Experimental characterization of anode heating by electron emissionfrom a multi-walled carbon nanotube

T. Westover a, T.S. Fisher a,*, F. Pfefferkorn b

aSchool of Mechanical Engineering, and Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USAbDepartment of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA

Received 18 January 2006; received in revised form 15 July 2006Available online 2 October 2006

Abstract

The steady-state temperature distribution in a thin anode bombarded by an electron beam field emitted from an individual multi-walled carbon nanotube is measured with an infrared camera, and this distribution is compared to that predicted by a numerical model.By assuming the electron distribution in the beam follows a Gaussian distribution, a good fit to the anode temperature profile is obtainedand this fit provides an estimate of the beam spreading radius. Results indicate the electron beam narrows as the emission currentincreases. A heat flux on the anode surface as high as 0.35 W/cm2 has been measured, corresponding to an electron beam radius ofapproximately 1.22 mm.! 2006 Elsevier Ltd. All rights reserved.

Keywords: Field emission; Carbon nanotube; Anode heating

1. Introduction

Efficient electron field emitters are becoming increas-ingly attractive for a wide range of applications, includingscanning probe tips, power electronics, and flat panel dis-plays [1,2]. Advances in microfabrication techniques nowpromise to extend the range of devices employing fieldemission by utilizing emitters arranged in patterned arrays.Cathodes consisting of materials such as carbon nanotubesand polycrystalline diamond have demonstrated high ratesof field emission at relatively low applied electric fields,although the emission generally occurs at nanoscale siteson the cathode where the local electric field is greatlyenhanced [3,4]. Emitted electrons accelerate under theapplied electric field as they traverse a vacuum gap and ulti-mately impact the anode. This energetic electron beam canproduce substantial heating in a localized region within theanode, causing thermal stresses and possibly failure [5].

Fisher et al. [6] used a Monte Carlo technique to determinethe spatial distribution of the electrons in a beam originat-ing from a point source cathode, and from this simulationestimated heat generation occurring within an anode due tothe penetration of the electron beam beneath the anodesurface. In a series of field emission experiments, Harriset al. [7] employed a polycrystalline diamond film as anelectron source to bombard a steel anode. By measuringthe anode temperature rise, they were able to estimate thetotal rate of energy absorbed by the anode and the approx-imate area over which the heating was confined. The pres-ent study utilizes an individual multi-walled carbonnanotube (MWNT) as the electron source in order to eval-uate the heating caused by electron emission from a singlenanoscale emission site.

Fowler and Nordheim [8] first described the physics gov-erning field emission from a flat surface. Since that time,elongated emitter structures have been shown to enhancethe local electric field greatly and thus enable emissionunder reduced applied electric fields [9]. With turn-on fieldsmeasured below 5 V/lm, carbon nanotubes are extremely

0017-9310/$ - see front matter ! 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2006.07.024

* Corresponding author. Tel.: +1 765 494 5627; fax: +1 765 494 0539.E-mail address: tsfisher@purdue.edu (T.S. Fisher).

www.elsevier.com/locate/ijhmt

International Journal of Heat and Mass Transfer 50 (2007) 595–604

efficient emitters [10]. Recent experiments report that car-bon nanotubes may support local current densities as highas 109 A/cm2 [11,12], and that emitter arrays may be able toproduce substrate-level current densities as high as 105 A/cm2 [12].

Experiments reveal that single-walled nanotubes andopen-ended multi-walled nanotubes can produce ring-likecurrent density patterns, indicating that emission occursprimarily from the nanotube ends [13,14]. Mayer et al.[15–17] employed an atomistic transfer matrix method tosimulate emission from various types of carbon nanotubesby assuming a constant electric field in the vicinity of theelectric tip. More recently, Walker et al. [18] performedsimilar simulations including Coulomb interactions amongthe electrons, non-axial field components at the nanotubetip, and random non-axial momentum components of theelectrons at emission, and confirmed the possibility thatring-type patterns can occur in the electron beam.

The present work provides an experimental extension oftheoretical anode heating studies. In the following sections,the experimental setup is first described, and then the elec-tron beam and anode heating models are introduced.Experimental data are then presented and analyzed to esti-mate of the electron beam radius and the maximum heat

flux on the anode surface for different applied electric fieldsand emission currents.

