On the calculation of melting temperatures for low-temperature phases of polymorphic metals

ON

THE

CALCULATION

OF

LOW-TEMPERATURE

MELTING

PHASES

OF

TEMPERATURES

POLYMORPHIC

FOR

METALS*

A. J. ARDELLT Methods for the calculation of experimentally indeterminable melting points are discussed. It is shown that these melting temperatures can be significantly lower than normal melting points and must be used in such parameters as the reduced temperature, T/T,, whenever this parameter pertains to the lowtemperature phase of a polymorphic metal. A principle for the accurate location of such melting points is rigorously established from the fact that changes in thermodynamic state functions are independent of the path connecting the initial and fmal states. An empirical rule is proposed which enables rapid estimation of these melting temperatures. and in addition greatly facilitates the application of the rigorous procedure. SUR

LE

CALCUL DES TEMPERATURES BASSE TEMPERATURE DE

DE FUSION POUR DES METAUX POLYMORPHES

PHASES

A

L’auteur fait la discussion des methodes de calcul de points de fusion qui ne peuvent btre d&ermines experimentalement. 11montre que ces temperatures de fusion peuvent 6tre nettement plus basses que les points de fusion normaux et doivent 6tre utilises dans des expressions telles que la temperature reduite T/T,, chaque fois que ce parametre se rapporte a la phase a basse temperature d’un metal polymorphe. Le fait que des changements de fonctions d’etat thermodynamiques sont independants du chemin parcouru entre l’etat initial et l’etat final, permet d’etablir dune fapon rigoureuse un principe valable pour la localisation precise de tels points de fusion. L’auteur propose une regle empirique qui vaut pour l’estimation rapide de CBSpoints de fusion et qui, de plus, facilite fortement le oalcul exact. UBER DIE BERECHNUNG VON TIEFTEMPERATURPHASEN

SCHMELZTEMPERATUREN POLYMORPHER METALLE

FUR

Methoden fur die Berechnung von experimentell nicht zu bestimmenden Sohmelzpunkten werden diskutiert. Es wird gezeigt, dam diese Schmelztemperaturen bedeutend niedriger sein konnen als normale Schmelzpunkte und in solchen Parametern wie der reduzierten Temperatur T/T, verwendet werden miissen, wenn sich diese Parameter auf die Tieftemperaturphase eines polymorphen Metalles beziehen. Ein Prinzip fiir die Festellung der genauen Lage solcher Schmelzpunkte wird streng hergeleitet aus der Tatsache, dass Anderungen von thermodynamischen Zustandsfunktionen unabhangig vom Weg sind, der Anfangs- und Endzustand verbindet. Es wird eine empirisohe Regel vorgesohlagen, die eine rasche Abschatzung dieser Schmelztemperaturen ermaglicht und zusatzlich die Anwendung der strengen Methode wesentlich erleichtert.

melting temperatures

INTRODUCTION

Thermodynamic thetical:

quantities

phase transformation

and practical

interest.

empirical correlations

absolute

of

parameters, of atomic

temperature

an example

diffusivity

by Sherby

and Sim-

It is not proper to correlate experimental

for a low-temperature

phase when T,

and data

is the normal

melting point of the metal, and it is then important know

the

hypothetical

temperature

at

which

to the

low-temperature phase melts. Since there is no known experimental procedure by which hypothetical

Consider exhibits

temperature interest

VOL.

solid

11, JUNE

1963

it is necessary

them.

polymorphic

phase,

metal

which

solid phase a, and a high /3.

The

the temperature

with the liquid, L.

temperature at which

of

a is in

One way to calculate

is to compute the difference in Gibbs free energy,

AGaeL, between ct and L as a function of the absolute temperature.

Obviously

The empirical equations

AG”-= = 0 when T = T,“. of Kelleyf2) are well suited to

the purpose

of calculating

extrapolate

properly.

