3492
Langmuir 1999, 15, 3492-3498
Microscopic Parameters of Sodium Dodecyl Sulfate Micelles from Optical Probe Studies and Quasi-Elastic Light Scattering Nicholas V. Sushkin and Delphine Clomenil Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609
John Ren Department of Physics, University of South Dakota, Vermillion, South Dakota 57069
George D. J. Phillies* Department of Physics and Associated Biochemistry Faculty, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 Received January 14, 1999. In Final Form: February 16, 1999 Quasi-elastic light-scattering measurements of the diffusion of sodium dodecyl sulfate (SDS) micelles in aqueous solutions, and the diffusion of mesoscopic optical probes through the same solutions, were carried out at 35 °C and multiple solvent ionic strengths. From nonlinear least-squares fits to both probe and mutual diffusion data, assuming a spherical micelle, we deduced the micelle radius, aggregation number, charge, and hydration. For SDS micelles the charge that we find is lower than that in the literature (Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983, 261, 1002. Triolo, R.; Caponetti, E.; Graziano, V. J. Phys. Chem. 1985, 89, 5743); the difference arises because we use an improved form for the intermicellar electrostatic potential. As further results of using the same potential, we find a smaller aggregation number and a larger micellar hydration than those reported in the literature.
I. Introduction In this paper we report a quasi-elastic light-scattering spectroscopic (QELSS) study of aqueous solutions of sodium dodecyl sulfate (SDS). We measured the mutual diffusion coefficient Dm of micelles and the diffusion coefficient Dp of optical probes diffusing through the surfactant solutions as a function of the surfactant concentration and ionic strength of the solution. We infer micellar microscopic parameterssaggregation numbers, degrees of hydration, and chargessfrom this experimental data and hydrodynamic theory.1-3 SDS has been subjected to extensive physical studies.4-17 Most studies report micellar molecular weights, aggrega* To whom correspondence should be addressed. E-mail:
[email protected]. (1) Batchelor, J. K. J. Fluid Mech. 1976, 1. (2) Felderhof, B. U. Physica A 1977, 89, 373. (3) Felderhof, B. U. J. Phys. A 1978, 929. (4) Cutler, S. G.; Meares, P.; Hall, D. G. J. Chem. Soc., Faraday Trans. 1 1978, 74, 1758. (5) Hayachi, S.; Ikeda, S. J. Phys. Chem. 1980, 84, 744. (6) Kale, K. M.; Cussler, E. L.; Evans, D. F. J. Phys. Chem. 1980, 84, 593. (7) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (8) Batchelor, G. K. J. Chem. Phys. 1984, 80, 6234. (9) Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983, 261, 1022. (10) Imae, T.; Kamiya, R.; Ikeda, S. J. Colloid Interface Sci. 1985, 108, 215. (11) Triolo, R.; Caponetti, E.; Graziano, V. J. Phys. Chem. 1985, 89, 5743. (12) Kumar, S.; Aswal, V. K.; Singh, M. N.; Goyal, P. S.; Kabir-udDin Langmuir 1994, 10, 4069. (13) Bezzobotnov, V. Y.; Borbely, S.; Cser, L.; Farago, B.; Gladkih, I. A.; Ostanevich, Y. M. J. Phys. Chem. 1988, 92, 5738. (14) Hayter, J. B. Langmuir 1992, 8, 2873 (15) Bales, B. L.; Almgren, M. J. Phys. Chem. 1995, 99, 15153. (16) Quina, F. H.; Nassar, P. M.; Bonilha, J. B. S.; Bales, B. L. J. Phys. Chem. 1995, 99, 17028. (17) Phillies, G. D. J.; Hunt, R. H.; Strang, K.; Sushkin, N. Langmuir 1995, 11, 3408.
tion numbers, and/or critical micellar concentrations. The most common microscopic model of a spherical ionic micelle is a charged hard sphere whose interior is composed solely of surfactant molecules; i.e., the interior is dry. These charged hard spheres interact via a screened Coulomb (Debye-Huckel) potential. Some models of micelles incorporate water molecules into the micelle interior. The level of micelle hydration can be deduced from measurements of the micelle partial volume. Small-angle neutron scattering studies report hydration levels and micelle charges deduced from the volumes of entrapped ions.9,11 Electrostatic and hydrodynamic interactions of micelles affect the micellar mutual diffusion coefficient. Quasielastic (dynamic) light-scattering experiments that measure mutual diffusion can therefore be used to infer a micellar charge. However, light-scattering spectra usually determine the micelle hydrodynamic radius, not the micelle aggregation number.7 In contrast to literature studies based solely on Dm, we combined quasi-elastic lightscattering experiments and optical probe studies to determine independently the micelle aggregation number N, the micelle hydration δ, and the micelle charge qm. A simultaneous analysis of Dm and Dp to determine microscopic micelle parameters has previously been qualitatively described18 and applied to solutions of nonionic surfactants.17 To use our approach with ionic surfactants, we had to derive a correct form for the electrostatic potential between charged dielectric spheres of unequal, nonnegligible radii in electrolyte solutions,19 including often neglected effects due to counterion cloud exclusion (18) Phillies, G. D. J.; Stott, J.; Streletzky, K.; Ren, S. Z.; Sushkin, N.; Richardson, C. Optical Probe Studies of Surfactant Solutions. In Structure and Flow in Surfactant Solutions; Herb, C. A., Prud’homme, R. K., Eds.; American Chemical Society: Washington, DC, 1994. (19) Sushkin, N. V.; Phillies, G. D. J. J. Chem. Phys. 1995, 103, 4600.
