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( d/
N -(p dv flc'(r)
dm
(3.10)
a
^)(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 "a/J.O. T- -- T(PS-aap>6) >
s = s(ps,aa/3,e),
K- = K(Ps:aa0,O,
Vfl),
where
(3.48)
hssV6 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 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 (