2. Experimental setup

Fig. 1 contains a schematic of the experimental appara-tus used to measure the heating effect at the anode pro-duced by field emitted electrons. The anode (a stainlesssteel disc having a diameter of 36.7 mm and thickness of0.025 mm) was fixed between a vertical plate and a metalring. A hole in the vertical plate with a diameter of18.9 mm ensured that the central portion of the anode’srear face was exposed to enable temperature measurementswith an infrared camera placed outside the vacuum cham-ber. The cathode consisted of an individual MWNTmounted on a tungsten tip that was fastened to the endof a metal rod. A groove in the cathode platform servedto align the tungsten tip along the anode axis. The verticalplate and the cathode platform were electrically isolatedfrom the base by means of ceramic spacers, ensuring thatboth the anode and cathode were electrically isolated fromthe other components in the vacuum chamber.

The tungsten tip was etched using the DC drop-offmethod commonly employed in scanning electron micro-

Nomenclature

b slope of infrared camera calibration curveI mean measured electrical currentkeff effective thermal conductivity of anodekp thermal conductivity of paintks thermal conductivity of stainless steelN number of data points used in the least-squares

fitq00 local heat fluxq00max maximum heat fluxQ energy deposition rate on the anode surfacer anode radial coordinateR anode radiusS estimate of the variability of the data taken by

the infrared camerat95,N!1 t estimator with 95% probability and N ! 1 de-

grees of freedomTdata,i mean measured anode temperature rise at loca-

tion riTpred,i predicted temperature at location riub uncertainty in slope of infrared camera calibra-

tion curveuCAM least-count uncertainty of infrared camerauqmax uncertainty in maximum heat flux on anode sur-

faceuQ uncertainty in total anode heatinguT total uncertainty in temperature measurement

with infrared camerauDz uncertainty in the anode thickness

ur total uncertainty in the electron beam radiusurfit uncertainty in the electron beam radius due to

least-squares fiturT uncertainty in the electron beam radius due to

temperature measurement uncertaintyurQ uncertainty in the electron beam radius due to

heating uncertaintyurDz uncertainty in the electron beam radius due to

anode thickness uncertaintyV applied voltage potential

Greek symbolsDzp width of paint layer on anode surfaceDzs width of stainless steel in anodehQ,r=0 local sensitivity coefficient at r = 0 correlating

predicted temperature rise to total anode heat-ing

hDz,r=0 local sensitivity coefficient at r = 0 correlatingpredicted temperature rise to anode thickness

hr,i local sensitivity coefficient at data point i corre-lating predicted temperature rise to electronbeam radius

/Q sensitivity coefficient correlating predicted max-imum heat flux to total anode heating

/r sensitivity coefficient correlating predicted max-imum heat flux to electron beam radius

re beam radius, a parameter characterizing thespread of the electron beam

596 T. Westover et al. / International Journal of Heat and Mass Transfer 50 (2007) 595–604

scopy (SEM) applications [19]. In this process, a tungstenwire of diameter 0.25 mm is dipped approximately 5 mminto a 2 N NaOH solution. A voltage potential of 10VDC is then applied to the tungsten wire while maintaininga graphite rod in the solution at zero potential. The result-ing electrochemical reaction etches the tungsten wire at thesurface of the NaOH solution, causing the tip of the wire todetach from the body and leaving a sharp point on the endof the tungsten wire.

Mounting of the nanotube to the etched tungsten tipwas performed using two micromanipulators (NewportM-460A-XYZ) aided by an inverted optical microscope(Nikon Epiphot 200) equipped with a 50"/0.55 objectiveto observe the process in darkfield at 750" magnification.A mesh of MWNTs was placed on a section of SEM tape,with individual nanotubes protruding from the edges of themesh. The MWNT mesh was synthesized using a previ-ously reported technique [20]. Before mounting a MWNTon the tungsten tip, a small amount of electrically conduc-tive adhesive was placed on the tip by carefully touching itto a clean portion of carbon tape (Ted Pella, Inc.) andremoving it. The tungsten tip with adhesive from the tapewas then moved to one of the protruding nanotubes andpositioned such that the nanotube became attached tothe tip. A MWNT attached to the etched tungsten tip inthis way withstood the applied electric fields encounteredduring field emission.

Fig. 2 shows a MWNT mounted on an etched tungstentip at 750" magnification. Subsequently, the anode washeld stationary, and the etched tungsten tip was fixed toa linear translation stage so that the vacuum gap distancecould be adjusted. This arrangement was necessary so thatthe vacuum gap distance could be determined after eachexperiment by carefully stepping the anode platform inmeasured increments until contact was made with the cath-ode. The entire experimental apparatus was placed within avacuum chamber that was maintained at a pressure ofapproximately 5 " 10!7 Torr during experiments.

