UCRL-JC-128692 PREPRINT

Analysis of the E-Beam Evaporation of Titanium - and Ti-6Al-4V

K. W. Westerberg, T. C. Meier, M. A. McClelland, D. G. Braun, L. V. Berzins, T. M. Anklam, J. Storer

This paper was prepared for and presented at the Electron Beam Melting and Refining State of the Art 1997 Conference

Reno, Nevada October 5-7,1997

February 11,199s

This is a preprint of a paper intended for publication in a jou

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 usefulness 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 product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or 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.

ANALYSISOFTHEEBEAMEVAPORATIONOF TITANIUM AND Ti-6Al-4V *

K. W. Westerbergt T. C. Meier

M. A. McClelland D. G. Braun L. V. Berzins

T. M. Anklam

Lawrence Livermore National Laboratory P.O. Box 808

Livermore, CA 94550 U.S.A.

J. Storer

3M 31211 Center, Bldg. 60-1NOl

Saint Paul, MN 55144 U.S.A.

Abstract

An experimental and finite element analysis was performed for the electron-beam evaporation of Ti and Ti-6Al-4V from a bottom-feed system. The bulk evaporation rate was measured by feed consumption, and the pool elevation was held constant by adjusting the feed rate in a closed-loop control system. The instantaneous titanium and aluminum evaporation rates were determined by laser absorption in the vapor plume. Water temperature rises in cooling water circuits provided heat flows, and post-run cross sections revealed the location of the solidification zone. The MELT finite element code was applied to model the steady-state two-dimensional fluid flow and energy transport in the rod.

There was good agreement between model and measured values of the heat flows and solidification boundaries for Ti. Measured bulk evaporation rates were similar for Ti and Ti-6-4 with greater variation observed for the Ti values. The model evaporation rates were higher than the measured values, but a similar linear dependence on e-beam power was observed in all cases. In a Ti-6-4 evaporation experiment with steady process conditions, laser absorption measurements showed much larger fluctuations in the evaporation rate for Al than Ti.

‘Work performed under the auspices of the U S Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405ENG-48. The authors wish to acknowledge the support of DARPA through the vapor phase manufacturing initiative (administered by Steve Wax) for supporting a portion of this work.

+Author to whom correspondence should be addressed

Ceramic fiber

Coated fiber 4

Lay-up and consolidate \W W J

Electron beam )- 41 ,/~~~pdo~oo,

13 p Water-cooled crucible

I-P- Solid ingot

Metal matrix -

composite

t Pusher

Figure 1: Ebeam deposition of metal on fibers is used to fabricate metal matrix composites.

1 Introduction

EBeam Physical Vapor Deposition (EBPVD) of refractory metals is a key step in the fabri- cation of metal matrix composites [l]. In a typical process, metal vapor horn a bottom-feed, e-beam evaporator is deposited on fibers advancing through a vapor plume (see Figure 1). In the evaporation of an alloy such as Ti-6Al-4V (Ti-6-4) from a single rod, it is impor- tant to rapidly approach steady-state and operate the feed smoothly in order to minimize fluctuations in the vapor composition. In addition, it is desirable to determine operat- ing conditions that maximize throughput. A detailed understanding of the e-beam vapor source would further these goals.

For the evaporation system of Figure 1, an e-beam impinges on the top surface of a rod confined in a water-cooled collar. Liquid metal is evaporated from the surface of a pool at the top of the rod. As material evaporates it is replenished by advancing the rod from below.

We are developing the MELT finite element code for fluid and energy flow in the ingot to calculate quantities such as evaporation rates, heat flows, and pool boundary locations [2, 31. We plan to use the MELT models to optimize source operating parameters such as e- beam power and footprint and crucible geometry. Modeling of these systems is a challenge due to the presence of moving phase boundaries which are coupled to the fluid flow and energy transport. In addition, the fluid 5ow is intense, interfacial behavior is complex, and material properties are difficult to determine.

