X-ray investigation of the segregation in Cu-Ti alloys

(1)

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X-ray investigation of the segregation in Cu-Ti alloys

H. Gürzing, V. Gerold

To cite this version:

H. Gürzing, V. Gerold. X-ray investigation of the segregation in Cu-Ti alloys. Journal de Physique I,

EDP Sciences, 1992, 2 (6), pp.1129-1144. �10.1051/jp1:1992201�. �jpa-00246593�

(2)

J.

Phys.

I France 2 (1992) l129-1144 JUNE 1992, PAGE 1129

Classification

Physics

Abstracts

X-ray investigation of the segregation in Cu.Ti alloys

H.

Giirzing

and V. Gerold

Max Planck Institut fur

Metallforschung, Stuttgart

and Institut fur Metallkunde,

University

of

Stuttgart, Germany

(Received J8 November J99J, revised and

accepted

26 March J992)

AbstracL The

segregation

of Cu-Ti

alloys

with various

compositions

_was

investigated using single crystals

and monochromatic

X-rays

(Cu~ and MOK«). The

developing

side bands

along (h00)

of the

reciprocal

lattice _were

investigated

_up to

(800).

The results _were

compared

with those from _a

simple

one-dimensional

segregation

model. The difficulties and limits of interpretation _are shown. No

principal

difference in the

segregation

behaviour _was found for

single

crystals

having

Ti concentrations between and 2.9 at.fb.

1. Introduction.

In the past ²⁰ years the

alloy

system

copper-titanium

has attracted many researchers since it seemed to be a

typical

_system for

spinodal decomposition.

In

addition,

the

steep

^increase ^of the

yield

stress in the

age-hardened

state is remarkable. The

precipitation

behaviour of these

alloys

has been

investigated

for _a

variety

of

compositions using

different

experimental

methods _as

X-ray diffraction,

transmission electron

microscopy (TEM)

_or field ion

microscopy (FIM). According

to the

theory

the kinetics of

precipitation depend

_on the

position

in the

phase diagram

relative _to the

spinodal.

Thus for

high temperatures

^and ^low titanium contents

(I.e.,

above the

spinodal)

_a normal nucleation and

growth

_process does

occur whereas for low

temperatures

^and

high

titanium concentrations

(I.e.,

below of

it)

a

spinodal decomposition

is

expected.

This

decomposition

is characterized

by

the

wavelength

L

of the concentration modulation

occurring mainly

in _one direction which is

typically

(100)

in fcc

alloys.

It is

expected

that this

wavelength depends

on both the

alloy composition

and the

aging

temperature.

The

typical X-ray powder

diffraction pattem of _a

spinodally decomposed alloy

shows _so- called side bands

(SBS)

^close to the

Debye-Scherrer (DS)

^lines. ^These side bands _are observed if

segregation

occurs in _a

supersaturated

solid solution.

They develop

because of _two _reasons.

The first _one is _a variation of the local electron

density.

In the well-known book

by

Guinier

Ill

this is called substitutional disorder _; it _causes also small

angle scattering.

This contribution

does _occur

only

if the difference of the atomic

scattering

factors of solvent

(f~)

and solute

~fB)

is

large enough, I.e.,

if _a considerable monotonic Laue

scattering (MLS)

of the solid solution does exist which is

proportional

to

(f~ f~

)~. ^The

intensity

contribution of the SBS

JOURNAL DE PHYSIOUEI -T 2. N'6, JUNE 1992 42

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1130 JOURNAL DE PHYSIQUE I N° 6

is taken out of the MLS which is reduced at the _same time. For the side bands of the DS lines

(H00)

with H=

2, 4,

the

intensity

varies

proportional

to

(f~ f~)~ii(Ah)

where H

= 2 ao sin

f§/A,

Ah

=

h H

=

2

ao(sin

8 sin

f§)/A

= 2 ao ^cos

f§ A8/A,

_ao the lattice

constant, 8 the

Bragg angle

and

f§

the

Bragg angle

of the

corresponding

DS line. The

function

ii (Ah )

describes the

profile

of _a

pair

of the side bands and is

expected

to be the _same function for all orders H.

The second _reason for the _occurrence of the SBS is _a variation of the local lattice constants due _to the

segregation (displacement

disorder

Ill coupled

with substitutional

disorder).

In that _case the

intensity

of the SBS increases with

increasing Bragg angle

and reads

approximately h~ f~i~ _{(Ah ).}

Now the

intensity

is taken _out of the DS

lines,

which intensities

are reduced _at the _same time

according

to a static

Debye-Waller

factor which

typically

increases with

increasing aging

time. For

spinodally segregating alloys

_as Cu-Ti this

contribution is the dominant _one if the

investigation

is made

by X-rays.

Most

segregation phenomena

_can be characterized

by

_a

length parameter

L which _can be evaluated from the distance of _a side band from the DS line. This

length

_can be

interpreted

either _as the modulation

wavelength

of _a one-dimensional

spinodal decomposition

_wave, or as an average distance between

neighbouring particles

in _a nucleation and

growth

_process. For a

copper

alloy containing

4.7 at.fb Ti

Miyazaki

et al.

