J. Phys. Chem. 1987, 91, 101-109 value of the order parameter and specific heat by using the relations"
( S 2 ) = a(T, - T)/2B
T < T,
C, - , C = a2T,/2B
(D3)
( g ) = -(kT/N) 1x1 g(T,p,O)
Qsub
- (kT/N) In
D(z) = D(0)
exp[-KN/kT)(AS2
+ BS4)11 d S
(D5) Applying the x = S2(2NB/kT)1/2substitution in the integral, one obtains
+
(2)
z
r=O
where D(0) =
= Jm
+
where D(z) = So"dx exp(-[zx x2/2]) and z = A(N/ (2BkT))'12. Close to the phase transition temperature, D(z) can be approximated by
(D4)
where Cpis the specific heat capacity at T < T, while Cpois at T > T,. Substituting eq D1 into eq 3 the specific Gibbs energy is
101
J m0
exp(-[x2/2])
dx = (i ) 3 / 4 r ( i) = 2.156 XI12
(2)r=o -[ =
& " x ' / ~ exp(-[x2/2]) dx] =
Substituting eq D7-D9 into eq D6, we get an implicit equation for the cooperative unit size N (g)
- g(T,p,O) = (kT/4N) In (2'NB/kT) (kT/N) In [2.156 - 1.03A(N/2BkT)112] (D10)
Registry No. DPPC, 63-89-8; DMPC, 18194-24-6; DSPC, 816-94-4.
Microscopic Modeling of Multicomponent Charged Interfaces Panaghiotis Nikitas Laboratory of Physical Chemistry, Department of Chemistry, University of Thessaloniki, Thessaloniki, Greece (Received: February 25, 1986; In Final Form: August 3, 1986)
A very general lattice model for organic electrosorption from multicomponent systems has been presented. The interface is treated as a sheet of point dipoles situated in vacuo. Every adsorbedmolecule may be present at the interface in various and distinct polarization states occupying one or more lattice sites. In each state, the adsorbed molecule experiences the field of the electrode and that of its surroundings. It also interacts with its nearest neighbors with short-range interactions. The equilibrium equations of the system are derived on the basis of the mean field approximation. The specific cases of the coadsorption of two or more organic adsorbates and the reorientation of an adsorbate are examined in more detail. Results suggest that the model describes, at least qualitatively, the experimental behavior of the systems. The model predicts a concentration-dependent adsorption maximum in all cases where there are more than two species with different dimensions simultaneously at the interface. This feature is in agreement with experimental results obtained in systems with reoriented adsorbate molecules, whereas it is open to further experimental study in coadsorption systems.
I. Introduction The theoretical study of the electrical double-layer properties in the presence of adsorbed species follows three basic approaches: Two are of phenomenological character and are based on the work of Frumkin-Dama~kinI-~and Mohilner-Karolczak,kE respectively, whereas the third is molecular and has its origin in the LBS modelg for the structure of a polarized interface in the absence of specific adsorption. (1) Damaskin, B. B.; Petrii, 0.A,; Batrakov, V. V. Adsorption of Organic Compounds on Electrodes; Plenum: New York, 197 I . (2) Frumkin, A. N.; Damaskin, B. B. In Modern Aspects of Electrochemistry; Bockris, J . , Conway, B., Eds.; Butterworths: London, 1964; Vol. 3. (3) Damaskin, B. B.; Kazarinov, V. E. In Comprehensive Treatise of Electrochemistry; Bockris, J., Conway, B., Yeager, E., Eds., Plenum: New York, 1980; Vol. 1. (4) Mohilner, D.; Nakadomari, H.; Mohilner, P. J. Phys. Chem. 1977,81, 244. ( 5 ) Mohilner, D.; Karolczak, M. J. Phys. Chem. 1982,86,2838. (6) Karolczak, M.; Mohilner, D. J. Phys. Chem. 1982,86,2840. (7) Karolczak, M.; Mohilner, D. J. Phys. Chem. 1982,86,2845. (8) Karolczak, M. Electrochim. Acta 1985,30,325. (9) Levine, S.; Bell, G. M.; Smith, A. L. J . Phys. Chem. 1969,73,3534.