The temperature profile of the anode was measured by aThermaCAM SC300 infrared camera placed outside thevacuum chamber. A germanium viewport in the vacuum

chamber wall allowed the infrared camera to view the backsurface of the anode, which was coated with flat blackpaint to achieve a surface emissivity of approximately0.94. Experiments have demonstrated that the germaniumviewport in the vacuum chamber wall affects infraredcamera measurements and increases the measurementuncertainty. Consequently, calibration experiments wereconducted to account for the effect of viewing objects invacuum through this viewport.

The calibration scale shown in Fig. 3 was generated byviewing a painted aluminum bar at a vacuum pressure ofapproximately 10!6 Torr. The temperature of the alumi-num bar was increased using an internal heater and wasallowed to reach steady-state before measuring the surfacetemperature at two locations with thermocouples. Calibra-tion of the infrared camera consists of determining theratio of the actual temperature change measured by thethermocouples to the temperature change registered bythe IR camera, which corresponds to the slope, b, of thecurve in Fig. 3. For the experiments in this study, the slopeof the calibration curve is 2.55, and multiple experimentsdemonstrated that this relationship was consistent during

Ceramic Spacer

Translation stage

Ceramic Spacer

Cathode platform

Metal rod

Metal ring Vertical plate

Anode

Ceramic bolt

Hole

IR camera Tungsten tip

Fig. 1. Schematic of the experimental apparatus.

Fig. 2. An individual multi-walled carbon nanotube mounted on anetched tungsten tip. Magnification is 750".

0

10

20

30

40

50

0 5 10 15Infrared camera temperature rise (°C)

Ther

moc

oupl

e te

mpe

ratu

re ri

se (°

C)

Fig. 3. Temperature rise measurements taken in vacuum with thermo-couples and an infrared camera viewing through a germanium viewport.The specimen was an aluminum bar painted flat black.

T. Westover et al. / International Journal of Heat and Mass Transfer 50 (2007) 595–604 597

the period in which the experiments were performed. Theuncertainty in the slope of the calibration curve, ub, maybe determined following the procedure outlined by Bowkerfor estimating the slope of a line when the offset is knownto be zero [21]. Using this procedure, a value of 0.024 isfound for ub.

The uncertainty, uT, associated with a temperature mea-surement using the infrared camera can then be estimatedas

uT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðuCAMÞ2 þ

ubbT

" #2

þ ðUTCT Þ2r

ð1Þ

where T is the actual temperature rise and uCAM is theleast-count uncertainty of the infrared camera viewingthrough a germanium viewport, approximately equal to0.4 "C. UTC is the percent uncertainty in the thermocoupletemperature rise measurement, which was estimated as±2%. This simple calibration is sufficient for the presentwork because the anode heating experiments depend prin-cipally on measured values of temperature changes, whichare related solely to the slope of the calibration line.

A schematic diagram of the experimental circuitappears in Fig. 4. A Kepco BKH1000-0.2 MG power sup-ply provided the electric potential bias to induce fieldemission, and a Keithley 6486 picoammeter measuredthe field emission current between the cathode and theanode. The uncertainties in the applied voltage and elec-tric current measurements are each less than 0.5%. Elec-tric current measurements were recorded through anIEEE-488 (GPIB) bus connected to a PCI-GPIB control-ler. A 2.82 MX shunt resistor was included in the circuitto stabilize the field emission current, and coaxial cableswere used to connect the instrumentation to reduceEMI and RFI signals.

3. Electron beam and anode heating models

Data recorded during field emission experimentsincluded the temperature field of the anode as a functionof three parameters: applied voltage, electric current, andvacuum gap distance separating the anode from the cath-

ode. Because temperature variations through the thicknessof the anode were negligible, a 1D finite-difference model ofthe anode was developed to predict the anode temperaturefield based on experimental conditions, and this tempera-ture field was compared to that obtained from infraredmeasurements in order to characterize the heating of theanode by the electron beam. The governing equation forthe anode temperature, T, as a function of radial location,r, may be written as

Zd

drkeff ' r

dTdr

$ %! 2 ' r ' r ' eðT 4 ! T 4

surrÞ þ r ' q00 ¼ 0 ð2Þ

where Z and keff represent anode thickness and effectivethermal conductivity, respectively. r is the Stefan–Boltz-mann constant (5.67 " 10!8 W/m2 K4) and q00 is the localheat flux on the anode surface produced by the electronbeam. e is the average anode emissivity, which was calcu-lated by taking the arithmetic mean of the emissivities ofboth sides of the anode. Assuming an emissivity of 0.94for the painted side of the anode and an emissivity of0.16 for the unpainted side, e was estimated to be 0.55.During experiments, the temperature at the outer edge ofthe anode was observed to increase above that of the metalring because of thermal contact resistance. Consequently, aDirichlet boundary condition was imposed at the outeredge of the finite-difference model, holding the temperatureat the edge of the anode equal to that measured by theinfrared camera. The inner boundary condition was im-posed by fixing the first derivative of the temperaturewith respect to r equal to zero at the center of the anode,which is equivalent to a Neumann zero flux boundarycondition.

