4390
J. Phys. Chem. B 1997, 101, 4390-4393
Determination of the Structure of Tetradecyldimethylaminoxide Micelles in Water by Small-Angle Neutron Scattering N. Gorski† and J. Kalus*,‡ Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research JINR, 141980 Dubna, Moscow Region, Russia, and Experimentalphysik I, UniVersita¨ t Bayreuth, 95440 Bayreuth, Germany ReceiVed: February 6, 1997X
By means of a small-angle neutron scattering experiment the radius R as well as the scattering length density of rodlike tetradecyldimethylaminoxide (TDMAO) micelles in D2O were determined. The micelles turned out to be homogeneous. As a function of the concentration c of TDMAO we found the mean aggregation number 〈N〉, which increases with c, and the value of two thermodynamic parameters describing the energy advantage by inserting a monomer in the end cap of the spherocylindrical micelles and in the cylindrical part, respectively.
1. Introduction The self-organization of surfactants in aqueous solutions is of current interest. In solutions of nonionic tetradecyldimethylaminoxides (C14H29NO(CH3)2O), abbreviated by TDMAO, in water, micellar aggregates are formed. From measurements of interfacial tensions a critical micellar concentration (cmc) of 0.12 mM was deduced in H2O at 25 °C.1 It was claimed from electric birefringence measurements that above 10 mM anisotropic micelles are present and below this concentration spherical micelles should be present.2 By a small-angle neutron scattering experiment (SANS) and at a concentration of 55.2 mM in D2O, the radius R of rodlike micelles turned out to be around 1.94 ( 0.03 nm and the length L was determined to be 14 ( 4 nm.3 For the evaluation of R and L it was assumed that the rodlike micelles are homogeneous. We tested this assumption by performing a contrast variation experiment, which is sensitive to the internal structure of micelles. Furthermore, we examined the micellar shape in the concentration range 0.61 < c < 53.4 mM in order to test whether a transition of spherical to rodlike micelles is present in this range. As we show later, no spherical micelles were found. The length increases with increasing concentration, which was expected according to the so-called ladder model we used for our data evaluation. 2. Experimental Details
by comparison with a corresponding H2O/D2O sample. The data treatment followed standard procedures.5 The contrast variation experiments were made with a 53.4 mM solution of TDMAO. The mole fractions of the D2O/H2O mixtures were X ) 1, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, and 0 (X ) [D2O]/([D2O] + [H2O])). The shapes of the micelles were determined for solutions of TDMAO in pure D2O. The concentrations were 0.61, 1.20, 3.44, 5.48, 9.86, 27.7, and 53.4 mM. For the concentrations below 5.48 mM, where the scattering intensities are becoming small, the same cell was used, avoiding systematic errors eventually related to the scattering of different cells. 3. Theory The scattering cross section dσ/dΩ of a homogeneous particle is given by
dσ/dΩ ) 〈F2〉(F - Fs)2V2S(Q)
where V is the volume of the particle and F and Fs are the scattering length densities of the micelles and solvent, respectively. For homogeneous cylinder-like micelles we get for the mean squared structure factor:6
〈F2〉 )
TDMAO was obtained as a gift from Hoechst AG, Gendorf, and recrystallized twice from acetone. D2O was obtained from ISOTOP in Moscow and had a 99.8% isotopic purity. The SANS experiments were performed at the time-of-flight spectrometer MURN4 of the pulsed reactor IBR-2 in Dubna, Russia. Neutrons were used in the wavelength range 0.07-1 nm, and the accessible range of momentum transfer in these experiments was 0.08-2.5 nm-1. Typically the neutron intensity at the sample position is 107 cm-2 sec-1. The samples were contained in 1 mm path length quartz cell and were thermostated to 24 ( 0.5 °C. Conversion of the scattered intensities into absolute differential cross sections was done by using an internal calibration standard (vanadium). Background scattering was substracted †
Joint Institute for Nuclear Research. Universita¨t Bayreuth. X Abstract published in AdVance ACS Abstracts, May 1, 1997. ‡
S1089-5647(97)00477-X CCC: $14.00
(1)
∫0π/2(sin(Q(L/2) cos β)/
