The concept of ideal pre-precipitation in supersaturated solid solutions

HAL Id: jpa-00236682

https://hal.archives-ouvertes.fr/jpa-00236682

Submitted on 1 Jan 1962

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The concept of ideal pre-precipitation in supersaturated solid solutions

T. Federighi

To cite this version:

T. Federighi. The concept of ideal pre-precipitation in supersaturated solid solutions. J. Phys. Ra- dium, 1962, 23 (10), pp.795-806. �10.1051/jphysrad:019620023010079500�. �jpa-00236682�

THE CONCEPT OF IDEAL PRE-PRECIPITATION IN SUPERSATURATED SOLID SOLUTIONS

By

^T.

FEDERIGHI,

Istituto Sperimentale ^{dei Metalli} Leggeri, Novara, Italy.

Résumé. ²⁰¹⁴ Un ensemble de résultats expérimentaux ^récents obtenus par des ^mesures de résis- tivité sur le processus de formation des ^zones G.-P. dans des alliages ^{riche en aluminium} (phéno-

mène de pré-précipitation), suggèrent l’opportunité ^{dans une} solution solide sursaturée, qui par

ses caractéristiques spéciales peut ^être appelée ^« pré-précipitation ^{idéale ».}

La pré-précipitation ^{idéale est un} processus durant lequel, ^tandis que la taille des zones croît, leur nombre et leur concentration interne varient continuement de telle sorte que l’on ait ^un minimum relatif de l’énergie ^{libre du} système.

Dans ce papier, après ^une définition de la pré-précipitation ^idéale, l’importance ^du modèle, ^ses conséquences ^et les conditions d’observation pratique ^{sont discutées.}

La notion de pré-précipitation ^idéale apparaît ^{comme une base très} utile pour interpréter ^la précipitation ^{dans les} alliages réels ; quelques résultats récents montrent que par exemple

dans les alliages ^{Al-Zn 10} %, le processus de pré-précipitation semble suivre le schéma idéal Abstract. ²⁰¹⁴ A set of recent experimental ^{results obtained} by resistivity measurements on

the process of formation of G.-P. ^zones in Al-rich alloys (pre-precipitation phenomenon), suggests

the opportunity ^of defining ^a particular transformation in a supersaturated ^solid solution, ^which

for its special characteristics can be called " ideal pre-precipitation ^".

Ideal pre-precipitation ^{is a} process during which, ^{as zones} grow in size, their number and their internal concentration is changing continuously ^{in such a} way ^to give ^a partial ^{minimum in the}

free energy of the system.

In the present work, ^{after a} definition of ideal pre-precipitation, ^the importance ^{of the} model, its consequencies and the conditions for practical observation of ideal pre-precipitation ^{are discussed.}

The concept ^{of ideal} pre-precipitation appears ^to be a very useful basis for the interpretation

of pre-precipitation ^{in real} alloys ; ^some recent results show, ^for example, that in Al-10 % ^Zn alloy ^the pre-precipitation process ^{seems to} follow the ideal pattern.

1. Introduction. ^- The first process of decom-

position occuring

^{at about room} temperature ⁱⁿ

most Al-rich

supersaturated

^{solid solutions is}

given by

^the

clustering

^{of solute atoms} in rather small

regions,

^{which, are} coherent with the

parent

^solid

solution

[1].

^These

regions,

^whose

shape

^is

depen- ding

^{on the nature of the}

alloy,

^are

usually

^called

"

zones " or,

extending

^{the name first}

given

^to

thosé discovered in Al-Cu

alloys, "

Guinier-Preston zones "

(G.-P. zones).

^The existence of zones when

they

have grown

enough

^{can be detected}

by X-ray analysis,

but many other

physical

^{and mechanical} parameters ^{can be}

employed

^to

investigate

^their

properties [2, 3].

It is well known for istance that

resistivity usually increases,

^{reaches a maximum}

and then

continuously

^decreases

during

^{the iso-}

thermal

aging

^{in most Al-rich}

alloys.

It may be usefull ^{to stress} that the

clustering

process is often called

pre-precipitation [1, 4] ;

^the

reason of this term is that the process of formation of zones is not a true

precipitation

process ^{in the} usual

metallographic

^and

crystallographic meaning.

