The Evaporation Rate of Filament Material from

Alternating Current Heated Filaments

A. D. Wilson

The operation of an incandescent filament on alternating current of the form, i = io sinwt, is reviewed.A model for the ac evaporation rate, in a vacuum of a uniformly heated straight wire, is presented interms of: (1) The known temperature fluctuations, and (2) The evaporation rate evaluated at the doand the ac mean filament temperatures. It is shown that the ac evaporation rate can be significantlygreater than the dc evaporation rate.spectral modulation is developed.

Introduction

The rate of evaporation of the filament material fora fixed filament temperature is well known' and will,in many instances, determine the "useful life" of anincandescent filament. The net rate of evaporationis a function of both the filament temperature and thepartial pressures of gases within the enclosing structure.It has been reported by Johnson2 that, in an analysisof several thousand "burned out" filaments, - investi-gators discovered that, in a vacuum lamp, 17% of thefilament mass had been evaporated while, in gas filledlamps, only 5% had been evaporated.

It is known that the heating of filaments until in-candescence with an alternating current causes a smallcyclic variation of the filament temperature at a fre-quency equal to twice that of the heating current. Thevariation in filament temperature causes a modulationin the radiant energy emitted by the filament. Therate of evaporation is an exponential function oftemperature, and thus, intuitively, we conclude thatfor small temperature variations about a mean value,the net rate of evaporation over one period of the sym-metrical temperature variation is greater than that rateof evaporation for the mean temperature. It is thisdifference which we shall calculate.

Elenbaas' has computed the ac rate of evaporation.However, this computation is based upon a mathe-matical representation for the rate of evaporation ofthe form R = A (T) B in which B was considered to beconstant over a temperature range AT; however, B isnot a constant. 4 Furthermore, only the first nonzero

The author is with IBM, General Products Division, Develop-ment Laboratory, Endicott, New York.

Received 5 July 1962.

A method of evaluating the ac evaporation rate in terms of the

term involving the temperature fluctuation has beenretained by Elenbaas in the expansion of the ac evapora-tion rate. The outcome of the present computation isin a closed form and thus is not limited bv this restric-tion. The exponent B can be approximated by therelation B vT, where v is a constant and T is thethermodynamic temperature of the metal which isbeing evaporated. A first-order estimate of the errorinherent in a computation based upon the assumptionB is equal to a constant over a temperature range ATis given in Table I. The error is defined as the dif-ference in the rates of evaporation at a temperatureTm + AT when B is constant, Bin, and when B is equalto v/T, respectively. The difference is normalizedwith respect to rate of evaporation evaluated with B aconstant. The normalized error is approximatelyequal to Bm(AT/Tm)2 . The error has been evaluated(Table I) for AT/T = 0.10 and the values of Bm havebeen taken from the work to follow below.

Nevertheless, Elenbaas' paper is thought by thisauthor to be the only quantitative discussion of theac rates of evaporation, and it is thus apparent that avery meager amount of attention has been focused onthis phenomenon.

Table I. First-Order Error in the ac Evaporation Rate When BIs Assumed to be a Constant

Tm( 0 K) Bm Error Bm (T°) (Percent)

2000 48.3 48.32400 40.2 40.22800 34.4 34.43200 30.1 30.1

October 1963 / Vol. 2, No. 10 APPLIED OPTICS 1055

Analysis

An equation which adequately describes the rate ofevaporation of tungsten and other metals, in a vacuum,as a function of temperature has been determined byLangmuir.5 The approximate form of Langmuir'scomputation which was found by Reimann6 to fit ex-perimental data closely is

R(T) 2A' exp [-qo/KTJ,

where

A'KR(T)q0

T

(1)

= constant, in units of g cm-2 sec-',= Boltzmann constant, 1.38 X 10-16 erg/OK,= rate of evaporation in g/cm2 see,= energy of evaporation per atom of metal at ' = 0,

erg, and= absolute temperature of the filament, K.

