J . Phys. Chem. 1993,97, 8082-8091
8082
A Novel Method for Determining Size Distributions in Polydisperse Micelle Systems Based on the Recovery of Fluorescence Lifetime Distributions? Aleksander Siemiarczuk Photon Technology International (Canada) Inc., 347 Consortium Ct., London, Ontario, Canada N6E 2S8
William R. Ware' and Yuan S. Liu Photochemistry Unit, Department of Chemistry, University of Western Ontario, London, Ontario, Canada N6A 587 Received: February 23, 1993; In Final Form: May 13, 1993
A method based on fluorescence lifetime distribution analysis is proposed for the determination of micellar size distributions in polydisperse systems. The exponential series method is employed to determine the lifetime distribution for a probe in the presence of quencher. This distribution is then analyzed by the maximum entropy method in order to recover the size distribution function. The method has been tested on sodium dodecyl sulfate and cetyltrimethylammonium chloride micellar systems in the presence and absence of added salt. Excellent agreement is obtained with other methods of size determination in cases where it is reasonable to make such comparisons. In both systems, the addition of salt is observed to bring about first a shift in the averageaggregation number to approximately double the zero salt value and then the development of a bimodal distribution. This latter phenomenon is interpreted as a transitional phase between spherical and rod-shaped micelles. Introduction The problem of determining the size and shape of micelles has received considerable attention during the past three decades.' It is generally agreed that, in many surfactant systems of low ionic strength with surfactant concentration close to the critical micelle concentration (cmc), micelles can be described as small spherical or near-spherical aggregates. However, at high ionic strength or high surfactant concentration this simple picture fails to describethe observed propertiesof a number of common micelle systems. Nonspherical, toroidal, ellipsoidal, and rodlike micelles have been suggested on numerous occasions to account for the propertiesof surfactant systemsin the presence of high electrolyte concentrationsor at high surfactant concentration, or both. Long rods have in fact been observed with electron microscopy in the case of cetyltrimethylammoniumbromide (CTAB) in the presence of NaBr.2 High salt concentrations have been found to increase the system viscosity dramatically in some cases,3 and abrupt changes in the internal micellar viscosity, as measured by the rate of photoinduced isomerization, have been associated with salt-induced formation of rods.4 The questionof polydispersity in the aggregationof surfactants has been approached mainly by fluorescenceprobe studies- and by the application of light scattering technique^.^ Small-angle neutron scatteringlo(SANS), NMR," and ESRIZhave also been employed to study micellar growth. While all of these techniques can provide evidence of the existence of polydispersity in the distribution of aggregation numbers, they are rather insensitive to the details of the shape of the distribution function and will not, for example, allow one to resolve shapes or bimodal distributionswith any confidence. Light scattering methods, when applied to polydisperse systems, require the assumption of a size distribution function as well as micelle shape factors in order to convert the results of scattering measurements into an average aggregation number. If two distinct shapes are present, each with a distribution of sizes, the number of adjustable parameters becomes excessive. Thermodynamic arguments, based on the requirements placed on the monomer-micelle equilibrium by the second law, yield ContributionNo. 491 from thephotochemistry Unit, University of Western Ontario.
distribution functions that are quite simple.I3-'5 For the rodlike micelle, an exponential distribution function is obtained. In this particular case, the position of the maximum and the width, either of the number or weight aggregation number distribution, are not independent;Le., the position of the maximum fixes theshape and thus the width. For spherical micelles, Gaussian-like distributionsare predicted with small but not insignificant widths. These thermodynamic models do not appear to have been tested critically, presumably because of the above-mentioned problems associated with experimentallyrecovering size distribution functions. In this paper we suggest a novel approach to the problem of determining micelle size distributions, including bimodal distributions, without any a priori assumptions as to the shape. The method, based on fluorescence quenchingmeasurementsanalyzed by recovering fluorescence lifetime distributions, is applied to measurements made on sodium dodecyl sulfate (SDS) and cetyltrimethylammonium chloride (CTAC) in the presence and absence of added salt. In both systems, there is considerable evidence' that the added electrolyte induces micellar growth and polydispersity. Of particular interest is the transition region where the micelle properties are changing rapidly, but before the salt concentration is high enough to induce what are thought to be long rods. "ry
Tachiyal6 and Infelta et ~ 1 .originally 1~ predicted that quencher molecules should be distributed among micelles according to the Poisson statistics. This is a direct result of the kinetic equilibrium scheme which assumes that the rate constant for the quencher entry into a micelle is independent of quencher content of the micelle. The fluorescence decay law for a probe emitter in the presence of quencher can be thus expressed as an exponential series with the Poisson coefficients as amplitudes
F ( t ) = F(0)
P(j,n) exp[-(k, + j k g ) t ] (1) P O where P(j,n) = d exp(-n)/j!, and n is the average number of quenchers per micelle. Exchange and exit rates are assumed to ~
0022-365419312097-8082$04.00/0 0 1993 American Chemical Society
Size Distributions in Polydisperse Micelle Systems
The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 8083
be negligible during the excited-state lifetime of the probe. l/ko is the unquenched lifetime of the probe, and kq is the quenching rate constant of a single quencher molecule in the micelle. The total quenching rate constant is explicitly assumed to be proportional to the quencher occupation number j, an assumption that will not be valid at high quencher concentration. In the latter case the Poisson distribution will also fail and should be replaced by the more general binomial distribution which assumes a finite maximum occupancy of the micelle by quencher molecules. Siemiarczuk and Ware19 demonstrated experimentally the validity of the Poisson model for SDS micelles by directly recovering Poisson patterns of amplitudesin eq 1 from fluorescence decay data by using distribution analysis without any a priori assumption about the underlying statistics for quencher partitioning. Their application of lifetime distribution analysis represents an alternative to a traditional approach, where the closed form of eq 1, namely
F ( t ) = F(0) exp(-kot - n[ 1 - exp(-k,t)]) (2) is used to fit fluorescencedecay data. Both methods are usually successful when micelles are monodisperse.19 If the solute molecules migrate between phases, the constants in eq 2 become functions of the exit and entry rate constants and the concent r a t i o n ~ .This ~ is a complication that is best to avoid when the goal is the recovery of size distributions. Polydispersity makes eqs 1 and 2 inappropriate for describingfluorescencedecays since each subset of micelles of equal size will generate its unique Poisson decay. The overall decay will thus be described by a series of individual Poisson distributions weighted by the distribution of micelle aggregation numbers. According to Almgren and L6froth,20the fluorescence quenching in polydisperse micelles can be described by the following decay function
F ( t ) = F(0)
(3)
( N ) q = (iV)++---+
Aq2 Eq3
...
