The asymptotic behaviour of the inclting curves for substances with closepacked
structures is analysed by means of Monte Carlo caIculations on
model systems of hard spheres. The Kraut--Kennedy melting law does
not show the correct asymptotic behaviour. On this basis some recent
results on the physics of the Earth's core are rejected

Boschi, E.

Earth-prints Repository

- 
Geophys. J.  R,  astr. Soc. (1974) 37, 45-50. 
On the Melting Curve at High Pressures 
Enzo Boschik 
(Received 1973 October 24) 
Summary 
The asymptotic behaviour of the inclting curves for substances with close- 
packed structures is analysed by means of Monte Carlo caIculations on 
model systems of hard spheres. The Kraut--Kennedy melting law does 
not show the correct asymptotic behaviour. On this basis some recent 
results on the physics of the Earth's core are rejected. 
Higgins & Kennedy (1971) have estimated that the melting temperature of iron at 
pressures equivalent to the inner core-outer core boundary is circa 4500°K with a 
melting point gradient of approximately 500" through the outer core. Their estimate 
is based on a hundred-fold linear extrapolation of the 30 kbar experimental results of 
Sterret, Klement & Kennedy (1965), by assuming that the melting point is a linear 
function of the volume of the solid (Kraut & Kennedy 1966). 
Birch (1972) has discussed the validity of such a procedure, stressing the fact that 
iron exists in four crystalline forins and that present knowledge of the thermodynamic 
properties of the two denser phases, which are those most likely to be applicable in 
the core of the Earth, is extremely rudimentary. Birch (1972) concludes that the 
melting temperatures of the y-phase may be some 700" above those estimated by 
Higgins & Kennedy (1971); the melting tempsratures of the &-phase may still be higher. 
Verhoogen (1973) has suggested that the Niggins-Kennedy estimate is grossly 
wrong on several grounds but mainly basing his views on significant structure calcula- 
tions by Leppaluoto (1973). The significant structure theory is based on the idea that 
molecules of liquids are more or less free to move about in a structure which is 
basically solid-like. This view is then translated into mathematical language, that is a 
partition function, from which all of the thermo-dynamic properties of the liquid are 
calculated (for a recent review in this subject see John & Eyring 1971). However the 
significant structure theory of liquids has been used by Tuerpe & Keeler (1967) to 
predict melting curves at high pressures obtaining anomalous results, which indicate 
that, although successful in predicting some thermal properties of liquids at moderate 
temperatures and pressures, the theory may not be suitable to describe the phenomenon 
of melting, as Kennedy & Higgins have pointed out. 
Jacobs (1971, 1973) has re-estimated the adiabatic gradient in the core, assuming 
that there is a linear relationship between the reciprocal of the thermal expansion 
coefficient and pressure in analogy with Bullen's compressibility-pressure hypothesis, 
finding that in the core the adiabatic temperature gradient is less than the melting 
point gradient of Higgins & Kennedy. Jacobs (1973) concludes that actual tempera- 
tures in the core are probably very close to those of the melting temperature. 
* On leave from Istituto di Geofisica, Universita di Bologna and Istituto di Scienze della Terra, 
Universita di Ancona. 
45 
In consequence it appears worthy of interest to speculate on the general 
asymptotic behaviour of the melting curve, which could constitute a necessary 
condition for the validity of every melting law at high pressures. At the present 
stage of our knowledge of the physics of phase-transitions, that is all what we may do. 
To this end we will use some results obtaiced by Monte Carlo calculations on model 
systems of hand spheres. I11 this context, the term Monte Carlo refers to a probablistic 
method of samp!ing configurations specified in various ensembles. Using configura- 
tions specified in this manner, estimates can be made for most equilibrium averages 
of interest in statistical mechanics and thermodynamics. 
The chief advantage of this approach lies in the fact that we can control the shape 
of the intermolecular potential energy to be used and the independent external 
variables with the desired accuracy. A complete review of this subject has been 
given by Ree (1971). 
It is reasonable to assume that at high densities a system of solid spheres can 
exist in two different states, namely, the crystal-like state (the spheres' centres are 
statistically located at the nodes of a periodic spatial lattice) and the non-ordered 
liquid-like state (the statistical spatial correlation of the sphere exists only over a 
distance of the order of several sphere diameters). 
Computer studies cannot provide the locations of the melting and freezing densities 
