J. Phys. Chem. 1983, 87, 1409-1416
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Comparative Study of Intermicellar Interactions Using Dynamic Light Scattering R. B. Dorshowlt C. A. Bunton,* and D. F. Nlcoll’t Department of Physlcs and Department of ChemlStty, Unlverslty of Californla at Santa Barbara, a n t a Barbara, Callfornla 93106 (Received: August 24, 1982; In Flnal Form: November 8, 1982)
Dynamic light scattering measurements are reported for two related surfactant/counterion systems, myristyltrimethylammonium bromide (MyTAB, Cl4HZ9N(CH,),Br)+ NaBr and cetyltrimethylammonium chloride (CTAC1, C16H,,N(CH3),C1) + NaCl. Diffusion coefficients were determined as a function of surfactant and salt concentrations at several temperatures. Using a linear interaction theory and DLVO pair potential, we obtained theoretical fits to D vs. [surfactant] over the region of relatively low salt concentrations where the net intermicellar interaction is repulsive. We estimate the “minimum-sphere”micellar radius, fractional ionization CY,and Hamaker constant A for these systems, comparing them to the values previously determined for cetyltrimethylammonium bromide (CTAB, Cl6H,N(CH3),Br) + NaBr. We relate variations in these parameters to changes in alkyl chain length and counterion physical properties. The behavior of SDS (ClzH2&304Na)+ NaCl is also discussed. According to this theoretical description, a shift of the net intermicellar interaction from repulsive to attractive is a requirement for micellar growth.
Introduction In an earlier publication’ we reported measurements and a theoretical analysis of micellar diffusivities obtained by dynamic light scattering for the system cetyltrimethylammonium bromide (CTAB) + NaBr. At sufficiently low salt concentrations (Le., [NaBr] 5 0.08 M at 40 “C), the diffusion coefficient, D, increases approximately linearly with increasing surfactant concentration, [ CTAB], a t constant [NaBr]. A plot of D vs. [CTAB] for low [NaBr] yields a “fan” of positive-slope lines which converge to a common diffusivity Do in the limit of zero micellar volume fraction (i.e., where [CTAB] equals the critical micellar concentration, or cmc). The slopes decrease with increasing [NaBr]. By contrast, a t higher salt content D decreases markedly with increasing [CTAB], and a t high [NaBr] the D vs. [CTAB] behavior is nonlinear. A representative summary of these results for CTAB + NaBr at 25 OC is shown in Figure 1. In the earlier study, we concluded that in dilute NaBr the solution contains essentially “minimum-sphere” micelles,2whose size (above the cmc) is approximately independent of both surfactant and salt concentrations at a given temperature. Hence, Do corresponds to the diffusion coefficient of these minimum-sphere micelles, whose hydrodynamic radius Rh is given by the familiar StokesEinstein expression3 (valid in the limit of infinite dilution)
Rh = kT/(G*qDo)
(1)
where k is Boltzmann’s constant, T the temperature (Kelvin), and q the shear viscosity of the solvent. The increase in D above Do with increasing surfactant concentration and/or decreasing salt concentration is therefore a consequence of changes in intermicellar interactions rather than in the micellar hydrodynamic radius Rk Because D increases with increasing [CTAB] at low [NaBr], the interaction is predominantly repulsive in this low-salt region. We obtained theoretical fits to our measured diffusivities using a variation of the procedure recently developed by Corti and Degiorgio4in their analysis of the system sodium dodecyl sulfate (SDS) + NaC1. This approach assumes a linear interaction theory, in which D depends on the Department of Physics. *Department of Chemistry. 0022-365418312087-1409$01.50/0
volume fraction q5 of the diffusing particles (in the limit of small 4) according to D = DOLI+ (Kt + Kh)6] (2) Perturbation coefficients Kt and Kh are due, respectively, to thermodynamic and hydrodynamic effects. Coefficient Kt is proportional to the well-known second osmotic virial ~oefficient.~Coefficient Kh, due to hydrodynamic perturbations of the friction factor in the generalized Stokes-Einstein formula, was evaluated independently by Batchelor6 and Felderhof.’ These coefficients can be conveniently expressed as integrals which involve the pair interaction potential energy. It is convenient to separate explicitly the contributions due to hard-sphere repulsion (which we refer to as KP and KF) from all other possible terms in the interaction potential energy V ( X )where , x is the normalized separation between two spherical diffusing particles (see ref 1)
where
KY = 8 G,(x) = 24 (1
+x
) ~
(3c)
KP = -6.44 (Felderhof) (3d) 15 27 G h ( X ) = -12 (1 + X ) + -(I + X ) - 2 - 64‘1 + X)-4 8 75 -(1 + x ) - ~ (Felderhof) (3e) 64 In the dilute-salt region, we modeled CTAB micelles as idealized, charged spherical particles of hydrodynamic radius a (=Rhof eq 1) and mean charge Q = aN, where CY is the effective micellar fractional ionization and N is the mean micellar aggregation number. The micelles (1) Dorshow, R.; Briggs, J.;Bunton, C. A.; Nicoli, D. F. J.Phys. Chem. 1982,86, 2388. (2) “Minimum sphere” is a convenient term applied to micelles at surfactant concentrations not greatly above the cmc. (3) Chu, B. “Laser Light Scattering”; Academic Press: New York, 1974. (4) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (5) Hill, T. L. “An Introduction to Statistical Thermodynamics“;Addison-Wesley: Reading, MA, 1960. (6) Batchelor, G. K. J. Fluid Mech. 1976, 74, 1. (7) Felderhof, B. U. J. Phys. A: Math. Gen. 1978, 11, 929.
