- Email: [email protected]
A generalized Gibbsian surface
SURFACE
SCIENCE
4 (1966) 125-140 0 North-Holland
A GENERALIZED
GIBBSIAN
Publishing
Co., Amsterdam
SURFACE
R. GHEZ Laboratoire de Physique Technique, Ecole Polytechnique de I’UniversitP de Lausanne. Lausanne,
Switzerland
Received 5 July 1965 In the case of equilibrium, the thermodynamical approach to the theory of capillarity and to surfaces of discontinuity in general is that of Gibbs. Using the methods of irreversible thermodynamics, the procedure is extended to the case of non-equilibrium and general balance laws are sought at the surfaces of discontinuity. In particular, this somewhat generalizes several known kinetic equations of moving surfaces.
1. Introduction In the study of surface phenomena, it may often be advantageous to construct a Gibbsian surface (ref. 1, p. 219). The procedure reduces the inhomogeneous transition layer separating two bulk phases to a geometrical idealization called the dividing surface. This surface is then endowed with : a) intensive physical quantities depending on the energetic organization, either defined a priori, or obtainable, in principle, through statistical mechanics (i.e. temperature, stress, chemical potentials etc.). b) extensive physical quantities arising from a global balance law (i.e. energy, entropy, mass etc.). It is then possible to give a precise meaning to the associated intensive quantity per unit area. Gibbs’ procedure, however, has three limitations : 1) the thermodynamic functions on the surface are taken as being homogeneous. 2) the stress defined on the surface is considered homogeneous and isotropic. This limitation is not severe and has been dropped recentlysps). 3) the model is valid at every instant of time, but nowhere does the state of motion of the surface appear. The model is seen to be inadequate for states of non-equilibrium, for instance, when one studies relaxation phenomena. It is the purpose of this paper to show that, using the methods of irreversible thermodynamics, a thermodynamics of non-stationary interfaces can be built. In fact, by specific reasoning, Scriven4) and Slatterys) have recently derived balance laws for mass and momentum. 125
126
R. GHEZ
2. Kinetics of moving surfaces Much work has been devoted to the subject (e.g. refs. 6 and 7, p. 498), but because of differences in notation and points of view, a brief outline is in order. A moving surface can be described by the mapping x = $(u’u2t)
3 $&‘z?),
(1)
whereby the U”S are the curvilinear coordinates in an arbitrary admissible parametrization, and t the time *. The image points form a one parameter set of surfaces C, and the mapping can be visualized as that of cylinders (u’u21) into R3 (see fig. 1).
Fig. 1.
The velocity
Description
of a moving surface.
of a surface point (u”) is w=
44,
(2)
and it can be seen (ref. 7, p. 499) that the normal component w. n, where n is the unit normal to C,, is independant of the coordinates U’ chosen. This may be why most authors choose a parametrization in which w is always parallel to n. However, in practical problems this may not always be convenient. The basis vectors are or in spatial
e, = GN, 9
(3)
XL = LJorl$Qt.
(4)
components
If gik is the metric tensor in R3, then %
=
i
k
e; ep = gikx,xg
(3
* Latin indices correspond to space-tensors and have a range of 1, 2, 3; Greek indices correspond to surface-tensors and have a range of 1, 2; a comma denotes the covariant derivative and the summation convention holds throughout.
A GENERALIZED
GIBBSIAN
127
SURFACE
is that of& (ref. 8, p, 195). From (1) and from the fact that R3 is Euclidean, we have Gctlu = &WI,. At this point,
let us mention
(6)
the formulas
b,p ,
edl = where b,, are the coefficients and
(7)
of the second fundamental n ,a =-
form (ref. 8, p. 200),
a@*bBar e 2.
(8)
(ref. 8, p. 202), where aa’alD =a;. Also rijkr riijk and rmsr, raflr are Christoffel’s symbols respectively based on gik and asp. Now every space vector X, defined on the surface, can be uniquely resolved according to the triad (e,, rz) X = nX’“’ + eOL _I@‘)’. For example
(appendix,
equation
(9)
(Al))
dram8= e; w,@ + eO. w,, = 2 (w{$,
- b,gw(“)),
(10)
where n$,,=+(w$+ w;,‘,) is the symmetrical part of w$. The second equality is but a straightforward application of formulas (9), (7) and (8). We now derive the often used expression (d,a*)/a”
z G = +(d,a)/a
= +a’8Zta,B = wy$jQ- 2Hw'"'
,
Uli
where a is the determinant la,,/ and 2H the mean curvature aaxSborS. The same formula was obtained earlier3) by means of virtual displacements. Thus far the surfaces considered were purely geometrical. We will now consider a material particle (e) on Z, whose motion is given by a= = @(5’c*t) and whose velocity
relative
to the basis vectors
(12) e, is
V” = a&1<. Its absolute
velocity
(13)
in R3 is
U = d,+ = &+I, + a,$_l,&cp”is = w + eorVa, where d, means
differentiation
at (e)
constant,
(14)
i.e. the material (total, in-
trinsic) derivative.
However independently V” and w are given, it is obvious ical problem will link them together.
that any dynam-
R. GHEZ
128
3. Stress and strain
Stress in a surface may be described elements
by internal
ds in C, (refs. 2, 3) and according
forces c ds acting on line
to (9) we can write
c = no@) + e,cr(f)a. Also the equilibrium
of an infinitesimal $0
triangle
= @)a
a(OP = @Pa
(15) yields3)
& lla
(16) (17)
where ,u~ is the unit normal to the line element and tangent to C, (see fig. 2). From (16) we observe that normal components of CJ are not necessarily absent.
Fig. 2.
Equation
The stress tensor defined on a surface.
(15) can then be expressed
as
J = (niO(n)a + X;6(f)Pa)P, s JaPL, = O’aC(kXa= &lCk.
(18)
It is also convenient to adopt the following sign convention for C. If 1, is the unit tangent to ds, then the plane (A, n) divides space into two regions: that for which p is > 0 and that for which p is < 0. We will count 0 negative when it represents forces acting from side p ~0 on side p >O. We may define the strain rate as dik
and the surface divergence
=
U(i.k)
(19)
(ref. 8, p. 206) of U is then
aa8xhxiUi,k = a”x~x~ dik = aaPx~gikU,~ = V,: + G .