0022-3654/87/209 1-0101 $01 S O / O
In the Frumkin-Damaskin theory'-3 the electrical double layer is visualized as two capacitors connected in parallel. Between the sides of one of these capacitors there are only solvent molecules, whereas the adsorbate molecules are between the sides of the other capacitor. A two-series-capacitor model'O*'las well as three- and n-parallel-capacitor modelsI2-l4has been developed in an attempt for a pure phenomenological description of the electrical double-layer properties in the presence of adsorbed organic substances. In the Mohilner-Karolczak approach4-* the interface is considered as a three-dimensional liquid mixture of two or more components. The equilibrium between adsorbed and bulk molecules is treated at a macroscopic level by considering the adsorption process as an heterogeneous equilibrium. The equilibrium conditions are determined directly from the law of mass action. (10) Parsons, R. J. Electroanal. Chem. 1964,7, 136. (1 1) Parry, J.; Parsons, R. J. Electrochem. Soc. 1966,113?992. (12) Damaskin, B. B. J . Electroanal. Chem. 1969,21, 149. (1 3) Tedoradze, G. A,; Arakelyan, R. A,; Belokolos, E. D. Elektrokhimiyo 1966,2, 563. (14) Batrakov, V. V.; Damaskin, B. B. J . Electroanal. Chem. 1975,65, 361.
0 1987 American Chemical Society
102
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987
By means of the Duhem-Margules equation, these conditions are expressed in terms of the excess electrochemical Gibbs energy of mixing, AGE. In the first papers,"-7 AGE is expanded as a power series of the surface mole fractions xi, whereas in recent publications Karolczakl59l6takes AGE from certain molecular models in an attempt to incorporate microscopic considerations into the theory . In the third approach the adsorption layer is limited to the first monolayer of solvent and adsorbate molecules. Thus, it is regarded as a two-dimensional system. According to the LBS model: the inner layer of a charged interface in the absence of specific adsorption may be considered as a sheet of point dipoles of the solvent molecules situated in vacuo. The dipoles are under the influence of the field of the electrode and that of their surroundings, and they are free to take a certain number of distinct polarization states characterized from the position of the dipole vector in relation to the field of the interface. The extension of this model to the case of adsorption of organic substances on a polarized electrode is almost In a recent paper24we have adopted the latter approach and have explored the properties of a charged interface where the solvent molecules are linked to form clusters. We found that the existence of clusters, as distinct entities, results in models which predict a concentration-dependent adsorption maximum, czaax, in disagreement with the e ~ p e r i m e n t . ~This ~ . ~erroneous ~ prediction seems not to be the result of the approximations involved in the treatment of the model. Every model, based on the assumption that the inner layer may be considered as a twedimensional system in which the solvent molecules may possess a number of distinct states with different dimensions, is expectedz4to predict a concentration-dependent agx.Therefore, the results of ref 24 may show the nonexistence of solvent clusters or that some basic assumption of the model-monomolecular inner layer or finite number of polarization solvent states-is not valid. The presence of clusters at an interface is equivalent to the presence of more than two species with different dimensions. Therefore, such interfaces are expected to be analogous with multicomponent interfaces, where the various species are adsorbed at geometrically different states (reoriented adsorbates) or there are more than one solvent or adsorbate (coadsorption). In these interfaces, if we choose properly the components and the experimental conditions, we can ensure the presence of more than two species with different dimensions. Thus, by comparing the experimental behavior of these interfaces with theoretical predictions, it is possible to get conclusions about the validity of the basic assumptions of the model. On the basis of these thoughts, we attempted an extension of our work in ref 24 to multicomponent systems. Particularly, in this paper, we study in detail the equilibrium conditions and the dielectric properties at the charge of maximum adsorption in the cases of the coadsorption, the adsorption from mixed solvents, as well as the reorientation of an adsorbate. The predictions of the model are compared with experimental results. 11. Thermodynamics of Multicomponent Interfaces
We consider an ideally polarized interface composed of NA adsorbates, say Al, A2, ..., A., Each of them may be present at the interface in nA, distinct states characterized from the dif(15) Karolczak, M. J. Colloid Interface Sci. 1984, 97, 284. (16) Karolczak, M. J. Electroanal. Chem. 1984, 181, 21. (17) Sangaranarayanan, M. V.; Rangarajan, S. K. Can. J. Chem. 1981, 59, 2072. (18) Sangaranarayanan, M. V.; Rangarajan, S. K. J . Electroanal. Chem. 1984, 176, 1. (19) Sangaranarayanan, M. V.; Rangarajan, - . S. K. J. Electroanal. Chem. 1984, 176, 29. (20) Sangaranarayanan, M. V.; Rangaraian, - - S. K. J. Electroanal. Chem. 1984, 176, 45. (21) Sangaranarayanan, M. V.;Rangarajan, S. K. J. Electroanal. Chem. 1984, 176, 65. (22) Levine, S.; Robinson, K.; Smith, A. L.; Brett, A. C. Discuss. Faraday SOC.1975, 59, 133. (23) Nikitas, P. Electrochim. Acta 1985, 30, 1513. (24) Nikitas, P. J. Chem. SOC.,Faraday Trans. 1 1986, 82, 971. (25) Trasatti, S. J. Electroanal. Chem. 1974, 53, 334.
