Thermomechanics of phase transitions in classical field theory

which can be put into the form
(1.36)

where we have introduced the notations

H=\K

, K=\det{bafi)=\.

(1.37)

Let us refer the space E2{r) to the basis «j , v2 of the unit eigenvectors of bag . With respect to such a basis, bag is diagonal so that the quadratic form (1.34) can be written as Xtn\(ty

=

Vi(^ ) + ^ j ( ^ ) - ^ X( n )W does no*'

vanish identically (i.e. if both tp-^ and y>2 do not vanish), three cases may be distinguished: a

) fv

f2 7^ 0 have the same sign. Then the quadratic form Xf \(ty —

y>1(A1)2 + y?2(^ )

nas a

definite sign at r and consequently the normal

curvature always preserves the same sign on varying A. All the points of !f lie on the same side of the plane E2{r). In this case r is called an elliptic point. b) ip1 , ¥2^® have opposite sign. This means that the curvature along the two orthogonal lines, defined by Vj , t>2 have opposite signs. Therefore,

13

CHAPTER 1. GEOMETRY OF SURFACES

the tangent plane £' 2 (r) intersects f in a neighbourhood of r which is called a hyperbolic

point.

Moreover, two directions \i),^to)

exist along which the

curvature vanishes. These lines, which are called asymptotic

lines,

divide

£' 2 (r) into four regions; in each of which the curvature is in turn positive or negative. c) cp^ ^ 0, f

= 2

0- In this case,

^ , ,(A)= ^ ( A 1 ) . Therefore the normal

curvature has the same sign along any direction different

from

vanishes when A is parallel to A2. Then we say that r is a parabolic T h e graphical representation of these cases is given in fig. 2.

fig.2

A2 and point.

14

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

The eigenvalues of ba/3, which we denote henceforth by Xny the principal principal

curvature

directions

X(2)>

are

ca

Ued

of f at r whereas the eigenvectors of bag are the

of $ at r. From Cartesio's rule we have the relations

below X(l) + X(2) =

2

#

. X(l)X ( 2 ) =

By definition, the scalar H is called mean Gaussian

curvature.

A

(L38>

'-

curvature

of if and K the

The relations (1.37) 2 and (1.31) lead to the

Gauss

egregium theorem which is expressed by the formula

A' = ^ P , which proves that : the Gaussian

curvature

(1.39) depends on the first

fundamental

alone. This theorem has many fundamental applications which we can

form

not analyze for the sake of brevity. We just mention that all the intrinsic geometry of surfaces and, more generally, of Riemannian manifolds is related to it.

1.5.

L i n e s of c u r v a t u r e .

A curve j on f is said to be a line of curvature if its tangent vector at all the points is a principal direction. In particular, if X m = X(2)

at

eac

^ Pomti

CHAPTER 1. GEOMETRY OF SURFACES

15

then all the coordinate curves are lines of curvature (planes and spheres). It is possible to prove t h a t in a neighbourhood of any point r of a surface of class C

(k > 2) a local coordinate system exists such that the

coordinate

curves at

r

corresponding

are principal lines. It is also evident

that

the

coordinate curves are principal lines if and only if

a

(1.40)

l 2 = 612 = °

In such a system the following relations hold

ds

,a/3

u

0

ll

0

= Ojifdu ) + a22(du

JL

3

6

bn

0

0

6 22

a/3 =

22

)

a

u

ll

0

a

^ 22

6n h

n U - I (m. X. -21\ V - U 22 v -111 v -- :22 a 2 Vall a227 ' ll a22 ' X l ~ all ' X 2 - a 22

w ^=-i«^=-S^(%^) &

(log, WiTn).

( t t ^ / ? , no s u m m a t i o n over a and /?) r

«/3 = 2 ^ ~ % , i 3 = ^ 3 ( / o < ^ w ) ( n o summation summation over over aa)) .

16

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

1.6. Surface gradient operator and Gauss theorem.

Let u = uaaa + u n be a vector field on the surface if , not necessarily tangent to if. It is easy to verify that (1.22), (1.23) imply that its derivative is given by the formula below

« a = («7a " 6 > 3 ) «7 + ("fa + K^)

» -

(1-41)

where

«7« = «7« + I > A

(1-42)

are the components of the covariant derivative on if of the tangent field us = uaaa which is obtained at any point r e if projecting w on the tangent plane to if at that point. In particular, if « is tangent to if (u = 0), (1.41) reduces to the following formula:

u

,a =

U

(1.43)

^ a 7 + 6a/3^n-

Similarly, let T = TaPaa ®a0 + Ta3aa ®n + T3an®aa

+ T33n® n ,

be a double tensor defined on if. Then (1.22), (1.23) allow us to derive the relation 3 3 ^T ^« =f Cr£j - T_ T^/ 3 3- 6 7T -T ^^6f) ) ^a / 3®®a ^7

(1.44)

1

3

3 3 33

+ (Tf3 ++ T ^ 6 7 a -T^b -T ja-T6^)a^®n bP)a0®n + (Tf 33 33 3 3 / 3»3 f e6 + T 36 T 3 3 ++T T T 3^/ 6 &£) + {Tfa + T * 6-crv 6g) r»® n ® a~ a, + ((T 7Q - T a/3 Q / 3i )„ W® » ,

CHAPTER 1. GEOMETRY OF SURFACES

17

where y/3*y _ -

T ^ 7 .i-r^T^

+r j ^ , (1.45)

/33 rnn{53

T->/33 ,■ p/3 p/3 y 7 3 _ -7-/33

,a ~

,a '

0a-y 7

In the sequel we are interested in the case of a tensor T satisfying the condition n-T = 0,

(T3o = T33 = 0),

(1.46)

so t h a t (1.44) reduces to

. T / 3 3 f c 7 ) «« J J a T , a„, = = l(T^2 ; a -- J ° a J a / 3®» a„ 7 37 3 T f ++ T ^ 6 7 6a 7Q n ®na 3 +^ T

(1.47)

Let us now define the surface gradient of T as follows

VeT = a Q ® T Q ,

(1.48) .48) (1

and divergence md the surface surface divt •rgence

Vs-T

= a«.T,a

.

(1.49) .49) (1

Under Inder the condition n-T n - T == 00 , we get from (1.41) and (1.47):

1 3 V f «u 3 , , U 7 7 --22t # v„a -• uU =---v>7

(1.50) (1 .50)

18

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Vs-T=(T?2-Ta3b2)a1

+

(T?* + T^b^)n.

(1.51)

In order to point out the meaning of the previous definitions, we consider first a rectilinear system of coordinates (V) in &3 and the associated frame of reference ( O , ^ ) . Then « a = (u'ei),a = u',aei = u\j XU Ci =

\ j ( e j • aa)

U

and we conclude that « aa

e

i =

a

c ■ («* jej

® e t) = aa ■

V

"

(1 - 5 2 )

represents the spatial gradient of « along the vector

which is tangent to If. Moreover, for any vector «, the following

decomposition holds « = « s + n (n-u) ,

(1.53)

where « s belongs to the tangent plane E2(r) to f at r e if. Let us introduce the identity tensor / of S 3 and the projector tensor P on the tangent plane £"2(r) defined by the condition Pu

= us.

Obviously P reduces to the identity / on E2(r). From (1.53) we derive (1.54)

P-u = (I-n®n)-u. Similarly, for any tensor field T on f we introduce the decomposition T=Ts + n®{n-T) ,

n-Ts = Q.

(1.55)

CHAPTER 1. GEOMETRY OF SURFACES

19

By adopting the notation (a®b)-T

,

(1.56)

,

(1.57)

= a®(b-T)

(1.55) can be written as P-T=(I-n®n)-T

where P -T = T

if n - T = 0. On the other hand, it is quite obvious that

coordinate independent representations of / and P are given by

P = aa®aa,

I = e'et ,

(1.58)

and by (1.54), (1.57) we a t t a i n the following relation

aa®aa

= e'®ei-n®n

(1.59)

.

The expressions (1.58) of P and / as well as (1.56), (1-52) allow us to write that

F • (Vu) = (aa ® aa) ■Vu = aa®(aa-Vu)

Hence, we can conclude that V s « represents We

want

now to prove that

= aa®u^a = Vsu .

the projection

the definition

ofVu

(1.49)

(1.60)

on E2{r).

leads us to a

generalization of the Gauss theorem to the case of surface vector or tensor fields which are not necessarily tangent to If. In fact, if w is a vector field on $, we have u? a
v-udl

,

(1.61)

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

20

where v is a unit vector which is tangent to f and orthogonal to the boundary df. By (1.50) we obtain [V s • u + 2H « - n) da - v - u dl

(1.62)

9$

Let T be a second order tensor field which is defined on f, satisfying the condition n-T = 0. By applying the result (1.62) to the vector field « =

T-v,

where v is an arbitrary constant vector , we obtain

[Vs-(T-v)

+ 2H n-T-v]

da =

vT-vdl.

Taking into account the condition n • T = 0 and the arbitrariness of v , we obtain the following formula:

V-Tdcr

, ( n - T = 0)

u-Tdl

=

(1.63)

df

For reasons that we make clearer in the sequel we need to extend the formulas (1.62), (1.63) to the case in which the surface $ is not regular along a curve T and the vector or tensor fields undergo jumps across T. We suppose that !f is divided by T into two parts $~, If"1" (see fig.3). By applying (1.63) separately to each of the two parts if+ and f ~ and adding the results we find that:

[V s ■ u + 2H n -«] da

v -u dl +

as

{vu}dl,

(1.64)

CHAPTER 1. GEOMETRY OF SURFACES

21

where

fig-3 {y •«} = 1/

■«

+ v

■u

and the meaning of other notations is self explanatory. If we denote by r = n x v the unit vector which is tangent to T and which determines an orientation on I\ we have v — r x n and then

i/"•" •«"•"+i/

■« —T^ -nT

x«T + r

•n

xu

22

But T+

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

= - T

= - T and we can then write:

v+

• u + +1/

provided that the sense of r

•«

=-r-[nx«]

is the same as the second term in the j u m p .

Finally, we can put (1.64) in the form below

[ V s - u + 2 # n - « ] da

v - u dl

(1.65)

T-lnxu}dl

dif Similarly, instead of (1.63), we have

V -T da =

v-T

dl +

{v-T}

(1.66)

dl

d$

or equivalently,

V-T

da

v-Tdl8$

T-lnxTjdl

.

(1.67)

Chapter 2

KINEMATICS OF SURFACES

2.1 Velocity of a moving surface. We call moving surfaces

embodied

surface in

the

a one parameter family of orientable regular Euclidean

three-dimensional

space

S3.

More

precisely, a moving surface $(t) is the locus of the points belonging to S 3 given by the equation

r=r(U\U2,t)

where the function

([/\t/2)eftc»2

,

r e C (fi x 3?) defines for every t g 5ft a regular

(2.1)

and

orientable surface of § 3 . In particular, these hypotheses imply that (see section 1.1): i) at every point of f(t),

the matrix {drljdU

) has rank equal to 2 so

t h a t the vectors S

A = ^

= f,A'

(

A = 1

.2).

(2-2)

which are tangent to the coordinate curves, are linearly independent, t h a t is,

23

24

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

they satisfy the following relation SjXo^O;

(2.3)

ii) the unit vector «=

"lX!2l ,

(2.4)

is normal to 3(t) and defines a continuous vector field on f(t).

All

geometrical objects relative to a moving surface will be functions of time. The metric coefficients a r A are given by a

rA — a r" aA ' whereas the reciprocal vectors are

(2.5)

a r = a r A aA ,

(2.6)

( 2

rA

) being the inverse matrix of ( 2p A ).

We remark that the parameters we have chosen in (2.1) are quite arbitrary and nothing prevents us from adopting at any instant t new parameters {u\u

) e Clt c 9? such that ua = ua(U\U2,t)

,

where the functions (2.7) are invertible for any t. If the moving surface

(2.7)

f(t)

represents the actual configuration of a bidimensional continuous system S, that is when it is material, we can preferably adopt (U ) as parameters to describe if(<). Then these parameters are called the material or lagrangian coordinates U

which correspond to a coordinate system in a reference

configuration $(t0) of $(t). Differently, when if(i) is a geometrical surface,

25

CHAPTER 2. KINEMATICS OF SURFACES

which is not connected with moving particles, the parameters U1,!/2 are quite equivalent to u\u

[ 1, 2, 3, 4, 5 ] .

Let ?(t) be a moving surface and (U ,U ) , (u , u ) two arbitrary parametrizations of lf(£) related to each other by a transformation (2.7). Then we have r=r{u1,u2,t)

= f(U1,U2,t)

.

(2.8)

It is possible to associate to the points of $(t) a velocity depending on the parametrization by the definitions

'=($„.

d

*

~\dt)uA-

(2.9)

From (2.8) and (2.9) we see that

C =

or

( W ^

\di)u<*

+ r a

'

VdTJjjA '

+ ^-—m v *" ($V*-

2 io

<- > (2.10)

If we apply the decomposition (1.53) to c and c, the relation (2.10) can be written as c„, +

(c.n)n=cs

+

(c.n)n

+

(^£)^« '[/ "

so that we finally obtain d

s

-A%-)u.'"

c = c - n = c - n = c„ .

(2.11)

26

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

The second of these equalities shows that the normal component of velocity of a moving surface is independent of parametrization.

On the other hand,

(2.11)j, which can also be written in the form ?A C

s

ML

a

duA

"a -

-c«a C

,§vZa

a

s

a

a + Qf

a >

shows that the surface velocity becomes purely normal, i.e. c = cnn , with respect to a parametrization ua(U , t) which solves the following partial differential equations dua

dt

8ua

_

- A /rrl C

~5t/A

*

[U

rr2x

'U

>■

For this reason this type of parametrization is called a normal

parametri­

zation for ^(<). If the surface if (i) is given by the equation f(x, t) = 0, where x = r(ua, t), we have

\dt)x

+ v/-|'6r\

KdtJua

and thus

<%b|

Vf\n-c--= 0 ,

3/

c

"=-p§5r-

(2-12)

Now we want to evaluate the time derivative of some kinematical quantities for a future use. We first have a

and this in turn results in

«

= f

!a

— c ,a '

27

CHAPTER 2. KINEMATICS OF SURFACES

,i

a/3 = c , a - a / 3 + a a - c , / 3 ■

When we take into account (1.41) , the previous relation assumes the form:

^ =2 ( { { ^ ) - W s 2 ^ '

(2-13)

We are now in condition to evaluate the time derivative of a = det(aaa). In fact, it is evident that da a0 ■ ■ a— ®a aa(} = aa a8 ^ aa0 . _ (9(7.

r

Thus recalling (2.13) and (1.50) we obtain a = 2a(c? a=2a(c? 2aVss.c■ c ==22arf < < .a. a-2HcJ a - 2Hcn) = 2aV

(2.14) (2.14)

From this formula we can now calculate the time derivative of the area element da and of the normal unit vector n to 'S(t). By (1.17) and (2.14) we have jfida)

= 7 ^ duldu2 = ^aVs-c

du^du2 ;

or jL(da) = Vs-cda

= 2r]aa.

(2.15)

Moreover, the time derivative of the orthogonality condition n-aa = 0, implies that n • aa = -n-c

a

c

and by (1.41) we are then led to the equation

n, a

= -n-aa-6^C/3.

28

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

But nn-n= ■n

=

a ,41)): 1 => = > n -•n n = 0=>-hh = (h-a-a)« ) aa so that we obtain (see (see (1 (1.41)): = 0=^ =

( * ■

= -- (( cc „n ,j 0„ ++^&%)«" ) a « = = ~ (»(»•«=,«) » . c J a «° °.»« =

2.2.

(2.16)

T h e Thomas derivative.

Let F — F(r,t)

be a time dependent field (scalar, vector, etc.) defined on

f (<). If r = r ( u a , t)

is the equation

of the surface,

we introduce

the

representation a F = F(r(u F(r(ua,t),t) ,t),t)

( ( = '§(u '§(u *,t), *,t),

(2.17)

where the function ?F depends on the chosen parametrisation of f(t). definition, the Thomas

derivative

or the displacement

derivative

By

of F is the

following limit

6F — Urn F ( r + c n n At,t + At)- -F(r, t) At fa At-»0

(2.18)

In other words, the T h o m a s derivative denotes the rate of the change of the field F with respect to an observer moving along the normal to $(t) with the velocity c n . W e note that the limit (2.18) does not depend on the choice of parameters ua owing to the independence of c n of this choice. It is evident t h a t (2.18) implies that

CHAPTER 2. KINEMATICS OF SURFACES

29

f = (f)r +C""-Vf-

<2-19)

On the other hand, it follows from (2.17) that

F -

<%.+<

■VF-- =

(dt)u«>

and consequently (2.19) becomes

f = (ft-<W*.

(2.20)

But (1.60) and (1.57) imply that

V , 5 = V F - n ® (n■ VF) => cg • V s 5 = c s - V f .

Finally, we may obtain a new expression for (2.20) in terms of parameters

TH^a-V^^'-V^ 5 -

(2-21)

In view of (1.49) this last relation can also be written as

f = (f)u«-<*.-

<2"22)

30

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

2.3 The velocity of a moving curve on a moving surface. Let T(t) be a moving curve which remain always in $(t). If $(t) has the parametric representation r = f(U , <), the parametric equation of

T(t)

becomes r = r(UA(ti,t),t)

=

ip(fi,t).

The velocity of T(t) in the parametrization we have chosen is given by

c=

t a r i „ = \§t.)VA + 5^ vw)^

so that in view of (2.9) and (2.2) it can be put into the form

c ==«♦(#)„ «A •

(2.23)

If we introduce different parametrizations for f(t) and T(t), that is if we use the new parameters ua = ua(V

,t) for $(t) and /J, = p(A,t) for T(t), we have

a different equation for T(t): r = r(ua(UA(fi(X, t), t), 0) - f(UA(fi, t), t) , and the velocity C of T(t) becomes now c-(dr\

(dr\ _(dr\

\dtJ =\dt) aa °<%\ x

uu

dr j (du<*\ ,_d^f(dv^

+ +

aa

a

+

dua ((dUA,du ) A

dt

(dUA\(d»\ \ ] ((dV±\.(dU±\(dtA\\

du \1\ \ dt \dp )\dt)J du dt 7y^A at/A\S^ dtb)»]\d}i) \dt)x) ( r [/A 0U

=-(¥u°«4ia+^(w)>-

31

C H A P T E R 2. KINEMATICS OF SURFACES

W h e n we note that ( dU

/dfi) a^ is the tangent vector f to the curve T(t) in

the parametric representation U

= U (n,t),

we obtain

A Ss A ++J°£\9.f . a JW \ a. c ==-(¥h°° dt (1)©/" ♦ ( # ) )„ . A

C =:eJ*£\

(2.24)

By comparing (2 .24) and (2.23) and taking into account (2. 11), we deduce that C-C = © r .

(2.25)

prof ection of This result resu It shows that the projection ofCC along the unit vector v tangent

m 'Ht)

and

and orthogonal

f(t) m

to

to T(t) T(<) is independent indepi indent of bi both ylh parametriza parametrizations tions of T(t) r(t)

Csauu

= C = cssu.- v

.

(2.26) (2.26)

Moreover, from (2.23), (2.25) we have Cn = C'n = cn in agreement with the condition T(t) c S(<).

2.4 Time derivative of a volume integral.

In this section and in the subsequent sections of this chapter we will state some general formulas to evaluate the time derivatives of volume and surface integrals.

32

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Let C be a continuous system and (2.27)

x = x(X,t)

its equation of motion, where X is any point in the reference configuration C+ and x is the corresponding actual spatial position in the configuration C(<) a t the instant t. We denote by V the velocity of the boundary dV(t) of any region V(t) enclosed in C(i), by v the velocity field of C(t) and by N the normal unit vector t o dV{t).

Our actual purpose is to evaluate the following

time derivative

dt V(t

(2.28)

f dv ,

where f(x, t) is any field ( scalar, vector, etc.) associated t o the continuous system C. Now, it is quite obvious that

A.

dt V(t)

f dv =

±(fdv)

v(t)

N-(V-v)f

+

da

(2.29)

dV(t)

W e note that the second integral on the right hand side represents the flux of / across dV(t) due to the combined effect of the^motion of C and dV(t).

On

the other hand it is well-known that

dt

(dv) =

(2.30)

V-vdv

and (2.29) can be written as follows

A dt V(t)

f dv =

[f + fV-v]dv V(t)

+ dV(t)

N- {V-v)f da

(2.31)

33

CHAPTER 2. KINEMATICS OF SURFACES

The following subcases of (2.31) are interesting: i) V(t) is material, that is VN = vN

; «) v(t)=-V

is fixed. In the former case we have: d

f dv = [f ++ [f dt *{'"•=, V(t) V(t) VV(t) V(t)

fV-v]dv. fV-v]dv.

(2.32)

V t)

whereas in the latter (2.31) becomes d dt

r

f dv == V

df dv dv Mdv --Tt

(2.32)'

V

In the sequel we need a generalization of (2.31) to the case in which / undergoes jumps across a surface £(£) moving with a normal velocity c . To +

attain this goal, we suppose that V{i) is divided into two parts V ~ ,V £(£), where y

+

by

is the part toward which the normal unit vector n to £(£) is

directed. By applying (2.31) to V _ a n d Vr + and adding the results, we have

[1]:

dv i\'V(t) =l V(t)

[f + + fV.v]dv+ fV-v]dv t) V V(t)

+

*

dV (t) dV(t)

f(VVN-vNV)daf ( N~ N)

IfU]IfUldcr, da ,

d(T

~

E(t) a *)

(2.33) (2.33)

where we have notations V V introduce 1 11 \j 1 KJK.I IJV_.(_ the U11V, 11 K.' UC* H U l l O = (V-v)-N U±=c=n-v± ((V VNj-v vN-)t . ^ ) = (K-i»)-JV , tf* « « - « „ * •. We want to emphasize that : i) in all the formulas we have deduced so far only the normal velocities Vjy and c n which do not depend on the parametrizations of dV(t) and E(t) appear ; ii) (2.33) is also valid in the more general case of a discontinuity surface dividing V(t) in many parts

34

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

provided that the boundary <9E(tf) lies on dV(t) (see fig. 4). To verify this it suffices to apply (2.31) to each of the aforesaid regions.

fig.4

2.5

T i m e derivative of a surface integral.

Let r = r{ua,t)

(2.34)

be the equation of a surface E(£) moving with velocity c and cr(t) a region of £(£) whose boundary da(t) is a curve which moves on E(<) with velocity W . In this section we want to evaluate the time derivative

35

CHAPTER 2. KINEMATICS OF SURFACES

F da

dt

(2.35)


where F = 5 ( u a , t) is a field ( scalar, vector, etc.) on E(<). Instead of (2.29) we now have

d dt

F da = = '

*{ t)

j-t{Fda) j-t{Fdo)^


v- {W-c)Fc)F dl .

(2.36)

da da(t) (t)

Owing to (2.15) and (1.50), this relation can be written as

d dt.

a(t)
F da =

-c -2Hc -2Hc IHcF] F]F] da + + f[($L+".-« s

ss

nn n

*(t)
- c)F dl . w{W-c)F v-{W-

fl_ 11\ dcr(t)

(2.37)

By recalling (2.21) we finally obtain

d F)-2Hc F da = [tg+Va.(ct®F)-2Hc nF]da+ nF)d
jf d
v{W--c)F v(W-

dl . (2.38)

It is very important to remark that the time derivative of the surface integral explicitly depends on the surface parametrization because of the presence of c s in both integrals on the right hand side of (2.38) . This is due to the fact t h a t the velocity cs is a priori physically meaningless. In other words, if we want to use (2.38) in physical balance equations, we are compelled to introduce a parametrization on the surface to which a physical velocity is associated. For instance, if T,(t) is material, i.e. if it is a twodimensional continuous system, we can use as parameters the Lagrangian coordinates of the particles of £(<) so that c e denotes their tangential velocity to E(<).

36

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Formula (2.38) can be extended to the case in which F undergoes jumps across a curve on E(<) which moves with velocity C and is singular for the surface E(£). We start with the hypothesis that £(£) is divided into two parts £~(£) and E + (<) by the curve T(t). If we apply (2.38) to each part and add the results, we obtain

[^

F da =

dt

Vs-{cs®F)-2HcnF]da

+

'(*)

+ j v(W-e)Fdl+

(2.39)

{u- {C-c)F}dl ,

r(t)n
a
where we have again used the notations of section 1.6. By the same procedure of that section, we can transform the last term of (2.39) as follows

{v {C-c)F} dl= r(«)n
r-lnx{C-c)F}dl

.

(2.40)

r(t)n
We want now to derive an interesting well-known consequence of (2.39). Let lf(f) be a material surface moving with a velocity field v and let F be a scalar field of the form u-N , where N is the normal unit vector to f(t). In this condition the integral u-N da , T{t) denotes the flux of the vector u across the surface 'S(t). From (2.21) , we find with W = c = v S

-£+Vs-(vs®F)-2HvNF

CHAPTER 2. KINEMATICS OF SURFACES

= F-v-v -F+ F(V F(Vss-v -vas-2Hv -2HvNN) ) s-V sFaF s-V

37

+ vvss.V -Vs3F.

If we take into account (1.50), we can immediately write the following equality 6

-£ +

Vs-(v3®F)-2HvNF =

F+FVsv.

But (1.41) and (1.52) imply that a a Vssv ■ v==aaav, V aa ■aVv ■=aatr[{a = tr[(a aa-Vva ® aaj ®■ajVv]■ Vw] a = Q■ v, =

and (1.59) allows to write W + + VVs-(v -F2HvNF ==FF++F(V-v-N-Vv-N) F (V • v - N ■ Vv ■ N) f£ ®F)-2Hv . .(2.41) (2.41) s • s(v s ® F) N .52)) (1.52)) If F = «• u ■NN we have (see (2.16), (1 d -^-(u-N) == u-N uii ■■NN + «u-N=ii-N-(N-v -(N■N) = ■N) v■TV N = = iiu ■N ■N- -(Na) dt (« = «u-N-(a = ■N-(aa--Vv-N) V v N) Uua

V y

a)

u ua aaa

and (2.41) becomes f,V

_

^+V &F + v s • (v, ®F)s-(v,®F)-2Hv NF

VvN++u-N 2HvNF ==ii-N-u ii-N-usu-N-u N -VvN ■-VvN Vv ■ N + u -u-NV-v TV V -Vw■ v sVvN-s-VvN-- uuNNN _= ( (^du + + vv-Vu-u-Vv V u --« • Vv + uV-uV-v)-N . ) . TV ~\ dt = ( | | +V Vx(uxt>) x ( u x t > ) + wV-u)-JV. wV-u)-TV.

(2.42)

38

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

On the other hand, the jump in (2.40) can be written as

-T-INX(C-V) -T-[NX{C-V)U-N]

u-N}= = -T-[UNNX(C-V)] -T-\UNNX(G-V)1

= -T-[{u-ua-r-l(u-u )x{C-v)] s)x(C-v)} = where « s is the projection of « on the tangent plane to f(t). By representing C-v

in the basis (r,v,N)

and noting that CN — vN , it is easy to verify

that -T-[Nx(C-v)

«.JV1= - r - [ u x ( C - v ) ] .

(2.43)

Collecting the results given by (2.42) and (2.43) we finally obtain d u- N dcr = [ | ^ + V x (w x w) + v V •«] ■ N da dt\ J(«) J(t)

r-[ux{C-v)]dl.

-

(2.44)

r (0

2.6 Two time derivation formulas derived from (2.39) and (2.44).

The interest of (2.39) and (2.44) has to be found in their use in the general balance law which we consider in the next chapter. However, (2.39) (2.44) cannot be used in their present form. In effect, to use them we need to make explicit the velocity W of the curve da(t) in (2.39) and the velocity C of Tit) in (2.44).

- 39

C H A P T E R 2. KINEMATICS OF SURFACES

Let V(t) be a material volume contained in the actual configuration C(t) of a continuous system, TV be the normal unit vector to dV(t),

£(<) c C(Z) be

a singular moving surface and T(t) C £(<) be a singular curve. If we want to apply (2.39) to the surface cr(t)

=

V(t)nT,(t)

velocity W of dcr(t) to the velocities v of dV(t)

, we have to relate the and c of S ( i ) (see fig. 5). T o

begin with, we remark that the vectors TV , n and the vector i / s , which is orthogonal to dcr(t) within the tangent plane to a(t)

are coplanar at any

points of dcr(t). In fact, TV is orthogonal to all vectors tangent to dV(t)

and

to the unit vector tangent r to dcr(t). But both n and J / S are orthogonal to r and to each other so t h a t the coplanarity of TV , n , J/^ along dcr(t) is held.

fig-5 Let f(x,t)

= 0 and g(x, t) = 0 be the equations of dV(t)

respectively. T h e equation x = r(\,t)

and

S(t),

of the curve dcr(t) satisfies the system

40

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

f(r(X,t),t)-- = 0, ff(KM),<) == 0. By differentiating with respect to t we have

v/.^+§H

v , . ^ + | = o.

On the other hand, owing to (2.12),

df dt

V

N ~ -

dg dt |Vff| '

C

[v/l

n~

and the previous equations imply WN = vN

W

= c

(2.45)

When we choose the basis (T,I/^,N) (r >"£> f) ,, we have W=W W=WTT TT + W W' v-^ ++ WW u *2 nnnn and it is very simple to verify that (2.45) leads us to the formula

w= =W ">

T ++{l^N ( ^

W

N)

c

„»-■ *fc++C n» ~--cCnn ^W)■NJ "E

(2-46)

c n n • AT )

(2-47)

Consequently, we find that (W

e) • * £ - ( « #

In view of this expression we can write

1^

c-i/E.

(2.39) in the situation we are

considering as follows d dt

fFdada=

f [7f+V =\^+Vs-(c 1Hc F] da nJnF]d<7 s®F)- s-(cs®F)-2Fc

" • ( < )

+ da

f

(0

F

KVN

c n n-JV)

" ■ ( * )

v

\

N

c-vx]

dl

r r(t) no-

rr-lnx{C-c)F]dl .[nx(C-c)f](f!

