SURFACE ORIENTATION IN ELECTROCAPILLARITY rotation perturbs the band shape, especially the wing parts, as is seen for Y6 of methyl iodide. I n concluding, we stress again that great care should be exercised in the calculation of time-correlation functions from observed band shapes, especially with regard to truncation effects. I n order to obtain a reliable time-correlation function, the extension of accurate ob-
4047 servations of the absorption coefficient to the wing region of the band is essential. Acknowledgments. We are grateful to the National Science Foundation for financial support of this research through Grant GP-3411. We are also grateful to Dr. Roger Frech for general help and Mrs. Charlotte Smith for assistance with the calculations.
Surface Orientation in Electrocapillarity by A. Sanfeld, A. Steinchen, and R. Defay Free University of Brussels (Faculty of Applied Sciences), Brussels, Belgium
(Received February 7,1969)
In the multilayer thermodynamic framework, the authors show the influence of the dipole orientation in electrocapillary systems. In the case of a low rate of orientation in the surface layer, the electrocapillarity equation of Lipmann must be modified. The new terms allow an approach of the electrocapillary phenomena when the orientation is delayed. Electrocapillary systems are usually treated as discontinuous media. I n this approach, the system is divided into well-defined spatial regions. I n each of these regions, electric and thermodynamic variables are continuous in space, but some of the intensive properties are discontinuous at the boundaries of the regions considered. This model has led to very good results in electrochemical kinetics, in colloid science, and in capillarity. It seems difficult, however, to describe quasi-microscopic discontinuous regions, as, for instance, interfacial layers with macroscopic variables. This difficulty disappears when the electrocapillary system is considered as a continuum, but the fundamental problem is then the mathematical formulation of the physical properties of this continuous system. If the layer is many molecules thick, its composition may vary with position within the layer; these circumstances make it physically consistent to use the multilayer model developed by Defay and colleagueslJa model one could call intermediate between the continuous and discontinuous models. The system is divided into uniform regions called phases. The nonuniform regions, such as the capillary layer, are subdivided into a number of laminae, each sufficiently thin to be considered homogeneous. Our purpose is to develop an electrocapillary theory based on the multilayer model with a view to deriving an explicit formulation of dipole orientation. Some results have been partially published in a previous papera4 I n fact, important properties of the interfacial layers
find their origin in this molecular orientation (ref 5-7). Excluding all microscopic fluctuation effects, this work deals only with systems for which orientation equilibrium occurs after the establishment of the diffusion equilibrium. All transport of matter from one region to another may be treated as a transfer of one or several components from one phase to another. The only entropy production sources are, on the one hand, the chemical reactions and the transport from one phase to another, and, on the other hand, the orientation of every component which occurs in the laminae. Thus, in addition to the classical electrochemical and transport affinities, we shall have orientation affinities. I n Defay’s work, l s 2 the orientation of the components is not treated as an independent variable. This means that the (1) R. Defay, I. Prigogine, and A. Bellemans, “Surface Tension and Adsorption,” D. H. Everett, trans., Longmans Green, New York, N. Y., 1966. (2) R.Defay, J. Chim. Phys., 46,375 (1949). (3) I. Prigogine and P. Mazur, Physica, 17,661 (1951);S. Nakajima, Proc. I n t . Conf. Theoret. Phys., Kyoto, Tokyo, Sept 1953. (4) A. Sanfeld, Koninkl. Vlaams, Acad. Wetenschap. Letter Schone Kunsten Belg. Colloq. Grenslaagverschijnselen Vloeistofllmen, 1966 (1966); “Introduction to Thermodynamics of Charged and Polarized Layers,” Monograph no. 10, “Statistical Physics,’’ I. Prigogine, Ed., Wiley Interscience, Dec 1968. (5) J. T. Davies and E. K. Rideal, “Interfacial Phenomena,” Academic Press, New York, N. Y., 1961. (6) J. Guastella, J. Chim. Phys., 44, 306 (1947); Mem. Sew. Chim. &at, (1947); J. Michel, ibid., 54,206 (1957). (7) A. N.Frumkin, ibid., 63, 786 (1966); B. B. Damaskin, Electrochim. Acta, 9,231 (1964). Volume 73, Number 18 December 1969
4048
A. SANFELD, A. STEXNCHEN, AND R. DEFAY
Table I (cos
Vertical upward Vertical downward Lying on random Lying and mutually parallel Nonoriented One half vertical upward, the other vertical downward Lying one half in one direction, the other half in the other direction
ad
(cos
sa)
a,)
0
0
0
1
0
0
0
0
Figure 1. F A is the unit vector on the axis of a dipole and pi are the direction cosines of
a.