0 1983 American Chemical Society
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The Journal of Physical Chemistty, Vol. 87, No. 8, 1983
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Dorshow et ai.
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(M)
Flgure 1. Diffusion coefficient D (cm2/s) vs. CTAB concentration (M) at 25 "C for a range of NaBr concentrations. The solid curves are our best theoretical fits (see text); the dashed curves are meant as guides for the eye only.
diffuse in an electrolyte characterized by the DebyeHuckel inverse screening length K . Intermicellar interactions consist of two opposing forces, repulsive and attractive: which depend on the charge Q and the Hamaker coefficient A, respectively. In our earlier paper we obtained theoretical fits to the measured micellar diffusivities using the above theory, treating the physical variables a and A as adjustable parameters. From best fits to the data we obtained relatively unambiguous estimates of the fractional ionization a and the attractive Hamaker constant A , as well as the minimum-sphere micellar radius a. Our theoretical fitting procedure differs from that of Corti and Degiorgio4in one essential respect. We assume that intermicellar interactions, both repulsive and attractive, dominate the behavior of micellar diffusivities only over the region of dilute salt, where D increases linearly with surfactant concentration (and with a common extrapolation point, Do, at the cmc) and not at higher ionic strengths, where D generally decreases monotonically (but not necessarily linearly) with increasing surfactant concentration. Corti and Degiorgio, on the other hand, applied linear interaction theory to the system SDS + NaCl over a large range of NaCl concentrations (0.1-0.6 M), where there is both positive- and negative-slope D vs. [surfactant] behavior. By this approach they assumed that decreases in D with increasing [SDS], observed for [NaCl] > 0.4 M (at 25 "C), are the consequence of a shift in the net intermicellar interaction from repulsive to attractive. Using this interaction model, Corti and Degiorgio concluded that there is little micellar growth even in high salt, and, at 25 "C, Rh increased from 25.3 A in 0.1 M NaCl to 27.9 8, in 0.6 M NaCl. This conclusion directly contradicts that of Mazer et aL9 and Missel et al.,l0 who assumed that intermicellar interactions are negligible in the high-salt region and that decreases in D below Doreflect substantial growth of SDS micelles (Rh 2 50 A). (Neglecting interactions, the above authors obtained the mean micellar hydrodynamic radius directly from D using eq 1.) Our interpretation of the role of intermicellar interactions in the physical interpretation of micellar diffusivities also differs quantitatively from that of Corti and Degiorgio. ~~
~
(8) Rosen, M. J. "Surfactants and Interfacial Phenomena": Wilev:
New York, 1978. (9) Mazer, N. A.; Carey, M. C.; Benedek, G. B. In 'Micellization, Solubilization and Microemulsions": Mittal. K. L... Ed.:. Plenum Press: New York, 1977; Vol. 1, p 359. (10)Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y. J. Phys. Chem. 1980,84, 1044.
