J. Phys. Chem. 1992,96, 8453-8461 that give rise to the model, are not predicted to be especially important in the overall energy transfer (in the same way that individual solvent molecule/particle collisions in macroscopic Brownian motion do not contribute much to the overall diffusive motion): indeed, in the most primitive version of the model, the average energy transfer is Seen to be completely independent of the highest-frequency modes of the substrate.1°
Conclusions Trajectory calculations and the BRW model agree with experiments on related systems: that there should be only a small isotope effect in the energy transfer between highly excited azulene and monatomic bath gases. This suggests that the low-frequency modes have an important role to play in determining the amount of energy transfer. In particular, the lack of any strong effect with helium suggests that classical calculations can provide an adequate representation, and the poor accord between experiment and trajectory results for light bath gas= is due to the poor description of the interaction potential. The results also predict that isotopic substitution should not have a strong effect on the fraction or the magnitude of supercollisions. Acknowledgment. The support of the Australian Research Grants Scheme is gratefully acknowledged. We also appreciative of interesting discussions with Professor John Barker and for his providing us with preprints of his work, as well as helpful interactions with Dr. Kieran Lim. Registry No. Xe, 7440-63-3; He, 7440-59-7; D2,7782-39-0; azulene, 275-51-4.
References and Notes (1) Tardy, D. C.; Rabinovitch, B. S . Chem. Reo. 1977, 77, 369. (2) Oref, I.; Tardy, D. C. Chem. Rev. 1990, 90, 1407. (3) Gilbert, R. G.; Smith, S.C. Theory of Wnimolecular and Recombination Reactions; Blackwell Scientific: Oxford, 1990. (4) Clarke, D. L.; Thompson, K. C.; Gilbert, R. G. Chem. Phys. Lerr. 1991, 182, 357.
8453
(5) Clarke, D. L.; Oref, 0.;Gilbert, R. G.; Lim, K. F. J. Chem. Phys. 1992, 96, 5983. (6) Lim, K. F.; Gilbert, R. G. J . Phys. Chem. 1990, 94,72. (7) Lim, K. F.; Gilbert, R. G. J . Phys. Chem. 1990,94,77. (8) Buck, U.; Kohl,K.H.; Kohlhase, A.; Faubel, M.;Staemmler, V. Mol. Phys. 1985, 55, 1255. (9) Buck, U.; Kohlhase, A,; Secrest, D.; Phillips, T.; Scoles, G.; Grein, F. Mol. Phys. 1985,55, 1233. (10) Gilbert, R. G. Inr. Reo. Phys. Chem. 1991, 10, 319. (11) Gilbert, R. G.; Zare, R. N. Chem. Phys. Leu. 1990,167,407. 112) Toselli. B. M.: Barker. J. R. Chem. Phvs. Lett. 1990. 174. 304. (13) Toselli,’B. M.;Brenner,.J. D.;Yerram, M:L.; Chin, W. E.; King, K. D. J . Chem. Phys. 1991.95, 176. (14) Toselli, B. M.;Barker, J. R. J . Chem. Phys. 1991, 95, 8108. (15) Date, N.; Ha=, W. L.; Gilbert, R. G. J. Phys. Chcm. 1984,845135, (16) Gilbert. R. G. J . Chem. Phvs. 1984.80. 5501. (17) Lim, K. F.; Gilbert, R. G. k . Chem. Phys. 1986,84, 6129. (18) Lim, K. F.; Gilbert, R. G.; Brown, T. C.; King, K. D. Inr. J . Chem. Kinet. 1987, 19, 373. (19) Lim. K. F. Ph.D. Thesis, University of Sydney, 1988. (20) Lim, K. F.; Gilbert, R. G. J. Chem. Phys. 1990, 92, 1819. (21) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics and Chemical Reacriuity; Oxford University: New York, 1987. (22) Brown, T. C.; King, K. D.; Gilbert, R. G. Inr. J . Chem. Kiner. 1987, 19, 851. (23) Hue, W. L.; Lim, K. F. Program package MARINER (a general
Monte Carlo classical trajectory computer program). Available from second author, Department of Chemistry, University of New England, NSW 2351, Australia. (24) Porter, R. N.; Raff, L. M. In Dynamics of Molecular Collisions; Miller, W.H., Ed.; Plenum Press: New York, 1976; Vol. B; p 1. (25) Yardley, J. T. Inrrduction to Molecular Energy Transfer, Academic Press: London, 1980. (26) Nakashima, N.; Yoshihara, K. J . Chem. Phys. 1983, 79, 2727. (27) Ichimura, T.; Mori, Y.; Nakashima, N.; Yoshihara, K. J. Chem. Phys. 1985,83, 177. (28) Ichimura, T.;Takahashi, M.;Mori, Y. Chem. Phys. 1987,114, 111. (29) Chao, R. S.;Khanna, R. K. Spectrochim. Acra 1977. H A , 39. (30) Hassoon, S.;Oref, I.; Steel, C. J . Chem. Phys. 1988, 89, 1743. (31) Lthnannsr6ben. H. L.; Luther, K. Chem. Phys. Lerr. 1988, 144,473. (32) vndvay, L.; Schatz, G. C. J. Phys. Chem. 1990.94, 8864. (33) Gilbert, R. G.; Smith, S.C.; Jordan, M. J. T. UNIMOL program suite (calculation of falloff curves for unimolecular and recombination reactions); 1991; available directly from the authors: School of Chemistry, Sydney University, NSW 2006, Australia.
