J. Phys. Chem. 1981, 85, 1173-1176
1173
Ampholyte-Micelle Equilibria. A New Method for Estimating Micelle Size and the Association Constant L. W. Nlchol” and A. G. Ogston Department of Physical Biochemistry, John Curtin School of Medical Research, Australian National Vniverslty, Canberra, A.C. T. 260 1, Australia (Received: April 2, 1980: In Flnal Form: January 7, 1981)
The formation of micelles is viewed as a simple association of monomeric ampholyte governed by single values of n (determining the size of the micelle) and of K , the thermodynamic association equilibrium constant. It is shown that values of n and K may be determined from a plot of any weight-averageproperty of the micellar system vs. total ampholyte concentration by utilizing three points on the sigmoidal curve, which are defined by equating the second and third derivatives with zero. The analysis does not require graphical location by differentiation procedures of these points but rather is based on the observation that weight fractions of monomer and of micelle as well as the normalized ordinate of the curve may be calculated as sole functions of n. Such calculations are tabulated to provide a ready means of finding equilibrium parameters for any micellar system of this type. The analysis procedure is illustrated by using numerical examples referring to the concentration dependence of the weight-average molecular weight of a micellar system.
Introduction The equilibria which are generally accepted to exist in solution between monomeric ampholytes and globular micelles1have been described in several ways. First, micelles have been regarded as a separate bulk phase, the transition from solution phase occurring necessarily at a sharply defined “critical micelle concentration”. As Tanford2has demonstrated, the predictions of this treatment are wrong, since the dependence of the base-molar entropy of ampholyte in the micellar state upon the volume concentration of micelles predicts a continuous (not discontinuous) relationship between monomer and micelle concentrations. This error is corrected in a second formulation where micelles are treated as a separate microscopic phase with allowance for the variation of the entropy of the system due to their varying c~ncentration.~The latter model predicts a narrow range of micelle size in spherical or spheroidal micelle formation, the relationship between the standard chemical potentials of monomer in free solution and in the micellar state being equivalent to an overall thermodynamic association constant governing micelle formation. Indeed, in the simplest case of the third type of description (mass action law models), a two-state system is envisaged in which micelles are regarded as the product of a chemical equilibrium involving n-merization of monomer, described by n and by a single association constant, K. Mukerjee4 and more recently Kegeles5have emphasized the observation that the two-state mass action law model must be regarded as an approximation, since in a mechanistic sense micelle formation must proceed by a set of multiple equilibria involving intermediates. Nevertheless, both workers in discussing the distribution of equilibrium concentrations of such intermediates have stressed that the two-state approximation is reasonable for the formal description of several surfactant micellar systems in aqueous solution. Thus, Kegeles5 provides a modified
distribution function for his shell model of micellar formation, which, with an appropriate value for the small nucleation factor, adequately expresses the behavior of many micellar systems by ensuring that the relative equilibrium concentrations of intermediates are small, while Mukerjee4points out that the size distribution index, a ratio of the weight-average degree of association to the number-average value, is close to unity at relatively high total concentrations for many nonionic and ionic systems capable of forming spherical or spheroidal micelles. It follows, therefore, for such systems that a formal description in terms of the two parameters, n and K , may provide a reasonable first approximation in determining the variation of solution composition with total concentration. Not only may such a description provide a basis for a more detailed interpretation of the operation of multiple equilibria, if required (for example, in relation to eq 6 of ref 5), but also it is necessary if quantitative interpretation is to be given to the interaction between certain ligand systems capable of micellar formation and macromolecular acceptors which arises in many situations.l+10 The purpose of this work is to present a simple method for obtaining values of n and K from measurements of the variation, with total ampholyte concentration, of any weight-average property, X,of the system, by use of three exactly defined points in the _sigmoidal relationship between total concentration and X. These points replace the variously (often inexactly1) defined “critical micelle concentration”.
(1)L. R. Fisher and D. G. Oakenfull, Chem. SOC. Reu., 6, 25 (1977). (2) C. Tanford, “The Hydrophobic Effect: Formation of Micelles and Biological Membranes”, Wiley, New York, 1973. (3)J. N. Israelachvili, D. J. Mitchell, and B. W. Ninham, J. Chem. SOC.,Faraday Trans. 2, 72, 1525 (1976). (4) P. Murkerjee, J. Phys. Chem., 76, 565 (1972). (5) G. Kegeles, J. Phys. Chem., 83, 1728 (1979).
