The relationship between the peak shape of a DTA curve and the shape of a phase diagram

Pergamon

Chemical Engineerin9 Science, Vol 50, No. 3, pp. 417 431. 1995 Copyright ~i'~ 1995 Elsevier Science Lid Printed in Great Britain. All rights reserved 0099 2509!95 t/.5(I + 0.00

0009-2509(94)00244-4

THE R E L A T I O N S H I P B E T W E E N THE P E A K S H A P E O F A DTA C U R V E A N D THE S H A P E O F A P H A S E D I A G R A M S1NN-WEN CHEN,* CHENG-CHIA H U A N G Department of Chemical Engineering, National Tsing-Hua University, Hsin-Chu, Taiwan 30043, R.O.C. and JEN-CHWEN LIN Surface Technology Division, Alcoa Laboratories, Alcoa Center, PA 15069, U.S.A. (Received 24 February 1994; accepted in revised form 8 October 1994)

Abstraet--DTA curves are simulated based on the heat transfer modeling of the DTA cells. The calculated results are compared with the experimental determinations, The rates of heat evolution or absorption of the specimens under the DTA scan are correlated with the shapes of phase diagrams. The relationships between the shapes of DTA curves and the shapes of phase diagrams are discussed.

I. INTRODUCTION Phase diagrams, known as the "road maps of materials", contain condensed information of the physical states of materials. Phase diagrams are essential for new materials development, failure analysis of materials, and materials processing. Differential thermal analysis (DTA) is a technique which measures the temperature difference developed between a sample cell and a reference cell under the same heating (or cooling) environment (Cunningham and Wilburn, 1970; Blazek, 1972; Pope and Judd, 1977; Wenlandt, 1986). Among other applications, DTA is often used for phase diagram determination (Cunningham and Wilburn, 1970; Blazek, 1972; Pope and Judd, 1977; Wenlandt, 1986; Fink, 1949; Noyak and Oelsen, 1969; Nedumov, 1970; Capelli et al., 1976; Bruzzone, 1985; Zhu and Devletian, 1991). As shown in Fig. I, the temperatures of the solidus and liquidus are determined from a DTA heating scan. Besides these reaction temperatures, the DTA curves also contain the information of the shapes of the phase diagrams, illustrated as follows. The shape of the DTA curve depends on the scanning rate and the temperature difference between the sample and reference cells. The temperature difference mainly originates from the heat evolution or the absorption of the phase transformations of the specimens which occurs during heating and cooling. When different materials are subjected to the DTA under the same scanning rate, they generate different shapes of DTA curves resulting from the difference of the phase transformation rates. With the assumption of the validity of the lever rule (Rhines, 1956), the phase transformation rates can be determined from the phase diagram and the predetermined

scanning rate. Phase diagrams of the two different hypothetical systems A-B and C-D are shown in Figs 2(a) and 3(a). Under a constant heating rate of lff'C/min, the phase transformation rates of the samples of the nominal compositions A-25 at% B and C-25at% D are shown in Figs 2(b) and 3(b), respectively. Since the transformation rates for the two samples of the different phase diagrams are varied, the different shapes of DTA curves can thus be expected. In this study, the relationships between the shapes of the phase diagrams and the shapes of the DTA curves are investigated. 2. THE PHYSICALAND MATHEMATICALDESCRIPTION OF THE DTA A schematic plot of the DTA is shown in Fig. 4(a). In the DTA, usually the temperature of the furnace is increased at a controlled constant rate. The DTA records the difference between the sample and the reference cells. A mathematical model is proposed by Gray (1968) to describe the heat flow of the DTA cells as shown in Fig. 4(b). They consist of the sample (or reference) and its container at a temperature, T, (or T,); a source of thermal energy at Tp, and a path having a certain thermal resistance, R, through which the thermal energy flows to or from the sample (or the reference). By having assumed that the specimen ternperature, Ts (or T,), is uniform and equals to that of the container, Gray gave the thermal balance equation for the DTA sample cell: dTs Cs d t =

( T p - Ts) ~- dh R dt

(t)

where t is the time, Cs the heat capacity of the sample cell (including the sample) and dh/dt, the rate of heat generation of the sample.

*Author to whom correspondence should be addressed. 417

SINN-WENCHEN et al.

