the kinetics of recovery and recrystallization of copper from hardness and thermoelectric-power...

THE KINETICS OF RECOVERY AND RECRYSTALLIZATION OF COPPER

FROM HARDNESS AND THERMOELECTRIC-POWER MEASUREMENTS*

T. LL. RICHARDS,7 S. F. PUGH,$ and H. J. STOKES?

Annealing of H.C. copper strip cold-rolled with reductions in thickness of 50 per cent and 96 per cent

has been studied as a rate-process by parallel observations of changes in hardness and thermoelectric

power. In accordance with earlier investigations, change in hardness is taken to be proportional to

fraction of metal recrystallized, while thermoelectric power is assumed to be directly related to residual

elastic lattice strain. While the hardness-isothermal annealing curve of strip rolled 50 per cent exhibits

a definite discontinuity indicative of two processes, such as recovery (or polygonization) and recrystalli-

zation, the thermoelectric power curve is, in agreement with Brindley’s earlier conclusion, pseudo

first order. The rate of decrease of average lattice strain during both polygonization and recrystallization

is apparently governed by similar processes of dislocation migration. The annealing of heavily rolled strip is more complex; hardness and thermoelectric power curves, however, are identical, and analysis

of the curves, outlined in an appendix, indicates that the dominant process is one of second order in

agreement with previous findings. The second-order character follows naturally from the concept that decrease in thermoelectric power and softening occurs by direct interaction and annihilation of

stress fields associated with stable arrays of dislocations at grain interfaces. It is,pointed out that this

concept is also consistent with recrystallization as a process of nucleation and growth.

CINETIQUE DE LA RESTAURATION ET DE LA RECRISTALLISATION DU CUIVRE

PAR MESURE DE DURETE ET DU POUVOIR THERMOELECTRIQUE

Le recuit de bandes laminees a froid en cuivre HC apres reduction d’epaisseur de 50 et de 96% a irt.6 Btudie par mesure des changements de durete et du pouvoir thermoelectrique. Conformement 8,

d’anciennes mesures, le changement de durete est proportionnel au pourcentage du metal recristallise,

tandis que le pouvoir thermoelectrique est suppose dependre directement des deformations reticula&x+

residuelles. Tandis que les courbes de durete apres recuit de bandes laminees It 50% montrent une

discontinuite bien deflnie caracteristique de deux processus tels que restauration (ou polygonisation) et

recristallisation, la courbe du pouvoir thermoelectrique est, conformement aux anciennes conclusions

de Brindley, du premier ordre. La vitesse de decroissance de la deformation reticulaire moyenne,

pendant la polygonisation et la recristallisation, est apparemment regie par un mecanisme analogue a

la migration des dislocations. Le recuit des bandes fortement laminees est plus complexe. Les courbes

de durete et du pouvoir thermoelectrique sont neanmoins identiques et leur analyse, faite dans un

appendice, indique que le mecanisme preponderant est du second ordre, en accord avec les mesures

anterieures. Ce caractere est la consequence naturelle de l’idee que la diminution du pouvoir thermo-

Blectrique et celle de la durete sont dues a l’interaction et it la destruction des champs de contraintes

associees aux repartitions stables de dislocations dans les joints de grains. 11 est remarqui: que cette

conception est en accord avec le mecanisme de germination et de croissance pour la recristallisation.

UBER DIE KINETIK DES ERHOLUNGS- UND REKRISTALLISATI~NSVORGANGES

BE1 KUPFER HERGELEITET AUS HARTEMESSUNGEN UND BESTIMMUNGEN

DER THERMOKRAFT

Durch gleichzeitige Beobachtung der Harteiinderungen und der ,&nderungen der Thermokraft wurde

die Zeitabhangigkeit des Anlassvorganges von Walzbiindern aus Hochleitfahigkeitskupfer mit einem

Verformungsgrad von 50 und 96% Dickenabnahme untersucht. In Ubereinstimmung mit friiheren Untersuchungen wird angenommen, dass die Hilrteiinderung proportional dem rekristallisierten Anteil des

Metalles ist, wahrend die Thermokraft in direkter Beziehung zu den verbleibenden elastischen Gitter-

spannungen steht. Wahrend die Hiirtekurve beim isothermen Anlassen des urn 50% verformten Bandes

eine deflnierte Unstetigkeit aufweist, die zwei Vorgiinge anzeigtdie Erholung (oder Polygonisation)

und die Rekristallisation-ist die Kurve der Thermokraft, in tibereinstimmung mit den friiheren

Folgerungen von Brindley, pseudo erster Ordnung. Die Geschwindigkeit des Abbaus der durchschnitt-

lichen Gitterspannungen wahrend der Polygonistion wie such wahrend der Rekristallisation wird offen-

sichtlich durch iihnliche Vorgange der Versetzungswanderung beeinflusst. Die Vorgange beim Anlassen der

stark verformten Bander sind wesentlich verwickelter; die Kurven der Hiirte und der Thermokraft sind

jedoch identisch. Eine Analyse dieser Kurven, die im Anhang umrissen wird, zeigt, dass der beherrschende

