Aggregation of Colloidal Particles Modeled As a Dynamical Process (Mass Action Idnetics/Phospholipid Vesicles/Na' Binding/Membranes/Intermolecular Forces) J

Proc. NatL Acad. Sci. USA Vol. 78, No. 3, pp. 1634-1637, March 1981 Biophysics

Aggregation of colloidal particles modeled as a dynamical process (mass action Idnetics/phospholipid vesicles/Na' binding/membranes/intermolecular forces) J. BENTZ AND S. NIR Department of Experimental Pathology, Roswell Park Memorial Institute, 666 Elm Street, Buffalo, New York 14263 Communicated by David Harker, November 19, 1980 ABSTRACT Aggregation kinetics of sonicated phosphatidyl- the process of Na'-induced aggregation of PS vesicles is re- serine (PtdSer) vesicles in NaCl indicate that the process is fully versible to both NaCl dilution and temperature changes. A dra- reversible and dynamical, involving the rapid formation and dis- matic illustration (8) has been that the dispersal of aggregates persal of aggregates. Accordingly, the general mass action kinetic equations are analyzed with respect to the equilibrium state and upon dilution of NaCl is a fast process, which means that the the formation of higher order aggregates. For a general class of backward rates (D.. in Eq. 1 below) can have a significant effect systems, the values for the mass average aggregate size at equi- on the outcome of an analysis of the kinetics of aggregation. librium are obtained from simple closed-form expressions. It is Hence, a consideration of the equilibrium distribution has to shown that an analysis of the aggregation equilibrium will yield supplement the analysis of aggregation kinetics. We will elab- estimates for the potential energy well that holds the aggregates orate on this point in discussing the question of rates and extent together. A fit to the experimental data for kinetics of Na -induced aggregation of the vesicles has been achieved by employing of aggregation. mass action kinetic equations that include the dissociation reactions. The threshold of NaCl concentration required for aggre- MASS ACTION MODEL gation involves the clear distinction between the rate and extent of aggregation. Consider a system of essentially identical primary particles, denoted X1. The aggregation process is modeled by the general Analysis of the kinetics and equilibrium of particle aggregation mass action reaction is one of the central problems of colloidal science and is a pri- CY. mary source for estimating interaction energies between these Xi + Xj = Xi+j, [1] colloidal particles (1, 2). The most useful approximate solutions Dij to the kinetics of aggregation were obtained by Smoluchowski where Xe is an e-mer aggregate (composed of e primary par- (3, 4), who treated the aggregation as an irreversible diffusional ticles), C.- is the rate constant for the rate at which i-mers and process. Similarly, very useful approximations to the equilib- j-mers form (i + j)-mers, and DU is the rate constant for the re- rium aggregate size distribution were obtained by Mukerjee verse reaction. (5, 6). In both cases, assumptions about the reaction rate con- Let [Xi(t)] denote the concentration of i-mers at time t. Typ- stants were made in order to obtain tractable analytical solutions ically the system is initially monodisperse-i.e., [Xl(O)] = [X,] to the kinetic and equilibrium equations. We have extended this and [X,(O)] = 0 for i - 2; for simplicity, we will assume that this analysis to a more general class of aggregating systems. For is the case. aggregation that follows the mass action model (7), we have found effective algorithms for obtaining both the time devel- Equilibrium distribution classification opment of aggregation and the equilibrium distribution of At equilibrium the principle of detailed balance (7, 9) requires aggregates. for each reaction of Eq. 1 Our interest has been to couple this general theoretical CY = framework to the study of aggregation of sonicated phospha- , [2] tidylserine (PS) vesicles (which have a net negative surface Do mlo.