Numerical models have been developed for metal processing systems with similar fea- tures such as crystal growth, welding, and casting. Numerical studies involving the explicit tracking of interfaces are reviewed by Westerberg et al. [2] and include the finite element analysis of Sackinger et al. (41 for Czochralski crystal growth and the finite difference

Vapor 4 , Electron beam

Skip electrons K I /

Thermal radiation t

I Buoyancy 1

and capillary -++ (Marangoni) 1 driven flow

I I

Melting point I I I

Symmetry axis

Solid

f

Contact line

Contact resistance

-Thermal radiation

r Contact resistance

r- t

Pusher

Figure 2: MELT model for e-beam evaporation of rod-fed metal.

approach of Lan and Kou [5] for floating-zone crystal growth. In an analysis of e-beam evaporation, Guilbaud [S] analyzed the 5uid flow and free sur-

face deformation of cerium in a two-dimensional trough. Westerberg et al. [2, 7] developed two-dimensional planar and axisymmetric models for the steady e-beam evaporation of alu- minum. Rotating spines were employed with a finite element method to track solid-liquid and liquid-vapor interfaces along with a tri-junction.

An initial experimental and numerical investigation was performed for the e-beam evap- oration of Ti from a bottom-fed vapor source [3]. The e-beam power and diameter of the ring footprint were varied. Measurements of the evaporation rate were made by feed con- sumption and laser absorption. A two-dimensional, steady-state finite element model was developed for 5uid flow and energy transport in which vertical spines were employed to track the solid-liquid and liquid-vapor interfaces. For the large diameter ring, the model, feed consumption, and laser absorption measurements of the evaporation rate were all in very good agreement. For the small diameter ring, the model evaporation rates were higher than the feed and laser values. The finite element model captured the important features of the pool boundary shape from the ingot cross section.

We extend this initial work to include multiple experimental runs with both Ti and Ti-6-4. The pool level is maintained with a feedback control system incorporating a video measurement of the pool level. Rotating, diagonal spines are incorporated in the finite element model to better track the deep pools formed by the “ring” e-beam profiles. Com- parisons are made between the model and feed-based evaporation rates for a range of e- beam powers and the large-ring pattern. Finally, in an evaporation experiment with Ti-6-4, fluctuations in the Ti and Al evaporation rates are characterized using laser absorption.

2 Melt Evaporation Model

A steady-state, axisymmetric model is used for the Auid 5ow, energy transport, and evap- oration of metal from a bottom-fed cylindrical ingot. The important features are described here, and the mathematical details for similar evaporation systems are described elsewhere [2, 71. A single species model is applied to both the pure metals and alloys. The model domain includes solid, liquid, and vapor phases (see Figure 2). The vapor phase is treated by applying boundary conditions at the top surface of the pool as described below. The buoyancy and capillary-driven flow is governed by the mass and momentum equations for incompressible flow. The energy balance in the liquid phase includes accumulation, con- vection, and conduction terms. A heat conduction model is employed in the solid region. Bulk material properties are treated as uniform in each of the material phases, except for the buoyancy term.

Conditions for mass, momentum, and energy transport are applied at the boundaries of the liquid and solid regions. The average level of liquid in the pool is specified. Since liquid circulation rates are large compared to evaporation rates, material 5ow rates through the top surface are taken to be negligible in treating the 5ow field.

The momentum balance at the liquid-vapor interface includes the normal stress result- ing from surface tension and the tangential stress generated by thermal gradients in the surface tension (Marangoni effect). The Marangoni effect drives liquid from the hot beam area to colder regions on the top surface. The momentum balance also incorporates the thrust of the departing vapor which is approximated as one-half of the vapor pressure. Liquid does not penetrate any boundaries, and the no-slip conditions are applied at solid surfaces.

At the axis of symmetry, the liquid-vapor interface is horizontal, and the elevation of the contact line is specified at the crucible wall. The latter condition approximates flow behavior at the crucible lip. For the case in which the feed rate exceeds the evaporation rate, the liquid overflows onto the top of the crucible, and the liquid level remains approximately constant near the lip.