Although

^{from the}

thermodynamic point

^{of view}

there is no doubt that G.-P. zones are a new

preci- pitating phase,

^{it is}

proposed

^{here to conserve the}

word

pre-precipitation

^to

distinguish

^the

clustering phenomenon (for

which the usual concept’ ^of

nucleation is not fundamental

(see

e.g. ref.

[5])),

from normal

precipitation

processes where the for- mation of a new

phase

with différent

crystallo- graphic

^{structure is}

occuring.

The

starting point

of this paper is

given by

^{a set}

of recent

experimental

^{results obtained}

during

resistometric

investigations

^{on the}

pre-precipi-

tation process

expecially

in the Al-Zn 10

% alloy [6],

but also in many other Al-rich

alloys

FIG. 1. ^- Maximum ’increase in resistivity ^{for the}

Al-10 % ^Zn alloy ^versus aging temperature [6].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010079500

(see e.g. ref.

[7]

^{and other not} yet

published results).

A first fundamental and

general

observation deri- ved from these researches is

that,

^{at least as}

judged by resistivity variations,

^the

pre-precipitation

pro-

cess is

strongly dependent

^{on the}

aging

tempe-

rature. An

important example

^is

given

ⁱⁿ

figure

1,

where the maximum increase in

resistivity

^obser-

ved

during

isothermal

aging

of the Al-10

%

^Zn

alloy

^is

plotted

^{versus the}

aging

temperature ; ^the

analytical law

^{which was found to}

interpolate

^the

experimental points

^{is also}

given.

^Another

impor-

tant

phenomenon

^{which has been}

expecially

^obser-

ved in the Al-10

%

^Zn

alloy (but

^{which can be}

observed in a less evident way also in other Al-rich

alloys)

is related to a

change

^{in the}

aging

tempe-

rature : if

during

isothermal

aging,

temperature ^is

suddenly

^{increased or decreased}

just

when resis-

tivity

^{has reached its maximum}

value,

^{one obser-}

ves a

sharp

^decrease or,

respectively,

^{an increase}

in

resistivity

^{as if the}

specimen

^would attempt ^te re-establish the values in

resistivity

^{which one}

would have observed

by aging directly

^{at the last} temperature. ^An

example

^{of this}

phenomenon

^for

the Al-10

%

^Zn

alloy,

^{is shown in}

figure

^2.

FIG. 2. ^- Isothermal and di-isothermal aging ^{at 0 and}

400C for the AI-10 % ^Zn alloy quenched from 300°C [6].

In the

previous quoted

paper the above

experi-

mental results were

tentatively interpreted by

sup-

posing

^that the .number of zones which forms per unit volume is

depending only

^on

aging

temperature

(with

^{the law}

given

ⁱⁿ

fig. 1)

^{as if it were a sort of}

thermodynamic

property.

By

^such

hypothesis

^it

is obvious that if the

aging

température ^is

changed,

also the number ^{of zones} has

correspondingly

^to

change

^{in such a} way ^to restore the correct

equi-

librium

number,

ⁱⁿ agreement with the results in

figure

^2.

Since the

proposed interpretation

^{had at that}

moment no theoretical

foundation,

^it

appeared

necessary ^to try ^to

develop

^a model for the pre-

precipitation

process, which could ^account for the observed

experimental

^results.

The aim of this paper is

just

^to

analyze

the pre-

precipitation

process from ^a

quite general point

^of

view,

^{and hence to} define and ^to discuss the ^con-

cept ^{of a} very

simple transformation,

called " ideal

pre-precipitation ",

^{which can furnish a} very useful tool in

interpreting experimental

results. Of course, many

single points quoted

^{in the}

following

are far to be _{new ;} it is

reputed

however that the

point

^{of view}

adopted

^{here in}

considering

^the

whole

pre-precipitation

process is

différent, although

^{in some}

aspect symilar,

from that assumed in other theories on the

subject [8, 9].

^{As a}

matter of

fact,

^no

employment

^has been’ made

a

priori

of the well-known

concepts

^{of "}

spinodal "

line and of "

instability "

^{of a}

supersatured

^solid

solution

against

^small concentration fluctuations (see e.g. ref.

