Frequency Domain Fluorescence Measurements of the Aggregation

Langmuir 2000, 16, 911-921

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Frequency Domain Fluorescence Measurements of the Aggregation Properties of CnEm Surfactants in Agarose Gels Stuart L. Nolan,† Ronald J. Phillips,† and Stephanie R. Dungan*,†,‡ Department of Chemical Engineering and Materials Science and Department of Food Science and Technology, University of California at Davis, Davis, California 95616-8598 Received March 9, 1999. In Final Form: September 14, 1999 The micellar properties of C12E6, C12E8, and C12E10 surfactants in agarose gels were investigated. The aggregation number and the aggregate polarity were determined as a function of temperature, gel fiber volume fraction, and surfactant type by using steady-state fluorescence and frequency domain fluorescence quenching techniques. Application of the latter technique to the study of micellar properties in solution is described, including a method for accounting for the excitation light scattered by the gel fibers. The aggregation behavior in gels of surfactants which form spherical micelles in water is compared to those which form cylindrical micelles, and the results are interpreted by using a thermodynamic model for micelle formation in a gel matrix. For surfactants which form small spherical micelles in solution, there is little difference between the aggregation number and micelle polarity in an agarose gel and in pure solution. For surfactants which form large cylindrical micelles in solution, the aggregation number decreases with increasing gel concentration while the micelle polarity remains constant. The thermodynamic model shows that, if the agarose gel restricts the micelle size distribution, then the weight average aggregation number decreases for large cylindrical micelles and remains unchanged for small spherical micelles. The model results are consistent with the experimental results for the aggregation number of C12E6 in an agarose gel.

1. Introduction A surfactant hydrogel material consists of an interconnected polymer matrix containing interstices that are filled with an aqueous surfactant solution. Under the proper conditions, the surfactant may aggregate into micelles, which in turn can greatly increase the aqueous solubility of hydrophobic compounds within the gel.1,2 Thus, the micelles can be thought of as microcontainers for hydrophobic solutes, while the polymer matrix can be thought of as a containment structure for the micelles. Since micelle diffusion is slowed within the gel, these surfactant hydrogels have potential in controlled release or solute extraction applications.3 Polymers and surfactants can interact through intermolecular forces, and thus the polymer fibers making up a hydrogel can alter the structure of any micelles contained within it. Due to these interactions and structural changes, the micelle solubilization and transport properties may also be altered. Variables such as temperature, salt concentration, polymer type, surfactant type, and concentration of surfactant and polymer dictate which interactions are most important and, consequently, could be used to control the micelle behavior within the gel. Thus, knowledge of how micelle-polymer interactions influence aggregate structure will be important in engineering surfactant hydrogel materials. There are a number of reviews of polymer-surfactant interactions;4-9 however, most previous studies have * To whom correspondence should be addressed. † Department of Chemical Engineering and Materials Science. ‡ Department of Food Science and Technology. (1) Mackay, R. A. In Nonionic Surfactants Physical Chemistry; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Vol. 23, Chapter 6. (2) Nakagawa, T. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1966; Vol. 1, Chapter 17. (3) Calvert, T. L.; Phillips, R. J.; Dungan, S. R. AIChE J. 1994, 40, 1449.

investigated solutions of surfactants and polymers that interact strongly with each other. Relatively little research has been performed on surfactants in gels, particularly for the case where the micelle is free to diffuse independently of the polymer fibers.10-13 Thus, although previous studies give valuable insights into the type of interactions that occur between polymers and micelles, they do not deal directly with a mobile micelle surrounded by fixed polymer fibers. In this work, the effect of such a gel on micelle structure is investigated by focusing on a system in which the interactions between the surfactant and polymer are predominately steric in origin. Nonionic surfactants in nonionic hydrophilic gels fulfill this requirement. This study focuses on poly(oxyethylene) monoalkyl ether surfactants (C12Em, m ) 6, 8, or 10) in agarose hydrogels. Holographic interferometry experiments,13 NMR selfdiffusion techniques,10 and recent theoretical calculations14 (4) Brackman, J. C.; Engberts, J. B. F. N. Chem. Soc. Rev. 1993, 22, 85. (5) Brackman, J. C.; Engberts, J. B. F. N. In Structure and Flow in Surfactant Solutions; Herb, C. A., and Prud’homme, R. K., Eds.; American Chemical Society: Washington, DC, 1994; Vol. 578, Chapter 24. (6) Li, Y.; Dubin, P. L. In Structure and Flow in Surfactant Solutions; Herb, C. A., and Prud’homme, R. K., Eds.; American Chemical Society: Washington, DC, 1994; Vol. 578, Chapter 23. (7) Saito, S. In Nonionic Surfactants Physical Chemistry; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Vol. 23, Chapter 15. (8) Piculell, L.; Lindman, B. Adv. Colloid Interface Sci. 1992, 41, 149. (9) Balazs, A. C.; Huang, K.; Pan, T. Colloids Surf. A 1993, 75, 1. (10) Penders, M. H. G. M.; Nilsson, S.; Piculell, L.; Lindman, B. J. Phys. Chem. 1993, 97, 11332. (11) Penders, M. H. G. M.; Nilsson, S.; Piculell, L.; Lindman, B. J. Phys. Chem. 1994, 98, 5508. (12) Johansson, L.; Hedberg, P.; Lo¨froth, J.-E. J. Phys. Chem. 1993, 97, 747. (13) Kong, D. D.; Kosar, T. F.; Dungan, S. R.; Phillips, R. J. AIChE J. 1997, 43, 25. (14) Clague, D. S.; Phillips, R. J. Phys. Fluids 1996, 8, 1720.

10.1021/la9902789 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/14/2000

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indicate that, under conditions where these surfactants diffuse as small spherical micelles in the absence of agarose, they also diffuse as comparably sized spherical micelles in an agarose gel. In previous work, it was shown that C12E8 and C12E10 micelles consistently exhibit this behavior.13 However, it was experimentally observed by Kong et al.13 that C12E6, a surfactant which forms larger prolate ellipsoid micelles in the absence of agarose, appears to diffuse in an agarose gel faster than expected on the basis of its hydrodynamic radius in water. In fact, C12E6 micelles diffuse as rapidly as the smaller, spherical micelles in the gel. Predictions from the hydrodynamic calculations of Clague and Phillips14 indicate that if an aggregate diffuses as a hard sphere in a bed of rigid cylindrical fibers, then the diffusion of the larger aggregates will be slowed relatively more than that of the smaller aggregates. Thus, the results of Kong et al.13 suggest that C12E6 may decrease in size upon incorporation within an agarose gel. Penders et al.10 have also studied the cloud point and diffusion behavior of C12E6 and C12E8 in agarose by using light transmittance and NMR self-diffusion techniques. However, these investigators added sodium thiocyanate (NaSCN) to the C12E6 agarose systems to elevate the surfactant cloud point. This addition was made since homogeneous gels are only formed when the cloud point of the surfactant/agarose solution is above the gel temperature. NaSCN/C12E6/water solutions result in smaller micelles than would be formed in the absence of NaSCN, and hence, their C12E6 results cannot be compared directly with this work. With regard to the diffusion of small spherical aggregates in agarose gels, the results of Penders et al. are similar to the results of Kong et al.13 Interestingly, Penders et al.10 found that as the concentration of C12E6 increases, the cloud point on heating and the melting temperature of the agarose gel decreases. This concentration effect could be indicative of an interaction between the aggregates and the agarose gel. Effects of the gel on micelle/micelle interactions have also been observed and explained theoretically by Buck et al.15 In the studies above, the size of the aggregates is determined from diffusion experiments and the assumption that the aggregates behave as hard spheres. Timeresolved fluorescence quenching (TRFQ) offers the possibility of determining the micelle structure and the dynamics of molecules added to the micelle both in solution12,16-27 and in hydrogels,12 independent of the effects of micelle/micelle or micelle/fiber interactions on diffusion and assumptions about micelle shape. Thus, this method could provide more direct information about the aggregation behavior of the CnEm-agarose gel system. In (15) Buck, K. K. S.; Phillips, R. J.; Dungan, S. R. J. Fluid Mech. 1999, 396, 287. (16) Zana, R. In Surfactant Solutions: New Methods of Investigation; Zana, R., Ed.; Marcel Dekker: New York, 1987; Vol. 22, Chapter 5. (17) Binana-Limbele´, W.; Van Os, N. M.; Rupert, L. A. M.; Zana, R. J. Colloid Interface Sci. 1991, 144, 458. (18) Frindi, M.; Michels, B.; Zana, R. J. Phys. Chem. 1992, 96, 6095. (19) Alami, E.; Kamenka, N.; Raharimihamina, A.; Zana, R. J. Colloid Interface Sci. 1993, 158, 342. (20) Alami, E.; Van Os, N. M.; Rupert, L. A. M.; De Jong, B.; Kerkhof, F. J. M.; Zana, R. J. Colloid Interface Sci. 1993, 160, 205. (21) Zana, R.; Weill, C. J. Phys. Lett. 1985, 46, L-953. (22) Danino, D.; Talmon, Y.; Zana, R. J. Colloid Interface Sci. 1997, 186, 170. (23) Binana-Limbele´, W.; Zana, R. J. Colloid Interface Sci. 1988, 121, 81. (24) Lo¨froth, J.-E.; Almgren, M. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 1, p 627. (25) Medhage, B.; Almgren, M.; Alsins, J. J. Phys. Chem. 1993, 97, 7753. (26) Almgren, M.; Alsins, J. Isr. J. Chem. 1991, 31, 159. (27) Almgren, M.; Alsins, J. Prog. Colloid Polym. Sci. 1987, 74, 55.