Thermodynamics of Micellization at Charged Interphases within the Framework of the Phaseseparation Model P. Nikitas,* S. Sotiropoulos, and N. Papadopoulos Loboratory of Physical Chemistry, Department of Chemistry, University of Thessaloniki, 54006 Thessaloniki, Greece (Received: September 16, 1991)
An application of classical thermodynamics is made in the case of surface phase transitions occurring throughout a charged
interphase. This kind of surface phase transitions characterizes the aggregation of micelle-forming surfactants within the interphase formed between electrolyte solutions and an ideally polarized electrode, at least at concentrations higher than the critical micelle concentration. On the basis of a multilayer model for the electrical double layer, the thermodynamic conditions for its stability and the reversibility of a pbseparation proc~sswhich extends along the double layer are established. It is further shown that a rigorous but sufficient thermodynamic criterion for a surface micellization to take place across a charged interphase is the existence of two deformed peaks in the differential capacity versus applied potential curves on both sides of the adsorption maximum. The peaks may have comers and/or abrupt vertical segments and they may be split into two or more parts. Moreover, due to hysterisis phenomena their shape may depend on the potential scan rate and scan direction. Some expected deviations from the above criterion are also discussed.
I. Introduction The thermodynamic description of the properties of micellar systems in bulk aqueous solutions has attracted much attention for many years. The most crucial points have been the prediction of a surface tension minimum and the calculation of the ther0022-3654/92/2096-8453$03.00/0
modynamic functions of micellization. The interpretation of the interfacial properties of micellar solutions is usually based on either the phase separation’+ or the mass model. The possibility of the formation of a new phase consisting of micelles, above the critical micelle concentration Q 1992 American Chemical Society
8454 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
Nikitas et al.
(cmc), was hinted by ReichenberglO and rigorously formulated to a pseudoseparation model by Shinoda and Hutchinson? The mass action model is an alternative approach, which in fact applies the mass action law to the equilibrium between monomer 'and micelle species in the solution. A more general treatment has been attempted by Hall and Pethica,' who applied the small-system thermodynamics developed by Hi1112 to micellar systems. An interesting conclusion drawn by Hall and Pethica is that the phase-separation model is a convenient approximation of satisfactory applicability, especially when the number of nilonomers in the micelles is great enough. For this reason, this model is adopted in the present work to describe the properties of charged interphases during micellization. The study of adsorption of micelle-forming surfactants from aqueous solutions on the Hg electrode shows two interesting points: First, substances which are already able to form aggregates in bulk solutions are very likely to form aggregates on the electrode surface as well, since their surface concentration is usually very high. Second, the field effect along a charged interphase may or may not favor micelle formation within the interphase. In addition, it may lead the system to other types of surface phase transitions. Up to now there are quite enough experimental data for the adsorption of micellar systems on the polarized Hg electrode. The majority of them concern differential capacitance measurements (see for example refs 13-19). In contrast, according to our knowledge, there does not exist a rigorous thermodynamic treatment of the properties of the electrical double layer in the presence of micellar surfactants. For this reason we believe that the interpretation of the capacitance data was rather empirical, usually directed toward the correlation of the cmc to changes in the differential capacity curve^.^^-^^ In recent publications we have attempted an extensive application of classical thermodynamics to phase transitions taking place at various This experience in combination with the lack of any relative study of micellization, led us to attempt in this paper an extension of that work to interphases formed between an ideally polarized electrode and an electrolyte solution of a micelle-forming substance.
,, ., ,
,
,
,
,
,
.
,
, ,
,
,
, , , ,
, , , ,
*
.
,
. . ' " *
e
*
a
,
,
*
.
. , . a
*
'
"
"
'
*
"
:
INTERPHASE
,, .., ,
,
,
* d
* .
*
'
"
* * *
' " '
" ' "
Figure 1. Schematic representation of the multilayer model of a charged interphase.
for bulk phases. At interphases though, intensive properties such as the inner potential, the potential drop, or the chemical potential of ionic species do vary perpendicular to the electrode surface. Thus, a rigourous and detailed study of surface phase transitions which extend several molecular diameters along a charged interphase should be based on the use of multilayer models.26 According to these models the interphase is divided into a finite number of sublayers (Figure 1). When these sublayers are very thin, they are not expected to behave like autonomous phases.26 That is, their properties are significantly affected by the interactions betwen their species and those of the adjacent layers. Therefore, in this case we must first examine which are the independent specific or extensive quantities of each sublayer and the expression of the fundamental thermodynamic equation for each of them. (2) The Fundamental Thermodynamic Equation. We assume that the adsorbate consists of neutral molecules and that the surface area A is equal to unity. We next divide the interphase into p sublayers of finite and arbitrary, for the moment, thickness. We shall return to the restrictions imposed on the dimensions of each sublayer, when we will try to inspect the structure of the interphase in section V. For the specific internal energy of the ith sublayer, i.e., the internal energy per unit area, we may write du' = du/
+ dui
where27*28
11. Multilayer Model of a Charged Interphase
dui -- =
(1) Generalities. Consider the system of an ideally polarized
