1794
J . Phys. Chem. 1992, 96, 1794-1805
Estimation of Micellization Parameters of Aqueous Sodium Dodecyl Sulfate from Conductivity Data P. C. Shanks? and E. 1. Franses* School of Chemical Engineering, Purdue University, West Lafayette. Indiana 47907 (Received: December 11, 1990)
The cmc (critical micelle concentration), counterion binding parameter (p), and aggregation number (n) of aqueous sodium dodecyl sulfate (SDS) are estimated from conductivity data at various temperatures and salinities. The conductivity models used are based on mass action micellization thermodynamics and the Debye-Hiickel-Onsager conductivity theory. With rigorous parameter estimation methods, the models are optimally fit to previous literature data. Moreover, new data were taken with an experimental design which is based on the precise solution of the inverse problem. The average j3 and n were determined to be 0.72 and 50, respectively. An increase in temperature from 25 to 60 OC results in a 5% decrease in j3 and an increase in the cmc from 8 to 10 mM. The estimated parameters compare well to those based on ion activity, surface tension, and light scattering. The results indicate strongly that SDS micelles contribute substantially to the conductivity but not to the effective ionic strength. Literature forcedistance measurements for surfaces immersed in aqueous SDS micellar solutions indicate effective ionic strengths which agree with those determined from the conductivity models.
Introduction Conductivity measurements of ionic micellar solutions have long been used to verify the presence of aggregates and to determine a critical micelle concentration (cmc or c*).’-~ Nearly all aqueous ionic surfactants have been observed to undergo a substantial slope change, or “break”, in the conductivity profile of K versus concentration (c) a t the onset of aggregation or cmc. The cmc is determined by assigning the location of this break at a single concentration value. By ignoring interionic interactions, Evans4 has estimated the counterion binding parameter /3 = 0.744 for sodium dodecyl sulfate, or SDS, from the ratio of slopes in the conductivity K versus concentration (c) above and below the cmc. With further assumptions, the average aggregation number (n) was estimated, also on the basis of the ratio of slopes, as n = 40 at 40 ‘C. Since his analysis relies on locating the cmc from the location of an apparent “breakn in the K vs c curve, it is not entirely objective, especially if such a break is ambiguous. Moreover, the conductivity is normally nonlinear with concentration, due in part to interionic interactions. Such an analysis of the conductivity of micellar solutions does not make full use of the conductivity data. Another method of estimating B from conductivity data involves observing the dependence of the cmc upon added salt concer~tration.~~~~~ This approach is based on the assumption that the counterion binding is independent of the ionic strength and that the micelle size is infinite (pseudo-phase separation model). Both of these assumptions are oversimplifications when the conductivity is to be calculated. An accurate model of conductivity for micellar solutions should incorporate at a minimum (i) a means of accounting for the interionic interactions, (ii) an objective definition of the cmc, and (iii) a means of using all of the available conductivity data to test for consistency. Interionic interactions are of great importance for interpreting data for simple electrolytes, such as NaCl,’ and must have considerable impact in systems which contain highly charged aggregates. The onset of aggregation in aqueous surfactants may be gradual or abrupt but cannot be treated as a discontinuity in the conductivity profile. The infinite micelle size implicit in the pseudo-phase separation model is inappropriate for interpreting the conductivities of solutions of micelles with finite charges and m~bilities.~-~** Finally, a realistic conductivity model should be consistent not only with the data near the cmc but with the entire data set. It is generally agreed that the cmc of aqueous SDS at 25 O C is about 8 mM. This has been found from conductimetry, surface ‘Present address: Clorox Company, Technical Center, P.O. Box 493,7200 Johnson Drive, Pleasanton, CA 94566.
0022-365419212096-1794$03.00/0
tensiometry, E M F and ion-specific electrode measurements, and light scattering data.”” Ion activity and electrophoretic mobility experiments show that B is about 0.7.11J7-20 The interpretation of light scattering experiments is widely varied, depending on which assumptions are used.21-26 The average aggregation numbers which have been reported for SDS range from 40 to 100, with a probably more reliable estimate for n being around 65. In considering the conductivity of micellar solutions, several key issues must be addressed. One should obtain from the data values of n and /3 which are both reliable and consistent with the results of other methods. The models used must be rigorous enough to describe the physics of the system and yet simple enough to be easily inverted and fit to the data. To test more than one model, an objective method of model discrimination is required. Objective determination of uniqueness of solutions and goodness of fit are important features. Simulations can answer the questions (1) Hartley, G. S.; Collie, B.; Samis, C. S. Tram. Faraday SOC.1936, 32, 795. (2) Hartley, G . S. Aqueous Solutions of Paraffin-Chain Salts. A Study in Micelle Formation; Librairie Scientifique Hermann et Cie: Paris, 1936. (3) Shinoda, K.; Nakagawa, T.; Tamamushi, B.-I.; Isemura, T. Colloidal Surfactants. Some Physicochemical Properties; Academic Press: New York, 1963. (4) Evans, H. C. J . Chem. SOC.1956, 579. (5) Kamrath, R. F.; Franses, E. I. I&EC Fundam. 1983, 22, 230. (6) Kamrath, R. F.; Franses, E. I. J . Phys. Chem. 1984, 88, 1642. (7) Fuoss, R. M.; Accascina, F. Electrolytic Conductance; Interscience Publishers: New York, 1959. (8) Moroi, Y . J . Colloid Interface Sci. 1988, 122, 308. (9) Williams, R. J.; Phillips, J. N.; Mysels, K. J. Trans. Faraday SOC. 1955, 51, 728. ( I O ) Mysels, K. J. J . Colloid Sci. 1955, 10, 507. (1 1) Mysels, K. J.; Dulin, C. 1. J . Colloid Sci. 1955, 10, 461. (12) Mysels, K. J.; Princen, L. H. J . Phys. Chem. 1959, 63, 1696. (13) Mysels, K. J.; Otter, R. J. J . Colloid Sci. 1961, 16, 462. (14) Goddard, E. D.; Benson, G . C. Can. J . Chem. 1957, 35, 986. (15) Phillips, J. M.; Mysels, K. J. J . Phys. Chem. 1955, 59, 325. (16) Elworthy, P. H.; Mysels, K. J . J . ColloidInterface Sci. 1966, 21, 331. (17) Sasaki, T.; Hattori, M.; Sasaki, J.; Nukino, K. Bull. Chem. SOC.Jpn. 1975, 48, 1397. (18) Cutler, S. G.; Meares, P.; Hall, D. G. J . Chem. Soc., Faraday Trans. I 1978, 74, 1758. (19) Kale, K. M.; Cussler, E. L.; Evans, D. F. J . Phys. Chem. 1980,84, 593. (20) Kamrath, R. F.Ph.D. Thesis, Purdue University, West Lafayette, IN, 1984. (21) Tartar, H. V. J . Phys. Chem. 1955, 59, 1195. (22) Tartar, H. V.; Lelong, A. L. M . J . Phys. Chem. 1955, 59, 1185. (23) Tartar, H. V. J . Colloid Sci. 1959, 14, 115. (24) Kushner, L. M.; Hubbard, W. D. J . Colloid Sci. 1955, 10, 428. (25) Huisman, H. F. Proc. K . Ned. Akad. Wet. 1964,867,361,388,407, (26) Parfitt, G . D.; Wood, J . A. Kolloid Z . Z . Polym. 1969, 229, 55.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1795
Micellization Parameters of SDS from Conductivity Data of uniqueness and model discrimination and can be used to show how the estimated parameters depend on experimental design and experimental errors. Moreover, it is important to determine how B depends upon salt concentration and temperature. The models presented here meet the criteria stated above by making use of an effective combination of the following: (i) a monodisperse mass action model of micellization to calculate the species inventories for a given set of parameters cmc (c*), j3, and n; (ii) the Debye-Huckel-Onsager conductivity equations to calculate the equivalent conductivity for each species; and (iii) the method of maximum likelihood combined with reduced gradient optimization to estimate optimal parameter values. This scheme has been tested extensively with simulated "data" to determine the effects of experiment design, experimental error, and sensitivity on each of the above three parameters. Also tested were various conductivity models, in order to gain insights into the general conductivity behavior of micellar solutions. Simulations show that for better estimating the aggregation number ( n ) with these models one needs data at concentrations higher than what are normally measured for determining the cmc. The models have been applied to data from the literature and have suggested the need for new data for unambiguously determining the effects of temperature and salt concentration on j3,the possible concentration dependence of n, and the question of how the presence of micelles affects the ionic strength of the solution. The models have also been applied to new data, obtained here according to an experimental design suggested by the simulations. The results are compared with the results from the literature data and with results by independent methods reported in the literature.
