T H E T H E O R Y OF MELTING By T. A. HOFFMANN CENTRAL RESEARCH INSTITUTE FOR PHYSICS, BUDAPEST

(Presented by A. K£

- - Received: 18. IV. 1961)

The energy of a finite real erystal is expressed as that of an alloy of the atoms and vacancies. This energy is a function of the vacancy concentration and of the size of the crystal block. The mixing entropy of the crystal is a function of the vacancy concentration. For eertain block sizes the energy-entropy curve shows two inflexions. In these cases the mehing temperature is determined by the common tangent of the curve and the melting entropy is the interval between the two points of eontact. We get asa necessary eondition for normal melting a certain block size. Results for alkali and noble metals and for diamond are in good agreement with experimental data.

1. Introduction The existing theories of melting are all u n s a t i s f a c t o r y in certain respects. One group of these theories uses the a n h a r m o n i c i t y of the potential curves, connecting la some w a y the melting properties with the tensile s t r e n g t h of the crystal [1]. These theories do not gire a coavincing e x p l a n a t i o n for the sudden melting, or t h e y gire a melting point 3 4 times too large (in Kelvia degrees). Other theories use the local and d i s t a n t order-disorder p h e n o m e n a to explain melting [2]. These ate in good agreement with the experiment, their insufficiency being their semi-empirical s t r u c t u r e only. There are f u r t h e r thermodynamic theories of melting, which do not give a direct connection between the properties of the a t o m s of the solid and the melting properties [3]. There has been a t e n d e n c y in v e r y recent times to develop the t h e o r y of melting in a direction which takes as the basis the non-ideal structure of the crystal [4]. The theories using order-disorder were in some respects a l r e a d y such theories, however, the theories m e n t i o n e d in this p a r a g r a p h extend the non-ideal b e h a v i o u r b e y o n d the order-disorder p h e n o m e n a to other properties too. The present t h e o r y also follows this direction. S u m m a r i z i n g the requirements which a m o d e r n significant t h e o r y of m e h i n g has to fulfil, we have collected the following list of p h e n o m e n a connected with the process of melting: 1. Melting occurs discontinuously at the melting point. 2. Melting is connected with a certain l a t e n t heat, i. e. an e n t r o p y change, the melting e n t r o p y . Acta Phys. Hung. Toro. X I I I . Fasc. 4.

382

T . A . HOFFMANN

3. At melting there is a discontinuous change in volume. 4. Melting occurs almost exclusively w i t h o u t the p h e n o m e n a of overheating. 5. The opposite process, freezing, is, on the c o n t r a r y , sometimes conn e c t e d with a considerable measure of undercooling. 6. There ate some smaller anomalies in the specific heat near the meltingpoint, however, these anomalies do not lead to ah infinite specific heat at the melting point. In addition to these the t h e o r y should show the following features : 7. Ir should reproduce the true q u a n t i t a t i v e connections between the t h e r m o d y n a m i c and n o n - t h e r m o d y n a m i c quantities occurring at melting. 8. The t h e o r y should make one hope, t h a t it - - o r a possible extension of it - - can give the absolute values of these quantities too. 9. The t h e o r y should be as nearly as possible a p u r e l y theoretical and n o t a semi-empirical one. In the present work we t r e a t the solid as a real crystal with vacancies, i . e . Frenkel-defects. We shall derive a relation between the v a c a n c y conc e n t r a t i o n and the e n t r o p y , f u r t h e r between the v a c a n c y c o n c e n t r a t i o n and the lattice energy of the crystal. The form of this last relation depends on the size of the crystal block a s a p a r a m e t e r . D e t e r m i n i n g this p a r a m e t e r we can t r e a t the energy a s a function of the e n t r o p y only. It is essential at this point to note t h a t when plotting this e n e r g y - e n t r o p y relationship, the curve has for certain block sizes two inflexions, and thus there exists one among the t a n g e n t s to this curve, which has two points of contact. As the slope of the t a n g e n t to a n y point of this curve gives the absolute t e m p e r a t u r e belonging to this point, this double t a n g e n t corresponds to the melting point, the point of c o n t a c t lying lower re˜ the solid state and t h a t lying higher the liquid state at the melting t e m p e r a t u r e . - - On these bases we get good agreement with the e x p e r i m e n t a l data for the m e h i n g point, for the melting e n t r o p y , for the v a c a n c y concentration in m e h i n g and we can f o r m a n idea of the p h e n o m e n a occurring in undercooling and overheating. The calculations were carried t h r o u g h in this work for one-valency metals and for diamond for atmospheric pressure only.

2. The energy of a real crystal block The first question which arises is: W h a t is the difference between an ideal a n d a real crystal? - - Usually there is one respect, which is emphasized as this difference: the ideal crystal has a strict geometrical a r r a n g e m e n t of the nuclei of the constituent atoms of ions, while in the real crystal there is the possibility of some deviation from this strict g e o m e t r y . These deviations A c t a Phys. Hung. Toro. X I I L

Fasc. 4.

THE THEORY OF MELTING

383

m a y be v a c a n t lattice points (vacancies) in the strict geometrical arrangement, or lattice points, which are occupied b y foreign atoms or ions (impurities), or t h e y m a y be atoms of ions located not at the lattice points defined b y the strict geometrical a r r a n g e m e n t (interstitials). There is a n o t h e r t y p e of deviation also, in which the g e o m e t r y does not repeat itself strictly periodically, i. e. there is a small d e f o r m a t i o n in the lattice. There is, however, a n o t h e r possibility, which is usually not emphasized as a deviation from the ideal case and this relates to the size of the crystal block. A crystal is ideal, ir besides its strict geomctrical a r r a n g e m e n t ir has infinite size in all directions. The calculations made for ideal crystals were related always to such infinite crystals. We consider therefore a finite crystal as a real crystal, even ir it has a strict geometrical a r r a n g e m e n t of the constit u e n t nuclei, in contrast to the infinite ideal one. In our model we t r e a t a crystal block which is finite and possesses vacancies, b u t no interstitials of impurities. The n u m b e r of vacancies in equilibrium is well d e t e r m i n e d b y the t e m p e r a t u r e . I f we denote the r m m b e r of atoms in the crystal block b y A and the n u m b e r of vacancies in the same b y V, we can consider the block a s a lattice of A + V points. The v a c a n c y concentration is thus: V p (1)

A+V

and the c o n c e n t r a t i o n of the atoms 1

p --

A

(2)

A+V The n u m b e r of vacancies in the block is from (1) V --

P 1--p

A.

(3)

The energy of the crystal block m a y now be calculated in a v e r y crude a p p r o x i m a t i o n as follows. In the lattice each lattice point has z nearest neighbours. This z, the coordination number, is given b y the crystal s t r u c t u r e under discussion. Ir is 6 for the simple cubic lattice, 8 for the b o d y - c e n t e r e d cubic lattice, 12 for the face-centered cubic lattiee and 4 for the diamond lattice. In our a p p r o x i m a t i o n we shall take into account additively only the contrib u t i o n of the binding energy of the nearest neighbours. In a simplified phraseology we m a y say t h a t all a t o m - a t o m bonds contribute EAA to the energy of the block and all a t o m - v a c a n c y " b o n d s " , i. e. all a t o m - v a c a n c y pairs of neighbouring position, EAV. I f the whole n u m b e r of the a t o m - a t o m bonds in Acta Phys. Hung. Toro. X I I I . Fase. 4.

384

T . A . HOFFMANN

the block is NAA and t h a t of the a t o m - v a e a n c y " b o n d s " of the block can be written in this a p p r o x i m a t i o n

E

= NAA

NAV, the whole e n e r g y

EAA + NAV Env.

(4)

We have to p a y a t t e n t i o n here to the circumstance t h a t NAv should contain also the a t o m - v a c a n c y " b o n d s " on the surface of the block, where the word v a c a n c y has lost its meaning. To determine NAA and NAv we suppose t h a t the vacancies are distrib u t e d u n i f o r m l y in the block, i. e. i f p is their concentration, each inner lattice point has pz v a c a n c y neighbours and (1 - - p ) z a t o m neighbours, irrespective of its being atom of v a c a n c y . I f we denote the n u m b e r of atoms on the surface b y A" and t h a t of the vacancies there b y V" supposing the c o n c e n t r a t i o n to be the same at the surface as in the inner parts of the bloek, we have similarly to (3)

v' -

P 1--p

A'.

(5)

We shall define a surface coordination n u m b e r as the n u m b e r of lattice points neighbouring a lattice point on the surface, counting only those which are within the block of on its surface. I f this is z', the n u m b e r of those points, which are outside the block, is z - - z'. These last z - - z' neighbours ate accordingly vacancies in a n y case. So each lattiee point on the surface has ( 1 - - p ) ( z - z') v a c a n c y neighbours in addition to the pz vacaneies located a r o u n d the inner points of the block. We have thus a t o t a l 1

1

N A A = ~ - ( 1 - - p ) Az-- --2 ( 1 - - p ) A ' ( z - - z ' ) ,

(6)

where the factor 1/2 p r e v e n t s the bonds from being counted twice once at each end a t o m connected b y t h e m . Similarly, we have for the a t o m - v a c a n c y " b o n d s " N ~v = p A z + (1 - - p ) A ' (z - - z'),

(7)

where the factor 1/2 is o m i t t e d since each bond is counted only once, at the a t o m end. P r e s e n t l y the various crystal structures will be t a k e n into account b y specifying A" and z'. Acta Phys. Hung. Toro. X I I I . Fasc. 4.

THE THEORY OF MELTING

385

3. NAA and NAV for various crystal structures We now proceed to determine the values (6) and (7) of the number of atom-atom and atom-vacancy bonds for the simple cubic, the body-centered cubic, the face-centered cubic and the diamond lattice. In Fig. 1 parts of these four structures can be seen together with the bonds occurring on the surfaces. It can be seen that for the simple cubic lattice z=6,

z'=5,

i.e.

5

z'=---z,

(8)

6

for the body-centered cubic lattice z:8,

z'=4,

i. e. z ' =

1

2

-z,

(9)

for the facc-centered cubic lattice z=12,

z'=8,

2 i. e. z ' = - - z 3

(10)

z'=2,

i.e.

1 z'=--z. 2

(11)

and for the diamond lattice z=4,

I f we consider crystallites of cubic shape only, the crystallite can be built up of n a unir cells, one of which is drawn with heavy lines in each figure as the right upper cube. A unit cell contains 1 lattice point in the simple cubic lattice, 2 lattice points in the body-centered cubic lattice, 4 in the face-centered cubic lattice and 1 in the diamond lattice. (In this latter case two neighbouring cells in Fig. 1 d contain 2 lattice points, so that we have an average of 1 lattice point per unit cell.) Thus the number A of atoms in the crystallite is determined by A = na(1 - - p )

(12)

A = 2n3(1 - - p )

(13)

in the simple cubic lattice,

in the body-centered cubic lattice, A = 4n a (1 - - p)

(14) Acta Phys. Hung. Tom. X l l l . Fase. 4.

3SG

T. A. HOFFMANN

/

/

ii

84

q 7

~

j

j

__72__~

.,,j: i6~##"0

_

I

/

5~ t / / / / / (o)

(c)

/ i

2

.-\

0.4"Ÿ

~,Y~,I~ .~+~~__,

/

/

i -

i

I

T

.,.~

~~- --

i

i

j"" (b)

-----0

.9-

i

/

(d)

Fig. 1. Sketch of the a) simple eubie, v) body-centered cubic, c) face-centered cubic and d) diamond lattice. The full ¡ lines sh0w the bonds directed inwards from the surface (the plane of the paper), the dotted heavy lines the bonds directed outwards from the surfaee. In each figure one unir cell is marked in the right upper corner by heavy lines

Acta Phys. Hung. Tom. X I I I . Fase. 4.

THE THEORY

387

OF M E L T I N G

in the face-centered cubic lattice and A ~- n a (1 - - p) in the diamond lattice. On the six surfaces of the cubes there are similarly A' =6n 2(l-p)

(15)

(16)

atoms in the simple cubic (s. c.) case, A" = 6n 2 (1 - - p )

(17)

atoms in the b o d y - c e n t e r e d cubic (b. c. c.) case, A" == 12n2 (I - - p )

(18)

atoms in the face-eentered cubic (f. c. e.) case and A" -- 3n 2 (1 - - p )

(19)

atoms in the d i a m o n d - t y p e lattice case. Eliminating n from equations (12)--(15) and the corresponding equations (16)--(19), we have 3

A ' = 6V1

-- p 9A2/3

(20)

3 V 2 ] ~ r - p . A2/a

(21)

for the s. c. lattice, A'= for the b. c. c. lattice, 3

A'

3

= 3 V~ V i -

p . A2/3

(22)

for the f. c. c. lattice, and 3

A' = 3V1

-

(23)

p . A2/3

for the d i a m o n d lattice. Substituting these values and (8)--(11) into (6) and (7) we obtain 3

NA A

Az 2

(1 -- p) 3

~~v ~z[~+(x , , V ~ ] 2

(25,

Acta Phys. Hung. Toro. X I l I . Fase. 4.

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T . A . HOFFMANN

for the s. c. lattice, 3

NAA

--

Az ~1 ~,i, 3V~}

(26)

2

3

NAv-=Az[p-t-3(1--p) V ~

]

(27)

for the b. c. c. lattice,

NAA -- Az 2

(X_p)[I_V

" 4 ( 1 A- - p ) ]

'

(28)

3

NAv=Az[p+(l_p)

V

4 ( l -]- pA)

(29)

for the f. e. e. lattiee and

NAA

--

Az 2

(1

--p)

[

( 3l~1-~/ 1

-- 2

A

. ,

3,1 ~,V1 A ~1

NAV = Az p -4- ~ -

(30)

(31)

for the d i a m o n d lattice.

4. D e t e r m i n a t i o n

of

EAA a n d EAV

T h e essential feature of the m e t h o d will be the d e t e r m i n a t i o n of EAA and Ezv as functions of the v a c a n c y c o n c e n t r a t i o n p and of the size n of the block. T h e usual theories of solids do not t a k e into account the finiteness of the crystal block. In fact the effect of this finiteness m a y c o n t r i b u t e a v e r y small corrective t e r m only to the t o t a l energy of the block. On the other h a n d , we ate n o t interested now in the t o t a l energy, b u t in the - - relatively small - change of this e n e r g y with p and n, and thus we have to use a model which can a c c o u n t for the finiteness of the block. In previous crude LCAO MO calculations we could build up such a m o d e l in the case of a linear chain [5]. According to t h a t work we have for the binding e n e r g y of a linear chain of k similar atoms

~=2~[ 1~ 11 sin

Acta Phys. Hung. Toro. X I I I . Fasc. 4.

2(k §

(32)

THE THEORY OF MELTING

3~~

if we suppose k to be even. ~ is the (negative) exchange integral between two nearest neighbours in the chain. For k odd we get a similar formula: E A = 2fl ctg

2(k+l)

1 .

(33)

(32) and (33) differ only v e r y little if k is not too small. F o r simplicity we shall use t h r o u g h o u t this article (32), i. e. we suppose k to be even. Now we consider the linear analogue of a finite crystal block with vacancies. The v a c a n c y means in the linear case t h a t the chain is b r o k e n at the place of the v a e a n c y . Suppose the t o t a l n u m b e r of atoms in the chain to be N and the n u m b e r of vacancies in it j. In this case the chain is b r o k e n into j -4- 1 parts. The i-th p a r t should contain ki atoms. Then, obviously, we have j+l

,~" k i = N.

(34)

i=1

In the nearest-neighbour a p p r o x i m a t i o n this s t r u c t u r e corresponds to

j § 1 i n d e p e n d e n t chains, each containing ki atoms. The t o t a l binding energy will be the sum of the energies of the j § 1 chains, i. e. from (32):

E -----2" Eki = 2~ -J'~ i=i

1 zt

- - J -- 1 "

(35)

i~1 sin 2(ki-4- 1)

The v a c a n c y c o n c e n t r a t i o n is here p -- - -1 ,

N+j

(36)

which we suppose in the linear chain to be the same as in the three-dimensional crystal. For given N and p, E has a well-determined range of variation, which is given b y the m i n i m u m and m a x i m u m of the expression (35) for all possible variations of the ki compatible with (34). According to the rules of calculating the conditional e x t r e m u m we have to solve the following system of equations: 0E --§ 91

(i=1,2,...,3"§

(37)

j+l

.~ k~ =

N,

(34I

i=1

where 2 is a Lagrange multiplier. 2*

Acta Pb?'s. Hung. Toro. X I I I . Fasc. 4.

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T . A . HOFFMANN

As we see from (35), E is a symmetrical function in all ki and so using (37) all ki are equal at the extremum. (34) gives now for the extremum

(3s)

N j+l

k i ~

and substituting this into (35): [

11

I ~(j A- 1)

E 1 = 2 f l ( j + 1)

sin

2(N §

1)

(39)

Introducing p instead of j with help of (36) we have E 1 : 2fin ( - 1 -_p P+

- 11" (40)

l_p ~rp+

N

si~T2 I --i-- -1-~) -~~ (40) gives one end of the range of variation of E. The other end of the range is given b y the end of the domain of the variables, i. e. with our convention that all k should be even, b y kl = k a . . . . .

kj=2,

ky+l=N--2j.