(2.48)

C H A P T E R 2. KINEMATICS OF SURFACES

41

where a{t) — E(<) D V(t) , V(t) any material volume and T(t) is a singularity line on £(£). Now we want to apply (2.44) to the case in which the moving curve T(<) on the material surface if(i) is obtained by intersecting $(t) with a moving singularity surface E ( i ) ( see fig. 6). First of all we note that r = i/ E x n , and then we have -r.[tix(C-»)]=

-i/E-[nx(«x(C-i»))]

where C is again given by (2.47).

fig.6

Chapter 3

BALANCE LAWS FOR A CONTINUOUS SYSTEM WITH AN INTERFACE

3.1 General balance laws.

The weight of balance laws in Continuum mechanics is well-known. T h e mass conservation and the balance laws of momentum, angular m o m e n t u m and energy constitute the fundamental principles to derive the corresponding local equations as well as the boundary and j u m p conditions of the fields which describe the evolution of the system C(t) we are concerned with. In their integral forms the balance laws are postulated to be valid for every fixed or material region of t(t)

(see, for instance, [1]). More recently [6,7,8,9],

it has been suggested that the above mentioned balance equations can be postulated only for the whole system C(<) but not for any of its arbitrary parts (Nonlocal Mechanics). In this chapter we propose to use Continuum Mechanics to describe the evolution of a system C(t) with an interface

E ( t ) . In such a system £(<) does

not reduce to a singular surface for the volume fields which C(i) carries in its

42

43

CHAPTER 3. BALANCE LAWS

evolution. On the contrary, £(£) has to be regarded as a nonmaterial which

is

able,

however,

means t h a t

£(£)

can foresee

that

to transport

thermomechanical

surface

properties.

This

is itself a physical system in interaction with C(t) . One this model is appropriate to describe phase

transition

phenomena; in fact, in such kind of phenomena two or more bulk phases are separated by very narrow layers in which the volume fields vary continuously but sharply. This layers can be regarded as discontinuity surfaces for the volume fields provided that some suitable surface fields are associated with these surfaces in order to take into account the mass, energy etc. contents of the layers they represent. It is quite obvious that the layers do not contain the same particles of the system during the phase transition and then the representative surfaces are not material. W h e n we try to use the formalism of Continuum Mechanics to describe the evolution of the system with an interface (C(<),E(<)) we are faced with nontrivial difficulties. In fact, we need to make clear: i) how a nonmaterial surface is able to transport thermomechanical properties , it) how we must modify the classical balance laws when there is a convective flux of quantities associated with £(£) across the curve £ ( / ) Pi dV(t) material

volume V{t).

on the boundary of any

In this section, taking for granted that

we can

associate physical quantities to a nonmaterial surface £(<), we propose a general balance law for the system (C(i),£(2)). W e postulate t h a t any balance law for the system (C(<),£(<)) has the following general form:

44

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

dt\

f dv +\-

V(t)

F da) =


8V{t)

Rda+j
N• (p d(T +

r dv +

V(t)

i/ s • * dl

d
F[(vN-cnn-N)L-±7f-c-vE]dI,

(3.1)

dcr(t)

where V(t) is any material volume,
fig-7 The following remarks about (3.1) are worthwhile:

due to the

45

C H A P T E R 3. BALANCE LAWS

i) the last term in (3.1) has never been considered in literature ( see, for instance, [3, 4, 5]) up to 1986. In paper [10] it is considered only in the case of a fixed material volume V(t) (i.e. vN — 0) and independently in [11] in the complete form; ii) if the interface is material, for instance in the case of two bulk phases of immiscible fluids, the last term in (3.1) vanishes since W = c = v (see (2.67)); Hi) if V(t) coincides with the whole volume C(i) of a system with an interface, the balance law (3.1) assumes the classical form:

^ ( j fdv+l C(t)

Fda) = | N-tpd
aC(t) • $ dl +

J/ E

9E(t)

r dv

C(t) i i da .

(3.2)

E(t

In fact, <9£(<) moves on 9C(i) so that again W = c on 3£(£)

and the

convective flux of F across 3E(<) vanishes. We will come back to this remark when we propose to employ the Nonlocal Mechanics to describe phase transitions in crystals. It is now possible to derive the partial differential equations and jump conditions which are equivalent to the integral law (3.1). To this purpose we begin with recalling the classical Gauss theorem for a field

N-
\V-
n-{ip}da. "

(3.3)

46

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

When we take into account (2.33), (2.48), (1.67) and (3.3) we obtain

[/ + / V - w - V y j - r ] dv V[t) 8

+ | [ -£+Vs-{cs®F)-2HcnF-Vs-*-R-UU

+

n-tp\]d

T-lnx({C-c)F

+ <S>)} dl = 0

lit) where y(t) = T(t) fl cr(t) and T(t) is the moving line of discontinuity on £(£) (see fig. 7). T h e arbitrariness of V(t) and the regularity of the fields in C(t) - (£(£) U T(t))

lead

us to t h e following

local equations

and

jump

conditions: / + / V - v - V y > - r = 0,

^ f + V s • (c3 ® F) - 2HcnF

mC(t)-E(t)

- V s • * - R - [fU + n.
T-[nx((C-c)F + 4)] = 0,onr(t).

(3.4)

Our axiom that all the balance laws of a continuum with a n interface have the form (3.1) is valid only in the absence of electromagnetic

fields.

When these fields are present, we have to take into account balance laws of the form (seefig.8)

j-A 1(t)

u-Nda=

| a-r dl + af(t)

b- N da + \
i(t)

(3.5)

r,(t)

where $(t) C C(<) is a n y material surface, rj(t) = E(<) n f(t) a n d r t h e

47

C H A P T E R 3. BALANCE LAWS

tangent unit vector to df{t).

Moreover, « and a are vector fields in the

volume C(t), 6 and ib vector fields defined on £(*). By noting that r =

"X

xn

and applying (2.44) and the generalized Stokes theorem we obtain

a if (t)

[a]•r dl ,

V xa- N da +

a-T dl =

f(t)

vU

we have

fig-8

and

m

(du + \7 x (u x v) + v Vu - V x a - 6) • N da

(nx{ux(W-v) Ldl=0, ( n x | u x ( W --v) + + aa] | ++*J • k)'V f £ dl =U,

-

M n(t)t)

(3.6)

48

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

where the velocity of r){t) is given by (2.46). The arbitrariness of the material surface f(£) leads us to the following local system | ^ + V x ( « x « ) + v V « - V x a - 6 = 0 , inC(<)-£(t) ,

(nxlux(W-v)

+ a} + k)-vE

=0,

on £(<).

(3.7)

This last condition is satisfied for every v^ tangent to £(t), however, it does not imply that the tangent vector which multiplies J/ S will vanish since W depends on i/ s ( see (2.46)).

3.2. Mechanical modelling of nonmaterial interfaces.

In this section we propose to make clear the model we will take into account in the sequel as well as to supply a possible interpretation of the mechanical properties of a nonmaterial surface. We have already said that a continuous system with an interface consists of two continua which occupy two adjacent regions Cj and C2 separated by an interface £ ( for the sake of simplicity we do not explicitly write

the dependence on time t). £ is

regarded in turn as a two-dimensional continuum with assigned proper constitutive equations and it is presumed either to be permeable or not and

49

CHAPTER 3. BALANCE LAWS

possibly adsorbing.

When E is permeable, we assume that the matter crossing

E, for instance from Cj to C 2 , is to be described by the same constitutive equations of C 2 . If E is not permeable, the model describes a system made by two solids or immiscible fluids occupying two adjacent regions. The task of the two-dimensional system E which represents the contact surface of the aforesaid regions is to take into account possible interfacial phenomena. In particular,

E could be a real two-dimensional continua (oil)

interposed

between three-dimensional continua (water and air). When E is permeable, the hypothesis t h a t m a t t e r changes its original constitutive equations into those of the continuum occupying the arrival region makes the model able to describe the state change. In order to complete the model (Cj,C 2 ,E) we must define its balance equations and the set of constitutive equations for each of its component. The former

are

momentum,

represented energy

by

mass conservation

balance

laws

as

well

as

and by

momentum,

angular

the

law

second

T h e r m o d y n a m i c s . In other words, we need to define the quantities

of f,F,...

which appear in (3.1), (3.4) for each of these laws. The most natural way to attain

this

objective

consists

in defining

the

volume quantities

as

in

C o n t i n u u m Mechanics and in associating corresponding quantities to E ( den­ sity, velocity, stress,...). However, two objections can be raised to this widely followed approach [2,3,4,5]. First of all, when E is not material, the meaning to a t t r i b u t e to the velocity of the particle which instantaneously lies on E is ambiguous. In the second place, the local angular m o m e n t u m equation which is derived in this way is not Galilean invariant. In order to overcome the above objections we want now to propose a model which leads in an almost

50

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

natural way to the proper quantities to introduce into the balance equations . We recall that the interface between two contiguous systems is really a very narrow layer formed by matter having different characteristics from the adjacent continua. This suggests to regard (
in details, we substitute the

layer with a surface E C AKy and then we associate surface fields to E which are averages of corresponding volume fields. This procedure lays itself open to a fundamental criticism. In fact, when we substitute the layer with a surface we extend, at the same time, the bulk phases to fill the layer which is now e m p t y . But this operation adds an amount of mass to the system which is of the same order of the mass we have concentrated on E. In other words, the system (C-[,C 2 ,E) is not equivalent to Cy. For this reason Gibbs [12,13] suggested to localize E at a suitable position in the layer and to put on it a surface mass distribution (possibly negative) by requiring that the total mass remains fixed. However, in dynamical situations we would localize E in such a way as to conserve also, besides the mass, the m o m e n t u m ,

angular

m o m e n t u m etc. and this proves to be impossible. To overcome this difficulty, we proceed as follows. i) Let C( be a family of not equivalent systems C(, where /-+0, including the system Cy we have started from, such that in each layer AV(

of C, the

content of mass, momentum,... is equal to the corresponding one in AVy . In this way, / determines the order of magnitude of the mass associated

51

CHAPTER 3. BALANCE LAWS

with the interface and moreover, when /-»0, the difference we have between Cj and the system with an interface obtained by concentrating mass, momen­ t u m , etc. on a surface S in AV ( goes to zero.

Then, we postulate that the

balance equations of our system (Cj,C 2 ,E) are equal to those we derive for C ; , when /-»0. It is evident t h a t this procedure rests on the assumption that, in the first approximation, the detailed distributions of mass, m o m e n t u m , etc. in the layer are not significant. If cl C C( is any material volume, a general balance equation for the system C ( can be expressed as

d

f dv = cI

N -cp da + r dv , c/ 9 c,

(3.8)

with the usual meaning for the symbols (see section 3.1). Concerning the system Cj we suppose that (see fig. 9): ii) C, = C 1 (/) U C 2 (/) U AVt parallel surfaces !f-(/) , i= 1,2,

, where the volume AV

t

is bounded by two

moving with the same normal velocity cn

and by the surface fy, belonging to 5Cj. Moreover, the distance / between the surface !f,-(/) is independent of time and 0 ( is orthogonal to both ^j(/)'s along the boundaries. Let Cj be any material volume such that ct n AVj ^ 0 and verifying the same conditions listed in ii). We denote by small letters the corresponding p a r t s of C[. Then we can write the first and the third integral of (3.8) as the s u m of the integrals over the three regions Cj(/), c 2 (/) and Az^ and apply Fubini's theorem to the integrals performed over Av ( , where Av{

can be

52

-PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

fig-9 represented by

A„, = U J(r) re
Here a C £ and £ is a surface which is equidistant from S-(l) whose unitary normal vector is n, while J(r) = (r + £n, - 1/2 < £ < 1/2). Let us now introduce the notations Ft(r, t) = j fj(r, 0 dZ, where

r

*( )

J(r,£)dtr{r)=dtr{r

rj(r, £) dl ,

R,= H*)

+ Sn).

We remark that assumption ii) implies that the limits of F[ and R{ are finite

CHAPTER 3. BALANCE LAWS

53

when /-»0. With these notations we have d dv = ft fdv+it dt ' f c(l) / ) U c 2 2( (0 /) Ccl x( (/)Uc e(l)

Ft da , '

(3.9) r dv = c(l)

rrft) +

ify dcr .

c x (/)Uc 2 (0

If we put <9c'(/) — dc(l) - u>(I) and note that

u(l) = ( J I(r) ,

AT = i/E on w(/)

the flux term can be written as

N -

( d/

N -(p dv flc'(r)

dm

(3.10)

a
where */(»■."£)

=

p(r + fn)J(f,r,i/ s ) d£

and dl = dl(r, v-^) is the line element on the curve da, i/ £ is the unit vector normal vector to da and tangent to E and finally J(£,r, i/j>)dl(r, i/ £ ) = dl(r+£n, i/ s ). We note that the

vector field
normal v-% when / ^ 0 so that the Gauss theorem is applicable to the second integral on the right hand side of (3.10) only in the limit /->0 (when /->0, J-*\ and $; is independent of i/g).

54

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

We now have to substitute (3.9), (3.10) into the formula (3.8) so that it now reads

ft]

fdv+^lr,***

C l (/)Uc 2 (7)

«

i/£ ■ % dl

N -

Qc (I)

3 <7

+

R{ da . R,d
rrdvdv +

(0U C el m ucc2 (m( )

(3.11)

"

It has to be remarked that in the above formula both cAl) and cdc' c' (/) are regions depending on the parameter /. To apply (2.33), (2.38) in (3.11)

we

must define the tangential velocity c gs of a. To this regard we assume tthhaatt Hi) each parallel surface to E contained in AVl c

has a tangential velocity

r

particles «s ( ' 0 which is equal to the tangential projection of velocity of the particles

lying on it at any instant. The condition that w ; is orthogonal to SAl)

and E £ implies t h a t the

all these these velocity projection Vss(r)— ( r ) = (w(r + ^n) • i/ s ) j / ^ is uniform along I(r) ; all Cs = cc^ = Vs and then we have considerations permit us to put in (2.38) C,

|

[f fV-v]dv[/ + / V-v] dv- |

cC

ll(( / ) U c 2 ( 0

J 2 c- vfx(cnJ [/ 2 (
- vln)Jt]

dcr da

Ss

«(0 i (')

). l)-HcnF [6-^+V -Hc + [-^+V s.(V s®Fll,®F nFt]da s-(V l]da

= =

c

N -

(I) dc dc'(l)

'

+ c x (0 IJc 2 (/)

r dv +

Rt da , r


(3.12)

55

CHAPTER 3. BALANCE LAWS

where fi

is the value of the field / on the surface 5 t ( / ) , c n is the normal

speed of S^l)

and E, Ji — dajda;

moreover, dai and da are the correspond­

ing surface elements on 5,(/) and E respectively, vin=vi

■ n and v- is the value

of velocity field on 5,-(/). If we now calculate the limit of (3.12) when /->0 (we suppose that for all functions F j ,
/ + / V v - V - p - r = 0,

6

-£+Vs-(V (V, ® F)nF-V-*-R --2Hc nF- -V . $ . R s •s®F)-2Hc St + v -[/E/

+

n^]

+

{ h ™ § - f } = 0.

(3.13)

In (3.13) F and R denote the limits of Ft and Rt respectively. Apart from the last term within brackets, equation (3.13) 2 coincides with (3.4) 2 but now we have the identification of F , R with suitable averages along the normal n to E of the volume fields / and r. To continue further, we now assume that iv) there exists a value /* of / such that F * = Ft for any / < /*. This assumption means that the family C(/) of systems which we are considering in our limiting process not only has the same amount of mass, m o m e n t u m , etc. in the layer AV[ (see assumption i) but also in each small "cylinder" da(r)x iv)

cannot

[-1/2,1/2].

be validated

In effect, we will show that the assumption

for all the averages Ft

because they are

not

56

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

independent of each other. In other words, iv) has to be employed under the condition that it does not lead to contradictions. If we refer to the mass conservation, /

is the mass density p and

consequently ps = lvy,l

= | pJ di .

(3.14)

I{r) Similarly, for the surface m o m e n t u m

P-lim

P we have

P.-Urn

(3.15)

pvJ df I(r)

so that we can define the surface velocity as that of the centre of mass of the particles instantaneously occupying the region da x I(r) :

(3-16)

V = Jl-

If we introduce the velocity v' of the particles in AVt

with respect to the

mass centre by the position vl = V + v'l ,

and recall t h a t the m o m e n t u m in this frame vanishes, we can write the total surface energy PsEtot

*

= Ps K ^ l = ' » " , P(e + 5 v2)J I(r)

as follows PsEtot

= lPsV2

+

PsE,

d£ i

57

CHAPTER 3. BALANCE LAWS

where P„E = pjim E, = Urn s s ;-K>

'

f-vo

p(e + ±v'2)J

(3.17)

d£.

Hr) We finally look for the surface angular momentum which is the limit of Ml =

(r + £n)x pvJ d£ = rx Pt + nx Hr)

£pvJ d£ ,

Hr)

which, owing Hi), can be written as M t = r x Pt + psd{ n x V s , where dt represents the distance between E and the centre of mass of the particles instantaneously lying in da x I(r) along the normal n to E. From this equation and (3.15), (3.16) we deduce that M = Urn M, = rx pV .

(3.18)

Now we have to evaluate the difference

6 Urn -^M \ H » M St)

(3.19)

where Ft should be identified as psl , Pl , psEt or Afj . It is quite obvious that assumption iv) implies the vanishing of the difference (3.19) these quantities except for the last one. In fact, we obtain for this case

for all

58

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

,. hm

l-*Q

6M

l —jr-iOt

-W='."xy'('te$)"

(3.20)

The limit appearing in the second member of this equation can be easily calculated when we recall that d[ dj represents the distance (along the normal to r) E) between E and the centre of mass instantaneously belonging to da x / (I(r) as

Sd, lim6-£-=Vn-cn. /-K) St n

n

(3.21)

To conclude, we can write

fe^-W = ^.-«J-»K..

(3.22)

system with an interface. interface. 3.3 Balance equations for a system We are now in the position to formulate formulate all the balance laws for a continuous (£, ,C 2 ,E) [15]. In the sequel C system with an ari interface (Cj,C c == Cj ca U u Cc 22 ,, KV CCCC isis V ffll EE . For the passage from integral laws and any material volume and a = = V the equivalent local equations we resort to (3.1), (3.4). Conservation JL(

of Mass

pdv + pdv \/

da pPssdaj= )=

cr

PsKvN-WW—Tpf-Vs-Vxldl ^^-Cn"-JV)^-^-"E]'« a
(3.23)

59

CHAPTER 3. BALANCE LAWS

where p p and ppss are the volume and surface surface mass densities respectively. T Thus hus

we locally locallv have pp + V ■ v == 00, , + PpV-u

in CC--EE

^*jf+V + sV, -(p•sV(sPs)-2Hc Vs) -nPs2Hc -lPU} = 00 ,, n Ps - [PU] = r-[nx(C-V)p,] r.[nx(C-V)p,]

Bt ilance of

dt\

pvdv

=O 01t

(3.24)

on o nEE- -Tr

on T. onT.

Momentum

+

n • £ da +

P,Vd
pfdv +

v^ ■ Tdl + pssFdcr, Fdcr,

av +

p^[(^-c„»-^)~af-*v*'E]<«

(3.25)

veloci ty, stress tensor and body force inside C --EE where w, v, t and and /f denote denote the velocity, T and and F fields on E. Hence we write respectively; V K, T F are the corresponding fields

locally locallv -^(pu) + (V fapv) ( V --v)pvv)pv-V V-•t*- -p/f» / = = 00,, in CC--EE #c „ '
(3.26)

C/ ++nn-t]-p ■ 0 -s-P F SF~-== 0, 0, on E -- T, 1',

r - [ n x ( ( C - V ) p , V + T)] = 0 ,

By noting &"> =

= pu + pi>

on T .

60

-PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

v Up* )>-=St6-irv+Pfw St 'St KHs

vs-(v,®Psv)=vvs-(psvs)+psvs-vsv

and taking into account (3.24), we can transform (3.26) as follows

pi> = V -t + pf ,

Pi

(8V \6t

+ vs ■vsv

)

■

■

+ T)] = 0, onT.

Momentum

■jjL(\prxvdv+ ( prxvdv+

\ppssrrxV x V da da)J + er

y

rx N t da + dv

rxV[(vNNp.rxV[(v Ps

c ) pss(V n P (Vn-n-c n)

c

rx pf dv + V

+

(3.27)

-V)U + n ■t} + PsF, on E- -r,

=vs •T+[p(v-

T-[nx((C-V)PtV

Balance of Angular

inC-E

d
nxV da nxV da

T

rxn-T

dl +

rx p F da
cnn-N) n - N) ,--^-V-u^dl. ^ - Vs • „ £ ] dl.

(3.28)

da

The presence of the third integral on the left hand side is due to the considerations of previous section. Locally (3.28) implies that

j-t{r x pv) + (V • »)(r x pv) - V • (r x t) - r x pf = 0 ,

£ ( r x p,K) + V s • (V. ® r x psV) - 2Hcn(r x psV)

(3.29)

61

CHAPTER 3. BALANCE LAWS

+ P.(Vn-- cn) n x V - Vs • (r x T) - [(r x /»«){/ + r x n - f ] - r x / ) / = 0 , r •[nx [(C--V)(rx Noting that rxn

■1 =

PSV) + rx T\] == 0 .

n r x 1 , the first of these equations, can be written as

(p + pV- v)(r x v) + prx v -- V ■ ( r x

0--rxpf

== 0 ,

consequently, owing and (3.27) e't ■ xe ing to to (3.24)! we have e'' t xt e,=; = ei (3.27)!, 0 0, ,where e(3.24), and x , we is a spatial basis and e* its reciprocal basis. It is quite obvious that this relation impl:ies the symmetry oft: t=tT,

in C - E .

(3.30)

Similarly, (3.29) 2 become: 65 nPf (( St

+ +

r + v s ■(P*V.)c nnxV V ■iPfV.)- -2Hct nPs)( ,/>„) (rxV) * ■V) ++ Pspc„nx

0 a rxp rxPs6s-?-+p -£+p -asa-a ®(rxV), ®(rxV), sVfsV a a

a -rxV T+a -Txa a- r x sV T+a"-Txa s a-

+ Psp(V --cJnxV s(Vnn--c n)nxV

-- rr x = 00 .. sF = x IpvU IpvU + n-t}-rx +n-tl-rxppsl<

On the other hand a ®(rxV), ■ K ®(rxV), P: Pt>v. y. ■(a a = -V Vxaa +

and id

V ■ K '®{a '®{aaa J =-- rpps V J= s r s s

a PsVs ■a rxV,a=

! xV)) + xV)) PsVs ■ K ® ( r x P, J) a)) +py. !

v,

a

- V x ^ + r x ^ ■° )v,a

62

-PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

^ „ » * Vs + PsVs ■ VSV = - Vnn x V + PsVs ■ VSV

so that the foregoing equation becomes

r X

{P'-W

+ P V

' » ■ V ^ - V « T - P'F

~IP{V- -V)U + n-t])+

a a - T x a a = 0,

and it follows from equation (3.27)2 that aa ■ Tx an = 0 , o n E - f .

This relation is equivalent to the conditions rpOt/3

r l

T (^Oi

rpa3

n

(3.31)

which imply that T is tangential to £ and its surface components are symmetric. Finally equation (3.29)3 can be written as

rxr [nx [(C- -V) T\h=0 rXT-lnx[(C-V)p + T}j=0 sVPsV + and it is identically satisfied in view of (3.26) 3 . Balance

of

Energy ■

dt\

p(^v2

+ e)dv

+

ps(\v2

+ E)da) =

n- (t-v-h) dv

da

63

CHAPTER 3. BALANCE LAWS

v • ( T )V- -K) da+^p + \pf-vdvv dv + \v(T-V-h dcr+ F 4-J,/. s

+

s

\psF-Vda V da

a

da

V

^ ^[(V - Nc -^ cn-n i •VN)) ~- i^ y- -^ V,- ^■ ]* s^] dl. . Ps Mfti ^V2 ++ E)

(3.32) (3.32)

a
where e and ft ft are the specific internal energy and

the heat flux flux vectors

respectively in in. C C -- EE, i? E and and hfts s have have similar similar meanings meanings when when we we refer refer to to EE ---r. T.

Wee now obtain from W from (3.32) that that 2 j^p P (( ±v e))++ pp((±v ±u 2 + + e) ej V s • vu - V V■ •{«■«(t ■ v ft) - ft)- pf-v - pf ■=v 0= , 0 ^ ^ 2 + e))

iMr

,

¥

£ -2Hc + VV*S ■(p ays V £ ('. ( 2 ^ ++ EE)) )) + • (P* s( Y + +E))» - 2HcnPs( nPs(Y\V+2 E) + E)

-WSs-{T-V-h F-V-{p ■(T V ■K)s)-Ps -v -PA"

2 V- -Ipfy {\v2 + e)U e)U + n (t n-{t-v-h)}={\ v --*)]==0

T-lnx[(C-V)p r • \n x [(C-s(±V V)2Ps{\v2

+ + E) E) ++(T.V-h (T.V-s)}}ft,)] ]== 00. .

(3.33) (3.33)

It is quite easy to verify that (3.33), can be written as follows: follows:

pe-tr(t®Vv) pe-tr(t®Vv)

(3.34)

+ V-h=0. V ft =0.

By the following simple calculations which employ (3.24) 2 we find that

)y ■v.v . '».(¥'+EE)Y

61 81 TA* ( * • "

= (lv*

+

ur ■(p. \\y>

^)(t + v •ip.v.)a

-2#c 2Hc „/>. ( ^ + +Ey0*.> + E)

A¥*

-2Hc nPs) + PsVs ■Jt+P'

E)

6E 'St

64

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

2 + Pspvsvs-v{±v* a-v(}V

++

E)yui+Psvs+.%+Ps E)=- .(lV2 E)=~(lv* + E)lpU}+PaV+s.6V Ps§§

2 + E) ■ V ^ + +P PsV 3VSs-V(±V E) ,,

a a -T-V-ajaa=-V -T-V^ -T.V!a, = -a".(T.V\aVs-(T.V)=-a = -Vs.(T.V) s-T.V-a a,

- VS-(T-V) ,

.

■

.

.

.

from which we obtain 6E

6V

VE + v sv- -v T ) . Vv ■VE P>yy* P.s6t ++p > (P,*6t + p*PvSV, />4f >■ +{p°Tt+ * •■v ^ - v *s •• T) ■

- Qv2

+ E) [pU]= p&

+ PsVs ■ VtE+paF

■V

2

a 2 E)lpUl -lp(v-V)U ~{p(v-V)U + + n-tj.V-an-q-V-a -Ta-T-V -V^-^V ta- (±V + EJlpU}.

Finally, (3.33)2 can be written as a (8E -V E)-a°.T.V VSE)-fa a T Ps{jt+V s v ss■ Ps \6t +

-lp(^(v-V)2

■V

.hs a + VVs sA

+ e-E)U + n-(t-(v-V)-h)}

=0 , on ES--TT..

(3.35) (3.35)

Now, it remains to consider (3.33)3. By subtracting the relation (3.26) 3 multiplied scalarly by C from the right we deduce that r■■lnx[(C-V)p - [ n x [ {C-V)sPs {\{V-C)2 2 (±(V-C)

+ E)) ++ T-{V-C)-\]\= 0 , on T. T.(V-C)-hs}}=0,onr.

(3.36) (3.36)

65

CHAPTER 3. BALANCE LAWS

3.4 T h e reduced dissipation inequalities. Together with the balance equations we have to consider the 2nd principle of Thermodynamics in the well-known form

M 1A

psdv +

v

psSda) > -

c

+

pss

■n h ■ nda -

av

(VN-Cnn-N

uL-N

lhs-Vdl

da dl s "EJ >

(3.37)

V

rr

where s and S denote the specific entropy in C - E and E - T respectively and 0 is the absolute temperature. This inequality is equivalent to the following set of local relations

j-t{Ps)

+ ps V-w + V - ( J ) > 0 , i n C - E ,

(3.38)

jl (PJS) + V s ■ (p,SV) - 2HpsScn + V s • ( ^ ) - [psU -1 h - n] > 0 , on £ - T , r.[nx[(C-t>sS-I/y]>0,onr.

For the sake of simplicity ( see [15] for the general case) we suppose that 8 is continuous across E; then by the same procedure used in the previous section we can put system (3.38) in the form

p8is + V • h - i& • V0 > 0 , P >eJt + p. w. ■V,5 + v

K K

' e

on C - E

V6- -lpO(s- -S)U- -h n j > 0

(3.39)

on E- ■ r ,

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

66

rT-lnx[(C-V)p -[»x[(C - *t]>0 >.S-lAj>0 tS-±h

n T .. ,, oon

By eliminating V • h between (3.34) and (3.39)j as well as V s ■ hs (3.35)

a n d (3.39) 2 ,

we obtain

the two following

between

reduced

dissipation

C- E , in C E

((3.40)

inequalities -p(i> + s9) -p(ij s6) + tr(t®Vv)-±h-V8>(} tr(t®Vv)-±h-V9>0

where %l> = e-6s,

(3.41)

is the specific free-energy in C and

-ps(V

2 2 + + [p[fyv-V) [p[fyv-V)

+ Se') +

aa.T-V,a

+ il>-V]U-n-(t-(v-V)-h)]>0 il>-9]U-n-(t-(v-V)-h)]>0 ,, on o nE E - rr , ,

(3.42) (3.42)

where y = E-9S

,

(3.43)

is the specific free-energy on E and we define

A' = Jf+V.-V,A.

(3.44)

All the considerations of the previous two sections could be generalized to the case in which the bulk phases are constituted by polar continua governed by Nonlocal Mechanics [16] or by a mixture [17]. Finally, in [18] the interface is supposed to be polar to take into account adsorption effects.

67

C H A P T E R 3. BALANCE LAWS

3.5 Constitutive equations.

We

concluded

the previous section

by remarking

that

the

balance

equations we derived were not general enough to permit the description of the wide phenomenology associated to interfacial phenomena; however, they are sufficient

to predict the most common aspects of phase changes. In this

chapter we propose a set of constitutive equations which permit, together with the balance equations and j u m p conditions, to formulate reasonable boundary value problems associated with phase transitions in materials which are described by those equations. In order to assign a continuous system with an interface (C-^CjjE), we have to give the constitutive equations describing the materials in the volume regions C,, C 2 and on the interface E. As is well known, the reduced dissipation inequality

(3.40) is regarded as a restriction for the volume

constitutive equations. In the same way, the inequality

(3.42)

supplies

restrictions for the surface constitutive equations and jumps. We shall show t h a t this last inequality leads to other constitutive equations. Let us suppose that Cj and C2 are elastic continuous systems; then their constitutive equations are simply given by 1> = iKF,0),

' = 5 =

" & •

89 '

(3.45)

68

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

h = h{F,9,V6)

,

where F is the deformation gradient with respect to a reference configuration and the heat flux vector h verifies the inequality

h-Ve<0.

