Langmuir 1995,11, 4564-4567
4564
Application of a Two-State Adsorption Isotherm to the Electrosorption of Dodecyldiphenylphosphine Oxide on Mercury Verginia Vacheva,?Milliana Kaisheva,*lt and Panagiotis Nikitad Faculty of Chemistry, University of Sofia, Sofia 1126, Bulgaria, and Department of Chemistry, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece Received October 14, 1994. In Final Form: August 9, 1995@ The adequacy of a molecular model to describethe adsorption of dodecyldiphenylphosphine oxide (DDPO) at a mercury electrode has been checked; the model explicitly treats the electrostatic and nonelectrostatic interactions in the adsorption layer on a molecular level and is based on a two-state adsorption isotherm derived by Sangaranarayanan and Rangarajan. Estimates ofthe adsorptionparameters have been obtained on the basis of a method applying nonlinear regression analysis and an experimental sample for the decrease in the interfacial free energy with the addition of surfactant, as a function of electrode charge density and surfactant concentration. A critical discussion of the obtained parameters has been made. Finally an important restriction which should be imposed on adsorption parameters when applying the model has been elucidated and the peculiarities of the model have been shown if this restriction is not fulfilled.
Introduction The adsorption of organic substances a t electrochemical interfaces is one of the intensively studied fields of electrochemistry. For this reason many experimental techniques for studying adsorption on electrodes have been deve1oped.l However, despite the improvements in this field, the molecular properties of a n adsorbed film are still examined indirectly by treating the experimental data on the basis of molecular models. During the last few years many sophisticated molecular models for the adsorbed layer have been presentedU2a However, the great number of adjustable parameters which usually appear in those models does not favor their direct application to experimental data by the use of simple statistical techniques. This problem may be overcome if an advanced method like the one proposed4 on the basis of nonlinear regression analysis is adopted. The present work is a n attempt to examine with the help of this technique4 the adequacy of the model of Sangaranarayanan and Rangarajan5g6to describe the adsorption of dodecyldiphenylphosphine oxide (DDPO) a t a mercury electrode.
Model The analytical model used in this work to calculate the decrease in the interfacial free energy 6with the addition of surfactant, as a function of the charge density omand surfactant concentration c , consists of two equations. One of them is the adsorption i s ~ t h e r m : ~ - ~
* To whom correspondence should be addressed. University of Sofia. Aristotle University of Thessaloniki. Abstract published inAduance ACSAbstracts, October 15,1995. (1)Frumkin, A. N.; Damaskin, B. B. In Modern Aspects ofElectrochemistry; Bockris,J. O M . , Conway, B. E.,Eds.;Butterworth: London, 1964. (2) Guidelli, R. In Trends in Interfacial Electrochemistry; Silva, A. F., Ed.; Reidel: Dordrecht, 1986. (3) Nikitas, P. Electrochim. Acta 1987, 32,205.. (4) Kaisheva, M.; Kaishev, V. Langmuir 1985, 1, 760. (5) Sangaranarayanan,M.V.;Rangarajan, S. K. J. Electroanal. Chem. 1984 176,1-27. (6) Nikitas, P. EZectrochim. Acta 1985, 30, 1513. (7) Nikitas, P. J. Chem. SOC.,Faraday Trans. 1 1986, 82, 977. ( 8 ) Nikitas, P. J. Phys. Chem. 1987, 91, 101. @
+ In c + {(a,- a,)[4n(d3/ze)(d" - 6",,)12}/ { 2 1 2 ~ [ d+ ~ /a, ~ ,+ 8 (a, - a,)12}+ 2 ~ e ~= I'n [e!
In B
(1 - e)] +A", (1) and the other is the equation of state:
4 = -RW, ln(1 - 0) + {RWm[4n(aa - a,)(d3/ze)6(8 - 6",,)12}/ { 2 k ~ ( d ~+/ .a,)[d3/ze , + a, + e (a, - a,)12}RWJaS02 ( 2 ) where
B = Bo exp[-(pa
- pJ2/2(a,- a,)k?1/24.56
Bo = exp[(p", - p", + Us- U,)/kTl
(3) (4)
p a and p s are the normal components of the permanent dipole moments pertaining to the adsorbate and the solvent, respectively, a, and a, are the polarizabilities of the adsorbate and the solvent, respectively, c is the molar bulk concentration of the organic adsorbate, 8 = TK, is the surface coverage of the organic adsorbate, d is the nearest neighbor distance (taken as equal to the inner layer thickness), ze is the effective coordination number, Aas is a constant connected with the interactions in the adsorption layer, omis the charge density of the electrode surface, omma, is the charge density a t maximum adsorption, pa and p, are the chemical potentials of adsorbate and solvent respectively, and Ua and Us are the specific interactions (with the substrate) of the adsorbate and the solvent, respectively. Equation 1of the present paper arises from eq 11of the work of Sangaranarayanan and Rangarajan5 by adding the termAas(l- 28) with the purpose oftaking into account the short-range interactions when the Bragg-Williams approximation is adopted and the adsorbed phase behaves as a two-dimensional lattice mixture of solvent and adsorbed molecule^.^-^^ Equation 1 can be obtained as well from eq 476 after performing the appropriate sim~
~~
~
(9) Guggenheim, E. A.Mixtures; Oxford University Press London, 1951.
