density and viscosity of molten zn-al alloys

Density and Viscosity of Molten Zn-Al Alloys

Y.H. LIU

Different Zn-Al alloys were applied in various hot-dip coating processes, such as galvanizing,GALFAN, GALVALUME, and aluminizing, for corrosion protection. Density and viscosity aretwo important parameters of the molten alloys in many studies, especially those involving fluidmechanics. However, the experimental data of these two important properties are scarce. Interpola-tion and extrapolation of experimental data are often necessary. Models that can describe the entiresystem are needed for the prediction of these properties at different temperatures and with differentalloy compositions. Various thermodynamic models were examined in the current study, and the onethat provided the best fit to the experimental data of the Zn-Al binary system was introduced.

I. INTRODUCTION

DIFFERENT Zn-Al alloys are applied in various hot-dipcoating processes to offer steel a combination of sacrificialand barrier protection against corrosion. The most commonprocess is galvanizing, which uses a Zn-based alloy con-taining about 0.2 pct Al (weight percentage is usedthroughout the article). GALFAN,* a proprietary process

*GALFAN is a trademark of International Lead Zinc Research Organ-ization, Research Triangle Park, NC.

promoted by the International Lead Zinc Research Organ-ization, uses the eutectic Zn-Al alloy containing 5 pct Al.GALVALUME,** a patented process invented by Bethlehem

**GALVALUME is a trademark of BIEC International, Inc., Vancouver, WA.

Steel, employs an alloy with a composition of 55 pct Al,43 pct Zn, and 1.6 pct Si. Processes using Zn-Al alloys withother compositions have also been developed to serve spe-cial purposes. Aluminizing, on the other hand, uses pure Alwith or without the addition of Si.

Density and viscosity are two material properties impor-tant to many studies involving fluid mechanics. Examplesof these studies include the computer simulation of flows ingalvanizing baths and mathematical modeling in coatingweight control using impinging air jets. Density and vis-cosity are also important to die casting studies. There isclearly a need in the industry for a means to predict thesetwo properties of Zn-Al alloys at various compositions andtemperatures.[1] Unfortunately, experimental measurementsof the viscosity of Zn-Al alloys are scarce. The most sys-tematic studies were carried out four decades ago and theresults are available mainly in internal reports.[2,3,4] Thedata processing performed in those studies was composi-tion specific, resulting in a group of equations that useddifferent parameters for different alloy compositions. Con-sequently, the prediction of properties of alloys with com-positions not covered by the experiments was unreliable, iffeasible at all. Advanced models, taking advantage of betterdescriptions of thermodynamic properties available afterthe widespread use of computers in thermodynamic assess-

ments, are able to predict viscosity at any composition inthe system. A reassessment of the experimental data usingthe advanced models is therefore justified.The density of a molten alloy is normally a linear func-

tion of temperature. Its modeling is a relatively easy task, asthe density of a binary alloy does not usually deviate sig-nificantly from the simple weighted average of the densitiesof two constituent elements. The modeling of density isdiscussed in the current study largely because its result isneeded in the modeling of viscosity. The modeling of vis-cosity, on the other hand, is more challenging. To start with,data collected from different sources may not agree verywell, probably due to the difficulties in accomplishing theexperiments. Data for Zn-based alloys are typically neededand available in a temperature range lower than the meltingpoint of Al. Extrapolating the viscosity of Al to temperatureslower than the melting point becomes necessary for modelingthis property of Zn-Al alloys. As the viscosity is a relativelycomplex function of temperature, such an extrapolation caninduce significant errors. Various thermodynamic modelswere examined in the current study, and the one that providedthe best fit to the experimental measurements is introduced.

II. THERMODYNAMIC MODELING OFLIQUID DENSITY

The modeling of density is essentially the modeling ofmolar volume. Once the molar volume is obtained, thedensity is calculated by dividing the molar weight by themolar volume. The molar volume of liquid metal Vi can bedescribed as a linear function of temperature T. The molarvolume of an alloy can be represented by the values of theconstituent elements and the excess Gibbs energy DGex ofthe liquid solution. For the molten Zn-Al alloys,

V 5 XAlVAl 1XZnVZn 1CDGex [1]

where C is an adjustable coefficient for curve fitting. Anegative DGex signifies affinity between two constituentelements that normally leads to a negative deviation ofthe molar volume from the weighted average. A positiveDGex, on the other hand, indicates repulsion between dis-similar atoms and, hence, a positive deviation in volume.The excess Gibbs energy DGex of the liquid Zn-Al solutionhas been evaluated by many researchers in their critical

