Volume 131, number 3 PHYSICS LETTERS A 15 August 1988

ESTIMATION OF THE bcc-fcc-LIQUID TRIPLE POINT OF IRON

Gang CHEN Center of Theoretical Physics, CCAST (World Laboratory) and Institute of Materials Science, Jilin University, Changchun, PR China

Received 26 April 1988; revised manuscript received 15 June 1988; accepted for publication 20 June 1988 Communicated by D. Bloch

The bcc-fcc-liquid triple point is estimated by means of the dislocation theory of melting. A simple relation between the triple point temperature and pressure is derived and applied to iron. The theoretical result is in agreement with experiment within an error of 10%.

The melting behaviour of materials can be af- fected by the presence of a solid-solid phase tran- sition. The presence of the solid-solid-liquid triple point makes the melting equations invalid, which are often obtained from experiment and used for ex- trapolation to higher pressure. For example, the melting curve of potassium shows a larger increase in slope above the bcc-fcc-liquid triple point [ 1 ]. I f we used the melting equation fitted from the exper- iment below the triple point, we would find that this equation is not capable of describing the melting be- haviour of potassium above the triple point. A direct investigation of the triple point is very important in studying the melting behaviour of materials.

The triple point is a fixed point in the P - T dia- gram, at which the free energy of different phases is equal and three phases coexist. To deal with the tri- ple point, we have to calculate the free energy for every phase. As is well known, it is difficult to give the free energy of a solid, and especially that of a liq- uid at finite temperature and pressure from first principles, and hence to calculate the triple point.

In the present paper we propose a new method to estimate the bcc-fcc-liquid triple point on the basis of the dislocation theory of melting. The advantage of this method lies in its avoidance of calculating the free energy of the liquid phase directly and the pres- ence of a simple form of the free energy of the solid phase at finite temperature and pressure.

The dislocation theory of melting developed by

Edwards and Warner [2] has been generalized by considering the effect of presure. This theory can be used to deal with the melting behaviour of materials at finite pressure [ 3 ]. As a function of temperature and pressure, the free energy of a crystal per unit vol- ume is [2,3]

flGa 2 kTp I n ( z - 1 ) + p a 2 p + F i , F ( T , P ) = ~ p - a

(1)

where G is the shear modulus, a the atomic distance, p the density of dislocations and z the coordination number of the lattice. The parameter fl is a constant related with the non-linear effects of the dislocation core,

Fi = (a3G~l.o/9"l~2d)t~. (2a)

is a factor representing the interaction effect of dis- locations, with which a complicated form of inter- action of dislocations can be simplified as discussed in ref. [2], d is the core radius of dislocations,

d = q a / 2 n , (2b)

where q is a number depending on the crystal type, q= 30 for close packed lattices and q= 15 for bcc lat- tices [2], and

~= 1 + Byo +In( 1 + Yo ) / 2yo

- x ~ o t an - l ( l /x~oo) , (2c)

190 0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

Volume 131, number 3 PHYSICS LETTERS A 15 August 1988

where Yo is the reduced density of dislocations at melting [ 2 ].

At the bcc-fcc-liquid triple point, the free energy of 8-Fe (bcc) is equal to that of y-Fe (fcc),

F, ( Tt, Pt)=F2( Tt, Pt) , (3)

where Tt and Pt are the bcc-fcc-liquid triple point temperature and pressure respectively. Subscripts 1 and 2 designate 7-Fe and fi-Fe respectively.

The bcc-fcc-liquid triple point is related to the polymorphic transition between 8-Fe and 3,-Fe brought by temperature. In general, when poly- morphic changes occur at normal pressure, even though near-neighbor distances often vary quite markedly from one allotropie form to another, the atomic volumes, and hence also the total energies are surprisingly similar [ 4 ]. So the difference of atomic volume between 8-Fe and y-Fe can be neglected.

On the basis of the fact that iron has almost the same atomic volume in the bcc and fcc lattice, we can use the expression of free energy per unit volume given by ( 1 ) instead of that in (3). From ( 1 ) and (3), one gets

( f lG/4n+Pt) (a 2 - a 2) + ~ 8(aZ/q~ -aZ/q2)

= k T t [ l n ( z ~ - l ) / a ~ - l n ( z 2 - 1 ) / a 2 ] . (4)

The parameters involved in (4), such as fl, ~, Yo, can be either directly estimated from an accurate cal- culation or by fitting them to the thermodynamic ob- servables. Edwards and Warner determined them by fitting the experimental value of the latent heat with the theoretical value [2 ]. These fitting results are also used in the following calculation.

I fa relation between a~ and a2 can be obtained, eq. (4) will be simplified further. Utilizing the condi- tion that 8-Fe and 7-Fe have equal atomic volume, one gets

a2/a, = ( 1 / 2 ) ' / 3 x / ~ . (5)

Making an accurate calculation from (4) and ( 5 ), one gets a more simplified form of (4)

(Pt +O.066G)a3=6.8kTt, (6)

where a is the interactomic distance of y-Fe and k the Boltzmann constant.