(3.46)

T o derive thermodynamical restrictions for surface constitutive equations, we suppose first t h a t £ is also thermoelastic, i. e.,

(3-47)

* = *(*».> "a/J.O. T- -- T(PS-aap>6) >

s = s(ps,aa/3,e),

K- = K(Ps:aa0,O,

Vfl),

where

(3.48)

hssV6<0. ■ ve < o .

Relations (3.47) are motivated by the following considerations.

The

surface quantities at a point of E depend on the geometric state of the surface since they are functions of metric tensor aag . In other words, they depend on metric

variations of £ ,

dependence

of ? ,

T

not on isometric

, etc. on

transformations.

mass density

The

is a consequence

explicit of

the

circumstance whether E can adsorb or emit mass under the same defor-

69

CHAPTER 3. BALANCE LAWS

mation. On the contrary, if the total mass m(
psda

=

p? do-

that (see (1.17)) (3.49)

Ps4a' =--P0s^°-

This equation shows that in the absence of adsorption there is a relation between ps and aan . Going back to the general case, we first note that owing to (1.46), (1.41), (3.31) we have

a<*.T.Vta

=

T°'P(Va.p--ba0Vn)

=

T"Paa0.

(3.50)

On the other hand, by adding and subtracting [ n - t - C ] to [n-f • ( « - V)] and recalling that Cs = Vs (section 3.2, in) , we obtain

in-t.(v-V)j

= lpUi-lpj(cn-Vn)

,

(3.51)

where p = - n-t-n

.

(3.52)

Consequently, introducing the specific Gibbs potential on E, <7 = ^ + - § ,

(3.53)

(3.42) can be written as

-/»,(*'+ SO') + T«V/3 2 2

V n, ), ) > + (v-V) + g- *]U}-[p](c -V) + ff-*]t/]-[p](c n- n-■ V > o0 . + IP[\ {p[\(v-

(3.54)

70

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Moreover, we have from (3.47) 1

,T,/ 9'

_- 69 89

~dfsPs

n,

. ,, 89 89

n„, ,

d%Tp a^

,89., ,89 a, +

Wte'

.

(OZK) (3.55) (3 55)

'

-

Before inserting (3.55) into (3.54) we must evaluate p's , a'aa . T h e following calculations are carried out in view of (3.44), (3.24) 2 and (3.50), (1.50)

P» == ^^ ++ V, V, ■■ V VsPs 2H 2H [pU] s s ■■VV s s++ Psc sPs== - -pspVsV Psn cn ++[pU] al3 c )) v = sa (
(3.56) (3-56)

lpUj. ■ +W

+

W i t h regard to the primed derivative of the metric tensor a, we deduce from (3.44), (2.21) and (2.13) that

2 e 2 V2 V 'c0= CnKp) = a/3= hKfi aP = ( af) ( ap-- "»'./i) = ^af} K r-((C„ n- -VJ^ )b^ap)) '.

a

(3.57) (3.57)

If we collect the results expressed by (3.55) , (3.56), (3.57) , we can transform (3.54) as follows . (o , 89\o,

, (Tal3 , „2 09

- Ps(S + %W.

+

-PS(S

+ (T

+ W)9S

(T-

+ P;

«

a/3

«„

an - - 2 P s ^

+Psd-p-s"

+ [p[5(«-^) 2 + ? - / i s r i

89

v

),„,

-2p.Q^)'ap

+ lp[^(v-V)2 + g-p,s}Ul - {(pg » a«0 _ 2 ^ ^ _ ) 6 a / 3 + [ p ] } ( C n _ yn) > 0 f ~ {(Pi §~ ^ - IPs J^)K0 + [Pl}(c„ -Vn)>0,

(3.58) (3.58)

*. = .♦,.£.

(3.59) (3-59)

where

Ps = * + Ps§f-e,

71

CHAPTER 3. BALANCE LAWS

is the specific surface chemical potential. A standard procedure can then be applied to inequality (3.58) to derive constitutive relations

j.a/3 T a/J _

_

S=-|f,

(3.60)

2|« a0 2 a«__ + Lp.s Qda • dps c0

((3.61) 3<61)

a P* s a— a

This last formula permits us to write the residual inequality in the form

\P[\ (v-V)2

+ g-ns]U]

+ [Ta\fi-

lpj[(cn-Vn)

> 0.

(3.62)

If we introduce the notation J = pU , we can regard inequality (4.17) as a function / of J

(3.63) +

, J ~ , (c n - Vn) which

has the property of taking the minimum value when the aforesaid variables vanish. Consequently, the derivatives of / with respect to these variables evaluated at J

+

= J~ = c n - V' = 0 vanish and then we have

( ?f -f -A/ OOOo+ + == °0. . (9-

(3-64)

" Ps)o - ° '

(Taa/3 6, \0-lPj)o

^-[Pl)o = 0O..

In particular, the first two expressions imply that

Mo = o •

(3.65)

72 ■

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Hence the Gibbs potential should be continuous across E at equilibrium. As is well known this is a classical result of phase equilibrium due to Gibbs. We note also that (3.64) 3 coincides with the result we derive from the balance of momentum equation (3.26)2 at equilibrium. It is quite natural to assume that if the system is far from equilibrium the first members of (3.64) are functions of J + , J ~ , (c n - Vn) : *

/*

(1/

.(vO).

22 2 I^pP ++ ((v+-V) (v+-V) «+-F ) + + (g-» p -s)^ +===//+++( (/(//+++, ,J, J-,{c - n-VJ-,(c + ((g-» s) ) + n)), n-VJ), 22

^p-(v--V) ^p-(v--V) >

(g-» (g-»)-)- " t>" - -«v/ 3y ++ (»-/*,)" ss

,,Q/3 Q / 63 o ((T T 6(/a/3 -[ P ]) 00 aa/3 / 3-[p]) (^ Q / 3 6a/3-|Pl)o 3-IP1)O

++ =f-(J ,./= / -(./ +

(3.66) (3.66)

,J-,(c ,J-,(cnn-Vn)), ".(«„-vj).

= ,,JJ~"- ,,(c„-v„)). (Mnc n --V^ „) )) ) . =: tt(J ^( . / +, ^ ++

From this constitutive equations for the jumps we also derive lg} = h(J + ,J-,(cn-V ,J-,(c n)). n-Vn)).

(3.67) (3.67)

It is evident that (3.66) and (3.67) vanish at equilibrium so that they are at least of the first degree in their arguments.

Chapter 4

PHASE EQUILIBRIUM

4.1 Phase equilibrium boundary value problems.

W e want now to formulate the boundary value problem relative to the phase equilibrium at uniform

temperature for conservative specific body

forces, t h a t is, for f — — VZ7, in the absence of specific surface forces. Starting from the local balance laws and j u m p conditions (3.24), (3.27), (3.30), (3.31), (3.35), (3.36), (3.65) the complete set of equilibrium equations can be written as:

V-t-pVU

= 0 ,

inCjUCj-E,

l ?A T; P| — = IL [*?]> s Jl i

o n E^'-- Ti

U11

rtarW, [g] = o , T

■ [n x T ] = 0 ,

73

on T ,

(4.1)

74

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

where p = -n-t-n

and tf = t■ n + pn .

The boundary conditions are represented by the prescribed external pressure pe on a part dC C 5(CX U C2) = <9C and by the contact force 7 on the line dY, n SC such that

t■n = - p N ,

T-i/ E = 7 ,

on dC

on<9Sn3C,

(4.2)

where, as usual, i/£ is the unitary vector tangent to E and orthogonal to <9E (see fig.10).

fig-io Let us verify that the surface nature of T and jump condition (4.1) 5 do not permit

E to have a geometrical discontinuity line T. In fact, with the

notations of section 1.6, condition (4.1) 5 is equivalent to T + -v£ +T~

1/2 = 0 ,

onf

(4.3)

75

C H A P T E R 4. PHASE EQUILIBRIUM

whereas the spatial character of T is expressed by the condition n ■ T = 0; if we denote by « the projection of T - i / s on the plane orthogonal t o the tangent vector r to the curve T, we have u = An ~ + fin+.

It is now a very

simple exercise t o verify t h a t the conditions ( « - n ) + =(u-n)~ the

angle

between

n~and

n"""

vanishes.

This

result

imply that

means

that a t

equilibrium £ may have a t most a discontinuity line for the T -T -v-^ ■ For this reason in the sequel we suppose that T is absent. W e shall also assume, that [< s ] = 0 in (4.2)-^ This hypothesis is appropriate to the case of fluid phases and it seems a reasonable assumption if the region Cj, for instance, is occupied by an elastic body. More generally, it could be substituted by a constitutive relation [< ] = / ( p

+

,F~ ,6) .

In the next three sections we will consider the phase equilibrium of fluids, so t h a t we adopt the following constitutive equations t = - p l where / , /

denote the unit

,

T=7 J S ,

tensors in the space and on the

(4.4) interface

respectively, and j is the surface tension. As is well known the isotropy of fluids implies t h a t 1> = rl>(p,6)

,

* = *O>„a,0j,

P = p2

^=-^;=p(v>e)

where p(v,0)

'

7 = 7(^,M)

(4.5)

(4.6)

is invertible with respect to the specific volume v. From (3.53)

and (4.6)j we obtain v = ^

= v(p,d).

(4.7)

76

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

T h u s equation (4.1)-[ becomes

dg , Vp + VU = V(g + U) = 0

Finally,

the

phase

equilibrium

system

for

fluids

under

the

action

of

conservative body forces can be written as

g(p,0) + U(x) = const. ,

(4.8)

y = const. , [p] = 0 ,

h(p .*)] ==

0

with the boundary d a t a p = pe

,

on 5 C

,

fvz

- 7 ,

on 5 E n 5C .

(4.9)

4.2 Equilibrium of fluid phases separated by a planar interface in the absence of body force.

We start with the elementary case of a fluid system ( C j , C 2 , E ) which is not subjected to forces and has a planar interface. Equations (4.8) then reduce to p(x) = p=Pe,

VxeC

fo(p,»)] = 0.

(4.10)

77

CHAPTER 4. PHASE EQUILIBRIUM

According to (4,10) 2 the equilibrium value p of the pressure is uniquely determined by the condition that the Gibbs potential is continuous across the interface for any value of temperature which permits the coexistence of two phases.

T h e variance of this system is one since there is only one free

variable ( for instance the temperature). W e can say t h a t the knowledge of the constitutive equation

g(p,0)

permits, at least in principle, to derive the function p = p (9) which supplies the equilibrium pressure corresponding to any temperature when the interface is a

plane. From

(4.10)„ 5 it is also possible to derive the

differential equation whose solution is just the function p(9).

Clapeyron

In fact, if we

put

f(pW,e)

= lg(p(9),6)]

= 0,

a n d suppose t h a t equation (4.11) admits at least a solution pQ,90

(4.11)

, we have

( see (4.7)):

%

(Po-*o) = [ff](Po.*o) = M ( P o A ) # 0 ,

since in a change of state we have a j u m p in the specific volume across E. Differentiating (4.11), we obtain dp _ d0~ On the other hand we have

ldg/89] [dg/dp]-

g(p,9) = ip(p,9) + pv = ip(v(p,9),9)

l4 1/j

'

+ pv and

78

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

therefore

8q d i > . + 86 ~--89 86+lJ

p

dv drpdv. dib 89 89'~" 8v dv 89 89++ 89 89

PP

dv

89~ 89 ~

~s ' '

provided t h a t we recall (4.6) 3 and (3.45) 3 . From (4.12) and (4.7) we obtain the Clapeyron

equation

dp \{6) dp___W)_ d6-$\v{p,6)\> d6-0[v(p,9)]'

(4U) (4.13) [q l,i>

-

where \(9) = 9ls} \{9) 9{s\ is the latent

(4.14)

heat.

Before advancing further, we need some general properties of the function g(p,9)

which will be suggested by the physical considerations in the next

section.

4.3

A b r i e f r e s u m e of p h e n o m e n o l o g y of s t a t e c h a n g e s .

It is well known that in the elementary Thermodynamics [19,20,21] the set of homogeneous equilibrium states of a pure substance ^B is described by a state equation having the form

pp = p{v,9) p(v,9). .

(4.15)

79

C H A P T E R 4. PHASE EQUILIBRIUM

It can be experimentally shown that (4.15) is not defined for all positive values of v and 9. More precisely, the qualitative behaviour of surface S given by equation (4.15) is represented in fig. 11 where the forbidden states are indicated by shading. This figure shows that the body can be solid, liquid or vapour. Moreover, it can pass from one state to another by (very slow) transformations which are geometrically represented by curves on surface S. Of

particular

transformations. planes p = const.

interest

are

the

isobaric,

isovolumic

and

isothermal

They are obtained intersecting S respectively with the , v = const, and 6 = const.

Further terminology connected

with the phase changes can be deduced from fig. 11. T h e most interesting property of (4.15) is represented by the possibility to put it into the form v = v(p,6)

only in the regions of S which correspond

to equilibrium states of ^ which are in the interior of Ss

, SL

, Sy.

Along

the curves l ' - l " , 2' - 2 " , 3 ' - 3 " the function v = v(p, 8) assumes two values representing

the specific

volumes of two phases coexisting at

a given

t e m p e r a t u r e and pressure. The curves 1, 2 and 3, that are obtained projecting respectively the curves l ' - l " , Clapeyron's

2 ' - 2 " , 3 ' - 3 " on the plane p, 6 are called

curves corresponding to sublimation, melting and vaporisation.

In particular, at the point A where, the curves cross each other, three phases coexist and have assigned values of specific volumes. Another remarkable aspect we deduce from the qualitative behaviour of function

(4.15) is represented by the existence of three values

Pc,8c,vc

corresponding to the m a x i m u m of the curve 3' - 3". For these values, the liquid and vapour phases become indistinguishable (opalescence phenom-

80

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

enon). Moreover, when 9 > 9c , no increase of pressure permits to obtain the liquid phase. Then one says that the point of the space p, 9, v corresponds to the gas phase. The values pc,9c,vc

are called critical values and the isotherm

9 — 9 is the critical isotherm. This curve exhibits a horizontal inflexion at the point pc, 9C, vc.

Finally, we observe that corresponding three critical

values of p, 9, v do not exist for the liquid-solid transformation.

fig. 11

CHAPTER 4. PHASE EQUILIBRIUM

81

A first theoretical a t t e m p t to determine (4.15) was made by Van der Waals who derived the following equation by statistical considerations

p

=

where r = R/M

r6 v -bb v - b

a v V

2 '

(4.16)

is the ratio between the universal constant R of gases and

the molar mass M whereas a and 6 are two constants depending on the substance. If a = b = 0 , (4.16) reduces to the equation of perfect gas p-r9 f — v •

P=r£.

(4.17) (4.17)

T h e presence of 6 is due to the finite molecular dimensions and the term a/v reflects the molecular forces due to the cohesion. W h e n p and 6 have prescribed values, equation (4.16) is a third degree equation in the unknown v

pv3 -—{pb (pb ++rd)v r9)v2 ++av av- - ab ab== 00. .

(4.18)

T h e critical isotherm intersects the straight line p = pc in a triple point at which the aforesaid isotherm has a horizontal inflexion. This means t h a t equation (4.18) a d m i t s a triple root so that we can put it into the form

(v-vvc-f vcf = Pc Pc( = 0.° •

(4.19)

equation with (4.19) we If we put v = -~-vVcc in (4.18) and compare the resulting equation

82

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

find t h a t

vc = = 3366 ,

Pc Pc

a

-= ^ ao 2, 2762 ' 276

8 a

(4.20) (4-20)

0cc = 4 - % . c ~ 27 rb ' 27 rb

These relations permit us to express vc ,pc

'

, 9C in terms of a, b, c and vice

versa. It is possible to give (4.16) a form

independent

of the

particular

substance and then to compare the obtained equation with the experimental behaviour of (4.15). In fact, introducing the nondimensional quantities

p == p/ Pc, ' P P/Pc

Tr

== e/e - v/v e/e c,c, vV = v/vcc, ,

equation (4.16) becomes pP _=

8r 3L . —pW - 1 VV22 ' W -I

(4.21)

This equation has a form independent of the substance since it contains only numerical constants ( principle of corresponding states ); moreover,

the

isotherms we derive from (4.21) are given by the curves in fig. 12. These

curves

agree

well

with

the

experimental

behaviour

of

real

isotherms in liquid, vapour and gas phases with two exceptions. First, they enter into the forbidden region, which is bounded by Clapeyron's curve, and there is no theoretical criterion to define this region. In the following section it will be proved t h a t this problem can be solved provided that the Gibbs

83

CHAPTER 4. PHASE EQUILIBRIUM

potential of material is known. Secondly, the states defined by the triplets p,9,

v

can go into the forbidden region when any of the phases consists of

one or more separate regions ( for instance drops of liquid or bubbles of

fig. 12 vapour) having very small diameters (10

-=-1 mm).

This circumstance will be explained in the next section by supposing the interface to be able to exert a surface (isotropic) tension. In this situation, the liquid drops can be at equilibrium with their vapour which higher t h a n the pressure p

is at a pressure

corresponding to the equilibrium with plane

interface (superheated liquid). Similarly, the vapour bubbles can be a t a pressure higher than p without becoming water (see fig. 13).

84

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

fig. 13

4.4 Equilibrium of fluid phases separated by nonplanar interfaces.

In this section we confine ourselves to a local analysis of the boundary value problem (4.8)-(4.9). To begin the analysis, appropriate hypotheses, which are suggested by remarks of the previous section, must be satisfied by the constitutive equations of the two fluids in order that a solution exists of the posited problem. More precisely, by omitting the dependence on 9 since

CHAPTER 4. PHASE EQUILIBRIUM

85

we are interested in equilibrium configurations at uniform temperature, we shall suppose that the specific Gibbs potential g^(p) of the two phases are such that the equation

92(P)- -9\(p) == 0 , is satisfied for at least one value p of the pressure (corresponding to a plane interface). Since (see (4.7)) d ^9i2 _ _ L1 ±i on

dp 5 ^2 = 7P2^ *0 ,>

w

V Pi,P2> VPl,p2,

we can say that the equation
(4.22) (4.22)

defines implicitly the function p2 = p2(Pi)> where pl varies in a suitable interval

[a,b]3p

and p2 in another interval

[c,d] $p

. On the other

hand, by (4.8) 1 we get Fil{(x,p X,pit)

) + U(x)-c = gil(Pi)+U(x)-c l = 0. i

(4.23) 0

If we fix the value p

0

e (a, b) of the pressure p1 at x e C 1 U C2 , a

corresponding value of Cj is determined by (4.23) which now defines implicitly Fl(x°,p~)

the function pj = px(x,p~)

because d F j / d p j = 1/pj / 0 and +

= 0. Moreover, a value p e(c,d)

can be associated

with

86

p

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

e (a, 6) by equation (4.22). If we assume that

Urn p2(x,c2)

= p+ ,

xgC2 we obtain a value of the constant c 2 and then a solution p 2 (x, p + ) of the equation (4.23) for i = 2. Let us now suppose that the equation

H

= 2 [P2( a : 'y' 2 r ( ; E 'y)'P

+

)-Pl(a;'2/'2:(:r'y)'P_)] '

with the boundary condition (4.9) 2 admits only a single solution containing

z(x,y)

the point x . It remains to prove that the condition (4.8) 4 is

satisfied at every point of z(x, y). This is obvious because, owing to (4.23) the difference [jr] is constant on E ; but it vanishes at x p

_

and p

+

because of the choice of

.

To conclude, we can state that the external pressure being given, the pressure p

+

is determined at a point x e C2 ; if p ~ is the corresponding

solution of (4.22), p 2 is assigned and hence

4.5

z(x,y).

E q u i l i b r i u m of fluid p h a s e s s e p a r a t e d b y s p h e r i c a l i n t e r f a c e s .

For the sake of simplicity, the body forces will be supposed to be absent so that the equilibrium system (4.8), in the presence of spherical interfaces,

87

CHAPTER 4. PHASE EQUILIBRIUM

can be written as p- = c, (const.)

,

in Cj U C2

(4.24)

j = const. , 2T

-=- = c (const.) , [p] = c > 0 ,

fo(p)] = 0, where R is the radius of the bubble or drop and 7 is a positive function of 9 (see [22]). To analyse system (4.24), we suppose the existence of a value cM such t h a t (4.24)4

5

a d m i t one and only one solution p~(c)

C

[ O I M ] - Owing to (4.24)j , p~(c)

, p

+

, p~*~(c) , Vc e

(e) coincide with the pressure fields in

Cj and C2 , respectively, i.e. Pl(vl)~

V2(v2) = P+

V~ ,

'

so t h a t it is not necessary to distinguish between either p x or p ~ and p 2 and p+. v+

When the external pressure is given, the specific volumes v ~ = v^ , = v2 can be evaluated. In order to guarantee the existence of one and

only one solution of (4.24) 4

5

in the interval [0,Cjy] , we suppose, in

agreement with the experimental results, that the function p(v,9)

is defined

in D — (6,00) x (6^,00) b > 0 , # + > 0 and satisfies the following conditions: i) a critical value 9 of 9 exists such that, for every 9 < 9c , the function p( ■ ,9) e C (6,00) ; moreover, dp/6v

> 0 in (6,00) and

lim p(v,0) = 00 , v-*b

Inn p(v,9) v—>oo

,

(4.25)

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

ii) for every 6> G (
C1[(6,i/1((9))U

(v2(9),oo)]

Vj(#) < v2(9), and the following relation holds

Umv 1 (^) = lirn , # ) ; c

c

Moreover, dp/dv < 0 in (6, Uj(0)) U (v2(9),oo), and $p lim -5^ = 1 ^

Pi = p(v-t,6) < p2 = p(v2,9) ,

dp -=- = 0 .

From now on, the temperature 0 will be discarded in all the formulas to simplify the notations. The hypotheses i), ii) imply that the function p(v) is invertible on (6,oo) for 9 > 9C and the inverse function u(p):(0,oo)-»(&, oo) is decreasing ; similarly, when 9 < 9C , by inverting the decreasing function p(v) on (bjV-^) and (t>2,oo) respectively, we obtain two functions

v1(p):(pvoo)

-* ( 6 , ^ ) ;

v2(p):(Q,p2) -» (v 2 ,oo) .

By recalling the thermodynamical relation (4.7), two functions g^(p) and <72(p) are obtained; these are defined to within linear functions 4>^{9) of 9:

9i(p) =

«,(?) dp + 4>i .

When we take into account the relations

g=*».

d29 _0v dp2 ~dp~

_(dp\-in \dv)

< U

'

(P^PI,P2) >

89

CHAPTER 4. PHASE EQUILIBRIUM

we conclude t h a t

the functions ffj(p), p e ( p j , oo) and v2,

pe(0,p2)

are

always increasing and present an upward convexity. Moreover, since

"i(p') < « 2 (P"),

Vp'e(pi,oo),

V P "e(0,p 2 ),

at every point, the function g-^(p) has slopes less steep than those of the curve 92(P) ■ It is now possible to determine the locations of these curves on the plane (p, g) because the functions fll1(p) and g2(p)

are not completely assigned

owing to the arbitrariness of the functions <^>j and 2 of the temperature. In order to reduce this indetermination

of g-^ and g2 , together with

the

properties of <7j and g2 above derived from i) and ii) , we assume that in) for every 9 < 9C , a value p 0 e (pvp2)

exists such that

ffi(Po) = 92(P0) -

(4-26)

and furthermore lim p - * oo

g^p)

= oo ,

lim g2(p) = - o o . p —► 0

It is obvious t h a t condition (4.26) is equivalent to the existence of a solution of (4.24) with a planar interface. It is also convenient to observe that (4.26) determines the difference <j>-y-(f>2; therefore, for 9 e (#*,# c ) , <7j(p) and

g2(p)

are determined up to an arbitrary parallel translation to the <j-axis. As it will be proved in the sequel, the remaining arbitrariness of g^ and g2 does not affect the physical results. Finally, we observe that (4.26) and the previous

90

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

considerations imply that (4.27) (4.27)

9922(P)<9I(P), (P) < 9i(p) ,

Pe{pvp0), pe(PvPo)

92(P) >9\{P) (p) > SI(P) .<

P£(P p e ( P0,P o 2' )P 2 )■-

>

In order to prove an existence and uniqueness theorem for the system (4.24), we start with the relations gi =
(4.28) (4-28)

g 2 = ^922(P ( ^2)2 ) '.

go = ffi(Po) ffl(Po) = 92(Po) (P0) >.

„ - „ „ f of „ f fthe i , „ following f ^ n „ „iTiriff rtheorem l-i A r t n # \ r v i and the proof

THEOREM

1. — The functions

are

g-, and g2

both invertihle

on the

interval [gi»gg]; moreover,<> on g 0 ] 92~[ (1g{g)-9r ) - ~9i (s) H

1

is positive,

decreasing

P R O O F ..— --The The

and vanishes at gQ. function

^<7j(p) ( p ) increases

in

[pj,oo) [p^oo)

and

{p) p j , o o ) . Similarly, the function function g22(p) invertible on [p 1 , p 0 ]j C [ [pj,oo). [Pi-Po'

then

it

is

increases on

i l is also invertible on [[p-pPg] (0,p 22 ]- Since ;p°i1 < p0 < P ^ P Q ] .■ By (4.27), (4.27), (4.28), < pP2,it < Po 2.

we have have g2{[p9v2p([P - [g2(P2)>So] 0)] we VPQ)]~= L 9 2 ( P 2 ) .Sol both

g^~ J (g) (g) functions g-^~

{d/dg)g-\g) (d/dg)g,r\g)

,,

cc

[gi-Sol- Therefore, Therefore, on interval [[g [gi-golon the interval g ia ,,g g 00 ]] g2~ (g) exist and are increasing because

= = l/vi 1/*,i > 0. Finally, from v

d " I " - 2
ff2_1(gi)

= P*

an<

^

recall that

ffi-1(gi)

91

CHAPTER 4. PHASE EQUILIBRIUM

= p 1 , t h e previous result allows us to say that the difference g^ (g) S' 1 _1 (g) a t t a i n s its m i n i m u m p* - p1 a t gv decreases in the interval [g 1 ,g 0 ] and vanishes at g 0 . By setting

*™ = ^ r '

(4 29)

"

Pv'Pl we are now in the position to prove the following theorem: T H E O R E M 2. —If the temperature

9 is fixed in the interval (O^.,0C) for

every value of c e (0, pv - Pj) the system (4.24) has one and only one R

(Pi> PV' )<

where

e

e

Pi [Pi ' Po\ • Pv [P* > Po)

and R

solution

e (Rm, oo) •

P R O O F . — Owing to the previous theorem, the equation

g2-1(g)-gr1(l)

(4.30)

= c,

has one a n d only one solution g if and only if c e ( O J P ^ - P J ) . On the other hand,
and g^

being invertible on [gj,g 0 ] ,

one and only one pair

corresponds to every solution g of (4.30) such that

9r\g)

= Pl,

F u r t h e r m o r e (4.30), (4.31) imply pv-pt

9i\l)

= Pv

(4-31)

= c > 0. Finally t o every e g

(0,

p * - p,) it can be associated a radius R for the spherical interface given by R = 2y/c which belongs to t h e interval [Rm, oo) since c = p * - p j is t h e greatest value of c (see fig. 14).

92

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

fig. 14

It is clear t h a t in a specific problem we have to take boundary conditions into account. If we suppose that the vapour is in the interior of spheres which are contained in the liquid ( bubbles), it follows from theorem 2 that T H E O R E M 3 . — If the temperature liquid

at uniform

pressure

Pi&[Pi,PQ)

6 is fixed in the interval is at equilibrium

bubbles if and only if the vapour is at a suitable pressure is contained

in bubbles of a fixed radius R e [i? m ,oo) .

(0^,6

), a

with its

vapour

p e [P*,PQ)

and it

93

CHAPTER 4. PHASE EQUILIBRIUM

It is worthwhile to observe that, when the functions p1(i<), P2iv)i 9\{p)i g2(p) are assigned (these last two being defined to within a constant), it is possible to evaluate the values of pv,R,vl,vv

for every pt e [PijPn]'

THEOREM A . — If the temperature is given in the interval (9^,9 ) , a vapour at a uniform pressure pv e [p 0 , p 2 ] is at equilibrium with drops of radius R of its liquid if and only if liquid has a suitable pressure p ( e

[PQ:P*]

and R e [2y/(p* - p 2 ), oo]. Theorems 3 and 4 lead to the same results that Serrin proved in [23] starting from the Korteweg theory of capillarity [24].

4.6 Equilibrium between isotropic solid and fluid phases.

In this section we consider an isotropic elastic solid which occupies the region C 5 and is at equilibrium with its melt or vapour in the region C^ [25] . We suppose that the body force is negligible, the condition [<SJ = 0 is satisfied (see section 4.1) and the external pressure p e is uniform.

From

(4.1), (4.2) we have V * = 0,

inC5-E,

Pp = const.= p e ,

in Cp - S ,

T?g = 0 ,

onS,

Ta\0

Pe-Ps'

=

(4.32)

94

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

9F where ps = -n-t-n

~ 9s = ° '

i and t-n

= -psn,

(4.33)

owing to the hypothesis [< s ] = 0 and tp — — peI. In the traction boundary value problem for an isotropic elastic solid Cs we specify a natural reference configuration C 0 , in which the mass density p0 of the solid is uniform,

and write (4.32)j and (4.33) in terms of the

Lagrangian coordinates. A displacement u(X),Xe£0,

which is a solution of

this problem, supplies the equilibrium configuration C s . In the case we are considering, the reference configuration CQ cannot be assigned completely. In fact, we can assume that C 0 is a homogeneous natural state at a uniform pressure p 0 , where p0 is of the order of the equilibrium pressure of the solid in the presence of its vapour or melt at a given temperature. However, we can not specify <9C0 because neither the total mass of the solid nor the region it occupies are known. On the contrary, as we shall prove, $C 0 is determined by (4.32) 3 4 . If we suppose that

C0 -* Cs is an infinitesimal deformation, the

stress tensor can be written ( see, [26], p. 249) as

t = t0 + Ht0 + t0HT - trHt0

+ C-E .

where H = Vw is the displacement gradient, E is the infinitesimal deforma­ tion tensor and C is the

infinitesimal elasticity tensor. By recalling that

10 = - PQI and the solid is homogeneous and isotropic in C 0 , we have C • E = XtrE + 2pE and the previous relation becomes

95

CHAPTER 4. PHASE EQUILIBRIUM

tt=-p = 0-I pQI + trE( trE(p + X)I A)/ + + 2(AI 2(/i -- P Po0 + P00)E )# •.

