The electric explosion of conductors by a high-power current pulse is a convenient object for funda-mental investigation of various physical phenomena athigh values of energy density [1–10]. At the same time,the electric explosion of conductors has numerous pro-cess applications in preparing artificial diamonds[8, 11, 12] and ultradisperse metallic powders [13], inelectric-explosion breakers [14], and so on. In view ofthis, it is of considerable interest to theoretically studythis phenomenon, including its initial stage at which adense cold core is formed, which exists throughout theprocess until the developed stage of metal explosion[15, 16].
This paper is devoted to the study of the dynamicsof nanosecond electric explosion of a conductor at theinitial stage. The conditions being treated in the processof evolution are characterized by the presence of alter-nating compression and expansion waves. In so doing,the state of metal at a certain moment reaches theparameters corresponding to its liquid–vapor binodal.The estimates indicate that the period of time duringwhich the stage of the process being treated occurs isinsufficient for the matter to separate into phases inaccordance with the conditions of equilibrium [17, 18].In view of this, the conductor melt may be in the meta-stable state. As was revealed by our calculations, thepossibility of realizing the states of the liquid phase ofmetal in the metastable region brings about results thatagree well with the experimental data of [15, 16]. At thesame time, the elimination of this possibility causesdisagreement with experiment. Note that, in the prob-lem being treated, the magnetic pressure decreases
from the conductor center to periphery; therefore, thewave of metal melting propagates from the surface toaxis, and, as a result, the overheating of the solid phasebecomes impossible [19, 20].
The adequacy of description of the thermodynamicproperties of a medium is of importance in the case ofnumerical simulation of unsteady-state hydrodynamicprocesses. The equation of state for matter largelydefines the reliability of the calculation results. We haveused the semiempirical equation of state for tungstenbased on the model of [21] allowing for the effects ofhigh-temperature melting and evaporation. The pre-dicted characteristics are compared with the availableexperimental data at high values of energy density.
Usually, the hydrodynamic description at the inter-face involves the use of the conditions of coexistenceobtained for the case of full thermodynamic equilib-rium in the absence of external forces. However, inaccordance with the methods of nonequilibrium ther-modynamics, when external fields affect matter in thestate of local thermodynamic equilibrium, the condi-tions of coexistence of phases must follow from theconservation laws. In this paper, the conditions of equi-librium on the liquid–vapor interface are discussed.
DYNAMIC EQUATIONS
The initial stage of electric explosion of a conductorof cylindrical shape by a high-power current pulse issimulated assuming the smallness of spatial perturba-tions of shape and the equality of the electron and iontemperatures. In this formulation, similarly to [22], one
Metastable States of Liquid Metal under Conditionsof Electric Explosion
S. I. Tkachenko, K. V. Khishchenko, V. S. Vorob’ev, P. R. Levashov,I. V. Lomonosov, and V. E. Fortov
IVTAN (Institute of High Temperatures) Scientific Association, Russian Academy of Sciences,Moscow, 127412 Russia
Received December 5, 2000
Abstract
—The paper deals with the simulation of the initial stage of tungsten wire explosion under the effectof a high-power nanosecond current pulse. The calculations involve the use of a semiempirical equation of statefor tungsten, which allows for the effects of melting and evaporation at high temperatures. The laws of conser-vation that take into account the presence of a magnetic field and relative motion of the media are written at theliquid–vapor interface. It is demonstrated that a description fitting adequately the available experimental datais possible if one takes into account the possibility of realizing metastable states of liquid metal in the processof evolution of the system.
THERMOPHYSICAL PROPERTIESOF MATERIALS
HIGH TEMPERATURE
Vol. 39
No. 5
2001
METASTABLE STATES OF LIQUID METAL 675
can write the following system of magnetohydrody-namic (MHD) equations:
(1)
(2)
(3)
(4)
where
m
is the mass,
ρ
is the density of matter,
ε
(
ρ
,
T
)
is the specific internal energy,
P
(
ρ
,
T
)
is the pressure,
υ
is the velocity,
T
is the temperature,
σ
w
(
ρ
,
T
)
is theelectrical conductivity,
κ
(
ρ
,
T
)
is the thermal conductiv-ity,
B
ϕ
is the magnetic field induction,
j
=(
µ
r
)
−
1
∂
(
rB
ϕ
)/
∂
r
is the current density, and
µ
is the mag-netic permeability. The last term in Eq. (3) simulatesthe radiation from the metal surface,
k
(
a
) = 1,
k
(
r
)
= 0at
r
≠
a
,
and
a
is the wire radius.
Because the time of development of MHD instabil-ity is, according to [23],
τ
m
~ 2(
ρ
/
µ
)
1/2
/
j
max
~ 0.2
µ
s, theone-dimensional formulation appears to be reasonablefor the characteristic times
τ
h
<
τ
m
, i.e., for the durationof the stage of the process being treated of
τ
h
~ 10
ns.The relaxation time of electron temperature in liquidmetal at a current density
j
~ 1
TA/m
2
is
τ
1
~ 10
ns [24];therefore, the use of the one-temperature approxima-tion in the nanosecond time range appears justified.
The initial and boundary conditions for this set ofequations may be written as
(5)
The thickness of Knudsen layer is assumed to benegligibly small. Then, in order to include the evapora-tion of matter from the wire surface, one can write
(6)
Here,
υ
b
is the velocity of the evaporation wave,
Λ
is the specific heat of vaporization,
R
is the specific gasconstant, and
T
a
is the surface temperature. The con-stant
a
k
depends on the back pressure of the mediumsurrounding the conductor. During evaporation intovacuum,
a
k
.