10.1021/la990035g CCC: $18.00 © 1999 American Chemical Society Published on Web 04/08/1999
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and due to the induced surface charge on each sphere. We treated SDS micelles as spheres; electrostatic as well as hydrodynamic interactions of ellipsoidal and rod-shaped micelles are hard to handle analytically. Dp and Dm depend on surfactant mass concentration c as
Dm ) Dm0(1 + kmc)
(1)
Dp ) Dp0(1 + kpc)
(2)
and
where Dm0 and Dp0 are the zero-concentration limits of the mutual and probe diffusion coefficients, respectively, and km and kp are the leading linear slopes. Theory gives expansions in powers of the micelle volume fraction φ
Dm ) Dm0(1 + Rmφ)
(3)
Dp ) Dp0(1 + Rpφ)
(4)
and
where, for spherical micelles having radius am and number concentration n,
4πam3n φ) 3
(5)
If the micelles and probes are adequately approximated as hard spheres, then Dm0, Dp0, and the Stokes-Einstein equation
D)
kBT 6πηa
(6)
can be used to compute the micelle and probe radii. Here kB is Boltzmann’s constant, T is the absolute temperature, η is the solvent viscosity, and D and a are the sphere diffusion coefficient and radius. When standard theoretical treatments of light-scattering spectroscopy20 are combined with expressions for hydrodynamic21 and electrostatic19 interactions, km and kp in eqs 1 and 2 may be interpreted in terms of the micelle and probe hydrodynamic radii am and ap, charges qm and qp, and the partial volume vj of micelles in solution. Depending on the strength of the hydrodynamic and electrostatic interactions, km can be negative or positive; kp is always negative. We present results of applying the above approach to SDS micelles. Section II describes procedures and materials used in the experiments. Section III summarizes experimental results. In section IV we derive the dependence of Dm and Dp on c, using the previous work of Carter and Phillies.20 We calculate the linear slopes km and kp, using the results of Mazur and van Saarloos21 for the translational hydrodynamic interaction tensor of spheres of unequal radii in a Newtonian solution and using our recent calculation of micellar electrostatic interactions.19 Section V discusses how the experimental data were analyzed to obtain microscopic parameters of micelles. (20) Carter, J. M.; Phillies, G. D. J. J. Phys. Chem. 1985, 89, 5118. (21) Mazur, P.; van Saarloos, W. Physica A 1982, 91, 290.
Closing sections compare our findings with the literature and give conclusions. II. Experiment For light-scattering studies, surfactant solutions were passed through 0.22 µm cellulosic filters (Micron Separations Inc.) into precleaned sample cells, which were then sealed to prevent evaporation and dust contamination. In optical probe measurements, 5 µL of probe solution was added after filtration of the surfactant to 2 mL of surfactant solution, giving a final probe volume fraction e10-4 of polystyrene latex spheres. SDS (Aldrich Chemical Catalog 86201-0) was used as received. SDS was guaranteed 98% pure and was 99.1% pure according to the manufacturer’s gas-liquid chromatography analysis of a similar batch. Optical probe particles for experiments in SDS solutions were 67 nm nominal diameter carboxylate-modified polystyrene latex particles (Seradyn, Lot A1-005/497, Pkg Lot 1R24). NaCl (Aldrich) was used without further purification. Water was purified using a Millipore Milli-Q Water System; the average resistance of the water was 15 MΩ cm. The mutual diffusion coefficient was measured for SDS concentrations of 4, 10, 20, 30, 40, and 50 g/L and NaCl concentrations of 0.1, 0.2, 0.3, 0.4, and 0.5 M. Probe diffusion was measured for SDS concentrations in the range 0-50 g/L and NaCl concentrations of 0.1, 0.2, 0.3, 0.4, and 0.5 M. Carboxylate-modified spheres were used to avoid charge-charge attraction of probes by negatively charged SDS micelles. All measurements were performed at 35 °C. Two different laser-correlator systems were used for QELSS measurements. To measure Dm of the micelles, a Spectra-Physics 2030 Ar+ laser (1.4 W maximum output at 514.5 nm) was used to illuminate sphere-free SDS/NaCl/water samples mounted in a Brookhaven Instruments photometer-goniometer. The intensity-intensity correlation function was obtained with a 268 channel Brookhaven Instruments 2030AT digital