Preliminary simulations modeled the heating of theanode surface as a uniform heat flux over a small circle cen-tered on the anode axis. However, those simulationsyielded poor fits to the measured data, and their resultsare not shown. A superior fit to the experimental datawas obtained by assuming that the electron beam (and con-sequently the heating occurring at the anode) followed anaxially symmetric Gaussian distribution. The spread ofthe electron beam was characterized by the standard devi-ation of the Gaussian distribution, symbolized by re andhereafter called the electron beam radius. The standarddeviation, re, was determined by minimizing the sum ofthe squared errors between the data obtained from theinfrared camera and the finite-difference model. With theabove assumptions and the constraint that the total heatingdue to the electron beam must equal the product of theapplied voltage, V, and the measured current, I, the localheat flux incident on the anode surface takes the form

q00ðrÞ ¼ VI2pr2

e 1! e!R2=2r2e& ' e!r2=2r2e ð3Þ

where R is the outer radius of the anode.The paint layer on the anode’s back surface was found

to have a significant effect on the anode temperature

Anode

Cathode

Shunt resistor2.82 M!

Keithley Picoammeter

Kepco power supply

Fig. 4. Schematic diagram of the electrical circuit for measuring current–voltage field emission characteristics.

598 T. Westover et al. / International Journal of Heat and Mass Transfer 50 (2007) 595–604

profile, and, therefore, it was included in the finite-differ-ence model. An effective thermal conductivity of thesteel/paint anode was calculated using [22]

keff ¼ksDzs þ kpDzpDzs þ Dzp

ð4Þ

where ks and kp are the thermal conductivities of stainlesssteel and paint with values of 16.3 and 0.65 W/m K, respec-tively. Dzs and Dzp correspond to the thicknesses of thesteel and paint layers, with mean measured values of0.025 and 0.031 mm, respectively. From these values, theeffective thermal conductivity of the anode was calculatedto be 7.76 W/m K. The accuracy of this effective conductiv-ity model was validated by comparison to a two-dimen-sional finite-difference model that included the distinctsteel and paint layers. However, because the one-dimen-sional model is computationally more efficient, it was em-ployed in the parameter estimation process to determinethe beam radius, re. Comparisons between these modelsand the dependence of the model results on the mesh sizeare discussed in the following section.

4. Results and discussion

Field emission from individual carbon nanotubes hasbeen shown to exhibit Fowler–Nordheim emission behav-ior at low currents in the range of 0.4–80 nA [23]. However,previous reports have indicated that field emission canbecome unstable at currents above 0.1 lA [24]. Fig. 5shows a typical current–voltage profile obtained in theseexperiments and indicates that the emission currentincreases exponentially with increasing voltage as expectedfor field emission, although some irregularities are presentat high values of the emission current. These irregularitiesare probably due to changes in the carbon nanotube emit-ter as the emission current increases. The inset in Fig. 5 dis-plays the Fowler–Nordheim plot of the same data in whichthe quotient (I/V2) are plotted as a function of 1/V on asemi-logarithmic scale. The resulting curve is usually linearwith a negative slope for metallic emitters. The change inslope that is evident in the inset data indicates a deviationfrom Fowler–Nordheim behavior, and is commonlyobserved for nanotube samples emitting over a large cur-rent range [9]. Each data point in Fig. 5 represents themean of approximately 200 individual measurements takenat 0.308 s intervals at a constant voltage.

Measured field emission currents above 0.1 lA regularlyexhibited significant fluctuations when the applied voltagewas increased or decreased. However, in many cases, thecurrent remained reasonably stable up to values as highas 70 lA, while the applied voltage was held constant. Atypical current response as a function of time is shown inFig. 6, in which a constant voltage of 446 V was main-tained. The emission current was observed to fluctuatebetween 34 and 40 lA for a period of time, and then atapproximately 38 s, the current abruptly rose to a meanvalue of 47 lA. This type of instability, where the emission

current is reasonably stable over moderate time intervals, istypical of high-current field emission.