0.77
; in the case of a high back pressure,
a
k
= 0.5 [25].
The value of total current
I
=
I
(
t
)
flowing in the con-ductor is defined by the circuit equation
(7)
dm/dt 0,=
ρdυ /dt ∂P/∂r– 2µr2( )
1–∂ r
2Bϕ
2( )/∂r,–=
ρdε/dt P∂ rυ( )/∂r–=
+ r1– ∂ κr∂T /∂r( )/∂r j
2/σw k r( )Ew,+ +
d µBϕ( )/dt ∂ σwr( ) 1– ∂ rBϕ( )/∂r[ ] /∂r,=
ρ r t0,( ) ρ0, T r t0,( ) T0, υ r t0,( ) 0,= = =
Bϕ r t0,( ) 0, υ a t,( ) da/dt, υ 0 t,( ) 0,= = =
Bϕ a t,( ) µI t( )/2πa, Bϕ 0 t,( ) 0,= =
∂T /∂r( ) r 0= 0.=
κ ∂T /∂r( ) r a= ρυb Λ akRTa–( ).=
d2
LI( )/dt2
d RlI( )/dt I/C+ + 0,=
in which L is the inductance, C is the capacity of thebank of capacitors,
is the wire resistance, and l is the wire length. The ini-tial conditions for Eq. (7) are preassigned at themoment of time t0 = 0,
where U0 is the initial voltage across the bank of capac-itors.
VAPOR PRESSURE AND CONDITIONOF PHASE EQUILIBRIUM AT THE CONDUCTOR
BOUNDARY
Usually, when performing a simulation at the con-ductor–vapor interface, use is made of the conditionPa = Ps , where Ps is the equilibrium vapor pressure at atemperature equal to the surface temperature Ta . It isassumed that the condition of equality of the chemicalpotentials of liquid and vapor is valid at the interface,i.e., µa(Pa) = µs(Ps). This corresponds to the approxi-mation of isothermal medium on condition of thesmallness of external stimulation and displacement ofthe medium. However, it is not always that the motionof the medium may be ignored in an expanding liquidconductor, and the escaping vapors may move, undercertain conditions, at a transonic speed. In addition, thematter is affected by the Ampere force caused by thecurrent flowing in the conductor. Therefore, the laws ofconservation of mass, momentum, and energy, with dueregard for the heat release during phase transition, mustbe valid at the moving interface.
In a first approximation, one can assume that thetemperature at the liquid conductor–vapor interfacedoes not vary (the effect of a possible temperature jumpdue to the presence of a Knudsen layer will be dis-cussed below). The law of conservation of mass definesthe well-known condition υb/Va = υg/Vg , where υg is thevapor velocity relative to the moving interface, and Va
and Vg denote the specific volumes of liquid and vaporat T = Ta , respectively. The integration of Eq. (2) over athin layer in the vicinity of the interface produces
By further integration, with respect to r, from zero toa, of Eq. (2) divided by density and by the use of thethermodynamic relation dµ = VdP valid under condi-tions of isothermal flow, as well as by performing
Rl t( ) l2π------ σwr rd
0
a
∫1–
=
I 0( ) 0, dI/dt( ) t 0= U0/L,= =
Pa υb2/Va+ Pg υg
2/Vg.+=
676
HIGH TEMPERATURE Vol. 39 No. 5 2001
TKACHENKO et al.
a similar procedure with respect to the gas layer, wederive
(8)
where P0 is the pressure on the conductor axis; µa , µg ,and µ0 denote the chemical potential of liquid at theinterface, of vapor, and of liquid on the axis of liquidconductor, respectively.
The set of equations (8) generalizes the conditionsof phase equilibrium to the case of moving media, ofwhich one is subjected to the effect of a buoyancy forcecaused by the magnetic field of flowing current. In thecase of absence of motion and field, this set of equa-tions is equivalent to the condition in accordance withwhich the pressure at the interface is equal to the satu-rated vapor pressure.
If we replace the chemical potential in relations (8)by the thermal function and ignore the contribution bythe magnetic field, we will have a set of equationsdescribing the shock adiabat. We eliminate the veloci-ties from the first of equations (8) to derive the correla-tion between the thermodynamic quantities on bothsides of the interface,
(9)
One can see the emergence of a shift of the parame-ters of phase equilibrium that is associated with themagnetic field and motion of the medium. The effect of themagnetic field was investigated in [26]. One can readilydemonstrate that the inequalities Pa > Ps and Pg < Ps mustbe valid. The degree of deviation from equilibrium inthe absence of motion depends on the rate of vaporflow. In the case of flow into a medium with a high backpressure Pe , when Ps ! Pe , the rate of vapor motion ismuch less than the velocity of sound CS , and the pres-sures Pa and Pg differ little from Ps . Most substantialdeviations are to be expected under conditions close toevaporation into vacuum, when Ps @ Pe and the flowvelocity becomes equal to CS . Both in the case of evap-oration into vacuum and in the case of evaporation intoa medium with a high back pressure, the inequality
(10)
is observed, where y is the function associated with thepresence of a Knudsen layer [25, 27] and dependent onthe Mach number M for vapor, y . 0.669 at M = 1; asthe Mach number decreases, y 1.
Inequality (10) places a restriction on the pressuredifference between the liquid and vapor at the interface.When the temperature decreases, the sound velocitydecreases, and the motion produces less effect on phase
µa Pa( ) VjB rd
0
a
∫υb
2
2-----+ + µg Pg( )
υg2
2-----+ µ0 P0( ),= =
µa Pa( ) VjB rd
0
a
∫ µg Pg( )– 12--- Va Vg+( ) Pg Pa–( )+ + = 0.
