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Quasielastic light-scattering studies of micellar sodium dodecyl sulfate solutions at the low concentration limit
Quasielastic Light-Scattering Studies of Micellar Sodium Dodecyl Sulfate Solutions at the Low Concentration Limit A X E L R O H D E AND E R I C H S A C K M A N N Abteilung Experimentelle Physik Ill, Universitiit Ulm, Oberer Eselsberg, D-7900 Ulm, West Germany Received November 6, 1978; accepted December 20, 1978 Correlation functions of scattered light intensity of carefully purified sodium dodecyl sulfate (SDS) solutions were measured as a function of tenside concentration and NaCI concentration of the aqueous phase. The correlation functions were analyzed by taking into account the influence of the Coulomb interaction between the micelle (macroion) and small electrolyte ions on the diffusion coefficient. Values of the hydrodynamic radius, the aggregation number, and the effective surface charges were obtained. The aggregation number increases from N = 27 to N = 95 upon increasing the NaCI concentration from 0 to 0.05 mole per liter, while it remains constant when the salt concentration increases further up to 0.2 mole per liter. The effective charge of the micelles decreases with increasing NaC1 content in the whole concentration region studied. These results could be interpreted qualitatively in terms of a model which relates the existence of an equilibrium size of the miceiles to the balance between hydrophobic and Coulomb interactions. Our results lead to the conclusion that at least up to an NaCI concentration of 0.2 mole per liter the SDS-micelles exhibit an oblate spherical shape rather than a cylindrical form. I. INTRODUCTION
work we performed experiments in the low concentration limit, that is in the neighborhood of the critical micelle concentration. It is customary to use high ionic strength aqueous solutions in order to minimize longrange Coulomb interaction between the aggregates and the ions of the electrolyte. In order to learn something about this interaction, we performed measurements as a function of the ionic strength o f the aqueous solution. The temporal fluctuation of the scattering intensity is primarily determined by the spatial diffusion of the aggregates. Therefore, the measured light-scattering spectra can be analyzed in terms of transport theories, which give a more detailed insight into such interaction mechanisms than the virial expansion method of phenomenological thermodynamics. For the interpretation of our data we applied the model of Stephen (5). This method allows us to take account of electrostatic interaction and, therefore, to determine the radius of the aggregates at any ionic strength. In addi-
The basic physical parameters characterizing micellar solutions of amphiphatic molecules are: (i) the critical micelle concentration (CMC); (ii) the size, shape, and distribution in size of the micelles; (iii) their electric charge; (iiii) the rate-constants of the dynamic equilibrium between micelles o f different size, as well as between micelles and monomeric units. Quasielastic light scattering is well suited to study micellar systems with respect to the second and the third point. With the same apparatus one can determine the critical micelle concentration. Finally, information concerning the chemical reactions rates, which are usually measured by p-jump or T-jump experiments (1), may be obtained under favorable circumstances by quasielastic light scattering. Recently dynamic light-scattering studies of sodium dodecyl sulfate (SDS) dispersions in aqueous solutions at high tenside concentrations were reported (2-4). In the present 494 0021-9797/79/090494-12502.00/0 Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol, 70, No. 3, July 1979
DYNAMIC LIGHT SCATTER OF MICELLES
tion, this method of evaluation also gives information about the surface charge of the micelles. II. MATERIALS AND METHODS
A. Preparation of highly purified SDS. Sodium dodecyl sulfate (SDS) was recrystallized several times from ethanol. As has been pointed out previously by other authors (6), further purification is necessary in order to remove traces of long chain alcohols. At sufficient high detergent concentrations the alcohols are solubilized in the micelles. However, in the vicinity of the critical micelle concentration the alcohols precipitate in the form of large aggregates of some hundred A diameter, thus disturbing the light-scattering spectra considerably in the CMC region. A foaming procedure, introduced by Elworthy and Mysels (7), was used in order to remove residual alcoholic impurities. The apparatus, used for the foaming purification, is shown in Fig. 1. It consisted of a glass flask F1 of 2 liters volume equipped with several supplies. The flask rested in a temperature-controlled water bath. A magnetic stirring bar was provided for mixing the solvent. To produce foam, the flask was evacuated with a filter pump, while air was introduced through a glass pipe (P1) with an to filter pump
VI~ r ~
.
bypass
, --t)
sc
3
F2
FI
FIG. 1. Apparatus for the purification of the solutions. FI, flask in which the foaming procedure was performed; F2, auxiliary reservoir; PU, peristaltic pump; MF, Millipore filter of 500 A mesh size; SC, scatteriag ceil.
495
opening below the liquid surface. A second glass pipe (P2), adjustable in height, was also attached to the filter pump and was used to draw off the foam. The rates of foam formation and removal were controlled by the valves V1 and V2. The ability to remove and to store solvent in a clean environment was also provided. For that purpose another flask (F2) was connected to the first flask (F1). A closed circuit led from F1 through a peristaltic pump, a Millipore filter, the scattering cell back into flask F1. In order to remove solvent for concentration measurements, an outlet valve (V3) was installed in this closed circuit. The Millipore filter of 500-A mesh size served to remove dust particles. All parts of the apparatus were built from glass or Teflon with the exception of the silicon pipe inside the peristaltic pump. Before use all parts were carefully cleaned. For the purification procedure 1 liter of aqueous SDS solution, at a tenside concentration slightly below the CMC, was placed into flask F1. The solution was foamed for about 20 hr, while there was a continuous flow through the scattering cell. The surface active long chain alcohols, which are the most disturbing impurities, are favorably distributed at the air-water interface of the foam. They are thus preferentially removed with the foam. During the foaming procedure, the solvent was changed several times between the flask F1 and the auxiliary reservoir F2 in order to clean this part of the assembly. When the foaming procedure was finished, the solution was cooled by introducing ice into the water bath. Some SDS precipitated and settled down. The supernatant was transferred into the auxiliary reservoir F2. Then the solution was rewarmed to room temperature. In this way the SDS concentration of the solution could be enriched above the critical micelle concentration up to about 30 mmoles per liter. The final concentration was adjusted b x siphoning back some of the low concentrated solution from the reservoir F2. Journal of Colloid and Interface Science,
Vol. 70, No. 3, July 1979
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ROHDE AND SACKMANN
At salt concentrations up to 20 mM the SDS concentration was determined by conductivity measurement. At higher salt concentrations the specimens were lyophilized and weighted. The sodium chloride concentration was adjusted by placing all the solution into the reservoir F2, weighing it, and adding the desired amount of salt. Samples with higher SDS concentrations could not be prepared by the freezing procedure. In that case the solution with the highest desired concentration was placed into flask F1 and the solution was pumped through the bypass (cf. Fig. 1). The concentration was changed by adding the appropriate amount of solvent. The success of the purification procedure was tested by measuring the scattered light intensity as a function of the SDS concentration in the region of the CMC. In the case of incomplete purification, the total scattered intensity exhibits a strong peak in the neighborhood of the CMC. This is attributed to the precipitates of long chain alcohols, formed when the SDS-micelles disappear. In contrast, sufficiently purified samples exhibited a step-wise shape of the intensity versus concentration plots. An example is shown in Fig. 2. B. The scattering apparatus. The scattering cell had a quadratic cross-section of 1 cm width. Two openings at opposite sides of the cuvette allowed the solution to be pumped through the cell, as described above. The cell rested in a brass block. The temperature was controlled by flowing water from a thermostat through the block. All measurements were performed at a temperature of 24°C. Light from an argon-ion laser, equipped with an etalon, was focused on the center of the cell. The laser was tuned to a wavelength of 488 nm and the output power was about 200 mW. In order to minimize the intensity of light scattered from the cell walls, all measurements reported here were performed at a scattering angle of 90°. The scattering region was imaged by a Journal of Colloid and Interface Science, Vol. 70, N o . 3, July 1979
=e >,
z
5
I
I
10 IJ5 2i0 25 SDS-CONCENTRATIONx 10 3 (moLe/L)
FIG. 2. Function of scattered light intensity versus SDS concentration. The stepwise shape demonstrates the success of the purification procedure.
lens to an aperture in front of the photocathode of a photomultiplier. A slit with variable width was situated in front of the lens. The slit and the aperture served to define both the opening angle of the scattered light and the probed scattering volume. They were adjusted to give a detection area of about one coherence area. An interference filter was situated just in front of the photocathode in order to remove stray light from the laboratory and fluorescence light from the solution. The output from the photomultiplier (EMI 9863) was amplified and fed into a discriminator, which was connected to a Malvern digital autocorrelator. The correlator was operating in the single clipped mode. It computed the autocorrelation function in 48 channels. The measured spectra were punched on paper tape and analyzed on a Nicolet BNC-12 computer. III. EVALUATION AND INTERPRETATION OF MEASURED CORRELATION FUNCTIONS
A. Analysis of Correlation Function in Terms of Cumulants For homodyne detection, the normalized clipped intensity correlation function g~2)(t) of a Gaussian process with a single correlation time is given by (8): g~2)(t) = 1 + C exp(-2Ft)
[1]
where C is an arbitrary constant and F is the inverse correlation time. If the scattered light arises from particles
DYNAMIC LIGHT SCATTEROF MICELLES of equal size undergoing Brownian motion, F can be expressed as: F = Dq 2 [2] where D is the diffusion coefficient of the particles and q is the scattering vector. In general, many dynamic processes with different correlation times F -1 contribute to the scattered intensity. An example is the diffusion of a polydisperse system of particles. Equation [1] has to be modified into:
g~2)(t) = 1 + C
[i0°G(F) exp(-Ft) 1 dF.
