E. Azaz
1636
to be any data in the literature with which to compare these trends. Conclusion I t has been found that a corresponding states formulation for the Hamaker constants of nonpolar fluids gives good numerical agreement with values obtained using optical data. This microscopic formulation, which utilizes a macroscopic experimental parameter, is also able to explicitly account for the temperature dependence of the Hamaker constant. Other surface free energy terms to be found in the literaturemwere investigated, along with the corresponding states formulation, but were found either to be impractical or to violate the assumptions inherent in the original derivation of the Hamaker constant. The application of the Prigogine corresponding states principle to this problem has been carried out in the same pragmatic spirit as that of the original derivation of Fowkes. References ‘and Notes (1) R. H. Ottewill, “Colloi Wince”, D. H. Everett, Ed., Specialist Periodical Reports, Vol. 1, The Chemical Society, London, 1973. (2) P. Richmond, “Colloi Science”, D. H. Everett, Ed., Specialist Periodlcal Reports, Vol. 2, The Chemical Society, London, 1975. (3) E. R. Smith, D. J. Mitchell, and B. W. Ninham, J. Colloid Interface Sci., 45, 55 (1973). (4) R. Evans and D. H. Napper, J. ColloMInterface Sci., 45, 138 (1973). (5) F. M. Fowkes, Ind. Eng. Chem., 56, 40 (1964). (6) D. Patterson and A. K. Rastogi, J. Phys. Chem., 74, 1067 (1970). (7) J. Gregory, Adv. Colloid Interface Sci., 2, 396 (1969).
and M. Donbrow
(8) D. Tabor and R. H. S. Winterton, Proc. R . Soc., London, Ser. A , 312, 435 (1969). (9) H. C. Hamaker, Physlca, 4, 1058 (1937). (10) J. Mahanty and B. W. Ninham, J. Chem. Phys., 59, 6157 (1973). (11) G. D. Parfilt, “Dispersbn of Powders in Liquids ’, 2nd ed, 0.D. Parfilt, Ed., Applied Science, London, 1973. (12) I. Prigogine (with the collaboration of A. Bellemans and V. Mathot), “The Molecular Theory of Solutions”, “ o k n d , Amstdam, 1957, Chapter 9. (13) D. Patterson and G. Delmas, Discuss Faraday Soc., 49, 98 (1970). (14) D. Patterson, Rubber Chem. Techno/.,40, 1 (1967). (15) R. Defay, I. Prigogine, A. Bellemans, and D. H. Everett, “Surface Tension and Adsorption”, Longmans, London, 1966, Chapter 11. (16) R.J. Roe, Roc. Natl. Acad. Scl. U.S.A., 56, 819 (1966). (17) K. S. Siow and D. Patterson, Macromolecules, 4, 26 (1971). (18) D. Patterson, S. N. Bhattacharyya, and P. Picker, Trans. Faraday Soc., 64 648 (1968). (19) P. J. Flory, R. A. Omoll, and A. Vrlj, J. Am. Chem. Soc., 86, 3507 (1964). (20) 8. Vincent, J. ColloU Interface Sci., 42, 270 (1973). (21) F. van Voorst Vader and H. Dekker, “Chemisty, Physical Chemistry and Applications of Surface Active Substances ’, Vol. 2, Carl Hanser Verlag, Munchem, 1973, p 735. (22) J. Visser, Adv. ColloidInterface Sci., 3, 331 (1972). (23) J. N. Israehchvili, J. Chem. Sm., Faraday Trans. 2,60, 1729 (1973). (24) B. Vincent, Adv. Colloid Interface Sci., 4, 193 (1974). (25) D. H. Napper, I d . Eng. Chem., Rod. Res. Devebp., 9,467 (1970). (26) M. J. Vold, J. Colloid Scl., 18, 1 (1961). (27) D. W. Osmond, B. Vincent, and F. A. Walte, J. ColbUInterface Sci., 42, 262 (1973). (28) J. G. Curro, J. Macromol. Sci. Rev. Macromol. Chem., C l l , 321 (1974). (29) H. Shih and P. J. Flory, Macromolecules,5 , 758 (1972). (30) S. W. Mayer, J. Phys. Chem., 67, 2160 (1963); T. S. Ree, T. Ree, and H. Eyring, J. Chem. phys., 41, 524 (1964); S. Wu, J. Macmmol. Sci. Rev. Macromol. Chem., ClO, l(1974); H. Schonhon, J. Chem. Phys., 43, 2041 (1965).
