J. Phys. Chem. 1993,97, 1153-1762
7753
Phase Structure of Poly(oxyethylene) Surfactants in Water Studied by Fluorescence Quenching Bo Medhage, Mats Almgren,’ and Jan Alsins Department of Physical Chemistry, Uppsala University, P.O. Box 532, S- 751 21 Uppsala, Sweden Received: January 20, 1993; In Final Form: April 27, 1993
Binary systems of water and nonionic surfactants of the dodecyl poly(oxyethy1ene) type, C I ~ H ~ ~ - ( O C H ~ CH&OH (ClzE,), have been studied by time-resolved fluorescence quenching (TRFQ). The interest has been focused on the diffusion-influenced quenching in liquid crystalline phases of the C12E4, C12E5, C12E6, and Cl2Es systems and the ability of the T R F Q method to discriminate between different quenching geometries. The relative diffusion coefficients of pyrene and 3,4-dimethylbenzophenonehave been measured in rod-shaped and lamellar aggregates in aqueous solutions as well as in neat surfactants. A new method, based on fluorescence quenching, for determination of layer thicknesses in lamellar systems is introduced and the results obtained are shown to be in close agreement with those from low-angle X-ray diffraction. In addition, we propose a method for determination of cylinder radii in rodlike micelles; even this method seems to produce values very similar to X-ray data. One attractive feature of the T R F Q method is that it does not require any liquid-crystalline order in the sample; it works equally well for aggregates in isotropic L1 phases, in which the occurrence of spherical, cylindrical, and disk-shaped micelles is demonstrated. For both rodlike and lamellar aggregates a slight increase in thickness with temperature is detected, which can be related to a decrease in effective surfactant head-group area, caused by a change in polarity of the ethylene oxide groups. The results of fluorescence quenching measurements in cubic phases show that the bicontinuous VIphase is essentially two-dimensional on the time scale of fluorescence, whereas for the I1 phase further evidence is given for the idea that the building stones in the I1 phase are small, slightly elongated, micelles with an axial ratio of approximately 1.51 for the C12Eg-water system. The quenching in the spongelike L3 phase proves to be more efficient than in the La phase, and the possible reasons for this are also discussed.
Introduction The phase behavior and phase structure in poly-oxyethyleneglycol dodecyl ether (ClzE,)-water systems have been intensely studied, and although some unknown or debated structures remain, the general features are well established.1-8 These systems are therefore the natural choice for the test of a new method which, with some luck, may give some clues to the unresolved structures. Recently, two of us presented time-resolved fluorescencequenching (TRFQ) measurements from the system C & ~ - H Z O . Since ~ then we have learned to handle diffusioninfluenced quenching in two-dimensional systems in an appropriate way,I0and we present a more comprehensive study of these systems as a test of the new method. The time-resolved fluorescence quenching method can monitor the diffusive behavior of small hydrophobic molecules confined to hydrophobic domains in microheterogeneous systems. From such experiments, besides from diffusion coefficients and quenching constants, also structural information about the host systems may be obtained, e.g., aggregate geometry, micellar size and shape, bilayer thickness, etc. An upper limit for the dimensions that can be determined with TRFQ is set by the diffusive displacement of the probe and quencher during the time-window of the measurement determined by the lifetime of the excited probe. The use of TRFQ to determine the aggregation numbers of micelles and other small fluid aggregates is well-known and is actually based on counting the number of discrete confinements. More recently decay curves resulting from quenching in quasione-dimensional (long rodlike micelles) and two-dimensional (lamellar bilayers) systems have been studied.9,11-16,45,46 In the one-dimensional (1D) case only one of two important parameters, the cylinder radius a and the diffusion coefficient D (=Dprob+ Dquenchcr) can be determined, whereas in two-dimensional (2D) systems both the bilayer thickness and the diffusion coefficient
* To whom correspondence should be addressed.
can be obtained, at least in principle. It should also be possible to discriminate between one- and two-dimensional systems.10 The use of an indirect method such as TRFQ, depending on the addition of extraneous probes to the system, for structural determination can hardly be the best alternative in well-defined systems where one or several of the powerful scattering methods may be applied. However, quite often microheterogeneous systems of current interest have such a complexity that scattering methods are difficult to apply or, more often, give results difficult to interpret. TRFQ is then a good complement and has proven its strength in for instance the studies of small aggregates in polymer-surfactant systems.17-19 It appears worth while, therefore, to examine the possibilities the TRFQ method offers in oneand two-dimensional systems as well. In this investigation we have tested TRFQ on various phases in binary systemsof nonionic alkyl poly(oxyethy1ene) surfactants, CnH~+1(OCH~CH2),OH, abbreviated to C,E,, and water. In these systems a rich variety of phases are found (see Figure 1); zero-dimensionalsmall micelles, one-dimensional rods, two-dimensional bilayers, and threedimensional homogeneous solutions. In addition, there are a number of regions in the phase diagrams, where the exact phase structure is not known or debated. Several models have been presented to account for the temperature-dependent solubility properties of the polyoxyethylene-water systems. The most fruitful approach has been presented by Karlstriim et ai. and is based on quantum mechanical calculations of the energies of different conformations of the -CHrCH+ segments in the ethylene oxide (EO) chain.21-z’ The conformations are either polar (with large dipole moments) or unpolar (with small or no dipole moments). It has been shown that the polar conformations have the lowest energies but low statistical weight; only 2 of 25 sterically possible conformers have appreciable dipole moments. Therefore, at low temperatures the polar conformations are dominating and hence the EO-water interaction is favorable. However, the unpolar conformations are entropically favored and become increasingly populated as
0022-3654/93/2091-1153%04.~0~0 0 1993 American Chemical Society
Medhage et al.
7754 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
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figure 1. Phase diagrams of some C12E,,,-HzO systems. (a) ClzEd-HzO (from: Mitchell et al. J . Chem. Soc., Faraday Trans. 1 1983,79,975). (b) CI~EJ-HZO(from: Strey et al. J . Chem. Soc., Faraday Trans. 1 1990,86,2253). (c) ClzES-HzO, the same as (b), but for a selected temperature range and with a logarithmic concentration scale to more clearly display the existence region of the L3 phase. (d) C&-HzO (from: Mitchell et al., see (a)). (e) C12EeH20 (from: Mitchell et al., see (a)). L1, L2, and L3 denote isotropic liquid solutions, H1 is a normal hexagonal phase, V Iis a bicontinuous cubic phase, and La is a lamellar phase. In regions denoted with S, solid surfactant is present.