0743-746319512411-4564$09.00/0 0 1995 American Chemical Society
Electrosorption of DDPO on Hg
Langmuir, Vol. 11, No. 11, 1995 4565
plifications and assuming that the number rs of lattice sites occupied by one solvent molecule is equal to one and equals the number rA of lattice sites occupied by one adsorbed molecule. Similarly eq 2 of the present paper is obtained from eq 51 of the work of Sangaranarayanan and Rangarajad by adding the term -AaW for the shortrange interactions or from eq 4g6 if we let rA = rs = 1.
Statistical Analysis The adsorption parameters in the analytical model, described by eqs 1and 2 were estimated using a method based on nonlinear regression a n a l y ~ i s .The ~ analytical model in the present work is given by eqs 1and 2 in which eq 1provides 8 as a function of the parameters P , of the bulk surfactant concentration c, and ofthe electrode charge density am. For the computation of q5 for currently given values of am,c, and P , the subroutine FUN was coded in PAR. In the subroutine FUN eq 1 was numerically solved with respect to 8. For that purpose the difference f was calculated for a given 0 between the left-hand side of eq 1, denoted by L
L = In B
+ In c + { ( a ,- a,)[4n(d3/z,)(d^- d",,)I2}/ {2m[d3/2,+ a, + e (a, - a,)12>+ ZA"' e
and the right-hand side of eq 1, denoted by R
+
R = ln[8/(1 - e>] A"' For the values 8*, for which eq 1has a solution, L(8*) should be equal to R(8*)or
After finding the solution 8* of eq 1for the given initial values ofam,c , and P , 8" is introduced in eq 2 and 4 is then computed with the same values of 8*, am,c, and P and is compared to the respective values of q5 in the experimental sample {&, amL, and ci}. The estimates P* of the parameters were found, for which the following condition holds: N
a, > 0, a, > 0, and d3/2, > 0 and
a, 5 3 x
(8)
cm3
Restriction 8 was introduced by us on the basis of the suggestion of Levine et al.13 to calculate molecular polarizabilities from the Laplas-Debye relation E
=1
+ 4n(M/A)a/l
(9)
assuming that the dielectric constant E in the inner part of the electrical double layer is about 6, the thickness of the inner layer is about 6 x loT8cm, and the number of adsorption sites per unit area MIA is about 8 x 1014 molecules/cm2. The necessity for restriction 7 will be discussed later in this work. Another consideration from a physical point of view might be that a, should be greater than a,,otherwise the adsorption maximum is shifted to infinity. In other words the polarizability of the solvent (methanol) should be greater than that of DDPO. We, however, did not restrict the parameters on the basis of this latter consideration and nevertheless results were in accordance with it.
Results and Discussion In the present work the analytical model given by eqs 1and 2 was applied for the estimation of the parameters for dodecyldiphenylphosphine oxide (DDPO) adsorption on a mercury electrode, using a n experimental sample of 60 values of 4 as a function of c and om. Electrocapillary measurements were performed14 in 0.1 M methanol solutions of LiCl with the addition of DDPO in different concentrations. The regression analysis carried out for four concentrations of DDPO (0.04,0.02,0.01,and 0.005 M) gave the following estimates of parameters of the Sangaranarayanan and Rangarajan's model:
P, = a, - a, = -2.99 P, = a, = 3.00 x
x
lop2, f 0.01 x lo-',
lo-',
P, = d3/2, = 0.4 x N
(7)
lo-',
cm3
f 0.01 x
lo-',
cm3
f 0.1 x
lo-',
cm3
P, = AaS= 0.7 f 0.3 P, = In B = 7.48 f 0.05
P, = R l T ,
= 3.27 f 0.01 erg/cm2
The estimates P* were computed using the program for nonlinear regression analysis PAR from the statistical package BMDP12 by minimizing the difference between predicted and experimental values of $I with respect to parameters. For that purpose the derivatives of the sum in eq 6 with respect to the parameters were numerically found. In the process of estimation the variation of some of the parameters was limited. So the following restrictions on the parameter estimates were imposed by us in the program PAR, based on ideas about their physical meaning:
To determine the adequacy of the model to describe the experimental sample the criterion, F, of F i s h e P was used. According to this criterion the model is adequate for use with the experimental sample with this set of parameters, although the mean square error Sre2,computed on the basis of the comparison between predicted and measured values of @, was 0.41 and was slightly higher when restriction 8 was imposed than in the case without such a restriction. The experimental error was Se2(25)= 0.25. Comparing the value of F
(10)Barker, J. A. Lattice Theories of the Liquid State; Pergamon Press: Oxford, 1963. (11)Prigogine, I.; Bellemans, A,; Mathot, V. Molecular Theory of Solutions; North-Holland: Amsterdam, 1957. (12)Dixon, W. J., Ed. BMDP Statistical Software; University of California Press: Los Angeles, 1981.