(41)

Substituting this into (35) we get for the other extremum of E:

E 2 = 2fil J - - 1 +

(42)

sin

7~

2(N-

2j + 1)

or expressed by p instead of b y j p

1

1 -- p

N

E 2 -- 2fiN

1

+ N sin

(43) zr

2~ l91~~ 1--p

+~}

The distribution of the vacancies in the chain is random and the energy of the system is established b y a suitable averaging. But E 1 and E z do not differ very much, if N is not too small, so that ir does not matter, in what w a y we perform the averaging. Thus we take the arithmetical mean of E 1 and E 2 as the average: Acta Phys. Hung. Toro. X I I I . Fasc. L

THE THEORY

E=

EI+E~ 2

391

OF MELTING

§

2fiN

2N sin 1--p B

(44) 1 N ~z{p + - 1 --- P N 2 sin 1--p P .§ 1--p

+

2{1 +

1 N

N

1 E x p a n d i n g (44) in aTaylor-series in powers o f - - - a n d N t e r m in the expansion, we have finally:

E =2fiN

retaining only one

1 -- 3p

P A2(1 - - p ) sin ztp zt(1 -- p) 2

(45)

1

N

zt

2 sin ~tp 2

4 s i n ~rp 2

This average energy can be interpreted accordiag to equation (4) in the following way. The n u m b e r of a t o m - a t o m bonds is in this linear case:

NAA =

~

i=~

(ki --

1) = N - - j -- 1 = N

1 -- 2p

(46)

1 --p

since each group of ki atoms contains k i - 1 a t o m - a t o m bonds. Similarly, each group has 2 a t o m - v a c a n c y bonds, one at each end, and so we have for the n u m b e r of the a t o m - v a c a n c y bonds

NAv=

j+l

~,'2=2(j 91 1)=2N i=1

P 1--p

~_ 1 N

(47)

Equation (4) is therefore in the linear case:

E=N( 1-2p 1--p

I }EAA ~-2N N

P ~ § --1 IEAv

1--p

N

.

(48)

Acta Phys, Hung. Toro. X I I I . Fasr. 4.

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T . A . HOFFMANN

Wc assume now that EAV, the interaction energy between an a t o m and a vacancy, does not depend on N and p. EAA, on the eontrary, depends essentially on both these variables. However, we can assume that in an ideal erystal, i. c. i f p = 0, the bonds are independent of the size of the crystal, i. e.

EAA(N, O) = EAA(OO,0).

(49)

With these assumptions we shall now compare (45) and (48) at the point p~0: 2

2

(1- -N--]EAA(~,O) + 9

2fll ~

1

N

[1-

~)]"

(50)

Since this equality holds for all N, it holds for the terms not containing

1

1

-- and for those eontaining 9 separately: N

4fl ,

EAA(o%0)_

--EAA(O~,O)+2EAv:--2fi(1--

(51) ~r2 ).

(51) is the well-known result for an ideal crystal. Eliminating (51) and (52) we get for EAV:

EAv__fl(4~

(52)

EAA(C~, 0) from

_ 1).

(53)

Substituting this value into (48) and comparing it now with (45), neglect1 ing powers of 9 higher than the first one, we have for EAa:

EAA = 2fl

P 2(1 -- 2p) sin ztp 2

+

P -4- 1 - 7 p 1 -- 2p ~r(1 -- 2p)

( 1

/ (1-- p)(2 + p)

N I

zt(1 -- 2p) 2

p(1--p) (1 -- 2p)2

~p(1 -- p)2 ctg z~p

(1 -- p)2 2(1 -- 2p) 2 sin z~p 2 .Acta Phys. Hung. Toro. X I I [ . Fase. 4.

+ 4(1 -- 2p) sin :rp 2

(54)

THE THEORY

393

OF bIELTING

We note t h a t in (54) N m e a n s the n u m b e r of a t o m s p r e s e n t in the chain. To c o n f o r m w i t h the previous n o t a t i o n (equ. (12)--(15)), where n was the n u m b e r of u n i t cells along an edge of the crystallite, we h a v e to express N in (54) in t e r m s of n in a w a y d e p e n d i n g on the crystal s t r u c t u r e .

5. Energies for the various crystal structures

W i t h the results of the last two seetions we can write d o w n the energies of the crystals as functions of n and p . I n the case of a simple cubic lattice the chains containing the nearest neighbours ate parallel to the edge of the cube and so the n u m b e r of a t o m s in one chain, N, is connected w i t h the n u m b e r of cells in the s a m e direction, n, b y N = n(1--p).

(55) 3

1

F u r t h e r , we h a v e to introduce - - i n s t e a d of n

V~

in (24) and

(25) using

e q u a t i o n (12). I n this w a y we o b t a i n b y s u b s t i t u t i n g (24), (25), (53) a n d (54) into (4) 1 a n d neglecting t e r m s of o r d e r - - o r higher, n 2

E = ~Az

1

p2 ._[_

1 -- 2p

(1 - - p)4/3 n0(1 - - 2p) 2

--

1

(1 - - 4p - - p 2 ) +

~r

(

1 - - 2p + 2p~

p(1 - - p) 2sin

1 -- 4p + 2p2 _

1

~rp 2

(1 - - 8p + 2p 2) - -

(56)

ztp(1 - - p)(1 - - 2p) ctg 7~p +

2 sin ztp 2

4 sin Jrp 2

for the s. c. lattice, where we i n t r o d u c e d no, the value of n at zero v a c a n c y concentration by 3

n o = nVl-p.

(57) Acta Phys. Hung. Toro. X l l l . Fase. 4.

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T.A. HOFFMANN

In the case of a body-centered cubic lattice the chains containing the nearest neighbours are parallel to the b o d y diagonal of the cube. The chains ate therefore of different length. As one can calculate easily, the average length 1 of a chain is ~ - t i m e s t h a t of the edge of the cube. Thus the average n u m b e r of atoms in the chain, N, is connected with the n u m b e r of cells along ah edge, n, by 2n

N =

3

(1 - p ) ,

(58)

where we have taken into account, t h a t the nearest-neighbour distance is V-~-times t h a t of the edge of the unir cell. So we have, r e p l a c i n g ~ 2 ~ A P ) by--

1

according to (13), substituting (26), (27), (53) and (54) into (4) and neg-

n

1 lecting terms of order ~-~, of tfigher:

E=flAz[p 1( 1(-p2"+" p )l~I(1-4p-p2)+ -2p n

2 s i n ~rp 2

3(1

p)41a

1 -- 4p 91 2p~ --

1

(1 -- 8p + 2p 2) --

2 n 0 ( 1 - 2p) 2

12~~2~ ~ 2 s~i n 2

(59)

~1 ~1 ~ct~)]4sin 2

for the b. c. c. lattice. Note t h a t (56) for the s. c. lattice and (59) for the b. c. c. 3 lattice differ only in t h a t the second term in the square bracket i s - - - t i m e s as 2 large in (59) as the corresponding term in (56). In the case of a face-centered cubic lattice the chains of nearest neighbours are parallel to the diagonals of the faces of the cube. The chains ate .4cia Ph?'s. Hung. Toro. N[I[. Fase. 4.

THE T H E O R Y OF MELTING

395

again of different length. A straightforward calculation shows t h a t the average 1 length of a chain is in this case ~ -times t h a t of the edge of the cube. The average n u m b e r of atoms in the chain, N, is therefore connected with the n u m b e r of cells along an edge, n, b y

N = n(1 - - p ) ,

(60)

where we have t a k e n into account, t h a t the nearest-neighbour distanee is 3

1 --P) r~-times t h a t of the edge of the unit cell. So we have, replaeing V4(1 A 1

b y - - aceording to (14), substituting (28), (29), (53) and (54) into (4) and neglectn

ing terms of order ¡

1

of higher:

E = flAzI 1 - -1- 2 p (P2A- zc1 (1 -- 4p -- p2) +

p(1 -- p) 2 sin ~rp 2

(1 -- p)4/a n0(1 _ 2p)2

(

1--2pA-2p 2 2 sin z p 2

1 -- 4p + 2p2 _

1

[

)

(1 -- 8p -4- 2p 2) --

7t

91

ztp(1 -- p)(1 -- 2p) ctg ztp 2 4 sin ~rp 2

w

(61)

~]

/1

for the f. e. c. lattice. Note t h a t (56) for the s. c. lattiee and (61) for the f. c. c" lattice ate entirely the same, e x c e p t t h a t the connection between A and n oecurring in these formulas is given b y (12) in the s. c. and b y (14) in the f. e. c. case. In the d i a m o n d lattice we cannot s e l e c t a linear chain transversing the block. However, we can consider b r o k e n chains as seen in Fig. 2. These chains contain the same n u m b e r of atoms as the chains in the f. c. e. lattice case. So the average n u m b e r of atoms in the chain, N, is connected with the n u m b e r Acta Phys. Hung. Toro. X I I I . Fase, 4.