Modified Potentiometric Method for Measuring Micellar Uptake of Weak Acids and Bases E. Azaz and M. Donbrow* School of Pharmacy, Hebrew University,Jerusalem, Israel (Received May 12, 1976; Revised Manuscript Received June 6, 1977)
The potentiometric method of measuring micellar uptake of weak acids or bases is reexamined. A modification simplifying and improving the method is proposed, based on measurement of the pH difference between paired so!utions identical in degree of neutralization and concentration of the weak function, one solution containing the surfactant. The theory is presented and its application assessed in relation to typical binding equilibria. Some experimental improvements and results for benzoic acid solubilization in a hexadecyl polyethylene glycol (24) ether are given.
Introduction The potentiometric determination of micellar binding of weak electrolytes, developed by Donbrow and Rhode~l-~ and independently by Evans: is based on comparison of pH values measured during the titration of the micellar solutions to those of the acid or base in surfactant-free solution. It is applicable only if the ionized species is not micelle bound, as shown for benzoate3 and c r e s ~ l a t eif, ~ other micelle-interacting acids, bases, or solvents are absent1 and if the surfactant does not influence the pK, or ionic strength, as is probable in the case of nonionic surfactants of low cmc. Ionic strength errors are introduced in Evans’ titration technique but can be avoidedS6 The method is applicable to surfactanb not fully saturated with solubilizate, a common situation for which few quantitative methods are available.’ The main advantage of the titration method over other methods is economy in The Journal of Physical Chemistry, Vol. 81, No. 17, 1977
time. Accuracy is however limited by the measurement of concentrations of the un-ionized and ionized species from slight buret reading differences, errors being greatly magnified near the start and end of each titration and further multiplied if comparison titrations are used. Extensive calculations are required, including corrections for phase volume ratio, which changes during the titration. The present authors have reexamined the method and found that greater accuracy, reproducibility, and speed can be achieved using a nontitration comparative technique. Experimental Section The pH values were measured for pairs of half-neutralized solutions identical in concentration of weak electrolyte and other respects, except for one of the pair containing the surfactant in the required concentration. The instrument and materials used were as previously
Measuring Micellar Uptake of Weak Acids and Bases
1637 1.2 r
described.6 Cetomacrogol 1000 B.P.C. had the average formula CH3(CH2)15(OCH2CH2)240H.6’7 However, instead of using freshly decarbonated water, (and excluding COz)as previously, which had been found to give falling and fluctuating pH values, air-equilibrated distilled water was used, which maintained its pH value for several weeks at 6.5 f 0.2. It was prepared by passing air, previously washed with 10% H2S04and then with distilled water, through distilled water for about 10 h.