υg Vg
Pa Pg–Vg Va–------------------ e yCS=
equilibrium. On the other hand, when the temperatureapproaches the critical value, the effect of motion islikewise unimportant, because the densities of the liq-uid and vapor phases equalize and, therefore, therespective velocities equalize as well. The motion mayhave the most effect on phase equilibrium in the below-critical temperature range, where the differencebetween the phase densities is even greater. Then, thepressure Pa may prove to be higher than Ps if the mag-netic forces predominate, or it may prove to be lowerthan Ps if the terms associated with the motion of liquidand vapor prove to be more important.
In order to solve the set of equations (1)–(4) with theinitial and boundary conditions (5)–(8), one must selectthe equation of state and transport coefficients (electri-cal and thermal conductivity) for the medium.
MODEL OF EQUATION OF STATE
We used the semiempirical model of equation ofstate for metals [21], which includes the melting andevaporation effects, in order to describe the thermody-namic properties of tungsten in a wide range of densityand temperature.
The free energy F, selected as the thermodynamicpotential of the solid and fluid (liquid and gas) phases,consists of three terms,
(11)
defining the elastic part of interaction at T = 0 ä (Fc)and the thermal contribution by atoms (Fa) and elec-trons (Fe); the first and last components of Eq. (11) havethe same form for both phases, while in the case of thesecond component these forms are different.
The volume dependence of the energy of elasticinteraction upon compression σc r 1 (where σc = V0c/V,V0c is the specific volume at P = 0 and T = 0 K) has atraditional form [28] of expansion in terms of degrees
of inverse interatomic distance ~ ,
(12)
which provides for normalization
(13)
In addition, supplementary conditions must be validfor pressure, bulk modulus, and its derivative withrespect to pressure at σc = 1, respectively,
(14)
(15)
(16)
Added to conditions (14)–(16) is the requirement ofa minimum of root-mean-square relative deviation of
F V T,( ) Fc V( ) Fa V T,( ) Fe V T,( ),+ +=
rc1– σc
1/3
Fc V( ) 3V0c
ai
i---- σc
i/31–( ),
i 1=
5
∑=
Fc V0c( ) 0.=
Pc V0c( ) dFc/dV– 0,= =
Bc V0c( ) VdPc/dV– B0c,= =
Bc' V0c( ) dBc/dPc B0c' .= =
HIGH TEMPERATURE Vol. 39 No. 5 2001
METASTABLE STATES OF LIQUID METAL 677
pressure in the range of σc = 30 to 5000 from the values
of calculated by the Thomas–Fermi model withcorrections [29]. This brings us to the problem on find-ing the conditional extremum of the functional [30, 31]
the solution of which enables one to find the values ofthe sought coefficients ai of the elastic compression
curve. In so doing, the parameters V0c , B0c, and areselected by iterations so that, under normal conditions of
P = –(∂F/∂V)S = 1 bar and T = 293 ä,
the tabular value of specific volume V = V0 be valid, aswell as the values of isentropic compression modulusand its derivative with respect to pressure, determinedby the data of dynamic measurements,
The energy on the cold curve in the expansionregion σc < 1 is preassigned by the polynomial
(17)
which helps provide for the tabular value [32] of thesublimation energy
and equality (14). The validity of conditions (13), (15),and (16) leaves two free parameters, n and l, in Eq. (17).
The contribution by thermal excitations of crystallattice into free energy is written in a quasiharmonicapproximation,
(18)
where R is the gas constant. The characteristic temper-
ature is defined by the approximation dependence
(19)
in which x = lnσ, and σ = V0/V. The constants Bs and Ds
are found by the experimental dependence of the Grue-
neisen coefficient = dln /dlnσ on the degree of
PcTFC
S ai λ µ η, , ,( ) gn 1 Pc ai Vn,( )/PcTFC
Vn( )–[ ]2
n 1=
N
∑=
+ λPc ai( ) µ Bc ai( ) B0c–[ ] η Bc' ai( ) B0c'–[ ] ,+ +
B0c'
BS V ∂P/∂V( )S– BS0,= =
BS' ∂BS/∂P( )S BS0' .= =
Fc V( ) V0c A σcm
/m σcn/n–( )[=
+ B σcl/l σc
n/n–( ) ] Esub,+
Fc V ∞( ) Esub=
Fas( )
V T,( ) 3RTθc
s( )V( )
T-----------------,ln=
θcs( )
θcs( )
V( ) θ0sσ2/3
=
× γ0s 2/3–( )Bs
2Ds
2+Bs
-------------------xBs
Bs2
Ds x Ds+( )+---------------------------------------arctan
,exp
γcs( ) θc
s( )
compression σ, obtained from the data of shock-wavemeasurements. The value of γ0s corresponds to the tab-ular value of the Grueneisen coefficient under normalconditions. Note that, in the limit of high degrees ofcompression of matter, Eq. (19) provides for the desiredideal-gas asymptotic correlation θ ~ σ2/3. The parame-ter θ0s is found from the requirement of zero entropy inthe normal state S = –(∂F/∂T)V = 0.
The melting is accompanied by transition from theordered lattice structure to the disordered state of atomsof liquid. This brings about a variation of both thevibration spectrum and the values of heat capacity uponmelting and further heating; these characteristics areaffected differently by the anharmonism effects. Whilethe vibration spectrum may vary abruptly in the case ofmelting (first-order phase transition), the jump in heatcapacity turns out to be not as pronounced because ofpreservation of the short-range order in the liquid.