396

T . A . HOFFMANN

Fig. 2. Sketch of the chains in a diamond lattice

o f cells a l o n g a n e d g e , n, b y t h e s a m e f o r m u l a (60) as in t h e f. c. c. l a t t i c e c a s e . 3

1;1 -A- P b y -n1- a c c o r d i n g t o (15), s u b s t i t u t i n g

Thus replacing r

(30), (31), (53) a n a

1 (54) i n t o (4) a n d n e g l e c t i n g t e r m s o f o r d e r n ~ o r h i g h e r , we h a v e :

~~~r 12~t~+ 114~ ~ 9 ~l:~~tsi 2~ (1 - - p)4,3

3 - - 11p + 6 p 2 - -

1

(5 - - 23p + 6p 2) - -

Yg

2no(1 - - 2p)~

~p ~rp(1 - - p ) ( 1 - - 2p) c t g - ~ --

2 - - 5 p + 6p 2 2 sin

~rp 2

for the diamond lattice. Acta Phys. Hurtg. Toro. X I I I . Fasc. 4.

+

2 sin

zrp 2

(62)

397

T H E T H E O R Y OF M E L T I N G

6. The mixing entropy of the crystal The entropy of ah atomic system can be built up of parts of various origin. These ate of two kinds. One part of the entropy is a function of temperature, varying relatively slowly. This part is independent, of quasi-independent of the various phases existing at the same temperature. The vibrational entropy for instance is of this kind. The other part of the entropy depends, on the other hand, significantly on the respective phases. This latter part may be only a very small fraction of the total entropy, but its change is much larger than the corresponding change in the other parts of the entropy in case of a change of phase. We consider here the vacancies as the causes of melting, thus itis obvious that we have to investigate that part of the entropy which has its origin in the vacancies. This is the same entropy of mixing which occurs in disordered alloys. In our notation this entropy is given by

S=klog

(A+V)! A! V!

,

(63)

where k is the Boltzmarm constant. In the Stirling approximation this gives S = k [ ( A + V)log(A-t-- V ) - A l o g A -

=Ak[

=--Ak

1 1 --p

log

1 1 --p

log(1--p) +

P 1 --p

Vlog V] = log

P ] 1 --p

(64)

1--p

where we have made use of equ. (3). Equation (64) gives the first approximation of the entropy only. Applying the next approximation in the Stirling formula to equ. (63) the second approximation is obtained which gives in the square bracket of equ. (64) a t e r m 1 1 proportional to ~ - , that is to %3 "As the whole calculation is carried out to the 1 first order in - - ordy, we can totally neglect in this approximation the changc n o

of the entropy with the block size. Similarly, the difference between the inner atoms of the block and the atoms on the surface is expressed by a term pro1 portional to ~-~, which we can again neglect in our approximation. n o

Acta Phys. Hung. Toro. X I I I . Fasc. 4.

T.A. HOFFMANN

398

7. The free energy of the block structure

In the preceding section we considered a block of the crystal consisting of a b o u t n 3 atoms. Now the question arises, w h e t h e r n o has of has not a finite value in the equilibrium state. In other words, the question is w h e t h e r in equilibrium the crystal has a finite block size of an infinite one. The straightforward way to decide this problem is to investigate the free e n e r g y of the system. I f one could show t h a t the free energy decreases b y b r e a k i n g the crystal into smaller blocks, the equilibrium state would be this latter. A n a t u r a l t r e a t m e n t of the problem would be to show t h a t the free e n e r g y of the crystal has a m i n i m u m at some finite value of n 0. There has been till now no possibility of showing this in general in the three-dimensional case. The present work is not able either to show this, since here all calculations ate 1 carried t h r o u g h in a p p r o x i m a t i o n s to the first power o f - - only so t h a t there no is no possibility of getting a finite extreme value for n 0. Instead, we show in a linear case t h a t the chain really has a m i n i m u m free e n e r g y if it is sliced into smaller chains of definite lengths and on the analogy of this we conclude t h a t the same is true in the spatial case. Let us suppose t h a t we have a linear chain of N atoms as in section 4. This chain be b r o k e n into j A-1 parts containing kl, k 2. . . . . kj+l a t o m s , respectively. Then equ. (34) holds and the t o t a l e n e r g y of the chain is given b y equ. (35). As the energy (35) depends on the sum of the energies of the j + 1 chains, the t h e r m o d y n a m i c probability, i. e. the n u m b e r of the micro-distributions is W =

N!

(j -f- 1)!,

(65)

kl! k2! ... kj+l! since the j § 1 chains m a y be interchanged. The free energy of the system is therefore from (35) and (65):

l-j+1

F=E--TS=E--kTlogW=2fl[i~=I

!

[

1 sin

~

--j--1

] -

-

2(kl §

J"

]

-- k T l o g N ! A-log(j-4- 1)! -- _~~'logk~ ! =

i=1

rj*l = 2flli=~1

- kT

Il o g F ( N

1

:r sin 2(kt A- 1)

--J--

(66) 1] _

l

+ 1) + l o g F ( j + 2) - - . ~ y l o g F ( k s + 1) ,

"i=l

where F(x) denotes the g a m m a funetion of a r g u m e n t x. .4cta Phys. Hung. Toro. X I I I . Fasc. 4.

]

399

THE THEORY OF MELTING

In the equilibrium state the energy has a minimum with respect to k , k 2, . . . , kj+i and with respect to j. First we consider the minima with respect to kl, k~ . . . . . kj+l. Here we have to solve the conditional minimum problem, where the conditŸ is given by equ. (34). We have thus in the usual manner, ir ~ is a Lagrange multiplier: ;g

0F

~_

ctg - 2(k~ + 1)

~/~

Oki

(k~+ 1)3

+

sin

2(ki+l) (67)

+ kT

~-/(ki) - - ]t :

( i = 1, 2 . . . . . j --[- 1),

O,

F(x +

where W(x) is the logarithmic derivative of the 1)-function. As the j + 1 equations in (67) are all of the same form, we easily obtain k i --

N

j+l

, (i = 1, 2,...,j + 1),

(68)

where we have made use of the condition (34). Substituting (68) into (66) we have: F = 2/5(j+ 1)

i

1

~t

sin

9

2{

j~l

1

-- 1 -+ 1}

(69)

-kT[l~176176

N~.+~+11]"

We seek the minimum of F given by (69) with respect to j, i. e. we have to determine j from the equation:

OF

-

1

2/~

2 ----+1 j+l

--1--

sin

sin 2{ ~ N j+l

-4-1/

2{ - - N - ~ j+l

+ 1)

(70)

--kTl~(j + l ) - l o g r { ~j +Nl

~N j+l

+1.} ~

(j + ~)

+ 1)+

m

j--~-] =0. Acta Phys. Hung. Toro. XITf. Fasc. 4.

400

T.A. HOFFMANN

Let us denote for the moment by N

j§

--n

(71)

the number of atoms in the short chains into which the long chain is broken. Equation (70) may then easily be solved for T and may be written 1 -- sin T =

ctg

28

2(n A- 1)

k

sin

2(n§

~

2(n + 1)2

2(n A- 1)

(72)

+nkU(n)--logF(n§

The temperature corresponding to any value of n may be easily calculated if we take into account, that the parameter N occurs only in one term in the denominator. The results of the calculation are shown in Fig. 3. To make the t/ 1000

5091 40~ 30~ 206

106

50 40

20

10

I £

Fig. 3. V a l u e s o f n for the m Ÿ

Acta Phys. Hung. Toro. X I I [ . Fasc. 4.

free energy at different temperatures f o r a linear chain built o f c o p p e r atoms

401

T H E T H E O R Y OF M E L T I N G

order of m a g n i t u d e of the t e m p e r a t u r e a n d n perceptible, we h a v e t a k e n in the Figure for fl the value - - 1 , 7 6 ev/mol. Considering the s i t u a t i o n at fixed t e m p e r a t u r e s t h e r e ate two points of the curves at each t e m p e r a t u r e . The lower point corresponds to a m a x i m u m of the free e n e r g y a n d the higher to the m Ÿ of the free energy, Moreover f r o m Fig. 3 one m a y see t h a t at not too high t e m p e r a t u r e s (lower t h a n 500 ~ K in the Fig.) each n corresponds to a v e r y n a r r o w t e m p e r a tute i n t e r v a l in a large i n t e r v a l of N. This means, t h a t a p p r o x i m a t e l y we h a v e a unique relation n ( T ) irrespective of the value of N. This would m e a n t h a t at a fixed t e m p e r a t u r e all chains are b r o k e n into small chains of lengths n(T) w h a t e v e r the length of the whole chain. These r e s u h s show t h a t in a linear chain the free e n e r g y has a m i n i m u m , ir the chain is b r o k e n into smaller chains. To m a k e the s a m e calculation for the p l a n a r of the spatial case would be, however, v e r y i n v o l v e d and so we restrict ourselves here to suggest t h a t b y a n a l o g y we m a y suppose t h a t the free e n e r g y is lowered b y the b l o c k f o r m a t i o n even in the p l a n a r of the spatial case. F o r the sake of simplicity we a s s u m e f u r t h e r t h a t the p a r a m e t e r of this block n o is i n d e p e n d e n t of the t e m p e r a t u r e . 8. T h e e n e r g y

--

entropy

curve

F o r m u l a s (56)--(62) a n d (64) allow us to i n v e s t i g a t e the b e h a v i o u r of the e n e r g y and e n t r o p y as fimctions of the v a c a n c y c o n c e n t r a t i o n . (56), (59), (61) and (62) can be w r i t t e n in a c o m m o n f o r m : E

e(p, no) -- - -

--f(p) + --

fl A z

1

g(p),

(73)

no

where f ( p ) is for all four lattice t y p e s the s a m e :

f(P)--

1

2p

p2§

--

(1--4p--p2)

§

~

p(1--p) 2sin ~p 2

,

(74)

