760
Langmuir 1985,1, 760-763
tension of "feetnfrom the particles; (2) detachment of some "feet" and their extension to form films around the particles or extension of the undetached feet; (3) coalescence in patches of the films; (4) further extension of the coalesced films. The main message of the paper is the observation that
redispersion does not occur merely via the emission of single molecules from the oxidized crystallites to the surface of the substrate but via a more complex process that involves wetting and spreading. Registry No. Ni, 7440-02-0; A1203, 1344-28-1.
Estimation of Adsorption Parameters in Two Models of Differential Capacity. A Comparison Based on Nonlinear Regression Analysis M. K. Kaisheva Department of Physical Chemistry, The University, Sofia 1126, Bulgaria
V. K. Kaishev" Laboratory of Computer Stochastics, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria Received April 19, 1985. I n Final Form: July 7, 1985 The application of nonlinear regression analysis is proposed to estimate the adsorption parameters in analytical models of differential capacity, to compare the models, and to select the one adequately describing the adsorption process. Two analytical models of differential capacity, based on Frumkin and Hill-de Boer isotherms, have been used for estimation of adsorption parameters. The ability of the two models to predict adequately the differential capacity dependenceon electrode potential and surfactant concentration for the system stationary mercury electrode-aqueous 0.05 M Na2S04solution of dodecyl hexaoxyethylene glycol monoether has been compared. The estimates of parameters for the adequate model, based on the Frumkin isotherm, are discussed.
Introduction One approach to modeling adsorption phenomena is that of fitting experimental data by a spline regression.' The spline model is linear in parameters and can be easily evaluated and conveniently applied for prediction, but its parameters do not carry physical information. Another possible approach to this problem is the construction, on the basis of physical ideas, of analytical models of the studied property. Using different adsorption isotherms, many analytical models of differential capacity can be derived as a function of electrode potential and surfactant concentration. It is proposed in the present investigation to use the method of nonlinear regression analysis in order to check the adequacy of capacity models and to choose the one adequately describing the investigated system. To illustrate this, two models, based on the isotherms of Frumkin2 and Hill-de Boer3 (representing two views on adsorption), have been chosen. The two analytical models of differential capacity, constructed on the basis of the two above mentioned isotherms, are systems of three equations, nonlinear in unknown parameters characterizing the electrical double layer. Since the adsorption of surfactants at the mercury-solution interface is not localized, the application of Frumkin isotherm to these systems has been put under q ~ e s t i o n . On ~ the other hand the isotherm of Hill-de Boer has been considered as better describing the adsorption of organic substances on mercury electrodes. It is important to know which isotherm describes better the studied adsorption process, supposing a set of experimental data exists. For this purpose the least-squares
* Author t o whom correspondence should be addressed. 0743-7463/85/2401-0760$01.50/0
estimates of the unknown parameters in the two analytical models have been found on the basis of a sample of differential capacity measurements for different electrode potentials and surfactant concentrations. Thus this work is an attempt to treat the models not deterministically,but as stochastic ones, (cf. ref 5, 6) and to investigate the influence of the measurements' error on the statistical quality of the estimates obtained. A statistical analysis, aiming to compare the ability of the two analytical models to predict adequately the studied differential capacity dependence on electrode potential and surfactant concentration, has been carried out. The investigated adsorption system is a stationary mercury electrode-aqueous saline solution of dodecyl hexaoxyethylene glycol monoether.' For the solution of the above stated problem the BMDP statistical package7is used. Physical Models The equations for the dependence of the differential capacity C on electrode potential E and surfactant con(1) Kaisheva, M.; Kaishev, V.; Matsumoto, M. J.Electroanal. Chem. 1984, 171, 111. (2) Frumkin, A. N.; Damaskin, B. B. In "Modern Aspects of Electrochemistry"; Bockris, J. OM., Conway, B. E., Eds.; Butterworth: London, 1964. (3)de Boer, J. H. "The Dynamic Character of Adsorption"; Calendon Press: Oxford, 1953. (4)Parsons, R.J. Electroanal. Chem. 1964,7, 136. (5)Kaisheva, M.;Kaishev, V. Ann. Sofia Uniu., in press. (6)Stenina, E. V. Elektrokhimiya 1982,18, 1349. (7) Dixon, W. J., Ed. 'BMDP Statistical Software"; University of California Press: Los Angeles, 1981.