Nolan et al.

this work, the frequency domain method of measuring fluorescence lifetimes is used to measure micelle aggregation numbers in hydrogels by fluorescence quenching. Previous TRFQ experiments to measure micelle aggregation numbers have employed pulse methods such as the single photon counting (SPC) technique.16,28 However, the instrumentation for SPC is expensive and can be difficult to obtain. As fluorescence frequency domain instrumentation is commercially available, relatively easy to use, and can be equipped with an inexpensive argon arc lamp for its light source (as opposed to the lasers used in most pulse methods), there are advantages which make this method attractive. As there are no publications describing the use of the frequency domain method to measure a micelle aggregation number by fluorescence quenching, this paper describes the theory and experimental details for this technique. The same series of C12Em surfactants studied by Kong et al.13 was investigated in the present work as a function of gel concentration and temperature in order to determine how the confinement of the agarose gel matrix influences their structure. C12E6, C12E8, and C12E10 are of interest in part because these three surfactants form different micelle structures in the absence of polymer. At 22 °C, the temperature studied in Kong et al., C12E8 forms small, monodisperse spherical micelles while C12E6 forms larger, prolate ellipsoidal micelles.12,21-27,29,30 Increasing the temperature promotes aggregate growth, and spherical to prolate ellipsoid transformations in micelle geometry are possible. Information about C12E10 is limited, but diffusion data from Kong et al.13 indicate its size should be between that of C12E8 and C12E6. Finally, a thermodynamic model for micelle aggregation in the presence of an interconnected polymer matrix was developed to give a clear physical interpretation of the experimental findings. The important feature that this theory captures is the excluded volume effect of the gel matrix on the micelle size distribution. This distribution depends on the free energy of transfer of a surfactant molecule from solution into a micelle, which is calculated using results from a method developed by Blankschtein et al.31 The influence of the gel fibers is captured through an activity coefficient based on Ogston’s solution for the distribution of space sizes in a fibrous network.32 Blankschtein’s model gives the micelle aggregation behavior of CnEm surfactants in both polymer-free solutions31 and in polymer solutions.33 Johansson et al.12 suggested using an activity coefficient to account for the effect of confining micelles in an interconnected polymer matrix on the surfactant partitioning behavior in κ-carrageenan gels. In this work, these approaches are combined to obtain predictions of micelle aggregation behavior in the presence of an interconnected polymer matrix. 2. Theory Frequency Domain Fluorescence Quenching (FDFQ). In a FDFQ experiment, a continuous but sinusoidally varying light intensity is used to excite the sample instead of the instantaneous pulse used in a SPC experiment. The sample responds with fluorescent light which is also continuous and sinusoidally varying. However, due to the finite lifetime of the fluorescent probe, (28) Gehlen, M. H.; De Schryver, F. C. Chem. Rev. 1993, 93, 199. (29) Carale, T. R.; Blankschtein, D. J. Phys. Chem. 1992, 96, 459. (30) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. (31) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. J. Chem. Phys. 1986, 85, 7268. (32) Ogston, A. G. Trans. Faraday Soc. 1958, 54, 1754. (33) Nikas, Y. J.; Blankschtein, D. Langmuir 1994, 10, 3512.

Aggregation Properties of CnEm Surfactants

Langmuir, Vol. 16, No. 3, 2000 913

the response lags the excitation and is demodulated. The latter change indicates that the amplitude of the timedependent fluorescence response, normalized by its steady or DC component, is reduced compared to the normalized amplitude of the excitation. Thus, if the excitation light intensity is expressed as

f(t) ) a + b sin(ωt)

I(t) )

∫-∞t f(t′)G(t - t′) dt′ + Rsf(t)

Φ ) arctan

N D

()

∫0∞G(z) sin(ωz) dz N) ∫0∞G(z) dz + Rs

(6)

∫0∞G(z) cos(ωz) dz + Rs D) ∫0∞G(z) dz + Rs

(7)

The integrals in the numerators of eqs 6 and 7 are the Fourier sine and cosine transforms, respectively, of the impulse response function G(t). If Rs ) 0, these equations reduce to the case of fluorescence emission only.34,36 For relatively simple forms of the impulse response function G(t), the integrals in eqs 6 and 7 can be evaluated analytically. They can also be evaluated numerically by using fast Fourier transform methods.43 It is convenient to rearrange eqs 6 and 7 by introducing the fractional scattered intensity ξs, defined as

ξs )

Rs

∫0 G(z) dz + Rs ∞

NA A - As

(9)

(D - ξs)A A - As

(10)

Nsc ) and

Dsc )

where Nsc and Dsc are N and D with the effects of scatter removed. In this case,

ξs ) (4)

(34) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum Press: New York, 1983; Chapter 3. (35) Jameson, D. M.; Gratton, E.; Hall, R. D. Appl. Spectrosc. Rev. 1984, 20, 55. (36) Weber, G. J. Chem. Phys. 1977, 66, 4081. (37) Patterson, M. S.; Pogue, B. W. Appl. Opt. 1994, 33, 1963. (38) Siemiarczuk, A.; Ware, W. R. Chem. Phys. Lett. 1990, 167, 263. (39) Szmacinski, H.; Lakowicz, J. R. Sens. Actuators, B 1996, 30, 207. (40) Lakowicz, J. R.; Szmacinski, H. Sens. Actuators, B 1993, 11, 133. (41) Sevick-Muraca, E. M.; Suddeath, L. F.; Burch, C. L. SPIE 1994, 2137, 673. (42) Sevick-Muraca, E. M.; Burch, C. L. Opt. Lett. 1994, 19, 1928.