24 where ( N ) w is the weight-average aggregation number
6
(5)
S
S
and E are parameters relating to the raw skewness and kurtosis. The series is usually truncated after two to three terms, and first few moments are determined by a numerical fit to a polynomial. The succc8s of this approach is limited by the precision of determining F,(O)/ F(0) and by the complexity of the distribution shape. If the distribution is strongly asymmetrical or multimodal, more terms have to be included in the expansion or assumed explicit analytical shapes of the distribution have to be introduced.21.22 In recent years, distribution analysishas been used successfully for analyzing fluorescencedecaysthat originate in heterogeneous systems, including micellesa23 The method utilizes a series of monoexponential terms in order to fit the experimental decay data and recover an underlying distribution of lifetimes. The decay function for polydisperse micelles described by eq 3 can also be expressed as a sum of monoexponentialterms. This can be accomplished by expressing eq 3 in the following equivalent form19 IJis the root-mean-square deviation, and A
I
,-”
F(t) = F(0)
(7)
However, for our purpose it is more convenient to use the following procedure. The terms in eq 7 can be grouped into two parts, one of which will contain all the terms with j-0 (“unquenched”probe emission) and the second part all the remaining terms O=l, ...). Thus
S
where A(s) is the distribution function of micelle aggregation numberss and n is now a function of s, i.e., n,. Severalassumptions have been made: (i) the mean number of probes and quenchers in a micelle is proportional to the aggregation number s; (ii) the number of quenchers in the micelle is independentof the presence of a probe; (iii) there is no significant exchange of amphiphiles during the excited-state lifetime of the probe; (iv) the quenching rate is proportional to the number of quenchers in the micelle. The treatment of micellarpolydispersity developed by Almgren and LBfrothZOinvolves the determinationof the so-called quencher average aggregation number ( N ) Qby extrapolating the longtime “quenched” and “unquenched” fluorescencedecay components to zero time. The result is
S i=m
If jk&) >> k ~distribution , analysis of the fluorescencedecay of the probe in the presence of quencher will result in lifetime distribution pattern where “quenched” and “unquenched” amplitudes are well separated, and the ratio of their integrated contributionscan be directly determined from experimentaldata. Introducing proportionality between n and s, as postulated by Almgren and L6froth:o i.e.
n, = qs S
S
where 7 = Q / ( C - cmc), and Q, C, and cmc are quencher, surfactant and critical micelle concentration, respectively, and it is also assumed that ns = 7s. The determination of ( N ) Qby this method is usually accomplished by graphical or numerical extrapolation of the decay data. An alternative approach involves using the Infelta equation as an approximation to the analytically intractable eq 3. The equation for the quencher average aggregation number is then expanded into a series as a function of q with the coefficients representing the central moments of the micellesize distribution. The result is
(9)
the ratior(q) of integrated amplitudesassociatedwith “quenched” and “unquenched”lifetimes can be derived from eq 8. The result is
For convenience another measurable R(7) can be defined:
R(q) has a form of an exponential series whose amplitudes are
Siemiarczuk et al.
8084 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 represented by the weight distribution of aggregation numbers. Once the variable R is determined as a function of q, distribution analysis can be applied in order to recover the function sA(s).A value of ( N ) Qcan also be obtained from the ESM derived R(q) or r(q). Since from Poisson statistics n = -In R(q),it follows that
(NQ = - ( 1 h ) In R ( d
(12)
If the sums in eq 11 are replaced with integrals, and the function sA(s) represented by some function F(s), then since CsA(s) is a constant R(q) = K-’
F(s) exp[-qs] ds = K-’L[F(s)]
LOG INT
(13)
or
F(s) = K L-’[R(q)]
(14) where K is a proportionality constant that includes the denominator of eq 11. That is, the desired distribution function SA(#) is the inverse Laplace transform of the function R(q). To recover F(s) is in fact well-known to be an inherently ill-conditioned problem in numerical analysisS23 The proposed protocol thus calls for double application of distribution analysis: first, for the recovery of lifetimedistributions from fluorescencedecays acquired at varying quencher concentration in order to construct R(q) and, second, to analyze R(q) in order to recover the weight distributionof aggregationnumbers. The exponential series method (ESM)24,25 was used for analysis of the decay curves, while the maximum entropy method (MEM)*5.26 was selected to recover micelle size distributions. The latter method, which is known for its stability and inherent lack of bias, is better suited to deal with relatively sparse data sets representedby R(q),as well as with the ill-conditioned nature of the inversion problem. Note that A(s) is recovered by this method without any assumptions as to the nature of the distribution. The ESM works reliably with more abundant and higher-precision fluorescence decay data.25 An important advantage of the approach proposed above for probe decay data analysis is that the method does not rely on any particular kinetic model. This is of particular importance when the unquenched portion of the probe decay is nonexponential.In fact, we have demonstrated elsewhere that the pyrene decay in small monodisperse SDS micelles in the absence of quencher is described by a distribution of lifetimes attributed to the inhomogeneous environment sensed by theprobe.2’ Neither the Infelta nor the Almgren and Lafroth equations make any provisions for a distribution of lifetimes of the unquenched probe. In fact, lifetime distribution analysis of high-precision decay data is necessary to reveal its presence. In such a situation, the use of lifetime distribution analysis is the indicated procedure for decay data analysis since it allows mapping out amplitude patterns for the “unquenched”and “quenched” lifetimes. Also, one need not introduce the functional form of k&). Almgren and Lafroth20 assume an inverse proportionality for which there is some experimental evidence.* The method proposed here avoids this problem altogether. Pyrene was selected as the fluorescent probe because of its long lifetime, which makes it ideally suited for the type of analysis contemplated. It was considered desirable to use a quencher that was immobile and caused the minimum perturbation of the properties of the micelle. Benzophenone was selected for this work because of its very small mobility between the micellar and aqueous phases. Almgren e? aL3 have in fact estimated the equilibrium constant for the exchange of benzophenone between the micellar and aqueous phases in the case of CTAC. They report values of 5.6 X lo4 for small micelles and 8.5 X 104 for large aggregates. The magnitude of this equilibrium constant is such that, in the case of monodisperse micelles, the interpretation of n, and k&) in eq 3 is unchanged to an excellent approximation
I
0
500
I
1000
I
1500
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2500
NANOSECONDS
Figure 1. Typical decay curves for ltJM pyrene in 0.07 M SDS containing0.4 M NaCl at 22 OC with various concentrations(M) of the quencher benzophenone: (A) 1.65 X 10-4, (B)6.60 X 10-4, (C)1.16 X 10-3, (D)1.65X lt3,(E)2.15 X lCr3. Alsoshownistheexcitationpulse from the dye laser.