In order to include melting, the thermal contributionby the lattice atoms into the thermodynamic potentialof the fluid phase is selected as the sum of two terms,
(20)
The first term allows for the anharmonism effectsand defines the vibration spectrum and the value of heatcapacity at high temperatures, and the second term pro-vides for the experimental values of density andenthalpy jumps during melting, the behavior of themelting curve, and the description of the thermodynam-ics of the liquid state region close to the melting curve.
The part of free energy Ft responsible for anharmon-ism is preassigned by the approximation dependencesimilar to that given by Eq. (18), but with the analog ofheat capacity Ct and the characteristic temperature θ(l)
that are variable with respect to density and tempera-ture,
(21)
The expression
(22)
describes a smooth variation of the heat capacity of liq-uid from a value of the order of 3R (close to the latticeheat capacity) to that of the heat capacity of ideal gas ofatoms 3R/2. The coefficients σt and Tt define the char-acteristic density and temperature of this transition. Thefrequency spectrum θ(l)(V, T) reflects a gradual varia-tion of the values of the Grueneisen coefficient of theliquid phase from the values of γ(l) . γ(s) typical of thecondensed state to its ideal-gas value of 2/3 in the lim-iting case of high temperatures or large specific vol-
Fal( )
V T,( ) Ft V T,( ) Fm V T,( ).+=
Ftl( )
V T,( ) Ct V T,( )Tθ l( )
V T,( )T
-----------------------.ln=
Ct V T,( ) 3R2
------- 1σTt
σ σt+( ) T Tt+( )---------------------------------------+=
678
HIGH TEMPERATURE Vol. 39 No. 5 2001
TKACHENKO et al.
umes. In view of the foregoing, the approximation for-mula
(23)
was selected for the characteristic temperature, inwhich
the parameters Tsa , Tca , Bl, and Dl are determined usingthe data on the shock compressibility of solid andporous samples; and the constant θ0l is determined from
the condition (V ∞) = Tca .
The term Fm of the potential given by Eq. (20) mustprovide for the experimentally obtained values ofentropy jumps ∆S = ∆Sm0 and volume jumps ∆V = ∆Vm0at the melting point under normal pressure and decreaseabruptly with increasing specific volume in the gasphase. As the degree of compression increases, the rel-ative contribution Fm decreases slowly, which corre-sponds to obliteration of differences in the propertiesbetween the solid and liquid phases with decreasingspecific volume. With due regard for the satisfaction ofthese requirements,
(24)
where σm = σ/σm0 is the relative density of the liquidphase on the melting curve. The constants Am , Bm, andCm are uniquely determined from the conditions ofphase equilibrium at P = 1 bar and T = Tm0.
The contribution by the thermally excited electroncomponent into the free energy is preassigned by theexpression
(25)
where
(26)
θ l( )V T,( ) Tsa
θcl( )
V( ) T+Tca T+
--------------------------σc2/3
=
θcl( )
V( )
= θ0l γ0l 2/3–( )Bl
2Dl
2+Bl
------------------xBl
Bl2
Dl x Dl+( )+-----------------------------------arctan
;exp
θcl( )
Fm V T,( ) 3R2σm
2Tm0
1 σm3
+------------------- Cm
2Am
5---------- σm
5/31–( )+
=
---+ Bm Cm–( )T
,
Fe V T,( ) = Ce V T,( )T 1Be T( )T
2Cei
------------------σγe V T,( )–
+
,ln–
Be T( ) 2
T2
----- β τ( ) τ T , Ceidd
0
T
∫∫ 3RZ2
-----------,= =
and Z is the atomic number of element. Formulas (25)and (26) contain generalized analogs of the heat capac-ity of electron gas
(27)
(28)
of the coefficient of electron heat capacity
; (29)
and of the Grueneisen coefficient for electrons
(30)
These approximation dependences are selected suchas to first of all satisfy the limiting relations for freedegenerate electron gas at low temperatures (T ! TFermi)and for ideal electron gas in the T ∞ limit.
The analog of electron heat capacity given byEq. (27) at high temperatures tends to the limiting valueof 3RZ/2 that characterizes an ideal electron gas in thecase of total ionization of atoms. In the plasma region,the heat capacity curve conveys the effects of primaryionization and the behavior of ionized metal, with cor-relation (28) allowing for the decrease in the ionizationpotential with increasing plasma density. The tempera-ture Ti has a value characteristic of primary ionization,and the parameter σi corresponds to the density ofmetal–dielectric transition. The coefficients Tz and σz,respectively, define the temperature dependence of tran-sition from singly ionized gas to fully ionized plasmawith the ion charge Z and the characteristic density oftransition of matter from the metallic to plasma state.
Relation (29) describes the characteristic singulari-ties of the coefficient of electron heat capacity, namely,the existence of extremum and of limiting values at low(β0) and high (βi) temperatures. The analog of the Grue-neisen coefficient for electrons given by Eq. (30) is pre-assigned with due regard for the asymptotics γe = γe0 atT = 0 K and γe = 2/3 at σ 0, σ ∞, and T ∞.The term with γm enables one to include the anomaly ofthe temperature dependence γe(V . V0, T), the details ofwhich are described by the coefficients σe and σd . Thefree constants Tb and Tg give the temperature depen-dences β(T) and γe(V, T).
Formulas (11)–(30) fully define the functionaldependence of the potential of free energy of metal in awide range of density and temperature. In what follows,we give the results of calculations by the equation ofstate for tungsten obtained within this model, in com-parison with the available experimental data at highvalues of energy density.