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0.70

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C

oo q,r,

1.00

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zxr ('c)

7

Fig. 1. Determination of the solidus and liquidus temperatures from a DTA heating scan.

850.00

800.00

750.00 O o

~-"700.00

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~650.00 Cz E 600.00 © 550.00

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A

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Fig. 2(a). Phase diagram of a hypothetical A-B system.

An equivalent equation can be written for the reference cell by assuming that the reference cell does not generate heat and the two cells have identical heat resistance R to the DTA environment:

(T~ - T,) = R dh

dT, + R(C, - C~) d-T

+ RC~

d ( T ~ - T~) dt

(3b)

C d r , = (Tp - r,) " dt

R

(2)

where C, is the heat capacity of reference cell (including the reference). By subtracting eq. (2) from eq. (1) and rearranging, the following equations [eqs (3a) and 3(b)] are obtained for the temperature difference of the two cells:

dh dT, _ RC dT~ (Ts-- T , ) = R - ~ + RCr ~ ~ dt

(3a)

3. R E L A T I O N S H I P BETWEEN T H E PEAK SHAPE OF A DTA CURVE A N D T H E SHAPE OF A PHASE D I A G R A M

A DTA cording to formation during the

curve can be divided into three parts acthe ongoing situations of the phase transof the samples, i.e. before the reaction, reaction, and after the reaction.

Before the reaction There is no heat generation (or consumption) from

Relationship between shapes of DTA curves and phase diagrams

419

O4O F ~ nominal composition A--25at~B [ heating r o t e 10 °C/rain

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T ME ,,re,n/ Fig. 2(b). Phase transformation rate of an alloy with the nominal composition A-25at% B under a 10°C/min heating rate.

0 50

025

~

nominal compos[tion A 25at~;B heating rate ' 10 °C/'min

0.00

\

'\,

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/

-0.50

-075

-1.30 60030

650.00

7 0 0 O0

7 ff 3,.06' ,o~,

?,:'0 : S

Somple Temperature ~ ~1

Fig. 2(c). Simulated DTA heating curve for an alloy with the nominal composition A-25 at% B.

the sample, i.e.

Since there is no reaction involved in the derivation, eq. (6) is valid for all kinds of phase diagrams.

dh --

dt

=

0.

(4)

Usually, in the D T A operation, efforts are made to adjust the baseline so that d ( T s - T~) dt

= o.

During the reaction An isothermal reaction and two hypothetical situations with constant or increasing phase transformation rates are discussed first.

(5)

Substituting eqs (4) and (5) into eq. (3) yields dL

L - L = n ( c , - C~)-T-. Of

(6)

Case I: isothermal reaction. The melting of a pure element and the eutectic reaction in the multicomponent system belong to this kind of situation. During the isothermal reaction, the temperature of the sample cell is fixed at the reaction temperature, Trxo. Substitu-

SINN-WEN CHEN et al.

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0.10

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ATOMIC FRACTION D

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0.90

1 O0

D

Fig. 3(a). Phase diagram of a hypothetical C-D system.

0.40

nominal composition C-25at~D heating rate • 10 °C/rain

c

E

/

-Z-o.5o ©

5 © ~ 020 (3

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~o.~o ©

£

0.00 0.00

,i

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1.00

,iLl,

2.00

,IILII

5,00

TIME (rain)

, ,11

, l l l , l l ~ l

¢.00

,11

500

Fig. 3(b). Phase transformation rate of an alloy with the nominal composition C-25at% D under a 10°C/min heating rate.

ting Ts = Trx. and dT~/dt = 0 into eq. (3a) yields dh

R ~t

=

T~,.- T , - RC, dT" dt

(7)

The temperature difference between the sample and the reference cells is determined by substituting eqs (8) and (9): dT, "~

The reference cell temperature can be expressed as dT, t

(8)

T'= T° + dt

where T ° is the temperature of the reference cell just before the reaction occurs in the sample cell. Equation (9) can be obtained from eq. (6) and the definition of

TO: Trxn

--

T ° =

--

R(C~- -

C,)

dT, dt

(9)

T s - T~= T**.= T° + - ~ - t / dT, = --R(C,--C,) dt

dT, dt t.