The physical assumptions which Langmuir made inhis original computation are: (i) the third law ofthermodynamics is valid in this problem, (ii) the re-flection of molecules from the surface of the metal takesplace to a negligible degree only; i.e., the "stickingcoefficient" is approximately one (1), and (iii) thevolume of the metal is very small as compared with thevolume of the vapor produced by evaporation. It isprobable that Eq. (1) represents the rate of evaporationof tungsten over a temperature range of 2000 to 3500'Kwith a considerable degree of accuracy when the propernumerical values of A' and qo are used. Langmuirfound for tungsten a value of 1.333 X 10-11 ergs forqo, the energy of evaporation at absolute zero per atomof metal. From the experimental data of Reimann, theexponent o is found to be numerically equal to the en-ergy of evaporation of Langmuir to three significant fig-ures. The thesis of this paper is not completely depend-ent upon the interpretation of the exponent o being theenergy of evaporation at absolute zero or the free energyof the system; however, it is assumed that q is constantover a temperature range AT, and that qo has beenjudiciously chosen so that Eq. () adequately describesthe temperature dependence of the rate of evaporationover a range in temperature AT. The numerical valueof q used here is that found by Langmuir and also byReimann.

To obtain an expression for the ac rate of evaporation,one first needs to determine the temperature of thefilament as a function of time. The dependence ofthe filament temperature on time, of an ac-heatedfilament, has been investigated by Corbino,7'8 Lang-muir,9 and others. A good approximation of theinstantaneous filament temperature, T is given by:

T Tm + AT, (2a)

AT = ATo sin(2wt + ), (2b)

where ATJ << T,.Equation (2) is derived for a straight wire of well-

behaved geometrical cross section which is uniformlyheated in a vacuum by a current of the form: i = iosinwt. It is assumed that the conduction and energyevaporation losses are negligible in comparison with theradiation losses. The magnitude of the peak-to-meanvalue of the temperature, ATo, is:

(3a)AT = - F PTm, 1L4(VI2 + 42)1/12j'

where

= 4T.mi3(Pfa/C'5A),a = Stefan-Boltzmann constant, 5.67 X 10-5 erg cm-2

sec'1 0K-4e = total emissivity,a = density of filament material g/cm3 ,C' = specific heat of filament, ergs/(g oK),p = periphery of filament cross section, cm,A = area of filament cross section, cm

2,

c = angular frequency, rad/sec, andT.e = absolute mean filament temperature, K.

The phase term 0 in Eq. (2) is: = tan-'(4/2w).If it is assumed that /2w < 10-1/2, then Eq. (3a)is approximated with an error of less than 5% by thesimple expression:

ATo = -[(toP/aA)(eTmi 4/C')(1/2w)J. (3b)

The validity of the assumption that V//2o < 10-1/2,can be established for a practical case by considering,for example, a filament operating at a mean thermo-dynamic temperature of 3000'K which has a periphery-to-cross section ratio (P/A) of 4000 cm-'. From theaccepted values for the physical constants of tungstenat 3000'K, the value of Ap/2 is computed: %I/2 101rad/sec. Therefore, for Eq. (3b) to be valid, the angu-lar frequency, w, of the heating current must satisfy therelation > (10-12) 101 rad/sec. A frequency of 60cps satisfies this condition. The restriction of 4'/2co <10-1/2 is equivalent to requiring (AToI/Tm) < (10-1/2/4),which is consistent with the derivation of Eqs. (2a),(2b), and (3a).

The negative sign in Eqs. (3a) and (3b) arises becauseof the choice of the sign of AT in Eq. (2a), and theparticular choice of sinusoid in Eq. (2b). The tempera-ture is observed to attain its maximum value, T +IATO1, at a time t + To, where t is the point in time ofmaximum current io and Tr = /w(i/4 -).

The calculation of the ac rate of evaporation is ac-complished by considering Eq. (1) above as the in-stantaneous rate of evaporation. This implies that theallowed temperature fluctuations are bounded above bya frequency so that the equilibrium between the vaporand solid metal is not disturbed. The substitution ofthe instantaneous temperature (Eq. (2)1, T = T +AT, into the expression for the instantaneous rate ofevaporation [Eq. (1)] yields:

R(Tmi, ATo, t) = R(Tmi) exp[(qo/K) ( TO°) sin(2wo + )] (4)

1056 APPLIED OPTICS / Vol. 2, No. 10 / October 1963

The term R(Tm1 ) is the evaporation rate at a tempera-ture Tmi. The temperature Tm is the mean tempera-ture of the ac heated filament. The relation of thistemperature (Tini) to the mean temperature of thefilament attained under the conditions of equal rmscurrent and zero temperature fluctuations (dc or"high" frequency ac current) will be given below.Equation (4) is the instantaneous rate of evaporationas a function of Tmi, ATo, and time t. Of more in-terest is the rate of evaporation averaged over onetime period of the temperature variation. Defining,a = (qo/K)(ATo/Tm1 2), and then expanding the termexp [a sin(2wt + ) ] of Eq. (4) in modified Besselfunctions, one obtains for the average rate:

(R(Tmi, AT,, t))0 = (1/T) R(T., AT,, t)dt (5)

(I?(Tm., A7 0 , t)), = [R(Tmf)/r I, {Io(0)

+ 2 E (j)2KI2K (a1) cos[2K(2wt)lK=1

-2 E (j)2KI (a) sin[(2K - 1)(2cot)] dt. (6)K=1

The two infinite series within the braces { } maybe shown to converge uniformly for all a; therefore,the operations of integration and summation in Eq. (6)may be interchanged. Performing this interchangeand then integrating term by term yields the simpleexpression below for the time average of the instantane-ous rate of evaporation:

(R(TmjATo)) = R(Tmi)Io(a), (7)

where Io(a) is the modified Bessel function of the firstkind of zero order. When there are zero temperaturefluctuations, a = 0 and o (0) = 1 ;10 therefore,(R(T. 1, 0)) = R(Tm1), as should be expected. Todetermine the ac rate of evaporation it is thereforenecessary to know only the physical properties of thefilament material, the geometrical parameters of thefilament, and the rate of evaporation at a temperatureTm. The parameter a may be evaluated from theexpression

a = [(qoa/K5)(P/A)( .,.2/C')( 1/2co) ], (8)

and thus Io (a) determined. a has now been taken tobe positive since there is no physical significance in theminus sign of Eq. (3a). Io(a) is a rapidly increasingfunction of the parameter a; thus, if a filament isselected with either a large ratio of periphery-to-crosssection, P/A, or a low current frequency, c, or both ofthese conditions simultaneously, a rate of evaporationwhich is considerably greater than the rate evaluated atthe filament mean temperature will occur.

To compare the evaporation rate of a filamentoperated on a "low-frequency" sinusoidal current to the

corresponding evaporation rate of a filament first oper-ated on a direct or "high-frequency" current, it is firstnecessary to determine a relation between the respectivemean temperatures. The two mean temperatures arenot equal. The difference is a second-order functionof the temperature variation, AT. Elenbaas" hasshown that, for two filaments operated at voltages sothat the rms value of the instantaneous voltage, v = Vosin wt, is equal to the dc voltage (or an rms ac voltageof a frequency so that ATo-0-), the temperature ratiois given by:

T effective = 1 _ (a + b (fTo/T1 )2,Tm2 voltages equal 4

(9)

where

Tm2= mean temperature of dc (or ac such that To - 0)heated filament,

a + u,u = (T/e)(de/dT),b = (T/p)(dp/dT), andp = specific resistivity.

Terms involving (To/Tm1)' with n > 2 have beenneglected in arriving at the approximate value forTmi/Tm2 in Eq. (9). One can also show, following anapproach similar to Elenbaas', that from a considera-tion of the energy balance equation (neglecting con-duction and evaporation losses) a similar approximaterelation for the ratios of the mean temperatures isobtained when the rms value of the current i = io sinwt is made equal (for purposes of comparison) to thefilament direct current (or an rms alternating currentof a frequency so that AT0 --o 0). The ratio of the meantemperature Tml of a filament with a temperaturefluctuation ATo to the mean temperature of the samefilament with no temperature fluctuations, T,2, underthe constraint of equal rms current is given by theapproximate equation:

Tm1AT'\ 2

- effective 1 - oTm2| currents equal \Tm]/

(10)

where

/a + b - 2b\4 ( a-b)

To demonstrate the difference in the rates of evapora-tion between a filament with temperature fluctuationsand one without, when operated at an equivalenteffective current, consider a tungsten filament in avacuum with a periphery to cross section ratio (P/A).

From Eqs. (3b) and (8), the quantities A To [f(P/A) ] anda [f(P/A) J, respectively, have been computed fortungsten at several temperatures and tabulated inTable II. To make these calculations, the specificheat C' was obtained from the relation C' = (4H'/67r)where H' is from the data of Jones and Langmuir.' 2

The total emissivity was obtained from the data of

October 1963 / Vol. 2, No. 10 / APPLIED OPTICS 1057

Table II. Normalized Temperature Fluctuations ATo and e and Corresponding lo(a) as a Function of Temperature for Tungsten

B ;:;

7Tm1 qo (ergs/cm, () ' PA) aKP/A) 10 3(P/A) X 102(P/A) X 10

(0 K) KT,.l e -OK) (g cm-' sec') (K-cm-cps) (cm-cps) (cm rad/sec) (cm-cps)