electrode/electrolyte solution, containing the micelle-forming surfactant A, the supporting electrolyte MX,and the solvent S. According to the phase-separation model, the formation of micelles in the bulk solution is a phase transition taking place when the solution becomes saturated with respect to the surfactant A. Then, if the temperature lies above the Krafft point of the surfactant, the latter, instead of precipitating as hydrated solid, turns into micelles? In a previous paper2' we have shown that thermodynamics predicts only two kinds of phase transitions at charged interphases, separation of the interphase into two new phases and two-dimensional condensation. The first kind of surface phase transitions occurs only at nonsaturated with adsorbate molecules interfaces and leads to the formation of either two saturated surface solutions of adsorbate in solvent and vice versa or a surface precipitate of adsorbate and a saturated surface solution of adsorbate in solvent. Since the precipitated adsorbate in the last case may exist in the form of micelles, surface micellization is a limiting case of the surface separation transitions. The thermodynamic criteria for a surface phase separation process to occur on an electrode surface are given in refs 20,21, and 23-25. However, the fact that the surface micellization is likely to be coupled with micelle formation in the bulk solution makes unclear whether those criteria can describe a surface micellization process. In addition, we should point out that this process is likely to extend throughout the interphase. In this case, we are facing the following problem: During a phase transition the values of the intensive variables of the system remain constant and uniform throughout the system, while those of the specific and extensive ones vary in an abrupt way?(t25This is strictly valid
,
, , ,, .. (. *. " " ' '
T
&' - P do' +
PA
dra
+
dI'& + &+ dI",+ + &-drk/*\
('1
and28 duL= SA(PdcrdA+ S p d p d V
a
(3)
Here cp is the inner potential at a certain point of the sublayer, is the volume charge density due to the presence of ions from the supporting electrolyte in the sublayer, cr is the surface charge density, and the other symbols have their usual meaning. Note that eq 3 is valid when a separation between surface and bulk charges has been adopted. The ith sublayer is not necessarily homogeneous with respect to the inner potential cp. Thus, it is neither homogeneous with respect to p i + and &-,since only the electrochemical potentials pM+ and px- are constant and uniform throughout the system at equilibrium conditions. In this case we must further divide the ith sublayer into m laminae of constant cp and consequently of constant pM+ and pr. Then, the last two terms of the right-hand side of eq 2 are written as p
-*
m
&+
dI",+
m
+ &-d&- = J'1E&+ d r h ++ C&- dr#-
(4)
j=1
In addition, the volume integral of eq 3 may be calculated from
Ivcp
m
m
m
j=l
j=1
j=1
'E#duJ = CFq4 dr&+- CFq' dr$-
dp dV '
In what concerns the surface integral of eq 3, if
d' and (8"
(5)
are
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8455
Micelle Formation at Charged Interphases
Figure 2. Schematic division of the interphase into parts A and B separated by the sublayer i with charge densities &, uM, and 8,respectively.
the inner potentials of the planes which contain the ith sublayer, then it is given by
SA~dudA=dd(oM+dP)+d+'daB=
d d(uM+ aA) - d+'d(uM+ uA+ 8) (6) where uMis the charge density on the electrode surface, 8 is that of the ith sublayer, and aA and uBare the charge densities of the part A of the interphase, which lies between the electrode and the ith sublayer, and the part B, which extends beyond that layer (Figure 2). Substituting eqs 2-6 into eq 1, we get du' = T ds' - P doi
+ Ad d(aM+ uA)- d+ld 8 + dr; + ps dI'i + pM+ dI",+ + px- d&- (7)
where
Ad = d - d+l
d Q + ps dr; + drL+ px- drk- (9)
jiM+
+
The expression for dus arises if eqs 1-3 are applied to the entire interphase. Note that in this case the surface integral of eq 3 is equal to @ d&. Similarly, for the parts A and B of the interphase we have dUA = TdrA- PdUA
+ @ duM- d d(aM+ aA) + p A dl?: + ps dI'$ + pM+ dr&++ px- dre- (10)
and dUB= TdsB-PdUB
+ d+' d(aM+ aA + 8)+ p A dr: + ps dr,B + pM+ dr&++ px- d r i - (1 1)
However, according to the properties of the multilayer the total differential of the specific internal energy dus may be written as P
P- I
dtJ
duS = j=1
+ jCdtJu+') = duA + duiA+ dui + duiB + duB =1
(12) where d is the specific internal energy of the jth sublayer without contributions from interactions with the adjacent layers and JU+l) the contribution to us arising from the j ( j + 1) subinterface. Therefore + du' + dU'B = duS - dUA - dUB = Tds' - P doi Ad d(uM uA)- d+'d 8 + d r i + ps dI'& + pM+ dI",+ + px- drk- (13)
+
where
+
i- 1
i- 1
j= 1
j=1
8 = P(rh+- rk-)
aA = F(Crb+- Cr&)
(15)
and consequently the intensive quantities T, P,Ad, pA?ps, pM+,and px- depend actually upon the ( 5 + 2i) variables SI, u', uM,FAi, r,i,ra+,rL+,..., rh+,rk-,rg, ..., rk. Thesevariables, though, are not independent. From eqs 7 and 15 we obtain
rsl,rh+,rk-) T~ = P(s2,u2, uM,FA2, rS2,rh+,rL+,rk-,ri-) T' = T~(s', d , uM,
FA1,
r;, &+, ..., rh+,rk-,..., rk-) (16) where P (j = 1, 2, 3, ..., i) denotes the temperature of thejth r' = T(s', vi, uM,F A i ,
sublayer. Similarly, general expressions can also be written for the quantities P,pA, ps, pM+,and px-. However, at equilibrium we have
(8)
is the potential drop across the ith sublayer. The interactions between the entities of the ith sublayer and its neighboring ones are not inherent in eq 7. To examine whether such interactions affect the expression of du' or not, we may work as follows: For the interphase as a whole, we may write duS = T ds" - P dos + @ duM+
It is seen that the only difference between eqs 7 and 13 is the definition of the specific entropy in the term T ds. That is, even if the contributions of duiAand duIBfrom the interlayer interactions of the ith sublayer with the two adjacent parts of the interphase are added to du', its expression does not change. (3) The Independent Variables. Equation 7 can be regarded as the fundamental thermodynamic equation of the ith sublayer. This equation shows that the intensive quantities T, P,Ad, +P1, pA, ps, pM+,and px- are, in general, functions of the specific variables si, vi, uM,uA, a', rAi,ri, F$+, I$-. However, uAand a' are given by
where n = A, S, W ,X.It is seen that there are 6(i - 1) equations with a total number of variables equal to 6i 1. Therefore, the number of the independent variables is 6i + 1 - (6i - 6) = 7. In this case we have a free choice of which particular specific quantities to select as independent variables. However, when we study the ith sublayer, it is quite reasonable to choose the independent variables among the specific quantities of this layer. Thus, we choose the variables si, ui, uM, FAi, rsi,rh+,and rk-, and PM+, therefore each intensive quantity, Pl = T, P,Ad, d", PA, 6, and px-, which describes the ith sublayer, can be expressed as
+
111. Stability Conditions and Conditions for Reversible Separation
When an infinitesimal change of the internal energy U of an open system is given by 9
V
i= 1
i= 1
6U = T 6s + CXidli + C p i dni
(19)
where Xi and li are the work coefficients and work coordinates, respectively, and we define the thermodynamic potential from
+
K
m
i= 1
i= 1
+ = U - TS - CXili- &ni
(K Iq, m I u)
(20)
then the criterion for equilibrium may be expressed as25927 6+10
(21)
provided that T, X1, ..., XK, IK+l, ..., 19, p1, ...,pm, n,+l, ..., n, are held constant. The equality sign of eq 21 corresponds to reversible processes of the system and the inequality sign to unnatural processes. That is, if the system is forced to proceed from its initial state to a neighboring one, then it returns spontaneously to the initial state. The initial state is then a state of stable equilibrium. Let us divide the interphase by means of a plane parallel to the electrode surface, the plane i of Figure 3, into two parts, say part A and B, and consider that a surface phase transition takes
Nikitas et al.