(13)Levine, S.; Robinson, K.; Smith, A. L.; Brett, A. C. Discuss. Faraday SOC.1976,59,133. (14)Nikitas, P.; Pappa-Louisi,A.; Jannakoudakis,D. J.Electroanal. Chem. 1984,162, 175. (15)Beck, J.; Arnold, K. Parameter Estimation in Engineering and Science; Wiley: New York, 1977.
4566 Langmuir, Vol. 11, No. 11, 1995
with the tabulated15 value F0.99 = 1.87 for a significance level = 0.01 and degrees of freedom V I = 40 and u2 = 25 it was shown that F < Fo.99, consequently the model is adequate to describe the sample. The hypothesis of normality of the error was confirmed by the graphs of distribution of residuals. The estimate obtained for the second parameter a,takes the highest permitted value and is about two orders of magnitude greater than the parameter a,. An obvious tendency is observed for the parameter a, to become infinitely small in comparison with a,, in other words to tend to zero. This result is in correlation with the physical assumption of a much greater polarizability ofthe solvent methanol in comparison to the very small polarizability of the adsorbate DDPO. When the restriction a, 5 3 x cm3 was not introduced, the obtained estimate of this parameter had a n inappropriate value, which was about two orders of magnitude greater. The regression description in the latter case was slightly better, the residual sum of squares being smaller than in the case with a restriction. Thus a conclusion could be made that a, has a weak effect on the treated model and might better be looked upon as a n adjustable parameter as suggested by Levine,16 not significantly influencing the adequacy of the model to calculate the data. The sixth parameter has almost the same value as the one obtained13 using the Frumkin adsorption isotherm, where rmequaled 1.32 x mol/cm2 and the area per molecule of DDPO was 1.26 nm2. The ability of the model to predict the dependence of 4 on c and omis illustrated for some of the experimental results in Figure 1. Points represent the experimental values of 4 for the respective concentrations of DDPO and solid lines predictions by the model. The dependencies of 8 on amcalculated by the model with the same parameters and for the same surfactant concentrations as in Figure 1 are shown in Figure 2. The positive value of the short-range interaction parameter A", implies that adsorbate-solvent attraction in the interface is weaker than half the sum of the adsorbate-adsorbate and solvent-solvent attractions. The estimates of the parameters for DDPO adsorption on a mercury electrode from methanol solutions, obtained in this work, are within the intervals outlined5 for such parameters on the basis of general considerations. Another purpose of the present work was to study the influence of the different parameters and the parameter interval of applicability of the model given by eqs 1and 2. It was shown in this study that the condition which should be imposed on the parameters for each value of 8 in the whole interval between 0 and 1 in order not to obtain incorrect parameter sets is that the denominator of L should not come close to zero, i.e.
otherwise, three solutions for 8 might be obtained for a given electrode charge density amand surfactant concentration c. Here 6 > 0 is the interval of denominator values, in which three solutions for 8 exist. These three solutions are obtained in the region of um,where 4 has a = -3.7ptC.~m-~).This is especially maximumvalue (omma true for the higher concentrations ofthe surfactant. Ifwe (16)Levine, S. Discuss. Faraday SOC.1975, 59, 170.
Vacheua et al.