(20)
The first equality in (20) results from the symmetry of aap, the second from the chain rule, and the third from an application of formulas (9), (7), (8) and
A GENERALIZED
GIBBSIAN
SURFACE
129
definition (11). Therefore d,x;x;
= V&)
+
wj& - bbBw(“) = da, - b,gw(n).
(21)
Comparing (21) and (lo), we see that di,x&f = +at~ab+ Vca,g)s
(2-a
This is just the expression obtained by Scrivena), and is consistent with Oldroyd’s Q)convective derivative. For future reference let us put
where the brackets denote the antisymmetrical
part.
4. The general balance equation To derive balance laws it will be necessary to consider closed lines 3, on Z;, fixed in the (Y) system, and to avoid superfluous symbols, by&, we will mean the whole surface at time t as well as the domain interior to -Y,. Let us consider surface integrals (A for area!) of the form
This is consistent with the fact that the quantity qA may depend on the us’s, for example, through +( u’u’~) (size effect or effect of position in space) or through n(u’u’t) (orientation effect, i.e. functions of Miller indices on a crystal surface), Referring to fig. 1 and to the transformation formula (32), it is seen that we are integrating over a fixed domain D in the (r’t2) plane and can therefore write
where .J is the Jacobian of the mapping (12). Then n
d,QA =
(a+ [.I(d,gA + dtgA d, IJI + qA IJI d, ti+) dt’ dg2.
(26)
J D
But it is easily seen that d,J = Jt?,V
(27)
130
R. GblEZ
Fig. 3.
The domains implied in the derivation of the general balance equation.
and it is known
(ref. 8, p. 182) that aa+=aJP a
(28)
S/J’
We also have d,d* = 8ta31y + V”d,a”. Using the definition of the covariant (26) finally becomes d,Q” =
1 &
derivative
(2%
and definition
(1 l), equation
(d,qA + &+(V$ + G)) dA .
(30)
As seen in fig. 3, every arbitrary closed line LG?~ will sweep out a surface S. In 9 interior to S we may formulate the fundamental hypotheses 10): a) Every extensive quantity Q in 9 can be written down as the sum (V for volume!) q”dV’+
Q(r)=_/ B b) The ordinary
balance
J qAdA.
(31)
&
equationI’)
atqY= -
div&
+ c?“
holds for points in 9 arbitrarily close to C,. Consistent with the general procedure in mechanics let us give the causes of any variation in Q: J,‘ot = total bulk flux, CT” = bulk source 1 JAa = surface flux 9 cTA = surface source.
(32)
of continuous
media,
A GENERALIZED
GIBBSIAN
131
SURFACE
Therefore d,e=So‘,qVdV+~~d,q”+q”(V:+G)~dA=
=~~~~~d~~g~~a~~d,,~~~d~+~~~d~. s
0
SPt
(33) &
The second term on the right hand side has apfgs sign because p is directed i~;~~rd~~,and may be transformed according to the divergence theorem for surfaces (ref. 8, p. 189) JAap, ds = i
-LPI
s &
.
(34)
The first term may be transformed according to the divergence theorem for domains including surfaces of discontinuity (ref. 8, p. 257) (35) By [q”] we will denote the discontinuity q;-q: at C, of any bulk field q’, where the subscripts 1 and 2 refer to the sides 1 and 2 of&, and by convention n is directed from side 1 to side 2. Substituting (34) and (35) into (33), recalling (32) and the fact that 5?, is arbitrary, we obtain d,qA=-qA(~;+G)-J,~-n*[J,V,,]+crA.
(36)
Equation (36) is similar in form to the “‘material equivalent” of (32) except that the term -II. [JL,] represents a source term resutting from the interaction with the adjacent phases, and the term -qAG means that C, is changing in extent. We could then call aA an ~~tri~~jc source because it stems from processes within Z;. Since the lines _fZtare material, there is no convective part to the flux JA”, but, calling u the velocity of a particle in the adjacent bulk phases, it is obvious that C, is not a material surface in the usual acceptation, for the fields w and u need not be equal. Then u -w is the velocity of a particle in R3 relative to Z;, and the total bulk flux may be split into convective and nonconvective parts Jy,, = q”(a - w) + JV. (37) Thus, we get for our final expression
132
R. GHEZ
Regarding our hypotheses (31) and (32), it may be said that they are but natural extensions of those formulated by Gibbs. There is still an arbitrariness in the choice of the dividing surface &,, and this might lead to certain conditions on the specific balance laws to be deduced in the following paragraphs. From now on, for brevity, systems.
we will restrict
ourselves
to one-component
5. Mass balance The total mass adsorbed MA=
on Z, can be written
as
pAdA.
s
(39)
&
Thus 4* = pA
(40)
and aA =
4"=p",
J*” =J”=O.
(41)
Then d,p* = - pA(G + v;) - nfp”(u
- w)].
(42)
The last term on the right hand side represents an adsorption reaction (ref. 12, p. 29), and could be written down in terms of the appropriate reaction rate parameters. Equation (42) can now be used to simplify (38) if all densities qA or q” are thought of as expressions of the forms e*q* and e”q” (q* and q” are now specific quantities, i.e. quantities per unit mass). Equation (38) then becomes pAdtqA = - n.[p”(u
- w)(q” - qA)] - n.[J”]
in which the explicit dependance
on the motion
- J,;“+
cA,
(43)
in C, and the latter’s change
in extent has disappeared. 6. Momentum and angular momentum balance The total momentum
is PA=
s &
pAUdA.
(44)
Thus qA = u
(45)
and q”=u,
J”=
JA”=O.
(46)
A GENERALIZED
The integrated following
source resulting
three terms
GIBBSIAN
133
SURFACE
from all the applied
forces is the sum of the
s
CtldA,
(47)
&
where t is the internal tensor
force per unit area expressible
in terms of the pressure
t’ = r’kn, ,
(48)
-
(49)
f
ads
pAf dA,
(50)
wheref are mass forces, for instance, due to external fields. Applying the divergence theorem (34) to (49) and recalling definition (18), equation (43) becomes pAdJJi = - n.[pV(u - w)(& - Vi)] + [t’] + of; + pAfi. iw In considering angular momentum, we will neglect any internal rotational motion (ref. 11, p. 305), and shall therefore write the balance law for LA =
s &
(x x pAU)dA.