g(p) is the same for the s. c., the b. c. c. a n d the f. c. c. lattice:

f

g(P) --

( l - - P ) 4'3 | (1 - - 2p)~

1 - - 4p A- 2p 2 . . . . 1_

1

(1 - - 8p + 2p 2) - -

(75) 1 - - 2p -4- 2p2 + 2 sin ~tp 2

~p(1 - - p)(1 - - 2p) ctg

~p 2

4 sin ztp 2 ~4cta Phys. Hung. Toro. X I I I . Fasc. 4.

402

T . A . HOFFMANN

We r e m a r k t h a t in the b. c. c. case n o has to be replaced b y

2n 0

3

9 For the dia-

m o n d lattice we have:

g(p)-

11p+6p ~ - 1

2(1 - 2z,)~ [

(5--23p+6p 2 ) -

7g

(76) 2 -- 5p + 6p 2

~rp(1 -- p)(1 -- 2p) ctg _2~r__ 2

+

2 sin ~p 2

2 sin ztp 2

For the sake of completeness we re-write here the formula for the e n t r o p y too, s(p) --

S Ak

_

log(1--p)

P

logp.

(77)

1 --p

Our t h e r m o d y n a m i c functions ate now the following: the internal e n e r g y of the s y s t e m e ( p , no), a funetion of the v a c a n c y eoneentration, p, only, if n o is held c o n s t a n t ; the e n t r o p y o f the system, s(p), again a funetion of p only if we disregard the slow change of the e n t r o p y with t e m p e r a t u r e . In our whole t r e a t m e n t we deal with the case of zero pressure. This is certainly admissible in the solid state and we suppose t h a t we do not m a k e a large error if we hold this supposition to be valid also for liquids in the n e i g h b o u r h o o d of the melting point, i. e. we t r e a t isobar melting at zero pressure. The volume of the crystallite can be expressed b y (12)--(15) in the form

-

V0

1--p

,

(78)

where v 0 is the volume at zero v a c a n c y cortcentration, i. e. at absolute zero t e m p e r a t u r e . (73) and (77) give in addition the internal energy and the e n t r o p y o f the system. The general t h e r m o d y n a m i c relations [7] give now the following results. The absolute t e m p e r a t u r e is given b y Oe

T = ,--~-/P=o

Os Op

Acta Phys. Hung. Toro. X I I I . Fasc. 4.

(79)

403

T H E T H E O R Y OF M E L T I N G

T h e specific h e a t at c o n s t a n t zero pressure has ah a d d i t i v e c o n t r i b u t i o n f r o m E (the m a j o r p a r t of it originating f r o m the v i b r a t i o n a l m o t i o n of the a t o m s , supposed to be t h e same in the solid a n d in the liquid phase). This contrib u t i o n is: 0E

[ OE }

Op = o [ oF. ~ O s ) ~p t 0 p / o e J

= ~cp = ~ -

.=0

(80) Op I-

= Ak

02 e

Os

Oe

02 s

Op2

Op

Op

Op2

Equs. (78)--(80) ate sufficient to describe the b e h a v i o u r of the s y s t e m . F o r a specified n o we can now plot the e--s curve. We g i r e h e r e a someWhat detailed discussion of the b e h a v i o u r of this curve, since it is of v e r y great i m p o r t a n c e for the considerations f u r t h e r below. According to (79) the absolute t e m p e r a t u r e is p r o p o r t i o n a l to the tangent of the e--s curve. We h a v e to note, t h a t the n o occurring in the f o r m u l a s a f t e r (73) are the same as the n o before this f o r m u l a b u t w i t h a reversed sign. This means t h a t one m u s t not a t t r i b u t e too great an i m p o r t a n c e to the i n t e r p r e t a t i o n of n o as the n u m b e r of cells in a specified direction. Indeed, the following discussion shows t h a t n o occurring in the f o r m u l a s a f t e r (73) t a k e s on positive values t h r o u g h o u t a n d this would m e a n a n e g a t i v e n u m b e r of cells in the s y s t e m . For the m o m e n t let us disregard t h e m e a n i n g of n o a n d m a k e the analysis with n o a s a simple p a r a m e t e r w i t h o u t a n y specific meaning. F u r t h e r we h a v e to note t h a t (79) holds only, if we suppose fl to be i n d e p e n d e n t of the t e m p e r a t u r e . This is n o t the case in reality. I n spite of this we m a k e our calculations i n d e p e n d e n t of the v a r i a t i o n of fl a n d for the sake of simplicity we m a k e a correction in t h e last step of the discussion only. As fl is n e g a t i v e , only those p a r t s of the curve are feasible in which Se 8s Se 8s is also n c g a t i v e . T h e p a r t s w i t h ~ ¥~245 positive ate not points o f t h e r m o i

-Jt-

ae Ÿ 9s

d v n a m i c equilibrium. Moreover, i f t h e curve has a p a r t , wllere ~ - : ~ - i s positive, " op/' ~p it m u s t h a v e more t h a n one m i n i m u m (the point s = 0 is e e r t a i n l y a m Ÿ belonging to p = 0). This would m e a n t h a t the crystal has a m e t a s t a b l e s t a t e at some v a e a n e y c o n e e n t r a t i o n different f r o m O. This we should exelude - - at least for o r d i n a r y p u r e metals. Sueh ah exelusion imposes the eondition 3

Acta Phys. Hung. Toro. X I I I . Fase. 4.

404

T.A.

HOFFMANN

~176191

0p

~ 0

(81)

along the whole curve. As Os

1

ap

(1 -p)~

logp

(82)

is a l w a y s positive, t a k i n g into a c c o u n t (73), this m e a n s , t h a t the i n e q u a l i t y Of + 91

1 no

0g < 0 8p

(83)

0 g ls~ n e g a t i v e e v e r y w h e r e , this con8p a f . ls n e g a t i v e (small p - v a l u e s ) . where - 0t,

should hold along the whole curve. Since - dition is fulfilled in the whole range,

of. ap ls

I n the range, w h e r e - - -

positive, n o m u s t be s u b j e c t to an additional condi-

tion, n a m e l y Og

n~ ~

0p

(84)

of @ everywhereinthisrange.--

0g /Of 0p / 0p is a function of p, which has a m i n i m u m in

this range. Therefore the condition (84) is fulfilled e v e r y w h e r e , if it is fulfilled at the m i n i m u m . The curve is given for the cubic a n d d i a m o n d t y p e lattices in Fig. 4. This curve has a m i n i m u m at some value of p . According to (84) this m e a n s , t h a t there is a m a x i m u m value for n o, n a m e l y the value at this m i n i m u m . This is n o = 11,08

(85)

no =

(86)

for the cubic lattices and 10,75

for t h e d i a m o n d lattice. I f one disregards the sign of n 0, one obtains, using equ. (12)--(15) and (57), t h a t the m a x i m u m value of a t o m s in a b l o c k of the crystallite is A :

1360

(87)

A = 2721

(88)

for the simple cubic lattice,

_h'to Phys. Hung. Toro. X I I L

Fase..I.

THE THEORY OF bIELTING

405

for the b o d y - c e n t e r e d cubic lattice, A = 5441

(89)

for the face-centered cubic lattice and A = 1242 (90) for the d i a m o n d lattice. F u r t h e r investigation of the e - - s curve shows t h a t f o r a certain range of n o it has two inflexion points, whereas for other values of n o it has no n o cubic

no d/arnofld

\

\ \

~/ //1,

\

\

\ J

11 0,35

0,36

0,37

0,38

0,39

S o.~o

o,41

0,~2 p

Fig. 4. The maximum value for no for the cubic and for the diamond-type lattices

inflexion points at all. I f the curve e - - s has two points of inflexion, it is possible to find a c o m m o n t a n g e n t to two points of the curve, whereas in the other case each t a n g e n t has only one point of c o n t a c t on the curve. We shall see later t h a t a necessary condition for the existence of a definite melting point is the occurrence of a double t a n g e n t to the e - - s curve. I n the case of no inflexion points we conclude t h a t the solid-liquid t r a n s f o r m a t i o n is a continuous one e x t e n d e d over a wide range of t e m p e r a t u r e s . This p r o p e r t y characterizes the a m o r p h o u s phases. The of n o not more and inflexion 3*

calculations show t h a t the two inflexion points existing in the case m u c h smaller t h a n given in (85) and (86) resp., a p p r o a c h each other more with decreasing n 0. We can find a value for no, where the two points coincide (where the second and third derivatives of e with Acta Phys. Hung. Toro. X I I I . Fasc. 4.

40~

T . A . HOFFMANN

respect to s v a n i s h at the s a m e p) and if n o is less t h a n this value, the c u r v e has no inflexion points at all. This value of n o is for the cubic lattices n o = 4,5

(91)

n o = 4,1.

(92)

and for the d i a m o n d lattice

Using equs. (12)--(15) and (57) we get for t h e m i n i m u m value of a t o m s in the c r y s t a l for a n o n - a m o r p h o u s melting process A = 91

(93)

A = 183

(94)

for the simple cubic lattice,

for the b o d y - c e n t e r e d cubic lattice, A = 365

(95)