2000 48.3 0.259 2.66 7.48 X 10-14 0.552 1.33 1.40 3.512400 40.2 0.294 2.87 2.32 X 10-10 1.22 2.02 2.54 6.372800 34.4 0.321 3.07 7.20 X 10-8 2.28 2.80 4.12 10.323000 32.2 0.334 3.18 7.14 X 10-7 3.02 3.23 5.07 12.753200 30.1 0.341 3.22 5.33 X 10-6 3.95 3.72 6.23 15.653400 28.4 0.348 3.38 3.14 X 10-' 4.88 4.06 7.25 18.20

Forsythe and Watson'3 and the evaporation rate fromthe data of Reimann.' 4 No corrections for possibledifferences in the experimental temperature scales havebeen made (i.e., the data of Table II are as found in theliterature). Table II includes computed values of^I/(P/A) from Eq. (3a) and fmin/(P/A) where f insatisfies the condition mi = (10-l/2 /2)I and fmin =27rcomin. Constructed in Table III are numericalvalues of To, a, and o (a) for several filament tem-peratures as a function of multiples of fmin/(P/A).If it is desired to know the ac evaporation rate forfrequencies below fin,,i, keeping (P/A) a constant, thenit is necessary to use Eq. (3a) rather than Table IIIsince Table III is derived assuming Eq. (3b) to bevalid. As an example in the use of Tables II and III,consider a ribbon filament at 3000'K heated by 60 cpscurrent whose width is very much greater than thethickness, thus P/A 2 9/thickness. Let the thicknessbe chosen so that f/(P/A) = fi.n/(P/A), that is, thick-ness = 4.25 X 10-3 cm which follows from Table II.From Table III, one finds that at 3000'K and 60 cps(fnin) ATo =237 0 K, a = 2.54 and Io(a) = 3.39. Onecan thus conclude that the rate of evaporation of thefilament with the temperature fluctuation of 2370K(peak) is 3.39 times as large as a filament with nofluctuations, both operating at the same mean tempera-ture of 3000'K; that is, (R(Tmi, ATo)) = Io(2.54)R(Tm1 ) = 3.39 R(Tmi). Since we are interested incomparing the evaporation rates of the filament heatedby the same effective current, it is thus necessary todetermine R(T. 2) in terms of T making use of Eq.(10). Hence,

R(T. 2 ) = R(Tmi) exp V(--mJ a2s]- (11)

An expression similar to Eq. (11) can be computed forthe condition of equal effective voltage by making useof Eq. (9), and thus, in general,

(R(Tmi, ATo)) /- yKTmi 21R(Tm2) = Io(a) exp [ ) j (12)

where the correct value of y has been chosen dependingupon the type of comparison: equal effective voltage orcurrent.

For the above examples, one therefore finds for theratio, (R(Tm1 , ATo))/R(Tm2) a value 3.39 e-0 z

2.78. It has thus been shown that the evaporationrate of the example filament experiencing symmetricaltemperature variations is 2.78 times greater than theevaporation rate of the filament when it is experiencingno temperature variations, yet heated with the sameeffective current.

The ratio of the rates of evaporation evaluated forequal effective (1) current and (2) voltage is greaterthan or equal to one (1). We can show from Eqs. (9),(10), and (12) that this ratio is given by the equation

(R(Tm, ATo))I(T..2)Ii =exp 2b \/KTmi\ )l .(R(Tmi, AT0 ))IR(Tm 2 )V e pa - b\ qo 2 (13)

Recently, two quantitative papers discussing themodulation of radiant energy have appeared in theliterature. 5 " There is an interesting relation for awhich can be derived from this modulation and usedexperimentally to assist in evaluating the ac rate ofevaporation. The spectral modulation defined as

Table IlIl. ATo, a, and Io(a) as Functions of Frequency, fmin, and Temperature for a Fixed P/A

AT (K) a Io(a)

Tm ( 0 K) fmin 2fin

3fm. fmai 2fmin 3fin fmin 2fin 3fin

2000 157 78.5 51 3.80 1.90 1.27 9.53 2.13 1.452400 191 95.5 63.5 3.17 1.59 1.06 5.56 1.72 1.292800 221 110 73.6 2.71 1.36 0.91 3.84 1.52 1.223000 237 118 79 2.54 1.27 0.85 3.39 1.45 1.173200 252 126 84 2.38 1.19 0.78 2.94 1.40 1.153400 268 134 89.5 2.23 1.12 0.75 2.66 1.32 1.14

1058 APPLIED OPTICS / Vol. 2, No. 10 / October 1963

Energy emitted at T -Energy emitted at Tm''' A- Energy emitted at T

was computed from Planck's radiation equation byTolles and Ihrig. Their approximation is

Att lC2 (ATO) (14)

where

X = wavelength of the radiant energy, con,C2 = second radiation constant, 1.4388 cm °1.