"ry
Mass Action Model of Micellization. The direct problem of predicting solution conductivities based on the concentrations of the surfactant ions, counterions, and micelles requires a realistic model of micellization equilibria. The simplest model which provides a quantitative description of certain real systems is the mass action model (MAM), for monodisperse micelle sizes. Although such a model ignores polydispersity in n and j3 and the dependence of the average aggregation number on concentration, it can describe several key aspects of conductivity, ion activity, and surface tension of certain simple surfactant systems, such as SDS. The expression for the monodisperse MAM of an anionic surfactant R-M+ is nR-
+ nBM+
The initial conditions at zero concentration (c, = 0) are CI = CM+ = c,
This system of equations is quite stiff for certain values of K',,, n, and 8, especially for large values of n 2 50, but can be solved numerically to a high degree of accuracy using the GEAR method.27 Details are given elsewhere.2s Conductivity Models. The conductivity equations for micellar systems treat the aqueous monomers and micelles as a solution of mixed electrolytes. The equivalent conductivities of the monomers plus counterions and the micelles plus counterions are calculated independently, with a common ionic strength. The limiting conductivity of each ion is based on a Stokes-Einstein expression for the mobility in a solvent of viscosity qo without interionic interactions (c 0). In such a case, the bulk conductivity of a solution containing only micelles and their dissociated counterions is expressed as follow^:^^^^^
-
where c, is the molar concentration of micelles, r, is the effective spherical radius of the micelles, n( 1 - B)c, is the molar concentration of counterions dissociated from the micelles, e is the elementary charge, and F is Faraday's constant. The molar conductivity of this solution per unit concentration of unassociated surfactant is
A,, Yn Cn yy-yff+ Cq&+
(5)
where the molar conductivity of the micelle is
R,,M$-@)-
The equilibrium constant is
Ktn =
iteration.6 For large values of n, this method becomes quite unstable. To avoid numerical instabilities and to facilitate the use of Dow Chemical's Simusolv modeling and simulation package (described later), the mass balances were transformed into a system of the following two ordinary coupled differential equations, with dependent variables being cl and cM+and the independent variable being the total concentration c,:
(1)
where the subscript 1 refers to the monomer, c's are concentrations, y's are activity coefficients, and j3 is the counterion binding parameter (0 I /3 I l). For dilute solutions, it is often assumed that the quantity Y,,/Y?Y$+ is unity. The mass balances for the monomers and counterions and a numerical definition for the cmc have been given previously.6 The definition of the cmc (c*) is combined with the mass balances and the expression for the equilibrium constant to give an explicit relationship between KL and c* in terms of n and j3:6 cc*[I-n(l+B)I nK', = (2) (1 - c)"(l - Be)"@ The equation which relates K',, to the cmc in the presence of salt at concentration c, is also available.6 The constant e (normally, we take e = 0.026) is the fraction of the total surfactant concentration c, which is aggregated when the cmc "point" is defined (c, = c * ) . ~In this manner, c* can be substituted for K'n as one of the three parameters which characterize the monodisperse MAM model. The mass balances lead to a system of two highly nonlinear algebraic equations, which can be solved by Newton-Raphson
eF n(1 = --
6 7 ~ 0 rn
and the molar conductivity of the dissociated counterions is (9)
The effective spherical radii of nonaggregated ions are normally found from the limiting ionic condu~tivities.~~ However, for the micelles this value is not directly accessible experimentally. An approximation is used to express rn in terms of the effective spherical radius of the monomer rl. This approximation is based on three assumptions: (i) The density of the aggregate is known (e.g. determined from densitometry). For simplicity it is approximated here to be equal to that of the monomer, since relative changes in the molar volume due to aggregation are often small. (ii) The monomer and the aggregate are considered as equivalent spheres; i.e. shape effects are ignored. (iii) The fraction of the effective spherical volume due to the solvation layer is the same for monomer and aggregate. Then the volume of the aggregate, (27) Hindmarsh, A. C. GEAR: Ordinary Differential Equation Solver. Lawrence Livermore Lab., [Rep.] UCID-30001 1974, (Revision 3). (28) Shanks, P. C. M.S.Thesis, Purdue University, West Lafayette, IN,
1990. (29) Robinson, R. A,; Stokes, R. H. Electrolyte Solutions; Butterworths Scientific Publications: London, U.K., 1955.
1796 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 TABLE I: Features of Conductivity Models Used contributions to I contributions to A n-mer interionic n-mer model n-mers counterions interactions n-mers counterions a a 1A no 1B no a a a a 1C yes 2A no no no no fractional 2B no 2C no no Yes no no 3A yes no fractional 3B yes no 3C yes Yes 3D yes Yes Yes
“Not applicable including the solvation layer, is n times the effective spherical volume of the monomer, and since both are taken to be spherical, r,, = n1/3r1. With this approximation, eq 8 is modified to yield
The correction factor (r(n)) can be used to account for the effects of the above assumptions. We have estimated that r(n)can differ from 1 by up to ca. 1096, except of course for cylindrical micelles, for which essential modification of the model may be needed. Here we take, however, r(n)= 1, for simplicity. We have examined various possible models (Table I), which are defined below. If one does not account for interionic interactions, the bulk conductivity of the micellar solution is the sum of the contributions due to monomers (K,)and micelles (K,,) (models 1A-C):
K = K1 + K,,
(1 1)
Interionic interactions reduce the molar conductivities by an amount which depends on the size, charge, and limiting conductivity of the individual ions, as well as the effective ionic strength. These interactions are generally due to hydrodynamic and electrostatic effects. In this paper, for dilute solutions, nonspecific interionic interactions are accounted for by the Debye-Huckel-Onsager (D-H-O) equation for each ~ p e c i e s . 7 9 The ~~ molar conductivity of a solution which contains monomers and micelles is
where K is Debye’s inverse length, a , and a, are the effective ionic radii, and I is the ionic strength. The interionic interactions are manifested via the constants A, and A,. These constants are determined independently for monomers and micelles and are functions of the valences and the limiting ionic conductivities. In cgs units and i = 1 or n
where q is a mobility function of the valence z and the limiting ionic conductivities Xo: Iz+z-I
xo, + 2
= lz+l + 12-1 IZ+lXO + IZ-lXo,
(15)
An important issue which arises at this point is that of the ionic strength (I). There is direct and indirect evidence described later that the highly charged micelles do not contribute to the ionic strength as the smaller unaggregated ions do. We have examined four possible expressions for the effective ionic strength of a micellar solution, each accounting for different individual ionic
Shanks and Franses contributions (Table I). Model 3A represents an extreme in that the ionic strength is determined only by the monomers and the monomeric counterions (the ions which balance the free monomers charge): I = c1
(16)
According to this model, the ionic strength of a micellar solution goes through a maximum at the cmc and decreases at higher concentrations (as shown below). Model 3C includes the contributions of only the “small” ionic species but does not include the micelles:
I = X(C1 + CM+)
(17)
Model 3D includes the contributions of all charged species in the solution: m
I = 1/2E~i~i’ = f / 2 ( ~ l + cM+) i= 1
+ )/zcnn2(1- 6)’
(18)
In such a model, the contribution of the micelles can be dominant over the other species in the solution. The previous models yield smaller ionic strengths than model 3D, because the micellar contributions are taken to be zero. The fourth model (3B) is intermediate between models 3A and 3C, in that it includes the monomeric ions and a fraction of the micellar counterions:
+
I = Y2[2c1 nc,(l - j3)2]
(19)
The above fraction was arbitrarily assigned to be (1 - j3) times the concentration of free micellar counterions.