correlator, with spectra being saved for later analysis. Sample temperatures were maintained to within (0.1 °C using a Neslab RTE-110 circulating bath. To measure Dp, light from a Spectra-Physics model 124B HeNe laser (30 mW at 632.8 nm) was passed through cells containing polystyrene latex spheres/SDS/NaCl/water. Light scattered from the equilibrium samples was detected with an RCA model 7265 photomultiplier tube. The resulting signal was analyzed using a Langley-Ford 144-channel digital correlator under remote computer control. Sample temperatures were maintained at 35 ( 0.1 °C by mounting the scattering cells in a massive thermoinsulated copper block through which a Neslab RTE-100 circulating bath pumped water. A digital correlator measures the intensity-intensity correlation function of the scattered light
S(q,t) ) 〈I(q,t) I(q,t+t)〉
(7)
where q is the scattering vector. S(q,t) is composed of a baseline value B (which is the t f ∞ limit of S(q,t)) and a time-dependent part. B may be determined either by direct measurement of S(q,t) for large values of t or by a calculation based on the average scattering intensity. From S(q,t), one may determine the field correlation function g1(q,t):
g1(q,t) ) xS(q,t) - B
(8)
To analyze all spectra, we applied Koppel’s method of cumulants,22 in which g1(q,t) is expressed as the Nth-order cumulant expansion 1
g (q,t) ) exp
(∑ ) N
Kn(-t)n
n)0
n!
(9)
Here N is the order of the fit, while Kn values are the cumulants. With an appropriate statistical weight for each data point, Kn may be obtained by a linear least-squares fit to ln(g1(q,t)). The cumulant representation is convergent in the limit N f ∞. In our (22) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814.
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Figure 1. Mutual diffusion coefficient Dm of SDS micelles at temperature 35 °C against surfactant concentration c at NaCl concentrations 0.1, 0.2, 0.3, 0.4, and 0.5 M. Lines represent linear least-squares fits to Dm and Dp to obtain microscopic properties of micelles (see text). probe diffusion measurements, the scattering from micelles was negligible, so that Koppel’s method was used without modifications. In a real system experimental noise limits the number of cumulants that can actually be determined. For our systems an optimal fit required N ) 2 or 3. K1 gives the diffusion coefficient (Dm or Dp, depending on the system) via
D ) K1/q2
(10)
For incident light of wavelength λ in vacuo
q)
4πn θ sin λ 2
()
(11)
where n is the sample’s index of refraction and θ is the scattering angle. The scattering angle was 90° in all experiments.
III. Experimental Results This section presents measurements on mutual and probe diffusion of SDS micelles. We first show the results obtained from mutual diffusion in surfactant/salt/water solutions and then turn to information obtained from probe diffusion. Figure 1 shows Dm of SDS micelles in water solution of NaCl at 35 °C and NaCl concentrations of 0.1, 0.2, 0.3, 0.4, and 0.5 M. Dm of the micelles increases with increasing SDS concentration for [NaCl] e 0.4 M; Dm decreases with increasing c in 0.5 M NaCl. The zero-concentration limit of Dm does not change by more than 20% of its average value over 0 e I e 0.5 M. Figure 2 shows the probe diffusion coefficient Dp of 67 nm diameter latex spheres in SDS/ NaCl/water solution at 35 °C, for NaCl concentrations of 0.1, 0.2, 0.3, 0.4, and 0.5 M. Dp decreases linearly both with increasing surfactant concentration and with increasing salt concentration. The slope varies little with salt concentration. Lines in Figures 1 and 2 are formal fits to eqs 1 and 2 and the theory in the following section. The free parameters in each fit are the micelle charge, the micelle and probe hydrodynamic radii, the micelle contact radius, and the micellar hydration. Fits in Figure 2 do not include points with CSDS ) 0. For NaCl concentrations of 0.1, 0.3, and 0.5 M, Dp at 0 g/L SDS and some salt concentrations is about 10% larger than values suggested
Figure 2. Probe diffusion coefficient Dp of 67 nm polystyrene spheres in solutions of SDS/NaCl/H2O micelles at temperature 35 °C against SDS concentration c at NaCl concentrations 0.1, 0.2, 0.3, 0.4, and 0.5 M. Lines represent linear least-squares fits to Dm and Dp to obtain microscopic properties of micelles (see text).