[2]

found _as _an

example

L _t ^"~ from the

beginning

of _an

aging

treatment at 400 °C without _a time _range where L _was constant. In

addition,

from the intensities of the side bands

they

calculated _a

steadily increasing

concentration

amplitude

with

aging

time. As it is discussed in the next

paragraph, however,

there exists

already

_a

segregation

in this

alloy immediately

after

quenching

from the

homogenization

temperature.

Obviously,

the observation of side bands in

polycrystals

is not sensitive

enough

to _see the

beginning

of the

decomposition

of this

alloy.

In _an

alloy containing

2.9 at. fb Ti Hakkarainen

[3]

could show

by

TEM

investigation

that _a modulated structure resulted

already

after

aging

for 8s at 500 °C. With

increasing aging

time

superstructure

spots ^of ^the ^metastable

phase p' (Dla

structure

Cu4Ti) develop.

The contrast of the

image depended

_very much _on the contrast conditions. Either _a modulated structure or

discrete

particles

could be _seen. Similar results _were found

by

Comie _et al.

[4]

and Datta et al.

[5].

For

alloys containing

3.8 _or 5.2 at.fb Ti modulated structures could

already

be _seen after

quenching

from 860°C. The

coarsening

of the modulated structure followed _a rule L

-- t ^"~

Recently, experiments

with field ion

microscopy (FIM)

contributed _a lot to the

knowledge

of

spinodal decomposition.

With this method the local

composition

within _a very small volume element

containing only

tens of atoms could be evaluated.

Using

this method Biehl and

Wagner [6, 7] investigated

the

aging

of _a 2.7 at.fb Ti

alloy

at 350 °C and found a

gradual

increase of the concentration fluctuation. After 50 min _a Ti content of 20 fb _was reached in the enriched

regions

which is the

equilibrium

value for the metastable

p' phase Cu4Ti.

In these

experiments

a continuous

growth

of the

wavelength

L was found from the

beginning according

to L _t ^"~ in accordance with the

X-ray

results mentioned above.

In contrast to these results Alvensleben and

Wagner [8]

found _a nucleation and

growth

behaviour for the metastable

p' phase

for _a 1.9 at.fbTi

alloy aged

at 350 °C. From these results the authors

judged

that at 350 °C the

spinodal

must lie between both concentrations 1.9 and 2.7 at.fb Ti. Table I

gives

_a _survey of the

positions

of the

spinodal

found

by

different

authors. In

spite

of the differences it _can be concluded that at 350 °C the

position

of the

spinodal

should lie between 1.3 and 2.7 at.fb Ti.

The present

experiments

have been undertaken with

X-ray

diffraction from

single crystals

of different titanium content. Since the former

X-ray experiments

have been

performed

^with

polycrystals

it is felt that the _use of

single crystals

_may enhance the resolution

especially

for

(4)

N° 6 X-RAY INVESTIGATION OF THE SEGREGATION IN Cu-Ti ALLOYS l131

Table I. The

position of

the

spinodal

in the

phase diagram

Cu-Ti

according

_to various authors.

Alloy Temperature

Reference

at,fb °C

1.0 350 9

~ l.3 and

~ 2.7 350

6,

7, 8

2.33 300 10

2.57 400 10

3.06 500 10

3.86 600 10

4.05 700 to 725 11

5.20 400 _to 500 5

short

aging

temperatures. ^The

experiments

_were

performed

with

alloys

on both sides of the

spinodal

in the

hope

to find characteristic differences in the

experimental

results because of the different

types

of

segregation

_processes. In

addition,

model calculations _were

performed

which _were used _to find

segregation _parameters

from the

experimental

results.

2.

Experimental procedure.

Two

single crystal

disks _were

provided by

Professor

Wagner

from GKSS

Forschungszentrum Geesthacht, Germany,

which contained 2.7 and 2.9 at.fb

Ti, respectively.

Both had _a flat

(200)

surface which could be used for the present diffraction

experiments.

In

addition, single crystals

with _a lower titanium content were

produced by

the

Bridgeman technique.

Their

compositions

_were 1.0 and 1.3 at.fb

Ti, respectively.

From the

crystals

disks _were

prepared

which had also _a

(200)

surface.

The

homogeneization

treatment occurred at 870 °C for 3 h in _a vertical fumace in _an argon

atmosphere.

To avoid oxidation the

crystals

were

wrapped

in _a niobium foil and

quenched

into

liquid nitrogen

after the heat _treatment. For the

aging

in _a temperature range between 200 and 450 °C the

crystals

were

encapsulated

in _a

glass

tube filled with argon. After

aging

for

a time t~

they

were

quenched

into _water where the tubes broke into peaces. For short

aging

times the

crystals

_were lowered into _a salt bath.

The diffraction

experiments

_were undertaken in _a 8 2 8 diffractometer

using

monochro- matized CuKa and MoKa

radiation, respectively.

The

quality

of the monochromator allowed

an

intensity

ratio of 30 _: between the aj and a~ radiation. With this

equipment

the sequence of

Bragg peaks (H00)

with H

=

2, 4, 6,

and 8

including

their side spots

(which

furtheron will be called side bands for convenience _as in the _case of DS

lines)

were measured

along

the line

(h00)

in the

reciprocal

lattice. The latter two

peaks (600)

and

(800)

could be measured

only

with the MoKa radiation.