(52)
Thus @ = xiuk _ XkUi
(53)
and 4V = xink _ XkOi,
J”=
JAa=O.
(54)
In the source term, besides the moments of the forces (47), (49) and (50), we may consider distributed surface couples c, as for example in an electric double layer. Substituting in (43), using (14) (51) and definition (18) we get a condition on the antisymmetry of the surface stress tensor @kl
zg
ik _
gki
=
$k.
(55)
7. Energy balance The total surface kinetic
energy is evidently (56)
134
from which (57) We then calculate pAdtet
= UipAd,Ui = - n*[pV(u
-
w)(z? -
Ui) ui] +
+ vi [t’] + uipy
+ @“a.
(58)
The last equality results from (51). Let us now transform the last term on the right UiOfZ= ( UiOi”),. - Ui,&Ti~= ( UiOim),a- t.Q’k = ( Qia),a
- #k) dik - &‘kQik
= (Qr’“),,
- Jik) dik - c. sz .
(5%
We must put (58) into the form (43) in which (ref. 11, p. 16) 4” = i”‘Ui
and
J”k zzz- t’kr$.
(60)
We then get J”“(et) cA(et)
= - Ui~ia,
= - acik)dik - c.sZ + pAU.f + [t(U
- u>J + n*[p”(u
(61) + - w):(U
- u)‘].
If we now suppose the existence of a potential energy function of time, f = - grad e,“(x),
(62)
independant (63)
it follows from (1) and (14) that pAd,et
= pAUidieg = - pACJ. f .
(64)
The total surface energy can be written EA =
pAeAdA. s Tit
(65)
Consistent with the First Principle, oA (e”) =O, and the system is subject to a superficial heat flux J,“” and a flux due to the mechanical work --be U ds (see sign convention fig. 2). Therefore JA”(eA) = - oiaUi + .I,““.
(66)
J,“”,
(67)
But we also have Jvk(ev)
= - likui +
where J,” stands for the bulk heat flux (ref. 11, p. 18). Thus we finally obtain pAdteA = - n.[p’(u
- w)(e”
- eA)] + [t-u - J,“en]
+ (cPUi - JyA”),,.
(68)
A GENERALIZED
Now the internal
GIBBSIAN
135
SURFACE
energy is defined as the difference U
Its bulk flux is known
A
A
=e
f -ep.
-e
A
(69)
to be (ref. 11 p. 18) J4y.
Substituting (69) into recalling (70), we get
(68), using
results
(70) (64), (61), (62), (43), (59) and
JA” (u”) = J4”“, &(U”)
= - [t.(U
- u)] + a’%,
+ a.[p’(u
- w){+(U
(71) + c*rl2 +
- u)’ + e; - e:}].
(72)
into an isotropic part For future reference, let ~7~‘~)be decomposed (appendix, equation (Bl)) a&x~x~ where c is a scalar, and a non-diagonal part eik. Expression (20) shows that a’ik’ dik = ~ ( V,~ + G) + ~ik dik .
(73)
Finally, as pointed out by Meixnerra), when Zt has no qA attributes, the equations of sections 4 to 7 reduce to discontinuity conditions across shocks and phase fronts 15). 8. Entropy balance To claim any resemblance with thermodynamics, some general physical assumption must now be made in order to derive the entropy balance law. We shall assume that local equilibrium exists in every surface mass particle, whence the surface can be considered as a locally autonomous system (see Prigogine’s terminology, ref. 12, p. 43). This assumption has already been discussed by Herring a) with a view to deriving boundary conditions for the chemical potentials. Mathematically, this means that uA can depend only on s A, the specific surface entropy, and on lIeA, and not on bulk state functions:
UA= uA(sA,l/PA). We now define the temperature
and stress distributions
TA2g, *
auA
d =a(llp”)
(74) within
the particle (75) (76)
136
R. GHEZ
It will also be assumed that CT*can be identified with the isotropic part cr of the stress tensor, although this is by no means obvious. From Euler’s theorem for homogeneous functions, and differentiating (74), we get respectively and pAdt uA =
au*
pA gd,sA
f -%!a(li#)
d,(l,‘pA)
d,pA = pATAdrsA - 0~1 P
(78)
Using the balance laws for mass and internal energy, as well as equation (77) and the corresponding bulk expression uv = TVsV - p/pv )
(7%
we may cast (78) into the form (43), remembering, however, that the bulk entropy flux is of the form JF/T” (ref. 11, p. 24). We then obtain the entropy
flux
= JF/TA,
(80)
The various balance laws just derived must be considered as part of the boundary conditions to be satisfied by the ordinary balance laws in the adjacent phases. There is, of course, no hope of solving most problems in closed form. 9. Some eiementary consequences 9.1. ABSENCEOF POLARIZATION Referring to equation (55) and definition (18), c =0 implies #Ia (nixt _ nkxt) = (e@)/J~_ &)~b)&$,
i82)
from which it is easily seen that the normal component o(“) is necessarily absent and that the tangential part asp is symmetric.
A GENERALIZED
GIBBSIAN
137
SURFACE
9.2. STATIC MECHANICALEQUILIBRIUM Neglecting
mass forces, equation
(51) reduces to
[t’] + Of = 0,
(83)
or [t’] + ni($)a
+ CFYQ,)
In the case when the t’s are ordinary [p] =
+ xfi(c$‘“” - (T(%Pb~J pressures,
= 0.
(84)
(84) reduces to
a:$)=+ c7(‘%aa.
(85)
Equation (85) has already been discussed briefly in ref. 3, and is found to yield the Laplace law under the assumption of isotropic tangential stress. 9.3. NORMAL AND TANGENTIAL PARTS OF THE MOMENTUMBALANCE All the balance laws, except that for momentum, tives of scalar quantities for which we have
involve total time deriva-
d,qA = iYtqAlu + aaqAlt V”.
(86)
But, to evaluate the acceleration d,U, we must be wary of the fact that tensors will generally not yield tensors when submitted to operations of type (86). Again, as we shall presently see, if the coordinates of R3 are curvilinear, in equation (86), it does not suffice to substitute the covariant derivative for the ordinary one. Let A: be a hybrid tensor, transforming as a contravariant space vector and as a covariant surface vector. Recalling (13) and (14), it can be shown (ref. 8, p. 199) that d,Ai is still a tensor with the following definition d,A; = atA& + I-;,,AJW”-
l-,$A$?
