UCRL-JC-118758 L-18866-1 PREPRINT

Prediction of Liquid Metal Alloy Radiant Properties from Measurements of the Hall Coefficent and

the Direct Current Resistivity

Mark A. Havstad Lawrence Livermore National Laboratory

Taiqing Qiu . Massachusetts Institute of Technology

This paper was prepared for submittal to the National Heat Transfer Conference

Portland, Oregon August 5- 9,1995

April 1995

Thisisa preprintof a paper intended for publication in a journal or proceedings. Since changes may be made before publication, this preprint is made available with the understanding that it will not be cited or reproduced without the permission of the author.

DISCLAIMER

This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or useful- ness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, OT favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

PREDICTION OF LlQUiD METAL ALLOY RADIANT PROPERTIES FROM MEASUREMENTS OF THE HALL COEFFICIENT AND THE DIRECT

CURRENT RESISTIVITY

Mark A. Havstad Laser Sciences Engineering Division

Lawrence Livermore National Laboratory Livermore, California

Taiqing Qiu Department of Mechanical Engineering Massachusetts Institute of Technology

Cambridge, Massachusetts

ABSTRACT The thermal radiative properties of high temperature solid

and liquid metal alloys are particularly useful to research and development efforts in laser cladding and machining, electron beam welding and laser isotope separation. However the cost, complexity, and difficulty of measuring these properties have forced the use of crude estimates from the Hagen-Rubens relation, the Drude relations, or extrapolation from low temperature or otherwise flawed data (e.g.. oxidized). We have found in this work that published values for the Hall coefficient and the electrical resistivity of liquid metal alloys can provide useful estimates of the reflectance and emittance.of some groups of binary liquid metal and high temperature solid alloys. - - - .

The estimation method computes the Drude free electron parameters, and thence the optical constants and the radiant properties from the dependence of the Hall coefficient and direct current resistivity on alloy composition (the Hall coefficient gives the free electron density and the resistivity gives the average time between collisions). We find that predictions of the radiant properties of molten Cerium-copper alloy, which use the measured variations in the Hall coefficient and resistivity (both highly nonlinear) as a function of alloy fraction (rather than linear combinations of the values of the pure elements) yield a good comparison to published measurements of the variation of the normal spectral emittance (a different but also nonlinear function) of cerium-copper alloy at the single wavelength available for comparison, 0.645 pm.

The success of the approach in the visible range is particularly notable because one expects a Drude based approach to improve with increasing wavelength from the visible into the infrared. Unfortunately, no other wavelengths or binary alloy systems have been identified for comparison. Details of the estimation method, the comparison between the calculation and the measured

emittance, and a discussion of what groups of elements may also provide agreement is given.

INTRODUCTION The thermal radiative properties of high temperature solid

and liquid metal alloys are needed for the development of many emerging technologies. Liquid lead-lithium alloy is being considered for the inner containment wall of fusion power reactors. Uranium-iron alloy films flow continuously in laser isotope separators under development for enriching uranium for reactor fuel. Gadoliniumcopper alloy may flow continuously in gadolinium enrichment plants (Gd feed to

-reador a r e s reduces the required-frequency of system shutdown for refueling). In laser cladding applications molten films solidify. on substrates to provide improved surface properties. For these processes, successful development to commercial feasibility requires modeling of the details of radiant absorption, emission and reflection. Unfortunately, the cost and complexity of obtaining a full angular, spectral and temperature dependent specification of the emitting, absorbing and reflecting properties of even a single binary alloy system are extreme and prohibabe to all but the most heavily funded efforts. Simple relations for the radiant properties in terms of more easily measured quantities are increasingly valuable as computer modeling of the details of radiant exchange becomes more able to assist process design and development.