The incident e-beam energy 5ux is given by a Gaussian distribution. The formation of skip electrons is included by taking the absorbed energy flux to be a fixed fraction of the incident flux. At the top surface of the pool, energy radiates to the cold surroundings fol- lowing a T4 expression. The heat loss associated with the phase change includes the latent heat and a translational contribution associated with vapor flow. The local evaporation flux is calculated using a Langmuir expression. The lower boundary of the pool is taken to have a uniform temperature which is the melting-point value for a single component material and an immobilization temperature for an alloy.

Thermal contact resistance between the liquid and the crucible wall is represented with a single heat transfer coefficient. The heat loss from the solid ingot to the crucible wall has

’ .1 .

.2 \

.l \ .

Figure 3: Finite element mesh deforms using pivoting and stretching spines.

contributions from contact and thermal radiation. The thermal contact resistance between the bottom of the ingot and the pusher is also modeled with a heat transfer coefficient. In order to reduce the effort associated with remeshing, the finite element domain is kept at a length shorter than the ingot. The lower portion of the ingot is modeled as a cooling fin since the temperature varies primarily in the axial direction. An effective heat transfer coefficient at the lower boundary of the finite element domain accounts for the thermal behavior in the lower section of the ingot.

The field and boundary equations are discretized using a Galerkin finite element method. The mesh deforms to track liquid-vapor and solid-liquid interfaces. For the deep pools of this investigation, we use a new type of mesh with rotating spines which makes use of earlier developments [2, 71. Coarse meshes are shown in Figure 3. There are two sets of rotating spines. Each spine in the first set (nos. l-3) begins at a common anchor point above and to the left of the pool. It enters the liquid along the axis of symmetry, bends at the solid-liquid interface, and proceeds to base points along the lower and right boundaries of the solid region. The first and last spines (nos. 1 and 3) in this set pass through the axis of symmetry at the junctions with the solid-liquid and liquid-vapor interfaces, respectively. Each spine in the second set (nos. 4,5) begins at an anchor point above the top of the ingot. It passes through the liquid-vapor interface, bends at the solid-liquid interface, and ends at anchor points along the outside radius of the ingot. The last spine (no. 5) in this set is vertical and passes through the contact line and liquid-solid interface at the crucible wall. As in the earlier studies, the interfacial mesh points move with the interfaces while the interior points move along the spines.

This meshing strategy helps with the tracking of nearly vertical liquid boundaries en- countered in deep pools. The use of vertical spines leads to excessive mesh deformation. The diagonal rotating spines of this study provide for less mesh deformation since spines are more orthogonal to the solid-liquid interface.

2

r 1.8 2 E 1.6 .P . . . . . ii 1.4 B p” 1.2

Ill) Ill\ Ill\ IllI IllI IllI III- -1.6 - s

I I I III III ! I I I I I I I I I I I I I I

11.2 11.2 11.4 11.4 11.6 11.6 11.8 11.8 12 12 12.2 12.2 12.4 12.4 12.6 12.6 Time [h] Time [h]

Figure 4: Feed controller maintains level within fl [mm] of setpoint for Ti-6-4 run 3.

As in earlier studies, the MELT computer code is used to solve the locations of the two interfaces simultaneously with the flow and temperature fields using the Newton- Haphson method. All results shown in this work are steady-state, but some were found by integrating from one steady-state point to another point using the false-transient method [S]. In this application, the backward Euler method was used with large time steps and error tolerances to damp strong transients and progress rapidly through the parameter space to the point of interest. This approach was particularly effective for traversing regions of the parameter space for which steady-state solutions were not available using traditional parameter continuation methods.

3 Experimental

Evaporation experiments were conducted in the e-beam Evaporation Test Facility (ETF) at Lawrence Livermore National Lab. The ETF chamber has a bottom-fed crucible with a side-mounted 50 [kW] e-beam gun which was operated at 33.3 [kV]. A beam spot of approximately 0.5 [cm] diameter is swept in a circular pattern at 3 (kHz] to form a ring pattern on the top of a 7.62 [cm] (3.0 [in]) d iameter feed rod of commercially pure Ti or Ti-6-4. A vapor enclosure consisting of water-cooled copper panels captures the vapor above the crucible and has openings to accommodate the e-beam, diagnostic laser beams and camera views. All water cooling circuits in the chamber have thermocouples and flow meters allowing heat flows to be computed using standard calorimetric methods.