-] charge density under the conditions considered in this work). where [Xi] denotes the equilibrium concentration of i-mers. Eq. In particular, the Na'-induced aggregation of PS vesicles is bf inductive to show that biological interest and is a system well-characterized with re- 2 is used in an argument and + spect to both the magnitude of the negative surface charge Ci_ F(i j) [3] the size of the vesicle. We were led to this work by the obser- F(i)F(j) vation that the rate of increase of light scattered from aggre- Dy gating vesicles (in NaCl) would decrease as the temperature was where raised (8). This implied that the overall rates ofaggregation were likewise decreasing, even though the frequency of collisions t-1 Clk F(e) =- fl F(1) = 1 [4] between the aggregates was increasing. This result could be I Dlk readily explained by assuming that the aggregation is dynami- k cal-i.e., that the dissociation reactions became significant as and the temperature was increased. This gave the impetus to extend the analysis of the kinetic [Xe] = F(C)[Xl]t. [5] equations for aggregation in a manner that would account for the dissociation of aggregates. The results (8) demonstrated that Hence, the concentration of monomers at equilibrium defines the equilibrium concentration of higher order aggregates. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- Abbreviations: PtdSer, phosphatidylserine; DLVO theory, Derja- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. guin-Landau-Verwey-Overbeek theory. 1634 Downloaded by guest on September 24, 2021 Biophysics: Bentz and Nir Proc. NatL Acad. Sci. USA 78 (1981) 1635 At equilibrium all possible aggregate sizes must be consid- Analysis of kinetics ered. A conservation of the total concentration of primary par- These considerations of the equilibrium allow a straightforward ticles, [X<], gives analysis of the rates of aggregation. A useful parameter of this [X0] = analysis is the average aggregate size at time t, (((t)), defined E [Xij [6] equivalently by Eq. 7 with [X,(t)] in the place of [Xi,]. In the early stages of the aggregation of small particles that are Rayleigh The mass average aggregate size at equilibrium may be de- scatterers, the relative increase in turbidity is given by (t(t)) fined by (11, 14). The mass action kinetic equations corresponding to the re- - Zoej,2[XRj/[XO], [7] actions of Eq. 1 may be written which is explained by noting that i[5i]/[X0] is the probability that d[Xe(t)] - - Mef(I + e)Cje[X)][Xe(t)] - De[X(i+e)(t)]} a randomly chosen particle will be found in an aggregate of size dt i. When the system is composed exclusively of j-mers, then + Ef'21{Ci(e.i)[X1(t)][XY(e.i(t)] - Dj(f-jXe(t)]} [X.] = [Xo]/j and (?) = j. Mukerjee (5, 6) considered this problem in its simplest form, [14] which assumes that the equilibrium constants CldDlk are all equal-i. e., where 3,e = 1 when i = eand 0 otherwise, and {(/2} denotes the largest integer s (/2. For the purposes of numerical in- a C111DI, = Cld/Dlk [8] tegration, it is necessary to fix a maximum-allowed aggregate size, denoted M. In practice M may be set sufficiently large Applying Eq. 8 to Eqs. 6 and 7 yields (20-30 in most cases that we treated) such that it does not ar- a[XI] []+ 1 - 9V tificially constrain the evolution of the solutions to the kinetic equations. Eq. 14 may be understood by noting that the first summation accounts for all reactions of the type XY + Xe - X+e and the factor (1 + Si,) accounts for the stoichiometry of the and reaction. The second summation accounts for all reactions of the type X, + X-i - Xe. These are the only two reaction types (e) = (1 + 4a[Xo])'2. [10] which affect the concentration of the species Xe. Hence, the average aggregate size (?) increases linearly with The numerical integrations of Eq. 14 have been performed [X0] when 4a[Xo] << 1, and it increases as [XO]12 when 4a[XO] by using several techniques, all of which gave essentially iden- >> 1. tical results (11). A straightforward Taylor series solution to Eq. In order to classify the behavior of other types of aggregating 14 is sufficient to accurately yield the concentration of each systems, we have chosen an empirical prescription for the species as a function of time from the initial condition t = 0 to higher order equilibrium constants. Suppose now that the ag- the final equilibrium state; the values obtained as t -- 00 are in gregation system may be represented by complete agreement with those calculated