Sample Calculations and Simulations: The Direct Problem The direct problem consists of specifying values for the model parameters and calculating the resulting conductivities. We have conducted simulations in this manner, using the models described above, in order to determine the effects of the individual micellization parameters on the conductivity profiles. The first sample calculations do not account for the interionic interactions. These calculations are used to show what estimates of j3 result from ignoring the interionic interactions and to show that the micellar conductivities must be accounted for, regardless of the size of the micelle. From eqs 9 and 10, it is evident that the micellar contribution to K can be comparable to or even larger than that of the counterions. Nevertheless, sample calculations show that if the micelles were nonconductive (models 1A and lC, Table I), despite their net charge, then j3 would be about 0.45 for the calculated conductivities to match the experimental data for SDSZ8 This value is substantially lower than that estimated from ion-selective electrode and electrophoretic mobility measurements. Without the micellar contributions, K would be due entirely to the monomers and counterions, and the excessively low value of fl would be necessary to boost the free counterion concentration. This value indicates that this model is unrealistic. Hence, the micelles should contribute to K. This is further supported by the fact that the micelles have a measurable electrophoretic mobility.l,zJ1 If a conductivity model does account for micellar conductivity and ignores the interionic interactions (model lB), the value of j3 which is required to approximate experimental molar conductivities is about 0.76.4*28In models 2A-C, interionic interactions are considered but micelle contributions to K are ignored.28 When micellar contributions to conductivity and the interionic interactions are included in the form of model 3D, the ionic strength above the cmc increases to levels much higher than that for an equal concentration of a 1:l electrolyte. This increase is due to the significant contributions of the micelles, which are treated as an [n(l - @)]:1electrolyte (see eq 18). For reasonable values of c* = 8 mM, @ = 0.70, and n = 65, model 3D shows that A would go to zero at a concentration of about 280 mM (see Figure 1). The data for SDS show, however, that A is nearly constant at A N 30 S-cmz.mol-l for concentrations between 120 and -900 mM,Zowhich argues strongly against model 3D (SDS data are shown later).
-
Micellization Parameters of SDS from Conductivity Data
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1797
1.o
0.8
0.8
% \ C
301
0.4
2o
0.2
0.0
0
100
200
c,,
300
400
t
01
500
0
I
I
I
0.02
0.04
0.06
mM
I 0.08
1
Conccnuation c, mo1.L-I
Figure 1. Calculated molar conductivity profile when all ionic species contribute to the ionic strength (model 3D): c* = 8 mM, p = 0.70, n = 65.
Figure 3. Molar conductivities for micellar system of Figures 1 and 2 for models 3A-C, from top to bottom.
100
I
17Mk 80
2
1Mo 1w)
eo
rdb
E
CI
40
lo00
7M
80 w)
0 0
20
40
c .
BO
eo
100
m M
Figure 2. Comparison of effective ionic strengths calculated for models 3A-D, from bottom to top; same parameters as in Figure 1.
If the micellar contributions to the ionic strength (Z) are neglected (models 3A-C), the ionic strength is much less than that for an equal concentration of a 1:l electrolyte (see Figure 2). This results in much more realistic values of A at concentrations above 100 mM (Figure 3). The ionic strength, according to model 3C, continues to increase slightly for concentrations above the cmc, resulting in a slight decrease in A with c,. However, because in model 3A the ionic strength depends only on the monomer concentration, which decreases above the cmc, A actually increases with c, at high concentrations. For model 3B, A is nearly constant with c,.
Solving the Inverse Problem The Method of Maximum Likelihood. The purpose of the models presented here is to determine equilibrium micellization parameters. It is necessary then to invert the models, i.e. to obtain the model parameters which best fit the data, by combining the model equations with an objective parameter estimation method. Such a method involves an optimization scheme which varies the parameters from an initial guess until an objective function (described below) is maximized. We have used the Dow Chemical Simusolv software (version 1.4) to accomplish this. This package allows for both simulation and optimization with the same (30) Steiner, E. C.; Blau, G. E.;Agin, G. L.Introductory Guide: SimuSoh Modeling and Simulation Software; Dow Chemical Company: Midland, MI, 1986.
0 0
0.01
om
om
0.0)
0.M
4. m0l-L-l
Figure 4. Fit of simulated ‘data” as generated from model 3C (c* = 8 mM, fl = 0.70, n = 65) with 1% random error added. The initial guesses were c * = 5 mM and n = 50. In the optimum solution shown, C* = 7.98 mM, p = 0.688, and n = 46.
Because the solution of the inverse problem is quite tricky and is presented here for the first time, we need to provide some details of the method used. This method is a lot more powerful than common least-squares methods. Moreover, because the method of maximum likelihood is not generally familiar to the reader, who needs to understand quantitatively the significance of the estimated parameters and their uncertainty, we describe it briefly below. The optimization scheme used here is a combination of a generalized reduced gradient search and the method of maximum likelihood. This method uses as an objective function a joint probability density function of all the errors in the measured variables.31 With proper experimental design, the measurements are independent of each other and the errors can be assumed to follow a normal distribution. The predictions of a model with a given set of parameters (0)and an independent variable (xi) yield the residuals y i -fi(x$), wherefi is the model-predicted value of the “true” value of pi. Iff(xi,O) is an accurate description of pi, then the likelihood function (LF) is the probability of obtaining m measurements with the values observed:
(31) Bard, Y.; Lapidus, L. Catal. Rev. 1968, 2, 67.
1798 The Journal of Physical Chemistry, Vol. 96, No. 4 , 1992 If this probability is very low and there are no systematic errors in the data, then it must be assumed thatf,(x,B) does not provide a good representation of the data. Maximizing L F by proper choice of model parameters 8 is the same as increasing the probability of obtaining the observed values from a system which is accurately modeled by the functionf. It is necessary to assume that the error variance can be modeled as a function of the magnitude of the measured values and some set of unknown model parameters P. The likelihood function then depends upon one or more independent variable ( x ) , a set of model parameters (e), and a set of error parameters (P):
Here, p is the joint probability of obtaining the residuals y -f, and y and x are the dependent and independent variables, respectively. The likelihood method involves simply finding those sets of values for 8 and P which result in a maximum value in LF for a given functional form of the modelf and the error model. Box and HilP2 proposed an error model which uses a power law relationship between si, which is the estimate of the true error variance ( u i ) , andf. as follows:
s; = a2f7
(22)
Here a is a constant of proportionality, and y is an adjustable parameter called the heteroscedasticity paramerer. Together, a and y make up the set of unknown error parameters (P).When y is zero, the variance is independent of the magnitude of the measured values (constant absolute error). A value of y = 2 corresponds to a constant relative error. Since a is a constant, there is one value of a which maximizes the joint probability (p) for a given heteroscedasticity parameter and set of residuals Qi -f,). By solving for a in terms of y, y , andf,, the set P can be reduced to a single parameter y. The estimate of the error ( s i ) is
The above error estimate is then substituted into the likelihood function. Normally, the logarithm of the likelihood function is used as the actual objective function to be maximized.30 The complete form of the logarithmic likelihood function, or LLF, is
m
LLF = --[log 2
+
( 2 ~ ) 11 -
Of less importance, but nevertheless useful, is the average error
sly, where j j is the average of all m observed values and
1
J
The number S2 is identical with the error estimate of eq 23 evaluated with y = 0. Model D k i ” t i 0 n . A comparison of the likelihood functions for two different models gives an idea of the validity or the relative plausibility of each model. A convenient method is to examine the ratio of the two likelihoods or the difference of their log (likelihood function) or LLF. This ratio, also known as the odds ratio, is a direct measure of these relative plau~ibilities.~~ When two models have the same number of estimated parameters, the ratio of likelihood functions is taken at the points of maximum (32)Box, G. E. P.; Hill, W. J. Technometrics 1974, 16, 385. (33)Reilly, P.J. Can. J . Chem. Eng. 1970, 48, 168. (34)Reilly, P.M.; Blau, G. E. Can. J . Chem. Eng. 1974, 52, 289.