Two pin-hole cameras with argon gas bleed are mounted inside the vacuum chamber to view the vapor source. A high angle view of the vapor source is used to position and monitor the beam on the ingot top, and a low-angle view is used to measure the liquid pool height.

The pool level is determined by applying an edge detection routine to images captured with the low-angle camera. In these evaporation experiments. the liquid pool surface was run at a high level with the top surface bulging above the crucible lip. This is possible given the relatively high surface tension and low density of titanium and the Ti-6-4 alloy. The liquid pool level is maintained by adjusting the feed rate using a feedback control system involving Internal Model Control [9]. In Figure 4, the feed rate, pool level, and associated setpoint are plotted versus time for the initial stages of a Ti-6-4 evaporation experiment. The e-beam power is constant, and the level controller is able to maintain the pool level within fl [mm] of the setpoint. This feedback control system makes it possible to quickly start the system and bring it to steady-state.

Evaporation rates are determined from feed consumption and laser sensor measure- ments. For the feed method, the advance of the feed ingot is recorded over a period of steady operation to determine the evaporation rate. Laser beams are passed through the vapor plume to measure the evaporation rates of titanium and aluminum simultaneously. For the titanium sensor, the laser beam is tuned to excite an atomic transition in titanium. Using Beer’s law and a model for the vapor plume, the absorption can be converted to an evaporation rate [lo]. Because aluminum is a strong optical absorber, the standard laser absorption techniques cannot be applied because the signal saturates. A laser-based phase-delay technique has been developed to monitor the aluminum evaporation rate Ill].

4 Comparison of Results

4.1 Conditions

Evaporation measurements were made in five runs with titanium and three runs with Ti- 6-4. In these runs, the e-beam ring diameter was approximately 4.7 [cm] and the power was varied from 38 to 46 [kW]. Each run was divided into periods in which conditions were held constant. Approximately 2.5 [cm] (0.5 [kg]) f o metal was fed in each period. At the end of each run the e-beam was quickly shut off to preserve the shape of the solid-liquid interface. A post-run sectioning revealed the shape of the solid-liquid interface during the final period of each run.

4.2 Properties

For the modeling of the these experiments with the MELT code. the physical properties of Ti and Ti-6-4 are assigned values for Ti (see Table 1). This is believed to be a reasonable approximation since Ti is the dominant constituent and V generally behaves similarly to Ti. All properties are assigned their physical values, except the viscosity which is a factor of 15 higher to accommodate the strong convection and meshes of this study.

Except for the e-beam footprint, the geometric parameters from the final period of Ti Run No. 3 were taken to be representative for the Ti and Ti-6-4 experiments of this study. The e-beam ring diameter and half-width were 4.8 and 0.24 [cm], respectively, for Run No. 3 and 4.7 and 0.3 [cm] for all other cases. As a result of operating the pool at a high level there was metal overflow onto the crucible by the end of each run. From the

Table 1: Physical properties of Ti metal.

Property

Melting point Molecular weight

Liquid properties Viscosity Density Coeff. therm expansion Thermal conductivity Heat capacity Surface tension Surface tension (temp. deriv.) Vapor pressure

Pl E

Heat of vaporization Thermal emissivity

a0

a1

Skip fraction

Solid properties Thermal conductivity Thermal emissivity

Value Units

1667 “C 0.0479 kg/gmol

3.200 x~O-~ 4130

5.570 x1o-5 31.0

8.842 x102 1.65

-2.400 x10-” PI~~P(EIW

2.317 x 101’ -4.355 XlOj 9.092 x106

UIJ + UlT 0.1810

2.533 x10-” 0.22

kg/m-s kg/m3 K-’ W/m-K J/kg-K N/m N/m-K N/m2 N/m2 J/gmol J/k unitless unitless K-1 unitless

31.0 W/m-K 0.4 unitless

metallographic cross section of the Ti Run No. 3 ingot, the level of the contact line and the average level of the pool were 0.55 [cm] above the crucible lip.

