Non equilibrium thermal and electrical transport coefficients for hot metals

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by  Juan-Luis Domenech-Garret

Non equilibrium thermal and electrical transport coefficients for hot metals
∗
J.L. Domenech-Garret
Departamento de F´ısica Aplicada, E.T.S.I. Aerona´utica y del Espacio.
Univ. Polit´ecnica de Madrid, 28040 Madrid, Spain.
(Dated: January 25, 2017)
This work discusses about the transport coefficients for non equilibrium hot metals. First, we
reviewtheroleofthenonequilibriumKappadistributioninwhichtheKappaparametervarieswith
7 the temperature. A brief discussion compares such distribution with the classical non equilibrium
1 function. Later, we analyze the generalized electrical conductivity obtained from the evolution of
0 theKappadistribution. Also,wereexaminetheconnectionbetweenamaterial-dependentcoefficient
2 whichcanbeextractedfromthethermionicemissionandthemeltingpointofthemetal. Weextend
previous studies by analyzing additional metals used as thermionic emitters. Finally, in the light
n
a of the Wiedemann-Franz Law, we present a new generalized thermal conductivity, which is also
J applied to several metals.

3
PACSnumbers: 05.90.+m72.15.Cz64.70.dj
2

]
l I. INTRODUCTION anexternalelectricfieldappliedtoit. Additionally, such
l
a metalisheatedby meansofaninner current. This heat-
h ing can be increased until the melting point. Besides,
- As it is well known, metals in equilibrium are de-
s scribed by means of the Fermi-Dirac distribution. How- from the evolution of the Kappa energy distribution has
e been derived a generalized Ohm law containing a gen-
ever there are many situations in which the non equi-
m
eralized electrical conductivity, [6]. Such conductivity
librium is present, and it is difficult to assume that this
. evolves with the temperature, and it drops when such a
t distribution depicts the electronpopulation ofsuchmet-
a melting temperature is reached. According to [1, 6], the
als. Thisisthecaseofmetalsrapidlyheatedbymeansof
m starting mechanism of melting can be originated from
dense currents, [1]. Also, the non equilibrium is present
- the enhancement of energy deposition on the metal lat-
in metals exposed to a high power laser, [2]. The de-
d
ticecausedbyincreasinghighenergyelectronpopulation.
n viation from equilibrium has been also investigated con-
Subsequently,it wouldincreasethe defectdensity onthe
o cerning the thermionic emission [3–5]. Moreover, there
lattice leading to the melting.
c are reported departures from equilibrium of metals near
[ Throughthiswork,wewillreviewthisgeneralizedelec-
their melting point [1, 6]. In addition, the departures
trical conductivity and we will apply it to other metals.
1 from equilibrium of particle distributions have been also
Moreover, as we shall see, from such a generalized coef-
v applied in wider contexts [3].
ficient, the Wiedemann-Franz Law can be extended in a
8
The Kappa distributions are used to a large extent
9 phenomenologicalmannerto thenon-equilibrium,giving
in statistical mechanics and other fields to study pop-
5 rise to a generalized thermal conductivity. The result-
ulations near and far from equilibrium. The classical
6 ing generalized law is applied later to study the thermal
0 Kappa is applied to nonequilibriumplasmas in space [7– conductivity of several metals.
. 9]. These kinds of distributions canbe castfrom general
1
formalisms. The non-extensive q-statistics [10, 11], and
0
7 the so-called Beck-Cohen superstatistics provide physi- II. THE KAPPA DISTRIBUTION IN METALS
1 cal meaning to these distributions [12]. Besides, Kappa
: distributions can be studied from the so called Kappa-
v The general purpose form for the generalized Kappa
deformedalgebras[13]. Thenon-extensivestatisticshave
i Fermi-Dirac distribution can be put as follows [9, 15],
X beenalsoappliedtoestablishgeneralpurposeKappadis-
ar tproibinuttioofnvsiefowr,btohseoKnsaapnpdafdeirsmtriiobnusti[o1n4,c1a5n].bFerroemgaarndoetdhaers fFD(T,E)= 1+ 1+ E−ǫF (κ+1) −1 (2.1)
thesolutionoftheFokker-Planckequationregardingcol- κ φ(κ)
" (cid:18) (cid:19) #
lective effects and collisional processes, [16].