[2]).

Due to the

complexity

^{of the}

subject

this paper has

been

^{limited to introduce}

only

^the

basic

^ideas related to the ideal process and ^to advance some

intuitive considerations

concerning

^a

simple

^model

of solid solution.

Quantitative analysis

^{of the}

theory

ⁱⁿ

particular significant

^{cases has been out-}

lined for future works.

2. The state variables. ^- Let us consider a

binary alloy

^{and let be N =} Na + Nb ^{the total} number of atoms of the two kinds A

(solvent)

^and

B

(solute)

for unit volume and _xo ⁼

Nb JN

^{the con-}

centration of the

alloy.

Concerning

^solute atoms, ^{one has of course to}

distinguish

^{between two different} situations since

they

^{can be} found inside zones or as free atoms in the matrix. To describe

satisfactorily

^{the state of} system ^{let us then introduce the}

following symbols :

z : number of zones per unit

volume ;

xs : residual solute concentration in the matrix

(outside

zones) ;

x., solute concentration inside _{zones ;}

m : total number of atoms

(of

^{kind A and}

B)

^inside

a zone.

Since we are interested to

give

the " state " of the system

(in

^a

thermodynamic

sense)

by only

^a

few

variables,

it is necessary ^to make some

simpli-

fications.

At this purpose it appears useful ^to define for the

following

^a very

simple

^{model which can be}

called the " basic model ". In a basic model it is

supposed

^{that at a}

given

^{time both the size m of}

zones and their internal concentration x. ^are the

same for all zones and that both the residual matrix concentration outside zones Xe. and the internal zone concentration xz ^are

homogeneous ;

furthermore it is

supposed

^{that the}

shape

^and

internal structure of zones are

strictly

^determined

by

^the

knowledge

^{of m and} xz ; the parameter m

can then be assumed as a

satisfactory

^{measure of}

the size of _zones, without

being

necessary ^to spe-

cify

^{their exact}

shape.

By

^{the above}

assumptions

it is clear that the

knowledge

^of m, ^z and x.

(besides

temperature

T )

^is

quite

sufficient to

give

^{the state of the} system ; ^the residual concentration in the matrix is

easily

^cal-

culated

by

^the

equation :

which can be considered as an

equation

^{of state of}

the system.

It is very useful for the

following

^{to introduce}

other

important

variables. Let

be the maximum number of zones which is

possible

to form

by collecting

all solute atoms into _zones, when their size is _{m ; then} the ratio :

will be

conventionally

called the " zone-fraction ".

It is easy ^to

verify that q

^{is also the} ratio between solute atoms inside zones and total solute atoms.

Finally

^another useful variable is

given by :

which can be called the "

precipitate

fraction ".

Since _m, Xzi z, Xs, zM, q and p are ^seven variables related

by

^four

equations,

^the

knowledge

of any three. of them is sufficient for the

knowledge

^{of the} system. ^Two additional sets of variables which will be sometimes

employed

^{in the}

following

^are

given by m,

xz, q and

by

m, xz, p.

Let us now state the range of the ^most

important

variables. Due to its

definition, q

^is

ranging

^from

zero to

unity (the

smallest not-null values is

given by q

⁼

mxzlxo N) ; concerning

^{the size m of} zones,

we can

conventionally

consider also small clusters

(m ⁼

2, 3,

⁴

...)

^as zones ; the

highest

^{value of m}

is

given by

^{mM =}

Nb/xz. Regarding

^{the number z}

of _zones,

obviously

^{it is}

ranging

^{between zero to} ZM, where _zm,

given by (2),

^can reach very

large

^values

when m is small.

Finally,

^x. can assume values smaller than

unity

When ^{zones are}

large enough ;

of _course, when zones are formed

by only

^{a few}

atoms x. will be assumed

equal

^to

unity.

¹

3. The

pre-precipitation

process. ^- Since in the

case of a basic model it is

possible

^to

give

^the

state of the system

by only

^{a few}

variables,

^for

example T,

^{m, z and xz, it must be}

possible

^to

exp; ess

thennodynamic

^functions

only by

^means

of

the.n ;

ⁱⁿ

particular

^one is interested in the free energy AF ⁼ AU - TAS :

where

Af_is

^the

free-energy

^variation per ^atom with

reference

to the

homogeneous

^{solid solution}

without zones (z ⁼

0)

^{and at the same} tempe-

rature T.