Poly(oxyethy1ene) Surfactants in Water
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7755
the temperature is increased, which has the effect that the EOEO interaction becomes progressively more advantageous. The effective head-group area is thereby slightly decreased, which may induce structural changes of the aggregates. The head-group area is largest at low surfactant concentrations and at low temperatures, and this region of the phase diagram is dominated by the optically isotropic L1 phase, which consists of micellar aggregates, usually spherical or rodlike micelles. As the surfactant concentration or the temperature increases, the head-group area decreases, which in turn leads to a tendency for a decrease in curvature of the aggregate surface, accompanied by a growth in size of the aggregate. At higher surfactant concentrationslyotropicliquid-crystalline phases, possessing both short-range and long-range order, are formed. Some examples are the cubic I1 phase, consisting of finite-size micelles arranged on a lattice belonging to the space group Pm3n, the normal hexagonal H1 phase, which consists of infinite cylindrical micelles packed on a two-dimensional hexagonal lattice and the L, phase, consisting of infinite lamellae arranged on a one-dimensional lattice. Between HIand L, the bicontinuous VIphase occurs, which is a bilayer with the midplane forming an infinite periodic minimal surface. The liquidcrystallinephases have typically very high viscosity. Furthermore, both the H1 phase and the L, phase are optically anisotropic, and their presence can be easily revealed by the use of crossed polarizers. Some regions of the isotropic phases (L1, Lz,and L3) are less well-characterized. In the LIphase, depending on concentration and temperature, the shape of the aggregates range from small globular micelles to long rodlike micelles and possibly also diskshaped aggregates. Between these characteristic shapes continuous structural transitions occurs, leading to intermediate forms. The Lzphase consists of reversed micelles and occurs at high surfactant concentrations, and is in some systems connected to the LIphase via a channel with bicontinuous structure. The exact appearance of the bicontinuous Ljphase, which is formed at low surfactant concentrations and always in theclose proximity to an L, phase, has been the subject of much debate but a now generally accepted model describes the L:,phase as a spongelike bilayer ~ t r u c t u r e . ~ . * ~ ~ ” ~ ~ In a recent paper we presented an approximatesolution, suitable for implementation on computers, to the fluorescencequenching behaviour in quasi-two-dimensionalgeometrylO-a case that had not been properly treated earlier because of intractable fitting functions in the fluorescence decay data analysis. This approximation not only allows us to determine diffusion coefficients without having to make the common, unsatisfactory, assumption of complete diffusion control (corresponding to the so-called Smoluchowski boundary c0ndition2~J~)but also makes the determination of a layer thickness possible. In the present study the new method is applied to real systems (in contrast to simulated data). It is examined to what extent it is possible to discriminate between different geometries-which works perfectly well with synthetic data, where no complications like fluorescent impurities are present. Diffusion coefficients in various geometries are determined and the layer thickness in some lamellar and bicontinuous cubic phases. Some of the less well-defined regions in the phase diagrams are also investigated.
The fluorescence decay following a &pulse excitation is given as In(F(t)/F(O)) = -A2t
+ A,(exp(-A,t)
- 1)
(1)
with A, = ko
k k + Ln kq + k-
(3) A, = k,
+ k-
(4)
where k, is the first-order rate constant for quenching in a micelle with one quencher, k- the exit rate constant for a quencher to leave the micelle, n the average number of quenchers per micelle, and ko the unquenched decay rate of the probe. The model is valid under several limiting assumptions; the set of micelles is supposed to be monodisperse and the probe and quenchers randomly distributed over the micelles (a Poissoniandistribution). The excited probe is assumed stationary in the micelle over the time window of measurement;the quenchers, however,are allowed to migrate between the micelles. If also the quenchers are stationary in the micelles, the final exponential slope of the decay will equal ko = 1/70; in other words, the observed fluorescence will finally emanate from probe molecules in micelles without quenchers. By fitting decay data to eq 1, a micelle aggregation number, N, can be calculated from
N = n [SI mic/ [QI
mic
(5)
where [ S ] ~and c [Q]fic denote the concentrations of micellebound surfactant and quencher, respectively. For the treatment of diffusion-influencedquenching in quasione- and -two-dimensional geometries the concept of a reaction zone is introduced. The probe molecule is thought to be surrounded by a cylindrical reaction zone with radius a, in which quenching takes place with the first-order rate constant k,. In the 2D case, a will also denote the half-thickness of the layer. The diffusion-influencedfluorescencequenching in one-, two-, and three-dimensional geometries (d = 1,2,3)can be described by ln(F(t)/F(O)) = -kot - a3c,Qd(ha,t/~,)
(6)
where a is the reaction distance or the length of the reaction zone, c, is the quencher concentration given as number density (molecules per volume unit), and h is a parameter that weights reaction against diffusion
h = k,a/dD
(7)
The dimensionality-characteristic Q d functions are
Theory The use of fluorescence quenching measurements for the determination of micellar aggregation numbers started out with the work of Turro and Yekta, who proposed a simple model, describing the dependence of the steady-state fluorescence intensity on the quencher concentration.29 The basic model for time-resolved studies, describing the fluorescencedecay resulting from quenching in small discrete confinements (like for instance micelles), was formulated by Infelta er al.30 and has thereafter been discussed and elaborated upon by several other a~thors.~I,32
Q2(hW/7q) = E(ha),Sdp 7r
dx [ x J l ( x )+ h ~ J ~ ( x+)[]x ~Y , ( x )+ haY0(x)l2x3 (9)
1756 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
where T , = aZ/Dand 0 = 1 + ha. J,,(x) and Y,,(x)are the Bessel functions of first and second kind, respectively, of order n. The natural decay rate, ko = 1/ T O , is measured in a separate experiment and is kept fixed in the analysis of quenching data. However, in the 1D case, ko can in principle be determined from a quenched curve (vide infra), because at long times the natural decay is linear in t , whereas Q1 decays as tilz. This cannot be done in 2D or 3D geometries, especially not in the latter dimension since also Q3 becomes proportional to t as t m. In the 2D case the longtime behavior is complex and a series expansion of Q2 would most likely contain a term linear in t . The integral in eq 9 must be numerically calculated, but the integrand is an extremely ill-behaved function and proper care must be taken when performing the integration. However, the intractable integral expression makes eq 9 impractical for curvefitting purposes due to the large CPU time required. We have therefore created an approximation to Qz especially suitable for computer implementation:
-
where C1-C4, a,and are constant parameters. Details on the creation and application of this approximationhave been presented elsewhere.10 The accuracy of the approximation is very good; it describes the true Qz function better than 1% at all times and ha-values. For the completely diffusion-coQtrolled case (ha = m), the approximation reduces to
in some similarity with an approximation previously suggested by Owen,” where a was assumed equal to unity (in our approximation a 0.9). However, a = 1 implies that an exponential decay will finally be developed, which is not true for the 2D case.