They estimated the parameters for the Coulombic term of the intermicellar interaction potential from diffusivity (and scattered intensity) data in low salt (the positive-slope region) and for the van der Waals term from data in high salt (the negative-slope region). As a result, their best-fit theoretical coefficients (Kt + Kh) agree reasonably well with the corresponding experimental values at the lowest values of [NaCl] (i.e., -0.1 M); however, these theoretical slopes become increasingly too small with increasing [NaCl] in both the positive- and negative-slope regions. By contrast, we found that, by confining the interaction model to the positive-slope (i.e., net-repulsive) region and including both the repulsive and attractive terms in the potential, we obtained quantitatively superior fits to the data over a wide range of temperatures and salt and surfactant concentrations for a variety of systems, including SDS + NaCl. These results suggest to us that the interaction model is valid only in dilute salt, where it is generally agreed that there is, at most, only modest micellar growth with increasing salt or surfactant concentration. By the procedure outlined above (explained in greater detail in ref l),we can characterize the intermicellar interactions which influence micellar diffusivities in dilute salt and establish the range of salt concentration for a given surfactant + salt system where the interaction approach is valid. Hence, we have established the usefulness of a simple, self-consistent technique for determination of the micellar fractional ionization a (i.e., 1 - p) when there is negligible micellar growth. It therefore becomes particularly interesting to establish the dependence of a , A , and a on the amphiphile structure and its counterion type. In so doing, we hope to improve our understanding of the physical forces responsible for micellization. In this paper we report light scattering measurements and interaction analysis for two systems containing cationic surfactants which are closely related to CTAB. The first of these, myristyltrimethylammonium bromide (MyTAB, C14H2,N(CH3),Br)+ NaBr, differs from CTAB + NaBr only in the length of the n-alkyl group (C14HZg instead of CI6H3J. The second, cetyltrimethylammonium chloride (CTAC1, C16H,3N(CH,)3C1)+ NaC1, differs only in the counterion. In addition, we show our theoretical analysis of the diffusivity data of Corti and Degiorgio for SDS + NaC1. Systematic variations in a, a , and A with surfactant/counterion composition are discussed. Materials and Methods MyTAB and CTACl were purified as previously reported.ll Sodium bromide and chloride (Mallinkrodt, AFt) were dried before use. The sample preparation and light scattering measurements were described in ref 1. Intensity-weighted mean diffusion coefficients were obtained from second- and third-order cumulant fits to 48-channel, 4-bit correlation functions with a long-delay measured base line (Nicomp Model 6864 computing autocorrelator). Results Our dynamic light scattering measurements of the mean micellar diffusion coefficients D (cm2/s)are summarized in Figure 2 for the system MyTAB NaBr and in Figure 3 for CTACl + NaCl in the range 0.01 5 [surfactant] 5 0.05 M and varied [salt]. The three parts in each figure correspond to the temperatures 40, 25, and 15 "C, respectively. For MyTAB + NaBr, the added [salt] ranges from 0.02 to 0.5 M, and for CTACl + NaC1, from 0.02 to 3 M. It is important to recognize that experimental mi-
+
(11) Bunton, C. A.; Romsted, L. S.; Thamavit, C. J . Am. Chem. SOC. 1980, 102, 3900.
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983 1411
Intermicellar Interactions
,
30-
1
T
I
30
' A '
40-C
T
~
'A
.
40'C
/i
i
I
, .-___.________.________---. -
,
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. 01
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i
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03
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'B
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=
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*\
- - - _ _ - - - ----.-? - - - - - ___.__ ..--__.__ _ _ - - _.- - - - - - - - -.--a_--
1
1
ob
01
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IMl
02
03
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(MI
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05
15 T = 15°C
-:5
t
0
0
01
[MYTAB]
04
05
-.--._______.________-*-201
0
(MI
Flgure 2. (A) Diffusion coefficient D (cm2/s) vs. MyTAB concentration (M) at 40 OC for a range of NaBr concentrations. The solid curves are our best theoretical fits (see text); the dashed curves are meant as guides for the eye only. The diffusivities have been corrected for increases in solvent viscosity due to the addition of NaBr. (B) Same as part A, but temperature = 25 OC. (C) Same as part A, but temperature = 15 OC.
cellar diffusivities are systematically lower in the concentrated salt solutions than in pure water because salt increases the solvent viscosity. Hence, the diffusivities plotted in Figures 2 and 3 have been normalized to their "water-equivalent" values by multiplying the observed diffusivities by ~ ( 2 OC)/~7~(20 0 "C),where ~ ( 2 OC) 0 is the viscosity of the appropriate salt water solution12at 20 "C and ~ ~ ( "C) 2 0 is the corresponding viscosity of pure water. The viscosity corrections at temperatures other than 20 "C are given, to first approximation, by the same ratio above. Variations in the corrected diffusivities with
+
(12) Weast, R. C., Ed. "Handbook of Chemistry and Physics";Chemical Rubber Publishing Co.: Cleveland, OH, 1971.
02
03
[CTACll
(Ml
04
05
Figure 3. (A) Diffusion coefficient D (cm2/s) vs. CTACI concentration (M) at 40 OC for a range of NaCl concentrations. The soli curves are our best theoretical fits (see text); the dashed curves are meant as guides for the eye only. The diffusivities have been corrected for increases in solvent viscosity due to the addition of NaCi. (B) Same as part A, but temperature = 25 OC. (C) Same as part A, but temperature = 15 OC.