InthisworkwewillapplytheFermi-DiracKappafunc- where T is the temperature, E, the electron energy, ǫF
tiontostudyinaphenomenologicalwaythebehaviourof being the Fermi energy. Here, φ(κ) is a linear function
the electronpopulationinmetalsoutofequilibrium. We proportionalto kBT, with kB being the Boltzmann con-
will focus on metals which are used as thermionic emit- stant. Such φ function can be extracted from a charac-
ters. We will consider a small volume of a metal with teristic velocity which is related with the mean square
velocity v2 , [4, 7]. This distribution, as it can be seen
in Figure 1, is able to develop high energy tails as the
(cid:10) (cid:11)
Kappa value becomes lower. The FD distribution is re-
∗Electronicaddress: [email protected] covered for high Kappa values. Moreover, the Fermi en-
2
modulus of v. The latter can be put in terms of the
evolution of f due to the temperature gradient and
FD
the electric field,
∂f e F ∂f
FD FD
g(x,v ,v ,v )=−τ + (2.4)
x y z ∂x m|v| ∂|v|
(cid:20) (cid:21)
where m is the electron mass, and τ stands for a relax-
ation time, which is related with the electron mean free
path, l = |v|τ, [6, 17, 19]. This perturbed term can be
relatedwiththeusualBGKcollisionoperator[6,20]. To
comparewith the first orderf distribution, we linearize
p
thef distribution,Eq.(2.2),bydevelopingitintheTay-
κ
lorseries. TheresultcanbeseeninFigure2. Bothdistri-
butions were evaluated at T = 3000K. As the input for
f , the electric field was also inserted. From this Figure
p
we realize the behaviour of both distributions is similar.
As it will be discussed later, this Figure also suggest the
Figure 1: (color on line) Behaviour of the κ-FD distri-
bution of Eq.(2.1) for different values of κ. Here, φ(κ) = Kappa parameter should also entangle the electric field.
(κ−3/2)kBT +ǫF. The equilibrium Fermi-Dirac distribu- Moreover,from the above distribution, the thermionic
tion is recovered as theKappa becomes higher. currentemittedfromthemetalsurfacealongtheperpen-
dicular direction pointed by the OX axis, can be calcu-
lated [4] using
ergies lie around few eV; much higher than the melting
temperature of real metals. Therefore, following refer- ∞ ∞ ∞
ence [4], from this distribution it can be derived an ex- Jκ(T)= e vx fκ(T,v) d3v
pression which is suitable for metals. The Generalized Zvox Z−∞Z−∞
FDdistributionbecomessmoothedforlowKappavalues Here v represents the minimum of velocity needed to
ox
when (E−ǫF)≪kBT or (E−ǫF)≫kBT. Under such overtake the potential barrier. This yields an expression
conditions, the following distribution has been built to which has the form [4],
describe the metals [4],
−κ+1
fκ(T,E)=Cκ(T) 1 + EE−+ǫǫF −(κ+1) (2.2) Jκ(T)= Bκ(T) ×(cid:18)1 + EκW+fǫF(cid:19) (2.5)
(cid:18) κ F (cid:19)
Here, B (T) is a factor, and W is the usual work func-
κ f
Here, the constant Cκ(T) is normalized to the electron tion. For low Kappa values this expression predicts
density of the metal, [4]. In this case, the φ(κ) fac- thermionic currents much higher than those calculated
tor of Equation (2.1) takes into account the velocity by the Richarson-Dushmann (RD) law. In the limit for
at the Fermi level ǫF = mvF2/2 within the calculation large κ, the above Equation (2.5) recovers the RD law.
of the mean square velocity. Then, according to refer- TotestEquation(2.5),inapreviouswork,[5], asetof
ence [4], the φ(κ) term transforms into Eκ +ǫF, where experimentshasbeendone. Briefly,itconsistsofaTung-
Eκ =(κ−3/2)kBT. Asexpected,forhighKappavalues sten wire merged into a vacuum chamber and heated
corresponding to equilibrium, fκ(T,E) recovers the FD with a DC current, J. By means of a sweep system,
distribution. the emitted current from the wire surface is measured.