Historically, the efforts to compute radiant properties from other quantities failed for two reasons. First. the surface conditions of the materials were far from ideal. For example, classic foundry operations with high temperature materials often involved highly oxidized (or otherwise contaminated) and rough surfaces. Theoretical relations for the radiant properties could not quantify these effects. Second, most or all efforts to predict radiant properties involved strictly classical physics. Quantum effects are

~

frequently present in the infrared and even more often in the visible. With the technologies noted above (and most others since the invention of the transistor), the critical processing occurs in a controlled environment (often high vacuum) without contamination which would be introduced by common ambient conditions. With laser and electron beam processing fresh, unoxidized metal is continually convected to the radiating and absorbing surface. In laser isotope separators, pure and smooth molten alloy films are continually condensing on square meters of surface in a device at temperatures in excess of 1000°C. Thus for liquid alloys in processing conducive to high purity the first cause for the failure of simple relations for radiant properties is eliminated.

The second problem noted above is also eliminated for some materials at elevated temperatures. The most conspicuous example being aluminum, where the interband absorption peak in the visible shrinks in magnitude as the metal is heated (Liljenvall, Mathewson and Myers, 1971 ). In the liquid state the peak is not observed and the classical free electron model can be fit to the data for the optical constants and the thermal radiative properties (Havstad, McLean and Self, 1993; Comins, 1972; Miller, 1969). In this work the Drude free electron model is applied to the cerium- copper system and computations of its optical constants and thermal radiative properties are compared to measurements. The objective is to begin to develop the capability to predict the radiant properties of the wide range of alloys whose properties are essentially unknown at present but are in use or contemplated in many new technologies. The present effort is encouraging for the agreement shown between the model and data but additional confirming work is required. The work has limited applicability because its classical approach neglects the quantum effects which complicate the radiant properties of many metals but there may be a significant group of metals where the molten alloys can be fitted to measured values of the Drude model parameters. .*

THEORETICAL BACKGROUND The Drude model can be considered as a special case of

the Lorentz model (Bohren and Huffman, 1983) for optical absorption by simple harmonic oscillators.

+ - - f - + + ..

m x + D x + K x = q E

Here m is the oscillator mass, D is a damping constant, K is the linear restoring force and qE is the driving force due to the electromagnetic radiation. When the restoring force applied to the oscillating charges of the Lorentz model is set to zero, the oscillators are unbound, Le. the valence electrons of the material are considered free. From this free electron concept is derived a two parameter model for the optical constants and radiative properties. In terms of the complex dielectric function one obtains (Miller, 1969):

E Nee T - = I - E, m E o ( l + a T )

2 2

2 2

(3) In this form the two parameters are Ne, the density of free

carriers, and T, the average time between collisions for the free electrons. The electron mass is m, e o is the permittivity of free space, e is the elementary charge and o is the frequency of radiation. The index of refraction n and the extinction coefficient k (the complex refractive index) are computed from the complex dielectric function:

2 1

(4)

Since the radiant properties are derived in terms of the Fresnel relations, the complex index form of- the opticat constants is more convenient to heat transfer. Similarly the Drude model can be formulated in terms of other pairs of parameters, an effective electron mass and the the conductivity for example. Lastly, the Hagen-Rubens relations, long applied in the absence of reliable radiant property data, are derived from the Drude result by setting the inertia term (electron mass) to zero in the Lorentz model (Strong, 1958).

The Drude model can be applied by computing Ne and T from measurements of the Hall coefficient and the direct current resistivity. The Hall coefficient is determined from an experiment with an electric field and a magnetic field applied at right angles to a material. A second electric field is produced in the third orthogonal direction (in order to maintain zero current in that direction) whose magnitude is dependent on the number density and mobility of the carriers present in the material (Bube, 1981). When only electrons arepresentfheHalLcoefficientis --- - - . -

- 1 H --

H - Nee

For both electrons and holes contributing to electrical conduction, the Hall coefficient depends on the carrier densities and mobilities:

(7) Mobility is p and hole density is Np. For materials with

electrical conduction due to multiple carrier types the computation of carrier densities is not simple. However, Hall coefficient measurements are preferable to direct radiant property measurements: the difficulties of maintaining and specifying surface condition are avoided and with the Hall work a single scalar is obtained rather than a property dependent on angle, wavelength and polarization. A survey

of experimental methods for the Hall coefficient of liquid which Hall coefficient, resistivity and radiant property data is metals and a listing of available results are given by Nyung available in the molten state and as a function of Lee and Lichter (Nyung Lee and Lichter. 1972). composition .