directly from the equilibrium distribution. These solutions also account for a sit- uation in which the degree of aggregation is reduced from its CldDlk= a( + k) [11] initial value by electrolyte.dilution or temperature changes. for some value of T, where a C111/D1. Setting 7 = 0 gives back the case of equal equilibrium constants. The format of Eq. THEORY OF DIMERIZATION AND THE 11 arose from Fuoss' (10) equilibrium analysis of cation-anion POTENTIAL ENERGY OF INTERACTION pairs; however, this extension to the argument is not rigorous, For two spherical vesicles of radius R (cm) which are held to- and we prefer to treat Eq. 11 as empirical. For more details on gether by a potential energy well of depth u (where u must be the results presented below, see refs. 11 and 12. Use of Eq. 11 negative if measurable dimerization is to occur), the equilib- in either Eqs. 6 or 7 requires that Ti . 0 for the series to con- rium dimerization constant C11/D11 is related to u (10, 15) by verge. For the case of T = 1, we obtain C11 a = NA x 10-3 [15] (?) = 1 + 2a[X1] [12] D11 -4(2R)33 exp{-u/kT} M'1, where a[FL] is given by solving where NA is Avogadro's number, kT is Boltzmann's constant a[X0] = a[XI] exp {2a[Xl]}. [13] times the absolute temperature and M-1 denotes the units of liters/mole. In principle, the potential energy of interaction When a[X0] is small, then this case becomes identical with Ti between two vesicles can be calculated as the sum of the Van = 0 because there would be essentially only monomers and der Waals attraction and the electrostatic repulsion (due to the dimers in the system. When a[X0] is large (greater than 5) then negative surface charges) at the equilibrium separation accord- (?) is quite accurately approximated by 1 + ln {a[XJ}; hence, ing to the Derjaguin-Landau-Verwey-Overbeek (DLVO) the- the mass average aggregate size will grow logarithmically with ory (1). The equations necessary to calculate the potential en- a[X)]. This functional dependence for large aXO is clearly dis- ergy of interaction between two PS vesicles as a function of tinct from the case of Ti = 0, where (Z) grows with the square separation have been presented elsewhere (16, 17). However, root of a[X0]. For the case of n = 2, the value of (Z) grows as in ref. 16 it was shown that the equilibrium separation between ln {(a[XO])'/2} when a[X0] is large. For instance, when a[X0] the sonicated vesicle surfaces must be less than 10-20 A, be- = 100, the value of (7) is close to 20 for nT = 0 and about 3 for cause there is no potential energy well occurring at larger sep- Ti = 2. Thus, the dependence of (Z) on changes in concentration arations that would support measurable aggregation. Unfortu- can yield an approximate value for Ti. A similar equilibrium anal- nately, it is not yet possible to extend the equations used to ysis for micelle formation has been presented (13). compute Van der Waals and electrostatic interactions to shorter Downloaded by guest on September 24, 2021 1636 Biophysics: Bentz and Nir Proc. Nad Acad. Sci. USA 78 (1981) separations, because both theories are macroscopic (17). On the ratio CII/D1, which equals a in Eq. 8. It has to be emphasized other hand, it is possible to obtain an estimate of the magnitude that current theories are not yet capable of providing estimates of u based upon an analysis of the kinetics of aggregation. for short-range interactions required in calculating D11. From the DLVO theory, we may obtain an estimate for the To provide a more rigorous test of the model and to obtain value of C11, the dimerization rate constant, which agrees quite a better estimate for the value of D11, we have chosen to sim- well with its experimentally measured value. On the basis of ulate the graph of the relative turbidity versus time. In calcu- predicting the concentrations of Ca2+, Mg2+, and Na+ needed lating the turbidity expected from the underlyingaggregate size to initiate fast aggregation of PS vesicles (11, 16, 18), we have distribution, we took explicit account of interference effects found that the value of C11 can be sufficiently well estimated (26). Ignoring the interference effects in calculating the inten- by the equation sity of scattered light would yield.an overestimate