(8)

Physically, ξs is the fraction of the sample emission intensity due to scattered excitation light. Equation 8 can be solved for Rs and substituted into eqs 6 and 7, causing ξs to be the unknown parameter instead of Rs. Alternatively, it is possible to show that the steady component A of the sample emission is proportional to the denominators of eqs 6 and 7. If N and D are calculated from experimental measurements of Φ and M by using eqs 4 and 5, then this proportionality can be used to correct N and D according to

(3)

where Rsf(t) accounts for the contribution to the detected intensity of the scattered excitation light. The scatter term takes this form provided that the scattering is instantaneous.37,38 Given the transmittance of the agarose gel10 and the lifetime of 350 ns of the probe used in this work, this assumption is appropriate.39-42 Substituting eq 1 into eq 3 and equating the result to eq 2 yields a phase lag Φ and demodulation factor M given by

(5)

and

(2)

where A is the steady component of the emission, B is the amplitude of the time-varying portion of the response, and Φ is the phase lag. In a frequency domain experiment, the experimental observables are the phase lag and the demodulation factor M ) Ba/Ab. Measurements of Φ and M can be used to probe the aggregation properties of a micellar solution by making use of kinetic models previously developed to describe the fluorescent response G(t) to an instantaneous δ-pulse of excitation light. The transformation of G(t) to equations that can be used to analyze the emission from the frequency domain method34-36 is given below. One of the difficulties that can be encountered in any fluorescence quenching method is the small amount of scattered light that cannot be physically filtered out. In gels, this scattered light is significant enough to affect the results. When carrying out SPC measurements, researchers have dealt with this problem by ignoring the very short lifetime region (below 10 ns) of the intensity decay. The derivation presented below shows how to account for the scattered light in a systematic fashion in FDFQ. The physical parameters in a model for G(t) can be related to Φ and M if the emission light intensity is expressed as

M ) (N2 + D2)1/2 where

(1)

where a is the steady component of the excitation, b is the amplitude of the time-varying portion of the excitation, ω is the angular frequency, and t is time, then the emission light intensity is

I(t) ) A + B sin(ωt - Φ)

and

As A

(11)

where As is the steady component of the scattered light in the sample emission. Since A is easily measured and ξs can be measured in an independent experiment with no quencher (see section 3), As is readily calculated. The parameters Nsc and Dsc can then be used to determine values of Φ and M that do not have a contribution from the scattered light. Three models for the impulse response function G(t) were investigated in this work. The first model, referred (43) Press: W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C, the Art of Scientific Computing, 2nd ed.; Cambridge University: New York, 1992; Chapter 15.

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to here as the “Infelta model”, was used in cases for which the exchange of probe and quencher between micelles was not observed.16,28,44 The second “Gehlen model” was used for situations in which exchange of the probe and quencher was thought to occur through micelle contacts or close encounters.28,45,46 Finally, the third “Almgren model,” which describes the one-dimensional diffusion of probe and quencher within a long cylindrical micelle, was used in this work to investigate the possibility that the micelles had formed long cylindrical structures.27,47,48 Thermodynamic Model. Previous models of surfactant-polymer systems have focused on specific hydrophobic, electrostatic, and hydrogen-bonding interactions between the surfactant aggregates and the polymer fibers.9,33,49-53 Such strong interactions lead to binding of surfactant to the polymer fiber, attended by substantial changes in the aggregate structure. In contrast, for the systems studied in this work, forces between the C12Em micelles and agarose fibers are thought to be weakly repulsive only, caused by steric interactions between aggregate and fiber.10 Thus, the polymer molecules are not expected to become incorporated within the surfactant aggregates. However, it is still possible that the gel matrix could affect the surfactant aggregate structure. The polymer matrix can be thought of as a collection of compartments which contain aggregates, and the nature of the gel is such that there is a distribution of sizes of these compartments. There is also a size distribution for the aggregates, with less space within a given compartment and potentially fewer compartments available to larger aggregates. Because such excluded volume interactions will be present even when other hydrophobic or electrostatic forces are also manifest, it is important to be able to determine their contribution. To capture this effect, a thermodynamic theory for micelle formation31 is used in conjunction with an expression for the space restriction imposed by the gel matrix.12 The micelle distribution is determined by requirements of equilibrium thermodynamics as described by Blankschtein et al.31 The chemical potential for a micelle of x monomers can be expressed as

µx ) µx° + kT ln(φxγ)

(12)

where µx is the chemical potential of a micelle with x surfactant molecules in solution and µx° is the standard state chemical potential (of a hypothetical phase of pure micelles of size x which behave ideally). Here kT is the Boltzmann energy, φx is the volume fraction of micelles with x surfactants, and γ is an activity coefficient which includes contributions from intermicelle interactions as well as micelle-polymer fiber interactions. By using eq 12 and the pseudo-phase approximation µ1 ) µx/x, an (44) Infelta, P. P.; Gra¨tzel, M.; Thomas, J. K. J. Phys. Chem. 1974, 78, 190. (45) Gehlen, M. H.; Van der Auweraer, M.; De Schryver, F. C. Photochem. Photobiol. 1991, 54, 613. (46) Gehlen, M. H.; Van der Auweraer, M.; Reekmans, S.; Neumann, M.; De Schryver, F. C. C. J. Phys. Chem. 1991, 95, 5684. (47) Almgren, M.; Alsins, J.; Mukhtar, E.; van Stam, J. J. Phys. Chem. 1988, 92, 4479. (48) Alsins, J.; Almgren, M. J. Phys. Chem. 1990, 94, 3062. (49) Ruckenstein, E.; Huber, G.; Hoffmann, H. Langmuir 1987, 3, 382. (50) Nagarajan, R. Chem. Phys. Lett. 1980, 76, 282. (51) Nagarajan, R. J. Chem. Phys. 1989, 90, 1980. (52) Wallin, T.; Linse, P. Langmuir 1996, 12, 305. (53) Wallin, T.; Linse, P. Langmuir 1998, 14, 2940.

expression for the micelle size distribution is found to be

φx )

(

)

(φ1)x xgmic exp γ kT

(13)

Here µ1 and φ1 refer to properties of the surfactant monomer and gmic ) µx°/x - µ1° is the free energy of micellization corresponding to the change in free energy which occurs upon taking a surfactant molecule out of solution and placing it in a micelle with x surfactants. The activity coefficient for unaggregated surfactant is assumed to be unity in this model.54 Puvvada and Blankschtein30 express gmic as

gmic ) gcyl mic +

xsph sph 1 (g - gcyl mic) + kT 1 x mic x

(

)

(14)

where gcyl mic is the change in free energy which occurs upon taking a surfactant molecule out of solution and placing it in an infinitely long cylindrical micelle and gsph mic is the change in free energy associated with taking a surfactant molecule out of solution and placing it in a spherical micelle with xsph surfactants. The last term on the right side of eq 14 results from the use of volume fractions in the free energy of mixing. Johansson et al.12 have suggested that the activity coefficient in eq 13 be written as γ ) γxγPY, where γPY accounts for intermicelle interactions in a polymer-free solution via a Percus-Yevick hard-sphere expression. The parameter γx accounts for the excluded volume in the polymer network that is unavailable to a micelle with radius rx. It is calculated by using Ogston’s32 result for the distribution of space sizes in a random array of fibers and is given by

((

γx ) exp φp

))

rx + r p rp

2

(15)

where φp is the polymer fiber volume fraction, rx is the radius of the sphere, and rp is the polymer fiber radius. Physically, eq 15 states that γx is the inverse of the probability of inserting a spherical micelle with x surfactants into a polymer network. As γPY is a function only of the total surfactant volume fraction, it does not contribute to changes in the micelle distribution in gels relative to pure solution and therefore does not need to be explicitly evaluated in this work.12,55 In eq 14, the micelle shape can range from being spherical to being cylindrical with spherical end caps. To account approximately for the interaction of cylindricalshaped micelles within a polymer fiber network, the equivalent sphere radius for the cylinder is calculated as

rx ) R

(

1 ar2 arcsin x1 - ar-2 + 2 2xar2 - 1

)

(16)

Here R ) lc + lhl is the short axis of the cylinder, lc is the length of the surfactant tail group, lhl is the length of the surfactant headgroup, and ar is the ratio (lx/R) of the long axis to short axis.12,56 The functional relationship between γx and x comes from the long axis of the cylinder (54) Desnoyers, J. E.; Caron, G.; DeLisi, R.; Roberts, D.; Roux, A.; Perron, G. J. Phys. Chem. 1983, 87, 1397. (55) Ben-Shaul, A.; Gelbart, W. M. J. Phys. Chem. 1982, 86, 316. (56) West, R. Biopolymers 1988, 27, 231.