Shanks and Frames TABLE 11: Simulations to Test the Effects of Number of Data Points, Concentration Range, and Initial Guess on the Optimum Parameter Values” cmc, mM B n LLF sly Example 1: 100 points, 0.5-50 mM 1 G 1 0.5 5 -608 0.11 2 0 1.7 f 0.5 0.64 f 0.04 1.8 f 0.3 -461 0.025 3 G 5 0.5 50 -738 0.68 59 f 1 -23.6 0.0005 4 0 8.0 f 0.003 0.697 f 0.001 5 G 15 0.9 150 -641 0.23 0.718 f 0.0004 152 f 8 6 0 7.9 f 0.05 -295 0.007 G 0 G 0 G 0
Example 2: 100 points, 1 0.5 2.2f 0.6 0.66f 0.01 5 0.5 8.01 f 0.002 0.70 f lo-’ 15 0.9 9.0f 0.2 0.7017f 0.0002
2-200 mM 5 1.9f 0.2 50 66.0f 0.02 150 65.0f 0.7
-781 -518 -856 -85 -837 -383
0.38 0.016 0.61 0.0002 0.60 0.004
13 G 14 0 15 G 16 0 17 G 18 0
Example 3: 25 points, 0.5 2fl 0.64 f 0.08 5 0.5 7.992f 0.003 0.7006 f 0.0003 15 0.9 7.89 f 0.02 0.719 f 0.002
2-50 mM 5 1.8 f 0.6 50 64.9f 0.6 150 151 f 20
-153 -1 17 -186 -19 -162 -76
0.11 0.027 0.68
8-200 mM 5 1.6 f 0.3 50 73 f 2 150 65 f 8
-197 -127 -216 -1 16 -21 1 -143
0.38 0.014 0.61 0.009 0.60 0.028
7 8 9 10 11 12
19 20 21 22 23 24
1
G 1 0 13.2 f 0.8 G 5 0 5.7 & 0.3 G 15 0 14 f 2
Example 4: 25 points, 0.5 0.67f 0.02 0.5 0.700f 0.001 0.9 0.705 f 0.003
0.0005
0.23 0.008
“The data were generated with model 3C using a cmc = 8 mM, B = 0.7, and n = 65. The inverse problem routine used model 3C and was given the initial guesses shown in the rows marked G. Rows marked 0 contain the optimum values corresponding to the initial guesses. Plus/minus values next to a parameter value are one standard deviation for that parameter and are calculated by the SimuSolv program. All data are without “noise”.
likelihood. If the difference in LLF is 1, then the model with larger LLF is favored. If the difference in LLF is close to zero, the results are statistically inconclusive. If the LLF is 2 (ratio of L F of 100) or more, then strong preference should be given to the model with the larger likelihood function. Simulations to Test the Solution of the Inverse Problem. The following important issues are addressed before the inverse problem is applied to real experimental data: (i) whether the optimum solution for a given case is in fact the best solution and how the initial guess of the model parameters affects the solution, (ii) the factors which should be considered in designing experiments and how these factors affect the performance of the inverse problem routines, and (iii) how the model discrimination capabilities are affected when there is a significant amount of error in the data. These issues are explored through the use of a series of simulations. The simulations are carried out using a conductivity model (direct problem) to generate artificial “data”. These data sets contain different numbers of points, which are either evenly or unevenly spaced over various concentration ranges. Certain “data” also contain various amounts of relative error (noise). The optimization routines are then used with variations in the model to check their effect on the likelihood ratios. The main criterion for determining the optimal solution is the maximum value of LLF. If different starting points result in different solutions, due to a relatively flat optimization surface, then the solution corresponding to the highest LLF value is chosen as optimal. Various starting points may be used to distinguish between local and global maxima in the optimization surface. The above discussion concludes the description of the method used. Some examples of the application of the method to simulated data sets follow. For each initial guess in example 1 of Table I1 (rows marked G), the LLF value is between -600 and -750. Only in rows 4 and 6 is the optimum value greater than -300. Simple comparison of the parameter values (c*, p, n) shows that these two solutions appear to be nearly identical. However, the LLF column shows that the parameters in row 4 are clearly more likely to be the real
Micellization Parameters of SDS from Conductivity Data solution of the inverse problem. In comparison of the LLF values, the solution in row 2 should correspond to a local maximum in LLF and the solution in row 4 to a global maximum. This conclusion is supported by the average errors. The S / J value in row 2 is 10-50 times larger than that for either of the other two solutions. In example 2 of Table 11, the concentration range is from 2 to 200 mM, while the number of data points is the same as in example 1. The LLF values for the three initial guesses are all around -800 and are slightly higher than the initial LLF's in example 1. This is due to the dependence of the LLF value on the range of the dependent variable. Comparison of the optimum LLF values shows that, as in example 1, the solution arising from initial guess ii (row 9) leads to the most likely set of model parameters. Furthermore, the solution due to initial guess iii, while much better than that in row 8, is not as good as that in row 10. Comparison of average errors confirms that row 8 contains a local solution (1 -6% average error when initial guess i is used vs less than 0.5% for the other two solutions). A comparison of the optimum parameter values in rows 4 and 10 shows that the extended concentration range of example 2 allows for a better estimation of n. In addition, the standard deviations of both n and @ in row 10 are improved by 2 orders of magnitude. The estimate of the cmc is unaffected by the extended concentration range. In each of the other two examples, there is a similar trend. Initial guesses ii and iii result in solutions which are similar to each other but are much different from the local solutions resulting from initial guess i, both in terms of the LLF and the average error. This indicates that the solutions which arise from initial guess i are only locally optimal and are "false solutions". False solutions are often nearly indistinguishable by visual inspections of curve fit from other solutions.28 These false solutions occur whether the maximum concentration is 50 or 200 mM. Hence, the existence of a false solution appears to be independent of the concentration range. False solutions would correspond physically to small aggregates with relatively low mobilities. The resulting profiles would lack a distinct break at the onset of micellization and would fit the data poorly at the cmc because n is small. The cmc is estimated to be lower than that of the actual value. The resulting estimate of @ is also low, in order to keep the counterion concentration high and compensate for the less mobile micelles. The estimate of the cmc is affected primarily by the number of data points in the immediate vicinity of the break in the K-c curve. This is best illustrated by comparison of the results of examples 3 and 4. The data set of example 4 contains few points in the vicinity of the cmc. Hence, the break is obscured, and the estimates of the cmc are close to those of the initial guess, except for the case of the false solution (row 20). The estimate of 6 depends on those points which immediately follow the cmc, up to the region where the monomer concentration becomes about constant. A reasonable estimate of 6 is obtained for several initial guesses which do not lead to a false solution in c* and n, regardless of the concentration-range of the data. When the estimate of n is not close to the actual value, small inaccuracies in 0 result, with the largest inaccuracies stemming from the very small values of n in the false solution. The effects of experimental errors were studied by adding to the simulated "data" a normally distributed random error having a mean of 1% of the dependent variable (the bulk conductivity). The results of these simulations are presented in Table 111. The first three examples demonstrate that the method is little affected by the addition of 1% relative error, even when the number of data points is reduced to 8, from 2 to 16 mM. In this range, however, the estimate of n is quite unreliable. When the data are limited to the transition region where the solution is changing from predominantly monomers to predominantly micelles, the conductivity profile depends strongly on 0 but is relatively independent of n. Only when the data extend well beyond the cmc can n be estimated with a substantial degree of accuracy. Figure 5 shows that even when n is varied by f15% the K-c profile changes little, except at the highest concentrations. Simulations illustrating the effect of nonuniformly spaced data
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1799 TABLE III: Simulations to Test the Effects of "Noise" on Parameter Estimation' cmc. mM B n LLF Slv Example 1: 25 points, 2-50 mM 1 G 5 0.5 50 -186 0.683 2 0 8.0 f 0.01 0.69 0.01 46 f 10 -92 0.014
*
Example 2: 13 points, 2-26 mM 3 G 5 4 0 8.0 f 0.01
0.5 0.69 f 0.05
50 47 f 3
-87 0.498 -35 0.0069
Example 3: 8 points, 2-16 mM 5 G 5 6 0 8.0 f 0.1
0.5 0.70 f 0.05
50 4 4 f 55
-47 0.287 -19 0.0058
Example 4: 100 points, 0.5-50 mM 7 G 5 8 9 10 11 12
0 5.2 i 0.4 G 5 0 8.0 i 0.1 G 15 0 7.9f 0.1
0.5 0.592 0.003 0.5 0.70 f 0.01 0.9 0.719 0.001
*
*
5 4.8 f 0.6 50 60 f 21 150 152 f 16
Example 5: 100 points, 2-200 mM 13 G 15 0.9 150 14 0 8.11 f 0.0001 0.7015 f 0.0003 67 i 1
-610 -406 -738 -348 -641 -373
0.111 0.014 0.229 0.015
-837 0.600 -449 0.0104
"The data were generated as in Table 11, with relative error normally distributed with 1% variance added to the conductivity.
2100
1800
Y
0 0
I
I
1
I
20
40
80
80
I 100
Figure 5. Conductivities for c* = 8 mM, /3 = 0.70,and n = 75, 65,and 55, from top to bottom.
reveal that it is desirable to have more data around the cmc for better determining c* and @ and more data extending to high concentrations to better estimate a2*The above method of parameter estimation has been used here for the first time to several previous literature data and to new data on SDS, obtained via experimental design suggested by the simulations.
Experimental Section Sodium dodecyl sulfate (SDS) was specially pure with a minimum assay of 99%. It was obtained from BDH Chemical Ltd., Poole, England and was used without purification. It was dried for 24 h or more at 45 OC under vacuum before use. Distilled water was treated in a Millipore Milli-Q four-stage cartridge system. Its average conductivity was 0.05&em-' at the exit of the system and increased up to a.0.5 pScm-' after contact with air. Sodium chloride was certified ACS from Fisher Scientific. Stock solutions were prepared by weight. Concentrations were precise to fO.O1 mM. Surfactant solutions were used within 10 days after preparation to minimize possible hydrolysis effects. Conductivities were measured using Jones cells from Metrohm and GenRad G R 1689 Precision RLC Digibridge, which measures pure conductance with an uncertainty of f l pS. The molar conductivity had an estimated uncertainty of better than 5
Shanks and Franses
1800 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
TABLE V: Fits of Models 3A-D to the Data of Mysels and Otter,I3 over the Full Range of the Data Set (3-50 mM)“ cmc, mM P n LLF f/J
1
8
r
Y
1 2 3 4
G
5 0 8.19 f 0.05 G 15 0 8.4 f 0.1
5 6 7 8
G 0
9 10 11 12
G
5 0 7.9 f 0.1 G 15 0 8.0 f 0.1
13 14 15 16
G 0
5 8.13 f 0.04 G 15 0 8.4 f 0.1
G, md-L-’
Figure 6. Fit of model 3B (optimum) to Kamrath’s SDS data (Table IV). The optimal parameters were C* = 8.0 mM, P = 0.71, and n = 44; the average estimated error is 1.9%.
TABLE I V Fits of Models 3A-D to the Data of Kamrath,Mover a Limited Concentration Ranee (3-88 mMP cmc, mM fl n LLF S/p Model 3A 1 2 3 4
G 5
0 8.1 f 0.1 G
15 0 8.3 f 0.1
0.5 0.714 f 0.001 0.9 0.764 f 0.005
50 42 f 1 150 152 f 22
-239.3 -128.2 -214.4 -144.9
0.023 0.048
Model 3B 5 6 7 8
G 5
0.5 0.71 f 0.02 G 15 0.9 0 8.18 f 0.05 0.75 f 0.01
0 8.0 f 0.1
50 44 f 18 150 151 f 49
-232.3 -120.8 -214.5 -132.7
0.019 0.022
Model 3C 9 10 11 12
G 5
0 7.8 f 0.1 G
15 0 7.9 f 0.1
0.5 0.69 f 0.01 0.9 0.703 f 0.003
50 48 f 11 150 72 f 10
-226.3 -128.5 -214.9 -134.0
0.042
G
5
0 6.8 f 0.1 G
15
0 16.3 f 0.1
0.5 0.58 f 0.01 0.9 0.9 f 0.1
50 10 f 1 150 143 f 33
-241.8 -139.3 -218.4 -218.0
50 43 f 8 150 148 f 2
Model 3B 0.5 0.709 f 0.001 0.9 0.76 f 0.01
50 -202.0 41.7 f 0.1 -80.8 0.009 150 -174.5 153 f 63 -94.5 0.016
Model 3C 0.5 0.70 f 0.03 0.9 0.735 f 0.002
50 46 f 28 150 145 f 37
-197.7 -88.0 0.019 -174.7 -97.8 0.029
Model 3D 0.5 0.59 f 0.01 0.9 0.9 f 0.1
50 10 f 4 150 148 37
-196.9 -112.6 0.028 -177.0 -176.8 0.261
*
-207.1 -87.0 0.017 -174.5 -101.9 0.018
‘Two initial guesses are used for each model, and the rows are each marked G or 0 corresponding to initial guess or optimal solution.
TABLE VI: Fits of Models 3A-D to the Data of Goddard and BensonI4a cmc, mM R n LLF SIP 1 2 3 4
G 5 0 8.4 f 0.1 G 15 0 10.1 f 0.6
Model 3A 0.5 50 0.72 f 0.01 44 f 1 0.9 150 0.95 f 0.10 150 f 24
5 6 7 8
G 0 G 0
Model 3B 0.5 50 -125.6 0.72 f 0.01 44 f 1 -57.6 0.9 150 -121.0 0.96 f 0.09 150 f 200 -90.8
5 8.4 f 0.1 15 10.1 f 0.3
-127.8 -57.9 -121.0 -91.0
0.009 0.043
0.009 0.044
Model 3C 0.051
Model 3D 13 14 15 16
5 6.9 f 0.3 G 15 0 15.8 f 0.5
Model 3A 0.5 0.72 f 0.01 0.9 0.765 f 0.001
0.065 0.430
OTwo initial guesses are used for each model, and the rows are each marked G or 0 for initial guess or optimum solution.
9 10 11 12
G 0 G 0
5 8.2 f 0.1 15 10.02 f 0.02
0.5 0.71 f 0.02 0.9 0.96 f 0.05
13 14 15 16
G 0 G 0
5 8.0 f 0.2 15 9.9 f 0.1
Model 3D 0.5 50 0.61 f 0.07 32 24 0.9 150 0.8 f 0.4 149 f 63
50 -123.5 45 f 33 -55.5 150 -121.0 150 f 500 -90.8
*
-123.1 -58.2 -121.0 -89.5
0.012
0.045
0.010 0.043
S-cm2.mol-l at 0.2mM of SDS and as low as 0.5 S.cm2.mol-’ at 100 mM of SDS. Calibration was done with standard aqueous NaCl or KC1 solutions.
‘The data set comprises concentrations from 3 to 15 mM. Two initial guesses are used for each model, and the rows are each marked G or 0 corresponding to initial guess or optimum solution.
Results and Discussion Three sets of SDS data at 25 OC were first analyzed by the inverse problem method previously described: (i) by Kamrath,zo (ii) by Mysels and Otter,13 and (iii) by Goddard and Benson.I4 The data from Kamrath range from 3 to 876 mM, with 36 out of a total of 42 points a t 3-90 mM. The pre-cmc data, extrapolated to zero concentration, yield a limiting molar conductivity of A? = 21.6 f 0.3 S*cm2.mol-l. There is a visual cmc “break” a t c, = 8 mM (Figure 6). The A-c profile shows simple 1:l electrolyte behavior up to 8 mM, above which A drops significantly. From 100 to over 876 mM, A N 31 S.cm2.mol-’. If models 3B-D are tit to the whole concentration range, the fit is quite poor, with c*, j3, and n being too low (c* C 7 mM, @ < 0.6,and n < 9) and $ / y exceeding 30%. Model 3A yields plausible values of c* = 8.1 mM, j3 = 0.72,and n = 41,but is 14%, which far exceeds the experimental error. Most of the error occurs at the higher concentrations, indicating that model 3A may fail due to
concentrated solution effects not accounted for in the model. The data from 3 to 88 mM were fit to the models. Model 3D fit very poorly (Table IV), yielding high fly, low LLF, and low values of cmc, @, and n. Models 3A-C fit the data much better than model 3D,with a slight advantage to model 3B (column 6, Table IV and Figure 6). The estimate of n normally had a large standard deviation and depended on the initial guess, indicating that the optimization surface is flat in the n direction. The best-fit model (3B) yields for c*, & and n the values 8.0 mM, 0.71,and 44,which appear quite reasonable. The value of f/jj = 0.019 slightly exceeds the experimental precision. Normally, model 3B yields lower c* and 0 than model 3A,and model 3C yields still lower values than model 3B. Model 3D has to be ruled out, suggesting that micelles cannot contribute to Z as the smaller ions do. Qualitatively similar trends were obtained by fitting two other independent sets of data from the l i t e r a t ~ r e . ~ The ~ , ’ ~data of Mysels and Otter,13 spanning the range from 3 to 50 mM, were
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1801
Micellization Parameters of SDS from Conductivity Data TABLE VII: Fits of Models 3A-C to SDS Data at Higher Concentrationsg cmc. mM B n LLF
TABLE VIII Fits of Models 3A-C to Partial SDS Data SeP cmc, mM B n LLF ffj Model 3A 1 G 5 0.5 50 -411.2 2 0 8.2 f 0.1 0.72 f 0.01 42 f 9 -210.4 0.016
SIP
Model 3A 1 G 5 2 0 8.1 f 0.1 3 G 15 4 0 9.3 f 0.1 5 G 5 6 0 7.5 f 0.2 7 G 15 8 0 8.0 f 0.1
0.5
50 0.72 f 0.01 39 f 4 0.9 150 0.778 f 0.002 159 f 10
-662.8 -321.3 0.030 -590.0 -385.8 0.024
Model 3B 0.5 0.70 f 0.02 0.9 0.757 f 0.004
-623.1 -401.0 -621.0 -405.6
50 44 f 14 150 154 f 24
3 4
G
5 0 5.4 f 0.4 G 15 0 5.4 f 0.4
50
0.5 0.59 f 0.01 0.9 0.59 f 0.01
-595.3 -420.8 -592.9 -420.7
7f2 150 8f2
5 6 7 8
0.072
I
I
I
0.05
.1
.15
Cp
15 8.6 f 0.1
G
0
0.9 0.77 f 0.01
150 131 f 20
-412.7 -236.7
50 56 f 16 150 142 f 25
-424.6 -192.2 -412.9 -200.7
G
0
5 8.0 f 0.1 15 8.2 f 0.1
0.5 0.72 f 0.01 0.9 0.76 f 0.01
5 7.4 f 0.1 15 7.4 f 0.1
Model 3C 0.5 50 0.69 f 0.01 45 f 7 0.9 150 0.67 f 0.01 33 f 7
0.095 9 10 11 12
0.110 0.096
'The data include 76 concentrations from 0.2 to 206 mM. Two initial guesses were used for each model, and the rows are each marked G or 0 corresponding to initial guess or optimal solution.