U
0
B I
L'
r.J7).lc.cz Figure 1. The decrease in the interfacial free energy 4 with the addition of surfactant as a function of electrode charge density ampredicted by the model based on the isotherm of Sangaranarayanan-Rangarajan (solid lines) for four concentrations of dodecyldiphenylphosphine oxide (DDPO): (1)4 x M, (2) 2 x M, (3) M, and (4) 5 x M. Each point represents an experimental4 value. The estimates of the cm3; a, = 3 x parameters are (a, - a,) = -2.99 x cm3;d3/2,= 0.43 x cm3;Aas = 0.7; In B = 7.47;R l T , = 3.27 erg/cmz. -L
-2
0
i
2
i G?$.cm-' Figure 2. Plots of surface coverage vs umcalculated by the model of Sangaranarayanan-Rangarajan for the following concentrationsof dodecyldiphenylphosphine oxide (DDPO): (1) 4x M, (2) 2 x M, (3) M, and (4)5 x M. The parameters used are as in Figure 1. -L
-2
0
2
denote the left side of eq 11byg(81, it can be shown after simple transformations that when d3/2,, a,, and a, are positive, 0 5 8 5 1, and a,< a,
+
+
(d3/2, aaY Ig(e) 5 ( d 3 / . , a,12 The left inequality combined with (11) leads to the being smaller conclusion that the quantity (d3/2, + than g(8), should not become smaller than a certain positive number 6 from eq 11in order to have a unique solution for the isotherm (1). In an analogous way it can be shown that if a, > a, with all other assumptions remaining the same, then
and now the quantity (d3/2,+ a,I2cannot accept values smaller than the certain small positive number 6from eq 11. An example of an incorrectly chosen set of parameters leading to three solutions for 8 of eq 1 is the following:
Electrosorption of DDPO on Hg
4
~
Langmuir, Vol. 11, No. 11, 1995 4567
15
I\
IC
'E i E
Gi 5
C
I Figure 3. Calculated plot off = R - L from eq 6 vs 8 for a concentration of DDPO c = 4 x M and a charge density am= -3 pC/cm2.The values of parameters used are (a,- a,) = -3.7 x cm3; a, = 2.8 x cm3;d3/z, + a, = 3.1 x cm3;Aas = -0.4; In B = 7.3; RZT, = 3.6 erglcmz. (aa- a,) = -3.7 x
cm3, a, = 2.8 x 10-23 cm3, (d31., cm3,Aa*= -0.4, In B = 7.3, and RZT, = 3.6 erglcmz. Calculations using these parameters lead to the results illustrated in Figure 3, where the dependence of the difference f between the right-hand side R and the left-hand side L of eq 1 on 8 at a charge density am= -3 pC/cm2 and concentration c = 4 x M is shown. Indeed, condition 11 is not fulfilled for this set of parameters, of amand c, since
+ a,>= 3.1 x
+ + 0 (a, - a,)]= 3.1 x
[d3/2, a,
- 3.7 x
Figure 4. The change +in surface tension vs electrode charge density om predicted by the model of SangaranarayananRangarajan (solid lines) for six concentrations of dodecyldiM,(2) M,(3) phenylphosphine oxide (DDPO): (1)5 x 5x M,(4) M,(5)2 x M,and (6)4 x M.Each point represents an experimental+value."he parameters used are the same as in Figure 3. transition; however, i t has nothing to do with it. We have carefully examined this problem and have found that the loops in the 9 us amplots of Figure 4 are associated with loops in the 8 us amplots, totally different in shape than the phase transition loops. Also no oscillatory phenomena were observed in the system as might happen if the latter case were relevant. It was shown in the present study that in order to avoid such unrealistic situations, the followingrestriction should be imposed on the estimates of the parameters:
+ a,) # O
8 and tends to 0 when 8 = 0.837 8378 .... This leads to a deep minimum in the dependence off on 8 as seen in Figure 3, which results in three solutions of eq 1 with respect to 8 for the given amand c. The same set of parameters leads to only one solution for 8 in the case of the lower surfactant concentrations. The three solutions for 6 of eq 1 respectively lead to three solutions for 9 of eq 2. This is illustrated in Figure 4 for the same incorrect parameters as in Figure 3. The regions of the three solutions of eq 1, resulting in the three values of 4 for the same c and am,can be easily seen near the maxima of the curves. These regions are symmetrical with respect to amma, and become broader with the increase of c. The peculiar behavior of the model given by eqs 1 and 2 in the region of amm,resembles the loops of a phase
(d3/2,
(a, - a,)
(12)
whenever using the model given by eqs 1 and 2. The fact that isotherm 1 has the peculiarity described above due to its denominator hampers to a certain extent the practical application of models based on this isotherm because of the necessity of imposing restrictions on the parameters. Nevertheless we consider it possible to use the physical meaning of the parameters in order to place narrower a priory limits on their values and be able to successfully apply such models in practice.
Acknowledgment. M.K. andV.V. wouldlike to thank the Bulgarian National Scientific Research Fund for the financial support of this work by Grant No. 334193. LA9408026
