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Estimation of the absolute values of cohesion energies of pure metals
G.Kaptay, G.Csicsovszki, M.S.Yaghmaee,
LIMOS R&D, Department of Physical Chemistry,
Faculty of Materials and Metallurgical Engineering, University of Miskolc,
3515 Hungary, Miskolc, Egyetemvaros
Abstract
Cohesion energy is a basic energetic property of pure metals, determining the majority of their
other physical properties. The reasons why the sublimation energy of metals cannot be a
proper estimation for their absolute cohesion energy values are discussed. The table of
absolute cohesion energies of pure solid and liquid metals has been constructed, based on the
surface tension of liquid gold, as a ‘reference value’. The melting point of metals was used as
an another correlation parameter. As a simplified result it has been found that the cohesion
energy of solid metals at 0 K (in kJ/mol) can be calculated by dividing the melting point of
the metal (in K) by the coefficient –3.5 ± 0.3. In the table presented in this paper more
accurate values are given, taking into account structural differences between solid metals. The
concept of absolute cohesion energies is suggested to use in education and in basic research.
Keywords: cohesion energy of metals, absolute scale, surface tension, correlations
1. Introduction
Cohesion energy (or binding energy) is a basic energetic property of metals, influencing
the majority of their other physical properties. The cohesion energy of solid and liquid metals
is the energy connecting atoms in the solid and/or liquid state. Although the usefulness of the
concept of cohesion energy is obvious for both education and research, its concept is used less
than expected. The major reason for that is that there are no straightforward experimental or
theoretical methods to obtain the absolute value of the cohesion energy. This is mainly due to
the fact, that chemical thermodynamics works with relative, not absolute values. Although
any change in the state of the matter is perfectly described by the (relative) Gibbs energy
change accompanying this process, the cohesion energy in pure metals would require an
absolute scale. In the present paper this question will be discussed, and some preliminary
results will be presented.
2. The State of the Art of the Problem of Cohesion Energy in Metals
According to chemical thermodynamics, the heat of formation of pure elements in their
most stable state is taken equal zero, by definition. Thus, the heat of formation of all solid
metals (and liquid mercury) is zero by definition, at 298.15 K and at 1 bar. Consequently, the
ideal gas of all metals at 298.15 K and at 1 bar is characterised by some positive values of the
heat of formation. Therefore, using formal logic, the heat of sublimation of metals should be
characteristic to the energy of broken bonds in the solid or liquid phase, as in ideal gas (by
definition) such bonds between gaseous atoms do not exist. Therefore, the classical equation
to calculate the cohesion energy in metals at 0 K can be written as:
o
o
o
U coh
( s ,0 K ) = ∆ f H s (0 K ) − ∆ f H g (0 K )
(1)
where ∆fH°g(0K) - the enthalpy of formation of gaseous metals at 0 K (kJ/mol),
∆fH°s(0K) - the enthalpy of formation of solid metals at 0 K (kJ/mol).
Eq.(1) is recognised by the scientific community in a contradictory way. On the one
hand, it is often used as a correlation parameter for different physical properties in the
scientific literature (see for example [1]). On the other hand, in the textbooks on materials
science cohesion energy or binding energy in metals is usually considered only qualitatively,
without the values of the binding energies [2-8], or with only some examples of binding
energies [2, 7] (the situation is similar in textbooks on physical chemistry [9-10]). However,
the various properties of metals are described mostly in a phenomenological way, without an
attempt to make the bridge between them, using the concept of cohesion (binding) energies.
Is the activation energy of diffusion in solid metals connected with the surface energy of
metals ? Are both of these quantities connected with the energy of vacancy formation ? The
answers to these questions are YES, and the same answer can be given to many similar
questions, as well. The key is the cohesion energy, connecting different properties of the same
material. Of-course, professors worldwide should not be blamed that the scientific description
of the discipline called materials science has not been written yet1, or that this task is passed
to quantum mechanics (which is unable to give consumable explanations to ordinary students
on materials science). It should be recognized, that the concept of cohesion energy is not
matured yet, and that is why at this moment it is not ready to be involved in textbooks. Today
it would rather confuse, than clear the minds of the students.
It should also be mentioned that the concept of cohesion energy is discussed in some of
the aforementioned textbooks, in relation to ionic crystals, where the concept has been found
more useful in explaining certain trends in properties of this class of materials. It is also
important to remember, that for ionic crystals the definition of the cohesion energy is more
sophisticated than that given by Eq.(1). Moreover, the values of cohesion energy calculated
from the thermodynamic Born-Haber cycle is in fairly good agreement with calculations
based on a simplest energetic model of ionic crystals (the Coulomb energy). Such a nice
connection is also missing in the world of metals.
