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Thermodynamics of continua with adsorbing interface
Is J En,qng Sci Vol. 23, No. 4. pp. 449-459. Printed in Great Britain
0020-7225/X5 $3.00 + .oO 8 1985 Pergamon Press Ltd.
1985
THERMODYNAMICS ADSORBING
lstituto
Matematico
Abstract-A model of a possibility of exchange of one. A thermodynamic equilibrium of two fluids
OF CONTINUA INTERFACE-f
DONATELLA IANNECE R. Caccioppoli dell’Universiti di Napoli, X-801 34, Napoli, Italy
WITH
via Mezzocannone,
continuous system with a structured interface is proposed considering the matter between the two-dimensional continuum and the threedimensional theory for such a system is developed and successively applied to the in a pipe.
I.
INTRODUCTION
thermodynamical theory already proposed [l-5] for a continuum with an interface, the surface S which separates the two phases is described as the simplest bidimensional continuum, i.e. as a membrane. On the other hand, the hypotheses that the interface is material [ 1, 2, 31 or its mass is conserved [4, 51 assure the validity of the scheme of a membrane for S at any instant when it is initially admissible. However, in chemical thermodynamics, in particular in the capillarity phenomena or in the phase transitions of solutions, the adsorbtion of matter by the interface has to be taken into account. It is evident that in these situations the identification of the interface with a membrane does not appear compatible with the adsorbtion hypothesis. Therefore, in this paper, it is proposed to describe S as a “shell” associating to it a couple tensor M, beside a stress tensor T,. After some kinematical preliminaries (Section 2) the balance equations of the considered system are written (Section 3) whereas in Section 4 the thermodynamical restrictions on the constitutive equations from the entropy principle are derived. In particular, some classical results relative to the open systems are extended to the case of inhomogeneous ones. In Section 5, the balance of angular momentum is employed in order to obtain further physically expressive restrictions on the constitutive equations. Finally, the equilibrium problem is analysed (Section 6) and in Section 7 it is shown that the proposed theory makes compatible some equilibrium problems of two fluids in a narrow pipe which can not be described when the interface is regarded as a membrane. IN ALL
2.
KINEMATICAL
PRELIMINARIES
Let S be a moving regular surface which divides the actual configuration of a continuous system C into two parts, C, and C,. It will be supposed that all the fields associated to C in its motion may exhibit first-kind discontinuities across S and finally S will be considered as a two-dimensional continuum whose unknown motion is connected with the motion of C. If C0 and S0 denote the reference configurations of C and S respectively, the motion of (C, S) is given by the equations: x = x(X, t), r = r(R, t),
(2.1)
where x denotes the point of C which corresponds to X at the instant t; similarly r E S is the actual position of R E S,,. Denoting by (rP)$ curvilinear coordinates on the reference configuration SO, the functions
t Work partially supported by G.N.F.M. of the Italian C.N.R. $ Here and in the sequel the Latin indices vary from 1 to 3 and they are relative to linear or curvilinear coordinates in C’a and C. The Greek indices vary from I to 2 and are referred to curvilinear coordinates on S. 449
450
D. 1ANNECE ~~~~(23, t) = a,
- a,
(2.2)
will represent, at the instant t, the coefficients of the first fundamental form of S. If S is an oriented surface and n is the unit external normal to S, the second fundamental form of S is defined by
where (a comma denotes partial differentiation with respect to u”) (2.4) Moreover, we recall the well-known Gauss-Weingarten ah
= %a,
formulas
+ b,,n,
n .Q = -b:a,,
V-5)
are the Christoffel symbols, and the mean curvature of S defined as follows H = jb:.
(2.6)
The surface element do corresponding to da0 is given by da = Sdrr,
(2.7)
where 9 = (u/uOy a = det [la,,]/
a$ = metric field on SO a0 = det I]a”,iiII.
(2.8)
If i = Pa, + (r * n)n is the velocity of a point of S represented in the basis (a,, n) and we introduce the notations (bar denotes the covariant derivative with respect to the curvilinear coordinates on 5’):
it is possible to prove by (2.5) the following identities
Letf’-(r) and f’(r) be the limits of a function J’(x) (regular in (‘7 U C!) when x goes to r E S from Ci and Cz respectively. Then [.f] = ,{’ -fis the jump off’across S.
Therm~ynamics
of continua
with adsorbing
interface
451
Moreover, for every tensor T, on S we have T, = T;%, @ a0 + Tz3a, @ n + TFn 63 a, + T23n 63 n. When we introduce the unit tangent vector to S v = v,ay and we recall the conditions ar.aB = S$, from th e previous relation we derive (2.12) where (2.13) Now we consider, for any material volume I/, a general balance law in the form
where G = V fl S, F, 4?, s are regular functions in C ’ - S having first kind discontinuities together with their derivatives across S and F,, +* and j’are regular functions on S. It can be proved (see ref [ 1J) that (2.14) is equivalent to the folIowing 1ocaI conditions:
(2.15) A corresponding 3.
result holds also when (2.14) is an inequality. THE
EQUATIONS LAW
OF OF
BALANCE
AND
THE
SECOND
THERMODYNAMICS
When the continuous systems occupying the regions C1 and C, are perfect fluids, the stress tensor T can be written T = -pl
in
(3.1)
G u cz
where p is the pressure and I denotes the unit matrix. For the reasons explained in the Introduction, we describe the mechanical behavior of S by adopting for it the model of a shell. (About the definition of a shell and the range of validity of this model see [7]).) This assumption permits us to define on S the following quantities: T, = surface stress tensor, M, = contact couple tensor.
