2017
Langmuir 1991, 7, 2017-2020
Articles Application of Phase Separation and Mass Action Models to Low Aggregation Number Micelles: A Comparative Study Manuel E. Moro and Licesio J. Rodriguez' Departamento de Qulmica fisica, Facultad de Farmacia, Universidad de Salamanca, Apdo. 449, E-37080 Salamanca, Spain Received November 13, 1990. I n Final Form: February 19, 1991 Interpretation of micellar behavior of low aggregation number micelles formed by aqueous tricyclic antidepressants hydrochlorides of imipramine, desipramine (iminodibenzyl derivatives), amitriptyline, and doxepin (dibenzocycloheptadiene derivatives), in aqueous solution at 25 "C, using two approaches, based on a phase separation model and a mass action model, has been carried out from conductivity measurements. Calculated micellar parameters obtained by using these two models compare acceptably well to each other and to previous literature values. Introduction The phenomenon of micelle aggregation has been interpreted thermodynamically either as an association equilibrium between the single ions and the monomer species' (mass action model, MAM) or as a phase separation process2 (phase separation model, PSM). Both approaches account for major features of micelle formation. In the first approach, the law of mass action is applied. While a multiple association equilibrium model is believed to be more r i g o r ~ u sthe , ~ simplicity of a single-equilibrium model makes it more useful in handling the algorithm to describe micellar behavior in s ~ l u t i o n .In ~ the latter approach, it is considered that micelles separate as a distinct phase at a particular critical concentration point, the critical micelle concentration (cmc), configuring a genuine phase transition. In both cases, the cmc may be defined as the maximum concentration of unassociated ions: and it represents a good micellar quantity to carry on the experimental test of the adequacy of a particular model. On these bases, to explore the way in which both approaches are interrelated, has became of interest. Empirical application of the PSM to physical measurements on micellar systems is amenable from a computational point of view.6 On the contrary, the high aggregation numbers involved in most of micelle equilibria make cumbersome the application of even the simpler MAM calculation to these systems. There are, however, surfactant compounds that form small micellar aggregates in solution with low aggregation numbers, less than 10 monomers per micelle, that can be used for this task with some advantage. To these belong some tricyclic antidepressant drugs, whose micellar properties have already been investigated to some extent by light scattering and conductivity techniques.s It makes them very appropriate as probes to carry out this application making, conse(1) Mitchell, D. J.; Ninham, B. W. J. Chem. SOC.,Faraday Trans. 1981, 77, 601. (2) Shinoda, K.; Hutchinson, E. J. Phys. Chem. 1961,66, 577. (3) Hunter, R. J. Foundations of Cotloid Science; Clarendon Press: Oxford, 1987; Vol. I, Chapter 10. (4) Burchfield, T. E.; Wolley, E. M. J. Phys. Chem. 1984,88, 2149. (5) Garcia-Maw, I.; VelBzquez, M. M.; Rodriguez, L. J. Langmuir 1990,6,1078. (6) Attwood D.; Gibeon, J. J . Pharm. Pharmacol. 1978,30,176.