(4.34)

ion we deduce that From this relation trt + 3p0 itrEr^ = 5-r—S- , + 3A ++ 2/z ~ pp 00 + 2p ' 3A + 2p

1

0

so that we have

we have Zr*+ +Zp3p 0 l1 trt P \ti I/ 7 - ((v i| AA\) 0 iJ [ +Pv (Ppo + )) [* P o + E [t+PoI {Po X) 3+ 2/z p + 33A ^ -=2 2i^pV) ( / , - p 0 ) " -" u 2 ^] - ' PPo o0 + A+ +% -

(4.35)

(4 35)

'

On the other hand, the specific free energy tf)g(H) up to terms of the second order is represented by the following expression:

i>s(H) = ^s(I)-^trE,

(4.36)

where V's(-0 is a constant owing to the homogeneity of CQ. Therefore, (4.32) 5 can be written as follows

iPs(I)-P<±trE

+^ ( l

+

trE) = gF(pe),

on E .

But we have also Pg = - n -1 -n = p 0 + terms of the I s order. Hence

MI)+jPj

= 9F(pe).

We recall t h a t rpg(I) is undetermined and gp{pe)

(4.37)

is known up to an additive

96

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

constant

(when the temperature is given). Moreover the quantity ps

expressed by the difference

P(,[gp(pe) - ips(l)]

is

which contains only one

additive arbitrary constant. This means that (4.37) permits to express ps as a function of p e when we experimentally know the value of ps corresponding to a particular value of p e . If we denote by p a plane interface,

the equilibrium pressure with

we deduce p s = p~e from (4.32) 4 . In conclusion (4.37) can

be p u t into the form PS~Pr

= Po[9F(Pe)-9F(Pe)]

>

on S

(4-38)

>

which does not contain arbitrary constants and shows t h a t ps is uniform on the whole boundary <9C5. -Psn

According to (4.33), t-n=

on E is a uniform

pressure and

therefore a solution of (4.32)^ is represented by the uniform stress tensor t = - p$I. This solution is acceptable because (4.35) shows that E is uniform and consequently satisfies the compatibility conditions V x V x E = 0. By taking all these results into account , (4.35) yields the form

£ _

Ps-Po p 0 + 3A + 2/z

f

Pol9F(Pe)-9F(Pe)] p0 + 3A +

+ Pe^

2/J

I = f(Pe)

I-

(4.39)

It is easy to verify that, up to a rigid displacement, (4.39) implies that

u = f(Pe)X,

XeC0.

(4.40)

97

CHAPTER 4. PHASE EQUILIBRIUM

On the other hand, by (3.64)1, gs = <7^(pe) = / i s ( a c*3' Ps)

anc

^

we can

'nen

express ps as a function of aQo so that the equation (4.32)4 becomes r""(a^)6Q/J=pe-Ps, where ps in turn

(4.41)

is a function of p e owing to (4.38). Consequently, the

pressure pe being given, the relations (4.32) 3 and (4.41) represent a system of three equations in the unknowns aao, bag. When we add the Gauss and Codazzi-Mainardi equations (1.28), (1.29), we obtain a system which, in principle, is able to determine the interface r = r(ua)

up to a rigid

displacement. In conclusion, recalling that the actual position x — X + u, we deduce the boundary r0(ua) of C0 corresponding to a given pressure as

r (

» "° ) = TT7TO'

4.7

<4 42

' >

Variational formulation of phase equilibrium.

In this section the equilibrium of a solid phase in the presence of its melt or vapour is discussed from a variational view point. In other words, we do not use the equilibrium system (4.1) but resort to the Gibbs principle according to which the equilibrium configurations are extrema of the total free energy with respect to variations at constant volume or of the total

98

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Gibbs potential with respect to variations at constant pressure [25, 27]. W i t h the notations of previous section, let

\p =

pFipF

pgi's dv +

c5

dv +

p^s

da ,

(4.43)

cF

be the total free energy of the system where ipF = ipp(pp)

, ipp — ipp(F)

,

$ = ^ ' ( p ( T ! i a g ) and E is a regular surface. This last hypothesis excludes that the following considerations can be applied to crystal planes. Moreover, we suppose that the solid and the interface are isotropic so that xps depends on the principal invariants of B = FF

and $ on the principal invariants of

aa0- The supposed isotropy of the solid phase is equivalent to require that the crystal structure does not manifest itself at a macroscopic level. This is possible if the dimension of Cs are large compared with those of the crystals which constitute the solid and if the crystals are oriented randomly. Let us consider the set of functions K = {Jfc:E —► k(E), regular , Jb(<9£) = <9E and conserving the surface mass}; R = {ps:i:—>ft

+

, regular}.

Moreover, we will denote by h :C —► C , C = Cs U C^ , a mapping such t h a t i) h is regular in C - E and exhibits finite discontinuities together with its derivatives across E; it) h.fo

: C s —► M ^ s ) ' ^ | C

' —* M ^ F ) '

are

diffeomorphisms and

conserve locally the mass. If we denote by H the set of these mappings, the quantity defined by (4.43) will be regarded as a functional over the set H x K x R which can be normed in an obvious way.

CHAPTER 4. PHASE EQUILIBRIUM

99

The Gibbs principle states that (he, ke, pes) is an equilibrium configuration if ^(h,k,ps)

has a conditional extremum with respect to all the arguments

h, k, p s satisfying the global constraint tp = tp =

pps sdv+ dv + r>5 C

Pp pFdv+ dv + pps sda-M — 0Q, da - M — , y rZ >F £

(4.44) (4.44)

M being the total mass of the system. As is well known, the extrema of (4.43) under the condition (4.44) coincide with the extrema of the functional *+ $ =y + Av?, \tp ,

(4.45)

where A is a constant. In order to evaluate the Frechet differential d<& of <&, we start by observing that the variations of i>s(F)dv' dv's^ss(F)

dtf 55 = rf* = C

5

F pPs^s( srps(F)

dv

^S

pss[iP [ii,ss(F') (F')-4> - ^sS(F)]dv(F)} dv - | ps^4> -n)Skn) da da s(6h s(Sh Sn Sn-8k

= =

E

C c?5

/ 6h dK, dv 6h Sn-6kn)6k PSips{6h da• . Ps PS -Qp-FdF jL./jL i, j3 dv ~ Ps^s( Sn - n) dcr

= = i>

e5

%±j

"

Ey.

By applying the Gauss theorem to the first integral and recalling that 6h = 0 on 9C , we obtain

100

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

fig-15

d ^ dF 9F,LL

di>s - L *W>« dada■ 9FiLiL **) 9F **) d p ' *S= iv.IPS-IP-FjLN^hs.da\(ps®pLFjL) d * ss

6h 6h

Si Si

dv dv

1

6hSidv

6kn) da da- . ~- J PS^s( Ps^s(6h6hSn s,, ---**„) E

Now we introduce the quantities t1 = Ps Ps^¥FT'

~

dF * ps — -n-t-n

' ,

t3 = t-n+ psn , t-n+psn<

iin n CC

s5 on E

(4.46) (4-46)

(ts-n === 0) 0) (*.••

ffs = ^+jj§» which respectively define the stress tensor t in C 5 , the pressure ps on E, the tangential stress and the specific Gibbs potential on E (see (3.53)). We remark that gs is meaningless in the volume. It is quite easy to verify that dWg can be written as d*s s == dtf

V - i - 5V't>6hdv-\p ft^\ps9 - <5fcn) da S(^Sn sgs{6h Sn--6kJd
E E

CHAPTER 4. PHASE EQUILIBRIUM

101

■ (Sh Sks) dcr - ps ps8k8k + ts-(6h + + ts t■ 6k6k s-6k s s-)das ■ n ndada s sdada. . vE

V. E

(4.47) (4.47)

m E

Similarly it can be proved that

d^F= =

» Vp ■ F8h Sk n) Vp-8h dv+ F dv + p F g F (6h Fn-6k PF 9F (6hFnn) E CF

p 8k 6knn da,

d<7 da+

+

(4.48)

E

where we have defined di>F P=J^, P = dPF>

(4.49)

iinn CCF

F + jp, 9gFF = ^^F -jTF-'

on o nEE ..

In order to evaluate the variation d'$a of the third integral in (4.43), we recall (see (1.17)) that da = -\[S du du ; consequently, 8{da) o(aa) = = k■-6aoadaaa. . Moreover, owing to (2.13), (2.14), (2.15) , we have af3 8{da) = i aaf} 6k 8ka/i a0 da ,

(4.50)

8a 8kapaj3 ., af) =
4 51 ((4.51) - )

where Sk

6k afi ap

= 6k +6kSk .n = 6k c;a.pd + 0;p.aa-2b ~ 2ba^k af}nSk

(4.52) (4.52)

•

According to hypothesis i), the surface variation 8k conserves locally the mass and then from psda = const, we obtain af, 6p.= -%P,a 6kalifi.6ka0. 6p,= -%PSf .

,

,i

,i

.

.,

(4.53)

When we bear in mind that there is another mass variation Sps due to the J,.

102

-PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

adsorption on the surface, (4.50)-(4.52) leads to the result

d*„ =

P's *(P>*fi)

d

d

°'- \ps nps^afi)

°

(Ps ~ \ Ps^K? + « * + §f- ( 4 ^"^a/3 + *P.)

+ a^-^](i+^M dff -

Ps*S

d(T

+6k i-htiiraaP+w-H'k°ie *«a/3

26

«*"»>d
+ ( +

I * "'^"-* r

By applying the Gauss theorem and introducing the notations 2 d^naP^0n

rpafi _

3*

(4.54)

^=4(^}' which define the surface stress tensor and the surface chemical potential, we have

d*s = - J T $ 6ka/3 aV- J V ^ 6knda- Pjp. da . E

(4.55)

E

It is now easy to prove that

Ps(6hSn-6kn)d
dtp = -

i:

PF (6hFn-6kn)

+ i:

d(T

5ps da .

+ i

(4.56)

103

C H A P T E R 4. PHASE EQUILIBRIUM

By collecting all these results, the condition d<3> = 0 can be written as

V • t ■ 6hs dv + ^S

Psi9s + x)(^Sn

Wp-8hFdv-

~ ^n)

da

^F

PF(9F

+ X)(6hFn

~ 6kn)

d u

(fif + \)6ps

+

da +

ts • (5hs - 6ks) da E

{T?l-

The

arbitrariness

£ ) da-

+ T"&bap)

(ps-p

6knda = 0

of the variations 6h, 6k leads us again

to the

phase

equilibrium system (4.32), (4.33). We conclude this section with the following remarks. i) When "t depends on a = det(aan)

(fluid interface) and the mass is

locally conserved in every deformation of E ( that is E does not adsorb), then the relation psda = const., conclusion that a = c/p2s

which implies ps~{a = const. = c, leads to the . Consequently, by putting ^!{ps, a) = $ ( p s ) and

recalling (4.54)j, we have

j.a/3 _

But

~P2s

<9# aa(3 + 2p 5 * 0 * aa(3 + 2p d± da _ s s da daa0~ - P I dPs da dp.

aa"*3

-o2(d±. 2c -da-)" a/3 ~P* d* = d¥ dl! ®Ps ®Ps da

da Ps

d

=

d* dPs

2c d
so t h a t na/3

Tap = 7 ( P >,«/3 -

2 . 3 * . „a/3

-P

(4.57)

104

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

This relation shows that the surface stress tensor is isotropic and the surface tension j depends only on p3 . Particularly, if \P = a ^ a , that is if E is a soap-like membrane, we have j = pg^ = const. const.

and the surface free energy is proportional to the area of E. it) Let us suppose that both the solid and liquid phases are at uniform pressures pg and p respectively. Then it is easy to verify that the equilibrium configurations of the surface E correspond to those closed surfaces which bound a volume having a fixed measure V and are extrema of the total surface energy provided that the surface mass is conserved. In fact, let us look for the extremal surfaces of the functional

$ =

Pa^iPs^afi) da < X 1 J

verifying the the constraint tp = \dv - V = 0 . tp = \dv - V = 0 . Owing to (4.55), (4.56), these surfaces satisfy the condition Owing to (4.55), (4.56), these surfaces satisfy the condition -

T?jJ6kadv-

(T°%,-A)M n
- 33T?j*6kade-^T°>\fi3-X)6knda

= 0,

£ E which leads to (4.32) 4 whenever the Lagrange multiplier A is identified with which leads to (4.32) 4 whenever the Lagrange multiplier A is identified with the difference p - Pg ■ the difference p- ps •

Chapter 5

STATIONARY AND TIME DEPENDENT PROBLEMS

5.1 A stationary problem and its nondimensional analysis.

In the previous chapters we have proposed a rather general model of a continuous system with an interface which can be applied to study the phase equilibrium. T h e investigation of stationary and is much more difficult.

time dependent phenomena

Therefore, we suppose in this chapter that

the

interface reduces to a pure moving surface of discontinuity for the volume fields. W e start with a model which is appropriate to describe the stationary continuous casting of an isotropic and incompressible solid. To be more precise, let us assume t h a t (C s , Cj, E) is a system with an interface which is formed by solid and liquid phases occupying respectively the regions C s and C,. Both materials are incompressible and inviscid and subjected to the

force

of gravity g. Moreover, we suppose that all fields are stationary, the interface

105

106

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

E is a surface of discontinuity at rest (c = 0) and finally the temperature 6 is continuous across E. Consequently, we have the following constitutive equations for the stress tensors t and specific internal energies e

ta=-PaI + t (1,11,0), t^-pfl, e s = e s (7,//,(?),

(5.1)

e, = e,(0),

where I, II are the first two principal strain invariants of the solid which deforms from a reference configuration C +s to Cs. Inserting these assumptions into (3.24), (3.27), (3.30), (3.31), (3.34), (3.35), (3.67) we obtain the following equations for the system we propose to discuss V-w = 0,

inC-E,

(5.2)

pv Vv = V • ( + pg, pv-Ve = t-Vv + IpvJ = 0,

V-(k(9)V6), on E,

(5.3)

[ p W n - t - n ] = 0, 2

[p ( i v + e) vn - v ■ t ■ n - kV6 ■ n] = 0, l±v2 + g} = 0, where k is the thermal conductivity. We have denoted fields which refer to different phases with the same symbol. This set of equations, though it is highly simplified as compared to the general system which involves a material interface, is still rather difficult to handle. Therefore, since we

107

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

intend to discuss the continuous casting, we suppose t h a t e, = e.(6),

(5.4)

and the velocity fields in both phases are uniform, that is vs = vsu,

vt = vp,

(5.5)

where u is a unit vector along the z-axis. Then the system (5.2), (5.3) becomes V t = pg,

in C,

p v c(0) 9,z =

PsVsn = PlVln J Plv lv} V lifJllnuu-[n-t} lnu-l"-H

[ i v22 + e\Pl Plvln ln

[l,2 +

hypotheses

considering; further

on

the

V(k(9)V0),

.

o n

E

(5-7)

>

= ~ 0, "'

- [vn • *t ]] • « -- [JfeVff] [JfeVff] ■■ nn= =0,0, V)W

j_n^n_g=0.

where c(9) = e'(9) is the specific heat. drastic

(5.6)

physical

This set is the result of the very properties

of

the

system

we

are

simplifying hypotheses would destroy its capacity to

describe the continuous casting. However not all terms appearing in (5.6), (5.7) have the same weight as could be recognized by a nondimensional analysis. In order to express (5.6), (5.7) in terms of nondimensional quantities, let us introduce certain scaling quantities for velocity, pressure and temperature (V, P, 0 ) a n d two length scales L and /. We calculate the derivatives of the t e m p e r a t u r e field with respect to z by scaling with the first length whereas variations of 9 with respect to x and y coordinates by the second one. In such

108

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

a way, the following analysis will be applicable to the case in which the system possesses dimensions of different order along the z axis and x, y axes, but comparable thermal variations along all the axes provided that the displacements are of the order of the ratio L/l. We treat the fluid velocity as scaling velocity U, whereas P and 0 are respectively identified as the atmospheric pressure and melting temperature. Moreover, if c = C/(0), k = &/(©) denote the specific heat and the thermal conductivity of liquid at temperature 0 we assume that "I < Ti c

? 0

(5

'

-8)

i.e., the specific kinetic energy is much smaller than the specific energy associated with the pressure and the latter, in turn, is much smaller than the specific internal energy. By denoting the nondimensional quantities with the same symbols of dimensional quantities, it is easy to verify that

system

(5.6), (5.7) can be written as in C,

^ t = p-p9,

(5.9)

1}c{e) 9>* = J^L ( f <*'••)■• + f <M'»)'y + (**»«)»*)' on E

Pvsn = vlw (vf-l)u—£j

»

[«.*] = 0, Plvl

[i« 2 + i #*i„=-- £ 5 [n-*]-i. —^TT ( 7 ^'*'' + M

+W

'»«] •» = 0.

(5-10)

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

109

[ l , M f *]_-£, («^3 + p l)as o, where /z = pjp\

and »', j are the unit vectors along the axes x, y. When we

take (5.8) into account, the previous system reduces to the following

V t = 0,

5*} c{9) e,z = a(^

in C,

(5.11)

{k9,J,x + £ (kd,y),y + (k9J,z),

l">»n = vlm

on S

'

(5-12)

[n.*J = 0, X(9)vln = -0[^k(0,xi

+ O,yj) + ke,xu]-n,

M = o, where we have introduced the notations: _

A *; PlVjcL

8a _- -a

Here A(0) is the latent heat. Equation

(5.12)4 supplies the melting

temperature on the interface which is independent of the velocity and pressure. This, under the hypotheses (5.8), is in full agreement with the experimental results. If z = ip(x,y) is the equation of the interface, the vector (
+ ^)1/2

110

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

and (5.12) 3 leads to the equation

\(6)vtu

■n = -

2

[ k k(9,J + 6,yj) + ke,2uj-(V
(5.13)

Pi T h e geometry of the problem we are considering is represented in fig. 14. On the section S (z = 0) of the cylinder Sx[0,L],

the pressure pe and

the

uniform entrance velocity VfU of the melt solid are given. Then equations (5.11)]^ and (5.12) 1

2

imply that t=

velocity of solid is v, = vj^i.

- peI

everywhere and the extraction

It remains to determine the temperature field in

S x [0, L] as well as the equation z(x, y) of the interface E. We are led to the following boundary value problem in the unknowns 9(x,y,z)

fig.16

, (p(x,y) :

111

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

V(x,j/,z)eSx[0,L]

-E,

^ } c{9) 9,z = a(^-

(k6,x),x

+£

(k0,y),y

+ (**„)„) ,

(5-14)

V(x,y,z)eS,

X(9)Vlu ■ n =

0(^,0) = ^,

L

[ i *(0 >jr i + 0 j ) + *0, z u] • (V

9(x,y,L) = 92,

kj^=f(x,y,z),

on

fi,

(5.15)

(5.16)

where # j > 0 , # 2 < ®i © ' s the melting temperature and $7 is the lateral boundary of S x [0, L], In [28] J. Rodrigues supplied a variational formulation as well as a weak existence theorem for the boundary value problem (5.14)

(5.16) for the case

L = I. In [29], L. Faria fe J. Rodrigues analyzed the aforesaid problem from the numerical point of view by assuming a rotational symmetry around the z axis. By supposing that the heat extraction from the lateral boundary Q. is represented by the formula / = a[9 - h(z)], they obtain the profiles in fig. 17. It appears that the m a x i m u m depth of the surface with respect to the plane x, y is affected by both the extraction velocity vs and the lateral cooling.

112

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

F,=0.5

V. = 0.7

fig. 17

5.2 On the approximate evolution of solid-liquid state changes.

The previous section was, in a way, dictated by the analysis of a stationary problem which describes satisfactorily the important process of continuous casting of metals. Now we want to analyze dynamical evolutions of state changes under the following hypotheses: i) the interface is a moving discontinuity surface for the bulk fields;

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

113

ii) the problems are one-dimensional. In this section we start with the solid-liquid

state changes. Let C be a

system formed by two phase C s and Cj of the same substance in the solid and liquid states, respectively. These phases fill the layers 0 < x < s(t) s(t) < x < L(t),

and

where s(t) denotes the location of the plane interface £ which

satisfies the condition i); moreover all the bulk fields are functions of the spatial variable x and of the temporal variable t. The solid in the region C s can

always be supposed

to be at

rest. Taking into account all

these

conditions the balance equations and j u m p conditions (3.24), (3.27), (3.30), (3.31), (3.35), (3.36), (3.65) can be written as

Pscse,t

ie[0,s(*)), *e[0,oo),

= kse,xx,

pH + (pv),x

= 0,

te[0,oo),

xe(s(t),L(t)],

(5.17)

(5.18)

p{vn + vv>x) = -P>x ' p(e,t + v e , J = -pv,x

,

p(v-s)=-pss

+ k6,xx

x = s(t) ,

,

ie[0,oo)

(5.19)

[v]J + [p] = 0, 2

[%{v-s)

+ e\J + [p(v-8)]

[i(t;-s)

where p3,cs,

2

=

[keim\.

+ ^ + | ] = 0,

k^ are three constants which represent respectively density,

specific heat and heat conductivity of C s and 0(x, t) is the temperature field; moreover,

to simplify the notations we have not affixed the subscript / to all

114

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

the quantities which refer to the liquid; finally, cn = s and (5.19) 4 is assumed to be valid also in dynamical conditions ( see (3.67) ). Equations (5.17)-(5.19) are valid also if the layer s(t) < x < L(t) is filled with vapour. We postpone the analysis of this case to the next section; at present we can further simplify

(5.17)-(5.19) by recalling the incompressi-

bility of liquid phase. More precisely, we assume that

p = const. ,

e = e{9) = c9+ eQ .

(5.20)

From (5.18) 1 and (5.20)j we at once deduce that v depends only on t; this conclusion, by taking into account (5.19)}, leads to

v = as ,

It is

where a = —p—- .

(5.21)

easy to verify that the boundary value problem (5.17)-(5.19) now

becomes 0.i = < » A « .

aps=-p,x,

xe[0,s(t)),

x e (s(t), L(t)] , Q,t + ase,x

lpj = apss2,

te[0,oo),

=

,

(5.23)

a6,xx,

x = s{t),

Pj e l* +^ a2pj3-aps

te[0,oo)

(5.22)

=

te[0,oo)

-mx].

(5.24)

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

M ♦l where a, = ks/pscs

as2- 2 -

and a = k/pc

P = "aP7 =

115

o,

are the thermal diffusibilities of solid and

fluid phases, respectively. We explicitly remark that the incompressibility of the liquid in C( imposes that the boundary L(t) moves with the velocity v of the particles of C(, i. e.,

L(t) = v(t) L(t) v{t) = as(t) as(t) ,

(5.25)

and therefore the motion of the free boundary of C; is determined by the evolution of the interface. Usually the densities of two phases have almost the same values so that a
remain

almost

the same during

the phase change.

The

boundary value problem (5.22)-(5.25), although highly simplified, remains quite difficult

to solve. However, by a nondimensional analysis we can

recognize t h a t not all the terms in equations (5.22)-(5.25) have the same weight. Let X be a length comparable with the dimension of the system C; as is well known, the quantity T = X / a conduction

phenomena.

denotes a time which is characteristic of

This means that

the interface

supposed to be comparable with the rate V = X/T.

velocity

can

be

Finally, let us prescribe

the initial and boundary d a t a for the temperature and let 0 be the difference between their m a x i m u m and minimum. By introducing the nondimensional variables T*

x _

X

X'

x

,

1

yi )' T

P

P

=

y~P'' >

* 0

V:'

* -— 6e— -' _

(5.26) (5-26)

116

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

s =

F'

e =

^e' * =7e'

where QM is the melting temperature at ordinary pressure, we obtain the following nondimensional system in which we have again employed the same symbols for nondimensional fields:

0> = 0,„ , ».xx = «,„ » p,xx= = --aAfis, aA/zs ,

t e[0,oo) , *e[0,oo),

«e[0,*(*)). «e[0,*(*)).

<e[0,oo) , *e[0,oo),

xe(s(t),L(t)], a: g (*(*), 1(01 i

ff,t + a«fi,a. = atf,a!ir

?

6,t + as6,x = a9,xx

,

[pl^a^s2, {pj-aAs2,

x = s(t), x = s(t),

*e[0,oo), <e[0,oo),

(5.27) (5.28)

(5.29) (5.29)

[e]s+Ia3s3-a|ps=(0,x)+-k(0,x)- , [e]s+Ia3s3-a|ps=(0,x)+-k(0,x)- ,

[ V - ] + i a 2 B s 2 - a | p = 0, W}+±a2Bs2-a%p

= 0,

where where

kk -=- * -

u-2**=£' »=f ,

f' k=A,

s

A-P*V2 A — ■"■

~

Pp

'i

aa -=- 2 a =

£' A,

5(5.30) ((5.30) -3°)

cs6'

and ( ) + denotes the limit of ( ) on the interface when we approach from the liquid phase. A similar meaning is attributed to ( ) _ . Tables 1 and 2

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

117

contain numerical values of the physical quantities we are considering and show that the numbers A, B and B/A are quite negligible with respect to a .

6.91

2.91

8.66

2.33

s

8.62

4.81

30.93

1

8.36

5.44

30.93

2.55

11.39

24.19

1

2.38

10.47

24.19

S

0.91

19.26

0.221

1

1.00

41.86

0.055

Aluminium

Water

k/io" erg/cms. D t

6.90

1 _

c/10* erg/8 D K

7.36

Iron

Copper

P g/ cm

0M

lat heat /10 =T8'8

1808

2.72

1356

2.13

930

3.93

273

3.35

Table 1

a

k

a

\

Iron

-0.067

0 80

0.68

2.18

2.1-10 -n

2.6-10-'6

n-io-4

Copper

-0.031

1.00

0.91

3.27

47-10" 10

8.5-10"14

1.8-10"4

Aluminum

-0.071

1.00

1.17

3.72

17-10" 10

6610" 1 4

38-10""

Water

0087

025

on

6.37

14-10" 14

3.0-10""

21-10" 3

A

B

B/A

Table 2

The numerical results contained in tables 1 and 2 permit us to neglect the terms containing the factors A,B or A/B

so that the system (4.27)-(4.29)

118

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

reduces to the following set of equations:

*.* = * . * » .

x e [ 0 , «(*)),

p = p(t) = Pe,

<e[0,oo), *e[0,oo),

xe(s(t),L(t)],

(5.31) (5.32)

d,t + as6,x = a6,xx , [p] = 0,

*€[0,oo],

x = s(t),

(5.33)

,

[e]i=(O,x)--k(0,t)+

W = o, where pe is the external pressure. We thus conclude that the pressure field is uniform and equal to pe; moreover, the condition [i/>] = 0 supplies the temperature on the interface since the specific free energy depends only on the temperature in both phases. Introducing the nondimensional latent heat A = [ e ] / c s 0 we arrive at the final formulation of the boundary value problem relative to solid-liquid state changes

»,* = « , „ ,

xe[0,a(0),

6,t + as6,x = &0,xx,

*e[0,oo), ie[0,oo),

xe(s(t),L{t)],

6=0,

x = s(t) ,

Ai = ( * , „ ) " - k ( 0 , J + , 9(x,0) = 9o{x),

xe [0,1(0)],

6(0,t) = 01(t),

«(0) = * OI

9(L(t),t) =

e2(t).

(5.34)

119

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

Carslaw and Jaeger determined a solution of the system (5.34) equipped with initial d a t a on the whole real line K, by employing the similarity method (see [30], p.209); C h a m b r e [31] found an exact solution on -ft for the state change of a viscous fluid taking into account the influence of convective phenomena. In [32] a solution of (5.34) is found in a semi-bounded domain illustrates the motion of the interface s(t) and the free boundary

which

L(t).

5.3 O n the approximate evolution of liquid-vapour state changes.

T h e liquid-vapour state change is a very complex phenomenon which is produced by the combination of conduction and convection. Inside the liquid, vapour bubbles form, they can unite with each other and give rise to larger bubbles. These reach the free surface where they release the vapour contained within. However, when the external temperature is only a few

degrees

different from t h a t of evaporation, the phenomenon takes place with simpler modalities. Let us consider, in a rigid container with a freely moving piston, a liquid mass subject to a pressure less than the critical one. Moreover, let us suppose t h a t the liquid is at rest and at saturation temperature. If the temperature of the piston is increased by a few degrees, a vaporization process begins across

120

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

the liquid surface near the piston (see [33], p. 491) and a vapour phase is formed which has a density much less than the liquid one; consequently, the total volume occupied by the system increases proportionally to the quantity of m a t t e r supporting a phase change. Under conditions on phase transition, we can apply the methodology we have followed in previous section with certain necessary modifications. In fact, with respect to the case of solidliquid state change there are two fundamental differences: i) the vapour phase cannot be supposed to be incompressible; it) the densities of the two phases are so different from each other that the vapour density can be totally neglected with respect to that of the liquid. In other words, the two cases differ

for

the

state

equations

describing

the

system

and

for

the

approximations we may introduce. W e again limit ourselves to the case of fields depending on one spatial variable x by

besides the time t. The interface is plane and its motion is given

the function

s(t);

finally

the liquid is incompressible.

Under

these

assumptions our system is again described by the equations (5.17)-(5.19) and it remains to find the further simplifications deriving from ii). From (5.19)j we obtain

P~ P-PlPi .

vv== ~±s, p *.

(5.35)

where the fields without suffix refer to the vapour phase. On the other hand, the vapour density is always negligible with respect to liquid one instance, at atmospheric pressure and temperature of 100°C, the density is 0.958 g / c m

3

whereas the vapour density is 0.596-10 ~

3

(for water

g/cm3).

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

121

Therefore, equation (5.35) can be written as

« = - £ * .

(5.36)

This last relation permits t o eliminate v from the other j u m p conditions (5.19) which become

lvl= -PiPi's\

(5-37)

5 * ( # ) 2 * 2 + p»(M+?)* = - [ * * „ ] .

H £ ) 2 * 2 + M + ? = o. As in the case of melting of a solid body, we can obtain a simplification of the evolution equations by a process of nondimensionalization. In choosing the reference values, the difference between the densities of the two phases is the determining factor. In fact,

it follows from it) that there exists a) a

corresponding significant variation of the linear dimension of the volume occupied by a given quantity of matter when it passes from one phase t o another; b) a noticeable difference between the vapour particle velocities and the interface velocity. These remarks impose the choice of scaling quantities for the length, density and velocity according to the phase. In particular, we scale the densities with respect to their values p ( and p a t atmospheric pressure and vaporization temperature. Owing to the incompressibility of liquid phase, the density will have the same value for x £ [ 0 , s ( i ) ) : p f = p ( .