Experimental Section Materials. The poly(oxyethy1ene glycol) mono-n-dodecyl ethers, C12E4, ClZES, C12E6, and from Nikko Chemical Co. (Japan) were stored in a refrigerator and used without further purification. 3,4-Dimethylbenzophenone(DMBP) (Aldrich) was used as received, pyrene (Aldrich) was recrystallized twice from ethanol. Sample Preparation. Pyrene (5 X 10-6M) and DMBP (10-20 mM) were dissolved in neat C12Em,and the sampleswere prepared by mixing the C12E,,,solution with water in an apparatusconsisting of a round-bottomed flask fused to a 1-cm quartz cuvette and with a vacuum stopcock with a ground joint for attachment to a vacuum line. The system was degassed by at least three freeze pumpthaw cycles; finally argon of approximately atmospheric pressure was added as a buffer gas. The solutionswere thoroughly mixed and heated if necessary and thereafter transferred to the cuvettecell while maintaining the argon atmosphere. The samples were thermostated for at least 30 min before measurement. The phases were checked between crossed polarizers before and after each experiment. The solutions were prepared by weighing the components H2O and (&E,,,. All surfactant concentrations will in the following be given in percent by weight, unless otherwise indicated. Time-correlatedsingle-photoncounting data were collected by using a cavity-dumped DCM dye laser operating at 400 kHz (Spectra Physics, Model 375,344S), synchronously pumped by a mode-locked Nd:YAG laser (Spectra Physics, Model 3800).
Medhage et al. The light from the dye laser was frequency doubled using a KDP crystal. The samples were excited at 330 nm and the emission was observed at 393 nm using a double monochromator (Jobin Yvon, Model H10). A microchannel plate (MCP) photomultiplier was used for detection of the emitted photons. To avoid pileup distortion, the excitation beam was attenuated so that the ratio of detected photons to excitation events was always kept below 0.5%. The outputs of the MCP and the photodiode (PD) monitoring the pulse train were routed through constant fraction discriminatorsinto the time-to-amplitudeconverter (TAC), which was operated in normal mode, Le., the start signal was provided by the PD and the stop signal was obtained from the MCP. The TAC output was converted to a number in an analogue-to-digital converter and was then stored in a multichannelanalyzer (Nuclear Data, USA)on an IBM/PS2. The number of channels were 5 12 and the width of one channel was typically 1.8 or 3.6 ns. The full width half-maximum of the instrument response function was less than 200 ps. Data Analysis. The fluorescence decay data were analyzed according to eqs 1-8, 10, and 11 with nonlinear least-squares programs using a modified Levenberg-Marquardt algorithm, which eliminates the need for explicit partial derivatives. The reduced xz was used to judge the quality of the fits. In previous work it has been shown that x2 can be used as a discriminatory indicator between different models for quenching in lD, 2D, and 3D geometries.1° However, a safer and more practical method is to examine the fitting parameters obtained using the different models. A physically unrealistic parameter is by far the best indication of that an inappropriate model is used. The general consequences the choice of an incorrect model has on the final results have been presented elsewhere.10 A disturbing short-lived impurity fluorescence in C~ZE,,, was compensated for by adding an extra term to the fitting function (eq 13) to be able to start as close to time zero (i.e., the moment of excitation) as possible. A satisfactory description of the perturbing fluorescence was given by a mono-exponential decay with a lifetime, T,, of approximately 6ns, determined by measuring the fluorescence decay from a sample without probe molecules.
In the analysis the amplitude, A,, of the extra fluorescence term was both allowed to vary freely and kept fixed. In the latter case the amplitude was estimated from a plot of the quenched decay curve. The influence of the impurity could also be completely neglected by starting the fitting sufficiently far from t = 0. All three methods were checked to produce similar results. However, the fitting parameter ha was found to be rather sensitive to how well the extra fluorescence was compensated for, while the parameters D and a seemed to be more insensitive in this respect. When fitting to the 2D-quenching equation (eq 9 or 1l), due to the very shallow minimum in the chi square hypersurface,lO proper care must be taken in the analysis to check that the nonlinear least-squares routine really reaches the minimum and not stops in the vicinity of it. In the analysis, the movements of the probe and quencher molecules in rods and lamellae were assumed to be restricted to the hydrophobic Clz domains. Thus, when calculating the quencher concentration, the volume ratio V(ClZE,,,)/ V(C12), typically around 2 for the surfactants studied here, must be considered.
Results and Discussion Fluorescence Lifetimes. The unusually long fluorescence lifetime and the hydrophobicity make pyrene a very suitable probe for diffusion-influenced fluorescence quenching experiments. However, due to the long lifetime pyrene is also highly susceptible to quenching by oxygen. Two factors make the removal of oxygen from the solutions essential: First, since some of the solutions
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7757
Poly(oxyethy1ene) Surfactants in Water
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Time I ns Figure 2. Illustration of the effect of oxygen consumption by DMBP Fluorescencequenchingdatafrom50wt %C&at 14 OC (corresponding to the hexagonal H1 phase). The solution was not deoxygenated before measurement. (a) Fit (and weighted residuals) to the decay data using a fluorescence lifetime of 345 ns, which is the TO value that yields the lowest x* in the analysis. (b) Fit (and weighted residuals) to decay data using a fluorescence lifetime of 186 ns as determined from an unquenched and undeoxygenated sample. Since the natural decay rate is faster(!) than the quenched decay rate, the fitting yields a diffusion coefficient which is zero and the weighted residuals are very bad-looking.
