Plasma-Producing Structures at Current Implosion

15 Pages·1.447 MB·English

by  Grabovsky E.V. и др.

Contrib.PlasmaPhys.45,No.8,553–567(2005)/DOI10.1002/ctpp.200510061
Heterogeneous Plasma-Producing Structures at Current Implo-
sion of a Wire Array
E. V. Grabovsky, K. N. Mitrofanov, S. L. Nedoseev, G. M. Oleinik, I. Yu. Porofeev, A. A.
Samokhin∗,andI.N.Frolov
SRCRFTRINITI,142190,Troitsk,MoscowReg.
Received23March2004,accepted7April2005
Publishedonline25November2005
Keywords Explodingwires,heterogeneousstructures.
PACS 52.80.Qj,52.70.La,64.75.+9
Characteristicproperties of the plasma production process have been considered for thecase of megampere
currentsflowingthrough hollow cylindrical wirearraysof theAngara-5-1 facility. In3-4nanoseconds after
voltage applying to the wire surfaces there appear a plasma layer. The system becomes heterogeneous, i.e.
consistingofakernel ofmetalwiresandaplasmalayer. Inseveralnanoseconds thecurrent flowgoesfrom
metaltoplasma,whichresultsinreducingtheelectricfieldstrengthalongthewire.
TheJoule heat energy delivered to themetal before the moment of complete current trapping by plasma
isinsufficientforthewholemasstransitiontoahotplasmastate. TheX-rayradiographytechniquesmadeit
possibletodetectandstudydenseclustersofsubstanceof∼1g/cm3atadevelopeddischargestage.Theradial
expansionvelocityof∼104 cm/smeasuredatthe70-thnanosecondafterthecurrentstartallowstreatingthe
densecoreatalatestageintheformofasubmicronheterogeneousstructurefromitsliquidandslightlyionized
gasphase.
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
1 Introduction
Theimplosionofcylindricalarraysfrommicron-sizetungstenandaluminumwires(liners)undertheactionof
nanosecondcurrentpulseshasbeeninvestigatedearlier[1,2]andisstillunderstudy[3,4]. Asaresultofcurrent
compressionofawirearraymaterialaZ-pinchdevelops. Atitsfinalstagethereappeara high-powerpulseof
soft X-radiation, that is of importancefor CTF [5], the investigationsinto fundamentalpropertiesof matter in
extremeconditions[6],etc.
Inearlyworkswithwirearraysaphenomenonofcarryingalow-denseplasmaandcurrenttowardsthecenter
of the system when affected by the Lorentz force has been revealed [1]. The process was considered to be
interferingin compactcompressinga wire liner by a collective magneticfield because of the transfer of a part
ofcurrentandmasstothecenterandthepre-pinchformation. Itwassupposed,thatforagreatnumberofwires
anynegativeconsequencescouldbemoderatedduetoafastformationofacontinuosandnearlyhomogeneous
shellofanimplodedwireplasma[7]. Itwasalsoconsidered,thattheself-magneticfieldofthecurrentflowing
throughthewirewashighenoughtocompensateaheatpressureandtolimittheplasmalayerradiusaroundthe
wireinthearray[8].
Abetterunderstandingofthespatial-timestructureofplasmafluxesandmaterialdistributioninthemultiwire
array has been gained in [9,10]. A concept of a prolonged plasma production process, as an alternative to a
0-dimensionalmodeland shellmodelsforwire arrays, is presentedas a dependenceof the plasma velocityon
thefixedinnersurfaceofthelineruponthefullcurrentJandthevelocityoftheplasmainflowdm/dttotheinner
volume[11]. Therelationexpressesthelawofconservationofmomentum,complimentedwithaconditionfor
Machnumber(MA=1)forathindissipativeboundarylayer(ohmic).