It is known that tungsten under normal conditions isin the body-centered cubic (bcc) phase [33, 34]. Theresults of isothermal compression in diamond anvils atT = 293 K demonstrate the absence of phase transfor-mations of this metal up to P . 4.5 Mbar [35]. Figure 1gives a predicted T = 0 K isotherm for tungsten. Onecan see that the cold curve obtained using the proceduresuggested in [30] at low degrees of compression up toσ = ρ/ρ0 . 10 (where ρ0 is the density of matter undernormal conditions) agrees well with the semiempiricaldependence of [36] and, at extremely high pressuresand degrees of compression up to σ . 100, with theresults of calculations by the Thomas–Fermi modelwith corrections [29]. An analysis of the compressibil-ity of tungsten at T = 293 K also gives an adequatedescription of the results of static experiments with dia-mond anvils up to P . 100 kbar [37] (Fig. 2).
A considerable discrepancy is found in the availabledata on the slope of the melting curve for tungsten: thevalue of dT/dP = 7.8 K/kbar is recommended in the ref-erence book [33], while Shaner et al. [38] obtained4.4 K/kbar in their experiments in rapid isobaric heat-ing of conductors. In this paper, we assumed an inter-mediate value of dT/dP = 6 K/kbar. Apparently, formore definite conclusions to be drawn on the positionof the melting curve for tungsten, additional experi-ments or theoretical calculations must be performed.
The thermodynamic properties of tungsten in thesolid and liquid phases at high values of energy densitywere studied in some detail in shock waves. In the
megabar pressure range, the dynamic compressibilityof this metal was studied using conventional explosivelaunchers [39–42] and a two-stage light-gas gun [43,44]. Experiments in shock loading of porous samplesperformed using both plane and spherical cumulativesystems up to pressures of P . 3 Mbar [40, 45–48] con-siderably extend the investigated region of the phase
105
1 10 100ρ/ρ0
P, Mbar
104
103
102
101
100
100
80
60
40
20
01.00 1.01 1.02 1.03
ρ/ρ0
P, kbar
Fig. 1. The pressure in tungsten at T = 0 K. The solid line indicates our results and the results of [30], the broken line indicates theapproximation curve of [36], and the points indicate the results of calculation by the Thomas–Fermi model with corrections [29].
Fig. 2. The pressure in tungsten at T = 293 K. The pointsindicate the experimental results of [37], and the brokenlines indicate the range of experimental error.
680
HIGH TEMPERATURE Vol. 39 No. 5 2001
TKACHENKO et al.
diagram towards higher values of thermal energy com-pared with the states of solid samples on the shock adi-abat. The data on the relative compressibility of tung-sten at pressures P . 60 Mbar attained under conditionsof underground explosion are given in [49]. Of specialinterest are the results of measurements (under condi-tions of underground explosions) of the compressibilityof porous tungsten at m = ρ0/ρ00 = 3.07 (where ρ00 is theinitial density of samples) [50, 51], which correspondto the states of weakly degenerate electron gas with anextremely high energy density ρE . 2 MJ/cm3 at a den-sity of the order of normal density and at a pressure P .10 to 20 Mbar. Figures 3 and 4 give predicted shockadiabats of solid and porous samples of tungsten incomparison with experimental data in kinematic anddynamic variables (D and U denote the wave and massvelocities, respectively, and P is the pressure). Thephase diagram for metal in the entire range of densityand pressure investigated under conditions of shock-wave loading is given in Fig. 5. An analysis of thesegraphs reveals a high accuracy of description of theresults of dynamic experiments within the obtainedequation of state for tungsten.
Of importance from the standpoint of checking theadequacy of the thermodynamic model are the data onisentropic expansion of shock-compressed samples oftungsten of initial porosity m = 2.17 [52]. The calcula-tion results demonstrate that liquid phase states withentropy sufficient for evaporation in the release waveswere realized behind the shock wave front in the case ofmetal loading under the experimental conditions of[52]. The curves of pressure dependence of the velocityof flow of matter under conditions of adiabatic expan-sion calculated using the equation of state suggested byus agree with the data of [52] within the experimentalaccuracy (Fig. 6); this proves the reliability of calcula-tion of the thermodynamic characteristics of liquidtungsten, as well as of the position of the line of liquid–vapor equilibrium at high pressures and temperatures.
In the region of lower-than-normal density, the ther-mophysical properties of the metal being treated wereinvestigated in experiments involving rapid heating bya high-power current pulse in an atmosphere of inertgas (argon) at P = 1 to 3 kbar [3, 53, 54] and in water atP = 1 bar [7]. A comparison of the results of calculationperformed using the developed equation of state with
24
16
8
0 8 16
m = 1
1
2
3
4
5
6
7
8
U, km/s
D, km/s
3.04
3
2
1
0 2 4
1
2
3
4
5
U, km/s
P, Mbar
1.44
2.17
1.8
4.19
m = 1
2.90
3.5
Fig. 3. The shock adiabats of solid and porous samples oftungsten; m is the initial porosity of the samples. Experi-ment: (1) [40], (2) [42], (3) [43], (4) [44], (5) [45], (6) [49],(7) [50], (8) [51].
Fig. 4. The shock adiabats of samples of tungsten with dif-ferent values of initial porosity m. Experiment: (1) [40],(2) [42], (3) [44], (4) [45], (5) [46].