In these hypotheses the equations of baIance of mass, momentum, inequality applied to the material volume Firesult in
p($i2 f c)d?’ + =
s
sd
(3.2) energy and the entropy
p&r2 + E,)do
(h - pi)*NdS+ c3Y
sV
pbexdV+
s do
(i-1, C ti*M, + h,)avdZ +
poi.b,dr, fc
452
D. IANNECE
(3.3)
where p = pO = b = b, = E= c, = h = h, =
q = q. =
mass density, surface mass density, specific body force, specific surface force, internal energy, surface internal energy, heat flux vector, surface heat flux vector. specific entropy, surface specific entropy.
We observe that, in the absence of volume quantities, the equation (3.3)3 coincides with the equation of energy balance proposed by P. M. Naghdi in its restricted theory of shells (see [7]). Moreover, as we said in the Introduction, the principle of balance of angular momentum is regarded as a further restriction on the constitutive equations and consequently it will be taken into account in Section 5 where we point out the restrictions deriving from the dissipation principle and the bafance of angular momentum. From (3.3) and (2.1 S), , in the regularity points of the fields we deduce the weIl-known local conditions b + p div ic = 0, pii = -grad p + pb, pk = -p div ir + div h, pt$ r div h -
he grad 19 0 .
(3.4)
Moreover, when we introduce the specific free energy + = t - 0~ relations (3.4)3.4 lead to the inequality --p(3, -t se) - p div X +
h-grad B 2 0. f?
(3.5)
If the temperature is continuous on S and we recall (2.12), (2.15h, the following surface equations can be deduced from (3.3) . [PM
.
-
r)l. n + P, -i-P& = 0,
[pi @ (i - i) + pi ] . n -t p,‘i + (p, + p&)i [p(&?
+ ~)(i - i) + pi
- h] -n + p,(i* ‘i + ;,) + (tj, + p,q:)($i2 - (Tb”‘.i
[pq(i
- T$ - p,b,
- i) - h] * n + p&
= 0,
+ to)
+ M’b”. %a - D.h,$ - p,b;i
+ (p, + p,~g)~~
-- V. h, - 9
Cl
= 0,
=_0.
(3.6)
Another useful form can be given to (3.6)3 and (3.6),. In fact, by subtracting to (2.6), the scalar product of (3.6), by i, we obtain the surface balance of internal energy [p($i2
-t- t - it a i)(i
- i) + p(i
- i) - h] - n + (4;” + ~,)(p, + p,qg) + p,;,
- VT). i,, - (Mb”‘+,,
3 Here C *h, denotes the surface divergence of h, on S.
- Va h, = 0.
(3.7)
Thermodynamics
of continua with adsorbing interface
453
Moreover, by introducing the density of surface free energy $c = E,,- 8,n,, and eliminating C- h, between (3.6& and (3.7), the following other condition, which will be called surface reduced dissipation inequality, is derived
[p(+* + rC,- X * i)(i- i)C p(X -
i)]-n + (4;’ + #o)(jb + p,&)
+pij (TV +pqS CO6 -P).id
,(Y- (M$$. i),,
-
h:e,, ,( 0.
(3.8)
In particular, when the interface S is material (X’ an = i. n = X-b n), the system (3.6)1,2 (3.7) and the inequality (3.8) can be written as follows
P, + P& = 0,
[p]n + p,?-
T3: - p,b, = 0,
p,t,-[h].n-Tb”).i,,-(Mb”).ir),,-C.h,=O, p,(\l, f TL,~,) - Tb”). i,m - (M?). ri)IU- &I,,,, 5 0.
(3.9)
The system (3.9) reduces to the system (2.17) of [4] when M, = 0 and T:D = ya”“. Moreover, when the material system reduces only to the material surface S, eqns (3.9) lead to the equations proposed in [7]. 4. CONSTITUTIV~
EQUATIONS
AND
THERMODYNAMICAL
RESTRICTIONS
Since the system occupying the regions C, and Cz are perfect fluids, their constitutive equations are in the form 1c,= \t(P> e>, T = T(p, 19)= -PI, 9 = dP> fl), h = hk, 8, o,i)*
(4.1)
As regard to the constitutive equations of the interface, the hypothesis it is a shell suggests to choose a,; n$; 8, and B,, as independent variables. More precisely, when we take into account the objectivity principle, these variables will attend through the combinations a,@, 6,,, 8, and 19,~(see [7]). In addition to the abovementioned variables, we assume as independent variable the surface mass density p,,. The following observation justifies this assumption: a connection between pV and a,, exists? when the law of conservation of surface mass is valid, therefore the presence of all these quantities into the constitutive equations appears reasonable when the surface can adsorb matter. The previous considerations suggest to assume for S the following constitutive equations
(4.2) Now, the entropy principle requires the constitutive relations to be such that every solution of the thermodynamic field equations satisfies the entropy inequality. In the case we are considering, this requirement corresponds to satisfy (3.5) and (3.8) for every choice of the fields t In fact From p-do = pcoduOand (2.8) it follows pm = p-9.
454
D. IANNECE Ptx,
0, %x,
0, W,
& dr,
0, i(r,
0, fL(r? 0
(4.3)
which verify the system (3.4), (3.Q, (3.7). It is well known the method of the Lagrangian multipliers introduced by I. S. Liu in [8] in order to determine the restrictions on the constitutive equations from the entropy principle when the fields associated to the three-dimensional continuum are regular. In ref. [4] it is proved that the Liu method is also applicable when the fields present discontinuities on a surface S. Then, if associate five Lagrangian multipliers A,,, A,?,, r\, to the balance equations (3.4),~~, we can prove (See [4], Appendix I) that the inequality (3.5) is verified in every process which is a solution of (3.4)i,2.Xif and only if the following conditions are satisfied
h . grad @2 0.