The heat transfer coefficient for the liquid in contact with the crucible is 6000 [W/m2-K] and for the pusher is 100 [W/ m2-K]. These values are consistent with the range of values reported for liquid and solid metals in contact with copper chills [12]. The heat-transfer coefficient for the solid contact with the crucible decreases linearly with temperature from the liquid value at the melting point to zero at 300 [“Cl below the melting point. The cooling fin model for the lower section of the ingot also gives an effective heat transfer coefficient of approximately 100 [W/ m2-K] for the lower surface of the shortened model domain. As the length of the model domain increases, the increase in the axial conduction resistance is balanced by the increase in ingot area at the outside radius and an associated increase in heat losses by thermal radiation.

r km1 -3 -2 -1 0 1 2 3

I I 1 I I I I

MELT model (fraction incident power)

38.5 kW

Tmax = 2419°C \

Measured heat flows (fraction- incident power)

Skip: 22.0% Vaporization:13.0% Radiation: 6.0% Shields total:41 .O%

Crucible: 59.2% C

J ‘r Pusher: 0.8% Pusher: O.% Total 101.0% Total 103.6%

Experiment /

hields: 39.7%

t Crucible: 63.9%

Figure 5: Comparison of model and measured heat flows and pool boundaries for Ti Run No. 3, final run period.

4.3 Cross section

For the last period of Ti Run No. 3 the predicted stream function, temperature contours, interface locations, and heat flows are shown in Figure 5 along with the measured heat flows and frozen solid-liquid interface location. The model captures two important features in the measured shape of the solid-liquid interface. The predicted and measured depths of the pools agree to within 2.3% as referenced to the crucible lip. The second feature is the the inflection caused by the counter-rotating flow cells. The cells near the symmetry axis are formed by Marangoni forces. Surface liquid flows horn the hot beam impact area with low surface tension towards the cooler symmetry-axis region with higher surface tension. Cooling at the symmetry axis is the result of thermal radiation to the surroundings. The deep pool is formed as the warm liquid is transported downwards along the symmetry axis.

For the final period of Ti Run No. 3, the heat flows predicted by the MELT model are compared with the measured values in Figure 5. The flow and temperature fields calculated by the model are well resolved since the incident e-beam power differs from the total heat removed by less than 1.0%. These same quantities agree to within 3.6% for the experiment. Approximately 40% of the incident energy is lost from the top of the

MELT model

0 Ti run 5 A Ti-6-4 run 1 - m Ti-6-4 run 2 l Ti-6-4 run 3

0 I I I I I I I Ii I I I I I I I I I I * 30 35 40 45 50

E-beam power [kVVj

Figure 6: Comparison of MELT model and measured bulk evaporation rate by feed ron- sumption for Ti and Ti-6-4.

ingot, and the difference between the predicted and measured energy flow is 1.3%. The calculated and experimental energy flows from the side of the ingot agree to within 4.7%. The predicted and measured heat flows to the pusher are both less than 1% of the incident e-beam power. These results suggest that the MELT model provides a good description of the energy transport in the ingot.

4.4 Comparison of evaporation rates

For the five Ti and three Ti-6-4 experimental runs, feed-based evaporation rates are plotted versus e-beam power in Figure 6. The results reported are for an e-beam ring of one diameter. Each measurement run is represented by a separate line, or in the case of two of the runs, a single point. For Ti, the run-to-run differences in the evaporation rate are significant with variations of &20% (Standard Deviation (SD)) at an e-beam power of approximately 38 [kW]. However, it is observed that measured Ti evaporation rates exhibit small variations of f5.4% (SD) for the same e-beam power repeated in multiple periods of the same run. The Ti vapor rate sensitivity to e-beam power is similar for all the runs. The results for Ti-6-4 evaporation rate show small run-to-run variations of f5.7% (SD) at 39.4 [kW].