for the f a c e - c e n t e r e d cubic lattice and A = 69 (96) for the d i a m o n d l a t t i c e . T h e final conclusion can be d r a w n t h a t the solids h a v i n g n o r m a l disc o n t i n u o u s melting p r o p e r t i e s h a v e subcrystallites of the size of 9 1 - - 1 3 6 0 a t o m s for the simple cubic lattice, of 183--2721 a t o m s for the b. c. c. lattice, of 3 6 5 - - 5 4 4 1 a t o m s , for the f. c. c. lattice a n d of 6 9 - - 1 2 4 2 a t o m s for the diam o n d lattice. This r e l a t i v e l y b r o a d range for the n u m b e r of a t o m s gives an u n c e r t a i n t y in the calculation of the m e l t i n g t e m p e r a t u r e , since each subc r y s t a l l i t e size is c o n n e c t e d w i t h a well-defined m e l t i n g point. T h e results of equs. (87)--(96) seem to give e x t r a o r d i n a r i l y low values for the size of the crystallite blocks. The idea of such intrinsic c r y s t a l blocks is f a m i l i a r in the l i t e r a t u r e . A p a r t f r o m the i n v e s t i g a t i o n s of G. W. Sr~.WA~T [8] a b o u t t h e c y b o t a c t i c groups t h e r e is some t h e o r e t i c a l and e x p e r i m e n t a l evidence for the existence of small intrinsic blocks, e v e n in the single crystals. M. BoRN [9] shows t h a t the consequent q u a n t u m m e c h a n i c a l t r e a t m e n t of the d y n a m i c s of a c r y s t a l lattice leads to difficulties which can~ot be solved unless an inner b l o c k s t r u c t u r e is a s s u m e d in the crystal. BoRN d e t e r m i n e s a critical n u m b e r of a t o m s , z 0, which is the m a x i m u m n u m b e r of a t o m s on one edge of a h ideal crystallite b l o c k at absolute zero t e m p e r a t u r e . He gets for this value z 0 = 500 for a l m o s t a n y s u b s t a n c e . For finite t e m p e r a t u r e the n u m b e r of a t o m s on one edge of a crystallite is according to BORN Acta Phys. Huag. Toro. X I I [ . Fase. 4.

THE THEORY OF MELTING

0 z = z0 lh - - 2T

,

407

(97)

where 0 is the Debye temperature of the substance. This gives e. g. for gold with 0 = 170 ~ K, at the melting point T m ---- 1336,2 ~ K about z = 30, which is of the same order of magnitude as the value (85). However, the meaning of the block in BORN'S work is not the same as in the present paper. Our block size namely is independent of the temperature, whereas BORN's block decreases with increasing temperature, further our block contains vacancies too, whereas BOR~'S block is ah ideal one, without any lattice defects. R. Fi~RTrI [10] has shown that any theory of melting, which takes into account ideal crystals and tries to explain the melting by the anharmonicity of the vibrations, gives unsatisfactory results for the melting point. He draws the conclusion t h a t the intrinsic block structure is essential f o r a theory of melting. In another paper [11] he connects BRAGr theory of the mechanical strength [12] with the theory of melting. BRAGr develops in his paper the theory of the mechanical strength of solids assuming the existence of ah internal block structure in the crystal and F~RTH, connecting the melting temperature with the existence of such a block structure concludes t h a t the evolution of this block structure is a strueture-insensitive, intrinsie property, as the melting point is also independent of the history of the sample etc. In contrast to this, the micro-crystalline mosaic structure (dealing withlarger micro-crystals than those mentioned above) is a structure-sensitive property depending very strongly on the previous mechanical, thermal, etc. treatment of the sample. As experimental evidence for the existence of subcrystallites of very small dimensions we may quote the investigations of W. A. WOOD [13]. WOOD and bis collaborators made X-ray structure investigations on quenched metals. They found t h a t deforming these metals the crystallite blocks can be broken up into smaller subcrystallite blocks, which have a well-defined minimal size, characteristic for the metal. This supports the view that this subcrystallite block structure is ah intrinsic property of the solid. Recently, SIMMONS and BALVFFI [14] have shown experimentally the existence of the subcrystallite blocks in aluminium and in silver. Considering the liquid state there is experimental evidence, evaluated by the E5tvi}s equation, showing t h a t there is a tendency of association in liquid metals near the freezing point [15]. A similar tendency of association was observed in melts of selenium by Prof. RICI~T~~ in Stuttgart with an X-ray diffraction teehnique [16]. In our opinion the small subcrystallite blocks given here do not constitute different units in the crystallite, observable by optical of X-ray methods, because the lattice planes of neighbouring subcrystallites are parallel and only a minute difference in the distance of planes in the same subcrystallite and of Acta Phys. Hung. Tom. X I I I . Fase. 4.

408

T . A . HOFFMANN

planes in neighbouring subcrystallites (at the surface) constitutes the deviation f r o m a larger crystallite. The subcrystallites m a y indicate in this sense t h a t the lattice distance at these planes is somewhat larger t h a n the n o r m a l one. This causes some difference in the binding e n e r g y and it is this e n e r g y which was t a k e n into account in our calculations. This s o m e w h a t larger lattice distance however, does, not affect a n y essential change in the crystal dimensions and thus it is not always observed b y the usual X - r a y techniques.

9. The melting properties I f we take a value for n o between (91) and (85) or between (92) and (86), the e - - s curve has the f o r m shown in Fig. 5. On the left of the Figure the tangent of the curve is parallel to the s-axis, i. e. the absolute t e m p e r a t u r e at this E

0E

!

milpoln(q "Mr /

.~~

,whd

~

i

pho~e

p,

tgc~=7"mi#,ng

,-;- ~[

fiqu/d

)*-3

phQse

Fig. 5. Typical E - - S curve for intermediate no

point is 0. As we proceed towards higher e n t r o p y values, the t a n g e n t becomes steeper, i. e. the t e m p e r a t u r e increases. Finally we reach a point, where the t a n g e n t to t h e curve has a second point of c o n t a c t on the curve. In this case b o t h points of c o n t a c t r e p r e s e n t phases at the same t e m p e r a t u r e (determined b y the c o m m o n t a n g e n t ) , the lower point representing a phase with a lower e n t r o p y , the higher one a phase with higher e n t r o p y . We a t t r i b u t e to the phase with the lower e n t r o p y the solid state and to the phase with the higher e n t r o p y the liquid state. The slope of the double t a n g e n t gives the melting t e m p e r a t u r e , the difference of the abscissae of the two points of contact the melting e n t r o p y , and the difference of their ordinates the latent heat of melting. Acta Phys. Hung. Toro. X I I I .

Fase. 1.

THE THEORY OF MELTING

409

By heating the solid the temperature increases and finally we arrive at the melting point, Pro" Here there are two possibilities for the system: By receiving heat the system wanders along the straight line PmP/, i. e. the temperature does not change until reaching the point P/. In the meantime the entropy of the system increases by AS. The phase with a higher entropy corresponds to the liquid phase, i. e. mehing occurs in the interval PmPi of the straight line. The second possibility is that the system proceeds on its way along the curve in the interval PmP/. In this case the temperature further increases right up to the point Pro with a relatively small change in the entropy of the system. This way would correspond to the overheating of the solid. We shall revert later to the choice between these alternatives. Right to the point PI the temperature is higher than the melting temperature and the system is in a state with higher entropy, in the liquid state. With decreasing temperature arriving at P/ we again have ah alternative. The system could continue along the straight line P/Pm and in this way ordinary freezing would happen at the melting temperature, of the system could proceed alortg the curve P/Pm"In the latter case the temperature would further decrease going left from point P / w i t h a small change of the entropy. This corresponds to the undercooling of the liquid. Presently we analyze under what circumstances the two alternatives occur. In the case of increasing temperature starting from the solid state it is clear t h a t the entropy change, accompanying the absorption of a certain heat energy, is larger in the case of melting than in the case of overheating. This may be seen in Fig. 5, since the double tangent lies always on the right side of the curve. To judge which of the two alternatives will occur, we have to corLsider the entropy change of art adiabatic system, namely the total system consisting of the melting (or overheating) solid and of the heating medium. According to the second law of thermodynamics, in a= adiabatic system always the process with a higher positive entropy change will be realized and never that with the lower one. Now heating is accomplished by transmission of the energy in some way from a system of higher temperature to one of lower temperature. Usually dE the entropy change of the heater system is then ~ ,where Tis the higher temperature of the heater and dE the transmitted energy. As the temperature of the heated system is always less than that of the heater system, the entropy chailge in the heated system is always larger than t h a t of the heater system and thus the net entropy change is always positive. If we use a specific heating system, its negative entropy change determined by the heater, will be nearly the same in any case, whereas as we have seen above, the positive entropy change of the heated solid will be larger in the case of melting than in case of Acta Phys. Hung. Toro. Xl11. Fasc. 4.