Figure

I shows _a

typical

_sequence of side band

profiles

at the

(200)

peak using

CuKa radiation. The first

sign

of side bands could be observed

already

after

aging

(5)

1132 JOURNAL DE

PHYSIQUE

I N° 6

normalized I/lm

0.002 CuT12.9at%

fi

T"350°C _i j

,/ i

) _~,

^'

l

o.ooi i,

o

1.86 1.93 2 2-07 2.14

rec. lattice vector h

Fig.

I.

Intensity profiles

of side bands (more accurately, side spots) in units of the peak

height

of the

Bragg peak

(200) for various

aging

times at 350 °C. CuKa radiation.

for 20 min _at 350 °C. The

experiments

were continued up to

aging

times of 100.000 min

(about

48

days).

The

following

_parameters were taken from the

experiments:

at

first,

the average

wavelength

L

(measured

in units of

ao)

^of ^the concentration fluctuation could be determined

according

to the

equation [12, 13]

Htan

(f§)

~

(A8)(H~

₊

K~

+ L ~)

where

(HKL)

_are the indices of the

Bragg peak, 8~

is the

corresponding Bragg angle

and he is the

Bragg angle

distance of _one of the two side bands from the main

peak.

Since there is

usually

_a

slight asymmetry

ⁱⁿ ^the ^distance ^of ^the two side bands _an average value is taken.

Parallel _to this method of L determination the _same parameter was tried _to find from line

broadening

measurements of the side bands. Corrections for

geometric broadening

have to be made in order to get ^the ^correct ^half ^width ^B. ^The

separation

from strain

broadening

occurred

by comparing

the widths of the

(200)

and

(400)

side bands

using

CuKa radiation

[14, 15].

In order to characterize the

intensity

of the side bands the

following

two parameters were

investigated

the

intensity

ratio

R~~m (I~)/(I~~~)

of the

peak heights

of the main

peak (I~)

and of the side band _at the

high angle

side

(I~~~),

^and ^the ^ratio

R~j

_m

(I~)/(Ij)

of the

peak heights

of the side bands on the

high

and low

angle

side of the main

peak

is taken.

3.

Experimental

results.

3,I INTRODUCTION. The present paper is restricted to results of the

aging

temperature 350 °C where _most of the

experiments

were undertaken. In all _cases the

as-quenched single crystals

did not show any

sign

of side bands which

developed only during aging.

Since there

were no

principal

differences in the results of different

alloy compositions, only

the measurements with the Cu-2.9 at.fb Ti

crystal

will be

reported

in detail.

3.2 THE MODULATION WAVELENGTH L. The variation of the modulation

wavelength

L

with

aging

time is shown in

figure

2. For the evaluation of L _an average of all Ah values of the

peaks (200)

and

(400)

has been taken. For the short

aging periods

_up to

(6)

