The surface tension calculated from the model decreases with increasing temperature and increases with increasing Sn concentration, which is in accordance with experimental results. Figure 22 presents the results of calculations using the Butler model [61] with data 123 Int J Thermophys (2014) 35:1725–1748 1745

520 923 K Sn-Sb SD, [58] MBP, [25] 480 -1 SD, [57] m . DC, [20]

mN Butler model

440

Surface tension, 400

360 0 0.2 0.4 0.6 0.8 1 Sb Sn XSb Fig. 22 Surface-tension isotherms of liquid Sn–Sb alloys obtained from the Butler’s model (continuous lines) against the background of experimental values, with literate data (symbols)[5,6,30]. Dashed line shows the linear changes of surface tension between pure Sb and Sn received from MBP, SD, and DC methods. Obtained values of the surface tension from these three methods are almost identical and have slightly lower values of calculated data from the Butler model.This may be due to the surface tension of pure Sb obtained from the DC method that has been used in the modeling is slightly higher than that obtained with MBP. Good agreement is observed between calculated data from the Butler 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 in earlier work [17]. A comparison of the experimental data of the viscosity of Sn–Ag alloys received from DC and CM methods with each of these models is presented in Fig. 23. The lack of a universal model and the difﬁculty in modeling the viscosity are apparent. The best agreement between measured and calculated viscosities is given by the 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. In such a case for alloys with a negative value of the enthalpy of mixing, the calculated viscosity will always be higher than that for the ideal solution, and for alloys with a positive value of the enthalpy of mixing, the viscosity will be lower than that for 123 1746 Int J Thermophys (2014) 35:1725–1748

15 14 13 T=823 K Sn-Ag 12 11 CM, [17] s . 10 DC, [17] 9 [64] 8 [65] [66] 7 [67]

Viscosity, mPa Viscosity, 6 [68] 5 4 3 2 1 0 0.2 0.4 0.6 0.8 1

XSn 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

1.6 923 K Sn-Sb DC, [20]

s OM, [23] . 1.2 [64] [65] [66] [67] [68] 0.8 [69] Viscosity, mPa Viscosity,

0.4 0 0.2 0.4 0.6 0.8 1

XSn 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 literature data of Sato and Munakata [23] for 923K

ideal solutions. The obtained values of the viscosity from the DC and OM methods are compared with the modeling data in Fig. 24. As can be seen, the model of Sichen et al. [65] gives the best ﬁt to the experimental data. 123 Int J Thermophys (2014) 35:1725–1748 1747

5 Conclusions

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

Acknowledgments This work was supported in part by the Natural Science and Engineering Research Council of Canada. The authors from IMIM PASwish to thank the Ministry of Science and Higher Education for ﬁnancing the projects.

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