RESULTS resistance measurements: m rno The Hall coefficient and resistivity of molten cerium-

copper alloy measured by Busch et. al. (6usch et al., 1974) 2 -- is replotted in Figures 1 and 2. The form of both curves is similar to that obtained with other rare earth (RE) to copper

pNee Nee2

binary systems. For all the light rare earths and gadolinium with copper the resistivity at elevated temperatures is a weak function of temperature (and increases with temperature) and to a concentration of 30% to 40% at. RE the curves have nearly the same values (Guntherodt and Zimmermann, 1973). The curves for Hall coefficient vary more widely with element alloyed with copper (Takeuchi and Murakami, 1974). Several alloys of transition metals also have compositions with positive Hall coefficients (conduction dominated by positive carriers) as does the cobalt-cerium system (Bus&, et ai., 1974). For the present work the complexities of energy band effects as a function of alloy composition are ignored (electrons near the top of nearly full bands behave as holes (Ashcroft and Mermin, 1976)). We seek to use the above simple relations and obtain a moderate match to measured radiant properties. This is a much less strenuous requirement than a good match to measured complex index of refraction and yet Jess strenuous than a good match to measured complex dielectric function (since E 2nk). Future work should address energy band effects and the present assumption that for positive RH all the carriers are holes and for negative RH all the carriers are electrons.

The average time between collisions T is obtained from

x=--

(8)

Here d is conductivity and p is resistivity. For this work no types of scattering mechanisms or details are described. Scattering is assumed to be simply a function of alloy composition, as is the Hall coefficient.

From the relations given above and the assumption that the Hall experiment gives carrier density for only a single type of carrier one then can compute the entire reflective Properties Of a binary alloy system. With KirChoff's laws (when applicable) the emissive and absorptive radiant Properties are also computed in full. There are many metals for which this procedure is unsuccessful in any temperature

- Or spectral region. Interband and other quantum effects (e4. SPin-orbit splitting) often make a Drude approach invalid, palticularly at ambient and moderate temperatures and in the visible region (Nomerovannaya, Kirillova and Noskov, 1971). A quantum based approach (lchikawa and KobaYashi, 1966) has been shown to match ~eas~~rements of the reflectance and optical constants Of highly ordered PYrolitic graphite (HOPG) from ambient to 1300K and from 0.4 to 8.45 prn (Havstad, Schildbach and McLean, 1993). However, for the present, one would like to know under what conditions and with which materials the classical relations are The present work the calculation described above to measurements for the only material for

'

E n2 - k2, E '/eo

- 10 I I 1 I (?

1

- r - 0 4 - - v

_ _ ._... _.-._ ..... ~ .___.__-__. -.-...---...-.---.-.- ...... - -

-

-10 I I I I

0 20 40 60 80 100

Atomic percent cerium FIG. 1. THE HALL COEFFICIENT AS A fUNCTlON OF ALLOY COMPOSITION FOR LIQUID CERIUM-COPPER. 'THE MEASUREMENTS (BUSCH ET. AL., 1974) ARE REPLOTTED HERE.

160

140

120

c5

c: a s -2 100

80

.- c u)

v) .- +- t

60

2 40 n

20

3 0

.-

0 2 0 4 0 6 0 80 100 I 1 I 1 1

Atomic percent cerium FIG. 2. COPPER. THE MEASUREMENTS (SUSCH ET. AL., 1974) ARE REPLOTTED HERE.