Missing a simple and widely known solution to a problem today, however, does not
mean that the problem cannot be solved, or that the concept of cohesion energy in metals
would not have sense, or value. It is probably accepted by all who works in materials science
of metals that at given T and p all metals with known composition and structure possess a
unique, absolute value of the cohesion energy, and this value largely determines various
mechanical, thermochemical, transport, etc. properties of metals. Thus, the impossibility to
measure or calculate these values is indeed frustrating.
One of the possible and elegant ways ‘around the problem’ is to scale the cohesion
energy by some macroscopic properties of the material [11-12], such as melting point and
molar volume at 0 K (and at 0 Pa). Although the absolute values of the cohesion energies are
lost in this way, the potential to apply the concept of cohesion is saved, using a relative scale.
The goal of this paper is to show the reasons why the classical definition Eq.(1) of the
cohesion energy fails for metals, and also to try to give in first approximation an absolute
scale for cohesion energy of metals. The authors realise that the subject is very much
controversial, and also that the table of absolute cohesion energies of metals to be suggested
in this paper is not the final, and not the most perfect table for this property. However, we
1
please, note: by definition - science = collection of empirical facts + mathematics
think that this subject is vital for our education efficiency and also for basic research, and
therefore it is worth to talk about, even if criticism is expected.
3. The reasons of imperfection of Eq.(1) as definition for cohesion energy of metals
Tm, K
Eq.(1) as definition for the cohesion energy in pure metals would be perfect, if
metallic atoms of different elements would be in the same energetic situation in a 1 bar ideal
gas. At first sight, it is the case, as in the ideal gas, by definition, all interactions between
atoms are negligible. However, the stability of gaseous atoms themselves of different
elements are different, due to the different stability of their outer electron shells. For example,
the vapour of Mg (with a relatively stable s2 configuration) is much more stable than the
vapour of Na (with an s1 configuration) or the vapour of Al (with an s2p1 configuration). This
difference is reflected in an exceptional behaviour of Mg (and other s2 metals) in the general
trends established between the cohesion energy (defined in a classical way) and different
properties of metals.
Let us present this controversies on the example of the dependence of the melting
points of metals on cohesion energy defined by Eq.(1) (see Fig.1), especially that melting
point was suggested and successfully applied by Beke et al [11-12] in correlation of some
properties of metals. Data are taken from [13, 14]. One can see, that although there is an
obvious correlation between the two quantities, the scatter in Fig.1 is quite large, with an R2 =
0.88. One can see, that two metals (Mg and Al), having practically the same melting points
(923 and 933 K, respectively), differ in cohesion energy values significantly (-146 kJ/mol and
-327 kJ/mol, respectively). Although the two metals differ also in crystal structure (hcp for
Mg and fcc for Al), this would explain a difference of much less than 10 kJ/mol [13].
Therefore, the ‘contradiction’ is obviously explained by different electron structures of metals
in the ideal gas. This is obvious also, if one compares the electron affinities of gaseous metals.
For ‘normal’ metals withot extra stability of gaseous atoms the electron affinity has a negative
value, while for metals with extra stable electron configuration in gaseous state the electron
affinity is positive (Be, Mg, Ca, Sr, Ba, i.e. the s2 metals, Cd and Hg, i.e. the d10s2 metals, Sc
and Y, i.e. the d1s2 metals, and Mn, i.e. the d5s2 metal [17]).
4000
3000
2000 Mg
1000
0
y = 4,0399x
2
R = 0,8851
0
Al
250
500
750
1000
Ucoh,s,0K, kJ/mol
Fig.1. Melting points of metals as a function of their cohesion energy defined by Eq(1)
4. Physical Properties for Scaling the Cohesion Energy
Our goal is to introduce corrections to the classical definition of the cohesion energy.
For that, however, we need some property (preferably as many as possible independent
properties), which surely will be proportional to the real cohesion energy. Moreover, we need
at least one property, for which also the constant of proportionality is known, and hence our
relative scale can be made absolute.
One of the scaling parameters will be the melting point, as suggested earlier by Beke
et al [11-12], and also by many researchers, searching for trends between properties of
materials (see for example [1]). However, the theoretical ratio between the melting point and
the absolute value of the cohesion energy is not known. The value of the slope in Fig.1 (-4.04)
is considered as a purely empirical value.