Theory and Discussion Designating the concentrations originally present in the aqueous blank as c, for the un-ionized and c, for the ionized acid, and with Ac, representing the amount of the unionized acid solubilized in the micellar phase of the solution containing the surfactant, it may be shown that under the conditions of validity of the Henderson-Hasselbach equation, the pH difference between the surfactant and aqueous solutions is given by ApH = - log C, - Ac, + log c-a CS
Hence ApH = - log ( 1 - Ac,/c,)
(2)
where A log y is the activity coefficient correction. It is generally negligible, being of the order of 0.005 in 20% nonionic surfactant solution (cf. Figure l).9 From eq 2, the unbound aqueous acid in the surfactant solution (c, - Ac,) is ~ , . 1 0 - ~and P ~ the micellar acid is c,(l - 10-ApH), hence the fraction of the total un-ionized acid bound is given by
(3) The ratio of bound to free acid at a given concentration is thus Ac,/(c, - Ac,) = IOAPH -1 (4) =
1-
This may be expressed as an apparent distribution coefficient KD’ for known volumes (or weights) of the micellar and aqueous phases,loVm,V,, given by KD’=
Q
4
CS
Equation 2 is applicable at any degree of neutralization, provided the paired solutions are at the same degree of neutralization with respect to the total un-ionized acid present. Identity of the ionic species and the ionic strength in the two solutions ensures that the apparent dissociation constant remains the same, enabling it to be eliminated from the calculation, together with activity coefficient corrections.6 Where the micelle volume is not negligible, the ionic strength is fractionally higher in the surfactant solution, in which case ApH - A log = - log (1- Ac,/c,) (2a)
Ac,/c,
r
Vm/Vw(lOAPH - 1)
(5) In the solubilization of the un-ionized form of a base, the equations corresponding to (4) and (5) are
Although half-neutralized solutions were found convenient for use in the modified method, the degree of neutralization does not control the ApH value, which is determined by the nature and parameters of the binding reaction. The equilibria are generally of the following types:
Ca m M Figure 1. Relation between ApH and total un-ionized acid, c,, at given K I K P values and varying K11K2 ratios. ApH is the pH difference between an aqueous solution and one which contains w grams of surfactant per liter both at identical concentration and degree of neutralization of the acid substrate. A, B, and C represent w = 1, 20, and 200, respective@. Numbers represent values of K1(L mol-‘). K1K2 values (L g-’) are as follows: 0.05 (-), 0.25 (---), 0.058 (-e-.). Experimental points are shown for benzoic acid at degrees of neutralization ranging between 25 and 90%, solubilized in 20% cetomacrogol (A).
(a) KD’ is virtually constant over the concentration range measured, c1 to c2. In this case, ApH is independent of concentration, eq 2 taking the form ApH = [log (1 + KD’)];;
(8)
(b) The micelles are unsaturated and binding follows a Langmuir type equation (9)
where K1 and K 2 are the Langmuir parameters representing the binding constant and saturation concentration, respectively, w is the weight of surfactant present in micellar form, and 4 is the volume correction factor for the aqueous acid concentration, defined by 4 = (V, + V,)/V,. Substitution into eq 2 yields ApH = log (1 -- @KlAc, - w $ K ~K2)
(10)
When 1 >> 4Kl(c, - Ac,) in eq 9 ApH = log (1 + @ w K ~ K ~ )
-
(11)
This defines the limiting ApH value as c, 0 and may also represent a constant value of ApH at higher concentration when K1 is very low. (c) When the micelles are saturated with substrate, Ac, is constant (4K,(c, - Ac,) >> 1 in eq 9),in which case AC, = wK2,yielding ApH = - log (1- w K ~ / c , ) (12) The forms of these ApH - c, relations are shown in Figure 1for some hypothetical parameters of similar order to the K1 and K2values measured experimentally for benzoic acid An actual set of experimental and phenol ~olubilization.~~~ points and the best matching theoretical curve are also The Journal of Physical Chemistty, Vol. 8 1, No. 17, 1977
E. Azaz and M. Donbrow
1038
I
1
15.0-
-
10.0-
c :4!
cn X
\ F
5.0
”
0.5
1.o
Ilk,-Ac,)rnM-l Flgure 2. Binding of benzoic acid by cetomacrogolat 25 OC: (0)data obtained by different methods and laboratories;' (0)present work. The plot represents a reciprocal form of Langmuir’s type equation l l x = 1/K, 1/KIK2(c, Ac,) where x i s the amount of acid bound to 1 g micelles (in mmd/g), c, - Ac, is the concentratkn of free ut+imized benzoic acid (in mM), K, is the binding constant (in L/mmol), and K2 is the amount of acid bound per 1 g of micelles at monolayer saturation (in mmol/g).