From Eq. (14) a is computed in terms of the modula-tion M,:

a = (qo/1c2)(XMX). (15)

The product (XMx) is easily determined experimentallyfor a given filament and operating conditions. There-fore, a may be evaluated without use of Eq. (8) byusing Eq. (15) and experimental data (XM,) instead.The ac rate of evaporation can then be computedfrom Eq. (7).

Conclusions

The average rate of evaporation of filament materialfrom an ac heated filament has been computed interms of the temperature fluctuations and in terms of(1) the rate evaluated at the filaments mean tempera-ture, and (2) the rate evaluated at the mean tempera-ture a filament would attain if no temperature fluctua-tions were present and the respective effective currentswere made equal. A relation has been presented to

determine the parameter a, which is a function of thetemperature fluctuations in terms of an experimentallymeasurable quantity, the spectral modulation. Theac evaporation rate has been found to be greater thanthe d evaporation rate under the constraint of eitherequal effective current or voltage. Furthermore, theac rate of evaporation, under the constraint of equaleffective current is greater than the ac rate of evapora-tion under the constraint of equal effective voltage.

The author wishes to thank H. B. Ulsh of this labora-tory for several helpful discussions.

References1. H. A. Jones, I. Langmuir, and G. M. J. Mackay, Phys. Rev.

30, 201 (1927).2. R. P. Johnson, Phys. Rev. 54, 459 (1938).3. W. Elenbaas, Physica 5, 929 (1938).4. W. E. Forsythe and A. G. Worthing, J. Astrophys. 61, 153

(1925).5. Ref. 1, p. 206.6. A. L. Reimann, Phil. Mag. 25, 834 (1938).7. 0. M. Corbino, Physik Z. 11, 413 (1910).8. 0. M. Corbino, Physik Z. 12, 292 (1911).9. I. Langmuir, Gen. Elec. Rev. 17, 294 (1914).

10. For tabulated values of Io(a) see E. Jahnke and F. Emde,Tables of Functions (Dover, New York, 1945), p. 226.

11. W. Elenbaas, Physica 5, 932 (1938).12. HI. A. Jones and I. Langmuir, Gen. Elec. Rev. 30, 31.3 (1927).13. W. E. Forsythe and E. M. Watson, J. Opt. Soc. Am. 24, 117

(1934).14. A. L. Reimann, Phil. Mag. 25, 842 (1938).15. W. E. Tolles and L. C. Ihrig, J. Opt. Soc. Am. 47, 101 (1957).16. W. J. Condell and F. T. Byrne, J. Opt. Soc. Am. 47, 1135

(1957).

November is the Coblentz Commemorative Issue

The Publications of W. W. Coblentz on Infrared Spectroscopy

The principal work of the late W. W. Coblentz on infrared spectroscopy originally appeared in three Publications of the Carnegie

Institution of Washington. Publication No.35, issued in 1905, dealtwith his investigations atCornell Universityof the infrared absorp-

tion spectra of some 135 organic compounds, with shorter supplementary sections on transmission and reflection spectra. Publica-

tion No. 65 (1906) and Publication No. 97 (1908) are concerned with later work on the transmission and reflection spectra of minerals

and other inorganic compounds carried out at the U.S. Bureau of Standards.These classical investigations form the basis of most modern work on analytical infrared spectroscopy. The journals have long

been out of print, but, with the kind permission of the Carnegie Institution, they have recently been reprinted in a single volume

under the joint sponsorship of the Coblentz Society and the Perkin-Elmer Corporation.All concerned with modern applications of infrared spectroscopy to problems of chemical structure and analysis should be con-

versant with this early work, which provides the basis for much of the modern technique. To maintain a properly balanced outlook

on the subject, this volume should accompany the standard modern texts on the library shelf of the spectroscopy laboratory.

A limited number of copies of the first reprinting are still available at $3.50 per copy on application to Dr. H. B. Kessler, The Secre-

tary, c/o The Perkin-Elmer Corporation, Norwalk, Connecticut.

October 1963 / Vol. 2, No. 10 / APPLIED OPTICS 1059