Now one has " a

priori "

^{two différent} expec- tations about this function. It may be that,

whichever are the values of _{m, z} and xz, ^the value of the

free-energy

^is

higher

^{than that one} compe-

ting

^{to the}

homogeneus

^{solid solution}

(z

⁼

0) ;

ⁱⁿ

this case the

homogeneus

solid solution is the stable state of the system and there it will be no

tendency

^{to a}

clustering

process.

On the contrary, ^{if the formation of zones} pro- duces a decrease in the

free-energy,

^{as one should} expect ^{in the case of a}

supersatured

^solid solutions,

there will be a

tendency

^{to a}

clustering

process. In this case the

knowledge

of the above function

(5)

will allows us to calculate for any temperature

T,

the

thermodynamic equilibrium

^values

m+,

^z+

and xZ (and

^{in the same time}

q+

^and

p+), namely

the values for which the

free-energy

^{of the} system obtains its minimum

value A f +.

From a

general standpoint

^{one should}

expect

that the above values should be obtained

by

^sol-

ving the equations (*) :

Keeping

^{in mind the above}

considerations,

^one

can define the

pre-precipitation

process in ^our basic model as a transformation of the state, of the

system

during which, starting

^from the super- saturated

homogeneus

solid solution (z ^-

0 ;

m ⁼

1),

the size of _zones,

namely

m, is

growing continuously

from the small values ^{of m}

competing

to the

homogeneus solid-solution,

^{to the value m+}

competing

^{to the}

thermodynamic equilibrium

state ;

correspondingly,

^{also z and x,, will}

change

ⁱⁿ

such a way ^to reach the final

equilibrium

^{values z+}

and

xz .

^{Therefore to describe}

satisfactorily

^the

transformation of the system ^it is necessary ^to know in which _way the number ^{of zones z} and their internal concentration zz are

changing

^{as m is}

growing, namely

^{it is necessary to}

give

^{the func-}

tions :

(*) ^It may be however that this is not always ^true since,

due the physical restrictions of the range of variables, ^the

mathematical minimum in the free

energy

^{could occur for}

values of the variables outside their physical range ; in this

case one can expect that the function Af obtains its physical

minimum value when at least one

independent

variable has reached a value on the boundary ^{of its} physical range of variation.

A

priori

^{one can} select almost

arbitrarily

^these

functions,

^since

thermodynamic gives only

^{a con-}

dition for their final values :

In other words from

thermodynamic

^conditions

only,

infinite "

paths "

^are

possible

^for

pre-preci-

pitation, starting

^{from the}

supersaturated

^and

homogeneus

^solid solution and

reaching

^{the final}

equilibrium

conditions.

Among

these infinite

paths

^for

pre-precipitation,

in the next section a

particular

^one will be selected which for its

peculiar properties

^{can be called}

" ideal

pre-precipitation

^".

Before

going

on, it may be useful to stress that in

equations (9)

^and

(10),

and in all the

following,

m and not time has been assumed as the funda- mental and

independent

^{variable of the}

process ;

this is due to the fact that the attention is focused

on the

thermodynamic

^{and not on the kinetic} aspects ^{of the}

phenomenon.

4. Ideal

pre-precipitation.

^{- To introduce the} concept ^{of ideal}

pre-precipitation

^{let us consider}

again

^{the first two}

thermodynamic

conditions

(6)

and

(7),

^{derived from the}

free-energy

^function

(5).

These are two

equations

^{in four}

independent variables ;

^{if we} consider T and m as

parametérs,

it is then

possible

^{to solve the}

equations

^{and to} get :

For any constant

temperature

^the above equa- tions are

depending only

^{on m. These}

functions,

considered in the range between ^{m =} 1 to m ⁼

m+,

define

conventionally, for

any ^constant temperature,

the ideal

pre-precipitation transformation.