Vorgang ein Prozess zweiter Ordnung ist, was mit friiheren Beobachtungen iibereinstimmt. Dass diesen Vorgang ein Prozess zweiter Ordnung ist, folgt naturgemiiss aus der Annahme, dass der Abfall der

Thermokraft und die Hlirteabnahme durch direkte Wechselwirkung und Vernichtung von Spannungs- feldern vorsichgehen unter gleichzeitigem Auftreten von stabilen Versetzungsgebieten an den Grenz- flachen der Khmer. Es wird gezeigt, dass diese Vorstellung mit der Rekristallisation als einem

Keimbildungs- und Wachstumsvorgang im Ubereinstimmung ist.

* Received October 27, 1954. t Research Department, Imperial Chemical Industries

z Formerly as above, now at Atomic Energy Research

Ltd., Metals Division, Birmingham, England. Establishment, Harwell, England.

ACTA METALLURGICA, VOL. 4, JANUARY 1956 75

76 ACTA METALLURGICA, VOL. 4, 1956

1. INTRODUCTION

From analyses of observations of the rate of

recrystallization, made directly or by following

changes of various mechanical and physical properties,

different investigators have concluded that recrystal-

lization simulates a rate process of either the flrst,(lp 2,

second,‘3) or thirdt4) order. The divergent conclusions

can be quite rationally explained on nucleation and

growth theories,(5) but it is difficult to express such

theories in terms of the more elemental and more

widely applicable dislocation theory.

Changes in the observed values of different pro-

perties of the same cold-worked metal on annealing

may occur at distinctly different rates, and, in

attempting to analyse annealing curves, such differ-

ences must be taken into consideration. Some pro-

perties, for example, may be purely dependent on

dislocation density, while others are affected by their

distribution; or alternatively, as has been suggested,(e)

while changes in one property may be associated

with behavior of dislocations, t’hose in other pro-

perties may be more directly related to lattice

vacancies. Also the analysis of the form of annealing

curves is highly dependent on the accuracy of the

curves t’hemselves, so that if such analysis is to have

real significance it is desirable to select for observation

a property which not only can be measured with

sufficient accuracy, but which is quite representative

of the average condition of the specimen examined.

Furthermore, in the analysis, the possibility that more

than one process is involved must be given considera-

tion, and the problem treated accordingly.

The work described in t’he present paper was under-

taken primarily with the object of studying recrystal-

lization as a rate process, but a secondary objective

was to explain the apparently conflicting observations

of Brindleyc2) and of Cook and Richardsc3) on the

recrystallizat,ion behavior of copper. The latter

determined recrystallization rate in heavily cold-

rolled copper strip by hardness measurements which

t’hey had first correlated with t’he proportion of each

sample recrystallized, estimated by direct observation

TABLE 1. Composition of the copper used

per cent

Tin Lead Iron Nickel Manganese Silver Antimony

0.0002 : <0.0002 / 0.0010

0.0005 / <0.0005

0.0022 < 0.0001

Arsenic Bismuth Selenium Tellurium Phosphorus Sulphur oxygen

< 0.0005 < 0.0001

Nil

Et: 0.0010 0.032

of changes in microstructure. From an analysis of

their results they showed that the softening rate

conformed to that of a second-order process, and

therefore proposed a two-stage mechanism involving

recovery and recrystallization. On the other hand,

Brindley’sc2) analysis of annealing curves determined

by Brandsma (‘) from thermoelectric-power measure-

ments indicated a single first-order process. It was

significant that, in their work on kinetics, Cook and

Richards(s) dealt specifically with recrystallization of

copper to cube-texture, while Brandsma’s experi-

ments(‘) were, in all probability, concerned with

recrystallization to random or to retained rolling

texture.

For the present investigation, it was clearly

desirable that thermoelectric-power and hardness

measurements should be made on the same specimens

and on material prepared so as to recrystallize in

characteristically different manners. Accordingly, two batches of cold-rolled H.C. copper strip were

produced in such a way that one recrystallized to

cube-texture and the other to a random structure.

2. PREPARATION OF MATERIAL

A cast slab of tough-pitch copper of H.C. quality

and composition indicated in Table 1 was hot-rolled

from an initial thickness of 3% in. to l&in. The

material was then annealed at 500°C for 3 hr and

cold-rolled to 0.75 in., t’he direction of cold-rolling at

this and all subsequent stages being transverse to that

of the initial hot-rolling. Strip for the experimental work at a thickness of 0.030 in. was then prepared in

the two following ways:-

(a) Strip of Random Structure

Strip, for the annealing experiments on material

recrystallizing to random texture, was prepared from

the 0.75 in. cold-rolled stock by annealing at 9OO”C,

and then cold-rolling to the final gauge of 0.030 in.

with intermediate anneals at 500°C at gauges of

0.150 in. and 0.060 in. The 0.030 in. strip was then

sheared to 4 in. widths. Some of the material was

annealed at 350°C for use as reference leads, while the

remainder was retained in the hard-rolled condition

for the annealing experiments.