0 1985 American Chemical Society
Estimation of Adsorption Parameters ~
Langmuir, Vol. 1, No. 6, 1985 761
centration c in both of the models treated in this work are based on the physical idea about the existence of two parallel condensors2 at the interface: u
= ~ o ( -l 8)
+ d8
(1)
Here u is the overall charge density, uo the charge density of the electrode, when not covered by surfactant, 8 the coverage, and d the charge density of the electrode when entirely covered by organic molecules. The electrode potential E is given with reference to the potential of zero charge in pure supporting electrolyte solution, equal8 to -0.47 V vs. normal calomel electrode. The energy of adsorption of 1mol of surfactant due to the electric field of the double layer is accepted2 to be
W = [C@/2 - C’E(E/2 - EN)]/^,
(2)
Co and C’in this equation are the differential capacities at 8 = 0 and 1, correspondingly, EN is the potential difference, arising as a result of the oriented adsorption of the organic dipoles, and rmis the maximum number of moles of adsorbed substance per unit area. The sign of E N in (2) depends on the orientation of the molecular dipoles and is chosen to be positive when the latter are turned with the positive pole to the electrode surface. E N coincides with the shift of the potential of zero charge when 0 changes from 0 = 0 to 1. Since C, is a function of E and its dependence on electrode potential can be precisely measured by a dropping mercury electrode; instead of C a 2 / 2 in (2), the integral . f f u o dE is usually usedg where uo = j f C o dE. .ffuo dE is the energy per centimeter squared of the electrical double layer in pure electrolyte solution. The influence of the electric field on adsorption is expressed2 by the bulk concentration of the surfactant, Cos, necessary to reach the same 8 as at E = 0 for a given E:
It will further be accepted that the charge density u is a function of electrode potential and coverage. By use of a procedure precisely described in ref 9, 2 the following analytical equation for the differential capacity is obtained C = Co(1 - 8) C’8 +
+
In such a way eq 4,6, and 7 form the first analytical model of differential capacity applied in this work for estimation of adsorption parameters, using a given sample of experimental data. As a basis of the secondanalytical model of differential capacity, the modified Volmer equation of state of the interfacial layer is used
which is a two-dimensional analogue of van der Waals equation (cf. ref 4). Here 4 has the meaning of a twodimensional interfacial pressure, a ’= RTI’,a, R is the gas constant, and T i s the absolute temperature. Equation 8 leads to an isotherm of Hill-de Boer type
BZ = 1/c0.5,~exp(a - 2) derived for the description of nonlocalized adsorption of gases on solid surface^.^ The combination of eq 3 with the corresponding expression for B2 in (9) gives for the dependence of the equilibrium constant B2on E the equation B, =
r
1
The tirst index of co8 denotes the value of 0 and the second the value of E. Frumkin’s adsorption isotherm2 (4) is used as the basis of one of the models treated in this study. B1 =
1/C0.5$
exp(a)
(5)
is the equilibrium constant of adsorption and a a constant, accounting for attraction or repulsion of the molecules. The combination of eq 3 and 5 gives the following expression for Bl B1 =
The concentration ~ 0 . 5 . 0in (5) and (6) is the bulk surfactant concentration a t E = 0, for which 8 = 0.5. ( 8 ) Grahame, D.C. Chem. Reu. 1947, 41, 441.
(9)
Hansen, R.S.;Minturn, R.E.;Hickson, D.A. J. Phys. Chem. 1966,
60, 1185.
x E u o dE - C’E(E/2 - E N )
1
Equations 9, 10, and the model of the two parallel condensors lead to the following expression for the differential capacity C = Co(1 - 8) C’8 +
+
[ J E C o dE + C’&
- E)]’
8(l - 8)’
(11) 1 - 2a8 + 4 ~ -82 ~~ 8 ~ Rmm derived in an analogical way as eq 7 but with different physical premises concerning the isotherm. In such a way (4), (6), and (7) on one hand, and (9)-(11) on the other, form two systems of equations describing in two different ways the differential capacity as a function of E and c. Unknown parameters in these two systems are ~ 0 . 5 , 0 ,C’, EN, I’,, and a. It is possible, using expressions 4 and 6 or 9 and 10, to calculate 8 for a given c, E , and certain initial values of the unknown parameters. By insertion of 8 obtained from (4) and (6) in (7) or 0 obtained from (9) and (10) in (11)the differential capacity according to the two different models can be evaluated.