(87)
But atA& = a,A& + Pa,A;
(88)
and U=w+V”e,.
(89)
Therefore d,A; = a,A& + V@A;,, + T&A,Pwq. This formula using (6),
may be generalized
(90)
to hybrid tensors of any order. In particular,
d,xa = amwi + rpiqx:wq + VP2 a,8 = Wfol+ VB&
(91)
and (A3) d,n” = - x~a”Bg,qnPw~z + V"nf, .
(92)
R.
138
Formula
(91) was first derived
Truesdell7).
Straightforward
GHEZ
by Slatterys),
calculation
(92) in a restricted
d,U’ = n’ {&w’“’ + Va8,w(“) + (wCt)’ + V)(&w(“)
+ !,Q(w(‘)~ + VP))> +
+ x; ((w(‘jm + V”)(w!$)P - 2U%A,W(“)) + a, (w(f)8 + P) + V” (w(‘)B + P),,
form by
then leads to
- W(“)CPalW(“)) .
+ (93)
If the direction of the vector fields, u, t,f, is known in the neighborhood of Z,, we can resolve equation (51) in the normal direction, giving the normal motion, and in a tangential part, giving the motion in the tangent plane. It is interesting to notice that the tangential part of d,U (see 93) contains functions of the normal component w(“) that are not immediately obvious. Also, equation (93) can lead to helpful linearizations. 9.4. STRESS AND ENERGY The internal
energy balance
pAd,UA = - n+?(u
- w)(u’
law was seen to be - u”)] - n$l,‘]
- J1”ya +
- [t(U - u)] + .(l’a, + G) + @dik + C.&J + - n*[pV(u - w){*(U - u)’ + e; - e$}].
(94)
This is the most general equation relating the surface energy to the surface stress. Although CJ and eauA are both measured in dyne/cm, they must certainly not be confused. In the static case, this distinction has already been made clear by Herring2) (see also ref. 16). 9.5. SURFACERELAXATION(~.~.THERMALETCH) By specific reasoning, Mullinsis) has deduced the equations of motion of a cylindrical surface resulting from each of the four usual elementary processes, viscous flow, evaporation - condensation, volume diffusion and surface diffusion. These equations can readily be deduced from the mass balance law (42). At the end of section 4 it was mentioned that some hypotheses must be made, in order to locate the dividing surface and to pay some consideration to its atomic structure. 1) Let L, be an arbitrary closed line on Zt fixed in the (u”) system. Now eA can be related to the number of atoms that make up the surface. For example, one may think of all atoms in phase (l), having a nearest neighbor relationship with those adjacent to phase (2), as belonging to the surface. In L, one must always count the same number of atoms, therefore the total mass on C,, interior to L, must be constant MA =
pAdA = const. s &
(95)
A GENERALIZED
Referring
to formula
(24’~~) plane)
GIBBSIAN
139
SURFACE
(1 l), (we are now integrating
on a fixed domain
in the
we get J,p*+p*G=O.
(96)
2) We will also assume that aapA = 0,
(97)
which means that each atom of the surface occupies is no orientation effect. Equation (42) then becomes pAv;“, + nfp”(u
the same area, i.e. there
- Iv)] = 0.
(98)
With Mullins’ further assumptions (cylinder-like surface, small slope approximation), (98) is seen to reduce to his equations of motion. The case has been illustrated in ref. 10 for surface diffusion. Acknowledgments I wish to thank Prof. J.-P. Bore1 for the interest he has shown in this work. I am also grateful to Prof. J. Meixner for a very stimulating discussion. Appendix A CALCULATION OF ataaS/u AND a,n/, a) Differentiating
(5), we get dtaorS= &$,g,
+ g,X&xi
+ gikx+?,xi.
But, (ref. 8, p. 152) 4Sik
=
wpapSik
=
WP(rikp
+
rkip)
and with (7) araap = gik (x-~w~~ + $wfa). b) Differentiating
the obvious
(AlI
formula g&rink = 1 )
we get
wpn’nkz;,, Likewise,
from g&.x:
+ gik&,n
= 0.
= 0, we get
W~rikpniX: +
gikx,ka,ni + gikniw;z = 0.
But giknk a,ni and as”gikxE a,n” are just the contravariant according to the triad (e,, n). Therefore a,nq = - nqwPflinkr. ,kp - x@“(gi,&?~
components
+ wp&&.,).
of B,n
140
R. GHEZ
For the last term on the right hand side, let us use the formula a@+!
(ref. 8, p. 197)
= @ - nqnk.
(A21
We then get
the last equality and (8).
resulting,
as usual,
from decomposition
(9), formulas
( 7)
Appendix B ISOTROPIC
TANGENTIAL
STRESS
We assume that a’“‘=0
and &)=(TP.
From (18) we get
(Bl) References 1) J. W. Gibbs, Scientific Papers, vol. I (Dover, New York). 2) C. Herring, The Physics ofPowder Metallurgy, ed. W. Kingston (McGraw-Hill, New York, 1951). 3) R. Ghez and F. Piuz, Physics Letters 4 (1963) 275. 4) L. E. &riven, Chem. Eng. Sci. 12 (1960) 98. 5) J. C. Slattery, Chem. Eng. Sci. 19 (1964) 379. 6) T. Y. Thomas, J. Math. Mech. 6 (1957) 311. 7) C. Truesdell ef a/., The Classical Field Theories, in: Handbuch der Physik m/l (Springer, Berlin, 1960). 8) A. J. McConnel, Applications of Tensor Analysis (Dover, New York). 9) J. G. Oldroyd, Proc. Roy. Sot. A200 (1950) 523. 10) R. Ghez, Helv. Phys. Acta 37 (1964) 619. 11) S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing Co., Amsterdam, 1962). 12) R. Defay et I. Prigogine, Tension Superficielle et Adsorption (Desoer, Liege, 1951). 13) W. W. Mullins, J. Appl. Phys. 28 (1957) 333; 30 (1959) 77. 14) J. Meixner, private communication 15) A. Jeffrey, Z. Angew. Math. Phys. 15 (1964) 68. 16) R. Ghez et F. Piuz, Compt. Rend. Acad. Sci. (Paris) 257 (1963) 2795.