N° 6 X-RAY INVESTIGATION OF THE SEGREGATION IN Cu-Ti ALLOYS 1133

wavelength L

CuT12.9at%

~ 6h (CuKa)

~~~

+ 6h (MoKa) B (CUKa)

g

i~

m.0.1941

10 loo 1000 10000 IOOOOO

aging

time t

[min]

Fig.

2. The

development

of the modulation

wavelength

L with

aging

time at 350 °C

showing

tllree ranges I to III of

wavelength growth.

The results from line

broadening

measurements are included.

100min the accuracy was limited to L

= 17±1. For the

higher

order reflections the

determination of the

peak positions

and therewith the accuracy of L _was _no _more _so

good.

The

resulting

_curves for both radiations _are _more _or less the _same.

They

_can be devided into three ranges. ^In range I the

wavelength stays nearly

constant ; it lasts up ^to 150 min

(2.5 h).

In range II _a

slope

of _m

= 0.19 is observed in the double

logarithmic plot

which increases _to

m = 0.36 in range III. The border between both ranges is about 900 min

(15 h) aging

time.

The latter

slope

value is close to _m ₌ 0.33 which is the

typical

value for Ostwald

ripening

where either the

particle

radius _or the _mean

particle

distance is used _as

parameter.

In

figure

2 the results for L from line

broadening

measurements are

plotted,

too.

They

show

agreement only

in range II. The

failing

in range ^I may ^be an

argument

^that ^the diffraction is caused

by segregation

without

sharp

boundaries. The latter is needed for _a correct

interpretation

of line

broadening.

It should be mentioned,

however,

that Eckerlebe et al.

[16, 17]

could detect discrete

particles already

after

quenching

of this

alloy.

3.3 THE _INTENSITY _RATios

R~~

AND

R~j.

^The next

question

is the

intensity development

of

the side bands

compared

to the

intensity I~

of the

Bragg peaks

which is

given by

the

intensity

ratio

R~~.

One has _to take into

consideration, however,

that

I~

itself is

decreasing

with

increasing aging

time and with

increasing

value of h while the

intensity

I~~~ ^of the side band is

increasing. Figure

3 shows the time

dependence

of all ratios measured with CuKa

(H

=

2 and

4)

and MoKa

(H

=

2, 4,

6 and

8)

in _a

double-logarithmic plot.

The ratio

decreases with time and order of reflection and reaches values of

nearly

_one. The h

dependence

of

R~~

is shown in

figure

4 for various

aging

times. For data from range II

(150

to

10.000min)

the

slope

of the _curves in the double

logarithmic plot

is about the _same for different times and _amounts to values of 2.5 _to 2.7. In range III the _curves start to flatten off when the

intensity

ratio reaches values close to _one. For the ratio

R~j

^{of the}

peak heights

of both side bands,

only

three results _are shown in

figure

8b where

they

_are

plotted

_as _a function of the order H.

At the late

segregation _stage

extra

spots (h'00)

with h' around 5.91 and 7.88 could be observed which could

belong

either to incoherent

p' (the

lattice constant in the

tetragonal

_c

direction is

enlarged by

^1.5

fb)

or to the stable

phase Cu~Ti.

It _can be

suggested

that the appearance of this incoherent

phase

determines range III.

(7)

l134 JOURNAL DE PHYSIQUE I N° 6

lm/lhsb

+- (2QQ)CU

-- (4oo)~cu

~~- (200)~MO

- 14oo),Mo

- 16001,Mo

+- (wol,Mo

T@350°C

i

io loo loco loooo looooo

aging time [min]

Fig.

3. The

development

of the

intensity

ratio

R~

= I~/I~~~ ^{of the}

peak

intensities of the main

Bragg

spot

(J~)

and of the

high angle

side spot ii_h~b) _as function of the

aging

time (at 350 °C) for various orders h of the (h00)

Bragg

spots.

lm/lhsb

loo ? iso mIn

+ 25O mIn

S 600 mIn

° 1600 mIn

X 6600 mIn

° 16700 mIn

1

6 325O0 mIn

Z 100000 mIn CuT12.9at%

T*350°C

o.

i io

Fig.

4. The h

dependence

of the

intensity

ratio R~~ ₌ i~/I~~~ (see

Fig. 3)

for various

aging

times at 350 °C.

4. Model calculations.

4.I THE ONE-DIMENSIONAL MODEL. In order to understand the

development

of the

intensity

data and _to try ^to

interpret

the results in terms of

parameters

^of the

developing

microstructure the

simple

one-dimensional

segregation

model shown in

figure

5 is used _as _a first step. ^In a

periodic region

of

length Lao

_a

segregation

of _a

supersaturated

solid solution

with solute concentration

Co

into two

phases (I)

and

(2)

with concentrations

Ci

and

C2

is assumed

(Fig. 5a).

As _a consequence of

this,

the average lattice

plane

distance

do

₌

ao/2 changes

to two different values

dj

=

(I

₊

ej) do

^and

d~

₌

(I

+

e~) do.

^This

change

(8)

II

L.ao

.

L2.a2 ^~~

al Liar

Co C2 Un

b)

x

Fig.

5.- One-dimensional

segregation

model where _a solid solution with solute concentration

Co

_segregates (a) into two

regions

with different concentrations C~

resulting

in (b) shifts u~ of lattice

planes

n.

can be described

by

_a shift u~ of the lattice

plane

_n from its undistorted

position (Fig. 5b).

The

resulting slopes

define the misfit parameters ej and e~.

Then,

the

following

relations hold L

j +

L~

=

L

(I)

Cj Li

+

C~L~

=

COL (2)

ej

Lj

₊

e~L~

=

0.

(3)

Since the concentration variations have

only

_a small influence _on the

X-ray

intensities

(the

atomic numbers of Cu

(29)

and Ti

(22)

_are close to each other and the maximum

segregation

AC amounts

only

to 20

fb)

there remain _two

independent parameters

ej and

Lj

besides the

wavelength

^L ^which ^have to be determined from

experiments.

Since the _amount of misfit

depends

_on the local

composition

an information on the latter _can be deduced from the data.

The model leads _to the

following

diffraction

amplitude [14]

_:

~

_

~~ sjjjjjji~+~jj iii

+

f, ~ii iii) _~iellll~ ~~"~ ~~~~~~~~~

~~~

in this

equation fj

^and

f~

are the atomic

scattering

factors of

regions (I)

and

(2)

which do not

differ very much from each other

and, therefore,

have been

replaced by

an average

f.

Parameter P is _an

integer

and describes the number of

periods

L in the

segregation.

From

equation (4)

the

intensity

I amounts to

1=

f~[A)+Aj+2AjA~cos (grLh)]I~(P) (5)

where A

i and A

~ are the two fractions inside the brackets

given

in

equation (4).

The last factor

I~(P )

in

equation (5)

is the square of the

corresponding

fraction in

equation (4)

and has its main

peaks

of

equal heights

at the ideal

positions

Ah

=

v/L from the

Bragg peak

with

v =

0, 1, 2,

...,

which _are broadened with

decreasing

number P of

periods

of

lengths

L. These

peaks

_are modulated

by

the other functions in the brackets. The terms

Al

_and

Al

have their main

peaks

at

positions

shifted from the

corresponding Bragg peak

H

by

Ah

= e~

h~.