[3]
G(F) is the light intensity scattered from the processes characterized by the correlation time F -~. The distribution G(F) is normalized to one. For sharply peaked functions G(F) the method of cumulants (9) may be applied in order to analyze gCe2)(t) in terms of valuable physical parameters. The measured correlation function can be expanded in a power series in time t. In (g~Z)(t) - 1) = In C - 2['t + Q2(f't) 2 - VaQa(['t) 3 + . . . .
[4]
where f" =
FG(F)dF
Q~ = l/f u I,~ (F - f')~G(F)dF.
[5]
is the average inverse correlation time, Q2 is a measure for the width, and Oa for the asymmetry of the correlation time distribution function G(F). Provided g~=~(t) can be measured with sufficient accuracy, these important quantities may be determined from the light-scattering experiments. For diffusing particles I" and Q2 yield information on both the average size and the size distribution of the particles.
B. Diffusion of Macroions in Electrolytes (10) For dilute solutions of neutral spherical particles, the particle diameter may be esti-
497
mated from the diffusion coefficient via the Stokes-Einstein relation: D = kaT/6~r~ro. If the viscosity 7/is measured by a capillary viscosimeter, the hydrodynamic radius r0 is readily obtained. For charged particles in electrolytes, the situation is much more complicated. Coulomb interaction between the various ionic species must be taken into account. The problem simplifies considerably if we are dealing with solutions containing one type of large charged particles (macroions) and one or more small ionic species. The small ions are called counterions, if they have the opposite polarity as the large ions, and byions, if they possess the same polarity as the large ions. If the concentration of the macroions is sufficiently small, the interaction between them can be neglected because their charges are screened by the counterions. Nevertheless, the diffusion of all ionic species is influenced by long-range Coulomb forces. These forces arise from the fast concentration fluctuations of the small ions, leading to nonspherical charge distributions about the micelles. The resulting dynamic forces lead to a faster random motion of the macroions, while the latter tends to retard the small ions (cf. Ref. 10). Stephen (5) treated this problem quantitatively by solving the Fokker-Planck equations for the large ions, the counterions, and byions in conjunction with the Poisson equation. He obtained the following expression for the diffusion coefficient of the large ions:
D(q) = D I ( I +
q~ q~ + ~ q~
)
[6]
i=2
q~ -
47'/"
%kBT
z~c~
[7]
The index " 1 " characterizes the macroions (micelles), ~0 is the dielectric constant of the solution, zi is the charge, and ct is the number density of species i. D~ is the diffusion coefficient in the absence of electroJournal of CoUoid and Interface Science, Vol. 70, No. 3, July 1979
498
ROHDE AND SACKMANN
static interactions, q is the scattering vector. Equation [6] shows that electrostatic interaction indeed gives rise to an increase in the diffusion coefficient of the macroions. The q~ are not independent variables. They are coupled by the condition of overall electric neutrality. At a given charge of the macroions, the largest increase of the diffusion coefficient is to be expected, if there are no byions present. This shows that the effect of the electrostatic interaction can be diminished by adding small ions of both polarity to the solution which leads to an increase in ~q]. The physical reason for this reduction of the electrostatic effect is the much higher mobility of the small ions. Local concentration variations in the counterion distribution can be compensated very fast by the high mobile byions.
C. Reaction Rates o f Dynamic Equilibrium and Correlation Function Besides the random motion, temporal fluctuations in the scattered light intensity due to chemical reactions may also affect the intensity correlation function. The reaction from a substance A to a substance B is characterized by the reaction rates kl and kz for forward and backward reaction, respectively. The reaction may influence the light-scattering spectrum because the diffusion coefficients and the polarizabilities of the two species may be different. In general the light-scattering correlation function of a system in which the described reaction occurs consists of the sum of two exponential functions. The exponents are functions of the two diffusion coefficients and the two reaction rates. Also the coefficients are complicated functions of the four time-determining parameters and of the polarizabilities and the equilibrium concentrations of the two species (11). McQueen and Hermans (2) treated the problem for the dynamic equilibrium between micelles and the corresponding monomers. Assuming that the reaction occurs Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
by bimolecular steps, and assuming further that only the concentrations of the monomers and the micelles are significant, the following rate equations are derived: 0Cmo
Ot OCmi
Ot
-- DmoV2Cnao - NklcNmo + Nkzcmi
-- DmiV2Cmi + klCSmo - k2cmi.
[8]
Cmo and Cmi are the concentrations of the monomers and the micelles, respectively, Dmo and Dmi are the diffusion coefficients, and N is the aggregation number of the micelles. From the fluctuations in concentration the correlation function of the scattered light intensity is derived. Using the fact that the rate of the fluctuation for micelle formation is of the order of N larger than that for micelle dissociation, the two exponents are Dmiq 2 a n d Dmoq z + N2klC. rnoN-l, w h e r e
Cmo
is the equilibrium concentration of the monomer and q is the scattering vector. The coefficient determining the contribution of the second exponential function is of the order of N smaller than the coefficient of the first exponential function. In the described approximation the reaction between micelles and monomers can hardly contribute to the correlation function of the scattered light intensity. First, the exponential function including the rate constant is much weaker than the function governed by the diffusive motion of the micelles. Second, the exponential function including the rate information always decays faster than the correlation function of the diffusive motion of the monomers. This time domain is out of the region obtainable by today's correlation techniques. IV. ANALYSIS OF EXPERIMENTAL DATA
Correlation functions of each sample were measured in several runs. Each run lasted from 2 to 8 min and was normalized separately. The time domains over which
DYNAMIC LIGHT SCATTER OF MICELLES
~2 x IL
÷ +
I l
,;
L
2'o
L
ZolgS)
FIG. 3. Values of F as a function of time domains, ~'0, of measurement. The different F were obtained for first-order (x), second-order (+), and third-order (O) fits of cumulant expansion (Eq. [4]).
the correlation functions were measured, were varied in several steps between about half the correlation time and about twice the correlation time. The total measuring time per sample was about 2 hr. According to Eq. [4], the measured points representing the functions In (g~2>(t)- 1) were fitted first to a linear function, then to a second-order polynomial, and finally to a third-order polynomial in the time. Values of both the average decay rates [" and the second cumulant Q2 were determined for each of the three orders of fit. This was done for the different time domains. Figure 3 shows a typical series of data determined in that way for the example of the mean decay rate F. Application of a method proposed by Brown et al. (12) yields values of the mean decay rates F with committment neither to one of the three orders of fitting, nor to a certain time domain. Provided the correlation function is measured in a time domain which is small compared with the correlation time, the linear term in Eq. [4] is significant only. Therefore it is possible to determine F by extrapolating the line through the data points of the linear fit to zero probing time. At larger probing times the higher order fits are more accurate, since only then do the higher terms contribute significantly to the correlation function. For most samples the values of F determined by the second-order fit at long probing times agree
499
very well with the decay rates obtained by the described extrapolation. The agreement was worse if the higher cumulants were large, or if the scattering intensity was low. Third-order fits gave no good accordance in most cases, because the accuracy of the data points was not high enough to allow a four parameter fit. The typical variance of the mean decay rate at medium tenside concentration was about 5%. This was tested by comparing several runs of the same sample. The second cumulants Q2 were determined by fitting a second-order polynomial to the measured points (cf. Eq. [4]). The values of Q~ reported here are the averages of the Q2 values obtained from measurements in the different time domains. The variance is not better than about 50%, so that only qualitative statements can be made concerning the width of the distribution of decay rates. V. EXPERIMENTAL RESULTS AND THEIR INTERPRETATION
In Fig. 4 values of the average correlation function decay rates F are plotted as a function of the tenside concentration. Plots are given for different NaC1 concentrations. The right side scale gives the average diffusion coefficients according to Eq. [2]. Figure 5 shows the values of F a n d / ) for a larger concentration range of the surfactant and for pure water as solvent. The bars in Fig. 5 indicate the variance of the experimental values as obtained from different experimental runs taken for the same sample. The interpretation of the data in Fig. 4 in terms of the Stokes-Einstein model would lead to the conclusion that the aggregation number decreases with increasing SDS concentration. This is highly improbable. The different behavior for different salt concentrations strongly suggests that it is the ionic character of the micelles which causes the increase in the diffusion coefficients. In the Journal of Colloid and Interface Science,
Vol. 70, N o . 3, J u l y 1979
500
ROHDE AND SACKMANN
x
x Water * 10 mM N e C t o 20raM NoEl A 50raM No.El " 0.1 M NQE[ v 0.2 M N0.E[
-- 15 x )1._
cJi ,( 2.5 3M ~_, _
/ xx
~×
10
+
.