SCIENCE
4 (1966) 125-140 0 North-Holland
A GENERALIZED
GIBBSIAN
Publishing
Co., Amsterdam
SURFACE
R. GHEZ Laboratoire de Physique Technique, Ecole Polytechnique de I’UniversitP de Lausanne. Lausanne,
Switzerland
Received 5 July 1965 In the case of equilibrium, the thermodynamical approach to the theory of capillarity and to surfaces of discontinuity in general is that of Gibbs. Using the methods of irreversible thermodynamics, the procedure is extended to the case of non-equilibrium and general balance laws are sought at the surfaces of discontinuity. In particular, this somewhat generalizes several known kinetic equations of moving surfaces.
1. Introduction In the study of surface phenomena, it may often be advantageous to construct a Gibbsian surface (ref. 1, p. 219). The procedure reduces the inhomogeneous transition layer separating two bulk phases to a geometrical idealization called the dividing surface. This surface is then endowed with : a) intensive physical quantities depending on the energetic organization, either defined a priori, or obtainable, in principle, through statistical mechanics (i.e. temperature, stress, chemical potentials etc.). b) extensive physical quantities arising from a global balance law (i.e. energy, entropy, mass etc.). It is then possible to give a precise meaning to the associated intensive quantity per unit area. Gibbs’ procedure, however, has three limitations : 1) the thermodynamic functions on the surface are taken as being homogeneous. 2) the stress defined on the surface is considered homogeneous and isotropic. This limitation is not severe and has been dropped recentlysps). 3) the model is valid at every instant of time, but nowhere does the state of motion of the surface appear. The model is seen to be inadequate for states of non-equilibrium, for instance, when one studies relaxation phenomena. It is the purpose of this paper to show that, using the methods of irreversible thermodynamics, a thermodynamics of non-stationary interfaces can be built. In fact, by specific reasoning, Scriven4) and Slatterys) have recently derived balance laws for mass and momentum. 125
126
R. GHEZ
2. Kinetics of moving surfaces Much work has been devoted to the subject (e.g. refs. 6 and 7, p. 498), but because of differences in notation and points of view, a brief outline is in order. A moving surface can be described by the mapping x = $(u’u2t)
3 $&‘z?),
(1)
whereby the U”S are the curvilinear coordinates in an arbitrary admissible parametrization, and t the time *. The image points form a one parameter set of surfaces C, and the mapping can be visualized as that of cylinders (u’u21) into R3 (see fig. 1).
Fig. 1.
The velocity
Description
of a moving surface.
of a surface point (u”) is w=
44,
(2)
and it can be seen (ref. 7, p. 499) that the normal component w. n, where n is the unit normal to C,, is independant of the coordinates U’ chosen. This may be why most authors choose a parametrization in which w is always parallel to n. However, in practical problems this may not always be convenient. The basis vectors are or in spatial
e, = GN, 9
(3)
XL = LJorl$Qt.
(4)
components
If gik is the metric tensor in R3, then %
=
i
k
e; ep = gikx,xg
(3
* Latin indices correspond to space-tensors and have a range of 1, 2, 3; Greek indices correspond to surface-tensors and have a range of 1, 2; a comma denotes the covariant derivative and the summation convention holds throughout.
A GENERALIZED
GIBBSIAN
127
SURFACE
is that of& (ref. 8, p, 195). From (1) and from the fact that R3 is Euclidean, we have Gctlu = &WI,. At this point,
let us mention
(6)
the formulas
b,p ,
edl = where b,, are the coefficients and
(7)
of the second fundamental n ,a =-
form (ref. 8, p. 200),
a@*bBar e 2.
(8)
(ref. 8, p. 202), where aa’alD =a;. Also rijkr riijk and rmsr, raflr are Christoffel’s symbols respectively based on gik and asp. Now every space vector X, defined on the surface, can be uniquely resolved according to the triad (e,, rz) X = nX’“’ + eOL _I@‘)’. For example
(appendix,
equation
(9)
(Al))
dram8= e; w,@ + eO. w,, = 2 (w{$,
- b,gw(“)),
(10)
where n$,,=+(w$+ w;,‘,) is the symmetrical part of w$. The second equality is but a straightforward application of formulas (9), (7) and (8). We now derive the often used expression (d,a*)/a”
z G = +(d,a)/a
= +a’8Zta,B = wy$jQ- 2Hw'"'
,
Uli
where a is the determinant la,,/ and 2H the mean curvature aaxSborS. The same formula was obtained earlier3) by means of virtual displacements. Thus far the surfaces considered were purely geometrical. We will now consider a material particle (e) on Z, whose motion is given by a= = @(5’c*t) and whose velocity
relative
to the basis vectors
(12) e, is
V” = a&1<. Its absolute
velocity
(13)
in R3 is
U = d,+ = &+I, + a,$_l,&cp”is = w + eorVa, where d, means
differentiation
at (e)
constant,
(14)
i.e. the material (total, in-
trinsic) derivative.
However independently V” and w are given, it is obvious ical problem will link them together.
that any dynam-
R. GHEZ
128
3. Stress and strain
Stress in a surface may be described elements
by internal
ds in C, (refs. 2, 3) and according
forces c ds acting on line
to (9) we can write
c = no@) + e,cr(f)a. Also the equilibrium
of an infinitesimal $0
triangle
= @)a
a(OP = @Pa
(15) yields3)
& lla
(16) (17)
where ,u~ is the unit normal to the line element and tangent to C, (see fig. 2). From (16) we observe that normal components of CJ are not necessarily absent.
Fig. 2.
Equation
The stress tensor defined on a surface.
(15) can then be expressed
as
J = (niO(n)a + X;6(f)Pa)P, s JaPL, = O’aC(kXa= &lCk.
(18)
It is also convenient to adopt the following sign convention for C. If 1, is the unit tangent to ds, then the plane (A, n) divides space into two regions: that for which p is > 0 and that for which p is < 0. We will count 0 negative when it represents forces acting from side p ~0 on side p >O. We may define the strain rate as dik
and the surface divergence
=
U(i.k)
(19)
(ref. 8, p. 206) of U is then
aa8xhxiUi,k = a”x~x~ dik = aaPx~gikU,~ = V,: + G .