222R (197x1. ____
0022-365418112085-1173$01.25/0
Theory Consider a system in which a self-associatingmonomer, A, forms an n-mer (micelle), C, according to nA e C, the relative amounts of any other polymers being assumed negligible. The components A and C are taken to include counterions. It is also assumed that the activity coefficient (6)E.F. Woods, G.G. Lilley, and M. A. Jermyn, Aust. J. Chem., 31, ~
(7)P.W. Kuchel, D. G. Campbell, A. N. Barclay, and A. F. Williams, Biochem. J., 169,411 (1978). (8)N. C. Robinson and C. Tanford, Biochemistry, 14, 369 (1975). (9)N. D.Hinman and J. R. Cann, Mol. Pharmacol., 12, 769 (1976). (10)L. W. Nichol, G. D. Smith, and A. G. Ogston, Biochim. Biophys. Acta, 184, l(1969).
0 1981 American Chemical Society
1174
Nichol and Ogston
The Journal of Physical Chemistry, Vol. 85, No. 9, 1981
*
TABLE I : Numerical Relationships Describing Critical Pointsa in the Plot of a Weight-AverageProperty for the System n A C
KcA"-' 2
CClE
CA/F
n 1 3 1 2 3 1 2 10 0.0251 0.1592 0.3608 0.9755 0.8627 0.7349 0.0245 0.1373 20 0.0174 0.1284 0.3007 0.9829 0.8862 0.7688 0.0171 0.1138 30 0.0130 0.1100 0.2643 0.9872 0.9009 0.7910 0.0128 0.0991 40 0.0103 0.0977 0.2399 0.9898 0.9110 0.8065 0.0102 0.0890 50 0.0086 0.0889 0.2221 0.9915 0.9184 0.8183 0.0085 0.0816 60 0.0073 0.0821 0.2083 0.9928 0.9241 0.8276 0.0072 0.0759 70 0.0064 0.0767 0.1972 0.9936 0.9288 0.8353 0.0064 0.0712 80 0.0057 0.0723 0,1881 0.9943 0.9326 0.8417 0.0057 0.0674 90 0.0051 0.0685 0.1803 0.9949 0.9359 0.8472 0.0051 0.0641 100 0.0046 0.0653 0.1737 0.9954 0.9387 0.8520 0.0046 0.0613 110 0.0042 0.0625 0.1678 0.9958 0.9412 0.8563 0.0042 0.0588 120 0.0039 0.0601 0.1625 0.9961 0.9433 0.8602 0.0039 0.0567 ' " 1denotes d 'X/dF3 = 0 (lower sohition); 2 denotes d'x/d/da2= 0 ; 3 denotes d3r/dF3=
ratio, yc/yAn, is unity,ll so that K and the total solute concentration, E , are given by eq 1, where CA and cc are the
K = cC/CA" E = CA + KcA" (1) equilibrium concentrations of monomer and micelle on the weight-concentration scale (g/L). The relationship between C, any weight-average property of the system X (for example, molecular weight, sedimentation coefficient, elution volume, or partial molar volume) and the corresponding values XA and X c for the equilibrium reactants is given by eq 2, where it is assumed that XA and X C , X = (XACA+ X ~ C ~ )=/ C (XACA+ XCKCA")/(CA + KcA") (2) referring to migration experiments, are independent of composition. This approximation suffices in illustrating the simple analysis of a two-state micellar system where the prime objective is to obtain first estimates of the basic parameters n and K , but more sophisticated procedures of analysis will be mentioned later. Successive differentiation of eq 2 with respect to cA, together with the relation dcA/dC = 1 / ( 1 + nKcA"-') from eq 1, yields dX/dC = KCA"(XA- x c ) ( l - n ) / [ ( C A + Kcff)2(1+ nKc~"-')] (34 d2X/dC2 = KcA"(XA - X c ) ( 1 - n ) ( n ( l K c A " - ~ -) ~ 2(1 ~ K C A " - ' ) ~ ] / [ ( CKCA")~(~ A ~ K C A " - ' ) ~(3b) ]
+ +
d3X/dc3 = KcA"(XA - x , ) ( 1 - ? I ) ~ / [ ( C A+ K C A " ) ~ ( ~nKc~"-l)'] (3c)
+
4 = n2(3n- 1)(2n - ~ ) ( K C A " + -~)~ 2n(3n + 1)(2n- ~ ) ( K C A " -- ~3n(3n ) ~ - l ) ( n~ ) ( K C A ' --~2n(n ) ~ - 2)(n+ ~ ) K C A "+- ~(?I - 3)(n - 2) (34 When XA = X Cor n = 1, eq 2 shows that 8 is a constant, and these trivial cases will not be considered fu_ther. Equation 3b shows that, when n > 2, the plot of X vs. C is sigmoidal with a single point of inflection (d2X/dC2= 0) defined by eq 4. Similarly, by placing eq 3c equal to
(6
6)
KcA"-' = - fi)/(nfi (4) zero, we find that the only acceptable solution is 4 = 0, which leads by eq 3d to a corresponding polynomial in (11) E. T. Adams, Jr., and H. Fujita in "Ultracentrifugal Analysis in Theory and Experiment", J. W. Williams, Ed., Academic Press, New York, 1963, p 119.
