On Static and Dynamical Young's Condition at a Trijunction

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On Static and Dynamical Young’s Condition at a Trijunction

C. Caroli, C. Misbah

To cite this version:

C. Caroli, C. Misbah. On Static and Dynamical Young’s Condition at a Trijunction. Journal de

Physique I, EDP Sciences, 1997, 7 (10), pp.1259-1265. �10.1051/jp1:1997122�. �jpa-00247453�

(1997)

OCTOBER 1997, PAGE

On Static and Dynamical Young's Condition at

_a

TYijunction

C. Caroli (~)

^and

^{C. Misbah}

_(~~*)

(~)

Groupe

de

Physique

des Solides

(**),

Universitds Paris 7 et 6, 2 Place

Jussieu,

75005 Paris, France

(~) Laboratoire de

Spectromdtrie Physique (***),

^Universitd Joseph

Fourier,

Grenoble I, BP 87, Saint-Martin

d'HAres,

38402

Cedex,

France

(Received

18 November

1996,

revised 20

February

1997,

accepted

10 June

1997)

PACS.68.45.-v Solid-fluid interfaces

PACS.81.10.-h Method of

crystal

growth;

physics

of

crystal growth

PACS.81.10.Aj Theory

and models of

crystal growth; physics

of crystal

growth,

crystal morphology and orientation

Abstract.

Using

_a consistent

purely

kinetic model _we show that

Young's

condition at _a

trijunction results from _a chemical and not a mechanical

equilibrium,

_as is often believed in the literature. In _an out-of equilibrium situation this condition is altered in _a way such that the triple point

acquires

_an effective

mobility.

The order of

magnitude

of the

mobility

is fixed

by

_a

velocity

scale

iatt

_{associated with} chemical attachment at the

interface,

and not

by

the sound

speed.

While for

ordinary

eutectics

ttt

_is

_{large (except}

_in

_rapid

solidification

experiments)

in

comparison to the

growth speed, implying

_a quasi-instantaneous

mobility,

finite

mobility

effects should show _up for faceted eutectics _as well _as for eutectoid transformations.

1. Introduction

This note is concerned with _a

question

which is in

principle

well understood

ill,

but in

practice

seems to

give

rise _to

systematic

confusion

namely

the

physical origin

^of

Young's equilibrium

condition at _a

trijunction,

I.e. a contact

point

between three

phases

of _a

binary

mixture.

This, typically, happens,

for lamellar eutectics

growing

from the

melt,

at the

triple point

where the interfaces between the two solid

phases

_a and

fl

_{emerge on} the

growth

front. The

point

which

we want to illustrate here with the

help

of _a

simple

model of isothermal

kinetically-limited

eutectics

growth

is the

following: contrary

to what is

usually stated, Young's

condition at the

triple point

^does not express mechanical

equilibrium,

but chemical

equilibrium. Consequently,

any

departure

from

thermodynamic equilibrium

leads to _a

velocity-dependent

correction to this

condition,

which _can be

interpreted

in terms of _an effective

mobility

of the

triple point.

While this effect is in

practice

small

enough

to be

neglected

in the _case of

growth

of _a metallic eutectics from the

liquid phase (except

under conditions of

rapid solidification),

it is

certainly important

when

dealing

with the

growth

^of faceted eutectics _as well _as for eutectoid

(* Author for

correspondence (e-mail: misbahflphase3.grenet.fr

_or

misbahflcoucou.ujf-grenoble.fr) (**)

Unitd associde _au CNRS

(*** Unitd associde _au CNRS

@

Les

#ditions

_de

Physique

1997

1260 JOURNAL DE

PHYSIQUE

I N°10

Fluid Phase Y

,' -' _' /

~ ,' ",

"

,'''

" Contact

angle

'

Contact

tingle

_,''

' ' , ' ,' '

_______-J>---'--- ~v~~~~~~~~~~~~~~~~~~~~~~

,

' '

A phase in the _B Phase

---><

j

C

Tfijunction

motion

Fig.

I. A schematic view of the

tri-junction.

transformations

(from

_a

homogeneous

mother solid

phase

into _a

two-phase daughter alloy),

due to the slowness of diffusion in solids

[2j.

