International Journal of Thermophysics, Vol. 11, No.

4, 1990

The Properties of Liquid Carbon 1

J. Steinbeck, 2'3 G. Dresselhaus, 4 and M. S. Dresselhaus 2'5

A model for the transport properties of liquid carbon based on Ziman liquid
metal theory with refinements for polyvalent liquid metals and Fermi surface
blurring is applied to calculate the electrical resistivity of liquid carbon at the
melting temperature. The thermal and electrical properties predicted by the
model are compared to experimental results using numerical heat flow calcula-
tions and found to be in good agreement with pulsed-current heating
experiments on the resistivity of carbon fibers.

KEY WORDS: carbon (liquid); electrical resistivity; high temperatures; laser

heating.

1. I N T R O D U C T I O N

The p r o p e r t i e s of the liquid phase of c a r b o n have been of recent interest

because e x p e r i m e n t a l results of r a p i d - p u l s e d laser heating suggest that two
different liquid phases of c a r b o n m a y in fact exist [1, 2]. The p r o p e r t i e s of
liquid c a r b o n are difficult to d e t e r m i n e by direct o b s e r v a t i o n due to the
extreme t e m p e r a t u r e s ( > 4 0 0 0 K ) required to melt solid c a r b o n [ 3 , 4 ] ,
t h e r e b y forcing a reliance on indirect m e t h o d s for the d e t e r m i n a t i o n of the
t r a n s p o r t a n d optical p r o p e r t i e s of m o l t e n carbon. Stevenson a n d Ashcroft
[ 5 ] have c a l c u l a t e d the electrical resistivity of metallic liquid c a r b o n in the

t Paper presented at the First Workshop on Subsecond Thermophysics, June 20-21, 1988,
Gaithersburg, Maryland, U.S.A.
2 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, U.S.A.
3Present address: RADC/EEAC, Hanscom AFB, Massachusetts 01731, U.S.A.
4 Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, U.S.A.
5 Department of Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, U.S.A.
789
0195-928X/90/0700-0789506.00/0 ~,~ 1990 Plenum Publishing Corporation
790 Steinbeck, Dresselhaus, and Dresselhaus

4 I I I

3- ~/~?:0.4

c; / =0.3
,,C~..

o 0.5 ~.o ~.5 2.0

k/2kf
Fig. 1. Liquidstructure factor for liquid carbon
using the Percus-Yevick theory for packing
fractions t/= 0.3, 0.4, 0.5.

high-density (>25 g . c m - 3 ) regime. In their work, the Ziman theory [6]

was applied using a structure factor (see Fig. 1) for liquid carbon calculated
from the Percus-Yevick theory [7], and the electrical resistivity was then
calculated using the Born approximation. We have extended this model to
the low-density limit and find that the predictions of this model are in good
agreement with an experimental pulsed-current heating determination of
the resistivity of liquid carbon [8].

2. CALCULATIONS
Recent pulsed-current melting experiments on carbon fibers indicate
that liquid carbon is a liquid metal [8]. On this basis, a model for the
properties of low-density liquid carbon (1.6 g-cm -3) has been constructed
using Ziman liquid metal theory, which accounts for resistivity experiments
on most liquid metals. The Ziman liquid metal theory requires calculation
of the electron mean free path A from the liquid structure factor a(k) and
a model potential v(k) for the liquid according to the relation

~1 - dkkSa(k ) Iv(k)l 2 F(k, kf, l) (1)

In the conventional Ziman liquid metal theory, the Fermi surface blurring
effect (discussed below) is neglected by setting F(k, Icy, l) = 1/k.
Of the various parameters, the electron mean free path for liquid
carbon is the most sensitive to the atomic density of the liquid [9].
Experimental work by Bundy [3] provides information about the density
of liquid carbon through use of the Clausius-Clapeyron relation. From
Bundy's data for dp/dT and for the heat of fusion [3], the density of liquid
carbon is determined to be ~1.6 g-cm -3.
Properties of Liquid Carbon 791

Since liquid metals have nearly spherical Fermi surfaces, various

parameters (such as the electron density ne, the Fermi wave vector kr, the
Fermi velocity vf, and the Fermi energy Er) may be calculated easily on the
basis of a spherical Fermi surface and the results are given in Table I [10].
We assume, as for other polyvalent liquid metals, that the electron mass is
the free electron rest mass, because of a lack of detailed experimental data
on the effective mass of the carriers in polyvalent liquid metals.
The liquid structure factor a(k) for liquid carbon is calculated by
applying the integral formula for a(k) developed by Ashcroft and Lekner
[-7] to liquid carbon. Figure 1 shows the structure factor for liquid carbon
assuming the packing fractions, r/=0.3, 0.4, and 0.5, which are typical
values of r/for liquid metals [7]. From Fig. 1 for the liquid structure factor,
it is seen that the intensity of the primary peak increases as the packing
fraction increases. This behavior is expected since as more atoms are
packed into a constant volume, their positions become more rigidly fixed
due to finite size effects of the atoms in the liquid.
The model potential used to calculate the transport properties of

