J . Phys. Chem. 1992, 96, 6442-6449
6442
Electrostatic Properties of Zwitterionlc Micelles Mauricio da Silva Baptista, Iolanda Cuccovia, Hernan Chaimovich, Mario J. Politi,* Departamento de Bioquimica, Instituto de Quimica, Universidade de SBo Paulo, P.O. Box 20780, Siio Paulo, Brazil
and Wayne F. Reed* Physics Department, Tulane University, New Orleans, Louisiana 70118 (Received: December 5, 1991; In Final Form: January 30, 1992)
The electrostatic properties of zwitterionic micelles of 3-(N-hexadecyl-N~-dimethylammonio)propanesulfonate (HPS) have been investigated by light scattering and conductivity,and a simple electrostatic model in the Debye-Hueckel approximation, containing no adjustable parameters, is presented which accounts qualitatively for the main experimental results on second virial coefficientsand conductivity,as well as for fluorescence quenching reported in earlier work. Additional photochemical data concerning reprotonation rates of 8-hydroxy-1,3,6-pyrenetrisulfonatein HPS micelles are also presented and interpreted by the electrostatic model. The model involves the superposition of solutions for concentric, oppositely charged spherical micelles. This leads to a broad maximum in the curve of the absolute value of surface potential and second virial coefficient vs ionic strength. It further implies a significant sequestering of mobile anions in the dipole layer (positive inner surface, negative outer) and low net mobile ion charge densities outside the micelle. These effects are observed experimentally.
Introduction Despite their importance, zwitterionic interfaces have peculiar properties still poorly understood. With aqueous zwitterionic micelles made from monomers such as betaines, lysolecithin (lys-PC), etc., a controversial picture remains. While previous light scattering datal reported a value of zero for the second virial coefficient A2 (CGS units for A2 are cm3.mol/g2 and are used throughout this work) and other data have reported insensitivity of zwitterionic micelle stability to ionic strength? anion enrichment and selectivity (in the order Br- > C1- > Fl-> OH-) were found for 3-(N-hexadecyl-N,N-dimethylammonio)propanesulfonate (HPS) micelles and l ~ s - P c . In ~ these latter cases no effects were observed for different cation species. It was argued in these studies that micellar aggregates having the positive layer closer to the micellar core than the negative one (monomers radially oriented) will present a higher surface positive charge density and might therefore be compared with a nondissociated cationic micelle.38 In order to obtain more details on the electrostatic properties of HPS micelles, we investigated A2 and ion distributions a t the interface by static and dynamic light scattering, ionic conductivities, and laser induced pH jump reprotonation kinetics. These phenomena may be best understood with reference to the distribution of electrostatic potentials and mobile ions around the zwitterionic micelles. A very simple electrostatic model for the zwitterionic micelle, which is equivalent to a double shell concentric spherical capacitor immersed in an electrolyte solution, is presented, and the consequences are investigated with respect to the various experimental results. This simple model is successful in accounting for qualitative and some quantitative aspects of the diverse data. It shows the electrostatic behavior of zwitterionic micelles to be fundamentally different from that of ionic micelles. Simple Electrostatic Model for Zwitterionic Micelles The simplest zwitterionic micelle model is a sphere with an inner positive charge layer of radius RA, and an outer negative charge layer with radius RB = RA + S, where S represents the thickness of the dipole layer (see Figure 2a for a graphical representation). All electrostatic calculations are in the MKSA (meterskilograms.seconds*amperes) system. Three regions are distinguished in the model of Figure 2a: The region outside the micelle, r 1 RE,whose potential and net charge are denoted as &(r) and pl(r),respectively. The second region is in the dipolar layer, RA Ir IRE,and its associated quantities To whom correspondence should be addressed.