quently, more feasible the proposed comparison between the two aforementioned models. This paper presents the results obtained in applying a phase separation model and a mass action model to conductivity measurements of hydrochlorides of imipramine, desipramine (iminodibenzyl derivatives), amitriptyline, and doxepin (dibenzocycloheptadiene derivatives), in aqueous solution, at 25 "C. Calculated micellar parameters obtained by using these two models compare acceptably well between to each other and to previous literature values.6 Description of the Algorithms Mass Action Model. This model is based on the simplest nonideal mass action treatment involvinga single 1:l surfactant electrolyte, (D+,X-),that aggregates according to the following equilibrium reaction: nD+ + pnX- s D,X, +n(*-8) (1) At equilibrium, the following mass balance of the surfactant is established
C=D+M C=X+bM
(2)
(3) and the thermodynamic equilibrium condition for reaction 1 can then be written as
-AGo/RT = In (M/n) - n In (D) - @nIn (X) + AG,,/RT (4)
where micellar concentration, [DnXng+n(1-8)], is represented by (M/n). An equilibrium solution of free surfactant monomers and micelles may be regarded as a mixed electrolyte formed by the 1:l electrolyte, (D+,X-1, and the The l:n(l - 0) electrolyte (n(1 - p)X-, DnXn,g+"('-@)). electrostatic excess free energy, AG,l f RT, can then be written as AG,,/RT = In f M - n In f D - pn In f x = -n(l+ 8) In f, (5) According to previous observations,2+ and due to the much lower charge density in a micelle than in a free ion, the model will assume ideal behavior for that species, Le., f M = 1. A mean ionic activity coefficient, f i ,expressed as
0743-7463/91/2407-2017$02.50/0 0 1991 American Chemical Society
Moro and Rodrlguez
2018 Langmuir, Vol. 7, No. 10, 1991
is then taken for the free electrolyte, (D+, X-), where In f* = -A11/2/(l
+ b,I'/')
(7)
is given by the Debye-Huckel equation, with A = 1.178 (L/mol)'/zand b+ = 1.5 (L/mol)'I2, as has been suggested by Guggenheim.' Ionic strengthof the electrolyte solution is determined only on the basis of the unassociated electrolyte concentrations
I = (D+ X)/2 (8) When the physical quantity used in the experimental measurements is conductivity, K , it can be expressed as K
= XXX
+ A@ + XMM
(9)
where the molar ionic conductivities are approximated by the limiting Onsager equations8
with values B1 = 0.2300 (L/mol)'l2 and B2 = 60.64 S cm2 L1/2 m01-3/2, in water as solvent at 25 "C. It is considered that the average molar conductivity of the micellized surfactant, AM, is fairly independent of ionic ~ t r e n g t h . ~ Phase Separation Model. This model considers that, after a given concentration, cmc, the solution is formed of two pseudophases: the aqueous and the micelle phases. Accordingly, a sharp change in the physical behavior of the system will take place around this concentration. Taking into account the mass balance of the surfactant, as in eqs 2 and 3, the following expression can be written5 for the degree of aggregation, y, of monomers into the micelle, M y = dM/dC = d[C
- D]/dC = (l/p)(d[C - X]/dC)
(12)
Defined in this way, this parameter can take values from 0 to 1 as the concentration of surfactant increases. Following the empirical definition of cmc as reported by Phillips,'* the value of y would change with concentration according to a sigmoidal profile: so that its slope will reach a maximum at C = cmc, i.e. dy/dC = d2M/dC2= -d2D/dC2 = maximum; C = cmc (124 d2y/dC2= d3M/dC3 = -d3D/dC3 = 0; C = cmc (12b) From a probabilistic point of view, the distribution of counterions or of monomers in the aqueous and micelle phases could be interpreted with the aid of a binomial distribution model, considering that each monomer can occupy two possible states, aggregated or free, and where the degree of aggregation, y, would represent the probability distribution function of the aggregated monomer. When the number of individual monomers is large enough, this would become a continuous distribution corresponding to the Gauss function11 (7) Kiesel, T. R. In Electrochemical Methods. Physical Methods of Chemistry; Rossiter, B. W., Hhilton, J. F., Eds.;Wiley-Interscience: New York, 1986; Vol. 11, Chapter 2. (8)Spiro, M. In Electrochemical Methods. Physical Methods of Chemistry; Rosaiter, B. W., Hamilton, J. F., Eds.; Wiley-Interscience: New York, 1986; Vol. 11, Chapter 8. (9)Kay, R. L.; Lee, K. S. J. Phys. Chem. 1986,90, 5266. (10)Phillips, J. N. Trans. Faraday SOC.1955,51, 561. (11)Papoulis, A. Probability, Random Variables and Stochastic Processes; McGraw-Hill: New York, 1965;Chapter 3.
