INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 39 (2006) 2124 2134 oi:10.1088/0022-3727/39/10/024 3D moelling of heting of thermioni thoes y high-pressure r plsms M S Benilov 1, M Crpij 2 n M D Cunh 1 1 Deprtmento e Físi, Universie Meir, Lrgo o Muniípio, 9000 Funhl, Portugl 2 Lighting Tehnology Institute, University of Krlsruhe, Kiserstr. 12, 76128 Krlsruhe, Germny Reeive 4 Novemer 2005, in finl form 7 Mrh 2006 Pulishe 5 My 2006 Online t stks.iop.org/jphysd/39/2124 Astrt Numeril investigtion of stey-stte intertion of high-pressure rgon plsm with ylinril tungsten thoe is reporte. A whole zoo of very iverse moes of urrent trnsfer is revele. Detile results re given for the first five (three-imensionl) 3D spot moes, four of them rnhing off from the iffuse moe n one from the first xilly symmetri spot moe. Divergenes in the generl pttern of solutions, whih hve een present in preeing works, re resolve. Hypotheses on stility of stey-stte solutions, ville in the literture, re nlyse. It is foun tht these hypotheses provie n explntion of the ft tht the trnsition etween iffuse n spot moes is iffiult to reproue in the experiment ut they o not explin the inition tht it is the low-voltge rnh of the first 3D spot moe tht seems to our in the experiment. Thus, the question of stility of stey-stte solutions remins open: n urte stility nlysis, s well s itionl experimentl informtion is require. (Some figures in this rtile re in olour only in the eletroni version) 1. Introution Current trnsfer from high-pressure r plsms to thermioni thoes my our in iffuse moe, when the urrent is istriute over the front surfe of the thoe in more or less uniform wy, or in spot moe, when most of the urrent is lolize in one or more smll res (thoe spots). In ertin r urrent rnge, either of the moes n our. In the se of n xilly symmetri thoe, the iffuse moe is xilly symmetri while spot moes n e oth xilly symmetri ( spot t the entre of the front surfe of the thoe) or three-imensionl (3D) (n off-entre spot or system of two or more spots). It ws suggeste some yers go [1] tht n equte theoretil esription of multiple moes of urrent trnsfer to thermioni thoes oes not neessrily involve essentilly ifferent physil mehnisms ut is rther mthemtil question of fining non-unique solutions: n equte theoretil moel of urrent trnsfer to hot thoes must in some ses llow ifferent stey-stte solutions to exist for the sme onitions, whih esrie ifferent moes of urrent trnsfer. It ws shown tht suh multiple solutions exist in the frmework of the moel of nonliner surfe heting n generl pttern of solutions esriing vrious moes ws suggeste on the sis of ifurtion nlysis n generl onsiertions. In the susequent yers, the moel of nonliner surfe heting ws vlite y n extensive omprison with the experiment [2, 3] n hs eome wiely epte tool for moelling of intertion of high-pressure r plsms with thermioni thoes. At present, stey-stte xilly symmetri moes on xilly symmetri thoes hve een (numerilly) stuie in etil n re unerstoo reltively well (e.g. [2, 4, 5]); this pplies to oth iffuse n xilly symmetri spot moes. Numeril results of 3D spot moes hve strte to pper only reently [6 8]. In [6], ifurtion points hve een lulte in whih 3D stey-stte spot moes on xilly symmetri thoes rnh off from xilly symmetri moes. It ws shown tht 3D spot moes n rnh off from oth the iffuse moe n xilly symmetri spot moes. The 3D trnsient spots on ylinril thoe were simulte in [7]. In [8], results of simultions were given for 3D stey-stte spot tthe to roune ege of ylinril thoe. 0022-3727/06/102124+11$30.00 2006 IOP Pulishing Lt Printe in the UK 2124
3D moelling of heting of thermioni thoes The physis of 3D spots shoul e similr to the physis of xilly symmetri spots, whih hs een stuie in etil in [5]. The importne of omputtionl spets of 3D simultions hs onsierly erese in reent yers ue to the pperne of powerful omputers n vne softwre. There is, however, n spet in the moelling of 3D spot moes whih is still of primry interest, nmely, fining the generl pttern of ifferent moes. Among other things, this pttern is ritil for nlysis of stility of vrious moes. It is lso neee in orer to permit the ientifition of prtiulr moe of urrent trnsfer oserve in simultions or in experiment. Note tht the ville works iverge s fr s this pttern is onerne: 3D spot moes reporte in [8] rek off t ertin urrent inste of turning k or joining other moes, whih oes not fit in the pttern [1, 6] n is not ehviour typil of multiple solutions. Estlishing the pttern of ifferent 3D moes of steystte urrent trnsfer to thermioni thoes represents the gol of this work. The outline of the pper is s follows. The moel of nonliner surfe heting is riefly esrie in setion 2. In setion 3, multiple solutions esriing vrious moes re given n generl fetures of 3D solutions re nlyse. The iffuse moe n the first 3D spot moe uner typil experimentl onitions re lulte n nlyse in setion 4. Informtion on stility of stey sttes elonging to ifferent moes, ville in the literture, is nlyse in setion 5. Conluing remrks re given in setion 6. 