by fits; similarly, the measured Dp at 0.5 M NaCl and no surfactant is 25% smaller than the value suggested by a fit. IV. Theory In this section we review the theory needed to analyze the above experimental data. We calculate the dependence of Dp and Dm on the volume fraction φ of the micelles. The theory is based on the previous work of Carter and Phillies,20 leading to eqs 3 and 4. Carter and Phillies provided a recipe for calculating the linear slopes km and kp in eqs 3 and 4 for spherical micelles, based on micellemicelle and micelle-probe hydrodynamic interactions and the corresponding two-particle distribution functions g(2)(r). In our model, each micelle has a spherical hydrated core of radius am and a dilute spherical corona of radius ac rising above the core. In terms of other models of micelles, the radius am includes both the anhydrous hydrophobic core and a hydrated outer layer of each micelle. am is thus the hydrodynamic radius, while ac is the micellar radius of closest approach. The chains in the coronal region are sufficiently concentrated that they present an effective entropic barrier, preventing other micelles from entering the region; the chains in the corona are sufficiently dilute that the solvent can flow through the corona. The micelle core has a fixed charge qm and dielectric constant i. The dipole and higher moments of the fixed charges are neglected. Each probe is a hard sphere of radius ap and charge qp; the probe charge is negative. The dielectric constant i of the probes is assumed to be the same as the dielectric constant of the micelles. The solvent is a Newtonian fluid of dielectric constant w. Hydrodynamic micelle-micelle and micelle-probe interactions are described by the hydrodynamic interaction tensors. Namely, for a set of spherical particles i subject to external nonhydrodynamic forces Fj, the induced velocities vi of the particles are determined by the particle mobility tensors µij as
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Langmuir, Vol. 15, No. 10, 1999 3495
N
vi )
µij‚Fj ∑ j)1
(12)
From the work of Batchelor,1 Felderhof,2 and Mazur and van Saarloos,21 the translational mobility tensor µij can be expanded as
1 µjj ) (I + f0
bil + ∑ biml + ...) ∑ i*l i*l,m;l*m
(13)
1 µij ) (Tij + Timj + ...), i * j f m*i,j
∑
(14)
Here f0 ) 6πηai and I is the identity tensor. Bij and Tij represent interactions between pairs of interacting spheres. Tij describes the displacement of particle i due to a force applied on particle j, while bij describes the retardation of a moving particle i due to the scattering by particle j of the wake set up by i. Timj and biml describe interactions between trios of interacting spheres. Timj describes the displacement of particle i by a hydrodynamic wake set up by particle j, with the wake being scattered by an intermediate particle m before reaching i. Biml describes the retardation of a moving particle i due to the scattering, first by m and then by l, of the wake set up by i. Equations 13 and 14 result in 3
bil ) -
15 aial 1 rˆ rˆ + 4 R 4 il il 16R il
6
{[24(5ai2 + 3al2)aial3 -
il
87aial5]rˆ ilrˆ il - 17aial5I} (15) Tij )
infinitesimal radii and dielectric constant discontinuities at the surface of each polyelectrolyte are included. As a result, this calculation obtains additional charge-induced dipole terms (in eq 19, following, the terms in qp2 and qm2, which at short range can dominate the familiar chargecharge (qpqm) terms). U(r) inside the spheres satisfies Poisson’s equation. The local charge density in the solvent is induced by the fixed charges according to the linearized Poisson-Boltzmann equation. The Poisson and PoissonBoltzmann equations are solved by spherical harmonic expansions and boundary condition matching. If the charge distributions in the two spheres are spherically symmetric, g(2) of charged dielectric spheres of radii ap and ac in electrolyte solution has the approximate form
g(2)(r) ) 0, if r e 2rg 1 g(2)(r) ) exp -βi-1 P0000qpqm + PB0000(qp)2 + 2 1 B′ P (q )2 , if r > 2rg (19) 2 0000 m
(
))
Here P0000, PB0000, and PB′0000 are given by eqs 49-62 in ref 19. We omit eqs 49-62 here because they occupy a journal page. g(2)(r) vanishes at r e ap + ac because there is a hard-sphere interaction and approaches unity at infinite r because limrf∞ U(r) ) 0. Dm and Dp are extracted from the dynamic structure factor S(q,t), which is an ensemble average of a function of the positions of the scattering particles at pairs of times. In QELSS, the mutual diffusion coefficient Dm is obtained from the initial slope of the dynamic structure factor S(q,t) via
