The Critical Temperature of Aluminum

Abstract

The critical temperature, T
c, of metals is a fundamental point when vaporization due to high energy exchanges occurs. Although aluminum is a metal often studied as a benchmark for theories, its critical temperature is not known with a high degree of accuracy. Its determination by experiment is difficult as a result of its high value. This paper reviews the existing data and proposes new ones resulting from recent measurements of particular physical properties and recent theoretical approaches. These new estimates lead to the recommended value of T
c = (6700 ± 800) K.

Full text

Int J Thermophys (2009) 30:1853–1863
DOI 10.1007/s10765-009-0671-6
The Critical Temperature of Aluminum
V. Morel ·A. Bultel ·B. G. Chéron
Received: 23 January 2009 / Accepted: 19 October 2009 / Published online: 4 November 2009
© Springer Science+Business Media, LLC 2009
Abstract The critical temperature, Tc, of metals is a fundamental point when vapor-
ization due to high energy exchanges occurs. Although aluminum is a metal often stud-
ied as a benchmark for theories, its critical temperature is not known with a high degree
of accuracy. Its determination by experiment is difficult as a result of its high value.
This paper reviews the existing data and proposes new ones resulting from recent mea-
surements of particular physical properties and recent theoretical approaches. These
new estimates lead to the recommended value of Tc=(6700 ±800) K.
Keywords Aluminum ·Cohesive enthalpy ·Critical point ·Critical temperature ·
Enthalpy of vaporization ·Equation of state ·Surface tension
1 Introduction
A relevant way for determining the in situ composition of a solid sample is to use
laser-induced breakdown spectroscopy (LIBS). This involves the irradiation of the
sample by a sufficiently powerful laser, leading to the formation of a plasma, followed
by analysis by time-resolved emission spectroscopy. This technique has already been
used in the case of tokamaks to probe the edge plasma [1], and developments are cur-
rently in progress to verify the carbon balance in fusion machines [2]. LIBS presents
an important problem which prevents the technique from being universal: it requires
a comparison of the experimental spectra with others performed earlier in controlled
conditions with known test sample compositions. The only way to avoid relying on a
representative databank is to model the complete situation. The model has to describe
V. Mo r e l ·A. Bultel (B
)·B. G. Chéron
CORIA, UMR CNRS 6614, Université de Rouen, 76821 Saint-Etienne du Rouvray, France
e-mail: arnaud.bultel@coria.fr
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1854 Int J Thermophys (2009) 30:1853–1863
the plasma formation, the kinetics mechanism, heating of the sample, its phase change,
and the interaction with laser light [3].
In spite of the evaporation of the material and the diffusion of energy inside the
sample, the surface temperature can reach the critical point. As soon as the critical
temperature, Tc, is reached, the behavior of the surface changes significantly: since
the liquid →gas transition does not occur anymore, a description of the edge of the
sample by using the concept of a surface with a discontinuity in terms of specific
mass is questionable. The critical point is therefore an upper limit of the traditional
treatment of the sample in an LIBS situation.
We are currently developing a complete model of the behavior of a sample under
nanosecond laser light irradiation based on our experience in modeling the thermal
and chemical behavior of non-equilibrium plasmas [4,5]. Aluminum has been cho-
sen as the sample material, since a lot of thermodynamic data have been determined
concerning this element up to high-temperature levels. The critical temperature is con-
versely less well known. As a result, the objectives of the present paper are to review
the existing estimates of Tcfor aluminium; to propose other estimates resulting from
recent data concerning the enthalpy of vaporization, the cohesive enthalpy, the surface
tension, and the equation of state; and finally to recommend a value.
2 Review of the Existing Estimates of Tcfor Aluminum
The critical temperature, Tc, for aluminum has never been experimentally determined.
The different estimates (see Table 1) range between 5400 K and 9500 K and have been
obtained using different methods. These methods can be divided into three groups.
The first one (denoted as 1 in Table 1) concerns the equation of state. The second one
(denoted as 2) is based on the use of rules assumed to be the same, more or less, for all
liquid metals. The third one (denoted as 3) consists of extrapolating thermodynamic
data experimentally obtained at lower temperature conditions. The related mean val-
ues are 8070 K for type 1, 7260 K for type 2, and 5950 K for type 3. The mean-type
value is 7090K whereas the mean value derived from all data of Table1is 7310K.