(4.4)
Likewise, every surface process which is solution of (3.6),,* and (3.7), must verify the inequality (3.8). This condition, if we introduce five Lagrangian multipliers X,, Xi,, h, and recall (2.9), (2.11)s and (2.12) is equivalent to the requirement that the inequality
is satisfied for every choice of the quantities p,, e, hap, quo, B,, 19,,~,II?,,,,, qja, %vP> b X&n3 2. This leads to the fohowing conditions
Tb~rm~ynamics
455
of continua with adsorbing interface
(4.4) as well as to the remaining inequality
p H
$ + i
X2 - i
0 i
+
i! (2 -
i) . n + X,[p(k - i)] e n - A:@,, =s0.
P>
(4.7)
1
It can be observed that, owing to (4k&,
eqn f4.6f6 can be written (4.8)
Moreover, if we introduce the chemical potentials p and p,,, relative to the volume and surface respectively, with the positions? (4.9) and we take into account (4&,
the remaining inequality (4.7) becomes
IP(P+ it;: - ;)*)(; - i)] * n -
5. RESTRICTIONS
(p, + P)[p(i
ON THE CONSTITUTIVE BALANCE OF ANGULAR
As we have pointed out in with the constitutive equations a thermodynamic process of regarded as a further restriction that the constitutive equations principle of balance of angular
(4.10)
- i)] * n - lzy,,, G 0.
EQUATIONS DERIVED MOMENTUM
FROM
THE
the previous sections, the system (3.4),,2,3, (3.6)~, (3.7) (4.1), (4.2) is sufficient, at Ieast in principle, to determine C. Then, the balance of angular momentum must be on the const~tutive equations. In other words, we assume (4. l), (4.2) must satisfy in every process the following momentum
d pxXidV+lp,rXido) z (S v zz-
s ar:
x XpNdSf
sY
px X bdV+
s JSJ
(r X ?‘?I + n X Mb”))p,dl +
pbr X b,da,
(5.1)
s (r
which in view of (2.1 5)2, on S implies r,* x
T’,“’ + n,, X I@?) + n X Map = 0
(5.2)
whereas, by (2.1.5)1 and (3.4),,, the local condition obtained from (5.1) in C1 U Cz is identically satisfied. Owing to (2.4), (2.5) and (2.12), the equation (5.2) in the reference (a,, n) leads to the system t The definitions (4.9) reduce to the usual definitions of chemical potential when the processes are homogeneous. Furthermore, owing to (4.4),, the specific volume chemical potential Jocoincides with the specific Gibbs potential g = $ + %. On the contrary, by (4.8) and (4.6)~, g0 does not coincide with the surface Gibbs potential in accordance with the circumstance that the surface is not a closed system.
456
D. IANNECE Qj{ r:”
+ g f%p)
M;f”, -
= 0,
Tax r = 0.
(5.3)
Since (5.3)* coincides with (4.6)3, two of the three conditions obtained from the balance of angular momentum are satisfied if the constitutive equations verify the objectivity and dissipation principles. Moreover (5.3), is equivalent to
and then, (4.8) becomes
In concluding, the dissipation and objectivity principles together with the balance of angular momentum lead to the following considerations: (a) In the proposed theory no role can be assigned to the normal component vectors Mb”)so that we can put &f3”
”
=
0.
,
of
(5.9
(b) (4.& implies that the surface free energy is a therm~ynamical potential only for the symmet~c part of M$j. However, when we assume A&C”]= 0 (see [7]), (4.6)3,s and (5.4) allow to derive also T:@and TZ3from the thermodynamical potential $,. Now, we wish to prove that the thermodynamical relations just obtained includes as particular case the results of 141. In fact, if M(,a)= 0, from (4.6)3 we have
Tz3= 0 and (5.4) reduces to
On the other hand, if the membrane is fluid, we must have (for the sake of simplicity we omit the argument 8,): $tlc(Pb% %@) = &&AT,a) from which it foilows?
Moreover, if the conservation of mass is satisfied for the surface it is also
so that the following identities subsist
from which we have (3.4) of 143. f Take into account the identity ff-
08
= uu”‘.
Thermodynamics 6. THE
of continua with adsorbing interface EQUILlBRIUM
457
SYSTEM
By the equations (3.4),, (3.6)2 we deduce the following equilibrium system? grad p = pb [p]
=
in
C, u
T:& + T:Ab,B + p-b, - n
c2
on S.
T:& = Tz3b: + p,b,. ay
(6.1)
Neve~heIess, the phase equilib~um is characterized by a further relation which can be derived from the residual inequ~ity. In fact, the expression on the left-hand side of (4.10) regarded as a function 3 of X = fir’, &, i, f?,,,) has its maximum at X = 0. Therefore the first derivatives of 3 must be zero at X = 0 and hence
hh,, aao,ho, @,,0) = 0.
