ThermophysicalpropertiesofundercooledliquidCo–Mo alloyslmms.sjtu.edu.cn/papers/03_PM.pdf ·...

PHILOSOPHICAL MAGAZINE, 2003, VOL. 83, NO. 13, 1511–1532

Thermophysical properties of undercooled liquid Co–Moalloys

X. J. Han and B.Wei{

Laboratory of Materials Science in Space, Department of Applied Physics,Northwestern Polytechnical University, Xian 710072, PR China

[Received 5 August 2002 and accepted in revised form 15 January 2003]

Abstract

Using electromagnetic levitation in combination with the oscillating droptechnique and drop calorimeter method, the surface tensions and specific heatsof undercooled liquid Co–10 wt% Mo, Co–26.3 wt% Mo, and Co–37.6wt% Moalloys were measured. The containerless state during levitation producessubstantial undercoolings up to 223K (0.13TL), 213 K (0.13TL) and 110 K(0.07TL) respectively for these three alloys. In their respective undercoolingranges, the surface tensions were determined to be 1895 � 0.31(T � 1744),1932 � 0.33(T � 1682), and 1989 � 0.34(T � 1607) mNm�1. According to theButler equation, the surface tensions of these three Co–Mo alloys were alsocalculated, and the results agree well with the experimental data. The specificheats of these three alloys are determined to be 41.85, 43.75 and44.92 Jmol�1 K�1. Based on the determined surface tensions and specific heats,the changes in thermodynamics functions such as enthalpy, entropy and Gibbsfree energy are predicted. Furthermore, the crystal nucleation, dendrite growthand Marangoni convection of undercooled Co–Mo alloys are investigated in thelight of these measured thermophysical properties.

} 1. IntroductionLiquids can be deeply undercooled below their freezing temperature TL, as has

been reported by Fahrenheit (1724) and Turnbull (1950). Owing to the metastablestate of undercooled liquid, the thermophysical properties, such as specific heat andsurface tension, are less understood than the normal liquid. Therefore, the thermo-physical properties in the undercooled regime are of great research interest (Fechtand Jonson 1988, Sommer 1990, Rajeswari and Raychaudhuri 1993, Egry et al. 1995,Wang et al. 2002). With the specific heat, one can calculate the enthalpy change�HLS, the entropy change �SLS and thus the Gibbs free energy difference �GLS

between the undercooled liquid and crystalline solid during the solidification process.Knowledge of C1

Pð�TÞ also gives some information about the velocity of crystalgrowth (Ivantsov 1947), the short-range order in undercooled liquid (Nelson et al.1982), the glass transition trend (Battezzati and Baricco 1987, Battezzati 1990) andthe glass transition temperature (Kauzmann 1948). The surface tension value and itstemperature dependence are essential for describing Marangoni flow, which affectsthe crystal–melt interface shape (Campbell et al. 2001) and contributes to the

Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2003 Taylor & Francis Ltd

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DOI: 10.1080/1478643031000090284

{Author for correspondence. Email: [email protected].

formation of the so-called ‘spoke pattern’ on the melt surface during the growth of asingle crystal by the Czochralski method (Przyborowski et al. 1995). Meanwhile, theviscosity of undercooled liquid can be theoretically predicted from the surface ten-sion. This together with the Gibbs free energy difference, makes the prediction ofnucleation rate as a function of undercooling possible.

So far, there have been only a few studies on the thermophysical properties ofundercooled liquid metals and alloys of high melting point because of theexperimental difficulties. At high temperatures, especially when the temperatureexceeds 1273 K, any physical contact with the container wall will induce immediatenucleation, and thus the metastable undercooling region is hard to access. In thiscase, the containerless processing technique and non-contact measurement methods,which will not disturb the metastable state of undercooling, have to be developed.Electromagnetic levitation is an elegant way to process liquid metals in a container-less way, which avoids surface contamination of the sample and can keep the under-cooling state for a long duration. Electromagnetic levitation in combination withoscillating-drop method can measure the surface tension of undercooled liquid, whileits combination with the drop calorimeter method provides possibilities to determinethe specific heat of undercooled liquid. In recent years, these two methods have beenwidely applied to determine the surface tension (Egry et al. 1995, 1996) and specificheat of undercooled liquid (Sommer 1990, Barth et al. 1993,Wang et al. 2002).

Co–Mo binary alloy is attracting research interest in that it is an importantconstituent subsystem of many Co-based superalloys, which can find many applica-tions in aerospace industry. This article deals with the measurement of the surfacetensions and the specific heats of highly undercooled Co–10 wt% Mo, Co–26.3 wt% Mo and Co–37.6 wt% Mo alloys. In addition, on the basis of the measuredthermophysical properties, the Gibbs free-energy difference and the relative nuclea-tion rate are calculated for a comparison with the calculated results by severalapproximate models for �GLS. Meanwhile, the dendrite growth and Marangoniconvection of the Co–Mo alloy system are investigated.