[salt] are then caused by changes either in the micellar hydrodynamic radius, a, or in intermicellar interactions, and not by changes in the solvent viscosity. For the most part, these solvent viscosity corrections were important only for CTACl NaCl in the positive-slope region. Both MyTAB + NaBr and CTACl NaCl show qualitative behavior similar to that observed in CTAB + NaBr.' That is, the D vs. [surfactant] behavior can be roughly divided into three regions, corresponding to low, moderate, and high salt concentrations. The salt concentrations which control these regions vary greatly with the surfactant/counterion system. First, there is always a low range of salt concentration over which D vs. [surfactant] consists of a fan of approximately straight lines of positive slope which converge to a common intercept
+
+
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The Journal of Physical Chemistry, Vol. 87, NO. 8, 1983
Dorshow et al.
Do in the limit of vanishing [surfactant] (Le., at the cmc). This is the region which we and others have interpreted as consisting of essentially minimum-sphere micelles of fixed hydrodynamic radius (given by eq and which we have analyzed using the theory summarized in eq 2 and 3a-e. Our best theoretical fits, shown as solid curves in Figures 2 and 3, and their quantitative implications will be discussed shortly. At higher salt concentrations, D in general decreases with increasing [surfactant]. (In both sets of figures these data points are connected by smooth, dashed curves to aid the eye.) This D vs. [surfactant] behavior can be roughly divided into two regions of added salt concentration. First, at intermediate salt concentrations D decreases approximately linearly with increasing [surfactant] over the entire measured range (0.01-0.05 M). Second, in relatively concentrated salt D falls nonlinearly (often precipitously) with increasing [surfactant]. Furthermore, in sufficiently concentrated salt D initially falls and then increases somewhat with added surfactant.13 This nonmonotonic behavior may be a consequence of strong intermicellar interactions which can be expected to occur if, for example, the micelles grow as rods with lengths which become comparable to the mean intermicellar ~eparation.’~-’~ An important qualitative result immediately emerges from the diffusivity plots of Figures 2 and 3. The salt concentration which is required to produce a net balance between intermicellar attractions and repulsions (resulting in a slope of zero for D vs. [surfactant]) varies with both the surfactant chain length and the counterion type. This “zero net interaction” ([salt]v=o) concentration is also moderately temperature dependent, increasing with increasing T. The approximate value of [ ~ a l t ]at ~= 40~“C is ~ 0 . 0 8M for CTAB + NaBr, r0.15 M for MyTAB + NaBr, and g0.5 M for CTACl + NaC1. As will be seen shortly, this balancing salt concentration varies with the surfactant/counterion system due to changes in both the micellar fractional ionization, a,and the Hamaker constant, A. A detailed description of the theoretical fitting procedure is given in our earlier paper.’ The pair potential energy V(x) between idealized charged spherical micelles is taken from DLVO theory.16 The attractive term V,(x) is given by the London-van der Waals expression for two dielectric spheres (eq 9 in ref 1); its strength is determined by the Hamaker coefficient A and the micellar radius a . The repulsive potential energy VR(X)is obtained from the solution to the double-layer problem for spherical particles as originally calculated by Verwey and Overbeek.l6 The algebraic form appropriate for VR(x) depends on the size of the product K U , where K is the well-known Debye-Huckel inverse screening length l)499J0
K
= [8~1e~/(&T)]”~
(4)
Here, I is the solution ionic strength (ions/cm3), e the electrostatic charge, and E the T-dependent solvent dielectric constant. In the positive-slope regions of both MyTAB + NaBr and CTACl NaC1, the repulsive interaction is best described by using the KU < 1 expression
+
34
t 22
\
L10 L I 2 d - J 30
20
40
50
60
T (“C)
Flgure 4. Micellar hydrodynamic radius R , (A) vs. temperature T (“C) for the surfactant systems investigatedthus far: closed circles, CTAB
+
+
+
Na& open circles, CTACl NaCI; closed triangles, MyTAB NaBr; open triangles, SDS NaCl (from data of Corti and Degiorgio‘). The R , values were determined from D o (eq 1).