(6.2)
Owing to (6.2)3, at equilib~um there is not heat conduction on S, whereas (6.2)1,2 express the circumstance that at equilib~um the chemical potential of volume, or equivalently the specific Gibbs potential, is continuous across S. So that we find the relation P
+=
FL-=
(6.3)
PC
which is usually employed in elementary thermochemistry. Moreover, if we associate to the equations (6.1),, (6.3), , the three compatibility conditions of Gauss and Codazzi-Mainardi, we attain to a system, on the interface S, of eight differential equations in the nine unknowns p-, p+, pF, aaD, b,,. On the other hand, when we assign the boundary conditions pi =
pe
a@c ac,
on
dC”=X-CK
I
are assigned
dS
(6.4)
in the region C,, by (6. 1)1 we deduce the function p; = pl(u”) and then, at least in principle, the problem is well posed. In other words, the variance of such a system is two (the temperature and the external pressure) in accordance with the Gibbs rule. 7. EQUlLlBRIUM
OF TWO
FLUIDS
IN A NARROW
PIPE
Let C be a system constituted by two fluids S1 and 5, in a narrow pipe C,. Let us denote by uiz, gz3, u13 the surfaces which separate two by two the three media. As it is weli known, in the classical approach to the ~uilib~um problem of C (see 19, 101) the interfaces between C,, C, and CJ are regarded as membranes. In this hypothesis, the following equilibrium condition is obtained aI3 cos
-
a23
B =
fflZ
(7.1)
where (~12,~~23, aI3 denote the surface tensions on 412, ~23, ~13 respectively and moreover B is the angle between u12 and the wall of the pipe or “contact angle.” Relation (7.1) implies that the surface tensions verify the condition t The ~pre~n~tion of ei into the bases (a,, n) can be obtained by taking into account (2.12) and the Gaul-W~nga~en formulas (2.5).
458
D. 1ANNECE lw3 -
a*31 <
(7.2)
a12.
However, an equilibrium configuration is experimentally realised also for those fluids whose surface tensions do not verify the previous conditions. We recall the arguments proposed by Landau and Lifshitz in [9, pg. 4711: It must be remembered that when three different substances are in contact there may in general be an adsorbed film of each substance on the interface between the other two, and this lowers the surface tension. The resulting coefficients Q: witi certainly satisfy the inequality (1492),t and such adsorbtion wiil necessarily occur if the
inequaIit~ would not be satisfied without it. In the scheme developed in the previous sections and with the adopted notations, the problem can be expressed as it follows. On the boundary of each surface u12, ~23, (131acts a force 1.v
= Tya, + T3a3.
(7.3)
If we recall (2.13) we have T3 = T3-vu.
T. = TTJa,
(7.4)
Let -C be the curve representing the common boundary of aforesaid surfaces and N the unit vector normal to the walls of the vessel in the genera1 point of _C Moreover, let us indicate by u2 the unit tangent vector to L and u1 the unit vector normal to the plane formed by u2 and N. Finally, let us consider the three following bases: (ur , u2, N) associated to ui3, (-Ul, -u2, N) associated to (Tag, (v, u2, n) associated to cL2 where v is tangent to @Izand n is normal to ar2. From the definition of v it follows Pi3
=
-lJt
f
v23
=
-ui,
VI2
=
y;
(7.5)
moreover, if B is the angle between n and N (the contact angie), we have v = (-cos
8, 0, sin O),
n = (sin 0, 0, cos 0).
(7.6)
In order to simplify the comparison with the classical approach, we suppose isotropic the tangential part of T: T = ara”“.
(7.7)
Equations (7.3)-(7.7) imply Tt3
=
(~13~1
+
T:3N
T23 = -(~23~1 +
T&N
T12= a12vf Trln = (--ai2 cos B + T:,sin B)ui + (cyrz sin 8 + Tf2cosff)N.
(7.8)
Owing to the rigidity of the wall of the vessel, the equilib~um condition can be written t
(149.2) = (7.2).
Thermodynamics
aI3 - cyz3 - at2 cos B +
If we put cy = aI3 -
0123,
459
of continua with adsorbing interface
T:2sin 0 = 0.
(7.9)
eqn (7.9) is equivalent to the following one
which can be written
(a* Therefore the discriminant
(T:2)‘)tg2tl
+
2a,2T:2@
+
(a2
-
42)
=
0.
(7.10)
A of (7.10) results A = ~CX’(C&+ (T:2)2 - (.u2).
Equation (7. IO) admits real solution if A 2 0; i.e. if LY’5 & + (T:z)~.
(7.11)
Condition (7.11) substitutes (7.2). In concluding, if (7.2) is verified, (7.11) is even more satisfied. On the contrary, the presence of a normal component of the surface stress permits some equilibrium configurations which are forbidden by (7.2). From (4.6)3 we deduce Ta3 # 0 only if on the surface M, # 0. This implies that the region separating the phases has an appreciable thickness. Consequently, the scheme of a membrane for this region is not acceptable and the hypothesis of Landau and Lifshitz is confirmed. REFERENCES
[I ] G. P. MOECKEL, Thermodynamics of an interface. Arch. Rut. Mcch. Anal. 57, 255 (1975). [2] M. E. GURTIN and I. MURDOCH, A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal 57, 29 I (1975). [3] K. A. LINDSAY and B. STRAUGHAN, A thermodynamic viscous interface theory and associated stability problems. Arch. Rat. Mech. Anal. 71, 307 (1971). [4] A. ROMANO, Thermodynamics of a continuum with an interface and Gibbs’ rule. Ricerche di Mutemafica 31, 277 (1982). [S] P. FERGOLA and A. ROMANO, On the thermodynamics of fluid and solid phases. Ricerche di Matematicu 32, 221 (1983). [6] D. IANNECE, On the phase transitions in fluid mixtures. Meccunica 19, 182 (1984). [7] P. M. NAGHDI. The theory of shells and plates. Hand. der Phis. Via/2 (1972). [8] 1. S. LIU, Method of Lagrangian multipliers for exploitation of the entropy principle. Arch. Raf. Mech. Anal. 46, 13 1 (1972). [9] L. D. LANDAU and E. M. LIFSHITZ, Sfafisfica/ Physics. Pergamon Press, Elmsford, NY (1970). [lo] 1. N. LEVINE, Physical Chemistry, an Advanced Treatise. (Vol. 1). McGraw-Hill, New York (1971).