HIGH TEMPERATURE Vol. 39 No. 5 2001
METASTABLE STATES OF LIQUID METAL 681
the experimental data of [3, 7, 53, 54] is given in Fig. 7.Also given in this figure are the curves for the lines ofliquid–vapor equilibrium [55], obtained using thesemiempirical model [56] in which the Monte Carlomethod was used [57] to approximate the results ofsimulation of a system of particles with a soft-sphereinteraction potential. Two versions of calculations (V1and V2) were performed in [55] using the data of [3] asthe reference points (V1) and the data of [7] (V2) withdifferent slopes of the experimental isobars (∂ρ/∂T)P .In both versions, the calculation results agree, withinthe experimental error, with the data on the velocity ofsound [3] and on the saturated vapor pressure [58, 59].However, the version V1 produces values of pressure atthe entry of the release isentropes into the two-phaseliquid–vapor region that are underestimated by twoorders of magnitude compared with the estimatesobtained using the equations of the version V2 and thewide-range equation of state developed by us andinvolving the use of the data of [52] (see Fig. 7). Notethat the value of critical volume calculated by the V1version exceeds considerably the known estimates
[60−66], while the critical parameters according to theV2 version take an intermediate position relative toother estimates and are close to our results: Pcr =11.8 kbar, Tcr = 15750 ä, Vcr = 0.206 cm3/g, and Scr =0.837 J/(g K). The predicted value of the evaporationtemperature for tungsten at atmospheric pressure, Tv0 =5766 K, is close to the value of 5643 K given in the ref-erence book [59].
TABULAR FORM OF THE EQUATIONOF STATE
It is difficult to directly apply the multiphase equa-tion of state in the initial analytical form in hydrody-namic calculations because of the large amount of com-puter time required to find the parameters of flow ateach space and time point. Therefore, during simula-tion, we used the equation of state for tungsten in theform of tables filled in as follows.
The plane of variables (V, T) is covered with a rect-angular mesh with points (Vi , Tj), i = 1, …, NV , j =1, …, NT , that covers the range of parameters of interest
10
1
0.10.6 1.0 1.4 1.8
123456
ρ/ρ0
P, Mbar
7
T = 10 kK
20
50
100
200
400
600
m = 4.19
1Pc
1.44
2.171.8
2.93.04
800 Hm
1000
10
0.10 2 4 6
U, km/s
P, kbar
123456
m = 2.17
S1
1
S2
Hm
Fig. 5. The phase diagram of tungsten at high pressures: Pcis the curve of cold compression at T = 0 ä, M is the meltingregion, broken lines T indicate the isotherms, and solid linesHm indicate the shock adiabats of samples with differentvalues of initial porosity m. Experiment: (1) [40], (2) [42],(3) [43], (4) [44], (5) [45], (6) [50], (7) [51].
Fig. 6. The shock adiabats Hm and release isentropes S oftungsten; m is the initial porosity of the samples. The solidlines indicate equilibrium curves, the broken lines indicatemetastable isentropes, and the wavy lines indicate the pointsof intersection of isentropes with the liquid–vapor interface.Experiment: (1) [40], (2) [42], (3) [43], (4) [44], (5) [45],(6) [52].
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to us. In the temperature range of Tm0 e T e Tcr, the sizeof mesh spacing is selected such that the boundaries ofthe liquid–vapor phase transition would not cross thecell sides but would only pass through their corners.With this method of constructing the mesh, any stablestate of liquid, gas, or two-phase liquid–vapor mixturehas a point (V, T) corresponding to it; this latter pointlies either inside or on the boundary of a triangular orquadrangular cell all of whose corners belong to theregion of stability of one and the same phase state. Thevalue of the function fi, j = f(Vi, Tj) is preassigned at eachpoint of the mesh, where f is one of the thermodynamicquantities used in the calculations, i.e., the internalenergy E = F + TS, the Gibbs potential G = F + PV, theentropy S, the pressure P, or the adiabatic velocity ofsound CS . Then, the value of f at an arbitrary point(V, T) is found by bilinear interpolation over the cor-ners of the cell in which it finds itself.
For example, for the case of a cell formed by fourpoints (Vi , Tj ), (Vi + 1, Tj ), (Vi + 1, Tj + 1), and (Vi , Tj + 1),one can write
where
For describing the properties of superheated liquidand supercooled vapor using the analytical relation ofthe equation of state for tungsten suggested by us, addi-tional tables were formed of the thermodynamic char-acteristics of metastable phases in two regions on the(V, T) plane. The first one of these regions is defined bythe curve of equilibrium evaporation of liquid and theliquid spinodal, and the second region is defined by the
f V T,( ) 1 t–( ) 1 u–( ) f i j,=
+ t 1 u–( ) f i 1 j,+ tu f i 1 j 1+,+ 1 t–( )u f i j 1+, ,+ +
tV Vi–
Vi 1+ Vi–----------------------, u
T T j–T j 1+ T j–----------------------.= =
1.0
0.6
0.2
0 8 16 24
1
2
3
4
5
6
7
8
9
10
11
T, kK
ρ/ρ0
10
30
+CP
P = 100 kbar
⟨σ⟩
ρ/ρ0B
M 4
2
0.8 1.0
Cs,
km/s
10–3 1 3
Fig. 7. The phase diagram of tungsten at lower-than-normal densities: M is the melting region, B indicates the liquid–vapor equi-librium curves (solid line, our results; dot-and-dash line, V1 [55]; dashed line, V2 [55]), ⟨σ⟩ is the half-sum of degrees of compres-sion of the liquid and gas phases on the evaporation curve, and solid lines P are isobars. Experiment: (1) P = 3 kbar [3], (2) P =1 bar [7], (3) P = 1 kbar [53], (4) P = 2 kbar [54]. Estimates of the parameters of the critical point (CP): (5) our results, (6) [60],(7) [61], (8) [62], (9) [63], (10) V1 [55], (11) V2 [55]. In the inset: the adiabatic velocity of sound in tungsten at P = 3 kbar; thepoints indicate the experimental results of [3].