Indeed,

in this latter

situation, growth

is

commonly

controlled

by

interface kinetics and

for

diffusion,

while diffusion

lengths

in the bulk _may become

comparable

^{with interface} thicknesses.

Then,

one expects on the _one hand

triple point mobility

effects _to become

important.

^{On the} other

hand,

stress

effects,

which

play

_an

important

role in solid

/solid transformations,

must be taken into account when

imposing

mechanical

equilibrium

of the

interfaces,

which

gives

rise to an

independent

condition.

2. The Model

Figure

I

displays

the

geometry

^of the

tri-junction.

^We use here the model of reference

[3j,

ⁱⁿ which:

. The

solid-liquid

interface is considered to be

sharp,

with

isotropic

surface tension ~.

. The

growing

solid eutectics is treated in the Cahn-Hilliard

representation. Namely,

the Helmholtz free _energy of _a volume V of the solid is

given by

F

"

/ _dvlflT,

C,

P)

+

&IVC)~l,

II)

NA

and

NE

will denote the numbers of A and B atoms in the

binary

mixture. For _a

homogeneous

solid the solute concentration _c and the number

density

_p of the mixture

~~~

~

NA~NB

^~

~~ ~~

~~~

Since _we

only

consider here isothermal

solidification, temperature

is _a

passive

parameter, which _we will omit from _{now on.}

. The two solid

phases

_are

assumed,

for the sake of

simplicity,

to have identical

physical properties f(c)

=

f(I c)j

and the

partial

volumes of the _two

species

_are taken

equal _[4j:

VA(C) "

UB(C).

. The

liquid

concentration is assumed to be constant and

equal

to

1/2.

These

assumptions

are consistent both with the situation of

global equilibrium

of the three

phases,

and with

kinetically

controlled

growth.

We shall consider _a linear kinetic law

JA

"

W(@FA

@SA)I

~B

_~

fl~(@FB /~SB) (3)

where

JA,B

are the _mass currents of the two substances _across the

solid-liquid interfaces,

W is _a

phenomenological

kinetic

coefficient,

and _{~tFA, tIFB are} the

(constant)

fluid chemical

potentials.

The _mass currents _are related to the normal

growth velocity in _by

J~

=

ii c)tn; _J~

=

ctn. _j4)

The chemical

potentials

_are related to the Helmholtz free _energy

by

JtsA =

-j(~()+(~)) ₁₅₎

T,p T,c

JtsB =

j

^~

_(~(

₊

_{~) (6)}

T,p T,~

Because _we consider _a

symmetric

^{model with the}

liquid

concentration

equal

to

1/2,

_we must have ~tFA = tIFB ^% tlF.

Equations (1-4)

describe

completely

the

growth dynamics

of _a diffusion- less

eutectic,

if

~ts(A,Bj(c)

are known _{as a} function of the 'order-like'

parameter

_c.

Therefore,

we must relate

explicitly

the chemical

potentials

^{with the solid}

composition

field. For _an

inhomogeneous

solid

phase,

and

taking

into account the

Laplace capillary

_{pressure, we} obtain from

(6) (where

derivatives _{are now}

interpreted

in the functional

sense)

/LsA -

II Ill

~

i7~Cl

+

Ill

~

+

~l~~ Ill

~

+ _vsA~~

17)

/LsB =

~

II Ill

~

i7~Cl

+

Ill

~

+

~l~~ Ill

~

+ vsB~~

18)

where _~ is the interface curvature, ^and vsA, vsB are the

specific

volumes of the two substances in the solid

phase.

Note at this

point

that

expressions (7, 8)

differ from the

corresponding

equations (3, 4)

ofreference

[3j. Indeed,in

that

work,

_a confusion _was made between Helmholtz and Gibbs

energies, leading

to _erroneous

expressions

^of the chemical

potentials.

While the

qualitative

conclusions of that work about the existence of

steady

and modulated solutions remain

valid,

the

quantitative

^{results about interface}

shapes,

_are incorrect.

The basic _source of the above mentioned confusion lies in the fact that

expression (I) implies

that variations such _as those needed to define chemical

potentials

_are made at constant volume

and _{not constant pressure.} One could of _{course recover} the correct result in terms of the

Gibbs free

enthalpy density (as

used in Ref.

[2]) provided

_one would enforce the condition of _a

constant volume

by

_means of _a

Lagrange multiplier, namely

the pressure.