(Received 10 April 1984)
0020-7225/X5 $3.00 + .oO 8 1985 Pergamon Press Ltd.
1985
THERMODYNAMICS ADSORBING
lstituto
Matematico
Abstract-A model of a possibility of exchange of one. A thermodynamic equilibrium of two fluids
OF CONTINUA INTERFACE-f
DONATELLA IANNECE R. Caccioppoli dell’Universiti di Napoli, X-801 34, Napoli, Italy
WITH
via Mezzocannone,
continuous system with a structured interface is proposed considering the matter between the two-dimensional continuum and the threedimensional theory for such a system is developed and successively applied to the in a pipe.
I.
INTRODUCTION
thermodynamical theory already proposed [l-5] for a continuum with an interface, the surface S which separates the two phases is described as the simplest bidimensional continuum, i.e. as a membrane. On the other hand, the hypotheses that the interface is material [ 1, 2, 31 or its mass is conserved [4, 51 assure the validity of the scheme of a membrane for S at any instant when it is initially admissible. However, in chemical thermodynamics, in particular in the capillarity phenomena or in the phase transitions of solutions, the adsorbtion of matter by the interface has to be taken into account. It is evident that in these situations the identification of the interface with a membrane does not appear compatible with the adsorbtion hypothesis. Therefore, in this paper, it is proposed to describe S as a “shell” associating to it a couple tensor M, beside a stress tensor T,. After some kinematical preliminaries (Section 2) the balance equations of the considered system are written (Section 3) whereas in Section 4 the thermodynamical restrictions on the constitutive equations from the entropy principle are derived. In particular, some classical results relative to the open systems are extended to the case of inhomogeneous ones. In Section 5, the balance of angular momentum is employed in order to obtain further physically expressive restrictions on the constitutive equations. Finally, the equilibrium problem is analysed (Section 6) and in Section 7 it is shown that the proposed theory makes compatible some equilibrium problems of two fluids in a narrow pipe which can not be described when the interface is regarded as a membrane. IN ALL
2.
KINEMATICAL
PRELIMINARIES
Let S be a moving regular surface which divides the actual configuration of a continuous system C into two parts, C, and C,. It will be supposed that all the fields associated to C in its motion may exhibit first-kind discontinuities across S and finally S will be considered as a two-dimensional continuum whose unknown motion is connected with the motion of C. If C0 and S0 denote the reference configurations of C and S respectively, the motion of (C, S) is given by the equations: x = x(X, t), r = r(R, t),
(2.1)
where x denotes the point of C which corresponds to X at the instant t; similarly r E S is the actual position of R E S,,. Denoting by (rP)$ curvilinear coordinates on the reference configuration SO, the functions
t Work partially supported by G.N.F.M. of the Italian C.N.R. $ Here and in the sequel the Latin indices vary from 1 to 3 and they are relative to linear or curvilinear coordinates in C’a and C. The Greek indices vary from I to 2 and are referred to curvilinear coordinates on S. 449
450
D. 1ANNECE ~~~~(23, t) = a,
- a,
(2.2)
will represent, at the instant t, the coefficients of the first fundamental form of S. If S is an oriented surface and n is the unit external normal to S, the second fundamental form of S is defined by
where (a comma denotes partial differentiation with respect to u”) (2.4) Moreover, we recall the well-known Gauss-Weingarten ah
= %a,
formulas
+ b,,n,
n .Q = -b:a,,
V-5)
are the Christoffel symbols, and the mean curvature of S defined as follows H = jb:.
(2.6)
The surface element do corresponding to da0 is given by da = Sdrr,
(2.7)
where 9 = (u/uOy a = det [la,,]/
a$ = metric field on SO a0 = det I]a”,iiII.
(2.8)
If i = Pa, + (r * n)n is the velocity of a point of S represented in the basis (a,, n) and we introduce the notations (bar denotes the covariant derivative with respect to the curvilinear coordinates on 5’):
it is possible to prove by (2.5) the following identities
Letf’-(r) and f’(r) be the limits of a function J’(x) (regular in (‘7 U C!) when x goes to r E S from Ci and Cz respectively. Then [.f] = ,{’ -fis the jump off’across S.
Therm~ynamics
of continua
with adsorbing
interface
451
Moreover, for every tensor T, on S we have T, = T;%, @ a0 + Tz3a, @ n + TFn 63 a, + T23n 63 n. When we introduce the unit tangent vector to S v = v,ay and we recall the conditions ar.aB = S$, from th e previous relation we derive (2.12) where (2.13) Now we consider, for any material volume I/, a general balance law in the form
where G = V fl S, F, 4?, s are regular functions in C ’ - S having first kind discontinuities together with their derivatives across S and F,, +* and j’are regular functions on S. It can be proved (see ref [ 1J) that (2.14) is equivalent to the folIowing 1ocaI conditions:
(2.15) A corresponding 3.
result holds also when (2.14) is an inequality. THE
EQUATIONS LAW
OF OF
BALANCE
AND
THE
SECOND
THERMODYNAMICS
When the continuous systems occupying the regions C1 and C, are perfect fluids, the stress tensor T can be written T = -pl
in
(3.1)
G u cz
where p is the pressure and I denotes the unit matrix. For the reasons explained in the Introduction, we describe the mechanical behavior of S by adopting for it the model of a shell. (About the definition of a shell and the range of validity of this model see [7]).) This assumption permits us to define on S the following quantities: T, = surface stress tensor, M, = contact couple tensor.