HIGH TEMPERATURE Vol. 39 No. 5 2001
METASTABLE STATES OF LIQUID METAL 683
curve of equilibrium condensation of vapor and thevapor spinodal. These two regions of existence of meta-stable states are divided into rectangular cells in thespace of variables (X, T), where
and Vb and Vsp denote the specific volume of the phasebeing treated on the binodal and spinodal at a giventemperature (below the critical), respectively. The valueof f(V, T) is determined by way of bilinear interpolationover the respective values of the function f at the pointsforming the cell in which the point (V, T) finds itself onthe coordinate plane (X, T). Triangular cells, for whichone of the corners is provided by the point (Vcr, Tcr), aretreated as a special case. The continuity of the tabularequation of state for the liquid and gas phases duringtransition from the region of phase stability to metasta-ble states is accomplished by the choice of one and thesame series of T = Tj isotherms for the mesh on eitherside of the line of liquid–vapor phase equilibrium.
Figure 8 shows the pattern of volume dependencesof pressure on the isotherms of the fluid phase of tung-sten with regions of stable, metastable, and unstablestates. One can see that, during transition from the vari-ables (V, T) to the plane of coordinates (V, P), the initialrectangular mesh becomes curved; however, the cellsremain regular. A similar statement is true of the (V, E)plane. In view of this, it is possible, for any pair of vari-ables (V, P) or (V, E) for which the condition P r Pc(V)or E r Ec(V) is valid, to find first the respective value oftemperature T and then the value of f(V, T) using thetabular equation of state.
TRANSPORT PROPERTIES OF TUNGSTEN
In this paper, the electrical conductivity of tungstenis described by the semiempirical equation [67]
where σ = V0/V; σw0 and T0 are the respective coeffi-cients for each phase state of matter; and β and δ denotethe thermal coefficient of electrical conductivity andthe exponent, which also depend on the phase state ofmatter. Under conditions of melting, the effective elec-trical conductivity is calculated in accordance with thevolume content of the phase σef = νσs + (1 – ν)σl (here,σs and σl denote the electrical conductivity of the solidand liquid phases, respectively, and ν is the volumecontent of the solid phase). After this, the electrical con-ductivity is calculated in accordance with the Wiede-mann–Franz law, κ = kwfTσw .
We performed a numerical simulation of the initialstage of electrical explosion of a tungsten conductor ofcylindrical shape (radius r = 7.5 µm, length l = 1 cm[16]). Figure 9 gives the time dependence of the pre-dicted values of current through the sample beinginvestigated, of the resistance of tungsten conductor, ofthe voltage drop across the wire, of the radius of the lat-ter, and of its density. At the moment of time t . 23 ns,the melting of the sample is completed, after which itsresistance exhibits a weak variation. A similar behaviorof the time dependence of the tungsten conductor resis-tance is observed in the case of microsecond modes ofenergy contribution [22]. By the moment of completionof melting, the current reaches its maximum value andthen decreases somewhat, as in experiments of [16].Note that a considerable increase in current wasobserved late in the experiment of [16]; this increasewas caused by the increase, because of breakdown, inthe conductivity of metallic vapor surrounding thedense cold core.
One can see in Fig. 9 that the sample continues toheat up after melting, and soon the parameters areattained that correspond to the liquid–vapor equilib-rium curve. In accordance with the theory of homoge-neous nucleation [17, 18], when the temperature rises ata rate of more than 106 K/s, the time of formation of astationary distribution of fluctuation boiling centers isτn ~ 10 ns. Because the characteristic rate of heating for
0.1
0
–0.1
–0.2
0.2 0.4 0.6 0.8 1.0ρ/ρ0
P, Mbar
T = 20 kK
B
SpPc
4 kK
0 K
M
Fig. 8. The state diagram of the fluid phase of tungsten: Mindicates the region of possible coexistence of the stable andmetastable solid and liquid phases; B indicates the curve ofliquid–vapor equilibrium with the critical point (circle); thebroken line Sp is the spinodal for liquid and gas; solidcurves T are the isotherms of the fluid phase T = 20, 18, 16,14, 12, 10, 8, 6, 4, and 0 kK, from top to bottom; and Pc isthe cold curve for crystal at T = 0 K.
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HIGH TEMPERATURE Vol. 39 No. 5 2001
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the mode of energy contribution treated by us is of theorder of 1012 K/s, the liquid phase of conductor mayremain in the metastable state for a period ∆t . τn afterthe parameters on the binodal are attained. In addition,as was indicated previously in [10, 18], in the processof electric explosion of metal under the effect of a high-power current pulse, the magnetic field has a stabilizingeffect on superheated liquid, because, under these con-ditions, the vapor phase nuclei must perform (in addi-tion to the work of surface formation) the work of curv-ing the lines of magnetic force. Therefore, in analyzingthe dynamics of electric explosion, one must take intoaccount the possibility of realization of metastablestates of the liquid phase in the initial stage of the pro-cess (for the mode of stimulation being simulated, theduration of this stage is τh ~ 10 ns).
The trajectories of the central and outer layers of thewire are shown on the pressure–temperature phaseplane (Fig. 10). One can see that the conductor heats up
to a temperature T . 14 kK (by this moment of time,about 5% of the total mass of the wire evaporates fromthe surface) without reaching the parameters corre-sponding to the liquid spinodal. Figure 11 gives the tra-jectories of the same wire layers on the density–temper-ature phase plane. The results of combined analysis ofFigs. 10 and 11 indicate that significant variations ofpressure in the compression and release waves propa-gating between the external surface and the conductoraxis of symmetry are accompanied by a slight fluctua-tion of the density of condensed phase. This behavior ofmatter is possible only with the emergence of metasta-ble states of liquid (at pressures lower than those on thebinodal, including at negative pressure) whose proper-ties differ considerably from those of the equilibriumtwo-phase liquid–vapor mixture at the same tempera-tures (see Fig. 8).