2 2 3 ai 3 ai(ai + aj ) 1 (I + rˆ ijrˆ ij) rˆ ijrˆ ij - I + 4 Rij 4 3 R 3 ij
(
)
4
-Dmq2S(q,0) ) lim tf0
ij
Here Ril is the distance between particles i and l, Ril ) ri - rl is the vector separation of i and l, and rˆ il ) ril/Ril. We are only interested in the corrections to the diffusion coefficients that are linear in particle concentration. The three-particle mobility tensors biml and Timl do not contribute to the linear corrections and thus are not considered further. The electrostatic interaction enters km and kp in the form of the two-particle distribution function g(2), which to O(c1) is exactly
g(2)(r) ) 0, if r e 2rg g (r) ) 1, if r > 2rg
(20)
A detailed evaluation of eq 20 by Carter and Phillies20 and Phillies24 shows
Dm )
1 q2S(q,0)
N
N
D0[q2 + ∑ ((q‚bil‚q) + ∑ i)1 l*i)1
〈
N
∑
N
exp(iq‚rˆ il)(q‚Til‚q) +
l*i)1
(18)
where 2rg is the contact distance; 2rg ) ap + ac for hard spheres of radii ap and ac. Sushkin and Phillies19 calculated the electrostatic interaction potential U(r) of a pair of spherical polyelectrolytes in a solution containing a simple electrolyte. This calculation differs from the familiar Debye-Huckel calculation in that both polyelectrolytes are given non-
∑ exp(iq‚ril)iq∇:(bil +
l*i)1
Til)]〉 (21) Evaluation of eq 21 with hydrodynamic tensors expressed by eqs 15 and 16 gives the slope Rm of the firstorder micelle volume fraction dependence of Dm, namely
(17)
where β ) 1/kBT and U(r) is the micelle-micelle or micelle-probe interaction potential. For hard spheres g(2) can be expressed as
(2)
dS(q,t) dt
3
75 ai aj rˆ rˆ (16) 4 R 7 ij ij
g(2)(r) ) exp(-βU(r))
(
Rm )
∫2a∞ {g(2)(r) - 1}
am3 4
93 4 r
c
} () () () () ()
am4 5
225 + 2 r
{
r 45 am r2 + -3 3 + 3 2 4 r2 am am
ac dr + 8 am
3
ac -6 am
31 am 32 ac
3
2
-
45 am + 8 ac
225 am + 128 ac
4
(22)
The calculation of a diffusion coefficient Dp of probes in a solution containing micelles is similar to the calculation of the mutual diffusion coefficient, except scattering from micelles is considered to be negligible. To the first order (23) Phillies, G. D. J. J. Chem. Phys. 1982, 77, 2623. (24) Phillies, G. D. J. J. Chem. Phys. 1995, 99, 4265.
3496 Langmuir, Vol. 15, No. 10, 1999
Sushkin et al.
in micelle partial volume, Dp may be written20,24 in terms of bij as
Dp )
〈
N kBT (I + b0l) f0 l)1
∑
〉
(23)
Here l proceeds over the N micelles, and subscript 0 refers to the probe. We obtain17
∫
p
{
c
15 15 a (a 2 - am2/8) {g (r) - 1} - apr-2 + ap+ac 4 2 p p 51 a a 2 r-4 dr (24) 16 p m (2)
(
) }
Our expressions for km do not agree with expressions used by Batchelor1 or Felderhof.2,3 References 1 and 2 correctly calculate a diffusion coefficient; however, as outlined in ref 25, they do not calculate the Dm measured by QELSS. V. Analysis In this section we analyze our experimental data. Using a nonlinear fitting algorithm, we deduce the micelle radius, micelle charge, and degree of hydration and aggregation numbers of SDS micelles. A separate fit to eqs 1 and 2 is done at each salt concentration. Coefficients Rm and Rp are calculated using eqs 22 and 24. The fixed inputs to the fitting program for each ionic strength include the inverse partial volume F˜ of the surfactant in g/L, the molar mass of the surfactant M in g/mol, and Dp vs c and Dm vs c data. F˜ was determined by weighing volumes of solution of known concentration. Free parameters of each fit are the micelle radius am, the probe radius ap, the ratio of the radius of closest approach to the micelle hydrodynamic radius ac/am, the micelle charge qm, and the micelle hydration number h, defined as the number of water molecules entrapped within a micelle. The charge qm is restricted to positive values during the fitting. Also, qm was limited above by the micelle aggregation number Na and by the PoissonBoltzmann model saturation limit qPBC (Bitzer et al.26), namely
4π0kBT
qm e qPBC ) 4am (1 + kam)
e2
(25)
where e is the elementary charge, is the solvent dielectric permittivity, and 0 is the dielectric permittivity of the vacuum. For our micelles qPBC is ∼50e at 0.1 M and ∼100e at 0.5 M; in all cases, qm < qPBC, so qPBC does not affect the final answer. The free parameters can be combined to calculate the micelle partial volume vj
vj )
h MH2O 1 + Fj FH2O M
4πam3
(26)
and the micelle aggregation number Na (25) Phillies, G. D. J. J. Colloid Interface Sci. 1986, 119, 518. (26) Bitzer, F.; Palberg, T.; Lowen, H.; Simon, R.; Leiderer, R. Phys. Rev. E 1994, 50, 2821.