of the expected turbidity. The morphology of the aggregates was as- Cii = E4kTNA (2d*)] exp{-VT*/kT} sumed to be a linear bead structure. This geometry is in accord with the electron micrographs presented in ref. 8 for vesicles. x 10-' M-' sec-', [16] Due to the fact that the sonicated PS vesicle radius (R = 150 where iq is the bulk viscosity of the medium, VT* is the potential A) is much less than the wavelength of the incident light (A energy of interaction between the vesicles when separated by = 4754 A in the medium), the interference effects are not sig- a distance d*, and d* = 2/K, in which K is the standard De- nificant during the initial stage of aggregation. Our calculations bye-Huckel reciprocal length. VT* > 0 for the systems treated (12) indicate that interference effects will reduce the turbidity here. (See refs. 1, 11, 16, and 17 for more details.) by about 5%, 10%, and 20% after 1, 3, and 6 sec of aggregation, For PS vesicles, a crucial ingredient in the estimation of. C11 respectively, in 650 mM NaCl. According to the analysis in refs. and D11 is accounting for the neutralization of surface charges 27 and 28, our calculated turbidities would not change dra- on the vesicles by cation binding-in this case, the neutrali- matically with other geometrical arrangements of these vesicles. zation arising from the binding of Na'. In the absence of sig- The curves in Fig. 1 give a comparison between experimental nificant binding, the value of VT* would be quite large because (8) and calculated changes in turbidity at a temperature of 200C, of the electrostatic repulsion (16, 18). Indeed, the tetramethyl- at which most of the binding studies (21, 24) were performed. and tetraethylammonium cations do not significantly bind to PS A more extensive comparison, at several other temperatures is vesicles (19); there is no observed aggregation with these cat- shown in ref. 12; however, the possible variation in Na' binding ions, even in cation concentrations exceeding 1 M (18, 20), to the PS as a function of temperature was not treated because which implies that u > -2kT. Na', on the other hand, has been of the lack of experimental data. From inspection of Fig. 1 at shown to bind to PS to a degree wherein more than two-thirds times below 1 sec, where the calculated turbidity values depend of the surface charges are neutralized when the concentration mostly on C11, it appears that our calculations slightly overes- of Na' exceeds 300 mM (21-23). In particular, the binding con- timate C1I. The value calculated by us (Eq. 16) is 8.7 X 107 M'1 stant of sodium to PS vesicles (0.8 ± 0.2 M-l) has been deter- sec', whereas a value of CII = 6.8 x 10 M-1 sec1 can fit the mined from binding competition experiments with Ca2" and experimental data (8) very closely. The difference in these C11 Mg2e (21, 24) and from 2Na NMR relaxation (25). A value of values amounts to an underestimate of VT* by -0.25 kT, which 0.6 M-1 was used to explain the electrophoresis of large mul- is a remarkably small difference. Fig. 1 indicates that the value tilamellar PS vesicles (19). Hence, it is the surface charge neu- of the potential depth, u in Eq. 15, is between -11. 1 kT and tralization due to the binding of Na' that accounts for its ability -8.8 kT. Thus, the experimental data provide an estimate for to induce sonicated PS vesicles to aggregate when the Na' con- the potential well holding two PS vesicles together, say 10 kT, centration exceeds 500 mM (8). Comparison of Eqs. 15 and 16 shows that D11 can be directly . u = -11.1 kT related to u and VT*. DI1 is reduced from the value it would take in the absence of any interaction energy between the vesicles 5 -.- by the factor exp{-(VT* - u)/kT}. The value of (VT* - u) is simply the height of the potential barrier-that must be overcome 4)> 4...