j
.
~ ° ~ o~ o A. .----t'A~------i--v 5
i
x
~ ~
.
~
~
o
x
.
"
i
-;"
v i
2.0
~
_
~
_ ,
_
1.5
~
~
1.0
t I
I
SDS-CONCENTRATIONx10~ (mote/I)
FK;. 4. Values of the average decay rates of the correlation function, f" (left scale), and of the average diffusion coefficients, /) (right scale), for aqueous SDS dispersions. Values of F and /9 are given as a function of tenside concentration for different salt contents of the aqueous phase. The values of I" were obtained by extrapolating the points of Fig. 3 to ~'0= 0. The curves are calculated from Eq. [9] as described below.
following we give an interpretation in terms of the Stephen model (cf. Section III, B): In order to calculate the concentrations and charges of the different ionic species present in the micelle solution, we proceed as follows (cf. Ref. 13). At SDS concentrations CT below the CMC the number of free dodecyl sulfate (DS-) ions equals the number of sodium ions. Above the CMC the concentration of free DS- ions does not change upon addition of further tenside, since the additional SDS aggregates. The
-_ ¢ 30
5
o,
2o
~=
number density of the micelles is equal to the number of DS- ions aggregated divided by their aggregation number N. The sodium ions are either free or bound within the Stern-layer of the micelles. The ratio of bound Na + ions to DS- ions aggregated is denoted by a. It is assumed to be independent of SDS concentration. The charge of a micelle is equal to the total charge of its DS- molecules minus the total charge of the bound sodium ions. Table I summarizes the concentrations and charges of the different species of the micelle solution. With the definitions of Table I the qi values (cf. Eq. [7]) of the different ionic species are easily calculated. According to Eq. [6] the diffusion coefficient of the micelles is given by D =D1
lo
2'o
io
6'0
s~
16o
SDS- CONCENTRATION,103(mote/I)
FIG. 5. E x t e n s i o n o f the r" v e r s u s SDS concentration plots to higher tenside concentration for the case o f pure water. Journal of Colloid and Interface Science,
Vol. 70, N o . 3, July 1979
1+
2 G + CMC + CT CT - CMC
[91 --
Ot
Upon deriving this equation from Eq. [6] the square of the scattering vector q2 has been neglected because it is always small compared to any of the q~.
501
DYNAMIC LIGHT SCATTER OF MICELLES TABLE I Concentration and Charges of Ionic Species Present in the Micellar Systema Species
Concentration
Charge
Micelles
C~r - CMC N
- N ( 1 - ,~)e
Free DS- ions
CMC
-e
Free Na + ions
CT -ct(Cr
+e
- CMC) Free Na÷ ions from solvent
C1
+e
Free Cl-ions
C~
-e
a N, aggregation number of micelles; Cr, SDS concentration; C1, salt (NaCI) concentration of aqueous solvent; e, elementary charge. All concentrations are measured in terms of number densities.
Equation [9] relates the measured diffusion coefficients with most interesting structural parameters of the micelles, i.e., the CMC, the aggregation number N, and the charge of the micelle. Now, the CMC has been measured either independently, or it has been taken from the literature (14). By the following procedure we succeeded in determining both N and a. Slightly above the critical micelle concentration, ( C T - CMC) is ver), small and therefore the second term in Eq. [9] which takes account of Coulomb interaction, can be neglected. The hydrodynamic radius of the micelles can then be obtained from the diffusion coefficient D~ via the Stokes-Einstein relation. The negligible influence of the electrostatic interaction slightly above the CMC may be explained as follows: The ratio of the number densities of free DS- ions to micelles is large in this concentration region. The free DS- ions act as small, highly mobile byions. They lead to a fast equilibration of the charge fluctuations about the micelle (macroion). Now, our model assumes that at increasing SDS concentration the concentration of free DS- ions remains constant. The ratio of free DS- ions to mi-
celles becomes smaller and the influence of the electrostatic interaction becomes significant. Following this argumentation, we extrapolated the plots of the diffusion coefficient versus SDS concentration to the CMC, and thus obtained the values of D1 for the different salt concentrations. The finding that the measured diffusion coefficient,/), is independent of SDS concentration Ca- at high salt content (cf. Fig. 4) strongly suggests that the aggregation number N does not depend on Cr at least in the concentration region studied (CT < 30 mM). N may then be determined from the hydrodynamic radius. In order to account for the Stern-layer thickness, a value of 4.6/~ is substracted from the hydrodynamic radius. The remaining volume is made up of the tenside hydrocarbon chains each of which has a volume of 350/~3 (15). By assuming a spherical micellar shape, the aggregation numbers given in Fig. 6 were determined. Now, both D1 and N are known. Therefore a may be determined by fitting the c u r v e s / ) versus CT as calculated by Eq. [9] to the measured values of the diffusion coefficients. The values of a obtained by this procedure are given in Fig. 7 as a function of NaC1 concentration. As seen in Fig. 4 good fit between calculated and measured/) versus CT curves is obtained. The agreement is less good for pure water as solvent at higher tenside concentrations. We conclude that another dynamic process besides diffusion contributes
~IOO
o----
I
i
i
01
i
0.2
NoEL-CONCENTRATION (mole/l)
FIG. 6. Aggregation number of SDS micelles as a function of NaCI concentration. Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
502
ROHDE AND SACKMANN
' 0.9
0.8
%
I
011
No.El-CONCENTRATION
'
012
(mote/t)
FIG. 7. Ratio c~ o f the n u m b e r s o f bound N a ions to DS ions aggregated as a function of NaC1 concentration. The charge o f the micelle is given by: N(1 - t~)e.
to the correlation function. This conclusion is suggested by the observed dependence of the second cumulant on the tenside concentration (cf. Fig. 8). For all samples the plots of the second cumulant versus SDS concentration decrease with increasing SDS concentration above the CMC. This behavior could be caused by the appearance of impurity aggregates about the CMC or could also be due to the lower accuracy of the measured data in that low concentration region. Samples with water as a solvent showed in addition a peak in Q~ with a maximum at a concentration of about 50 raM. We assume that the dynamic equilibrium between micelles and monomers is responsible for this finding. The approximation given in Section III, C does not support this suggestion. But we assume that by taking into account the full expression for the correlation function in the presence of chemical reactions, the experimentally observed behavior may be explained. It is obvious that the most prominent contribution to the correlation function comes from the diffusion of the micelles. In order to observe chemical reactions it is necessary that the coefficients determining the exponential functions including the chemical relaxation time are of appropriate magnitude. Furthermore, it is necessary that the inverse relaxation time of the chemical reaction is of the Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
order of 2Dq 2, because we did measurements only in the time domain determined by the diffusive motion of the micelles. Figure 9 shows this function of inverse relaxation time l/r1 versus SDS concentration, as has been measured by pressurejump experiments (16). Also the function 2Dq 2 .is included. The two functions are of equal magnitude in the concentration region where the described incompatibilities with our diffusion model occur. VI. D I S C U S S I O N
A comparison of the aggregation numbers, N, determined in the present work with values from other authors who used static light scattering (17-19) or quasielastic light scattering (2-4) shows that good agreement exists at higher salt concentrations of the solvent. At low ionic strength only results obtained by static light scattering are available for comparison. The aggregation numbers obtained by that method are by about a factor of two larger than the values of N determined by us. But we believe our results to be reliable for the following reasons: (a) the tenside was purified very carefully, (b) our measurements agree very well with the literature at high ionic strengths of the solvent, and (c) in quasielastic light scattering one always measures the lower limit of the diffusion coefficient. Uncertainty may 04 Q2 0.3
0.2
0.1 x
, 210
x
~ 1 l I 0 60 80 100 $D$'EONCENTRATION xl0 3 (mole/I)
FIG. 8. Second c u m u l a n t s Q2 for SDS in pure water (O ©) and in 0~1 M NaCI solution ( x - ×) as a function o f tenside concentration.