(b) Hard-Rolled Strip for Recrystallization to Cube

Texture

Part of the 0.75 in. cold-rolled stock was annealed

at 350°C to a grain size of 0.015 mm, and then rolled

without further annealing to 0.030 in. Since this

material recrystallizes slowly at room temperature,(*)

the rolling was interrupted at a thickness of 0.15 in.,

RICHARDS, PUGH, AND STOKES: RECOVERY AND RECRYSTALLIZATION 77

rolhng to final gauge being carried out immediately

prior to the annealing experiments.

3. EXPERIMENTAL PROCEDURE

(a) Thermoelectric-Power Measurement

The thermoelectric power measurements were made

with a Paschen astatic galvanometer, the control

magnets being adjusted so as to provide a sensitivity

of 3.4 x 1OW volts/mm deflection. Since, with a

temperature difference of 5°C between the junction of

a cold-worked annealed copper thermocouple, an e.m.f.

greater than lo-’ volts is developed, the error in its

measurement is thus about 1 per cent. The galvano-

meter and leads were contained in a constant-

temperature enclosure, on a heavy mounting to reduce

vibration. The 4 in. wide strip sample formed the

detachable short element of a differential thermo-

couple, being bolted to annealed copper reference

leads so that it could be removed for annealing

treatment. A temperature difference was maintained

between the junctions by immersing one of them in

an oil-bath at room temperature, and the other in a

bath about 5°C higher, the exact difference in tempera-

ture being measured by means of a chromel-alumel

thermocouple calibrated against a Beckmann thermo-

meter.

(b) Determination of Isothermal Annealing Curves

The progress of isothermal annealing of the strip

reduced 50 per cent by cold-rolling, which recrystal-

lized to a random crystal orientation, was followed by

measuring thermoelectric power with respect to the

fully annealed reference leads of random crystal

orientation, and by D.P. hardness measurements after

successive ten-minute annealing periods in an air

circulation oven at 180°C.

In order to maintain a steady and uniform temper-

ature and to hasten the attainment of thermal

equilibrium, the specimen was inserted in a narrow

space between two slabs of copper maintained

continuously at a steady temperature in the oven.

At the end of each annealing period the specimen was

water-quenched. The cold-rolled strip, bent into a

U-shape with arms 5 in. long, had holes drilled at the

ends in order to bolt it to the reference leads, the thin

oxide film formed during annealing being first removed

at the contact points by rubbing lightly with metal

polish. The sole purpose of the polishing was to ensure

good electrical contact, any mechanical working of

the surface having no effect, provided the whole

junction was at uniform temperature. Hardness

measurements were also made on the same sample

after each annealing period.

TABLE 2. Diamond pyramid hardness and thermoelectric- power values on annealing at 180°C for H.C. copper strip cold-rolled 50 per cent reduction in thickness

Diamond pyramid Thermoelectric hardness power

Observec (10-B volts)

0 123

:: 120 117 30 ’ 115

100 5.38 96.2 5.00 91.8 4.50 89.0 ’ 4.78

40 117 91.8 50 116 90.4 60 70 80 90

100 110 120 130 150 170 190 220 250 280 310 350 410 470 530 590 680 800 980

117 ~ 91.8 114 1 87.6 113 86.3 114 87.6 113 111 105 108

94.9 83.6 79.6 76.3 70.5 70.5 69.0 61.8 59.5 60.0 55.5 56.7 52.1 50.8 50.4

86.3 83.5 75.3 79.4 61.4 45.4 40.3 35.6 27.8 27.9 25.0 15.7 12.5 13.2

7.0

;:; 0.6 0

4.52 4.42 4.26 3.96 3.66 3.40 3.22 3.00 2.90 2.78 2.56 2.34 2.24 1.90 1.72 1.48 1.42 1.38 1.22 1.08 1.08 1.02 0.96 0.96 0.88*

Per cent total

change on annealing

5.38 - 0.925)

100 92.5 80.8 87.4 81.4 79.2 75.9 68.8 62.0 57.0 52.0 45.8 44.8 42.0 37.0 32.2 29.8 22.2 17.6 12.6 11.1 10.3

6.7 3.5

&5) 0.8 0.8

-1.0

* The asymptotic end value of thermoelectric power obtained from the smoothest curve through experimental points was estimated to be 0.925 x lo-* volts.

Strip reduced 96 per cent by rolling, which recrystal-

lized to cube-texture on annealing, was also made into

a U-shaped specimen, and thermoelectric power again

measured against random-orientation copper reference

leads. Annealing of this strip was effected in boiling

water, measurements being made after successive

five-minute annealing periods until no further change

in properties took place. Towards the end of annealing

the periods were increased, since the rate of change

was slower.