It is easy ^to understand the

physical meaning

^of

ideal

pre-precipitation :

^{this is a} process

during which,

^{at constant} temperature, for any value of _m, the number

of

^{zones z and their internal concen-}

tration Xz are such ^to

partially

^{minimize the}

free-

energy

of

^the system.

We will call î and xz the "

pseudo-equilibrium "

values for the number and internal concentration of zones

respectively. In a

similar way it is pos- sible to define

q, p

^and

Ai ;

^{the last one is the}

pseudo-equilibrium

^or maximum decrease in the

free-energy

^{of the} system ^{when m is} constant, ^and

can be obtained

by putting

^{into the}

expression (5) of Af

^{the values}

of z

^{and xZ}

^{given by (13)}

^and

^(14).

Of course,

also Ai is depending only

^{on m and T :}

As it will be better understood in the

following,

the

shape of AJ(m)

^{at constant}

temperature

^{T is}

of fundamental

importance

ⁱⁿ

establishing

^the

behaviour of the process.

It is

possible

^to introduce the

concept

^{of ideal}

pre-precipitation

^{in a} way which is ^not

explicitly dépendent

^on

equations (6)

^and

(7).

To this pur- pose let ^us first suppose that the internal ^concen- tration xZ of zones is contant so that it can be discarded in the discussion. Then for any ^cons- tant

temperature T

^the

free-energy

^{is a function}

only

^{of m and z. Consider now m =} ml ^{as a} para- meter and let us

imagine

^{that an}

increasing

^number

of _zones, all

of

^{the same size} ml ^are

forming.

^Since

we have

supposed

^{that the}

homogeneus

solid solu- tion is

supersaturated,

^the formation of an increa-

sing

^{number of zones will}

produce

^{at the}

beginning

a decrease in the

free-energy ;

however if we ima-

gine

^that the process of formation of ^zones is pro-

gressively going

^{on, till to} form the maximum pos- sible- number of _zones,

given by (2),

^{one can} expect ^{that the}

free-energy

^will

finally

^increase

again,

since the solute concentration in the matrix will be lowered of the end below that one com-

peting to

^the solid solution in

thermodynamic equi-

librium ^with

particles

^{of size} ml.

FIG. 3. ^- Expected free energy variation in function of the number z of zones for different sizes of zones (scheme).

The situation is sketched in

figure 3,

^curve m1.

In

conclusion,

^{one can} expect ^{that for a}

given

size _{of zones,} m ⁼ mx, ^a

pseudo-equilibrium

^num-

ber of

zones -Z, exists, producing

^a maximum de-

crease

Af,

in the free energy ; for différent values

of z, Ai

^{is not minimum. One can} repeat ^the above considerations for zones of

higher

^{size m2 ;}

let us suppose that ^curve m2 is the

corresponding

variation in the

free-energy ;

^the

equilibrium

^num-

ber of ^zones is

given

^now

by z2,

^which

usually

^will

differ from

z1. Since,

^{as it is}

supposed

ⁱⁿ

figure 3,

the maximum decrease in the

free-energy

^for

m2 is larger

than for ml,

following

^{the ideal} process there it will be the

tendency

^{to the}

growth

^{of zones}

from ml ^to m2 and to a

change of

their number

from Zl to Î2’

^{Just the same} situation is

occuring

for _m3 and _{m4 ;} let us now suppose however that _m4 is

coincinding

^{with the}

thermodynamic equilibrium

value

m-1 ;

then for m. > m4, ^the maximum ^de-

crease A/g

will be smaller than that observed _{for m4,} and the

growth

^{of zones will} stop ^when

they

^have

reached the size m+ ⁼ m4.

If one

imagines

^to represent ^the

surface Ai

^versus

variables m and _z, it is obvious from the above considerations on

figure

^{3 that}

during

^{the ideal}

process the

index-point

^{of the} system ^{on the sur-}

face

given by Ai (m,

z), ^is

running

^{on the bottom}

of a " wrinkle " and is

continuously decreasing toward A/+

^{as m is}

growing

^{to m+.}

The above situation is not

substantially

^différent

if also the variable xZ is

considered ;

^of course, for any value of m ^{one has to} suppose ^to

change

pro-

gressively

^{the value xz and to find out the}

pseudo- equilibrium

^{value xZ which}

together

^{with Z}

gives

the lowest value in the

free-energy.