In addition, we can test the suitability of the Kappa From these experiments, a departure from the RD ex-
distribution to describe a nonequilibrium metal. We can pressioncan be observedwhen the wire temperature be-
comparethefκdistributionwiththeclassicallylinearized comes higher. The high values of the measured emitted
distribution function usually employed to describe the current at high temperature can not be recovered from
outofequilibriumelectronpopulationofametalmerged the classical expression. The deviation from equilibrium
within an electric field, F, [6, 17, 18]. Under these con- wascalculatedusingEq.(2.5) asa function ofthe Kappa
ditions, the distribution function as altered by the field parameter. Asanexperimentalresult,theKappaparam-
can be written in terms of the equilibrium Fermi-Dirac eter was found to be temperature dependent, κ(T). The
distribution. The non-equilibrium distribution reads as, possibility of a κ(T) dependence has been also pointed
out in a more general study on suprathermal distribu-
tions, [15]. Using Equation (2.5) and the experiments,
f =f + v g(x,v ,v ,v ) (2.3)
p FD x x y z a linear Kappa law was found through a pure fit of the
current along the available range of temperatures,[5].
inwhich,thegfunctiondescribestheperturbedpartand
it is a function of the velocity components through the κ(T)=a − b T (2.6)
3
III. GENERALIZED THERMAL AND
ELECTRICAL CONDUCTIVITY OF METALS
In this section, we will analyze the impact of the κ(T)
lawwhentheKappadistributionevolvesaccordingtothe
Boltzmann Transport Equation (BTE).
∂f ∂f dT ∂f
κ κ
= v + e F (3.1)
x
∂t ∂T dx ∂E
(cid:20) (cid:21)coll (cid:20) (cid:21)
InwhichweusetheBTEwithf (T),Eqs.(2.2)and(2.6).
κ
ThecalculationsleadtoageneralizedOhmLawJ =σ F
κ
with an electricalconductivity, σ , whichcan be written
κ
as, [6],
σ = σ 1+C (T,E) (3.2)
κ κ
Figure 2: (color on line) Comparison between the usual (cid:20) (cid:21)
perturbeddistributionofmetalsfp (dot-dashedline)andthe Here, σ stands for the usual, equilibrium, electrical con-
linearized fκ(solid line) : T =3000K; F =1.9kV ductivity. The corrective term is labeled as C (T,E),
κ
and it becomes zero for large κ. Therefore the usual
Ohm law is recovered for metals in equilibrium. The
explicit form of σ can be read in Equation (2.20) of
κ
[6]. As the temperature approaches the melting point,
T , this correction begins to be important. When T
m m
is reached, the σ becomes singular and subsequently
κ
it drops. The responsible for this vanishing conductiv-
ity is the singular behaviour of the Digamma function,
ψ[κ(b,T)], contained within the corrective term. Such
Digamma depends on the temperature and the material
properties through Kappa. As the temperature rises the
Kappavaluedecreasesanditiscontrolledbytheslopeb.
Since such a coefficient is material dependent, this sug-
gests that it contains specific informationabout how the
particular metal departs from equilibrium, [6].
If there is available experimental data about the slope
ofκ(T)ofametalfromitsthermionicemission,itcanbe
usedwithinequation(3.2)inorderto predictits melting
point. Conversely, using the value of the melting points
ofmetals,T ,asaninput,byfindingrootsfromequation
m
(3.2) the b coefficient can be obtained, [6].
Figure3: (coloronline)Electricalconductivity. Comparison In this work we provide the study about the σκ and b
of σκ with theexperimental data: Case of Tantalum. forTantalum,sinceitisalsousedasathermionicemitter,
[21]. Figure 3 shows the comparison of the experimen-
tal behaviour of the electrical conductivity of Tantalum
withthetemperatureandtheobtainedvaluesfromequa-
tion (3.2). The experimental data relevant to this study
was collected from [21–23]. The calculated coefficient is
The b slope was found to be material dependent. For b = 0.02 K−1 from its melting point T (Ta)= 3293K.
m
thermionicmetals,thisvalueisb≃10−2(K−1)[5,6]. By The typical Kappa values far from the melting point are
inserting this κ(T) law into the emitted current J (T), κ(T = 1500K) = 39; a high value which would corre-
κ
the experimental results are recovered with a very good spond to a distribution almost in equilibrium. Near the
agreementoverthe entiretemperature range. Therefore, melting κ(T = 3180K) = 5, it shows the transition to-
the experiments and this later law, indicate that the de- ward non-equilibrium. These results are similar to those
parturefromtheequilibriumofmetalsisgovernedbythe found in other thermionic metals in reference [6]. The
temperature modulated by the b coefficient. The scope electronpopulationdevelopshighenergytailsinteracting
and the suitability of such a linear κ(T) law will be dis- with the metal lattice and increases the disequilibrium
cussed later. until it reaches the melting point. This interaction leads
4
Figure 5: (color on line) Thermal conductivity , χκ of Plat-
Figure 4: (color on line) Thermal conductivity,χκ of Tung- inum for different energies in Fermi energy units, EF.
sten for different energies in Fermienergy units, EF.
calculated from the experimental melting point. Both
to the enhancement of energy deposition on the particu- cases agree with the available experimental extrapola-
lar metal lattice. Subsequently this absorption increases tions. Theseextrapolationsarebasedondifferentexper-
the defect density in the lattice [1], leading to melting. iments on thermal conductivity near the melting point,
On the other hand, we can use the Equation (3.2) to [24]. As a new feature, the predicted thermal conductiv-
generalize the Wiedemann-Franz law (WFL). First, we ityofthesolidphaseofthemetaldropswhenthemelting
define the generalized thermal conductivity, χ , point is reached.