DYNAMIC L I G H T SCATTER OF MICELLES
503
of the determined values of ~ is strongly dependent on the applicability of the Stephen theory to our system. To justify the approxi< ~3 q2 mations of the Stephen theory it is necessary that the average electrostatic energy of a micelle and a counterion is smaller than k BT. This condition is fulfilled in our measureSI]S-CON[ENTRATION xlO~(mota/t) ments except in the two measurements with FIC. 9. Twice the decay rate ([') of the correlation the highest tenside concentrations with pure function and the reaction rate (ti-1) of micelle-monowater as solvent. mer equilibrium taken from the literature (16) as funcThe behavior of the measured functions tion of SDS concentration. of aggregation number N and a versus NaC1 concentration may be understood in terms lay in the extrapolation to the CMC, but that of a counterbalance of hydrophobic and cannot fully explain the discrepancy. electrostatic interactions (22): First measurements of micellar self-diffuA micelle is composed of three different sion at low ionic strength are from Stigter regions: et al. (20). Comparison with our results (a) The hydrocarbon core. It is composed show: (a) The reported value of the diffusion of the hydrocarbon chains of the amphicoefficient for SDS-micelles at the CMC is phatic molecules. The chains are in a rather 1.099 × 10-6 cm2/sec, whereas our meas- fluid, disordered state. urements give a value of 1.325 +_ 0.1 x 10-6 (b) The Stern-layer, enclosing the core. cm2/sec. (b) A decrease of diffusion coef- It consists of the hydrated headgroups of ficient with increasing SDS concentration the amphiphatic molecules and of the bound is reported, whereas we observe just the counterions. The outer surface of the Stemopposite behavior. The diffusion coeffi- layer is the shear surface. Its distance from cients were determined by the diffusion of the center of the micelle determines the hymicelles, tagged with dye, through porous drodynamic radius, which is measured by glass disks. The method includes several quasielastic light scattering. hazards. Mysels and Princen (19) pointed (c) The Gouy-Chapman diffusive layer. out that the measurements may be disturbed It contains the unbound counterions and byby hydrolysis catalyzed by the glass sur- ions, and its thickness is determined by the faces. We further assume that it is difficult Debye screening length. to compare diffusion measurements in bulk Amphiphatic molecules aggregate bewater with measurements in capillaries, es- cause the reduction of hydrocarbon-water pecially if charge effects come into play. interface is energetically favored. If the An indication for the reliability of our re- aggregation would be determined by this hysults are the measurements of other authors drophobic interaction alone one would ex(3, 6), which observe the same increase of pect phase separation rather than micelle diffusion coefficient with increasing SDS formation. The size limiting parameter in ionic micelles are the length of the hydroconcentration as we do. The ratio of bound counterions and sur- carbon chains and the Coulomb interaction factant ions in the miceUes, a, is normally between the headgroups. For spherically shaped micelles the maxiassumed to be about 0.5 in pure water as solvent (21). Our measurements give a value mum value of its core is equal to the length of a = 0.81 for pure water, while a in- of the fully extended hydrocarbon chain. creases up to about 0.96 at a salt concen- Therefore the aggregation number of spheritration of 0.2 mole per liter. The reliability cal micelles is limited. A larger aggregation Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
504
ROHDE AND SACKMANN
number can be obtained only by changes of the micelle shape. The second size limiting factor is the area per headgroup in the Sternlayer. By increasing the aggregation number of the micelles, this area per headgroup reduces and the Coulomb interaction between the headgroups increases. Due to this counterbalance between hydrophobic and electrostatic interaction micelles exhibit a rather narrow size distribution. The Coulomb interaction between the headgroups can be reduced by introducing more counterions into the Stern-layer. This Can be accomplished for example by raising the ionic strength of the solution. Figure 7 shows that the ratio a of bound counterions to DS- ions increases with increasing salt concentration, as expected by the above considerations. Figure 6 shows an increase in aggregation number with increasing ionic strength up to an NaC1 concentration of about 0.05 mole per liter. N is constant above this limit. In contrast, a increases continuously above 0.05 mole per liter. Our results lead to the suggestion that the micelles do not exhibit a spherical shape in the region of high salt content (>0.05 mole per liter). This follows from the measured hydrodynamic radius of the micelles which has a value of 24.5 A. The corresponding core radius is 19.9 A which is significantly greater than the value of 16.7/~ for the length of a fully extended hydrocarbon chain of SDS. Two nonspherical shapes are usually considered: hemisphere capped cylinders and oblate spheroids (23). The different behavior of a and N above an ionic strength of 0.05 mole per liter strongly favors the existence of the oblate spheroid. Consider first the case of the cylindrical micelle. Addition of NaCI increases a (cf. Fig. 7). Now, the area per tenside molecule would not change if the cylinder grows in length. Therefore a continuous growth with increasing NaCI concentration would be expected which is in contrast to Fig. 6. Consider now the oblate micelle. Each incorporation of a furJournal of Colloid and Interface Science. Vol. 70, No. 3, July 1979
ther molecule would decrease the area per head group. Therefore an increasing amount of counterions with each addition of a surfactant molecule would be necessary for a growth process. A limit in size of an oblate micelle may be reached if no more counterions can be introduced into the Stern-layer. This seems to be the case in our system. The Stern-layer is filled as far as possible with counterions. However, the screening is not good enough to allow the micelles to exhibit a cylindrical shape which would permit a further increase of the aggregation number. VII. SUMMARY
Quasielastic light scattering is a very powerful method to study structural features of micelles of both charged and uncharged amphiphatics. A prerequisite for such studies is the preparation of highly purified tensides. For charged micelles the electrostatic interaction between the charged species has to be considered. One then obtains detailed information on the effective charge of the micelles, the aggregation number, and the micelle shape. There is some indication that the measured correlation functions also include information concerning chemical reaction rates of the micelles. REFERENCES 1. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, I., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 (1976). 2. McQueen, D. H., and Hermans, J. J., J. Colloid Interface Sci. 39, 389 (1972). 3. Corti, M., and Degiorgio, V., Chem. Phys. Lett. 53, 237 (1978). 4. Mazer, N. A., Benedek, G. B., and Carey, M. C., J. Phys. Chem. 80, 1075 (1976). 5. Stephen, M. J.,J. Chem. Phys. 55, 3878 (1971). 6. Corti, M., and Degiorgio, V., Chem. Phys. Lett. 49, 141 (1977). 7. Elworthy, P. H., and Mysels, K. J., J. Colloid Interface Sci. 21, 331 (1966). 8. Jakeman, E., in "Photon Correlation and Light Beating Spectroscopy" (H. Z. Cummins and E. R. Pike Eds.), p. 75. Plenum Press, New York, 1974.
DYNAMIC LIGHT SCATTER OF MICELLES 9. Koppel, D. E., J. Chem. Phys. 57, 4814 (1972). 10. Berne, B. J., and Pecora, R., "Dynamic Light Scattering," p. 207. John Wiley and Sons, New York, 1976. 11. Berne, B. J., and Pecora, R., "Dynamic Light Scattering," p. 91. John Wiley, New York, 1976. 12. Brown, J. C., Pusey, P. N., and Dietz, R.,J. Chem. Phys. 62, 1136 (1975). 13. Tanford, C., "The Hydrophobic Effect." John Wiley, New York, 1973. 14. Mukerjee, P., and Mysels, K. J., Nat. Stand. Ref. Dara Ser., Nat. Bur. Stand. (U. S.) 36, 1971. 15. Stigter, D., J. Phys. Chem. 79, 1008 (1975).