There still exist an uncertain role of the electron emission at a fast Joule heat up in both arising a primary
plasma on the wire surface [12, 13] and subsequent producing of plasma from the products of electric wire
∗ Correspondingauthor:e-mail:[email protected]
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
554 E.V.Grabovsky,K.N.Mitrofanov,andS.L.Nedoseevetal.:Heterogeneousplasma-producingstructuresat...
implosion. Astothelatter,naturalenergysourcestoionizeacoldsubstanceare: athermalenergy(transferred
byheatconduction)ofaplasmaflux[11],anenergyofpre-pinchradiation[15]andanenergyoftheelectricfield
inthepointofthewirelocation. Themodelwithheatconductiongivesadependenceofdm/dtonJ,thatistrue
onlyforaninitialdischargephase. Aconstantofproportionality,whichincludestheenergyvalueofionization
(priceofplasmaproduction),canbedeterminedfromthecomparisonwiththeexperiment.
Forthefirsttime,theheterogeneousstructureoftheexplosivewirecorewasdetectedusinganX-raydiagnos-
tics with an X-pinchtechniquefor botha single wire [16]andan array[17]. The spatial(r, z)-structureof the
plasmashellandthecoreoftungstenandtitaniumwiresinaself-magneticfieldissimulatednumericallywithin
thesystemofequationsoftwo-dimensionalradiationmagneto-hydrodynamics,onpassingashockwavethrough
adesigneddiscontinuityofthedensity(1:50)onthedisturbedcylindricalborderoftheplasmaandtheone-phase
liquidcore[18].
Recently,theelectricexplosionofasinglethinwireatahighrateofcurrentrisehasbecomeasubjectofan
active experimentalstudy [19, 20, 37]. The expansionvelocity of the explodingwire rangingfrom 0.2 µm/ns
(titanium)to5µm/ns(zinc)isgivenin[19],dependentonthespecificresistanceofthematerialandtheenergy
inputatthestageofJouleheatingin∼50-60nsatacurrentdensityriseof∼4MAcm−2/ns. Forasignificantly
fasterdischarge(∼50MAcm−2/ns)theexpansionvelocityofthetungstenwire0.6µm/ns[20]isnearlytwice
aslargeasthatin[19](0.35µm/ns).
Nevertheless,ontheAngara-5-1facilityintheexperimentswithmultiwiretungstenarrays[21]forfirst10ns
thecurrentdensityderivativeisaround500MAcm−2/nsthatisbyanorderofmagnitudegreaterthanin[20].
Fromthecomparisonofthedatain[19and20](inthelogarithmicscale),itfollowsthat,foroperationconditions
of[21],theexpansionvelocityshouldincreasetwotimesandachieve∼1.2µm/ns. Averificationofthisestimate
ofthecoreexpansionvelocityseemstobeespeciallyurgentduetothepresenceofastrongcollectivemagnetic
field,whichdisturbsacylindricalsymmetryofasinglewire. Notonlyquantitativedifferencescanbeexpected,
but, first of all, qualitative ones, since the interaction between the core and the plasma that drifts towards the
centerundertheLorentsforceprovestobemuchslighterthanthatintheexperimentwithasinglewire.
AtthecurrentcompressionofnestedwirearraysanattainableZ-pinchstateischaracterizedbyregularityof
itsreproducibilityandabetter,asagainstasinglearray,uniformity[22-25]. Atpresent,attentionisfocusedon
the study of the inner and outer array interaction[26]. The magnetic field measurementstaken outside, inside
andbetweenthearraysontheAngara-5-1facilityenableustoelucidatesomedetailsofthisinteraction,whilethe
visualizationofwirekernelsusinganX-rayradiographytechniquemakesitpossibletoelicittheirinnerstructure.