y = 1 / 2 + erf[(C - (C))/u] (13) where (C) and u represent the mean and the standard deviation, respectively. Taking the degree of aggregation as defined by eq 12, one has that y = dM/dC
+
= -dD/dC = -(l/B)(dX/dC) = 1 / 2 erf[(C - (C))/ul (14)
and dy/dC = d2M/dC2= -d2D/dC2 = -(1//5)[d2X/dC2] = ( l / a ( 2 ~ ) ' /exp[-(C ~) - ( C))2/2u2] (15) If the cmc is considered as the maximum concentration of free monomers2 in the aqueous phase, one has that this concentration coincides with the mean of the Gauss distribution, Le.: cmc = (C),such that eq 15can be written as follows: d2D/dC2= - ( l / a ( 2 ~ ) ' / ~exp[-(C ) - cmd2/2u2] (16) By integration of this differential equation, the surfactant concentration in the aqueous phase, D, can be obtained in terms of the mean, cmc, and the width, u, of the Gauss band, as fitting parameters. The two consecutive integrations may be carried out numerically by using, for instance, and Euler method,12 taking as first integration constant, (dD/dC)(c=o) = 1 and as second integration constant, (D)(c=o)= 0. When the physical quantity used in the experimental measurements is conductivity, K , eqs 8-11, along to the double integration of eq 16, may be used in the fitting procedure of experimental data to the model.
Experimental Section The reagents, hydrochlorides of imipramine, desipramine (GeigyPharmaceuticals),amitriptyline(Merck,Sharp & Dohme), and doxepin (Pfizer), were sufficiently well characterized and purified by the manufacturers to be used without further purification.
The solutions were prepared with water obtained after treatment with a Milli-Q system from Millipore. In order to attain high statistical confidence in the numerical fits, a large number of experimental results are required, so that a technique of conductometric titration was employed. Thus, each curve has more than 50 conductivity/concentration values. Conductivity was measured with a Crison 522 conductivity bridge with an Ingold cell of 0.991 f 0.001 cm-l cell constant calibrated with solution of KCl of known concentration.1s All measurements were performed at 25.0 f 0.1 O C . The numerical procedures used to fit eqs 2-11 (MAM, with fitting parameters n, 8, AM, -AGoIRT) or eqs 2, 3, and 8-11, along to the double integration of eq 16 (PSM, with fitting parameters: cmc, u, 8, AM), to the experimental results, were based on a Monte Carlo search in parameter space, in which the fitting parameters were varied in a random chain as long aa the chi-squared value, x2 = C ( K .-. ~K ~ J Z , were decreasing." The Onsager limiting slopes, and, consequently, the values of the ionic limiting conductance of the surfactant monomeric ionic species, A D O , were separately determined by using data up to a concentration around 2u below the cmc. A value of Axo = 76.32 S cm2 mol-' was used as ionic conductance for the C1- ion.8
Results and Discussion Application of the PS Model. The conductivity/ concentration results for aqueous solutions of the sur(12)Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry, 2nd ed.; Van Nostrand Princeton, NJ, 1956; Chapter 13. (13)Lind, J. E.;Zwolenik, J. J.; Fuoss, R. M. J. Am. Chem. SOC. 1959, 81, 1557. (14)Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969; Chapter 11.
Langmuir, Vol. 7, No. 10, 1991 2019
Low Aggregation Number Micelles
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Figure 1. Conductivity of aqueous solutions of tricyclic antidepressant surfactants at 25 O C : (a-d) experimental data (open circles) and PSM calculated values (line)of conductivity vs concentration; (e-h)experimental data vs MAM calculated values of conductivity (open circles). Upper line represents percentage of deviation. factants, hydrochlorides of imipramine, desipramine, amitriptyline, and doxepin, are shown in Figure 1. When these data are analyzed via the PSM numerical procedure, the plots in parts a-d of Figure 1show good concordance between the experimental values of conductivity and those calculated by the proposed PS model. Fitting parameters, cmc, u, 0,and AM, that give the best fit for each system, are presented in Table 1. The deviation between the
calculated and experimental values is also shown. It can be seen that cmc and values are in acceptable agreement with those previously reported in the literature,6 determined from the change of slope in plots of ionic conductance vs total surfactant concentration (cmc) and from light scattering measurements (p), in aqueous solutions of these compounds. Except for desipramine, Gaussian bandwidth values, u, seem to follow the changing trend of
2020 Langmuir, Vol. 7, No. 10, 1991
Mor0 and Rodriguez
Table I. PSM and MAM Micellar Parameters for Aqueous Tricyclic Antidepressants at 26 OC PSM MAM AD", s
mol-'
ref 6 lo%, AM, s XM,S lpcmc, lOacmc, % mol-' mol % mol 102cmc, mol mol-' molL-1 L-' @ cm2 RMSa n @ ACo/RT cm2 L-l RMS' L-1 n 4.8 7.5 6.28 0.26 33.01 5.881 1.2b 0.7879 37.06 0.22 7.298 0.7636 -32.54 0.22 5.7 7.2 35.51 8.64 6.65; 0.9% 0.743; 33.0; 0.22 6.98s 0.7952 -28.38 0.31 4.4 6.7 5.20 34.86 4.619 0.5115 0.7740 39.61 0.30 8.6Ot3 0.816 -42.90 7.78 0.15 6.8 6.6 27.74 7.826 1.46 0.8209 39.47 0.13 7.266 0.7998 -30.92
surfactant cm* himamine HCl 29.h desipramine HCl 26.G amitriptyline HCl 28.76 28.38 doxepin HCl a Percent root mean square deviation.