Nikitas ferent orientation of its dipole vector in relation to the electric field of the interface. We denote the ith adsorbate in its J state by A,. We also accept that there are Ns solvents in ns, states each. We use the symbol SI, to denote the ith solvent in its J state. If during the adsorption process of one molecule A, r''/ molecules of the solvent s k i are replaced from the surface solution with probability E"'!, then the following equilibrium equations are valid A,@) Sdb)
+ Cr'Lij?LiSk/(ads) kJ
+ Zr'L/P?/sk(b)
7= A,(ads)
+ EmifiPfMads) + W a d s ) + C m i P i ' M b ) kJ
(1)
k,l
kJ
(2)
where mi{ is the number of solvent s k i molecules which are replaced from one s,, molecule with probability p;,, the symbols b and ads denote the bulk and the adsorbed phase, and s = 1, 2, ..., Ns - 1. Therefore, when thermodynamic equilibrium is established, the chemical potentials ( w ) of the various species fulfill the following relationships: hA,,(ads) - Er'L/&MS~,(ads) = MA,(b) - xr?/pL/h,(b) kJ
kJ
ws,,(ads) - Cmi\PSkiws,,(ads) = Pss(b) - CmiV%s,(b) kJ
k,i
(3)
(4)
In the simple case of adsorption of a substance A from a binary system of solvents S, and Sz with almost spherical molecules, the equilibrium relationships may be written in the form
- (rA/rl)Pl&l,(ads) - (rA/r2)p2PS21(ads) = PA@) - (rA/ri)PIPsl(b) - ( r A / r d P 2 ~ s ~ ( b()5 ) PsJads)
- P1ws,,(ads) - (r1/r2)P2ws2,(ads) = PSI@) - P,Ps,(b) - (h/r2)P211s2(b) (6) Ms,,(ads) = ms,,(ads), Ps,(ads) =
ads)
(7)
where rAadsorbate molecules replace 7 , molecules of the solvent S1 with probability P, and r2 molecules of S2with probability P2 = 1 - Pi. Equation 6 can be written as Ps,,(ads) - (r1/r2)lLs2,(adS) = PSI@) - (r1/r2)11s2(b) (8) If we now substitute eq 8 into eq 5 , we find PA(ads) - (rA/r2)&21(ads)
- (rA/r2)k2(b)
(9)
Therefore, we can regard that the adsorption process which takes place in the system under consideration is a coadsorption of the substances A and S, from the solvent S,. We can show, more generally, that the system of eq 3 and 4 has as a solution the following equations HA,,(ads) - r'%hSnn(ads)= Ps,,(ads)
- m:LPs,.(ads)
- r'%k3,(b)
= &b)
(lo)
- ~ L P s , ( ~ ) (1 1)
where S,, is one of the possible surface states of the solvent S,. From eq 11 we have Ps,,(ads) - n:/Pss,(ads) = PSI.@)- n:/Ps,(b)
(1 2)
If we multiply eq 12 by mGPLi and add over all the permissible values of k and I , we obtain Cmi\fi%s,,(ads) - C(mi',nf,')fihs,(ads) = Cmi$fi\ws,(b) - C(mi'd/)fi\Ps,(b)
( 1 3)
which results in eq 4 since
Zfi$=
1 and
mi:nf; = 1
(14)
It is similarly proved that eq 10 is a solution of eq 3. Therefore, we can assume that the processes which take place in a multicomponent ideally polarized interface may be represented by A,(b) r",,S,,(ads) A,(ads) + r;,S,(b) (15)
+
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987 103
Multicomponent Charged Interfaces
+ m$,,(ads)
* %(ads) + m%S,(b),
(k,
I)
Z (n, n)
(16) 111. Chemical Potentials The equilibrium conditions for the adsorbed phase may be obtained from eq 10 and 11, if we first calculate the chemical potentials of the adsorbed species. A significant contribution in this direction has been made in previous publications on the basis of the lattice In this approach, the adsorbed phase is considered as a two-dimensional layer of point dipoles which follow the geometry of a regular hexagonal lattice. Each adsorbed molecule may occupy either one lattice site (monomer) in the center of which the (ideal) dipole of the molecule is situzted or more (say r ) than one site (r-mer). In the latter case, if P is the dipole moment of the molecule which occupies r lattice sites, then 5t every site we accept the presence of an ideal dipole equal to P / r . Every adsorbed molecule is under the influence of the field of the interface, 4 m M ,as well as of the field X of the surrounding dipoles. In addition, the adsorbed molecules interact with the boundaries of the interface and between themselves with shortrange interactions. In the lattice model adopted here the molecules are distributed over the sites completely in random. The chemical potentials of the adsorbed species may be determined if, on the basis of the above model, we calculate separately the contributions arising from the entropy of mixing, the Coulombic interactions due to the dipolar nature of the adsorbed species, and finally the short-range interactions. Entropy Effects. In previous publications we have developed an approximate method for the calculation of entropy effect^.