2 Experimental design
Cylindricalsingleanddoublewirearrays8-20mmindiameterand10-15mmhighwereusedaswirelinersinthe
presentwork. Thediameterofthetungstenwiresrangedfrom4to10µm. Thearrayswereplacedinavacuum
chamberoftheAngara-5-1facility[17]. Thechamberwaspumpeddownto10−5torrwithadiffusionoil-vapor
pump. Apre-pulseofthegeneratorwassuppressedbyapre-pulsesparkgaplocatedjustinfrontoftheliner. At
thepre-dischargestagethefieldstrengthonthearraywasnotinexcessof10V/cm,thetemperatureincreaseof
thewireswaslessthan10oCforthistime.
Theresultsoftwoexperimentalrunsaregiveninthework.Inoneoftherunsamagnitudeoftheelectricfield
onthewiresurfacewasstudiedforthefirst∼5-15nsperiodofthecurrentflowthroughthearray. Inthesecond
runoftheexperimentsdensecoreswerestudiedin60-80nsafterthecurrentstart.
2.1 Initialcurrentflowstage
Thelinercomprising8(or16)wireswasincludedintheAngara-5-1facility. Withtheaimofstudyinganinitial
discharge phase current and voltage were simultaneously measured directly in the array. A shunt with a time
resolutionof5nswasusedtotakecurrent.Tomeasurevoltagearesistivedividerwascoaxiallyconnectedtothe
anodeand,byuseofametalrod0.1cmindiameterplacedstrictlyalongthelineraxis,tothecathode(Fig. 1).
Thetimeresolutionofthedividerwas2ns.
The voltage on the divider differs from that on the wire in question by having an inductive componentfor
the contour formed by this wire and conductors connecting the divider. The peculiarity is that the inductive
componentissmallandmaybeneglected. TheopticframesystembasedonanMKPcameraintendedtodetect
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
Contrib.PlasmaPhys.45,No.8(2005)/www.cpp-journal.org 555
thewirearrayimagehada10nstimegate.Thespatialresolutionontheobjectwas40µmwithadepthoffocus
ofabout2mm. Thedynamicdetectionrangeofthesystemwas∼20.
10 mm
Fig.1 Circuitdiagramofthewirearrayandthevoltagedivider.
1-anode,2-wires,3-metallicrodtoconnectvoltagedividerto
cathode,4-cathode,5-cathodeelectrodeofwirearray.
2.2 Wireradiographyatthedevelopedstageofimplosion
In the second run of the experimentsthe probingof the nested arrayswas performedby makinguse of the X-
pinchradiationontheAngara-5-11facilityatacurrentdischargeof∼3MA[28]. Adoublecoaxialarrayfrom
identicaltungstenwires6µmindiameterwasunderstudy. Thediametersoftheouterarrayof40wiresandthe
innerarrayof120wireswere20mmand12mm,respectively.
The scheme of the experiment on probing multiwire nested arrays is shown in Fig. 2. Passed through the
objectinquestiontheX-pinchradiationwasdetectedbyaphotographicfilm. Theadjustmentwasmadeinsuch
away,thatanimageoftheinnerandouterareasoftheinnerarrayappearedonthefilm. Thefilmwas1.2mfrom
theX-pinch.
Test wires
X-pinch
Arrays Shields Films
Fig.2 Layoutoftechniquesforradiographicmeasurements.
The X-pinch looked like four symmetrically crossed in the middle molybdenum wires 20 µm in diameter.
The X-pinchsubstitutedone ofthe eightreversecurrentleadsat the distanceof 45 mmfromthe axis andwas
connectedinserieswiththeliner[29]. TheenergyoftheX-pinchradiationintheX-rangeexceeded200mJper
pulse. The radiation was detected by films RF-3, UF-SC. A 10 µm aluminumfilter was placed in frontof the
films. Animagewasformedinquantawithanenergyof>1.5keV.
For the purpose of transiting from film blackening to the density of the tungsten liner a step attenuator of
tungstenfilterswiththeirdensityfrom300to1300µg/cm2 wasplacedinfrontofthefilm. Theshadowsfrom
thefiltersweredetectedonthesamefilmsastheonesfromthelinerintheX-rayquanta. Thespatialresolution
ofthetechniquewaslimitedbyadiffractiondivergencyandequaledto4µm. Withtheaimofcontrolsometest
wires5µmindiameterwithnocurrentwereplacednearthelinerandreliablydetected(Fig. 2).