It should also be mentioned that among physical properties of solid metals it is hard to
find a property, connected with the absolute cohesion energy with theoretically supported
proportionality constant. The reason is that the properties of solid metals depend largely on
structure and also structure imperfections, which make solid state physics a very complicated
subject. On the other hand, the majority of these structure differences between different
metals disappear upon melting. Although the crystal structure and long-term order is lost, all
irregular imperfections and strains are lost, as well. Therefore, the structure and energetics of
different liquid metals just above their melting points are much more similar to each other,
even if the mathematical description of their structure is obviously more complicated than that
of ideal (and therefore not real) solid crystals. For example, the average coordination number
in all liquid metals just above their melting point is about 11 [1], although the experimental
determination of this quantity is not very accurate. Nevertheless, large coordination
differences, such as between bcc and fcc (or hcp) structures (8 against 12) disappear upon
melting. As an example, let us remind the cases of Ge and Si, with 4-coordination and nonmetallic properties in solid state, becoming true metals with the coordination of about 11
above their melting points. Therefore, we suggest to search an absolute scaling parameter for
the cohesion energy in metals among the properties of liquid metals. If the cohesion energy in
liquid metals is known, it can be easily extrapolated back to solid through the well known
value of enthalpy of melting and heat capacity of solids.
To our opinion, the best, theoretically supported correlation between the cohesion
energy in liquid metals and their properties is provided by the surface tension of liquid metals.
The enthalpy part of the surface tension (converted from J/m2 into the J/mol scale)
characterises the certain ratio of the ‘broken bonds’ at the surface compared to bulk, i.e. the
certain ration of the cohesion energy. This ratio depends on the structure of the bulk and
surface liquid metals. If bulk liquid metals are taken with an average coordination number of
11 [1], and surface liquid metals are structured similar to the (111) plane of the fcc lattice
with the surface coordination number 9, this ratio can be found theoretically2. According to
our earlier results [15,16], the following quantity should be proportional to the cohesion
energy in the liquid metal at its melting point, with the proportionality constant of α = -0.172:
(
Y ≡ 0.001 ⋅ σ lg ⋅ Vm2 / 3 ⋅ N 1Av/ 3 + 5.43 ⋅ Tm
)
where σlg is the surface tension of liquid metals at their melting points (J/m2),
Vm – is the molar volume of liquid metals at their melting points (m3/mol),
NAv = 6.02 1023 mol-1 – the Avogadro number,
2
this follows from the intention of liquids to gain minimum surface energy by ensuring the highest
possible packing at the surface
(2)
Tm – the melting point of metals (K),
the constant 0.001 is used to have the unit of Y in kJ/mol.
The following relationship exists between the cohesion energy of liquid metals and the
cohesion energy of solid metals at 0 K:
o
o
U coh
( l ,Tm ) = U coh ( s , 0 K ) + ∆U s , 0 K − l ,Tm
(3)
Tm
∆U s ,0 K −l ,Tm ≅
∫C
p,s
⋅ dT + ∆ m H
(4)
0K
where Cp – is the heat capacity of solid,
∆mH – is the enthalpy of melting of the metal at its melting point.
For metals with allotropic transformations between 0 K and Tm the heats of those
transformations should also be added to Eq.(4).
The correlation between parameter Y and the cohesion energy of liquid metals
calculated by the classical definition (Eq.1) is presented in Fig.2. Values for surface tension
and molar volume of liquid metals are taken from [1]. From Fig.2 one can see, that the
correlation clearly exists, with R2 = 0.90, and with a similar to the theoretical (-0.172) average
coefficient α = -0.179. Mg is again situated ‘out of the correlation’ (α = -0.298), due to the
error in Eq.(1), i.e. due to the special electronic structure of gaseous Mg.
The only problem with Fig.2 and with values for the surface tension of liquid metals
is, that its measurement for many metals is difficult due to surface contamination by oxygen,
and due to high temperatures. The correlation in Fig.2 is not perfect partly because of that,
and partly because of Eq.(1).