(10)

In most cases, dT,/dt is the constant scanning rate and R ( C s - C,) can be regarded as a constant. The calculated D T A curve for this type of reaction is shown in Fig. 5(b). The time required for the completion of the isothermal reaction is determined from the experimental results. It is worth noting that al-

Relationship between shapes of DTA curves and phase diagrams

421

3 58:

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nominal comsosition S - 2 5 a t z O heating rate ' 1,9 °~',/min '~

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j r - '!

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\,,\ / \

-375

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i i h t t i , , t L , , ~ 65000

t I I I I

700.00

I , I I £00.

750.00

Sample Temperature

L

(°C'I

Fig. 3(c). Simulated DTA heating curve for an alloy with the nominal composition C-25 at% D.

A T and the X-axis is Ts. Scal~ple

cRlflr

Furnace Case II: constant rate reaction. Figure 6(a) indicates a situation in which a constant rate reaction occurs. In this case,

dh R - - = constant t = C~. dt

Tor ,ng un,t

From eqs (3b), (5), and (11), the following equation is obtained: dAT

Fig. 4{a). A schematic diagram of the DTA instrument.

(1 I)

dT,

AT+ RC~-~=C~ q - R ( C , - C s ) ~ i - = C

2 (12)

and AT = T~ - 7",. Solving eq. (12) yields AT=C2-exp

-~+C3

.

(13)

Figure 6(b) shows a DTA curve of the system with a constant-phase transformation rate as shown in Fig.

6(a/. Case III: increasin9 rate reaction. Most reactions in phase equilibria belong to this case. If a linearly increasing rate is assumed, as shown in Fig. 7(a), then

dh dt

(14)

R -- = Cj.

Fig. 4{b). Gray's (see Myles et al., ]976) description of the DTA cell.

Combining eqs (3), (5), and (14) yields dAT A T - - RC~ ~ -

though AT's in eqs (10), (13), (16), and (19) are correlated with time, t, for the experimentally determined DTA curves, A T is most often correlated with the temperature of the sample cell, T,. Thus, for all the D T A curves shown in this study, both from experimental determinations and simulations, the Y-axis is

(15)

= C4t + C5.

Equation (16) is obtained by solving eq. (15): AT=CJ+C6-exp

-~+C.~

.

(16~

The DTA curve for this situation is calculated and plotted as shown in Fig. 7(b).

422

SINN-WENCHEN et el. 050

nominal composition AJ-25.Sat% Li nearing rate ' 2.0 °C/rain 0.00

-

-

LD o v m-

-0.50

<~

-1.00

--1.50

' ' ' l l l l ' k l l l l l l ~ ' l l L l l l l l ' l h J l l ' h l L I J l ~ l l ' l l L b l ' ~

595.00

600.00

605.00

610.00

615.00

620.00

Sample Temperature (°C)

Fig. 5(a). The experimentally determined DTA heating curve ~ r t h e AI-Lieutectic.

050

nominal composition Ai-25.8at~Li heating rate : 2.0 °C/min RCs : 0.408 rain 0.00

o v

-0.50 <:~

-1.00

I -1.5O 595, O0

L L l l , l l L h + l t , , l l l l l l , J k ' l J l l ' ' ' ' ' ' l ' ' h ' ' ' l ' l ' ' '

60000

605.00

610,00

6 t 5.00

620.00

Sample Temperature (°C) Fig. 5(b). Simulated DTA heating curve for the AI-Li eutectic.

For the realistic materials systems, the phase transformation rates are usually more complicated. For example, the phase transformation rates of AI-8 at% Cu (Chen et al., 1991) and AI-8 at% Li (Chen et el., 1989) are shown in Figs 8(b) and 9(b), respectively. The transformation rates are neither constant nor linearly increasing. The pedagogic equations discussed in the last section cannot be applied to most of the realistic materials systems. The DTA curves, of the materials systems without simple phase transformation rates under constant scanning rates, can be simulated by using the finite difference method. Equations (3a) and (3b) are modified to the finite difference form. The initial condition is that the specimen has 100% solid at the solidus temperature and the boundary condition is that it has 100% liquid at the liquidus

temperature. The solidus and liquidus temperatures are determined from the phase diagram. The calculated DTA curves for alloys with nominal compositions A1-8.0 at% Li, Al-15.0 at% Li and AI-20.7 at% Li are shown in Figs 9(c)-(e), respectively. After the reaction

When the reaction is completed, there is no longer any reaction heat input or output in the sample cell, i.e. dh dt 0. (17) Inserting eq. (17) into eq. (3b) yields dAT

AT + RCs di-

= R ( C , --

C

dT,

s) dr-"

(18)

Relationship between shapes of DTA curves and phase diagrams

423

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-

beating

rate

1.0

--

N

c(}/r-r-,[o

i

E l

o

s 3 {3

? ÷ ]

:~ 22

3

;£

050

~ ]';3

' [

_

Z

Fig. 6(al. A constant phase transformation rate for a hypothetical system.