are denoted with the subscript 11. The impenetrable micelle core is region 111. The boundary conditions which the potentials must satisfy are
-e1
a91+ €1, a611= -Q ar ar
4?r~B2
at r = RB
(IC)
Here Q = Ze and represents the absolute magnitude of the charge on each dipole surface, where Z is the micelle aggregation number and e the elementary charge. Z is given by M / m , where M is the micelle molecular weight and m is the monomer weight (Z is the same on both surfaces). Here tI and cIIin each region are the permittivity of free space (8.85 X C2/(N.m2)) times the dielectric constant in media I and 11, respectively. We use the bulk water dielectric constant of 78 in both regions, so that q1 = €1. At this point numerical solutions to the Poisson-Boltzmann (P-B) equation could be generated with the boundary conditions la-c. The focus of this work, however, is to obtain simple analytical expressions which allow a wide variety of experimental data to be rationalized. Hence, the DebyeHueckel ( B H ) linearization of the P-B equation is assumed. The use of the D-H approximation must be approached cautiously, however, since potentials on the inner surface reach Q / h t R A 1100 mV at no added salt (the outer surface potential is zero in this case) and remain well above kBTeven as electrolyte is added. Direct application of the D-H approximation to the zwitterionic micelles whose dipole layer is freely permeable to mobile ions hence should lead to overestimates of the potentials within the dipole layer and underestimates of the net charge density, which in turn will lead to underestimates of surface potentials, second virial coefficients, and differential conductivities. The solution for the D-H approximation where the dipole layer is freely permeable to ions and satisfies eqs la-c is presented in the Appendix, and illustrative curves resulting from those potentials are indicated with dotted lines on certain curves in the text. There is no contradiction with the uniqueness of the solution to Poisson's equation here because, even though the solutions to be presented below and those in the Appendix satisfy eqs la-c and the Poisson equation, the values of the potential on the surfaces RA and RB are different in each case. These surface values seem to be in 0 1992 American Chemical Societv
The Journal of Physical Chemistry. Vol. 96, No. 15, 1992 6443
Electrostatic Properties of Zwitterionic Micelles better accord experimentally with the potentials now presented, and thus the D-H solution to the freely permeable problem is placed in the Appendix. More direct experimental determinations of surface potentials would be desirable in further judging the two approaches. To avoid the problems of the D-H approximation within the dipole layer, and the lack of analytical insight from numerical solutions of the P-B equation, a conceptual device, which is entirely valid mathematically and yields good agreement with the data, is now employed. Namely, the potentials within and without the micelle are considered as the superposition of solutions for a separate cationic micelle of radius RA and an anionic micelle of radius RE. The conceptual device is that each isolated micelle is considered to be impermeable to mobile ions. The inner positive surface contributes the potential and charge distribution of a cationic micelle &(r), whose core is impenetrable to mobile ions. In MKSA units (meters.kilograms.seds.amperes) the potential r#&), which decays to zero as r a, is
-
Here K is the Debye-Hueckel shielding parameter, which in water at 25 OC is computed conveniently by
where Cs is the millimolar (mM) concentration of added univalent electrolyte. The outer negative surface acts as a spherical anionic micelle with an impenetrable core at constant potential according to the superposition model explained above. The potential outside the anionic micelle &(r) is given by
The constant potential within the anionic micelle is
= 4m(l
-Q
+ KRB)RB
OIrIRB
Adding the solutions of the two spherical layers to obtain the net zwitterionic micelle potential 4 ( r ) = &(r) + r$a(r)yields
(34 i.e. the presence of the outer negative layer simply adds the constant negative term in brackets to 4(r) within the dipole layer. Outside the micelle
In the impenetrable micelle core, the potential is constant and given by
Equations 3a,b predict that as electrolyte is added ( K increases), the magnitude of the outer surface potential #(RB)of the zwitterionic micelle will start at zero at no added electrolyte ( K = 0), then rise, and pass through a maximum before decaying back toward zero at very high K . (Above Cs = 0 the outer surface potential is always negative.) Equations 3a,b are seen to recover the simple limits for a spherical shell capacitor in the limit of K = 0. Figure 1 shows +(RE)as a function of ionic strength Cs. The parameters used for the figure were m = 391 (HPS monomer), S = 4 A (from the thermodynamic rotameric average discussed below), RB = 27 A from dynamic light scattering, and
::v \:: 0 0 100
1.000
10.000 C,
100.000
1000.000
0
(mM)
Figure 1. Solid lines are the negative value of the zwitterionic micelle surface potential -$(RB) from eqs 3a,b, and the anion concentration factor, CF, from eq 6. The dotted lines are the negative of the surface potential according to s Ala,b and A2a-c and CF using eq A4. Model parameters: R A = 2 3 1 , RB = 21 A, S = 4 A, M = 52 400. These are the same for all figures involving them unless otherwise stated.