During the intervals in which the field emissionremained reasonably stable, the anode temperature profileexhibited no noticeable temporal change and was charac-terized by taking the mean of five temperature data samplescollected with an infrared camera at intervals of one sec-ond. For the experiment shown in Fig. 6, images weretaken by the infrared camera during the time interval from10 to 14 s. Fig. 7(a) shows the infrared image at 10 s. Theanode temperature profile was measured along four linesextending radially from the point of maximum temperatureon the anode surface, as shown in Fig. 7(a). Data lines 1aand 1b were aligned in the vertical direction, and data lines2a and 2b were aligned in the horizontal direction. Thetotal heating rate, Q, in Fig. 7(a) was determined to beapproximately 16.4 mW by multiplying the applied volt-age, 446 V, by the mean electric current measured in theinterval 0–14 s shown in Fig. 6 approximately 36.7 lA.The value calculated for the average electric current wasfound to depend on the time interval used in the calcula-tion, causing an uncertainty in the average current ofapproximately 0.3 lA. The uncertainty in the total heatingwas then found to be approximately 0.16 mW or 1%.

0

5

10

15

20

25

30

100 150 200 250 300Voltage (V)

Curr

ent (

µA)

0.003 0.006 0.009

1/V (V-1)

I/V 2

(A. V

-2 )

10-9

10-11

10-13

10-15

Fig. 5. I–V characteristics for field emission from an individual MWNT.The vacuum gap is 2.6 mm.

0

10

20

30

40

50

0 10 20 30 40 50 60Time (s)

Curr

ent (

µA)

Fig. 6. Emission current plotted with time at an applied voltage of 446 Vand an electrode gap of 2.6 mm.

T. Westover et al. / International Journal of Heat and Mass Transfer 50 (2007) 595–604 599

Fig. 7(b) displays the mean temperature profilesresulting from averaging the data taken by the infraredcamera during the time interval 10–14 s in Fig. 6. Thepeak anode temperature rise, located approximately 1 mmdirectly below the geometric anode center, is 15.1 "Cwith an uncertainty of approximately ±0.4 "C as calcu-lated from Eq. (1). The mild asymmetry in the anodetemperature profile observed in Fig. 7(b) indicates thatthe electron beam is slightly asymmetric with a greaterdensity of electrons impacting the right side of theanode.

In the data presented in this study, the point of maxi-mum temperature rise occurred within approximately1 mm of the geometric anode center, thus ensuring thatthe observed asymmetry in the anode heating was dueprincipally to asymmetry in the electron beam and notmisalignment between the electron beam and the anodecenter. In additional experiments, however, the point ofmaximum temperature rise was observed to shift by asmuch as 3 mm as the applied voltage was adjusted. Also,in some cases, the location of peak temperature rise wasseen to shift slowly in time before becoming stationary,indicating that the electron distribution in the beamdepends on conditions at the nanotube tip in addition tothe applied electric potential. Shifting of the location of

the anode peak temperature rise due to field emission hasalso been reported by Harris et al. [7].

Similar experiments were conducted for different appliedvoltages, and maximum temperature rises of the anode areplotted as functions of the total heating rate in Fig. 8 forvacuum gaps of 2.2 and 2.6 mm. The error bars in Fig. 8indicate the uncertainty in the temperature measurement.Table 1 summarizes the pertinent experimental resultsand includes the measurement uncertainties. As reportedelsewhere [7], the peak temperature rise appears to increasenearly linearly with increased heating. Interestingly, how-ever, the maximum temperatures for the experiment withthe larger vacuum gap appear higher for similar heatingvalues, indicating that conditions at the carbon nanotubeemitter tip had a more significant effect on the size of theelectron beam than did the vacuum gap separation. Thisresult is consistent with electron trajectories in the vacuumgap determined by Monte Carlo simulations, which indi-cate that for the vacuum gaps listed in Table 1, the broad-ening of the electron beam is independent of the vacuumgap for the same level of power input. The approach uti-lized in the Monte Carlo simulations has been reportedelsewhere [18].

As noted previously, the concentration of electrons inthe beam was assumed to follow an axially symmetricGaussian distribution, and the electron beam radius wascharacterized by the standard deviation, re, which wasdetermined by comparing the 1D finite-difference modelresults to the mean anode temperature profiles calculated

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Radial location (mm)

Tem

pera

ture

rise

(ºC)

data line 1adata line1bdata line 2adata line 2b

data line 1a

data line 2a

data line 1b

data line 2b

14˚C

13

12

11

10

9

8

7

6

5˚C

a

b

Fig. 7. (a) Infrared camera image of anode heating experiment. (b) Anodetemperature rise obtained by averaging five data samples. The totalheating on the anode surface was approximately 16.4 mW.