:3 5 0

hea~ing rate " .S < 3 ' m : " RCs " 1.5625 rain RA, H " o n or ,

0 80 I / k

i

-050 -

<

/

t,

s

//

~

/

L

,,

f

,\ 'd /

I

/

i

f

i 595 03

. 60000

Somoe

635 X

Temserctu"e

6'

'

<1,

r

s-

Fig. 6(b). A simulated DTA heating curve for a hypothetical system with a constant-phase transformation rate.

Equation (19) can be derived by solving eq. (18) with the boundary condition, A T = ATm,x, when t=0. A T = ATmaxexp ( - ~ ) .

(19)

Since the derivation ofeq. (19) involves no reaction, it is valid for all kinds of materials systems. 4. DISCUSSION Based on the DTA modeling, the DTA curves can be simulated or predicted with the information of phase diagrams and has been illustrated in the previous section. Dtte to the differences of the phase diagrams for the two hypothetical materials systems

shown in Figs 2(al and (b), different DTA curves as indicated in Figs 2(c) and 3(c) can be expected. ]'he liquidus and solidus temperatures are identical; however, the shapes of the DTA curves are different as a result from the different shapes of the phase diagrams. This suggests that besides the temperatures of the phase transformations, the experimentally determined DTA curves bear the information of the shape of the phase diagrams. The information of the shape of the phase diagrams is very useful, especially when the accurate temperatures of the reactions are difficult to determine. For example, in the literature (Chen et al., 1989), experimentally determined solidus and liquidus curves of the APLi systems are scattered owing to the difficulty of the inevitable and frequent

424

SINN-WENCHEN et al. 1.00 o

/

C

E ~"~-o.8o q)

0.60 c

o 8 0

0.40

0.20 O3, 0 C-

G_ 0.00 V 0.00

L k l l l l l l

~,

0.50

~

J'

100 TIME ( m i n )

L,

J

1~0

k l l J

I

2 oo

Fig. 7(a). A linearly increasing phase transformation rate for a hypothetical system.

0.50 h e a t i n g r a t e : 1.0 ° , C / m i n RCs : 1 . 3 6 2 5 m i n o RAH : - 2 . 4 0 C*mln 0.00

L.P ov -0.50 <3

-1.00

-1.50 595.00

600.00 Sample

605.00 Temperature

61 o.oo

615.oo

(°C)

Fig. 7(b). A simulated DTA heating curve for a hypothetical system with a linearly increasing phase transformation rate.

lithium loss of the samples (Chen et al., 1989, 1991; Mikheeva et al., 1960; Ueda et al., 1985; Anyalbechi et al., 1988; Myles et al., 1976) during the course of experiments. Based on the experimental and theoretical determinations, Chen et al. (1989) and Myles et al. (1976) as well, proposed two different AI-Li phase diagrams as shown in Figs 9(a) and 10(a), respectively. Based on the two different phase diagrams, simulations of the DTA curves are shown in Figs 9(c)-(e) and 10(c)-(d). The DTA experiments were conducted in a Perkin-Elmer DTA 1700 system for which the procedures and sample preparation techniques have been described previously (Chen et al., 1989). The parameters used in the DTA modeling are determined by an optimization procedure. With the given initial parameters, DTA curves can be simulated

with the proposed phase diagram according to the procedures discussed in the previous section. The new parameters are obtained by comparing the calculated results with the experimental determinations, and the final parameters are determined after iterations. The comparisons, as shown in Figs 9(c)-(e) and 10(c)-(e), clearly indicate that the phase diagram proposed by Chen et al. (1989) is a better assessment. The Gray's (1968) model adopted in this study is a rather simplified model. According to Gray's (1968) model, the temperature of the sample cell is constant during the reaction for the isothermal-type reaction, such as the melting processes of pure elements and eutectics. However, the experimental results always indicate that the temperature of the sample cell increases during the melting reaction. For instance, Fig.