505
16. Folger, R., Hoffmann, H., and Ulbricht, W., Bet. Bunsenges. Phys. Chem. 78, 986 (1974). 17. Tartar, H. V., J. Colloid Sci. 14, 115 (1959). 18. Kushner, L. M., and Hubbard, W. D., J. Colloid Sci. 10, 428 (1955). 19. Mysels, K. J., and Princen, L., J. Phys. Chem. 63, 1696 (1959). 20. Stigter, D., Williams, R. J., and Mysels, K. J., J. Phys. Chem. 59, 330 (1955). 21. Stigter, D. ,J. Colloidlnterface Sci. 23, 379 (1967). 22. Stigter, D. ,J. Colloidlnterfaee Sci. 47, 473 (1974). 23. Leibner, J. E., and Jacobus, J., J. Phys. Chem. 81, 130 (1977).
Journal of Colloidand Interface Science. Vol.70, No. 3, July 1979
work we performed experiments in the low concentration limit, that is in the neighborhood of the critical micelle concentration. It is customary to use high ionic strength aqueous solutions in order to minimize longrange Coulomb interaction between the aggregates and the ions of the electrolyte. In order to learn something about this interaction, we performed measurements as a function of the ionic strength o f the aqueous solution. The temporal fluctuation of the scattering intensity is primarily determined by the spatial diffusion of the aggregates. Therefore, the measured light-scattering spectra can be analyzed in terms of transport theories, which give a more detailed insight into such interaction mechanisms than the virial expansion method of phenomenological thermodynamics. For the interpretation of our data we applied the model of Stephen (5). This method allows us to take account of electrostatic interaction and, therefore, to determine the radius of the aggregates at any ionic strength. In addi-
The basic physical parameters characterizing micellar solutions of amphiphatic molecules are: (i) the critical micelle concentration (CMC); (ii) the size, shape, and distribution in size of the micelles; (iii) their electric charge; (iiii) the rate-constants of the dynamic equilibrium between micelles o f different size, as well as between micelles and monomeric units. Quasielastic light scattering is well suited to study micellar systems with respect to the second and the third point. With the same apparatus one can determine the critical micelle concentration. Finally, information concerning the chemical reactions rates, which are usually measured by p-jump or T-jump experiments (1), may be obtained under favorable circumstances by quasielastic light scattering. Recently dynamic light-scattering studies of sodium dodecyl sulfate (SDS) dispersions in aqueous solutions at high tenside concentrations were reported (2-4). In the present 494 0021-9797/79/090494-12502.00/0 Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol, 70, No. 3, July 1979
DYNAMIC LIGHT SCATTER OF MICELLES
tion, this method of evaluation also gives information about the surface charge of the micelles. II. MATERIALS AND METHODS
A. Preparation of highly purified SDS. Sodium dodecyl sulfate (SDS) was recrystallized several times from ethanol. As has been pointed out previously by other authors (6), further purification is necessary in order to remove traces of long chain alcohols. At sufficient high detergent concentrations the alcohols are solubilized in the micelles. However, in the vicinity of the critical micelle concentration the alcohols precipitate in the form of large aggregates of some hundred A diameter, thus disturbing the light-scattering spectra considerably in the CMC region. A foaming procedure, introduced by Elworthy and Mysels (7), was used in order to remove residual alcoholic impurities. The apparatus, used for the foaming purification, is shown in Fig. 1. It consisted of a glass flask F1 of 2 liters volume equipped with several supplies. The flask rested in a temperature-controlled water bath. A magnetic stirring bar was provided for mixing the solvent. To produce foam, the flask was evacuated with a filter pump, while air was introduced through a glass pipe (P1) with an to filter pump
VI~ r ~
.
bypass
, --t)
sc
3
F2
FI
FIG. 1. Apparatus for the purification of the solutions. FI, flask in which the foaming procedure was performed; F2, auxiliary reservoir; PU, peristaltic pump; MF, Millipore filter of 500 A mesh size; SC, scatteriag ceil.
495
opening below the liquid surface. A second glass pipe (P2), adjustable in height, was also attached to the filter pump and was used to draw off the foam. The rates of foam formation and removal were controlled by the valves V1 and V2. The ability to remove and to store solvent in a clean environment was also provided. For that purpose another flask (F2) was connected to the first flask (F1). A closed circuit led from F1 through a peristaltic pump, a Millipore filter, the scattering cell back into flask F1. In order to remove solvent for concentration measurements, an outlet valve (V3) was installed in this closed circuit. The Millipore filter of 500-A mesh size served to remove dust particles. All parts of the apparatus were built from glass or Teflon with the exception of the silicon pipe inside the peristaltic pump. Before use all parts were carefully cleaned. For the purification procedure 1 liter of aqueous SDS solution, at a tenside concentration slightly below the CMC, was placed into flask F1. The solution was foamed for about 20 hr, while there was a continuous flow through the scattering cell. The surface active long chain alcohols, which are the most disturbing impurities, are favorably distributed at the air-water interface of the foam. They are thus preferentially removed with the foam. During the foaming procedure, the solvent was changed several times between the flask F1 and the auxiliary reservoir F2 in order to clean this part of the assembly. When the foaming procedure was finished, the solution was cooled by introducing ice into the water bath. Some SDS precipitated and settled down. The supernatant was transferred into the auxiliary reservoir F2. Then the solution was rewarmed to room temperature. In this way the SDS concentration of the solution could be enriched above the critical micelle concentration up to about 30 mmoles per liter. The final concentration was adjusted b x siphoning back some of the low concentrated solution from the reservoir F2. Journal of Colloid and Interface Science,
Vol. 70, No. 3, July 1979
496
ROHDE AND SACKMANN
At salt concentrations up to 20 mM the SDS concentration was determined by conductivity measurement. At higher salt concentrations the specimens were lyophilized and weighted. The sodium chloride concentration was adjusted by placing all the solution into the reservoir F2, weighing it, and adding the desired amount of salt. Samples with higher SDS concentrations could not be prepared by the freezing procedure. In that case the solution with the highest desired concentration was placed into flask F1 and the solution was pumped through the bypass (cf. Fig. 1). The concentration was changed by adding the appropriate amount of solvent. The success of the purification procedure was tested by measuring the scattered light intensity as a function of the SDS concentration in the region of the CMC. In the case of incomplete purification, the total scattered intensity exhibits a strong peak in the neighborhood of the CMC. This is attributed to the precipitates of long chain alcohols, formed when the SDS-micelles disappear. In contrast, sufficiently purified samples exhibited a step-wise shape of the intensity versus concentration plots. An example is shown in Fig. 2. B. The scattering apparatus. The scattering cell had a quadratic cross-section of 1 cm width. Two openings at opposite sides of the cuvette allowed the solution to be pumped through the cell, as described above. The cell rested in a brass block. The temperature was controlled by flowing water from a thermostat through the block. All measurements were performed at a temperature of 24°C. Light from an argon-ion laser, equipped with an etalon, was focused on the center of the cell. The laser was tuned to a wavelength of 488 nm and the output power was about 200 mW. In order to minimize the intensity of light scattered from the cell walls, all measurements reported here were performed at a scattering angle of 90°. The scattering region was imaged by a Journal of Colloid and Interface Science, Vol. 70, N o . 3, July 1979
=e >,
z
5
I
I
10 IJ5 2i0 25 SDS-CONCENTRATIONx 10 3 (moLe/L)
FIG. 2. Function of scattered light intensity versus SDS concentration. The stepwise shape demonstrates the success of the purification procedure.
lens to an aperture in front of the photocathode of a photomultiplier. A slit with variable width was situated in front of the lens. The slit and the aperture served to define both the opening angle of the scattered light and the probed scattering volume. They were adjusted to give a detection area of about one coherence area. An interference filter was situated just in front of the photocathode in order to remove stray light from the laboratory and fluorescence light from the solution. The output from the photomultiplier (EMI 9863) was amplified and fed into a discriminator, which was connected to a Malvern digital autocorrelator. The correlator was operating in the single clipped mode. It computed the autocorrelation function in 48 channels. The measured spectra were punched on paper tape and analyzed on a Nicolet BNC-12 computer. III. EVALUATION AND INTERPRETATION OF MEASURED CORRELATION FUNCTIONS
A. Analysis of Correlation Function in Terms of Cumulants For homodyne detection, the normalized clipped intensity correlation function g~2)(t) of a Gaussian process with a single correlation time is given by (8): g~2)(t) = 1 + C exp(-2Ft)
[1]
where C is an arbitrary constant and F is the inverse correlation time. If the scattered light arises from particles
DYNAMIC LIGHT SCATTEROF MICELLES of equal size undergoing Brownian motion, F can be expressed as: F = Dq 2 [2] where D is the diffusion coefficient of the particles and q is the scattering vector. In general, many dynamic processes with different correlation times F -1 contribute to the scattered intensity. An example is the diffusion of a polydisperse system of particles. Equation [1] has to be modified into:
g~2)(t) = 1 + C
[i0°G(F) exp(-Ft) 1 dF.