Langmuir, Vol. 16, No. 3, 2000 915

lx, which is calculated from the micelle and surfactant geometry via

xVhc

lc + lhl lx ) + 2πl 2 3

(17)

c

Here Vhc is the volume of the surfactant tailgroup.29 In calculating the activity coefficient from eqs 15 and 16, we are neglecting any effects of micelle flexibility. Micelles can be assumed to be rigid for ratios of the contour length to persistence length much less than unity,26,57 a constraint which should be readily satisfied for most of the experimental systems studied here. One exception may be C12E6 micelles at high temperature for which this ratio is of order unity,26,57 and therefore, these micelles may possess limited flexibility. Carale et al.58 describe in detail a method for calculating gmic in eqs 13 and 14 and demonstrate its efficacy for the C12Em surfactants studied here. A brief summary of the method follows. The free energy of micellization has four contributions: the transfer of the hydrocarbon tail groups from water to the micelle core (gw/hc); the formation of the hydrocarbon core-water interface (gσ); the packing of the hydrocarbons in the micelle core (gpack);59 and the formation of the PEO headgroup layer (ghl). These parameters are functions of the surfactant and micelle geometry. In particular, the contribution ghl is determined from the unfavorable confinement of the PEO headgroups at the micelle interface. To quantify this effect, Carale et al. use enthalpy of solution data for PEO solutions, critical micelle concentration (CMC) data for C12E6, and a Monte Carlo calculation to evaluate the thickness of the headgroup layer. These calculations are performed at 25 °C and can be used only at that temperature, given the sensitivity of the PEO headgroup conformation to temperature changes.30,60,61 The sum, ggeo mic ) gw/hc + gσ + gpack + ghl, is minimized with respect to lc for spherical and cylindrical cyl micelle geometries to obtain gsph mic and gmic, respectively. After one obtains gmic, the weight average aggregation number can be calculated by specifying the volume fraction of surfactant φs, choosing an initial guess for φ1, and using eq 13 to calculate iteratively the first moment of the distribution, where

Mj )

∑x xj-1φx

(18)

until M1 ) φs. The weight average aggregation number is then given by

aw )

M2 M1

(19)

and the standard deviation σ of the micelle distribution is

σ)

x

M3 -1 M2

(20)

The value of φ1 at which M1 ) φs is the CMC. (57) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Langmuir 1998, 14, 6013. (58) Carale, T. R.; Pham, Q. T.; Blankschtein, D. Langmuir 1994, 10, 109. (59) Naor, A.; Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 7830. (60) Karlstro¨m, G. J. Phys. Chem. 1985, 89, 4962. (61) Nilsson, P. G.; Lindman, B. J. Phys. Chem. 1983, 87, 4756.

3. Materials and Methods Materials. Polyoxyethylene 6 lauryl ether (C12E6), polyoxyethylene 10 lauryl ether (C12E10), cetylpyridinium chloride (CPC), glycogen (type VII), and agarose (type VII: Low Gelling Temperature) were used as received from Sigma. Type VII agarose was chosen so that the samples would gel below the cloud point of the combined agarose-surfactant solution. Octaethylene glycol monododecyl ether (C12E8) was used as received from Fluka. Reported critical micelle concentrations for C12E8 and C12E6 at 25 °C are 8.1 × 10-5 and 6.6 × 10-5 M, respectively,58 while the CMC for C12E10 at 22 °C is 8.4 × 10-5 M.62 Pyrene was obtained from Aldrich and was recrystallized twice from ethanol. Ethanol (200 proof, dehydrated) was used as received from Quantum. Sample Preparation. Dissolved oxygen in aqueous solutions is known to be an efficient fluorescent quencher of pyrene, and could affect the aggregation number measured in a fluorescence quenching experiment under the conditions of this study.25 As a result, it was necessary to remove the dissolved oxygen. The water used to prepare the aqueous solutions was deoxygenated by bubbling argon through it while vigorously stirring for at least 0.5 h. The deoxygenated water was then transferred to tared vials containing a known amount of surfactant or agarose. To prevent the reentry of oxygen into the samples and to remove any oxygen which may have diffused back into solution during transfer, argon was circulated over the top of the solutions. Evaporative losses were monitored by measuring the weight of the sample vials before the solutions in them were transferred. The agarose-surfactant samples were prepared by making concentrated stock solutions of surfactant and agarose. These solutions were diluted with each other to obtain the proper surfactant and agarose concentrations. The agarose stock solutions were prepared by mixing agarose powder with the deoxygenated water and then boiling the solution under argon. The concentrated agarose solution was then cooled to 40-50 °C and held at this temperature. The surfactant stock solutions were prepared by adding deoxygenated water to sample vials containing surfactant. To these vials 5 µL of a stock solution of pyrene in ethanol and 5-10 µL of a stock solution of CPC in ethanol were added. The concentration of pyrene (1.0-4.0 µM) was chosen so that the average number of pyrene molecules per micelle was approximately 0.01. The concentration of CPC (5-26 mM) was chosen so that the average number of CPC molecules per micelle was approximately unity. The small amount of ethanol added to the sample, approximately 0.2-0.3 vol %, did not significantly affect the measured aggregation numbers. The surfactant stock solutions were then heated to 40-50 °C and mixed under argon for at least 0.5 h. The agarose stock solution was then transferred to the surfactant stock solutions, and the resulting samples were allowed to mix for at least 1 h at 40-50 °C. After the samples had been allowed to mix, they were transferred to cuvettes (Starna type 23-SOG-10) and sealed with septum taken from Vacutainer vials (tube size 10.25 × 64 mm). Argon was circulated over the solution in the cuvette for several minutes via hypodermic needles passed through the septum. The samples prepared in this way remained oxygen-free for at least 1 month. The solutions were then allowed to cool below the gelation temperature (ca. 30 °C) and left to stand overnight at room temperature. Steady-State Fluorescence. The steady-state emission spectra from the samples were recorded by using a Spex Fluoromax in the front face geometry (22° detection angle) and with a slit width of 0.3 mm. The samples were excited at 335 nm and the fluorescence emission intensity was recorded from 360 to 500 nm. The ratio of the first to third peaks of the pyrene emission spectra has been shown to correlate with the polarity of the environment in which the pyrene molecule is located.16,63,64 FDFQ. An introduction to the frequency domain method and instrumentation can be found in Lakowicz34 and in Jameson et al.35 An ISS K2-Analogue phase fluorometer with a crosscorrelation frequency of 40 Hz was used to measure the (62) Shi, Z.; LaTorre, K. A.; Ghosh, M. M.; Layton, A. C.; Luna, S. H.; Bowles, L. Sayler, G. S. Water Sci. Technol. 1998, 38, 25. (63) Dong, D. C.; Winnik, M. A. Can. J. Chem. 1984, 62, 2560. (64) Kalyanasundaram, K.; Thomas, J. K. J. Am. Chem. Soc. 1977, 99, 2039.

916


Nolan et al.