In these hypotheses the equations of baIance of mass, momentum, inequality applied to the material volume Firesult in
p($i2 f c)d?’ + =
s
sd
(3.2) energy and the entropy
p&r2 + E,)do
(h - pi)*NdS+ c3Y
sV
pbexdV+
s do
(i-1, C ti*M, + h,)avdZ +
poi.b,dr, fc
452
D. IANNECE
(3.3)
where p = pO = b = b, = E= c, = h = h, =
q = q. =
mass density, surface mass density, specific body force, specific surface force, internal energy, surface internal energy, heat flux vector, surface heat flux vector. specific entropy, surface specific entropy.
We observe that, in the absence of volume quantities, the equation (3.3)3 coincides with the equation of energy balance proposed by P. M. Naghdi in its restricted theory of shells (see [7]). Moreover, as we said in the Introduction, the principle of balance of angular momentum is regarded as a further restriction on the constitutive equations and consequently it will be taken into account in Section 5 where we point out the restrictions deriving from the dissipation principle and the bafance of angular momentum. From (3.3) and (2.1 S), , in the regularity points of the fields we deduce the weIl-known local conditions b + p div ic = 0, pii = -grad p + pb, pk = -p div ir + div h, pt$ r div h -
he grad 19 0 .
(3.4)
Moreover, when we introduce the specific free energy + = t - 0~ relations (3.4)3.4 lead to the inequality --p(3, -t se) - p div X +
h-grad B 2 0. f?
(3.5)
If the temperature is continuous on S and we recall (2.12), (2.15h, the following surface equations can be deduced from (3.3) . [PM
.
-
r)l. n + P, -i-P& = 0,
[pi @ (i - i) + pi ] . n -t p,‘i + (p, + p&)i [p(&?
+ ~)(i - i) + pi
- h] -n + p,(i* ‘i + ;,) + (tj, + p,q:)($i2 - (Tb”‘.i
[pq(i
- T$ - p,b,
- i) - h] * n + p&
= 0,
+ to)
+ M’b”. %a - D.h,$ - p,b;i
+ (p, + p,~g)~~
-- V. h, - 9
Cl
= 0,
=_0.
(3.6)
Another useful form can be given to (3.6)3 and (3.6),. In fact, by subtracting to (2.6), the scalar product of (3.6), by i, we obtain the surface balance of internal energy [p($i2
-t- t - it a i)(i
- i) + p(i
- i) - h] - n + (4;” + ~,)(p, + p,qg) + p,;,
- VT). i,, - (Mb”‘+,,
3 Here C *h, denotes the surface divergence of h, on S.
- Va h, = 0.
(3.7)
Thermodynamics
of continua with adsorbing interface
453
Moreover, by introducing the density of surface free energy $c = E,,- 8,n,, and eliminating C- h, between (3.6& and (3.7), the following other condition, which will be called surface reduced dissipation inequality, is derived
[p(+* + rC,- X * i)(i- i)C p(X -
i)]-n + (4;’ + #o)(jb + p,&)
+pij (TV +pqS CO6 -P).id
,(Y- (M$$. i),,
-
h:e,, ,( 0.
(3.8)
In particular, when the interface S is material (X’ an = i. n = X-b n), the system (3.6)1,2 (3.7) and the inequality (3.8) can be written as follows
P, + P& = 0,
[p]n + p,?-
T3: - p,b, = 0,
p,t,-[h].n-Tb”).i,,-(Mb”).ir),,-C.h,=O, p,(\l, f TL,~,) - Tb”). i,m - (M?). ri)IU- &I,,,, 5 0.
(3.9)
The system (3.9) reduces to the system (2.17) of [4] when M, = 0 and T:D = ya”“. Moreover, when the material system reduces only to the material surface S, eqns (3.9) lead to the equations proposed in [7]. 4. CONSTITUTIV~
EQUATIONS
AND
THERMODYNAMICAL
RESTRICTIONS
Since the system occupying the regions C, and Cz are perfect fluids, their constitutive equations are in the form 1c,= \t(P> e>, T = T(p, 19)= -PI, 9 = dP> fl), h = hk, 8, o,i)*
(4.1)
As regard to the constitutive equations of the interface, the hypothesis it is a shell suggests to choose a,; n$; 8, and B,, as independent variables. More precisely, when we take into account the objectivity principle, these variables will attend through the combinations a,@, 6,,, 8, and 19,~(see [7]). In addition to the abovementioned variables, we assume as independent variable the surface mass density p,,. The following observation justifies this assumption: a connection between pV and a,, exists? when the law of conservation of surface mass is valid, therefore the presence of all these quantities into the constitutive equations appears reasonable when the surface can adsorb matter. The previous considerations suggest to assume for S the following constitutive equations
(4.2) Now, the entropy principle requires the constitutive relations to be such that every solution of the thermodynamic field equations satisfies the entropy inequality. In the case we are considering, this requirement corresponds to satisfy (3.5) and (3.8) for every choice of the fields t In fact From p-do = pcoduOand (2.8) it follows pm = p-9.
454
D. IANNECE Ptx,
0, %x,
0, W,
& dr,
0, i(r,
0, fL(r? 0
(4.3)
which verify the system (3.4), (3.Q, (3.7). It is well known the method of the Lagrangian multipliers introduced by I. S. Liu in [8] in order to determine the restrictions on the constitutive equations from the entropy principle when the fields associated to the three-dimensional continuum are regular. In ref. [4] it is proved that the Liu method is also applicable when the fields present discontinuities on a surface S. Then, if associate five Lagrangian multipliers A,,, A,?,, r\, to the balance equations (3.4),~~, we can prove (See [4], Appendix I) that the inequality (3.5) is verified in every process which is a solution of (3.4)i,2.Xif and only if the following conditions are satisfied
h . grad @2 0.