Relationship between shapes of DTA curves and phase diagrams

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Fig. 8(a). Phase diagram of AI-Cu system proposed by Chen et al. (see Chen et al., 1991).

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nominoJ compositior A!-8.0at~C,.: heating rate " 1.C °S,/m[n ?dZw

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40,00

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TIME (rain} Fig. 8(b). Phase transformation rate of an AI-8.0at% Cu alloy.

L

7000

426

SINN-WEN CHEN et al.

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,

05o

Li

Fig. 9(a). Phase diagram of AI-Li system proposed by Chen et al. (see Chen et al., 1989).

025 nominal compostion AL-8.0atHLi heating rate •: 1.0 ~C/min o • ~

E

~'020

015 c?

8

010

0

C

o 005 6C C

C C:C, 0 80

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......... ! 80

i ......... 2 O0

JJ,,,,,,,,~ 500

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.......

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600

.... 7 00

TIME (rain) Fig. 9(b). Phase transformation rate of an A1-8.0 at% Li alloy.

,1111

8.00

Relationship between shapes of DTA curves and phase diagrams

427

OAO nominal composition At-8.0at~L; h e a t i n g rate • 1.0 ° C / r a i n RCs : 0 . 7 2 0 rain RAH " - 1,960 oC,ml.n

020

-

exp. cal.

+

~-~ 0.00 o o._,

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,%

'<,q

i

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J,l+J,t~llJ~,,,,t,l,,~Jh,,J,l,,,,,tlJ,l,.,,,,,

5900

6000

tJ,,,,

610.0 6200 630.0 Sample ~emperature

,,

6400 (°C::,

, LI

i i i~

650.8

6600

Fig. 9(c). A simulated DTA heating curve, based on Chen's proposed AI Li phase diagram, superimposed with the experimentally determined curve for an Al-8.0at% Li alloy.

1 O0 n o m i n a l c o m p o s i t i o n AJ-15,0at~oL[ h e a t i n g rate : 2.0 ° C / r a i n RCs 0,8 rain RAH -2.28 °C*mn 0.50 ._

CC

.

0 00 _

+['

-,S,.50

_1.0C,

~-~-

i

I

~ i

J , J ~ L I J

560 0 570.0 580.0

~,ll

i

,

I

,

5900 6008 6:3.0 6 2 Sampie -emoerct~-e

!:

4,

s

6-.(

;:

Fig. 9(d}. A simulated DTA heating curve, based on Chen's proposed AI-Li phase diagram, superimposed with the experimentally determined curve for an Al-15.0at% Li alloy.

428

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nominal composition AI-20.7at%~_i heating rate • 2.0 °C/rain RCs • 0 . 9 2 rain RAH • - 2 . 2 0 ° C , r ~ q" l N 0.50 ¢XD. ....

Ca'.

(D

o v

0.00

<3

-0.5o

545.00

565.00

585.00

Sample

605.00

625.00

Temperature

64500

(°C)

Fig. 9(e). A simulated DTA heating curve, based on Chen's proposed AI-Li phase diagram, superimposed with the experimentally determined curve for an AI-20.7 at% Li alloy.

700.00

50.00 ._

/

/

~

,

,_ 6 o o . o o

E @ /

550.00

/ /

5oo.oo 0.00

Al

.........

'.........

' / ........ O. 10

0.20

ATOMIC

' ....... 0.30

FRACTION

LITHIUM

,, ~,,,,,,h

,~

0.40

0 50

LI~

Fig. 10(a). Phase diagram of AI-Li system proposed by Myles et al. (see Myles et al., 1976).

Relationship between shapes of DTA curves and phase diagrams

429

0 !3 c

E _

E

nominal heating

composition AI-8.0ot~L] r a t e " 1.0 °C,/mFn

< 0,5

E b

k

0

0,3 6

/ 0

i

o

i

c 004

Q

b9 C O

/

/

J

//

//jj

ru 002 ,Ln c', L 000

r, L , ,.LL._~

o oo

J , , , ~ ~ ~ , , , l .......

s.oo

~ooo

~J,,,,

,,,,,L,,,,,,~,,,,,,,,

15.oo TIME ( r a i n )

....

