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Universality of melting and freezingindicators and additivity of melting curvesYaakov Rosenfeld aa Nuclear Research Center-Negev , P.O.B. 9001, Beer-Sheva, IsraelPublished online: 23 Aug 2006.

To cite this article: Yaakov Rosenfeld (1976) Universality of melting and freezing indicators and additivityof melting curves, Molecular Physics: An International Journal at the Interface Between Chemistry andPhysics, 32:4, 963-977, DOI: 10.1080/00268977600102381

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MOLECULAR PHYSICS, 1976, VOL. 32, No. 4. 963-977

Universality of melting and freezing indicators and additivity of melting curves

by YAAKOV ROSENFELD

Nuclear Research Center-Negev, P.O.B. 9001, Beer-Sheva, Israel

(Received 10 March 1976)

The assumption of universality of some melting and freezing indicators, in the context of approximate models of the liquid and the solid, gives rise to simple semi-empirical melting equations that describe additivity of melting curves. The melting equations are exact in the high temperature limit and give reasonably accurate results also near the triple point. They find many applications including quantum corrections to the melting curve.

1. INTRODUCTION

From a statistical point of view the melting problem is already solved. The thermodynamic conditions for melting require Ts= TL, Ps=PL, Gs(P, T)-- GL(P, T), corresponding to thermal, mechanical and chemical equilibria between the solid and the liquid. However, the mechanisms for melting comprise a complex problem and can depend on many microscopic details. Much pro- gress towards understanding the physical nature of melting was achieved by computer experiments [1, 2]. In this paper we want to deal with the simplest sort of question that can be raised with respect to melting. Given a classical system of particles, interacting via a temperature and density dependent pair potential ~(r; p, T), calculate the melting curve and locate the triple point. Present computer simulation techniques can give quite an accurate answer to this question. However, computer experiments are very time-consuming (the computer time needed to answer the above question is measured in decahours), and it is desirable to follow one of the following paths : (i) Use approximate models of the liquid and the solid to do the free energy comparison. (ii) In the computer experiment use some structural criterion to locate the melting transition. (iii) Use some structural criterion in the context of approximate models.

The method presented below is a special blend of these possibilities. We express the melting characteristics of the given system in terms of those of other systems, by means of simple semi-empirical melting equations.

In w 2 we present two approximate models of the liquid and the solid. Their application in connection to melting and freezing criteria is discussed in w 3. Computer melting experiments and their comparison with approximate models suggest the assumption of universality of the melting and freezing indicators (w 4). This assumption is incorporated in the approximate models to give the melting equations (w 5). The melting equations find many ap- plications (9 5) which include Lennard-Jones-type potentials, analysis of the Simon and Krout-Kennedy ' l aws ' and quantum corrections to the melting curve. The situation is summarized in w 7.

M.P. 3 T

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964 Y. Rosenfeld

2. A P P R O X I M A T I O N MODELS OF THE L I Q U I D AND THE SOLID

In this section we give a short account of two approximate models of the liquid and the solid, emphasizing those features which are relevant to our main topic. We consider systems of particles interacting via a stable two-body potential r :

r ; p, T)= • ~i(T)q~;(r ; p) (1) t

which is a linear combination of the stable potentials q~ with temperature- dependent coefficients. The potentials may be density dependent.

2.1. Variational hard-sphere theory The Gibbs-Bogoliubov inequality [3-5] for fixed N, V, T, with the system

of hard spheres as reference, takes the form

FtrueE(p, T ) ~ - TSoE(B)+ UoE(p, T, n ) - F(1)(P, T, "q),

where

and

UoS(p, T, 7/)= ~ ~,(T)Uo, iE(p, ~) z

pE(p, T) - p2 ~Fn(p,~p T) [

the excess entropy per particle

(2)

(3)

~FE(p, T)[ (8) SE(p, T) - ~T p'

~c

u0, iE(p, n)= 12n I r ; p)go(x ; n)x 2 ax. (4) 1

The symbols are defined as follows : F is the free energy per particle, S is the entropy per particle, ~=(Tr/6)pd a is the hard-sphere packing fraction, while p =N/V is the particle number density and d is the hard-sphere diameter. The superscript E denotes excess properties over the ideal gas values, and the subscript 0 denotes hard-sphere properties.

The variational approach [6, 7] consists of minimizing F (1) with respect to ~, i.e.