by the almost insignificant exchange of quencher between the two phasesn3s5This will hold true for the polydisperse cases as well since the equilibrium constant shows very little variation with aggregate size. The only effect will be a small decrease in the average unquenched lifetime as the quencher concentration is increased, and this may be accompanied by a slight increase in the broadening of the lifetime distribution for the unquenched species. These effects have no influence on the validity of the proposed method for recovering size distributions.
Experimental Section Materials. Pyrene (Aldrich) was recrystallized from methanol. Sodium dodecyl sulfate (BDH, AnalaR Biochemical, 99%), cetyltrimethylammoniumchloride (Aldrich), benzophenone (Aldrich), sodium chloride (BDH, Analytical grade), and sodium chlorate (Aldrich)wereused as received. Deionized triply distilled water was used to prepare micellar solutions. Fluorescence Lifetime Measurements. Fluorescence decays were measured with the time-correlated single photon counting technique utilizing a laser-based system. The excitation source was an Nd:YAG mode-locked laser (Antares 7 6 4 ) synchronously pumping a Coherent dye laser (Model CR-599) equipped with acavitydumper (Model7200). Thedyelaseroutput wasdoubled with a KDP crystal. The remainder of the system was conventional. Samples were excited at 295 or 310 nm and decays monitored at 385 nm. The width of the instrument response function at half-maximum was about 300 ps. The data were collected over 512 channels. All samples were purged with nitrogen gas for about 20 min prior to fluorescence measurements. The experiments were conducted at 22 OC. This procedure gave the same results as obtained by the freeze-pumpthaw method. Data Analysis. Decay curves were subjected to distribution analysisusing the exponential series method softwarefrom Photon Technology International (PTI). This program combines deconvolution with ESM analysis. The fitting function consisted of up to 100 exponentialterms with fixed, logarithmically spaced lifetimes and variable preexponentials. The probe function typically spanned a range of 1-1000 ns. A shift parameter was incorporated into the fitting function as a constant. The purpose of using the shift parameter was to eliminate or minimize any artifacts usually occurringin the short lifetimedistributionregion due to such effects as a color shift between excitationand emission
The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 8085
Size Distributions in Polydisperse Micelle Systems 1.01
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Figure 2. ESM resultsgiving pyrene lifetime distributionsin 0.07 M SDS containing0.4 M NaCl at 22 O C for various concentrations(hi)of the quencher benzophenone: (A) 1.65 X lV,(B) 6.60X l e , (C) 1.16 X lW3,(D)1.65 X lW3, (E)2.15 X le3.
pulses, finite channel width errors, or temporal instability of the detection system.28.29 In the few cases where the "quenched"and
"unquenched"lifetime distributionswere not fully separated,the minimum between them was taken as the integration boundary.
8086 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993
Siemiarczuk et al.
TABLE Ik Results for 0.07 M SDS in 0.4 M NaCI Quenched by Benzophenone
TABLE I: Results for 0.07 M SDS Quenched by Benzophenone in the Absence of Added Salt
t)
r
distribution analysis
graphical analysis
infelta equation
0.003 76 0.007 51 0.011 27 0.015 02 0.018 79 0.022 54 0.026 29 0.030 05 0.033 80 0.037 56 0.041 31 0.045 06 0.048 82 0.052 57
0.312 0.645 1.134 1.865 2.790 3.690 4.395 6.43 10.1 1 11.69 19.23 25.39 35.23 42.37
72 66 67 70 71 68 70 67 71 68 73 73 73 72
70 62 57 65 59 58 61 61 61 57 63 63 62 62
74 73 71 75 72 72 72 71 72 71 73 74 74 73
The error involved in not using more sophisticated methods for separating overlapping distributions was considered to be negligible. In most cases the ESM distribution analysis of the decay curves in the absence of quencher revealed the existence of a very small contribution from a short-lived component at 80-1 00 ns in addition to the main pyrene component at 300-400 ns. This extra component was treated as an impurity and was subtracted from the integrated "quenched" amplitudes obtained from the ESM analysis of the decays with quencher present. Some decay data were also analyzed in terms of the Infelta equation (eq 2) using a software package from PTI. The recovery of micelle distributions was carried out with the maximum entropy method software fromPTI, which was modified to deal with the size distribution problem rather than with the recovery of distributions of lifetimes. The fitting function was in the form of an exponential series given by eq 1 1. The input data, R(9),for the MEM treatment were calculated from the ESM recovered lifetime distributions according to eqs 10 and 11. Testsofthesizerecoverymethod wereconducted with simulated data. Simulated decay data files were created by numerical convolution of eq 3, which describes the fluorescence kinetics in polydisperse micelles, with an experimental laser pulse profile. Various shapes for the micelle size distribution function A(#) were assumed. The data were subjected to the ESM lifetime distribution analysis, the function R(9) was calculated, and then MEM analysis was carried out in order to test the success of recovery of the micelle size distribution A ( s ) .