molecules, while moving from one phase to another, are always supposed to be in instantaneous orientation equilibrium with the dipole structure of the successive laminae. If, on the contrary, the orientation equilibrium is reached after the diffusion equilibrium, orientation variables independent of diffusion variables should clearly appear in the thermodynamic formalism. This will enable us to show how the Gibbs formula may easily be extended to these systems and, for illustrative purposes, we shall discuss a very simple example of surface orientation.
Orientation Variable We consider an electrocapillary system composed of undeformable neutral polar and nonpolar molecules with charged molecules. We suppose that there are 1. . . y., . c constituents in 1 . . . a phases. The orientations of the components in each phase are directly related to the chemical and electrical interactions. Let be a unit vector on the axes of a dipole (Figure 1) 3
cos2 pt = 1
(1)
m.
where the cos 0%are the direction cosines of We suppose that rotations of F A around the coordinate axes are free. We shall introduce now the mean value of cos p defined by
The!Journal of Physical ChCmhtTy
(cos1
0 0 0 Oor # O 0
x2
1
an)
0 0 0 Oor 2 0 0
1 -1 0 0 0
7
..i
(cos
1 1 0 0 20
(cos1
a*)
0 0 #O # O or 0 #O
0
#O
(cos9 pa)
0 0 0 # O or 0 #O 0
#O
where f(P) is the distribution function on the angles and where the denomiriator is the normalization integral. On the other hand, it is easy to see that the three mean cosines are insufficient to describe the orientation. If in addition we introduce the value of the mean-square direction cosine (cos2 pi), we then obtain for different experimental situations the values listed in Table I. We conclude that the six mean values (cos pi) and (cosz P,) are at least necessary to describe all orientations. Generally, the dipole length is not equal t o unity, and thus we replace variables (cos p,) and (cos2 pi) by the mean projection on xi, (mxira),and the mean-square projection on xi, (mxira2)of the dipole moment, per mole of y in the phase a. The set of all mean variables and of all mean-square variables corresponding to all different values of a = 1,2. . . of y = 1,2,. . . c and of x1 = 1,2,3 are respectively, represented by the abbreviated notations (mxiyu)and (mz,ya2).
Remark. I n fact, the distribution function of the orientation may not be described by only six projections (m,,) and (mxia). We assume that the classical approximation of a Gaussian distribution is valid (the first projection gives the more probable value and the latter gives a way to calculate the dispersion around this value, ie., the fluctuation). Thermodynamics of a Closed Electrocapillary System We adopt the multilayer model in which each lamina within the surface layer is considered as a homogeneous phase of infinitesimal thickness. Let us consider now a system a t uniform temperature and in mechanical equilibrium, unable to exchange molecules, ions, and electrons with the surrounding world. We include in the definition of the system all forms of electric energy including that of the field which the system creates around itself. The system is not subject to the influence of external charges since we include all relevant charges within the boundaries of the system. The system cannot, therefore, interact electrically with the surrounding world. In particular, for a system
SURFACE ORIENTATIONIN ELECTROCAPILLARITY containing only one plane interface, the work done on the system by its surroundings is dW = -pedV
+ UdQ
(3)
where p , is the external (uniform) pressure acting on the system, V = Eva is the total volume, equal a
to the sum of the volumes of the individual phases, u is the surface or interfacial tension, d0 is the increase of the area of surface 0. The equations, derived from the first and second principles of thermodynamics, have the form d o = dQ - x p e d V a a