~F(Z)(p,= T, n)[ =0, (5) c~/ I p, T

taking the minimized expression as representing the approximate free energy of the given system. Equation (5) definds a correspondence with the hard-sphere system by a density and temperature-dependent packing fraction, ~/(p, T), and the approximate free energy per particle is given by

FE(p, T)=F(')(p, T, ~(p, T)). (6)

With FE(p, T) one can use the usual thermodynamic relations to get the excess pressure

=p~ ~UoE(p, T, ,7) , (7) T ~P ~/= 7/(p, T)

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Melting curves 965

and the excess energy per particle

UE(p, T ) - FE(p, T)+ TSE(p, T). (9)

When the coefficients ~i are temperature independent (i.e. real constants) we get

sn(p, T)=SoS(n(p, T)) (10)

and

UE(p, T)= UoS(p,-q(p, T)). (11)

In (7) and in order to get (10) and (11) from (8) and (9) we made use of (2) and (5). For future purposes we rewrite (5) and (7) in the following form :

where

and

T= Y. oq(T)T~(p, ~), (12) t

P(p, v(T, p))= ~., ~x,(T)P,(p, v(T, p)), (13 / t

~UoE(p, ~) /CS:(~),~ E (14)

Pi(P, ~l)=kBpTi(P, 71)+P2 ~U~ : iE(p' r/) cp

(15)

The variational hard-sphere theory proved successful for simple liquids and can be used with considerable accuracy even for potentials as soft as the inverse 4th power [8]. There are indications that the same approach will be adequate also for simple solids near melting, but this was not studied due to lack of a good representation for the angle-averaged radial distribution function of the hard-sphere solid.

2.2. Harmonic cell model In the cell model [9] each atom is confined within its cell and moves in the

potential field #(x) of its neighbours, where x is the displacement of the central atom from its static equilibrium position. We denote by z z and a I the number and the position of the lth neighbours in the given lattice, and put a t = ),tat and v - V/N= )~at 3 (y = l/V/2 for the f.c.c, lattice).

The static lattice energy per particle is given by

~(0) Ust atie(p, T)= �89 - t 3 . p, T ]= (16) ' 2 i

= Z% ' (T ) { � 89 - 1 3 ; P]} -= Ze~ i (T)Uis ta t ie (p) " (17) 1 l t

The spherically averaged potential field, in the harmonic approximation, is

Ag(A) _ g ( h ) - ~(0) _ �89 ' (18) kBT kBT

3T2

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966 Y. Rosenfeld

where

K(p, T) (yp)-21a V = 3kBT

and

2; l [ 26'(r, ; p, T) 6"(r, T)-] + P, r I ] r t = y t ( y p ) i,a

1 K * I E - ~ (p, T)= T , e~,(T)K~*(p) (19)

= ~ e ~ 4 , x = x / a 1 . @'(r; p , T ) ~q~ ,@"(r; p, T)=~r--- T o,T p ,T

The reduced free volume has the form

~ m a x

v,*=v~=4rr I exp [ - AN(A)/kBT]A 2 aA, (20) v Y 0

where /~max is a constant of the order of unity (usually Area,:=1). For K ~ I one may take Amax = oo and obtain for the Lindemann ratio 3 :

32__( A2)= 4rr ~ : e x p ( _ K A 2 / 2 ) M d A = 3 / K , (21) y v f * o

i.e. (Ao~'(A)/kBT)=IK3==I.5, as expected. The free energy per particle is given by

F = F I + U Static - kB T ( l n vf* - 1 ), (22)

where F I is the ideal gas contribution, and the pressure is obtained from

~F = k B p T + p 2 ~Ustatie I k = ~K*(p, T)[ + Rup" 32 (23) p = pZ 7p T ~P T 2 cp T

We rewrite equations (19) and (23) in the form

T= ~, a~(T)T,(p, 3), (24) z

P(p, 3)= Z a,(T)P,(p, 3), (25) I

where

T,(p, 3)= K~*(P) a2 K 3 K,*(p), (26)

and ~Uistati('(p) hBp282 ~Ki*(p)

P~(P' 3)=kBpT'(P' 3)+P2 20 -~ 2 ~p (27)

The cell model is an excellent model of the simple solid, its validity tested against Monte Carlo results. Since we use the harmonic approximation, our results apply only for cases where anharmonic contributions are small.