122

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Let

us

introduce

the

nondimensional

P
rate

~Pi

the

water

a = 0.622-10 ~ 3 ) ; the ratio of the linear dimension of the regions occupied by a given mass of m a t t e r in the liquid phase to t h a t in the vapour phase is about \i. Therefore we fix the values Xt and X of the reference values in such a way t h a t Xl coincides with the initial dimension of liquid phase and X =

Xxlii. If we assume that the quantity T — X^/a

, where al = kj/pjCj

diffusibility of liquid, as the time scale, we obtain two velocities Vt = and V = X/T; velocity

is the XJT

we refer the vapour velocity to the former and the interface

to the latter.

Finally, we obtain

the following

nondimensional

variables * — ±, rri , t* =

x x

*»

s(t) Sit)

~ x,

xx

*e[o,«(<)) * e [0,5(0) ;

X -x-s(t) s(t)

r . . , , ., xe[s(t),L{t)],

*= - jt'x,'

+

x

(5.38) (5-38)

+

1T ^T~^ ' *e[«(0.i(0]. P ■ * s * V v =F, P ~J> V,' *

9

P ~ p >

- ee ''

pP*-

s p

pClo'

v

~Vl' e _

C / e'

~v

* ~ c ,e '

where 9V is the vaporization temperature at atmospheric pressure, 0 is the difference between the m a x i m u m and minimum of the boundary temperature and C; is the specific heat of liquid. The choice of the reference pressure is justified by the consideration that the constant pclQ is of the same order as the atmospheric pressure P (for water when we put 0 = 40°C, we have P/pClO~ ~ 1.1). P/pc,Q 1.1).

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

123

Simple calculations lead us to the following system where, for the sake of simplicity, the nondimensional fields have not the asterisk: p= = p(t), p(t), zx€[0,s{t)), e[0,s(<)), 9

H

=

d

>xx

Pt + (pv), ( H .x x = = 0Q,' Pt + Pix~ P'x=

£t e6 [[0,00) 0,oo), '

xe(s(t),L(t)], xe{s(t),L(t)],

-Ap(v, -AP(vrt t

Ve P(eH + ve 'x)= 'x)=

(5.40)

+ vv, x) i, + VV >x)

-P^'x + ^ ' x x

v=-j, » =-*«

'

x = s(t) x= s(t), ,

bl =

(5.39)

«

(5.41)

p 1

1 A^ +(lej ±A°l ±AJ£ (le]++P)s P)s=(9, -/*(*,,) +f(W - ^ ( ^x)+ x ) ++ ,. +0* =(9,x)- x)- -pk(9, 2 =

i 2

(

«

•

.

.

)

■

^4+ w+f= : 0 •

Here we employed the notations of (5.29)2 and A is defined by 4

V2 Cj
The number A is of the order of the ratio between the kinetic energy and heat energy per unit volume and it is very small (for instance, for water, when we put X, = 102 cm., 9 = 40°C, we have /1=6.10 • 10 " 1 3 ) . In view of

124

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

(5.40) 2 and (5.41)2 we see that the pressure field is uniform throughout the whole system, is continuous across the interface and is equal to the external pressure pe. Finally, the system (5.39), (5.41) assumes the approximate form

»„ = » , „ , P,t

xe[0, «(*)),

+ (pv),x = 0,

te[0,oo),

xe(s(t),L(t)},

(5.42) (5.43)

P(e> t + ve, x) = ~ Pv, x + ake> xx i v= - 1 ,

x = s(t) ,

(5.44)

([e]+ £)* = (*, ,)--(<**„)+ ,

W+f = o. This system should be supplemented by the appropriate initial data for the density, velocity and temperature as well as

boundary data for the

temperature and the continuity of this field across the interface. When we assign constitutive equations for liquid and vapour phases we obtain a boundary value problem in the unknowns p(x,t),

v(x,t),

9(x,t) and s(t).

5.4 The case of a perfect gas.

In this section we analyze the system (5.42)-(5.44) under the condition

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

125

that the vapour can be considered as a perfect gas. More precisely, we assume that the nondimensional specific internal energy e and entropy s of liquid are given by the following constitutive equations (^ + 0 ) 9)-e - e +m,, e= = (8 )t +

«, = l n ( 0* .. ++ 0 ) - «, .. ,,

(5.45) (5.45)

where 8^ = 0V/Q and e t , s ± are constant whereas for the vapour phase we have ec = c(fl. + 0 ) ,

S S

(5.46) (5.46)

= cln(9* + 0)-J^]np, 0)-J^]np,

R is the universal constant of gases, M the molar mass and c=c/ct.

The

additive constants have been omitted in (5.46) since it is always possible to assume that they vanish in one of the two phases. We need the constitutive equation of pressure in order to complete the description of vapour:

pP = = ^p{e J% By recalling the definition

* ' .*++•(>)■ )■

(5.47) (5.47)

of specific free energy ip — e - (6^ + 0), the

condition (5.44) 3 assumes the form RE>

(MCA /Mc,\

/Mc-M /Mc-Mc, Cl

+ R\ iiv,

, ,. „v1 e -ln(9^ + 6)}-8^ + ^

= 0.

This last equation can be solved with respect to p to obtain Mcle^

(e^ ++e)--<e~ pp = = CC{$. 0)-fe

R{e R(K ++ e) 6)

*

(5.48) (5.48)

126

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

where 7 = - (Mc - Mct - R)/R

and C = Rexp[y +

(Mc,sjR)yMc[.

The inverse function of (5.48) defines the vaporization curve which associates the vaporization temperature to any external value of pressure. T h e density is then given by (5.47) and we can conclude that the temperature and density on the interface are known functions / and h of external pressure pe as follows 0(s(t),t)

= f(Pe),

p(s(t),t)

= h(Pe).

(5.49)

These functions do not depend on the motion of the interface. If we define the nondimensional latent heat by A = ( 0 , + #)[s] we deduce from equation (5.44) 3 the following relation:

A= [e]+£,

(5.50)

which, by taking into account (5.45), (5.46), (5.47), becomes

A=

/Mc-Mc,

+ R\/n

(—w4—r*+(?)+e*-

<5-51)

This relation shows that A is a linear function of temperature on the interface and therefore a function of external pressure pe. Figures 16 and 17 show, respectively, the vaporization curve (5.48) and the latent heat curve (5.51). The characteristic values relative to the vapour-water case have been assigned to the parameters; moreover, the constants e„ and s 4 have been fixed by imposing t h a t the vaporization curve contains the point corresponding to 100°C at one atmosphere where the latent heat assumes the experimental

127

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

value 2.26 • 10 _

10

erg/g.

It is interesting to note that condition (5.44) 3 can be put into the form

g(e,p)-4>,(e) = o where

g

is

the

thermodynamical

Gibbs relations

potential

of

the

(5.52)

vapour.

By

(3.45) 3 and (4.7), we obtain

equation (see (4.13)) dp

p\{9)

de

'* + <

fig-18

recalling

the

the Clapeyron

128

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

llg. 19

We are now ready to make clear the boundary value problem which we are considering to determine the liquid-vapour phase change. We suppose that the initial and boundary temperatures in the liquid phase coincide with the vaporization temperature at a given external pressure; in this hypothesis the temperature field in the liquid admits this constant value on the whole liquid phase. Consequently, we need to solve the problem only in the vapour phase. We start with the remark that (5.46), (5.47) imply Mc,p Mc lPee R

pe = —g—- = pt-— const. const. ,, '

and then , taking into account (5.43)a

2

>> we find we find

CHAPTER 5. STATIONARY AND TIME DEPENDENT PROBLEMS

(pe),

t

+ [(pe [{pe ++ p pe)v> -- ,ik6, nk8, J , x = 0 ..

129

(5.53) (5.53)

This equation can be integrated with respect to the variable x and the resulting arbitrary function depending on time only is determined by (5.44) r In this way we obtain kR i r XR M -vu - (Mc^+ R)pB e,xn(Mc ]i1"+ R)p P = W^P ^W^WP^ -

(5.54) (5 54)

"

By employing (5.47), (5.54) and (5.43)2 we attain the equation eeuH + [Ae,x + [A9,

+Bs}9, Bs}e,xx == A 6A9, , x xxx ,,

+

(5.55) (5.55)

where pkR ~(Mc(Mc + R)pe ''

A

R—

^^

(Mc R)pe (Mc ++ R) Pe

R(8m + 6) Mcp Mc ■ Pee

l(5.56) Dj

°-°

It is quite clear that equations we have not yet used, that is (5.44)2 and (5.55), are not sufficient to determine the temperature field in the domain Q. = fi = {(as, {(x,t):s(t) t):s(t) < <x
(5.57)

because the boundary L(t) itself is an unknown. Therefore we need a further condition which is supplied by the global conservation of mass: 1(0 s(t)++ s(t) S(t) 0 »(

p(x, =J Jht ,, p(x,t)t) dx =

(5.58)

130

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

where Jk is the constant mass per unit area and p is expressed by (5.47). Finally, to describe the evaporation of a liquid we have to find the unknowns 6(x,t),s(t)

of the following boundary value problem in the domain

(5.57) 0,t + [A9,s + B8]6,x

x(e)k = n McMr s(t) + -Pe l

i(t) dx

em+&(x,t)

s(t)

=9 ,

(5.59)

(x,t)en,

-fike,x(s(t),t),

R

6(s(t),t)

,

= A0,xx

-Jt,

)

0(L(t),t)

= 01(t)>0

,

where A and B are given by (5.56) and 6 is the nondimensional vaporization temperature implicitly defined by (5.48) in terms of external pressure p .

Chapter 6

PHASE CHANGES IN MIXTURES

6.1 Balance laws i n classical mixtures.

In this chapter we propose to extend all the results obtained in previous chapters to binary mixtures of classical fluids. This further generalization of the model of a continuous system with an interface which we have proposed so far, will permit us to attain the Gibbs rule of phase equilibrium in a mixture, to describe the evaporation of a component of a mixture into a gas, etc. Moreover, it will be employed in the next chapter to supply a model of crystal growth. T o formulate the new model of a continuous system with an interface ( C 1 , C 2 , E ) which we are looking for, we start with modifying the balance equations we derived in chapter 3. In fact, those equations were valid for a system formed by one constituent and now we want to take into account a binary mixture of fluids. If we limit ourselves to classical mixtures (see, for instance, [34, 35, 36, 37, 38]) it is sufficient to add: i) the mass balance of one constituent to the other balance laws; ii) an extra-flux of energy due to the diffusion to the heat flux. More precisely, we suppose that, besides the mass conservation, the following mass balance for one constituent (and then

131

132

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

for the other) is valid:

A -2- ( dt

pv dv+ pi/dv+ Vr

• da) — q-N dapgsvi> q-N da- q -vq^v^dl s sda) s Edl
* Vv Vv n n N- # ) - L dlP v,VVs ■ "E dldl > + », s(.i NN — + Ps ~ C:n„ ■ ) ~TN dl ~ j Ps '"* ' ' *E ' 9

(6.1) (6J )

d9(7

C7
where V is any material volume, a = E n V, v and vs denote the concen­ trations of one constituent and q, qs are the diffusive flux vectors in V - E and on

£ , respectively, n and JV are the normal unit vectors to E, J> S is the

normal unit vector to da in the tangent plane to E and finally the last two integrals are suggested

by the considerations of section 3.1. T h e

local

equations, which we derive from (6.1) by adopting the usual procedure (see (3.4)) result in

{pv)' + pv V-w {pv)' V - v ++ V V-?g = 0 ,

iinn C - EE, ,

(6.2)

6 2Hc nPs ^ s + St (Ps '.) + V,-(*V, v.)~ -IpvU-q-nj ==
%

T>lnx({C-V)p,v T- n x {{C-V)p svfm-qJ]

= Q, 0, -1s)l =

on T,

where C = Cj U C 2 . By taking into account the local implications (3.24) of mass conservation for the whole mixture, we have

pv+V-q pv + V ■ q==0,0 ,

(6.3)

133

C H A P T E R 6. PHASE CHANGES IN MIXTURES

Psv'+ V,-1.-lP{v-".)U-*■"] T-[nx((C-V)pava-qa)}

onS-T,

= 0, = 0,

onT.

where we have used the derivative operator given by (3.44). With respect to the balance of energy, we recall that in classical theory of mixtures the extra flux of energy due to the diffusion is - p^q , where the coefficient p1 is called the reduced chemical potential . Consequently, it will be sufficient to substitute - h with -(h+p^q)

and - hs with - (hs + Pisqs)

in (3.34), (3.35), (3.36) to obtain the new local equations of energy balance. Moreover, since we deal with binary mixtures of fluids the stress tensors in bulk phases and on the interface should be represented by

t=-pl

+ td,

(6.4a)

T = yl, + Td,

where I = (5-) , Is = (aag) are the identity tensors in the volume and on the interfacial surface E; while p and j denote the pressure and surface tension. Moreover, p, td ,y ,Td are isotropic functions of their arguments, which are listed in the next section. On the basis of these considerations we can write the local energy balance equations of a binary mixture of fluids in the form

pe+pV-v-tr(td®Vv)

+ V-(h+ii1q) = 0 ,

inV-S,

(6.4b)

134

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

paE>.-\p(\{v-V?


yaa^afi

- Tfa{Q0)

+ Va • (A, +

Hsqs)

n

+ e - £ 7 + £ ) ^ - ( A + / i i g ) . n ] + [ P ] ( c n - 7 „ ) = 0,

2

T-lnx[(C-V)pa(±(v-V)

+ E) + T.(V-C)-(hs

onE-T,

nsq8)}}=0,

+

on T. In order t o write (6.4b) 2 we have again used the derivative operator (3.44), the constitutive relations (6.4a), the obvious identities (see (3.50))

«a'(T/s + Trf)T,a =( K / 3 + ^

)

V

)

,

and finally (3.51).

6.2 Constitutive equations and dissipation inequalities.

In the classical theory of fluid mixtures the variables appearing i n the constitutive equations are t h e velocity gradient Vw, t h e density concentration

v, t h e temperature

p, t h e

0 a n d t h e gradients of density a n d

temperature. For the sake of simplicity we assume that they depend only on the first gradients of density, concentration and temperature. Similarly, we suppose t h a t the constitutive equations relative to the fluid mixtures on the interface depend on
and V s #. The

dependence of a is introduced t o take into account the adsorption which might take place on the interface (see section (3.5) and remark i) in section

135

CHAPTER 6. PHASE CHANGES IN MIXTURES

(3.7)) which is now fluid. In order to derive the restrictions on the constitutive equations imposed by the second principle of Thermodynamics [39] we have first to find t h e reduced dissipation inequalities. By eliminating V • h between (6.4) and (3.39) we have -(V> + s$)-pV

■ v + tr(td ®Vv) + V-(ntq)-lh-V0

> 0.

On the other hand, by mass conservation (3.24) we obtain

V-v=

-lp;

moreover, from (6.3)2 it follows that

V • (MX«) = q ■ V/ij + /i a V • q = q ■ Vp1 -

ppxv.

Then the previous inequality becomes

_p^

+ sb-£-p-p1v)

+ tr{td®Vv)-q-Vp,l-\h-V9

P

= 0 .

(6.5)

°

In this inequality ip , s , p , ^ , tj , q and h are functions of Vw , p , v , 9, Vp, V ^ , V0 and consequently

dtp-0, dip . dtp dp dv

' = dv

♦ & ■ < * • > ■

♦ & • < " > ■

♦ & ■ < * " > ■

♦ & • < * • > ■ ■

136

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

By substituting this expression into (6.5) we obtain an inequality from which it is possible to derive with a standard procedure the following relations

tl> = rl>(p,v,e), l

a\

dip

(6.6) t

a\

2 dip

s = s(p,i/,e) = —jjfg, p = p[p,v,o) = p j - , Pi = t*i(p,v,8) = -fa » tr(td ® Vw) - q ■ V/ij - i h ■ V6» > 0 . We can also extend the procedure which we applied to inequality (3.62) to (6.6)5. T °

t m s en

d'

w e re

gar(l

tne

left hand side of (6.6) 5 as a function /

of Vw, q, V9, provided that we assume an explicit constitutive relation q — q ( V ^ , . . . ) with respect to Vpj. Function / assumes its minimum when all these variables vanish, that is, at equilibrium of bulk phases. Consequently all the partial derivatives of / vanish when they are evaluated at equilibrium:

t°d = (V/Mo = h° = 0 .

(6.7)

It remains to proceed in a similar way with the surface energy balance (6.4) 2 . By eliminating [A] ■ n and V s • hs between (3.39)2 and (6.4)2 , we obtain - p . ( * ' + S9>) + 7 a % + T A

M )

- V • (plsqs) - i h,- V.0

+ [p(j(i'-T02 + s - * - / i 1 > - O ) ^ ] -[pKcn-VJ-I^gJ-n^O,

(6.8)

CHAPTER 6. PHASE CHANGES IN MIXTURES

137

where g is the specific Gibbs potential (3.53). On the other hand, by recalling (6.3)2 we have V V l» -«■ . % - ■VV •. ((MiA) ^Pu l s - ■-^ Hs MIA) = = --9% s "■ P V -v y)U-qU = -9-9 = Hx.P„v' lp(^ - s s) - « ■»], •»]. l *l s ++ A*I,P, " ia- -P (h l sSM S-S ' ^V/i V

and then (6.8) becomes a CT - /»,(«' S0' - ^Hyl sX />.(*' + SO' ) ) + ya ^afi 0a/3 + • Vfl TO<*P ^ 1 lsS -- \1H*fc, •V
+w|

-^ls( '- 'S)]

+ (Pi S - M l ) «

(6.9)

-[P](c )>0. n-Vn-V n)>Q. -[?](*„ On the other hand according to (3.44 ), (2.21) and (2.14) we derive a' =22aaaa««VVMM))--((cc„„--VVnn))66a a/ /33)) a ' = da=

(6.10)

so that, we write in view of (3.56) a a P's= -VnC))n - ^ n ) ) + /»:= --Ps^ PS% « afi)-Kfi(c V M ) - nM + [pU] I ^ l -.

(6.11) (6.11)

Collecting the results expressed by (6.9), (6.10), (6.11) as well as taking into account our constitutive assumption r af}, V $tf == «(o, p„• v"«»*»« , ^s ,. VV/ )/ ) «(a,/>. s, 6, cr a/»»VsPs PS>s ,vVs^.

we have g = 22 St ** '' = da

aaa/

{Q0r(c n-Vn)baf)) V( a /aa^(a -VJKp) 3 ) - (*n

138

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

+ §- ( " Pfia\afi)

+ PsH{cn - Vn) + [pU])

•s

+

+§^<+§0' s

9^) ^) + dv7s{VsPs)

+

5v^ (v ""' ) + 5 v 7 ( v ^ '

When we substitute this expression into (6.8) we deduce, again in the standard way, the following thermodynamical restrictions on the constitutive equations 9 = 9(a,Pa,va,6),

S = S(a, Ps,v„6)

= - | |

7 =

T

d\afi)

,

(6.12)

/ i l s = nls{a,p„ vt,*) = g^-

2 5* , o

<9*

-PsWS2^^*'

~ Is ■ V s ^ l s - i A. • Vs6> + ( M - 2F 7 )( C „ - V J

+ [p[5("-^)2 + 3 - ^ - / i 1 > - ^ ) ] ^ ] + [ ( / i l s - ^ 1 ) ? ] - n > 0 ,

(6.13)

where we have introduced the surface chemical potential (see (3.59))

".=* + '«^ = =4" (p '* ) -

(6 14)

-

Now, in the usual way, we regard the left hand side of (6.14) as a function / of (T(a/3y qs, V s 0, ( c „ - F J , J7 + , t / ~ , q+,

g~, provided that

we assume an explicit constitutive relation qs = 9 s (V s ^ l s ,...) with respect to

139

CHAPTER 6. PHASE CHANGES IN MIXTURES

V s /x ls . Since / assumes its minimum at thermochemical equilibrium (see [35]), all its partial derivatives vanish when they are evaluated at this case. Consequently, we have [39]

on E at thermochemical equilibrium :

T° = 0 ,

(Vs^f

=0 ,

[I fP JJ(J ] o - 2—7 1t — f = 0 ," +

h°s=0,

?

[9

-fis-^ls(l/+

-^JJO^0 '

[9-

-p,-ih,(v~

-(/s)]o = ° .

(Mi + )o-(^i s )o = 0 '

(6.15) (6.16)

\~-~~/ (6.17)

(6-18)

(fh )o - (//i*)o = ° ■ The meaning of (6.15) t

3

is quite obvious; condition (6.15)2 imposes that the

reduced chemical potential of the interface of the mixture is uniform on E. Owing to (6.18)'s , which imply

[/*1lo = 0 ,

onE,

(6.19)

it is found that the reduced chemical potentials of bulk phases are uniform on E. Finally from (6.17), (6.18) we derive

[ j - ¥ l o = 1'

(6-20)

At non equilibrium we can proceed as at the end of section 3.7, that is,

140

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

we can suppose t h a t the coefficients of v
q

U +,

U~,

, q~ , in (6.13) are functions at least of the second order of these variables.

This implies

that

we can accept

(6.15)-(6.20)

also at

non-equilibrium

provided t h a t the aforesaid variables are not too large.

6.3

Phase equilibrium in a binary fluid mixture.

In this section we want to analyze the equilibrium boundary problem and to verify t h a t the Gibbs rule is satisfied also for nonhomogeneous fields. As we did in the case of phase equilibrium of pure substances, we suppose that the body force admits a potential U, that is / = - VU. On the other hand, dp /dp > 0 and

consequently

the function

p = p(p, v)

is invertible

with

respect to p ; then we can adopt p and p as independent variables. Moreover, from the relation

g(p,v)

= ip(p(p,i/),v)

+

p{v,v) '

we derive

dg__dj)_dp_ dp~dpdp dg _ dip dp dv dp dv

+

dip dv

i _p_dp__\ P p2dp~pp. dp _drp _ pi dv ~ dv ~

l

,fi91, ° lj

CHAPTER 6. PHASE CHANGES IN MIXTURES

141

Finally, it follows t h a t I V p = V
.

(6.22)

In the absence of surface body force, from (3.27) 1 ; (6.22), (3.27) 2 , (6.7) and (6.15)-(6.20) we derive the equilibrium boundary system

F

i(*> Pi> vi) = 9i(Pi, f ,0 - P-i^i + U{x) - c ■ = 0 , in C- , i = l , 2 Ft(x, p . , i/.) = gt(Pl, i/,.) - / p_ , ^.. +\ £/(*) - nc, = 0 , in C- , i = 1,2 _

[p] = 2 T # ,

[p] = 2 T # ,

7 = const. , on £ 7 = const. , on E

h-wl P\ ++

[a l9

(6.24) (6.24)

=o i

- t*i,(7ivt)

-^-^is(^

(6.23) (6.23)

+

-

- ^" )J] ]o0 = °° .>

pP = pPee, ,

on 3<9C C ,,

(6.25)

j/s = « ,

on <9E ,

(6.26)

where a and c are constant, <9C is that part of <9C on which the external uniform pressure p e is acting and « is an assigned unit vector. We have omitted in all the formulas the temperature 9 which is supposed to be uniform and given. From (6.23), and (6.24) 2 we derive that the reduced chemical potential p. is uniform on the whole region C; consequently / i l s is constant on E and, in view of (6.24)j, the surface concentration v$ is also constant on E. The 11

142

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

unknowns of this system are represented by the fields p t , u-, the constant c^, ai in both phases, the constant 7 and the field vs on £ and by the equation z(x,y) of this surface. Since the number of equations is 10, provided that p e is given, we can conclude that the variance or the degrees of freedom of the phase equilibrium is 3, namely, p e , 6 and, for instance, v. More precisely, let us suppose that the constitutive equations of g and n1 are such that the system

±

±

/ 1 (( p p ± .l .i ^ / M ) == [^ = 0o,, A b1i] l =-(p±,u± ±)

ffo(P 2 admits at least a solution p

+

(6.27)

==19-^1 0, , == o {g-- P i 1/] = >»*)-= = p~,T>+,u~

(corresponding to equilibrium

with a planar interface); then, if the jacobian determinant of functions / i t / 0 at this point is different from zero, we can say that the system (6.27) defines p~ ,u~ +

as functions of p

,f

+

at least for {p

,v

) in a neighbourhood /

+

of (p ,Z7 ) . When we assume that at a point x0 e E the values of pressure and concentration (PT IVZ)

(p*, v'J) e / ,

we can find a corresponding

couple

such that (6.27) 's are satisfied. In this way we find the constants

ci,ai and then the fields of pressures and concentrations in Ca and C2. The equation (6.24)j with the boundary condition (6.26) defines the only surface containing x 0 .

To prove that (6.27)'s are satisfied not only at x0 but

everywhere in E it is sufficient to note that, owing to (6.23) , the differences on the left hand sides of (6.27) are constant in E . Finally, we derive 7 and »/, from (6.24) 4

5

.

143

C H A P T E R 6. PHASE CHANGES IN MIXTURES

6.4

The influence of mass adsorption on surface tension.

Let us consider the phase equilibrium with planar interface in a binary mixture and in the absence of body forces. The equilibrium system reduces to the following one

FfaPfiVi)

= 9i(Pi^i)-Ci

= 0 ,

fhi(Pi>''i)-ai

=

0

in C- , i = 1,2

(6.28)

'

[ p ] = 00 ,, on E [p] iL A * 1 JI

(6.29)

1

h -(M1V} == 0o,, Iff-/*!"] ..+ v [g+ -n [
p= = ppee, ,

All the

10 unknowns

1..

,. \

-I/I/ss)]o )]o = ° on dC dC .. on

pt-, //,-, 7, v , a^, c ; are constant and are

(6.30)

completely

determined by the 10 equations (6.28)-(6.30), that is the variance of phase equilibrium with planar interface is 2 since it is determined, for instance, by 8 and p . Now we want to take 6 and i/j as fundamental variables and omit in the formulas 6 which is held constant. In this way the system (6.28)-(6.30) defines the surface tension 7 as a function of u1. Now it is experimentally verified (see [40, 41]) that can be an increasing or decreasing function of vv

The first circumstance occurs when vs{v\)

<

v

\

144

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

(negative adsorption), while the other one occurs when vs(v-y) > vx (positive adsorption). In order to obtain this result theoretically,

Gibbs assumed t h a t

it was possible to substitute the actual system constituted by bulk phases separated by a narrow interfacial layer with a fictitious one in which the quoted layer is substituted by an internal suitably chosen plane and the bulk phases are assumed to fill homogeneously the remaining part of the same layer. This plane is assumed to be placed in such a way that Gibbs' relative surface excess (see for instance [41, 12]) of one of the constituents of t h e mixture

vanishes.

Under

these assumptions

Gibbs derived

the relative

adsorption equation (see [13], p.25) dy= -Tdpu,

(6.31)

where T is a coefficient, dip $) 2i. = - g ^ = ^ + ( l - « ' > i . and ^{P\s,

P2S)

ls a

(6-32)

function of the densities of constituents of mixture. The

last equality is an obvious consequence of the following identities

Ps = Pis + P2s

The differential djils

'

*(*>1»> P2s) = * ( P » . "») •

in (6.31) is relative to the function /*i s (tO which is

obtained by expressing ps and v in terms of the variable v,. Our aim is to deduce the relation (6.30). T o start with, we regard all the equilibrium quantities as functions of vv omit the index 1 since we refer only to the liquid phase a n d neglect the dependence on a in the constitutive

CHAPTER 6. PHASE CHANGES IN MIXTURES

145

surface equations. Moreover, we remark that if we introduce the volume quantity d(pil>) J/ Pl=-%r + (l-v)l* " 1 - dp, = 9=9+ (l- - )1^ l ,, we derive from (6.29) 2

(6.33)

3 4

(6.34) (6-34)

h£i = = Khs>. and we can write (6.31) as follows

and we can write (6.31) as follows dj = - r

dp,.

(6.35) (6.35)

d1=-Tdpl. On the other hand, we obtain from (6.24) 5 , (4.54), (6.12) 4

On the other hand, we obtain from (6.24)5, (4.54), (6.12)4 7 ^ P . ^ - f f + PiC*'-".))•

(6-36)

As all the quantities in both members of (6.36) are to be regarded as functions of v, we can calculate their total derivatives as -".)) £ = ^ - ' + "i("-0) £=^w-«+M"dv x „ ((99dV ddP d$ du s* dg dg dp Ps,dV dp s + + K( dl/ \ dPa sdv dv dpdv f dv *{Wp~ ^ dv d^^~d^~dp~

+ + pPs

dp, , u - dv (Ms -Vu s

./d^

dp

d^x

/

dus\\

S ■'J$*+%)+4-T® WT W + ftr¥/i

7 v\ (( g + fi1is + 7 ) + Ps (Ps

[dip dp\ (9±_j fdrp L p i l^\dp dP\ dp ( (djP iidp) -{fo + P dvJ'Kdp f? + Pdp)dv + ^~

°^« 77 dp* 'PI"s

dv ^s

dv

.dj^

(

dj^\\

-J&W'-S))-

146

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Owing to (6.12) 4 and (6.24)5 , we finally obtain dj d-1

d

1i

Pl v» dP-i

Pas dp dp P

= -r-P.\v.~v) rsy dv dv ' -j dv

7r-r-P dv

IR Q 7 N v(6.37)

'

In order to compare (6.37) and (6.35) we observe that (6.33) implies d

px -+(1_j/) dJh

dji H 1=dlddg dg R dp + dg__ d

dv rfi/

3/? dp dv dz/

~\dp~

<9JV dv

' dv df

V , 1 dP \dp * P dp) dv"

'^~dT V ■8)£+®l+*!S)-*+p■ /i a + ( i -

d^ ' /I P dv; ++Kl( l -■^-arThis relation allows us to put (6.37) into the form 1 - - ^ 5 dp

-v dp,

d7 d

U Ps P l 7r Ps=1Ps -f^¥-£- 1/-i^¥- -v dv P 1dv

dv

dv

rs

1 -v dv

P 1 - v dv

y

But in ideally diluted mixtures it is (see [41], p. 333) p ( l -- i /i/)) ddp . (i <€ p rfPl

K-

(6-38)

(6.38)

" 1 s

and moreover (see [53], eq.(96.7), p.319) d£

fdi/ la>>00 ;; dv consequently the function y(v) has the experimentally observed behaviour.