0
0
0.026
Model 3B
Model 3C 9 10 11 12
G
G
0 G
0
0.013 0.018
-41 1.2 -239.2 0.048 -414.6 -232.8 0.029
"The data include 5 5 concentrations from 0.2 to 99 mM. Two initial guesses were used for each model, and the rows are each marked G or 0 corresponding to initial guess or optimal solution.
2
mol L-1
Figure 7. Optimum fit to new SDS data, from 0.2 to 206 mM, model 3 A c* = 8.1 mM, fi = 0.72, n = 39; average error 3%; most error above 160 mM (Table VII).
*
fit to all four models (Table V). Their value of A; = 22.9 0.3 Scm2.mol-' exceeds that based on Kamrath's data by 6%. Model 3B had a slight edge over 3A and 3C,based on both the LLF and S / y (0.9%), yielding c* = 8.13 mM, 0 = 0.709, and n = 42. Goddard and Benson's data (from 3 to only 15 mM) yield (model 3C) c* = 8.2, 8 = 0.71, and n = 45 f 33. (Table VI). The large uncertainty in n is due to lack of data at high concentrations. Models 3A and 3B fit the data fairly well. Even model 3D yields a reasonable cmc, perhaps because the differences in the models appear less pronounced the lower the concentration range used. Here A'& = 23.1 S.cmz.mol-'. A set of 76 new data (0.2-206 mM) was obtained and fit to models 3 A X . The larger number of data was designed to allow optimum determination of Ao, c*, 8, and It. The whole set of data is best represented by model 3A (Table VII, row 2, and Figure 7). A value of A& = 22.9 S.cm2-mol-' was found. The average error of 3% was due mostly to the higher concentration data. Fitting the data up to 99 mM decreased the error to 1.3% (Table VI11 and Figure 8). In this range, model 3B provided a better fit and yielded c* = 8.0 mM, 0 = 0.72, and n = 56 f 16. Such a large uncertainty in n should be expected, as discussed earlier. It is i n f d , nonetheless, that the monodisperse mass action model represents the data quite well, as does the assumption that the micelles contribute to the conductivity but not to the effective ionic strength (Le. model 3D does not apply).
Figure 8. Optimum fit to new SDS data, from 0.2 to 99 mM, model 3B: average error 1.3% (Table VIII).
c* = 8.0 mM, @ = 0.72, n = 56;
The larger number of the new data set allowed the following test. The cmc was fixed at 8.1 mM, and model 3A was applied piecewise to three overlapping concentration ranges, 40-120, 80-160, and 120-205 mM. The following values were obtained: 0 = 0.73 f 0.02,0.73 f 0.02, and 0.72 f 0.01; n = 45 f 17,44 f 18, and 45 f 7. These values are the same within experimental error as the values obtained over the whole concentration range. Similar results were obtained with model 3B.** We infer that 0 does not change with concentration and that if the average n changes with concentration the change is too small to be detected by conductimetry. It seems that the monodisperse MAM model represents the data quite well. Data on aqueous SDS in the presence of NaCl are considered next. The added electrolyte increases the effective ionic strength (0 and also the concentration cM+. For 30 and 100 mM NaCl, the visually-determined, conductivity-based cmc of SDS is known to decrease to 3.1 and 1.5 mM, respxtively? If the pseudo-phase separation model is used and if 0 is taken to be independent of salt concentration (cs), then one can easily derive that the cmc (c*(c,)) in the presence of salt is given by
c*(c,)[c*(c,) + c,]@ = c*'+@
(26) A plot of In [c*(c,)] vs (In [c*(c,) + c,] - In c*) would have a slope of -8. Fitting the above data yields 6 0.77. This treatment ignores the finite micelle size, which affects the conductivity and hence affects the conductivity-based cmc. The inverse problem method was applied to the data by Williams et al.9 and to new data of our laboratory. The results are
1802 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
Shanks and Franses
with New Data for SDS/NaCI/H,O Systems'
TABLE I X Comparison of tbe Data of Williams et
Williams et aL9
parameter c, c* (visual) c* (num)
B n
LLF
JJJ c* (num)
B n
LLF fJJ
c* (num)
B n
LLF SJJ
this work
30 3.1
100 1.5
3.0 f 0.1 0.71 f 0.07 46 f 62 -22.95 0.0073
1.08 f 0.04 0.5 f 0.8 42 f 43 -27.28 0.0121
2.90 f 0.03 0.7 f 0.1 44 f 99 -23.24 0.0077
1.20f 0.06 0.5 i 0.4 46 f 10 -25.81 0.0116
2.9 f 0.1 0.70 f 0.07 43 f 70 -23.03 0.0078
1.11 f 0.08 0.5 f 0.4 40 f 23 -26.68 0.0121
30 3
100 1.5
Model 3A 8.2 f 0.1 0.72 f 0.01 42 f 9 -210.4 0.016
2.5 f 0.2 0.66 f 0.08 87 f 68 -167.25 0.0048
0.78 f 0.01 0.5 f 0.8 44 f 47 -76.35 0.0015
Model 3B 8.0 f 0.1 0.72 f 0.01 56f 16 -192.2 0.013
2.0 f 0.4 0.66 f 0.03 30 f 54 -1 80.6 0.0072
0.58 f 0.01 0.57 f 0.07 32 f 17 -81.11 0.0012
1.7 f 0.1 0.64 i 0.02 21 f 18 -190.1 0.0075
(0.5) 0.59 f 0.01 13 f 18 -132.4 0.0010
0 8
Model 3C 7.4 f 0.1 0.67 f 0.01
33 f 7 -232.8 0.029
'The values of cy (visual) were obtained by determining the position of the "break" in the K-c profile. All concentrations are millimolar.
-
lax,
'€
rd
I/ 600-
m
3ooo
0
0.02
OM
0.06
0.08
01' 0
.1
q, mol L-' Figure 9. Fit of model 3A to the new SDS data: c, = 30 mM, c* = 2.5 f 0.2mM, fl = 0.66 f 0.08,n = 87 f 68;average error 0.48% (Table IX).
summarized in Table IX. The visual cmc's of the two sets of data agree well. For c, = 30 mM, the numerical cmc is 3.0 mM for the Williams et al. data and 2.5 mM for our data. For the former data, which are mostly close to the cmc, all three models fit the data comparably. Because of too few data, the determinations of j3 and n have more uncertainties, especially at c, = 100 mM (Table IX). Model 3A fits the Williams et al. data Our data (Figure 9 and Table IX) show that for all three models there is a trend for j3 to decrease with salinity. Based on the LLF values, we infer that model 3A works best. The results on estimation of n are quite uncertain and have to be considered inconclusive. The effect of temperature on the micellization parameters is considered in Table X. All three models yield similar results for the data of Goddard and Benson,14because the data range is small (Figure IO). The cmc increases by about 5% from 25 to 40 O C , and j3 decreases. For our data, which span a larger range (Figures 11 and 12), model 3B is superior. The cmc increases a bit with temperature up to 40 O C and a lot more at 60 O C . There seems to be a clear and consistent trend for @ to decrease with temperature. Moreover, n decreases with temperature for models 3A and 3B (but not for 3C, which fits more poorly). Data were obtained at the higher salinities and the higher temperatures and are summarized in Tables XI-XIII. For models
I
I
1
OmoS
0.01
0.015
I
n
c,, mol * L-'
Figure 10. Fit of model 3B to the SDS data by Goddard and Benson at 40 "C: c* = 8.64 mM, 6 = 0.699,n = 45 (Table X).