3. ONE-PHASE MELTING THEORIES

Since melting is a first-order-phase transition, there is no simple relation between the potential energy and the transition temperature, so that the transition

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Melting curves 967

point is determined by the equality of the thermodynamic potentials. However, simple relations characterizing melting can be found for the inverse power potential systems, c~(r)=E(e/r) n, due to their homogeneity. For the inverse powers any structural characteristic can serve as an indicator of melting or freez- ing. Thus, knowing a single point on the melting (freezing) line, one can draw the entire melting (freezing) line. The philosophy behind the so-called one- phase theories of melting is that even for non-homogeneous systems one can find some structural characteristic, which although not exactly constant along the melting (freezing) line, is very nearly so.

Lindemann formulated his melting criterion taking the view that each density along the melting line the average arrangement of the atoms in some suitably reduced dimension is always the same. He stated [2, 10] that a solid melts when the mean square amplitude of vibrations of a given atom from its equilibrium position reaches a certain fraction of the mean interparticle distance. The same criterion for crystallization will require an unvarying scaled radial distribution function along the freezing line. Ross generalized Lindemann's law by looking for similarity in configuration space rather than real space. Ross's melting rule [11] states that the reduced free volume is constant along the melting line.

The application of the one-phase theories to the calculation of melting curves is done by means of the approximate models. The Lindemann law for hard spheres is applied in the context of the variational theory : the melting (freezing) line is characterized by a single packing fraction r/S(~L)" The melting tempera- tures and pressures T(pL, r/L), T(ps, ~s), P(PL, r/L), P(Ps, r/s) can be calculated from equations (12) and (13). Since in practice one has to use a specific hard- sphere model, the application of the above is restricted, nowadays, to the calcula- tion of the freezing line. This is because there is no good representation for the hard-sphere solid. For the hard-sphere liquid one uses the Carnham-Starling [12] equation of state and the Percus-Yevick [13] or Verlet-Weis [14] radial distribution function. The melting line can be calculated by the Ross criterion of v**= constant in the context of the cell theory. When anharmonic contribu- tions are small one can use the harmonic approximation and calculate the melting line via 82= Ss2= constant. With the appropriate 8 s, the melting temperatures and pressures T(ps, 6s), P(Ps, 3s) are given by equations (24) and (25). Examples of one-phase calculations are given in references [1, 15-17]. For short-range potentials one can further simplify the calculations and get a simple analytical expression for the melting temperatures in terms of the melting densities (see the Appendix).

4. C OM P UT E R MELTING RESULTS AND THE APPROXIMATE MODELS

UNIVERSALITY OF MELTING PARAMETERS

Computer simulation experiments of the melting characteristics of the inverse power potentials [8, 18-22] (r- ' , n= 1, 4, 6, 9, 12, oe) and the Lennard- Jones (L-J) system [22, 23] confirmed the Lindemann point of view. It was found that along the melting line the Lindemann ratio, 5, and the reduced free volume, v**, are nearly the same for all the inverse power potentials. The principal maximum of the structure factor, S(ko), exhibits the same behaviour along the freezing line. These parameters vary by no more than 5 per cent along the entire melting curve of the L-J system. The anharmonic contributions to the

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968 Y. Rosenfeld

free energy at melting are less than a few per cent (except, of course, the extremely anharmonic hard-sphere case).

The calculation of the melting characteristics of the inverse power potentials by free-energy comparison with the approximate models for the liquid and the solid gave reasonable agreement with the computer experiments [17]. Again it was found that melting and freezing are chracterized by nearly the same 8 and ~7, respectively, independent of the interaction.

There have been attempts [25-30] to correlate the transition of the hard spheres [24] and the melting transition of simple systems. If we regard the melting problem as a problem of packing of the effective hard spheres, then, because of the above-mentioned correlations, one can expect that the melting curve of a given system will correspond to a narrow interval of r/in the vicinity of the hard-sphere melting (~/~ 0-54) and freezing (~ ~ 0-494). Restricting our- selves to systems with soft repulsion, we would like to explore the outcome of, what now seems to be a reasonable approximation, the assumption of universality of the melting indicators ~ and ~.

To feel on more solid ground we do some more calculations, reversing the usual procedure. With the computer melting results for the inverse powers and the L-J system we use equations (12) and (13) to solve for ~q that corresponds to the freezing line [31]. and equations (24) and (25) to solve for 8 that cor- responds to the melting line [32]. We find that both ~ and 8 keep their nearly universal attitude. The accuracy of the approximate models is demonstrated by the fact that one gets nearly the same ~ or 3 from the temperature equations ((12), (24)) and the pressure equations ((13), (25)).