(4.4)
Likewise, every surface process which is solution of (3.6),,* and (3.7), must verify the inequality (3.8). This condition, if we introduce five Lagrangian multipliers X,, Xi,, h, and recall (2.9), (2.11)s and (2.12) is equivalent to the requirement that the inequality
is satisfied for every choice of the quantities p,, e, hap, quo, B,, 19,,~,II?,,,,, qja, %vP> b X&n3 2. This leads to the fohowing conditions
Tb~rm~ynamics
455
of continua with adsorbing interface
(4.4) as well as to the remaining inequality
p H
$ + i
X2 - i
0 i
+
i! (2 -
i) . n + X,[p(k - i)] e n - A:@,, =s0.
P>
(4.7)
1
It can be observed that, owing to (4k&,
eqn f4.6f6 can be written (4.8)
Moreover, if we introduce the chemical potentials p and p,,, relative to the volume and surface respectively, with the positions? (4.9) and we take into account (4&,
the remaining inequality (4.7) becomes
IP(P+ it;: - ;)*)(; - i)] * n -
5. RESTRICTIONS
(p, + P)[p(i
ON THE CONSTITUTIVE BALANCE OF ANGULAR
As we have pointed out in with the constitutive equations a thermodynamic process of regarded as a further restriction that the constitutive equations principle of balance of angular
(4.10)
- i)] * n - lzy,,, G 0.
EQUATIONS DERIVED MOMENTUM
FROM
THE
the previous sections, the system (3.4),,2,3, (3.6)~, (3.7) (4.1), (4.2) is sufficient, at Ieast in principle, to determine C. Then, the balance of angular momentum must be on the const~tutive equations. In other words, we assume (4. l), (4.2) must satisfy in every process the following momentum
d pxXidV+lp,rXido) z (S v zz-
s ar:
x XpNdSf
sY
px X bdV+
s JSJ
(r X ?‘?I + n X Mb”))p,dl +
pbr X b,da,
(5.1)
s (r
which in view of (2.1 5)2, on S implies r,* x
T’,“’ + n,, X I@?) + n X Map = 0
(5.2)
whereas, by (2.1.5)1 and (3.4),,, the local condition obtained from (5.1) in C1 U Cz is identically satisfied. Owing to (2.4), (2.5) and (2.12), the equation (5.2) in the reference (a,, n) leads to the system t The definitions (4.9) reduce to the usual definitions of chemical potential when the processes are homogeneous. Furthermore, owing to (4.4),, the specific volume chemical potential Jocoincides with the specific Gibbs potential g = $ + %. On the contrary, by (4.8) and (4.6)~, g0 does not coincide with the surface Gibbs potential in accordance with the circumstance that the surface is not a closed system.
456
D. IANNECE Qj{ r:”
+ g f%p)
M;f”, -
= 0,
Tax r = 0.
(5.3)
Since (5.3)* coincides with (4.6)3, two of the three conditions obtained from the balance of angular momentum are satisfied if the constitutive equations verify the objectivity and dissipation principles. Moreover (5.3), is equivalent to
and then, (4.8) becomes
In concluding, the dissipation and objectivity principles together with the balance of angular momentum lead to the following considerations: (a) In the proposed theory no role can be assigned to the normal component vectors Mb”)so that we can put &f3”
”
=
0.
,
of
(5.9
(b) (4.& implies that the surface free energy is a therm~ynamical potential only for the symmet~c part of M$j. However, when we assume A&C”]= 0 (see [7]), (4.6)3,s and (5.4) allow to derive also T:@and TZ3from the thermodynamical potential $,. Now, we wish to prove that the thermodynamical relations just obtained includes as particular case the results of 141. In fact, if M(,a)= 0, from (4.6)3 we have
Tz3= 0 and (5.4) reduces to
On the other hand, if the membrane is fluid, we must have (for the sake of simplicity we omit the argument 8,): $tlc(Pb% %@) = &&AT,a) from which it foilows?
Moreover, if the conservation of mass is satisfied for the surface it is also
so that the following identities subsist
from which we have (3.4) of 143. f Take into account the identity ff-
08
= uu”‘.
Thermodynamics 6. THE
of continua with adsorbing interface EQUILlBRIUM
457
SYSTEM
By the equations (3.4),, (3.6)2 we deduce the following equilibrium system? grad p = pb [p]
=
in
C, u
T:& + T:Ab,B + p-b, - n
c2
on S.
T:& = Tz3b: + p,b,. ay
(6.1)
Neve~heIess, the phase equilib~um is characterized by a further relation which can be derived from the residual inequ~ity. In fact, the expression on the left-hand side of (4.10) regarded as a function 3 of X = fir’, &, i, f?,,,) has its maximum at X = 0. Therefore the first derivatives of 3 must be zero at X = 0 and hence
hh,, aao,ho, @,,0) = 0.