We _now

specify

for

f

the usual

generic

form:

f=fo-h4~+lj4; j=c-j. _j9j

1262 JOURNAL DE

PHYSIQUE

I N°10

The three coefficients

b, I,

& _are functions of the

density

_p

(and

of the

passive

_parameter

T,

which _we

omit).

Our

assumption

^of a

completely symmetric

model

entails,

in

particular,

that the

partial

volumes _{v~ are}

equal.

It is shown in the

appendix

that this is true

provided

that

b, I,

are

proportional

to the

density

_{p, so}

that,

for

example: fi&/tip

=

Alp.

So

finally,

we

get

^for the chemical

potentials

_on the solid side of the curved L

IS

interface

~tsA =

'

_~~

12di

_411i3 +

iiv211

+

[)

₊

_pi1-d1i2

₊

_i14

+

( _iv~)2j

+

~p-i~ jio)

/lsB =

~~

~~12d4 4iil~

₊

_&i7~41

₊

~~

+

p-~ l-d4~

₊

_l1j4

₊

^(i7J~)~j

₊

~p-i~. iii

Note that fi

lo lap

=

~ts(c

₌

1/2).

We find it convenient to use, rather than

equations (3) separately,

the two linear combinations

[(I c)3(a)

+

c3(b)j

and

[3(a)-3(b)j,

which

yield,

_{upon use} of

equations (10, II),

l/" +

2l/ 2l/~

_{= l/}

/~ _l12)

+ y

~~ ~ ~

~,,

~r

~

~~

~ ~

~°

~

(i

+

y,2)3/2

^{~ ~ ~}

fi~~~~

^~

^~~

~' ~~~~

~~~~~

~

i d~~)(, _lc~ (, V~fi

~~~~

~J

(j /2i)~/2

a C

and the dimensionless

undercooling

b is

given by

b

=

~(lJts(1/2) ^{JtL@/2)1. lis)}

Note that _we

specialize

_our calculation to the _case where _a one-dimensional front

[ji(f)j

is

growing steadily along §

at the

physical velocity I.

_In

_equations (12, 13)

the a-dimensional

quantities

are measured in units of

tc.

3.

Equilibrium

_versus

Moving Trijunction

We first consider the _case where the

system

is in

thermodynamical equilibrium. Equations (12, 13)

then reduce to

1/" + 21/ 21/~ = 0

(16)

,,

~'~ 2~~

+

~~ do ji

~j,2)3/2

+ /~ ₌ °.

Ii?)

When each solid

phase

is in

equilibrium

with the

liquid

_one with _a

planar

interface equa- tions

(16, 17)

possess the

following

solutions

1/ = +I and b

= I. When the three

phases coexist,

a

planar

front is _no

longer

_a solution

(because

of the

tri-junction). Equation (16)

admits the

following

solution

1/(~)

₌

tanh(x). (18)

Plugging

this solution into

equation (17), integrating

from 0 to cc

(this

is taken to _mean from 0 to distances much

larger

than

tc;

recall that

lengths

_are reduced

by t~),

and

taking advantage

of the

symmetry

of

y(x)

_we

readily

obtain

where 6

designates

the usual contact

angle.

Note that this relation relates the contact

angle

to the

liquid-solid

surface tension

(the

surface tension ~ ^enters in the definition of

do)-

What remains to be shown is that the contact

angle

is related to the solid

/solid

wall energy as

well,

and this leads

precisely

to

Young's

^condition. For that _{purpose we} calculate the ~wall'

(or surface)

_energy

starting

from the Gibbs

definition,

and it will _appear

clearly

that the

capillary

condition is

intimately

related to the fact that

tc

is finite. Let

fw

denote that _energy. The Gibbs definition reads

(in physical units)

fw

=

~lim / _{dilf~o h4~}

+

'4~

₊

)4'~l ^Llf14eq)

⁺

^{fl-4eq)1 120)}

where the last term

represents

^the _energy in the _case where each solid

phase

would be alone

ire-

member that ijeq =

/~). _Upon

substitution of

expression (18)

into

(20)