+
+
(eq 7, ref 1) for VR(x), as was the case for CTAB NaBr. (See Figure 1, ref 1, and the related discussion.) The remaining parameters which determine VR(x) are the micellar spherical radius, a , and the total charge, Q. However, the former is not a free parameter in the fitting procedure. Its value is determined with relatively little uncertainty (wk0.5 A) from Do (eq l),which is obtained by extrapolating the plots of D vs. [surfactant] to their common intercept at the cmc (indistinguishable from [surfactant] = 0 on the scale of the figures). It is important to appreciate that the minimum-sphere micellar radius, a, emerges from the D vs. [surfactant] plots independently of the details of the linear interaction fitting procedure. As for CTAB + NaBr, the micellar hydrodynamicradius for the two new systems incrcases upon cooling in the range 15-40 “C. In Figure 4 we compare the T-dependent micellar radius for MyTAB NaBr (closed triangles) and CTACl + NaCl (open circles) with the previous findings for CTAB + NaBr (closed circles). (The open triangles represent our estimates of the micellar radius for SDS + NaCl using the data of Corti and Degi~rgio,~ to be discussed later.) In dilute salt, where the behavior of D is consistent with the assumption of negligible micellar growth as a function of [salt] or [surfactant], there is a modest increase in mean micellar hydrodynamic radius upon cooling. It is instructive to compare a(T) for MyTAB + NaBr and CTAB + NaBr, where the only difference is the alkyl group chain length. If one naively scales the radius for the former system by 16/14, the ratio of the extended alkyl group chain lengths for CTAB (c16)and MyTAB (C14),one arrives at radii which are within 1-2 A of the values previously determined for CTAB + NaBr. If one considers the experimental error brackets (indicated in Figure 4) associated with the determination of Do and the naivete of the model, the agreement is reasonable. We see from Figure 4 that the corresponding radii for micelles of CTAC NaCl lie between those of the above two systems. While the radii are larger than those of MyTAB + NaBr, as expected, further detailed predictions become difficult because different counterions may significantly affect both the Stern layer thickness and the intrinsicmicellar packing/size due to a difference in fractional charge (to be discussed below). The remaining unknown which determines the strength of VR(x)is the micellar charge Q, which can be expressed
+
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(13) Nicoli, D. F.;Ciccolello, R.; Briggs, J.; Dawson, D. T.; Offen, H. W.; Romsted, L. S.;Bunton, C. A. In “Scattering Techniques Applied to Supramolecular and Non-Equilibrium Systems”; Chen, S. H., Chu, B., Nossal, R., Eds.; Plenum Press: New York, 1981; p 363. (14) Kalus, J.; Hoffman, H.; Reizlein, K.; Ulbricht, W.; Ibel, K. Ber. Bunsenges. Phys. Chem. 1982,86,31. (15) Hoffman, H.; Rehage, H.; Platz, G.; Schorr, W.; Thurn, H.; U1bricht, W. Colloid Polym. Sci. 1982,260, 1042. (16)Verwey, E. J. W.; Overbeek, J. T. G. “Theory of the Stability of Lyophobic Colloids”; Elsevier: New York, 1948.
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Intermicellar Interactions
as Q = aN, where N is the mean micellar aggregation number. As discussed earlier,' N (at a given temperature) is assumed to be approximately constant in the positiveslope, net-repulsive region, leaving a as our adjustable parameter. This assumption is consistent with the observation in our systems of linear fans of D vs. [surfactant] which converge to a single diffusivity value Do at the cmc. Furthermore, recent st~diesl'-'~of the dependence of N on [NaCl] for the system SDS NaCl indicate that N increases little with increasing [NaCl] in the range 0.1 I [NaCl] < 0.45 M, which corresponds to the positive-slope D vs. [SDS] region. (These studies indicate a significant increase in N for [NaCl] > 0.45 M; this demarcation is consistent with the observations of Mazer et al? and Missel et al.1° using dynamic light scattering.) For MyTAB NaBr we chose a value of N consistent with our choice for CTAB NaBr (ref 1). That is, by analogy with the literature values20for SDS NaC1, we assumed that N at low ionic strengths is approximately 50% larger than the value at zero salt concentration. Recently, Lianos and Zana2' (from the fluorescence decay of pyrene), Dorrance and Hunter22(from pyrene excimer formation), and Jones and Reed23 (using classical light scattering) found values of N for MyTAB in the absence of salt in the range 60 5 N ,< 70. Jones and Reed also report N r 92 for [NaBr] = 0.05 M. Hence, we have chosen N = 90 for our fits for MyTAB + NaBr. In the case of CTACl NaC1, we were unable to find reliable measurements of N in NaCl solution. Consequently, we estimated N in the positive-slope region by assuming that CTACl micelles have roughly the same surface area per head group as CTAB micelles, for which we previously estimated N = 120. Since their respective hydrodynamic radii at 40 "C (Figure 4) are 27.0 and 29.2 A, we arrive at N = (120)(27/29.2)2= 103. Hence, we adopted N = 100 for our fits for CTACl + NaC1. (As a check, we verified that application of this procedure to MyTAB + NaBr