2. The moel n its numeril reliztion A summry of equtions for the theory of intertion of thermioni thoes with high-pressure r plsms, se on the moel of nonliner surfe heting, is given in reent pper [9]. (Some itionl theoretil mterils n n online tool for simultion of the iffuse moe of urrent trnsfer evelope in the frmework of this theory n e foun on the Internet [10].) The moel n e riefly esrie s follows. A stey-stte temperture istriution in the oy of thoe is onsiere. Joule het proution in the oy of the thoe is neglete. The se of the thoe is mintine t fixe temperture T y externl ooling n the rest of the thoe surfe is in ontt with the plsm or the ol gs n is hete or oole respetively. Mthemtilly, the prolem mounts to solving the therml-onution eqution with the ounry onition (κ T)= 0 (1) T = T (2) t the se n with the nonliner ounry onition κ T n = q(t w,u) (3) t the rest of the thoe surfe. Here κ is the therml onutivity of the thoe mteril, n is iretion lolly orthogonl to the thoe surfe n irete outsie the thoe n q = q(t w,u)is given funtion of the thoe surfe temperture T w n of the ner-thoe voltge rop U, whih esries the ensity of the energy flux to the prt of the thoe surfe tht is in ontt with the r plsm n the ol gs. Funtions q = q(t w,u)n j = j(t w,u)esriing the ensity of eletri urrent to the thoe surfe re lulte y mens of equtions summrize in [9] (see lso [2, 6, 11]) n re not isusse here. We only nswer frequently ske question of erivtion of expressions given in the pper [11] for the ensities of fluxes of the kineti energy elivere to, or remove from, the thoe surfe y ifferent plsm speies. These expressions re otine y mens of verging the prout of the prtile kineti energy times the prtile veloity; etile erivtion hs een poste on the Internet [10]. Note tht, sine the verge vlue of prout is not equl to the prout of verge vlues of multipliers, the ensity of flux of kineti energy of prtiles whose istriution is ssume to e Mxwellin (whih is the se of plsm eletrons, the emitte eletrons n neutrl toms leving the thoe surfe) is equl to 2kT J rther thn 3kT /2J s one oul expet intuitively (here T is the temperture of the speies in question, J is the numer ensity of the flux of this speies n k is the Boltzmnn onstnt). Eqution (1) is written without ounting for Joule het proution in the thoe oy. This pproximtion is justifie provie tht the eletri power issipte insie the thoe is muh smller thn the power supplie to the thoe surfe or, equivlently, tht the voltge rop insie the thoe is muh smller thn U. If the thoe opertes in the iffuse moe then the voltge rop insie the thoe my e estimte y orer of mgnitue s Iρh/A, where I is the r urrent, ρ is the eletril resistivity of the thoe mteril, h is the thoe height n A is the re of the thoe ross setion. Let us ssume for the estimte tht the r urrent is 10 A, the thoe height is 1 m n the thoe ross setion is irle of rius 1 mm. Eletril resistivity of tungsten inreses from 5.3 10 8 m t room temperture to 94 10 8 m t 3000 K [12]; let us ssume ρ = 10 6 m for the estimte. Then Iρh/A = 32 mv n the Joule heting in the thoe oy my e sfely neglete. If the thoe opertes in the spot moe then there re two omponents of the voltge rop insie the thoe: the voltge rop in the expnsion zone ( omin insie the thoe jent to the spot in whih the urrent ensity ereses from vlues typil for the spot to those of the orer of I/A) n the voltge rop in the ulk of the thoe oy, whih is of the orer of Iρh/A. In orer to get resonle estimte of the first omponent, let us ssume tht the urrent expnsion is semi-spheril, then the orresponing voltge rop equls Iρ/2πr, where r is the rius of the spot. Assuming r = 10 µm, one fins Iρ/2πr = 160 mv n the Joule heting in the thoe oy my e neglete in the spot moe s well. The oe evelope for numeril solution of the oveesrie prolem onsists of two moules. The first moule lultes funtions q = q(t w,u) n j = j(t w,u) y solving equtions esriing the ner-thoe lyer in highpressure r plsm whih re summrize in [9]. Note tht this moule provies t in wie rnge of vlues of surfe temperture (up to 5000 K or higher) n ner-thoe voltge rop (up to severl hunre volts or higher) n for vrious plsm-prouing gses (most of the pure monotomi gses 2125
M S Benilov et l n numer of mixtures; see mnul of the moelling tool [10]). This moule is written in Fortrn. The seon moule lultes the temperture istriution insie the thoe oy n t the surfe y solving the nonliner ounry-vlue prolem (1) (3) in the thoe. This moule is relize using the ommeril finite element softwre FEMLAB. The two wys of onneting the two moules re use in ifferent versions of the oe. One wy onsists of running the moules sequentilly. In the frmework of this pproh, the first moule genertes files with t on funtions q = q(t w,u) n j = j(t w,u) n the seon moule mkes use of these files. The other wy onsists of invoking the first moule from insie the seon one. In the frmework of this pproh, the first moule is relize either s n exeutle file or, preferly, s Mtl-omptile ll file. 3. Generl fetures of 3D solutions 40 30 20 10 iffuse moe entrl spot entrl spot n two ege spots four ege spots three ege spots two ege spots one ege spot Numeril results given in this work refer to tungsten thoe in the form of right irulr yliner ( ro) operting in n rgon plsm. Dt on therml onutivity n emissivity of tungsten hve een tken from [13] n [14], respetively; the vlue of 4.55 ev ws ssume for the work funtion of tungsten. While eling with temperture istriutions possessing plnr symmetry, one n restrit the lultion omin to hlf of the ro. In ition to svings in RAM n CPU time, this fixes the zimuthl position of the spot system n hene improves the onvergene. A finite element mesh utomtilly generte y FEMLAB ws lolly refine in the viinity of eh spot in orer to otin goo ury. The numer of finite elements routinely use in omputtions ws roun 40 000, whih ws presumly suffiient to otin solution to n ury of few per ent t worse. In this setion, results re given n isusse for (tungsten) thoe of rius R = 2 mm n height h = 10 mm; geometry onvenient for the illustrtion of generl fetures of multiple solutions to the prolem onsiere. The (rgon) plsm pressure is ssume to e tmospheri. 3.1. Current voltge hrteristis Current voltge hrteristis of vrious moes of urrent trnsfer to tungsten thoe of this geometry operting in tmospheri-pressure rgon plsm re shown in figure 1. A mgnifition of the entrl prt of figure 1 is shown in figure 2. Current voltge hrteristis of the iffuse moe (whih is xilly symmetri) n the first xilly symmetri spot moe (whih orrespons to spot t the entre of the front surfe of the thoe) shown in figure 1 oinie with those lulte in [5] n re not isusse here. We only note tht the urrent voltge hrteristi of the iffuse moe hs two rnhes, one flling n the other rising, seprte y point of minimum, while the urrent voltge hrteristi of the first xilly symmetri spot moe hs two rnhes, low-voltge one n high-voltge one, seprte y turning point. Note tht the iffuse moe n the first xilly symmetri spot moe hve een lulte in this work y mens of the finite-element softwre FEMLAB n in [5] y 10 1 10 2 10 3 10 4 I (A) 10 5 Figure 1. Current voltge hrteristis of ifferent moes of urrent trnsfer. R = 2 mm, h = 10 mm. Soli line: iffuse moe. Dshe line: first xilly symmetri spot moe. Dotte line: 3D spot moes whih rnh off from the iffuse moe. Dshe-otte line: 3D spot moe whih rnhes off from the first xilly symmetri spot moe. Cirles n squres: ifurtion points. 20 16 12 4 1.3 2 4 3 k = 1 I (A) 0 125 250 375 500 Figure 2. Trnsition etween xilly symmetri moes n 3D spot moes. R = 2 mm, h = 10 mm. Full irles: ifurtion points t whih 3D moes rnh off from the iffuse moe. Squres: ifurtion points t whih 3D moes rnh off from the first xilly symmetri spot moe. k etermines the numer of spots t the ege of the front surfe of the thoe existing in eh moe. Open irles: sttes of 3D spot moes whih re shown in the susequent figures. mens of Fortrn oe implementing n itertive pproh se on finite-ifferene numeril sheme. Results given y the two oes oinie to very high ury, whih ttests to urte opertion of oth oes. Also shown in figures 1 n 2 re ifurtion points positione on the iffuse n first xilly symmetri spot 2 2126
3D moelling of heting of thermioni thoes moes, lulte s esrie in [6]. We remin tht, oring to the generl pttern estlishe in [1, 6], 3D spot moes on n xilly symmetri thoe rnh off from xilly symmetri moes. In other wors, vlue of eletri urrent exists for every 3D spot moe suh tht t this urrent the thoe temperture istriution orresponing to this 3D moe turns xilly symmetri n extly oinies with the thoe temperture istriution orresponing to n xilly symmetri moe. Suh violtions of unity of solutions re well known in mthemtis n re terme ifurtions; in these terms, the ove-mentione vlue of eletri urrent (s well s the orresponing vlue of U) my e lle ifurtion point. The zimuthl epenene of temperture istriutions orresponing to 3D spot moes is esrie in the viinity of ifurtion points y the ftor os kφ, where k = 1, 2, 3,... n φ is the zimuthl ngle. Vlues of k orresponing to eh ifurtion point re inite in figure 2 (the ifurtion points orresponing to k = 1 n k = 3 positione on the first xilly symmetri spot moe oinie with the grphil ury). Also note tht, stritly speking, wht rnhes off t eh ifurtion point is fmily of 3D spot moes rther thn single moe; however, these moes re ientil to the ury of rottion n n therefore e onsiere s single moe with n ritrry zimuthl position of the spot system. In ontrst to the xilly symmetri moes n ifurtion points, 3D spot moes shown in figures 1 n 2 hve not een lulte previously. Eh one of these moes rnhes off from n xilly symmetri moe, in or with the generl pttern suggeste in [1,6]. Five 3D spot moes re represente in figures 1 n 2, four of them rnhing off from the iffuse moe n one from the first xilly symmetri spot moe. Points t whih this rnhing ours oinie extly with ifurtion points tht re preite y ifurtion theory [6] n re shown in figures 1 n 2 y full irles n squres. At lrge istnes from the ifurtion points, ll spot moes ten towrs the region of smll urrents n high voltges, i.e. pproh the xis of voltges. Generl onsiertions explining this ehviour hve een given in [1]. On the other hn, ehviour of ifferent moes in the viinity of the ifurtion points is essentilly ifferent: while the first two 3D moes ifurte from the iffuse moe with positive erivtive U/I, the thir n fourth 3D moes ifurte from the iffuse moe with negtive erivtive; the first 3D moe ifurtes from the first xilly symmetri spot moe with negtive erivtive. In the viinity of ifurtion points, ner-thoe voltges n therml regimes of 3D moes re lose to those of xilly symmetri moes. On the other hn, the ft tht ner-thoe voltges of two moes re lose oes not neessrily men tht therml regimes of these moes re lso lose. For exmple, the urrent voltge hrteristi of the low-voltge rnh of the first xilly symmetri spot moe in figure 1 is very lose to the hrteristi of the iffuse moe; however, therml regimes of these two moes re ifferent. 