-,. u= -8.8 kT if the two particles are to separate. Given an estimate of DI1, we may directly estimate the value of u. ANALYSIS OF AGGREGATION OF PtdSer VESICLES A3 This section illustrates how the mass action model can simulate the experimental results (8) of the kinetics of Na'-induced aggregation~of PS vesicles. The first qualitative test for the model of reversible aggregation was the demonstration that aggregates can be rapidly dispersed by either diluting the Na' concentration or raising the temperature (8). The amount of available data 0 1 2 3 4 5 (8) dictated the employment of the simplest analysis and, hence, Time, sec the analysis was limited to the case of equal rates CY = C11 and FIG. 1. Relative turbidity over time for the aggregation of soni- Di, = DI1, thus employing only two parameters. Our procedure cated PS vesicles in 650 mM NaCl at 20TC. A, Experimental values is to fit the experimental data by values of C11 and DI,, hence (8); ---, fit obtained by arbitrarily setting C11 = 6.8 x 107 M-1 sec-' obtaining values for the height of the potential barrier and the and D11 = 0.015 sec', which corresponds to u = -11.1 kT. All of the depth of the potential well. We also undertook to test the power calculated curves set CQ = Cl-l andD# = D11 for the integration of Eq. 14. The other two curves were obtained using the value of C11 = 8.7 of the DLVO theory (1) in providing estimates for the forward x 1o7 M1 secl, which was calculated from the DLVOtheory. -,Dl rate, C11. This test is feasible because the outcome of the ear- = 0.19 sec-' (u = -8.8 kT); - - -, D11 = 0.019 sec.1 (u = -11.1 kT). liest stages in the aggregation process is primarily determined The turbidity curves were calculated assuming a linear morphological by C11. The fit provides the backward rate DI, and, hence, the relationship (26, 28).[X0] = 16 nM. Downloaded by guest on September 24, 2021 Biophysics: .Bentz and Nir Proc. Natl. Acad. Sci. USA 78 (1981) 1637

Table 1. Comparison of the rates and extents of aggregation* presence of 400 mM Na, the rate is not slow (if the aggregation A B were strictly irreversible, it would require just over two min C11 = 2.5 x 107 M'1-sec-1 C11 = 2.5 x 106 MW1-seC- for the relative turbidity to double), but the equilibrium dis- D11 = 0.1 sec1 DI, = 0.001 sec' tribution corresponds to little aggregation. With 500 mM Na, the threshold for both the rate and the extent of aggregation is t, i[XiMIA]/Xo] i[XIMt)]AX0] reached. We note that this case would yield a time required for sec i = 1 10 (e(t))t i = 1 10 (e(t))t a doubling in scattered light of approximately 25 sec, whereas 1.0 0.552 -t 1.61 0.917 -t 1.08 the case of 650 mM Na' yields a doubling time of =-0.5 sec. 10.0 0.169 0.0129 3.76 0.529 -t 1.65 In summary, this work is focused upon an analysis of aggre- 100.0 0.152 0.0177 4.12 0.074 0.0359 5.48 gation that simultaneously accounts for the kinetics and the OO§ 0.152 0.0177 4.12 0.0213 0.0515 12.69 equilibrium state. Useful estimates for the forward rate constants of may be obtained from DLVO * = aggregation the theory For both case A and case B, [X0] 16 nM. All forward rate constants (1, 2, 16) when it is supplemented to account for specific cation are set equal to C11, and the reverse constants are set equal to D,1. For each time, the maximal aggregate size M was set sufficiently binding (16, 18, 23). There is still significant work needed be- large, so that effectively M = cc. fore good theoretical estimates for the short-range potential t The average aggregate size at time t; see Eq. 7. energies between vesicles (or other macroscopic particles) may t In these cases, i[Xj(t)]/[XoI < 10-6. be made-i.e., the value of u in Eq. 15. The expressions pre- § This refers to the equilibrium values computed from Eqs. 9 and 10. sented here can provide much of the needed information when as an average between the two extremes. The percent uncer- they are applied to the analysis of experiments of aggregation tainty in u is about the same as the uncertainty in the binding kinetics and equilibria. constant of Na' to PS (19, 25). The expert typing of Ms. D. Ross is acknowledged. Grants GM 23850 Whereas the potential barrier for close approach of particles (National Institutes of Health) and CA 17609 (National Cancer Institute) may be small (implying a fast forward rate of aggregation), it can supported this work. happen that the extent of aggregation need not be significant. Table 1 illustrates this effect for two (hypothetical) cases of rate 1. Verwey, E. J. & Overbeek, J. Th. G. (1948) Theory ofthe Stability constants. In case A we observe the effect of high cation con- of Lyophobic Colloids (Elsevier, Amsterdam), pp. 164-185. centrations on negatively charged particles-i.e., a low poten- 2. Overbeek, J. Th. G. (1977)J. Colloid Interface Sci. 58, 408-422. tial barrier and rapid progress to the equilibrium state. How- 3. Smoluchowski, M. (1917) Z. Physik Chem. 92, 129-168. ever, because the depth of the potential well also depends upon 4. Chandrasekhar, S. (1943) Rev. Mod. Phys. 15, 1-89. short-range molecular interactions, it may happen that u re- 5. Mukerjee, P. (1972)J Phys. Chem. 76, 565-570. 