The plot derived from Table 1illustrates how Tchas evolved with time (see Fig.1).
This shows that the early estimates belong to three groups: group A (before 1980),
group B (between 1990 and 2000), and group C (since 2006). The mean values are
7530 K, 7400 K, and 6550 K, respectively, and therefore, in the vicinity of the global
mean value. With time, we can conclude that Tccorresponds to lower and lower values.
Recently, a series of experiments has been performed by Korobenko et al. [24]to
determine the electrical resistivity of aluminum passing the critical point. This work
is based on the observation of the explosion of a wire suddenly heated by a pulse
of current. Underlying the difficulty of performing experiments in very high temper-
ature conditions, this determination necessitates an equation of state: the SESAME
library has been used [25]. As expected, the curves given in the paper emphasize
an important increase of the resistivity passing the critical point (7000K to 8000K).
Unfortunately, an equation of state being used, an evaluation of Tccannot be derived
from this experiment.
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Tab le 1 Early estimates of the critical temperature Tcfor aluminum (type of method and group: see Sect. 2)
Value (K) Year Reference Method Type of method Group
7400 1964 Morris [6] Corresponding states 1 A
7740 1967 Kopp [7] Enthalpy of vaporization 2
7151 1971 Young [8] Equation of state 1
6900 1973 Jones [9] Model of pair potential 2
8000 1975 Fortov [10] Kopp-Lang rule 2
6040 1976 Lang [11] Surface tension 2
5410 1976 Martynyuk [12] Wire explosion 3
9502 1977 Boissière [13] Equation of state 1
7543 1977 Hohenwarter [14] Planck-Ridel vapour pressure 3
8560 1977 Lang [15] Enthalpy of vaporization 2
8550 1977 Young [16] Equation of state 1
5654 1993 Blairs [17] Sound velocity 2 B
8556 1993 Celliers [18] Equation of state 1
5726 1993 Celliers [18] Isobaric measurements 3
8860 1996 Likalter [19] Equation of state 1
5754 1998 Hess [20] Guldberg rule 2
8232 1998 Hess [20] Kopp-Lang rule 2
7499 1998 Hess [20] Goldstein rule 2
8944 1998 Hess [20] Vapor pressure curves 2
5115 2006 Blairs [21] Surface tension 3 C
6557 2006 Blairs [21] Guldberg rule 2
7917 2008 Gordeev [22] Equation of state 1
6595 2008 Povarnitsyn [23] Equation of state 1
The same remark has to be made concerning the paper of Wu and Shin [26] which
treats the determination of the absorption coefficient αλnear the critical point. Using
the Drude model to determine αλand starting from the results of Korobenko et al.,
the use of the equation of state for the determination of the temperature prevents an
estimate of Tc.
Finally, another estimate of Tcis given in the work of Mazhukin et al. [27]. The
value proposed (8000K) belongs to the domain of high values, but its source is not
cited and has not been taken into account in this review.
3 New Estimates of Tcfor Aluminum
In the following, four methods based on recent measurements of physical properties
and recent theoretical approaches are used to estimate Tc.
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1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Year
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
11000
12000
13000
14000
15000
Tc, K
Morris [6]
Kopp [7]
Young [8]
Jones [9]
Fortov [10]
Martynyuk [12]
Lang [11]
Boissiere [13]
Lang [15], Young [16]
Hohenwarter [14]
Blairs [17]
Celliers [18]
Likalter [19]
Hess [20]
Hess [20]
Blairs [21]
Gordeev [22]
Povarnitsyn [23]
Celliers [18]
Group B
Group C
Group A
Fig. 1 Evolution in time of estimates of the critical temperature Tcof aluminum. The dashed line is located
at the mean value 7310 K. Circles,triangles,andstars are related to the results obtained from methods
1 (mean value 8070K), 2 (mean value 7260K), and 3 (mean value 5950 K), respectively
3.1 Enthalpy of Vaporization
Considering the equilibrium vapor pressure of aluminum, Hess [20] discusses the
validity of the interpolations allowing the matching of the experimental data up to the
boiling point. The general equation of Dupré is adopted:
ln (pv/p0)=A+B/T+Cln (T/T0)(1)
with p0= 1 bar and T0= 1 K. The constants A,B, and Care listed in Table2.