+ ud0
(4)
4049 Usually, in electrocapillary theory, orientation and diffusion are treated as independent variables, i.e., the orientation and diffusion occur simultaneously.2 We consider here the case of independent variables. Applying the de Donder and Prigogine m e t h ~ dthe , ~ entropy production dig can be written for all the possible reactions Tdig =
+ C xz,y~ordEz,y~a +
C
XY,
P
where the summation symbol
-
dQ dX = T
+ dQ' = dex" + d$ T ~
with dQ'/T = d,S" 3 0. The function is the total entropy including electrochemical effects and dQ' = Td$ the Clausius "uncompensated heat." Let us remember that if the orientation of the components is a t each time in statistical equilibrium with the field (a reversible process), then x" may be replaced by S.s Because of the way in which the system has been defined (no thermal flux, no hydrodynamical motion), the only possible sources of entropy production are the chemical (including electrochemical) reactions, the diffusion of molecules or ions from one part of the system t o another, and the orientation variations in each lamina within the surface layer. The variations in orientation are due to the macroscopic electric field and to interactions between molecules or atoms belonging to the same lamina or to neighboring laminae. We define now the degree of advancement introduced by de Donderg for all possible reactions (matter transport, Le., passage of one or more components from one phase to another, chemical and orientation reactions). We have
where and are, respectively, the degrees of advancement of the transport "reaction" of constituent y from phase N - 1 to phase a, and from a to a 1. Index 1" refers to the chemical reactions, superscript 0 t o the time t = 0 (origin of E ) , subscripts 0 and Q refer, respectively, to (moiya)and (mxlyaZ), and vya is the stoichiometric coefficient of y in a and in the chemical reaction. If the molecule orientation varies during t,he crossing from one lamina to another, then variables or-lEya, EXtyoa,and EXlyQa vary together during the crossing.
represents the triple
cxxa.The coefficient xpis the electroXY.
summation
a
and
c ~x,yQordEx,Y&a(9)
2 7%
yz=l
chemical affinityg of the reaction p ; subscript p , here, refers to all reactions except orientations (i.e., passage and chemical reactions). The orientation affinities and are related, respectively, to moments (mzbya) and (mxzyOL2). Combining (5), (9), and (4), we obtain
xXlyQa
d F = -x"dT -
pedV" a
xx,yOa
+ ad0 -
d&z,yOa-
APdEpP
ax,r&"dEz,yQa
(lo)
where P=O-Tg
(11)
The free electrochemical energy becomes thus the function where a = 1,2. . . ; y = 1 , 2 . . . c, and i = 1,2,3. The derivatives of k with respect to one variable all others remaining constant, have for example the form bijl _ -- -5 bT
bF ~bVa
-"e
+
(8) A. Sanfeld and R. Defay, Physica, 30,2232 (1964). (9) Th. de Dander, "L'affinitB," P. Van Rysselberghe, Ed., GauthierVillars, Paris, 1936; I. Prigogine, "Introduction to Thermodynamics of Irreversible Processes," Charles C. Thomas, Springfield, Ill., 1955; R. de Groat, "Thermodynamics of Irreversible Processes," NorthHolland, Amsterdam, 1951; P. Van Rysselberghe, "Thermodynamics of Irreversible Processes," Hermann, Paris, 1962.
Volume 73, Number 13 December 1969
4050
A. SANFELD, A. STEINCHEN, AND R. DEFAY
This description implies the assumption that the relative disposition of the phases remains uchanged. The relative position of the phases and the magnitude of the charges (ions and electrons) that they carry determine the electric field in the system. Therefore the energy of the system, including the energy of the electric field, will for a fixed arrangement of the phases be described by T , Val8, t,, t x 2 y ~& ?aQ,~ .
Extension to Open Systems The previous discussion leads to the conclusion that F is a function of the variables which determine the physico-chemical state of the phases, the mode of repartition of the components among the phases and their orientation within each phase
as the electrochemical potential of component y in the phase a for a state where the mean orientations have given values. When these mean orientations take their equilibrium values, the electrochemical potential reduces to the classical electrochemical potential. A derivative, in which only nya varies, means that component y, added to phase a, takes in this phase the preexistent orientation. From (14) and ( 6 ) , we have
Equation 10 can thus be written
dk
= -gdT
-
p,dVa
+ ad8 +
0
(17) where the symbol nYa represents n:: : :, the number of moles of each component in each phase of the system. The same will be true for an open system, provided that when a charged particle is removed from the system, it is taken to infinity so that the system is never subjected to a field of force. Similarly, when a charged particle is added to the system, it must be brought from infinity. Let us remark now that the orientation variables are intensive. If the system is subjected to a transformation in which all nya, (mzlra), and (mZlya2) remain constant, the free energy will vary in exactly the same way as it would in a closed system where all ( are constant. Equation 13 can thus be written
On the other hand, if the system is subjected to a transformation in which only (mwya)vary, the free energy will vary in exactly the same way as it would in a closed system where all Z: are constant except tx;,oa. From (7) we obtain
dP
(w
-
~T,V',fi,n~u,(mziyu2)
The same conclusion is valid for (mZ,ya2), and from (8) we have
Furthermore, we define the quantity
The Journal of Physical Chemistry
pyadnyaa"/
For an electrochemical system to be in equilibrium, all electrochemical affinities, all phase-transfer affinities, and all orientation affinities, which can actually take place, must be zero
K,
= 0 0; KXi70= 0; Plane Interphase between Two Electrically Conducting Phases Consider two phases I and 11, each one becoming homogeneous at a sufficient distance from their interface (Figure 2). 1
=
---I
n
"
I
I
Figure 2. Two systems, on the same interface, of same height but of different interfacial areas i and ii.
If the phases are electrically conductive (ionic or electronic) and are a t thermodynamical equilibrium, potentials (PI and pI1 must be uniform in the homogeneous regions of these phases. If they were not, then charges would continue to move in the resultant field. I n general, however, (PI and (PII will not be equal. The two homogeneous regions are separated by a heterogeneous region which constitutes the surface layer. We consider this layer as being made up of a number of laminae each of which can be treated as an infinitesimally thin homogeneous phase. The surface layer is in general a region in which there exists an intense electric field because in a very short distance (for example a few tenths of an angstrom in aqueous solutions) the value of cp changes form (PI to (PII. For a sufficiently large plane interface, the lines of force are normal to the surface. This is not
405 1
SURFACE ORIENTATION IN ELECTROCAPILLARITY strictly true a t any instant on a molecular scale, but will be so on the average. Although the successive surface layers may have finite charges associated with them, from Gauss's theorem it is easy to see nevertheless that the interface is, as a whole, neutral.'t2 This property is connected to the conductivity of the bulk of phases I and I1 (the field is zero in the bulk of these phases). If we cut out, on the same interface, two systems defined as in Figure 2, having the same height but different interfacial areas i and ii; and if area ii is k times area i, then the extensive variables values V", 0 and nYain system ii will be k times their values in system i. The kinetic, structural, and chemical contributions to the free energies for the two systems will also be in the same ratio. Furthermore, provided the dimensions of the surface are large compared with the thickness of the interfacial region, the electrostatic edge effects will be negligible.1t2 Consequently, the electrostatic contributions from the surface will be proportional to t#hearea. In particular, we conclude that the function $' defined by eq 20 is a homogeneous function of the first degree in the variables V", 0, nY* and thus, from Euler's theorem and from eq 18 and 21
P
-Ca
peV'
+ a0 +
*Y
nnyUf7iya
(24)
where the electrochemical potentials p,* depend on the orientations (see eq 21) and where the summation over a includes all bulk of both phases I and 11, and all the surface layers. By differentiating this expression and subtracting (23), we obtain
=
&I
pyol
=
(26)
PYII
although the orientation affinities are different from zero. The superscripts to f7iy, may then be dropped. Equation 25 reduces then to
+ Vdpe -
Qdr = -gdT
Y
nydpy -
E,xxirOPd(mziya) - ayd xi[,yQ'd(mx;~a2) (27) *Yl.
where
C nra = n,
(28)
This case better lends itself to the Gibbs treatment in which the interfacial layer is replaced by a geometrical division surface defining the volumes V' and V". I n the bulk of phases I and 11,each element of volume is neutral, without any electric field and in orientation equilibrium. The potentials (PI and (PII must be uniform in the homogeneous regions of these phases and since the vector displacement obeys Poisson's electrostatic equation, the average net charge density must be zero in each element of volume in the homogeneous regions of phases I and I1; hence Y
X,C,I
=0
XsYCYII= 0
Y
(29)
and
C Z,C,IFaIdP
= 01
Y
zyC,IIFadp = 0 Y
I
where 2, is the elementary charge of the component y and Fa the Faraday. We have thus the Gibbs-Duhem equation
+ CC,Idp, = 0 dpe + CC,I1d,ii~ = 0
s,IdT - dp, stldT
1
(31)
These two equations are now added to (27), remembering that for the Gibbs model The above proof does not assume the existence of equilibrium with respect to the distribution of components among the surface layers, but does assume mechanical and thermal equilibrium; p, is the pressure applied externally and is equal to that in the bulk of the two homogeneous phases (cf. eq 4). Equation 25 can thus be applied both to systems in equilibrium and to others, such as polarized electrodes, which are not in a true equilibrium state.
Partial Equilibrium in Systems Where Each Component May Be Present in All Surface Layers I n the case where orientation is much slower than diffusion, we can reach a partial equilibrium state and for each component y,
+ V"C,I1 + r,o 27 = v's: + C"S21 + 02
(32)
v = 8' + v"
(34)
n, = C'C,I
(33)
while
Dividing by 0, we obtain d a = -TadT
-
r,dp,
-
Y
c
1 1 ~x,oUd(mx,,*) - 0 ~,YQad(mxiya2) (35) 0 UT* ayi This is an extension of the Gibbs equation to electrochemical systems where orientation reaches the equilibrium after a long time after the diffusion equilibrium. -
c
Volume 73, Number 1.9 December lQ8Q
4052
A. SANFELD, A. STEINCHEN, AND R. DEFAY
At the true equilibrium (diffusion and orientation), 0, and (35) reduces to the classical equationltz
zXiroa = 0 and AxiyQa = d a = -3"dT
If the system is uncharged, the electrochemical functions (p,, z X , , O U , ~ x l y Q U )reduce to the corresponding chemical functions ( p y , AXlyOU, AXiyQU) and, at uniform temperature, we have dg =
-E r
rYdpy -
where 1
2 C Jx,y~ad(mxiya) -
Aa0 and f2
JXi0* = -
a-0
1
6 c Ax,YQ"d(mxi,*z)
(37)
UYi
Let us now suppose that we maintain a constant temperature, pressure, and composition of the bulk phase, ie., dp, = 0. The evolution of the surface from a state of partial equilibrium (ie., from a state where the orientation is not in equilibrium) would be given with the aid of (35) =
to the orientation of the adsorbed molecule, has to be taken into account in the evolution rate to the equilibrium sta'te. Now we suppose that only one component of the upper lamina orientates itself at the interface. Equation 38 may be then rewritten
-g1 C xxcyoad(mx,ziyP)- 21 C Jx,Y&Ud(mxi"P2) ayi
(38) This formula implies that each py is constant during the transformation, i.e. , that diffusion occurs quickly enough to ensure continually the equality (26), by balancing the influence of the change in orientation on the local p,. For uncharged components, we obtain
Let us suppose that initially (out of equilibrium) onehalf of the undeformable or rigid dipoles are directed vertically upwards and the other half vertically downwards, while a t equilibrium all dipoles turn vertically downwards. If axis 2 1 is perpendicular to the surface of the layer, the rigid moments (m2J and (mZl2)may be written as COS 01) and m2(coszPI), where m is the arithmetic value of the dipole moment. Furthermore, let us suppose that A, varies only slowly with (m) in such a way that contribution of the second integral
L1
rl",*d(m2) can be neglected.