'

147

CHAPTER 6. PHASE CHANGES IN MIXTURES

6.5 T h e Gibbs principle for phase equilibrium in fluid mixtures.

In this section we extend the results of section 4.3 to the equilibrium of a liquid binary mixture C^ in the presence of a mixture of its vapour Cy. We start again from the Gibbs principle according to which the equilibrium configurations are extrema of the total free energy with respect to variations at constant mass. Let

da

(6.39)

be the total free energy of the system where ifiF = ^f{pp,i^p)

, 4>y =

*=

Pvi^v + U) dv + p^s

pF(ipF + U) dv + C^r

Cy

il>y{py,Vp) and \PS = ^s{ps,a,v

s)

and E is a regular surface. Let us introduce

the set of functions K = {k : E —► Jfc(E) ,regular, and conserving the surface mass }, R

= {Ps> vs'-

E

—* *

+

> re g u l ar }»

7V = {z/: C — » » + } , where C = C^ U Cy. Moreover, we will denote by ft : C —► C, a mapping such that: i) it is regular in C - £ and exhibits finite discontinuities together with its derivatives across E; ii) h,p

: C^i —► h(Cp), h,~

: —► /»(Cy), are diffeomorphisms and

locally conserve the mass. If we denote by H the set of these mappings, (6.39) can be regarded as a functional defined over the set H x K x Rx N which can be normed in the

148

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

usual way. The Gibbs principle states that {he,

ve,

ke, pea, ves) is an equilibrium

configuration if *(A e , v€, ke, pes , v\ ) is a conditional extremum with respect to all the arguments h, v, k, ps , vf

$ =

pL dv +

pLvL

$ 1

dv+

satisfying the global constraints

py dv +

PyVy dv +

(6.40)

Au

Ps

p „v „ da IS

S

As is well known, the extrema of (6.39) under the conditions (6.40) are defined by the condition

(6.41)

d(^r + A$ + A 1 $ 1 ) = 0 ,

where A and Xx are numerical Lagrangian multipliers. By proceeding as we did in section 4.3 and choosing the unit vector n normal to E directed from C^ to Cv , it is easy to verify t h a t (6.41) can be written as follows

( V p + pVU)

p(p1 + Aa) bv dv

-6hdv+ C

p + (g

p~(g

+

U + \ + \1)

(6hn-6kn)+

da

+ U + X + X1)~

{Shn-6kn)~

da

+

(6.42)

CHAPTER 6. PHASE CHANGES IN MIXTURES

149

+ 17, a &kada + | ([p] - 2Hj) 6kn da + (Vs + Xl)

6

"sd
E

^ ( p A +A+ A l ) V s d ( T = °

+

where p =

p

Jp-' „_0V

^-^7' / , P

9 = ^P + j ,

2 SI'

7 =

^,

It is now quite evident that we again

9

„ „ 3*

(6.43)

„ _ d#

^ - 5IT ' ,T,

3*

find

fl(p.*) " s "

the equilibrium system (6.28)-

(6.29).

6.6 The evaporation of a fluid into a gas.

T h e previous considerations are relative to phase equilibrium in a binary fluid mixture. In this section we want to study the evaporation of a fluid into a gas [42]. T h e system with an interface we use is made up of a pure liquid phase Cj and of a gaseous phase C2 which, in turn, is a mixture of the vapour of the substance filling Cj and of another gas (for instance the evaporation of water in the presence of air). The dynamics of this process is carried out by supposing t h a t the following hypotheses are verified: a) the phenomenon

150

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

exhibits plane symmetry; b) the pressure is less than the critical one; c) the interface

has

no

material

characteristics.

These

hypotheses

imply

the

following approximations: i) the fields depend on the temporal coordinate t and on one spatial variable x; ii) the liquid phase is incompressible and at rest; Hi) the mixture is made up of perfect gases; iv) the mixture density is negligible with respect to that of pure phase. In these conditions the region Cj is represented by the interval [0,

s(t))

and the region C2 by the interval (s(2),/(£)], where s(t) is the abscissa of the planar interface and l(t) denotes the abscissa of the free planar boundary of vapour mixture. Moreover, the balance equations of mass (3.24), (6.3), of m o m e n t u m (3.26) and of energy (6.4) assume the following form

w = 0,

P,x = 0,

* e[0,«(<)),

0, t -

a

i


<e[0,oo),

(6.44)

'

where aj is the liquid thermal diffusibility (see (5.22)) and

P,t + M . * = 0 .

* 6 («(*),/(<)],

*e[0,oo),

p(uH + w,x)=

~q,x

,

p{v,t + vv,x)=

-p,x

,

p(e, t + ve, x) = - pv,

+ kd, xx - (// 1 ? ),

,

(6.45)

151

CHAPTER 6. PHASE CHANGES IN MIXTURES

where k is the thermal conductivity of vapour and

p(s-v)

= pls,

x = s(t) ,

pv(s-v)-pls-q

<e[0,oo),

(6.46)

=Q ,

pv(s - v ) - [ p ] = 0 , 2

P(^v

+ e + ^) (s -v)-

Pl(e+^)s

1 v2 + [V> + ?]"-pxv2

+{ke,J-

p,q = 0 ,

=0 .

In writing (6.46) we have recalled that in the liquid phase v = 0, v = 1 and g = 0. Since p j ( see section 5.3), from (6.46)1 we have

r ~

.; s ,

(6.47)

P

and (6.46)2_5 become g = p/*(i/-l),

(6.48)

ip]=-4*a. />/«

/IP?

\2 „2

s2 + W-

►?)*■[**„]--HlPls(u. -1) =, 0 , !

! ( # *

+M+J--Hv-

=0.

Now we can repeat the nondimensional analysis of section 5.3 without any modification and attain the following system

152

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

v = 0,

p,s = 0,

* e [ 0 , *(*)),

eH = />.t + ( H - x = ° .

te[0,oo),

(6.49)

°1 XX '

*e(«(*)i'(*)].

p{y,t + vvtx) =

<e[0,oo),

(6.50)

-9>i .

P»x = 0 . P( e . t + »e> x) = - P v ' x + a ^ ' xx - (M),

v= - i ,

* = «(*).

X

,

*e[0,oo) ,

(6.51)

[p] = 0 ,

(w+J+a-"))* = (»,«)"-(«w,x)+, T V* -")}°

—

\"ixJ

— 1 U«/l/ ,

M+J+("-i)/h = o , where a is the ratio of the vapour density at atmospheric pressure and vaporization temperature to p ( ; moreover, we have used the same symbols for nondimensional quantities. To the previous equations we have to add the initial data and boundary conditions:

s(0) = s0 , v(x, 0) = w0 ,

P(x,

1(0) = lQ ,

0) = p 0 , v(x, 0) = v0 ,

0(o,t) = * i ( 0 . p(l(t),t = pe,

9(x, 0) = 0O ,

o(i(t),t) =

v(l(t),t) = l ,

(6.52) xe [0,1(0)} ,

e2(t),

q(l(t),t) = 0.

pe being the external uniform pressure. The boundary value problem (6.49)-(6.52) in the unknowns p(x,t),

i/(x,t),

0(x,t),

v(x,t),

s(t) and l(t) is very complicated; however, it is still

CHAPTER 6. PHASE CHANGES IN MIXTURES

153

possible to derive interesting results from it. First of all, we recognize at once that p(x, t) = pe for any x and t. On the other hand, when we look at hypotheses

«), Hi), the free energies of liquid and vapour phases are

respectively expressed by the following constitutive equations

i> ^(8) 8)[l -- ln(0* ln(0*++8)) 8))-- ee00 ++ (8* (8* + 8)s00 ,, t(8) = (8* + 8)[1

(6.53) (6.53)

V>(0, P,v) p, u) == [cu + c (1-i (l- »w i/)](0*+*)[! + 8)[1-ln(0* ln(0*++0)} 8)} m A

+^ ^v{8* + 8)\npu ^npV + ^ R- (- 0 l -- ■u v){9*. W ¥8)\np{\ + 8)lnp(l-u), + -»), + c,M 4 R

where R is the gas universal constant, M and M A the molecular masses of the evaporating substance and of the mixture respectively, c = c/ct and cA — cA/ct

are the ratios of specific heat of vapour and evaporating substance

and liquid; finally, 9* has already been introduced in section 5.4. By the thermo-dynamical relations , . e = if> + 8s , p p

-w

d\p 2dip

dtp dip s — 88 '

^

=

dtp dip

dU'

and (6.53) we can put (6.51) 3 into the form

P , ==/ where A,j

M

' A'

M 1 *"+ g M - "l -"\e( W+»)-<,->'+» ) - ^

(6.54)

and 6 are constants. The external pressure being fixed, Eq. (6.54)

supplies a relation v = v(8) between the temperature 8 and the concentration v on the interface. In particular, at ordinary pressure (p = 1), the evapor-

154

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

ation temperature corresponding to a given concentration v is obtained by intersecting the curve 7 „ y = A{6* + (?) ~ 7 e

e* + (

(6.55)

with the straight lines MAv

MAv + M{\-v)

'

(6.56)

In fig.20 the curve (6.55) is represented together with the straight lines (6.56) for different values of v under the assumption that the physical

Fig. 20 parameters refer to the water evaporation in the air. The figure is, in agreement with experiments such that the evaporation temperature decreases

155

CHAPTER 6. PHASE CHANGES IN MIXTURES

with the vapour concentration on the interface. If we define the (nondimensional) latent heat

A = <0*+ * ) [ « ] , condition (6.51) 3 becomes

(6.57)

{e] + % + (\-i>)fi1=\,

and (6.51) 2 can be written as

\s=(9,x)-

-(ak9,x)+

.

This relation is formally similar to Stefan's condition that we have already deduced in the case of melting (see (5.34)). However, A in (6.57) is not constant but depends on the unknown temperature on the interface. Now we want to show that the boundary value problem (6.49)-(6.52) can be further simplified if the gases in the mixture are both monoatomic or diatomic. T o verify that, we begin by writing the balance of energy (6.50) 4 in the vapour phase in the following form

(pe), t + [(pe + p > ] , x = (akO - / i l 9 ) , x .

But p and e are expressed by the relations

(6.58)

156

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

_ MMAc, p ~ R [MAv + M(l-v)](e*

p

+ 6)>

l

°-°9j

e = [cv + cA{l-i>)]{8* + 8), which can be easily derived from (6.53). Consequently, if the gases in the mixture are both monoatomic or diatomic, the specific heats c and cA inversely proportional to the molecular masses M and MA

are

and then the

energy E = pe and the enthalpy H — pe + pe per unit volume are constant and uniform. From these remarks and Eq. (6.58) we derive

v=

ak9,x-p1q H

+f

(6.60)

where / is a function of time which we must determine starting from the interfacial data. Relations (6.59) and (6.60) can be introduced into the mass conservation (6.50)j and the partial mass balance (6.50)2; simple calculations lead us to the following system of nonlinear parabolic equations in the unknowns 9 and v

E\MAv

+ M (1 - v)}6, t + H(MA - M){9* + 9)v, t

+ (ak9, x-piq

+ f)[(MAu

+ M{\ - v){9* + 8)},x

= [MAv + M{\ - v)}{8* + 0){ak9, x -

%g)

,x (6.61)

HvH + {ak9,x-q HR[MAv

+ f)v,x

+ M(l-v)}(9* MMActpe

+ 9) ~q,x

'

157

CHAPTER 6. PHASE CHANGES IN MIXTURES

Finally, we need a further equation in order to determine the free boundary l(t); such an equation is supplied by the global mass balance that can be written (see (5.58)) as l(t) s(t)+

p(i>(x,t),0(x,t))

dx = const ,

(6.62)

s(t)

where p(is, 6) is expressed by (6.59)1. To

summarize

we

can

state

that

the

temperature

9(x,t),

the

concentration v{x, t), the interface abscissa s(t) and the free boundary position l(t) represent the unknowns of the following system

i£[0,s(()),

te[0,oo), 9, t =

xe(s(t),l(t)}, re(*(0,/(0].

o, XX

(6.63)

'

t* e6 [[0,oo) 0 , o o ) ,, r r r ?IT ,. . s ^ / 1 ,.\1a , TJI H[M Av + M(l - v)]6, t + H(MA - M){6* + B)v, t

+ (ak9,x-li1q

+ f)[{MAv + M(l-v)(B*

(6.64)

+ e)],!t

= \MAv + M{1 - v)](8* + 6)(ak6, x - Hq) , x ,

H»,t x = ${t) ,

+

, ,„ i«ke,x-q

+

,, f>,x=

HR[MAv + M(l-i/)](6* MMAclPe

+ e) "«». •

t e [0,oo) , \s=(e, ake,x)+ ,, x)+ A* = (x)--(akO, * „ ) - -(

(6.65)

158

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

with boundary and initial conditions

q(s(t),t)

= Pls(v-l),

q(l(t),t)

0(0,0 = 01,

= 0,

(6.66)

*(J(t).0 = W .

= Wh- *

/("(«(<). t),9(s(t),t))-; tf(ar,0) = ff0(r) ,

+


-J/)^

= o,

=

v ( i , 0 ) = i/„(x) ,

«(0) = s0 ,

/(0) = / 0 ,

and the integral condition l(t) s(t)+

p(v(x,t),0(x,t))

dx = const.

(6.67)

s(t)

In (6.66) 5 ip is given by (6.53) and p in (6.66) 5 , (6.67) is obtained by inverting the constitutive equation p = p(p,v,9)

= pe which, in turn, is given

by the thermodynamic relation

P

2dip

= pJp

Finally, we have to give the constitutive relation q(9,v,9,

,i>, ).

Chapter 7

CRYSTAL GROWTH

7.1

Gibbs' and WulfPs equilibrium laws.

In this chapter we are faced with the difficult problem to describe the equilibrium and the growth of a crystal in its melt or in a mixture by resorting to the model of a continuous system with an interface. We shall see t h a t this approach has to be modified if we want to derive from the model the experimental evidences. W e begin with deriving the classical results

about equilibrium due to

Gibbs and Wulff [12, 43] although the proofs we propose to obtain them are much more formal [28]. Let C be a polyhedric crystal having a volume V, n faces
and m edges /

(p = l , . . . , m ) . We denote by /■■ the

edge between the faces cri and a ■ and by i/- the unit vector which is normal to /•

and lies in the face a{. Finally, a uniform surface energy Ei per unit

area is associated to any face ai of C. Moreover, E^ is supposed to depend only on the orientation of c^ with respect to the crystal cell. Let C be a polyhedron and [C] the class of all polyhedrons having the same volume V, the same number of faces and edges of C and faces which are

159

160

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

parallel to those of C. The Gibbs principle postulates that a polyhedron C represents an equilibrium configuration for the crystal if total surface energy

**=

71 n

i1l

£,. **

(7.1)

has an extremum at C in the set [C]; moreover, this configuration is stable if it corresponds to a minimum in [C]. To determine the conditions which characterize the polyhedron C, we need to give the procedure to pass from C to any polyhedron in the class [C]. To this end, we introduce the normal displacements 8hni (i = 1,...,n) along the unit normal n, to the faces ai as well as the displacements 6K-

of any edge I-

of C. Of course, these

displacements will define a new closed polyhedron in [C] if the following conditions are satisfied (see fig. 21)

Fig. 21

161

CHAPTER 7. CRYSTAL GROWTH

6KtJ.ni

= 8kni,

6KXJ.nj

(7.2)

= 8kn3

It is quite evident that the class [C] is obtained by varying arbitrarily 6kni and 6K— provided that these displacements verify conditions (7.2) and leave the volume V constant. According to the Gibbs principle we can say that C is an equilibrium configuration for the crystal if the total surface energy (7.3)

*(ttm.Stf,-j)=E,-^*i has an extremum at (0,0) for any (Sk^SK-)

satisfying (7.2) and the global

condition

*(«*„,, tf*y) =

(7.4)

dv - V = 0 , C

V C e [C]. This is equivalent to say that (0,0) is an extremum for \P - A$, where A is a Lagrangian multiplier; that is, (7.5)

d(tf-A$) = 0 , for any (6k itSK:A

verifying (7.2). On the other hand, by Gauss' theorem we

have dv — i (

rf$ = \ dvC

C

aC

(r + 5knn) ■ n da dC

r ■ n dv) = ^6kn dv , aC

162

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

where r is the position vector of any point on the crystal surface. By recalling that 6kn is uniform on any crystal face o^-, we can write

d*=±±iHni'i-

(™)

Owing to this last result, (7.5) is transformed to the form m

n

E PiEivij + V i < ) • 6Kij ^ - x E i 6km °i = o .

(7-7)

for any (6kni, SKiA verifying (7.2). In (7.7) A = A/3 and the indices i and j depend on the edge index p since they are relative to the faces ai and awhich form the p-th edge. If we put cosa • =nt-• n , it is an easy exercise to verify that v

ij

~ ~ cotaij

n

i +

csca

n

ij

j

■

(7-8)

By taking into account (7.2), (7.8), we can write (7.7) as follows m

n

E piAJK,

+ AJt6knj) ltJ - A E i *t Skni = 0 ,

(7.9)

where A

ij

=

E

j

CSCa

ij

- Ei

C0ta

ij

■

(7-10)

The first summation in Eq. (7.9) contains the same number of terms as pairs of adjacent faces and it t O CX11U l b becomes l_»C»-.tJlllCD n

1r■ (V

A i

- A
163

CHAPTER 7. CRYSTAL GROWTH

where Yl 'j indicates that the summation is extended only on edges which form the boundary of the face a^ From the arbitrariness of Skni previous relation

we finally

derive the Gibbs rule for the

in the

equilibrium

configuration of a crystal ([12, formula 666])

£-iE'jAijlij

To

recognize

which kind

= \,

« = l,...,n.

of conditions

Eq.

(7.11)

(7.11) imposes on t h e

equilibrium form of the crystal, we analyze with greater attention to the case of a polyhedron such that there exists an internal point r 0 the perpendicular from which t o a face intersects it in an internal point of this face. If the distances from r 0 to adjacent faces cri and a ■ are hi and see from fig. 22 t h a t

fig. 22

h- we immediately

164

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

hh n fH

a

a v

-V>i = o.-

jihji3nJ i i + aijvij-aij''ijjiVji-

=0

Thus we have a ij

a

ji y + °j*

sina -, = h*j i sma ij = ij tJ>

COSQ COSQ-

cosa-j sina.j= h^i sina ji + afly ij cosa ij = ij <, i« +

a a

or cota^■ csca = ji csca ij (J -~--h ^i cota ij >>o, 0, h csca a ta- > 0 . csca aa ji = ij tJ -~- -hhji coc^o ij > 0 ■ H = Ki

a

ijij

a

= hh

7 ((7.12) -12)

We then deduce that /i1 and hJ• should satisfy the conditions ^hj «way < j - ,

cosa.j < j ^ ,

h h ii cosatJ < Y , cosatJ < -^3 ,

which have to be met if two adjacent faces are formed. From (7.12) it follows that °i = \ Z'i « y hj = k E 'i (^ csc a ! J -fcf.cota,-,-) I.. .

(7.13)

Similarly, it can be proved [44] that /■ ■ is a linear combination of h^ h-, h.-_!, ^y + i- Introducing these quoted relations and (7.13) into (7.11) we get a system of n equations in the unknowns ft- which in principle supplies the equilibrium configuration of a crystal for which we have given a priori the number n of faces and the angles a ■ ■ between their normals. The value of A is determined by inserting the values of h^ which depend on A, into (7.4). Moreover, if we substitute (7.13) into (7.11) we obtain

"I*.

Ot 2£ ij [(^[ (E3 -1 cscay ■-- ((££, . ,. - | />, )) Ccota^.] = 0, « , j ] '/,, ij = 2 h^ Vcsca,j

165

CHAPTER 7. CRYSTAL GROWTH

so we conclude that for any crystal there exist possible equilibrium configurations in accordance with the Wulff law 2E- ^ = A,

i = l,...,n.

(7.14)

We can easily find the constant A by introducing (7.14) into (7.3) so that we have *

_

A n 2 i

K °\ ~ 23

AV,

where V is the crystal volume, and thus A = 2\t/3V and has dimensions of a pressure. We can also obtain the equilibrium configurations of a crystal by mi­ nimizing the total free energy at constant total mass. For the sake of simplicity, we suppose that the crystal is completely surrounded by its melt or vapour and the pressure is everywhere uniform. Then the total free energy can be written as (see (4.43))

* = Ei i £,•*< + pi>{F) dv +

pF^F{pF)

dv ,

(7.15)

C-F

where C^ is the region occupied by the melt or vapor. It is an easy exercise to verify, by resorting to sections 4.7 and to the calculations relative to (7.1), that an equilibrium configuration of the whole system is characterized by the following conditions:

p = const

in C ,

pF = pe ,

in CF ,

(7-16)

166

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

i> + p = w-iE'i^ijUj

4>F

+ PF -

= p-PF>

«' = !»•..,"•

In particular, the Wulff law becomes IE-j^-P-PF^

i=l,...,n.

(7.17)

We conclude with the following remarks: i) In the formulas one finds in classical papers or textbooks there is no difference between surface tension j : of the crystal surface a^ and its surface energy E±. This is due to the hypothesis that ai is like a soap film (see section 4.7, remark i)). ii) Owing to the very small values of the surface energy Ej, we have an appreciable pressure jumps on the crystal faces only for very small crystals (germs).

7.2

About WulfPs construction.

To apply Wulff law we have to give the number n of faces of the crystal. This means that we can have many equilibrium configurations for a fixed jump pressure depending of n. Wulff suggested that there is only one

167

CHAPTER 7. CRYSTAL GROWTH

equilibrium configuration which is in accordance with (7.17) and in addition represents an absolute minimum for the surface energy. This configuration should be obtained with the following Wulff construction. Let us consider the function E(n), which gives the surface energy per unit area relative to a crystal face whose unit normal is n. If we put /(n) = 2E{n)/(pc

- pL), the

equilibrium configuration of the crystal is represented by the spatial region W which is defined as follows W = {re3?3:r-n
(7.18)

where S denotes the unit sphere (see fig.23).

fig.23 The previous considerations are not quite satisfactory for several reasons. First of all the Gibbs formula characterizes an infinity of equilibrium confi­ gurations which could be stable or not and in this latter case they would not

168

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

be observable. Moreover, an equilibrium configuration C is an extremum for the total free energy only in the class [C] of suitable polyhedrons and not with respect to arbitrary variations of size and form of crystal C. Finally, Wulff construction of equilibrium configuration is not justified by proving that it corresponds to a global minimum for the total surface free energy. A correct formulation of crystal equilibrium consists in finding the closed surface E which represents a minimum for the functional *(E)

(7.19)

E{n)da,

under the constraint that the volume V internal to E is assigned and in the hypothesis that the surface energy per unit area E(n) : S2 —► R

+

is a

continuous and a.e. differentiate function (see fig. 23, and [45], [46], [47]). This variational problem is discussed, for instance, in [48], [49]. We remark that this approach to crystal equilibrium attributes to surface tension a dominant role in crystallization but does not permit to relate this process to temperature and pressure. A more complete variational formulation of equilibrium is obtained by requiring that the displacement field « in the elastic crystal C, its boundary E and the density p in the fluid phase £p represent a minimum for the functional

* ( « , £ , p)

ipc(Vu)dv+ C

I if>F(p)+ CF

E{n)da,

(7.20)

Z

under the constraint that the total mass is constant. It is evident that for macroscopic crystals we can neglect the surface integral since the total surface

CHAPTER 7. CRYSTAL GROWTH

169

energy is very small with respect to the energies associated with the volume phases. Moreover, in the case of a crystal which is not in the presence of its melt or vapor but is itself composed of phases, we have to consider only the first integral (see [53, 54]). We want to conclude this section by noting that these last variational formulations do not describe the equilibrium of a crystal in its melt or vapor since they foresee that the pressure jump becomes zero when the crystal faces are planar. This result does not agree with Gibbs' or Wulff rules which in turn justify the supercooling or the superheating of small crystals. In the next section we propose an approach which is based on a new interpretation of nonlocal thermomechanical theory.

7.3 An introduction to nonlocal thermomechanical theory.

The basic assumptions of standard continuous thermomechanics can be summarized as follows i) the general balance laws are valid for any seabed V of the body C, however small it is ( localization principle); it) the physical response of a material particle X e C depends on the state of C at X; this means that only interactions having a very short range are effective;

170

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Hi) the boundary values can be prescribed on any part of the boundary dC independently of the values one assigns on other parts of <9C. We shall call a field theory nonlocal if some or all of these assumptions are relaxed. We do not want to discuss here the motivations which suggest us to leave one of the previous hypotheses; for a deep and exhaustive analysis of the foundations of nonlocal thermomechanics we refer to the quoted references [54, 55, 56]. We are only interested in putting in evidence the particular viewpoint we adopt in this essay which is essentially concerned with the assumption i); to this purpose we begin with discussing some examples. Let us suppose that any particle of a continuous system C interacts with the others in an arbitrarily large neighborhood of it and let / be the relative specific body force. Moreover, let us suppose that this interaction satisfies an action-reaction principle or, more generally, the condition / dv = 0 .

(7.21)

C It is quite clear that / does not appear in the integral momentum balance for the whole system; however, its contribution is present in this balance law when it is referred to any part V of C, V ^ C, or when it is written in local form. This conclusion is trivial but it may be read in a more interesting way. Let us suppose that we do not know in details the physics of the system C so that we ignore the presence of the internal interaction we are speaking of. If we suppose that the momentum balance is valid only for the whole

CHAPTER 7. CRYSTAL GROWTH

system

171

and not for any part of it, we reduce the damage of our ignorance

since a body force / , which satisfies (7.21), appears again when we localize the momentum balance. We shall call / a localization residuals. Let C be a rigid conductor of heat which is initially at a nonuniform temperature 9 and is bounded by adiabatic walls so that it does not exchange heat with the external world. If we do not know that C is able to conduct heat by previous experiences and apply the balance of energy to any part of it we arrive at the wrong conclusion that the temperature remains uniform at any point of C. However, if we suppose that the balance of energy is valid only for the whole system 4d dt

e(6) dv = 0 ,

(7.22) 22) (7

(C 3

we locally have, e= f ,

where

r dv = 0 , C 3

where f is a localization residuals, so that the temperature is not uniform. If, in addition, we suppose that r = - V • h, we obtain again the right equation. This example shows that our loss of information about the internal nature of system C is only partially compensated by the hypothesis that the energy balance law is valid for the whole system and not for its parts. In fact, we have to assign a constitutive equation for f and it is difficult to imagine a priori, for instance, that f — - V • h — V • (k- V#), where k is the thermal conductivity.

172

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Let us go back to the considerations we have developed in section 3.2 to justify the model of a continuous system with an interface (Cj, C2, £). In that section we considered a general balance law for some fields / , (p, r which were defined over the whole system but undergo continuous and fast variations across a thin layer A. Suitable averages on this layer led to fields associated with a surface T, in the layer. Let us now consider the case in which ip and r can be neglected in all the region C except for an internal narrow layer A in which they are prevalent. The same average process leads again to surface quantities $, R on E but now these fields do not appear in the global balance law since they verify the condition

$ ■ vv dl + Rdcr = 0

and they will be called again localization residuals. Once more, if we ignore the presence of these internal interactions we can mitigate the consequences of this loss of information by the introduction of localization residuals. It is quite evident that the introduction of such localization residuals as well as their constitutive equations can be justified only a posteriori by the results they permit to derive. This is what we have in our mind when we try to apply the nonlocal thermomechanics to crystal equilibrium and growth. This approach, provided that it leads to concrete and physical results, represents the simplest way to describe these phenomena without resorting to complicate models which, on the other hand, supply a deeper understanding of these processes.

CHAPTER 7. CRYSTAL GROWTH

173

7.4 Nonlocal balance laws.

Let (Cj,C 2 ,S) be a continuous nonpolar system with an interface and let C = Cj U C2; we suppose that the general balance law (3.2) is valid only for the whole system [55]

da) == N-
(7.23) (7.23)

When we localize this equation we have, instead of (3.4), the following equations

f + fVv-V
in in C C --EE, ,

(7.24) (7.24)

+ n -=y>]J R, = R on , Eon- E T -, T , + V ss -■( (F F ® V V) ) - 2- #2Hc c n nFF- V- ,V-s* ■- [$/ t- / [fU + n-v>] -$,, fr-[nx((C-V)F+*)]= [ n x ( ( C - F ) F + f ) J = -§,,

on T, onT,

where the localization residuals f , R , 1R> verify the integral condition £d<7 fi dcr++ % dd// = 0 ..

r? ddv u ++ C

1£:

(7.25)

ir

In the sequel we suppose that one or both of the phases Cx, C2 of the phases we are considering are occupied by binary mixtures. By applying (7.23)(7.25) to mass conservation, instead of (3.24) we derive

174

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

p + pV-v

= p,

+ VVss-(p .(psV)-2Hc -lpU} p''ss + -lpUj nPnP P sV)-2Hc

inC-E,

(7.26)

== p sps,, on on E E - TT ,,

r.[nx(C-V>J=-?,

onT,

where ( )' is defined by (3.44) and the localization residuals p, ps, r satisfy the global condition

p dv + \ps da +

r dl = 0 .

(7.27)

E

Equations. (7.26), (7.27) clearly imply that although the mass of the system is conserved globally there are local creations or annihilations of masses which are compensated by annihilations or creations elsewhere. This situation is not comparable to the one which is encountered in mixture theories where local mass transfers are allowed between species occupying the same point. To our knowledge there exists no physical phenomena within the classical framework which may lead to such mass residuals. So they should be eliminated. Moreover, if we keep them we would obtain in the other local balance equations some non objective terms (see [57]). For the same reasons the mass balance for one constituent assumes the ordinary form (6.3)

pv + V-q = 0,

inC-E,

V i +v , - « . - K " - , / . ) y - « - n ] = 0 , r-lnx(C-V)psV-qJ

= G,

(7.28) on E - T , onT.

When we apply (7.24)-(7.25) to the balance of momentum, instead of (3.27),

CHAPTER 7. CRYSTAL GROWTH

175

we have pv Pv = V-t + pf + pf ,

iinnCC--EE,,

(7.29) (7.29)

p.v PsV' = VVss-T -T r+[ + + lp(v-V)U lp(v-V)U + + n-tj n-t] + + ppssFF + + pr* pssF, F, on on EE -- TT ,, P(v- - V)U + n ■H + P,f+P. T-lnx(C-V) V Psv + + T} T]== -- ## ,, co n T , [»x(<7-Ps-v) where, as usual,

pps5FF doda++

/>/ d« cfo + /?? C

E

<J $ d/ = 0 .