Y, kJ/mol
150
100
50
0
0
y = 0,1786x
2
R = 0,901
200
400
600
800
Ucoh,liq,Tm, kJ/mol
Fig.2. Parameter Y as function of cohesion energy of liquid metals calculated by the ‘classical
definition’
5. The absolute scale of Cohesion Energy
In order to make the absolute scale for the cohesion energy of metals, the task will be
solved in two steps:
i.
first, one single liquid metal should be chosen with known well measured value of the
surface tension and molar volume at the melting point, i.e. with the known value of
parameter Y. Using the theoretical coefficient 0.172, the absolute value of the
cohesion energy for this metal can be found. From this value, the proportionality
constant between the cohesion energy of liquid metals and the melting point of metals
can be found.
ii.
as the melting points of all other metals are known with high accuracy, the absolute
values of cohesion energies of all liquid metals are estimated using the same
proportionality constant. The absolute cohesion energy of solid metals are recalculated using Eq.(4).
i.
ii.
iii.
iv.
First, a liquid ‘reference’ metal should be chosen, having the following properties:
not corrosive,
not sensitive to oxidation,
‘normal metal’, i.e. having an fcc structure in solid state, with no allotropes at 1 bar,
having well established data for surface tension and molar volume in liquid state.
Liquid gold has been chosen by us as a ‘reference metal’. It has an fcc structure with
no allotropes at 1 bar, Tm = 1,338 K, Uocoh(s,0K) = -368.0 kJ/mol (see Eq.(1)) [13, 14], ∆Us,0Ko
-6
3
l,Tm = 47.86 kJ/mol [13, 14], U coh(l,Tm) = - 320.14 kJ/mol, Vm = 11.3 10 m /mol, σlg = 1.169
J/m2 [1], Y = 56.97 kJ/mol. Parameter α = -0.178, i.e. it is not far from the theoretical value.
If the theoretical value of α = -0.172 is used, the corrected values for Au will be as follows:
Uocoh(l,Tm) = - 331.2 kJ/mol, Uocoh(s,0K) = -379.1 kJ/mol. This value will be taken as the absolute
cohesion energy of solid Au at 0 K.
The theoretical proportionality constant between the melting point and the cohesion
energy of the liquid at its melting point can be found from the value for gold as: 1,338 K / 331.2 = -4.04 Kmol/kJ. In the first approximation, the same slope can be used to find absolute
values of cohesion energies of other liquid metals at their melting points, as well:
o
U coh
( l ,Tm ) ≅
Tm
4.04
(5)
with Tm in [K] and Ucoh in [kJ/mol].
Once the cohesion energies for liquid metals are estimated by Eq.(5), the cohesion
energies of solid metals can be re-calculated by Eq.(4). In this way, the cohesion energy of
two solid metals with the same melting point, but different structures will be different, as the
melting entropy (being the part of Eq.(4)) is a structure sensitive property.
As an example, for Al one can find: Uocoh(l,Tm) = 933/-4.04 = -230.9 kJ/mol, Uocoh(s,0K)
= -263.6 kJ/mol. This value is more positive by 20 %, compared to the classical value of -327
kJ/mol. For Mg: Uocoh(l,Tm) = 922/-4.04 = -228.2 kJ/mol, Uocoh(s,0K) = -259.6 kJ/mol. This value
is more negative by 78 % compared to the classical value. Hence, the value for Mg with extra
high stability of the gaseous atom was shifted to the negative direction much more, than the
value of Al was shifted to the positive direction, with obviously somewhat higher than the
average in-stability of gaseous atoms.
In Fig.3 the dependence of parameter Y on the corrected values of the cohesion
energy is shown. One can see that for low-cohesion energy metals the data points are situated
around the ideal line (drawn with a slope of 0.172), with a scatter normal to surface tension of
metals. For high cohesion energy metals, however, the ideal line is significantly above the
measured points. This might be the indication that the surface tension of high melting point
metals is still known with not a sufficient accuracy, due to difficulties of measurements and to
the role of adsorbed oxygen.
Y, kJ/mol
200
150
100
50
0
0
250
500
750
1000
Ucoh,liq,Tm,corr, kJ/mol
Fig.3. Dependence of parameter Y on the corrected cohesion energy of the liquid metals (data
points: experiment of [1], line: ideal line with the slope of 0.172)
Tm, K
In Fig.4 the correlation between the corrected cohesion energy at 0 K for solid metals
and the melting point is shown. The correlation is ‘too good’ (with R2 = 0.999), but this is due
to Eq.(5). However, it can be seen that the scatter of maximum 5 kJ/mol is still present, due to
the structural differences between different solid metals. From Fig.4 one can see that any
correlation given in the literature between properties of metals and their melting point, can be
converted into the more sensible correlation between those properties and the cohesion energy
in metals. The ‘theoretical’ proportionality constant of –3.5 ± 0.3 has been established
between the melting point and the cohesion energy of solid metals at 0 K (see Fig.4).