Integrating (40) from (cos PI) = 0 to (cos 01) = -1 and (cos2PI)= 1to (cosz 61)= 1, one has then
(XO*)= m ACJ
where (KO*) is the mean value of the surface orientation affinity in the integration and Au the variation of 1 1 the surface or interfacial tension due to the change of = -Jx,,o"d(mx,,*) - 6 ~xiy~ad(mx,~z) orientation. Q a,, uy; If by way of a perturbation, all the rigid dipoles are (39) initially directed vertically upwards and if , while Examples equilibrium takes place they turn downwards, then after integration eq 40 becomes We consider a system in a real equilibrium state. A short-impulse, transitory field is applied so as to avoid diffusion (in the bulk phase the temperature, the pressure, and the composition are constant and we suppose all the p, uniform in the medium). Nevertheless, this A measure of the surface or interfacial tension for the field is able to turn over the molecules' orientation in two extreme positions of the orientation can give us a certain laminae. When the field cancels out, the system value of the mean affinity of orientation, as long as returns to equilibrium in agreement with (38). the dipoles are rigid and their moments are known. We could also consider the return to equilibrium after Remark a. If we assume that only one of variables a perturbation due to friction or a motion laying down &.;ya varies, all others being kept constant in eq the molecules. This case may be related to the vis10, and that moreover (m,,) varies from 0 to cosity flow of monomolecular solutions. lo The return -(m,,) ((cos p ) varies from0 to -1)) there comes to equilibrium follows eq 38 or 39. On the other hand, Defay and Roba-Thilly,ll Defay (10) M. Joly, I11 Internationale Vortragstagung fiber grenzflachenand Hommelen,ll and PBtrB and DebellelZ show that aktive Stoffe, Mar 1966, Berlin. the rate of adsorption of sebacic acid, azelaic acid, and (11) R. Defay and J. Roba-Thilly, J . CoZZoid Sci., SuppZ., 1,48(1954); R. Defay and J. R. Hommelen, ibid., 13, 553 (1958); ibid., 14, 401 diols a t the air-aqueous solution interface is not only (1959). diffusion controlled. A barrier of potential energy (12) G. PQtrQand P. Debelle, Vth International Congress on Surface Activity, Barcelona, Sept 1968. between the substrate and the surface phase, related The Journal of Physical Chmistry
4053
SURFACE ORIENTATION IN ELECTROCAPILLARITY A? =
cz'
&,,d(m,,) = (&o)(~,,) = Q(A*>(m,S (43)
da =
Putting (43) in (41), we get
(44)
(A?) = QAa
Remark b. It is easy to compare now the mean affinity RT/(m), where I' is the number of moles per cm2. At 15", RT N 2.3 X 1O'O erg/mol. Let us choose, for example, a binary liquid whose surmolecule/ face is covered by 10-10 mol/cm2 (6 X of the surfactants. We suppose now that if onehalf of the surface molecules of the surfactant are turning, the experimental value of Aa is 5 dyn/cm. The variation of the mean affinity per mole being only due to the surfactant in the upper lamina (we exclude the orientation variation of the other component), it is easy to see, from eq 47 that @*)/I' is of the order of magnitude of 2.2RT/(m).