Eq. (6) describes the relation between the triple

point temperature and pressure. But as the triple point cannot be determined only from (6), another equation of the triple point is needed. At the bcc-fcc- liquid triple point, the free energy of 8-Fe must be equal to that of liquid iron,

F2 ( Tt, PO = FL ( Tt, P, ) (7)

where the subscript L designates the liquid iron. As metioned previously, it is hard to derive the free en- ergy of the liquid phase. But the dislocation theory of melting releases us from the need of calculating it for the liquid state.

The condition (7) can be fulfilled by all the values of temperature and pressure along the melting curve of 8-Fe under pressure, i.e., all the values of tem- perature and pressure coinciding with the melting equation of ~-Fe are capable of making (7) valid. Thus the second equation of the bcc-fcc-liquid tri- ple point can be obtained from the melting equation of ~-Fe, which has been derived from the dislocation theory of melting previously [ 3 ]. Using this and (7), one obtains

Tt = T ° ( 1 +4nPt/otflG) . (8)

Eq. (8) reveals another relation between Tt and Pt. Now, using (6) and (8), we can determine the

bcc-fcc-liquid triple point temperature and pressure of iron. The predicted result is compared with ex- periment in table 1. The two results are in good agreement with a relative error within 10%.

For the bcc-fcc-liquid triple point, it seems that there is a simple relation between the triple point temperature and pressure,

(Pt +Po)a3=ckTt, (9)

where a is the interatomic distance of the bcc phase and c is a constant defined as

Table 1 Theoretical and experimental values of the bcc-fcc-liquid triple point.

Pt (kbar) Tt (K)

theoretical a) 47 1952 experimental b} 52 1991 deviation - 10% - 2%

a) From the calculation by (6) and (8). b~ From ref. [5].

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Volume 13 l, number 3 PHYSICS LETTERS A 15 August 1988

ln(zl - 1 ) - (al/az) l n ( z 2 - 1 ) c= (al/az) 2 - 1 ' (10)

where z is the coordination number o f the lattice [ 2 ], which is invariable. Po is defined as

( 11 ) 2yG~ (cll/a2)2/ql 1/q2

P o = ~ 9n (a~/a2)2--1

The important information about the bcc-fcc- l iq- uid triple point can be obtained from (9) as

Pt /T ,=cka-3-po/Tt , (12)

namely, the larger a is, the smaller the ratio of P~/Tt. From (10) and ( 11 ), we know that c is a positive constant depending on al/a2, which is governed by the volume change from one allotropic form to an- other, and that Po may be either positive or negative depending on the parameters fl, 7, ~ and at~a2. In general, parameters are different for different ma- terials, so that c and Po vary with materials. For this reason, the strict linear relation between Pt/Tt and a - 3 may not be fulfilled, but the tendency that Pt/ T~ varies with a is correct.

Table 2 gives the bcc-fcc- l iquid triple point tem-

perature and pressure as well as the ratio It~ Tt of po- tassium, rubidium and cesium. It is evident that the value o f Pt/Tt decreases with the increase o f inter- a tomic distance indeed, which suggests that eq. (12) reflects a qualitative relation between the triple point and the lattice.

The calculation is based on the dislocation theory of melting, which is a single phase theory. Within this theory, the melting is associated with an instability of the crystal with respect to the dislocations. In other words, the dislocation theory of melting only says that the crystal is unstable at the melting point. Such a treatment neglects the description o f the free energy of the liquid phase. On the one hand, it shows its ad- vantage o f avoiding a description o f the liquid state, on the other hand, it neglects certain amounts o f val- uable and more accurate information of the liquid state. That is why the calculation has a deviation from the experiment.

The calculation presented in this paper shows that the dislocation theory o f melting is capable o f esti- mating the solid-solid-l iquid triple point. In this sense the present calculation is a natural extension of the previous theory [2,3].

Table 2 The bcc-fcc-liquid triple point and the ratio P~/Tt of potassium, rubidium and cesium ~).

References

Pt (kbar) Tt (K) Pt/Tt

K 110 568 0.194 Rb 97.5 505 0.193 b) Cs 22 450 0.049

a) From ref. [ 11. b) Average value from ref. [ 1 ].

[ 1 ] R. Boehler and C.-S. Zha, Physica B 139/140 (1986) 233. [2] S.F. Edwards and M. Warner, Philos. Mag. 40 (1979) 257. [3] G. Chen, Phys. Lett. A 123 (1987) 82. [4 ] C.S. Barrett and T.B. Massalski, Structure of metals, 3rd Ed.

(Pergamon, Oxford, 1980). [5] H.E. Strong, R.E. Tuft and E. Hanneman, Met. Trans. 4

(1973) 2657.

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