From this equality, we can derive the boiling point Tb, the enthalpy of ebullition hb
(i.e., the enthalpy of vaporization under atmospheric pressure) as well as its derivative
with respect to temperature in the form,
dhb/dT=CR
g(2)
by using the Clausius–Clapeyron equation, Rgbeing the universal gas constant. These
data are also given in Table 2.
Tc,Tb,hb, and dhb/dTare linked together by the Meyra et al. [30]orWatson
[31] equations, leading to the following form:
Tc=Tb−Zchb/(dhb/dT)(3)
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Int J Thermophys (2009) 30:1853–1863 1857
Tab le 2 Critical temperatures derived from the treatment of the enthalpy of vaporization.
Constants of Reference Tb(K) hbdhb/dTTc(K) Tc(K)
Eq. 1(J ·mol−1)(J ·mol −1·K−1)Eq. 3Eq. 4
A=21.75 [28] 2730 290.88 –8.314 12946 7626
B=−37716 K 16025
C=−1
A=13.62 [29] 2740 310.34 0 ∞8899
B=−37327 K ∞
C=0
A=24.37 [29] 2800 293.84 –10.919 10658 7504
B=−39020 K 13026
C=−1.3133
The two values given for Tccorrespond to the use of Eq. 3with Zc= 0.292 (Meyra et al. [30]) and Zc=
0.38 (Watson [31])
with Zc= 0.292 for Meyra et al. and Zc= 0.38 for Watson.
On the other hand, Fish and Lielmezs [32] have worked on liquid metals and have
extracted from experimental results a correlation which can be presented as follows,
after some algebra:
Tc=Tb{2[1−Tb(dhb/dT)/hb]}/{1−2Tb(dhb/dT)/hb−n+m}(4)
where n= 0.20957 and m= – 0.17464.
We observe that Tb,hb, and dhb/dTderived from Eqs.1and 2and the Clausius–
Clapeyron equation can be used to calculate Tcwith Eqs. 3and 4. Table 2presents
the values of Tcobtained by this way. It is clear that Eq. 3overestimates the critical
temperatures obtained in the past and displayed in Table1. Therefore, Eq. 3has to
be rejected. Conversely, the Fish and Lielmezs equation is better: the values obtained
(7500 K to 8900 K) belong to the range of high values and are in good accord with
previous determinations. Alcock et al. [29] indicate that the most precise set in Table2
is the third one; therefore, the value of 7504K has to be particularly considered. The
first set is less precise, but the value derived for Tcis very close to the previous one.
The second set leads to a constant hbnear Tb: this behavior does not obey the usual
law concerning the enthalpy of vaporization. This set has to be therefore also rejected
as well as the related critical temperature.
Notice that Roman et al. [33] modified by Velasco et al. [34] give a correlation
formula different from those of Meyra et al. and Watson, but it leads to the same form
given by Eq. 3for Tc: the resulting values are therefore the same as those of Table2.
3.2 Cohesive Enthalpy
Recently, Kaptay [35,36] has given a unified model linking the cohesive enthalpy of a
liquid hc
l(T)with the melting point Tmof alkali metals. This model also yields fairly
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good results for other metals. The cohesive enthalpy is defined as the energy existing
in the liquid resulting from the mutual atomic attraction, which vanishes when the
critical temperature is reached. As a result,
hc
l(T)=hl(T)−hl(Tc)(5)
Kaptay gives its general evolution towards Tm;
hc
l(Tm)=−q1RgTm−q2RgTm2(6)
with q1=26.3×10−4and q2=−2.62 ×10−4mol ·J−1.
Over the range 933 K ≤T≤4000 K, Gathers [37] has derived the evolution of the
specific enthalpy with temperature from experiments. It leads to an evolution of hl
according to
hl(T)=1319.7+28.881T+6.2284 ×10−4T2(7)
The measurements of Gathers were performed under rather high-pressure conditions
(p=0.3 GPa). Nevertheless, the same measurements performed under lower pressure
conditions ( p=0.2 GPa) have given similar results [38]: the pressure seems to have a
negligible influence on hl. As a result, Eq. 7has been used directly for calculating hc
l
(Tm)from Eqs. 5and 6, the latter having been established under atmospheric pressure.