yields N = (120)(24.7/29.2)2= 86, in good agreement with our choice, N = 90.) Although these choices for N are only approximate, we previously showed that the theoretical fits to the D vs. [surfactant] data are relatively insensitive to the choice of N. This is because the net interaction coefficient, Kt Kh (eq 3a-e), turns out to be nearly linear in N (over a physically plausible range), while the micellar volume fraction rp (eq 2) is inversely proportional to N , resulting in almost complete cancellation of the N dependence. The fitting procedure (see ref 1) consists of adjusting a and A at a particular temperature (typically 40 "C) to produce the best theoretical agreement with the D vs. [surfactant] data over the positive-slope region. One then calculates the resulting theoretical curves for the remaining temperatures, keeping a, A , and N constant. However, at each temperature a different micellar radius, a, is used (eq l),consistent with Do, as summarized in Figure 4. While the quality of the resulting fits is essentially independent of N , as discussed above, the best-fit value for the Hamaker constant, A , depends on the choice of N . The resulting theoretical fits in the positive-slope region are shown in Figures 2 and 3 (solid curves). The slight
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1413
TABLE I: Parameters of Surfactant t Salt System surfactant t salt
a
CTAB + NaBr MyTAB t NaBr CTACl + NaCl SDS + NaCl
0.22" 0.22 0.27 0.36b
AC:N
[saltIv=,, M
15":120 0.05-0.10 7:90 0.15-0.20 2:lOO 0.5-1.0 4':95 0.40-0.45 O1 Best-fit parameters from ref 1. Best-fit parameters using data of ref 4. Hamaker constant in units of hT (40 "C).
nonlinearity (downward curvature) of the curves at lowest salt and highest surfactant concentration occurs because the ionic strength I in the expression for K (eq 4) contains a second-order contribution due to the surfactant: I = [salt] + '/2a[surfactant]. The relative contribution of the latter term becomes negligible except in the most dilute salt. We obtained good agreement of the theoretical D vs. [surfactant] fan with the experimental data over the entire repulsive region for both systems investigated. The fact that the fits are equally good at all three temperatures for fixed parameters a, A , and N is particularly encouraging, given the substantial T dependence of the slopes for each salt concentration and the large range of salt concentrations spanned (especially for CTACl + NaC1). Obviously, the temperature dependence of Kt + Kh is nontrivial, given the form of eq 3a-e. The best-fit values of a and A for the two new systems, together with those for CTAB + NaBr, are shown in Table I. The assumed aggregation number N and the observed crossover salt concentration, [salt]v=o, are also given for the three systems. The Hamaker constant, A , is given in units of kT (40 "C). Interestingly, for CTAB + NaBr and MyTAB + NaBr we obtain the same estimate for a (0.22), but values for A which differ by a factor of 2. For the third related system, CTACl NaC1, we find that a is 20% higher (0.27) and A is much smaller than the corresponding parameters for the other two systems. A detailed discussion of these results is reserved for the next section. Because of the quality of these theoretical fits, we applied the same fitting procedure to the recently published diffusivity data of Corti and Degiorgio4for SDS + NaCl (confining our analysis to the positive-slope region). The resulting best theoretical fits to their data are shown in Figure 5. The fits are, again, quite good, considering the relatively large experimental scatter evident in the data (particularly for [NaCl] = 0.1 M at 40 "C). (It should be noted that the range of surfactant concentrations covered by their data is twice that which we have generally used.) The resulting best-fit parameters are listed in Table I. As is evident from Figure 5, we obtain better fits to the diffusivity data than those originally reported by Corti and Degiorgio, primarily because we have confined our interaction analysis to the net-repulsive, positive-slope region. Interestingly, the estimated a (0.36) for SDS + NaCl is considerably larger than the values found for our three cationic micellar systems; SDS micelles appear to be relatively highly charged at low salt concentrations. Finally, we recently reported24on a "universal" behavior observed for the diffusivities of several micellar systems, including CTAB NaBr and MyTAB + NaBr. For each system the entire family of positive-slope D vs. [surfactant] curves can, to first approximation, be compressed into a single linear plot of D vs. [surfactant]/[salt]. Since the micellar volume fraction 4 (eq 2) is proportional to [sur-
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(17)Hayashi, S.;Ikeda, S. J. Phys. Chem. 1980,84,744. (18)Lianos, P . ; Zana, R. J.Phys. Chem. 1980,84,3339. (19)Zachariasse, K.A.;Phuc, N. V.; Kozankiewicz,B. J.Phys. Chem. 1981,85,2676. (20)Kratohvil, J. P.J. Colloid Interface Sci. 1980,75, 271. (21)Llanos, P.; Zana, R. J. Colloid Interface Sci. 1981,84, 100. (22)Dorrance, R.C.;Hunter, T. F. J.Chem. Soc., Faraday Trans. 1 1974,70,1572. (23)Jones, M.N.; Reed, D. A. Kolloid Z . Z . Polym. 1969,235,1196.
(24)Briggs, J.; Dorshow, R. B.; Bunton, C. A.; Nicoli, D. F. J. Chem. Phys. 1982,76, 775.
Dorshow et ai. I
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Flgure 5. (A) Diffusion coefficient D (cm2/s) vs. SDS concentration (M) at 40 "C for a range of NaCl concentrations. The solii,curvesare our best theoretical fits (seetext). The d W i have been corrected for increases in solvent viscosity due to the addition of NaCI. (B) Same as part A, but temperature = 25 "C.
factant], this observed dependence is consistent with linear interaction theory (eq 1)provided that K (=K, + Kh) is approximately proportional to [salt]-'. Using eq 3a-e and the DLVO potential, we have calculated K vs. [salt]-l for spherical micelles of radius 30 A at 40 "C over a range of total charge, 10 I Q I40. (Assuming N = 100, this corresponds to the physical range 0.1 I CY 5 0.4, which encompasses the values in Table I.) In Figure 6A we show the resulting predicted K vs. [salt]-' curves, assuming A = 0 (i.e., no attractive interactions). For a given value of Q, the dependence does not greatly deviate from linearity. In Figure 6B we show the corresponding predictions assuming A = lOkT (40 "C), a Hamaker value intermediate to those found for CTAB + NaBr and MyTAB + NaBr. The resulting plots follow even more closely a linear dependence in [salt]-'. Given the analytical complexity of VR(x)and VA(x)and of the integral representations of Kt and Kh, it is perhaps surprising that there should be such a simple relationship between K and [salt].
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'j
+
Figure 6. (A) Theoretical interaction coefficient K(=K, K,,eq 3a-e) vs. [salt]-' (M-') for a range of micellar charge Q and Hamaker constant A = 0 (i.e., no attractions). (Micellar radius a = 30 A and temperature = 40 "C.) (B) Same as part A, but Hamaker constant A = lOkT(40 "C).
Only for the simplifying case V(x) Rcl-,we would (33) Ketelaar, J. A. A. ‘Chemical Constitution”;Elsevier: New York, 1958; p 102. (34) Anacker, E. W. In “Solution Chemistry of Surfactants”; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. 1,p 247. (35) Visser, J. In “Surface and Colloid Science”; Matijevic, E., Ed.; Wiley: New York, 1976; Vol. 8, p 3. (36) Visser, J. Adu. Colloid Interface Sci. 1972,3, 331. (37) Ingold, C. K. “Structure and Mechanism in Organic Chemistry”; Cornel1 University Press: Ithaca, NY, 1969; p 142. (38) Bauer, N.; Fajans, K.; Lewin, S. Z. In ‘Technique of Organic Chemistry”; Interscience: New York, 1960; Vol. 1, part 11, p 1182. (39) Ingold, C. K., ref 37, p 146.
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The Journal of Physical Chemistty, Vol. 87, No. 8, 1983
expect "A > AmACI, all other contributions being equal. This is the case (Table 11). The stability of a colloidal dispersion, defined as the ratio of the rate constants for coalescence of particles in the absence and presence of an energy barrier, is related8 to the maximum V,, in the interaction potential, V , VR. (See Figure 1, ref 4,for a typical plot of V ( x )vs. x . ) As discussed earlier,l for CTAB + NaBr V - J k T ranges from 3 (15 "C) to 9 (55 "C) for [NaBr] = 0.02 M. Given these relatively large values, the micelles are highly stable against collisions. That is, intermicellar repulsions are so great at this low salt concentration that it is statistically improbable that two micelles can come close enough to each other to overcome the barrier V,, and thereby permit the short-range van der Waals attraction to take over, bringing them into essential contact. However, as the salt concentration increases, V,, decreases. For the CTAB system when [NaBr] = [NaBr]v=o= 0.08 M (40 "C), V,,/kT drops to 11.5. Comparable values are obtained for the MyTAB and CTACl systems. For MyTAB NaBr, V,,,/kT ranges from 5 (15 "C) to 8 (40 "C) at [NaBr] = 0.02 M and becomes approximately 0.5 at [NaBrIvzo. For CTACl + NaC1, V,,/kT varies between 12 (15 "C) and 16 (40 "C) at [NaCl] = 0.02 M and is very close to zero at [NaC1]v=o. We think it highly significant that, in the several systems which we have examined thus far, we have never observed evidence of substantial micellar growth until the slope of D vs. [surfactant] becomes negative-i.e., until [salt] > [ ~ a l t ] ~ That = ~ . is, all positive-slope D vs. [surfactant] curves extrapolated at the cmc to the same value, Do, and linear interaction theory explains the data on the assumption of a basically unchanging micellar size. On the other hand, when [salt] > [ ~ a l t ] ~ the = decrease ~, in D with increasing surfactant and/or salt concentration seems to be consistent with substantial micellar growth, as described by Missel et al.'O and others.42,43 It therefore appears that the salt concentration which marks the approximate beginning of substantial micellar growth coincides with the concentration at which the micelles are able to overcome the repulsive barrier and begin to collide with significant frequency. The above arguments are consistent with the qualitative trends related to counterion type recently reported by Missel et al.44 These authors performed dynamic light scattering measurements on dodecyl sulfate micelles in solutions of alkali chlorides in the high-salt region, where rodlike growth is postulated and electrostatic repulsions
+
+
(40)Ketelaar, J. A. A., ref 33, p 91. (41)Gordon, A.; Ford, R. "The Chemist's Companion: A Handbook of Practical Data, Techniques and References": Wiley: New York, 1972: p 154. (42) Ikeda, S.;Hayashi, S.; Imae, T. J. Phys. Chem. 1981, 85, 106. (43)Porte, G.;Appell, J. J. Phys. Chem. 1981, 85, 2511. (44)Missel, P. J.; Mazer, N. A.; Carey, M. C.; Benedek, G. B. In "Solution Behavior of Surfactants"; Mittal, K. L., Fendler, E. J., Ed.; Plenum Press: New York, 1982;Vol. 1, p 373.
are assumed to be negligible. They noted that micellar growth for the system XDS + XC1 depends on the alkali metal cation, X+, in the following decreasing order: Cs+ > K+ > Na+ > Li+.& Because Li+ has the largest hydrated radius of the series (as noted by Missel et al.), we would argue that micelles of LiDS + LiCl would possess the largest CY in the low-salt, net-repulsive region. Hence, a relatively large concentration of added salt would be required to screen the intermicellar repulsions (i.e., so that VR = VA),thus permitting micellar growth. However, given the relatively small hydrated radius of Cs+, and the correspondingly small value expected for a,the system CsDS + CsCl should require the least amount of added salt to reach V, r VA,where micellar growth can commence. The contribution of the counterion polarizability to the Hamaker constant for each of these systems should follow the trend in molar refractivities (R) r e p ~ r t e dfor ~ ~these ,~~ cations: RC8+ > RK+> RNa+> RLi+.Hence, we expect A to decrease in the same order; Le., the system CsDS + CsCl should possess the largest VA, requiring the least amount of salt to obtain V , V,. Therefore, considerations of the effects of counterion type on both the repulsive and attractive terms in the intermicellar potential are consistent with the observations of Missel et al.44on relative micellar growth for these related systems. In conclusion, the balance between intermicellar repulsions and attractions seems to be intimately related to the onset of micellar growth. For all of the systems that we have investigated, there is a salt concentration large enough to cause the net intermicellar interaction to shift from repulsive to attractive, at which point the diffusivity typically decreases with increasing salt and surfactant concentrations; these concentration effects have been ascribed to micellar growth by a number of authors. This "critical" concentration varies with the surfactant/counterion system because of differences in the fractional ionization CY and the Hamaker constant A. This description, based on the existence of intermicellar interactions, must ultimately be compatible with equilibrium models of micellar growth at high salt concentrations. These models predict the distribution of micellar sizes as a consequence of the free energy cost of incorporating an additional surfactant monomer into a micelle of a given aggregation number and do not explicitly take into account the presence of neighboring micelles.
Acknowledgment. We gratefully acknowledge support by the National Science Foundation (Chemical Dynamics Program) and communications with H. Hoffmann and co-workers. Registry No. Trimethyltetradecylammonium bromide, 1119-97-7; cetyltrimethylammonium chloride, 112-02-7; NaC1, 7647-14-5; NaBr, 7647-15-6. (45)Mean hydrodynamic radii R h for micelles were obtained directly from diffusivities D by using the infiite-dilution Stokes-Einstein relation (eq I), assuming interactions are negligible at high salt concentrations.