6. Mukerjee, P. (1974)J. Pharm. Sci. 63, 972-981. mains shallow and the equilibrium provides little aggregation. 7. Boyd, R. K. (1977) Chem. Rev. 77, 91-119. Quite roughly, this case corresponds to -=600 mM Na' at the 8. Day, E. P., Kwok, A. Y. W, Hark, S. K., Ho, J. T., Vail, W. J., higher temperatures, "=25-300C (8, 12). Case B shows the ef- Bentz, J. & Nir, S. (1980) Proc. Natl Acad. Sci. USA 77, fect of a factor-of-ten-smaller forward rate constant and a factor- 4026-4029. of-ten-larger value of a (i.e., much slower progress to an equi- 9. Bak, T. (1963) Contributions to the Theory of Chemical Kinetics, librium state that shows substantially greater aggregation). (Benjamin, Reading, MA), pp. 31-47. 10. Fuoss, R. M. (1958)J. Am. Chem. Soc. 80, 5059-5061. Table 2 shows the values of C11 and u that were obtained for 11. Bentz, J. & Nir, S. (1981)J. Chem. Soc. Faraday Trans. 1, in press. the sodium-induced aggregation of sonicated PS vesicles. In the 12. Bentz, J. (1979) Dissertation (State University of New York at Buffalo, Buffalo, NY). Table 2. Kinetic and equilibrium parameters for sodium-induced 13. Missel, P. J., Mazer, N. A., Benedek, G. B., Young, C. Y. & aggregation of PS vesicles Carey, M. C. (1980)J. Phys. Chem. 84, 1044-1057. Na, C11,* _u't 14. Kerker, M. (1969) The Scattering of Light and Other Electro- M-1-sec' magnetic Radiation (Academic, New York), pp. 36-39. mM (kT) EX/[Xo]t (?)t 15. Petrucci, S. (1971) Ionic Interactions, ed. Petrucci, S. (Academic, 650 8.7 x 107 10 0.033 9.8 New York), Vol. 2, pp. 101-106. 500 1.7 x 106 8.2-6.4 0.29 1.9-4.1 16. Nir, S. & Bentz, J. (1978)J. Colloid Interface Sci. 65, 399-414. 400 3.0 x 105 6.4-3.6 0.83 1.0-1.9 17. Nir, S. (1977) Prog. Surface Sci. 8, 1-58. 18. Nir, S., Bentz, J. & Portis, A. (1980) Adv. Chem. 188, 75-106. * C11 is obtained from methods (16-18) that assumed the~binding con- 19. Eisenberg, M., Gresalfi, T., Riccio, T. & McLaughlin, S. (1979) stant for sodium-PS binding to be 0.8 M-1. The temperature is 200C, Biochemistry 18, 5213-5223. kT = 4.05 x 10-14 erg, and [Xo1 = 16 nM. 20. Hauser,-H. ,-Phillips, M. C. & Marchbanks, R. M. (1970) Biochem. t The estimated depth of the potential well which' holds the dimer; see J. 120, 329-335. Eq. 15. The value of u = - 10 kT for 650 mM Na' is an average be- 21. Nir, S., Newton, C. & Papahadjopoulos, D. (1978) Bioelectro- tween the values shown in Fig. 1. For the other two Na' concentra- chem. Bioenerget. 5, 116-133. tions, the lower boundary on the range for u is a plausible overesti- 22. Bentz, J. & Nir, S. (1980) BulL Math. Biol 42, 191-220. mate of the well depth and is obtained by adding AVT* e VT*(Na+) 23. Bentz, J. (1981)J. Colloid Interface Sci., in press. - VT*(650) to - 10 kT, where VT*(Na+) is the DLVO-calculated po- 24. Newton, C., Pangborn, W., Nir, S. & Papahadjopoulos, D. (1978) tential for the vesicles in the chosen Na+ concentration. The more Biochim. Biophys. Acta 506, 281-287. shallow well depth is estimated by adding 2AVT* to - 10 kT. We have 25. Kurland, R. J., Newton, C., Nir, S. & Papahadjopoulos, D. (1979) used this procedure only to indicate theprobable effectwhich the Na+ Biochim. Biophys. Acta 551, 116-133. concentration would have on the depth of the potential well. 26. Benoit, H., Ullman, R., DeVries, A. &Wippler, C. (1962)J Chim. t See Eqs. 9 and 10. For 0.5 M Na', [Xl]/[Xo] was calculated by using Phys. 59, 889-895. the average u = -7.3 kT = -(8.2 + 6.4)/2 kT, whereas the range 27. Lips, A. & Levine, S. (1970)J Colloid Interface Sci. 33, 455-464. of (?) represents the range of u-= -8.2 kT to u = -6.4 kT. The same 28. Lips, A. & Willis, F. (1973)J. Chem. Soc. Faraday Trans. 1 69, procedure was used for 04 M Na+. 1226-1236. Downloaded by guest on September 24, 2021