x vs. C
-
X / X A (with XCJXA = n ) 3 1 2 3 3.386 0.2651 1.220 2.236 0.2312 1.325 3.162 5.392 7.063 0.2090 1.372 3.874 8.546 0.1935 1.398 4.471 9.906 0.1817 1.418 5.001 0.1724 1.428 5.476 11.171 0.1647 1.439 5.916 12.366 0.1583 1.448 6.327 13.508 0.1528 1.452 6.706 14.595 0.1480 1.453 7.068 15.651 0.1437 1.456 7.412 16.662 0.1398 1.462 7.746 17.634 0 (higher solution).
KcAn-l. By Descartes' rule of sign, this polynomial may have two positive real roots for any given value of n, and these roots, values of KCA"-~, may be determined by numerical solution of the polynomial. The solutions correspond to the maximum rates of change of slope (positive and negative, respectively) of the plot of X vs. E , and they establish, together with eq 4,three points on the plot12at each of which KcA"-' is uniquely related to n. Values of KcA"-l have been calculated for a range of values of n for each of the three points (1,2, and 3 in ascending order of C) and are given in Table I. Since from eq 1 cA/c = 1 / ( 1 + KcA"-~)and cc/E = 1 - (cA/C), values of the weight fractions of monomer and of micelle may also be calculated for each of the three points as shown in Table I. Values of cC/care in all cases relatively small, making it likely that the assumption yc/yA"= 1 is a fair approximation. It is now noted that the three points defined in Table I may be difficult to locate in an experimental plot by difference procedures. However, this difficulty may be overcome by writing eq 2 as
X / X A = [1 + ( X C / X A ) K C A " - ~ + ] / KcA"~) (~ (54 When XA and X c are known, eq 5a may be used to calculate X / X A for the three points as a function of n. For example, combination of eq 4 and 5a yields for the point of inflection (d2X/dt2= 0) X/xA = [nfi + ( x C / ~ A ) ( f -i fi11/[1& - 111 ( 5 ~
6
Analogous expressions applicable when d3X/dC3= 0 cannot be written explicitly; but nevertheless numerical solution of eq 3d in c3mbination with eq 5a also permits determination of X / X A as a sole function _of n for these critical points. In the particular case when X is a weight-average molecular weight, X c / X A = n and thus values of x/xA may be calculated for the three points even without knowledge of Xc, the molecular weight of the micelle; such values are shown in the last column of Table I. Applications Consider first a situation where X is a weight-average molecular weight so that Xc/XA = n. Figure 1 shows plots of X / X A vs. C simulated by using eq 1 and 5a with n = 60 and arbitrarily selected values of K (g'-" L"-l) = (a) 5 X 1013, (b) 5, and (c) 5 X 10-13. The arrows denote the (12) J. N. Phillips, Trans. Faraday SOC.,51,561 (1955). Phillips has used a third derivative obtained by assuming micelle concentration to be negligible to define a "critical micelle concentration" and has employed this sole point together with a known or assumed value of n to calculate K.
The Journal of Physical Chemistry, Vol. 85, No. 9, 198 1 1175
Ampholyte-Micelle Equilibria
TABLE 11: Illustration of the Evaluation of the Association Equilibrium Constant Governing the Reaction n A 2 C when n is Unknown
P
CA
n 1 2 3 1 2 1.34 1.60 1.43 10 1.47 1.55 1.67 1.45 1.41 20 1.48 1.59 1.74 1.47 1.46 30 1.49 1.62 40 1.49 1.64 1.79 1.47 1.49 50 1.49 1.66 1.85 1.48 1.52 1.55 1.90 1.48 60 1.49 1.68 70 1.50 1.69 1.95 1.49 1.57 1.59 2.01 1.49 80 1.50 1.71 1.61 2.06 1.49 90 1.50 1.72 100 1.50 1.74 2.11 1.49 1.63 110 1.50 1.75 2.16 1.49 1.65 120 1.76 2.22 1.49 1.66 1.50 a Values interpolated from Figure ICon the basis of assumed value of n.