(7.30)

r

For the balance of angular momentum we derive (see (3.30) too): ee1l -<xe-txei + + rrxx ppf/ == p? pi ,, iinn C C -- EE, ,

(7.31)

aa aa ■ • TT xx aaaa ++rr xxppssFF == ppssLL ,, on on EE-- TT ,, - r x #? = -- I£, ,

on onT,

and the global condition

pi pi dv dv++ C e

c* r id/ = = o0 pssLda+ X do- + Ldl : i E r 5

should be satisfied. We remark that rxpf

and rxpsF

(7.33) (7.33) are not objective so

that we can introduce the objective quantities

pi* = pi -rxpf, pi* pi-rxpf

,

PsL* psL* = pPssL-rxp L-rxp33FF

which permit to write (7.31) and (7.32) as follows

,

(7.33)

176

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

= pl* ,

e'-txe{ aa-Txaa

= p.L* ,

(7.34)

inC-E, onE-T,

c p(l* + rxf)

dv +

c ps(L* + rxF)

da +

c s

L dl = 0

(7.35)

r

The stress tensor i £ C - E and T on E are represented by t = f \ ®e■ ,

T = Taf3aa a„ + T a a a ® n ,

(7.36)

due to the constraint n - T — O on E. Introducing (7.36) into (7.34) we see that Hihii3*H + Pi* = 0 •

^ ^ a / 3 » + ^ ^ V + PSX* = 0 .

(7.37)

Equations (7.37) are objective and this suggests that /* and L* are physical quantities for which we must give constitutive equations. On the other hand, (7.35) looks non objective owing to the presence of r x / and rx F ; however, it is easy to recognize that (7.30) implies the objectivity of (7.35). In fact, under a change of frame of reference I -* V given by

i ' = x o (0 + x ,

(xn = x'0 + Q)x^,

Q'jhiQ1 )] = &))

due to the objectivity of / , / * , . . . we have /' = / ,

r ' = r , . . . , r ' = x0 + :

177

CHAPTER 7. CRYSTAL GROWTH

and thus from (7.30) and (7.35) it follows that

p(l*' + r'x / ' ) dv + [ ps(L*' + r' x F " ) d
p(l* + rxf)

dv+

ps(L* + r x F) da +

C

ee

psF da + £

L' dl r

s£

+ x0x(\pfdv x M pf dv + +

L' dl

# d/) = 0 . i

Moreover, when applied to the balance of energy, eqs. (7.24)-(7.25) yield

pe-tr(t®Vv)

+ V-(h + n1q) = p(e-f-v)

,

in C - E ,

(7.38)

^ ' - ^ ■ T - V ^ +V - ^ + ^ g J

-[/»(£(*- - ^ ) 2 + e _ ^ ) C f

+ n.(f.(v_v)_ft_/iig)] =

^ _ F . y ) , onE-T,

r ■[nx [(C- -V)p,( I ( T ^ - C ) 2 + £ ) + T- (V--Oon

= - ( § - # C), pe dv+

psE da + E

H~vM'

(7.39)

g dl = 0 . r

we get

w

'j = v[},t}

-VulsW

r,

On the other hand, if we introduce the notation

d

K

= ^jhw

>

178

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

tr(t ® Vt>) = t>vjt i = t^d4j

+ < M W y = t^di}

+ t^%hwh

.

(7.40)

Similarly, on the surface we have

VSV = a"®V,a

= (yp.

a

- Vnbaf})aa ® a? + (V B| „ + 6 a / 9 v V ® n

= D+W , where D = aapaa ® aP + ffa{aa ®n + n®aa) , a

a

W = Wafia ®aP

a

+ (Ta{a ®n-n®a )

(7.41)

,

and a

ocH -

V

(fl; a) - Vnbal3

"a^hfY^ W

afi

'

(7.42)

+ b^Vfi), =

V

[0,a]-

We introduce the surface curl of V by

Q = V,xV = ««xK, a = ( ^ ; « - V n ) « " ^ + ( f n , a + *a/^ / 3 ) a" x n

=
> / 3

whence it follows that

«„ = f"%; « = ^ ^ - «a = 2f *% . or

^

= K/3 f i "' ' - = 5 ^

(7-43)

CHAPTER 7. CRYSTAL GROWTH

179

It is then straightforward to verify that the difference of velocities of two points on E which are apart by a vector dr is given by dV = dr-D+[ dr-D+[^Q±n + nQaaaaaa]xdr ]xdr nn nn

(7.44) (7.44)

.

Hence the angular velocity on the surface is the vector n = nQaaaaaa + Q Q nnnnn. .

(7.45) (7.45;

It is now easy to verify that (7.41)-(7.43) imply {al3) aa ■ • TT ■■ V, V,aa = =TT{a0) aaa/3a0 ++ T^W^ T^a^Wap + + 2T 2TaaD Daa

=

Tt«n*afi+ica$T"enn+tafF°aP. (a

= T ^a0

+ i ca0r*iin

+ tal?*&

.

(7.46) (7.46)

By recalling (7.37), (7.40), (7.46) we get By recalling (7.37), (7.40), (7.46) we get

tr(t V«) = fi^dy -pl*-u>,

(7.47)

{ij)

(7.47)

tr(t $ V«) = t di;j -pl*-u>, aa

■T.V, aa .T.V, a a a

a .T.V,a

t

= = =

Jl(l

T^a -PsL*.n. ■ n. a0-Ps a0L* a/3T^a 1

T^aa0-PsL*.n.

Introducing (7.47) into (7.38) we find the final form of the local energy balance equations pe-t-{i3) dtJtJ pe t^d

) + V-(h V • (A++fiHiqq) +

= i nC C - E ,, = p(e-f-v-r-uj) p(e - / » - ? * ■ ») = = pr pe* , in

(7.48)

180

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

P,E'-

j(a

B )

° a ^ s ■(*» + P i . O +ee-E)U n-{t-{v-V)-h-n -E)U ++n •(*•(* -F)-ft -Plfl)] + iq)}

-lp{\{v-Vf -[/»( \{v-Vf = PpSs(E-F (E-F-V-L*-Sl) psE* , oonnSS- -•r, r, ■V - L* ■ Q) = =/>,£*, T.lnx[(C-V)Ps(±(V-C)2

+ E) + T-(V-C)-hs-p.uqs}}

= -(e-#-c)= -s*,

onr

.

Finally, (7.30) becomes

p(e* + f-v + l* ■ u) dv + ps(E* + F-V +

L*-n)da

(&* + <J-C)dl = Q.

(7.49)

The quantities e*, E* and 8* are objective in view of the condition (7.49). In fact, under a change of reference we have

L(e'* + f'-v' + r*-L>') dv = \p(e* + f-v + l*-u) dv + x0- \fdv + t

e

c

{rxf

+ l*)dl,

c

since t/ = t> + i + u > x r , LJ' — u+w . By applying the same considerations to the other terms of (7.49) and adding the results, we derive the invariance of (7.49) as a consequence of (7.30), (7.35).

CHAPTER 7. CRYSTAL GROWTH

181

7.5 Nonlocal reduced dissipation inequalities.

We postulate that the following well-established inequality is valid for all admissible processes which take place in the whole system d_( psdv+ ps dv + ■4j( C

da)> psSS d(A>

±h-Nda\h-N da-

aC

1£

\h \h,.*zdl s-Vlidl s dl

(7.50)

as as

(see (3.37)). If we localize (7.50) we obtain

(see (3.37)). If we localize (7.50) we obtain

A» + V - gjf) = ps+V-(£j

/» ,

in C - E inC-E

= ps ,

(7.51) (7.51)

«v

-lp(s-S)U-ih-nj = p,S psS ,, on E - T , - M « - S ) t f - 1 A.n] = PSS' + VS( *) & PsS' + Vs-fy)-lp(s-S)U-ih-n} = psS , o n E - T ,

r-[»x[(C-K)p,5-iy =

-» , r.[nx[(C-V)pa5-lA,]]= -5 ,

on T , on I \

where ps dv \p$ dv++

cC

S£z

psSS da da++ J* dd// >> 00. .

(7.52)

ir

However we will now show that (7.51), (7.52) can be transformed to a form which is more suitable for applications. Let us partition localization residuals arbitrarily in such a way that

= s S+ + Ss* * ,, sS =

S5 =§ = 5 ++ §* 5* ,,

f£ ==?? ++?* ?*, ,

where S s > 00 ,

in C - E ,,

55>>0 0 , , on o nEE-- T ,

t? >0 > 0 ,,

on on TT ..

182

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Let us now denote the left-hand side of (7.52) by R(t). Of course depends on the thermodynamically

process and the only

R(t)

information

available about its properties is that it is non negative for all times. Since s, S, 1 are chosen to be positive, we can suppose that

psS da+ \$ dl = R(t)>0

ps dv + C

S

.

r

For instance, a possible choice is provided by s , S , !f subject only to the constraint sMc + SM^ + f/ = R(t) > 0 , where M*> and M s are the masses of C and / is the length of T. Hence relations (7.51) can be expressed in the form

ps+V-(£\-ps*

PSS' + Vs-fy)-lp(s

r-lnx[(C

= ps > 0 ,

-S)U -± h-nj-

-V)PsS

inC-E

(7.53)

psS* = psS >0 ,

-j: hs]} + r = - ? < 0 ,

onE-T,

onT

We call the fields s*, S*, !f* as localization residuals. They satisfy the global condition ps* dv +

psS* da +

r di = o.

(7.54)

If we use (7.38) in (7.53) to eliminate V • h and V s • h , we arrive at the

183

CHAPTER 7. CRYSTAL GROWTH

nonlocal form of the reduced dissipation inequalities

(i])

-p(^ -p(i>++s0)80)+ + t(ij)tdij-'V-(ii dtj 1q)-^h-V9 - V • Oilff)

- ps(V

+ S6> -

- 1 A • +V0 p^*>O, + p#* > 0in i ,n C-E C - E ,,

HgV's)

+ T^Da(i

+ [p[ \ (« - V? + g - * - nls(u

-

9s

(7.55)

■ VsMls

- vs))V + (Pls -fijq-

+ [n-t-(»-V)]+/),**-ift,.V,»>0 ,

n]

on E - T

T.l S-±h 9*== - ? < 0 ,, s}]) ^ - U j ] + J* r . n[x[(C-V)p n x [ ( aC -V

o nT T, on

where g = ip + p/p is the specific Gibbs potential and we have put V>* = e * - 6 > r ,

$* =

E*-6S*.

W e could here choose a class of constitutive equations for all the

fields

which appear in our balance equations and try to derive thermodynamically restrictions

for

these

constitutive

equations

including

the

localization

residuals. W e are not interested in developing so wide a nonlocal theory of phase transitions; on the contrary, we prefer to treat a restricted class of materials in order to test the effectiveness of this approach.

7.6 Preliminary considerations on crystal growth.

In this section we want to recall some fundamental aspects of crystal growth to have in our hands elements to control the results of our model. It

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

184

is well known t h a t a crystal of a substance C can be produced i) in the melt or vapor of C; ii) in a solution containing C as a constituent; Hi) by the presence of impurities in the melt, vapor or solution. It is easy to understand t h a t the first case is very difficult to realize experimentally. On the contrary, the

second

one

is

much

more

practicable

and

allows

a

control

of

crystallization process. The third is almost spontaneous and consequently it is very difficult to describe and govern. Whatever is the environment in which the crystal grows, the process always starts from a germ.

This is an agglomerate of few hundred

or

thousand molecules which are temporarily linked to each other as a result of statistical fluctuations. When the temperature and pressure are given, the destiny of the germ depends on the ratio between its surface and volume. If this ratio is below a critical value, the agglomerate dissolves; in the opposite case, it grows up and attains a configuration which is governed by Wulff rule (see sections 7.1, 7.2). Starting from a germ, the crystal increases under suitable temperature and pressure conditions. From now on we suppose that these external conditions guarantee the regularity of the growth process so that

the

crystal

faces

are

planar

and

advance

remaining

parallel

to

themselves. Two fundamental theories have been proposed to interpret the crystal growth. Classical

theory.

According to this theory the growth is due to three

main factors: i) Adsorption:

The atoms and molecules on the crystal surface

are

captured, from the surrounding phase, by the molecules already stored in the

CHAPTER 7. CRYSTAL GROWTH

185

crystal. Some of these new collected molecules, owing to thermal oscillations,

Fig.24

go back to the surrounding phase after a mean time which depend on the substance and varies from 10

sec. to 10

centuries. Other molecules, due

to the greater number of interactions they are subjected to, definitively fit into the steps which are present on the crystal face (fig. 24). ii)

Surface

diffusion

: the captured molecules migrate toward the steps

(fig. 24). Hi)

Surface

nucleaiion

: to allow the face growth after a molecular layer

has filled the face, it is necessary the creation of a two-dimensional nucleus which plays the part of a crystal germ for the new layer.

186-

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Fig.25

Fig.26

187

CHAPTER 7. CRYSTAL GROWTH

Burton, Cabrera and Frank [58] showed that the probability that a new two-dimensional germ is created on a face has appreciable values only if the vapor or liquid supersaturation rj = (p - p0)/p0,

where p is the pressure in

the vapour or fluid and p 0 the equilibrium pressure, is greater than 25%. On the other hand, it is possible to obtain significant growth velocities for values of 77 ~ 1%. This gap between experimental and theoretical d a t a requires a new model to describe crystal growth. Dislocation

growth. This theory foresees the presence of dislocations (figs.

25, 26) which generate one or more steps on the crystal faces whose height is a few molecular layers. These steps are stable since the molecules are stored on the face without modifying the distribution of the steps. During the growth, which is a consequence of the installation of new molecules along one step, the latter makes a spiral around the dislocation. The

mentioned

Authors evaluated the number of dislocations, which could be present in a crystal, and very high values were found. For instance, in 1 cm

there could

be dislocations for a total length of about 10 cm. These growth spirals can be observed by interferencial and phase contrast microscopy.

7.7

A

mathematical

model of crystal growth in a

binary

nonreacting mixture.

In this section we apply the general nonlocal balance laws to a particular

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

188

continuous system with an interface (CC,C, E) which is able to describe the main characteristic of crystal growth. More precisely, from now on we suppose that: i) the crystal phase Cc is rigid and at rest whereas the phase C is filled with a perfect classical binary mixture; consequently, the unknown stress tensor tc in the crystal has to be determined by the evolution equations; it) the phase Cc is a convex polyhedron so that the interface

E = 0ec=U',-.

J = l,2,...,n,

(7.56)

t' = 1

is the union of n planar faces ai of the crystal; the broken line

is the union of n planar faces a^ of the crystal; the broken line m

r =PUm= i/„, T = U L,

p == 1,2.....,m ,

(7.57)

p = l,2,...,m,

(7.57)

which represents the union of the edges / of Cc , is a discontinuity line for the fields defined on E. The only physical attributes of each ai are represented by a surface stress tensor Ti and a surface energy density E^ Moreover, on each cr^ , the surface stress tensor Ti is a function of the values on
189

CHAPTER 7. CRYSTAL GROWTH

iv) the localization residuals are absent in the bulk phases but are present on £ and T ; in other words, the nonlocal effects are related only to surface phenomena (see section 7.3). Finally, we neglect the external body and surface forces. When we write all the equations (7.26)-(7.55) in these hypotheses we get

p +pV-v=0,

in C ,

(7.58)

pv + V • q = 0 , pi, = - V p , pe = -V-(h

+ niq) ,

pc = const. ,

in Cc ,

(7.59)

V-< r = 0 , t= Pckc= -

[[pU] 0, PU] = Q,

tT, V

9c .

oonnEE--TT,,

(7.60; (7.60)

lpUv]-q-n IpUvJ-q-n^O, = 0, -Vs-T-lpvU rxF=L 2

^-lp(±v

+ n-q = F , ,

+ e)U + n-t-v-n-(h

+ ti1q)} = E ,

where in C , e is the partial time derivative since the solid phase is at rest. Finally, we have (see (1.64)'):

190

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

r - [ r » x T ] = -{v-T}=

-9,

on T ,

(7.61)

rx§ = £ , f [ n x £ ; C ] = -{EC-v}=

-I,

r - [ n x 5 C ] = -{£C-i/} =

-If.

Here C is the velocity of T, v = rxn

and the localization residuals satisfy the

integral conditions ,Fdcr + f 5 ( f l = 0 , E f rxF

(7.62)

r x 5 d / = fl,

da + r

E

LB


J E

r S da +

f d/ = 0 .

We remark that we have neglected the dissipation on the edges. We are interested in those crystal growth processes in which each face moves remaining planar and parallel to itself so that the final result is a regular crystal. We assume, to guarantee this kind of evolution, in agreement with experimental procedures, that: i) all the fields are uniform on each crystal face; ii) the stress in the crystal reduces to a pressure; Hi) the growth velocities are very low ( ~ 1 0 - 6 m/sec.) so that the

191

CHAPTER 7. CRYSTAL GROWTH

process lasts many minutes or some hours. This last circumstance permits to neglect some terms in the equations (7.58)-(7.62). In fact, in ordinary conditions of pressure ( ~ 10 5 N / m ) and crystal dimensions (10 ~~ -r 10

m), by using a nondimensional analysis

(see sections 5.2, 5.3), it is simple to derive that the pressure is uniform in the mixture and the terms pvll,

pvU/2

are negligible with respect to the

others. Moreover, if O is an internal point to the convex crystal and hi the distance between O and the face ai we have

C

ij{p)H(p)-nnii=Cni = Cn. = ilhii> '

(7.63) (7-63)

where C- ■, •> is the velocity of the p-th edge /,-•»_% between the faces a., cr ■; finally, the T h o m a s derivative vanishes in (7.60) 5 since E depends only on n and n remains parallel to itself during the whole process. When we take into account (7.8) as well as all the previous remarks, we can write equations (7.58)-(7.61) as follows

p + pV-v

= 0,

in C ,

pi/ + V ■ q = 0 , P = Pe>

pe = - V - ( A + / * , « ) ,

(7.64)

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

192

pc = const. , tc=

(7.65)

in Cc ,

-Pe{t)I,

pch= - V A > vn.=—p-^h^phi

,

(7.66)

on or,. , i = 1,2,....n ,

9"wi = / ' c ( y - 1 ) ^ . T, = const. ,

F, = Ftn{ ,

[p] = F, ,

rxF,.=2,., P c ^ +^ l + ^ C i - ^ ^ - W - " , - - ^ - , (7.67)

-(Ty»i + ^ » i + ^ o - ) = - ( % " ; + V » i + % * - ) , on / t j ( p ) ,

p = 1,2,..., m ,

-(V^A'}=-%-' r

here where • T > • ni - va ■ Tj ■ nj

*u = ^"iji^ji «?y = CSC

^

2

a i J [# t J

«i =

E

j

ij]

.

(7.68)

• n,- - # ( i • Ti^ COBOy] ,

= («/ IJ -T 1 + „ J 1 - T J ) - r l J , A

cosa

CSCQ

ij

- Ei

^■= # , r r . i l coia

i]

■

To derive (7.68) 1 . 4 it is sufficient to represent both sides of (7.61) 1 in terms of the basis (n,-, n-, T-) which we fix along the edge / • ■. Equation (7.68) 5 is obtained by (7.8), which allows to express vi ■ and u •■ in (7.61) 3 in terms of the aforesaid basis, and taking advantage of (7.63). For the sake of simplicity, in the sequel we suppose that

193

CHAPTER 7. CRYSTAL GROWTH «P. • = <J'- • = 0 .

To obtain the field equations describing the crystal evolution it remains to introduce constitutive equations, initial and boundary conditions. First of all, (7.55) lead to the following reduced dissipation inequalities

-p(V> + s f l ) - p V - t > - V " ( / i 1 f ) - i f t - V 0 > O ,

-p(^c

+

sJ)-lhc.V9>0,

PclV' + § + ^ 1 ( l - ^ ) ] A l

+

in C ,

inCc,

*I + F t / l l > 0 ,

(7.69) (7.70)

on<7 t ,

(7.71)

where if) = e-6s is the specific free energy and $,- = Ei - 9iSi . In the bulk phases, according to the classical mixture theory and the reduced dissipation inequalities (7.69)-(7.70), we adopt constitutive equations of the form (see section 6.2):

iPc = iPc(6), , ,/ m lP = 1P(P,1S,0) ,

s = - ^ > dtp S=—QQ,

(7.72)

hc=-k-Ve, 2H P = P-T^,

dip V1=-fo,

h= -kve, q = q(p,v,e,vp,vv,ve), ^ - v ^ - ^ w f ^ o , (7.73) where k is the thermal conductivity tensor in the crystal and k(p,v,9)

is the

thermal conductivity of mixture. It is evident that we must also consider

194

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

constitutive equations for F t , Ei , ?,- ■ , 84-j but we discuss about t h e m in the next sections.

7.8 On the crystal equilibrium.

At equilibrium the system (7.64)-(7.67) reduces to the following set of equations p(p,0,v)

=

inC,

Pe,

pc = const. ,

(7.74)

in C r ,

[ p ] = Fi > on o-,- , ?' = l , . . . , n ,

- (**««,• + V » i ) = - @Hni + V 1 * ) '

on

W ) 'P = 1 - " m '

to which we have to add the integral conditions (7.62) 1 Now an equilibrium configuration

2

(7J5)

.

of the crystal in the mixture is

characterized by the uniform values of 8, pe, p, v and pc as well as by the orientation n, of the normal to the faces ai and their distances hi from a point O internal to the crystal. When we assign constitutive equations for F-, 9- ■ as functions of the aforesaid variables, (7.74), (7.75) and (7.62) x

2

represent a system o f 2 + n + m + 2 = 4 + n + m equations with 5 + 3n + n + 2n = 5 + 6n unknowns pe ,pc , p ,0 ,v , Ti ,h , n , , where h = (hv...,/in).

In

the next section we show that the reduced dissipation inequality on E allows one to derive another condition on the faces in the same unknowns. This

195

CHAPTER 7. CRYSTAL GROWTH

means

that,

in general,

t h e equilibrium

configuration

is n o t uniquely

determined. However, b y taking into account the crystal symmetries, we could determine some possible configurations which are compatible with these symmetries and the equilibrium conditions. For instance, for a cubic crystal, we have T , = yj where 7 is a surface tension which does not depend on the face. T h e n we can ask whether a cube represents a possible equilibrium configuration for the crystal. In these hypotheses we have

p{p,vt8)

=

Pe,

pc = const. , [ P ] =Fi 1

o na

i .

-7(n,. + n i ) = - 5 ( « , . + n i ) ,

inC, in C r , i = 1,...,6 ,

on l.j{p) ,

„A

p = l,...,12,

), - * * 21 ("* ++n»;),

PFEi"i=-I*%.P(n P i 1

(7.76)

~ 12 12 ,. .. rr s,_ d
(7.77)

(7 (7.78)

i

r .11

x

1

'« where / is the length of the cube edge, r is the position vector with respect to the center O of the cube and rs is its projection on the crystal face. It is clear that both sides of (7.78) 1 vanish separately for any choice of /, F, 7 . Moreover, the integral on the left hand side of (7.78) 2 represents the center of mass of cri so t h a t rs = 0 and each term of the summation is zero. Similarly, the integral on the right hand side of (7.78) 2 denotes the center of mass of the edge /

and consequently the right hand side also vanishes. We can

196

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

conclude that system (7.76)-(7.77) reduces to four equations with unknowns pe,pc,p,v,0,y,

h = 1/2. If we add to those the other conditions on the crystal

boundary, which we derive from thermodynamics, we attain the conclusion that the variance of these cubic equilibrium configurations is two. Similar considerations can be repeated for the symmetric polyhedrons. We note that the only a priori information about the form of the constitutive equations of Fi and ^i ■ are represented by the restrictions which are imposed by the crystal symmetries. However, by experiment we have that [p] = 0 for large crystals so that lam F- - 0 .

(7.79)

This, in turn, implies that the equilibrium configurations of large crystals have to satisfy the equations p{P,6,v) = pe, Pc = Pe>

in

inC, c

(7.80)

c

so that these configurations are arbitrary in form and have again variance two. In the case of small crystals, the integral condition (7.62)j can be written as

t ,•*>.•*,• = - E P@iilp)*i +

*ji(P)ni)lij{p)

p

= - s Pc*ij(p)ni+*

jnp)njhj(Py

The last two terms of this equation contain the same number of terms as pairs of adjacent faces. Therefore these terms can be combined as it is shown

CHAPTER 7. CRYSTAL GROWTH

197

in the relation

t i?Wi = ~ E i E ■ IH*^ = - t i E 'i hFi,**

(7-81)

where the notation E i means that the summation is extended only to the edges which form the boundary of or.. When we suppose that i) the condition (7.81) is satisfied as a consequence of the following hypothesis $Wi

= ~ E ■ hpiFi = - E • ijij^i,

(7.82)

H) Tt = 7,-J». it is easy to verify that (7.74) 3 and (7.81) lead to the relation

VC~V = \ E j V . j ' where ij = 7y csc Q j i - 7,- cota t J .

This relation coincides with the Gibbs rule (7.16)3 when j i = Ei (see remark it) in section 7.1). The following considerations supply a justification of hypothesis (7.82). Let us regard the crystal surface as a set of planes which are constrained to remain parallel to themselves during the phase transitions and in contact along the crystal edges. The crystal faces and the edges interact with the

198

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

volume particles which are contiguous to them and these interactions are represented by the localization residuals of surface and edge, respectively. Consequently, the edges are subjected to the forces 5 , and the crystal faces are subjected to the parallel and uniform forces F,n t , which are independent of the face. We have no phase transition if and only if each face is at equilibrium. Now, it is easy to recognize that the face ai is at equilibrium if and only if (7.82) is satisfied. In fact, let us consider a virtual displacement owing to which the crystal reaches a new configuration in which only the face ai is translated by the amount 6h^ along the normal n,-; this implies that Sh ■ = 0, j ^ i, whereas the edges of the contiguous faces move to remain in contact with
iW\=-£;',A-•*•.;>

( 7 - 83 )

where <5r-■ denotes the aforesaid displacement of the edge /•• and therefore verifies the condition Sr-■ ■ n = 0. Since it is straightforward to verify that

Sr

ij =

csc2

% K -

cosa

«jnj).

equation (7.83) reduces to (7.82) when we take into account (7.68) 2 . Hence, (7.82) could represent a restriction to be satisfied by the constitutive equations of F-•, 'i- ■.

199

CHAPTER 7. CRYSTAL GROWTH

7.9

The

energy equation a n d t h e reduced dissipation inequality.

If we introduce the latent heat

A= [ 6 + ^ 1 + ^(1-1/),

(7.84)

the energy equation (7.66) 5 becomes

AA, + £ i

=

[fcJ.n,.

(7.85)

As is well known, the latent heat is the amount of heat which the unit volume of the substance must adsorb to pass from the solid state to fluid or vapor state. This adsorbed heat generates a j u m p of heat flux. However, (7.85) shows t h a t only a part of the heat flux j u m p is used as adsorbed heat because of the presence of the energy localization residual E'-. We have to justify the presence of this additional term in (7.85). First of all, we remark that in the absence of E^ the growth velocity of the crystal faces would depend on temperature 9, concentration v, external pressure pe as well as on the orientation of the face owing to the heat conductivity tensor, which in the crystal is not isotropic. This dependence generates differences among the velocities

of

crystal

faces

which

are

too

small

compared

with

the

experimental velocities. Therefore, we could think of assigning to Ei the task to describe this strong dependence of the velocity on the crystal face. In order to understand the meaning of E^

we take into account (7.61) 3 to write

global condition (7.62) 3 in the following way [59, 60, 61]:

200

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

n

m

(7-86)

E i E, *t = E tj(p) (E,Vij + V / . 0 •
The left hand side is the total excess heat adsorbed by the whole crystal surface during the growth and it is equal to the total energy associated to the creation of new surface parts per unit time. This consideration supplies a remarkable meaning of Ei but, at the same time, would lead to the wrong conclusion that the presence of Ei is meaningful only for small crystals for which the energy per unit area necessary to increase the surface of the faces is comparable with the energy adsorbed by the unit volume in a phase transition. In fact, (7.86) is a global not local assertion about E±. What follows is to make this point clear. The integral condition (7.62)3, when we take into account (7.67) 3 and repeat the considerations relative to (7.81), becomes

£< V.-=-E,-E'V.-i A .-« 1

1

(7-87)

3

so that we derive f

.-=4E'Wi + ^

(7-88)

3

E,-^i«r,- = 0.

(7.89)

With these notations the energy balance equation (7.85) 5 can be written

( 4 E '

'AijLij), ^ = [A] • n, - E\. 3

(7.90)

201

CHAPTER 7. CRYSTAL GROWTH

It is easy to verify that (7.86) reduces to (7.88), in which we put E\ — 0, for a growth process corresponding to hi ^ 0, h = 0, j / i. In other words, to suppose

E'; = 0 is equivalent to say that the localization residuals Ei of the

face <Xj depends only on the growth

velocity of
this

assumption only for small crystals, the quantities E\ have to satisfy the property Urn E'- = 0

(7.91)

and therefore it describes the growth process for large crystals since it is obvious t h a t when E\ = 0 .Urn

^ - ^ E ^ / , / , - 0 .

| h | - » oo

«

(7.92)

j

It is possible to give a physical uinterpretation for E^. In section 7.6 we have already noted t h a t the theory which better fits the experiments on crystal growth is t h a t of spiral growth. According to this theory, a condensation nucleus of suitable radius is formed at a point O of the face and it gives rise, as a stable dislocation, to the expanding spiral on the face. T h a t suggests the introduction of a surface energy flux j associated with this expansion and depending on the face. By applying the Gauss theorem to the whole crystal with planar faces we obtain n

E, i

m * V s -j da-- ? P


■

{ii'Vij +

i / J *j (p)

T h e previous expression can be put into the form

iyVjddl-^.

202

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY n

=o i dcr E * ) * < ) = o, E,(^(jv ' s -i^-E'ji i -?'/'' *r*^0^ 3

'y '.«•

where the notations were previously introduced. Consequently, the quantities

^ ^ ( [ V ivA r - E ' U - hi ^■tvaf )dl,) s -jrf
(7.93)

hi

associated with the face
e,h,k).