must be heated to enable efficient mixing, part of the dissolved oxygen will be driven off. The equilibration with air after heating will be slow due to the very high viscosity and slow convection in samples containing liquid crystalline phases. The reproducibility of the oxygen concentration in the samples will be bad therefore. Second, DMBP seems to consume oxygen when irradiated by ultraviolet light or excited by energy transfer from pyrene. This effect is especially pronounced in highly viscous or solid systems, because then a very small domain in the sample can be irradiated by a largeamount of UV light and consequently, become totally depleted of oxygen. The effect is illustrated in Figure 2: analyzing the decay curve shown using a lifetime 70 = 186 ns, as determined from an unquenched (and nondegassed) sample, produces completely erroneous results-in fact, the unquenched curve decays faster than the quenched one, even though the two samples were prepared in exactly the same way! Since the quenching in this case correspondsto a one-dimensional geometry (50% C12E6, 14 "C, H1 phase), the unquenched fluorescence lifetime could be extracted from the decay curve, yielding a value TO = 345 ns, which is within limits of error the same as for a degassed sample. The quenching parameters obtained using TO = 345 ns are the same as those for a degassed sample. In Figure 3 the fluorescence lifetimes for some concentrations and temperatures are shown for degassed samples of the C12Em systems studied here. The presence of a water-soluble quenching impurity inC12Emisevident.The fluorescencelifetimewas always kept fixed in the analysis, even if it in principle can be determined from OD- (without migration of quenchers) or 1D-quenching curves (but never from 2D or 3D geometries). Fluorescence Quenching Experiments. The differences in quenching behavior in 0-3-dimensional geometries are illustrated in Figure 4. For the same concentration, the quenching is more
efficient the higher the dimensionality. The final slope in the zero-dimensionalcase equals the slope of the unquenched curve, because at sufficientlylong times only probe. molecules in micelles without quenchers contribute to the fluorescence emission. Lamellar (La)Phases. The temperature dependence of the diffusion-influenced quenching in lamellar phases can be monitored in C12E4, C12E5, and C& at surfactant concentrations around 70%. From the analysis of 2D-quenching data, diffusion coefficientsas well as layer thicknessesare obtained. The diffusion coefficients (see Figure 5 ) are very similar in C12E4 and CnES and are a bit higher in C12E6, and the activation energies are very similar in all three systems. The opposite trend is observed for the layer thickness; the layers in C&6 seem to be slightly thinner than in the other two (see Figure 6). This is consistent with the fact that the head groups in C&6 are somewhat larger than in C12E4and C12E5. On the other hand, no difference between the latter two can be detected. It is therefore doubtful whether the difference in thickness is significant. The change in thicknesswith temperature, most clearly observed in CI2Es,is probably more significant. It has been shown that an increase in temperature leads to dehydration of the head groups.536 The effective head-group area is thereby slightly decreased, which by necessity is accompanied by an increase in layer thickness. The results obtained in this study are compatible with X-ray data,3 which show that the hydrophobic layers are very thin in these systems, actually even shorter than the length of a stretched Clzchain. This means that in the bilayer, the alkyl chains of the amphiphilic molecules are "interdigitated", which in turn may be attributed to the bulky hydrophilic head groups. The halfthicknesses reported by Clunie et al. range from 7.1 to 7.8 A for 68-85 wt % C12Es at 22 oC.3 The lamellar phase of C I ~ has E ~been studied by static lightscattering and small-angle neutron scattering by Strey et ala7 After an area correction, necessitated by the thermally induced undulations in the bilayer, a layer thickness of 30 A was reported. From this value the half-thicknessof the C12 layer can be calculated to be 7.5 A, since in C12ES the volume of the Cl2 domain is approximately equal to the volume of the E5 domain. The ha values observed in the lamellar phases are typically around 2-3 and should be compared to the completely diffusion controlled case, for which ha = 01. It should be stressed that the reaction is still strongly diffusion-influenced; in fact the error made in Owen's approximation (vide supra) makes it to be close to the correct case with ha 3 (see ref 10). In ClzEs the lamellar phase can be swollen to approximately 98.8 wt % of water (see Figure 1). Experiments performed at 2 and 70 wt 5% C I ~ Eproduce S very similar results, while the intermediate concentrations 10 and 45% yield somewhat peculiar results. As shown in Table I, the results obtained at 10%and 55 "C (and even at 57 "C, data not shown) are rather close to the ones for 2 and 70% C12E5. However, in the temperature range 57-40 "C a change in the decay behaviour occurs and the quenching is characterized by relatively large a values and low diffusioncoefficientswhen fitted to the 2D model. The experiment at 10% was also performed with a small amount of SDS present (molar ratio C12Es:SDS = 100:1) to make the bilayers negatively charged and therefore well separated by electrostatic repulsion, but the SDS addition gave no drastic change in the quenching behavior. As demonstrated in Table I, neither the 1D nor the 3D model provides satisfactory descriptions of the decay behavior as judged by the unphysical diffusion coefficients obtained. We can only speculate about the physical origin of these anomalies, which are reproducible, however. With the results from fluorescencequenching in the L3phase at hand (see below), which when analyzed with the 2D model yield very large a and very low D values, it is tempting to invoke the explanation that in the Laphase a gradual structural transition towards L3 phase behavior takes place as the temperature is increased. In more general terms it can be noted that the estimates of D and a are
I758 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 400
Medhage et al. 400
a
C 375
300 0
2 5 o " ' " ' " " ' " " ' ' " ' " ' ~
0
1 0 2 0 3 0 4 0 5 0 6 c
30
20
10
40
I
50
T I T
T I T
40 375
I 325
300--"', ' ' " '." ,,, ,, ' " 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0
T I T
Figure3. Fluorescencelifetimesatsomeconcentrationsandtemperaturesfor (a) C12E4, (b) C12E5, (c) C&, and (d) Cl2E8. Thesurfactant concentrations are displayed in the figures. Note that the deviating points for 100% ClzEs and 100%Cl& at 14 OC correspond to the solid phase. It is quite possible
that the probe molecules aggregate in the cooling process.
0
500
1000
3.0
1500
Time I ns Figure 4. Synthetic fluorescence decay data, illustrating the difference in quenching behaviour in various quenching geometries. The influence of the natural decay rate has been eliminated by multiplication with the factor exp(k0r). (a) Unquenched sample; (b) OD quenching according to eq 1 without migration of quenchers; (c) 1D quenching (eqs 6 and 8); (d) 2D quenching (eqs 6 and 9 ) and (e) 3D quenching (eqs 6 and 10). The following parameters, typical for the ClzE,,,-H20 systems studied here,wereused todescribethediffusion-influencedfluorescencequenching in lD, ZD, and 3D geometries: D = 15 X m2 s-I, ha = 3, U ~ D= 15 A (cylinder radius), IIZD = 8 A (bilayer half-thickness) and a3D = 8 A (encounter radius). For the OD case, a quenching constant kq = 1 X lo7 5-1 and an aggregation number N = 100 were chosen. All data correspondto the samequencher concentration(33mM in the C12 phase).