8456 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
(i)
!--
QA
BULK
W
4
W
Figure 3. Schematic representationof a phase separation process in part B, which leads to the subparts a and b with charges Q" and Qb,respectively.
place either in a certain sublayer of part B or throughout this part. For the internal energy of part B we may write (see eq 11) d V = T d S B - P d P + rBdA+ d d ( Q M+ QA) + pA dni ps dn: + p M + dnL+ + px- dng- (22)
+
where yBis the value of the surface tension which corresponds to part B, QMis the total charge on the electrode surface, and QA is the charge in part A. If we define the potential $B from $B
= V - TSB+ P P - pAni- p s g
+
+
dUA = T dSA- P d P dA p dQMd d(QM+ QA) + d& + ps d& + PM+ d&+
(23)
+ P d P + yBdA-
d d Q B- n i dpAn! dps - n&+ dpM+- ng- dpx- (24)
+ Px- d&(32)
and if @ is defined from
v=
VA - TSA + P P - #QM - pA& - psn& - pM+&+ - jix-4-
- pM+ n&+- px-ng-
then d$B = -SBdT
The last equality shows that during a reversible surface phase transition in the part B of the interphase the inner potential d remains constant when the charge of this part and consequently the electrode charge density fl vary. Moreover, since the location of plane i is arbitrary, we may identify it with the electrode surface, i.e., we may take 48 = p. Thus, we have reached the interesting conclusion that when a surface phase transition occurs reversibly somewhere in the interphase, the electrode inner potential p remains constant. Let us now suppose that a phase transition takes place in a certain sublayer of part A. Then we may write
(33) then d@ =-SA d T + P dP + YA dA- QMd# d(dQM+ dQA) - & dpA- ne dps - nb+ dpM+ - 4-dpx(34) and
because
(25) = -(eM + e") Now, consider that the surface phase transition in part B occurs under constant T, P, A, QB,pA, ps, and pMxconditions and it a2@ SA(&@ 6QM) (35) results in the instantaneous separation of part B into the subparts dA aQB a and b with area and charge equal to (A - 6A)/2, Q" = (QBprovided that the instantaneous phase separation occurs at constant 6QB)/2 and (A + 6A)/2, Qb = (Q"+ 6@)/2, respectively. Then the total change of $B due to this process is given by22*25929930 T, P, #,A, @ @, pA, h,and pMx values. However, we have already proved that when a reversible surface phase transition 1 a2+B 1 a2$B occurs within the interphase at constant T, P, A, pA, h,and pMx, s$B = - -(6A)2 - -(6QB)2+ -6A6QB then p remains constant. Therefore, a reversible phase separation 2 dA2 2 a(QB)2 dA aQB of part A or a certain sublayer of part A is described by (26) QB
+
+
+
According to (21) the separation process cannot proceed spontaneously when 6t,bB > 0. This inequality leads to the following criteria for stable
(5)
or
>o (37)
T,P,@#
where p denotes the set of chemical and electrochemical potentials pA, ps, pM+,and px-. However, from eq 24 we obtain
because d is independent of the area A. Therefore, the criteria for stable equilibrium of part B may be expressed as
which shows that d remains constant when fl and aA vary. Thus, we have proved that irrespective of where a reversible phase transition occurs inside a charged interphase, the inner potential of the electrode and the inner potential at any point of the interphase remain constant, whereas the electrode charge density uM and the charge densities of all the sublayers vary.
IV. Phase Transitions at the Multilayer Model Consider the ith sublayer of the interphase. The intensive properties of this layer are given by eq 18. Therefore, they may be expressed in general as PI = P,(xf, x i , ..., xf) (38) where 4 = si, vi, aM,I?;, I?i, I$+, rk and PI = T, P, Ad,pA, ps, pM+, px-. The differential change in PI, which is caused by an infinitesimal change in the values of the specific quantities is given by
4,
r dP,
The criteria for the reversible phase separation result from the equality sign of (21) and are given by
r
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8457
Micelle Formation at Charged Interphases The system of eqs 39 can be resolved with respect to 62,and we obtain21&27,31
3 6pM++ - 6px- (40) 0: oi, Dp7
where Di is a determinant of the elements ub and dp,is the cofactor of the element ui. In addition, if the system is studied at constant T, P, and pMx,then eq 40 is reduced to
I
(B) C
I
7
which yields
E
Figure 4. Plots illustrating the variation of (A) $ (=#, I$, PM+, rfr)and (B) C as a function of E, when a reversible surface phase transition is restricted to the inner layer.