It remains now to consider the reduced dissipation inequality (7.71). If we extend the decomposition- (7.88) to the entropy localizations V •""y

s,= -ilZ'Bijtijhi + Fi, SJ. £'s;-^=o, : ' % '« A,;+ '

3

1

and recall that $ ; =

E--6SJ

3 3

, we can write (7.71) as follows

p E' % PcM+Tl - i ^- ^-) -h^ S - * 'Vt Jy)) ^* + i +**; }>>o0,. c ( ^ + ^ ]++^ M

(7.94) (7-94)

where DiS =

, VAij-*ii-0B -V ij

£),■*; *,- = o.

The left hand side of (7.94) can be regarded as a function / • of h- (i = 1,..., n) which has a minimum when h; = 0 (i = l,.,.,n) so that at equilibrium we have

r«£n oh^ y

h

==0 0

= /»c(W-►?]- f / i j ( l - - " ) ) - -E f j

£»■■

J

,a*j J

^ dhi \

L )

= Q=

0

i

CHAPTER 7. CRYSTAL GROWTH

203

± i, (wl \dh-h.n =(#)■ V a / , , 4 - n =•■ i'*'• v

dhi

J

h

= Q

v a/ij ih _ o

$i Ai0»,v ,.,n), $J = ■**(/ >, M1fl)h. ) \ - ++*J' *J' (,-(i == !,.. l,.-.,n),

(7.95) (7.95)

where the functions ty" are at least of the second order with respect to h^ Consequently, when we insert (7.95) into (7.94) we obtain the following j u m p conditions on the crystal faces

PC(N> + ^ 1 + ^ i U - ^ ) ) - £ % ' , , +A, = 0 ,

(7.96)

3

as well as the thermodynamical restrictions on the functions

$'.' %'{ >0 > 0 .. In the sequel we also accept (7.96) in dynamical conditions since we consider here only very slow processes. In order to derive an interesting consequence of (7.96), we introduce the notation a M to denote the value of the quantity a for a large crystal. In these conditions (7.96) leads to the relation

P ^] + M / i i*( l -~- ""))oo PcW ) ) « , + ( A oo), = ^oc(Pe> o c ) , = 0° ,. ^oo(Pe> 0oo^oo- O "oo) + ( A oo)t c(lV> + £

(7.97) (7.97)

which allows to evaluate the equilibrium temperature 6^ as a function of pe and v. But at equilibrium the temperature is uniform and this imposes that the quantities (A )■ do not depend on the crystal face. Moreover, we can put

204

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

(^oo)i = °

since

in

tne

absence of this term in (7.97) we obtain values of

equilibrium temperature which fit the experimental results. On the other hand, to within terms of the 2nd order, we have A:

=mj°- •o=

-•Boo =


(e- - ' « , ) .

and thus (7.96) can be written as 6- -*oo =

*oo " S oOO o

[

1 *

E 'Di3 hi-- A J . J

j

This relation shows that the quantities within the brackets must not depend on the face and that, for a small crystal, the equilibrium temperature 8 could be smaller than 6^ (supercooling).

7.10

The free boundary value problem describing the crystal

growth.

The evolution of a crystal is described by the system (7.64)-(7.67) which is a system of partial differential equations in the unknowns p, v, 6, pc, v. A set of reasonable boundary conditions associated with this system can be derived by supposing that the mixture is contained in a vessel. The walls which are in contact with the fluid phase will be denoted by S. If the free

205

CHAPTER 7. CRYSTAL GROWTH

surface of the mixture is 5 ' and the crystal is far apart from the walls, we can give the following boundary conditions

onS ,

6, vn, q-n = 0, Q, P = Pe> q-n-0,

(7.98)

on S ' ,

0, vn, q • n, Pc,

on E.

We remark t h a t the nonmaterial character of E allows us to assign on it a velocity vn and a pressure. Now, the d a t a (7.85) 3 are given in terms of j u m p conditions (7.66)j

2

3

on

the unknown surface Sigma whose evolution is in

turn described by (7.66) 5 (generalized Stefan's problem). Finally, another condition on E is represented by the inequality (7.71) which we assume to be satisfied in the form (7.96) (slow processes), in what follows. W e are now able to give the precise mathematical formulation of the boundary problem which describes the crystal growth. We have to find the fields p, v, v, 0 in C and the field 9 and the quantities hi in C c such that

p + pV-v

= 0,

in C ,

(7.99)

pv + V • q = 0 , pe = -V-ih

PcK=-V-qc,

jB. = ^ i

f

H j 5 i j ,

+ fitf) ,

inCc,

on a,-, i = l , 2 , . . M n ,

(7.100)

(7.101)

206

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

qn (A- 1

= p c ( j / - l ) h{ ,

£''4i ' « K = [*]• n, - ^ . ) , ■

'

3

pc(W+^] + M i - ^ ) ) + E * ^ ^ + ^ = oi

(7.102)

In the previous equations the functions A-;' s are defined by (7.68) 4 ,

L r csca csca LL cota
D D

i.,

(7.103) (7.103)

where the Li are, together with the quantities A-, given constitutive equations of the variables p,v,9,E

■,hi and the E\ 's are functions of p,i/,6,E

-,/i ,A .

Moreover, E\ and A ■ satisfy the global conditions

E i ^ " i = 0, l

E , \ ^ T = o,

E . A . ^ ^t ^ O . il

(7.104)

W e have also defined (see [44])

« = 5 £ '(ftj c s c a ^ - &,. c o t a ^ ) li3 , i


= *«i = hi

V

V

- 11

ccsca. sca

o t ot * , \3-i --11 ccoifi ^ i -_- 1l,. J' j - -hftii ccsca M O f*^( «c *o /t ?^, -_. -l1f . ii J++ ccot/?^ ^ - . , ^. +j +x )x)

csca csc csc/^+ h i[coto,-, ( c oi_thj^ . . , , -+r C + ** i; + O t ^h}. i ++ 1i)) cotf3 lii{cotp + 1i csca-i ,,ii + + 1l Pj,j■ j ++ 1l + hi[eoto -li

csc -- cota cotaj csc/? i : ,.i ++ iJ' i, , jj --1 ^ i _- -ii,-i- -j "c-o ct o t,a^; ,,-y++ ji csc/?^ J, - l cscf3j

CHAPTER 7. CRYSTAL GROWTH

T T w h e r e cos/? cos/?^ j■++ i~ where 1 = r ij' . - i ' rii , j+1 i +l

207

an( a n^ d Trij

denotes the the unit unit vector vector along along the the »i denotes

edge formed by the faces ai and a •. This problem is very difficult but it is possible to simplify it substantially if we are dealing with large crystals (i.e. having linear dimensions (i.e . crystals having greater than 10 _ "' m). In this case we have

p+pVv=0,

in C ,

(7.105)

pi/ + V • q = 0 , pe = -V*(h

+ nxq) ,

PA=-V-AC,

vn,=^frhl

= l3hl,

inCc,

on*,., t = l , 2 , . . . , n ,

V l n q-n{ = {v-l)h = pcPcV - U i i, « • « , ■

Xhtz

= [*]•"*- ^ > [^+7?I+Pi(i-»') = 0.

(7.106)

(7.107)

Chapter 8

SYSTEMS WITH INTERFACES AND FERROELECTRICITY 8.1 A b r i e f s u r v e y of f e r r o e l e c t r i c i t y .

In a dielectric crystal C the polarization per unit volume P is a function of the external electric field E

which depends on the nature of the substance.

When the electric field is very strong, P tends to a constant value and we say t h a t the dielectric is close to the saturation because almost all the molecular dipoles have been oriented by E. In particular, P is parallel to E if C is isotropic. A dielectric crystal C is said to be ferroelectric if very high values of polarization, in particular the value which corresponds to the saturation, are obtained by applying very low electric fields (curve 1 in fig. 28). Moreover, if we reduce E after the saturation has been reached, the polarization

assumes

the values corresponding to the curve 2 and reduces to zero when we apply an opposite field -Ec

which is said to be a coercive field. When we again

increase the values of E, we obtain the values corresponding to curve 3. The set of curves 1, 2, 3 represents an hysteresis loop.

208

209

CHAPTER 8. FERROELECTRICITY

fig.27

This anomalous behaviour of ferroelectric crystals has to be found in the domain structure of polarization. More precisely, even in the absence of external electric fields, the polarization in a ferroelectric crystal is represented by a vectorial field P = P0p,

where P0 is a characteristic scalar of the crystal

and p is a unit vector field. This latter is piecewise constant in almost the whole crystal C except in thin layers across which it varies from one constant direction to another. In general, the regions Da, a = 1,2,...,n, in which p is constant (Weiss domains)

have at least one microscopic dimension (a few

hundred microns) whereas the others are comparable to those of the crystal. Moreover, the aforesaid transition layers (domain

walls) have a thickness of a

few microns. T h e theoretical justification of this complex distribution is to be found in the combined effects of three factors: the electric energy due to

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

210

the anisotropy of the crystal, the dipole-dipole exchange q u a n t u m forces and the form of the crystal. The presence of an external field can only modify the weight of the effect of these three factors on the distribution of the domains. F r o m a classical viewpoint one takes the first two factors into account by associating with the crystal an energy e(p,Vp)

per unit volume in which the

contribution due to the presence of V p is significant only for very large values of || V p || . Then the evolution equations of the crystal are so complex t h a t it is very difficult to deduce some general results from them and very few information on the distributions of domains, on their evolution or on the energy contained in a transition layer was obtained only by resorting to rather drastic approximations [62-73]. In order to obtain a description sufficiently accurate and also to simplify the situation illustrated above, in [74, 75] it was suggested t h a t the domain walls are to be substituted by surfaces of discontinuity Sb, 6 = 1,2,...,3 of p and to take into account the energy contained in a transition layer with a surface energy E(n,p)

depending a priori on the orientation of the normal

n with respect to crystallographic axes and on the constant vectors p

in

two domains adjacent to the domain wall under consideration. However, in addition to this surface energy we still go on to consider an energy per unit volume of the crystal depending on V p besides p. In fact, except for very particular forms of C related to the symmetry class of C, the satisfaction of the boundary conditions on <9C may require the formation of small regions adherent to the external walls where the polarization field is not uniform. The scheme we are adopting suggests to use again the model of continua with interfaces to describe the evolution of a ferroelectric crystal. In next sections

CHAPTER 8. FERROELECTRICITY

211

we show t h a t this approach is fruitful.

8.2 Maxwell's equations for a moving continuous system.

Let C be a charged and electrically conducting continuous system which moves in the inertial frame I. We denote by V C C any material volume and by I f c C any material surface; moreover, we suppose that there exists a moving (not necessarily material) surface E C C across which the volume fields undergo discontinuities. Then the Maxwell equations for C become (see for instance [76] and the general balance equations (3.1), (3.5))

d dt

B-Ndcr=BN da=

jj

j

j-\D-Nd(j= D Nda=

Ttl ' f

-

1

(E+VXB)-T (E+VXB)-T

(8.1) (8.1)

dl ,

af

\df

(H-vxD)-Tdl(B-VXD)-T

JN daJ-Nda-

dl-

If

D-N da = D-N =

aavV

K-v^dl K-v^dl,, I

dv++ uif Uf da , Pf dv

v\

"c r

r

| B BN N d
av av j- [ M t

dt\ v

JN

ujfdcr) w fdcr}== -

Pfdv dv+ + Pf

"

av --

da +

[uf(vN cfn(vn-N) \Ndl J-Nd*+j[u N-cnn-N)-j-Ljjdl

aa (wfV, + K)-vfi dl

s

a

212

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

where the meanings of n, N, T, I/ S , vn and cn are the same as in the general balance equations (3.1), (3.5) and a = E n V, I = i fl E. In these expressions, D and B are the electric and magnetic induction fields, E and H the electric and magnetic fields, J is the volume current density, p* denotes the volume free charge and finally K and w± are respectively the surface current density and surface free charge on E and Vs denotes the transport velocity of the free charges along E. The local conditions which correspond to the integral laws (8.1) can be derived by resorting to (3.4) and (3.6). We thus have VxE V x £ == - dB ^ , dt '

in i n C - EE, ,

(8.2)

VxH VxH = v+<^, = JJ ++ Pfvpf+ §, VVD D = pPf> f ,

V • B5 == 00 ,,

dpf dp f „ pfv)--= 0, -^-+V-(J ■{J++Pfv) = 0,

v£z-nxlE-BxW} • n x \E - B x W\ == 0, 0, vTi -nx\H E-nxlH

+ DxWl DxW}

[D)-n = 8ui 8uj f

8u!ff

oonn EE --TT, ,

(8.3)

= K ,

uf,

\B\ ■ nn == \B\ ■ [B] = «00),, IL-"JI " n—

v s ■WfVs)-

-2Hcnuf

+

Vs- K- -[pfU + J-•n] == 0 ,

- fC)u T-[nx((V,-C)LJ T-[nx((V +1-*)] K)] ==== 0,00, oon o nnTT,, T-[nx({V,-C) K)] f -+ a-C)u f Uf (OV

(8.3a)

CHAPTER 8. FERROELECTRICITY

213

where the velocity W of the line / = E D if is given by (2.46) in terms of t; and the normal velocity c n of E and C is the velocity of discontinuity line T on E.

8.3

A system of a rigid ferroelectric crystal in the presence of

conductors and quasi-static approximation.

It is well known t h a t Maxwell's equations are covariant under Lorentz transformations. Since our goal is to use them together with the classical balance equations, which are Galilean invariant, we search for the conditions under which Maxwell's equations are also approximately Galilean invariant. F r o m now on we refer our analysis to a particular continuous system C [75]. Let D be a ferroelectric rigid crystal which is at rest in the frame of reference / . Let us suppose t h a t around D there are Sj fixed rigid conductors, Cj,

i = 1 , . . . , Sj, which are connected to a pole of battery, as well as s 2

isolated rigid conductors C , j = l , . . . , s 2 ,

each of them carrying a total

charge Q . Let C the system constituted by D, the s = Sj + s 2 conductors and by the (eventually unbounded) region V among the crystal and conductors, to which we a t t r i b u t e the electromagnetic characteristics of vacuum.

214

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

fig. 28 The electric and magnetic fields produced by C are discontinuous across the boundaries of each part of C as well as across the domain walls Sb, b = l,...,q,

which in our model are represented as discontinuity surfaces of

electric and polarization fields on D. D is partitioned by these walls in p domains Da, a= 1,...,p. It is evident that the Maxwell equations are valid over all the volumes occupied by C with the exception of the aforesaid surfaces of discontinuity on which the j u m p conditions are held. Since the whole system is at rest in the frame of reference / , the Maxwell equations and j u m p conditions become

V x £ = - f ,

inC-E,

(8.4)

CHAPTER 8. FERROELECTRICITY

215

V x E = J + dD ~J+ dt ' V-D = pf Pf,

V f■lfl==00, ,

vzE-nx{S-BxW} • n x I E -- £B> x:WJ W] = == (Q, 0,

£ - IT\ , oonn E

(8.5)

fD WDxW]+K)=0, + iV2-nx(lH / E - n x ((Lff[F + D xx W ] + «') = 0 )

[D] wf ,, [ £ ] .- nn == w, [B] ■ nn = 0 , IB] SLO, 8uif

^

+v V s -•(w/VJ( u V V s ) --2Hc 2 t f cs n1 W // s

+T

V sS . tA: f - "[ ptP/ -•n] n ] === 00,, / cCnn,++. /J*

T-lnx((V K)} T-{nx((Vs~C)u 0, f a-C) f + K)] K == G, *VUf-C)w

on o nT ,

(8.6) (8.6)

where, owing to (2.46) and the condition v = 0 , W W

W = r^~c + c^nn.n^f^ =W r'r -^^f^ +

(8.7) (8-7)

In what follows we assume that i) == 00 in CC -: Uu C- ; 0 p. P/ = &

•*

■

>

t := 1

w) C evolves in a neighbourhood of an electric equilibrium configuration and all the variations connected with this evolution are of low frequency. It is possible to prove that under this assumption the nondimensional analysis leads to a drastic simplification of Maxwell's equations.

216

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

Let us denote by I a scaling length and by T a scaling time; moreover, let A be the scaled version of the filled A and e,n,cr are respectively, the dielectric and magnetic permeability and the electric conductivity. If we denote by the same symbol a dimensional and a nondimensional field, it is an easy exercise to verify that (8.4)-(8.6) in nondimensional form can be written as * B L dB , VxE VxE = = -M-kQB, %T at

iinn C C --EE, ,

(8.8)

*

aEaL x€HEL i c-t*E LdE dE nV xB y R= _ ^ —£— JT + % 1T dt di ' B B PL n V V-EE = ?±p., J ! — cE Pi

*PL 5 p / ,v dt + V o-TE*, 0t

V -BB== o, 0, V-

■E

= 0 , 1

* vL-nxlE-£-^BxW} = 0, onE-T, E1 * » "x-nx(lH + ^±ExW}+^K) = 0, B J B

[JB].„ = 4 « / , [-»] • n = 0 , **

+ v s •(w*V.)-2Hcu*+2£v. >,v s )- -2J5fc, i i i , + vs iKf

7^

(8.9)

217

CHAPTER 8. FERROELECTRICITY

Lp * u

IPfn

* E ,aET * JL P

•n] == 0 ,

r-[nx ( 0 V C ) W / + KT 4 f ff)] = 0,

»T.

(8.10)

It is well known that, at equilibrium, the electric and magnetic fields are zero in all the internal points of each conductor C,-, i= l,...,s,

of C. This

implies that, in dynamical conditions the electric fields are produced by temporal variations of magnetic fields and vice versa so that (see (8.8) 1 2 )

£~^B, Therefore, E ~ L2fieE/T2,

B~^fi€E.

E{\ - fieL2/T2)

(8.11)

~ 0; but neL2/T2 ~ 0 and then

E ~ 0. This result and (8.11)2 allow us to conclude that H ~ 0. Finally, by eliminating V JE between (8.8) 3 and (8.8) 5 we derive /> ~ 0. Equations (8.4) become D= E = B= H = J = 0 ,

in C , ; , i = l,...,s.

(8.12)

On the other hand, owing to i) and the condition cn = 0 since the conductors are at rest, we have

nxE=0,

on dC,-, i= l,...,s nx(£T + Ji') = 0 , D-n

= uif ,

J5-n = 0 , ^ + Vs • K - 0 ,

,

(8.13)

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

218

where n is the external unit normal to the conductor boundaries <9C, and the fields are evaluated on dCi coming from the empty space V or from the crystal D. In the empty space V among the conductors and the ferroelectric crystal D we have p, = a = 0 and on dV n 3D it is uij = 0, K = 0. Moreover, in statical conditions, there is an electric field which is due to the charges carried by some conductors and to potentials of the others. Consequently, (8.11)2 is no longer valid. On the contrary, (8.11)2 still holds since the magnetic fields are produced by the temporal variations of electric fields. Then, we have from (8.4)

VxJ5 = 0,

in V,

(8.14)

v x u

~ €° dt ' V-E = Q , V-J3 r = 0 ,

where e0 is the dielectric permeability of the vacuum. Similar considerations, referred to the ferroelectric crystal D, in which B = (ioH,Pf=Ljf=0,K

= 0 ,lead to the following set of differential equations

and jump conditions:

V x £ = 0,

i n £ > - U Sb, b= i

V §& V x J = x a

~ dt ' V ' z> == o , V - f f == 0 ,

(8.15)

CHAPTER 8. FERROELECTRICITY

n x [ £ ] = 0,

219

nx[lf] = 0,

»•[!>] = 0, « x [ £ ] = 0, n-[D] = 0,

on dD ,

(8.16)

n-[H} = 0,

i / r n x [ f + D x l F ] = 0, n.[ir] = 0,

(8.17)

o n S j , 6 = 1,...,?.

By the expression (8.7) of W, condition (8.17)2 becomes

* E • n x [if] + i/ s • (WJDJ - IDJW) = 0 so that, taking into account (8.17) 3 , we have

" E ' ( » x I*} + WJDJ

) = * £ > * ([1T| + [J5] x cnn)) = 0.

Finally, by noting that i/ s is arbitrary and the vector within the parentheses is independent of i/ £ , (8.17)2 can be written as

n x ( [ 5 + Dxc„n]) = 0,

on Sb , b = 1,.. .,n.

(8.18)

In the sequel we will be interested in the jump of Poynting's vector flux Ex H. First of all, from (8.13) and (8.16) we derive the continuity of E and H across dC:, i = l , . . . , s , and dD; therefore, it follows that

n-[ExH]

=0 ,

on

Q 8Ct \JdD. i =

l

(8.19)

220

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

On the other hand, (8.17)j shows that the tangential components of E are continuous across any surface S^ so that we can write

.[£xf n-\ExE\ n

f l = n-\E n - f f£xB\ , x i f ] ,=r ln-E . £ t1x{H\= x [ i r l = - I E, [B\. ? , -■n nx x[IT].

From this equality and (8.18), (8.17) 3 we obtain

n -\E x H\ = = {DSlDj.E \ -Escns,cn, on Sb6 ,, 6 == 1,...,
(8.20) (8.:

We note that in a ferroelectric crystal the vectors D and E are related by

D = e0E + P0p,

(8.21)

where P0 is a constant, which expresses the polarization intensity and p is a unit vector (see section 1). The constitutive equation (8.21) allows us to write (8.20) as follows

n n-lExHj •[#x

s. • « . • ff] = cC nP^0lpj.E 0 [P]'

(8.22)

It is very easy to verify that (8.12)-(8.22) are invariant under a Galilean transformation if E, D, w*, K are invariant and H1 = H + « x D, where « is the uniform velocity of the frame of reference / ' with respect to / .

CHAPTER 8.

8.4

221

FERROELECTRICITY

The balance of energy and the entropy inequality.

W e suppose t h a t the following energy balance equation is valid for the whole ferroelectric crystal D :


e dv +

dt\

E d ■)=

D

n(T-p-ExH-h)

dc

(8.23)

3D

where £ = (J Sb \J dD, e and E are respectively the energies per unit volume and surface and r-p

is an extra-flux of energy due to the polarization [75].

The other symbols have the usual meaning. In addition to (8.23) we assume the following entropy inequality for the whole crystal:

dt

I

s dv +

S da)>

h-N

-

da

(8.24)

3D

D

where 9 is the absolute temperature, s and S are respectively the entropies per unit volume and surface. In order to find the local implications of (8.22) and (8.23), we begin by noting t h a t V-(ExH) E- VxH V ■(E xH) = E •V xE-H-VxE-E-VxH 2 2 ! ■0EoE -±e -E-P 0p. - -E

= - -E-D= - ED

P0P-

(8.25)

By the Gauss theorem (3.3) as well as by (8.22), (8.25), (2.33) we obtain

—

aD

V-(ExH)

n- Ex H da = I1

n-\ExH\da

dvI

222

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY • 1 td\t0E•2dv 2 = ^£ + l^le20E2}cnd
+ \c}-E J-E + \E-P JE.P 0pnP0lP 0p-jcnP0lPa sd tr.sd
D

This relation m a y be employed to transform (8.23) to the equivalent form [76]

A( j-( dt\

edv+ e dv + E

D

n- ( r -- p-h)

E da\ do\ =

da +

dD

•

P E-pdv P0E Q- p dv

D

2 + \l^0E2-P0P-E^l±e s}d
(8.27) (8.27)

E where we have introduced

ee = = e-±e e - l €0E0 2£ 2 . .

(8.28)

By comparing (8.27) and (8.24) with the general balance law (7.23) and the equivalent local conditions (7.24) we have the following system of partial differential

equations and j u m p conditions

eE = P P00E-p E - p+ + (V(VT)-p ■ T) ■+p tr(T®Vp)-V-h, + tr(r Vp) - V • hin, D ,,

s ++

09

-*-,y»>8. e2

6 b

2

2 -§ -§ -- 2HEc 2HEcn n --l [e e ++^10eE , ]s}c 0E S S ••pp c „n 0 E - - PP 0E

-n-lT-m-h}

(8.29)

= E ,

-K-

— -2/7SY f-2HSc n-lscn-^nj>S, 6t "

Sb6 ,b .,q , on 5 ,6 = = 1,.. l,...,g

-^]>5,

(8.30) (8.30)

CHAPTER 8. FERROELECTRICITY

-EiE whh-v-ui(h)i{h) -ZiEmmW =-i=-i, t

223

/ l ) -

- 8

0 on Thh,h=l,...,r, , h=l,. onT

'

(8.31)

■,r ,

-Z,si{h)i{h) ww <-9, -EiS -v
Eda Eda+Ylh +£h

f

g8rf/ dl = - 0 ,,

(8.32)

r

v.

? d/ == 00.. *dl

Sda+J2 S da+J2hh s

rh

By eliminating V ■ h and [/»] • n between (8.29) and (8.30)]^ 2 and supposing [0] — 0 across any discontinuity surface, we are led to the inequalities -(■j> - (V» + s6) s9) + + PoQE-p E ■ p++ (V • r) • p(V-T)-p+tr(T®V + tr(r Vp) P-)-!±?l>0, ^ ^ > 0, 22 (6V + S^) + 22H*c f QE -(^ - -PQE. # * cnn ++ lTP 0Es.ps}cnPSlCn [V> ++±±e o £ -P

+4!)+

++ [lT-p} T ■P]

where iP == e-0s s-8s , , ip

tfV== .E-
$* == EE-6S - BS ..

(8.32a) (8.32a)

0 , ++$ >$>(>,

224

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

8.5 Constitutive equations and thermodynamical restrictions.

Since in every Weiss domain Da the polarization has a constant modulus P0

which is independent of electric field (saturation), p and E are

independent variables. This implies that we have to specify constitutive equations relating ip, s, T to the polarization field p if we wish to have a complete description of the system we are considering. In full agreement with what is usually assumed, we propose here constitutive equations of the following kind (8.33)

A = A{p,Vp,0), where A denotes any of the quantities ip, s, r.

In agreement with the well-

known dissipation principle, the inequality (8.32a)j has to be regarded as a restriction on the constitutive equations. To apply this principle, we need to evaluate %j> by employing the constitutive equation (8.33), to substitute the resulting expression into (8.32a)j and to impose that the inequality (8.32a)j is satisfied in every process verifying the condition P-P=l.

(8.34)

We note that (8.34) implies also the following relations which we use in the sequel p - p = 0,

V p - p = 0, V P - p + V p - p = 0.

Thus we first obtain from (8.31)

(8.35)

CHAPTER 8. FERROELECTRICITY

225

• dip dtp i i,-. ~ de

dip • , <¥ T7 Vp dp

since the spatial and temporal coordinates are independent when v = 0. By substituting this expression into (8.32a)j and taking into account the constraints (8.34), (8.35) by Lagrangian multipliers A,i/,/*, we attain the inequality - ( § £ + s)6»+(P ») » + ( F00£7+V-r £ + V . r - g - A-\pp - pH-Vp) . V p ) . jp.

+{T*

Wi-a*.)*^-

_h-V9>Q 9 -

(8.36)

which, in agreement with the dissipation principle, has to be satisfied in every process, i.e. for all the values of p,p,Vp,Vp,9,V9.

From (8.36) we

obtain dip 09 '

s= d^ T

>j

P0E,

d

~ P;j

dip f dip

= dpi-{apii

+

(8.37)

; + f*jPi, \

,t

WiJ'i+Xp*

h/ i ■ve V0<
,

+ ,t p

iU

'

226

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

p E

+p +p "°°^wA£t^> -=wA£r,)-' °Xp°"Xpwhere we have put P0\

= - u

( (8.38) 8>

"

,- + A. The relation (8.38) shows t h a t it is

always possible to assume ft = 0 to evaluate E; we attain the same conclusion for r , which appears in the integral balance law through the term r - p . This term, owing to the constraint (8.35)j, is independent of p. On the other hand, it is straightforward to verify by a simple calculation that (8.32a)j is independent of p. Therefore, (8.37) 2 becomes

Uj-

av

(8.39)

~dP',j'

In order to make explicit the consequences of the surface inequality (8.37) 2 , we choose for \t, S and $ constitutive equations of the following form

tf == V(p * ( p ±±,n,n,0), 9 = t6)t where p

(8.40)

are the values of p on both sides of a domain wall. Differentiating

(8.40) we obtain

SV

<9tf St ' ~ dpd+P+

S +

P St st

. <9# ' dp~' dP-

sPSt st

, <9tf Sn.dV S6 dn 8n St ++ 89 St

+ +

where, according to (2.19) Sp - =

st -p

P ++ cCn Vp. nn n. Vp.

Taking into account these last relations as well as the constraint (8.35) 1 , we

CHAPTER 8. FERROELECTRICITY

227

have the inequality

\d9+^) ♦(■

■T +

aP+

, a* 6n I^+i

+

8n

-A + P + >

st +,

<5p + -r «5« - ( -

a* a P - •+ A p "

-

)

■

6p~ St

e0E2 + ^o * V P S - n - r «'A;,fcnfc + 2fftf] c n + * > 0

which implies for arbitrary processes

5=-%|,

(8.41)

"•r+--=7^ +A+p 3p" • j y

T»' T

a* aP-

-A

p

,

ff •ft+ ^ + 5 e° ^ + P°E*' p* - n « r 'i p i, A"* + 2^*1 c„ + * > where the Thomas derivative of n, with the help of (2.16), (2.21), (1.23), can be written as

%=h-V,-Vsn=h-Vs-a°®n,a=-a°%cnja.

(8.42)

Here ^

is the metric tensor in the curvilinear coordinates (ua) on E and a ,

a =1,2,

is the holonomic basis associated with these coordinates. The

formula (8.42) allows us to write (8.41)4 as follows

228

PHASE TRANSITIONS IN CLASSICAL FIELD THEORY

-(Uacn);a +

U + ^e0E2

+ P0ES ■ p3 - nirijPJ!

hnh

+ 2H 0 .

(8.43)

where Ua is the surface vector which is tangent to Sb and has components given by Ua = ^-aa\.

(8.44)

Finally, from (8.31) we have

Ei*Hh)"nhywh>(s-9*) = *'

(8-45)

8.6 Structure of the Weiss domains.

We now assume that the polarization field P = PQp(x) is a constant vector in each of the Weiss domains [74]. This implies that the polarization gradient vanishes throughout the union of all Weiss domains. If p is constant then the field equations in the Weiss domains and the relevant jump conditions on their walls become simply (see (8.38), (8.15)a 3 , (8.17)3) -^- + P(jVip + PoXp = 0 ,

in Da, a = l,2,.,.,m

A^ = 0 ,

(8.46)

229

CHAPTER 8. FERROELECTRICITY

[f0Vv> - P0pj ■ n = 0 ,

onSfc, 6 = 1 , 2 , . . . , , ,

(8.47)

where E = - Vtp. We remark here that the Lagrange multiplier A(x) is in general a different function for each domain. We can obtain V