3 1 2 3 1 2 3 -1.95 -1.10 0.42 -2.95 1.18 0.04 0.21 -2.55 -4.75 -3.73 1.28 0.18 0,39 0.03 -4.64 -6.72 -5.73 0.36 0.02 0.16 1.38 -7.75 -6.79 0.35 -8.39 1.44 0.02 0.15 -9.95 -9.42 -10.51 0.34 0.01 0.14 1.51 -12.24 -12.22 -12.31 0.33 1.57 0.01 0.13 -14.63 -15.35 -14.12 0.32 0.01 0.12 1.63 -17.03 -18.73 0.32 -15.85 1.69 0.01 0.12 -19.57 -22.38 0.31 -17.59 0.01 0.11 1.75 -22.18 -26.04 0.31 -19.32 0.01 0.11 1.80 -24.92 -29.90 -21.05 0.31 0.01 0.10 1.85 -34.23 -22.78 -27.41 0.31 0.01 0.10 1.91 ordinate values taken from the last column of Table I for each
-35b
01 0
' I
c (giliter)
2
log K
CC
I 3
Flgure 1. Dependence on total concentration of the weight-average function X / X , calculated from eq 1 and 5a for the s stem nA $ C with n = X c / X A = 60 and K(g'-"L"-') = (a) 5 X,'ol (b) 5,and (c) 5 X The arrows denote positions on the curves for which d2X/dE2 = 0 (central) and d3R/dE3 = 0 (upper and lower). The approach of the sigmoidal plots to the upper limiting value of 60 is not shown; the curves reach 95% of this value at concentrations of E (g/L) of (a) 12.10,(b) 20.13,and (c) 33.40.
positions of the three critical points on each curve, taken from X / x A ( n = 60) values in Table I. It is evident that these points could not be satisfactorily identified by inspection of these curves. n Is Known. In certain experimental situations (such as that exemplified by curve a in Figure l),it may be possible to encompass a sufficiently large range of E to determine XA and X c as limiting values of the sigmoidal plot of weight-average molecular weight vs. E and hence to determine n as X , / X k Values of x/xAfor this n may then be taken from Table I (which may readily be extended, if required) to locate the positions of the three critical points and hence to determine the values of E at which they arise. For the particular n, values of C A / E and of cc/C are also available in Table I, for each critical point, which leads to three sets of correspondingvalues of C A and C C , from each of which K may be calculated by using eq 1. Use of this procedure in relation to the curves shown in Figure 1correctly regenerated the assigned values of K. n Is Unknown. In other experimental situations (for example, curve c of Figure 1) it may not be feasible to determine X C because the sigmoidal plot may only approach this limiting value at unrealistically high values of
20
40
O\
80
100
1
Figure 2. Determination of the parameters n and Kin the analysis of curve c of Figure 1 by the procedure, described in the text, which
utilizes the three critical points d2x/dE2 = 0 (A)and the two solutions of d3X/dE3 = 0 (0, 0).