3.2. Therml regime of the thoe Evolution of the temperture istriutions long the ifferent moes represente in figures 1 n 2 is shown in figures 3 9. Evolution of xilly symmetri istriutions, shown in figures 3 n 4, hs een nlyse in [5] n is not isusse here (these figures re shown here for the ske of ompleteness). Evolution of 3D temperture istriutions is shown in figures 5 9. The first stte shown for eh moe orrespons to the ifurtion point t whih this moe rnhes off from the iffuse moe or from the first xilly symmetri spot moe, the seon n thir sttes re inite in figure 2 y open irles n the fourth stte for eh moe orrespons to the mximl voltge rop vlue shown in figure 1, i.e. to U = 40 V. The first stte shown in figure 5, eing ifurtion point, elongs to the iffuse moe n is hrterize y nerly onstnt temperture of the front surfe of the thoe. The seon stte, eing still lose to the ifurtion point, is hrterize y smooth 3D perturtions. As the istne from the ifurtion point grows (the thir stte), the perturtions evolve into well-efine spot t the ege of the front surfe of the thoe. As the istne from the ifurtion point grows further n the urrent voltge hrteristi tens towrs the region of smll urrents n high voltges (the fourth stte), the spot shrinks n eomes righter. Evolution of the temperture istriution long the seon, thir n fourth 3D spot moes rnhing off from the iffuse moe (figures 6 8) is similr, the ifferene eing tht these moes re hrterize y two, three n four spots (rther thn one) positione t the ege of the front surfe of the thoe. The first stte shown in figure 9, eing ifurtion point, elongs to the first xilly symmetri spot moe n is hrterize y spot t the entre of the front surfe of the thoe. As the istne from the ifurtion point grows, two itionl spots emerge positione opposite eh other t the ege of the front surfe. Sine the spots shrink when the urrent voltge hrteristis ten towrs the region of smll urrents n high voltges, one shoul expet tht in this region the intertion of eh spot with istnt prts of the thoe n with other spots (if they exist) eomes less importnt n the spot strts to ehve like solitry one. Let us hek whether this n e oserve in the moelling results. Aoring to generl theory [5], the temperture insie solitry spot is lose to the limiting temperture T 2, whih is the vlue of the surfe temperture strting from whih thermioni ooling exees the omine heting y the ions n plsm eletrons n funtion q turns negtive. In this onnetion, the mximum temperture of the thoe surfe n the orresponing limiting temperture T 2 for the first two 3D spot moes re shown in figure 10. Note tht the mximum temperture ours t the ege of the front surfe t φ = 0; if the istne from the ifurtion point is not too smll n spot(s) re lrey well efine, this temperture my e interprete s the mximum temperture insie the ege spot(s). Sine T 2 = T 2 (U), vlues of T 2 shown in figure 10 for eh moe were lulte with the use of the urrent voltge hrteristi, U = U(I), of the respetive moe: T 2 = T 2 [U(I)]. Also shown in figure 10 re the mximum temperture for the iffuse moe n the ifurtion points in whih the 3D moes rnh off from the iffuse moe. The mximum temperture ttine y the thoe in eh moe remins elow the upper limit of the thoe temperture s it shoul. The ifferene etween these two tempertures is smll, n inition tht the spots inee strt ehving like solitry ones s the urrent ereses. 2127
M S Benilov et l 5000 K 4500 K 4000 K 3500 K 3000 K 2500 K 2000 K 1500 K 1000 K 500 K Figure 3. Temperture istriution for vrious sttes of the iffuse moe. R = 2 mm, h = 10 mm. () I = 15 A. () 110 A. () 10 ka. () 67 ka. Figure 5. Temperture istriution for vrious sttes of the first 3D spot moe whih rnhes off from the iffuse moe. R = 2 mm, h = 10 mm. () ifurtion point. () U = 12 V. () 15V.()40V. The temperture r is shown in figure 3. Figure 4. Temperture istriution for vrious sttes of the first xilly symmetri spot moe. R = 2 mm, h = 10 mm. () I = 15 A, low-voltge rnh. () 110 A, low-voltge rnh. () 110 A, high-voltge rnh. () 41 A, high-voltge rnh. The temperture r is shown in figure 3. Temperture n eletri urrent ensity istriutions long the line φ = 0 on the front surfe of the thoe re shown in figures 11 n 12, respetively, for the first two 3D spot moes t U = 40 V. (In these figures, r is the istne Figure 6. Temperture istriution for vrious sttes of the seon 3D spot moe whih rnhes off from the iffuse moe. R = 2 mm, h = 10 mm. () ifurtion point. () U = 12 V. () 15V.() 40 V. The temperture r is shown in figure 3. from the entre of the thoe, so r = 2 mm orrespons to the ege. U = 40 V hs een hosen s the highest vlue of the ner-thoe voltge rop shown in figure 1, i.e. U = 40 V orrespons to the lowest vlue of eletri urrent for eh 2128