(6.2)
Owing to (6.2)3, at equilib~um there is not heat conduction on S, whereas (6.2)1,2 express the circumstance that at equilib~um the chemical potential of volume, or equivalently the specific Gibbs potential, is continuous across S. So that we find the relation P
+=
FL-=
(6.3)
PC
which is usually employed in elementary thermochemistry. Moreover, if we associate to the equations (6.1),, (6.3), , the three compatibility conditions of Gauss and Codazzi-Mainardi, we attain to a system, on the interface S, of eight differential equations in the nine unknowns p-, p+, pF, aaD, b,,. On the other hand, when we assign the boundary conditions pi =
pe
a@c ac,
on
dC”=X-CK
I
are assigned
dS
(6.4)
in the region C,, by (6. 1)1 we deduce the function p; = pl(u”) and then, at least in principle, the problem is well posed. In other words, the variance of such a system is two (the temperature and the external pressure) in accordance with the Gibbs rule. 7. EQUlLlBRIUM
OF TWO
FLUIDS
IN A NARROW
PIPE
Let C be a system constituted by two fluids S1 and 5, in a narrow pipe C,. Let us denote by uiz, gz3, u13 the surfaces which separate two by two the three media. As it is weli known, in the classical approach to the ~uilib~um problem of C (see 19, 101) the interfaces between C,, C, and CJ are regarded as membranes. In this hypothesis, the following equilibrium condition is obtained aI3 cos
-
a23
B =
fflZ
(7.1)
where (~12,~~23, aI3 denote the surface tensions on 412, ~23, ~13 respectively and moreover B is the angle between u12 and the wall of the pipe or “contact angle.” Relation (7.1) implies that the surface tensions verify the condition t The ~pre~n~tion of ei into the bases (a,, n) can be obtained by taking into account (2.12) and the Gaul-W~nga~en formulas (2.5).
458
D. 1ANNECE lw3 -
a*31 <
(7.2)
a12.
However, an equilibrium configuration is experimentally realised also for those fluids whose surface tensions do not verify the previous conditions. We recall the arguments proposed by Landau and Lifshitz in [9, pg. 4711: It must be remembered that when three different substances are in contact there may in general be an adsorbed film of each substance on the interface between the other two, and this lowers the surface tension. The resulting coefficients Q: witi certainly satisfy the inequality (1492),t and such adsorbtion wiil necessarily occur if the
inequaIit~ would not be satisfied without it. In the scheme developed in the previous sections and with the adopted notations, the problem can be expressed as it follows. On the boundary of each surface u12, ~23, (131acts a force 1.v
= Tya, + T3a3.
(7.3)
If we recall (2.13) we have T3 = T3-vu.
T. = TTJa,
(7.4)
Let -C be the curve representing the common boundary of aforesaid surfaces and N the unit vector normal to the walls of the vessel in the genera1 point of _C Moreover, let us indicate by u2 the unit tangent vector to L and u1 the unit vector normal to the plane formed by u2 and N. Finally, let us consider the three following bases: (ur , u2, N) associated to ui3, (-Ul, -u2, N) associated to (Tag, (v, u2, n) associated to cL2 where v is tangent to @Izand n is normal to ar2. From the definition of v it follows Pi3
=
-lJt
f
v23
=
-ui,
VI2
=
y;
(7.5)
moreover, if B is the angle between n and N (the contact angie), we have v = (-cos
8, 0, sin O),
n = (sin 0, 0, cos 0).
(7.6)
In order to simplify the comparison with the classical approach, we suppose isotropic the tangential part of T: T = ara”“.
(7.7)
Equations (7.3)-(7.7) imply Tt3
=
(~13~1
+
T:3N
T23 = -(~23~1 +
T&N
T12= a12vf Trln = (--ai2 cos B + T:,sin B)ui + (cyrz sin 8 + Tf2cosff)N.
(7.8)
Owing to the rigidity of the wall of the vessel, the equilib~um condition can be written t
(149.2) = (7.2).
Thermodynamics
aI3 - cyz3 - at2 cos B +
If we put cy = aI3 -
0123,
459
of continua with adsorbing interface
T:2sin 0 = 0.
(7.9)
eqn (7.9) is equivalent to the following one
which can be written
(a* Therefore the discriminant
(T:2)‘)tg2tl
+
2a,2T:2@
+
(a2
-
42)
=
0.
(7.10)
A of (7.10) results A = ~CX’(C&+ (T:2)2 - (.u2).
Equation (7. IO) admits real solution if A 2 0; i.e. if LY’5 & + (T:z)~.
(7.11)
Condition (7.11) substitutes (7.2). In concluding, if (7.2) is verified, (7.11) is even more satisfied. On the contrary, the presence of a normal component of the surface stress permits some equilibrium configurations which are forbidden by (7.2). From (4.6)3 we deduce Ta3 # 0 only if on the surface M, # 0. This implies that the region separating the phases has an appreciable thickness. Consequently, the scheme of a membrane for this region is not acceptable and the hypothesis of Landau and Lifshitz is confirmed. REFERENCES
[I ] G. P. MOECKEL, Thermodynamics of an interface. Arch. Rut. Mcch. Anal. 57, 255 (1975). [2] M. E. GURTIN and I. MURDOCH, A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal 57, 29 I (1975). [3] K. A. LINDSAY and B. STRAUGHAN, A thermodynamic viscous interface theory and associated stability problems. Arch. Rat. Mech. Anal. 71, 307 (1971). [4] A. ROMANO, Thermodynamics of a continuum with an interface and Gibbs’ rule. Ricerche di Mutemafica 31, 277 (1982). [S] P. FERGOLA and A. ROMANO, On the thermodynamics of fluid and solid phases. Ricerche di Matematicu 32, 221 (1983). [6] D. IANNECE, On the phase transitions in fluid mixtures. Meccunica 19, 182 (1984). [7] P. M. NAGHDI. The theory of shells and plates. Hand. der Phis. Via/2 (1972). [8] 1. S. LIU, Method of Lagrangian multipliers for exploitation of the entropy principle. Arch. Raf. Mech. Anal. 46, 13 1 (1972). [9] L. D. LANDAU and E. M. LIFSHITZ, Sfafisfica/ Physics. Pergamon Press, Elmsford, NY (1970). [lo] 1. N. LEVINE, Physical Chemistry, an Advanced Treatise. (Vol. 1). McGraw-Hill, New York (1971).
(Received 10 April 1984)