deviating from the expected in the same way as if a model with too high a dimensionality had been used. It should be expected, therefore, that the system is somewhat restricted in one further dimension. This could either be due to avoided regions-holes in the membrane-or to that probes and quenchers preferentially get concentrated in certain regions-as if there were "frozen" undulations in the bilayer. It is not possible to determine what the reason is from the TRFQ method, but the change in the behavior is a reproducible one, and it could be profitable reexamining the phase structure in these regions of the L, phase by other methods, even if the study of Strey et a1.7 gave no indication of departures from the expected. L1 Phasesabove the Hi L1 PhaseTransition. Table I1 shows a and D values obtained in the L1 phase above the HI L1 transition in 45% C12Es. At 25 OC, immediately above the melting point of the H I phase, neither the 1D nor the 2D model provides an adequate description of the decay data. Upon increasing temperature, both models produce better fits as judged by the x2
-
-
3.1
3.2
3.3
3.4
3.5
1000 T-* I K-'
Figure 5. Relative2D diffusioncoefficientsversus reciprocaltemperature for pyrene-DMBP in some lamellar liquid crystalline phasea: 70 wt 96 C12E4 (filled triangles),70 wt 5% ClzE5 (filled squares), 63 wt 96 C1& (open circles), and 75 wt W C& (filled circles). Note that the two points correspondingto the lowest temperatures for 63 wt 96 C& belong to the cubic V Iphase. The lines represent Arrhenius type of fits to the data for C12Ed. C12E5, and 75% C&.
values, whereas the values of the diffusion coefficients indicate a transition from 1D to 2D. The reason for the melting of the hexagonal phase is probably not that the thermal motion of the rods becomes too large; if there are rods present, they must be rather well aligned. More likely, it is the packing constraint and structure that are changed. When the temperature is increased, the effective headgroup area decreases and a transition from rods to a structure with less curvature occurs; disk-shaped aggregates could become favorable. The appearance of these aggregates is difficult to assess; our results indicate that a possible structure could be some kind of elongated oblates, being an intermediate form between rod- and disklike micelles. As the temperature is raised the situation in the L,-phase discussed above is approached; both the 1D and 2D models yield good xz values but strange parameter values. It is interesting to compare the results for 45% C12ESwith those obtained for 50%Cl2E6 (see Table 111). In the latter case the melting point for the hexagonal phase is higher (-37 oC),becausetheheadgroupislarger. Thetendencies in the L1 phase are rather similar in both cases, however, with the difference that in C&6 a quenching behaviour very similar to the quenching in an L, phase (as judged by the a and Dvalues) is finally developed a t 55 OC. This implies that the aggregates present in 50% C I Z Ea t~55 OC can be described as disk-shaped
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7759
Poly(oxyethy1ene) Surfactants in Water
TABLE II: Bilayer Half-Thickness and Diffusion Coefficients Obtained Using One- and Two-Dimensional Models to Analyze Ll Phase Data from 45 wt % C12W temp/°C a z ~ / A 10"&o/m2 5-l X ~ D ' 10"DID/m2 S-l X I D ~ 25 33 45
22.2 16.7 11.8
0.50 1.7 6.7
1.441 1.252 0.975
29.3 70.0 22 1
1.768 1.589 1.044
The x2 values demonstrate the gradual change in fitting quality with temperature. A cylinder radius a = 14.6 A was assumed for the 1D analysis.
TABLE IIk Bilayer Half-Thickness and Diffusion Coefficients Obtained Using One- and Two-Dimensional Models to Analyze L1 Phase Data from 50 wt % C&f temp/'C a 2 ~ / A 1O1'D2~/m25-I X2D2 10"D1D/m2 5-I 25 33 40 45 55
(-45) (-60) 11.4 7.5 8.5
(-0.02) (nO.02) 4.9 13.7 27.3
(1.039) (1.098) 1.134 1.250 1.080
(11.0) (20.8) 121 284 91 1
x
1
~
(1.034) (1.108) 1.179 1.358 1.136
a The values in parentheses correspond to data from the hexagonal H1 phase, where the 2D model is not expected to be applicable, which is clearly demonstrated by the unphysical values of the a and D values.
TABLE Iv: Illustration of How TRFQ Can Be Used To Discriminate between Different Quenchinn Geometries' 0 1 0 2 0 3 0 4 0 5 0 6 0 TlOC
I
20 C
T/OC
phase
ha
28 31 45 50
Vi Vi
0.031 0.026 0.020 0.017
L, L,
1D 10LID 215 290 623 850
x2 1.75 1.70 1.65 1.64
ha 1.9 2.0 1.7 1.1
2D 10"D 9.0 10.8 16.4 19.0
x2
1.31 1.35 1.20 1.23
a The data correspondto 63 wt % C 1 2 band have been analyzed using one- and two-dimensionalmodels. The cylinder radius in the 1D model was assumed to be 14.6 A. The diffusion coefficients, D, are given in units of m2s-'. The bilayer half-thicknessesobtained using the 2D model are displayed in Figure 6c.
0
1
0
2
0
3
0
4
0
5
0
6
0
TlOC Figure 6. Bilayer half-thicknessand cylinder radius determined for some concentrations and temperatures in three C12E,,,-H20 systems. (a) L, bilayer half-thickness in 70% C&. (b) L, bilayer half-thickness in 70%C12E5 (filled circles) and cylinder radius for rodlike micelles in the L1 phase at 10%C12E5 (filled squares). (c) Bilayer half-thicknessin 75% Cl2& (filled circles) and 63% C& (open circles). Note that the temperatures 28 and 31 OC correspond to the cubic V1 phase. Also shown is the cylinder radius for 50% C&6 (filled squares).
TABLE I: L,Bilayer Half-Thicknessand One-, Two-, and Three-Dimensional Diffusion Coefficients Obtained from Analysis of Data for Four Different CI& Concentrations at 55 and 60 OC. wt % 1O1'D2~/ 1O1'Dio/ 1O1'D,D/ C12E5 2 10 10 lob 45 70
T/OC 60 55 60 60 60 55
A
9.6 8.4 15.5 14.5 14.5 8.6
m2s-1
25.6 19.0 10.5 13.7 6.9 27.7
m2 5-1
m2 5-1
1500 560 620 850 340 1100
7.5 6.4 7.1 11.8 6.2 7.3
4 A cylinder radius of 14.6 A was used for the 1D analysis and for the 3D case complete diffusion control (ha = m ) and an encounter distance a = 8 A was assumed. bA trace of SDS was added to further stabilize the lamellar phase.