[3]
G(F) is the light intensity scattered from the processes characterized by the correlation time F -~. The distribution G(F) is normalized to one. For sharply peaked functions G(F) the method of cumulants (9) may be applied in order to analyze gCe2)(t) in terms of valuable physical parameters. The measured correlation function can be expanded in a power series in time t. In (g~Z)(t) - 1) = In C - 2['t + Q2(f't) 2 - VaQa(['t) 3 + . . . .
[4]
where f" =
FG(F)dF
Q~ = l/f u I,~ (F - f')~G(F)dF.
[5]
is the average inverse correlation time, Q2 is a measure for the width, and Oa for the asymmetry of the correlation time distribution function G(F). Provided g~=~(t) can be measured with sufficient accuracy, these important quantities may be determined from the light-scattering experiments. For diffusing particles I" and Q2 yield information on both the average size and the size distribution of the particles.
B. Diffusion of Macroions in Electrolytes (10) For dilute solutions of neutral spherical particles, the particle diameter may be esti-
497
mated from the diffusion coefficient via the Stokes-Einstein relation: D = kaT/6~r~ro. If the viscosity 7/is measured by a capillary viscosimeter, the hydrodynamic radius r0 is readily obtained. For charged particles in electrolytes, the situation is much more complicated. Coulomb interaction between the various ionic species must be taken into account. The problem simplifies considerably if we are dealing with solutions containing one type of large charged particles (macroions) and one or more small ionic species. The small ions are called counterions, if they have the opposite polarity as the large ions, and byions, if they possess the same polarity as the large ions. If the concentration of the macroions is sufficiently small, the interaction between them can be neglected because their charges are screened by the counterions. Nevertheless, the diffusion of all ionic species is influenced by long-range Coulomb forces. These forces arise from the fast concentration fluctuations of the small ions, leading to nonspherical charge distributions about the micelles. The resulting dynamic forces lead to a faster random motion of the macroions, while the latter tends to retard the small ions (cf. Ref. 10). Stephen (5) treated this problem quantitatively by solving the Fokker-Planck equations for the large ions, the counterions, and byions in conjunction with the Poisson equation. He obtained the following expression for the diffusion coefficient of the large ions:
D(q) = D I ( I +
q~ q~ + ~ q~
)
[6]
i=2
q~ -
47'/"
%kBT
z~c~
[7]
The index " 1 " characterizes the macroions (micelles), ~0 is the dielectric constant of the solution, zi is the charge, and ct is the number density of species i. D~ is the diffusion coefficient in the absence of electroJournal of CoUoid and Interface Science, Vol. 70, No. 3, July 1979
498
ROHDE AND SACKMANN
static interactions, q is the scattering vector. Equation [6] shows that electrostatic interaction indeed gives rise to an increase in the diffusion coefficient of the macroions. The q~ are not independent variables. They are coupled by the condition of overall electric neutrality. At a given charge of the macroions, the largest increase of the diffusion coefficient is to be expected, if there are no byions present. This shows that the effect of the electrostatic interaction can be diminished by adding small ions of both polarity to the solution which leads to an increase in ~q]. The physical reason for this reduction of the electrostatic effect is the much higher mobility of the small ions. Local concentration variations in the counterion distribution can be compensated very fast by the high mobile byions.
C. Reaction Rates o f Dynamic Equilibrium and Correlation Function Besides the random motion, temporal fluctuations in the scattered light intensity due to chemical reactions may also affect the intensity correlation function. The reaction from a substance A to a substance B is characterized by the reaction rates kl and kz for forward and backward reaction, respectively. The reaction may influence the light-scattering spectrum because the diffusion coefficients and the polarizabilities of the two species may be different. In general the light-scattering correlation function of a system in which the described reaction occurs consists of the sum of two exponential functions. The exponents are functions of the two diffusion coefficients and the two reaction rates. Also the coefficients are complicated functions of the four time-determining parameters and of the polarizabilities and the equilibrium concentrations of the two species (11). McQueen and Hermans (2) treated the problem for the dynamic equilibrium between micelles and the corresponding monomers. Assuming that the reaction occurs Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
by bimolecular steps, and assuming further that only the concentrations of the monomers and the micelles are significant, the following rate equations are derived: 0Cmo
Ot OCmi
Ot
-- DmoV2Cnao - NklcNmo + Nkzcmi
-- DmiV2Cmi + klCSmo - k2cmi.
[8]
Cmo and Cmi are the concentrations of the monomers and the micelles, respectively, Dmo and Dmi are the diffusion coefficients, and N is the aggregation number of the micelles. From the fluctuations in concentration the correlation function of the scattered light intensity is derived. Using the fact that the rate of the fluctuation for micelle formation is of the order of N larger than that for micelle dissociation, the two exponents are Dmiq 2 a n d Dmoq z + N2klC. rnoN-l, w h e r e
Cmo
is the equilibrium concentration of the monomer and q is the scattering vector. The coefficient determining the contribution of the second exponential function is of the order of N smaller than the coefficient of the first exponential function. In the described approximation the reaction between micelles and monomers can hardly contribute to the correlation function of the scattered light intensity. First, the exponential function including the rate constant is much weaker than the function governed by the diffusive motion of the micelles. Second, the exponential function including the rate information always decays faster than the correlation function of the diffusive motion of the monomers. This time domain is out of the region obtainable by today's correlation techniques. IV. ANALYSIS OF EXPERIMENTAL DATA
Correlation functions of each sample were measured in several runs. Each run lasted from 2 to 8 min and was normalized separately. The time domains over which
DYNAMIC LIGHT SCATTER OF MICELLES
~2 x IL
÷ +
I l
,;
L
2'o
L
ZolgS)
FIG. 3. Values of F as a function of time domains, ~'0, of measurement. The different F were obtained for first-order (x), second-order (+), and third-order (O) fits of cumulant expansion (Eq. [4]).
the correlation functions were measured, were varied in several steps between about half the correlation time and about twice the correlation time. The total measuring time per sample was about 2 hr. According to Eq. [4], the measured points representing the functions In (g~2>(t)- 1) were fitted first to a linear function, then to a second-order polynomial, and finally to a third-order polynomial in the time. Values of both the average decay rates [" and the second cumulant Q2 were determined for each of the three orders of fit. This was done for the different time domains. Figure 3 shows a typical series of data determined in that way for the example of the mean decay rate F. Application of a method proposed by Brown et al. (12) yields values of the mean decay rates F with committment neither to one of the three orders of fitting, nor to a certain time domain. Provided the correlation function is measured in a time domain which is small compared with the correlation time, the linear term in Eq. [4] is significant only. Therefore it is possible to determine F by extrapolating the line through the data points of the linear fit to zero probing time. At larger probing times the higher order fits are more accurate, since only then do the higher terms contribute significantly to the correlation function. For most samples the values of F determined by the second-order fit at long probing times agree
499
very well with the decay rates obtained by the described extrapolation. The agreement was worse if the higher cumulants were large, or if the scattering intensity was low. Third-order fits gave no good accordance in most cases, because the accuracy of the data points was not high enough to allow a four parameter fit. The typical variance of the mean decay rate at medium tenside concentration was about 5%. This was tested by comparing several runs of the same sample. The second cumulants Q2 were determined by fitting a second-order polynomial to the measured points (cf. Eq. [4]). The values of Q~ reported here are the averages of the Q2 values obtained from measurements in the different time domains. The variance is not better than about 50%, so that only qualitative statements can be made concerning the width of the distribution of decay rates. V. EXPERIMENTAL RESULTS AND THEIR INTERPRETATION
In Fig. 4 values of the average correlation function decay rates F are plotted as a function of the tenside concentration. Plots are given for different NaC1 concentrations. The right side scale gives the average diffusion coefficients according to Eq. [2]. Figure 5 shows the values of F a n d / ) for a larger concentration range of the surfactant and for pure water as solvent. The bars in Fig. 5 indicate the variance of the experimental values as obtained from different experimental runs taken for the same sample. The interpretation of the data in Fig. 4 in terms of the Stokes-Einstein model would lead to the conclusion that the aggregation number decreases with increasing SDS concentration. This is highly improbable. The different behavior for different salt concentrations strongly suggests that it is the ionic character of the micelles which causes the increase in the diffusion coefficients. In the Journal of Colloid and Interface Science,
Vol. 70, N o . 3, J u l y 1979
500
ROHDE AND SACKMANN
x
x Water * 10 mM N e C t o 20raM NoEl A 50raM No.El " 0.1 M NQE[ v 0.2 M N0.E[
-- 15 x )1._
cJi ,( 2.5 3M ~_, _
/ xx
~×
10
+
.
j
.