(42) Note that for the derivation of eq 41 we have taken into account the validity of the Gibbs-Duhem equation &6pA
+ n:Sp:
= 0 (T,P,pMx= constant)
C
XS
(43)
for the bulk solution. When a reversible phase transition takes place in the ith sublayer at constant T, P, pA, pMX,then, according to the treatment presented in the previous section, A# remains constant, whereas x' varies from an initial to a final value. Therefore, a reversible pkase transition in the ith sublayer is described by
I
I
I
I
E
E
D; = 0 Moreover, during a reversible surface phase transition, @' and +4at every point of the interphase remain constant and therefore the applied potential E is also constant. In addition, the changes of 9; at constant E are very likely to cause changes in 4, Le., in the specific quantities of the entire interface. Therefore, the first of relationships (44)yields
When a micellar system is present in the bulk, then 6WA
=0
(46)
and therefore relationships 45 are still valid, provided that the chemical potential pA is eliminated from them. The experimental study of phase transitions at charged interphases has shown that the majority of them is restricted to the first monolayer, which is adjacent to the electrode surface, because there the adsorbate surface concentration is very high.32 In this case, relationships 45 show that the characteristic features of such a surface phase transition are the appearance of abrupt vertical steps in the plots of x' or $ vs E (Figure 4A). When 4 = uM, the vertical step in t i e plot of uMvs E makes the differential capacity of the interphase, C = duM/dE, tend to infinity during the phase transition. The general shape of the plot of C vs E at the transition region is given in Figure 4B. Now consider the case when a surface phase transition occurs at the successive layers of the interphase (Figure 5). According to (44),a phase transition at a certain sublayer takes place only when the potential drop across this layer attains a certain value. Thus, a surface phase transition is extremely unlikely to occur at the same value of the applied potential within all the sublayers.
2 :
INTERPHASE
BULK
Figure 5. Schematic representation of a surface phase transition extending to four successive sublayers of the interphase with the variation of the applied potential and the corresponding plots of (A) 4 vs E and (B) C vs E.
Note that since (PM and d , for every i, remain constant, not only Ad but also E is fmed during a reversible phase transition in the ith sublayer. If the interphase has been divided intop sublayers, then relationships 45 are valid not only at a single but for a maximum of p values of E, corresponding to Ad values which satisfy relationships (44). Thus, the plot of 2,vs E exhibits a stepwise pattem, where each step corresponds to a surface phase transition at a certain sublayer of the interphase (Figure 5A). In this case, the differential capacitance C vs E curve acquires the shape depicted in Figure 5B. If the number of sublayers at which a phase transition can extend is great and/or the value of Ad at which the transition of the ith sublayer occurs varia in a continuous manner from layer to layer, then the plot of YPvs E is expected to be quite smooth without vertical steps (Figure 6A). In this case, the existence of an inflection point in the uMvs E curve leads to the formation of a characteristic peak in the plot of C vs E (Figure 6B). Note that capacitance peaks are encountered in the stable region of a charged interphase. Finally, if there is a mixed variation pattem for Ad, for example, if Acp' differs significantly from Ad, i L 2 and Ad vary contin-
8458 The Journal of Physical Chemistry, Vol. 96, No. 21, 1992
Nikitas et al.
x s
UM
C
E Figure 7. Expccted uM vs E plots, according to the thermodynamics of small systems, when the miccllization is restricted to the first sublayer onto the electrode surfacc and the micclles consist of a finite number N of monomers increasing from curve 1 to 3.
UM
C \
L E
Figure 6. Plots illustrating the properties of .$ and C as a function of E during a reversible surface phase transition, which extends to a great number of sublayers with (A), (B) continuous variation of Ad and (C) a discontinuity between ApI and Ad (i 1 2).
uously across the sublayers which lie on top of the first one, then the capacitance peak is expected to be deformed, characterized by corners and/or abrupt vertical segments (Figure 6C). We should point out that plots similar to those depicted in Figures 5 and 6 have been experimentally detected. Thus, the surface crystallization of TI+ on a polarized Hg electrode proceeds discontinu~usly.~~ In contrast, the formation of a polylayer film by n-hexanol and n-heptanol molecules on the Hg electrode is continu~us?~ whereas deformed capacitance peaks are frequently observed during the adsorption of micelle-forming surfactants on Hg.13-I9 It is seen that the appearance of deformed capacitance peaks or a stepwise variation of the capacitance curves are criteria for a surface phase transition to extend along the interphase, forming polylayer or micellar films. Unfortunately, these criteria are not necessary for a multilayer surface phase transition to occur, since such a transition may also lead to the appearance of bell-shaped capacitance peaks if it continuously proceeds along the interphase.
V. Micellization at Charged Interphases Thermodynamics is an exact but phenomenological science. Thus, the treatment presented above is valid irrespective of the nature of the successive phase transitions at the sublayers of the interphase. It is valid when each layer is separated into two immiscible solutions of adsorbate in solvent and vice versa or when the adsorbate is expelled in the form of a surface precipitate or micelles. Moreover, from a thermodynamic point of view, micelles cannot be distinguished from aggregates or clusters of adsorbate molecules. For this reason we use the term micelles in this paper without a strict physical content for all the adsorbed entities composed of monomer adsorbate molecules. Note that the term micelle adsorption has already been used in adsorption studies of micelle-forming surfactants.19+3s-36 We should point out that the treatment presented in the sections above applies to micelles composed of a great number of monomer units. This is a prerequisite for the phase-separation model to be applicable. When the number of monomers in the micelles
UM
C
E
Figure 8. Plots of (A) u M vs A@ when the interphase consists of two discrete sublayers and the composed plots of (B) @ vs E and (C) C vs E.