~10

T . A . HOFFMANN

overheating. So we conclude t h a t by normal t herm odynam i c heating melting occurs in all cases and never overheating. In the case of decreasing temperature starting from the liquid state the (negative) entropy change accompanying the development of a certain heat energy is in the case of freezing always less t h a n in the case of undercooling, since the double tangent lies always right to the curve and so the entropychange on the straight line is less than t hat on the curve, reaching in the two cases the same ordinate. In a cooling process the cooler system absorbs heat energy from the cooled

dE

system. The entropy of the cooler increases therefore by an am ouat ~ - , where T is the lower temperature of the cooler. The ent ropy change of the total system is therefore always positive, as it has to be. At first sight the freezing procedure seems to be t hat with a larger ent ropy change in contrast to the undercooling procedure, since in this case the negative change in entropy is smaller. The detailed investigation of the system shows, however, the following situation. To examine the behaviour of the melting or the freezing system during the process, we proceed on the straight line connecting Pro with Pf. Any point of this line corresponds to a state, where a certain amount of the whole substance is in the liquid phase and the remainder in the solid phase, both at the same temperature. In Pro 1 0 0 ~ is in the solid phase and 0~o in the liquid one and in P j just reversed. The system needs a small, but in any case finite time to get from Pro to P ! or from Pf to Pro. In the case of melting the presence of the heat needed for the process is the only necessary condition for its realization. Therefore this can be achieved homogeneously in any part of the system, however small this part is chosen. In this case the total entropy of the system discussed above m ay be regarded as the q u a n t i t y determining the process. In the case of freezing, however, the situation is somewhat ahered. In this case the possibility of extraction of heat is only one necessary condition for the realization of the process. The other one is t h a t the particles should reach their regular positions in the crystal. This is connected with the formation of crystallization grains. In the liquid just over the melting point there exist already some such grains distributed relatively uniformly over the whole liquid. Considering this we see t hat these grains have a finite average distance from each other. I f we pick out of the whole system, these grains only, without the surrounding grain-free parts of the system we must conclude t h a t in the neighbourhood of the grains all the conditions for freezing ate given, whereas in the grain-free regions t hey ate not. Any point of the straight line PmP! corresponds therefore to a state, where some percents of the total system ate already frozen and the others ate not. But this is achieved in such a way t h a t the frozen parts ate concentrated on the grains and reversed. But ir we Acta Phys. Hung. Toro. X I I I . Fasc. 4.

T H E T H E O R Y OF MELTING

~] |

pick out a frozen region, this region is not adiabatically shut off from the other parts of the system and therefore the energy balance cannot be made for this part alone. However, the entropy of the individual parts of the total system can be discussed. This discussion shows that the entropy change in the frozen region of the system must have a very large negative value, whereas in the neighbouring undcrcooled regions ir has less negative values, so that in total the entropy change (together with t h a t of the cooler) is positive. That would mean that in the regions of the crystal grains the process is - - at least locally - not submitted to the law of largest positive entropy change. We conclude therefore, that the natural process Inust be that of undercooling. The situation is somewhat ahcred, ir we provide in some other way for the fulfilment of the second condition for freezing. We also have to furnish the freezing system with the necessary atoms in the proper positions. This is made easier if the atoms have a larger chance of finding the right place in the system, either by stirring of shaking. These operations supply always a positive entropy change and so the large negative entropy change by freezing can be counterbalanced, and freezing accomplished. Undercooling is therefore possible only under totally undisturbed conditions. According to the above, undercooling can be accomplished in principle down to a temperature which corresponds to the tangent in the upper point of inflexion. Therefore if the crystal structure and the binding energy are given, we have two relations between the three quantities: virtual subcrystallite size (n0), mehing temperature (Tm) and maximum undercooling temperature (T,c). Ir any one of the three is given, the other two can be calculated. A first approximation of the mehing temperature can be calculated with the assumption t h a t the substances may be undercooled right down to the absolute zero temperature. This assumption is not justified experimentally, but i t i s a well-known fact that sometimes a system can be undercooled even down to very low temperatures. The condition for such an undercooling is t h a t the tangent in the upper inflexion point is horizontal. The condition for the horizontal tangent is et

-- 0,

(98)

SP

where the sign denotes differentiation with respect to p. The condition for the inflexion point is similarly e" s" -- e" s"

= 0.

(99)

S'3

Sinee s'=/= 0, these give the simultaneous conditions Acta Phys. Hung. Toro. X I I L

Fasc. 4.

4 12

1'. AI HOFFMANN

e' ---- 0

(100)

e" = 0.

(101)

and

According to (73) equs. (100) and (101) can be w r i t t e n in the forro 1 f ' -f . . . . g" = 0

(102)

no

and 1 f " A - - - ~ , -" ---=0 9

(103)

n o

E l i m i n a t i n g n o f r o m (102) a n d (103) we h a v e g"f" --g'f"

---- 0.

(104)

As we s t a t e d a f t e r equ. (84), the curves of Fig. 4 give the n o r e s u l t i n g f r o m (84), ir we use t h e r e t h e e q u a t i o n sign i n s t e a d of the i n e q u a l i t y . (104) d e t e r m i n e s the m i n i m u m p o i n t of these eurves a n d (102) the values of n o c o r r e s p o n d i n g to these m i n i m u m values. These are the values (85) and (86); t h u s we conclude t h a t the a s s u m p t i o n of the possibility of tmdercooling to t h e a b s o h t e zero t e m p e r a t u r e implies the choice of the largest possible values of n 0. Our a s s u m p t i o n m e a n s Tuc = 0. Finally, T m can be c o n s t r u c t e d f r o m the e - - s d i a g r a m s with these m a x i m u m n 0. As we h a v e s t a t e d in section 7, to be e o n s e q u e n t we would h a v e to determine n o b y minimizing the free energy of the s y s t e m with respect to the variation of n 0. This p r o c e d u r e would be, h o w e v e r , v e r y involved, so t h a t we do not follow this w a y . I n the p r e s e n t t r e a t m e n t we a p p r o x i m a t e d b y t a k i n g into 1 a c c o u n t the d e p e n d e n c e on n o only to the first p o w e r o f - - in a p o w e r series rt 0

1 in powers o f - - . So in this a p p r o x i m a t i o n the m i n i m i z a t i o n w i t h respect to n o no is meaningless. Consequent m i n i m i z a t i o n could be carried out in a higher a p p r o x i m a t i o n only. T h a t is w h y we h a v e followed the i n c o n s e q u e n t w a y of d e t e r m i n i n g n o f u r t h e r a b o v e f r o m some o t h e r considerations. Fig. 6 represents the e - - s curves for the cubic a n d for the d i a m o n d s t r u c t u r e s c o n s t r u c t e d w i t h t h e m a x i m u m n o values (85) and (86). The melting e n t r o p y is the difference of the abscissae of the t w o points of c o n t a c t of the double t a n g e n t . According to (77) we m a y e v a l u a t e this value in cal/molgrade. I n T a b l e I we gire the e x p e r i m e n t a l values of the m e h i n g e n t r o p y t o g e t h e r w i t h the theoretical values calculated here. .4eta Phys. Hung. Toro. X I I I .

Fase. 4.

THE THEORY

OF MELTING

413

Table I

Calculated and experimental mehing entropies in cal/molgrade AS experimental

AS

calculated

Li ............

2,43

~a

...........

1,70

K

...........

Ag . . . . . . . . . . . Au . . . . . . . . .

1,72 1,70 1,66 2,29 2,19 2,27

average . . . . . .

2,00

2,16

C (diamond) ..

--

2,28

Rb . . . . . . . . . . Cs ............ Cll

...........

2.16