~ ° ~ o~ o A. .----t'A~------i--v 5
i
x
~ ~
.
~
~
o
x
.
"
i
-;"
v i
2.0
~
_
~
_ ,
_
1.5
~
~
1.0
t I
I
SDS-CONCENTRATIONx10~ (mote/I)
FK;. 4. Values of the average decay rates of the correlation function, f" (left scale), and of the average diffusion coefficients, /) (right scale), for aqueous SDS dispersions. Values of F and /9 are given as a function of tenside concentration for different salt contents of the aqueous phase. The values of I" were obtained by extrapolating the points of Fig. 3 to ~'0= 0. The curves are calculated from Eq. [9] as described below.
following we give an interpretation in terms of the Stephen model (cf. Section III, B): In order to calculate the concentrations and charges of the different ionic species present in the micelle solution, we proceed as follows (cf. Ref. 13). At SDS concentrations CT below the CMC the number of free dodecyl sulfate (DS-) ions equals the number of sodium ions. Above the CMC the concentration of free DS- ions does not change upon addition of further tenside, since the additional SDS aggregates. The
-_ ¢ 30
5
o,
2o
~=
number density of the micelles is equal to the number of DS- ions aggregated divided by their aggregation number N. The sodium ions are either free or bound within the Stern-layer of the micelles. The ratio of bound Na + ions to DS- ions aggregated is denoted by a. It is assumed to be independent of SDS concentration. The charge of a micelle is equal to the total charge of its DS- molecules minus the total charge of the bound sodium ions. Table I summarizes the concentrations and charges of the different species of the micelle solution. With the definitions of Table I the qi values (cf. Eq. [7]) of the different ionic species are easily calculated. According to Eq. [6] the diffusion coefficient of the micelles is given by D =D1
lo
2'o
io
6'0
s~
16o
SDS- CONCENTRATION,103(mote/I)
FIG. 5. E x t e n s i o n o f the r" v e r s u s SDS concentration plots to higher tenside concentration for the case o f pure water. Journal of Colloid and Interface Science,
Vol. 70, N o . 3, July 1979
1+
2 G + CMC + CT CT - CMC
[91 --
Ot
Upon deriving this equation from Eq. [6] the square of the scattering vector q2 has been neglected because it is always small compared to any of the q~.
501
DYNAMIC LIGHT SCATTER OF MICELLES TABLE I Concentration and Charges of Ionic Species Present in the Micellar Systema Species
Concentration
Charge
Micelles
C~r - CMC N
- N ( 1 - ,~)e
Free DS- ions
CMC
-e
Free Na + ions
CT -ct(Cr
+e
- CMC) Free Na÷ ions from solvent
C1
+e
Free Cl-ions
C~
-e
a N, aggregation number of micelles; Cr, SDS concentration; C1, salt (NaCI) concentration of aqueous solvent; e, elementary charge. All concentrations are measured in terms of number densities.
Equation [9] relates the measured diffusion coefficients with most interesting structural parameters of the micelles, i.e., the CMC, the aggregation number N, and the charge of the micelle. Now, the CMC has been measured either independently, or it has been taken from the literature (14). By the following procedure we succeeded in determining both N and a. Slightly above the critical micelle concentration, ( C T - CMC) is ver), small and therefore the second term in Eq. [9] which takes account of Coulomb interaction, can be neglected. The hydrodynamic radius of the micelles can then be obtained from the diffusion coefficient D~ via the Stokes-Einstein relation. The negligible influence of the electrostatic interaction slightly above the CMC may be explained as follows: The ratio of the number densities of free DS- ions to micelles is large in this concentration region. The free DS- ions act as small, highly mobile byions. They lead to a fast equilibration of the charge fluctuations about the micelle (macroion). Now, our model assumes that at increasing SDS concentration the concentration of free DS- ions remains constant. The ratio of free DS- ions to mi-
celles becomes smaller and the influence of the electrostatic interaction becomes significant. Following this argumentation, we extrapolated the plots of the diffusion coefficient versus SDS concentration to the CMC, and thus obtained the values of D1 for the different salt concentrations. The finding that the measured diffusion coefficient,/), is independent of SDS concentration Ca- at high salt content (cf. Fig. 4) strongly suggests that the aggregation number N does not depend on Cr at least in the concentration region studied (CT < 30 mM). N may then be determined from the hydrodynamic radius. In order to account for the Stern-layer thickness, a value of 4.6/~ is substracted from the hydrodynamic radius. The remaining volume is made up of the tenside hydrocarbon chains each of which has a volume of 350/~3 (15). By assuming a spherical micellar shape, the aggregation numbers given in Fig. 6 were determined. Now, both D1 and N are known. Therefore a may be determined by fitting the c u r v e s / ) versus CT as calculated by Eq. [9] to the measured values of the diffusion coefficients. The values of a obtained by this procedure are given in Fig. 7 as a function of NaC1 concentration. As seen in Fig. 4 good fit between calculated and measured/) versus CT curves is obtained. The agreement is less good for pure water as solvent at higher tenside concentrations. We conclude that another dynamic process besides diffusion contributes
~IOO
o----
I
i
i
01
i
0.2
NoEL-CONCENTRATION (mole/l)
FIG. 6. Aggregation number of SDS micelles as a function of NaCI concentration. Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
502
ROHDE AND SACKMANN
' 0.9
0.8
%
I
011
No.El-CONCENTRATION
'
012
(mote/t)
FIG. 7. Ratio c~ o f the n u m b e r s o f bound N a ions to DS ions aggregated as a function of NaC1 concentration. The charge o f the micelle is given by: N(1 - t~)e.
to the correlation function. This conclusion is suggested by the observed dependence of the second cumulant on the tenside concentration (cf. Fig. 8). For all samples the plots of the second cumulant versus SDS concentration decrease with increasing SDS concentration above the CMC. This behavior could be caused by the appearance of impurity aggregates about the CMC or could also be due to the lower accuracy of the measured data in that low concentration region. Samples with water as a solvent showed in addition a peak in Q~ with a maximum at a concentration of about 50 raM. We assume that the dynamic equilibrium between micelles and monomers is responsible for this finding. The approximation given in Section III, C does not support this suggestion. But we assume that by taking into account the full expression for the correlation function in the presence of chemical reactions, the experimentally observed behavior may be explained. It is obvious that the most prominent contribution to the correlation function comes from the diffusion of the micelles. In order to observe chemical reactions it is necessary that the coefficients determining the exponential functions including the chemical relaxation time are of appropriate magnitude. Furthermore, it is necessary that the inverse relaxation time of the chemical reaction is of the Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
order of 2Dq 2, because we did measurements only in the time domain determined by the diffusive motion of the micelles. Figure 9 shows this function of inverse relaxation time l/r1 versus SDS concentration, as has been measured by pressurejump experiments (16). Also the function 2Dq 2 .is included. The two functions are of equal magnitude in the concentration region where the described incompatibilities with our diffusion model occur. VI. D I S C U S S I O N
A comparison of the aggregation numbers, N, determined in the present work with values from other authors who used static light scattering (17-19) or quasielastic light scattering (2-4) shows that good agreement exists at higher salt concentrations of the solvent. At low ionic strength only results obtained by static light scattering are available for comparison. The aggregation numbers obtained by that method are by about a factor of two larger than the values of N determined by us. But we believe our results to be reliable for the following reasons: (a) the tenside was purified very carefully, (b) our measurements agree very well with the literature at high ionic strengths of the solvent, and (c) in quasielastic light scattering one always measures the lower limit of the diffusion coefficient. Uncertainty may 04 Q2 0.3
0.2
0.1 x
, 210
x
~ 1 l I 0 60 80 100 $D$'EONCENTRATION xl0 3 (mole/I)
FIG. 8. Second c u m u l a n t s Q2 for SDS in pure water (O ©) and in 0~1 M NaCI solution ( x - ×) as a function o f tenside concentration.