Results and Discussion Typical decay curves are shown in Figure 1 for SDS in 0.4 M NaCl at several quencher concentrations. The results of ESM analysis areillustrated in Figure 2. It isclear that theunquenched and quenched regions of the lifetime distributions are well separated. The width of the unquenched peak is about 30 ns, a phenomenon that has been previously reported.2' The lifetime scale used here is too broad to properly display this width. As mentioned above, as long as the quenched and unquenched regions are well separated, the width of the unquenched distribution is not relevant to the subsequent size distribution analysis. Tables I-IV indicate the range of quencher concentrations (in terms of 7) used for each salt concentration. The decay data were analyzed by ESM which yielded R(q),which was then analyzed by MEM to recover SA(#). For purposes of comparison, the decay curves were also subjected to graphical analysis to establish values for F,(O) and F(0)and thus ( N ) Q . In addition, the 4 9 ) values from the ESM analysis were used to calculate values of ( N ) Q . The values of ( N ) Qfrom these two methods of analysis of the decay data were then analyzed with eq 5.
n
r
distribution analysis
graphical analysis
0.00267 0.00534 0.00801 0.0107 0.0134 0.0160 0.0187 0.02 14 0.0240 0.0267 0.0294 0.0321 0.0347
0.413 0.945 1.104 1.870 2.606 3.562 4.725 6.461 8.014 10.56 12.89 15.71 19.61
129 125 93 98 96 95 93 94 92 92 90 88 87
118 118 116 122 119 118 117 114 112 112 113 111 110
TABLE 111: Results for 0.07 M SDS in 0.5 M NaCl Quenched by Benzophenone t)
r
distribution analysis
graphical analysis
0.001 70 0.003 41 0.005 11 0.006 82 0.008 52 0.010 22 0.011 93 0.013 63 0.015 33 0.017 04 0.018 74 0.020 45 0.022 15 0.023 85 0.025 56
0.437 0.984 1.674 2.476 3.288 4.405 5.465 6.951 8.74 11.18 12.98 16.28 20.25 25.99 28.21
213 20 1 192 183 171 165 156 152 148 147 141 139 138 138 132
157 200 194 189 166 170 169 157 150 155 151 146 150 143 134
TABLE I V Results for 0.07 M SDS in 0.6 M NaCl Quenched by Benzophenone (NQ t) r distribution analysis graphical analysis 0.000 801 0.002 404 0.003 206 0.004 007 0.004 808 0.006 41 1 0.008 815 0.009 6 17 0.010 42 0.011 22
0.335 1.218 2.027 2.587 3.507 5.721 9.944 11.40 12.89 16.10
360 33 1 345 319 313 297 27 1 262 253 253
360 312 318 296 316 306 286 26 1 262 25 1
For SDS, the size distributions, expressed as weight-average aggregation numbers, are shown in Figure 3. The qualitative features are of considerable interest. As salt is added, the distribution shifts to larger aggregations numbers, and between 0.4 and 0.6 M we observe a transition region indicated by the presence of a bimodal distribution, which leads to a broad unimodal distribution a t 0.6 M NaCl. At 0.4 M NaCl the most probable aggregation number is about twice that of the zero salt case, whereas for the 0.6 M NaCl it is about 5 times the value a t zero salt. The results of the analysis with eq 5 are given in Table V, along with the average aggregation numbers and widths recovered by MEM. For the MEM results in the bimodal case, the average given in Table V is over the entire distribution. For zero salt, the use of eq 5 with either graphical or ESM generated R(9) yields quencher aggregation numbers that are constant within experimental error, indicating a very small width. This is entirely consistent with the MEM results. The average aggregation numbers obtained by the two approaches are considered to be in good agreement. For the case of 0.4 M salt, the MEM method and eq 5 with graphical data agreevery well. The ESM generated
The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 8087
Size Distributions in Polydisperse Micelle Systems
LOG INT
I
Figure 3, Micelle size distributions recovered by MEM for SDS at 22 OC at various NaCl concentrations. (A) 0.00 M NaCl, (N)w = 67.5, u = 1.4; (B) 0.4 M NaCl, (N)w= 139, u = 22; (C) 0.5 M NaCl, (N1)w = 114, ul = 22, (Nz)w= 380, uz = 38; (D) 0.6 M NaC1, (N)w = 374, u = 140. TABLE V Weight-Average Aggregation Numbers and Distribution Widths from MEM Analysis and Polynomial Fit to Ea 5 for SDS Micelles polynomial fit ESM graphic MEM (N)w 0 (N)w u [NaCIl(M) (N)w u 0.0 0.4
67.5 139
0.50 0.6
235 374
1.4 22.1 136 140
70.1 131e 14gd 228d 365d
80.8 127 133 132
61.5 l2W 233d 352e
24.2 144 141
a At 0.5 M NaCl the distribution recovered with MEM analysis is bimodal. b Values constant. Linear fit. d Cubic fit. Quadratic fit.