M = 52 400 (from static light scattering, presented below). The bulk water dielectric constant of 78 is used throughout. The dotted lines are for +(RE) from eqs A l a and A2a and for CF from eq A4. The net mobile ion charge densities inside and outside the zwitterionic micelle are determined according to p(r) = -tV2qi(r). This yields
pll(r)
-""[
= 4?r r ( l
]
+ KRA)
RA Ir IRB
(4a)
The net mobile ion charge density within the dipole layer (eq 4a), then, is the same as that for the inner positive surface in the absence of the outer negative one, since this latter one contributes only a constant negative potential within the dipole layer; i.e. the ion concentration in the dipole layer can be substantially higher than the bulk Cs and is equal to what would exist for a cationic micelle with the 2 and RA of the inner positive surface. The net mobile ion charge density outside the micelle (eq 4b), however, is significantly lower than for the case of the isolated negative surface, being strongly attenuated by the term due to the inner positive surface. Thus, both the potentials and net mobile ion distributions outside the zwitterionic micelle are much less than in the corresponding ionic micelle case. The net mobile negative charge in the dipole layer is found by integrating eq 4a over the dipole layer volume. This yields
The net concentration factor of anions within the dipole layer, CF, can be defined as the ratio of net negative charge concentration in the dipole layer to the bulk Cs (in mM, with RA and R B in meters). CF =
QD
(4./3)(RB3
- RA~)~?NAC,
(6)
Figure 1 also shows CF according to eqs 5 and 6 as a function of cs. Figure 2a is a schematic of the zwitterionic micelle with the model parameters indicated. It qualitatively indicates that the mobile ion charge density compared to the bulk is slightly higher just outside the micelle and significantly higher within the dipole layer. Figure 2b shows 4(r) for selected Cs. Figure 2b gives the net mobile ion concentration, p(r), for selected Cs.
6444 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
da Silva Baptista et al. cluded volume between spheres of radius Rg, and the second term represents the binary cluster integral for interparticle interactions outside the hardcore excluded volume. rI2represents the spheres' center-to-center separation, and U(r12)is the interaction energy. A priori U(r12)can include attractive terms (e.g. van der Waals interactions) as well as repulsive terms. Since the zwitterionic micelles are thermodynamicallystable, the attractive (negative) contributions to A2 in the second term will be ignored, and U(r12) will be considered to be dominated by the repulsive electrostatic interaction. Approximate expressions for the electrostatic interactions between charged spheres have been developed by Verwey and Overbeek,' and elaborated upon by Hogg et aL8and Stigter and Hd19 (again in MKSA units here). Dorshow et al.IOand Corti and Degiorgio" have employed these expressions, in addition to hydrodynamic interaction terms, in analyzing the concentration dependent diffusion coefficients of ionic micelles, and we follow their precedent here. For KR < 1 Wr12)
=
277fR~4(Rg)~e-~~~~~ +
f
(8)
where f is a complicated function of Rg, rI2,and K and is 51.' f becomes =l for K(rI2- 2RB) >> 1. We have used the tabular representations off from ref 7 for calculations of A*. x is a convenient transformation given by For KRB>> 1
--
0
-50 --
---50
-1007
0
2C
10
30
40
50
60
70
r-100
E0
r (Angstroms)
i
-25004
I
"
20
30
40
50
r (Aigstroms)
Figure 2. (a, top) Schematic of the concentric sphere model for the linear electrostatic zwitterionic micelle model, showing the dimensions R,, RB, and S and giving a qualitative notion of the ion sequestering and greater than bulk concentration in the dipole layer and the lesser effect in the external region. (b, middle) Potential $(r) for selected Cs from eqs 2a-c. Same model parameters as for Figure 1. (c, bottom) Net mobile ion charge density pnc,(r)from eqs 3a,b. Same model parameters as Figure 1.
Test of Electrostatic Potentials via A The light scattering data allow the potentials of the linear model to be tested indirectly via the second virial coefficient A2, which is given by6
I
NA 32:RB3 + 4.1, 2M2
[ l - e ~ p ( - U ( r , ~ ) / k ~ Z " drI2 )]r~~~
(7)
The first term in the brackets corresponds to the hard-core ex-
Asesment of Iimitatim of the Model Concerning the overall conceptual basis of the model, several potentially important effects must be commented on: The appropriateness of a continuum description for the electrostatic potentials in a dipole layer only a few angstroms thick is highly debatable. Such a description ignores not only any possible quantum mechanical effects, but also any local structuring of the water and microscopic geometrical asymmetries. Furthermore, use of a bulk aqueous solution dielectric constant, or indeed any scalar polarizability, is necessarily a coarse oversimplification. Because the counterions are treated as pointlike, the model cannot account for ion specificity, which is observed experimentally4and is probably due to a combination of discrete size, hydration, local solvent anisotropies, and quantum mechanical effects. Furthermore,the concentric sphere model, even in a continuum picture, is obviously a gross oversimplificationof the dynamic nature of the micelles. The micelle is not constrained to be spherical and in fact must undergo rapid fluctuations in shape and mass distribution as monomers enter and leave the micelle, jostle about thermally in the micelle, and undergo conformational changes of their long carbon chains.s Hence, not only do the inner and outer radii RAand RBrepresent thermodynamically averaged quantities, but, probably more importantly, the dipole layer thickness S, represents the thermodynamic average of a quantity subject to large fluctuations. Indeed, in a "snapshot" of the micelle in any moment, it might be hard to recognize anything resembling a spherical shell dipole layer. The only aspect of these various dynamics which can be treated simply in an approximate fashion is the root mean square positive to negative charge distance of the zwitterionic moiety determined by statistical weighting of the different rotameric states of the intervening chain. This will be discussed briefly below. Despite these oversimplifications, the simple electrostatic model makes several experimentally observable qualitative predictions and, besides the dipole layer thickness which can be reasonably approximated from the surfactant's chemical structure, contains no adjustable parameters. Materials and Methods 3-(N-Hexadecyl-N,N-dimethylammonio)propanesulfonate (HPS) was prepared as described in the l i t e r a t ~ r e . ' ~ JIt~ was purified by triple recrystallization from acetone/methanol (90/10
. . - - . . . .-. .. Electrostatic Properties of Lwitterionic Micelles _.