AC”-’

provided

Experimental

heat

for solid and liquid phases are generally the relation C, = a + 2bT -

t Department of Materials Science, Stanford University, Stanford, California. z The word“ hypothetical” connotes transformations which are thermodynamically possible in principle, but which do not normally occur. METALLURGICA,

typical

is T,“,

equilibrium T,”

a

a low temperature

* Received November 26, 1962.

ACTA

can be determined,

to find some way to calculate

are often of academic

in which a reduced temperature

correlation

melting

a hypo-

This is true, in particular

T/Tm is one of the important being a recent nad.(l)

characterizing

where

a,

b, and

c are

c/T2

temperature

that they capacities

expressed by (1) independent

constants. From the relationships dH/dT = C, and dX/dT = C,/T, (we are considering a closed system 591

ACTA

592

maintained phere)

at the constant

integration

METALLURGICA,

VOL.

11,

1963

pressure of one atmos-

of equation

(1) is readily

per-

formed to yield AG”-L = (HL -

H&s.,,)-

(H” -

T(SL-

H;,,.,&

where pure 01 at 298.15”K

S&,.,,)

+ T(Sa -

S&m,)

(2)

S

is taken as the reference

state for the liquid as well as M. This

method,

analytical

which

method,

will

involves

be referred

to

as the

the extrapolation

of data

for the stable u and L phases into a temperature where

they

are both

where p is stable. performed

unstable,

i.e. into

The extrapolation

with the aid of equation

range

the range

1

is, of course,

Tt

(l), but there is no

guarantee that this equation properly extrapolates

I

T

I

T;

Tfi

1. Schematic curve of entropy vs. absolute temperature for a typical polymorphic metal.

PIG.

the

data for the stable phases. In

order

analytical

to

eliminate

method,

tion is established whereby

T,”

the

uncertainties

the correct method and an empirical

of

the

of extrapola-

rule is proposed

can be rapidly found. THEORETICAL

H

Establishment of correct extrapolation procedure Consider

the schematic

absolute temperature, a typical polymorphic Let us first calculate

S, vs.

G$*o -

G,$, over

and then over the dashed path

in Fig. 1. G$,D is the Gibbs free energy

the liquid at T,‘, equilibrium a +

of entropy,

the difference

the solid path (ABCD) (AEFD)

curves

T, and enthalpy, H, vs. l/T for metal shown in Figs. 1 and 2.

etc., and T,, T,”

temperatures

L and /3 +

tionship

L, respectively.

dG/dT = -S

and T,”

are the

of the transitions the

Utilizing

calculation

1

I

of

CI+ /3,

I/T,

path

In

an analogous

dashed

since the isothermal differences

= -

-

G& = G&a G&

GgmB -

A

-

Tma (@ -

T,

Sa) d’j-’

I

s

SL dT. s TllIE

(XL TWIU

Sp) dT.

l/T, s l/T,”

(HP -

2 by

(4) of the

(5)

over using

calculate the

solid

the and

the relationship

This leads to the equation

H”) Q/T)

=s According

path, equations (3) and (4) may be equated, and after rearranging the integral in equation (3) the resulting relationship is

s

in Fig. = H.

we may

G$,lT,

to equations

WIP 1,T p (HL m

correct extrapolation the correct location

HP) 41/T)

(6)

(5) and (6) the shaded areas

in Figs. 1 and 2, respectively,

TllP

Gf, is independent

T9l?

paths

similar

G& + AGGmiL + G&B

T,O = S”dT s Tt

Since the quantity

(3)

AG$;a and AGJm~L are

zero at the equilibrium temperatures. calculation over path AEFD yields G$J

SPdT,

Tt

manner

G$mp/T,fi -

d(G/T)/d(l/T)

s T4

I/T

FIG. 2. Schematic curve of enthalpy vs. reciprocal absolute temperature for a typical polymorphic metal.

quantity

AG,P,L m

I

l/T+

the relaover

A BCD yields

+

I

I/T;

must be equal.

procedure

is established

Thus the in that

of T,= must be such that proper

extrapolation of H and S for the liquid and u phases fulf?lls the requirements demanded by equations (5) and (6). It should be mentioned that equations (5) and (6) are not restricted to solid-liquid transitions but are applicable to any first-order phase change. It would now be possible to utilize equations (5) and (6) by calculating an approximate T,OL from the

ARDELL:

analytical

method,

and

ON

THE

then

CALCULATION

adjusting

the

and new T, u’s until these equations

satisfied.