3D moelling of heting of thermioni thoes Figure 7. Temperture istriution for vrious sttes of the thir 3D spot moe whih rnhes off from the iffuse moe. R = 2 mm, h = 10 mm. () ifurtion point. () U = 12.7V.() 15V. () 40 V. The temperture r is shown in figure 3. Figure 9. Temperture istriution for vrious sttes of the first 3D spot moe whih rnhes off from the first xilly symmetri spot moe. R = 2 mm, h = 10 mm. () ifurtion point. () U = 15.2V.() 20V.() 40 V. The temperture r is shown in figure 3. T w (10 3 K) 4.8 upper limit of the thoe temperture 1st 3D spot moe 4.4 2n 3D spot moe iffuse 4 10 2 10 3 I (A) Figure 10. Mximum temperture of the thoe surfe in ifferent moes. R = 2 mm, h = 10 mm. Cirles: ifurtion points. Figure 8. Temperture istriution for vrious sttes of the fourth 3D spot moe whih rnhes off from the iffuse moe. R = 2 mm, h = 10 mm. () ifurtion point. () U = 13.4V. () 15V.() 40 V. The temperture r is shown in figure 3. moe.) While within the spots the temperture istriutions re lose mong themselves, they iffer ppreily t lrge istnes from the spot. The ifferene in the istriutions of the eletri urrent ensity within the spots is more pronoune thn the ifferene in tempertures, whih is onsequene of strong epenene of funtion j on T w. One n onlue tht the spots still ontinue to e ffete y the thoe geometry uner the onitions onsiere, i.e. they re not fully evelope solitry spots yet. In ll the moes epite in figures 4 9 spots re positione either t the entre of the front surfe, i.e. on the xis 2129
M S Benilov et l 5 T w (10 3 K) 4 3 1st 3D spot moe 2n 3D spot moe 40 30 iffuse moe, h = 10 mm 1st 3D spot moe, h = 10 mm iffuse moe, h = 10.2 mm 1st 3D spot moe, h = 10.2 mm 2 20 1 0 1 r (mm) 2 Figure 11. Distriution of the temperture within the spot. R = 2 mm, h = 10 mm. U = 40 V. 6 1st 3D spot moe 2n 3D spot moe 4 2 0 j (10 8 A/m2) r (mm) 1.5 1.6 1.7 1.8 1.9 2 Figure 12. Distriution of the urrent ensity within the spot. R = 2 mm, h = 10 mm. U = 40 V. of symmetry of the thoe, or t the ege of the front surfe. The ft tht the ege of the front surfe of ro thoe, eing point where onitions for therml-onution het removl re the worst, is one of the preferre points of spot tthment is well known from the experiment; see, e.g. [7, p 61], [15, p 1649]. On the other hn, one n expet tht stey-stte spots n lso tth somewhere etween the ege n the entre; however, orresponing moes hve een left eyon the sope of the present pper. For exmple, the 3D moe rnhing off from the first xilly symmetri spot moe t the ifurtion point ssoite with k = 1 is likely to e moe with two opposite spots, one of them eing positione t the ege n the other etween the ege n the entre [6]. 10 0 10 20 I (A) 30 Figure 13. Current voltge hrteristis of the iffuse moe n the first 3D spot moe. R = 0.75 mm. Cirles: ifurtion points t whih the first 3D moe rnhes off from the iffuse moe. 4. Solutions for thin thoes 4.1. Clultion results Cthoes est stuie in the experiment (e.g. [3]) were ppreily slimmer n higher thn those onsiere in the previous setion n were operte t rgon pressure higher thn 1 r. As representtive exmple, let us onsier tungsten thoe of rius R = 0.75 mm n height h = 20 mm operting in the rgon plsm uner the pressure of 2.6 r. Current voltge hrteristis of the iffuse moe n the first 3D spot moe on thoes of rius R = 0.75 mm n heights 10 mm, 10.2 mm n 20 mm, operting in the rgon plsm uner the pressure of 2.6 r, re shown in figures 13 n 14. One n see from figure 13 tht n inrese in the thoe height from 10 to 10.2 mm proues no notiele effet on the urrent voltge hrteristi of the iffuse moe: the orresponing urves oinie. The effet proue on the hrteristi of the first 3D spot moe is more onsierle. First, the turning point is slightly shifte in the iretion of smll urrents. Seon, there is onsierle shift in the position of the ifurtion point t whih the first 3D spot moe rnhes off from the iffuse moe: it hnges from 5.9 to 2.9 A. The urrent voltge hrteristis of the thoe of height h = 20 mm, shown in figure 14, re qulittively similr to those shown in figure 13; the ifferene is tht the first ifurtion point hs move into the region of very high voltges (well in exess of 100 V) n is sent from the grph, while the turning point hs move further into the region of lower urrents. One n view the first 3D spot moe s onsisting of two rnhes, high-voltge rnh whih exists in the urrent rnge 0 <I I t (i.e. from zero up to urrent I t orresponing to the turning point) n low-voltge rnh whih exists in the urrent rnge I I I t (i.e. etween the first ifurtion point n the turning point). In the se of wie thoe, shown in figures 1 n 2, the first ifurtion 2130
3D moelling of heting of thermioni thoes 50 iffuse moe 1st 3D spot moe ege spot high [8] 40 ege spot low [8] pyrometry [8] proes [8] 30 25 20 15 10 20 5 10 0 2 4 6 8 10 12 I (A) Figure 14. Current voltge hrteristis of the iffuse moe n of the first 3D spot moe. R = 0.75 mm, h = 20 mm. Points: experimentl t on ner-thoe voltge otine y mens of pyrometri n proe mesurement. point n the turning point re not very istnt n the lowvoltge rnh is not well pronoune (i.e. the urrent rnge I I I t in whih this rnh exists is rther nrrow). As the thoe eomes thinner, the urrent rnge etween the first ifurtion point n the turning point of the first 3D spot moe expns n the low-voltge rnh eomes well pronoune. The greter prt of the urrent voltge hrteristi of the low-voltge rnh on thin thoe is virtully inistinguishle from the hrteristi of the iffuse moe. The urrent voltge hrteristi of the first 3D spot moe on thin thoe shown in figure 14 is very similr to the hrteristi of the first xilly symmetri spot moe on wie thoe lulte n isusse in [5] n shown lso in figures 1 n 2 of the present work. An nlysis of the temperture