-
'I
rd
I
4. mol * L-1
Figure 11. Fit of model 3B to the new SDS data at 40 "C: c* = 8.06 mM, @ = 0.682,n = 41 (Table XII).
3A and 3B, which show the better fits, one sees that the cmc increases with temperature also for c, = 30 and 100 mM. There is also a trend for @ to decrease with both salinity and temperature;
The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1803
Micellization Parameters of SDS from Conductivity Data
TABLE X: Comparison of the Data of Goddard and Benson with Data Obtained in This Work" Goddard and Benson14
parameter c* (visual)
25 OC 8.4
9.1
c* (num)
8.4 f 0.1 0.72 f 0.01 44f 1 -57.9 0.009
8.7 f 0.1 0.60 f 0.03 44 f 24 -64.4 0.013
8.4 f 0.1 0.72 f 0.01 44 f 1 -57.6 0.009
8.64 f 0.03 0.69 f 0.03 45 f 32 -64.9 0.016
8.2 f 0.1 0.71 f 0.02 45 f 33 -55.5 0.012
8.7 f 0.1 0.68 f 0.02 46 f 34 -64.7 0.01 1
40
this work 8.4
60 OC 10.1
8.5 f 0.1 0.70 f 0.01 41 f 7 -227.9 0.069
10.3 f 0.1 0.68 f 0.01 35 f 5 -239.9 0.016
8.06 f 0.06 0.682 f 0.005 35 f 3 -197.1 0.035
9.7 f 0.1 0.666 f 0.001 34.5 f 0.4 -219.4 0.013
7.2 f 0.1 0.65 f 0.01 26 f 5 -240.1 0.025
5.07 f 0.05 0.646 f 0.005 49 f 5 -314.8 0.041
25 OC
O C
8 .O
40 OC
Model 3A
P n
LLF SIP
8.2 f 0.1 0.72 f 0.01 42 f 9 -210.4 0.016
Model 3B c*
P
(num)
n
LLF
SlY
8.0 f 0.1 0.72 f 0.01 56 f 16 -192.2 0.013
Model 3C c* (num)
P
n
LLF
fly
7.4 f 0.1 0.67 f 0.01 33 f 7 -232.8 0.029
"The values of c* (visual) were determined from the "break" in the K-c profile. All concentrations are millimolar. TABLE XI: Comparison of New SDS/NaCI/H,O Results at Various Salt Concentrations and Temperatures according to Model 3A"
parameter
25 OC
40 OC
60
O C
c, = 0 c*
P
'I zd
n
LLF
TIJ
8.2 f 0.1 0.72 f 0.01 42 f 9 -210.4 0.016
8.5 f 0.1 0.70 f 0.01 41 f 7 -227.9 0.069
10.3 f 0.1 0.68 f 0.01 35 f 5 -239.9 0.016
c, = 30 c*
P n
LLF
SIP
2.5 f 0.2 0.66 f 0.08 87 f 68 -167.3 0.0048
2.8 f 0.3 0.64 f 0.04 57 f 56 -137.0 0.13
4.0 f 0.6 0.64 f 0.03 55 f 48 -152.0 0.22
c, = 100 c*
P Ct,
mol L-'
Figure 12. Fit of model 3B to the new SDS data at 60 OC: c* = 9.7 mM, j3 = 0.666, n = 34 (Table XII).
n
LLF sly
0.78 f 0.01 0.5 f 0.8 44 f 47 -76.35 0.0015
1.0 f 0.02 0.5 f 0.6 38 f 40 -72.31 0.13
2.07 f 0.04 0.53 f 0.08 31 f 11 -68.17 0.22
"All concentrations are millimolar.
B is as low as 0.55 at 100 mM and 60 OC. Comparison of SDS Results Based on Conductivity to Those Based on Other Methods In nearly all references, light scattering yields average aggregation numbers which are larger than those estimated from conductivity.21-26 Moreover, light scattering data indicate that n increases from 58 to 91 as c, increases from 0 to 100 mM.25 For the same salinities, Mysels has reported an increase from 95 to 117.1° We feel that the light scattering method is surely better than conductimetry in determining aggregation numbers, at least close to the cmc, where intermicellar interactions are minimal. If such interactions are i m p ~ r t a n t , ~then ~ " ~one needs to incorporate them in the model to reliably determine n at higher concentrations. With the parameters c* (or K'"),4, and n as estimated from conductimetry, the concentrations of surfactant cR- = c- and counterions cM+ = c+ can be calculated. These values along with the mean concentration c* = (C+C..)~/~ are compared in Figure 13 to two sets of ion activity data from the l i t e r a t ~ r e . ' ~There J~ is a good agreement with one set of data and a fair agreement
TABLE XII: Comparison of New SDS/NaCI/H,O Results at Various Salt Concentrations and Temperatures according to Model 3B"
parameter
M. J . Colloid Sci. 1955, 10, 507.
40 OC
60 OC
c, = 0 C*
P n
LLF
fly c*
P n
LLF Sly c*
P n
LLF
SIP (35) Mysels, K.
25 OC
8.0 f 0.1 0.72 f 0.01 56 f 16 -192.2 0.013 2.0 f 0.4 0.66 f 0.03 30 f 54 -180.6 0.0072 0.58 f 0.01 0.57 f 0.07 32 f 17 -81.1 0.0012
8.06 f 0.06 0.682 f 0.005 35 f 3 -197.1 0.035
c, = 30 2.3 f 0.1 0.62 f 0.01 50 f 13 -153.7 0.13 c, = 100 0.7 f 0.1 0.6 f 0.1
24 f 31 -76.3 0.16
"All concentrations are millimolar.
9.7 f 0.1 0.666 f 0.001 34.5 f 0.4 -219.4 0.013 3.6 f 0.3 0.62 f 0.01 5 3 f 11 -168.1 0.22 2.0 f 0.3 0.53 0.17 27 f 31 -70.2 0.28
*
1804 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992
Shanks and Franses
TABLE XIII: Comparison of New SDS/NaCI/H,O Results at Various Salt Concentrations and Temperatures according to Model 3C” parameter 25 OC 40 OC 60 OC c, = 0 C* 7.4 f 0.1 7.2 f 0.1 5.07 f 0.05 B 0.67 f 0.01 0.65f 0.01 0.646 f 0.005 49 f 5 n 33 f 7 26 f 5 LLF -232.8 -240.1 -314.8 3IY 0.029 0.025 0.041 ~
~~
1.7 f 0.1 0.64 f 0.02 21 18 -190.1 0.0075
C*
B n
LLF 317
c, = 30 2.1 f 0.3 0.60 f 0.01 17 f 12 -156.7 0.13
c, = 100 (0.5) 0.7 f 0.2 0.59 f 0.01 0.55 f 0.03 13 f 18 17 i 24 -132.4 -78.1 0.0010 0.15
C*
P n
LLF
SlY
2.9 f 0.3 0.59 f 0.01 I9 f 1 1 -171.5 0.22 01
0
1.8 f 1.2 0.54 f 0.06 16 f 45 -71.1 0.28
I
I
1
I
I
0.01
O M
0.03
0.04
OM
c,, mol-L-’
Figure 14. Comparison of Debye lengths for aqueous SDS calculated by ~ (0)to those calMarra and Hair39 (A) and Stigter and M y ~ e l s ’data culated via models 3A-D.
“All concentrations are millimolar.
0.01
0.005
01‘ 0
I
I
I
I
0.02
om
0.06
om
I .I
c,, mOl*L-’
Figure 13. Ion concentrations in aqueous SDS. Solid curves show species inventories from the best fit of model 3B to new SDS conductivity data: c* = ( C ~ ~ + C & A, ~ ; EMF data of Sasaki et al.;” 0, EMF data of Cutler et al.;’* 0,calculations of Chang36 based on surface tension data from Elworthy and Mysels.16
with the second. The values of c+ agree even better, probably because they are less sensitive to errors. Also shown are the values of surfactant mean concentration above the cmc as estimated from tension data.36 The method involves using the data from 1 to 7 mM in the Szyskowski equation3’ to calculate the two parameters in the adsorption isotherm, followed by using the isotherm and the tension data to estimate monomer concentrations. The agreement gives further support to the micellization parameters as estimated from conductimetry with model 3B. From the ion activities and activity coefficients without intermicellar interactions, Sasaki et al. determined @ = 0.73” and Cutler et al. determined 6 = 0.80.18It is unclear whether the differences are significant. Based on electrophoretic mobilities (by dye tracer technique) and micelle size based on light scattering (n = 80), Stigter and Mysels estimated 6 0.71 and K - I = 35 A.38 Using tracer electrophoresis with radio-labeled 22Na to determine transport numbers, Mysels and Dulin determined @ = 0.72.l’These numbers are in good agreement with our results. (36) Chang, C.-H., Purdue University. Personal communication, 1989. (37) Szyskowski, B. von, 2.Physik. Chem. 1908, 64, 385. (38) Stigter. D.; Mysels, K. J. J. Phys. Chem. 1955, 59, 45.