(27)
3(vs1 + hv1w)
where MH2O ) 18 g/mol is the molar mass of water, vs1 ) M/p j NAv is the molecular volume of the surfactant, vw 1 is the molecular volume of water, and NAv is Avogadro’s number. A further calculation gives the effective thickness of the micelle water shell Rw
(4π3 N v ) s a 1
Rw ) am -
ap (5ap2 - 11am2/4) 15 ap + + Rp ) 4 ap + a c 2(a + a )3 ∞
Na )
1/3
(28)
and the mass fraction δ of water in a micelle,
MH2O
g)h
M
(29)
It is not claimed in quoting Rw that the water is actually a spherical shell external to the micelles. Numerical methods were used to extract best values of fitting parameters from our data and the above theoretical results. We used the Romberg algorithm27 to evaluate the integrals in eqs 22 and 24. The simplex nonlinear leastsquares algorithm27 was used to obtain parameters by minimizing the root-mean-square (RMS) error of the fits. The simplex algorithm was set to stop when the minimum bracket became less than 1.0 × 10-12. The final RMS errors for our fits are in the range (0.3-1.5) × 10-2. Table 1 summarizes results of simultaneous fits of Dm and Dp for SDS solutions to eqs 1-5 and 15-24. The fits are represented in Figures 1 and 2 by the solid lines. Dp at zero SDS concentration was not included in the fit. The probe charge qp was fixed at 10 000e, which corresponds to 1 charge/140 Å2 of surface area, based on the probe manufacturer’s determination. The fitting result is not sensitive to the value of qp. Fixing the probe charge at 200e changes the fitted values of the probe and micelle radii by less than 1% and decreases the aggregation number and micelle charge (while increasing the water shell width and micellar partial volume) each by less than 6%. A dependence of the probe charge on the pH of the solution will therefore not affect micelle parameters substantially. Titration of latex spheres28 shows that the surface charge density of spheres does not change in the acidic pH region by more than 50%. Table 2 serves to demonstrate the importance of using our improved electrostatic potential rather than the Debye-Huckel potential. The table summarizes results of fits to equations and data used to obtain the results in Table 1, except that in generating the numbers in Table 2 we used the Debye-Huckel electrostatic interaction potential and not our19 improved electrostatic potential. The micelle and probe radii am and ap are the same as those in Table 1. There is little difference between Tables 1 and 2 except for the inferred micelle charge. The micelle charges obtained using the Debye-Huckel potential are in general 2-10 times larger than the charges obtained by fitting the same data while using our19 improved electrostatic potential. VI. Discussion We have combined light-scattering spectroscopy and optical probe methods to determine the size, charge, (27) Press, W. H.; Teukolsky, S. A.; Vettering, W. T.; Flannery, B. P. Numerical Recipes in C: The Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1992. (28) Gisler, T.; Schulz, S. F.; Borkovec, M.; Sticher, H.; Schurtenberger, P.; D’Aguanno, B.; Klein, R. J. Chem. Phys. 1994, 101, 9924.
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Langmuir, Vol. 15, No. 10, 1999 3497
Table 1. Analysis of SDS Dm and Dp Data Using Our Sphere-Sphere Electrostatic Potential and Equations 1-5 and 15-24a I (M)
am (Å)
ap (Å)
qm (e)
a c/ am
N/a
h
δ*
R/w (Å)
vj × 103 (L/g)
0.1 0.2 0.3 0.4 0.5
29.3 25.3 28.6 28.6 32.0
333 345 360 370 367
7.0 10 6.8 7.4 6.9
1.45 1.24 1.27 1.17 1.03
76 59 79 67 64
31 23 26 33 56
1.9 1.5 1.7 2.1 3.6
9.0 6.7 8.1 9.1 13
2.9 2.4 2.6 3.0 4.5
a An asterisk indicates an inferred parameter. I is the salt concentration, am and ap are the micelle and probe radii, qm and qp are the corresponding charges (for all ionic strengths, qp ) 10 000), ac is the micellar radius of closest approach, Na is the micellar aggregation number, h is the micellar hydration number in water molecules per surfactant molecule, δ is the micellar hydration in grams of water per gram of surfactant, Rw is the equivalent water shell thickness, and vj is the micellar partial volume.