(v/v)) and was shown to be pure by elemental analysis and by IH NMR spectroscopy. Conductivity of 0.1 M HPS in pure water was about 10 $3. Cetyltrimethylammonium bromide (CTAB) was triply recrystallized from acetone/MeOH/H20 (80/ 10/5 (v/v/v)). CTAB concentrations were determined by bromide titration.14 All the inorganic salts (analytical grade or better) were ovendried at least 24 h before use, and their stock solutions were titrated by standard methods. The prototropic probe 8-hydroxy-l,3,6-pyrenetrisulfonate (POH) (trisodium salt from Eastman Kodak) was triply recrystallized from acetone/H20 (90/10 (v/v)) and showed no impurity spots on TLC plates.I5 Deionized water doubly distilled from an all-glass apparatus had conductivity around 1 pS and was used throughout. pH determinations were performed in a glass combined electrode using an InciBras (Brazil) pH meter. Conductivity measurements were made in a Digimed (Brazil) conductimeter, with 1-cm2electrodes in a thermostated cell calibrated with standard KCl solutions.I6 Fluorescence emission spectra were obtained in a Perkin-Elmer LS-5 spectrofluorimeter(fixed 2.5-mm slits for both emission and excitation monochromators). Laser pH jump reprotonation kinetics were obtained in an Applied Photophysics fast kinetics transient photometer operating at around 20 mJ/pulse with a p proximately 15-ns fwhm of the Nd:YAG third harmonic (A = 366 nm) used for producing the transient ApH. A pulsed xenon lamp was used as the monitoring beam at a 90" observation angle. The reprotonation kinetics were monitored following the disappearance of PO- (510 nm) with a Tektronix 2230 100-MHz digitizer. Digitized signals were transferred to an IBM microcomputer and analyzed by first order decay kinetics. Static and dynamic light scattering results were obtained on custom-built equipment, both in SBo Paul0 and New Orleans. The Brazilian system utilized a vertically polarized 10-mW He-Ne laser (Hughes), a Thorn EM1 head-on photomultiplier for detection, and a Brookhaven Instruments BI2030 autocorrelator. The dynamic light scattering system in New Orleans used a variable power vertically polarized argon ion laser (A = 488 nm), had a similar detection and signal processing section, and has been fully described earlier.17 Scattered intensity autocorrelation curves were analyzed under the standard cumulant method within the approximation of Gaussian statistics for the scattered electric field. For rapid and detailed static light scattering measurements in New Orleans, it proved more efficient to use a Wyatt Instruments Dawn-F scattering photometer. This instrument uses a vertically polarized He-Ne laser and has an array of 15 photodiodes from 8 = 26O to 128O for simultaneous scattered intensity measurements. Custom written software was used to control the Wyatt instrument, collect normalization and calibration data, collect scattering data from the micellar samples, and analyze the data in the form of a Zimm plot. The results for molecular weight and A2 were within error bars of those obtained from the argon ion laser based system, so that the two systems could be used interchangeably for static light scattering. All solutions for light scattering measurements were prepared in deionized distilled water having conductivity of less than 1 pS, withdrawn from the still reservoir through a 0.22-pm membrane filter attached to the outlet. Surfactant solutions were filtered through 0.22-rm filters directly into 2.5-cm selected Pyrex cylindrical scattering cells. The standard static light scattering analysis was made according to 1 K ( c - ccmc)/zR = ~ ( +1q2(R2)/3) + 2A2(c - Ccmc) (1 1) where ZR is the absolute Rayleigh scattering ratio, obtained from the relative scattering counts from the absolute Rayleigh ratios at 632 for toluene of 3.96 X at 488 nml* and 1.408 X nm. c represents the concentration of surfactant (g/cm3), and c,,, is the surfactant critical micellar concentration. For HPS c, < M,Ivl2 and for CTAB,c , < M at all Cs.I9 c,