R(q) gives a high width, but the aggregation number still agrees
well with the MEM determination. For salt concentrations of 0.5 and 0.6,the two approaches agree very well indeed, but only the MEM analysisreveals the presence of the bimodal distribution. A large number of studies have been reported concerning the aggregation number for SDS in both the presence and absence of added salt. Comparison of the results obtained in this work with previous determinations is made difficult by virtue of the following. Prior to about 1980,some of the studies reported were conducted with impure SDS which gave incorrect aggregation numbers. In addition, some studies were at 25 OC and the aggregation numbers must be corrected for temperature in order to compare with 22 OC data. Some papers fail to quote the temperature. Also, not only have studies been conducted over a wide range of SDS concentrations but also some papers fail to specify the concentration and some studies report only values extrapolated to the cmc. In addition, both light scattering9 and small-angle neutron scattering experiments require a model in order to calculate the aggregation number from the observed scattering data. SANS is frequently conducted in DzO, and in this solvent, SDS appears to exhibit larger aggregation numbers than in ordinary water.30Jl Aggregation numbers for SDS determined by fluorescence techniques, light scattering, and sedimentation for'the period up to about 1982 have been summarized by Almgren and Swarup32 for SDS for both saltfree systems and for NaCl concentrations up to 0.8 M. Recent determination~3S3~ of the aggregation number of SDS in the absence of added salt give an average value at 25 OC of 67f 2fortherangeofSDSconcentrationsof0.05-0.1M.Results at 25 OC were adjusted using the data of Malliaris et ai.@ and that of Croonen et al.37 Uncertainties in the numbers make it difficult to determine whether there is a significant difference between 0.05and 0.1M SDS as regards the aggregation number,
1
1000 1500 NANOSECONDS
-
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Figure 4. Typical decay curves for 10-5 M pyrene in 0.04 M CTAC containing 0.042 M NaClO3 at 22 OC with various concentrations(M) of the quencher benzophenone: (A) 8.34 X l t 5 , (B) 3.34 X lv,(C) 5.84 X l v , (D) 8.34 X l e . Also shown is the excitation pulse from the dye laser.
but if there is, it is small. Light scatteringresultsgenerally appear to be somewhat higher. Two results based on SANS, one4' giving 65 for 0.125M SDS and theothelJ2 giving 68 for deuterated SDS in H20, are of interest in this context. For NaCl concentrations in the range 0.4-0.6M, the spread of literature values is greater. At approximately 0.4 M NaC1, r e p ~ r t e d aggregation ~~l~~ numbers average 142f 8 whereas we obtain 139. Waar, Grieser, and Evans43reported a value of 162 for 0.42 M NaCl obtained with eq 5. Above 0.4 M there appears to be a transition region since at 0.5 M salt we observe a bimodal distribution. A wide range of aggregation numbers are reported in the literature, not only at 0.5 M but also at 0.6 M, most determined without regard for the polydispersity known to be present. At 0.5 M salt, fluorescence measurements43*# give about 210 whereas light scattering m e a s u r e m e n t ~ ~are a ~lower. ~ At 0.6 M salt, there is some agreement between light scattering and fluorescent techniques, with an averageUsa of around 300. It is interesting to compare our result at 0.6 M with recent light scattering results of Briggs, Nicole, and Ciccolello.4* They assumed a model involving polydisperse prolate ellipsoids with a distribution controlled by a constant surface area per monomer and obtained a value for the number-average aggregation number at 25 OC of 315. If we use our value for CT of 140,this is equivalent to a weight-average aggregationnumber of 377,in excellent agreement with our value of 374. Typical decay curves and ESM recovered lifetimedistributions for CTAC are shown in Figures 4and 5. The unquenched lifetime distributions are broader than observed for SDS. The full width at half-maximum (fwhm) ranges from 63 ns at 5 mM CTAC to 56 ns at 50 mM CTAC.49 The addition of sodium chlorate has a small but measurable effect49on the width of the unquenched lifetime distribution; the fwhm increases smoothly from 56 ns at zero salt to 70 ns for 0.11 M chlorate for [CTAC] = 0.05 M. Since the unquenched region is still well separated from the quenched region at all salt and quencher concentrations used, this rather large lifetime broadening of pyrene in CTAC is not a problem. It is noteworthy that the broadening exhibited by CTAC is much greater than that observed and previously reported for SDS.27 Tables VI-IX provide an indication as to the range of q values and the ( N ) Qvalues obtained from graphical and ESM analysis. MEM recovered size distributions are shown in Figure 6 for various NaClOS concentrations. The behavior of CTAC as the salt concentration is increased is somewhat different than that observed for SDS. At 0.034M, the size distribution is found to be narrower than for the zero salt case. As the salt concentration
8088 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993
Siemiarczuk et al.