^If the

region

enriched

by

Ti is called

(I)

then ej ~ 0 and A contributes

mainly

to the

low-angle

side band. The third term

(T)

in the brackets is _an interference term which

can take

positive

and

negative

values.

As _an

example, figure

6 shows the three terms A

), Al

and Tat the

(600) peak position

^for a

(9)

1136 JOURNAL DE

PHYSIQUE

I N° 6

L~55.5, L1.18.5,E1.0.00155,E2.-O.oo077 Intensity

[arbitrary

unitsl

12

8

)

4

o 6

4 Ai A2

2 /

-2 -4

-0.04 -0.02 O O.02 o.04

deviation ah

Fig.

6. The

intensity

contribution of the tllree terms in the brackets of

equation

(5) at the

(600) Bragg peak.

Below the tllree terms

A), Al

and T, ^their peaks ^marked

by

_arrows. Above the _sum _curve I~~ ^of all tllree terms. Model parameters used _: L

=

55.5

Lj

=18.5 (both in units of the lattice constant) _; _ej

= 1.55

x10~~

_The

positions

of I~ are marked by ^vertical ^lines.

typical

set of model

parameters.

^Their

peak positions

_are marked

by

_arrows. The

peak positions

off

~

are

given by

vertical lines. For _v

= 0

(the Bragg peak position)

all three _terms in the brackets

give

a

positive

contribution which results in _a

large Bragg peak intensity.

As

shown in the upper part ^of ^the

figure

the _sum _curve

I~~

of the bracket _terms demonstrates the interference of all terms. In

addition,

the maxima of this _curve

I~~

have different

positions

as

compared

to the maxima of

I~, I-e-,

the last factor in

equation (5).

The

peaks

of

I~~

are determined

by

intemal

strain,

their

positions

_vary ^with ^the ^order H of the

Bragg peaks

while the

peaks

off

~

are determined

by

the

segregation wavelength

L. The latter dominate for the

intensity

distribution I of

equation (5).

With

(I) increasing

order H and

(it) increasing wavelength

L the

overlapping

of the functions A

j and

A~

decreases because of

(I) increasing peak separation

and

(it) increasing sharpness

of the

intensity profiles. Therefore,

the interference term T decreases in

intensity

which results in _a decrease of the

corresponding Bragg peak.

This model differs from that used

by Tsujimoto

et al.

[18] by

the interference term T which

they ignored.

For the further discussion it is assumed that the model does

give

also _a _correct

peak height

for the

Bragg peaks.

^This would _mean that the

experimentally

measured

intensity

is the _sum of the intensities of many domains of dimensions FL _as it is the _case for the side bands. The fact that there exist three different orientations for the one-dimensional

segregation

is

ignored

for the

following

_reasons. At

first,

there exists

only

_one _set of side spots at

Bragg peaks (hoc)

since the

corresponding

intensities for side spots in the k and

f

_directions

are

negligible.

Secondly,

_as it will be discussed

lateron,

the matrix has

locally segregated

into three

dimensions which will influence the

interpretation

of the parameter set of the _one-

dimensional model.

Thus,

the whole

segregated

volume of the

crystal

will contribute to the side bands _as well _as _to the

Bragg peaks.

4.2 THE _DEPENDENCE _OF _INTENSITY RATIOS ON THE MODEL PARAMETERS. in order to

study

the influence of the

independent

^model parameters

P, L, Lj

and _ej _on the

intensity

ratios

R~~

^and

R~j

_one ^of the parameters was varied while the other three _were

kept

constant.

(10)

The first

example

shown in

figure

7

gives

the influence of the number P of

periods

_on the ratio

R~~

as a function of the reflection order h. This

diagram

_may be

compared

with the

experimental

results in

figure

4.

Only

the _curves with P

= 10 and 50 _are close _to _one of the

experimental

_curves

(t~=250min). Therefore,

all models have been calculated with P

= 50. There is

only

_a

slight

difference in the

slope

of the

corresponding

curves which is 2 for the model _curve and 2.5 for the

experimental

_one.

The

following figures

_represent the influence of the distortion

parameter

ej

(Fig. 8)

and _a

combination of _ej and

Lj (Fig. 9)

on the ratios

R~~(h)

and

R~,(h).

With

increasing

ej the ratios

R~~

decrease while the

slope

of the _curves remains the _same in the double-

logarithmic plot (Fig. 8a).

For the ratio

R~,(h

an increase is observed, too, but the _curves _are

no more

straight (Fig. 8b).

In the latter

figure

_some

experimental

results _are shown also which will be discussed later.

In

figure

9 the

product

^of _e,

Lj

_was

kept

constant while both

parameters

were varied

simultaneously.

^For ^the ^ratio

R~~ only

_a small variation of the _curve is observed

(Fig. 9a)

while _a

large

variation _occurs for the other ratio

R~j (Fig. 9b).

Thus it _can be concluded that the ratio

R~~(h

^will be useful _to determine the

product

_F, L

provided

^the

experimental

data

show _a similar h

dependence

than the model data.