(&,*)/rwith
wz)
Ideal Polarized Electrode For some metal solution interfaces, the surface can be regarded as consisting of two parts, only one of which is accessible to the constituents of the solution. As an example, we consider mercury in contact with an aqueous solution (ideal polarized electrode). The thermodynamical properties of an ideal polarized electrode may be discussed conveniently by dividing the surface layer into two parts which we imagine to be separated by a surface impermeable to all components of the system as indicated below.l3 Here the plane B separates the surface layer of the mercury (AB) from the surface layer of the solution (BC). The fundamentally important feature of an ideal polarized electrode is that its potential drop cpl - cpI1 can be notified without changing the concentration of the bulk of the phase. We denote by h the components which can exist in mercury and in its surface layer (that is Hg, Hg+, and e-, together with any other metals dissolved in the mercury and their ions) and by 1 the components of the solution and its surface layer. The index 0 is used to denote surface layers in the bulk mercury and its surface, while /3 denotes those associated with the solution of the bulk and surface. If there is a partial equilibrium between each phase bulk and the corresponding surface layer in such a way that :
BI
AI Phase I Mercury
Mercury surface layer
Complete phase'
C 1
Solution surface layer
Phase 11 Solution
Complete phase"
We choose plane B as the Gibbs dividing mrface, so that
where, as before, surand s V I rrefer to entropies per unit volume in the bulk of the phases and 8" is the surface entropy per unit area. Applying the Gibbs-Duhem equation to the bulk phases which are neutral and are not subjected to an electric field, one has, combining (46), (477,and the expression of py Fir = P-/
+ z,Fucp
(48)
where cp is the Galvani potential
da = -S"dT
-
rhdph' h
-
I'ldhl'I 1
-
where q M is the charge per unit surface area of the metal surface layer. Because of the over-all neutrality of the system, qM is equal but of opposite sign to the charge of the solution surface layer and is given by rhZhFu
qM
h
=
-E1
I'iZiF,
(50)
At the real equilibrium, all the orientations affinities are zero and (49) reduces to the classical electrocapillary equation. Particularly, for equilibrium displacements keeping the temperature and the composition of the bulk phases constant, we find again the Lippmann equation (13) D.C. Grahame, Chem. Rev., 41, 441 (1947); F. G.Koenig, J . Phys. Chem., 38,339(1934).
Volume 73, Number 1.9 December 1968
4054
A. SANFELD, A. STEINCHEN, AND R. DEFAY
If, in the case of a partial equilibrium (45), without orientation equilibrium, (PI - (PI’ can be varied without changing the temperature, the composition of bulk phases and the orientations in the layers (49) reduces also to Lippmann eq 51. Nevertheless, qM has not necessarily the same value in both cases. If the composition of the bulk remains constant but not the orientation in the layers, then a t constant temperature eq 49 reduces to
component in the considered laminae have then the following values (taking the x axis perpendicular to the surface and lml = 1) (mJ = 0, (m,) = 0, (m,) = 0, (mx2)= a,,
(m,2)
=
am, (mz3 = a, (55) where a, is t3, positive mean value lying between 0 and 1. When all the molecules will have orientated perpendicularly to the surface in the direction ( - z ) , we will have
(mx>= 0,(mu) = 0,(m$ = -1, (mx2)= 0, (mu2)= 0, (mZ2)= 1 (56)
Let us suppose now that only the orientation affinity of one constituent 1in one interfacial lamina layer 6 is different from zero. We denote this affinity by symbol A0 and the mean dipole moment by (m). From (52), we get
.ZxJ
du = -qMd(pl
- pI1)
- &*d(m)
(53)
I n the neighborhood of orientation equilibrium, let us adopt the classical hypothesis that the orientation ,,~ffinityis proportional to the orientation rate. Equation 53 now reads
Variables (m,) and (m,) have remained equal to zero, but the four others have changed. Rigorously, in (53) terms related to the evolution of the three mean quadratic moments should have been added. As these mean quadratic moments are invariable under a complete rotation of 180’ of all the molecules it is expected that, in this case, they will not play an important role in an evolution towards a determined direction and so we can neglect them in (53). However, these mean quadratic moments may not be neglected in other phenomena as when a chain molecule, having two identical chemical ends, has a tendency to lie flat on the surface. One then goes from a three-dimensional disorder to a two-dimensional one which state is characterized by (mz) = 0,(mu) = 0, (m3 = 0,b X 2 )= a,’,