5. UNIVERSALITY OF MELTING PARAMETERS AND ADDITIVITY OF MELTING CURVES

If one assumes the universality of the melting parameters ~ and ~ in equations ((12), (13), (24), (25)) then the melting curve of the potential defined in equation (1) takes the form :

TL(S)(p r(s)) = ~ ai(TL(S))TiL(S)(pL(S)), (28) 1

pL(S)(pL(S)) = ~ ~i( TL(S))piL(S,(pL(S)), (29)

where TiL(S)(pL(S)), PiL(S)(p L(s)) are the melting temperature and pressure, respectively, of the ~i potential system (see equation (1)), as functions of the melting density pL(S), along the liquidus (L) and solidus (S). For obvious reasons we refer to equations (28) and (29) as describing additivity of melting curves and in what follows call them the melting equations.

The melting equations, being of approximate nature, lead to different P(T) results, depending on whether one uses the equations for the liquidus or the solidus. However, the difference between pL(T L) and p s (T s) can serve as some measure to the accuracy of the equations. One can have a good estimate of the triple-point temperatures and densities by solving the melting equations for P = 0 . Again, the difference between TLtriple(Ptr L) and TStriple(Ptr s) can give an estimate of the approximations involved.

Since there are established methods for calculating melting curves for given

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Me l t i ng curves 969

potentials, the main practical advantage of the melting equations lies in their simplicity and relatively high accuracy.

The melting equations are very general, but since they demand knowledge of some melting curves in order to construct another, their scope is limited at present due to lack of sufficient computer melting results. Extensive computer melting data are available only for the inverse powers and the L-J system, which will be our test case to judge the capability of the present method. We take the view that the L-J potential can represent a general reasonably behaved potential.

1 Potential - u (r/a) o (A) E (K)

E

L-J 4 [(~/r) ' 2 - (r 3-405 119-8 11-6-8 (7 = 2) 2 (o/r) H - (o/r) 6 - 2 (o/r)" 3"754 137 11-6-8 (7=3) 2'4 (a/r)U-0"4 (a/r)~- 3 (a/r) "~ 3"669 153

Table 1. Simple potentials for argon [33-35].

6. APPLICATIONS or THE MELTING EQUATIONS

6.1. L - J - t y p e potent ia ls and the mel t ing o[ argon

Consider the systems with the following two-body potential :

r = e ~] %(or/r)", (30) n

i.e. linear combination of inverse powers. This sort of potential, of which the L-J is one, can serve as good effective potentials for argon and other rare gases [33-35]. Some potentials of this sort, with the appropriate parameters for argon are given in table 1. In reduced units, p* = pa a, T * = k B T /e and P* = Paa/e, the melting equations for these potentials take the form

T*L(s) = ~ % ( O * L ( S ) / a L ( S ) ) , ~a, (31 ) n

p*L(S) = ~ %,rr L(S)(p*L(S))I+. a. (32) n

T h e melting constants of the inverse powers, a . I'(s), ft. l'(s), as calculated by computer experiments, are given in table 2. Th e results for the L-J system are

n p n / ' / 2 anS.' \ "2 anL/~/2

oc 8.3 0-736 0"667 12 16 0"844 0"814 9 22 0-971 0"943 6 61 1'56 1"54 4 426 3'94 3"92

Table 2. Melting constants of the inverse power, r ", potentials [2]. The constants rrn s, rrn s (see text) are defined by ~vn L(s) = pn/(anL(S)) 1-n'a and pn = P r n e u * / T m e l t 1+3 n.

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970 Y. Rosenfeld

T* P* pL* T12 L T6 L TIlL-T6 L P12 L P6 L P12L--Pn L

0'66 0'0 0"862 1-26 0"63 0'63 21"31 21-40 -0"09 1"35 9"0 0"964 1'97 0'78 1-19 37"27 29-92 7"35 2"74 32"2 1"113 3"50 1.04 2"46 76'47 46'06 30-41 5 86 1-279 6"10 1.38 4'72 153"24 69"89 83"35

20 590 1'765 22"10 2'63 19"47 766'48 183"67 583-22 100 4800 2"601 104"25 5-71 98"54 5329'77 587"79 4741 "98

Table 3. 'Repulsive' and 'attractive' contributions to the melting curve of the L-J system. T12 L = 4 (pL/al2L) 4, T6 L = 4(pL/aaL) 2, PI2 L = 4rrl2LpL 5, Pe L = 4rrebpL3.