The Journal of Physical Chemistry, VO~. 96, NO.15, 1992 6445 is taken as negligible compared to c in eq 11, since the largest c in the dilutions was always >>c,,. In eq 11, q [=(47rn/A) sin (6/2)] is the scattering vector, and (&2) is the mean square radius of gyration, which, however, was unmeasurably small for the micelles studied. K is an optical constant given by
K=
4?r2n2(dn/dc)' NAX4
Since the Wyatt instrument automatically collects data for 15 angles simultaneously, Zimm plots were constructed according to eq 11, which allows noise and other random errors to be smoothed out by fitting to many points. When measurements were made on the other light scattering systems, scattered intensities were generally taken at 8 = 70", 9O0, and 1lo", and the results were averaged. Interferometric differential refractometers designed and built by the authors in both SHo Paulo and New Orleans were used for determining the refractive index increments, dnldc, of the micellar solutions. The refractometers are absolute systems and require no calibration. Measurements of An vs [NaCl] gave values within 2% of those listed in section VI1 of ref 20. All experiments were performed at 25 "C.
,
ReSultS Light Scattering. The main object of the static light scattering experiments was to measure A2 vs Cs for HPS micelles, and for cationic CTAB micelles for comparison, and to obtain micellar molecular weights for computing 2 in the model. For CTAB dn/dc = 0.1585 over the range of Cs of 5-50 mM NaBr. For HPS dn/dc = 0.1622 from Cs = 0 to about 100 mM. At 500 mM dn/dc = 0.1534, and it is 0.1313 at 2000 mM. The main object of the dynamic light scattering was to obtain estimates for the hydrodynamic radius to use in the electrostatic model. The mutual diffusion coefficient, D, obtained from the autocorrelation decay function is a strongly increasing function of micelle concentration for CTAB, whose slope increases with decreasing Cs. In contrast, D was only a very weak function of [HPS] at any Cs. Much effort has been devoted to understanding the dependence of D on Cs and colloid concentration for neutral and charged ~ y ~ t e m ~ ,and ~ ,it~is-generally ~ ~ , ~ thought ~ , ~ ~that D depends on both Az and hydrodynamic interaction terms. For our purposes we use the extrapolation of D to zero micelle concentration, Do, and simply use this to calculate an effective hydrodynamic radius, RH, by the Stokes-Einstein equation. (RH = kBT/6?rqDo;q = 0.0089, is the viscosity of water at 25 "C.) RH = 27 A for HPS and 25 A for CTAB. The molecular weight M of HPS was found to be fairly constant and around 53 000 f 5% from Cs = 0 to 100 mM, after which it increased to 60 000 f 5% at 500 mM and to 68 000 f 5% at 2000 mM. With a monomer molecular weight of 391, this gives aggregation numbers in the range of 130-170. M for CTABr was found to be around 65000 over the range of Cs = 5-50 mM. Figure 3a shows A2 (in cm3.mol/g2) vs Cs for the two types of micelles. A2 for CTAB shows almost 2 orders of magnitude of variation with Cs, ranging from 7.7 X lob5at Cs = 70 mM to 6.18 X at Cs = 2 mM. For HPS at Cs = 0 the Kc/I vs [HPS] data were quite flat and gave, alternately, slightly positive and slightly negative slopes. The average was around 5 X 10". A2 vs Cs for the HPS micelles rises to a plateau value of about 2.5 X lo4 at Cs = 15 mM, which remains roughly constant until between 100 and 500 mM, after which there is a descent to around 1.6 X 10-4 by 2000 mM. Thus, the variation in A, for HPS, while small but measureable, is drastically less than for the ionic CTAB micelles. Because the osmotic compressibility of charged colloids (polyelectrolytes) is very low at low ionic strength, such low Cs solutions scatter very little light above the solvent background scattering level, and it is usually not possible to gather data appropriate for analysis by eq 11 (the validity of eq 11 breaks down at extremely low Cs whenever long range interparticle correlations and more complicated scattering structure functions
6446 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
da Silva Baptista et al.
1 OE-2
A2
2500
1 OE-3
1
OE-4
l a
1 OEM!
0
100
10
1000
Cs (mM)
0
,..,.....""