Table 1. I1/I3 Ratios for C12E8, C12E10, and C12E6 (1 wt %)a temp °C 12 22 40 60 a

C12E8 ((1.5%)

C12E10 ((1.5%)

1.22

1.21 1.14 1.05

C12E6 ((1.5%) 1.22 1.19 1.14

Values are independent of agarose gel concentration.

fluorescence lifetimes. The samples were excited at 335 nm by using a 300 W xenon arc lamp. The excitation wavelength was selected by an ISA model H1061 monochromator with 2 mm slits (16 nm band-pass). The fluorescence emission was detected at 390 nm by using an Oreil interference filter (model number 53425) with a bandwidth of 10.5 nm. A glycogen solution was used as a scattering reference. The Hamamatsu (model R928) photomultiplier tubes supplied with the fluorometer are of a type for which color errors resulting from using a scattering reference should not affect the measurements.35 On 3-10 separate samples, measurements of the phase lag and the demodulation factor were made at 10-15 logarithmically spaced frequencies over a range of 0.5-5 MHz. Curve fits of eqs 4 and 5 to the measured phase and modulation given one of the assumed models for G(t) were made by using a Levenberg-Marquardt method adapted from a published algorithm.43 The Appendix gives analytical expressions for eqs 6 and 7 for the Infelta and Gehlen models. For the Almgren model, eqs 6 and 7 were evaluated numerically. Errors were calculated by using 95% confidence intervals. The scattering component of the emission was either a fit parameter or it was measured in a separate experiment. When one is fitting for ξs in samples containing both the fluorescence probe and quencher, the fit results tend to overestimate the quenching kinetic parameter kq while underestimating ξs if the amount of scattered light is small (i.e., if ξs e 0.07 as measured in an experiment without quencher, as discussed below). As a result, it is best in these cases to subtract out the scatter from the data or fix ξs in the fit procedure by using a measured value. It is possible to estimate As, and hence ξs, via eq 11, by using data from an experiment with the fluorescent probe but no quencher. The phase lag and demodulation factor are measured on samples containing a fluorescent probe of a single lifetime. In this case, G(t) is given by a single term of eq 31 given in the Appendix where R0 is set equal to one and the lifetime of the unquenched fluorophore τ0 ) 1/k is a fit parameter. The other fit parameter is ξs, obtained from substituting eq 8 into eqs 6 and 7. The steady component of the scattered light As is then determined from eq 11 and used to adjust results from experiments in the presence of quencher for the effect of scattering (eqs 9 and 10). No extra work is involved in this procedure since measurements of k from solutions without quencher are required in any case. Direct measurement of ξs circumvents the degeneracy between quenching components and the scattering component. For the ISS K2 fluorometer, this procedure works well when the amount of scattering is small (ξs < 0.07) as measured in a sample without quencher. If the scattering component is large (ξs > 0.07) in a sample without quencher, then this procedure does not yield accurate enough values of ξs to obtain a good fit to the data. Fortunately, when the scattering is large ξs is well resolved as a fit parameter in experiments with quencher.

4. Results Steady-State Fluorescence. Table 1 gives the micelle polarity as measured by the ratio of the first (I1) and third (I3) peaks of the pyrene emission spectra. These values are independent of gel concentration and indicate that micelles are present in the gel at all temperatures studied. The apparent decrease in polarity with increasing temperature is partly due to the change of the pyrene polarity scale with temperature.65 However, this decrease could also reflect the increasing hydrophobicity of the poly(ethylene oxide) headgroups with increasing tempera(65) Nivaggioli, T.; Alexandridis, P.; Hatton, T. A.; Yekta, A.; Winnik, M. A. Langmuir 1995, 11, 730.

Figure 1. Variation of n (2), k (b), and kq (9) with quencher concentration for a 1 wt % C12E8 solution at 22 °C. The error in k is approximately the size of the symbol. Table 2. Comparison of Fluorescence Quenching Parameters for a 1 wt % C12E8 Solution as Determined by Frequency Domain (FDFQ) and Single Photon Counting (SPC)a technique

T/°C

a

k/106 s-1

A2/106 s-1

FDFQ SPC22 SPC66 SPC23 SPC21

22 25 22 22 22

86((3) 99 ∼85 ∼80 ∼105

2.82((0.06) 2.75

2.80((0.08) 2.86

kq/106 s-1 6.8((0.3) 7.70 ∼6.5

a A is the value of k measured in the presence of quencher. 2 Reported errors for the SPC technique are k ( 3%, A2 ( 10%, n ( 10%, and kq ( 10%.21

ture.19,60,61 In general, the I1/I3 ratios for all of these surfactant micelles are similar to I1/I3 ratios for other CnEm surfactants.18,19 FDFQ. Single photon counting (SPC) has been used to determine the aggregation number of C12E8 in water in a number of studies.21-23,66 Consequently, it is an excellent test surfactant for the frequency domain method. Figure 1 shows the variation with respect to quencher concentration of the average number of quencher molecules per micelle n ) [Q]/[M], where [M] is the micelle concentration and [Q] is the quencher concentration. Also shown in Figure 1 are the first-order rate constant k for the decay of an unquenched fluorescent probe and the first-order rate constant kq for the decay of a fluorescent probe in a micelle with one quencher molecule. As k determined in the absence of quencher equals k determined in the presence of quencher within experimental error, the migration of quenchers can be neglected in the Infelta model.20,21,23 Thus, the fluorescence quenching model predicts that n should vary linearly with CPC concentration while k and kq should be independent of it. Within experimental error, the data in Figure 1 are consistent with this prediction. Table 2 compares averages over all quencher concentrations from the FDFQ technique shown in Figure 1 with literature values from the SPC technique. The agreement between the FDFQ experimental data and the SPC values from literature is excellent. Aggregation numbers a for C12E8 and C12E10 as a function of temperature and gel concentration are shown in Figure 2. These values for a are obtained from measured n results together with the known surfactant concentrations. The Infelta model was used with eqs 6 and 7 to analyze the data since, within experimental error, k determined in the presence of quencher was again equal to the value of k determined in the absence of quencher at all temperatures and gel concentrations. Although there (66) Anthony, O.; Zana, R. Langmuir 1994, 10, 4048.


Figure 2. Variation of aggregation number a with agarose gel concentration for 1 wt % C12E8 at 22 °C (2) and 1 wt % C12E10 at 22 °C (×), 40 °C (1), and 60 °C (b). The errors in aggregation numbers for C12E10 at 22 and 40 °C are approximately 10%.

is relatively little published information about C12E10, the aggregation numbers in water reported here are consistent with size measurements from holographic interferometry and atomic force microscopy, which indicate that C12E10 micelles are somewhat larger than C12E8 micelles.13,67 A larger C12E10 micelle has been attributed to polydispersity in the number of ethylene oxide monomers of the C12E10 headgroup.67 For C12E8, there is no change in the aggregation number with increasing gel concentration at 22 °C. For C12E10, at 22 °C there is a small decrease in aggregation number with increasing gel concentration, at 40 °C there is no change, and at 60 °C there is a small increase with increasing gel concentration. For C12E6 in agarose gels at all temperatures investigated, and in water at 40 °C, fitting the Infelta model to the data revealed that k determined in the presence of quencher was not equal to k determined in the absence of quencher. Two possible interpretations of this result are that the gel contains very large, polydisperse micelles or that moderately sized micelles are present which exchange quencher by micelle collisions during the time scale of the fluorescence experiment.16-28 From the fluorescence data alone, it is not possible to distinguish with certainty between these two cases.27,47,68-72 If the former case holds, this suggests substantial micelle growth to form large cylindrical micelles which interpenetrate the gel network. Such growth should be accompanied by a significant decrease in surfactant diffusion, contrary to the gradient diffusion measurements of Kong et al.13 Moreover, the cloud point measurements of Penders et al.10 suggest that the gel suppresses the formation of large micelles, preventing subsequent phase transition at the cloud point of agarose-surfactant solutions. Hence, it seems likely that it is the mechanism of quencher exchange that is causing the disparity between values for k measured in the presence and absence of quencher. However, for completeness both possibilities are considered. The model proposed by Almgren et al.27,47,48 can be used to investigate systems consisting of very long, cylindrical micelles. Medhage et al.25 have used this model to investigate the hexagonal phase of long cylinders formed (67) Patrick, H. N.; Warr, G. G.; Manne, S.; Aksay, I. A. Langmuir 1997, 13, 4349. (68) Almgren, M.; Lo¨froth, J.-E. J. Chem. Phys. 1982, 76, 2734. (69) Binana-Limbele´, W.; Van Os, N. M.; Rupert, L. A. M.; Zana, R. J. Colloid Interface Sci. 1991, 141, 157. (70) Malliaris, A.; Lang, J.; Strum, J.; Zana, R. J. Phys. Chem. 1987, 91, 1475. (71) Luo, H.; Boens, N.; Van der Auweraer, M.; De Schryver, F. C.; Malliaris, A. J. Phys. Chem. 1989, 93, 3244. (72) Makhloufi, R.; Hirsch, E.; Candau, S. J.; Binana-Limbele, W.; Zana, R. J. Phys. Chem. 1989, 93, 8095.