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Size Distributions in Polydisperse Micelle Systems
The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 8089
TABLE VI: Results for 0.04 M CTAC Quenched by Benzophenone with No Added Salt ~~
TABLE M: Results for 0.04 M CATC in 0.045 M NaC103 Quenched by Benzophenone
~~
(N\n
(N)Q
n 0.002 86 0.005 73 0.008 60 0.011 46 0.014 33 0.017 19 0.020 06 0.02292 0.02579 0.02865
r 0.293 0.646 1.11 1.65 2.38 3.20 4.15 5.48 7.28 8.81
distribution analysis
graphical analysis
infelta eauation
89.6 88.5 86.8 85.2 84.9 83.5 81.7 81.6 82.0 79.7
123.2 108.3 108.8 95.5 98.5 94.1 91.3 95.2 81.4 95.3
1 45.3 123.9 117.6 113.1 111.5 105.5 104.8 106.8 106.2 103.3
TABLE MI: Results for 0.04 M CTAC in 0.034 M NaCI03 Quenched by Benzophenone n 0.002 50 0.00500 0.007 50 0.010 00 0.012 50 0.015 00 0.01750 0.02000 0.02250 0.02500 0.02750 0.03000 0.03250 0.03500
r 0.399 0.907 1.606 2.629 3.989 5.679 8.347 11.87 17.66 24.48 33.24 47.07 66.25 93.50
distribution analysis
graphical analysis
134 129 128 129 129 127 129 128 130 129 128 126 129 130
129 133 130 133 130 127 128 127 132 130 132 131 132 130
TABLE WI: Results for 0.04 M CTAC in 0.042 M NaC103 Quenched by Benzophenone (N)o r)
0.00216 0.00432 0.00648 0.00864 0.010 80 0.012 96 0.015 12 0.017 28 0.019 45 0.021 61 0.023 77 0.025 93 0.028 08
r 0.555 1.356 2.424 3.957 5.828 7.844 10.88 13.83 18.87 26.06 35.22 44.89 65.39
distribution analysis
graphical analysis
204.4 198.3 189.9 185.3 177.9 168.2 163.7 156.0 153.7 152.6 151.0 147.6 149.4
190.4 205.3 202.3 190.2 182.6 176.7 168.0 159.9 164.2 162.8 159.6 159.0 154.7
Comparative values taken from the literature5"53 for CTAC aggregation numbers with [CTAC] 0.04 M at zero salt concentration average to 93 f 4, which agrees well with our results. Of particular interest is the result of Almgren et ai.) They obtained 90.5 with a width u of 29 from an analysis based on eq 5 . This is to be compared with 88.8 (a = 20.5)for MEM and 90.1 ( u = 28) from eq 5 obtained in this work. At a chlorate concentration of 0.034M, our results for the average aggregation number disagree by about a factor of 2 with those reported by Almgren et al.,3 whose results are based on recovering ( N ) Qfrom the Infelta equation or from F(O)/F,(O). When the salt concentration was 0.04 M, Almgren et ai. encountered difficulties in data analysis requiring modification of the Infelta equation to facilitate extrapolation to zero time. It is of course at this point that we observe the bimodal distribution. Part of the reason for the disagreement at 0.034M salt may be due to errors in the analysis of ( N ) Qvs 1. The extrapolation to (N)w depends strongly on the values of (N ) Qat the low quencher concentrations. A small background in the decay data can have a large effect on this extrapolation.
=
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distribution analysis
graphical analysis
0.001 81 0.00362 0.005 32 0.007 23 0.009 04 0.010 85 0.012 66 0.014 47 0.016 28 0.018 09 0.01990 0.02351 0.02532
0.616 1.497 2.785 4.440 5.686 7.836 10.56 12.75 17.99 21.69 27.02 43.77 51.34
265 253 245 234 210 200 193 181 181 172 167 162 156
267 243 250 216 222 206 210 194 193 192 184 184 174
3
Figure 6. Micelle size distributions recovered by MEM for CTAC at 22 OC at various NaClO3 concentrations. (A) 0.00 M NaClO3, ( N ) w = 88.8,u = 20.5;(B)0.034M NaClO3, ( N ) w = 129,u = 5.9;(C) 0.042 M NaClOp, ( N ) w = 223,u = 100;(D)0.045 M NaClO3, (N1)w 113, 01 17.4, (Nz)w = 376, u2 29.0.
-
TABLE X Weight-Average Aggregation Numbers and Distribution Widths from MEM and Polynomial Fits to Eq 5 for CTAC Micelles polynomial fit MEM ESM graphic [NaCKhIW ( N ) w u (N)w u (N)w u 0.00 0.034 0.042" 0.045'
88.8 129 223 293
20.5 90.1' 5.9 13oC 101 217' 125 285d
27.0 11.0 97.3 137
129e 13ob 21P 283d
82.1 78.8 145
a At 0.042and 0.045 M NaC103 the distributions recovered by MEM were bimodal. Valuesconstant. Linear fit. Cubicfit. Quadratic fit. We have simulated narrow and broad size distributions, calculated the expected pyrene decay functions, added noise, and then performed the standard ESM-MEM analysis. The results for both unimodal and bimodal distributions are shown in Figure 7. It is clear that the methods used recover bimodal distributions with satisfactory accuracy. Thus, we consider both the unimodal and bimodal distributions observed in both CTAC and SDS to be real and that the true widths are approximately reflected in the observed widths. As an additional check on the observation that a bimodal distribution is required for the transition region, the ( N ) Qdata for CTAC obtained with eq 5 at 0.042 and 0.045 M salt were converted intoR(1) values and submitted to MEM analysis. These values of R(7) are much less reliable than those obtained from the ESM analysis but nevertheless, when analyzed, gave bimodal distributions similar to those shown in Figure 6. For 0.042 M salt, the maxima were at 133 and 423 with u of 16 and 37. For 0.045 M salt, the maxima were at 138 and 324 with u of 19 and
8090 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 ,nl
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S Figure 7. MEM results from simulated decay data for pyrene in monodisperse and polydisperse micelles (A) input ( N ) w = 100, u = 0.0, rccovered ( N ) w = 96.3,u = 4.5. (B)input (N1)w = 100, UI = 0, ( N z ) w = 500, uz = 0,recovered (N1) = 101, uI = 14, (N2)w = 500, 6 2 = 31. ( C ) input ( N ) w = 370,u = 100,recovered ( N ) w = 365,u = 103.