The observed values of hardness and thermo-

electric power are recorded in Tables 2 and 3, from

which it should be noted that the end value of the

thermoelectric power in Table 3 is a small negative

quantity. For the purpose of comparison of rates of

change in hardness and thermoelectric power, the

values of these quantities are also expressed in Tables 2

and 3 as the percentages of the total change on

annealing from initial value to an asymptotic end


TABLE 3. Diamond pyramid hardness and thermoelectric- power values on annealing at 100°C for H.C. copper strip cold-rolled 96 per cent reduction in thickness

Time of isothermal annealing at 100°C

(min)

0 5

10

;I? 25 30 35 40 45

z 70

:: 100 110 120 130 140 155 175 200 230 215 350 420

1140

Diamond pyramid ’ Thermoelectric hardness power

0

-

bserved (I&P.

5Kg load)

126 125 123 125 123 123 124 121 118 121 106 110 92 82 87 72 59 60 58 54 48 49 48 47 46 45 44 44

L

Per cent total

change on annealing per cent

(126 - 44)

100 98.8 96.3 98.8 96.3 96.3 97.6 93.8 90.2 93.8 75.6 80.5 58.5 46.3 52.4 34.2 18.3 19.5 17.1 12.2 4.9 6.0 4.9

?Z 1:2 0 0

, 1 Per cent

Observed total

(10-E / change on

volts) , annealing per cent

(10.34 + 0.475)

10.34

9.06 8.78

9.40

8.46 8.58

9.30

8.18 7.76

9.50

6.72 5.64

9.16

4.30 3.30 2.50 1.78 1.36 0.98 0.68 0.40 0.14

-0.03 -0.17 -0.28 -0.36 -0.43 -0.52*

88.12

100

85.57 82.58

91.26

83.68 80.00

90.35

76.12 66.39

92.19

56.52 44.14

89.04

34.93 27.50 20.84 16.96 13.45 10.68 8.09 5.12 4.13 2.84 1.80 0.95 0.40

-0.41

* The asymptotic end value of thermoelectric power was estimated to be -0.475 volts.

value derived from a smooth curve through experi-

mental points. Isothermal annealing curves have been

drawn in Fig. 1 from both the hardness and thermo-

electric-power measurements on strip, rolled 50 per

cent. Since, apart from a small initial lag in hardness

values, the corresponding curves for strip, rolled 96

per cent, are exceedingly close to each other, in order

to avoid confusion only the more accurate thermo-

electric-power curve is plotted in Fig. 2.

4. ANALYSIS

Analysis of the recrystallization process by reference

to changes in a secondary property depends on either

the assumption of a linear relation between the

measured value of the property and the corresponding

fraction of metal recrystallized at any stage or on the

experimental determination of the precise relationship.

For a partly recrystallized metal, considered as a

mixture of a residual cold-worked portion with that

which has already transformed to the fully annealed

0 % EMF

. % D.P. HARDNESS I EXPERIMENTAL VALUES

- THEORETICAL CURVE 6 =&, ewat 0 \a I I (HALF LIFE 105min) " -

b0 \

TIME OF ISOTHERMAL ANNEALING AT 180°c- mm

FIG. 1. Isothermal annealing curves of H.C. copper strip cold-rolled 50 per cent reduction in thickness.

state, a linearly related property conforms to the

simple rule of mixtures, so that an observed value is in

simple proportion to the quantities of the two com-

ponents present.

It has already been establishedt3) for the recrystal-

lization of heavily cold-rolled H.C. copper strip, that

apart from a small initial lag, during which no apparent

change in average hardness of the strip occurs with

the first microscopically observed signs of recrystal-

lization, the relation between hardness and fraction

recrystallized is, in fact, linear. No direct calibration

of thermoelectric power was made for the purpose of

the present investigation, although the close agreement

between changes in this property and in hardness

during isothermal annealing of heavily rolled copper

indicates that, in this instance at least, it also is

linearly related with the fraction recrystallized. This

view is in direct opposition to one previously held,tg)

namely that thermoelectric power might actually

indicate the state of recovery of the continuous

unrecrystallized matrix quite independently of the

presence of discrete and therefore disconnected

islands of recrystallized metal.