0

5

10

15

20

25

0 5 10 15 20 25 30 35Total rate of energy deposition (mW)

Max

. Tem

pera

ture

rise

(°C)

Gap = 2.2 mm

Gap = 2.6 mm

Fig. 8. Maximum anode temperature rises measured by an infraredcamera in experiments with vacuum gaps of 2.2 and 2.6 mm, respectively.

Table 1Experimental electron emission data from experiments with vacuum gapsof 2.2 and 2.6 mm, respectively

V (V) I (lA) Q (mW) DT ("C)

Vacuum gap: 2.2 mm349 ± 2 43.2 ± 0.3 15.0 ± 0.1 10.0 ± 0.4351 ± 2 52.8 ± 0.3 18.5 ± 0.2 12.5 ± 0.4405 ± 2 79.8 ± 0.5 32.3 ± 0.2 23.4 ± 0.4

Vacuum gap: 2.6 mm286 ± 2 26.1 ± 0.2 7.47 ± 0.08 4.7 ± 0.4446 ± 2 36.7 ± 0.3 16.4 ± 0.2 15.1 ± 0.4

600 T. Westover et al. / International Journal of Heat and Mass Transfer 50 (2007) 595–604

by averaging the four data lines recorded during eachexperiment. Fig. 9(a) and (b) shows the mean anode tem-perature profiles for the experiments in which the appliedvoltages were 286 and 446 V, respectively. Applying aleast-squares fitting process, values of 1.64 and 0.94 mmwere determined for re under applied voltages of 286 and446 V, indicating that the electron beam narrowed as theelectric field increased. The anode temperature profiles pre-dicted by the 1D finite-difference model using these valuesfor re are also plotted in Fig. 9, where a good fit is demon-strated between the data and the finite-difference modelpredictions.

In order to verify the results obtained from the 1D finite-difference model, a grid refinement study was performed,and the results were compared to those obtained from a2D heat diffusion solver using commercial software. Theresults presented in Fig. 9 were obtained using 200 controlvolumes clustered more densely near the anode center andwere determined to be grid independent to five decimalplaces. A similar grid refinement study performed for the2D model showed that the results obtained from thatmodel were grid-independent to three decimal places for10 control volumes in the axial direction and 1000 controlvolumes in the radial direction. Although not shown inFig. 9 for the purpose of clarity, the temperature profilespredicted by the 2D model almost exactly match those pre-dicted by the 1D model for both applied voltages. The

maximum differences between the models occur at the peaktemperatures and were less than 0.01 "C in Fig. 9(a) andless than 0.03 "C in Fig. 9(b). The close correspondencebetween the 1D and 2D models is expected because thetemperature variation through the anode thickness is verysmall.

One additional comment is in order regarding Fig. 9.Contrary to what is usually expected from a least-squaresfit, the model predictions in Fig. 9 are not equally balancedabove and below the measured anode temperature profiles.The reason for this behavior is that the temperature in theouter region of the anode is determined almost exclusivelyby the temperature at the anode’s outer edge. The distribu-tion of electrons in the beam only affects the anode temper-ature profile inside the region directly impacted by thebeam, and has negligible effect on the anode temperatureprofile outside this region. Because nearly the entire elec-tron beam is concentrated on the anode surface within4 mm from the point of maximum temperature, only datalying in this region affect the least-squares fit. Anotherconsequence of this phenomenon is that weighting theleast-squares fit more heavily near the anode center isunnecessary because the discrepancy between the tempera-ture profiles given by the experimental data and the finite-difference model is naturally largest in the central portionof the anode. Consequently, for this study the least-squareserror terms were equally weighted regardless of position onthe anode surface.

In order to validate the conclusions drawn from thedata, confidence intervals for the electron beam radiusare necessary. The procedure that was followed to calculatethe confidence intervals is demonstrated here by calculatingthe uncertainty associated with the data in Fig. 9(a). Theuncertainty in the electron beam radius can be estimatedfrom

ur ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðurfitÞ2 þ ðurT Þ2 þ ðurQÞ2 þ ðurDzÞ2

qð5Þ

where urfit, urT, urQ, and urDz are the respective uncertain-ties in the electron beam radius due to uncertainties in theleast-squares fit, the infrared camera temperature measure-ment, the total heating on the anode surface, and the anodethickness. An estimate for the uncertainty in the electronbeam radius associated with the least-squares fit may beobtained from

urfit ¼ t95;N!1S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

PNi¼1ðhr;iÞ

!2

sð6Þ

where N is the number of data points used to determine thevalue of re, and t95,N!1 is the t estimator corresponding to95% probability and N ! 1 degrees of freedom. hr,i is thelocal sensitivity coefficient (the partial derivative of the pre-dicted anode temperature rise with respect to re at eachdata point i) associated with re in the least-squares fit.Lastly, S is an estimate of the variability of the data takenby the infrared camera and may be calculated using [21]