micelles, so large that they are effectively infinite on the length scale of diffusion-influencedfluorescence quenching. The rootmean-square displacement of the probe and quencher molecules at 5 5 OC during the time of observation is approximately 150200 A, which means that the disks should be larger than this. At
lower temperatures the aggregates are smaller, and the effective number of quencher molecules available for quenching of an excited state becomes lower. The smaller the aggregates, the more pronounced becomes this finite-size effect and the lower become the apparent diffusion coefficients. Bicontiouous Cubic (VI) Phases. The quenching in optically isotropic bicontinuous cubic phases is best characterized by a two-dimensional model, which is demonstrated in Table IV for 64 wt % C12Ea. The table is a good illustration of how different models of fluorescencequenching can be distinguished from each other. Evidently, the 1D model does not work in this case: the Dvalues are unphysicallylarge and the ha-values are unreasonably small; even the x2 values strongly indicate that the 2D model is the appropriate one. The layer thickness in the cubic phase is alsovery similar to the thickness of the L, bilayer at corresponding temperature (see Figure 6c). L3 Phase. In the isotropic L3 phase the local structural unit is supposed to be a bilayer. Several different models have been suggested for the large scale structure. In recent papers the L3 phase is described as a spongelike random surfactant bilayer network that divides space into two interpenetrating solvent labyrinth~.~J',2'26 A continuous web of bilayer in a spongelike structure is formed to avoid energetically unfavourable edges of disk-shaped micelles. The fluorescence quenching in such a structure (0.6% C12Es at 55-60 "C) proves to be more efficient than in an L, bilayer. A possible explanation to this behavior is that in the crumpled bilayer, due to the high local layer curvature more or less isolated compartments are formed, in which probe andquencher molecules are gathered together. The local quencher concentration in these compartments will be somewhat higher than on average. Figure 7 illustrates the slight difference in quenching behavior between the L3 phase and the La phase. Analyzing the L3 phase data with the 2D model results in
~
7760 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
Medhage et al.
TABLE VI: Results from Fluorescence Quenching in ClzES: One-Dimensional Diffusion Coefficients, Qb Calculated Using a Cylinder Radius BID = 14.6 A and Calculated Cylinder Radii, a'lb Determined Using the Assumption QD = U m AS Discussed in the Text' 2 ~t % (LI) 10 wt % (Ll) 45 wt % (HI) T/OC
10IlD2~ 10"D1~
14 20 25
5.0 6.5 7.0
5.8 7.1 11.0
U'ID
10"Dl~
U'ID
10llD1~ a ' l ~
15.2 14.9 16.3
7.2 9.3 11.6
16.0 16.0 16.6
8.3 11.6
16.6 16.9
D ~ is D the 2D diffusion coefficient determined from measurements . diffusion coefficients,D, are given in units of m2 at 70 wt % C I ~ E SThe s-l and the cylinder radii in angstroms. (I
Time I ns Figure 7. Difference in quenching behavior between the L. phase (70 wt % C12E5, represented as dots) and the L3 phase (0.6% C I ~ E Sfull , line) at 55 OC. The quencher concentration was the same in both phases (cq = 2.0 x 10-5 A-3).
TABLE V Activation Energies and Interpolated Relative Diffusion Coefficients at 25 OC for Pyrene-DMBP in C12Em Systems' surfactant wt % phase 10I1D25/m2s-l EJkJ C12E4 C12ES
C12E6
CI~EB
70 2 10 45 70 100 50 75 100 50
La
L1 LI Hib La
neat HI La
neat HI
7.4 10.3 11.5 14.9 7.6 8.6 12.3 10.6 8.1 14.6
32.7 40.7 30.8 39.0 30.8 26.8 43.8 30.7 27.4 44.3
In the data analysis, the 1D model was used for HI and LI phases, the 2D model for La-phasesand the 3D model for neat phases. The HI phase melts at 24 OC, but HIdata were used to calculate the extrapolated diffusion coefficient at 25 OC.
TABLE W: Results from Fluorescence Quenching in C1&: One-Dimensional Diffusion Coefficients, Qb Calculated Using a Cylinder Radius a1D = 14.6 A and Calculated Cylinder Radii, d i m Determined Using the Assumption 40 = UD, As Discussed in the Text (50 wt % CtzEd' TIOC
Dm
14 20 25 33
6.6 9.1 10.5 16.8
Din ~
~~~
a'in ~~~
6.2 9.0 11.0 20.8
14.4 14.6 14.8 15.4
D ~ is D the 2D diffusion coefficient determined from measurements at 75 wt % C1&.
lamellar phases (DzD)and assuming those to be equal to the ones in rodlike phases (DID),a size determination is made possible. The two independent parameters K I = a3f ha and K2 = haD112f a are obtained from analysis of 1D quenching data. The calculated cylinder radius, a'ID, is then given by
0
unphysical values of the diffusion coefficient and the layer thickness. For instance, fitting the L3 data in Figure 7 using eq 11 yields a = 32 A and D = 2.9 X 10-11 m2s-1. Thecorresponding values for the La phase are a = 8.5 A and D = 27 X 10-11 m2 s-I. Hexagonal (HI) Phases. In hexagonal phases or in rodlike micelles in L1 phases, the quenching behavior follows the onedimensional model. Since this model contains only two independent parameters related to quenching, it is necessary to know either the diffusion coefficient or the radius of the rodlike aggregates. The radii in the hexagonal phase of Cl2E6 are available from X-ray diffraction experiments and are reported to be 14.6-15.2 A for 40-60 wt % C&6 a t 22 0C.3 A slight increase of the radius with surfactant concentration is observed. N o data for other temperatures have been presented to our knowledge. We used a = 14.6 A throughout in the analysis of 1D-quenching data. The diffusion coefficients thus obtained show a slightly faster increase with temperature than in the lamellar phases (see also ref 9). From an Arrhenius plot (see, for instance, Figure 5), an activation energy, Ea, can be determined from the slope of the linear correlation between In D and 1/T. Table V summarizes the results obtained from various phases in the different surfactant systems. The larger (as compared to lamellar and neat phases) activation energies seen in the hexagonal phases of C I ~ EC12E6, ~, and Cl2E8 can be due to an increase in micellar radius caused by a decrease in effective head-group area with temperature. Furthermore, the one-dimensional diffusion coefficients in C12Es are significantly larger than the D values in the lamellar phases. This could also be an effect of that the radius is actually slightly larger than 14.6 A. Since the head groups are smaller in CIZES than in C1& this might very well be the case. It is therefore tempting to reverse the discussion and make the a priori assumption that the diffusion coefficients are similar in rods and lamellar phases. So by using diffusion coefficients for
Results for C12E5and ClzEs are presented in Tables VI and VII. Two interesting points can be made. First, a gradual increase of the radius with temperature is observed. Second, in C12E5 an increase with surfactant concentration is apparent. Both these effects are expected and in agreement with X-ray and N M R data.3,5 For C12E8we have no data for lamellar phases. Hence, a radius determination is not possible. Another observation is that the quenching behavior in 2 wt % is best described by a 1D model, while at corresponding concentrations in Cl2E6 and C&, a OD model must be applied. Thus, thesmaller headgroupsof C12E5 seem to favor the formation of long rodlike micelles, whereas in C& and Cl2E8small globular micelles are preferentially formed. This is in accordance with results from IH N M R studies of the ClzEs-water system: where rather broadmethylenesignals of the Clzchain have beenobserved even a t low temperatures and low concentrations of C12E5. Since the long rotational correlation times for large surfactant aggregates result in broadening of the NMR signals (as compared to solutions with small micelles), this implies that the aggregates present are considerably larger than spherical micelles. In CIZEB, the N M R signals from the C12 methylene protons are much more narrow and show only a slight increase in width with increasing C12Es concentration. This means that the ClzE8 micelles a t low temperature remain relatively small throughout the whole concentration range 0-30%, whereas in C12E5 a dramatic broadening of the signals and hence an aggregate growth are observed with increasing surfactant concentration. This is also in agreement with the Occurrence of a cubic I1 phase between the aqueous micellar solution and the hexagonal phase in C12E8. Discrete Cubic (11) Phases. The results from analysis of fluorescence quenching in 37% C12E8, using zero- and onedimensional models (eqs 1-8), are summarized in Figure 8. The data for 12 and 17 OC correspond to the cubic 11 phase, which is usually supposed to consist of close-packed small, possibly elongated, micelles; X-ray diffraction has shown the space group of the 11phase to be Pm3n, alternatively P43n.3g35 However,
-
The Journal of Physical Chemistry, Vol. 97, No. 29, I993 7761
Poly(oxyethy1ene) Surfactants in Water 700
z
I
m
Axial ratio=
Figure 9. Schematic picture of a possible surfactant aggregate building up the cubic 11 phase. The axial ratio is 1.52 (as found for the 11 phase in C12Eg).