summarized in table 3 and in reference [31]. The melting equations are exact in the high tempera ture-high density limit. It is important here to emphasize that the r -6 term of the L-J potential gives appreciable contribution to the thermodynamic properties of the system up to very high temperatures (see table 3). The melting equations are very accurate at temperatures above the critical, but tend to underestimate the solid densities and overestimate the liquid densities at low melting temperatures. The triple-point densities and tempera- tures are estimated to better than a few per cent. For a given melting tempera- ture p . L > p . s , and having no better suggestion we take

P*(T*) = �89 + p . s ( T . ) ) (33)

and obtain the triple point from

P*(T*t,ivle) = P*( T*(oIL t riple)) = n*( T*(p*st ~iv,,)) = 0. (34)

The melting characteristics of argon as calculated by the melting equations for the potentials of table 1 are summarized in table 4 and in reference [31 a].

T(K) P(kbar) expt l . P (kba r ) l e -6 p(kbar)ll-6 8(7=2) P(kbar)ll-~ s(7=a)

83.81 0.001 0.025 - 0.05 - 0'21 100.76 0-721 0"74 0"68 0,52 120'85 1.674 1"68 1"62 1,47 140"88 2"708 2.69 2'61 2'51 156'39 3"574 3'52 3"46 3'35 181 "28 5'061 4'94 4'83 4.77 221 "41 7'684 7.00 7"30 7.25 253.49 9.960 9.48 9"39 9.35 322 15'840 14.32 14'16 14-16

Table 4. Melting characteristics of argon calculated by the melting equations. experimental data is compiled in reference [2].

6.2. A n a l y s i s o[ the S i m o n equat ion

For the inverse nth-power potential the melting curve is given by

p * = p , , T .1+3 "

Argon

(35)

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Melting curves 971

(see table 2). Taking a reference point (Po*, To*) on the melting curve, equation (35) can be rewritten in the form

a - - \ T o . } - 1 (36)

with a = PnZo .1+3/n and c = 1 + 3/n. Equation (36) is of the well-known Simon [37] form which is exact for the inverse power potentials. However, the simpler Simon form

P = A T c + B (37)

is exact for the inverse powers only for B = 0. As pointed out by Stishov [2], for real materials or when attractive forces are present, the constants of equation (37) can be regarded only as pseudo constants, and their numerical values depend on the region of temperatures that is investigated. In fact the Simon equation even in the form of equation (36) cannot be rigorously justified for real materials. This can be demonstrated easily.

Define the effective power of the melting curve in the following way :

in T (38) n~f T - 3 ?, In p '

net~ e - 3 In P

Taking the L-J potential as an example we get from the melting equations

nenT= 12 + 6/((a82/a124)p . 2 - 1), (40)

n~t e = 12 + 6/((Tr~e/Tr6)p*z -- 1), (41)

where the coefficients a62/a124~. 27r12/Tr 6 are of the order of unity. Thus the at- tractive part of the interaction makes nef f a monotonically decreasing function of the density. That is why the Simon equation, if fitted to a certain temperature range (i.e. certain density range), cannot be trusted at other temperature ranges of the melting curve.

On the other hand, data fit with equations (31) and (32) produces parameters of direct relation to the prescribed potential. A least square fit of the L-J melting data to equations (31) and (32) reproduces the inverse 12 and 6 melting constants to better than 1 per cent and 10 per cent respectively.

6.3. The Krout-Kennedy relation and density-dependent power potentials

Krout and Kennedy [38] proposed an approximate T(V) melting relation

T/T o = 1 + C( A V/Vo), (42)

where (To, Vo) is a reference point on the melting line, AV= V 0 - V and C is a constant. For the inverse power potentials the relation (42) represents only the first two terms of the full expansion :

T / T ~ ~ + 18 \ V0J + .... (43)

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972 Y. Rosenfeld

This is a very poorly convergent series even for small n, and equation (42) should fail at high compressions. For the alkalis, however, relation (42) holds up to relatively high compressions (2xV/Vo~0.4), which is associated with a density dependent Gruneizen parameter [39]. If the power n is density de- pendent compensations can occur in (43) to make up for the omission of higher terms. For the alkali metals this can be demonstrated by fitting the following simple model to the experimental melting results.