Crussard and Aubertin,oO) however, have shown

that change in thermoelectric power of tensile

specimens on unloading varies directly with applied

stress and therefore with residual elastic strain. It

thus appears that thermoelectric power provides a

/ 11 . EXPERIMENTAL VALUES

b0

0 50 100 150 200 25O x30 350 400 450 500

TIME OF ISOTHERMAL ANNEALING AT IOO’C - mm

Fm. 2. Isothermal annealing curve of H.C. copper strip cold-rolled 96 per cent reduction in thickness.


t STRAIGHT LINE THROUGH EXPERIMENTAL POINTS INDICATES HALF LIFE VALUE OF IOSmin

sa zz” 2fc VZujl’O EOa 5

M$ $y BU zg ’ ;,‘g 05

a;; 22

loo 200 300 400 500 600 700 800 9m 1000

TIME OF ISOTHERMAL ANNEALING AT 180°C - min

FIG. 3. Isothermal annealing curve of H.C. copper strip cold-rolled 50 per cent reduction in thickness.

means of following changes in lattice strains during

isothermal annealing, and being truly representative

of the average condition of the specimen, observation

of thermoelectric power is ideally suited for rate-

process analysis. The localized nature of the diamond-

pyramid hardness test and the heterogeneous character

of partly annealed metal are responsible, on the other

hand, for a wide statistical fluctuation in results for a

single specimen, which is reflected in the greater

scatter of the observed hardness values about the

estimated annealing curve in Fig. 1. Because of the

relative inaccuracy of hardness determinations, full

mathematical analysis of the kinetics of the isothermal

annealing process is carried out only with the thermo-

electric data.

(a) Recrystallization of Strip Rolled 50 per cent

The plot of residual thermoelectric power of strip

rolled with reduction of 50 per cent against time of

isothermal annealing at 18O”C, Fig. 1, appears to

follow a simple exponential curve, apart from a small,

but significant, deviation in the initial stages. This

deviation is probably connected with the more marked

discontinuity in the hardness curve. To derive the

exponential coefficient of decay of thermoelectric

power with time, the experimental values are plotted

in Fig. 3 on a semi-logarithmic scale. The points fall

almost on a straight line, confirming the exponential

nature of t’he relationship. From the slope of the line,

the decay coefficient is 6.6 x 10e3 min-l, correspond-

ing to a half-value period of 105 minutes. The full

curve in Fig. 1 is in fact the exponential curve of

this half-life. Change in thermoelectric power is

therefore either a simple first-order process or at

least a pseudo first-order process. The significance

of this finding, and the difference in character of the

thermoelectric and hardness isothermal curves, will

be discussed later.

(b) Recrystallization of strip rolled 96 per cent

The thermoelectric-power and hardness curves of

strip rolled with reduction in thickness of 96 per cent

and isothermally annealed at 100°C are, apart from an

initial lag in hardness values, coincident. The curve,

Fig. 2, is clearly more complex than the thermoelectric-

power curve, Fig. 1, for strip rolled 50 per cent. As a

first attempt at analysis, the experimental values of

thermoelectric power were plotted on a semi-loga-

rithmic scale, as in Fig. 4. The appearance of this curve

is suggest’ive of three successive processes. The first is

a rapid process accounting for an initial fall of about

10 per cent of the total change: the order of the

second, accounting for the major proportion of the

100

- EXTRAPOLATION OF LINEAR TAIL OF

50 x x ARITHMETIC DIFFERENCE (Q)-(6)

0.3 0 10 40 60 80 100 120 140 lb0 I80 200 220 240 260 280 300 320 340 360 380 400 420 440

TIME OF ISOTHERMAL ANNEALING AT 100 ‘c - min

FIG. 4. Isothermal annealing curve of H.C. copper strip cold-rolled 96 per cent reduction in thickness.


total change, is greater than unity, while the third and

last is first order. The procedure for analysis of suc-

cessive first-order reactions, which is not strictly

applicable in the present instance, has been applied in

order to obtain an approximate solution. The extra-

polation (b) of the linear tail EzePBt of the full curve (a)

in Fig. 4 is first subtracted arithmetically from the

experimental curve yielding the thermoelectric-power

values indicated by points marked x These points lie

close to the theoretical parabola (c) expressed by E = EIe-a(t-@

t Th e second process, accounting for

the major proportion of change in thermoelectric

power, thus simulates one of second order.

The above approach is only valid for successive

first-order processes, but now that it is apparent that

the second process is second order, a more rigorous

mathematical treatment is possible. The first stage is

so rapid that it is assumed to be a fast exponential

decay process which is virtually complete in a few

minutes, giving rise to an incubation period for the

second stage. The complete mathematical treatment

for a second-order process followed by one of first-order

is set out in an appendix, the effect of the first process

being represented by a zero shift or incubation period.

Assuming values of the constants derived from analysis

of Fig. 4, more exact values can be obtained by

successive approximation. The equation derived for

the complete annealing process is rather complex, but

the constants can be determined fairly easily; and a

comparison of experimental results with theoretical

values, which are recorded in Table 4 and plotted in

Fig. 2, shows agreement well within experiment’al

error.

5. DISCUSSION

The possibility of interpretation of rate of change of

a secondary property in terms of recovery and

recrystallization naturally depends on there being a

definite correlation between that property and fractions

of metal in the recovered and recrystallized states.