is relatively small, then an improved picture about the structure of the interphase can be gained from the thermodynamics of small systems developed by Hill.l* According to it, when the number N of monomers in a micelle is not very large, then the abrupt vertical step in the plot of uM vs E (Figure 4) must be replaced by an %shaped segment with slope which decreases as N reduces (Figure 7). This means that (a) the micellization of small micelles is not completed at a single potential and (b) during this process the inner potentials at every point of the interphase do not remain constant. S-shaped curves of uM vs E (or Ad) entail bell-shaped capacitance peaks. However, if the number of sublayers, in which the transition takes place at different potentials, is small, then the final # vs E curve, which is made up from the sum of the partial uM vs Ad curves, may not be smooth enough. In this case deformations of the bell-shaped peaks are expected. For example, in the extreme case where the micellization can occur in two
The Journal of Physical Chemistry, Vol. 96,No. 21, I992 8459
Micelle Formation at Charged Interphases
C
C
E
E
Figure 10. Schematic plots of C vs E when a micellization process is a
restricted to the first sublayer on the electrode surface and the micelles have a finite (1) and infinite (2) number of monomer units.
b C
d e I h
h I
I k INTERPHASE
BULK
POTtNTl AL
Figure 9. Schematic representation of micelle formation throughout a charged interphase and the corresponding plot of C vs E when there is one discontinuity in the variation of the Ad values between i = 1 and i 2 2.
sublayers and at substantially different potentials, the capacitance peak of micellization is split into two parts (Figure 8). When the micellization can take place in more than two sublayers, then the merging of the individual peaks which correspond to micelle formation at each sublayer, may give rise to a split and/or deformed capacitance peak. In fact there are two deformed capacitance peab on both sides of the potential of maximum in adsorption, since the adsorbate surface concentration is a doublevalued function of E and therefore eqs 44 are valid at two different values of Ad. In this case in the potential region between these two peaks micelles dominate on the electrode surface. In contrast, outside the peaks the adsorbate can be adsorbed only in the form of monomers. Figure 9 shows the successive formation of micelles at a charged interphase and the expected shape of the capacitance curve for the entire polarization range. In this figure, (A, S) denotes a surface solution of A in S, which may contain ions of the supporting electrolyte, and M,, M denote micelles. The structure of the micelles which lie on the electrode surface is likely to be different from that of the micelles at the subsequent layers.37 Note that, according to the thermodynamic treatment presented in the previous section, when the number N of monomers in the micelles tends to infinity and a micellization process takes place along aa' (Figure 8) in the first sublayer, the potential drop A$ across the second sublayer remains necessarily constant. Thus, the vertical straight line cc'does not correspond to a simultaneous phase transition of the second layer. The second sublayer undergoes a reversible micellization along the path dd'. During this transition the potential drop Aq' of the first layer remains constant at the value which corresponds to the line bb'. Finally, we must point out that at concentrations well below the cmc an interfacial micellization process may be restricted to just one sublayer on the electrode surface. If the micelles consist of a small number of monomers, the capacitance curve is expected to exhibit two bell-shaped peaks (Figure lo), resembling those of the adsorption of a single organic substance in the stable region. When the number N of monomers in the micelles increases, the peaks become sharper and fmlly in the limit N a capacitance pit is formed (Figure 10). We should point out that the height and shape of the capacitance peaks of Figure 10 depend upon the number N. Therefore, the bulk adsorbate concentration is ex-
--
E
Figure 11. Hysterisis phenomena during a monolayer (A) and multilayer
(B)phase transition at an electrified interphase. pected to have a weak effect on them. The picture given above for monolayer and multilayer micellization at charged interfaces is strictly valid when these processes occur reversibly. However, when the controlled electrical variable is the applied potential E, via a potentiostat, a surface phase transition can hardly take place reversibly. Suppose that the interface undergoes a phase transformation following the scheme of Figure 4. When the interface reaches the state a, it jumps irreversibly to state a', because we are unable to control this process and make the system pass from the initial state a to its final one a' slowly. This process would be reversible if the charge density uM was controlled. In this case the transition process would proceed along aa' on both directions at any rate and extent. Therefore, the irreversibility of transition processes at charged interfaces is not the exception but the rule when the controlled electrical variable is the applied potential. The irreversibility of a surface phase transformation is closely related to the existence of metastable states predicted by thermodynamic~*~ at each sublayer of the electrical double layer. Due to these states a surface transformation rarely follows the path aa' (Figure 4) but other vertical paths bb' and cc', the position of which on the E axis depends upon the potential scan rate and scan direction. It is evident that along the metastable bb' and cc' paths eqs 44 are still valid, though at a value of Ad which m a y be smaller of greater than that of the reversible transformation. This has an interesting consequence. The conclusions drawn above for reversible phase transitions are valid also for irreversible processes. However, we should take into account the following: When a surface phase transition is restricted to the first monolayer on the electrode surface, the metastable paths bb' and cc' (Figure 4) result in the hysterisis loop bb'cc' in the capacitance plot of Figure 11A. The extent of this loop depends in general upon the potential scan rate. Very low scan rates give enough time for
8460 The Journal of Physical Chemistry, Vol. 96, No. 21, 199'2
Nikitas et al.
nucleus to grow, and therefore the vertical steps approach the reversible path aa'. When the surface transformation of the first monolayer extends to subsequent layers, the validity of eqs 44 indicates that a deformed and/or split capacitance peak will be developed in the place of the abrupt vertical step of Figures 4 and 11A. However, depending on the scan rate and scan direction, the values of Ad which satisfy eqs 44 are different. Therefore, in this case the hysterisis is reflected by the appearance of deformed capacitance peaks, the shape of which in general depends on both the scan rate and scan direction.