25. While this does not represent exact agreement with the superior ESM-MEM analysis, nevertheless, it adds considerably to the confidence that can be placed in the assertion that there arein fact bimodaldistributionspresent. In addition, this alternate approach is independent of overlap or spillover between the quenched and unquenched portions of the lifetime distribution. It is of interest to compare our results with those predicted by thermodynamics. For the zero salt case, the CTAC distribution width is approximately equal to that predicted by T a n f ~ r and d~~ by Israelachvili et al.14 The SDS distribution width is narrower than expected. For SDS in 0.6 M NaC1, we observe a broad distribution. The thermodynamic prediction for rod-shaped micelles is a broad distribution, the width of which is fixed by the position of the maximum (the exponential distribution). We predict on the basis of our data a somewhat broader distribution than observed at this salt concentration. However, at ( N ) w z 400, micelles that are rod or prolate ellipsoidal in shape have a length that is only about 3 times the diameter. It is also possible that this is not the correct shape. Israelachvili et al.14 have discussed the allowed shapes of large micelles. They conclude, on the basis of packing considerations, that the ellipsoidal shape is not favored, whereas a toroidal shape is possible. However, Porte has argued convincingly against the toroidal shape on the basis of configurational entropy." It is tempting to associate the micelles having aggregation numbers approximately 2 times the minimum sphere with the globular aggregate. Those with aggregation numbers approximately 4-5 times the minimum sphere would be associated with the prolate ellipsoid or very short rod. This is consistent with the fact that the highest salt concentrationsused in this work appear to be only at the threshold of the conditions favorable for long rod formation. Thus, the transitional distribution may simultaneously include large globular micelles as well as prolate ellipsoidal shapes or a short rods. The method we have used assumea that the quencher molecules aredistributed in themicellesaccording toPoisson statistics where there is no upper limit on the occupation number. The binomial distribution is a natural choice if one wishes to set an upper limit, but it is mathematically a much less convenient distribution function in the context of the formulation of methods for size distribution analysis. It is necessary to vary the quencher concentration and thus the occupationnumber in order to recover size distributions by our method, or aggregation numbers and distribution widths by the use of eq 5 . In fact, for SDS our measurements involvean averageoccupation number n that ranges from 0 to 3 whereas for CTAC the range is 0 to 4. A comparison of the Poisson and binomial distributions for values of n of 3 or 4 also requires one to assume a maximum occupation number.
Siemiarczuk et al. If the maximum occupation number is set between 8 and 16, and the binomial and Poisson distributionsfor n of 3 or 4 are compared, it is found that for n of 3 the distributions approximately superimpose independent of the limit set on the occupation number. For n of 4, a limit of 8 in the occupation number results in a binomial distribution that is more sharply peaked than the Poisson distribution and slightly shifted to higherj, but otherwise superimposes. As the limit is increased,the superposition becomes better. Thus, it appears that the two distributions are more or less equivalent in the range of parameters in question, especially since most of the micelles in this study have aggregation numbers of 90-300. With the larger micelle, there will be a larger upper limit on the occupation number and thus better correspondence between the two distribution functions. If the failure of the Poisson assumption was a problem, one might expect to see evidence for this in the SDS experiments in the absence of salt. Here the micelle is small and the effect of an upper limit might be seen. However, flat plots of ( N ) Qwere obtained out to a value of corresponding to n of about 3 . The quenching aggregation number calculated is thus independent of the quencher concentration, and the value obtained, as indicated above, is in excellent agreement with recent determination by fluorescence and other methods. However, the situation is more complex than just the choice of a distribution function. On the addition of a probe and a significant number of quencher to the micelle, changes may take place in the size, and the assumption that the average number of probes and quenchers per micelle is proportional to the size will at some point fail.56 There is some evidence to indicate that aromatic additives increase the size of micelles. In the case of SDS, adding toluene appears to increase the aggregation number by about 1.4 units per added molecule.57 If toluene and benzophenone can be considered equivalent,then we would expect an increase in size of the SDS micelles of about 4 parts in 70 as n goes from 0 to 3. For CTAC the effect would be smaller because the zero salt micelle has a larger aggregation number, although here we must assume that the size increase per molecule of quencher in CTAC is similar to that for SDS. Once salt is added and themicellesize expands, the effect of the presence of quenchers on the size should be drastically reduced and become insignificant.
Conclusions We have proposed a method for determining size distributions in polydisperse systems that requires only the assumption that the probe and quencher molecules are distributed according to Poisson statistics and that the mean occupation numbers are proportional to the micellar aggregation number, assumptions that are also essential to other fluorescencemethods currently in use. In cases where is reasonable to compare with other methods the agreement is excellent. The fact remains that for the polydisperse case all such comparisons are with methods that require the specification of the distribution shapeor simply recover an average aggregation number and width without regard for the possibility of multimodal distributions. The method proposed in this paper has revealed bimodal distributions in the transition region between spherical and rod-shaped micelles in SDS and CTAC without the a priori assumption of such distributions. Simulations have provided reason for confidence that distribution shapes are being recovered with reasonable fidelity. Much additional work is clearly required in order to fully test this proposed method and examine experimentally each assumption on which it is based. Such work is currently in progress. Acknowledgment. Financial assistance from the National Scienceand EngineeringResearch Council (Canada) is gratefully acknowledged. References and Notes (1) Kalyanasundaram, K. Phorochemisrry in MicroheterogeneousSysrems; Academic Press: Orlando, 1987.