Table 2. Analysis of SDS Dm and Dp Data Using the Debye-Huckel Potential and Equations 1-5 and 15-24a I (M)
qm (e)
ac/am
N/a
h
δ*
R/w (Å)
vj × 103 (L/g)
0.1 0.2 0.3 0.4 0.5
1 23 51 38 11
1.43 1.08 1.02 1.01 1.03
71 56 76 66 63
34 25 28 34 58
2.2 1.6 1.8 2.2 3.7
9.5 7.0 8.4 9.2 13
3.1 2.5 2.7 3.1 4.5
a
Symbols are as in Table 1.
aggregation number, and degree of hydration of SDS micelles in NaCl/water over a range of salt concentrations. In our model of a micelle, each micelle has a dense spherical anhydrous core, a dense, thick hydrated, evenly charged spherical surface layer, and a dilute spherical corona of protrusions (of unspecified form) rising above the core. The protrusions are rare enough that they do not obstruct solvent flow appreciably but are substantial enough to prevent two coronae from overlapping. The surface of other micelles and probes cannot come closer to a micelle than the coronal radius ac, where ac g am. From the viewpoint of hydrodynamics, micelles are hard spheres of radius am. From the viewpoint of electrostatics, micelles are dielectric hard spheres of radius am and dielectric permittivity 2.5. The screened Coulomb potential describes the interaction of micelles with each other and with probes. The potential that we use takes into account the difference in the sizes of interacting charged spheres, induced dipole effects, and counterion exclusion effects.19 We assumed that the aggregation number depends only on the salt concentration and does not vary with surfactant concentration. Some SANS and fluorescence quenching measurements13,15,16 suggest that the micelle aggregation number varies as the one-fourth power of the surfactant concentration c. To test the consistency of our SDS data with the c0.25 growth, we analyzed our Dm and Dp using the fitting algorithm described in the previous section, modified by a constraint that the aggregation number increases with c as
Na ) N0(1 + k1c1/4)
(32)
Here N0 and k1 are free parameters of the fit. In our modified fit the micellar hydration number is independent of c so that the micellar radius increases with increasing Na. Also, the micellar charge grows proportionally to Na, keeping the fractional charge constant. As Na, am, and qm vary with increasing surfactant concentration, so do Dm0 and km. From our model, a plot of Dm against c curves up with increasing c if k1 is positive (i.e., if the micelles grow
with increasing c). However, our Dm data vary linearly with the surfactant concentration, or even curve down. Thus, our data are not consistent with the occurrence of observable concentration-dependent micelle growth at the low surfactant concentrations studied here. We therefore do not discuss other growth models. Dp0 and kp are not sensitive to the variation in c, because ap . am and qp . q m. Using the analysis described in the previous section, we found the aggregation number for SDS micelles at 35 °C to be 67 ( 9.4 for [NaCl] ) 0.1-0.5 M (Table 1). Other light-scattering studies found values of 85-1077 for [NaCl] ) 0.1-0.5 M at 40 °C and 70-1505 for [NaCl] ) 0.01-0.5 M at 25 °C. Small-angle neutron scattering gives N ≈ 70-7515 at 25 °C, 7011 at 25 °C, and 117-1269 for [NaCl] ) 0.1-0.2 M at 40 °C. Fluorescence methods give 6529 at 20 °C and 91-10416 for [NaCl] ) 0.01-0.25 M at 30 °C. Our aggregation numbers for SDS micelles are modestly lower than those in the literature. We find hydration number h of SDS micelles to be 34 ( 11.7 water molecules/surfactant molecule, increasing from h ≈ 23 for 0.2 M NaCl to h ≈ 56 for 0.5 M NaCl. Hayter and Penfold9 obtained a hydration of 9.5-9.7 water molecules/surfactant molecule. It should be emphasized that their measurement is a static determination based on neutron scattering, while ours is a hydrodynamic determination that counts any water molecule that has been entrained by a moving micelle. It is reasonable that our values are substantially larger than Hayter and Penfold’s. Hayter and Penfold14 assume that all methyl groups of the surfactant molecule form a physically smooth dry core; their anhydrous core is a sphere having a radius no larger than the length of the surfactant hydrocarbon chain. The surface of the outer, hydrated shell is also assumed to be spherical. Hayter and Penfold’s micelle is physically smooth but may be chemically rough; the outer shell is filled, but headgroups may be found at varying distances from the center. In our view, the micelle is rough both chemically and physically. Some parts of the micelle (surfactant molecules or trapped ions) protrude from the micelle by ∼13 Å ()ac - ap). These protrusions allow solvent to penetrate the outermost shell but exclude other micelles from the region r e ac. Recent computer simulations30 suggest that a micelle is a highly irregular aggregate of surfactant and water molecules, consistent with our findings that each micelle has very large amounts of hydrodynamically entrained water. Our analysis indicates that the micelle charge is 0.10 ( 0.04e/surfactant molecule. In contrast, Hayter and Penfold9 give 0.20-0.24e/surfactant molecule. Hayter14 shows self-consistency between charges assessed by SANS and derived from theory using the nonlinear PoissonBoltzmann equation. Hayter’s9 experimental values for micellar charges at 0-40 g/L SDS are 10-30% higher than the values predicted by Hayter’s theory of dressed micelles.14 Triolo et al.11 obtained a charge of 0.29e/SDS molecule in a small-angle neutron-scattering (SANS) study at 25 °C. The charge that we inferred is substantially smaller than the charge historically reported in the literature. A significant part of this difference in calculated qm arises from using our improvements19 on the Debye-Huckel potential, which appreciably reduces (relative to the Debye-Huckel potential used by Hayter and Penfold9 or (29) Lianos, P.; Zana, R. J. Colloid Interface Sci. 1981, 84, 100. (30) Karaborni, S.; van Os, N. M.; Hilbers, P. A. J. Langmuir 1993, 9, 1175.