Consider a system with a density-dependent power potential

q~(r; p)=E(~/r)"(p). (44)

From equations (12) and (24), with the assumption of universality of ~7 and 3, the melting temperatures are given by

p*L(S) In (p*L(s','3 T.L,s,(p.L,s,) = L a , , , s , ( 4 5 )

i.e. the same type of equation as for a density-independent power. The melting equations for the pressures are somewhat modified. From equations (13) and (25), with the assumption of universality, we get

P*L(S)(p~aL(S))=~(n(p'I'L(S)) ; p'~L(S))(paL(S))I+n(P *L(s))/a, (46)

where

and

, ~p-----~ l n p * + 3 ? c n - ' (47)

A = rr(n) - a(n)-" z (48)

Under circumstances when one can ignore the correction term proportional to ~n/~p*, one has

~(n ; p*)=Tr(n) (49)

The approximation (49) which can be checked a posteriori, is justified for the alkalis, and is particularly useful, since then

P V Zm~tt =(-~--k-T)m~,t=rr(n)a(n) ~'3, (50)

i.e. function of only the power n. To make the model more realistic (i.e. have a triple point) one can add to the pressures a Van der Waals type attraction term, Part.* ~:p*~. The results for the alkali metals are given in reference (40).

One finds that the Krout-Kennedy relation is associated with the softening of the potential as the compression increases. Moreover, within this model the maximum that occurs in the melting curves of Rb and Cs is also associated with this softening of the effective power of the repulsion.

6.4. Temperature-dependent potentials and quantum corrections to the melting curve

Consider the first Wigner correction to the free energy [41] of a system

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Melting curves 973

characterized by the pair potential r Eu(r/~),

where

F l Falassi c N k B T N k B T ~- 2~p* f (u"(x)+ u (x))gcla~<,.(x)x dx,

h 2

y = 12mo2E,

and m is the mass of the particle (see references [41-42]).

U-- u i, i ~j

V = { ~ (ui/,+__2 uii ) = �89 %, i ~j r i j i ~

where u<i = u(rij ) = u( lri - rjl ). Equation (51) can be written in the form

Denote

(51)

(52)

(53)

(54)

F 1 = Fclassi c + ~ ( V)classic" �9 T .

Let F~ be the free energy of the system with the following pair potential :

(55)

(56) r ~ U~j + ~ 7~'ij

and use the obvious notations where Fdassic = F. , (V}(qa~< e = (V} . . The double Gibbs-Bogoliubov inequality reads :

F Y Y ( V}u = F ~, (57) u+~--~ (V}~, ~< F,o ~< F, ,+ T--- ~

that is, if

then

(v>~-(v>~ 41 (v>,,

(58)

F~ ~ F 1. (59)

The approximation (59) holds for neon down to the triple point. To the extent that F,o = F 1, the melting characteristics of the classical system

with the first quantum correction (i.e. F 1) are identical to those of the classical system with the temperature-dependent effective potential (i.e. Fo,). The melting equations for the F~ system are

Y (s)(pL(S)), TL(S)(pL(S))= TuL(S)(pL(S)-~ TL(S)(pL(S)) T, "L (60)

pL(S)(pL(S))_pL(S)(pL(S))+ Y P L (S)(,nL (S)~ (61) -- , TL(S)(pL(S)) ~" ~r l,

where (Tu, P.), (T~., P,,) are the melting curves of the systems F., F,. respectively. For F . = FLennara_Jone s we have

u(x) = 4(x -12 - x-6), (62)

v(x) = 4(132x -14 - 30x -s) (63)

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974 Y. Rosenfeld

and the melting equations take the form

- 3 0 - - 3 / , ~ (cl'-(Cl l T*(P*)=g[ka,2 / ka6/ J + ~ \a,4/ kas/ (64)

43, P*(p*) = 417r,2p*5 - rr6p*3 ] + - - [1327rlap *~7 3 _ 307rsp*,x a]. (65) T*(p*)

The neon melting characteristics are given in table 5 where they are compared to the more fundamental results of Hansen and Weis [41]. We predict correctly the shift in the triple point densities and temperature as well as other differences between the classical and quantum cases.

T* pvl~S pq#S pel'll'L pq~,L Pel* Pq*

(" 0-75 0"973 0.935 0"875 0-843 0.96 0-94 Hansen-Weis [41] I 1.15 1"024 1"000 0'935 0.909 5.68 5-91 L-J potential { 1.35 ! "057 1.026 0.964 0.933 9.00 9.09

0-66el 0.96 0.862 0"62q 0"92 0.826

Triple. point

0'75

Melting equations t. 15 1.35

L-J potential Triple f 0"70e1 point \ 0.66q

Argon triple point 0"70 Neon triple point 0.668

0-927 0'898 0.888 0.859 0"65 0.96 1.000 0-978 0"959 0-937 6'03 6.16 1.031 1-009 0-988 0.969 9.1 9-2 0.92 0"88

0.88 0.84

0"968 0.841 0.933 0.813

Table 4. Quantum corrections to the melting characteristics of neon. The indices cl and q denote classical and quantum values, respectively.