Langmuir, Vol. 16, No. 3, 2000 917

Figure 3. Variation of the aggregation number a (9) and the exchange rate km (]) of the quencher (or probe) between micelles with agarose gel concentration for 1 wt % C12E6 at 12 °C.

Figure 4. Variation of the aggregation number a (9) and the exchange rate km (]) of the quencher (or probe) between micelles with agarose gel concentration for 1 wt % C12E6 at 22 °C. The line is the theoretically predicted exchange rate in the absence of intermicelle interactions (cf. eq 21).

by C12E6 at high surfactant concentration. At 12 °C, the Almgren model does not fit the FDFQ data. At 22 °C, unphysical values for the quenching rate constant, kq, were obtained from this model. Thus, the analyses at 12 and 22 °C support the conclusion that large cylindrical micelles are not present in these gels. At 40 °C, the Almgren model yields reasonable fits to the FDFQ data, and additional experimental techniques, such as electron paramagnetic resonance to measure changes in micelle aggregation73 and relaxation methods to measure the surfactant dynamics,74 would be useful to completely rule out the presence of very large micelles. However, the balance of FDFQ, gradient diffusion, and cloud point behavior at all three temperatures indicate that the Infelta or Gehlen models are probably more appropriate for fitting the FDFQ data. Aggregation numbers and rates of exchange km of pyrene and CPC for 1 wt % C12E6 in agarose gels at 12, 22, and 40 °C are shown in Figures 3-5. The measured aggregation numbers for C12E6 micelles in water are consistent with those from other TRFQ measurements at the temperatures studied.21,24 At all temperatures, the aggregation number decreases upon incorporation into the gel, with this decrease quite pronounced at 40 °C. The decrease upon increasing gel concentration is small. The rate of exchange increases with gel concentration at all temperatures. If it is assumed that exchange occurs due to micelle collisions, and on the basis of TRFQ studies where exchange of pyrene is observed, it is likely that both probe and quencher are migrating between ag(73) Bales, B. L.; Stenland, C. Chem. Phys. Lett. 1992, 200, 475. (74) Herrmann, C.-U.; Kahlweit, M. J. Phys. Chem. 1980, 84, 1536.

918


Nolan et al. Table 5. Results from Eqs 19 and 20 for C12Em Micelles with m ) 6, 8, and 10 at 25 °C (rp ) 19 Å 13) surfactant

φp

φ1/10-6

aw

σ

C12E6

0 1 2 3 4 0 4 0 4

1.604 1.606 1.607 1.609 1.610 2.036 2.049 2.314 2.332

174 171 169 167 165 52 52 50 50

12.3 12.1 12.0 11.9 11.8 10.1 10.1 10.2 10.2

C12E8 C12E10

Figure 5. Variation of the aggregation number a (9) and the exchange rate km (]) of the quencher (or probe) between micelles with agarose gel concentration for 1 wt % C12E6 at 40 °C. Table 3. Quenching Rate kq/106 s-1 as a Function of Temperature and Gel Concentration for C12E8 and C12E10 surfactant

[agarose]/wt %

22 °C (( 21%)

40 °C (( 35%)

60 °C (( 14%)

C12E8

0 1 2 3 4 0 1 2 3 4

7 6 9 9 9 5 7 7 6 6

7 8 6 10 10

9 7 7 6 6

C12E10

Table 4. Quenching Rate kq/106 s-1 as a Function of Temperature and Gel Concentration for C12E6 surfactant

[agarose]/wt %

12 °C (( 10%)

22 °C (( 14%)

40 °C (( 25%)

C12E6

0 1 2 3 4

3 6 6 5 6

3 6 7 5 9

2 8 15 7 21

gregates.20,21,23,27 As a result, the Gehlen model was used with eqs 6 and 7 to analyze the data. To reduce the number of fit parameters in the Gehlen model, the exchange rates for the probe and quencher were assumed to be equal.2 [If the fluorescence data are fit with a model in which only quencher is assumed to exchange, values for km are approximately a factor of 2 greater than for the model in which both probe and quencher are allowed to exchange. Results for the aggregation number and kq are the same from both models.] Tables 3 and 4 show the variation of kq with temperature and gel concentration for each of the surfactants in this study. For C12E8 and C12E10 at 22 and 40 °C, there is a slight variation in kq with increasing gel concentration which is probably not significant. As mentioned in section 3, scattering can influence the value of the quenching rate, and the error in kq increases from less than 15% in water to 20-35% in gels due to the increase in scattering from the gel. For C12E10 at 60 °C the scattering decreases (probably due to partial melting of the gel), and more accurate measurements of kq are obtained. In this instance, the quenching rate appears to decrease slightly. For C12E6, the quenching rate in the gel is larger than the value in water and generally increases with increasing gel concentration. In these quenching experiments, ξs is a fit parameter. While errors for kq (ca. (15-20%) are smaller than those for C12E8 and C12E10, scatter may still be influencing these values. However, even after the ac-

counting for an apparent increase due to the scattering and experimental error, the increase in kq is significant. As discussed below, an increase in quenching rate is expected with decreasing micelle size.23,75,76 Thermodynamic Model. Table 5 shows the results of eqs 19 and 20 applied at a temperature of 25 °C, as described in section 2. For C12E8 and C12E10 the model predicts spherical aggregates in water. In an agarose gel, these micelles are unchanged. The effect of the polymer fiber network on the more polydisperse cylindrical C12E6 micelles is to increase the unmicellized surfactant volume fraction φ1, decrease the aggregation number aw, and decrease the standard deviation σ. Examination of eqs 15-17 shows that the large aggregates in a polydisperse size distribution will have a low probability of finding space in the gel large enough to accommodate them. This low probability results in a large activity coefficient for these micelles. The net result on the micelle distribution (cf. eq 13) is that it is shifted to smaller, more monodisperse micelles. 5. Discussion Both the experimental and theoretical results show that the aggregation numbers of small monodisperse micelles are unaffected upon incorporation into an agarose gel matrix. While there may be a weak repulsive interaction between the surfactant and agarose,10 these results indicate that such an interaction does not result in changes in the aggregation number upon incorporation in agarose gel. The FDFQ results are consistent with gradient diffusion measurements in agarose gels.13 The gradient diffusion measurements were interpreted by using a hydrodynamic theory that treats the C12Em aggregates as hard spheres with a radius equal to their hydrodynamic radius in water, diffusing within an aqueous medium containing stiff, randomly distributed fibers.14 The good agreement between these calculations, which have no adjustable parameters, and the diffusion data for C12E8 suggest that the aggregates do behave approximately as hard spheres and that their size is unchanged from that in water. Moreover, the aggregation number of C12E8 micelles also changes little in carrageenan gels, further supporting the idea that the aggregate structure is unaffected by steric interactions in gels.12 The aggregation behavior of C12E6 is different than C12E8 in that it forms larger micelles at any given temperature. This is probably the result of its smaller headgroup, which in a spherical micelle cannot shield the hydrophobic core effectively. On the basis of the packing arguments of Israelachvili,77 a cylindrical geometry would allow a smaller headgroup to protect the micelle core more (75) Van der Auweraer, M.; Dederen, J. C.; Gelade´, E.; De Schryver, F. C. J. Chem. Phys. 1981, 74, 1140. (76) Van der Auweraer, M.; De Schryver, F. C. Chem. Phys. 1987, 111, 105. (77) Israelachvili, J. Intermolecular & Surface Forces, 2nd ed.; Academic Press: London, 1992; Chapters 16 and 17.