Experimental Sample The results of the experimental investigation precisely described earlier’ are used in the present study. The differential capacity of the electrical double layer has been measured on a stationary mercury electrode for solutions of dodecyl hexaoxyethylene glycol monoether (C12E6)in
762 Langmuir, Vol. 1, No. 6, 1985
Kaisheua and Kaishev
different concentrations. The supporting electrolyte was aqueous 0.05 M NaZSO4.It has been established' that for potentials more positive than +0.3 V and more negative than -0.7 V adsorption-desorption phenomena are taking place in some of the solutions of C12E6.The maxima on the C,E curves, connected with these phenomena, are not equilibrium ones. That is the reason why in the experimental sample only the values of C in the interval -0.7 < E < 0.3 V are taken into consideration. The concentrations of C,,E, solutions varied from to M; i.e., some of the higher concentrations were above the critical micelle concentration (cmc), which for the studied system is approximately 7 X mol dm-3.1 Thus an experimental sample S = (Ci,Ei, ~ containing ~ l N~ = 110 = experiments ~ ~ was obtained. Since according to the physical ideas of the discussed models the latter correspond to the investigated phenomena only for concentrations of the surfactant lower than the cmc, only part of the data, 66 experiments, was used for parameter estimation. In the final sample the concentration interval was 5 c 5 5 x lo+ mol dm-3. As an unbiased estimate of the "pure error" variance, arising from the experiment, the mean square Se2was applied and determined to be 0.109.' Se2was estimated from the repeated observations in 24 points of the experimental design.
Parameter Estimation In order to estimate the unknown parameters of the double layer, the following approach is used both physical models are treated as stochastic ones; i.e., the random nature of the experimental error in capacity measurements is taken into consideration. Let us rewrite the equations of both models, namely, (71, (4), and (6) and (111, (91, and (10) respectively as I
C = f l ( E ,0, p , Cot1') + e B1c = f & P)
Bi
(12) (13)
= f3(E,p , I z )
(14)
I1
C = cel(E,0, p , Co, Ill+ e
(15)
Bzc = d o , 8)
(16)
B2 = ' ~ 3 0 3 8, , 12)
(17)
The system of equations I corresponds to the model based on the Frumkin isotherm and the system I1 to the model based on the Hill-de Boer isotherm. The functions f l , f z , f 3 , PI, 'p2, and 'p3 in the above equations are given by (7), (4), (6), (10,(9), and (lo), respectively, while e N(0, u,2) is supposed to be a normally distributed random variable, correspyding to the error, with zero mean and a variance u,2. P is the vector of unknown parameters: 9 = ( c ~ . ~C',, ~ E, N , rm,a)'
-
= .f&, dE and I2 = J f u 0 dE are known functions of E. 0 is an implicit function of E and c, which is given by formulas 13 and 14 in I or 16 and 17 in 11.
c_loseness_ofmodel to data. The two vectors of parameters Pland p F 2 should be found, for which
and
are retched. Here Oi is the value of 0 for Ei, ci, and a fixed P. The least-squares criterion has been accepted, since the LS estimates possesslo such good properties as consistency and efficiency, are unbiased, and can be comparatively easily found. However, some other criteria for closeness of model to data could be used as well.
Computation of Estimates of The estimates $l and &z, for which the conditions (18) and (19) hold, are computed using the program for nonlinear regression analysis PAR from the statistical package BMDP.7 In the program PAR a pseudo-Gauss-Newtonian algorithm for nonlinear optimization is realized. The program does not need explicit information about the derivatives of the response function with respect to the parameters. For the computation of capacity in each of the models, I and 1%for currently given values of E , c, Co, 11,Iz,and vector P,the subroutine FUN is coded in PAR. In the two subroutines, corresponding to the two models, the equations In Blc
0 + 2a0 = In 1-0
and In B,c
0 1 + 2aO = In +1-0 1-0
(21)
are numerically solved in respect to 0, B1, and B, being calculated from (6) and (10). Taking into consideration that 0 I0 I1and a < 5 it is not difficult to conclude that (20) and (21) have unique solutions. These solutions are found by a simple algorithm.
Discussion of Results The analysis of results obtained by computations using the program PAR shows that: 1. The hypothesis of normality of the error is confirmed by the graph of distribution of residuals, which is a straight line in the probability scale. 2. Model I is adequate and does not contradict the sample. This conclusion can be made by comparing the value
11
Let the sample S = {Ci, Ei, ~ ~ containing l ~N experi, ~ mental observations be given. Since Co, 11,and Iz are known, we can include in S the corresponding data values and Iz and obtain S' = (C,Ei, ci,Co,i,Il,i, IZ,i)i=lN. of Co, 11, The problem-to be solved is to find such estimates of the parameters P that the models I and I1 will be as close as possible in a certain sense to the capacity values from the sample S'. In order to make the above statement more definite we shall admit the least-squares (LS) criterion of
~with the tabulated" value Fo.w = 2.4 for significance level a = 0.01 and degrees of freedom u1 = 61 and uz = 24. Here Sm2is the mean-square error computed on the basis of the predicted values of capacity. Since F < Fo.99, model I is adequate. (10)Jennrich, R. I. Ann. Math. Stat. 1969,40,633. (11)Beck, J.; Arnold, K. "Parameter Estimation in Engineering and Science"; Wiley: New York, 1977.