100
260
3bO
460
Cs (mM)
A2
Figure 4. Experimental Au vs Cs for different salts in 0.1 M HPS solution: (m) NaBr, ( 0 ) CsCI, (A) NaC1, (A) KCl, (0) NaI, ( 0 ) NaOH. Solid line is the prediction according to eqs 15 and 5 using u, = 75 pS/mM. Dotted line is Au using eq A3 for QD in eq 15.
2.OE-4
1.OE-4
a 11 1
,....."
n
3.OE-4
0.0
......." '
i
i""' 10
lo0
1do0
Cs (mM)
Figure 3. (a, top) Experimental values of A2 for HPS (A)and CTAB ( 0 ) . The solid line is the prediction of A2 (eq 6) for the ionic CTAB micelles using q 7 for U(q2),with R = 25 A, M = 65 500, m = 361, and u = 0.24. (b, bottom) Experimental values of A2(0)on an enhanced scale. Solid line uses q 7 and the potential from (3a) in q 8 for prediction of A2. S = 3 A. The dotted curve uses q 7 with the potential of eq Ala in eq 8, with S = 4 A.
become important). Furthermore, since the free counterions of the ionic micelles contribute to the solution ionic strength over and above the level of any added salt, it is not a simple matter to make isoionic dilutions at low Cs. For CTAB concentrations well above the cmc, for example, the Br- counterions contribute approximately 0.28 mM of ionic strength per mg/mL, where a headgroup ionization factor, a = 0.24,is assumed.4J2 Cs = 2 mM was consequently the lowest ionic strength studied for the case of CTAB. In contrast, HPS micelles scattered strongly at Cs = 0 and in fact showed comparatively little change in scattering intensity from Cs = 0 to 2000 mM. This is an immediate qualitative difference between ionic and zwitterionic micelles and is easily understood by the fact that at, for example, Cs = 0, the ionic micelle should have a very large surface potential, whereas the zwitterionic micelle should have zero outer surface potential (Figure l), and hence a minimum A2 and maximum scattering. The implication of the slight variation of A2 vs Cs is that the intermicellar electrostatic forces are far weaker for zwitterionic than for ionic micelles. In the case of CTAB the strong dependence of A2on Q is clear evidence of the intermicellar electrostatic repulsion, or electrostatic excluded volume. The solid line in Figure 3a shows A2 for CTAB micelles vs Cs according to eq 7,using the intermicellar interaction energy of eq 8, with 4(R) given by
where R is the radius of the ionic micelle. The line shown is the linear fit to theae calculations. The slope is -1.03. The calculations provide a good semiquantitative match to the actual A2 data. The molecular weight is taken as 65 500, and R is taken as 25 A, from the dynamic light scattering value for the quivalent hydrodynamic radius. The hardcore part of A2 in eq 7 is 3.67 X lW5cm3-m01/g2. Figure 3b shows A2 vs CS for HPS zwitterionicmicelles on an enhanced scale. The values of A2 with the current parameters are somewhat larger than the experimentalvalues. The solid line gives the values of A2 calculated by eq 7 with the interaction
potentials of eqs 8 and f from ref 7. The hard-core part of Az in eq 7 is 7.27 X The parameters are the same as in Figure 1 except here S = 3 A instead of 4 A. The dotted line in Figure 3b shows the values of A2 calculated according to eqs 7 and 9 with the potentials from the Appendix, (Ala), (A2a-c). The broad maximum is apparent, but it is much lower than both the data and the predictions of the current model. The simple electrostatic model predicts the qualitative features of the data: Zero micellar surface potential and interaction energies at Cs = 0, leading to a minimum A2 which corresponds to the hard excluded volume of the micelle. Rising surface potential and interaction energies with increasing Cs, leading to a slight increase with a broad maximum for Az, followed by a decline in potential and Az at high Cs. Conductivity. Figure 4 shows Au, the difference between the bulk and micellar solution conductivities, vs Cs for different salts. [HPS] was 39.1 mg/mL (0.1 M) in these curves, which are roughly linear from Cs = 0 to 10 mM. A definite degree of ion specificity is seen, which of course cannot be accounted for by the simple electrostatic model. Since the volume fraction of micelles is quite low at this concentration (- 1%) it is assumed that the micelles themselves do not lower conductivity by presenting obstacles to the small mobile ions. To estimate the conductivity of the micellar solution at a given Cs u,(Cs), it is assumed that anions and cations "sequestered" into the dipole layer are responsible for the majority of the loss of conductivity a&,)
z UOCS - UiC,,,(CS)
(14)
where uo is the specific conductivity of the particular salt used; e.g., for NaCl and CsCl uo = 148 fi/mM and for NaI and NaBr uo = 138 @/mM.l6 C,,(Cs) represents the concentration of sequestered ions and is equal to the total number of ions sequestered in a single micelle layer times the micelle concentration, C, (=[HPS]/M). Since virtually all the ions in the layer are anions, the conductivity of the anion ui is used in the computation. In terms of the model the difference in conductivity between the electrolyte solution with and without micelles Au is given by Au = uiCmQD(Cs)/e