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
556 E.V.Grabovsky,K.N.Mitrofanov,andS.L.Nedoseevetal.:Heterogeneousplasma-producingstructuresat...
3 Experimental Results
3.1 Break-downalongthewiresurface
ThetemporalvoltageprofilesontheaxisaregiveninFig. 3forseveralshotswithtungstenwirearraysandone
no-loadrun(withnoarray). Intheabsenceoftheloadfor50nsthevoltageisequaltotheoutputvoltageofthe
generator. Detectedforfirst20nsthemaximumvoltagevaluevariedwithin5to18kVforvariousliners,which
differedintheirdiameterandthenumberanddiameterofwires. Thevaluesofmaximumvoltageprovedtobe
considerablyless,andthepulselengthsshorter,thanthosemeasuredatnoload.
U, [V]
0
-2000
-4000
-6000
-8000
-10000
-12000
-14000
-16000
Without wires
-18000
(no-load shot)
660 680 700 720 740
ns
Fig.3 VoltageonthelineraxisU(t)forseveralshotswiththewirearrayandforoneshotofno-loadcondition.
Fig.4 MaximumamplitudeofvoltageU
onthelineraxisagainstdiameteroftung-
stenwiresinthearray.
Fig. 4showsthemaximumvoltageonthearrayagainstthetungstenwirediameter. Ifouterupperpointsbe
takenintoaccount,onecanstate,thatmaximumvoltageamplitudesincreasewiththewirediameter.
In all the experimentsperformedthe load resistance for first 10 nsprovesto be a on monotonicfunctionof
time. Itsmaximumvaluecalculatedintermsofthevoltageandcurrentmeasuredwasalwaysseveraltimesless,
thanthelinerresistanceatthetungstenmeltingtemperature. Thehighconductivitycanbeeasilyexplainedbya
shuntingeffectofanear-surfaceplasma[12].AtypicaltemporaldependenceofthevoltageU(t)ontheaxisand
thecurrentJ(t)referredtoasinglewireisgiveninFig. 5(thelinerofeighttungstenwires∅10µm,1cmhigh,
and8mmindiameter). ThebottompartofthisFigureshows,intermofasinglewire,theloadresistanceU/J,
thecalculatedwireresistanceRw andthenear-surfaceplasmalayer(shunt)resistanceRpl=(J/U-1/Rw)−1. The
temporalwireresistanceRw(t)wascalculatedfortheinitialradiusandlengthofthewireintermsofa known
temperaturedependenceofthespecificresistanceoftungstenandthemeasuredvoltageU(t).
Thecurrentandvoltageontheloadmakeitpossibleto calculatethe energyinputtothe wiredueto Ohmic
heatingforfirst2-3ns.Thetungstentemperatureobtainedatthetimemomentdeterminedbydivergencybetween
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
Contrib.PlasmaPhys.45,No.8(2005)/www.cpp-journal.org 557
Fig.5 (a) Waveforms of the voltage U(t) across an 8-
mm diameter wire array consisting of eight 10-µm tung-
stenwiresandthecurrentthroughonewireJ(t).(b)Time
evolutionoftheresistances(peronewire)calculatedfrom
thevoltageandcurrentwaveforms:thetotalloadresistance
U(t)/J(t),thewireresistanceRw(t),andtheresistanceof
thesurfaceplasmaRp(t).
the load resistance and the calculated one does not exceed 2000oC for the majority of the shots. The given
temperatureislowerthanthatoftungstenmelting(3380oC).