^'*^^ However, we have also shownz4that this method cannot be extended to the general case of a multicomponent interface, except for the two limiting cases where (a) the monolayer is composed of monomers and r-mers of just one type and (b) all the multisite species at the interface are flexible. In these interfaces the contributions from entropy effects to the chemical potential of the Ith species may be calculated from the following relationship^.^^ (a) monolayer with monomers and one type r-mers:
of allowed distinguishable ways in which an r-mer can be placed in the lattice after one of its end elements has been placed. If of the r sites which can be occupied by a single r-mer the 1, 2, ..., i ( i < r ) sites are already occupied by monomers and the i 1 site is occupied by an element of an r-mer, then the number of the available sites for the r-mer in the lattice is not equal to p but to p - b,. We denote b: = b,/p. ~p~ is referred to the pure state of 1 species at the interface. Finally, the symbol k # m denotes that the summation, in eq 17 and 18, is performed over all the species k = 1, 2, ..., N with the exception of the monomers. In the case where the adsorbed phase is composed of small molecules, eq 19 may be used without serious problems, since this equation may be considered as an acceptable approximation even in the case of rigid molecules.27 Alternatively, one may use the statistics developed by DiMarzioZswhich, in a multicomponent system, is extended as follows. We consider N completely rigid rods which are going to be placed onto a lattice. The permissible orientations are restricted to three mutually orthogonal orientations (j= 1,2, 3). We denote by NIJthe number of molecules of type i (i = 1, 2, ..., m) which are going to be placed in the direction j (= 1, 2, 3) and by r, the number of lattice sites occupied by the ith molecule. In order to calculate the number of ways, g(NIJ, No), to pack N molecules such that N,, of them lie in the directionj and there are No holes, we place the molecules in the following order. We place first the N,, molecules, one at a time, in direction 1, then the N12molecules in orientation 2, and lastly we place the remaining NI3molecules in orientation 3. In each orientation all the available molecules are placed following the order i = 1, 2, ..., m, and afterward we start to place molecules in the next direction. Suppose that we have already placed all the available molecules up to the k - 1 direction and in the kth direction up to i ( = 1 1)-th type. In addition, n/k molecules of lth type have been placed in the kth direction. The number of the available lattice sites for the first segment of the next lth type molecule is given by
+
m
G-1
I- I
, p i- fi,O,ent
1
1
r l
The expectancy that a site is unoccupied when an adjacent site in the kth direction is also unoccupied is
1
I- I
Therefore, the number of ways to place the (n/k + 1)-th molecule onto the lattice is (b) monolayer with flexible r-mers:
i= 1
(19) Here, the various symbols have the following meaning: ri denotes the number of lattice sites which are covered from the ith species at the interface. ai is the surface coverage of the ith species. If M is the total number of the adsorption sites per unit area and Ni the number of the ith type species, also per unit area, then we have i?i= riNi/M
(20)
qr is defined as follows. The product zq, denotes the number of pairs of neighboring sites of which one is a member of the group occupied by an r,-mer and the other is not, z being the coordination number of the lattice. For flexible r-mers we have q, = (zri - 2ri + 2 ) / z (21)
b{ is a constant depending upon the geometrical characteristics of the r-mer. It is defined as follows. We denote by p the number (26) Nikitas, P.J . Chem. SOC.,Faraday Trans. I 1984, 80, 3315.