(15)
where Q D is given by eq 5 and C,,, is the millimolar concentration of micelles. The solid line in Figure 4 is the prediction of eq 15 using the same model parameters as in the other figures. ui is taken as 75 p+S/mM. The predicted w e lies above all the data points,except those for NaI, for which the curve matches well. The differences among the ions are presumably due to ionic specificity effects, for which the model takes no account. It is important to point out that the order of decreasing sequestering (I- > Br- > C1- > OH-)follows previous observations of selectivity by other metho d ~ The . ~ dotted curve is from eq A3 for Q D in eq 15. It underestimates the lass of conductivity. Alternatively,use of eq A4
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6447
Electrostatic Properties of Zwitterionic Micelles
- :s
lo
10 8
10.6
1 0
0
50
100
150
200
1
250
Cs (mM)
Figure 5. log &/(MT1 SI)) vs Cs in bulk water with NaCl or NaF (0) and in 2.5 mM HPS in NaCl(0) and in NaF (A). [POH]= 3 X IOd M.
to get CF, whence QD is calculated by CF*CS.VD,where VDis the dipole layer volume in m3 [(4?f/3)(RB3 - RA3)],strongly overestimates ACT(curve not shown). It is noted that the contribution to the conductivity due to the micelles themselves has been neglected. The contribution of a particle to the solution’s conductivity is proportional to the particle’s net charge and inversely proportional to its radius. Over a certain range of Cs the net charge of the zwitterionic micelle divided by its radius might give a conductivity comparable to that of a small mobile ion, but the concentration of micelles for the data in Figure 4 is less than 1 mM; Le., [micelle]
#
C,
(mM)
F i 6. ( 0 )Quenchingdata (l/I, arbitrary units) vs bulk Cs for NBN in HFS micelles,from ref 3g,h. Solid line is average dipole layer concentration, CF-CS (in mM);CF from eq 6.
the factors controlling the recombination event may be quite complicated, if it is assumed that the proton recombining with the PO-arrives from a distance away of very roughly this order, then the proton has spent most of the recombination time in the bulk water phase. As long as there is no substantial electrostatic barrier for cation diffusion where the PO- resides, then the environment over the last few angstroms of the proton's diffusion will not have much effect on the reprotonation rate. The prediction that the potentials outside the zwitterionic micelle are small for all Cs are born out, then, by the experimental observation that the recombination time is not measurably affected by PO- residing in an HPS micelle. As only a qualitative to semiquantitativedescription of a wide variety of experimental data on zwitterionic micelle behavior is sought here, no attempt is made to improve the model at this point. Solutions to the full P-B equation with freely permeable boundary layer conditions might prove fruitful, for example, as well as more accurate expressions for the intermicellar potential energy used to calculate A2. It is also not known at this point if there is any significant attractive portion to A2 which lowers their values at low Cs below those of the predictions. Due to rotational degrees of freedom between the inner positive and outer positive charge constituting the micellar dipole and a decreasing attraction due to shielding in the dipole layer as Cs increases, S would be expected to vary with Cs. Such an average might be approximated by
(S)=
zhiN(hi) exp(-U(h,) / kBT) CJV(hi) ex~(-U(h,)/k~T)
(16)
where h, represents the end-to-end length from the N+ to SO; group, N(hJ is the number density of conformational states with end-to-end length h,, and U(h,) is the interaction energy between an SO; group and the electrostatic potential produced by the inner positive charge layer (i.e. -e4(r), where 4(r) is from eq 2a, that is, without the second constant term due to the outer negative shell; this ignores the interactions between the negative charges of the outer layer). The sum is over all possible end-to-end lengths. We note that (S)rather than ( S2)is calculated, as is normally done for randomly oriented polymers in solution, since the zwitterionic moiety is constrained to start with the N+ moiety at RA and then zig-zag radially outwards. For the four links involved in the R,N+-(CH2),-SO< moiety it is easy to Calculate (S),since there are only four rotamerically generated end-to-end lengths: 2.59, 3.63,4.44, and 5.1 1 A, which, with equally weighted rotational isomeric energies, yield respective fractional occupanices of JV = 2/9, 2/9, 4/9, and (We note that a single zwitterionic moity does not have enough rotational freedom for SO< to 'bend around" and neutralize the N+.) Calculating (S) from eq 17 shows that the dipole layer thickness would start at 4 A at Cs = 0, reach a minimum of around 3.52 A at Cs = 160 mM, and then increase to about 3.9 A at high Cs. Thus, the effect on (S)of Cs due to rotameric freedom is probably negligible. The effect of correlations between neighborhoad headgroups and the dynamic nature of the micelle may have a significant effect on S, however,