There is, however,

the transformation,

more information

of an approximate

were about

nature, which

from what is proposed

below.

the entropy

Sgtns = Sg, and Hgm8 before.

AND

DISCUSSION

In order to test the proposed

empirical

rule, it is

necessary to show that the righth and sides of equations (7) and (8) are nearly equal to zero. In many cases C, data show considerable scatter (see Darken and Smithc3) on the specific heat of y-iron, for example)

Proposed empirical rule Let us compute

593

TEMPERATURES

RESULTS

extra-

polations

may be obtained

OF MELTING

and enthalpy

changes

Hg, over the same paths as

It is readily shown that

that graphical evaluation

(7) and (8) can be unreliable.

Equations

can best be tested by comparing obtained

so

of the integrals in equations

by the analytical

(9) and (10)

the values of T,”

method and equation

(11).

This is entirely equivalent to evaluating the right hand sides of equations equation

-I

T,”

TWP

Tm”

v&F-

C$? dTIT

(7)

=s

It would be particularly on the right in equations for then directly

AS&iL

and

allotropic

formation

AH&L - AH”,+t m TWIa (C,@ - CBa) dT Tt T’,B (CDL s TmE convenient

(10)

in cobalt are excluded.

cannot

be

applied

transformations

which occur of iron lies

of the u + y transformation,

of magnetic

et a1.f4) there is

ordering

to the entropy

and enthalpy

temperature.

are considered

to be the same phase, there is prac-

AH$m;L

could

be calculated

tically

= AH$+jL

+ AH;+,

(10)

of paramagnetic

no contribution

entropy and enthalpy point.

Therefore

except

state of virtually

/?-cobalt p-cobalt

yields = AH;m;L/AS;m;L.

(11)

If it can be shown that equations

(9) and (10) are

correct, two things are accomplished:

u and &iron

of magnetic

that

the

to the

The case of cobalt

u + B transformation

is below the Curie temperature.

at the transformation complete

at the normal

state of complete

Tmu was calculated

That is,

temperature

magnetic

melting

magnetic

is in a

order whereas

temperature

is in a

disorder.

by the analytical

the aid of the Burroughs equation

ordering

of &iron at the normal melting

the normal melting temperature. is similar

from

AGgmiL = 0 which

at the M+ y

u-iron at the u + y transformation

temperature

condition

u-iron Although

is not strictly the same phase as &iron at

since the terms on the right are known from experiThe value of Tmu is then readily calculated

ment.

approximately

(9) and

transformations

transformation

temperature

T,”

Equations

these

if the integrals

and

equilibrium

The

(7) and (8) summed to zero,

(9)

the

to

but according to the data of Hofmann, a contribution

with

known.

The Curie temperature

below the temperature (8)

associated

are

in iron and the cc + /l trans-

because of the magnetic

from the equations

AH&;=

quantities

transformations

in these metals. C$) dT.

220 computer

method

The values of AGaP= were plotted as a of T, T,’ being the temperature at which

function

AGuPL = 0. These values of Tma are compared

facilitating the extrapolation procedure; (2) by virtue of equation (11) we will have access to a rapid

those of equation

T,”

with the same

degree of accuracy afforded by the analytical method. Equations (9) and (10) are not subject to thermodynamic proof. Rather, it will be shown that they constitute an empirical rule which is useful either by itself in estimating T,“, or by greatly simplifying an accurate location of T,” through equations (5) and (‘3).

to

(2).