7. CONCLUSION

The melting equations for the liquidus and the solidus were derived from the variational hard-sphere theory and the harmonic cell theory, respectively. Formally one can derive the liquidus melting equations also from the cell model, but in this case we have no justification to assume universality of the Lindemann parameter 8 L. On the other hand, there are indications that the variational hard-sphere model is equally adequate to describe the simple solid near melting, and the assumption of universality of rl~ is justified. Thus one can regard the melting equations as a special type of a hard sphere melting model. Also, for the solid, instead of the usual cell model, one can use the variational theory with the harmonic cell model as reference system [43], which will be equally accurate and leads to the same melting equations if universality is assumed. In view of the above, the situation is summarized as follows.

Computer simulation experiments of melting and freezing of simple systems demonstrated that certain structural averages are insensitive to the details of the interaction. This fact is incorporated in some reasonably good models of the liquid and the solid that contain the appropriate structural averages as the variational parameter. One obtains simple semi-empirical melting equations

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Mel t ing curves 975

that describe the melting curve of a given potential system in terms of melting curves of other potentials, in a form that can be termed additive.

The melting equations, although of approximate nature, are exact in the high temperature limit, and proved successful even at low temperatures near the triple point. Their usefulness was manifested by a wide spectrum of applications, their main advantage being the easiness of computation and analysis of the results. A more fundamental approach to melting is a free-energy comparison between the two co-existing phases, using approximate models of the liquid and the solid. However, small errors in the free energy may lead to large errors in the transition density and pressure. In view of this the melting equations, which are nearly as accurate, are of particular importance. Their simple form makes these equations the high density counterparts of the second virial coefficient in analysing effective two-body potentials for various substances.

get

A P P E N D I X

Approx ima te analy t ica l representation of melting curves

Consider the two-body potential d?(r)=~u(r/~). From equation (26) we

a2 I-2u'(r,) . . . . ] T * = ~ x 2 ~ t z~ l + u t r o i , (A1)

L rt J,, = y,,

where x = (7p*) -lla. If the Laplacian of u(r) falls sufficiently fast, one retains contributions only

from the nearest neighbours to get

T* = -~ x = . z l + u"(x) , (a 2)

where z, is the number of nearest neighbours (z l= 12 for the f.c.c, lattice). Thus, the melting line is given by (for the f.c.c, structure)

T* = ]8=[x%'(x)] ' (A 3)

and the parameter 3 is chosen by fitting this equation to some experimental point on the melting line. For demonstrating the accuracy of (A 3) we choose the exp-6 potential, i.e.

6 a u(x)= _ 6 e x p [ a ( 1 - x ) ] a - - ~ x -6 (A4)

with the argon parameters ~ = 122 K, r = 3'85 A, a = 13. Equation (A 3) takes the form

T*=82 8 a x a [ ( ~ ) I (X'2"~la "~_-'Z---6 e x p [ a ( 1 - x ) ] a - - 5 x -8 , x = \ - ~ l . (A5)

We fix 82 by the point Vs=19-40 cma/mole, T=322 K, and get 82=0.01107. The predictions of (A 5) are compared to Ross's calculations [16] in table 6. Equations of this type can serve as simple fits to experimental data, with para- meters of direct relation to the prescribed potentials.

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976 Y. Rosenfeld

T(K) Ross [16] Vs(cm~/mole)

Equation (A 5) Experimental

201 "3 21.74 322 19"40 19"40 500 17"22 17"25 700 15,61 15"65

1000 14"00 14"02

1200 13-17 13"21

21 "69 19'40

T~ble 6. Calculated melting line of argon.

REFERENCES

[1] HOOVER, W. G., and Ross, M., 1971, Contemp. Phys., 12, 339. [2] STISHOV, S. M., 1975, Soviet Phys. Usp., 17, 625. This is a recent review article that,

among other things, includes computer melting results and experimental argon and sodium melting data.