(20)
The first equality in (20) results from the symmetry of aap, the second from the chain rule, and the third from an application of formulas (9), (7), (8) and
A GENERALIZED
GIBBSIAN
SURFACE
129
definition (11). Therefore d,x;x;
= V&)
+
wj& - bbBw(“) = da, - b,gw(n).
(21)
Comparing (21) and (lo), we see that di,x&f = +at~ab+ Vca,g)s
(2-a
This is just the expression obtained by Scrivena), and is consistent with Oldroyd’s Q)convective derivative. For future reference let us put
where the brackets denote the antisymmetrical
part.
4. The general balance equation To derive balance laws it will be necessary to consider closed lines 3, on Z;, fixed in the (Y) system, and to avoid superfluous symbols, by&, we will mean the whole surface at time t as well as the domain interior to -Y,. Let us consider surface integrals (A for area!) of the form
This is consistent with the fact that the quantity qA may depend on the us’s, for example, through +( u’u’~) (size effect or effect of position in space) or through n(u’u’t) (orientation effect, i.e. functions of Miller indices on a crystal surface), Referring to fig. 1 and to the transformation formula (32), it is seen that we are integrating over a fixed domain D in the (r’t2) plane and can therefore write
where .J is the Jacobian of the mapping (12). Then n
d,QA =
(a+ [.I(d,gA + dtgA d, IJI + qA IJI d, ti+) dt’ dg2.
(26)
J D
But it is easily seen that d,J = Jt?,V
(27)
130
R. GblEZ
Fig. 3.
The domains implied in the derivation of the general balance equation.
and it is known
(ref. 8, p. 182) that aa+=aJP a
(28)
S/J’
We also have d,d* = 8ta31y + V”d,a”. Using the definition of the covariant (26) finally becomes d,Q” =
1 &
derivative
(2%
and definition
(1 l), equation
(d,qA + &+(V$ + G)) dA .
(30)
As seen in fig. 3, every arbitrary closed line LG?~ will sweep out a surface S. In 9 interior to S we may formulate the fundamental hypotheses 10): a) Every extensive quantity Q in 9 can be written down as the sum (V for volume!) q”dV’+
Q(r)=_/ B b) The ordinary
balance
J qAdA.
(31)
&
equationI’)
atqY= -
div&
+ c?“
holds for points in 9 arbitrarily close to C,. Consistent with the general procedure in mechanics let us give the causes of any variation in Q: J,‘ot = total bulk flux, CT” = bulk source 1 JAa = surface flux 9 cTA = surface source.
(32)
of continuous
media,
A GENERALIZED
GIBBSIAN
131
SURFACE
Therefore d,e=So‘,qVdV+~~d,q”+q”(V:+G)~dA=
=~~~~~d~~g~~a~~d,,~~~d~+~~~d~. s
0
SPt
(33) &
The second term on the right hand side has apfgs sign because p is directed i~;~~rd~~,and may be transformed according to the divergence theorem for surfaces (ref. 8, p. 189) JAap, ds = i
-LPI
s &
.
(34)
The first term may be transformed according to the divergence theorem for domains including surfaces of discontinuity (ref. 8, p. 257) (35) By [q”] we will denote the discontinuity q;-q: at C, of any bulk field q’, where the subscripts 1 and 2 refer to the sides 1 and 2 of&, and by convention n is directed from side 1 to side 2. Substituting (34) and (35) into (33), recalling (32) and the fact that 5?, is arbitrary, we obtain d,qA=-qA(~;+G)-J,~-n*[J,V,,]+crA.
(36)
Equation (36) is similar in form to the “‘material equivalent” of (32) except that the term -II. [JL,] represents a source term resutting from the interaction with the adjacent phases, and the term -qAG means that C, is changing in extent. We could then call aA an ~~tri~~jc source because it stems from processes within Z;. Since the lines _fZtare material, there is no convective part to the flux JA”, but, calling u the velocity of a particle in the adjacent bulk phases, it is obvious that C, is not a material surface in the usual acceptation, for the fields w and u need not be equal. Then u -w is the velocity of a particle in R3 relative to Z;, and the total bulk flux may be split into convective and nonconvective parts Jy,, = q”(a - w) + JV. (37) Thus, we get for our final expression
132
R. GHEZ
Regarding our hypotheses (31) and (32), it may be said that they are but natural extensions of those formulated by Gibbs. There is still an arbitrariness in the choice of the dividing surface &,, and this might lead to certain conditions on the specific balance laws to be deduced in the following paragraphs. From now on, for brevity, systems.
we will restrict
ourselves
to one-component
5. Mass balance The total mass adsorbed MA=
on Z, can be written
as
pAdA.
s
(39)
&
Thus 4* = pA
(40)
and aA =
4"=p",
J*” =J”=O.
(41)
Then d,p* = - pA(G + v;) - nfp”(u
- w)].
(42)
The last term on the right hand side represents an adsorption reaction (ref. 12, p. 29), and could be written down in terms of the appropriate reaction rate parameters. Equation (42) can now be used to simplify (38) if all densities qA or q” are thought of as expressions of the forms e*q* and e”q” (q* and q” are now specific quantities, i.e. quantities per unit mass). Equation (38) then becomes pAdtqA = - n.[p”(u
- w)(q” - qA)] - n.[J”]
in which the explicit dependance
on the motion
- J,;“+
cA,
(43)
in C, and the latter’s change
in extent has disappeared. 6. Momentum and angular momentum balance The total momentum
is PA=
s &
pAUdA.
(44)
Thus qA = u
(45)
and q”=u,
J”=
JA”=O.
(46)
A GENERALIZED
The integrated following
source resulting
three terms
GIBBSIAN
133
SURFACE
from all the applied
forces is the sum of the
s
CtldA,
(47)
&
where t is the internal tensor
force per unit area expressible
in terms of the pressure
t’ = r’kn, ,
(48)
-
(49)
f
ads
pAf dA,
(50)
wheref are mass forces, for instance, due to external fields. Applying the divergence theorem (34) to (49) and recalling definition (18), equation (43) becomes pAdJJi = - n.[pV(u - w)(& - Vi)] + [t’] + of; + pAfi. iw In considering angular momentum, we will neglect any internal rotational motion (ref. 11, p. 305), and shall therefore write the balance law for LA =
s &
(x x pAU)dA.