Takingintoaccount,thatthetungstenresistanceatapowerfulelectricheatupofthewiredependsuponnon-
controlledandinsufficientlyinvestigatedprocesses,onecanpositivelystatethefactofplasmaappearanceatthe
momentoftheresistancemaximumachieved(Fig. 5). Theplasmaappearanceisduetoevaporationofsurface
impurities with a small atomic weight [31], and, likely, with presence of oil traces on the wire surface [9]. A
preliminary liner heat up in vacuum up to red glow did not practically affect the wire array impedance at the
dischargestart.
Abreak-downoftheformedgaslayercanbeinfluencedbyelectronemissionamplifiedbythefactthatona
certainpartofathinwireforacertainelectrodeconfigurationtheradialelectricfieldcanbe10-100timesashigh
astheaxialfield[32].
3.2 Wireradiographyintheprocessofimplosion
Fig.6presentssynchronizedsignalsofcurrentthroughliner,softX-radiationofZ-pinchandX-pinchflash. The
flash of the probing radiation occurred 80 ns before the maximum Z-pinch radiation. The results of probing
the wire arrayby X- pinchraysare givenin Fig.7. In its upperpartFig.7showsa fragmentof a shadowgraph
obtainedasaresultofthisprobing.ThinslopinglightlinesinrightbottomcorneroftheFigureisashadowfrom
thetestwire6µmindiameter.
Fig.6 Oscillograms: 1 - discharge current (1 MA/div),
2 - profile of Z-pinch SXR (a.u.), 3 - profile of X-pinch
radiation(a.u.).
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
558 E.V.Grabovsky,K.N.Mitrofanov,andS.L.Nedoseevetal.:Heterogeneousplasma-producingstructuresat...
Theshadowgraphsofthestepattenuatorfromtungstenwithaknownarealdensityallowtheplasmadensity
invariouslinerpointstobeevaluated.Further,therewillbeusedsuchacharacteristicofthemassdistributionas
thearealdensityρl[µg/cm2],thatisequaltotheintegralofvolumetricdensityalongtheprobingray.
Fig.7 Radiography data. At the top - shadow-
graph of part of theouter and inner arrays Atthe
bottomleft:plotofintegralalongtheprobingbeam
plasmadensity.
InthebottompartofFig.7thetreatmentresultsofthewirearrayshadowgraphintheprocessofimplosionare
presented. Theradialprofileof thearealdensityρlfora straightline a)was calculatedusingthe detectedfilm
blackeningdensityintheupperpartofFig. 7. TheabscissainthebottompartofFig. 7isorientedfromleftto
rightontheradiusfromthesystemcenter. The0-coordinatecorrespondswiththeborderoftheinnerarrayand
thecoordinateofitsaxisis-0.6cm.
Thesharppeaksinthedistributionofthearealdensity(Fig.7)correspondwithdensecores.Totherightfrom
theinnerarrayborderasmallpartoftheouterarrayplasmaisseen. Thegivendetectedplasmaextendsfromthe
wires of the outer arrayin the directionof the axisat a distance of ∼800µm. The plasma borderof the inner
arrayisdistantnotmorethan150µmfromthewires(inaverage)intheradialdirectiontowardsthecenter.
Fig.8givesanamplifiedimageofthecoresoftheouterandinnerarrays.Thesecoreshavethesamediameters.
However,in contrastwith the wire coresof theinnerarray, the corefromthe outerarrayhasa more“spongy”
structure.Thecoreoftheouterarrayarecutintheaxialdirectionwithanintervalof∼20µm.Thesamestructure
ofcoreswasobservedatimplosionofsinglearrays. Adetailedprofileofthearealplasmadensityinthevicinity
ofoneofthedensecoresoftheinnerlinerisshowninFig. 9. Themajorfractionofthesubstancefromtheinner
arraywires(∼80%)isinsidethedensecore. Theother20%aredistributedintheregion∼80-90µmthick. As
totheouterarray,thecorecontainsabout60%oftheinitialwiremass.