4000
3000
2000
1000
0
y = 3,4721x
2
R = 0,9993
0
400
800
1200
Ucoh,s,0K,corr, kJ/mol
Fig. 4. Correlation between the corrected cohesion energy values of solid metals at 0 K and
melting point of metals
In Table 1 the suggested values for the cohesion energies of solid metals at 0 K and
liquid metals at their melting point are collected. It is suggested to use these tables to search
for correlations between different properties of metals and cohesion energy. Empirical
constants found in this way between different properties and the cohesion energy can be
rationalised and compared with ‘absolute’ theories.
Metall
Li
Na
K
Rb
Cs
Be
Mg
Ca
Sr
Ba
Al
Ga
In
Tl
Si
Ge
Sn
Pb
Sb
Bi
Te
Cu
Ag
Au
Zn
Cd
Hg
Sc
Y
La
Ti
Zr
Hf
V
Nb
Ta
Cr
Mo
Table 1. The suggested values of absolute cohesion energies of metals
Tm
-Uo(coh,s,0K)
-Uo(coh,l,Tm,corr )
-Uo(coh,s,0K,corr)
K
kJ/mol
kJ/mol
kJ/mol
453.69
157.703
112.30
124.08
370.98
107.498
91.83
102.98
336.35
89.889
83.25
93.86
312.65
82.208
77.39
87.65
301.55
78.088
74.64
84.64
1560
319.725
386.14
432.76
923
145.643
228.47
259.86
1115
177.694
275.99
317.35
1050
164.239
259.90
330.80
1000
182.842
247.52
289.71
933.45
327.279
231.05
263.75
302.92
270.605
74.98
85.93
429.3
243.083
106.26
119.89
577
181.578
142.82
161.90
1685
445.783
417.08
502.00
1211
371.304
299.75
359.98
505
300.848
125.00
144.04
600.58
195.754
148.66
169.02
903.89
261.728
223.74
265.90
544.5
209.839
134.78
158.82
722.7
209.354
178.89
215.95
1358
336.236
336.14
383.66
1234
283.484
305.45
348.75
1338
368.034
331.19
379.05
692.7
129.467
171.46
194.97
594
111.000
147.03
167.07
234.29
62.418
57.99
65.70
1812
376.130
448.51
523.39
1799
422.891
445.30
514.57
1193
430.823
295.30
335.56
1939
470.674
479.95
555.55
2125
608.703
525.99
606.87
2500
618.629
618.81
726.78
2190
510.865
542.08
626.85
2750
718.113
680.69
789.79
3258
781.332
806.44
934.05
2130
395.091
527.23
612.95
2896
656.661
716.83
838.90
W
Mn
Re
Fe
Ru
Os
Co
Rh
Ir
Ni
Pd
Pt
Gd
Dy
Yb
Lu
U
3680
1519
3453
1809
2523
3300
1768
2233
2716
1728
1825
2045
1585
1682
1097
1936
1405
849.515
281.818
772.877
412.927
648.438
785.870
426.573
550.803
667.495
427.828
375.594
563.427
395.281
290.014
152.402
427.169
522.215
910.89
375.99
854.70
447.77
624.50
816.83
437.62
552.72
672.28
427.72
451.73
506.19
392.33
416.34
271.53
479.21
347.77
1 068.5
445.35
998.87
528.81
724.48
938.76
511.58
643.48
784.35
497.76
520.03
584.70
455.79
485.93
309.86
559.49
406.86
Conclusions
In the present paper the reasons why the sublimation energy of metals cannot be a proper
estimation for the absolute cohesion energy in metals are discussed. The table of absolute
cohesion energies of pure solid and liquid metals has been constructed, based on the surface
tension of liquid gold, as a ‘reference value’. The melting point of metals was used as an
another correlation parameter. It is suggested to use these tables to search for correlations
between different properties of metals and the cohesion energy. Empirical constants found in
this way between different properties and the cohesion energy can be rationalised and
compared with ‘absolute’ theories. This method is suggested to use in education and in basic
research.
Acknowledgements
The authors are grateful for prof. P.Bárczy of the University of Miskolc and to prof. D.Beke
of the University of Debrecen for their helpful discussions of the matter discussed in this
paper.
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