THE DIRECT CURRENT RESISTIVITY AS A FUNCTION OF ALLOY COMPOSITION FOR LIQUID CERIUM-

Predictions (based on the method outlined above) of the normal spectral emittance at 0.645pm for cerium-copper alloy are compared to measurements made by (Ookko and Bautista, 1980) in Figure 3. All samples were liquid, sample temperature ranged from 1 1 18K to 181 1 K. The dashed

curve is the polynomial fit to their data reported by Dokko and Bautista:

= 0.2920 - 0.0650X + 0.2242~2 + 1.1381~3 - 1.4773~4 (9)

,

0 20 40 60 80 100

Atomic percent cerium FIG. 3. COMPARISON OF PREDICTION AND MEASUREMENT OF THE NORMAL SPECTRAL EMllTANCE OF LIQUID CERIUM-COPPER ALLOY AT 0.645 pm. THE DATA POINTS WITH ERROR BARS ARE THE MEASUREMENTS (DOKKO AND BAUTISTA, 1980) THE SOLID LINE IS A PREDICTION BASED ON THE DRUDE MODEL AND MEASUREMENTS OF THE HALL COEFFICIENT AND THE DIRECT CURRENT RESISTIVITY (SUSCH ET. AL.. 1974). THE DASHED LINE IS A POLYNOMIAL FIT TO THE MEASUREMENTS.

The mole fraction of copper is X. Taking the measured emittances as exact and error free, the computed results are in error by less than 3% out to 65% cerium and less than 10% to 53% cerium. At 41% cerium the computation is 16% low. Agreement is better again in the 5 to 25% cerium range. The comparison suggests that a simple Drude approach is satisfactory for this liquid alloy down to the visible (0.645 pm). Since the classical approach improves as photon energy declines and is usually invalid until wavelengths greater than 5 to 10 pm, the agreement here is probably either coincidence or indicative of agreement all the way out into the infrared. Experimental work to test for this agreement and obtain a more complete comparison with the gadolinium-copper system are in progress.

The normal spectral emittance of liquid Ce-Cu alloy as a function of wavelength is predicted in Figure 4. The magnitude and rate of decline of emittance with wavelength

are similar to that observed with many smooth clean metals. The curve for pure copper resembles that noted earlier for pure aluminum and that for the high Ce concentrations is typical of that observed for many transition metals (Edwards and de Volo, 1965). The dependence of the components of index of refraction on wavelength (Figs. 5 and 6) is also consistent with that observed for many common pure metals, at elevated and ambient temperature. Past about lpm, k is greater than n but their difference is decreasing. The difference between n and k is greatest for pure copper but for all compositions n and k are converging beyond 10 pm. For pure molten copper the complex dielectric function determined by an ellipsometric technique (Comins, 1972 and Miller, 1969) agrees well with a Drude approach to wavelengths as low as 0.6 pm, for a free electron density of 0.76 of the atom density.

t

40% Ce &-- 70% Ce -*c--

100% Ce -0--*

Wavelength, h urn) FIG. 4. PREDICTED DEPENDENCE OF THE NORMAL SPECTRAL EMllTANCE AT 0.645 pm VS. WAVELENGTH WITH ALLOY COMPOSITION AS A PARAMETER FOR MOLTEN CERIUM-COPPER.

45

40 - cu -8- 10% Ce -+--

35 20% Ce -R-- 3 0 % Ce ...x ..... 40% Ce 4.- t 30

r' -2 25 70% Ce -*-- 3 100% Ce -e--- * 20 e

15 X PJ 2 10 -

5

0 1 Wavelength, A (pm) 10

FIG. 5. PREDICTED DEPENDENCE OF THE INDEX OF REFRACTION VS. WAVELENGTH WITH ALLOY COMPOSjTlON AS A PARAMETER FOR MOLTEN CERIUM-COPPER.

4 5

4 0

y 35

.a, 30

% 25

= 20

.- E 15 10

5

0

i c

0

0 0

0

._

.- Y

cu 10% Ce 20% Ce 30% Ce 40% Ce 7 0 % Ce

1 Wavelength, 3c (pm) l o FIG. 6. PREDICTED DEPENDENCE OF THE EXTINCTION COEFFICIENT VS. WAVELENGTH WITH ALLOY COMPOSITION AS A PARAMETER FOR MOLTEN CERIUM-COPPER.

The computational results given above can be summarized by the dependence of free electron density and average time between collisions on alloy fraction, Figures 7 and 8. Because the Hall coefficient goes through zero at -18.8% at. cerium the Drude model gives the non-physical result of infinite free electron density at this composition. At all other compositions more than -0.20% at. away from this singularity the model gives results which are both physically reasonable and in approximate agreement with existing

data. One interpretation of the RH=O condition is rather than a state of zero free carriers there is a balance of hole and electron conduction (eqn. 7) which causes the measured Hall coefficient to go to zero. The assumption of a single carrier type may be the reason the predictions are poorer fo: compositions near 19% at. cerium in Fig. 3. Analysis including the mobility in eqn. 7 or a more detailed treatment of band effects has not been done.

I I I I f 0 20 40 60 80 100

Atomic percent cerium

I I I I I f 0 20 40 60 80 100

Atomic percent cerium FIG. 7. FREE CARRIER DENSITY AS A FUNCTION OF ALLOY COMPOSITION FOR LIQUID CERIUM-COPPER ALLOY.

h

W v)

CD 0

X r

Y

.-

I I I I

-

-

-

-

I I I

.- E c

25

20

15

10

5

0

Atomic percent cerium

FIG. 8. AVERAGE TIME BETWEEN COLLISIONS AS A FUNCTION OF ALLOY COMPOSITION FOR LIQUID CERIUM- COPPER ALLOY.

The dependence of collision time on alloy fraction shows that small amounts of cerium rapidly change the alloy from a very good conductor to an average or poor one. The minimum at 18.8% cerium is a consequence of Ne in the denominator of the expression for T (eqn..-8). Both the radiant property data and the Drude model computations indicate that the alloy is essentially unchanged from 40% cerium to 100% cerium. Less-radiant property data will be required in the cerium rich range than in the copper rich range to confirm the usefulness of the Drude approach for this binary system.

The carrier density, the collision time, the resistivity and the conductivity do not vary linearly with composition. Computed values of the optical constants and radiative properties from interpolation between values for the two pure constituents give results that are very different from those shown here and are quantitatively and qualitatively in error. This condition, the need for measurements of the Hall coefficient and the resistivity as a function of composition, makes the described approach less easily applied but also shows that the Drude approach is a simplified interpretation of the interaction between this liquid alloy and electromagnetic radiation.

-

CONCLUSION The similar dependence of both Hall coefficient and

resistivity on alloy composition for several rare earth metals

with copper may indicate that the approach described here works well for other high temperature alloys of copper and the rare earths. The similarities among the highly conducting metals (Au, Ag, Cu, AI) suggests these alloyed with rare earths are also well fit. For low concentrations of transition metals in copper, gold, silver and aluminum the method described above may also provide good estimates. These are particularly important to a range of emerging technologies, laser cladding principally, so that further work with these materials is warranted. Measurements under representative pure conditions at elevated temperatures-are - - - - -

required. The theory presented here is clearly a simplification of operative absorption, emission and reflection mechanisms but the agreement with the measurements warrants more attention.

ACKNOWLEDGEMENTS The authors wish to thank Janice Butler for many

sessions of careful and skilled searching a: the library in the science and engineering databases.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.

ST94-20QlIMAH:dlb L-18866-1

REFERENCES Ashcroft, N.W. and Mermin, N.D., 1976, "Solid State

Physics," Saunders College, Philadelphia. Bohren, C.F. and Huffman, D.R., 1983, Absorption and

Scattering of Light by Small Particles, Wiley-lnterscience, New York.

Bube, R.H., 1981, Electrons in Solids, Academic Press, New York, Second Edition.