istriutions on the first 3D spot moe on thin thoe lso revels suh similrity: the spot is hot n well pronoune on the high-voltge rnh n is oler n somewht iffuse on the low-voltge rnh, similrly to wht ws foun in lultions of the first xilly symmetri spot moe on wie thoe [5] n is shown in figure 4 of the present work. In [8], simultion results hve een reporte for onitions similr to those of figure 14. Three moes hve een foun: iffuse moe n two ege spot moes terme lowtemperture spot moe n high-temperture spot moe. Current voltge hrteristis of the spot moes tken from figure 7 of [8] re epite in figure 14. Both moes ese to exist (rek off) s the r urrent exees ertin vlues, whih re out 4.5 n 5.5 A for low n high-temperture spot moes, respetively. The pttern of spot moe solutions foun in [8] oes not onform to the pttern estlishe in [1,6] n is not typil for multiple solutions (normlly solution oes not just ispper ut rther turns k or joins nother solution). Results of the present work o not onfirm this pttern: figure 14 suggests 0 0 1 2 3 4 5 6 7 I(A) Figure 15. Current voltge hrteristis of the iffuse moe. R = 0.3 mm, h = 20 mm. Points: experimentl t on ner-thoe voltge otine y mens of pyrometri n proe mesurement [8]. Dshe line: moelling of the present work. tht spot moe solutions hve een foun in [8] in only prt of their existene regions; these solutions o not represent seprte moes ut rther re prts of the high-voltge n low-voltge rnhes of the first 3D spot moe; the rek off of spot moe solutions reporte in [8] is numeril n not physil effet. In ition to the ove-isusse ifferene in pttern, the spot moe solutions of the present work n of [8] lso mnifest quntittive ifferenes. In prtiulr, one n see from figure 14 tht vlues of the ner-thoe voltge rop orresponing to eh rnh of the first 3D spot moe, lulte in the present work, re somewht higher thn those reporte in [8]. This ifferene my originte in low ury of numeril simultion n/or ifferenes in funtions q(t w,u) n j(t w,u) use in the present lultions n in [8]. In orer to illustrte the role of ifferenes in funtions q(t w,u)n j(t w,u), we reproue in figure 15 t on the iffuse moe shown in figure 12 of [8]. The experimentl points re onnete y the soli or otte lines n represent t on the thoe fll erive from pyrometri temperture mesurements or t otine y ifferent proe mesurements performe immeitely efore n fter the temperture mesurements. The urrent voltge hrteristi lulte y the uthors [8] oinies extly with the hrteristi given y pyrometri mesurements; moelling results of the present work re epite y the she line. One n see tht the ner-thoe voltge rop lulte in the present work gin is somewht higher thn tht lulte y the uthors [8]. Sine the (xilly symmetri) numeril simultion of the iffuse moe is quite urte, one shoul onlue tht this ifferene stems from ifferenes in funtions q(t w,u) n j(t w,u) use in the lultions. It shoul e emphsize, however, tht oth the urrent voltge hrteristi of the present work n tht lulte y the uthors [8] onform to the experimentl 2131
M S Benilov et l t within experimentl error, whih s reiility to the pproh onsiere. 4.2. Comprison with the experiment A quntittive greement of the present moel with the experiment for the se of iffuse moe hs lrey een estlishe [2, 3]; figure 15 in this sense represents just one more exmple. As fr s the spot moe is onerne, quntittive greement n hrly e expete. On one hn, this is onsequene of limittions of the moel of nonliner surfe heting, in whih only the voltge rop in thin ner-thoe lyer is onsiere. The ltter is likely to e resonle pproximtion for the iffuse moe; however, in the se of spot moe the eletron temperture in the spot is quite high n perturtions re not neessrily onfine to thin ner-thoe lyer. There re lso iffiulties on the experimentl sie: even suh si quntity s the nerthoe voltge rop nnot e mesure iretly; thoe spot is iffiult ojet for experimentl investigtion ue to its smll imensions n ility to pper t n ritrry position; itionl iffiulties re use y the poor reprouiility of the iffuse-spot trnsition. The sene of quntittive greement etween the moel of nonliner surfe heting n the experiment in the se of the spot moe n e immeitely seen from the ft tht the ner-thoe voltge rop mesure in the spot moe is lower thn tht for the iffuse moe n the ifferenes re out 6 V t 1.5An1Vt6A [15]. No suh ifferenes hve een etete either in the present moelling or in [8]: the voltge rop on the low- or high-voltge rnhes of the spot moe is lose to or, higher respetively, thn tht in the iffuse moe. On the other hn, the ifferene etween moelling n the experiment is not rmti, whih n e seen from figure 14 where experimentl t on the ner-thoe voltge rop in the spot moe tken from figure 17 of [8] re epite. One n see tht the urrent voltge hrteristi of the low-voltge rnh of the first 3D spot moe is in resonle greement with the experiment, while vlues of the ner-thoe voltge rop orresponing to the high-voltge rnh exee the experimentl vlues. It is interesting to note tht the ltter oes not pply to results of the moelling [8]: one n see from figure 14 tht the urrent voltge hrteristis of oth the low- n highvoltge rnhes lulte in [8] re in resonle greement with the experiment. However, the uthors [8] elieve tht it is the low-voltge rnh of the spot moe whih is oserve in the experiment sine the thoe surfe temperture whih they lulte for the high-voltge rnh is fr eyon the melting temperture of tungsten. 