(52)
Thus @ = xiuk _ XkUi
(53)
and 4V = xink _ XkOi,
J”=
JAa=O.
(54)
In the source term, besides the moments of the forces (47), (49) and (50), we may consider distributed surface couples c, as for example in an electric double layer. Substituting in (43), using (14) (51) and definition (18) we get a condition on the antisymmetry of the surface stress tensor @kl
zg
ik _
gki
=
$k.
(55)
7. Energy balance The total surface kinetic
energy is evidently (56)
134
from which (57) We then calculate pAdtet
= UipAd,Ui = - n*[pV(u
-
w)(z? -
Ui) ui] +
+ vi [t’] + uipy
+ @“a.
(58)
The last equality results from (51). Let us now transform the last term on the right UiOfZ= ( UiOi”),. - Ui,&Ti~= ( UiOim),a- t.Q’k = ( Qia),a
- #k) dik - &‘kQik
= (Qr’“),,
- Jik) dik - c. sz .
(5%
We must put (58) into the form (43) in which (ref. 11, p. 16) 4” = i”‘Ui
and
J”k zzz- t’kr$.
(60)
We then get J”“(et) cA(et)
= - Ui~ia,
= - acik)dik - c.sZ + pAU.f + [t(U
- u>J + n*[p”(u
(61) + - w):(U
- u)‘].
If we now suppose the existence of a potential energy function of time, f = - grad e,“(x),
(62)
independant (63)
it follows from (1) and (14) that pAd,et
= pAUidieg = - pACJ. f .
(64)
The total surface energy can be written EA =
pAeAdA. s Tit
(65)
Consistent with the First Principle, oA (e”) =O, and the system is subject to a superficial heat flux J,“” and a flux due to the mechanical work --be U ds (see sign convention fig. 2). Therefore JA”(eA) = - oiaUi + .I,““.
(66)
J,“”,
(67)
But we also have Jvk(ev)
= - likui +
where J,” stands for the bulk heat flux (ref. 11, p. 18). Thus we finally obtain pAdteA = - n.[p’(u
- w)(e”
- eA)] + [t-u - J,“en]
+ (cPUi - JyA”),,.
(68)
A GENERALIZED
Now the internal
GIBBSIAN
135
SURFACE
energy is defined as the difference U
Its bulk flux is known
A
A
=e
f -ep.
-e
A
(69)
to be (ref. 11 p. 18) J4y.
Substituting (69) into recalling (70), we get
(68), using
results
(70) (64), (61), (62), (43), (59) and
JA” (u”) = J4”“, &(U”)
= - [t.(U
- u)] + a’%,
+ a.[p’(u
- w){+(U
(71) + c*rl2 +
- u)’ + e; - e:}].
(72)
into an isotropic part For future reference, let ~7~‘~)be decomposed (appendix, equation (Bl)) a&x~x~ where c is a scalar, and a non-diagonal part eik. Expression (20) shows that a’ik’ dik = ~ ( V,~ + G) + ~ik dik .
(73)
Finally, as pointed out by Meixnerra), when Zt has no qA attributes, the equations of sections 4 to 7 reduce to discontinuity conditions across shocks and phase fronts 15). 8. Entropy balance To claim any resemblance with thermodynamics, some general physical assumption must now be made in order to derive the entropy balance law. We shall assume that local equilibrium exists in every surface mass particle, whence the surface can be considered as a locally autonomous system (see Prigogine’s terminology, ref. 12, p. 43). This assumption has already been discussed by Herring a) with a view to deriving boundary conditions for the chemical potentials. Mathematically, this means that uA can depend only on s A, the specific surface entropy, and on lIeA, and not on bulk state functions:
UA= uA(sA,l/PA). We now define the temperature
and stress distributions
TA2g, *
auA
d =a(llp”)
(74) within
the particle (75) (76)
136
R. GHEZ
It will also be assumed that CT*can be identified with the isotropic part cr of the stress tensor, although this is by no means obvious. From Euler’s theorem for homogeneous functions, and differentiating (74), we get respectively and pAdt uA =
au*
pA gd,sA
f -%!a(li#)
d,(l,‘pA)
d,pA = pATAdrsA - 0~1 P
(78)
Using the balance laws for mass and internal energy, as well as equation (77) and the corresponding bulk expression uv = TVsV - p/pv )
(7%
we may cast (78) into the form (43), remembering, however, that the bulk entropy flux is of the form JF/T” (ref. 11, p. 24). We then obtain the entropy
flux
= JF/TA,
(80)
The various balance laws just derived must be considered as part of the boundary conditions to be satisfied by the ordinary balance laws in the adjacent phases. There is, of course, no hope of solving most problems in closed form. 9. Some eiementary consequences 9.1. ABSENCEOF POLARIZATION Referring to equation (55) and definition (18), c =0 implies #Ia (nixt _ nkxt) = (e@)/J~_ &)~b)&$,
i82)
from which it is easily seen that the normal component o(“) is necessarily absent and that the tangential part asp is symmetric.
A GENERALIZED
GIBBSIAN
137
SURFACE
9.2. STATIC MECHANICALEQUILIBRIUM Neglecting
mass forces, equation
(51) reduces to
[t’] + Of = 0,
(83)
or [t’] + ni($)a
+ CFYQ,)
In the case when the t’s are ordinary [p] =
+ xfi(c$‘“” - (T(%Pb~J pressures,
= 0.
(84)
(84) reduces to
a:$)=+ c7(‘%aa.
(85)
Equation (85) has already been discussed briefly in ref. 3, and is found to yield the Laplace law under the assumption of isotropic tangential stress. 9.3. NORMAL AND TANGENTIAL PARTS OF THE MOMENTUMBALANCE All the balance laws, except that for momentum, tives of scalar quantities for which we have
involve total time deriva-
d,qA = iYtqAlu + aaqAlt V”.
(86)
But, to evaluate the acceleration d,U, we must be wary of the fact that tensors will generally not yield tensors when submitted to operations of type (86). Again, as we shall presently see, if the coordinates of R3 are curvilinear, in equation (86), it does not suffice to substitute the covariant derivative for the ordinary one. Let A: be a hybrid tensor, transforming as a contravariant space vector and as a covariant surface vector. Recalling (13) and (14), it can be shown (ref. 8, p. 199) that d,Ai is still a tensor with the following definition d,A; = atA& + I-;,,AJW”-
l-,$A$?