_a

simple

calculation

provides

_us with

~

fw

=

~~~~

=

2~sin(6) (21)

3b

where _use has been made of the definition of

do together

with relation

jig).

This is

precisely

the

Young

condition for _a

symmetric

model. Had _we taken _a

non-symmetric

model _we would

then have obtained the

general

form of

Young's

condition: ~i

sin(61)

+ _~2

sin(62)

=

fw,

and

~i

cos(61)

_{= ~2}

cos(62),

where _~i and ~2 are the surface tension between the

liquid phase

and the two solid ones, 61 and 62 are the two contact

angles.

For _a tilted

pattern

^{the extension}

should be feasible

along

the _same lines.

We _now consider the _case of _a

growing

solid and derive the

dynamical Young

condition. The nonlinear terms due _{to y in}

equations (12, ¹³⁾ preclude

_an exact

analytical

solution. If

y'2

<

1, however,

_an exact solution is obtained

for1/ (see

also Ref.

[3])

1/ =

lltanh(xfij

(22)

2 2

with the condition V _< 2. Far away from the

triple point

_a

planar

front solution exists with

composition

_{1/o =}

+(I V/2),

and b

= 1

V2 /4

+

2V[1 2(h16) (1 V/2)]. Equation (13)

can be rewritten _as

doY" ~

ll'~ 21~~ ~()

+

l~~ ~(). 123)

Integrating

from 0 to _cc and

using equation (22),

_we obtain the

dynamical Young

condition

2~sin(6)

=

fwfi(1+

^~

₍₂₄₎

2 4

where

fw

is the

eq~tilibri~tm

wall _energy

(which

enters

Eq. (21)).

For small V _{we can} write

2~sin(6) fw

_ci

-fwV2/16.

This is _a

simple dynamical equation

for the motion of the

trijunction.

Remember that the

physical velocity

is reduced

by

Wh

/p,

and from _a dimensional

analysis

this is _a

velocity

related _to the molecular

attachment,

and not to the sound

speed

which involves the

compressibility

constant.

1264 JOURNAL DE

PHYSIQUE

I N°10

4. Discussion

We have shown that

. The two solid

phases

at

thermodynamic equilibrium

with the

liquid

at the eutectic _com-

position

have front

profiles

which

satisfy asymptotically Young's condition,

and that this

simply

follows from

imposing

^{the condition of chemical}

equilibrium only. By asymptotic

we mean for distances

larger

than l~.

.

Although

it is clear in _some papers

ill

that

Young's

condition follows from chemical

equilibrium,

confusions _are still

present

ⁱⁿ the literature. It must be

emphasized

that

expressing

chemical

equilibrium

does not

imply

at

all,

that mechanical

equilibrium

is

satisfied. To make this

point stronger

let _us consider the

following

_case. For

example,

in the _presence of _a mechanical

stress,

surface energy does not affect mechanical

equi-

librium

(its

contribution is

negligible except

for _very

large

stress values which _are of the order of the

Young

modulus~ that is close to the fracture

threshold).

Thus at a

trijunc- tion,

the mechanical condition in the _presence of stress would

simply

involve the stress

tensor and not surface

energy!!

The surface energy would in turn be essential in the

chemical

eq~tilibTi~tm (chemical

and mechanical forces _or stresses have different nature!

).

A

typical example

is the Asaro-Tiller-Grinfeld

problem.

For further

considerations,

_see reference

ill.

At finite

growth velocities,

the

Young

condition is altered

by

kinetic

effects;

the

trij

unction

acquires

_a finite

mobility. Moreover,

when diffusion in the

liquid phase

is the

limiting factor,

_{we expect}

Young's

condition to be modified

by

a contribution which is of the order of the Peclet number. This reinforces the fact that

Young's

condition is affected

by

_any

departure

from

global

chemical

equilibrium.

The

non-equilibrium

modification of

Young's

condition is not

only

_an

interesting

^fact in itself _on the

conceptual level,

but also is

expected

to

play

_a

practical

role at least in the three

following

situations:

(I)

at

large enough growth

velocities where kinetics

together

with finite Peclet effects _are

decisive, (it)

at small

growth

rates in faceted

eutectics, (iii)

in eutectoid transformations.

. A