5. Thé free-energy ^{function. -} To go further into the

analysis

^of

ideal-pre-precipitation

^{it is}

necessary ^to

give

^an

appropriate expression

^{for the}

free-energy which,

^on the other

hand,

^{will be}

strictly dependent

^{on the nature of the} system.

It

is

yet

possible

^{however to advance some} gene- ral

considerations,

^{which can} be very useful for

applications

and calculations and

which,

^{as it will}

be shown in the

following section,

^{allow to} get

some

important

^and

general

results when

applied

to a

single

^case of solid solution.

As it was stressed in the

introduction,

^{from a}

thermodynamic point

^of

view,

^{zones can be consi-}

dered as a new

growing phase ;

it is then

possible

^to

apply

^to

pre-precipitation

many concepts

usually employed

^{in the}

theory

^of

precipitation.

^{Let us}

consider to this purpose ^the

free-energy diagram (see fig. 4)

in function of

composition x

^{and let}

IA(x)

be the

free-energy

per ^atom of the

parent

^solid

solution and

f B(x)

^the

free-energy

per atom of the

precipitate (it

^{is to stress that}

f B(x)

^is

concerning

the solid solution of A in

B,

^{which is considered}

now as a

solvent,

^{and B}

having

^{the same}

crystal- lographi,

^{structure as}

A).

From the above

diagram

^{it is}

possible

^{to calcu-}

late as it is well

known,

the decrease in the free- energy per ^atom

0 f 1,

in function ^{of the} fraction of

precipitate

p ⁼

mzin,

^{in the}

hypothes

^is

that

^the

precipitate phase

^{is all collected and " outside "}

the solid solution. In other

words, A/,

^{is the}

"

volume " variation ^{of the}

free-energy.

^Since

(1- p)

^{is the} fraction of the

original

^solid

solution,

whose

composition

xs, for

equations (1)

^and

(4)

^is

given by

^.

one has

simply :

It is necessary ^stress

again that Õ.¡ 1

^is

depending only (apart

^from

xz)

^{on the}

product

^{mz but not}

separately

^{on the number or size of zones.}

From the

geometrical interpretation

^{of the above}

formula,

it is well known

that Af 1

^is

given by

^the segments OP between

fA(xo)

^{and the}

interception

with x ⁼ x. of the line between

fA(xs)

^and

f B(xZ),

^as

it is shown in

figure

4 for several values of x. and

FIG. 4. ^- Calculation of 0394 f x ⁱⁿ function of xg(p) ^{for a}

constant composition ^{of zones.}

one value of zz. Therefore if

A/f1

^{were the}

only

contribution to the variation of the free energy, ^one could

easily

obtain the maximum decrease in the free energy,

OP", by drawing

^the tangent ^{at the}

two curves and

intercepting

^it

by x

⁼ xa.

One has however to consider other

important

contributions ^to the variation in the

free-energy

^of

the system, ^{which are}

arising

from the fact the the second

phase

^is

dispersed

^{in z}

equal particles

of size _m, inside the parent ^{solid solution. These}

contributions will be

conventionally

^{indicate as}

"

dispersion

contribution ". In

particular,

^one

will have contribution to the internal _{energy and} contribution to the entropy.

The contribution ^to internal energy have been

recognized

^{from a}

long

^{time and are}

substantially

two :

a)

^the interfacial

energy AU.

^and

b)

^the

strain

energy AUe.

Concerning

^the variation in entropy ^{it is}

possible

to

distinguish

^three

contributions,

^{that is :}

c)

^the

mixing

entropy

AS., d)

^{the local} entropy ^{ASt and}

finally

e) the orientation entropy

ASO.

^{These con-}

tributions

have just

^the

same meaning

^as in the well known case of

point defects ; namely

^the

mixing

entropy ^is

arising

from the fact that each zone can

be formed in many

equivalent places

in the matrix and it has been considered for the first time

by Dehlinger

^and

Knapp [8] ;

^{the local} entropy ^{is due}

to the

change

^{in the} vibrational entropy ^{inside the,}

matrix and inside the G.-P. zone when the zone

is

put

^{into the}

matrix ; finally

the orientation

entropy ^is

arising

from the fact that ^{a zone can} have several

equivalent

orientations.