κ
χ = χ 1+C (T,E) (3.3) IV. CONCLUSIONS
κ κ
(cid:20) (cid:21)
where χ stands for the usual thermal conductivity of In this work we have studied the previously modeled
metalinequilibrium. TheC withinbracketscorresponds ad-hoc Kappa distribution for metals, which comes from
κ
to the same corrective term of Equation (3.2). the generalKappa Fermi-Dirac. This fκ function is nor-
The classical WFL reads as, malized to the electron density of the metal. By ana-
lyzing the thermionic emission of several metals, it was
found that the Kappa parameter depends on tempera-
χ
=LN (3.4) ture. From the experiments, this κ(T) law was found to
σ T
be linear. At this point it must be noticed that this lin-
where, again, σ is the equilibrium electrical conductiv- ear law was extracted in a phenomenologicalway within
ity, and L is the Lorentz number. We can rewrite the alimitedtemperaturerange,notfromtheory. Therefore,
N
WFL in terms of the generalized electrical conductivity. the possibility to find this law as an effective expression
Hence, by inserting Equations (3.2) and (3.3) into the constrainedtoasmalltemperaturerangecannotbe dis-
above expression we attain, carded.
The b slope actually measures the rate of change of
χ κ with respect to the temperature and it could entangle
κ
=LN (3.5) the physics of such a mechanism. In addition, to ana-
σ T
κ
lyzethe suitability off for metals,we comparedit with
κ
Therefore, the same WFL holds by replacing the usual the classical perturbed distribution function for metals,
transport coefficients with their respective generalized findingasimilarbehaviour. Thislatterfunctiondepends
expressions. Also, from Equation (3.5) it is possible to on the electric field applied to the metal. As it is well
calculateχ throughaknowledgeofthegeneralizedelec- known such a field is a source of disequilibrium [17, 18].
κ
trical conductivity of a particular metal. Therefore it supports the idea the κ(T) law is an effec-
In Figures 4 and 5, we can observe the results of χ tive expression. It is expected the Kappa parameter de-
κ
obtainedforTungstenandPlatinumusing Eqs.(3.2) and pends notonlyonthe temperature,butitshouldbe also
(3.5) for different energies, in ǫ units. In the first case a function of the applied electric field. Looking at the
F
the σ were obtained from the experimental b coefficient linearκ(T)law,itsuggeststhe b coefficiententanglesin-
κ
for Tungsten. In the Pt case the corresponding σ were formation not only about the features of the metal, but
κ
5
also the conditions leading to nonequilibrium. Hence, ousstudies byapplying the generalizedexpressionofthe
it is mandatory to obtain an explicit expression of this electricalconductivitytothe caseofTantalum. This lat-
κ(T,F,...) law. This work is in progress. ter metal is also used as a thermionic emitter. We found
Concerningthe mechanismofthe developmentofhigh all the values extracted are consistent with previous re-
energytails,againit isentangledwithin the b coefficient sults from other metal emitters. Finally, in light of the
of the κ(T) index. The phase transition is mainly de- Wiedemann-Franz Law, we presented a new generalized
scribedbytheDigammatermswithinC (T,E). Asmen- thermal conductivity. This Kappa transport coefficient
κ
tioned, the Kappa distributions develop through the ad- ledustogeneralizetheWFLtonon-equilibrium. Aswell,
justable Kappa parameter. Such an index is modeled on this generalized WFL allowed us to calculate the ther-
thetermsofthewave-electroninteractionsandstrengths, malconductivityvaluesnearthemelting pointofseveral
[8]. Near the phase transitionit is expected that the lat- metals. These values are consistent with the experimen-
tice oscillations will become larger. Their interactions tal extrapolations of the thermal conductivity near their
with the electron-latticewavescould be the originof the melting points. In addition, as a new feature, such an
developmentofsuch(low-Kappa)highenergytails. Sim- expression makes the thermal conductivity vanish when
ilareffectsanalyzingtheinteractionbetweenthephonons the melting point of the solid phase is reached.
and electrons in heterolayershave been reported [25]. In
this latter case the cause of the increasing energy of the
electron distribution comes from the reabsortionof non-
equilibrium phonons. In our case, as the temperature
Acknowledgments
rises, the increasing defect density in the metal lattice
could play the role of the heterointerfaces. Again, more
effort is needed to clarify such a topic. This work was funded by the MINECO, Spanish Min-
On the other hand, in this work we extended previ- istry, under Grant ESP2013-41078-R.