Langmuir, Vol. 16, No. 3, 2000 919

completely. However, such cylindrical micelles will be larger (i.e., prolate ellipsoid micelles) and more polydisperse.77 The thermodynamic model shows that the distribution of distances between the unyielding fibers of the gel network discourages the formation of large micelles since they will experience more unfavorable surfactantpolymer steric forces than smaller micelles. Interestingly, the average space size in a 4 wt % agarose gel is about 6 nm according to Ogston’s distribution.32 The experimentally measured aggregation number for C12E6 micelles in 4 wt % agarose gels at 40 °C is 152, which corresponds to an equivalent sphere radius (rx in eq 16) of about 5 nm. By comparison, the equivalent sphere radius for a C12E6 micelle in water, calculated by using eq 17 and assuming an aggregation number of 400, is 11 nm, substantially larger than the average gel pore spacing. Thus, this comparison supports the idea that significant rearrangement of C12E6 must occur in the gel relative to its solution properties. The FDFQ results for C12E6 in this work also are qualitatively consistent with the diffusion results of Kong et al.13 In a 4 wt % gel, the rate of diffusion is somewhat higher than would be expected on the basis of the aggregation numbers in the gel. However, this comparison assumes the micelles diffuse as hard spheres, which may not be accurate for cylindrical micelles. The C12E6 results indicate there are significant differences between how this surfactant behaves in agarose gels compared with its behavior in the absence of agarose.21,24,78,79 At 12 °C, where C12E6 aggregates are about the same size as C12E10 aggregates at 22 °C, there was a detectable increase in the exchange rate km between aggregates in gel as compared to aggregates in water. As either the temperature or the gel concentration increased, km also increased. Penders et al.10 used cloud point and self-diffusion measurements to show that there is a weak repulsive interaction between C12Em surfactants and agarose. This weak interaction can lead to an increased concentration effect in which the micelles are forced to be closer together than they normally would be in water.12,80,81 This enhanced concentration effect would increase the probability of collisions between micelles, as well as augmenting attractive interactions between micelles. Thus, the increase in exchange rate for C12E6 could be due to the aggregates being in closer proximity to each other within the gel. Significant exchange was not observed between C12E8 or C12E10 micelles. This suggests that the larger headgroups of these surfactants are able to slow the transfer of probe and quencher upon micelle collision relative to C12E6. To investigate further this C12E6 micellar exchange phenomenon, the diffusion results of Kong et al.13 can be combined with Smoluchowski’s82 theory for particle collisions in the absence of interparticle interactions to calculate a theoretical exchange rate. Assuming that the probe and quencher always exchange during a micellar collision, an estimate of km is

km ) 8πDrxno

(21)

where D is the diffusion coefficient of the micelle, rx is the (78) Balmbra, R. R.; Clunie, J. S.; Corkill, J. M.; Goodman, J. F. Trans. Faraday Soc. 1964, 60, 979. (79) Herrington, T. M.; Sahi, S. S. J. Colloid Interface Sci. 1988, 121, 107. (80) Feitosa, E.; Brown, W.; Vasilescu, M.; Swanson-Vethamuthu, M. Macromolecules 1996, 29, 6837. (81) Feitosa, E.; Brown, W.; Swanson-Vethamuthu, M. Langmuir 1996, 12, 5985. (82) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, U.K., 1986; Vol. 1, Chapter 7.

equivalent radius of the micelle, and no is the number density of C12E6 aggregates. As discussed above, the effective concentration of aggregates is expected to be higher in the gel than for the same amount of surfactant in the absence of the gel. This effect may be approximately determined by following the reasoning of Johansson et al.,12 who used the activity coefficient given by eq 15 to calculate the effective concentration of aggregates in a κ-carrageenan gel. This effective concentration φw is given by ∞

φw )

( ( ))

∑φx exp φp

x)1

rx + r p rp

2

(22)

At 22 °C, the gel also decreases the size of the C12E6 micelles, effectively increasing no. The volume fraction of surfactant φw is approximately equal to the mass concentration cs of surfactant in g/mL; therefore, as an estimate no ) csNa/(MWsaw), where Na is Avogadro’s number and MWs ) 450.7 g/mol is the molecular weight of C12E6. The results for km as calculated by eq 21 are shown by the solid line in Figure 4, and this straightforward calculation agrees surprisingly well with the experimentally measured exchange rates. Taken as a whole, the fluorescence quenching results and gradient diffusion measurements on C12Em surfactants in agarose gels suggest that constraining effects of the gel fibers and an enhanced concentration effect could both play a role in these surfactant hydrogels. Further, these two phenomena could influence the aggregation properties of the surfactant micelles in opposite ways. Enhanced micelle concentration effects in the gel could strengthen attractive interactions between micelles, leading to an increase in aggregate size. On the other hand, the thermodynamic theory presented in this work shows that if the aggregates are not allowed to grow beyond the bounds of the agarose fibers, then the aggregation number will decrease. Results from both FDFQ and holographic interferometry suggest that, for stiff agarose gel fibers, the latter influence predominates at temperatures up to 40 °C. At a higher temperature (60 °C) near the gel melting point, the gel fibers loosen and swell, and effects of enhanced surfactant concentration become more important, as they are in more flexible gels12 and in polymer solutions.10,80,81,83 Enhanced concentration effects may also promote higher rates of intermicellar exchange within agarose gels. The interplay between these two phenomena is consistent with the observed clouding behavior observed by Penders et al.10 Agarose gels have been observed to suppress surfactant phase separation at the cloud point, while that phase separation is facilitated in agarose solutions.10,11 The experimental results for C12E10 at 60 °C indicate some micellar growth with increasing agarose gel concentration. This trend is in apparent contrast to the predictions of the proposed thermodynamic theory. However, the latter accounts only for changes in micelle structure due to the presence of the gel, whereas at 60 °C changes in the gel structure also must be considered. The agarose gel melts at approximately 65 °C, and the gel structure begins to change as it approaches its melting point.84 The gelling process of agarose polymer chains is thought to occur by formation of double helixes and then (83) Li, X.; Lin, Z.; Cai, J.; Scriven, L. E.; Davis, H. T. J. Phys. Chem. 1995, 99, 10865. (84) Clark, A. H.; Nishinari, K.; Ross-Murphy, S. B.; Watase, M. Macromol. Symp. 1995, 93, 187.

920


aggregation of these helixes into stiff bundles.85,86 If the gel melts by the reverse of its process of formation, then the bundles of agarose fibers must come apart during this process. At a fixed volume fraction of fibers, a medium of thin fibers has less space available to a spherical micelle than a medium of thick fibers. Consequently, as the fibers separate, they exclude more volume within the gel as well as becoming more flexible. Thus, the gel becomes more like an agarose solution and the micelles are concentrated further by the melting process. At 60 °C, C12E10 micelles in water grow relative to their structure at lower temperatures and most likely become more cylindrical or prolate ellipsoid in shape.30 Since the aggregation number of cylindrical micelles should increase with an increase in the surfactant concentration,77,87 an increase in the aggregation number should be expected if the gel is concentrating the micelles. This explanation is consistent with studies on CnEm surfactants in polymer solutions which exhibit segregative phase behavior.66,80,81,83,88-93 As the agarose fiber bundles come apart, it seems reasonable that they would also be more flexible, allowing the aggregates to push them away. In this case, there would be less restriction on aggregate growth and the thermodynamic theory presented in this work is not applicable. The situation of C12E10 in agarose gels at 60 °C is similar to that of C12E6 in carrageenan gels studied by Johansson et al.12 While agarose and carrageenan polymers are both polysaccharides with similar molecular structures, the carrageenan polymers are charged. The charges impede the aggregation of carrageenan polymers into stiff fiber bundles.94-96 Thus, the fibers of a carrageenan gel are more flexible and occupy more space than agarose fibers.13 Johansson et al. found that self-diffusion of C12E6 aggregates is slower in carrageenan gels than the value measured by Kong et al.13 in agarose gels and that the aggregates are larger. The thermodynamic model developed here would be more appropriate for an agarose gel well below its melting point than for κ-carrageenan, since the gel fibers are more rigid and compact in the former.