TheTeafter,

_a

separation

of both

parameters _e, ^and

L,

is

possible by comparison

of the ratio

R~j. Again,

a similar h

dependence

is necessary.

lm/lhsb

io

4~ P.3 L.55.5

n p.5 L1.18.5

° P.10 E 1. 0.00155

P.50 E2.-o.000755

1

10 rec. lattice vector h

Fig.

7. The h dependence of the

intensity

ratio R~ on the number P of

periods

L for the _one- dimensional model.

5.

Comparison

between

experiment

and model.

5.I THE ALLOY CONTAINING 2.9 at.fbTi. Before the

comparison

is undertaken _a few

remarks should be

given

what

parameters

^could ^be

expected.

The metastable

miscibility

_gap

(11)

II 38 JOURNAL DE

PHYSIQUE

I N° 6

lm/lhsb

ei _. o.oo i

e i

/

O[O014 /

.~~~~

E1.O.O022 O-O018

~"~~~

L.55.5.c0nst. L1.18.s~const.

10

al

_rec. _iatt;ce _vector _h

lhsb/llsb

w---_

Experiment iO,COO

~

1.8

t~,

^min +

--, _~

~

l,200

~ ^°

1.6 ~ 350 ~

___~ 2.2

~

)_

~

l.8

Model ^*~

1.4

L . 55.5

L~

^.18.5

~~

~

l

,

2 4 6 8 10

b)

Fig.

8. The h

dependence

of the

intensity

ratios la)

R~

and (b) R~j on the distortion parameter ej of the one-dimensional model. In

figure (b)

tllree

experimental

data sets are included.

at 350°C has the

approximate

concentration limits

C~

= 0.3fb and

C~

= 20fb for the

depleted

matrix and the

precipitate

in the later

stage, respectively (all

data

given

in at.fb

Ti).

From these concentrations the volume fraction of the

precipitate

is

expected

to be

f~

= 13.2 fb. From the

composition dependence

of the lattice constant

(ao/nm

=

0.3616 ₊

0.044.C)

the

following

distortion

parameters

can be

suggested:

_e~

=19.6x10~~

_and

e~=

-3.2x10~~

_The _latter _data _have _to be

compared

with the model parameters ej and e~.

The

comparison

between

experimental

^results ^and ^model calculations _came out to be very difficult. For the

product

_{ej L}

j it _was

possible

to reach _an

agreement

^for most

aging

times with the

exception

of _very

large

_ones. The results _are shown in

figure

10 where the

experimental

data

points

from

figure

3 _are

compared

with calculated model _curves. For each

aging

time another

product

_ej

Lj

was found in this way

(see Fig. 12).

The difficulties of

interpretation

arise for the

separation

of both

parameters

ej and

Lj, I.e.,

the

interpretation

of the

intensity

ratio

R~j

^of ^both ^side ^bands. ^There were two

problems

which could not be solved

reasonably

well. At

first,

the H

dependence

of

R~j

^did not agree with that of the model _curves for the

higher

orders H

= 6 and 8. This is

(12)

lm/lhsb

ioo

Li,ei

L1.23.5, E1 _. -O.O012

a)

10

lhsb/llsb

L.55.5.const. ~~

Lie~.

L1~L2,

e1,

E2 _. variable

10

b)

_rec. _lattice _vector _h

Fig.

9. -The h

dependence

of the

intensity

ratios (a) R~~ ^and (b) Rhi _on the variation of ej and

Li

under the condition ej

Lj

=

0,0285.

shown in

figure

8b where _some

experimental

data _are

given

as a function of H.

They

can be

approximated by

the model

only

if

R~j

is

steadily increasing

with H which is found

only

^for

short

aging

times below 1000 min. Thus the

comparison

was

only

undertaken for

H

=

2 and 4. In table II _a few model data _are

given

which have been evaluated from

diffraction

experiments

for different

aging

times. The numbers for L

j, ej and e~ are the best

fits found in this way.

The second

problem

is the

interpretation

of these

parameters. They

were derived from _a one-dimensional model where the ratio

Lj/L

should

correspond

to the volume fraction

f~

of the

segregated phase

enriched in Ti. The

corresponding

values in table II _are much too

high

and

they

decrease with

increasing aging

time.

Since the

segregation

does _occur in three dimensions the obtained data for

Lj

and ej have to be discussed in this way. As the _next

possibility

the model could be

interpreted

_as _a

one-dimensional

projection

of _a three-dimensional

periodic

_array of cubic

particles

of size L j in _a cubic

primitive

arrangement as it is sketched in

figure

II. The

particles

_are located in

region (I)

where the distortion parameter _ej ^is a

weighted

_average of the distortions in the

precipitate (e~)

^and ⁱⁿ ^the ^matrix

(e~).

In this model e~ is then identical with e~. In that _case

(Lj/L)~

should

give f~

in _an

approximate

_way, which is _now too

large

(13)

1140 JOURNAL DE

PHYSIQUE

I N° 6

lm/lhsb

o calculation

% ~

experiment: ^°

O (200)

° (400)

~'~ ~ (800) MO

z j800) MO o.oi

io ioo 1000 ioo00 iooo00

aging time [mini

Fig.

IO. The time

dependence

of the ratio R~~ ^for the four orders of reflection for

experimental

data

(symbols)

and for the model (curves)

resulting

in _an evaluation of the

product ejLj

_as _a function of

aging

time.