which, after some algebra, results in (see ref 27)
In the case of a two-dimensional lattice with permissible orientations 1 and 2 parallel to the lattice and orientation 3 normal to it, eq 25 is reduced to
H(M - 2 ( r i - l)Nii)!
This case corresponds to adsorption of N molecules on a planar surface. The No holes may be considered as vacant lattice sites (27) Guggenheim, E. A. Mixrures; Oxford University: London, 1951. (28) DiMarzio, E.J . Chem. Phys. 1961, 35, 658.
Nikitas
104 The Journal of Physical Chemistry, Vol. 91, No. 1, 1987
when the adsorption process takes place from a gas phasezgor as monomer solvent (or even adsorbate) molecules when we have adsorption from solution-as in the case we are studying. If we regard that the distribution of the adsorbed molecules over the lattice sites corresponds to the most probable distribution, then eq 26 results in
z
mOij
E E-
Ojj
m
- + Oo In bo+ c8,,In
Oi3 (27) ri ri i= 1 with AAgk being the entropy contribution to the free energy of mixing. From eq 27 results that the entropy contribution to the chemical potentials of the adsorbed molecules is given by ,=li=1
..
U,,ent .
- U,,O,ent . ..
kT
In
291,
= In
/r/
(33) Here, X,,, is the electric field normal to the adsorbing surface at a site occupied by a monomer, caused by the surrounding dipoles, Pi is the normal component of the permanent dipole moment of the ith type species taken as positive when the positive pole of the dipole faces the electrode, a, is its polarizability, d is the nearest-neighbor distance, and z , is an effective coordination number. In previous publications, a value equal to 15.2 has been used for z,. 9~19*21-31-39 However, we have shownMthat this value is erroneous and ze is equal to 10.6, Le., close to Topping's value of 1 1.03.40 Short-Range Interactions. The contribution from these interactions may be calculated from the corresponding change of the free energy of mixing40 (34)
which results in p/sr
- rP."'
kT
kT (1
- E?OjZ)(
1-
+$,)
= -zqiNiAi' -4, 4r CNiNjqiqjAiJ (35)
Nqi+/
Under equilibrium conditions we have yLII= p12. Therefore, it is reasonable to expect that yrlent= ylzentis valid, resulting in Or1 = 812. Here, for simplicity reasons, we have used eq 19 for the analysis of the model. The use of other equations (17, 18, 28-30) is straightforward. Moreover, tests, performed at an indicative level, showed that qualitatively this choice does not affect the results. Coulombic Interactions. The contributions to the chemical potentials arising from dipolar Coulombic interactions have been studied in ref 18-21, 23, 24, and 30. These interactions are the resultant of the interactions between the adsorbed dipoles with the field of the interface and the dipole-dipole (permanent plus induced) interactions between the adsorbed molecules. Because at a multicomponent interface the dipoles of the adsorbed molecules may have a normal and a parallel component to the electrode, at every lattice site the field of the surrounding dipoles may be split into a normal and a parallel one in relation to the electrode surface. Thus, the dipole-dipole interactions may be considered as the resultant of the interactions of the adsorbed dipoles with these two fields. However, we have shown that the predictions of a lattice model become unsatisfactory as "parallel" interactions are taken into c o n ~ i d e r a t i o n . ~This ~ is attributed to a probable random distribution of the dipoles in real interfaces which is favored by the existence of energetic barriers. Thus, it seems realistic to take into account only interactions due to the normal component of the adsorbed dipoles. Under this approximation we obtain24
+
f/za/(4aaM)2 d3 r12 hX?( ze 41 +
i