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6Radial location (mm)

Tem

pera

ture

(°C)

Mean of data lines

0

1

2

3

4

5

0 1 2 3 4 5 6Radial location (mm)

Tem

pera

ture

(°C)

Mean of data lines

v=286 v

v=446 v

= 1.64

= 0.94

"e

"e

Fig. 9. Comparison of the mean anode temperature profiles with thetemperature profiles predicted by the 1D finite-difference model. Theapplied voltages are 286 V (a) and 446 V (b), and the vacuum gap is2.6 mm.

T. Westover et al. / International Journal of Heat and Mass Transfer 50 (2007) 595–604 601

S ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N ! 1

XN

i¼1

ðT data;i ! T pred;iÞ2vuut ð7Þ

where Tdata,i is the mean measured anode temperature riseat location ri, and Tpred,i is the predicted temperature rise atthe same point.

Typically, approximately 45 measurements were takenalong each data line to characterize the anode temperatureprofile. Employing Eq. (7), the variability in the data inFig. 9(a) may be estimated as 0.08 "C as listed in Table 2.This estimate of the variability is inflated by the relativepoorness of the model fit at intermediate radial locations;however, it does provide an upper limit to the variabilityand hence is useful in approximating the uncertainty inthe electron beam radius due to the least-squares fit. Settingthe t estimator equal to 2.021 [21], the uncertainty in thebeam radius due to the least-squares fit is estimated fromEq. (6) as 0.04 mm.

The uncertainty in the electron beam radius due to theinfrared camera temperature measurement, urT, may beestimated by dividing the uncertainty in the anode peaktemperature rise as given in Table 1 by the local sensitivitycoefficient, hr,r=0, at r = 0. Approximate values for all ofthe sensitivity coefficients were obtained using the 1Dfinite-difference model as described by Figliola [25], andin this process a value of 1.5 "C/mm as found for hr,r=0,yielding a value of approximately 0.26 mm for urT as givenin Table 2.

The uncertainty in the beam radius due to the total heat-ing uncertainty, urQ, may be estimated from

urQ ¼ uQhQ;r¼0

hr;r¼0ð8Þ

where uQ is the uncertainty in the total anode heating as gi-ven in Table 1 and hQ,r=0 is the sensitivity coefficient corre-sponding to the partial derivative at r = 0 of the predictedtemperature rise with respect to total heating. A value of0.23 "C/mW was found for hQ,r=0, giving an estimate of0.03 mm for urQ.

The uncertainty in the beam radius due to the anodethickness uncertainty, urDz, may be estimated from

urDz ¼ uDzhDz;r¼0

hr;r¼0ð9Þ

where uDz, is the uncertainty in the anode thickness, takenas 0.0013 mm, and hDz,r=0 is the sensitivity coefficient corre-

sponding to the partial derivative at r = 0 of the modeltemperature rise with respect to anode thickness. The valueof hDz,r=0 found from the 1D finite-difference model is56.9 "C/mm, yielding a value of 0.05 mm for urDz. Substi-tuting the values of urfit, urT, urQ, and urDz from Table 2into Eq. (5), the total uncertainty in the electron beam ra-dius in Fig. 9(a) is estimated as 0.27 mm. Comparing theuncertainty terms in Table 2, it is found that the uncer-tainty associated with temperature measurement dominatesthe overall uncertainty in the electron beam radius, and thesame trend is seen for all the measurements.

Fig. 10 displays the relationship between the electronbeam radius and the emission current for all the experi-ments listed in Table 1 and indicates that the beam radiusdecreases as the emission current or applied voltageincreases. The error bars represent the uncertainties in elec-tron beam radius calculated for each point using the proce-dure outlined above. The first two data points for theexperiments having a vacuum gap of 2.2 mm are of partic-ular interest because of the overlap in their confidenceintervals. As shown in Table 1, the applied voltages forthese experiments are nearly equal; thus, the overlap inthe confidence intervals of the electron beam radii is notsurprising.