DYNAMIC L I G H T SCATTER OF MICELLES
503
of the determined values of ~ is strongly dependent on the applicability of the Stephen theory to our system. To justify the approxi< ~3 q2 mations of the Stephen theory it is necessary that the average electrostatic energy of a micelle and a counterion is smaller than k BT. This condition is fulfilled in our measureSI]S-CON[ENTRATION xlO~(mota/t) ments except in the two measurements with FIC. 9. Twice the decay rate ([') of the correlation the highest tenside concentrations with pure function and the reaction rate (ti-1) of micelle-monowater as solvent. mer equilibrium taken from the literature (16) as funcThe behavior of the measured functions tion of SDS concentration. of aggregation number N and a versus NaC1 concentration may be understood in terms lay in the extrapolation to the CMC, but that of a counterbalance of hydrophobic and cannot fully explain the discrepancy. electrostatic interactions (22): First measurements of micellar self-diffuA micelle is composed of three different sion at low ionic strength are from Stigter regions: et al. (20). Comparison with our results (a) The hydrocarbon core. It is composed show: (a) The reported value of the diffusion of the hydrocarbon chains of the amphicoefficient for SDS-micelles at the CMC is phatic molecules. The chains are in a rather 1.099 × 10-6 cm2/sec, whereas our meas- fluid, disordered state. urements give a value of 1.325 +_ 0.1 x 10-6 (b) The Stern-layer, enclosing the core. cm2/sec. (b) A decrease of diffusion coef- It consists of the hydrated headgroups of ficient with increasing SDS concentration the amphiphatic molecules and of the bound is reported, whereas we observe just the counterions. The outer surface of the Stemopposite behavior. The diffusion coeffi- layer is the shear surface. Its distance from cients were determined by the diffusion of the center of the micelle determines the hymicelles, tagged with dye, through porous drodynamic radius, which is measured by glass disks. The method includes several quasielastic light scattering. hazards. Mysels and Princen (19) pointed (c) The Gouy-Chapman diffusive layer. out that the measurements may be disturbed It contains the unbound counterions and byby hydrolysis catalyzed by the glass sur- ions, and its thickness is determined by the faces. We further assume that it is difficult Debye screening length. to compare diffusion measurements in bulk Amphiphatic molecules aggregate bewater with measurements in capillaries, es- cause the reduction of hydrocarbon-water pecially if charge effects come into play. interface is energetically favored. If the An indication for the reliability of our re- aggregation would be determined by this hysults are the measurements of other authors drophobic interaction alone one would ex(3, 6), which observe the same increase of pect phase separation rather than micelle diffusion coefficient with increasing SDS formation. The size limiting parameter in ionic micelles are the length of the hydroconcentration as we do. The ratio of bound counterions and sur- carbon chains and the Coulomb interaction factant ions in the miceUes, a, is normally between the headgroups. For spherically shaped micelles the maxiassumed to be about 0.5 in pure water as solvent (21). Our measurements give a value mum value of its core is equal to the length of a = 0.81 for pure water, while a in- of the fully extended hydrocarbon chain. creases up to about 0.96 at a salt concen- Therefore the aggregation number of spheritration of 0.2 mole per liter. The reliability cal micelles is limited. A larger aggregation Journal of Colloid and Interface Science, Vol. 70, No. 3, July 1979
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number can be obtained only by changes of the micelle shape. The second size limiting factor is the area per headgroup in the Sternlayer. By increasing the aggregation number of the micelles, this area per headgroup reduces and the Coulomb interaction between the headgroups increases. Due to this counterbalance between hydrophobic and electrostatic interaction micelles exhibit a rather narrow size distribution. The Coulomb interaction between the headgroups can be reduced by introducing more counterions into the Stern-layer. This Can be accomplished for example by raising the ionic strength of the solution. Figure 7 shows that the ratio a of bound counterions to DS- ions increases with increasing salt concentration, as expected by the above considerations. Figure 6 shows an increase in aggregation number with increasing ionic strength up to an NaC1 concentration of about 0.05 mole per liter. N is constant above this limit. In contrast, a increases continuously above 0.05 mole per liter. Our results lead to the suggestion that the micelles do not exhibit a spherical shape in the region of high salt content (>0.05 mole per liter). This follows from the measured hydrodynamic radius of the micelles which has a value of 24.5 A. The corresponding core radius is 19.9 A which is significantly greater than the value of 16.7/~ for the length of a fully extended hydrocarbon chain of SDS. Two nonspherical shapes are usually considered: hemisphere capped cylinders and oblate spheroids (23). The different behavior of a and N above an ionic strength of 0.05 mole per liter strongly favors the existence of the oblate spheroid. Consider first the case of the cylindrical micelle. Addition of NaCI increases a (cf. Fig. 7). Now, the area per tenside molecule would not change if the cylinder grows in length. Therefore a continuous growth with increasing NaCI concentration would be expected which is in contrast to Fig. 6. Consider now the oblate micelle. Each incorporation of a furJournal of Colloid and Interface Science. Vol. 70, No. 3, July 1979
ther molecule would decrease the area per head group. Therefore an increasing amount of counterions with each addition of a surfactant molecule would be necessary for a growth process. A limit in size of an oblate micelle may be reached if no more counterions can be introduced into the Stern-layer. This seems to be the case in our system. The Stern-layer is filled as far as possible with counterions. However, the screening is not good enough to allow the micelles to exhibit a cylindrical shape which would permit a further increase of the aggregation number. VII. SUMMARY
Quasielastic light scattering is a very powerful method to study structural features of micelles of both charged and uncharged amphiphatics. A prerequisite for such studies is the preparation of highly purified tensides. For charged micelles the electrostatic interaction between the charged species has to be considered. One then obtains detailed information on the effective charge of the micelles, the aggregation number, and the micelle shape. There is some indication that the measured correlation functions also include information concerning chemical reaction rates of the micelles. REFERENCES 1. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, I., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 (1976). 2. McQueen, D. H., and Hermans, J. J., J. Colloid Interface Sci. 39, 389 (1972). 3. Corti, M., and Degiorgio, V., Chem. Phys. Lett. 53, 237 (1978). 4. Mazer, N. A., Benedek, G. B., and Carey, M. C., J. Phys. Chem. 80, 1075 (1976). 5. Stephen, M. J.,J. Chem. Phys. 55, 3878 (1971). 6. Corti, M., and Degiorgio, V., Chem. Phys. Lett. 49, 141 (1977). 7. Elworthy, P. H., and Mysels, K. J., J. Colloid Interface Sci. 21, 331 (1966). 8. Jakeman, E., in "Photon Correlation and Light Beating Spectroscopy" (H. Z. Cummins and E. R. Pike Eds.), p. 75. Plenum Press, New York, 1974.
DYNAMIC LIGHT SCATTER OF MICELLES 9. Koppel, D. E., J. Chem. Phys. 57, 4814 (1972). 10. Berne, B. J., and Pecora, R., "Dynamic Light Scattering," p. 207. John Wiley and Sons, New York, 1976. 11. Berne, B. J., and Pecora, R., "Dynamic Light Scattering," p. 91. John Wiley, New York, 1976. 12. Brown, J. C., Pusey, P. N., and Dietz, R.,J. Chem. Phys. 62, 1136 (1975). 13. Tanford, C., "The Hydrophobic Effect." John Wiley, New York, 1973. 14. Mukerjee, P., and Mysels, K. J., Nat. Stand. Ref. Dara Ser., Nat. Bur. Stand. (U. S.) 36, 1971. 15. Stigter, D., J. Phys. Chem. 79, 1008 (1975).
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16. Folger, R., Hoffmann, H., and Ulbricht, W., Bet. Bunsenges. Phys. Chem. 78, 986 (1974). 17. Tartar, H. V., J. Colloid Sci. 14, 115 (1959). 18. Kushner, L. M., and Hubbard, W. D., J. Colloid Sci. 10, 428 (1955). 19. Mysels, K. J., and Princen, L., J. Phys. Chem. 63, 1696 (1959). 20. Stigter, D., Williams, R. J., and Mysels, K. J., J. Phys. Chem. 59, 330 (1955). 21. Stigter, D. ,J. Colloidlnterface Sci. 23, 379 (1967). 22. Stigter, D. ,J. Colloidlnterfaee Sci. 47, 473 (1974). 23. Leibner, J. E., and Jacobus, J., J. Phys. Chem. 81, 130 (1977).
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