Nolan et al.

not in the time domain. Consequently, models describing fluorescence quenching in micelles which have analytical solutions only in the frequency domain can be applied directly to the experimental data, without further approximations or numerical transformations. The aggregation behavior of three nonionic surfactants in a nonionic hydrogel was investigated as a function of temperature and gel concentration. For small spherical micelles such as C12E8, an agarose gel was found to have no influence on the micelle aggregation number over the range of 1-4 wt % gel concentration and at 22 °C, whereas in the case of C12E10 there appeared to be a small decrease at 22 and 40 °C. These results are in agreement with measurements of the micelle diffusion coefficient in agarose gels as well as measurements on C12E8 in other nonionic polymer solutions. For C12E10 in agarose gels at 60 °C, the gel promoted a slight growth of the surfactant aggregates. This was explained by the combination of the temperature-induced geometric changes of the C12E10 micelles that would occur in the absence of gel and the melting of the agarose gel which occurs near this temperature. Like C12E10 at 22 °C, the aggregation number of the comparably sized C12E6 micelles at 12 °C was relatively unaffected by the gel; however, the rate of exchange of probe and quencher was increased in the latter. At 22 and 40 °C, the aggregation number of C12E6 was found to decrease with increasing gel concentration while the rate of exchange was found to increase with increasing gel concentration. These results were explained by using the idea of an increased concentration effect and an assumption that the agarose gel network limits micelle size. A thermodynamic model incorporating both these ideas and the micelle size distribution predicted that small, spherical micelles would be unchanged in an agarose gel network while the weight average aggregation number for large cylindrical micelles would decrease with increasing gel concentration. These predictions are consistent with the experimental findings at temperatures low enough so that the structure of the agarose gel remains stable.

6. Conclusions A frequency domain fluorescence quenching technique for measuring micelle aggregation numbers was presented. This technique yielded results for C12E6 and C12E8 micelles in water which are in excellent agreement with those from pulse (SPC) techniques. The FDFQ method was also used to investigate the aggregation properties of surfactant micelles in a hydrogel. One advantage of the FDFQ technique over pulse methods is that its data are (85) Arnott, S.; Fulmer, W.; Scott, W. E.; Dea, I. C. M.; Moorhouse, R.; Rees, D. A. J. Mol. Biol. 1974, 90, 269. (86) Djabourov, M.; Clark, A. H.; Rowlands, D. W.; Ross-Murphy, S. B. Macromolecules 1989, 22, 180. (87) Nagarajan, R. Langmuir 1994, 10, 2028. (88) Piculell, L.; Bergfeldt, K.; Gerdes, S. J. Phys. Chem. 1996, 100, 3675. (89) Drummond, C. J.; Albers, S.; Furlong, D. N. Colloids Surf. 1992, 62, 75. (90) Zhang, K.; Jonstro¨mer, M.; Lindman, B. Colloids Surf. A 1994, 87, 133. (91) Zhang, K.; Jonstro¨mer, M.; Lindman, B. J. Phys. Chem. 1994, 98, 2459. (92) Zhang, K.-W.; Karlstro¨m, G.; Lindman, B. Colloids Surf. 1992, 67, 147. (93) Wu¨stneck, R.; Buder, E.; Wetzel, R.; Hermel, H. Colloid Polym. Sci. 1989, 267, 516. (94) Watase, M.; Nishinari, K. Carbohydr. Polym. 1989, 11, 55. (95) Watase, M.; Nishinari, K.; Williams, P. A.; Phillips, G. O. Polym. J. 1990, 22, 991. (96) Nishinari, K.; Watase, M.; Miyoshi, E.; Takaya, T.; Oakenfull, D. Food Technol. 1995, 49, 90.

Acknowledgment. We gratefully acknowledge the help with the ISS instrument provided by Bruce Schyler, Professor August Maki, and the UCD Chemistry department. The excellent technical support of Dr. Martin vandeVen of ISS is also appreciated. This work was supported by NSF Young Investigator Award CTS-9358508 to S.R.D., NSF CAREER Award BES-950663 to R.J.P., and a Jastro Shields Graduate Research Scholarship Award from UC Davis to S.L.N. Support from Procter and Gamble Corp. to S.R.D. is also gratefully acknowledged. Appendix Starting from eq 3, the objective is to obtain eqs 6 and 7. The excitation intensity f(t) ) a + b sin(ωt) can be expressed as

f(t) ) a + Re{-jbejωt}

(23)

where j ) -11/2. Substitution of eq 23 into eq 3 yields

I(t) )

∫-∞t aG(t - t′) dt′ + ∫-∞t Re{-jbejωt′}G(t - t′) dt′ + Rsf(t)

(24)


Langmuir, Vol. 16, No. 3, 2000 921

Making the substitution z ) t - t′ gives

I(t) )

identified as

∫0∞aG(z) dz + ∫0∞ Re{-jbejω(t-z)}G(z) dz +

Rsf(t) (25)

Reexpressing eq 25 in terms of sine and cosine functions and simplifying yields

∫0∞G(z) dz + Rs) + b(∫0∞G(z) cos(ωz) dz + ∞ Rs) sin(ωt) - b(∫0 G(z) sin(ωz) dz) cos (ωt) (26) A ≡ a(

∫0∞G(z) dz + Rs)

(27)

then eq 26 can be written as

b I(t) ) A + A (D sin(ωt) - N cos(ωt)) a

τi )

b B)A M a

(30)

(36)

∫0∞G(z) sin(ωz) dz ) (1 - kp D′)N′2 + kp2N′2

where N and D are given by eqs 6 and 7. Equation 28 can therefore be expressed as

(29)

1 (A2 + iA4)

where A2, A3, and A4, are fit parameters and functions of n, k, kq, and km.16,28,44 In the sums of eqs 32 and 33, increasing the number of terms beyond 10 does not significantly affect the calculated value of the phase and modulation. For the Gehlen model, a method presented by Seborg et al.97 can be used to obtain

(28)

I(t) ) A + B sin(ωt - Φ)

(35)

and

I(t) ) a(

If A is defined as

A3ie-A3 i!

Ri )

m

∫0∞G(z) cos(ωz) dz )

(37)

m

D′(1 - kpmD′) - N′2kpm 2 (1 - kpmD′)2 + kp2 m N′

(38)

where and ∞

and Φ and M are given by eqs 4 and 5, respectively. While eqs 6 and 7 can be evaluated numerically by fast Fourier transform methods for any function G(t), the fitting procedure is considerably faster if analytical expressions are used. If G(t) is given by a sum of exponentials of the form

( )

∞

G(t) )

t

Ri exp ∑ τ i)0

(31)

∫0∞G(z) dz )

∫0 ∫0

G(z) cos(ωz) dz )

∞

∑ i)0

1 + ω2τi2

∞

∑ i)0

Riτi

1 + ω2τi2

(32)

A3ie-A3 i!

(40)

1 (kpm + A2 + iA4)

(41)

∫0

G(z) dz )

τi )

Here, kpm is the probe exchange rate between micelles, assumed to be equal to km in this study, and A2, A3, and A4 are the same as used in the Infelta model.

∞

Riτi ∑ i)0

and

(33)

and ∞

Riτi ∑ i)0

i

Riωτi2

∞

(39)

where N′ and D′ are given by the right sides of eqs 32 and 33, respectively. In this case, Ri and τi are

Ri )

G(z) sin(ωz) dz )

∞

1 - kpm

One can show that ∞

Riτi ∑ i)0

(34)

The Infelta model is equivalent to eq 31 if Ri and τi are

LA9902789 (97) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; Wiley: New York, 1989; Chapter 14.

Frequency Domain Fluorescence Measurements of the Aggregation

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