Y.H. LIU, Senior Scientist, is with the Product Technology Centre, TeckCominco Metals Ltd., Mississauga, ON, Canada, L5K 1B4. Contacte-mail: [email protected]

Manuscript submitted January 30, 2006.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 37A, SEPTEMBER 2006—2767

JOBNAME: MTA 37A#9 2006 PAGE: 1 OUTPUT: Thursday August 10 08:30:52 2006

tms/MTA/123238/ETP0642A5

assessments of the Zn-Al system. The values suggested bythe latest assessments[5,6] were adopted in this study. Inthose latest assessments, DGex was expressed in the Red-lich–Kister polynomial:

DGex 5 XAlXZn½L0 1 L1ðXAl � XZnÞ� [2]

where L0 5 10,288 � 3.035 T and L1 5 �810 1 0.471 T.The density of molten Al was measured by Magnusson

and Arnberg[7] using an indirect Archimedean method. Themeasured densities of Al at different temperatures wereused in the current study to derive the molar volume of Al,VAl. The linear regression of data points, as shown in Figure 1,yielded VAl 5 10.058 1 0.001397 T.

The density of molten Zn and 13 Zn-Al alloys with Alcontents ranging between 0.15 and 15 pct were measuredby Thresh et al.[2–4] using a pycnometric method, which wasconsidered most suitable for Zn-based alloys due to theirhigh vapor pressures. The measured densities of Zn at dif-ferent temperatures were used to derive the molar volumeof Zn, VZn. The linear regression of data points, as shown inFigure 2, yielded VZn 5 8.982 1 0.001400 T.

The values of Zn-Al alloys were used in the determina-tion of the constant C in Eq. [1]. A value of C 5 7.73 10�5

yielded the best overall fit to experimental data. The differ-ence between the measured density rm and the predicteddensity rp fell within a narrow range between �0.26 and10.25 pct, as shown in Figure 3. The group of data locatedin the lower-left corner in Figure 3 was for Zn-Al alloyscontaining 15 pct Al. The spread within the group was dueto the difference in temperature. The next group with thesecond lowest viscosity was for those alloys containing8 pct Al. The rest of the data points in the chart formed acontinuous band because of the proximity of alloy compo-sitions. The standard deviation of the prediction,

s 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi+ðrp � rmÞ2

n� 1

s[3]

was 0.0084 g/cm3 for a total of 214 data points for Zn-Alalloys. The standard deviation decreased to 0.0075 g/cm3

when the 50 data points for pure Zn and Al were alsoincluded, as the model fit the pure element data even moreprecisely.

The advantage of a model such as Eq. [1] is the ability topredict the property at compositions not covered by experi-ments. Figure 4 demonstrates an example of its application.The density of Zn-Al alloys at 460 °C (733 K) was calcu-lated as a function of alloy compositions and was plottedas a solid line. The line stopped at a composition whoseliquidus temperature was 460 °C. Alloys with Al contentshigher than this composition would be in semisolid or solid

Fig. 1—The molar volume of Al determined from the density valuesmeasured by Magnusson and Arnberg.[7]

Fig. 2—The molar volume of Zn determined from the density measure-ments carried out by Thresh et al.[2,3]

Fig. 3—The predicted densities of Zn-Al alloys fit well with the exper-imental measurements.[2-4]

2768—VOLUME 37A, SEPTEMBER 2006 METALLURGICAL AND MATERIALS TRANSACTIONS A



states at this temperature. Also presented in Figure 4 aredata points at 14 different compositions (closed circles)where experimental measurements were carried out. Theactual measurements were performed not exactly at 460 °Cbut in a temperature range that encompassed 460 °C. Thedata for 460 °C were then calculated using the linearregression of data points for each composition. The stand-ard deviation s for each composition was very small. If the3s had been plotted as error bars in Figure 4 along with thedata points, they would have been too short to be clearlyvisible. In other words, those data points shown in Figure 4could be viewed as a good representation of experimentalmeasurements. The dashed line plotted in Figure 4 showsthe density values calculated using the weighted average ofAl and Zn, without considering the interaction between Aland Zn atoms. Although the difference between the aver-aged value and the values predicted by the model was rel-atively small, it is apparent that the model provided a muchbetter fit to experimental points.