(1) we will establish useful information concerning the at T,” in Figs. 1 and 2, thereby greatly

of calculating

with

applied

ordinates

and simple method

of

for all the metals for which the

thermodynamic

cc + y transformation

and AHGmiL -

was calculated

relevant the

(7) and (8) by direct substitution

(1) into the integrals.

the calculations

with

(11) in Table I. The data used in all are from Kelley.(2)

As can be seen from Table I, the agreement in T,” resulting from the analytical method and equation (11) is very good for most of the metals, the principal exceptions being u-neodymium, y-ruthenium, uu-titanium, and u-zirconium. For uthorium, ruthenium the analytical method yields an unreasonable value of T,*, i.e. one which is higher than the normal

melting

point,

so the value is not reported.

ACTA

594

METALLURGICA,

VOL.

11,

1963

TABLE 1. The melting temperatures of the low-temperature phases of polymorphic metals as calculated by the anslytical method and by equation (11). Normal melting temperatures are included for comparison T,,,a (OK)* Metal

Phase

Barium Calcium Cerium Iron Manganese Manganese Manganese Neodymium Praseodymium Ruthenium Ruthenium Samarium Strontium Thallium Thorium Titanium Uranium Uranium Ytterbium Zirconium

c( c( ; “B Y a CL ; ci CL a a cc i c. c.

Analytical method 948 1051 1069 1812 1411 1479 1501 1294 1200 2614 1316 1022 570 1939 1785 1239 1294 1092 1900

Equation (11) 943 1054 1068 1809 1415 1487 1506 1276 1195 2601 2629 1312 1028 569 1922 1728 1225 1289 1096 1858

Normal melting temperature (“K) 983 1123 1077 1812 1517 1517 1517 1297 1208 2700 2700 1325 1043 577 1968 1940 1406 1406 1097 2130

* These melting temperatures are referred to in the paper as !!‘,a though in some cases the low-temperature phase is labelled p or y.

In the case of a-T& a-Zr and y-Ru the analytical method involves extrapolations over very large temperature ranges, thus for these phases this method is probably unreliable. The only case in which the Tma’s do not agree well, and for which an explanation does not seem to be readily available, is that of a-Nd. These exceptions notwithstanding, the general agreement in T,” is a convincing demonstration that equations (9) and (10) are approximately correct, and that equation (11) may therefore be applied whenever a quick estimate of T,” is desirable. CONCLUSIONS

The proper procedure for extrapolating data for the purpose of calculating hypothetical melting points (T,“) is derived rigorously from the thermodynamic principle that state functions are independent of the path. In addition, useful information is presented which can be said to constitute an empirical rule. This information, embodied in equations (9) and (lo), is useful in itself for rapidly estimating T,“. When used

in conjunction with the rigorous equality of areas expressed by equations (5) and (6), the empirical rule is of great aid in accurately locating T,“. Reference to Table 1 reveals that T,* is often significantly lower than the normal melting temperature, so that calculations of T,” for many metals are certainly warranted. ACKNOWLEDGMENTS

The author wishes to thank his supervisor Dr. 0. D. Sherby for his encouragement and advice throughout the preparation of this paper. The helpful discussions of Dr. D. A. Stevenson and Dr. P. van Rysselberghe are also appreciated. Thanks are due to Mr. W. D. Nix for preparing the computer program. REFERENCES 1. 0. D. SHERBY and M. T. SIMNAD Tram. Amer. Sot. Metal8 54, 227 (1961). 2. K. K. KELLEY U.S. Bureau qf Mines Bulletin 584, U.S. Government Printing Office, W&hington, (1960). 3. L. S. DARKEN and R. P. SMITH, Industr. Engng. Chew 48, 1815 (1951). 4. J. A. HOFMANN A. PASKIN. K. J. TAUER end R. J. WEISS, J. Phys. C&+-n.Solids 1, 45 (1956).