A key element in assessing anode performance is themaximum heat flux on the anode surface produced bythe electron beam. An estimate of the maximum heat flux,q00max, in each experiment may be obtained by substitutinginto Eq. (3) the respective values of the electron beamradius and total heating. The estimated maximum heatfluxes are listed along with the electron beam radii in Table3. The uncertainties in q00max listed in Table 3 were calculatedusing [25]

uqmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðuQ/QÞ

2 þ ður/rÞ2

qð10Þ

where ur is the uncertainty in the electron beam radius gi-ven in Table 3. /Q and /r are the sensitivity coefficientscorresponding to the partial derivatives of q00 at r = 0 withrespect to Q and re, respectively. We note that the uncer-

Table 2Estimated uncertainty parameters involved in calculating the 95%confidence interval for electron beam radius in Fig. 9(a)

S ("C) 0.08urfit (mm) 0.04urT (mm) 0.26urQ (mm) 0.03urDz (mm) 0.05ur (mm) 0.27

0.0

0.5

1.0

1.5

2.0

20 30 40 50 60 70 80Emission current (mA)

Beam

radi

us (m

m)

Gap = 2.2 mmGap = 2.6 mm

Fig. 10. Predicted electron beam radius versus emission current for theexperiments listed in Table 1.

602 T. Westover et al. / International Journal of Heat and Mass Transfer 50 (2007) 595–604

tainties in Fig. 10 and Table 3 assume the electron beam isperfectly symmetric and must be regarded simply as first-order approximations.

Plotting the heat flux profiles predicted by the finite-dif-ference model illustrates electron beam narrowing at highelectric fields because the electron flux at a given locationon the anode surface is proportional to the heat flux at thatpoint. Heat flux profiles for the experiments in which theapplied voltages were 286 and 446 V are shown on a loga-rithmic scale in Fig. 11. The heat flux profile (or electronflux profile) associated with the higher potential is clearlynarrower and actually exhibits a lower flux for radial loca-tions exceeding 2.5 mm from the point of peak temperaturerise.

The data in Table 3 also indicate that the relationshipbetween Q and q00max is nonlinear. For the experiment inwhich the vacuum gap was 2.6 mm, as the applied voltageincreased from 286 V to 446 V, Q increased by a factor of2.2 while q00max increased by a factor 6.8. If re were constant,then Q would be proportional to q00max for all experiments;however, because the electron beam narrows as the appliedvoltage increases, q00max grows much faster than Q. This phe-nomenon becomes particularly important when consider-ing anode performance at high rates of energy deposition.The resulting local heat fluxes on the anode surface canlead to significant local temperature rises in the anodeand possibly anode failure.

5. Conclusion

Significant anode heating resulting from electron fieldemission in vacuum has been observed. Experiments indi-cate the thermal energy deposition distribution can beapproximated by a Gaussian profile, and the electron beamradius becomes narrower as the electrical potential increasesand depends on emission conditions at the emitter tip. Thenarrowing of the electron beam radius may be at least par-tially explained by observing that higher electric potentialsaccelerate the electrons more rapidly toward the anode,thereby decreasing the electron transit time and reducingthe spread of the electron beam at the anode surface.

Several approximations were made to estimate the valueof re for the experiments in this study. As indicated by thedata in Fig. 7(b), the electron beam is not perfectly sym-metric. This effect was neglected in the present work butcould be included in future work to obtain more accuratemodels of the actual electron beam distribution. Anotherfactor not investigated in this work involves shifting ofthe location of the anode peak temperature with timeobserved in some experiments at a constant applied electricpotential. This phenomenon indicates that the electronbeam is affected by emission conditions at the emitter tip.Future experiments could study a possible correlationbetween total emission current and shifts in position ofthe electron beam over time.

Acknowledgments

The authors wish to thank numerous colleagues for theirassistance in this research. From the Nanoscale Thermo-Fluids Laboratory at Purdue University, Matt Maschmannand Sungwon Kim provided the nanotube samples, andMike Peterson and Jun Xu assisted in mounting the nano-tubes on the etched tungsten tips. The authors also appre-ciate Dr. Ronald G. Reifenberger and the NanoscalePhysics Laboratory (Purdue University) for the use ofequipment in their laboratory, including the Nikon Epip-hot 200 microscope.

We wish to acknowledge the use of Fluent Inc.’s solverFLUENT/UNS and its mesh generator GAMBIT in thiswork. We also gratefully acknowledge funding by theNational Science Foundation through the NIRT andCAREER programs.

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Table 3Summary of observations for experiments with vacuum gaps of 2.2 and2.6 mm

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