III. THERMODYNAMIC MODELING OFVISCOSITY

The modeling of viscosity is a rather challenging task. Athorough overview of various models was given by Brookset al.[8] Different models were evaluated in the currentstudy. It was found that the model developed by Duet al.[9] at the Royal Institute of Technology, Sweden (here-after referred to as the KTH model), provided the best fit tothe experimental results of Zn-Al alloys.

The viscosity measurements of Zn-Al alloys are scarce.The most systematic studies available to the undersignedwere those carried out by Thresh et al.[3,4] in the mid-1960susing the oscillating cylinder method. Unfortunately, Threshdid not measure the viscosity of pure Al. Because viscositydata from different sources may not agree well, cautionmust be exercised when looking for the replacement set ofdata for pure Al. The experimental work carried out byRothwell[10] was found to be a satisfactory choice because

it was performed in the same period of time (1961) usingthe same type of viscometer (oscillating sphere). Theresults of Rothwell were plotted in Figure 5 together withfitting curves using the Arrhenius equation and the KTHmodel. The Arrhenius equation is commonly used to depictthe viscosity dependence on temperature:

h 5 h0 expE

RT

� �[4]

where h0 is a pre-exponential constant and E is the activa-tion energy. The value of h0 and E are constants determinedby the regression of experimental measurements. The KTHmodel[9] has a similar formula except that the pre-exponen-tial constant is determined using the absolute reaction ratetheory as hN0/Vm. The activation energy becomes the onlyterm to be determined using the experimental data. Itacquires a new name, the Gibbs energy of activation, andis denoted as DG*. It can usually be described as a functionof temperature:

DG� 5 a1 bT1 cT ln T [5]

where a, b, and c are constants that can be optimized fromexperimental data. The optimal values for a, b, and c werefound to be 1328.04, 101.290, and �10.5558 for pure Al, asshown in Figure 5. The optimal values for pure Zn weredetermined in the same way as 9586.77, 63.6023, and�6.00523. It is apparent in Figure 5 that the Arrheniusequation and the KTH model both offered an adequate fitto the experimental data, with standard deviations of 0.0053and 0.0052 mPa�s, respectively. However, the extrapola-tions of these two models to lower temperatures differedsignificantly. Although there was no means to judge whichmodel provided better extrapolation to temperatures lowerthan the melting point of pure Al, this difference was car-ried over into the final modeling of viscosity where theKTH model provided more accurate predictions, at leastfor Zn-Al alloys.

Fig. 4—The density of Zn-Al alloys at a bath temperature of 460 °C (733K). Data points shown as closed circles represent experimental measure-ments. The density predicted by using Eq. [1] is shown as a solid line,whereas that calculated from the simple weighted average of Al and Zn isshown as a dashed line.

Fig. 5—Different models provide the same good fit to experimental meas-urements[10] but lead to different predictions when extrapolated to lowertemperatures.




The Gibbs energy of activation for the Zn-Al alloys,according to the KTH model, should be described as

DG� 5XAlDG�Al1XZnDG

�Zn 1RTðXAl ln XAl 1XZn ln XZnÞ

1DGex [6]

The prediction made by this model compared favorably toother models. The predicted viscosity using the KTH modelwas plotted against the experimental measurements, asshown in Figure 6. It is evident in Figure 6 that the KTHmodel has a tendency to underestimate the viscosity ofZn-Al alloys. It is, therefore, reasonable to believe thatthere is room for further improvement. Close examinationof the model raised questions about the role that DGex playsin Eq. [6]. Further to the discussion in Section II, a negativeexcess Gibbs energy signifies the affinity between dissimilaratoms, which tend to agglomerate, leading to an increase inviscosity and a decrease in diffusivity. In that sense, the signpreceding the excess Gibbs energy term in Eq. [6] should benegative rather than positive. In accordance with the changein the enthalpy term, introduction of an adjustable coefficientA to the entropy term becomes necessary in order to achievea good fit to the experimental data. The modified equationthus becomes

DG� 5XAlDG�Al1XZnDG

�Zn1ARTðXAl ln XAl1XZn ln XZnÞ

� DGex [7]

Note that Eq. [7] did not follow the strict Gibbs energyexpression of the original KTH model and, in this sense,represents a semiempirical modification. Nevertheless, usingthe modified model (Eq. [7]) resulted in a significant improve-ment in accuracy, as shown in Figure 7. All predictions werewithin 63.1 pct of the measurements, and the standard devia-

tion was reduced almost by half, from 0.070 mPa�s (Figure 6)to 0.039 mPa�s (Figure 7).

The viscosity of molten Zn-Al alloys at 460 °C (733 K)is shown in Figure 8. Each data point in the figure repre-sents a set of experimental measurements[3,4] carried out ina temperature range that encompassed 460 °C. Each set ofexperimental data had the same composition and, therefore,could be readily modeled using the Arrhenius equation.The standard deviation s between the experimental meas-urements and the Arrhenius model was evaluated. The errorbar associated with each data point plotted in Figure 8represents 63s. The data shown in Figure 8 demonstratea significant deviation from the simple weighted average ofAl and Zn (dashed line). The original KTH model (Eq. [6])captures the general trend, but the predictions fell outside3s for most data points. The modified model (Eq. [7])

Fig. 6—Viscosity modeling using the KTH model (Eq. [6]).

Fig. 7—Viscosity modeling using the modified KTH model (Eq. [7]).

Fig. 8—Viscosity of Zn-Al alloys at 460 °C (733 K) demonstrates a sig-nificant deviation from the weighted average of Al and Zn properties. Themodified model (Eq. [7]) shows a notable improvement over the originalKTH model (Eq. [6]).

2770—VOLUME 37A, SEPTEMBER 2006 METALLURGICAL AND MATERIALS TRANSACTIONS A



provided much more accurate predictions, as shown in thesolid line in Figure 8.

It is interesting to note that data for 4.5 pct Al alloy(XAl 5 0.103) showed a much larger error band than others,attributable mainly to four scattering measurements carriedout at temperatures only about 10 °C above the eutectictemperature. Data for 5.0 pct Al alloy (XAl 5 0.113, theeutectic composition) did not show any anomaly at lowertemperatures. It is, therefore, believed that the unusuallylarge error at 4.5 pct Al was probably an isolated incidentand should not be regarded as indirect evidence for theexistence of some sort of eutectic structure in the melt.

IV. SUMMARY

Thermodynamic models using the excess Gibbs energyof mixing were successfully used to calculate the densityand viscosity of molten Zn-Al alloys. The models offeredreliable and accurate predictions of these two importantmaterial properties at compositions and temperatures notcovered by experiments.

ACKNOWLEDGMENT

The author thanks Dr. X.G. Lu, Thermo-Calc SoftwareAB, Sweden, and Dr. N.-Y. Tang, Teck Cominco MetalsLtd., Canada, for constructive discussions, and acknowl-edges the assistance of Mrs. Angeline M. Prskalo in thepreparation of the manuscript.

NOMENCLATURE

DGex excess Gibbs energy of mixing (J/mol)DG* Gibbs energy of activation (J/mol)

L0, L1 thermodynamic interaction parameters (J/mol)N0 Avogadro’s number, equals 6.022142 3 1023 mol�1

R gas constant, equals 8.31447 J/mol/KT absolute temperature (K)Vm molar volume, in cm3/mol (10�6 m3/mol)XAl molar fraction of Al in molten Zn-Al alloyXZn molar fraction of Zn in molten Zn-Al alloyh Planck’s constant, equals 6.6260683 10�34 kg�m2/sn number of experimental data points used in model-

ing or regressionh viscosity, in mPa�s (10�3 Pa�s)r density, in g/cm3 (103 kg/m3)

REFERENCES

1. M. Dubois: R&D Center of Arcelor Cockerill, Liege, Belgium, privatecommunication, 2005.

2. H.R. Thresh, D.W.G. White, J.O. Edwards, and J.W. Meier: InternalReport No. PM-R-64-26, Mines Branch, Canada Department of Minesand Technical Surveys, Ottawa, Canada, 1964, pp. 1-47.

3. H.R. Thresh: Internal Report No. R-133, Mines Branch, CanadaDepartment of Mines and Technical Surveys, Ottawa, Canada, 1964.

4. H.R. Thresh, D.W.G. White, J.O. Edwards, and J.W. Meier: InternalReport No. PM-R-64-27, Mines Branch, Canada Department of Minesand Technical Surveys, Ottawa, Canada, 1965, pp. 1-64.

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density and viscosity of molten zn-al alloys

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