+
results were generally measured between ApH 0.2 and 1.2 and their scatter about mean readings or about the best curve for a range of substrate concentrations was within the predicted values above. Some examples of the effect of surfactant concentration are illustrated in Figure 1 for different solubilization parameters. ApH values are maximal at low c,, where micellar uptake is highest, converging to the eq 11 value as c, 0 for all K1:Kzratios having a common KIKzvalue. However, reproducibility may fall off at very low concentrations, due to the low buffer capacity of the solution. All curves tend toward ApH = 0 at high enough c, in accordance with eq 2 and 12, the ApH value and convergence depending on the degree of saturation of the micelles. Where the K1:K2ratio is high, it is advantageous to work at a low c, value. The method has been applied to benzoic acid over a range of concentration up to 12 mM free acid (equilibrium value) in 2% cetomacrogol, yielding KD’ values close to those obtained by the titration technique and with less scatter. Since the value of KD’ varies with free acid concentration, it is difficult to make a point to point comparison of accuracy. However, linearization of the relation KD’ - free acid concentration by means of a reciprocal Langmuir equation plot7 led to slopes of similar values in the two methods which were also in agreement with those obtained by other methods and in other laboratories (Figure 2). From the slopes of individual tiL/g,7 trations, KlK2values ranged from 4.4 to 6.9 X where K1K2is the product of the two Langmuir equation parameters for the system benzoic acid-cetomacrogolwater. The statistical slope of the line obtained from Figure 1 gives a KIKz value of 5.2 X while the modified method yields 4.9 X
-
included to illustrate the sensitivity. Since the experimental error resides mainly in pH measurement, error is minimized by raising the ApH value, which can be achieved by increasing the surfactant concentration and selecting a suitable c, range. The experimental error in assessment of bound substrate due to inaccuracy in pH readings may be estimated by differentiation of eq 5. Neglecting changes in the phase volume ratio, this gives dKD’/d(ApH)= 2.303V,/V;1OApH (13) The fractional error in conveniently expressed as AK,’/KD’ = 2.303( 1 pH, (14) where pH, is the experimental error in measurement of ApH. Using equipment of 0.002 pH accuracy6p8with the modified method, ApH should be measurable to 0.005 pH or better. The corresponding errors in KD’at this limit calculated from eq 14 are 5.6, 3.1, 1.7, and 1.3% at ApH values of 0.1,0.2,0.5,and 1.0, respectively. Experimental
The Journal of Physical Chemistry, Vol. 87, No. 17, 1977
References and Notes (1) M. Donbrow and C. T. Rhodes, J. phann. h m c o l . , 15, 233 (1963). (2) M. Donbrow and C. T. Rhodes, 23rd F.I.P. Conference, Munster, 1963, Govi-Verlag GMBH, Frankfurt am Main, 1964, pp 397-404. (3) M. Donbrow and C. T. Rhodes, J. Chem. SOC. Suppl., 2, 6166 (1964). (4) W. P. Evans, J. Pharm. Pharmacol., 16, 323 (1964). (5) E. Azaz and M. Donbrow, unpublished data. (6) M. Donbrow and C. T. Rhodes, J. pharm. h r m a c o l . , 17,258 (1965). (7) M. Donbrow, E. Azaz, and R. Hamburger, J. Pharm. Sci., 59, 1427 (1970). (8) E. Azaz and M. Donbrow, J. ColbMInterfaceSci., 57, 11, 19 (1976). (9) Where ionic strength affects the binding equilibrium through alteration of solubility and/or micellar properties, adjustment to known ionic strength is required, as in other methods. The paired solutions of the modified method are particularly convenient for such adjustment and no correction was needed in the mathematical treatment.’,* (10) V , is defined and determined as in A. T. Florence, J. Pharm. Pharmacol., 18, 384 (1966).