Inspiteofastrongaxialnon-uniformityanddiversedegreesofmassexhaustion,thedetected(byquantawith
energies∼3keV)coresoftheinnerandouterlinershavethesamediameter∼20µm.Anincreaseinthediameter
from6µmto20µmcorresponds,atleast,a10-foldfallofthedensity.
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
Contrib.PlasmaPhys.45,No.8(2005)/www.cpp-journal.org 559
Fig.8 Shadowgraphofpartofwirecores: left-fromthe
innerarray,right-fromtheouterarray.
Fig.9 Profileofarealplasmadensitynearonecoreofthe
innercascadehighlightedfromimageinFigure7.
3.3 Spatialnon-uniformityofbreakdownandheterogeneityofcores
Toinvestigatetheuniformityoftheinitialdischargestage(breakdown)framephotographyofasingletungsten
linerwastakenintheX-ray(Fig. 10a,[43])andinthevisiblespectrum(Fig.10b).
The earlier wire glow detected in the X-ray range at the ∼2-nd nanosecondis non-uniformboth in the az-
imuthalandaxialdirection(Fig. 10a). Onlyabout5wiresof48witha5µmdiametercanbeseeninthearray.
Atthismomentthecurrentvalueforasinglewireinthearrayis100A,andforoneglowingwireitequals1kA.
The characteristicsize of the plasma formationaroundthe wire in the azimuthaldirectionis ∼300µm, thatis
less,thanthespacebetweenthewiresinthearraywhichisequalto800µm.
Theimageoftheeightwirearraywiththewirediameterof7,5µmdetectedatthemomentwhenthecurrent
attained∼3kA/wireispresentedinFig. 10b. Acharacteristicpropertyisaparticularlypronouncedlarge-scale
axial non-uniformityof emittance distribution along the wires. The images of separate wires are also greatly
different. A non-uniform energy input to a wire at an initial cold stage predetermines, later, a not uniform
exhaustionofvariouspartsofthecore.Suchacorewitha∼1mmsizeofitsnon-uniformityisgiveninFig. 11.
Thecoreexhaustionnon-uniformitydevelopsnotonlyalongtheaxis,butalsofordifferentwiresinthearray.
So,Fig.12demonstratesanobtainedin[33]profileofthearealplasmadensityρlnearanouterradiusofasingle
array. TheX-axisin Fig. 12is radiallydirectedfromrightto leftto thearraycenter,X=0atthe arraysurface.
In the middle on the plot of the areal density ρl (Fig. 12) one can see two narrow peaks (at X=220 µm and
X=550µm)around25µmwide,whichcorrespondwiththeshadowsfromtwodensecores.Totherightfromthe
dense corestwo feeblymarkedhumps, which arecorrespondstwo othercoreswith substancein a low-density
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
560 E.V.Grabovsky,K.N.Mitrofanov,andS.L.Nedoseevetal.:Heterogeneousplasma-producingstructuresat...
(a)
(b)
Fig.10 (a)X-rayimageat2ns.Linerwithdiameter12mmandheight15mm.FortyeightwiresØ5µm.Resolutionis200
µmforhv∼200eV.(b)Imagedetectedbyopticframechamber. Currentis24kA.Linerwithdiameter8mmandheight10
mm.EightwiresØ7.5µm.
Fig.11 ShadowgraphinX-rayofthepartfrominnerarray. Fig.12 Densityofplasmanearthesurfaceofasinglearray.X-
Attime40nsbeforemaximumofX-radiation. Energyof axisisorientedalongradiusfromrighttolefttowardstheliner
X-pinchquantais1-1.5keV. axis.X=0onthewirearrayradius.
plasmastate.Thewireexhaustionatonemomentoftimeisdifferent,thatalsoevidencesthedifferenceintherate
ofplasmaproductionandislikelyduetoanon-uniformcurrentdistributionoverwiresinthelinerattheinitial
stage.