There is good agreement between the results of all three Ti-6-4 runs and three of the Ti runs. It is possible that systematic changes in the e-beam evaporation system occurred during the initial stages of testing, since the largest vapor rate variations occurred during

- 0.08 B : 0.06 3 L 0.04 z

2 ,,,,,,,I ',,,,,,,,,,,,,,,,,,,,

0 IIII'~~l~'~llI'~ll~'l~ll'lll~ 12 12.1 12.2 12.3 12.4 12.5 12.6

Timelh]

Figure 7: Feed rate, pool level, and Ti and Al evaporation rates from Ti-6-4 Run No. 3.

the first two runs which involved Ti. It is possible that the surface characteristics of the crucible was altered by wear, changing the thermal contact resistance between the ingot and crucible. It is also possible that there were changes in the transport of the e-beam giving increased stray e-beam energy on the top of the crucible. This would be difficult to detect since a single water circuit cools the crucible top and interior surface which contacts the ingot.

The MELT model prediction for the Ti and Ti-6-4 evaporation rate curves is also shown in Figure 6. The model provides a single curve for the two materials since the properties are taken to be the same as described above. The evaporation rate curve predicted by the MELT model for Ti and Ti-6-4 is higher than the experimental curves. However, the e-beam power sensitivities of all the model and experimental curves are similar. The nearly linear dependence given by the model is confirmed by Ti Run No. 2 which has points at three power levels.

The question remains as to why the model curve for the evaporation rate is higher than the experimental curves for Ti and Ti-6-4. One factor is the choice of skip fraction. In an earlier work (McClelland et al. [3]), the MELT model predictions for Ti evaporation rate compared quite favorably to the results of Ti Run No. 1. In the earlier calculations a power skip fraction of 0.29 was selected based on the results of Archard [13] for back- scattered electron current. In the present work, a power skip fraction of 0.22 is used based on measurements of back-scattered power (Reichelt [14]). The good agreement between the

model and measured heat flows supports the latter value (see Figure 5). However, the lower skip fraction leads to the higher model evaporation rates in this study. Other contributions to the model-experiment discrepancies are under investigation. It is important to note that no model parameters were adjusted to better represent the measurements. In addition, only a few model parameters such as the skip fraction and vapor pressure curve have a strong influence on the calculated evaporation rate.

4.5 Laser absorption measurements of Ti and Al evaporation rates

Laser absorption measurements of the Ti and Al evaporation rates are shown along with the pool height and feed rate (pusher rate) during a 0.5 [h] period of Ti-6-4 Run No. 3 (see Figure 7). During this period the e-beam power was constant at 39.4 [kW], and fluctuations in the pool level and feed rate were small. The titanium evaporation rate varies by only &ll% (peak-to-peak) while the aluminum rate varies by a much larger f32% (peak-to- peak). Compositional variations of this type are undesirable for the fiber coating process, and methods to minimize this effect are needed.

The fluctuations in Al evaporation rate can be interpreted in terms of results from the MELT model. The model shows that a variation in the thermal contact resistance has little affect on the bulk evaporation rate, but has a much stronger influence on the location of the solidification zone [2]. It is likely that the contact resistance varies significantly in a stick-slip cycle as the ingot advances. In the stick mode, there is considerable mechanical resistance to movement and the ingot makes good thermal contact with crucible. As the ingot slips, the thermal contact resistance increases and feed material is melted and mixed into the pool. The evaporation rate of the volatile Al increases rapidly on the time scale of the melting and mixing process. It then decays as the aluminum from the pool is depleted. The cycle repeats when the next thermal-mechanical disturbance occurs.

5 Conclusions

Experimental measurements for the e-beam evaporation of Ti and Ti-6-4 were analyzed with a finite element MELT model for fluid and energy flow in the ingot. In the finite element model, rotating diagonal spines were used to track solid-liquid and liquid-vapor interfaces of deep pools. In the experiments, bulk evaporation rates were measured by feed consumption with the pool level held constant using a feedback control system. The results show that Ti and Ti-6-4 generally have similar bulk evaporation rates. Although the model evaporation rates are higher than the experimental values, all of the evaporation rates have a similar linear dependence on e-beam power. For Ti, the model provides a good representation of heat flows measured by cooling water circuits and melting line locations taken from ingot sections. In an experiment with Ti-6-4 at steady process conditions, laser absorption measurements show much larger fluctuations in the Al evaporation rate than the Ti evaporation rate.