Temp I OC
.i._J
P
10
5 0
10
3
20
0
4
0
5
0
6
-
the structure of this phase has been debated (refs 36 and 37 and references therein), but both N M R self-diffusion38 and TRFQ39 data show that the aggregates, whatever their shape, must be of a closed structure. The quenching rate in a micelle is characterized by the firstorder rate constant k,,, basically a measure of the frequency of quenching encounters between the excited probe and a quencher. Some theoretical calculations have been pre~ented~~,@q~' which relate the first-order quenching constant to the micellar radius, a, and the intramicellar diffusion coefficient, D
(15) where y is a constant of the order of unity. In the cubic 11 phase, the size of the micelles shows only a slight increase with temperature. Since also the diffusion coefficient increases with temperature, the quenching constant is somewhat higher at 17 OC than at 12 O C . Above the I1 L1 transition the micelles grow more rapidly in size and a gradual transition to long rods, accompanied by a decrease in k,, occurs as demonstrated in Figure 8. The OD model does not work a t high temperatures, where the 1D model is applicable and vice versa. In the intermediate
-
temperature interval 30-45 OC neither model is appropriate, because in that region rods of finite length are dominating, which leads to underestimated diffusion coefficients in the 1 D case. The effective quencher concentration is lower than anticipated; an expanding correlation hole which extends beyond the edge of the micelle is created around the excited probe. At -50 OC the 1D diffusion coefficients obtained are similar to the ones typically observed in hexagonal C12Emphases at that temperature. The aggregation number for a spherical micelle, Nsphere, can be estimated by assuming the hydrocarbon core radius to be that ofan extended C12chain,Le., 16.68,,42andcalculatingthevolume contribution from each C12 chain using the group volumes V(CH2) = 28 8,' and V(CH3) = 49 8,3.43944 By doing so, an aggregation number of 54 is obtained. Hence, the aggregation numbers observed in the I1 phase, N = 96 at 12 O C and N = 102 at 17 OC, indicate that the micelles are not spherical, but slightly elongated, which previously also has been observed by Johansson andsaderman for the 11phase in the dodecyltrimethylammonium chloride/water system.39 Assuming the aggregates to be small rod-shaped micelles with hemispherical ends as depicted in Figure 9,an axial ratio can be calculated under the assumption that the density is the same in the cylindrical and the hemispherical parts. Making use of the volume ratio between the cylindrical and the spherical parts, we get
0
Temp I OC Figure 8. Results from analysis of fluorescencequenching data from 37 wt Z C&g at three different quencher concentrations ([Q]&[S]mic = 1:301 (triangles), 1:202 (squares), 1:151 (circles)). The vertical bars in the figures indicate the 11 LI phase transition temperature. (a) Aggregation numbers Nand (b) first-order quenching rate constants k, obtained using eq 1 without migration of quenchers. (e) Diffusion coefficientsobtainedusing the one-dimensionalmodel (eq 8) in the decay data analysis.
k, = r ( W 2 )
2R+I 2R
or
with parameters as defined in Figure 9. Using Naphere = 54 (vide supra) the axial ratios obtained are 1.52 at 12 O C and 1.59 at 17 O C . Homogeneous Solutions. From quenching experiments in homogeneous solution it is possible to extract both ha, u, and D from the decay curve. However, since the transient behavior constitutes only a small part of the overall decay, it is absolutely necessary to start the fitting from the very first channel, which due to the fluorescent impurity is difficult in the C12Emsystems, even if the extra fluorescence compensation is rather good. Therefore, we have instead chosen to use a fixed encounter distance a = 8 8, and assumed the reaction to be completely diffusioncontrolled. The diffusion coefficients (interpolated to 25 "C) obtained from the final exponential parts of the decays and the activation energies are displayed in Table V and are found to be similar to those in lamellar phases. Conclusions
We have demonstrated that time-resolved fluorescence quenching can be a useful tool for the determination of structural properties in various phases of surfactant systems. A new method is shown capable of reporting on the bilayer thickness in lamellar phases of some binary ClzE,,,-water systems. The thickness of the layers in the La phases and a bicontinuous cubic VIphase are
7762 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
found to be in the range 7-8 A at 25 OC. A slight increase in thickness with temperature can be observed, which can be explained by a decrease in effective head-group area due to less favorableinteractionsbetween water and the ethyleneoxidegroups at higher temperatures. The relative diffusion coefficients obtained for the p y r e n e DMBP pair are typically around 8 X 10-1' m2s-1 in both lamellar and neat surfactant phases at 25 OC; for these phases theactivation energies of diffusion were found to be around 30 kJ mol-'. The rather high activation energies of approximately 40 kJ mol-' observed for diffusion in rodlike aggregates, for which a temperature-independent cylinder radius was assumed in the 1D analysis, stimulated us to propose a method for determination of cylinder radii in microheterogeneous systems, based on the assumption that the diffusion coefficientsare similar in rods and lamellae. The cylinder radii thus obtained are compatible with X-ray data (where available) and show a slight temperature dependence in analogy with the lamellar thickness. The fluorescence deactivation has also been studied in two cubic structures (Vl and 1'); the quenching in the VI phase is, in consistency with the bicontinuous structure, best described by a 2D model and the bilayer thickness and the diffusion coefficients obtained are very similar to those for Laphases. The quenching in the I1 phase excludes the presence of a bicontinuous structure and add further strength to the discrete-aggregate modePs for this phase. The axial ratio of the elongated micelles in the I1 phase of C12Es-water is found to be around 1.5:l. Thequenching in the spongelikeL3phase exhibits an anomalous behaviour by being more efficient than in the lamellar Laphase. The only reasonable explanation for this seems to be that the crumpling of the bilayer leads to the formation of waists and bellies, where the probe and quencher molecules tend to concentrate in the latter. The TRFQ method has proven its strength by being able to discriminate between different quenching geometries in aqueous micellar solutions as well as in liquid-crystalline phases. In the L1phase, the existence of both spherical, rodlike and disk-shaped micelles has been demonstrated.