(87)
But atA& = a,A& + Pa,A;
(88)
and U=w+V”e,.
(89)
Therefore d,A; = a,A& + V@A;,, + T&A,Pwq. This formula using (6),
may be generalized
(90)
to hybrid tensors of any order. In particular,
d,xa = amwi + rpiqx:wq + VP2 a,8 = Wfol+ VB&
(91)
and (A3) d,n” = - x~a”Bg,qnPw~z + V"nf, .
(92)
R.
138
Formula
(91) was first derived
Truesdell7).
Straightforward
GHEZ
by Slatterys),
calculation
(92) in a restricted
d,U’ = n’ {&w’“’ + Va8,w(“) + (wCt)’ + V)(&w(“)
+ !,Q(w(‘)~ + VP))> +
+ x; ((w(‘jm + V”)(w!$)P - 2U%A,W(“)) + a, (w(f)8 + P) + V” (w(‘)B + P),,
form by
then leads to
- W(“)CPalW(“)) .
+ (93)
If the direction of the vector fields, u, t,f, is known in the neighborhood of Z,, we can resolve equation (51) in the normal direction, giving the normal motion, and in a tangential part, giving the motion in the tangent plane. It is interesting to notice that the tangential part of d,U (see 93) contains functions of the normal component w(“) that are not immediately obvious. Also, equation (93) can lead to helpful linearizations. 9.4. STRESS AND ENERGY The internal
energy balance
pAd,UA = - n+?(u
- w)(u’
law was seen to be - u”)] - n$l,‘]
- J1”ya +
- [t(U - u)] + .(l’a, + G) + @dik + C.&J + - n*[pV(u - w){*(U - u)’ + e; - e$}].
(94)
This is the most general equation relating the surface energy to the surface stress. Although CJ and eauA are both measured in dyne/cm, they must certainly not be confused. In the static case, this distinction has already been made clear by Herring2) (see also ref. 16). 9.5. SURFACERELAXATION(~.~.THERMALETCH) By specific reasoning, Mullinsis) has deduced the equations of motion of a cylindrical surface resulting from each of the four usual elementary processes, viscous flow, evaporation - condensation, volume diffusion and surface diffusion. These equations can readily be deduced from the mass balance law (42). At the end of section 4 it was mentioned that some hypotheses must be made, in order to locate the dividing surface and to pay some consideration to its atomic structure. 1) Let L, be an arbitrary closed line on Zt fixed in the (u”) system. Now eA can be related to the number of atoms that make up the surface. For example, one may think of all atoms in phase (l), having a nearest neighbor relationship with those adjacent to phase (2), as belonging to the surface. In L, one must always count the same number of atoms, therefore the total mass on C,, interior to L, must be constant MA =
pAdA = const. s &
(95)
A GENERALIZED
Referring
to formula
(24’~~) plane)
GIBBSIAN
139
SURFACE
(1 l), (we are now integrating
on a fixed domain
in the
we get J,p*+p*G=O.
(96)
2) We will also assume that aapA = 0,
(97)
which means that each atom of the surface occupies is no orientation effect. Equation (42) then becomes pAv;“, + nfp”(u
the same area, i.e. there
- Iv)] = 0.
(98)
With Mullins’ further assumptions (cylinder-like surface, small slope approximation), (98) is seen to reduce to his equations of motion. The case has been illustrated in ref. 10 for surface diffusion. Acknowledgments I wish to thank Prof. J.-P. Bore1 for the interest he has shown in this work. I am also grateful to Prof. J. Meixner for a very stimulating discussion. Appendix A CALCULATION OF ataaS/u AND a,n/, a) Differentiating
(5), we get dtaorS= &$,g,
+ g,X&xi
+ gikx+?,xi.
But, (ref. 8, p. 152) 4Sik
=
wpapSik
=
WP(rikp
+
rkip)
and with (7) araap = gik (x-~w~~ + $wfa). b) Differentiating
the obvious
(AlI
formula g&rink = 1 )
we get
wpn’nkz;,, Likewise,
from g&.x:
+ gik&,n
= 0.
= 0, we get
W~rikpniX: +
gikx,ka,ni + gikniw;z = 0.
But giknk a,ni and as”gikxE a,n” are just the contravariant according to the triad (e,, n). Therefore a,nq = - nqwPflinkr. ,kp - x@“(gi,&?~
components
+ wp&&.,).
of B,n
140
R. GHEZ
For the last term on the right hand side, let us use the formula a@+!
(ref. 8, p. 197)
= @ - nqnk.
(A21
We then get
the last equality and (8).
resulting,
as usual,
from decomposition
(9), formulas
( 7)
Appendix B ISOTROPIC
TANGENTIAL
STRESS
We assume that a’“‘=0
and &)=(TP.
From (18) we get
(Bl) References 1) J. W. Gibbs, Scientific Papers, vol. I (Dover, New York). 2) C. Herring, The Physics ofPowder Metallurgy, ed. W. Kingston (McGraw-Hill, New York, 1951). 3) R. Ghez and F. Piuz, Physics Letters 4 (1963) 275. 4) L. E. &riven, Chem. Eng. Sci. 12 (1960) 98. 5) J. C. Slattery, Chem. Eng. Sci. 19 (1964) 379. 6) T. Y. Thomas, J. Math. Mech. 6 (1957) 311. 7) C. Truesdell ef a/., The Classical Field Theories, in: Handbuch der Physik m/l (Springer, Berlin, 1960). 8) A. J. McConnel, Applications of Tensor Analysis (Dover, New York). 9) J. G. Oldroyd, Proc. Roy. Sot. A200 (1950) 523. 10) R. Ghez, Helv. Phys. Acta 37 (1964) 619. 11) S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing Co., Amsterdam, 1962). 12) R. Defay et I. Prigogine, Tension Superficielle et Adsorption (Desoer, Liege, 1951). 13) W. W. Mullins, J. Appl. Phys. 28 (1957) 333; 30 (1959) 77. 14) J. Meixner, private communication 15) A. Jeffrey, Z. Angew. Math. Phys. 15 (1964) 68. 16) R. Ghez et F. Piuz, Compt. Rend. Acad. Sci. (Paris) 257 (1963) 2795.