M. Micelles and Polylayers The thermodynamic study of a micellization process extended across a charged interphase, enriched with elements from the small system thermodynamics, shows that the most striking feature of this process is the appearance of two p i b l y deformed capacitance peaks on both sides of the adsorption maximum. The same feature may also characterize a polylayer formation. Thus, from a thermodynamic point of view, we can hardly determine whether a deformed capacitance peak is related to micellization throughout the interphase or to polylayer formation. This is because both processes consist of successive phase transitions occurring in different sublayers. Therefore, we must carefully examine this point, Le., the distinction between micellar and polylayer films, necessarily within the frames of extrathermodynamic assumptions. Such a reasonable assumption is that a micellar film is less compact than a polylayer film of closely packed monomers. Therefore, a polylayer formation is expected to lead to differential capacities smaller than those of a multilayer micellar film, especially in the case that the latter introduces a net charge into the electrical double layer. In particular, the differential capacity immediately after a deformed peak, must be less than ca. 3 pF/cm2 when a polylayer film is formed, whereas higher values are expected to characterize the formation of micellar films. The precise values of the differential capacity depend upon the specific characteristics of the system and can be determined only experimentally. Another criterion which may be used for the distinction between micellar and polylayer films is the split of a deformed capacitance peak. We have proved that this split arises from the fact that a micelle, unlikely a compact layer, is a microphase and the transitions in such small systems do not exhibit abrupt vertical steps when a specific quantity, like u', is plotted as a function of an intensive one, such as E (Figure 7). Therefore, the split of a deformed peak is also an indication of a micellar film extended across the interphase.
MI. Some Other Properties The treatment presented in the previous sections describes strictly the properties of interfacial micellization of neutral surfactants. However, it is easy to show that the micellization of ionic surfactants exhibits precisely the same features. Consider the ionic surfactant AM consisting of anions A- and cations M ' , which are common to those of the supporting electrolyte. Then, the fundamental equation 7 is still valid but with the electrochemical potential pA-instead of pA. In addition, each intensive quantity of a certain sublayer is a function of seven independent variables, which may be the specific quantities of this sublayer and the charge density. That is, eq 18 and consequently relationships 44 and 45 are valid irrespective of the ionic character of the adsorbate. This entails that the charge of the monomer units of an ionic surfactant does not affect the micellization properties described in the previous sections. We have proved above that the shape of the C vs E C U N ~ S and in particular that of the peaks may give useful information about the Occurrence of an interfacial micellization process. At concentrations below the cmc the location of the capacitance peaks depends upon the adorbate bulk concentration cA, as it clearly arises from the presence of the chemical potential wA in relationships 45. Here, we will show that the same is expected to be valid even at concentrations above the cmc, despite the fact that in this region pAis constant and independent of the amount of
CA
Figure 12. Plot of y vs cA at constant E when bulk micellization takes place irrespective of surface phase transitions.
the adsorbate in the bulk solution. If the entire interphase is subdivided into p layers, then the applied potential may be written as F-1
E=
EAd + cp"' - (pb + constant
i-I
(47)
*
Note that in this model each of the A,S M transitions occurs at a fixed value of Ad, determined by relationships 44, provided that lA has been eliminated. When micelles are present in the bulk solution and we add more surfactant A, then the micellar phase increases with respect to the saturated solution phase. This obviously causes a change in the inner potential cpb of the bulk solution which, according to eqs 44, 45, and 47, means that a new value of E is needed for A 4 to take the certain value needed for a phase transition to occur. Therefore, both above and below the cmc the location of the phasc transition peaks in the C vs E plots is expected to depend upon the surfactant bulk concentration, ck However, we should point out that the change of cpb with cA is not expected to be significant. That is, in general, there will be a rather weak dependence of C on cA at concentration above the cmc. Another thermodynamic parameter which plays an important role in adsorption studies is the interfacial surface tension y. The properties of this parameter during a micellization process result from the Gibbs adsorption equation and the treatment presented in section IV. Thus, we have the following: The Gibbs absorption equation26s3829 -dy = uM dE +
(rA
- rsnAb/nsb) dPA (T,p , PMX constant) (48)
in combination with eq 46 shows that when a micellization process takes place in the bulk solution, then irrespective of the simultaneous or not occurrence of a surface phase transition, the surface tension y attains a constant value when E is held constant. That is (87 / ~ c A 7,P,pMX.E ) =0
(49)
The expected plot of y vs cA is depicted in Figure 12. The variation of y with the adsorbate bulk concentration in the premicellar region is described by (ay/aCA)T,P.rMx,E
(50)
( ~ ~ / ~ P A ) T , P , ~ ~ ~ s ( ~ P A / ~ c A ) T , P , ~ M ~ P
However, eq 52 is valid only for pAvalues which differ from the transition one. At the transition value, which is defined from the second of relationships 44,the differential of eq 52 does not exist, since r A and rschange abruptly from an initial to a final value. Therefore, the plot of y vs cA (at constant E) is expected to exhibit an abrupt change of its slope at the value of cA which corresponds to the transition value of wA, since at that point the differential of eq 52 changes abruptly its value (Figure 13A). If the surface phase transition extends to several sublayers toward the solution, as surface micellization does, then the change of the slope in the y vs c curve is expected to be smoother (Figure 13B).
Micelle Formation at Charged Interphases
The Journal of Physical Chemistry, Vol. 96, No. 21, 1992 8461 metastable states may be observed. They appear with the dependence of the shape of the capacitance peaks on the potential scan rate and scan direction. In general, the differential capacitance is expected to be a sensitive parameter with respect to surface phase transitions. For this reason the shape of the capacitance curves may give useful information about micelle formation at charged interphases for a variety of experimental systems which undergo phase changes on electrodes.