20 oo

2s oo

sooo

Fig. 10(b). Phase transformation rate of an AI-8.0at% Li alloy.

0.40

020

nominal composition AI-8.0at~Li h e a t i n g r a t e • 1.0 ° C / r a i n RCs : 0 . 7 2 0 m i n RAH : - 1 . 9 6 0 °C*mrn ____ ....

exp. cal.

~,, 0.00 O •

,

,

it

\\,

/

i

"


',..

'~\.

'

0.40

-0.60 Lu, 590.0

6000

610.0

Sample

6200

630.0

Temperature

6400

6530

660

0

(°C)

Fig. 10(c). A simulated DTA heating curve, based on Myles' proposed AI-Li phase diagram, superimposed with the experimentally determined curve for an AI-8.0 at% Li alloy.

430

S1NN-WEN C~tEN et al.

1.00 I

050

nora;hal c o m p o s i t i a p "Al-15.0at%Li heGting rate • 2.0 ° C / r a i n RCs 0 . 8 m,in RAH ' - 2 . 2 8 °C*mir,

t-

exp. cal.

...

O" 0.00 <~

-0 50

-1.00 ' ' L * ,

560,0 570.0 580,0 590.0 6000 610.0 620.0 6-}00 6400 S a m p l e T e m p e r a t u r e (°C)

6500

Fig. 10(d). A simulated DTA heating curve, based on Myles' proposed AI-Li phase diagram, superimposed with the experimentally determined curve for an AI-I 5.0 at% Li alloy.

1.00 nominal composition Al-PO.7at%Li heatirg rate' 2.0 ° C / r a i n ~Cs " C'92 mir, RAH ' 2.20 °C*rr,;n

t 0 50 -

exp. _

o oo

-'05C

--"

.

.

.

.

.

.

.

.

.

cal.

.

~i

, ~L I

00

545 OS

_

'''

56500

~

'

58500

'

E

605.00

Sample Temperature

625.00

645 ~,3

(°C)

Fig. 10(e). A simulated DTA heating curve, based on Myles' proposed AI-Li phase diagram, superimposed with the experimentally determined curve for an A1-20.7 at% Li alloy.

Relationship between shapes of DTA curves and phase diagrams 5(a) which is an experimentally determined D T A curve of an isothermal reaction, i.e. the melting of the A1-Li eutectic, shows a melting range of 5°C. The DTA sample cell contains the cell material, the specimen and other materials, such as quartz glass and alumina powder in the A1-Li studies (Chen et al., 1989, 1991). Except for the specimen, all other materials in the sample cell do not encounter phase transformations. Thus, the temperature of the sample cell is not constant during the melting reaction. This explains the above-mentioned discrepancy. In this study, the heat capacities were assumed to be a constant. However, in the D T A experiments, the heat capacities of the sample cells, including the specimens, are continuously changing during the phase transformation reaction. This phenomenon causes the baseline shift (Pope and Judd, 1977; Chen et al., 1989). Modification of Gray's model, including a better description of the DTA cells, the changing heat capacities, and others, such as, segregation (Chen et al., 1992), kinetic factor and the discussion of the validity of the lever rule, are needed for future work. 5. CONCLUSIONS (a) The shapes of the D T A curves are related with the shapes of phase diagrams. (b) Modeling of the DTA can be used to simulate the D T A curves. (c) Besides the reaction temperatures, information of shapes of phase diagrams can also be obtained from the experimentally determined D T A curves. Acknowledoements- The authors wish to acknowledge the valuable discussions with Professor Y. A. Chang and the financial support of the National Science Council of Taiwan, R.O.C. through Grant No. NSC 82-0405-E-007-130.

NOTATION Ct C2 C3 C,, C5 C6 C7 C, Cs h R

t 7',

CES 50-3-F

constant constant constant constant constant constant constant heat capacity of the reference cell heat capacity of the sample cell heat of reaction thermal resistance of the path from the sample cell and reference cell to the source of thermal energy time temperature of the reference cell

Tr° Tr~n

T~ T~

431

temperature of the reference cell just before the reaction occurs reaction temperature of the isothermal reaction temperature of the sample cell temperature of the source of thermal energy

Greek letters AT temperature difference between the sample cell and reference cell AT,~a~ temperature difference between the sample cell and reference cell at the end of the reaction REFERENCES

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