cmc values. Unfortunately, a simple relationship between u and other micellar parameters, such as cmc, aggregation number, n,etc., is not clear from these results. Therefore, to make a deeper analysis on the only basis of these results seems to be not easily approachable. Nevertheless, the goodness of the fit of the experimental data to this model might endorse it with, at least, some empirical validity. Application of MA Model. The results obtained from the application of the MAM algorithm to the conductivity data are displayed in parts e-h of Figure 1,and the values for the fitting parameters that give the best fit to the model, n, 8, AM, -AGo/RT, are presented in Table I. Also a calculated cmc value is reported. The determination of this parameter, not a fitting parameter for the case, has been carried out by computing the maximum concentration of free monomeric surfactant in the micellar system, using eqs 2-8 and Table I. It is to be noted that the concentration of unassociated surfactant, calculated following this procedure, begins increasing with total surfactant concentration, then it reaches a broad maximum, which in this case is taken for the calculated cmc, and then starts slowly to decrease.lS This fact might incorporate some limitations to the model. It would have been desirable that this maximum concentration were better a saturation value than just a broad maximum, as it is largely known from experimental evidenceale However, micellar parameters calculated by using the MAM approach, seem to compare reasonably well to those obtained from the PS model, as well as to those reported previously in the literature: i.e., n and 8, as is shown in Table I. Differences between the PSM and MAM values for the fitting parameters 8, AM, and cmc lie in ranges from less than f1 % to a maximum of around f15%. Concerning the standard free energy for the transfer of a monomer from the aqueous to the micellar state,it takes the values -4.46, -4.06, -4.99, and -4.26 RT units, for imipramine, desipramine, amitriptyline, and doxepin, respectively. These results agree qualitatively with the (15) Attwood, D.;Florence, A. T. Surfactant System; Chapman and Hall: New York, 1983,Chapter 3. (16) Lindmann, B.;Wenneratrom, H.Top. Current Chem. 1980,87, 1.
@ 0.77 0.79 0.85 0.83
minor hydrophobiccharacter differencesthat exist among these compounds.6
Conclusion It has been made evident that micellar behavior may be modeled by simple numerical methods, being able to account for the value of such important micellar parameters as average aggregation number, counterion binding ratio, or micellization free energy, from a fast and simple experimental procedure as solution conductivity measurements. This is a far reaching accomplishment that may have notable applications among ionic surfactants, as a convenient alternative of other more elaborated techniques, such as light scattering, fluorescence, The developed models can easily be extended to higher aggregation number micelles with little additional computing effort.
C X D M n
P AGO fi
Z K
A
Glossary total concentration of surfactants free counterion concentration free surfactant monomer concentration micellized surfactant monomer concentration average number of monomers aggregated per micelle micelle counterion binding parameter micellization reaction standard free energy ionic activity coefficient of species i ionic strengthof free surfactant electrolytesolution conductivity ionic conductance
Acknowledgment. Thanks are due to Geigy Pharmaceuticals, Merck Sharp & Dohme, and Pfizer Laboratories for generous gifts of samples. Registry No. Imipramine HCl, 113-52-0; desipramine HC1, 58-28-6;
amitriptyline HCl, 549-18-8; doxepin HC1, 1229-29-4.