3498 Langmuir, Vol. 15, No. 10, 1999
Triolo et al.11) the charge needed to create a given intermicellar repulsion at short distances. Fits to our experimental data using our treatment of Rm and Rp and the Debye-Huckel potential yield micellar charges 2-10 times higher than the corresponding fits, to the same data, using our19 improved electrostatic potential. From SANS experiments, Hayter,14 using the Debye-Huckel potential for 0.1 M SDS and 0.2 M NaCl, obtains a surface charge density of 0.39e/nm2. If we use the simple Debye-Huckel potential to treat Dp and Dm in 0.2, 0.3, and 0.4 M NaCl, as opposed to using the Sushkin-Phillies potential, we deduce surface charge densities of 0.29, 0.49, and 0.37e/ nm2, respectively. That is, when we use the Debye-Huckel potential, our estimates of qm agree with those of ref 14. Our charges (Table 1) are lower than the literature (refs 9 and 11) values purely because we use our improved19 electrostatic potential. Kale et al.6 and Cutler et al.4 obtained a fractional charge on the SDS micelle of 0.85 ( 0.01 using surfactant ion selective electrodes in aqueous SDS solution, which is higher than the fractional charge in other literature. To deduce the micellar charge reliably, the activities of Na+ and dodecyl sulfate anions DS- have to vary significantly with surfactant concentration. In the presence of added salt, Kale and Cutler report that this variation is small and explain that the lack of variation in the activities results in a challenge to establishing the micellar aggregation number and charge accurately. The probe radius in SDS/NaCl/water solutions of 0.1, 0.2, 0.3, 0.4, and 0.5 M increases with increasing NaCl concentration from 333 to 370 Å. The increase in probe size is comparable to the micelle radius (30 Å), implying that the surfactant is probably binding to the surface of the probe. Fits to our data using our improved electrostatic potential and the micellar growth assumption of eq 30 also yield a high micellar charge. If we fix the micellar growth parameter k1 at 8.76 (g/mol)-1/4, which corresponds to N0k1 ) 114 molecules/mol1/4 reported by Bales and Almgren,15 we deduce fractional micellar charges in the range of 0.3-0.9e/surfactant molecule (Table 3). Our charges deduced under the assumption of no micellar growth (Table 1) are in the range of 0.09-0.17e/surfactant molecule. Micellar growth suggests higher charges because the decrease in Dm with increasing c has to be compensated
Sushkin et al. Table 3. Analysis of SDS Dm and Dp Data Using Our Sphere-Sphere Electrostatic Potential and Micellar Growth Model Described by Equation 30a I (M)
N0
a0m (Å)
Rm
N0
k1 [(g/mol)1/4]
Rm
0.1 0.2 0.3 0.4 0.5
21 15 20 15 16
19.0 16.0 18.0 17.4 20.0
0.9 0.5 0.8 0.3 0.3
92 95 26 1 62
-0.6 -1.3 6.0 306 0.09
0.4 0.0 1.0 0.0 0.2
a N and k are defined in eq 30, R is the micellar fractional 0 1 m charge, and a0m is the micellar radius at zero surfactant concentration. Columns 2-4 of the table summarize results of fits with k1 fixed at 8.76 (g/mol)1/4. In fits described by the last three columns, k1 is a free parameter.
by an increase in the repulsion between the micelles via an increase in the micelle charge. If, in addition to N0, we also treat k1 as a free parameter, our fits become numerically unstable (Table 3). VII. Conclusions Measurements of probe and mutual diffusion of SDS micelles in aqueous solutions of varying ionic strength were carried out at 35 °C. As presented in Tables 1 and 2, we deduced the micelle radius, aggregation number, charge, and hydration from a simultaneous analysis of probe and mutual diffusion coefficients and their dependence on concentration. We find a fractional ionization for SDS micelles of 0.10 ( 0.04e/surfactant molecule, which is lower than literature values.9,11 We demonstrate that the difference between our results and the literature arises because our electrostatic potential includes chargeinduced dipole terms19 omitted in previous calculations. For SDS micelles, our aggregation numbers are smaller, and our hydration values are higher, than numbers reported previously. If we analyze our data using the potential used by previous workers, we get about the same value for the fractional ionization as they did. Our data do not indicate that SDS micelles grow appreciably at the low surfactant concentration studies here. Acknowledgment. The support of this work by the NSF under Grant DMR94-23702 is gratefully acknowledged. Registry No. SDS, 151-21-3. LA990035G