} 2. Experimental procedure

Experiments were performed with an electromagnetic levitation facility, as illu-strated in figure 1. A 30 kW high frequency generator operating at 300–450 kHz isused for power supply, which is connected to a conically shaped coil. The levitationcoil was placed in a vacuum chamber, which is evacuated to about 1 � 10�4 Pa andbackfilled with a gas mixture of Ar, He and H2. Co–Mo master alloys were preparedfrom 99.997% pure Co and 99.97% pure Mo in an ultrahigh-vacuum arc meltingfurnace and each sample has a mass of 0.8–0.9 g. During experiments, the samplewas levitated and melted by inductive heating. Then He with an addition of 25% H2,which flowed through liquid N2, was blown towards the sample in order to achievehigh undercooling. When the liquid was undercooled to an expected temperature, theflow rate of the blowing gas was adjusted to keep this undercooling for 15–40 s. Thetemperature was recorded with an infrared pyrometer. The surface oscillations aredetected with a photodiode focused on the top side of the sample through a spectro-scope. Having passed a low-pass filter with a threshold of 110 Hz and an amplifier,the signal was recorded with a sampling rate of 800 Hz to avoid aliasing. Afteranalogue-to-digital conversion with a sampling rate of up to 100 kHz, fast Fouriertransformation is performed off line in order to obtain the oscillation spectra.

1512 X. J. Han and B. Wei

The surface tension is calculated according to the famous Rayleigh (1987) equa-tion:

� ¼ 3pM!2R

8: ð1Þ

Under terrestrial levitation conditions the sample is distorted from its perfect sphe-rical shape owing to gravity and the existence of magnetic field. As a result, theexpected single Rayleigh frequency !R will shift and split into five peaks in thespectrum. Cummings and Blackburn (1991) considered the influence of the magneticfield and derived the following frequency sum rule to obtain the Rayleigh frequency:

!2R ¼ 1

5

X2

m¼�2

!22;m � 2!2

t ð2Þ

with

!2t ¼ 1

3ð!2

x þ !2y þ !2

zÞ; ð3Þ

where !2;m are the frequencies of the surface oscillation for the l ¼ 2 mode; !x, !y

and !z are the translational frequencies of the droplet’s centre of mass along thedirection of the axis x, y and z axes respectively and !t is the mean translationalfrequency, which is often 4–7 Hz according to the shape of coil, the mass of thedroplet and the levitation condition.

Figure 2 is a typical oscillation spectrum of a levitated droplet. Apparently, theRayleigh frequency is split into five peaks around 43 Hz. The lower-lying set of peakscorresponds to the translational frequency. Such spectra were obtained at differenttemperatures, the surface tension being determined according to equation (2).

Having recorded the oscillation signal, the power of rf induction heating wasswitched off and the sample was dropped into an adiabatic Cu calorimeter, whosemass is 1317.5 g. The details of the adiabatic calorimeter has been described else-where (Barth et al. 1993). Under adiabatic condition, after determining the tempera-ture change dT of the calorimeter block, the enthalpy change of the undercooledmelt is given by:

Thermophysical properties of undercooled liquid Co–Mo alloys 1513

Figure 1. Schematic diagram of the experimental facility for surface tension and specific heatmeasurement.

HðTÞ �Hð293 KÞ ¼ M

mCCu

PS dT þ C293KPS ðTe � 293Þ þQlost

m; ð4Þ

where HðTÞ is the enthalpy of the levitated droplet, H(293 K) the reference enthalpyat 293 K, M the mass of calorimeter block, CCu

PS the specific heat of calorimeter blockwhich is calibrated as 0.386 Jg�1 K�1; C293 K

PS the specific heat of the sample at 293 K,m the mass of the sample, Te the final equilibrium temperature of calorimeter blockand Qlost the heat loss of the droplet during its fall into the calorimeter. Consideringthe convection and thermal radiation, Qlost can be expressed as

Qlost ¼ A"h�ðT4 � T40 ÞtD þ AðT � T0Þ

ðtD0

�c dt: ð5Þ

Here, A is the surface area of the sample, "h the emissivity, � the Boltzmann con-stant, T the temperature before the sample dropping, T0 the ambient temperature, tDthe drop time, about 0.2 s, and �c the heat transfer coefficient of inert gas. When theenthalpy change of the sample is determined from equation (4), the specific heat ofthe sample can be derived directly from:

CLP ¼ o HðTÞ½ �

oT¼ o HðTÞ �Hð293 KÞ½ �

oT: ð6Þ

} 3. Results and discussion

Figure 3 is the upper left part of the Co–Mo binary alloy phase diagram, inwhich the locations of the selected compositions are indicated. Co–10 wt% Mo andCo–26.3 wt% Mo are single-phase alloys and Co–37.6 wt% Mo is a two-phaseeutectic alloy. During experiments, these three Co–Mo alloys were undercooled byup to 223 K (0.13TL), 213 K (0.13TL) and 110 K (0.07TL) respectively. Withintheir respective temperature ranges, the surface tensions and the specific heatswere measured.


Figure 2. A typical oscillation spectrum of levitated drop obtained by fast Fouriertransformation (a.u., arbitrary units).

3.1. Surface tension of undercooled Co–Mo alloysThe measured surface tensions of Co–10 wt% Mo, Co–26.3 wt% Mo and

Co–37.6 wt% Mo alloys are demonstrated in figure 4. Obviously, the surface tensionincreases continuously and linearly with decreasing temperature. Data analysisshows that

�CoN10 wt%Mo ¼ 1895 � 0:31ðT � 1744ÞmN m�1 ð7Þ

�CoN26:3 wt%Mo ¼ 1932 � 0:33ðT � 1682ÞmN m�1 ð8Þ

�Co�37:6 wt%Mo ¼ 1989 � 0:34ðT � 1607ÞmN m�1: ð9Þ

Figure 5 illustrates the relationship between the surface tension and the weightfraction of Mo in these three alloys. To reveal the changing tendency of the surfacetension with increasing Mo content, the surface tension data of liquid pure Co (Hanet al. 2002) is also added to figure 5. From the �NC curves at different temperatures,it is seen that the surface tension does not increase linearly with increasing the weightfraction of Mo but follows a two-order polynomial function relationship.

To evaluate the measured surface tension data, the Butler (1935) equation wasapplied to calculate the surface tensions of the Co–Mo alloys. Butler equation isexpressed for the surface tension � of any A–B binary liquid solution as follows:

� ¼ �A þ RT

AA

ln

�1 �NS

B

1 �NBB

�þ 1

AA

GE;SA ðT ;NS

BÞ �1

AA

GE;BA ðT ;NB

BÞ

¼ �B þ RT

AB

lnNS

B

NBB

þ 1

AB

GE;SB ðT ;NS

BÞ �1

AB

GE;BB ðT ;NB

BÞ ð10Þ

with

AX ¼ 1:901N1=30 V

2=3X ð11Þ

where R is the gas constant, T the temperature, �X is the surface tension of pureliquid X ðX ¼ A or B), N0 is Avogadro’s constant, VX is the mole volume of liquidX, NS

X and NBX are mole fractions of a component X in the surface phase and the

bulk phase, and GE;SX ðT ;NS

BÞ and GE;BX ðT ;NB

BÞ are partial excess Gibbs energies of X


Figure 3. Selected compositions and obtained undercoolings designated in the Co–Mobinary alloy phase diagram.

in the surface phase and in the bulk phase respectively. For GE;SX ðT ;NS

BÞ, differentmodels have been proposed, but they can be summarized as (Tanaka and Iida 1994)

GE;SX ðT ;NS

BÞ ¼ GE;BX ðT ;NS

BÞ; ð12Þ

where is a parameter corresponding to the ratio of the coordination number Z inthe surface phase to that in the bulk phase, ZS=ZB. is often chosen as 0.75 or 0.83


Figure 4. Measured and calculated surface tensions of liquid Co–Mo alloys versustemperature.

in the literature for liquid alloy. GE;BX ðT ;NB

BÞ can be derived from GEðT ;NBBÞ, which

is often obtained from database or calculation of the phase diagram. For the Co–Mobinary alloy system (Davydov and Kattner 1999),

GECoNMo ¼ XCoXMo½ð�7020:2 þ 43:036TÞ þ ð6523:4 þ 2:012TÞðXCo � XMoÞ�: ð13Þ

From equations (10)–(13), and with the parameters listed in table 1, the surfacetensions of Co–10 wt% Mo, Co–26.3 wt% Mo and Co–37.6 wt% Mo liquid alloyswere calculated. It should be mentioned that in the literature there are two differentsurface tension values for liquid Mo at its melting point, namely 1.915 and2.25 N m�1. During calculations, these two values are both used for calculating thesurface tension of Co–Mo alloys. The calculated results of these three alloys aresuperimposed on their respective �NT curves. Apparently, the determined surfacetension of Co–10 wt% Mo agrees well with the calculated result, and the experimen-tal results of Co–26.3 wt% Mo and Co–37.6 wt% Mo are a little lower, but within1% deviation of the calculated results using �Mo ¼ 1:915 � 0:0003ðT � Tm), andwithin 2% deviation of the calculated value using �Mo ¼ 2:25 � 0:0003ðT � TmÞ:

Having obtained the surface tensions of Co–10 wt% Mo, Co–26.3 wt% Mo andCo–37.6 wt% Mo alloys, the viscosities of these three alloys can be predicted fromthe following expression (Egry 1993):

¼ 16

15

�M

kT

�1=2

�; ð14Þ

where M is the absolute atomic mass and k the Boltzmann constant.


Figure 5. Surface tension of Co–Mo alloys versus Mo content.

Table 1. Thermophysical parameters of liquid Co and Mo used for the calculations.

Tm

(K)�

(kgm�3)�m

(N m�1)d�=dT

(Nm�1 K�1)

Co 1766 7760 � 0:988ðT � TmÞ a 1.875b �0.000 322b

Mo 2880 9350 � 0:5ðT � TmÞ c 1.915d ; 2.25c;d �0.000 30c

a Campbells (1983).b Han et al. (2002).c Iida and Guthrie (1993).d Keene (1993).

The predicted viscosities of these three alloys are illustrated in figure 6. To becompatible with the generally accepted formula for the temperature dependence ofviscosity, namely ðTÞ ¼ 0 exp ðA=RTÞ, a fit with an Arrhenius dependence ofviscosity on temperature is used. Data regression shows that

10 wt%Mo ¼ 1:948 exp

�10947

RT

�; ð15Þ

26:3 wt%Mo ¼ 1:967 exp

�11387

RT

�; ð16Þ

37:6 wt%Mo ¼ 2:035 exp

�11456

RT

�: ð17Þ

Apparently, the addition of the element Mo to a Co–Mo alloy will increase itsviscosity. Owing to the lack of the viscosity data in the literature, deviation of thepredicted viscosities from experimental data cannot be evaluated.

3.2. Specific heat of undercooled Co–Mo alloysFigure 7 presents the relationships between the enthalpy change and the

temperature of Co–10 wt% Mo, Co–26.3 wt% Mo and Co–37.6 wt% Mo alloys.Within their respective temperature ranges, the enthalpy change shows a good linearrelationship with the temperature. The linear regressions of the data for these threealloys are given by

½HðTÞ �Hð293 KÞ�10 wt%Mo ¼ �5303:4 þ ð41:85 1:51ÞT Jmol�1 ð18Þ

½HðTÞ �Hð293 KÞ�26:3 wt%Mo ¼ �7261:0 þ ð43:75 1:79ÞT Jmol�1 ð19Þ

½HðTÞ �Hð293 KÞ�37:6 wt%Mo ¼ �8174:4 þ ð44:92 1:67ÞT Jmol�1 ð20Þ

As an example, table 2 listed the parameters necessary to calculate the enthalpychange of Co–26.3 wt% Mo alloy. Apparently, the heat loss due to the convectionand thermal radiation during free fall of a droplet can account for 0.7–1.05% of theenthalpy change according to the different gas environment.


Figure 6. Viscosities of Co–Mo alloys derived from the measured surface tension.

Thus, according to equation (5), the specific heats of these three alloys are

C10 wt%MoPL ¼ 41:85 1:51 Jmol�1 K�1; ð21Þ

C26:3 wt%MoPL ¼ 43:75 1:79 Jmol�1 K�1; ð22Þ

C37:6 wt%MoPL ¼ 44:92 1:67 Jmol�1 K�1: ð23Þ


Figure 7. Enthalpy changes of Co–Mo alloys versus temperature.


Table

2.

Para

met

ers

nec

essa

ryfo

rca

lcula

ting

the

enth

alp

ych

ange

of

the

Co–26.3

wt%

Mo

alloy.

Tem

per

atu

reT (K

)

Sam

ple

mass

m (g)

Tem

per

atu

rerise

dT (K)

Bala

nce

tem

per

atu

reT

e

(K)

Hea

tlo

ssQ

lost

(%)

Enth

alp

ych

ange

HðT

Þ�Hð2

93

KÞ

(Jm

ol�

1Þ

1942

0.7

98

1.8

59

38

288.8

60.7

97

78034.5

1795

0.8

51.7

81

25

288.7

80.7

45

70132.7

1622

0.8

51.5

79

69

290.5

80.8

32

62285.3

1560

0.8

51.5

70

31

287.5

70.8

08

61821.3

1468

0.8

52

1.4

53

13

288.4

50.7

58

57057.2

1747

0.8

51.7

65

63

288.7

70.9

93

69690.1

1879

0.8

45

1.9

06

25

288.9

11.0

48

75740.9

1666

0.8

46

1.6

56

25

289.6

60.8

45

65601.2

1689

0.7

91.5

62

50

291.5

60.8

09

66290.7

1520

0.8

01.4

29

69

288.4

30.7

92

59821.8

Figure 8 illustrates the dependence of the specific heats of Co–Mo alloys on theweight percentage of Mo. To show the tendency more clearly, the specific heat ofundercooled liquid Co (Wang et al. 2002) is also added to figure 8. Apparently, thespecific heat increases linearly with increasing Mo content. Data analysis shows that

CPL ¼ 40:63 þ 0:115 53CMo J mol�1 K�1: ð24Þ

3.3. Thermodynamic functions �HLS, �SLS and �GLS

Having obtained the specific heats of undercooled liquid Co–Mo alloys, one canrigorously calculate the dependences of the enthalpy difference �HLS, entropy dif-ference �SLS, and thus the Gibbs free energy difference �GLS on the undercoolingaccording to their respective expressions:

�HLS ¼ �Hm �ðTm

T

ðCPL � CPSÞ dT ; ð25Þ

�SLS ¼ �Sm �ðTm

T

CPL � CPS

TdT ; ð26Þ

�GLS ¼ �HLS � T �SLS; ð27Þ

where CPS is the specific heat of the crystalline solid and �Hm the melting enthalpy,both of which can be determined conveniently with a commercial differential scan-ning calorimeter. Figure 9 illustrates the calculated �HLS, �SLS and �GLS values ofthese three Co–Mo alloys. The parameters used during calculation are listed intable 3. It can be seen that, with increasing undercooling, �HLS and �SLS decrease,whereas �GLS increases. Meanwhile, at the same undercooling level these threethermodynamic functions all increase with increasing Mo content.

Previously, owing to the lack of specific heat data on the undercooled melt,especially on alloy melts of high melting temperature, several models have beenproposed to estimate the Gibbs free energy difference. Turnbull (1950) assumed azero specific heat difference �CP between the undercooled liquid and the crystallinesolid which resulted in the linear approximation:


Figure 8. Specific heat of Co–Mo alloys versus Mo content.

�GLS ¼ �Hm �T

Tm

: ð28Þ

Jones and Chadwick (1971) suggested that �CP could be represented by a ‘suitable’value, which is mostly often chosen as the difference �Cf

P at the melting temperature.This yields the following relation:


Figure 9. Enthalpy change �HLS, entropy change �SLS, and Gibbs free-energy �GLS ofCo–Mo alloys versus undercooling.

�G ¼ �Hm

Tm

�T ��CfP �Tð Þ2

Tm þ T: ð29Þ

Following some empirical considerations, Singh and Holz (1983) approximated�GLS as

�G ¼ �Hm

Tm

�T7T

Tm þ 6T: ð30Þ

On the basis of the free-volume model of liquids, Dubey and Ramachandrarao(1984) proposed another expression for �GLS:

�G ¼ �Hm

Tm

�T ��CfPð�TÞ2

2T1 ��T

6T

� �: ð31Þ

To confirm whether these models can describe the thermodynamic functionsquantitatively, �GLS for Co–26.3 wt% Mo alloy was calculated as an example bythe aforementioned four models and compared with that calculated by its rigorousexpression on the basis of the experimentally determined specific heat data. Thecalculated results are shown in figure 10. Obviously, at small undercoolings, thesefour models agree well with rigorous expression for �GLS on basis of theexperimentally determined specific heat. However, with increasing undercooling,deviation occurs. Moreover, the higher the undercooling, the larger is the deviation.When undercooling reaches 600 K, the deviation amounts to 4.57%, 2.19%, 3.09%and 1.81% respectively for these four models. Of these approximation models, theDubey–Ramachandrarao model depicts �GLS the best.


Table 3. Thermophysical parameters used for calcuation of �GLS

�Hm

(Jmol�1)CPS

(Jmol�1 K�1)

Co–10 wt% Mo 16 784 37:57 þ 0:002 04TCo–26.3 wt% Mo 19 112 33:28 þ 0:005 73TCo–37.6 wt% Mo 20 929 29:13 þ 0:008 62T

Figure 10. Gibbs free energy difference �GLS of Co–26.3wt% Mo alloy calculated bydifferent approximate models.

3.4. Crystal nucleation rateWith the Gibbs free-energy difference calculated rigorously and the viscosity

derived from the measured surface tension data, we can go one step further tocalculate the nucleation rate according to the classical nucleation theory (Turnbull1956, Spaepen and Turnbull 1976):

I ¼ 1036

exp

�16p�3

3kBT�G2LS

f ð�Þ !

; ð32Þ

where is the viscosity, kB the Boltzmann constant, f ð�Þ the heterogeneous nuclea-tion factor given by f ð�Þ ¼ ð2 þ cos �Þð1 � cos �Þ2=4 and � the solid–liquid interfacialenergy, which can be calculated from the interfacial model of Spaepen (1975) andSpaepen and Meyer (1976).

In order to determine the influence of the deviation in �GLS between equations(28)–(31) and its rigorous expression on the nucleation rate, log ðIcal=IexpÞ is definedas the relative nucleation rate, where Ical is the nucleation rate based on �GLS

calculated by the aforementioned four models and Iexp is obtained by �GLS fromits rigorous expression using the specific heat determined experimentally. Obviously,the relative nucleation rate reflects the deviation degree of approximation modelsfrom the experiment. Figure 11 shows the calculated relative nucleation rate ofCo–26.3 wt% Mo alloy with a heterogeneous nucleation factor f ð�Þ ¼ 0:5 using theaforementioned approximation models. Since the Turnbull, the Jones–Chadwick,and the Dubey–Ramachandrarao models always show positive deviation from therigorous expression of �GLS, their relative nucleation rates are also positive. Withinthe undercooling range 0–400K, log (Ical=IexpÞ is larger than 4.2 and smaller than�3:7 for the Turnbull and the Singh–Holz models respectively. This means that thenucleation rates calculated from these two models deviate at least four orders andthree orders respectively of magnitude from the experiment. Even for the Dubey–Ramachandrarao model, log (Ical=IexpÞ > 1. This indicates that a minor deviation of�GLS will lead to a major deviation in the calculated nucleation rate. Therefore, inorder to obtain the accurate nucleation rate, the specific heat of undercooled liquidmust be measured quantitatively.


Figure 11. Relative nucleation rate of Co–26.3wt% Mo alloy calculated with differentapproximate models for �GLS.

3.5. Rapid dendrite growthApart from the crystal nucleation, the thermophysical parameters also influence

calculations in the rapid crystal growth under the high undercooling condition. Inorder to evaluate this influence, the crystal growth of Co–26.3 wt% Mo alloy wasinvestigated with the specific heat determined experimentally and that extrapolatedfrom the data of pure elements. The dominant crystal growth mechanism ofCo–26.3 wt% Mo alloy in undercooled melt is dendrite growth. So far, the Lipton–Kurz–Trivedi (LKT) (1987) and Boettinger–Coriell–Trivedi (BCT) (1987) have beenthe most successful models to predict the rapid dendrite growth from highly under-cooled liquid.

The LKT–BCT theory consists of two principal equations. The first equation isconcerned with the relationship between the partial undercoolings and the bulkundercooling.

�T ¼ �Tc þ�Tt þ�Tr þ�Tk: ð33ÞHere, �Tc is the solute undercooling, �Tt the thermal undercooling, �Tr thecurvature undercooling and �Tk the kinetic undercooling. Of these four partialundercoolings, �Tt is related to the specific heat CPL by

�Tt ¼�Hm

CPL

IvðPtÞ; ð34Þ

where Pt is the thermal Peclet number and IvðPtÞ the thermal Ivantsov function.The second equation relates the dendrite tip radius to the marginally stable

wavelength of perturbations at the solid–liquid interface.

R ¼ G=�*

ð�H=CPLÞPt�t � f2m 0l C0 1 � kð ÞPc=½1 � 1 � kð ÞLv Pcð Þg �c

; ð35Þ

where G is the Gibbs–Thomson coefficient, �* is the stability constant equal to1 4p2�

, Pc is the solute Peclet number, �t and �c are the thermal and solute stabilityfunctions, m 0

l ¼ mlð1 þ fke � kv½1 � ln ðkv=keÞ�g=ð1 � keÞÞ and ml are the effectiveand equilibrium liquidus slopes respectively, C0 is the alloy concentration, IvðPcÞis the solute Ivantsov function, kv ¼ ðke þ V=VdÞ=ð1 þ V=VdÞ is the effective solutedistribution coefficient, ke is the equilibrium partition coefficient, Vd ¼ DL=a0 is theatomic diffusive speed, DL is the diffusion coefficient and a0 is the characteristicdiffusion length.

Another parameter related to the specific heat is thermal diffusivity �, which isgiven by

� ¼ �

�CPL

: ð36Þ

� influences the crystal growth through the thermal Peclet number Pt:

Pt ¼VR

2�: ð37Þ

In the above two equations, � is the density, V is the dendrite growth velocity, R isthe dendrite tip radius and � is the thermal conductivity, which is closely related tothe electrical conductivity �e via the Wiedemann–Franz law (Iida and Guthrie 1993)

��e

T¼ 2:45 � 10�8 W�K�2: ð38Þ


Owing to the lack of data, the density and electrical conductivity of Co–Mo alloyshave to be extrapolated linearly from the data of pure elements. From equations(37)–(39), the thermal diffusivity of liquid Co–26.3 wt% Mo alloy is given by

�26:3 wt%Mo ¼ 3:5 � 10�7 þ 3:78 � 10�9 T m2 s�1: ð39Þ

Using the same method, the thermal diffusivities of Co–10 wt% Mo and Co–37.6 wt% Mo are calculated as

�10wt%Mo ¼ 3:73 � 10�7 þ 3:64 � 10�9 T m2 s�1: ð40Þ

�37:6 wt%Mo ¼ 3:18 � 10�7 þ 3:93 � 10�9 T m2 s: ð41Þ

The calculated growth velocity V of �-Co dendrite in undercooled liquidCo–26.3 wt% Mo alloy is shown in figure 12, in which the solid curve is the VN�Tcurve using the specific heat CPL ¼ 43:75 Jmol�1 K�1 determined experimentally,and the broken curve represents the calculated results using the specific heatCPL ¼ 38:4 Jmol�1 K�1 extrapolated linearly from the data of pure elements. Thephysical parameters used during calculation are listed in table 4. The growth velocityof �-Co dendrite determined experimentally (Wei and Herlach 1994) are also pre-sented in figure 12. Apparently, the growth velocity calculated fromCPL ¼ 43.75 Jmol�1 K�1 agrees with the experimental result much better than thatcalculated from CPL ¼ 38:4 J mol�1 K�1. This again indicates the great importance ofdetermining the thermophysical properties of undercooled metallic melts experimen-tally.

The effective solute distribution coefficient kv is a very important parameterduring the crystal growth. With the specific heats from different sources, kv duringthe rapid crystal growth of undercooled liquid Co–26.3 wt% Mo alloy is calculatedand shown in figure 13. Obviously, kv increases monotonically with increasingundercooling and, at an undercooling of about 80K, there exists an anomalouspoint, which corresponds to the transition of �-Co dendrite from solute-diffusion-controlled growth to thermal-diffusion-controlled growth. The calculated effectivepartition coefficient with the specific heat measured experimentally is a little higher


Figure 12. Experimentally determined and calculated growth velocity of �-Co dendrite inCo–26.3 wt%Mo alloy.

than that with the approximated specific heat. Moreover, this deviation is mainlyfocused on the stage of thermal diffusion controlled growth. This is understandable,since the specific heat is mainly related to the thermal undercooling �Tt.

3.6. Marangoni convectionMarangoni convection is a fluid flow phenomenon, which is driven by surface

tension differences along the surface. Together with other fluid flows, Marangoniconvection can influence crystal growth by modifying the boundary layers ahead ofthe solid–liquid interface. In a microgravity environment, Marangoni convection isthe main convection mechanism, thereby exerting great influences on the manufac-turing of the new materials in outer space.

The surface tension differences can arise from the differences in temperature or insolute concentration. The former is called thermal Marangoni convection, and thelatter solute Marangoni convection. These two Marangoni convections are mainlycharacterized by the thermal Marangoni MaT and the solute Marangoni numberMaC. These are defined as follows:


Table 4. Parameters used for the calculations of Co–26.3wt% Mo alloy

Liquidus temperature (K) 1682Heat of fusion (Jmol�1) 19 112Interfacial tension (Jm�2) 0.48Prefactor of diffusion coefficient (m2 s�1) 2:05 � 10�7

Activation energy for diffusion (Jmol�1) 67 640Gibbs–Thomson coefficient (Km) 4:02 � 10�7

Liquidus slope K (wt �1) �6.55Length scale for solute trapping (m) 7 � 10�10

Sound velocity (m s�1) 2000ke CS=CL

CL ¼ 30 422:5304 � 55:46394T þ 0:033 83T2

�6:895 59 � 10�6T3

CS ¼ 18 027:419 46 � 32:667 82T þ 0:019 83T2

�4:024 72 � 10�6T3

Figure 13. Calculated effective partition coefficient of Co–26.3wt% Mo alloy with differentspecific heats.

MaT ¼ rT L2

�

o�

oT; ð42Þ

MaC ¼ rCL2

D

o�

oC; ð43Þ

where rT is the temperature gradient, rC is the concentration gradient, L is thecharacteristic length of the system, D ¼ D0 exp ð�Q=RTÞ is the solute diffusioncoefficient, which can be approximated from empirical expressions, o�=oT is thesurface tension temperature dependence and o�=oC is the surface tension concentra-tion dependence.

The o�=oT values for Co–10 wt% Mo, Co–26.3 wt% Mo and Co–37.6 wt% Moalloys have been determined to be �0:31, �0:33 and �0:34 mN m�1 K�1

respectively. Compared with o�=oT , o�=oC is more complex. From figure 5, it canbe seen that � at any temperature can be well fitted with a two-order polynomialfunction of solute concentration CMo. For example, � at 1600 K can be fitted with

� ¼ 1933:8 � 0:3087CMo þ 0:04335C2Mo: ð44Þ

Thus, at a given temperature, o�=oC is a linear function of Mo content. The relation-ship of o�=oC, C and T is presented in figure 14. For clarity, only o�=oCNC curves at1400, 1600, 1800 and 2000 K are illustrated.

Assuming a temperature difference of 5K and a 0.5 wt% difference in Mocontent in a levitated droplet of 5 � 10�3 m in diameter, MaT and MaC arecalculated and presented in figure 15. Apparently, MaT increases and MaC decreaseswith reduction in temperature. Moreover, an increase in the Mo content increasesMaC whereas it decreases MaT at a given temperature.