4 Plasmasurrounding ofwireinthe linerataninitialdischargestage
The temporal dependencies of the current and voltage on the liner make possible the estimation of produced
plasma parameters. Let us consider a cylindrical plasma column around a wire with a Spitzer conductivity
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
Contrib.PlasmaPhys.45,No.8(2005)/www.cpp-journal.org 561
dependenton a temperatureTe and an average ion charge Z∼T0e.5. Assume also that the plasma expandsinto
vacuumwithavelocityofionsound. Thismodelwellagreeswiththedetectedvoltageandcurrentthroughthe
arrayfor∼10nsforthetungstenplasmawithatemperaturefallingintimefrom30to20eV.Fig. 13showstwo
resistancevaluesdividedbyaninitialresistanceofthelinerdependingontime. Oneofthem,theexperimental
one,iscalculatedonthebasisofthedetectedvoltageandcurrent(shownbycircles),thesecondisobtainedfrom
the given model for the tungsten plasma with Te∼24 eV.The plasma channel resistance is equal to the initial
resistanceofthewireatthetimemomentof∼5.5ns. Inthemodelpresentedthevelocityofplasmaexpansionis
0.9·106 cm/s. Accordingly,in10nsafterplasmaappearanceonthesurfacetheplasmacolumnradiusachieves
∼100µm. Note,thegivenvaluesareindependentontheplasmamassofthecurrentchannelandnotrelatedwith
theequationofenergy,thatcorrespondswiththemodelofpointexplosion.
The counterpressureshould be taken into account, which is of importancewhen the plasma radius exceeds
the skin-layerthickness. However,sincethe modelconsideredagreeswiththe experimentupto 1kAperwire
onecanobtainalowerborderfortheplasmamassanddensitybyequatingtheexpansionvelocityofthevisible
glowborder(0.9·106cm/s)totheAlfvenvelocity. Thenin10nsafterbreakdowntheplasmamassperonewire
exceeds∼10−8g/cmandtheconcentrationismorethan∼1017cm−3.
The collective magnetic field in the array is much lower than that of a separate wire on its surface: their
ratio is Nrw/R (cid:3)1, where N is the number of wires, rw is the wire radius and R is the liner radius. As the
pressure of the plasma exceedsthat of the self magnetic field, the collective magnetic field does not affectthe
plasma columnexpansion,atleast, forfirstten nanoseconds. In accordancewith theprobemeasurements[19]
thecurrentappearsonthelineraxis(prepinch)in30-40nsafterthedischargestart. Thevelocityoftheplasma
flowtowardsthecenter,∼107cm/s,attributedtothecollectivemagneticfieldisbyanorderofmagnitudegreater
thanthatoftheplasmacolumnexpansionofasinglewire[32].
What is the cause of a large-scale non-uniformplasma distribution over wires detected in the optical range
andaviolatedazimuthalsymmetrybetweenthem(Fig. 10)isnotclear. Nevertheless,themodelgivenmakesit
possible,onthebasisofthemeasuredcurrent-voltagecharacteristics,toevaluatethetemperature,thelowermass
borderandthetungstenionconcentrationinthenear-wireplasma(r∼0.01cm).
Fig.13 Resistanceof cylindrical plasma layer near wire
relatedtoresistanceofwireat300K.Circlesisexperiment,
solidline-plasmamodelwithTe varyingintimefrom30
to20eV.
5 Dense cores ata developeddischarge stage
The data on the core expansionvelocity, 104 cm/s, obtained by X-ray probingthe wire array with an X-pinch
technique [21,23], permits drawing conclusions about the structure of a dense part of tungsten at a rather late
time moment(40-70ns) afterthe currentstart. The valueof the averagedensity ρK of the coresubstanceis 1
g/cm3, thatislessthanthecriticaltungstendensityρCP =4.5g/cm3 [34].Thesmallexpansionvelocity∼104
cm/spreventsconsideringthecoreasahomogeneousstateofavapour,asthecorrespondingpointinthephase
diagramhasatemperatureof>12000Kandathermalvelocityof>8·104 cm/s[34]. Undertheseconditionsit
isnaturaltoconsiderthecoretobeaheterogeneousmedium,i.e. amixtureofliquid(drops)andvapour. Note
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
562 E.V.Grabovsky,K.N.Mitrofanov,andS.L.Nedoseevetal.:Heterogeneousplasma-producingstructuresat...
thevelocityofcoreexpansion,observedinthewirearray∼0.1µm/ns,lessthanintheexperimentswithasingle
wiremorethanone[19,20].