Moreover, assuming that Eq. 7is correct for a range of temperature extending up to
Tc, the resulting value for the critical temperature is
Tc=6548 K (8)
This value is remarkably close to that of Povarnitsyn [23].
3.3 Surface Tension
Molina et al. [39] have performed measurements of the surface tension of molten
aluminum in particularly clean conditions of the surface, i.e., by minimizing the influ-
ence of adsorbed oxygen by working at high temperature. They have brought to light
an increase of the surface tension when the influence of oxygen is reduced. The val-
ues obtained are γ(1375 K)=850 mN ·m−1withaslopedγ/dT=−0.185 mN ·
m−1·K−1. Assuming a constant slope over the range [Tm, 1375 K], they have deduced
γ(Tm)=940 mN ·m−1.
The surface tension being zero at the critical temperature Tc, linking the surface
tension and Tccan provide an estimate of the latter temperature. A lot of work has
been devoted to the problem of the link between γand Tc. Since the work of Binder
[40], a general trend,
γ(T)=γ0(1−T/Tc)n(9)
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Int J Thermophys (2009) 30:1853–1863 1859
with n= 1.26 is more or less accepted. In the case of hydrocarbons, the exponent
is 1.244 [41] and for nitrogen, n=1.27[42]. For quantum fluids such as helium,
n= 1.289 [43]. In the case of liquid metals, Eustathopoulos et al. [44] have proposed
a model of dγ/dT: their results are very well correlated with the experimental ones
obtained by Molina et al. Unfortunately, their model is based on assumptions which
are valid only for temperatures around the melting point: consequently, their model
cannot be applied up to Tc.
In more recent work, Digilov [45] has developed another model of the evolution
with temperature of the surface tension γleading to the following equation:
γ(T)=γ(Tm)[1+0.13(1−T/Tm)]1.67 (10)
Therefore,
(dγ/dT)Tm=−0.217γ(Tm)/Tm(11)
Considering the reference value γ(Tm)=940 mN ·m−1,Eq.11 leads to dγ/dT=
−0.219 mN ·m−1·K−1which is in good agreement with the work of Molina et al. By
equating the surface tension to zero when T=Tc, the critical temperature Tc=8.69
Tm= 8114 K can be deduced from Eq.10. This value belongs to the high range of
critical temperatures displayed in Fig. 1. Digilov adopts assumptions which are valid
only in the vicinity of the melting point: these assumptions explain why the slope
dγ/dTis so well correlated with the experiment and cannot be used to derive Tc.
Assuming that n=1.26inEq.9, a critical temperature of
Tc=7164 K (12)
is deduced from the results of Molina et al. obtained for T= 1375 K.
3.4 Equation of State
Different equations of state for aluminum have been established recently, and we have
tested the more reliable among them for the purpose of evaluating the critical temper-
ature. The equation of Chiew et al. [46], initially developed for the case of chain-like
fluids and polymers, has to be mentioned. It is based on a first-order variational per-
turbation theory applied to molecules considered as chains of atoms interacting with
each other according to a Lennard–Jones potential (called the PLJC method for the
perturbed Lennard–Jones chain method). The resulting equation of state is
p/(nkBT)=f1(η)+f2(η)(13)
where nis the number density of atoms, f1is the hard-sphere reference part, f2is the
attractive perturbation part (proportional to n3), and ηis the hard-segment packing
fraction depending on the number density nand the temperature T. The parts f1and
f2depend on the usual two Lennard–Jones parameters σand ε.
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1860 Int J Thermophys (2009) 30:1853–1863
Tab le 3 Early estimates of the critical density ρcand pressure pcfor aluminum
ρc(kg ·m−3)pc(GPa) Reference Year Type of method
600 0.415 Morris [6] 1964 1
690 0.546 Young [8] 1971 1
1030 – Jones [9] 1973 2
640 0.447 Fortov [10] 1975 2
895 0.938 Boissière [13] 1977 1
– 0.246 Hohenwarter [14] 1977 3
420 0.182 Young [16] 1977 1
685 – Celliers [18] 1993 1
280 0.468 Likalter [19] 1996 1
430 0.473 Hess [20] 1998 2
660 0.467 Gordeev [22] 2008 1
698 0.399 Povarnitsyn [23] 2008 1
This approach gives results in good agreement with the experiment and has been
consequently applied recently by Mousazadeh and Ghanadi Marageh [47] to the case
of liquid metals. At critical conditions, the first and second derivatives of the pressure
with respect to the density at constant temperature are equal to zero. For aluminum,
we have therefore deduced
Tc=5698 K (14)
and ρc=566 kg ·m−3and pc= 0.393 GPa. These values for critical density and
pressure are in good agreement with those reported in Table 3.
Other approaches have been proposed combining a hard-sphere chain equation of
state perturbed by a van der Waals attraction term and applied to some liquid met-
als, for instance, by Eslami [48]. In this case, the equation of state still presents the
form of Eq. 13, but with a repulsive part proportional to n. The parts f1and f2also
depend on two parameters: a reference radius and a reference energy denoted σand
ε, respectively, as in the case of the Lennard–Jones approach.
The metals Li, Na, K, Ru, Ce, Hg, Sn, Pb, and Bi have been treated by either the
PLJC method or the perturbed van der Waals method. The comparison of σand ε
between both methods shows that the reference energy εis approximately the same,
whereas the reference radius σis systematically lower when using the van der Waals
method. The latter has not yet been tested on aluminum: we have consequently chosen
to adopt the value of εgiven in [47]. The value of σhas been evaluated by using the
equation of state (Eq. 13) such that the specific mass of liquid aluminum at the melting
point derived from the equation of state is equal to the reference value (2377kg ·m−3)
recommended by Assael et al. [49]. We have obtained σ=2.52×10−10 m which is
lower than the value proposed by [47] as for the other metals.
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Int J Thermophys (2009) 30:1853–1863 1861
Tab le 4 New estimates of the
critical temperature Tcfor
aluminum
Tc(K) Method Type of method
5698 Equation of state 1
6063 Equation of state 1
6548 Cohesive enthalpy 2
7164 Surface tension 3
7504 Enthalpy of vaporization 3
7626 Enthalpy of vaporization 3
Finally, the critical point estimated with the perturbed van der Waals method cor-
responds to the following conditions:
Tc=6063 K (15)
ρc= 556 kg ·m−3and pc= 0.373 GPa. The coordinates of the critical point are also
well correlated with the data given in Table 3.
4 Discussion
In Sect. 3, we have estimated Tcfor aluminum with different methods. Table 4sum-
marizes these estimates ordered in increasing values. We see that passing successively
from a method of type 1 to type 3 leads to an increase of Tc, contrary to the case of the
early estimates. Therefore, we cannot conclude that one method is more reliable than
another: all methods have consequently to be developed in order to refine the estimate
of Tc. This conclusion is reinforced by the analysis of Table 3: except for the results
of Jones [9], Boissière and Fiorese [13], Hohenwarter et al. [14], Young [16], Celliers
and Ng [18], and Likalter [19], our estimates of ρcand pcare in good agreement with
the existing values derived not only from the use of method 1 (equation of state), but
also from method 2.
The mean value obtained in this work is 6770K, confirming the estimates of the
recent group C and the decrease of the estimates of Tcwith time. Our work is based
globally on recent data: we can consequently combine our results with those of group
C and deduce a value of 6680K. For lack of other estimates, and waiting for a direct
measurement of Tc, the recommended value is therefore Tc=6700 K.
The uncertainty of the previous value has to be given. The standard deviation of
the group of values obtained in Table 4and in group C is 800 K. We assume that this
standard deviation is a good estimate of the uncertainty even if these values are not
related statistically. As a conclusion, we can state that Tc=(6700 ±800) K.
5 Conclusion
The critical temperature of aluminum has not been measured directly so far, and it has
been the subject of a lot of estimates in the past. In this work, we have reviewed the
existing values and shown that the later the estimate, the smaller the value. By analyz-
ing recent data concerning aluminum’s enthalpy of vaporization, cohesive enthalpy,
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1862 Int J Thermophys (2009) 30:1853–1863
surface tension, and equation of state, we have deduced six new estimates of Tccon-
firming the global evolution with time put forward in our review. In addition, they are
in good agreement with the values obtained for the last 3years. Finally, the recom-
mended critical temperature of aluminum, obtained as the average of our own results
and the latter, is
Tc=(6700 ±800)K
Acknowledgment The authors wish to thank the “Région Haute-Normandie” in France for its financial
support (contract no 922/2000A/00112).
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