where ROis the “resistance” coefficient. Experimentally, it is possible to keep constant in time an arbitrary value of the imposed tension (pl - VI’). If the orientation is slow, we could watch the time evolution of u towards its value in the orientation equilibrium state. The evolution in time of function d&d(m)/dt being thus known the variation with time of the orientation rate vo = d(m)/dt is indicated. Integral Jflo(d(m)/dt)dt is thus computable between two limits, and we could, for instance, plot function ( d z ) ( ( m ) e q - (m),,) for each value of the imposed tension (0’ - (PI’) [(m}eq is the value of the moment a t orientation (m),, its value a t the initial time]. However, such a conclusion is only possible when the orientation rate is much slower than the transfer rates, in such a way that after an evolution due to the dflusion and to the transfer, the system reaches a state of partial equilibrium which is then controlled by the orientation rate. The question now is what is the meaning of a single orientation affinity? I n a state of three-dimensional space, we consider a random orientation; the related variables of a single The Journal of Physical Chemiatrg
(mu2)= am’,(ma2)= 0 (57) By comparing (55) and (57), one sees that this phenomenon depends essentially on the evolution of (m2,). Here the 8,, affinities can be neglected. The approximation thus depends on the special studied cases. Electrocapillary Maximums in the Case of Partial Equilibrium’* Let us now consider the case of slow orientation and keeping the approximation leading to (53). By following the evolution from partial equilibrium to complete equilibrium, keeping the temperature and the compositions of the bulk phase constant, it could be possible, for each constant value of (PI - (PI’, to build the curve u(t) where t represents the age of the surface. By taking all the u values corresponding to a same t (14) C. J. F. Bottcher, “Theory of Electric Polarization,” Elsevier, Amsterdam, 1952. (15) J. Barriol, “Les Moments Dipolaires,” Gauthier-Villars, Paria, 1957. (16) H. Frohlich, “Theory of Dielectrics,” Clarendon Press, Oxford, 1959. (17) M. Mandel, Bull. Soc. Chim. Fr., 7,1018 (1955). (18) A. Sanfeld, A. Steinchen, and R. Defay, Vth International Congress on Surface Activity, Barcelona, Sept 1968.
4055
SURFACE ORIENTATION IN ELECTROCAPILLARITY
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value, one can build a u(pl p I I ) curve at constant t. It will be the electrocapillary curve of a given age surface. The last of these curves, i.e., corresponding to t = a i s the equilibrium electrocapillary curve. It is probable that, a t least for states neighboring complete equilibrium, the curves correspond to a given age, will also have an electrocapillary maximum of the same age. I n order to analyze these curves, one could suppose as a first approximation that the surfaces having the same age have the same orientation. The curves of a given age will then be curves at constant (m) and following (53),their slope
Remarks
will give the charge per unit area of the metal surface layer at the considered time. By studying all these curves, one could then study the evolution of the charge us. time. If the surfaces of same age are in the same orientation state, the charge will be zero at each electrocapillary maximum. One could then also compare the points of zero charge corresponding to the maximum of the various curves. On a line joining the maximum, one will have
(b). If there is a coupling between the various orientations, one should introduce interference coefficients. (e). A more careful analysis of the influence of orientation on the electrochemical potential would perhaps separate the terms related to pure orientat’on and to pure diffusion. The orientation in an external field could then be studied with the aid of the relaxation polarization effect. (d). From the experimental point of view, the influence of orientation in the mercury electrode process has been developed by VolkeZ0and by Smolders.21
dumax =
(””) d(m)
d(m) +,I-~II
+
( a ) . Following the classical hypothesis, d(m)/dt is proportional to the orientation affinity A0*; eq 53 may be rewritten
where Lois the “straight” phenomenological coefficient. The knowledge of the instantaneous values of cpl - c ~ I I permits us to calculate the time variation of the orientation affinity. Equation 60 also becomes
Acknowledgment. The work was performed under the auspices of the Fonds de La Recherche Fondamentale Collective. where the second term is zero. Thus, a t each point of this extreme curve
(19) S. R. de Groot and P. Mazur, “Non-equilibrium Thermodynamics,” North-Holland, Amsterdam, 1962,p 349. (20) J. Volke, Chem. Listy, 62,497 (1968). (21) C.A.Smolders, Thesis, Utrecht, 1961,pp 41-46.
Volume 73,Number 18 December 1969