A simple and reasonable interpretation can be

developed if it is assumed that, as Crussard and

Aubertin(‘O) have shown, thermoelectric power varies

directly with applied stress and therefore with residual

elastic strain.

The first-order character of the thermoelectric curve

of strip rolled 50 per cent is therefore equivalent to the

simple exponential decay of internal stresses on iso-

thermal annealing.(ll). These stresses would, in

moderately deformed metal, be expected to vary more

or less continuously throughout the processes of

recovery (or polygonization) and subsequent recrystal-

lization. The hardness curve of strip rolled 50 per cent,

TABLE 4. Calculated values of the terms A and l3 together with their sum S* and the corresponding observed values

Time of isothermal annealing A

t (min)

0 5

10 15

f: 30 35 40 45 50 60 70 80 90

100 110 120 130 140 155 175 200 230 275 350 420

1140

100 91.26 90.35 92.19

90.0 0 89.04 89.5 0.08 88.12 87.9 0.14 85.57 85.0 0.50 82.58 82.7 83.68 78.6 ::: 80.00 74.0 2.0 76.12 62.5 3.0 66.39 52.5 4.8 56.52 40.6 30.8 :::

44.14 34.93

22.6 6.6 27.50 15.0 6.9 20.84 9.8 6.9 16.96 6.2 6.7 13.45 4.0 6.4 10.68 2.7 5.8 8.09

:::7 4.8 5.72

4.13 0 ;:: 2.84 0 0 A::

1.80 0.95

0 0.4 0.40 0 0 -0.41

100 91.7

1 90.3 90.04 90.0 89.58 88.04 85.5 83.7 80.2 76.0 65.5 57.3 45.9 36.8 29.2 21.9 16.7 12.9 10.4

8.5

K7 2.9 1.8 0.9 0.4 0

* Values of S for t in the range O-20 min are estimated on the assumption that initial fall from 100 to 90 per cent corresponds to a first-order process of 2 min half-life. A = _&e-act-r)*

Observed values (From

Table 3)

B = 2 e-B(t-7) 1 - e--a@-# _ $ (t _ 7) 1

S = A + B, GC = 2.2 x 10d4 (min)-a,

/3 = 9.9 X 10d3 (min)-I, 7 = 20 min, E, = 90 per cent,

Ez = 23 per cent, K = 1.9, & = 22.5 min.

on the other hand, shows a definite discontinuity

indicative of two such processes. These may well be

the two processes which Stroh(12) envisages in ex-

plaining annealing behavior of cold-worked metal,

namely a short-range rearrangement of dislocations in

piled-up groups which is associated with appreciable

change in stored energy and specific heat(13) but not in

hardness, and with a long-range interaction of the

st,ress fields of neighboring groups which is associated

with a further liberation of stored energy and with

appreciable softening. On this basis, it would seem

that thermoelectric power is a measure not only of

strain associated with dislocations, but also of stored

energy which in moderately deformed metal varies

continuously throughout the annealing process.


In rolling, individual crystals are deformed in direct

proportion to the strip itself, so that, wit’h increasing

reduction, initially equi-axed crystals are rolledprogres-

sively thinner, each crystal taking up the approximate

orientation of one component or other of the standard

rolling textures. The small initial fall in thermoelectric

power on annealing of heavily rolled material is, on the

present thesis, equivalent to a decrease in general

lattice distortion with the establishment of dislocation

arrays at the int’erfaces of adjacent crystals. Disloca-

tions of opposite sign would be annihilated at the

interfaces leading to an equilibrium density of disloca-

tions of one sign corresponding to the definite

orientation difference between the various components

of texture, while the dislocations in neighboring

interfaces would be of opposite sign. Further struc-

tural change can now only occur by direct interaction

and cancellation of the dislocation stress fields across

the thickness of individual crystals leading to an

abrupt change in lattice orientation from rolling-

texture to cube-texture. Such a mechanism would

account both for the large change in thermoelectric

power associated with the second process evident from

the t’hermoelectric-power curve, and for its second-

order character.

It has in fact been establishedc3) that, for heavily

rolled copper, a linear relationship exists between

hardness and fraction recrystallized to cube-texture.

The similarity between the hardness and thermo-

electric power curves of strip rolled 96 per cent

indicates that, in this instance, the change in thermo-

electric power is also proportional to fraction recrystal-

lized. The change from rolling-texture to cube-

texture must, of necessity, be one of localized atomic

rearrangement such as nucleation and growth, other-

wise large dimensional changes would be involved. The

formation of cube-texture by annihilation of high-

energy interfaces, however, is equivalent to two-

dimensional growth from predetermined nuclei, which

as Evans(14) has shown is a process of second order.

The final first-order process is merely relief by grain growth of residual stresses at boundaries of cube-

texture grains in slightly different orientations.