(Tab. II). Therefore,

_one has to increase L

j

considerably

in order _to fit

f~

which results in the data shown _as _curves PM in

figure12.

As _a next

possibility

_one has to assume that the cubic

particles

_are _more _numerous in

region (I)

of

figure

11. As it has been found from _neutron small

angle scattering experiments by

Eckerlebe et al.

[16, 17]

the

particles

in the later stages are rod-like and have _an aspect

ratio which increases from 1.5 _to 4.5 with

increasing aging

time. If L

j is

supposed

to be the diameter of the rods, then for _two of the three orientations

(rods parallel

to y and

z in

Fig.

II

a

projection

_on the _x axis would contribute to the distortion in this direction. This would increase the volume fraction and therewith allow _a decrease of

Lj (resp. Lj/L) compared

_to

the cubic

particle

model and would result in the _curves _« PM ₊ asp.r. » in

figure

12.

However,

this still would _not

satisfy

the model value

R~j

as shown in

figure

13 for the

Bragg positions (a)

H

= 2 and

(b)

H

=

4.

Table II. The parameters

L, Lj

^and _ej ^and

functions of

them.

ta L El L

j

L

I ~l ~2 L

l~~ (L

l~L)~

min _nm

10~~

nm nm 10~~ 10~~ fb fb

20 5.7 2.66 2.2 1.2 0.8 39 5.8

120 6.4 6.01 2.2 2.7 1.4 34 4.0

000 9.6 16.24 2.4 6.8 2.3 25 1.6

10 000 14.8 26.37 3.7 7.2 2.4 25 1.5

30 000 21.2 38.95 4.4 9.5 2.4 21 0.9

(14)

3-dim. PM

L

hl

Fig.

ii. Tllree-dimensional

particle

model with

regions

(I) and (II) normal to the _x direction.

L,

L1

Inml L1*eljnml

Ll.el

L

PM

PM _. asp. r PM

a wra a+

r~

CuT2.9at% Tz350°C

lo loo IOOO IOOOO IOOOOO

al

_aging _time _[mini

strain El

,

e2

T=350°C ^' ^~'"~'

jj

_PM

+ asp. r.