Figure 12 gives the predicted distribution of densityon the wire radius at different moments of time. Two
0.3
0.2
0.1
0 10 20 30 40
I
R
(a)
t, ns
I, kA; R, kOhm
20
10
0 10 20 30 40
r
U(b)
U, kV; r, µm
20
10
0 10 20 30 40
ρ
T
(c)
t, ns
T, kK; ρ, g/cm3
Fig. 9. (a) The current I and the wire resistance R, (b) the voltage U on the conductor and the radius r, and (c) the temperature T andthe density ρ of the sample being investigated as functions of time.
HIGH TEMPERATURE Vol. 39 No. 5 2001
METASTABLE STATES OF LIQUID METAL 685
versions of calculations were performed, namely, thosewith due regard for metastable states and those subjectto the requirement of inertialess equilibrium liquid–vapor phase transition. The simulation results demon-strate that, if the possibility of superheating is ignored,the obtained distributions of the parameters of matterand field on the conductor radius are highly nonuni-form. However, according to the experimental data of[16], in the modes of electric explosion with character-istics preassigned in the calculation, the condensedphase of sample (core) remains uniform on the radiusuntil late in the process.
One can see in Fig. 12 that the distributions of metaldensity, obtained with due regard for metastable states,may be assumed to be uniform over the cross sectionwith a high degree of accuracy. A similar inference isvalid for the predicted distributions of temperature andcurrent density in the condensed phase.
PARAMETERS OF VAPOR
Equation (8) enables one to find the pressure, spe-cific volume, and mass velocity of metallic vapor sur-rounding the liquid conductor and, in accordance withthese data, estimate the degree of ionization [4],
ne2
2 g1/g0( )n0
2πkmeTh
--------------------
3/2 I–kT------exp=
and the electrical conductivity [68]
where n0 and ne denote the concentration of atoms and
σv
32nee2τe
3πme
---------------------,=
4
0
–4
4 8 12 16
P, kbar
T, kK
1
2
B
g CP
l
20
16
12
8
0 4 8 12 16T, kK
ρ, g/cm3
1
2
Sp
B
20
16
12
8
0 2 4 6 8 10
ρ, g/cm3
r, µm
18
24
26 28
2935
45
Fig. 10. The trajectory of (1) the axial and (2) the outer lay-ers of conductor on the P–T plane: B indicates the curve ofliquid–vapor equilibrium with the critical point (CP); g andl indicate the gas and liquid spinodals, respectively.
Fig. 11. The trajectory of (1) the axial and (2) the outer lay-ers on the ρ−T plane: B indicates the curve of liquid–vaporequilibrium, and Sp indicates the liquid spinodal.
Fig. 12. The radial distribution of the density ρ of con-densed phase of sample at different moments of time: thesolid lines indicate the results of calculation with due regardfor metastable states, the broken lines indicate the results ofcalculation disregarding metastable states, and the numeralsat the curves indicate the time in nanoseconds.
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electrons, respectively; me is the electron mass; I is theionization potential of matter; g1 and g0 denote the sta-tistical weight of ion and atom, respectively; and τe ~0.1 ps is the electron relaxation time. In so doing, onemust take into account the fact that the evaporation ofliquid in a gas medium with low back pressure isaccompanied by the formation of the liquid phase–vapor interface of a nonequilibrium Knudsen layer[25, 27], in which the relaxation of Maxwellian veloc-ity distribution occurs. With the Mach number M = 1,the temperature jump in the Knudsen layer may be esti-mated at ∆T . 0.33Ta [25].
The electrical conductivity of liquid metal under heat-ing differs little and amounts to σax . 5 mOhm–1 cm–1 atT . 10 kK. At the same time, with the liquid tempera-ture T . 10 kK, we will have σv . 0.05σax; with T .12 kK, σv . 0.6σax; and with T . 14 kK, σv . 2σax .Therefore, when the system is heated to T . 1 eV, theconductivity of the liquid column becomes comparablein magnitude with that of the surrounding vapor, and,with further energy contribution, the current from thecentral, condensed part of the sample will be redistrib-uted to the periphery. An abrupt rise of current isrecorded, as well as intense radiation accompanying themetallic vapor breakdown [16]. The model suggestedby us does not describe the effects observed after thismoment of time. In the subsequent stages of the pro-cess, one must treat the multiple ionization of matterand take into account the kinetics of relaxation of thetemperature of the electron subsystem, as well as pos-sible disturbances of spatial homogeneity of the cylin-drical conductor.
CONCLUSION
Therefore, the possibility of realization of metasta-ble states of liquid has a considerable influence on thedynamics of the initial stage of nanosecond electricexplosions of conductors under the effect of a high-power current pulse. The effects, which influence thephase stability of the conductor and are due to the pres-ence of a body force, as well as of mass displacement,prove to be insignificant in the case of parameters pre-assigned in the calculation. It is to be expected that theinfluence of these effects will be appreciable with muchlonger durations of the processes. The redistribution ofcurrent to the external vapor envelope is possible whenthe conductor is heated to T . 1 eV, and it is only afterthis that one must expect the separation of matter intophases in the core that remains cold.
ACKNOWLEDGMENTS
This study received support from the Russian Foun-dation for Basic Research (grant nos. 00-15-96529,99-02-16596, and 99-02-19916).
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