An interesting question about the thermal and solute Marangoni convectionprocesses is which is the dominating mechanism. The answer depends on the thermaland solute profiles and the dependence of surface tension on the temperature andsolute content. If the temperature and the concentration gradients have differentsigns, these two Marangoni convection processes are additive; otherwise they actin opposition. As for the latter case, which dominates the convection at a givencoordinate r depends on the value of the combined surface tension gradient:


Figure 14. Concentration dependence of surface tension for Co–Mo alloys.

o�

or¼ o�

oT

oT

orþ o�

oC

oC

or: ð45Þ

The o�=or values of Co–10 wt% Mo, Co–26.3 wt% Mo and Co–37.6 wt% Mo atdifferent temperatures are presented in figure 16. Obviously, in a temperature range1400–2000 K, the o�=or of these three Co–Mo alloys were negative. This indicatesthat thermal Marangoni convection dominates the total Marangoni convection pro-cess.

For a sphere-like geometry of a liquid drop, the flow velocity induced byMarangoni convection is approximated by (Malmejac and Frohberg 1987)

VM ¼ ��

8; ð46Þ

where �� is the difference in surface tension between the ‘cold’ and the ‘hot’ part ofthe drop and is the viscosity of the liquid.

With the derived viscosity from the surface tension, the flow velocity induced bythermal Marangoni convection and solute Marangoni convection are calculated and


Figure 15. Thermal Marangoni number MaT and solute Marangoni numbers MaC versustemperature.

Figure 16. Combined surface tension gradient of Co–Mo alloys versus temperature.

illustrated in figure 17. Apparently, the flow velocity induced by Marangoni convec-tion is of the order of centimetres per second. Comparing the flow velocity inducedby Marangoni convection with the growth velocity of Co–26.3 wt% Mo alloy shownin figure 12, Marangoni convection is important at small undercoolingsð�T < 100 KÞ in the mass transport in front of the solid–liquid interface. Withincreasing undercooling, this role becomes smaller and smaller. At high undercool-ings ð�T > 200 KÞ, this role can almost be neglected.

} 4. Conclusions

The oscillating-drop technique and drop calorimeter method together with elec-tromagnetic levitation were used to measure the surface tensions and specific heats ofundercooled liquid Co–10 wt% Mo, Co–26.3 wt% Mo and Co–37.6 wt% Mo alloys.These three alloys were undercooled by up to 223, 213 and 110 K, respectively. Theirsurface tensions were determined to be 1895 � 0:31ðT � TLÞ, 1932 � 0:33ðT � TLÞ,and 1989 � 0:34ðT � TLÞmN=m. The surface tension of Co–Mo alloy was found tobe a two-order polynomial function of Mo content. Based on the Butler equation,the surface tensions of these three Co–Mo alloys were calculated, and the resultswere in good agreement with the experimental data. Within a wide temperatureregime around the liquidus temperature, the specific heats of these three alloys variedvery little and were determined to be 41.85, 43.75 and 44.92 Jmol�1 K�1 respectively.The specific heat of Co–Mo alloy is found to be a linear function of Mo content. TheGibbs free energy differences and the relative nucleation rates of Co–10 wt% Mo,Co–26.3 wt% Mo and Co–37.6 wt% Mo alloys were calculated according to theapproximate models of Turnbull, Jones–Chadwick, Singh–Holz and Dubey–Ramachandrarao and compared with experimental data. Comparison shows thatdeviation exists between these four approximate models and experimental results. Aminor deviation of the Gibbs free energy will cause a major deviation in the nuclea-tion rate. Furthermore, the dendrite growth and Marangoni convection of Co–Moalloy were investigated on the basis of the determined specific heats and surfacetensions. The growth velocity of �-Co dendrite in Co–26.3 wt% Mo alloy calculatedfrom the specific heat determined experimentally agrees with the experimental result


Figure 17. Flow velocities induced by thermal and solutal Marangoni convection processesversus temperature.

much better than that from the specific heat approximated linearly from the data onthe pure elements. The thermal Marangoni number MaT increases and the soluteMarangoni number MaC decreases with reduction in temperature, and an increase inMo content increases MaT but decreases MaC at a given temperature. Whether thethermal or the solute component is the dominating mechanism of Marangoni con-vection depends on the profiles of temperature and concentration and the surfacetension dependences of temperature and concentration. The flow velocity induced byMarangoni convection is of the order of centimetres per second, which indicates thatMarangoni convection exerts great influence on mass transport at small undercool-ings. However, at large undercoolings, this role can be neglected.

ACKNOWLEDGEMENTS

This work was financially supported by the National Natural ScienceFoundation of China (grants 59901009, 50101010 and 50221101), the HuoYingdong Education Foundation (grant 71044) and the Doctorate Foundation ofNorthwestern Polytechnical University. The authors are grateful to Dr N. Wang, DrC.D. Cao, Dr X. Y. Lu, Dr W. J. Xie and Dr W. J. Yao for helpful discussions.

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