but there is no simple way of calculating this on the basis of current experimental parameters. The semiquantitative nature of the agreement between model predictions and data obviates the need of introducing phenomenological parameters in the model. There is no evidence, for example, that there is a measurable excluded volume effect involving hydrated ions and hydrocarbon chains in the dipole layer. The excluded volume might be estimated by considering the mobile ions to have effective hydrated radii of around 1.5 A and the hydrocarbon chains having the same radii and lengths of around 4 A. Then with an aggregation number around 140, the total volume available to hydrated ions in a micelle with RB = 27 A and S = 4 A would be around 31 500-4000 = 21 500 A'. This would allow around 2000 hydrated counterions in the dipole layer, corresponding to a molar concentration on the order of 100 M. Since the CF of eq 6 does not predict concentrations above a few molar even at high Cs (around 4 M), volumetric saturation of the layer by hydrated ions probably is not a limiting or determining factor in the model. The problem of ion selectivity is not addressed by the model. Although direct volumetric saturation of the dipole layer is probably not a problem, the different hydrated sizes of the different ionic species,as well as water structuring and other highly local effects, may determine the selectivity phenomenon. In light of the magnitudes of the potentials within the dipole layer resulting from the D-H linearization, further model studies should include numerical solutions to the full P B equation. More directdeterminations of the surface potentials, e.g., electrophoretic mobility measurements, potential sensitive dyes, or binding of polycations, might indicate if outer surface potentials of eq la are realistic, as the A2 behavior suggests, or if they are lower, as predicted by eqs Ala,b. Studies on zwitterionic surfactants with greater spacing between the positive and negative moieties should show larger effects on Au, surface potentials, values of A2, and ion sequestering. Similar experiments could also be performed on surfactant vesicles with zwitterionic headgroups (e.g., phosphatidylcholine2*)for which the divergences between the models in the text and the Appendix should be even greater. Acknowledgment. We acknowledge support from U.S. NSF Grants DMB-8803760, MCB-9116605, and INT-9101058 and from grants from the Brazilian agencies CNPq, PADCT, FINEP, and FAPESP. W.F.R. and M.J.P. acknowledge travel support from the BID/USP exchange program.
Appendix It is assumed that the dipole layer is freely permeable to mobile anions and cations. The micellar core, r IRA, is assumed impenetrable to all mobile ions. The potentials have the following form in each region: =
aCKr 7
r 1 RB
&& = constant = &(RA)
r IR A
(Alc)
The boundary conditions which determine the constants a, b, and c are eqs la-c from the beginning of the text. Applying these boundary conditions to the potentials of eqs Ala,b yields
Electrostatic Properties of Zwitterionic Micelles The potentials la-c and the same values for the coefficients a, b, and c in eqs A2a-c could equivalently be found by superimposing (i) the solution to #(r) for an impenetrable cationic micelle of RAwith #I(r) of the form (Ala) for r 1 RA and obeying (1b) at r = RA and (ii) the solution of an anionic micelle with radius RB, penetrable to depth RA, with potentials obeying (la) and (IC) as they stand and (lb) with Q = 0. Equations Ala,b, together with the constants a, b, and c from eqs AZa-c, predict the same trends as eqs 2a,b concerning the behavior of outer surface potential vs Cs, but the magnitudes for eqs Ala,b are smaller (dotted line, Figure 1). Equations Ala,b recover the simple limits for a concentric spherical shell capacitor in the limit of K = 0; i.e., a = 0, b c = Q/4m. The net charge in the dipole layer, QD is found by calculating pII(r) = -V2#11(r)and integrating from RA to RB with the result
+
QD
+
+
-4aeK[b(R~ - K-’)eRB b(K-’ - R A ) ~ ~ ~ C(K-’ RA)e“RA- c(RB K-’)CrRB] (A3)
+
+
Calculation of A u using (A3) in eq 15, for example, underestimates the net charge, as seen in the dotted line in Figure 4. Indeed, within the D-H approximation CF cannot be legitimately much larger than unity, since C-(r)/Cs = 1 e411(r)/kT
+
and e+/kBTmust be