Table I: A Summary of the Properties for the Metallic Liquid

Carbon Phase [ 1 0 ]

Symbol Property Value Model

n~ Electron density 3.2 x 1023 c m 3 electrons/atom

4
kf Fermi wave vector 2.1 x 108 c m 1 Fermi gas
vf Fermi velocity 2.45 x 108 c m 9s 1 F e r m i gas
Ef Fermi energy 16.9 eV F e r m i gas
Pm Mass density 1.6 g . c m - 3 Clausius-Clapeyron
a M e a n C - C distance 2.06/~ --
Tm Melting point 4450 K Disorder depth calculation
Tv Boiling point 4700 K Vaporization calculation
Hf Heat of fusion 105 k J - m o l - 1 Thermodynamic models
St Entropy of fusion 2.63 J . m o l - 1 . K i Thermodynamic models
p~ Electrical resistivity 30 p Q . c m Pulsed-current experiments
to(T) Thermal conductivity 2.9 W - c m - 1 - K 1 Wiedemann-Franz
Cp(T) Heat capacity 26 J . m o l - 1 . K - 1 Fermi gas
rio Compressibility 6.4 x 10 - 12 P a - i Model calculation
A Mean free path 3.4 ~ Ziman liquid metal m o d e l
Rf Reflectivity 0.82 2 = 694 nm (Drude)
Rf Reflectivity 0.7 2 = 248 nm (Drude)
c~ Absorption coefficient 1.4 x 106 c m - 1 2 = 694 nm (Drude)
ct Absorption coefficient 1.7 x 106 c m - 1 2 = 248 nm (Drude)
D1 Liquid diffusivity 10 -4 e m 2 . s - 1 Liquid segregation model
Vp Interface diffusive speed 26 m . s - 1 Liquid segregation model
k Segregation coefficient 0.17 Liquid segregation model
792 Steinbeck, Dresseihaus, and Dresselhaus

liquid carbon is the Heine Abarenkov model potential [11, 12]. Using the
parameters given in Table I the model potential for liquid carbon is
calculated directly. The constants Ao, A1, and A2 needed for calculation of
the Heine-Abarenkov model potential were calculated using the energies of
the carbon 3s, 3p, and 3d orbitals and tabulated values for the Coulomb
wave functions [13] as reported by Animalu [-14]. The constants'are given
by Ao -- 2.05, A1 = 2.44, and A2 = 2.65.
In Fig. 2 the model potentials calculated for carbon and silicon [-14]
are shown and compared for 0 < k/2kf < 2, illustrating the rapid decrease in
the potential as k becomes smaller. Note the qualitatively similar behavior
of the two potentials (for C and for Si) with the nodes at similar values of
k/2kf. Comparing Figs. 1 and 2 for liquid carbon, the node in the
calculated model potential is seen to occur at a k value which is near the
center of the primary peak in the liquid structure factor. Thus the contribu-
tion to the scattering [-see Eq. (1)] will not be large even for k values where
the structure factor is greatest. Initial calculations of A in liquid carbon
using Eq. (1) with F(k, kf, l) =l/k show that the mean free path is only a
few interatomic spacings. Therefore, the Ziman model calculations were
refined to include the blurring of the Fermi surface due to electron scatter-
ing using the Ferraz and March formula [-16] for Fermi surface blurring.
The results for F(k, kf, l) used in the calculation of Eq. (1) are shown in
Fig. 3, where k and kf denote the two wave vectors involved in the scatter-
ing event. In the limit l -~ oo (where l is the electron mean free path for the
unblurred Fermi surface model), the difference between the initial and the
final k vectors cannot exceed 2kf.

0.2
I I I
0f ~ X C
si
Loo.Ts
~1
Carb
0.1-

"~ 0

-0.1 -

-0.2
0 0.5 1.0 1.5 2.0
k/2kf

Fig. 2. The model potential (eV) for carbon

compared with the model potential for Si [15]
calculated using the Heine-Abarenkov model
[12]. The numbers in the inset are the values of
the potential at k ~ 0.
Properties of Liquid Carbon 793

o o.5 ~.o ~.5 2.0

k/2kf
Fig. 3, The calculated functions F(k, kf, l ~ co )
and F(k, kr, l = 3.2 ~.) plotted for liquid carbon
illustrating the effect of a finite electron mean
free path. Note that the integral for the mean
free path must now be extended beyond 2kf.