Analysis of changes in thermoelectric power and

hardness purely as rate processes, making the basic

assumptions that thermoelectric power is directly

proportional to lattice strain and hardness to fraction

recrystallized, has led to the following general picture

of the annealing behavior of rolled copper strip. From

the hardness curve, annealing of strip rolled with

reduction of 50 per cent occurs by two distinct pro-

cesses, polygonization and recrystallization. Thermo-

electric power, and therefore lattice strain, however,

6

varies more or less continuously in an exponential

manner throughout, since both polygonization and

recrystallization involve essentially similar processes of

migration of dislocations. Annealing of heavily rolled

strip may again be regarded as occurring by poly-

gonization and recrystallization, although polygoni-

zation, being restricted within the boundaries of the

now thin initial grains, is associated with only a small

change in thermoelectric power and lattice strains,

and none in hardness. Most of the change in hardness

and thermoelectric power occurs by a second-order

process which is interpreted as a process of annihi-

lation of stable arrays of dislocations at t’he grain

interfaces by direct interaction of their stress fields.

The findings of the present investigation thus confirm

the earlier observation of Cook and Richardsc3) that

recrystallization of heavily rolled copper is mainly a

second-order process, and establish that the difference

in order found by these authors and by Brindleyc2) is

due to the fact Ohat the findings of the former refer to

heavily rolled copper which recrystallizes to cube-

texture, and of the lat’ter to copper processed in such a

way that it recrystallized in a different manner on

annealing, probably to a retained rolling texture or

perhaps a random texture. Furthermore, the picture

of recrystallizat’ion derived by rate-process analysis is

quite consistent with the structural observations and

nucleation and growth concepts.

ACKNOWLEDGMENTS

The authors wish to thank several of their colleagues

for much valuable discussion, and Dr. N. P. Inglis,

Research Director, for his interest and encouragement.

1.

2.

3.

4.

5.

6.

ii:

9.

:::

12. 13.

14.

REFERENCES

A. KRUPKOWSKI and M. BALICKI, Ann. Akad. Sci. Tech. Vareowie, 4, 270 (1937). G. W. BRINDLEY, Phys. Sot. Bristol Conference Report, 9.5 (1948). M. COOK and T. LL. RICHARDS, J. Inst. Metals, 73, 1 (1946). B. F. DECKER and D. HARKER, TTans. Amer. Inst. Min. Met. Eng9x, 188, 887 (1950). J. E. BURKE and D. TURNBULL, Progress in Metal Physics, 3, 220 (1952). L. M. CLAREBROUGH, M. E. HARGREAVES, and G. W. WEST, Phil. Mug., 44, 913 (1953). W. F. BRANDSMA, 2. Phys., 48, 703 (1928). M. COOK and T. LL. RICHARDS, J. Inst. Metals, 70, 159 (1944). T. LL. RICHARDS. Phvs. Sot. Bristol Conference R~DoI%. 105 (1948). ’ ”

I ,

CH. CRUSSARD and F. AUBERTIN, Rev. Met., 45,402 (1948). W. G. BURGERS, Proc. K. Ned. Akad. Wetensch., 50, 452 (1947). A. N. STROR. P~oc. Rev. Sot., A218. 391 (1953). T. SUZURI, &i. Rel. ‘kee. inst. T’ohoku’ University, 1, 193 (1949). U. R. EVANS, Trans. Faraday Sot., 41, 365 (1945).


Appendix

SUCCESSIVE RATE PROCESSES

The change in thermoelectric power of H.C. copper- strip cold rolled with a reduction in thickness of 50 per cent, on isothermal annealing, conforms to a first- order process, that is, the thermoelectric power E, at time t can be expressed by the equation

E, = E,,ewmt

where E, is the initial thermoelectric power at time t = 0, and a a constant, the value of which is temperature-dependent.

Recrystallization of heavily rolled copper is complex, and an attempt was first made at analysis of the isothermal annealing curve, by following the standard procedure for successive first-order processes, such as a radioactive series for which a product A trans- forms to a product B, and B to C, and so on, the rate of decay of a product at any instant being proportional to the amount present at that instant. The analysis indicated that the recrystallization process of heavily rolled copper did not conform to successive first-order

processes, but rather to a second-order process, followed by one of first order.

The mathematical analysis for successive first-order processes has long been established, but the authors are unaware of a similar treatment for a second-order process followed by one of first order, and t’hey have accordingly developed their own treatment.

ANALYSIS OF SECOND-ORDER FOLLOWED BY

FIRST-ORDER PROCESS

Let product A transform to B, and B to C, and let x, y, z be the amount of each present at time t, so that

x+y+z=1

Let transformation A --f B be a second-order process, so that

5= lee-d2 (1)

where u is a constant. Let B -+ C be a first-order process such that, as a

completely independent process, unit quantity of B

decays according to the relation

y= 1 -e-Bt

where B is a constant.