5. Stility onsiertions 5.1. Generl resoning It hs een shown in the preeing setions tht the prolem of urrent trnsfer to thermioni thoes hs whole zoo of very iverse stey-stte solutions. A question rises s to whih of these solutions re stle n n e oserve in the experiment. Unfortuntely, theory of stility of solutions to the prolem onsiere is sent. 16 14 12 10 iffuse moe, stle setion 1st ifurtion point iffuse moe, unstle setion 1st 3D spot moe, stle setion turning point 1st 3D spot moe, unstle setion 0 200 400 I (A) 600 Figure 16. Trnsition etween the iffuse moe n the first 3D spot moe. R = 2 mm, h = 10 mm. The uthors [8] ssume, on the sis of rguments stemming from Steenek s priniple of minimum voltge (power) for ishrges with fixe urrent (e.g. [16, pp 184 5]), tht moe with the lowest ner-thoe voltge rop is the preferre one. However, Steenek s minimum priniple ers no reltion to funmentl physil priniples; lthough some uthors (e.g. [17 19]) seem to elieve tht it n e prove y methos of thermoynmis, we hve not enountere ny suh proof in the literture. Eqully, Steenek s minimum priniple ers no reltion to stility theory. In summry, this priniple is just n ritrry ssumption n rguments se on this priniple re not onvining. In [1], hypotheses onerning stility of ifferent steystte solutions hve een put forwr on the sis of generl trens typil for nonliner issiptive systems. These hypotheses n e summrize s follows. The iffuse moe is stle ginst smll perturtions eyon the first ifurtion point (t I>I, I eing the urrent orresponing to the first ifurtion point) n unstle t lower urrents (t I<I ). The ifurtion whih ours in the first ifurtion point is suritil uner the onitions onsiere, i.e. the first 3D spot moe rnhes from the iffuse moe into the rnge I > I in whih the iffuse moe is stle. By nlogy with other prolems in whih suritil ifurtions our (e.g. [20]), one oul expet tht the initil setion of the first 3D spot moe is unstle. One oul lso expet tht the hnge in stility of the first 3D spot moe ours t the turning point: this moe is stle eyon the turning point n unstle t lower voltges. The seon n ll the susequent 3D moes re expete to e unstle. In the frmework of these hypotheses, the trnsition etween the iffuse n spot moes uner onitions of figure 1 ours s shown in figure 16. At smll urrents, the ishrge urns in the spot moe. With n inrese of urrent, trnsition to the iffuse moe ours t I = I t (in the turning point). At high urrents, the ishrge n urn only in the iffuse moe. With erese in urrent, trnsition to the first 3D spot moe 2132
3D moelling of heting of thermioni thoes Tle 1. Rnges of existene n/or stility of the iffuse moe n the high-voltge rnh of the first 3D spot moe. R = 0.75 mm. Stle high-voltge rnh Stle high-voltge rnh Cthoe height of the first 3D spot moe, of the first 3D spot moe, Stle iffuse ( mm) unstle iffuse moe(a) stle iffuse moe(a) moe(a) 10 I<5.9 5.9 <I<27.0 I>27.0 10.2 I<2.9 2.9 <I<26.1 I>26.1 20 I<11.1 I>11.1 ours t I = I. Sine I <I t, the trnsition etween the iffuse moe n the first 3D spot moe mnifests hysteresis in the urrent rnge I I I t. 5.2. Comprison with the experiment When pplie to thin thoes, the ove hypothesizing suggests tht the low-voltge rnh of the first 3D spot moe is unstle n the high-voltge rnh is stle. For onveniene, rnges of existene n/or stility of the iffuse moe n the high-voltge rnh of the first 3D spot moe uner onitions of figures 13 n 14 ( tungsten thoe of rius R = 0.75 mm operting in the rgon plsm uner pressure of 2.6 r) re summrize in tle 1. For the thoe of height h = 20 mm, whih is representtive experimentl exmple, the iffuse moe is stle. (More preisely, the iffuse moe is stle in the whole urrent rnge in whih it is shown in figure 14; it loses stility t very low urrents orresponing to very high voltges well in exess of 100 V.) Thus, there re two moes in the rnge I I t = 11.1 A whih re stle ginst smll perturtions, the iffuse moe n the high-voltge rnh of the first spot moe. Only the iffuse moe is possile t I>11.1A. Experiments on iffuse-spot trnsition re usully performe in limite urrent n voltge rnge, sy I 10 A n U 100 V. On the sis of the ove hypothesizing, one shoul ssume tht two stle (ginst smll perturtions) moes exist in the whole rnge of onitions of suh experiments, the iffuse moe n the high-voltge rnh of the spot moe. If the experiment is performe uner well-ontrolle qusi-sttionry onitions, moe whih hs ourre immeitely fter the ignition of the ishrge will e mintine uring the whole experimentl run. The moe hnge n our only ue to finite perturtions. For exmple, if experimentl prmeters (e.g. the r urrent or the plsm pressure) hve hnge rpily, it is possile tht the ishrge will our, fter the stey-stte hs een reovere, not in the originl moe ut in the other one. If the moe hnge is systemtilly oserve in suh n experiment uner qusisttionry onitions, it mens tht the experiment is not wellontrolle. For exmple, non-uniformities of the thoe surfe (e.g. protrusions) n provoke moe hnges whih re not esrie y the ove resoning. The ove hypothesizing explins the generl tren tht the trnsition etween the iffuse n spot moes is iffiult to reproue in the experiment. On the other hn, thishypothesizing preits