PM

~~

~

-O.O04

~~~ IOOO

lOOf°

^1°°°°~

lo

~~;ng time [mini

Fig. 12. Variation of the model parameters ^with ^the

aging

time of the Cu-2.9 at.9b Ti

crystal

for tllree different models _: a) the parameters L, _ej

L,

and

Lj

_; b) the parameters e, and e~.

(15)

1142 JOURNAL DE PHYSIQUE I N° 6

lhsb/llsb

experiment: _CuT12.9at% T«350°C

° (200) Cu

2.5 ^~ (200) _Ma

calculation=

2 PM + asp, r.

PM + CIUSt.

~ ~ ~ ~

x

~'~

~

~w

^I

~

~ ^~

l

lo loo 1000 10000 100000

a)

aging time lmin]

lhsb/llsb

experiment:

° (40O) Cu

~ (40O) MO ^~

o o

calculation; _o

o ~ ~

PM ₊ _asp. _r. '

~

xi

+ ~ o a

x

~ ~o

oo

o °

I

io loo loco loo00 looooo

b)

aging time lmin]

Fig,

13. The time

dependence

of the ratio R~, ^for

experimental

data

(symbols)

and for _two models (curves) for tile side bands (a) at (200) and (b) at (400). The resulting parameters Lj, _ej and e~ are shown in

figure

12.

In order to get a better fit _one still has to decrease L

j/L

which would _mean that _now still

more

panicles

have _to be concentrated in

region (I)

of

figure

I I in order to reach the

expected

volume fraction

f~

=

13.2 fb. So to

speak

it is _a model in between the one-dimensional _one where the whole volume of

region (I)

is enriched in Ti and that _one where

panicles elongated

in the y and z directions with

large

spaces ^of

depleted

matrix in between _are

filling

this

region.

This model is called cluster model and the

corresponding

curves « PM ₊ dust. _» _are found in

figure12

and the

corresponding

values in table II.

Only

these data fit the model

parameter R~,

for H

=

2 and 4 _as shown in

figure

13 but _no fit could be obtained for the

higher

^orders 6 and 8. At

higher aging

^times ^this ^fit ^fails also for the order H

=

4 _as it is shown in

figure

13b.

It must be admitted that in these estimates the different contribution of the third

orientation

(rods

in the _x

direction)

has _not been taken into account. A _more _accurate calculation would need _a much

higher

effort but would contain _so many ^not well known

parameters

^that ^it ^has not been

performed.

5.2 THE ALLOY CONTAINING 1.0 at, fb Ti. For this

alloy

_a nucleation and

growth

behaviour

is

expected

for the

precipitation

kinetics.

However,

all diffraction

experiments

with this

alloy

(16)

L,

Ll

Inml

Ll*ei

inml

L

+ asp, I

+ clust.

Ll* El

i

CuTilat% T«350°C

o-i

too 1000 10000 IOOOOO

al

_aging _time _lm;nj

strain E2

PM , clust.

CuTilat% T.350°C 0.006

PM _. asp- r.

~i

PM

~~

L2

0.00~

^~~

loo 1000 lO000 100000

b)

aging time lmin]

Fig. 14. Variation of tile model parameters ^with ^tile

aging

time of the Cu-1.0 at.9b Ti

crystal

for three different models _: a) tile parameters L, _ej

Lj

and

Lj

; b) the parameters ej and e~.

single crystal

did not show remarkable differences in the

development

of side bands

except

a

slower

segregation

rate

(Fig. 14). Therefore, only

the ranges I

(up

to 1000 min

aging time)

and II _can be _seen for the

development

of the

segregation wavelength

L. At the transition

between both _ranges _a kink in the _curve

Li

_ej ^is deduced which results from

reproducible

kinks in the

R~~(t)

curves for all orders of reflections.

Otherwise,

the _same arguments ^hold for the model

adaptions

_as for the former

alloy crystal.

6. Conclusions.

The

present investigations

have shown that there is _no marked difference in the

development

of the side band

profiles

for both

alloys

discussed _so far _as for the other

alloys

studied in the range ^between ^I ^and 2.9 at,fbTi. Thus it _cannot be

proved

if for the

higher

concentrated

alloys

the

segregation

_process is

govemed by spinodal decomposition

as it has been found

by

FIM

techniques [6, 7]

for _a similar

alloy (2.7

at.fb

Ti)

for the first 50 min of

aging

at 350 °C.

(17)

l144 JOURNAL DE

PHYSIQUE

I N° 6

Obviously,

it _was also

impossible by

neutron small

angle scattering,

to find any prove for this mechanism

[16, 17].

In that _case the diffraction _curve is caused

by changes

of the local Ti concentration and _not

by

local

changes

of the lattice

parameter

as in the present

experiments (which

makes the

interpretation

of the results _even _more

difficult). Obviously,

diffraction

profiles

from different

segregation

_processes as nucleation and

growth

or

spinodal decomposi-

tion do _not differ very much from each other which makes it

impossible

to differentiate between these both mechanisms.

Acknowledgments.

The authors _are very much

obliged

to Prof. R.

Wagner

of the

Forschungszentrum

Geesthacht, Germany,

who

provided

them with slices of

single crystals having higher

titanium

contents

(2.7

and 2.9

at,9b).

The critical comments

by

him and

by

the referees of the

joumal

have

improved

the

manuscript

and _are

gratefully acknowledged.

References

[Ii GUINIER A., X-ray Diffraction in

Crystals, Imperfect Crystals

and

Amorphous

Bodies (Freeman and

Company,

San Francisco, 1963).

[2] MIYAzAKI T., YAJIMA E, and SUGA H., Trans. JIM 12 (1971) l19.

[3] HAKKARAINEN T., PhD Thesis,

University

of

Technology,

Helsinki, Finnland (1971).

[4]

CORNIE J. A., DATTA A. and SOFFA W. A., Metall. Trans. 4 (1973) 727.

[5] DATTA A. and SOFFA W. A., Acta Metall. 24 (1976) 987.

[6] BIEHL K. E. and WAGNER R., Proc. Int. Conf, _on Solid-Solid Phase Transformations, H. J.

Aaronson et al. Eds. (TMS/AIME,

Pittsburgh,

Penn., 1981)

p.185.

[7] BIEHL K. E. and WAGNER R., Proc. 27th Int. Field Emission

Symp.,

Y. Yashiro and N.

Igata

Eds.,

Tokyo, Japan

(1980), _p. 267.

[8] ALVENSLEBEN L. V. and WAGNER R.,

Decomposition

of

Alloys

the

Early Stages,

P. Haasen

et al. Eds., 2nd

Acta-Scripta

Conf. Series

(Pergamon

Press, Oxford, 1984) _p. 143.

[9] SAXLOVA M. and BALIK J., Czech. J.

Phys.

B 31 (1981) 215.

[10] DUTKIEWITZ J., Bull. Acad. Polan. Sci. 22 (1974) 323.

[I

Ii

LAUGHLIN D. E. and CAHN J. W., Acta Metall. 23

(1975)

329.

[12] DANIEL V. and LIPSON H., Proc. Roy. Sac. A181 (1943) 368.

[13]

DANIEL V. and LIPSON H., Proc.

Roy.

Sac. A182

(1944)

378.

[14] DEHLINGER U. and KOCHENDbRFER A., Z. Kristall. A101 (1939) 134.

[15] GURzING H., Doctoral Thesis,

University

of Stuttgart,

Germany

(1990).

[16] ECKERLEBE H.,

Diploma

Thesis, University of Hamburg-Harburg (1985).

[17] ECKERLEBE H., KAMPMANN R. and WAGNER R.,

SANS-Investigation

of

Early Stage Precipitation

Kinetics in Cu-2.9 at,9b Ti _; Atomic

Transport

and Defects in Metals

by

Neutron

Scattering,

C.Janot, W.

Petty,

D. Richter and T.

Springer

Eds.

(Springer-Varlag,

Berlin

Heidelberg,

1986) _p. 66.

[18] TSUJIMOTO T., HASHIMOTO K. and SAITO K., Acta Metall. 25 (1977) 295.