Letusconsideramodeltakingaccountofevaporationfromthesurfaceoftheliquidphaseandthemassloss
byvapouroutflowfromthecoreborder.TheohmicheatingbycurrentJ,whichflowsthroughanionizedvapour
in the core at a specified electric field on the liner “surface”, will be taken as an energy source. The electric
fieldstrengthisevaluatedintermsofthemodelofprolongedplasmaproductionwithathindissipativeboundary
layer [11], whichseparatesthe dense coresand the hotacceleratedplasma of ideal√conductivity. Assume, that
withinthislayertheelectricfieldstrengthisconstantandequalsE =2·10−9VJL/ 3RL ∼50(kV/cm).Here,
JL ∼ 2(MA)isthefullcurrent,RL ∼ 0.5(cm)thelinerradiusandV∼ 107(cm/s)theplasmavelocity.
Nowweconsideracylindricalcloudofliquiddrops,characterizedbyaverageradius rLandadensityρLwith
anumberofdropsNL perunitlengthofcore. Thelineardensityofaliquidphaseshouldnotbeinexcessofthe
correspondingcoredensity.Alimitationonthedropconcentration nL =NL/πrK2 followsfromtheopacityof
a heterogeneousstructureforX-rayprobing: thepathlengthlimitedbyabsorptionindropislessthanthecore
radius-NLrL2 > rK. So,assumingρL > 10(g/cm3),weobtainarelationbetweensizesanddensitiesofthe
dropsandthecore:
rLρL < rKρK = 9(µg/cm3) ⇒ rL < 0.9(µm),NL > 1.1·105 (1/cm).
In terms of the initial wire with its initial radiusrw = 2.5 µm the volumetricconcentrationof liquid drops,
0.55·1012(cm−3),iscomparablewiththatofdefectsinmetal,i.e. 1012(cm−3)[36].
Thebalanceequationsofmass-mL,gas-mG andenergy-mLεL+mGεG oftheliquidinthecylindrical
core,ontheassumptionoftheequalityofliquidandvaportemperatures,areasfollows:
(cid:1)
mL/ρL + mG/ρG = πrK2 , rL = 3 0.75mL/(πNLρL), (1)
(cid:1)
uT = kBT/2πM, V˙ =2πrKuT, (2)
dmL/dt=−4πrL2 NLuT(ρV(T)−ρG), d(mL+mG)/dt= −ρGV˙, (3)
mL (dεL/dt) + mG (dεG/dt) − (dmL/dt) (εG − εL) + PGV˙ = JE. (4)
Here,ρV(T)isthedensityofthesaturatedvapourinthecore.Thegivensystemiscompletedwiththeequation
of state [34,35]. The equilibrium degree of gas ionization was calculated with due account of corrections for
plasma non-ideality [37], and the core conductivitywithin the model of an effective Coulomb logarithm[38].
The influence of the liquid disperse phase on the currentflow in the tungsten plasma [39] was neglected, that
imposes,inparticular,restrictionontheliquidandgasvolume-mL/ρL < mG/ρG.
Fig.14 Massofliquidandgasphase(left),temperatureandcurrentofcore(right). NL=3·105 cm−1,rK=10µm, E=30
kV/cm.
Fig.14presentstheresultsofsimulationthedynamicsofaheterogeneouscoreintheframesoftheequation
system (1-4)at constantvaluesof radiusrK=10 µm and field strengthE = 30 kV/cm and at initial conditions
(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim