3408
Langmuir 1996,11, 3408-3416
Aggregation Number and Hydrodynamic Hydration Levels of Brij-35 Micelles from Optical Probe Studies George D. J. Phillies,*it R. H. Hunt, K. Strang, and N. Sushkin Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 Received January 23, 1995. In Final Form: May 31, 1 9 9 P Light-scattering methods were used to study the physical properties of micelles of Brij-35 (poly(oxyethylene(23))lauryl ether; C12H2s(OCH2CH2)230H).A minimal model of Brij-35 micelles describes them as hydrodynamichard spheres enclosingan anhydrous core and a hydrated outer shell. An adequate model of spherical Brij-35micelles embodiesan inner hydrodynamiccore (radiusam),containinganhydrous and hydrated zones that block solvent flow, and an outer coronal zone (radius ad, permitting solvent flow but obstructingthe closer approach of two micelles. By combiningquasi-elasticlight-scatteringspectroscopy and multiple-size mesoscopic optical probes, we determined simultaneously the aggregation number (M, micelle hydrodynamic radius (am),micellar radius of closest approach (ac),and degree of hydration (6). The simple hard-sphere model finds am 44 A and N R= 40 and (at T 5 25 "C) overestimates 6 x 4 g of H2Olg of Brij-35. With increasing T,N increases and 6 falls, reaching N R= 63 and 6 1.4 at T = 70 "C. The better micelle model finds a coronal thickness ac-am * 13 A at 10 "C, falling progressively to ~ = A7 at 70 "C. With this model, 6 is e 3 at 10 "C, falling to ~ 1 . at 5 70 "C.
-
Introduction Brij-35 (poly(oxyethylene(23))lauryl ether) is a typical nonionic surfactant having the n-alkyl poly(oxyethy1ene) monoether (H(CH2)i(OCH&H2)jH or C,El) structure. Nonionic surfactants have had significant biological interest' because they are useful for the solubilization ofmembrane proteins. Nonionic surfactants are also of physicochemical interest because they show a wide variety of aggregation and phase behaviorsa2 Brij-35 (C12E23)and its homologues have been the subject of extensive physical studies'-9 involving light-scattering photometry, quasi-elastic light scattering, viscometry, and low-angle neutron scattering. Physical studies3of Brij-35 micelles using light-scattering photometry imply a n aggregation number of approximately 40, with a reported3-8 critical micellar concentration in the range 0.05-0.2 g/L,typically at 25 "C. Physical studies on nonionic amphiphiles are potentially somewhat easier to interpret than are physical studies on ionic amphiphiles, because with the latter systems long-range Coulombic interactions can obscure data interpretation. Bechel.3 used light-scattering photometry to examine micelles of various lauryl- and nonyl-phenol poly(oxyethylenes), obtaining micellar molecular weights, aggregation numbers (N), and critical micellar concentrations (CO) at room temperature. For a series of homologous C12Ej surfactants with j E (8, 231, with increasing j , the aggregation number fell, while the critical micellar concentration increased. These trends are explained by
* To whom communications may be addressed. EMail: phillies@ wpi.wpi.edu (Internet). + Also Associated Biochemistry Faculty. * Abstract published inAdvanceACSAbstracts, August 15,1995. (1)Tanford, C.; Nozaki, Y.; Rohde, M. F. J . Phys. Chem. 1977,81, 1555.
(2)Degiorgio, V.Nonionic Micelles. In Proc. Intl. School ofphysics 'Enrico Fermi', Course XC; Physics of Amphiphiles: Micelles, Vesicles, and Microemulsions; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985;p 303. (3)Becher, P. J . Colloid Sci. 1981, 16, 49. (4)Schefer,J.; McDaniel,R.; Schoenborn, B. P. J. Phys. Chem. 1988, 92,729. ( 5 )Johnson, K.A.; Westermann-Clark,G. B.; Shah, D. 0.J . Colloid Interface Sci. 1989,130, 480. (6)Levitz, P. Langmuir 1991,7,1595. (7)Ross, P.; Olivier, J. P. J . Phys. Chem. 1959,63, 1671. (8)Sharma, B. G.;Rakshit, A. K. J . Colloid Interface Sci. 1989,129, 139. (9)Nagarajan, R.; Ruckenstein, E. Lungmuir 1991,7,2934.
the increase in monomer hydrophilicity attendant to extending the poly(oxyethy1ene)tail. Equivalent behavior was found with the homologous nonylphenol derivatives. For Brij-35, Becher reports N E 40 and co = 0.11 g/L. From proposed geometric constraints, Becher suggested that a rod-shaped micelle model is preferable to a sphere as the poly(oxyethy1ene) chains become long. Tanford et al.l studied sedimentation equilibrium, intrinsic viscosity, and gel chromatography of C12E8and C16E6 micelles in 0.1 M NaCl at 25 "C. No dependence of micelle molecular weight on c was found for c < 10 g/L. Tanford et al. proposed a simple model of micelles and applied it to a variety of n-alkyl poly(oxyethy1ene glycol) monoethers. In this model, the n-alkyl groups form a central anhydrous core, while the poly(oxyethy1ene)chains make a potentially-hydrated shell around the core. Except for very small aggregation numbers, N < 20, if the core of a C12Ej micelle is free of water and poly(oxyethy1ene) residues, then geometric constraints prevent a liquidlike core from being spherical. For Brij-35 micelles with N = 40 and assuming an oblate ellipsoidal core, Tanford et al.' computed a core radius of 16 A and half-thickness of 12 A. For an aggregation number as small as 40, the n-alkyl core calculated by Tanford et a1.l is very nearly spherical. The poly(oxyethy1ene chains are predicted to fill a shell of thickness R e 25 outside the core. Is the micelle core of a CiEj micelle anhydrous? For Brij-58 (C16E20) at weight fractions of 0.2% and 0.4% at 7 "C, Schefer et aL4report low-angle neutron scattering measurements. Their data are consistent with a spherical model with a ca. 25 A hydrocarbon core surrounded by a 25-A-thick oxyethylene shell. The central core is very close to the size expected if it were anhydrous. The outer shell is much larger than expected from the molecular volume of the oxyethylene groups, implying a high degree of hydration of the outer shell. Schefer et al. infer ca. 4 water molecules per oxyethylene group or ~ 1 .g4of H2Olg of surfactant. We have previously reported severall0J1 studies of nonionic surfactant micelles, using quasi-elastic lightscattering spectroscopy (QELSS) and optical probe diffusion as our primary experimental methods. In a QELSS
A
(10)Phillies, G.D.J.;Stott, J.; Ren, S. Z. J . Phys. Chem. 1993,97, 11563. (11)Streletzky, K.;Phillies, G . D. J. Langmuir, in press.
0743-746319512411-3408$09.0010 0 1995 American Chemical Society
Bro-35 Micelle Size and Hydration
Langmuir, Vol. 11, No. 9, 1995 3409
experiment, an equilibrium water/surfactant mixture is illuminated with laser light, and the temporal fluctuations in the light-scattering intensity are statistically characterized. This experiment permits determination of the mutual diffusion coefficient (D,) of the micelles; D, describes the relaxation of micellar concentration fluctuations. In an optical probe experiment, mesoscopic probe particles (in our experiments, polystyrene latex spheres) are added to the water/surfactant system. Light-scattering spectroscopy is then used to measure the diffusion coefficient (D,) of the probe particles amidst the micelles. Interpretation of probe spectra is greatly simplified if the probe species is dilute and dominates scattering by the micelles. D, and D, are affected by micelle-micelle and micelleprobe interactions. By examining how these diffusion coefficients depend on surfactant mass concentration IC), one may infer physical properties of the micelles. In particular, one has
D, = Dmo(l+ k,c) and
always negligibly small). From c , a,, 4, and physically reasonable assumptions as to the partial volumes of surfactant and solvent molecules within a micelle, the micellar aggregation number (N) and degree of hydration ( 6 )of a micelle can be separately determined. It should be stressed that 6 is here obtained from a hydrodynamic, not a thermodynamic, measurement, so the water of hydration determined by this approach refers to water entrained by the micelle within its hydrodynamic radius, not to water thermodynamically bound to the micelles. Here we present the results from applying the above approach to Brij-35micelles. Solutions were studied over a wide range of temperatures (10 < T < 70 "C)and concentrations (0 5 c I 100 g/L). In order to obtain complete agreement with our measurements, it was necessary to extend previous models ofmicelles to provide for the possibility that the micelle hydrodynamic radius and the micelle distance of closest approach are unequal. The followingsections present our experimental methods, the underlying theory, our data and its interpretation, and a discussion.
Experimental Methods
D, = Dpo(l+ kpC) where D,o and DN are the zero-concentration limits of the mutual and probe diffusion coefficients,respectively, and where k , and k, are the leading linear slopes. Theoretical treatments of diffusion coefficients more naturally lead to expansions in powers of the micelle volume fraction (4); namely,
(3) and
(4) where, for spherical micelles having radius a , and number concentration n,
If the micelles and probes are adequately approximated as hard spheres, then Dm0,Dpo,and the Stokes-Einstein equation
can be used to compute the micelle and probe radii. Here k~ is Boltzmann's constant, Tis the absolute temperature, 17 is the solvent viscosity, and D and a are the sphere diffusion coefficient and radius, respectively. Equation 6 is only applicable to isolated (noninteracting,highly dilute) spheres diffusing through a Newtonian solvent. If the sphere concentration is high enough that the solvent and solution viscosities are distinguishable, then eq 6 is not applicable, as discussed further below. By applying standard12-14 theoretical treatments of light-scattering spectroscopy to compute D, and D, and their concentration dependences, k, and k , may be interpreted in terms of the micelle and probe hydrodynamic radii ( a , and a,) and the volume fraction (4) of micelles in solution. (The concentration of the probes is (12)Phillies, G.D.J. J. Colloid Interface Sci. 1987, 119, 518. (13)Carter,J. M.;Phillies, G. D. J. J.Phys. Chem. 1985,89, 5118. (14)Phillies, G.D.J. J. Phys. Chem. 1996,99, 4265.
Brij-35 ( C ~ ~ H Z ~ ( O C H ~ C H Janssen ~)~~O Chimica) H, was used as received. Surfactant solutions were prepared using filtered, deionized water from a Millipore (Bedford, MA) Milli-Q water system. Preparations of carboxylate-modifiedpolystyrene latex spheres having nominal diameters of 67 nm (Seradyn) and 38, 87, 189, and 282 nm (Interfacial Dynamics) were used without further cleaning of the probe particles. Individual samples were made by volumetric serial dilution from stock solutionshaving concentrationsof 50 or 100g/L.Lightscattering sample cells were rigorously cleaned with dust-free deionized water and dried using filtered (0.02-pm)nitrogen. All surfactant solutions were cleaned by passing them through 0.22pm polycarbonate filters as the solutions entered the sample cells. Two laser-light-scatteringtrains were employed in our QELSS measurements. The first apparatus, used for most studies of optical probe diffusion, used a 30-mW HeNe laser (632.8 nm). Samples were held in disposable plastic cells, all sides clear; temperature control was performed by fixing the cells in a massive copper block through which was pumped thermostatted water. Light scattered to the side (usually through 90") was isolated by a pair of irises and detected with an RCA 7265 photomultiplier tube. The photon intensity-intensity autocorrelation function was then obtained with a Langley-Ford Instruments 144channel digital correlator. The correlator and temperature control bath were under automatic computer control, permitting multiple spectra to be taken a t each of a series of temperatures without human intervention. In general, three spectra were taken of each probdsurfactant sample under given conditions so that the statistical scatter in the data would be apparent from a comparison of repeated experiments. In later experiments, the HeNe laser was replaced with a Coherent DPS-532 Nd:YAG frequency-doubledsolid-statediode laser, which provides 50 mW of CW coherent light. Preliminary experiments on nonfluctuating targets showed that the Nd:YAG laser had significantly less coherent noise (largely 60-Hz ripple) than the HeNe laser that it replaced. For measurements of D,, probe-free solutions of Brij-35/H20 were filtered into precleaned glass fluorimeter cells, four sides polished, and studied in a system incorporating a 3-W Spectraphysics Series 2000 Ar+ laser, Brookhaven Instruments goniometer-light-scattering spectrometer, and BI Model 2030AT 262-channel multi-r digital correlator. The correlator and goniometer were under full computer control, permitting serial multiangle or multi-r studies to be done without human intervention. Temperature control was provided by circulating temperature-controlled water through a heat-exchange coil in a perhydronaphthalene-filled index matching vat. In most cases, our light-scattering spectra were analyzed with
Phillies et al.
3410 Langmuir, Vol. 11, No. 9, 1995 Koppel's method of ~umu1ants.l~ This technique is specifically appropriate for a light-scattering spectrum S(k,t) whose field correlation function GW,t) can reasonably be assumed to be written as a sum of exponentials
G"'(k,t)
[S(k,t) - B1'1/2'=
s;
d r A ( r ) exp(-rt)
s.
(7)
d r A(r)
such as the scattering spectrum of a solution of not-quitemonodisperse probe particles. Here, B is the baseline, so B = S(k,-). For freely diffusing particles having diffusioncoefficient D, r = Dk2 is the decay rate. A(T) is the scattering intensity associated with all objects having a decay coefficient r. In the method of cumulants, the measured G[l)(k,t)is fit to a truncated polynomial N
log[G'"(k,t)] =
K,(-t)'
I!
where the Ki are the cumulants and N is the truncation order ofthe cumulant expansion. "he log function distorts the relative noise in different parts of G'l)(k,t),so a weighted linear leastsquares fit is required to obtain the KL.Here, KOis the zero-time intercept ofG(l),while K1= Dk2_determinesthe intensity-weighted average diffusioncoefficient (D). With different samples, D may be either D, or D,. In practice, eq 8 is applied by fitting each field correlation function forN = 1,2,3,4, ...,while calculating several goodnessof-fit parameters for each value of N. The goodness-of-fit parameters are found to improve rapidly with increasingNuntil the spectrum is fit by the series to within the available signalto-noise ratio. Further increases in N lead to little or no improvement in goodness-of-fit parameters such as the rootmean-square error ( R )
(9) or the quality parameter (&I
Here, M is the number of correlator data channels used in the analysis; the analysis excludes all data taken within 2 ps oft = 0 (this is generally the first channel),Di is the baseline-subtracted measured spectrum in the ith channel, and Tiis the theoreticallycalculated (using eqs 7 and 8) spectrum a t the delay time of the ith correlator channel. Comparison of D, and D, with various theoretical forms was made via a nonlinear least-mean-squares program using the simplex algorithm. In general, errors in determining c were far smaller than errors in determining D, so comparison of data and theory was made withc assumed to be noise-free. Measurements of D generally have an error that is some fixed fraction of D, rather than being an absolute error of some size, a circumstance reflected in the least-squares programs by seeking to minimize the fractional root-mean-square difference between data and theoretical forms. "he program was highly flexible; if several parameters appeared in a fitting program, for each parameter, it was independently possible either to constrain the parameter to some forced value or to allow that parameter to float (be a free variable whose value was optimized by the fitting program).
Theory This section presents the hydrodynamic theory for interpreting the measurements ofD, andD,. Probes and micelles are approximated as noninterpenetrating spheres with longer-range hydrodynamic interactions. Many micellar systems are expected from their aggregation (15)Koppel, D . E.J. Chem. Phys. 1972,57,4814.
numbers and the shapes of the hydrophobic, sections of their component molecules to assume an ellipsoidal form; in these systems, approximating micelles as spheres would require careful tests. However, from the calculations of Tanford et al.,l it appears reasonable to assume that Brij35 micelles are very nearly spherical. Is a hard-sphere model adequate? The micelles are uncharged, so electrostatic intermicellar interactions can be ruled out. The effect of shape on D and excluded volume interactions is not trivial; we are able to proceed because the micelles are very nearly spherical. A spherical hydrodynamic body has two fundamental radii, namely, a hydrodynamic radius (aH)and a radius of closest approach (a,). Previous hydrodynamic treatments16 of interacting spheres have generally used a hydrodynamic radius (aH)and implicitly assumed that a pair of spheres can approach as closely as twice their hydrodynamic radius so that a~ = a,. In a system having aH > a,, the hydrodynamic calculation seen in ref 16 is not applicable, because this treatment implicitly assumes that interacting spheres are separated by solvent. The16 treatment of hydrodynamics used here does, however, remain self-consistent for hydrodynamic spheres whose radii of closest approach are larger than their hydrodynamic radii. Indeed, constraining a, > aH avoids convergence issues related to series expansions for the hydrodynamic interaction tensors. In a probe-matrix (e.g., polystyrene sphere: micelle) system, two hydrodynamic radii arise; micelle and probe hydrodynamic radii are denoted a, and a,, respectively. In previous work on Triton X-100 micelles,10J1y20we aH treated micelles as simple hard spheres having a, = a,. In order to obtain a quantitative interpretation of data on Brij-35 micelles, as seen below, we found it necessary to allow the radius of closest approach (a,) of the micelles to be larger than the hydrodynamic radius (a,,,). For Brij-35 micelles, this behavior is physically reasonable. Each micelle has a hydrocarbon core surrounded by heavily hydrated poly(ethy1eneoxide) chains. Each polyethylene chain functions as a tethered (to the hydrocarbon core) polymer. Entropic considerations prevent pairs of spheres coated with tethered chains for approaching each other closely, even if the number of tethered chains is relatively small. The degree of extension of a tethered polymer, like the end-to-end distance of a free polymer, fluctuates. Since even a few tethered chains prevent contact, a, is determined by the length of a few most-fully-extended chains between which solvent can flow. In contrast to a,, a, is the smaller radius within which the density of poly(ethy1eneoxide) chains becomes substantial enough to prevent solvent flow. In our model, the potential energy of two bodies (two micelles, or a micelle and a probe) satisfies
where re and rb are the radii of closest approach of the two bodies. Hydrodynamic micelle-micelle and micelle-probe interactions are described by the hydrodynamic interaction tensors given by Mazur and van Saarloos.16 Namely, for a set of particles i subject to external nonhydrodynamic forces Fj, the induced velocities vi of the particles are determined by the particle mobility tensors pij as (16)Mazur, P.;van Saarloos, W . Phys. A: Amsterdam 1982, 115A, 21.
Langmuir, Vol. 11, No. 9, 1995 3411
Brij-35 Micelle Size and Hydration N
N
p,*Fj
vi =
S(k,t)=
(13)
(Eexp(-ik*[r,(t+ z) - rj(z)l))
The mobility tensors are related to the diffision tensors via a generalized Einstein diffusion equation
D, = kBTp,
(14)
For an isolated sphere,pii = (6n7a)-' leads to the StokesEinstein equation. Mobility tensors for hydrodynamicinteractions between spheres of different sizes were obtained, e.g., for Kynch,17 Batchelor,18 and Mazur and van Saarloos.16 It is useful to separate pg into its self (i =j ) and distinct (i ' j ) parts. Self terms can be written as
+
Here, k is the scattering vector and ri(t z) and rjct)refer to the locations of scatterers i andj at the indicated times. In a probe experiment, i and j refer to the positions of probe spheres, the relatively weak scattering from the micelles being approximated as not contributing to S(k,t). If the probe species is dilute (the case here), only the i = j terms contribute significantly to S(k,t),while K1 yields the probe diffision coefficient (Dp). In a description of light-scattering spectra of probe-free micelle solutions, i and j refer to some of the N micelles, i = j and i tj both being allowed, the first cumulant K1 of S(k,t)yielding D, of the micelles. D,is the single-particle (self) diffusion coefficient of a probe through a solution of nonscattering micelles. To first order in micelle concentration, D,is written in terms of bg as
where f o = 6n7a is the Stokes' law drag coefficient and bil and bilmdescribe two- and three-particle contributions to the mobility of i. The distinct terms are p,
1
=-(Tu+ fo
(20)
i j=l
j= 1
C Ti4+ ...)
(16)
m+ij
where Tij and T i , describe two- and three-body hydrodynamic interactions, respectively. The b and T terms have expansions in powers of the interparticle separations. To lowest order, for two spheres of the same radius, the two-particle interaction tensors are the Oseen tensor
(17) and its self-term equivalent
(21) In this equation, the index 1 proceeds over the N micelles, subscript 0 refers to the probe, and the brackets (...)provide an ensemble average. Combining eqs 2,5,19, and 21, for micelles whose radius of closest approach is their hydrodynamic radius (a,), one has
a,=-[-15 -4
ap
N
k2S(k,0) i = l
(18)
5
apam -171 - 8 7 M ) (19) 6R6
Here, a, and a, are the radii of the particle i of interest and a particle 1 in the surrounding medium, respectivelx, while R is the distance between particles i and 1 and R is a unit vector pointing from i toward 1. The above interactions may be used to calculate the diffusion coefficients determined by light-scattering spectroscopy. These diffusion coefficients are obtained from the first cumulant K1, which is the initial slope of the logarithmic derivative of the dynamic structure factor (17) Kynch, G. J. J.Fluid Me&. 1959,5, 193. (18)Batchelor, G. K. J.Fluid Mech. 1976, 74, 1.
8(~,
As shown by Carter and Phillies13 and in more detail by Phillies,14 1
-(
+
+a m
D, = - - - f - - ( [ ~ D , [ k 2+
where rg and 9~are the distance and unit vector between particles i and j . Mazur and van Saarloos16give expressions correct through order rij-7 for interactions between two spheres having unequal radii. We only need their results for bil; namely,
2 0 4 - llapa,
ap
N
+
(k*bitk) l*i=l
For diffusing hard spheres, Carter and Phillies show13 that eq 23 can be reduced to eq 3 with a, = -115/128 x -0.90. In this paper, micelle-micelle interactions remain hard-sphere so that the radial distribution function for two micelles is stillg(r) = 0, r I2ao andg(r) = 1 at larger r. However, we consider both the standard hard-sphere case (in which a, = aH a,) and the case a, > aH = a,. In the latter case, at their distance of closest approach, two sphere centers are separated by a distance 2a, > 2a,. In terms of previous treatments of a,, an increase in the hard-sphere distance of closest approach (so that a, > a,) is incorporated into the model by dividing the intermicellar potential into an inner hard-sphere part and a perturbation part. Correspondingly, the radial distribution function g(r) has an inner hard-sphere part g H S ( r ) and a perturbation part dg(r). The simple hard sphere part is g H & ) = 0 for r 5 2a, and g&) = 1 elsewhere. In general, 6g(r) = 0 for r I2a, and dg(r) = g(r) - 1 for r
=- 2a,.
For the first-order concentration dependence of D,, eq 23 gives
Phillies et al.
3412 Langmuir, Vol. 11, No.9, 1995
(
37.2 am= -0.9+LZmdr -,+---+a,
3r
45a,
93am3
am2
4r2
4r4
+
i
the integral converging through the short range ofg(r) = 1. While dg(r) is referred to as a "perturbation" and while eq 24 is linear in dg(r), eq 24 is complete to first order in concentration c. There is no implicit assumption in eq 24 that the perturbation dg(r) is locally small. In the case needed here, namely, hard spheres having a, =- a,, dg(r) = -1 for 2a, I r I 2a, and dg(r) = 0 elsewhere. On defining Y = ada,, one has
I
I
20
"0
60
80
100
c (g/L)
If a, # a,, one must also modify eq 22; namely,
Data This sectionpresents our measurements on mutual and probe diffusion involving Brij-35 micelles. We first show the results obtained from surfactantlwater solutions and then turn to information obtained from probe diffusion. Figure 1shows measurements ofD, against surfactant concentration (c) at temperatures of 10,30,50, and 70 "C. At all temperatures, D, increases gradually with increasing surfactant concentration. Solid lines in the figure are fits to eq 1with D,o andk, as free parameters. Physically, these solid lines correspond to a low-concentration linear approximation with no assumption about either the nature of intermicellar interactions or the effect of these interactions on D. The fit and lines are used only to extrapolate D, to the limit c 0. Table 1 shows D,o from these fits and from fits to additional data sets at 25 and 40 "C.The lower limit on concentration is instrumental. With decreasing c, micelle spectra become weaker and weaker, accurate measurements on D, being impractical at concentrations below a few gramsfliter of Brij-35. Figure 2 shows D,o (from Figure 1)plotted against T/?) where q is the viscosity of pure water at the same temperature. The straight line is a least-squares fit;D,o is consistent with
40
Figure 1. Mutual diffusion coefficient of Brij-35 micelles at 30 (O), 50 (01, and 70 "C (B) against temperatures 10 (O), surfactantconcentration (c). Solid lines represent linear-leastsquares fits to the data for extrapolation to c = 0 to obtain D,o (see text). Dashed lines represent fits of the data here and in Figure 4 to eqs 25 and 22, yielding parameters in Table 5 (10 and 70 "C) or Table 4 (30 and 50 "C).
-
00
Figure 2. Zero-concentration limit (D,o) of D , as a function of T/q.To good approximation, D,o is linear in T/qand vanishes in the Tlq 0 limit. The c 0 limit of the micelle radius is thus nearly independent of T.
-
The quadratic term is negligibly small. Over this temperature range, Brij-35 micelles in the limit of low concentration behave to first approximation as diffusing particles with a temperature-independent hydrodynamic radius of 44 A. The scatter in D,o and in the calculated a, in Table 1 arise from the scatter in the original measurements of D, which is largest at high T. Does the constancy of micelle radius with increasing temperature persist to elevated surfactant concentrations? Measurements of the temperature dependence of probe diffusion at different surfactant concentrations give some information on this question. We measured D, for 67-nm polystyrene latex probes for c of trace (elg/L),20,25,30, 40,60,80, and 100 gL. The data cover 10 5 T I70 "C
-
Table 1. Zero-Concentration Limiting Diffusion
Coefficient (D-0) and Apparent Hydrodynamic Radius (a,) of Brij-36 Micelles a8 Functions of Temperature (!l'') 10 25 30 40
50 70
3.63 5.41 6.17 7.92 9.33 14.78
43.6 44.1 44.9 44.1 46.1 41.9
in 5 "C intervals and were measured automatically in a single series of spectra on a single sample. Representative results, plotted against TI?,appear in Figure 3. At low surfactant concentrations (c 5 25 gL),
BrG-35 Micelle Size and Hydration
Langmuir, Vol. 11, No. 9, 1995 3413
I
0 . 0 V '
0
/ I
'
200
'
'
400
'
'
600
'
'
I
800
T/v W c P )
Figure3. Diffusion coefficient (Dp)of 67-nm polystyrene latex spheres in Brij-35 solutions, plotted against T/qfor temperatures 10 s T s 70 "C and Brij-35 concentrations of 25,).( 40 (O), and 100 g/L (m), Walden's rule is followed at c s 25 g/L.
-
Walden's rule type behavior is seen; D, is linear in Tlv and extrapolates properly to zero in the T/v 0 limit. Above 30 g/L of Brij-35 (Figure 3 shows 40 g/L),D, begins to deviate from Walden's rule behavior, D, being larger than expected at large T. At the highest Brij-35 concentration that we examined (Figure 3,100 g/L), deviations from Walden's rule are quite marked. The lowestconcentration data show the behavior expected if probes and micelles had temperature-independent radii, the temperature dependence of D, arising purely from the temperature dependence of the solvent viscosity. In contrast, at larger surfactant concentrations, Figure 3 is not consistent with the probe radius, the micelle radius, and the probe-micelle interactions all being independent of T. The probe radius (ca. 340 A) is much larger than any reasonable estimate for a plausible change in effective probe radius attendant to changes in surfactant adsorption by the probe, so it is difficult to ascribe the behavior seen in Figure 3 and elevated c to a variation in the probe radius with T. However, changes in micelle size and interactions with temperature at large c are plausibly expected in a system known to have complex phase behavior at large c and elevated T; these changes could reasonably lead to the visible deviations from Walden's rule. From Figure 3, we infer that the micelle radii a , and a, are independent of c for c 5 30 g/L. At higher concentrations, the micelle radius or the intermicellar interactions become temperature- and concentrationdependent. Our morg detailed analysis, below, of the concentration dependence ofD, and D, is therefore limited to the region c I30 g/L in which the micelle radius is apparently nearly constant. Figure 4a-d presents measurements of D,against c at temperatures of 10,25,40, and 70 "C and probes having a variety of radii. For each system, D, decreases linearly with surfactant concentration. This behavior of D, is expected from eqs 21 and 22, which explain the concentration dependence of D, in terms of the hydrodynamic interactions between probe spheres and micelles. A quantitative interpretation follows. Analysis We used two general approaches to fitting the data to hydrodynamic theory. The first model treats micelles as simple hard spheres with a , = a,. In this model, a , is obtained by extrapolating D, to c = 0; the fitting process is applied only to measurements of D,. The second model
retains a , and a, as independent parameters. With this model, the fitting process can be applied to D, as well as DP. How are these models related to the classical image' of a Brij-35 micelle as an anhydrous hydrophobic core surrounded by a hydrated hydrophilic shell? In our first model, the radius a , = a, encloses the anhydrous and hydrated regions ofthe micelle, a , being an effective radius within which solvent is entrained by the micelle and beyond which solvent and other micelles are free to move. In the second model, the micelle interior is divided into three zones, namely, (1)a central anhydrous region, (2) an inner hydrated region containing concentrated surfactant chains and entrapped solvent, and (3) an outer corona containing at least a few surfactant chains but through which water is free to move. The chains in region 3 are sufficiently concentrated as to present an effective entropic barrier preventing other micelles from entering the region but are sufficiently dilute that solvent flow through the region is allowed. The radius a , encloses regions 1 and 2, while the radius a, encloses all three regions; a, - amis the thickness of the coronal region. The important advantage of the first model is that it has fewer free parameters, increasing the stability of any fit. The disadvantage of the model is that it does not explain the dependence of D , on c. We find that D, increases substantially with increasing c , contrary to eq 25 for hard spheres with a , = a,. On the other hand, there is a controversy13J8J9in the literature as to the , . By using only D, extrapolated to c correct form for a 0 in our analysis, we avoid both the model's disadvantage and the controversy. The important advantage of the second model is that it can be applied simultaneously to our measured D, and D,, not only to D, or D, separately. Our fits reveal that a,, a,, a,, and N have different numerical sensitivities to D, and Dp. As seen below, measurements ofD, at a series of concentrations do a good job of determining a , and a, but are not effective at determining N . Fits with N anywhere in the wide range indicated in the table, with somewhat different a , and a,, have very nearly the same % rms errors. In contrast, fits that include D, and D , measurements give stable values for N . That is, the use of optical probe methods as a supplement to quasi-elastic light-scattering spectroscopy permits reliable determination of a parameter (N) that could not have been reliably inferred from QELSS spectra of probe-free micelles. Our analysis is summarized by Tables 2-5 and represented by lines in Figures 1and 4. Experimental data were acquired for c up to 50 or 100 g/L, but parameters in Tables 2-5 are taken from fits with D, confined to c 5 30 g/L. The degree of hydration 6 (the weight ratio: grams of water per gram of Brij-35) of the micelles was computed for a sphere ofradius a , whose interior contains N molecules of Brij-35 (at density 0.95 g/cm3) and water (at density 1g/cm3). Water in the outer corona ( a , 5 r Ia,) is not included in 6. This model neglects the volume of mixing effects, which are small relative to other approximations in our study. Table 2 shows the outcomes of fits of D, to eq 22 for each probe and temperature, using the first model. a , was calculated from D,o using eq 6; fitting parameters were N and a probe radius a,. At 10 "C,N was -37, trending upwards toward 40 at 25 "C and 44 f 5 at 45 "C. At high temperatures (70 "C),N rose substantially further, to N
-
(19) Felderhof, B. U.J. Phys. A 1978,11, 929. (20) Phillies, G.D. J.; Stott, J.; Streletzky, K.; Ren, S. 2.;Sushkin, N.; Richardson, C. A. In Surfactants and Association Colloids: The RheologyofSurfwtant Solutions; Herb, C.A., Prudxomme,R. K., Eds.; American Chemical Society: Washington, DC, 1994.
Phillies et al.
3414 Langmuir, Vol. 11,No.9, 1995
t
; o l l , l , c . > ~
0.1 0.1
10 10
20 20