Y
References and Notes Stainby, G.; Alexander, A. E. Trans. Faraday Soc. 1950. 46, 587. Matijevic, E.; Pethica, B. A. Trans. Faraday Soc. 1958, 54, 587. Hutchinson, E.; Inaba, A.; Bailey, L. G. Z . Phys. Chem. 1955,5,344. Shincda, K.; Hutchinson, E. J. Phys. Chem. 1962, 66, 577. (a) Jones, E. R.; Bury, C. R. Philos. Mag. 1927,4,841. (b) Bury, C. R. Philos. Mag. 1927, 4, 980. (6) Hartley, G. S.Aqueous Solutions of Paraffin Chain Salts; Hermann & Cie: Paris, 1936. (7) Murray, R. C.; Hurtlay, G. S. Trans. Faraday SOC.1935, 60, 202. (8) Tanford, C. J. Phys. Chem. 1974, 78, 2469. (9) Philip, J. N. Trans. Faraday Soc. 1955, 51, 561. (10) Reichenberg, D. Trans. Faraday SOC.1947, 43, 467. (1 1) Hall, D. G.; Pethica, B. A. Nonionic Surfactants: Schick, M., Ed.; Marcel Dekker: New York. 1967: Chabter 16. (1 2) Hill, T. L. Thermodynamics o i Small Systems, Benjamin: New York, 1963-1964; Vols. 1 and 2. (13) Eda, K. J. Chem. SOC.Jpn. 1959,80, 349; 1959,80,708; 1960,81, (1) (2) (3) (4) (5)
I
cA
or
E
Figure 13. Schematic plots of y vs cA (or E) when surface micellization takes place just at the first monolayer (A) and when it extends throughout the interphase (B) in the absence of bulk micellization in both CaSCS.
Similarly, it can be proved that in the premicellar region the plot of y vs E (at constant pA), exhibits an abrupt change of its slope at the transition potential, provided that the surface phase transition is restricted to a certain sublayer of the interphase. For a surface micelliition, which extends to a considerable thickness of the interphase, the corresponding change of the slope of the y vs E curve is expected to be quite smooth (Figure 13). It is seen that the plots of y vs cA (at constant E) and y vs E (at constant pA) can hardly give any useful information about a surface micellization process. Only the constant values of y as a function of cA at constant E can serve as a criterion for bulk micellization.
VIII. Conclusions Up to now the presence of micelles in the electrical double layer was supposed intuitively. There was not any rigorous criterion for a micellization to take place within an interphase, and for that reason the main effort was directed toward the empirical examination of capacitance curves and the relation of the correspondence peaks to the cmc.l6-l9 The present thermodynamic treatment does not relate the interfacial micellization to the corresponding process in the bulk solution. It may take place at concentrations below or above the cmc, and it is characterized by the appearance of two capacitance peaks on both sida of the adsorption maximumwhich may exhibit serious deformations. Thus,they may have corners and/or abrupt vertical steps. They may also be split into two or more peaks. These features of the capacitance plots and especially the split of a capacitance peak indicate undoubtedly the formation of micelles within the interphase, although well-formed, bell-shaped capacitance peaks may also correspond to an interfacial micellization process. Hysterisis phenomena due to the existence of
689. (14) Damaskin, B. 8.;Nikolaeva-Fedorovic, N. V.; Ivanova, R. Russ. J . Phys. Chem. 1960,34, 894. (15) Volhard, D. Colloid Polym. Sci. 1969, 229, 61. (16) Volhard, D. Colloid Polym. Sci. 1976, 254, 64. (17) Dofler, H.-D.; Muller, E. Tenside Deterg. 1976, 13, 322. (18) Dofler, H.-D.; Muller, E. Tenside Deterg. 1977, 14, 75. (19) Muller, E.; Emons, H.; Dorfler, H.-D. J. Colloid Interface Sci. 1980, 79, 567. (20) Nikitas, P. Electrochim. Acta 1991, 36, 447. (21) Nikitas, P. J. Electroanal. Chem. 1991, 300, 607. (22) Nikitas, P. J. Colloid Interface Sci. 1991, 144, 548. (23) Nikitas, P. J . Electroanal. Chem. 1991, 308, 63. (24) Nikitas, P. Electrochim. Acta 1992, 37, 81. (25) Nikitas, P. J . Electroanal. Chem. 1991, 326, 23. (26) Defay, R.; Prigogine, I.; Bellemans, A. Surface Tension and Adsorption (translated by D. Everett); Wiley: New York, 1966. (27) Munster, A. Classical Thermodynamics; Wiley-Interscience: London, 1970.
( 2 8 ) Kirkwood, J. G.; Oppenheim, I. Chemical Thermodynamics; McGraw-Hill: New York, 1961. (29) Guggenheim, E. Thermodynamics; North-Holland: Amsterdam, 1951. (30) Landsberg, P. Problems in Thermodynamics and Statistical Physics; Pion: London, 1971. (31) Tisza, L. Generalized Thermodynamics; MIT Press: Cambridge, MA, 1911. (32) De Levie, R. Chem. Rev. 1988, 88, 599. (33) Elliot, C. M.; Murray, R. W. Anal. Chem. 1976, 48, 259. (34) Baikericar, K. G.; Hansen, R. S.J . Colloid Interface Sci. 1985,105, 143. (35) Kaisheva, M. K.;Gyrkov, T. D.; Damaskin, B. B. Elektrokhimiya 19a5,21,832. (36) Naficy, G.; Vanel, P.; Schuhmann, D.; Bannes, R.; Tronel-Peyroz, E. J . Phys. Chem. 1981, 85, 1037. (37) Somasundaran, P.; Fuerstenan, D. W. J . Phys. Chem. 1966,70,90. (38) Bockris, J. 0. M.; Reddy, A. Modern Electrochemistry; Plenum Press: New York, 1970. (39) Aveyard, R.; Haydon, D. A. An Introduction to the Principles of Sutface Chemistry, Cambridge; University Press: Cambridge, UK, 198 1.