(2)

Consider now the successive transformations A -+ B --+ C. The rate of decay of A and therefore of formation of B at time t is obtained by differentiation of equation (1). The amount of B formed in time- interval dt, at t, is therefore

2at, * e+@- dt,. (3)

The amount of B formed at t, which subsequent time t is, by combination of

= 2&r * e-&la - e-fl(t-tl) s &,,

remains at

(2) and (3),

and the total amount of y present at time t is therefore

= I

sat, . e-Q . ,-8&t,) . at,,

0

where t, is the variable.

t y= 2ae-pt

s

t,e-b+Bh) . &, (4) 0

Completing the square for the variable exponential coefficient,

s

t y = &. e-Pt. ,P/4a t, . e--cc(t, -B/2ctY . &,

0

and changing the variable from t, to d/b: (tl - #I/2a) = x

J-BI~QG

=e -IQ . ,imaa

The last integral term in this expression corresponds to the error function, or integration of the Gaussian equation for normal distribution of errors, and can be written in a form, such that its value can be derived from standard tables. So that, on replacing

I& (t -/3/2a) for z,

y = ,BV4a . e -I% _e -act - B/2a)8 1 t

0

B

+5 u ,/- [P(t 6 -/3/2&J + P(@&)]] (6)

where P(B/2 v’%) and P(t I&---_B/2 I.&) can be

evaluated from tables. After simplification, equation (6) may be written in the form

y = ,-pt ( 1 _ e-a(tx-;t)

+ t J

i * eBe/4c( [P(t l/O: - j3/2 l/Cr) + P(b/22/Cr)] (7) a 1


But since, as indicated in the text, it is reasonable to

assume that thermoelectric power due to any product

is linearly related with y, the quantity of that product

present, equation (7) may be rewritten

E = E2e-@t 1 _ e-a(t2-;t) (

(8)

where E is the contribution to the observed thermo-

electric power of the product B at any time T, and

E, that of a thermocouple element composed wholly of

product B with respect to the one composed wholly

of the end product C.

It now remains only to demonstrate how the

constants of the equation may be evaluated. By

inspection, it is evident that the expression within

brace brackets of equation (8) increases with time,

and finally approaches a constant value K, when E subsequent’ly decreases according to the relation

E = KE,eeBt.

Identifying this relation with the linear tail of curve (a)

in Fig. 4, the slope of line (b) corresponds to a /I value

of 9 * 9 x 10-a min-l. Some difficulty now arises in

the evaluation of the remaining constants. Firstly, it

is clear from Figs. 2 and 4 that a rapid decay process

precedes the successive second-order and first-order

processes to which the present analysis refers. For

lack of sufficient data, this is presumed to be a simple

first-order process of half-life of about two minutes.

The complete decay of the initial product reduces the

observed thermoelectric power from 100 per cent of

the initial value to 90 per cent; that is, to a value E, corresponding to the thermoelectric power of the

product A, which decays according to the relation

E = E, . e--a(t--7)e,

an incubation period 7 being introduced to represent

the delay caused by the first process. The constants

other than /? of equation (8) are now evaluated by the

method of successive approximation. First, the value

of r is assumed to be approximately 20 min, then a

value for a is calculated for which the value of the

expression reduces to a half in the appropriate time

interval of (t - T) = 80 min. K * E, is now the

thermoelectric power corresponding to the time

r = 20 min on line (b) of Fig. 4. The R.H.S. of

equation (8) can now be evaluated for various times

over the whole isothermal annealing range. K is the

value of the expression in brace brackets of equation

(8) corresponding to very long times. Bearing in mind that the annealing times are taken from t = -r (20 min)

as zero, equation (8), thus evaluated, is plotted as

curve (B) in Fig. 5, and then subtracted arithmetically

from the experimental curve (X) to give curve (A). New values of u and T are now found which satisfy

curve (A), and the procedure repeated until a satis-

factory agreement is obtained. The value of 20 min for 7 was found to be sufficiently accurate, but the

ultimate values of the other constants are, a = 2.2 X 10e4 min-z, /3 = 9.9 X 1O-3 min-l, E, = 90 per cent’, E, = 23 per cent, K = 1.9, and

b/Zcl= 22.5 min. The terms,

A = Eie-“(t-‘? (9)

B = $ * e-p(t-7) (t _ ‘12 _ $ (t _ T)

(10)

are evaluated in Table 4 for times corresponding to

actual annealing periods after allowing for the incuba-

tion period r = 20 min, during which a simple

first-order process of 2-min half-life is assumed to

account for the initial 10 per cent change in thermo-

electric power.

The sum of t’he terms

A+B=S (11)

is plotted as the full curve in Fig. 2, the initial rapid

decay being shown as a broken curve. Apart from

some initial scatter, agreement with observed values

is satisfactory.

00 2s

TIME OF ISOTHERMAL ANNEALING AT IOO’C- min

FIG. 5. Analysis of isothermal annealing curve of H.C. copper strip cold-rolled 96 per cent reduction in thickness.

the kinetics of recovery and recrystallization of copper from hardness and thermoelectric-power...

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