6 Acknowledgments

Rich Palmer is acknowledged for his efforts in developing, operating, and maintaining the electron gun as well as implementing and maintaining the feeder controller and data

acquisition systems in ETF. Paul Gronner is acknowledged for his efforts in operating and maintaining the vacuum and mechanical systems in ETF.

References

[l] J. Storer. Electron beam deposition for the fabrication of titanium MMCs. In R.-Bak- ish, editor, Electron Beam Melting and Refining State of the Art 1993, pages 235-245. Bakish Materials Corp., Englewood, NJ, 1993.

[2] K. W. Westerberg, M. A. McClelland, and B. A. Finlayson. Finite element analysis of flow, heat transfer, and free interfaces in an electron-beam vaporization system for metals. Int. J. Numer. Methods Fluids, in press. 1998.

[3] M. A. McClelland, K. W. Westerberg, T. C. Meier, D. G. Braun, L. V. Berzins, T. M. Anklam, and J. Storer. Experimental and numerical study of e-beam evaporation of titanium. In R. Bakish, editor, Electron Beam Melting and Refining State of the Art 1996, pages 151-162. Bakish Materials Corp., Englewood, NJ, 1996.

[4] P. A. Sackinger, R. A. Brown, and J. J. Derby. A finite element method for the analysis of fluid flow, heat transfer and free interfaces in Czochralski crystal growth. Int. J. Numer. Methods Fluids, 9:453-492, 1989.

[5] G. W. Lan and S. Kou. Heat transfer, fluid flow and interface shapes in floating-zone crystal growth. J. Crystal Growth, 108:351-366, 1991.

[6] D. Guilbaud. Effect of melt surface depression on the vaporization rate of a metal heated by an electron-beam. In R. Bakish, editor, Electron Beam Melting and Refining State of the Art 1995, pages 227-242. Bakish Materials Corp., Englewood, NJ, 1995.

[7] K. W. Westerberg, M. A. McClelland, and B. -4. Finlayson. The interaction of flow, heat transfer, and free interfaces in an electron-beam vaporization system for metals. Paper 124e, 1994 Annual AIChE Meeting, San Francisco, CA, 1994.

[8] P. M. Gresho, R. L. Lee, and R. L. Sani. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In C. Taylor and K. Morgan, editors, Recent Advances in Numerical Methods in Fluids, volume 1, pages 27-79. Pineridge Press, Ltd., Swansea, U.K., 1980.

[9] M. Morari and E. Zafiriou. Robust Process Control. Prentice Hall, 1989.

[lo] L. V. Berzins, T. M. Anklam, F. Chambers, S. Galanti, C. A. Haynam, and E. F. Worden. Laser absorption spectroscopy for process control - sensor system design methodology. Surface and Coatings Technology, 76-77:675-680, 1995.

[ll] L. V. Berzins, J. Benterou, J. B. Cooke, T. C. Meier! and T. M. Anklam. Group velocity delay technique for monitoring the density and composition of optically thick vapors. Paper Ul-0320, 191st Electrochemical Society Meeting, Montreal, Canada, 1997.

[12] R. S. Ransing, Y. Zheng, and R. W. Lewis. Potential applications of intelligent pre- processing in the numerical simulation of castings. Xn R. W. Lewis, editor, Numerical Methods in Thermal Problems, volume VIII, Pt. 2, pages 361-375. Pineridge Press, Swansea, 1993.

[13] G. D. Archard. Backscattering of electrons. J. Appl. Phys., 32:1505-1509, 1961. _

[14] W. Reichelt. Produktion und kondensation von metalldampfen in grofien mengen. Angew. Chem., 87:239-243, 1975.

Technical Inform

ation Departm

ent • Lawrence Liverm

ore National Laboratory

University of C

alifornia • Livermore, C

alifornia 94551