[1] S.V.LebedevandA.I.Savvatimskii,Metals during rapid [14] A. Algin and M. Senay, High-temperature behavior of a
heatingbydensecurrents Sov.Phys.Usp.27,(1984)749. deformedFermigasobeyinginterpolatingstatistics Phys.
[2] B.Y. Mueller and B. Rethfeld, relaxation dynamics Rev.E 85,(2012)041123.
in laser-excited metals under nonequilibrium conditions, [15] R.A. Treumann,Europhys. Lett. 48 (1),(1999) 8.
Phys.Rev.B87, (2013) 035139. [16] A. Hasegawa, K. Mima and MinhDuong-van, Plasma
[3] V.E. Zakharov, V.I. Karas Nonequilibrium Kolmogorov- Distribution Function ina Superthermal Radiation Field
tipeparticledistributionsandtheirapplications,Physics- Phys. Rev.Lett. 54, (24), (1985) 2608.
Uspekhi56(1) (2013) 49. [17] B.G. Lindsay, Introduction to Physical Statistics.
[4] J.L. Domenech-Garret, S.P. Tierno and L. Conde, En- (J.Wiley and Sons,New York 1962) Chapter 10.
hanced thermionic currents by non equilibrium electron [18] J.M Ziman, Electron and phonons: The theory of trans-
populations of metals, Eur. Phys. J. B 86,(2013)382. port phenomena in solids. (Oxford University Press,
[5] J.L. Domenech-Garret, S.P. Tierno and L. Conde, Non- 1960).
equilibrium thermionic electron emission for metals at [19] J.M Ziman, Electrons in metals: A short guide to the
hightemperatures arXiv:1505.05378 [cond-mat.mes-hall]. Fermi Surface (TaylorandFrancis, London1963),Chap-
J. of Appl.Phys.118, (2015)074904. ters I and IV.
[6] J.L. Domenech-Garret, Generalized electrical conductiv- [20] P.L. Bhatnagar, E.P. Gross, and M. Krook, A model for
ity and the melting point of thermionic metals, J. Stat. collisionprocessesingasesI.Phys.Rev.94(3),(1954)511.
Mech. (Vol.2015), (2015),P08001 [21] F. Cardarelli, Materials Handbook (Springer, N.Y.,
[7] J.J. Podesta, NASAReports, CR-2004-212770, (2004). 2008), Chapter 9.
[8] B.D.Shizgal,Suprathermalparticledistributionsinspace [22] P.D. Desay, T.K. Chu, H.M. James, C.Y. Ho Electrical
physics: Kappa distributions and entropy. Astrophys. ResistivityofSelectedElementsJ.Chem.Phys.Ref.Data
SpaceSci. 312, (2007) 227. 13(4),(1984) 1069.
[9] G. Livadiotis, D.J. McComas, Astroph.J. 741:88 (2011). [23] D.R.Lide,ed., CRCHandbookofChemistryandPhysics
doi:10.1088/0004-637X/741/2/88. (CRC Press LLC 1995,2005), Section 12.
[10] C. Tsallis, Possible generalization of Boltzmann- Gibbs [24] R.W. Powell, C.Y. Ho, and P.E. Liley, Thermal Con-
statistics, J. Stat. Phys.52 (1988) 479. ductivity of Selected Materials (NSRDS-NBS-8National
[11] C. Tsallis, Nonextensive physics, Phys. Lett. A 195, Bureau of Standards, U.S. Government Printing of-
(1994)329. fice,1966), Category 5.
[12] C. Beck, E.G.D. Cohen, Superstatistics, Physica A 322 [25] K.KimandC.Hess,MonteCarlostudy ofelectron heat-
(2003) 267. ingandenhanced thermionicemissionbyhotphonons in
[13] G. Kaniadakis, Non-linear kinetics underlying general- heterolayers Appl.Phys. Lett. 52(14), (1988) 1167.
ized statistics Physica A 296, 405 (2001).