Langmuir, Vol. 1, No. 6, 1985 763
Estimation of Adsorption Parameters
Table I. Estimates of Parameters, Characterizing the Adsorption of CI2E,at the Mercury-Solution Interface according to Model I and Their Standard Deviationsn EN,V r,, mol cm-2 co ”, mol dm-3 a C‘, wF cm-2 0.284 f 0.008 8.5 f f 0.5 X 2.6 X & 0.2 X lo4 (3.5 X lo-*) 0.26 f 0.06 (0.4) 6.07 f 0.03 (6.2) “The values in parentheses are obtained in ref 1 by applying approximate methods and are given for comparison. No values for EN and
rmare obtained by these methods.
3. Model I1 is not adequate, since the estimated mean-square error in this case was found to be S,,2(61) = 1.3700, therefore
and
F > Fo.9, = 2.4
It should be noted that the difference in the estimates of the mean-square error obtained from the two models is substantial, which allows a definite conclusion in favor of model I. The ability of the two models to predict the dependence of C on c and E is illustrated in Figure 1. In this figure the capacity value? for both models are computed, using the estimates of Prl and Pr2 found on the basis of experimental data according to models I and 11, correspondingly, as described in the previous paragraph. It should be emphasized that the statistical analysis described in this work gives a new possibility for distinguishing between the adequacy of several models of differential capacity. The fact that the model based on the Hill-de Boer isotherm is not adequate to the experimental data indicates that the adsorption layer could not be an analogue of a real gas, for which van der Waals equation holds. On the other hand, in order to derive the Frumkin isotherm, it is not necessary to suppose a localized adsorption. Frumkin has treated the interfacial layer as a concentrated solution and the decrease in interfacial pressure by surfactant adsorption he has explained as a two-dimensional analogue of osmotic pressure of concentrated solutions.2 There are some other approaches for deriving Frumkin isotherm as well. A significant result of this investigation is the conclusion that assumptions of the model based on the Frumkin isotherm could be accepted as adequate to the studied system, in contrast to the model based on the_Hill-de Boer isotherm. 4. The estimates of P for the adequate model (I) and their standard deviations are given in Table I. As s h o w in ref 1, approximate methods give estimates (in parentheses in Table I) of only three from all five adsorption parameters. As seen in Table I, the values of the parameters ~ 0 . 5 , 0 ,C’, and a, determined by the two methods, do not differ significantly. Still, the difference is big enough to show the necessity of applying the proper statistical method. The estimated value for ~ 0 . 5 , oillustrates the great surface activity of C12E6,a = 0.26 indicates attraction between adsorbed surfactant molecules, and the compar-
I
I
I
I
I
I
I
I
I
I
0,o -q2 -0,L -0,6 E/V Figure 1. Differential capacity vs. potential curves predicted by the model based on Frumkin isotherm (solid lines) and on Hill-de Boer isotherm (broken lines) for two concentrationsof &Es: (a) M; (b), 5 X M. Each point represents an experimental capacity value.
02
atively high value of C’is close to those obtained for adsorbed monolayers. An important result from applying nonlinear regression analysis is the possibility of finding correct estimates of the maximum number of moles of adsorbed substance per unit area rmand the potential difference, arising as a result of the oriented adsorption of the surfactant dipoles EN The surface area per adsorbed molecule CIZEB calculated on the basis of rm is S = 19.5 A2. This value correlates well with the data obtained by the Langmuir trough method for the insoluble higher homologues of saturated fatty acids (about 20 A2). The cross-sectional area of the polyoxyethylene chain in the zigzag configuration, found by M. Rosch,12is 19 A2 and the cross-sectional area of the paraffin chain 18.5 A2. A conclusion can be made that for a 5 X lo4 mol dm-3 solution of a close packing of the organic molecules is reached and the latter are vertically oriented at the electrode-solution interface. A twodimensional condensation might be supposed at that and higher concentrations of the surfactant. The positive sign of EN indicates an orientation of the adsorbed molecules with the positive pole of the dipole toward the mercury surface. Registry No. ClzEB,3055-96-7; Hg, 7439-97-6. (12) RBsch, M. In “Nonionic Surfactants”;Schick, M., Ed.; Dekker: New York, 1967.