since they do not give any information on the entropy constant which would allow the 
solid state entropy to be evaluated from the same reference state used for the entropy 
of the fluid branch. If an ideal gas state is chosen as the reference state, entropies of 
the fluid can be measured from a low density state. This is achieved by integrating 
on the fluid branch the relatio~ship 
where B = ( k ~ ) - ' ,  T is the temperature, P  is the pressure, p is the number of 
particles in the unit volume, A is the free energy and k is the Boltzmann constant. 
We get 
P 
A = a i d c a ,  + / [ ( P P / p )  - l ~ / P ) " P  
0 
where Aid,,, is the free energy for the ideal gas at g. However a similar approach 
cannot be used in the case of the solid branch because this branch terminates at a 
dense state. Hoover & Ree (1967; see also Ree 1971) have overcome this difficulty by 
using the single-occupancy system, i.e. a similar system with a constraint in the volume, 
in which a particle can move around. This constraint assigns each particle to its own 
Wigner-Seitz cell within which the centre of the particle is confined. For the single- 
occupancy system at high densities, a particle with a repulsive core would pre- 
dominantly collide with neighbouring particles, and on!y very rarely it would collide 
with the wall of its own cell. Since the particle cannot feel the pressure of the wall, 
thermodynamic properties at high densities could be satisfactorily approximated by 
those without the wall constraint. This fact plus the fact that the artificial wall forces 
the single-occupancy system to remain in a solid phase at all densities offers a practical 
way to extend the real isotherm to lower fluid densities by using the isotherm of the 
artificial solid. The free energy has an expression similar to that of the fluid branch, 
i.e. 
P 
A* = A i d m l +  r { [ ( P ~ * / P ) -  'IIpIdp 
On the melting curve at high pressures 47 
where the asterisk means single-occupancy thermodynamic quantities. In this way a 
reversible path which links the solid and the fluid braiiches is achieved. The melting 
pressure and the CO-existence densities of solid and fluid are then determined by a 
systematic use of a Maxwell double-tangent constructjon drawn using A and A*. By 
the method here outlined Hoover 8r Ree (1967) conclude that hard spheres systems 
melt at (0.736+0.003) p, and freeze at (0.667+0.003) p, and the pressure of the 
CO-existence region is (8.27$0-13) pok T,, where p, is the density of the system in the 
state of closest packing, 2nd T, is the melting temperature. 
Let us now collfine our attention to substances with close packed structures. 
It is well established that the melting point of a substance with close-packed 
structure rises as the pressure increases. In other words 
where P, is the melting pressure. It is well established also that 
in other words, the slope of the nlelting decreases as the pressure increases. This leads 
to a more rapid decrease in the volume jump AV during the melting than the corre- 
sponding entropy jump AS. It can be stated that Awl/, ,  where V ,  is specific voluine 
for the solid at melting, decreases very rapidly at low pressures. This has been experi- 
mentally shown for some metals by Urlin (1965). The fact is also substantiated by 
some observations made by Bridgman (1931) who established that the compressibility 
of liquids under low pressures greatly exceeds the compressibility of the corresponding 
solids, but, already at moderately high pressures, it becomes very near to the com- 
pressibility of solids. Urlin's experimental data show also that the rate of variation in 
AVT/, diminishes greatly as pressure increases and it can be assumed that this ratio 
tends to a non-vanishing constant. Corresponding conclusions apply to the entropy 
discontinuity at melting, which rapidly reaches a nearly constant value as the pressure 
is increased. 
Thus the available experimental data lead to the conclusion that 
lim (AV/K)  = const 
P - t m  
lim AS = const 
P--)m 
For the melting of model systems of hard sphere particles, we have the following 
relations 
AC//I/, = const 
A S  = const 
Thus it is reasonable to suppose that close packed systems can be represented by 
hard sphere models at sufficiently high pressures. 
As already mentioned the equation of the melting curve for a system of hard 
spheres is (Hoover & Ree 1957) 
where V. is the volume of the model system in the state of closest packing and N is 
the number of particles. 
48 E. Boschi 
Assuming an identical melting behaviour in both model and real systems, we can 
re-write equation (2) for the real system in the following manner 
p,VOR = const X kTm (3) 
where VoR is the volume of the real system in its closest packing. VoR is temperature 
dependent and can be derived from the energy conservation principle: 
where +(r) is the interaction potential. If we consider power-law repulsion only, as is 