point

worth of

mention,

^{in view of the} recent

development

of

phase

field models

[5, 7,

8]

is that _we derived

Young's

condition withofit

resorting

to the

sharp

interface

limit,

in the

spirit

of what may be termed the

physicists' point

of view

[6, 7].

^{In this}

approach,

one

keeps

in mind the fact that interfaces associated with _a first order

phase

transition have

finite, though, usually, atomically

small thickness. This underlies the very Gibbs

definition of surface

(extensive) thermodynamic quantities

such _as the surface _energy. In this

perspective,

^the so-called

sharp

^interface

equations,

_one of which is

precisely Young's condition, only

make

physical

_{sense on} "outer"

length

sales much

larger

than the

physical length

l~, and do _not

depend structurally

_on the details of the interface

shape

_on the inner l~ scale. The mathematical

question,

formulated in the frame of

phase

field

models,

of

formally recovering

the

sharp

interface

description

via _an

asymptotic (multi-scale)

expansion

in the l~ ~ 0 limit

is,

from this

point

^of

view,

irrelevant.

Appendix

In this

appendix

_we derive the condition under which the

specific

volumes in both solid

Phases

_are

equal.

The

specific

volume is related _to the chemical

potential by

_{v~ = fi~t~}

/tip)T,c Ii

₌

A,B).

Because

~t~ ⁼

~t~(T,p,c)

_{we can} write that _v~

=

fi~t~/tip)T,clip/tip)T,c. ^Using

the definition of the _pressure P

=

f

+

pa f lap,

we can rewrite the

specific

volume _as

a~l~ ¹

~~

aP T,cPa~flaP~'

^~~~~

The relation between the chemical

potential

and

f

is

given

in

equations (7, 8),

_so that the

expression

for the two

specific

volumes take the form

~~

p2

^tic _p

ficfip pfi2 f/fip2

^~ _p ^~~~~

"~

P~ tic ^~ _P

icfiP~

Pfi~f/fiP~

^~ _P ^~~~~

Imposing

that _VA

= vB, then

implies

~

=

H(c, T), (28)

P C

where H is _a function which

depends

_{on c} and T

only. Using

the definition of

f (Eq. (9),

_we

immediately get

that

h,

b, ^& -J p. In order that _v be

independent

_{on c we} must then have p =

p(T, P). Using

the above definition of the pressure and the fact that the

phenomenological

constants that enter

f

_are

proportional

to p, we obtain P

=

-fso

₊

pfifso/tip,

which is _a function of p and T

only.

Since P does not

depend

on c, and that the system ^is

isothermal,

_we

conclude that _p does not

depend

_{on c.} This

completes

_our

proof.

References

ill

Nozibres

P.,

In solid far from

equilibrium,

in

"Shapes

and forms of

crystals",

C.

Godrbche,

Ed.

(1991).

[2] Indeed in this _case

growth

is

commonly

controlled

by

interface kinetics

and/or diffusion,

while diffusion

lengths

in the solids _may become

comparable

with interface thicknesses.

Mechanical

equilibrium along interfaces,

which must be written _{as an}

independent

set of

conditions,

must take elastic effects into account.

[3] Misbah C. and Temkin

D-E-, Phys.

Rev. E 49

(1994)

3159.

[4] These conditions which _are

implicit

in all current models of eutectic

solidification,

_ensure

that the

liquid

and solid concentrations _on the

front,

in the absence of

kinetics,

lie _on the

phase diagram.

If

they

would not

hold,

the

corresponding cs(cL)

relation would contain _a curvature correction.

[5]

Kobayashi R., Physica

D 63

(1993) 410;

Mc Fadden

G-B-,

Wheeler

A-A-,

Braun

R-J-,

Corriel S-R- and Sekerka

R-F-, Phys.

Rev. E 48

(1993) 2016;

Wheeler

A-A-,

Mc Fadden G-B- and

Boettinger W-J-,

Proc. R. Soc. Lord. A 452

(1996) 495;

Karma

A., Phys.

Rev.

E 49

(1994)

2245.

[6]

Langer

J.S. and Sekerka

R.F.,

Acta Metall. 23

(1975)

1225.

[7j ^A different

analysis,

free of the

problems

mentioned

here,

has

recently

been

performed by:

Karma A. and

Rappel W.-J., Phys.

^Rev. E 53

(1996)

3017.

[8j After this paper was submitted the referee drew _our attention _{on a} paper

by

A-A- Wheeler and G-B- Mcfadden

(preprint

n°

256)

which sheds

light

_on

questions

related _to those clarified in this _paper.