Table I shows t h a t there is a connection between the crystal structure and the melting e n t r o p y . The melting entropies of the b. c. c. alkali metals (except Li) are fairly c o n s t a n t and so ate the melting entropies of the f. c. c. noble metals. The exception, Li, shows p r o b a b l y a transition to molecular binding and this explains its high value. The difference in the b. c. c. and the f. c. c. structures c a n n o t be given b y the present simplified t r e a t m e n t . The m e h i n g t e m p e r a t u r e can n o w be calculated k n o w i n g the proper values of/~. Here we deal only with the simphficd t r e a t m e n t outlined with the a s s u m p t i o n of the possibility of undercooling to the absolute zero. In a following paper we shall discuss the complete connection between n o, T m and T~ c. I n our p a r t i c u l a r case, Tuc ~ 0, we now have to fix the values we shall use in the calculations. For an ideal crystal HOr•MANN [17] could correlate t h e / / z - - values with the binding e n e r g y of the crystals. According to this we have in our n o t a t i o n : E~ = 2,0048 ~~z,

(105)

where E b is the binding energy per a t o m and we write in our n o t a t i o n / ~ z for in the paper quoted. The A/30z - - values determined from the experimental values of the binding energies at r o o m t e m p e r a t u r e are given in Table I I . The experimental values were t a k e n from the compilation of BICHOWSKY and

RossiNi [18]. The values of Table I I are r o o m t e m p e r a t u r e values and since most of the melting t e m p e r a t u r e s ate far from r o o m t e m p e r a t u r e , we are for this reason compelled to make a correction in the /~ value. This is in contradiction Acta Ph~s. Hung. Toro. XITI. Fase. 4.

T.A. HOFFMANN

414

Table II .4fioZ values in kcal/mol according to (97) Li

19.45

Na

K

Rb

Cs

Cu

Ag

Au

12,92

9,88

9,43

9,38

40,50

33,92

45,89

[

C (diamond)

84,80

to our previous a s s u m p t i o n of c o n s t a n t ti, b u t the error m a d e in this w a y is smaller t h a n t h a t c o m m i t t e d b y using the fl0-value for r o o m t e m p e r a t u r e . To establish the t e m p e r a t u r e correction we r e m a r k t h a t fl is essentially the e x c h a n g e integral of two n e i g h b o u r i n g a t o m s (see e. g. reference [5]). T h e w a v e f u n c t i o n oecurring in this exchange integral is of the forro ~,~e

(106)

-rr ,

where 7 is a c o n s t a n t , the Slater e x p o n e n t . T h e exchange integral is therefore in a crude a p p r o x i m a t i o n of the f o r m fl ,~ e -2ra ,

(107)

where d is the distance o f t w o n e i g h b o u r i n g nuclei. If/30 is the exchange i n t e g r a l in the case, where the two n e i g h b o u r i n g nuclei are at the distance d 0, we conclude f r o m (107): /3 = q e - z : ' ( a - a , )

.

(108)

N o w the t e m p e r a t u r e correction consists in the correction of fl for the change of the i n t e r a t o m i c distance caused b y the change of t e m p e r a t u r e . T h e l a t t e r is connected with the linear expansion coefficient, a of the s u b s t a n c e b y d - - d o --~~d o. T -- T O

(109)

(108) can be written with the aid of (109) as fl ~

floe-2;,,,a ( r - r , )

(110)

The e x p o n e n t in (110) is so small t h a t we can e x p a n d the e x p o n e n t i a l in powers of the e x p o n e n t in a Taylor-series and neglect th~ t e r m s of orders higher t h a n the first one. So we h a v e instead of (110): fl = rio [1 -- 27ado(T - - To) ] . Acta Phys. Hung. Toro..u

Fasc. .1.

(111)

THE THEORY OF MELTING

415

W e h a v e for the m e l t i n g t e m p e r a t u r e using (73) a n d (77) dE . . ds

Tm.

.

A f l z de . . Ak ds

Aflz Ak

tge =

(112) -

[1 - - 2 y a a o ( T , , , - T o ) ] t g e .

Afl~ Ak

-

F r o m (112) we c a n express T m a s follows:

A/? o z (1 + 2yad o To) t g e T., -~

R

~ R 1 -[- A ~ ' ~

R

,

(113)

9 2~,r~do t g e

w h e r e R = A k is t he gas c o n s t a n t , R = 1,9864 c a l / m o l g r a d e . F o r t h e cubic c r y s t a l s wc h a v e fr o m Fig. 6 t g e =- 0 , 0 8 7 5 ,

(114)

0.48

0.•

cub~c-

tgc=0,0.876 po= ~46% A.s=l,Og

I

d/amond: /g E =0.,0.904 flo= 0.8 % A,s=~i5 0,62

I

0,5~

r----

0,56

,

0,68 . . . .

,tŸ237 ,~

0,60

0.62--

0.64

0.66

0,68

r

/~/"~/ dlomond

0.70

0.2

F i g . 6. e - - s

0,4

a~

i I

0.8

~o

1,2

Z4

1.6,,'

curve for the m a x i m u m n0-values for cubic and d i a m o n d s t r u c t u r e s Acla Ph?'s. Hung. Toro, X I I I . Fasc. 4.

41{5

T.A.

HOFFMANN

and fora diamond-type erystal t g e = 0,0904.

(115)

I n T a b l e I I I we s u m m a r i z e t h e v a l u e s of t h e e o n s t a n t s n e e d e d t o e v a l u a t e T m f r o m (113) w i t h T O :

293 ~ K (20 ~ C).

Table Ill The Slater exponent, ~, the thermal expansion coefficient at room temperature, a, the nearestneighbour distanee at room temperature, do, and the value of 2yctdo 7 {l/A)

a ( 1 0 - t/grade)

do(~ )

3,040

4,3315

3,707

7,2942 8,7134

Li

...........

1,2283

~a

...........

K

...........

1,3857 1,1236

58 71 84

1,0393 0,9898

90 97

..........

1,8896 1,7479 1,6647

16,5 18,7 14,2

C (diamond) ..

3,0718

Rb . . . . . . . . . . Es ............ Cu

...........

Ag . . . . . . . . . . . Al1,

1,18

2~.ad, (10 '/grade)

4,616 4,867

9,1049 10,0600

5,239 2,553 2,885 2,878

1,5920 1,8860 1,3606 0,1116

1,540

T h e m e l t i n g t e m p e r a t u r e s c a n n o w be c a l c u l a t e d u s i n g (113), (114), (115) a n d t h e d a t a of T a b l e I I a n d I I I . T h e s e v a l u e s a n d t h e e x p e r i m e n t a l ones a t e g i v e n i n T a b l e IV. Table I r

Caleulated and experimental melting points T,n ealeulated ( K ~

T m e x p e r i m e n t a l {K ~

Li

............

704,3

575,0

Na

............

K

............

488,1 396,O 381,7

370,9 336,7 312,2 301,7 1357,2 1233,7 1336,2

Rb . . . . . . . . . . . ES . . . . . . . . . . . . . Cu

............

Ag . . . . . . . . . . . . Au

............

C (diamond) . . .

Acta Phys. Hung. Toro. X I I I . Fase. 4.

377,8 1454,3 1230,3 1655,0 3712,0

3973

THE THEORY OF MELTING

d17

The next approximation in this theory would take into account the finite undercooling. However, as s~ffficient experimental data for the maximum undercooling temperature are not available, the following discussion seems to be of use. From the known melting temperature values we can calculate the virtual subcrystallite size and the m a xi m um undercooling temperature at the same time. With these data the maximum undercooling temperature can be compared with the available lmdercooling temperatures. On the other hand, if we know the maximum undercooling temperatures exactly enough, we can calculate from these values the melting temperature and the virtual subcrystallite size. These calculations will be given Ÿ anotber paper, however, we can state, th at the melting points given here for Tuc : 0 will in any case inerease in the case Tuc > O. One can see from Fig. 6 and equ. (77) t hat the vacancy concentration p increases at melting from a value of 1,5--2% to a value of about 40%. This would mean according to (78) a similar change in the volume at mehing. However, this change in volume seems to be too large and the high percentage of vacancies in the liquid phase calls also for an explanation. The high vacancy concentration would mean in a linear chain the total disintegration of the chain. In the planar, of even in the spatial case, however, tkis high vacancy concentration does not bring about the disintegration of the system. The remaining atoms, namely, m ay preserve a skeleton, which contains a v ery high number of vacancies and which aH the same does not fall into pieces. The reason for this lies in the high coordination number arising in the planar and spatial cases. The large increase in volume on account of the high vacancy concentration is partly compensated by the local shortening of the nearest-neighbour distance in the case of diminishing coordination number. The experimental data for the nearest-neighbour distance in metal and in molecules built up from two metal atoms show for instance for the alkali metals t hat the latter is about 0,85 tbat of the former. This is connected with the fact t hat in the metal the coordination number is 8, wbereas in the molecule ir is 2. This shortening would mean a decrease in the volume, i. e. 0,853 ~ 0,614, i. e. of about 40%. In the case of mehing the effective decrease in the coordination number is not so great as the decrease from 8 to 2, but it is still considerable, the decrease in volume will be therefore less than 40%. This and the increase caused b y the large vacancy concentration can explain the 10--15~/o increase in volume at melting. We conclude with the assertion t h a t the t heory describes quite correctly the observed anomalies in the specific heat of the alkali metals near the melting points. This will be discussed in anotber paper. Veta Phys. Hung. Toro. X I I I . Fase. 4.

418

T.A. HOFF~ANN 10. Acknowledgements T h e a u t h o r g r a t e f u ] l y ackrtowledges t h e h e l p of Mr. A. K o R o D t , Mrs.

HEDDA HOFFMANN, Mrs. MAGDALENE KŸ n Ÿ R a n d Miss MONICA GROFCSIK i n carrying out the nttmerieal calculations.

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Acta Phys. lfung. Toro. X I I I . Fasc. 4.

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