[3] L u g s , T., and JONES, R., 1968, J. Phys. A, 1, 29. [4] ISHIHARA, A., 1968, J . Phys. A, 1, 539. [5] SNIDER, N. S., and KNUDSON, S. K., 1973, J. chem. Phys., 58, 2223. [6] MANSOORI, G. A., and CANFIELD, F. B., 1969, J. chem. Phys., 51, 4958. [7] RASAIAH, J., and STELL, G., 1970, Molec. Phys., 18, 249. [8] HoowR, W. G., GRAY, S. G., and JOHNSON, K. E., 1971, J. chem. Phys., 55, 1128. [9] LENNARD-JONES, J. E., and DEVONSHmE, A. F., 1937, Proc. R. Soc. A, 163, 53.

[10] STISHOV, S. M., 1969, Soviet Phys. Usp., 11, 816. [11] Ross, M., 1969, Phys. Rev., 184, 233. [12] CARNAHAN, N. F., and STARLING, K. E., 1969, J. chem. Phys., 51, 635. [13] PERCUS, ]. K., and YEv,cK, G. L., 1958, Phys. Rev., 110, 1. [14] VERLET, L., and WETS, J. J., 1972, Phys. Rev. A, 5, 939. [15] Ross, M., 1974, J. chem. Phys., 60, 3634. [16] Ross, M., 1973, Phys. Rev. A 8, 1466. [17] YOSmDA, T., and KAMAKURA, S., 1974, Prog. theor. Phys., 52, 822. [18] HoovER, W.G.,andREE, F.H.,1968, J. chem. Phys.,49,3609. ALDER, B.J.,HoovER,

W. G., and YOUNCL D. A., 1968, J. chem. Phys., 49, 3688. [19] HOOVER, W. G., Ross, M., JOHNSON, K. E., BARKER, J. A., and BROWN, B., 1970,

J. chem. Phys., 52, 4931. [20] HANSEN, J. P., and SHirr, D., 1973, Molec. Phys., 25, 1281. [21] POLLOCK, E. L., and HANSEN, J. P., 1973, Phys. Rev. A 8, 3110. [22] HANSEN, J. P., 1970, Phys. Rev. A 2, 221. [23] HANSEN, J. P., and VERLET, L., 1969, Phys. Rev., 184, 151. [24] ALDER, B. J., and WAINWRIGHT, T. E., 1960, J. chem. Phys., 33, 1439. [25] LONGUET-HIGGINS, H. C., and WIDOM, B., 1964, Molec. Phys., 8, 549. [26] GUG~ENHmM, E. A., 1965, Molec. Phys., 9, 43. [27] CRAWFORD, R. K., and DANtELS, W. B., 1968, Phys. Rev. Lett., 21, 367. [28] CARNAHAN, N. F., and STARLING, K. E., 1970, Phys. Rev. A 1, 1672. [29] JENA, P., and SMITH, W. R., 1973, Chem. Phys. Lett., 21, 295. [30] CRAWEORD, R. K., 1974, J. chem. Phys., 60, 2169. [31] ROSENFELD, Y., 1975, J. chem. Phys., 63, 2769. [31a] ROSENFELD, Y., 1976, Chem. Phys. Lett., 38, 591. [32] ROSENFELD, Y., 1976, J. chem. Phys., 64, 1248. [33] KLEIN, M., and HANLAY, H. J. M., 1970, J. chem. Phys., 53, 4722. [34] HANLAY, H. J. M., BARKER, J. A., PARSON, J. M., LEE, Y. T., and KLEIN, M., 1972,

Molec. Phys., 24, 11. [35] HANLAY, H. J. M., and WATTS, R. O., 1975, Physica A, 79, 351.

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[36] WETS, J. J., 1974, Molec. Phys., 28, 187. [37] See the review articles (references [1, 2, 10] and references therein. [38] KROUT, E. A., and KENNEDY, G. C., 1966, Phys. Rev. Lett., 16, 608 ; 1966, Phys. Rev.,

151, 668. [39] GROVER, R., 1971, J. chem. Phys., 55, 3435. [40] ROSENFELD, Y., 1976, J. chem. Phys., 64, 500. [41] HANSEN, J. P., and WEts, J. J., 1969, Phys. Rev., 188, 314. [42] HmSCHFELDER, J. O., CuRxIss, C. F., and BIRD, R. B., 1954, Molecular Theory o[ Gases

and Liquids (Wiley). [43] MANSOORt, G. A., and CANFIELD, F. B., 1969, J. chem. Phys., 51, 4967.

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