Since the potential, v(k) falls off more rapidly than k 4, the integral
may be evaluated over a finite region (e.g., the first two diameters of the
Fermi sphere) and a good estimate of the electron mean free path A is still
obtained. Then evaluating the full integral in Eq. (1) by numerical techni-
ques, we obtain for the electron mean free path for liquid carbon A = 3.4 ~,
close to the value obtained with the simple calculation which neglected the
Fermi surface blurring effect.
From the calculated A, the electrical resistivity of liquid carbon p~
immediately follows as
mvf 39.5 ~ts'2 - cm (2)
Pl = nee2A

where we have again used the free electron gas properties [ 1]. This value
of p~ compares favorably with the experimental value for the electrical
resistivity of liquid carbon ~30___ 8/~f2 .cm [8]. It is also found that the
calculated electrical resistivity is relatively insensitive to the packing density
or hard-sphere radius for carbon.
Since all liquid metals can be approximately modeled as degenerate
Fermi gases, we may use this fact to calculate most of the thermal proper-
ties of liquid carbon (see Table I). The specific heat of the liquid at
constant pressure Cp(T) is given by
, 7rZkB T
Co(T ) = 3R + ~ g (3)

where R is the gas constant. The first term on the right-hand side of Eq. (3)
is the atomic contribution to C p ( T ) , while the second term on the right-
794 Steinbeck, Dresselhaus, and Dresselhaus

hand side is the electronic contribution. The electronic contribution to the

thermal conductivity ~c can be calculated using the Wiedemann Franz
relation

ZC2 kB2 T
K 3e2 Pl (4)

where p~ is the electrical resistivity calculated above.

The specific heat for liquid carbon has been calculated by Leider et al.
[173 and the agreement between the liquid metal theory prediction and the
prediction by Leider et al. [17] for liquid carbon agrees to within ~ 10%.
The temperature dependence of the specific heat in both models is linear.
The salient feature of the thermal conductivity is the large increase in
the thermal conductivity when graphite melts. The thermal conductivity
increases to ~ 3 W- cm- 1. K - 1 when graphite becomes molten. This value
is ~ 4 times larger than the in-plane thermal conductivity of graphite at the
melting temperature, while the liquid thermal conductivity is ~ 100 times
the c-axis thermal conductivity of graphite at the melting temperature,
The optical properties of liquid carbon may also be calculated based
on the Drude theory [9]. Using the standard relations for the dependence
of the optical constants on the complex dielectric function, the reflectivity
R(~o) normal to the surface and the absorption coefficient, c~(o~), can be
calculated. See Table I for the values of ~(o) and R(o) for frequencies o3
of the pulsed lasers used to melt graphite. [1 ]

3. DISCUSSION AND CONCLUSIONS

Since liquid carbon is a polyvalent liquid metal, the temperature

dependence of the electrical resistivity is very weak for temperatures
between the melting point (Tm~4450K) and the boiling point
(Tv~4700 K). The reason for the insensitivity to the temperature is that
changes in the structure factor at low k have little effect on changing the
integral in Eq. (1) since the low k values are suppressed by the k 5 weighting
of the integral. Also, the primary peak in the structure factor falls close to
the node in the model potential, and further, a(k) ~ 1 for k > 2k r so that
changes in a(k) do not significantly alter the results for A. The weak
temperature dependence of the calculated electrical resistivity of liquid
carbon is also in good agreement with the experimental measurements by
Heremans et al. [8], where the temperature of the liquid carbon (super-
heated) is known to have gone above 5000 K, but no increase in the
electrical resistivity is observed. Therefore, the temperature dependence of
Properties of Liquid Carbon 795

to(T) and Cp(T) for liquid carbon is given by Eqs. (3) and (4), assuming Pl
and Er to be constant in the temperature range between Tm and T~.
The most convincing proof that the liquid metal model provides the
correct description for liquid carbon comes when the model is tested in
heat flow calculations for pulsed-laser [10] and pulsed-current [8] heating
experiments. Figure 4 shows the good agreement between the resistance
versus time experimental traces for pulsed current melting of two very
different graphitic fibers with a numerical simulation of the experiment
using the liquid metal model for liquid carbon. This particular calculation
tests both the electrical resistivity and the thermal properties predicted by
the calculations above.

UO
E
48~.52
80

64

16
~,
I I I
THT 1700~
- - Data
...... Simulation
I

9
(~ 0 .............................
o 50 l I I
.u0 THT 2850 %
40 .F'I," r----,
E;" tf [i'i I ["v" I . ] ~Graphite
30 / q! Ill I V(t)~ ~J fiber _

!
10 ~ ~ .........................
0 i I i I i I i t J
0 5 10 15 20 25
T i m e , /.zs
Fig. 4. Comparison of the measured (solid curve)
resistance versus time traces for two carbon fibers
heated by a pulsed current [8] with the correspond-
ing traces calculated using the liquid metal model
for liquid carbon. The carbon fiber with a heat
treatment temperature THT= 1700~ is highly dis-
ordered, while that for THx=2850~ is highly
graphitic. The schematic for the experimental
arrangement used to heat and melt the graphite
fibers is shown in the inset to the lower trace.

840/'11/4-13
796 Steinbeek, Dresselhaus, and Dresselhaus

Refinements to the model such as improving the model potential for

atomic carbon and understanding the molecular nature of the liquid near
the melting temperature may provide improved agreement with experimen-
tal results.

ACKNOWLEDGMENTS
We thank NSF Grant DMR 88-19896 for support of the work. We
also thank Drs. T. Venkatesan and J. Heremans for many useful discus-
sions.

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