tht the low-voltge rnh of the first 3D spot moe is unstle n the high-voltge rnh is stle, whih ontrits the inition tht it is the low-voltge rnh of the first 3D spot moe tht seems to our in the experiment (see isussion in setion 4.2). This ontrition sts outs on the prt of the ove hypothesis onerning (in)stility of the two rnhes of the first 3D spot moe n suggests tht in relity the low-voltge rnh is stle while the high-voltge rnh is unstle. It shoul e emphsize tht suh suggestion, while resolving the ontrition, oes not interfere with the ove onlusion in the sene of reprouile trnsition etween iffuse n spot moes uner typil experimentl onitions if the experiment is wellontrolle. One n onlue tht the ville theoretil informtion on stility of ifferent moes is lerly insuffiient n n urte nlysis of stility of stey-stte solutions is require. Another step whih oul prove highly useful for unerstning the generl pttern n stility of ifferent moes woul e n experimentl investigtion of iffuse n spot moes uner onitions when the intervl I I I t is insie the urrent rnge eing investigte. 6. Conlusions Numeril investigtion of stey-stte urrent trnsfer from high-pressure r plsm to ylinril thermioni thoe hs revele whole zoo of very iverse moes. Detile results re given for the first four 3D moes rnhing off from the iffuse moe n for the first 3D moe rnhing off from the first xilly symmetri spot moe. Eh 3D moe is lulte in the whole omin of its existene, from the ifurtion point in whih it rnhes off from n xilly symmetri moe own to very low urrents. One n lerly oserve the evolution of temperture istriution long the thoe surfe from rther smooth one in the viinity of the ifurtion point to one with well-efine spots t lrge istnes from the ifurtion point. At very low urrents, the orresponing ner-thoe voltge rop infinitely inreses n eh spot shrinks n its ehviour pprohes tht of solitry spot. The physis of 3D spots is similr to the physis of xilly symmetri spots. The numeril results otine onfirm the generl pttern of urrent voltge hrteristis of vrious moes suggeste in [1, 6] on the sis of ifurtion nlysis n generl onsiertions. It is shown, in prtiulr, tht 3D spot moe nnot just terminte ut rther turns k or joins n xilly symmetri moe. It is lso shown tht the urrent rnge in whih 3D spot moes exist is limite from ove: I I t, where I t is the urrent vlue orresponing to the turning point of the first 3D spot moe. Vlues of the ner-thoe voltge rop lulte for the low-voltge rnh of the first 3D spot moe onform to experimentl t for the spot moe ville in the literture. Hypotheses on stility of stey-stte solutions se on generl trens typil for nonliner issiptive systems provie 2133
M S Benilov et l n explntion of the ft tht the trnsition etween iffuse n spot moes is iffiult to reproue in the experiment: if thin thoe is operte in limite urrent rnge, it is likely tht oth the iffuse moe n one of the rnhes of the first 3D spot moe will e stle ginst smll perturtions in the whole rnge of experimentl onitions. On the other hn, these hypotheses o not explin the inition tht it is the lowvoltge rnh of the first 3D spot moe tht seems to our in the experiment. Thus, the question of stility of steystte solutions remins open: n urte stility nlysis is require. Another step whih oul prove highly useful woul e n experimentl investigtion of iffuse n spot moes uner onitions where oth the first ifurtion point n the turning point of the first 3D spot moe re insie the urrent rnge eing investigte. Aknowlegments The work ws performe within tivities of the projet POCI/FIS/60526/2004 Moes of urrent trnsfer to thoes of high-pressure r ishrges n their stility of FCT, POCI 2010 n FEDER n of the tion 529 of the progrmme COST of the EC. MC knowleges finnil support grnte y the tion COST-529 towrs his sty t Universie Meir. Referenes [1] Benilov M S 1998 Phys. Rev. E 58 6480 94 [2] Benilov M S n Cunh M D 2002 J. Phys. D: Appl. Phys. 35 1736 50 [3] Nnelstät D, Rewitz M, Dringhusen L, Luhmnn J, Lihtenerg S n Mentel J 2002 J. Phys. D: Appl. Phys. 35 1639 47 [4] Böttiher R n Böttiher W 2000 J. Phys. D: Appl. Phys. 33 367 74 [5] Benilov M S n Cunh M D 2003 J. Phys. D: Appl. Phys. 36 603 14 [6] Benilov M S n Cunh M D 2003 Phys. Rev. E 68 056407 [7] Böttiher R, Grser W n Kloss A 2004 J. Phys. D: Appl. Phys. 37 55 63 [8] Dringhusen L, Lngensheit O, Lihtenerg S, Rewitz M n Mentel J 2005 J. Phys. D: Appl. Phys. 38 3128 42 [9] Benilov M S, Cunh M D n Niis G V 2005 Plsm Soures Si. Tehnol. 14 517 524 [10] http://www.r thoe.um.pt [11] Benilov M S n Mrott A 1995 J. Phys. D: Appl. Phys. 28 1869 82 [12] White G K n Minges M L 1997 Int. J. Thermophys. 18 1269 327 [13] Touloukin Y S, Powell R W, Ho C Y n Clemens P G 1970 Therml Conutivity. Metlli Elements n Alloys (Thermophysil Properties of Mtter vol 1) (New York Wshington: IFI/Plenum) [14] Yih S W H n Wng C T 1979 Tungsten: Soures, Metllurgy, Properties, n Applitions (New York: Plenum) [15] Lihtenerg S, Nnelstät D, Dringhusen L, Rewitz M, Luhmnn J n Mentel J 2002 J. Phys. D: Appl. Phys. 35 1648 56 [16] Rizer Yu P 1991 Gs Dishrge Physis (Berlin: Springer) [17] Hir J 1999 J. Phys. D: Appl. Phys. 32 263 72 [18] Shmitz H n Riemnn K-U 2002 J. Phys. D: Appl. Phys. 35 1727 35 [19] Hntzshe E 2003 IEEE Trns. Plsm Si. 31 799 808 [20] Benilov M S 1992 Phys. Lett. A 169 57 61 2134