Size Distributions in Polydisperse Micelle Systems (2) Toyoko, I.; Kamia, R.; Ikeda, S.J. Colloid InterfaceSci. 1985,108, 215. (3) Almgren, M.; Alsins, J.; van Stam, J.; Mukhtar, E. Prog. Colloid Polym. Sci. 1988, 76, 68. (4) Miyagushi, S.;Matsumura, S.;Asakawa, T.; Mishida, M. J. Colloid Interface Sci. 1988, 125, 237. (5) Almgren, M. In Kinetics and Catulysis in Microheterogeneous Systems; GrHtzel, M., Kalyanasundaram, K.,Eds.;Marcel Dekker: New York, 1991. (6) Malliaris, A. In?. Rev. Phys. Chem. 1988, 7, 95. (7) Zana, R.; Lang, J. Colloids Surf. 1990, 48, 153. (8) Reckmans. S.: De Schrwer. F. In Frontiers in Suoramolecular Orgunic Chemistvhrd Photochemistry;Schneider, H., Dtirr, H.,Eds.; VCH Publishers: New York, 1991. (9) Mazer, N. In Dynamic Light Scattering; Ptcora, R., Ed.; Plenum: New York, 1985. (10) Chen, S.H.; Lin, T. In Methods of Experimental Physics;Academic Prcas: New York, 1987; Vol. 23B. (1 1) Kalyanasundaram, K.;GHatzel, M.; Thomas, J. J. Am. Chem. Soc. 1975,97, 3915. (12) Wikander, G.; Johansson, B. Lungmuir 1989, 5, 728. (1 3) Moroi, Y. Micelles, Theoreticaland Applied Aspects; Plenum: New York, 1992. (14) Israelachvili, J.; Mitchel, D.; Ninham, B. J. Chem. Soc., Faraday Trans. 1976, 72, 1525. (15) Missel, P.; Mazer, N.; Benedek, G.; Young, C.; Cary, M. J. Phys. Chem. 1980,84, 1044. (16) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (17) Infelta, P.; GrHtzel, M.; Thomas, J. J. Phys. Chem. 1974, 78, 190. (18) Tachiya, M. In Kinetics of Nonhomogeneous Processes; Freeman, G., Ed.; Wiley: New York, 1987. (19) Siemiarczuk, A.; Ware, W. Chem. Phys. Len. 1989, 160, 285. (20) Almgren, M.; L6froth, J. J. Chem. Phys. 1982, 76, 2734. (21) Lang, J. J. Phys. Chem. 1990,94, 3734. (22) Waar, G.; Grieser, F. J. Chem. Soc., Faraday Trans. 1 1986, 82, 1813. (23) Ware, W. In Photochemistry in Organized and Constrained Media; Ramamurthy, V., Ed.; VCH Publishers: New York, 1991. (24) James, D.; Ware, W. Chem. Phys. Lett. 1986, 126, 7. (25) Siemiarczuk, A.; Wagner, B.; Ware, W. J. Phys. Chem. 1990.94, 1661. (26) Skilling, J.; Bryan, R. R. Mon. Not. R. Astron. Soc. 1984,211, 111. (27) Siemiarczuk, A.; Ware, W. Chem. Phys. Lett. 1990,167, 203. (28) Birch, D.; Imhof, R. In Topics in Fluorescence Spectroscopy; Lakowicz, J., Ed.; Plenum: New York, 1991; Vol. 1, Chapter 1. (29) OConnor, D.; Phillips, D. Time-CorrelatedSingle Photon Counting; Academic Press: London,,j984.
The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 8091 (30) Miyagishi, S.;Asakawa, T.; Nishida, M. J. Colloid Interface Sci. 1987, 115, 199. (31) Chang, N.; Kaler, E. J. Phys. Chem. 1985, 89, 2996. (32) Almgren, M.; Swarup, S . In Proceedings of the 4th International Symposium on Surfactants in Solution; Mittal, K.,Lindman, B., Eds.; Plenum: New York, 1984; Vol. 1, p 613. (33) Boens, N.; Luo, H.; Van der Auweraer, M.; Reekmans, S.;De Schryver, F.; Malliaris, A. Chem. Phys. Lett. 1988, 146, 337. (34) Luo, H.; Boens, N.; van der Auweraer, M.; De Schryver, F.; Milliaris, A. J. Phys. Chem. 1989,93, 3244. (35) Almgren, M.; Swarup, S.J. Colloid Interface Sci. 1983, 91, 256. (36) Liana, P.; Viriot, M.; Zana, R. J. Phys. Chem. 1984,88, 1098. (37) Croonen, Y.; Gelade, E.; Van der &gel, M.; Van der uwAeraer, M.; Vandendriessche, H.; De Schryver, F.; Almgren, M. J. Phys. Chem. 1983,87, 1426. (38) Reekmans, L.; Luo, H.; Van der Auweraer, M.; DeSchryver, F. Lungmuir 1990, 6, 628. (39) Almgren, M.; Swarup, S.J. Phys. Chem. 1983,87,876. (40) Malliaris, A.; Le Moigne, J.; Strum, J.; Zana, R. J. Phys. Chem. 1985,89, 2709. (41) Beuobotnov, V.; Borbcly, S.; Czer, L.; Farago, B.; Gladkih, I.; Ostanevich, V.; Vass, S . J. Phys. Chem. 1988,92, 5738. (42) Cabane, B.; Duplcssix, R.; Zcmb, T. J. Phys. (Paris) 1985,46,2161. (43) Waar, G.; Grieser, F.; Evans, D. J. Chem. Soc., Faraday Trans. 1 1986, 82, 1829. (44) Almgren, M.; Mfroth, J. J . Colloid Interface Sci. 1981, 81, 486. (45) Waar, G.; Grieser, G. Chem. Phys. Lett. 1985, Zl6, 505. (46) Hayashi, S.;Ikeda, S . J. Phys. Chem. 1980,84, 744. (47) Ikeda, S.;Ozeki, S.;Tsunoda, M. J. Colloid Inreflace Sci. 1980,73, 27. (48) Briggs, J.; Nicoli, D.; Ciccolello, R. Chen. Phys. Lett. 1980,73, 149. Ware,W. Unpublishcdresults. (49) LaPalme,K.;Siemiarczuk,A.;LiyY.; (50) Roelants, F.; De Schryver, F. Lungmuir 1987, 3, 209. (51) Makhloufi, R.; Hirsch, E.; Candau, S.;Binana-Limtcle, W.; Zana, R. J. Phys. Chem. 1989, 73, 8095. (52) Malliaris, A.; Lang, J.; Zana, R. J. Chem. Soc., Faraday Trans. I 1986, 82, 109. (53) Roelants, E.; Gelade, M.; van der Auweraer, M.; Croonen, Y.; De Schryver, F. J. Colloid Interface Sci. 1981, 96, 288. (54) Tanford, C. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 1811. (55) Porte, G. J. Phys. Chem. 1983,87, 3541. (56) Almgren, M.; Gricaer, F.; Thomas, J. J. Am. Chem. Soc. 1979,101, 279. (57) Almgren, M.; Swarup, S.J. Phys. Chem. 1982,86, 4212.