One has therefore for the variation of the free- energy per ^atom

A/2,

^{due to the}

dispersion

^{of zones}

in the matrix :

where the various contributions are functions of _m, z

and

eventually

^{of other variables as x. and T.}

The total variation of the

free-energy,

^{due to the}

formation of z zones, each

containing m

^{atoms and}

of internal

composition

xz, is therefore

given by :

This

equation corresponds strictly

^{to the free-}

energy

expression (5),

^{and can be}

employed

^{to cal-}

culate the

free-energy variations,

in function of

number, sizé

^and

composition

^of zones, and there- fore the whole ideal

pre-precipitation

^transfor-

mation.

Of ^course the above formulae

(17), (18)

^and

(19)

are yet

quite general equations ;

ⁱⁿ

practice,

^to

make calculations it is _necessary to

specify

for any

physical

system ^{both the functions}

f A(x)

^and

f B(x) (to evaluate Af1,),

and the five

dispersion

^contri-

butions to internal energy and entropy

(to

^eva-

luate

0394f 2).

It is to stress

however,

^{that from a}

general point

of

view,

^the

dispersion

contributions have ^not all the same

importance.

^{More in}

detail,

it appears that AU. and AS. ^{are to} be considered as " funda- mental"

contributions,

^since

they

^are

always present

in any system. On the other

hand,

^{even if the}

remaihing

contributions A

Ue,

^{ASt and}

AS,

^{can be}

very

important

in many cases,

they

^{are not to be}

considered ^as fundamental _ones, since it is

possible

to

imagine

^an

alloy

where there is not strain- energy, ^no local variation in entropy ^{and where}

also the orientation

entropy

^is

negligable (this

should _occur, e.g., in ^a systeln where the solute and solvent atoms have the same ’radius and where zones, ^at least when

they

^{are not too}

small,

^are

spherical

ⁱⁿ

shape).

6.

Properties

^{of ideal}

pre-precipitation.

^{- To} try

to obtain a

general insight

^{into the features of the}

ideal process, it appears useful to select a

simple

but

représentative

^{case ;} for this purpose it will

now be

supposed

^{that the}

^parent

solid solution is

an ideal and dilute

(e.g. xo 0.05)

^{solid solu-}

tion,

^{and that}

precipitating

^{zones are formed}

only by

^{solute atoms}

(xz

⁼

1).

If E is the solution enei gy of B into A and Sf ^the local entropy ^of

solution,

^{one can then assume as}

an

appropriate expression

^of

fA(z) :

where the

entropy

^{term in the}

right

^{member due}

to the diluteness of the solution has been

approxi-

mate from the well-known

complete expression

(see e.g. tef.

[10]).

Concerning f B,

^the

only

condition for it is that it has to reach its minimum value for xz ⁼ 1. Let

us suppose ^that this minimum value is

given by :

where c is a parameter,

eventually depending

^on

temperature. Figure

5 illustrates in a schematic way the

shape,

^{at several} temperatures, ^of

JA(x)

FIG. 5. ^- Calculation of Afi at several temperatures (scheme).

and also of

f B(x) (the parameter

^c

being supposed

constant).

By

these functions it is ^now

possible by (17)

^{to calculate}

Af 1.

Concerning

^the

dispersion

contributions

only

^the

fundamental ones and the local entropy ^{will be}

considered. Let us suppose, for

simplicity

^{that m}

is

large enough

^{that it is}

meaningful

^{to define the}

surface of a zone

(e.g.

m >

10),

^and furthermore that zones are

spherical in shape. ^Then,

^{for a f. c. c.}

metals of lattice constant a it is

straightforward

^to

calculate the total surface of zones in function of _{m ;} hence if _y is the

specific

interfacial energy between zones and

matrix,

^{one has :}

where

Of course y ^can

depend

^{on the}

composition

^of

zones and

matrix ;

however it will

supposed

^here

that it is

practically

^constant.

Coining

^{now to the}

mixing entropy,

^{one can}

obtain a

satisfactory approximation

^{of it}

by

sup-

posing, following ^Dehlinger

^and

Knapp [8],

^that