Busch, G., Guntherodt, H.J., Hauser, E., Kunzi, H.U., and Muller, R., 1974, "Electronic Properties of Liquid Rare Earth Metals and Their Alloys," Proc. Rare Earth Res. Conf., 11 th, Haschke, J.M. (Editor), NTIS, Springfield, VA, pp. 1003- 1013.

Comins, N.R., 1972, "The Optical Properties of Liquid Metals," Phil. Mag. 25, pp. 81 7-831.

Dokko, W. and Bautista, R. G., 1980, "The Normal Spectral Emissivity of Liquid Cerium-Copper Alloys," Metallurgical Transactions 6 1 1 B, pp. 309-31 2.

Edwards, D.K. and De Volo, N.B., 1965, "Useful Approximations for the Spectral and Total Emissivity of Smooth Bare Metals," Advances in Thermophysical Properties at Extreme Temperatures and Pressures, ASME, New York.

Guntherodt, H.J. and Zimmermann, Z., 1973, "d- Resonance Scattering in the Resistivity of Liquid Cu-La, Cu- Nd and CuGd Alloys," Phys. kondens. Materie 16, pp. 327- 333.

Havstad, M.A., McLean I I , W., and Self, S.A., 1993, "Apparatus for the Measurement of the Optical Constants

and Thermal Radiative Properties of Pure Liquid Metals from 0.4 to 10 pm," Rev. Sci. Instfum. 64 (7), pp. 1971 -7 978.

Havstad, M.A., Schildbach, M., and McLean I f , W., 1993, "The Optical Constants and Spectral Specular Reflectivity of Highly Oriented Pyrolytic Graphite (HOPG),"ASM€ HTD-Vol.

Ichikawa, Y.H. and Kobayashi, K., 1966, "Optical Properties of Graphite in the Infrared Region," Carbon 3, pp.

Liljenvall, H.G., Mathewson, A.G., and Myers, H.P., 1971, "The Temperature Dependence of the Optical Constants of Aluminium," Solid State Communication 9, pp. 169-1 70.

Miller, J.C., 1969, "Optical Properties of Liquid Metals at High Temperatures," Phil. Mag. 20, pp. 11 15-1 132.

Nomerovannaya, L.V., Kirillova, M.M., and Noskov, M.M., 1971, "Optical Properties of Tungsten Monocrystals," Soviet

Nyung Lee, 0. and Lichter, B.D., 1972. "Relation Between Thermodynamic and Electrical Properties of Liquid Alloys," ch. 3 in Liquid Metals, Beer, S. (editor) Marcel Dekker Inc., New York.

Strong, J.D., 1958, Concepts of Classical Optics" Freeman and Co., San Francisco.

Takeuchi, S. and Murakami, K., 1974, "On the Hall Coefficient of Liquid Copper Alloys," Journal of the Japan Institute of Medals 38 ( l ) , pp. 29-36.

244, pp. 85-95.

401 -406.

Physics JETP 33, pp. 405-409.

NOMENCLATURE

D

e

damping constant (newton . second/meter)

elementary charge (1.6 x 10-1 9 coulomb)

electric field vector (volts/meter)

extinction coefficient (-)

linear restoring force (newton/meter)

mass of electron (kilogram)

index of refraction (-)

number density (#/meters)

q electronic charge (coulomb)

RH Hall coefficient (meter3lamp-second)

X displacement from equilibrium (meter)

X

I

.-

mole fraction of copper (-)

GREEK & emissivity (-)

h wavelength (meter)

P T

0 conductivity (l/(ohm meter)

P carrier mobility (meters2lvolt second)

direct current resistivity (ohm . meter)

average time between collisions (second)

.- frequency (radianlsecond)

real part of complex dielectric function (faradlmeter) E

* E' imaginary part of complex dielectric function (faradlmeter)

EO permittivity of free space (8.85 x 10-12 faradlmeter)

SUBSCRIPTS

e electrons

n normal

0

P holes

free space value or resonant value