Marra and Hair39estimated the effective ionic strength ( I ) and the Debye length ( K - I ) based on surface force measurements of charged surfaces immersed in aqueous SDS and using the DLVO theory.40 Their measured values of K - ~are shown in Figure 14, along with calculated values of K-’ based on models 3A-D. Model 3D underpredicts I ’ (overpredicts I) by over 6W0,lending support to the hypothesis that in the context of the D-H-O theory micelles do not contribute to the effective ionic strength. The data fall closer to model 3C than to model 3B. Nonetheless, the differences among models 3A-C are minor compared to the difference of all these models to model 3D. The above hypothesis is further supported by the work by Pashley and Ninham on an aqueous cationic surfactant CTAB.4’ Finally, Burchfield and W o ~ l e have y ~ ~derived equations for activity coefficients and ionic coefficients of ionic micellar solutions based on measurements of vapor pressures, partial molar volumes and enthalpies, and ion activities. Their equations are based on the Debye-Hiickel theory and the use of a “shielding factor” (6) to account for the fact that the micelles contribute to the ionic strength (Z) differently from other monovalent ions.
Their equation is identical to the one used in model 3C if 6 = 0 or to model 3D if 6 = 1. Their model is therefore an intermediate case between our models 3C and 3D. Using n = 64 and /3 = 0.77, they estimated 6 = 0.52.Had they used 0 = 0.70,6would turn out to be smaller than O S . Unlike our models, where we ignore activity coefficients, Burchfield and Wooley’s model uses Debye-Hiickel (D-H) activity coefficients (ri # l). Using c* = 8.0 mM, = 0.70,and n = 45, with D-H activity coefficients, we have repeated the calculation of Figure 13. We found that the calculated values c-, c+, and c* decrease by up to 595, with c+ changing the most.28 Using D-H activity coefficients in the inverse problem would make only a slight difference in the values of the parameters estimated and would of course make our models more rigorous, but at the expense of much increased complexity.
Conclusions (1) A method has been developed for estimating c*, 0, and n of aqueous micellar solutions from conductivity data. (2)Results for c* and (3 compare well with those by other methods; determining n is more uncertain. (3) SDS data on c*, & and n are (39) Marra, J.; Hair, M. L. J. Colloid Interf. Sci. 1989, 128, 511. (40) Verwey, E J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (41) Pashley, R. M.; Ninham, B. W. J . Phys. Chem. 1987, 91, 1901. (42) Burchfield, T E.; Wooley, E. M. J . Phys. Chem. 1984, 88, 2149.
J. Phys. Chem. 1992, 96, 1805-1809 most consistent with models in which the charged micelles contribute substantially to the conductivity but not to the effective ionic strength, in the context of the Debye-Huckel-Onsager model. Although this hypothesis is supported by many lines of evidence, no clear theoretical explanation is known. (4) The monodisperse mass action model of micellization is consistent with the SDS data for various temperatures and salinities. ( 5 ) The counterion binding parameter (8) decreases slightly with increasing temperature; 8 decreases more strongly with increasing salt concentration, perhaps because of increased charge screening at the higher ionic strength.
Nomenclature a size parameter for electrolytes coefficient in the Debye-Huckel-Onsager conductivity AI equation 'C cmc CM+ concentration of counterion concentration of salt with common counterion cs ct total surfactant concentration elementary charge of an electron e function of predicted values in model A F Faraday constant I ionic strength K conductivity K' micellization equilibrium constant LF likelihood function natural logarithm of the likelihood function LLF m number of measurements n aggregation number
1805
mobility parameter in the Debye-Hiickel-Onsager conductivity equation effective spherical radius of species i estimate of variance average standard deviation temperature independent variable dependent variable average value of dependent variable valence of species i
4 ri Si
s
T Xi
Yi
P Z
Greek Letters a
B Y Yi
r(n)
6 €
9 90
e K
x
A ai
Q
constant of proportionality, eq 22 counterion binding parameter heteroscedasticityparameter in the likelihood function activity coefficient of species i correction factor for nonsphericity shielding parameter for micellar contributions to ionic strength dielectric constant of the solvent viscosity solvent viscosity set of parameters inverse Debye length molar conductivity of an individual ionic species molar conductivity of solution error variance set of error parameters
Registry No. SDS, 151-21-3; NaCI, 7647-14-5.
High-Temperature Interaction of Vanadium Pentoxide with H-ZSM-5 Zeollte. ESR and I R Study Marek PetrG and Blanka WichterlovP* The J . Heyrovskg Institute of Physical Chemistry and Electrochemistry, DolejSkova 3, 182 23 Prague 8, Czechoslovakia (Received: February 1, 1991)
ESR, IR, UV-vis spectroscopies and temperature-programmed desorption of ammonia (TPDA) have been used to study the high-temperature interaction of vanadium pentoxide with the H-ZSM-5 zeolite at 720 K in a nitrogen stream. For comparison, the ESR parameters of the VOz+ion exchanged ZSM-5 zeolite,were investigated. It appears that part of the vanadyl ions formed as a result of VzOs reduction moves predominantly into the zeolite cationic sites, replacing strong acid OH groups, the other part is supported on a zeolite via terminal Si-OH groups. Both types of vanadyl complexes exhibit a square pyramid (C4J coordination with IB,) as the ground state. Adsorption of water or ammonia causes changes of the ligand field symmetry of V02+ ions. The resulting square bipyramid complexes produce elongation and thus weakening of the V=O bond and, on the other hand, shortening and strengthening of the V-0 bonds in the basal plane. The bonding coefficients @,*and e** obtained from the ESR spectra revealed that the average ionic character of the vanadium bonding in VOz+complexes increases with the content of vanadium in the mixtures, approaching that of ion-exchanged V@+ complexes.
Introduction Studies of the behavior of composite metal oxide-zeolite materials have recently attracted attention because of their promising application as bifunctional zeolite catalysts for processes of oxidative dehydrogenation of paraffins and for denitrification and hydrogenation processes et^.'-^ The vanadium oxide-zeolite systems are of particular interest, especially for selective oxidation proces~es.~ The relatively simple optical and magnetic properties ( I ) Beran, S.;Wichterlovl, B.; Karge, H. G. J. Chem. Soc. Faraday Trans. 1 1990,86, 3033. (2) Wichterlovl, B.; Beran, S.; Bednliovl, S.;Nedomovl, K.; Dudikovi, L.; Jiru, P.Srud. Surf. Sci. Caral. 1988, 37, 199. (3) Fornes, V.; Vazquez, M. I.; Corma, A. Zeolites 1986, 6, 125. (4) Zatorski, L. W.; Centi, G.; Lopez Nieto, J.; Trifiro, F. Srud. Surf. Sci. Coral. 1989, 49, 1243.
0022-365419212096-1805$03.00/0
of the V4+ ion, which arise from the single unpaired d electron and the nuclear spin of 7/2 have enable detailed studies of vanadium coordination in these systems. The most useful information has been obtained from the ESR spectra. In a study of vanadium in zeolites Martini et al.5 and Willigen and Chandrashekar6 established V4+ coordination at the cationic sites in V02+-X and -Y zeolites, exhibiting C4, and D4,,ligand field symmetry for the dehydrated and hydrated states, respectively. Kucherov and Slinkin' employed the ESR spectra to observe formation of isolated V4+ ions in heated V,05-H-ZSM-5 (5) Martini, S.;Ottaviani, M. F.; Seravalli, G. L.J . Phys. Chem. 1975, 79, 1716. ( 6 ) van Willigen, H.; Chandrashekar, T. K. J. Am. Chem. Soc. 1983.105, 4232. (7) Kucherov, A. V.; Slinkin, A. A. Zeolites 1987, 7, 38.
0 1992 American Chemical Society