reasonable for very high pressures, +(r)  can be written as 
and we get 
Then from the relations (3) and (5), it follows that 
Pm is proportional to Tm'+3'n, 
dlnP, - 3 
- l+-, dln T, n 
which gives the asymptotic expression for the melting curve. 
The Kraut-Kennedy relation (Kraut & Kennedy 1966) relates the melting tem- 
perature linearly to the compression 
T, = T,O l + -- ( "F) 
where TmO and V 0  are the melting temperature and the volume at room conditions 
and m is a constant characteristic of the considered substance. Equation (3) implies 
that the volume at a given temperature is proportional to VoR. We may therefore 
replace V in the Kraut-Kennedy relation by a constant times VoR and so obtain a 
linear dependence of vOR on T,, which is clearly in contradiction with our relation (5). 
Incidentally we may observe that the asyinptotic behaviour of the melting curve 
here discussed, is formally consistent with the Simon empirical equation 
where T, is the temperature of the triple point, P, and c are empirical parameters. 
The experimental values of c generally lie between 1.2 and 1-5 for pressures up to 
103 atm, but this pressure is not high enough for quantitative comparisons. 
Concluding we can state that the Kraut-Kennedy equation does not behave in the 
asymptotic manner to be expected from hard sphere models and the geophysical 
results based on it about the core (Higgins & Keenedy 1971) must be open to 
question. 
Birch's (1972) assumption that the melting curves of each phase of iron are 
straight lines in the temperature-volume plane following the Kraut-Kennedy relation 
also cannot give the correct asymptotic behaviour. 
The values of the resistivity of liquid iron at high pressures calculated by Evans 
& Jain (1972) in the basis of Higgins-Kennedy melting curve should be revised. 
On the melting curve at high pressures 49 
Higgins & Kennedy (1971) have also pointed out that the temperature gradient is 
much less than the adiabatic gradient in the outer core ~rhich would then be thermally 
stably stratified, thereby inhibiting radial convection which is necessary to drive the 
geomagnetic dynamo. Various ways have been developed to circumvent this difficulty. 
BuIlard & Gubbins (1 971) have pointed out that a stable Auid can have internal wave 
nlotions and their calculations iildicate that a body of fluid could act as dyrramo, 
even when its motion is purely oscillatory. Busse (1972), Malkus (1972) and Eisasser 
(1972) have all indepe~ldently proposed that the liquid in the outer core can have a 
temperature distrib- iti ion along a melting point curve, but they propose that the 
liquid is a slurry made up of extremely fine iron particles suspe~ldecl in 311 iron-rich 
liquid. They show that such a liquid, given the proper distribution of solid particles, 
can behave as an adiabatic fluid. At the light of present knowledge all these sug- 
gestions, although ingefiious, appear no more than speculations. 
The author wishes to express his appreciation to Professor A. H. Cook, FRS for 
his help and advice. Thanks are due also to Professor J. Verhoogen for having read 
the manuscript, and to Professor E. Vero~dini for valuable discussions. This work has 
been performed dr~ring a tenure of a C.N.R. fellowship. 
Cauenrlish Laboratory, 
University of Cambridge. 
References 
Birch, F., 1972. The melting relations of iron and temperatures in the Earth's core, 
Geophys. J. R. a~11.. Soc., 29, 373-387. 
Bridgman, P. W., 1931. The physics of high pressures, Bell, London. 
Bullard, E. C. & Gubbins, D., 1971. Geomagnetic dynamos in a stable core, Nature, 
232, 548-549. 
Busse, F. N., 1972. Comment on the paper ' The adiabatic gradient and the melting 
point of the earth ' by G. Higgins and G. C. Kennedy, J. geophys. Res., 77, 
1589-1 590. 
Elsasser, W., 1972. Thermal stratification and core convection, paper presented at 
the Core-Maiztle interface coulference, NASA, Melbourne, Fla. April 1972. 
Evans, R. & Jain, A., 1972. Calculations of electrical transport propertics of liquid 
metals at high pressures, Phys. Earth Pkunet. Int., 6, 141-145. 
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melting transition and ca!culate the entropy in the solid phase, J. Chew. P!~ys., 
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Jacobs, J. A., 1971. Boundaries of the Earth's core, Nattlre, 241, 170-171. 
Jacobs, J. A., 1973. Physical state of the ear~h's core, Nature, 24413, 113-1 14. 
John, M. S. & Eyring, H., 1971. The significant structure theory of liquids, in 
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50 E. Boschi 
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melting of iron, J. geoplzys. Res., 70, 1979-1984. 
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On the Melting Curve at High Pressures

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On the Melting Curve at High Pressures

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