Acknowledgment. We thank Dr. Emad Mukhtar for skillful technical assistance. Financial support from the Swedish Natural Science Research Council and the Knut and Alice Wallenberg Foundation is gratefully acknowledged. References and Notes (1) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDonald, M. P. J. Chem. Soc., Faraday Tram. 1 1983, 79, 975. (2) Sjdblom, J.; Stenius, P.; Danielsson, I. In Nonionic surfactants. Physical Chemistry; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; Vol. 22, p 369.
Medhage et al. (3) Clunie, J. S.;Goodman, J. F.; Symons, P. C. Trans. Faraday Soc. 1969, 65, 287. (4) Nilsson, P.-G.; Wennerstrbm, H.; Lindman, B. J. Phys. Chem. 1983, 87, 1377. ( 5 ) Nilsson, P.-G.; Lindman, B. J. Phys. Chem. 1983,87, 4756. (6) Zulauf, M.; Weckstrdm, K.; Hayter, J. B.; Degiorgio, V.; Corti, M. J. Phys. Chem. 1985.89, 341 1. (7) Strey, R.;Schom&ker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. SOC.,Faraday Trans. I 1990,86, 2253. ( 8 ) Jonstrbmer, M.; Strey, R. J . Phys. Chem. 1992, 96, 5993. (9) Almgren, M.; Alsins, J. Isr. J. Chem. 1991, 31, 159. (10) Medhage, B.; Almgren, M. J. Fluorescence 1992, 2, 7. (1 1) Almgren, M.; Alsins, J.; Mukhtar, E.; van Stam, J. J. Phys. Chem. 1988, 92,4479. (12) Alsins, J.; Almgren, M. J. Phys. Chem. 1990, 94, 3062. (13) Owen, C. S.J. Chem. Phys. 1975.62, 3204. (14) Vanderkooi, J. M.; Fischkoff, S.;Andrich, M.; Podo,F.; Owen, C. S . J . Chem. Phys. 1975,63, 3661. (15) Kano, K.; Kawazumi, H.; Ogawa, T.; Sunamoto, J. J. Phys. Chem. 1981, 85, 2204. (16) Miller, D. D.; Evans, D. F. J. Phys. Chem. 1989, 93, 323. (17) van Stam, J.; Almgren, M.; Lindblad, C. Prog. Colloid Polym. Sci. 1991, 84, 13. (18) Almgren, M.; van Stam, J.; Lindblad, C.; Li, P.; Stilbs, P.; Bahadur, P. J. Phys. Chem. 1991, 95,8975. (19) Thalberg, K.; van Stam,J.; Lindblad, C.; Almgren, M.; Lindman, B. J . Phys. Chem. 1991, 95, 5677. (20) Andersson, M.; Karlstrdm, G. J. Phys. Chem. 1985, 89, 4957. (21) Karlstrdm, G. J. Phys. Chem. 1985,89,4962. (22) Karlstrdm, G.; Carlsson, A.; Lindman, B. J. Phys. Chem. 1990, 94, 5005. (23) Lindman, B.; Carlsson,A.; Karlstrdm,G.; Malmsten, M. Ado. Colloid Interface Sci. 1990, 32, 183. (24) Anderson, D.; Wennerstrdm, H.; Olsson,U. J. Phys. Chem. 1989, 93,4243. (25) Strey, R.; Jahn, W.; Porte, G.; Bassereau, P. Lungmuir 1990,6,1635. (26) Roux, D.; Codon, C.; Cates, M. E. J. Phys. Chem. 1992,96,4174. (27) Szabo, A. J. Phys. Chem. 1989,93, 6929. (28) Smoluchowski, M. 2. Z . Phys. Chem. 1917, 92, 129. (29) Turro, N. J.; Yekta, A. J. Am. Chem. Soc. 1978, 100, 5951. (30) Infelta, P. P.; GrHtzel, M.; Thomas, J. K. J. Phys. Chem. 1974, 78, 190. (31) Tachiva. M. Chem. Phvs. Lett. 1975. 33.289. (32j Van der'Auweraer, M.-Dederen, J. C.; Gelade, E.; De Schryver, F. C. J. Chem. Phys. 1981, 74, 1140. (33) Balmbra, R. R.; Clunie, J. S.;Goodman, J. F. Nature 1969, 222, 1159. (34) Tardieu, A,; Luzzati, V. Biochim. Eiophys. Acta 1970, 219, 11. (35) Fontell, K.; Fox, K. K.; Hansson, E. Mol. Cryst. Liq. Cryst. 1985, I , 9. (36) Lindblom, G.; Rilfors, L. Ado. Colloid Interface Sci. 1992,41,101. (37) Fontell, K. Ado. Colloid Interface Sci. 1992, 41, 127. (38) Bull, T.; Lindman, B. Mol. Cryst. Liq. Cryst. 1985, I , 9. (39) Johansson, L. B.-A.; SBderman, 0. J . Phys. Chem. 1987,91,5275. (40) Sano, H.; Tachiya, M. J . Chem. Phys. 1981, 75, 2870. (41) Tachiya, M. In Kinetics of Nonhomogeneous Processes; Freeman, G. R., Ed.; Wiley: New York, 1987. (42) Tanford, C. J. J . Phys. Chem. 1972, 76, 3020. (43) Jdnsson, B. Thesis, University of Lund, 1981. (44) Gallot, B.; Skoulios, A. Kolloid Z . Z . Polym. 1966, 208, 37. (45) van der Auweraer, M.; De Schryver, F. C. Chem. Phys. 1987,111, 105. (46) van der Auweraer, M.; Reekmans, S.;Boens, N.; De Schryver, F. C. Chem. Phys. 1989, 132, 91.