c. In these cases, it is possible by use of Table I and the procedure described above to calculate apparent values of K corresponding to a range of values of n for each of the three critical points (Table 11). These three values of K should agree, for the correct value of n: for example, by inspection of Table I1 we see that the assigned values of n (=60) and K (=5 X have been correctly regenerated, the latter within reasonable precision. Alternatively, the values of log K may be plotted against n, as in Figure 2, giving an intersection at the correct value of K . Inspection of Table I1 and Figure 2 suggests that, with reasonably accurate experimental data, n can be estimated within *5 and log K within f2 by this procedure. X , / X , # n Which Is Known. Consideration is now given to situations where the concentration dependence of a weight-average property other than molecular weight has been determined experimentally. Such sigmoidal plots can only be analyzed in terms of the theory under discussion when XA and X c can be reliably estimated as limiting values of the plots. It is then possible to utilize in eq 5a the values of KcAn-' given in Table I to calcylate for a range of values of n corresponding values of x/xA for each of the three critical points. These values would replace those tabulated in the last column of Table I, and the analysis could proceed as described when n is unknown. The procedure is valid since the solutions of the equations d2X/dC2= 0 and d3X/dC3= 0 are seen from eq 3b and 3c to be independent of X Aand X c , which means
1176
The Journal of Physical Chemistry, Vol. 85, No. 9, 1981
that the values of KcAn-l, c A / c , and cc/E given in Table I pertain to the three critical points regardless of the weight-average property measured. Discussion A simpler method for determining n and K would appear to be provided by conversion of eq 1 and 2 into the form of eq 6, which should provide a linear plot utilizing log ( C ( X - XA)/(XC - XA)) = log K + n log {C(Xc - X ) / ( X c - X A ) )(6) all experimental (X, e) results in the determination of n from the slope and of K from the zero intercept. The use of eq 6 is, however, limited to situations in which XA and X c may both be determined. Moreover, even for these cases, when n is large, points will be so grouped and subject to error (inherent in the difference terms in eq 6) as to render hazardous the determination of the slope and the extrapolation to find log K. In addition, nonlinearity of the plot may arise from thermodynamic nonideality of the system additional to that assumed by setting yC/yA” equal to unity, or from inadequacy of the n-merization model. In contrast, the method presented here offers the following advantages: (1)When weight-average molecular weights are determined, it is applicable regardless of whether or not X c may be determined. When Xc is available, the method is applicable generally. (2) Estimates of n and K are made from experimental values obtained at relatively low total solute concentration, where the weight fraction of micelle is small. It is in this region that the assumption concerning activity coefficients will likely be met and where aggregates of the n-mer micelles (if they occur) will be present in relatively small amount. (3) The method does not involve any extrapolation step. Moreover, the values of n and K obtained in this way may be used in eq 1and 5a to calculate a predicted curve of X / X A vs. C over the whole experimental range of c for comparison with the experimental curve. In this way, the appropriateness or otherwise of the two-state model on which the present analysis is based may be tested, although, in the high range of E , deviations may arise either because of the inadequacy of the model or because of the manifestation
Nichol and Ogston
of more extensive nonideality effects or both. For more complicated systems involving appreciable concentrations of intermediates4y5or where explicit information on nonideality coefficients is sought,13there is no doubt that a more sophisticated analysis procedure would be required. Such procedures are a~ailable’~-~* particularly in the treatment of molecular-weight averages derived from equilibrium methods to elucidate thermodynamic interaction parameters for associating systems. Nevertheless the weight of e v i d e n ~ e suggests ~-~ that for globular micelle formation in many systems a formal description in terms of the two parameters n and K is entirely reasonable. Such a simplification is of considerable advantage in treating the important problem of elucidating the binding of ampholytes capable of micelle formation to which may themselves be capable of self-interaction. As discussed in ref 10, knowledge of the micellar formation constants, n and K (even with the neglect of second-order effects, such as relatively small concentrations of intermediate micellar species), forms a vital component in the interpretation of such binding results. It is hoped that the present approach for obtaining estimates of these thermodynamic parameters may prove to be a useful addition to existing procedures which, in contrast, frequently rely on the determination of an arbitrarily defined critical micelle concentration.lJg Acknowledgment. One of us (A.G.O.) thanks the Australian National University for a Silver Jubilee Visiting Fellowship of University House and a Visiting Fellowship at the John Curtin School of Medical Research. ~~
(13)C. J. Biaselle and D. B. Millar, Biophys. Chem., 3, 355 (1975). (14)E. T.Adams, Jr., W. E. Ferguson, P. J. Wan, J. L. Sarquis, and B. M. Escott, S e p . Sci., 10, 175 (1975). (15)B. K.Milthorpe, P. D. Jeffrey, and L. W. Nichol, Biophys. Chem. 3, 169 (1975). (16)H.Kim, R.C. Deonier, and J. W. Williams, Chem. Rev., 77,659 (1977). (17)E.T.Adams, Jr., L.-H. Tang, J. L. Sarquis, G. H.Barlow, and W. M. Norman in “Physical Aspects of Protein Interactions”, N. Catsimpoolas, Ed., Elsevier/North-Holland Publishing Co.,New York, 1978,p 1. (18)P. R. Wills, L. W. Nichol, and R. J. Siezen, Biophys. Chem. 11, 71 . - (1980). \----,-
(19)D.Attwood, A. T.Florence, and J. M. N. Gillan, J.Pharm. Sci., 63,988 (1974).