(Q(L/2) cos β) 2J1(QR sin β)/(QR sin β))2 sin β dβ (2) β is the angle between the axis of symmetry of the micelle and the vector of the momentum transfer Q B . J1 is the Bessel function of the first kind and of order 1. L and R are the length and the radius of the micelle, respectively. The value of |Q B | is given by Q ) 4π sin(φl)/λ. The wavelength of the neutrons is λ and 2φ is the scattering angle. S(Q) is the structure factor and is related to the interaction between the micelles. For a sufficiently high value of Q or for noninteracting micelles, S(Q) goes to 1. Indeed, in our case we did not observe a so-called correlation peak in the intensity distribution of the scattered neutrons, and we assumed therefore, that S(Q) is equal to 1 in the studied Q-range. This assumption is strengthened by the results of static light scattering experiments, where the correlation between the micelles becomes severe above a concentration of about 100 mM.2 © 1997 American Chemical Society
Structure Determination of TDMAO Micelles in Water
J. Phys. Chem. B, Vol. 101, No. 22, 1997 4391
Figure 2. The square root y of the scattering intensity as a function of the mole fraction X of D2O. Figure 1. Scattering intensities of contrast variation experiments on a 53.4 mM solution of TDMAO in water. The molar fraction X of D2O is given from top to bottom by 1 (10), 0.8 (8), 0.7 (6), 0.6 (4), 0.5 (2), 0.4 (0), 0.3 (-2), 0.2 (-4), and 0(-6). The number in parentheses indicates the number of units the curves are shifted down or up with respect to the X ) 0.4 curve. Notice the ln(IQ) versus Q2 plot. The solid lines are due to a fit described in the text. Missing points have negative values.
For cylindrical micelles with an inhomogeneous structure eq 2 has to be altered.7 But according to our experimental facts, this is not necessary for our system within the experimental errors. 4. Experimental Results and Discussion 4.1. Contrast Variation Experiments. In Figure 1 we present the scattering intensities I for the different contrasts X as a function of Q. Notice that in the figure ln(IQ) versus Q2 is shown. In this representation a long linear section is found, if rodlike micelles are present. The solid lines are due to a fit where L and R are two common fit parameters, which were used in a fit where all scattering curves were treated simultaneously. We mention again that within our model the micelles are homogeneous. Notice that the curves are shifted for convenience by multiplication factors, indicated in the figure caption. The fact that the shape of all scattering curves is the same within the statistical error is a proof that the model of homogeneous micelles is sufficient to explain the data. We mention that the intensity distribution for the X ) 0 curve was so weak that a conclusion about the real shape of this curve is indeed not possible. The value of the radius and the length are: R ) 1.85 ( 0.02 nm, L ) 14.3 ( 0.3 nm. Notice that for this fit no length distribution of the micelles, as introduced below, was used. In Figure 2 we present the values of the square roots of the fitted intensity curves at a certain Q-value of Figure 1 as a function of X. This intensity is named VF. These VF factors characterize the absolute intensities of each of the curves. According to eq 1, y ) (dσ/dΩ)1/2 ) prop(VF)1/2 is proportional to (F - Fs). Fs itself is proportional to X. We see, as expected, a straight line. A fit and extrapolation to y ) 0 gives for X at the position y ) 0 the so-called matching point, a value of XMP ) 0.06 ( 0.01. Using this experimental result, we can calculate the scattering length density in the interior of the micelles, which is given by F ) (-0.15 ( 0.06) × 1010 cm-2. The scattering length densities of D2O and H2O were assumed to be 6.270 × 1010 and -0.562 × 1010 cm-2, respectively. Using the known scattering length lM of the molecule (-0.933 × 10-12 cm), we now can extract an experimental value of the volume V0(exptl)
Figure 3. Scattering intensities for TDMAO solutions in D2O. Notice the ln(IQ) versus Q2 plot. From top to bottom the concentrations are 53.4, 27.7, 9.86, 5.48, 3.44, 1.20, and 0.61 mM. The solid lines are due to a fit described in the text.
) lM/F of the TDMAO molecule in the micelle. We found V03 (exptl) ) 0.62+0.42 -0.18 nm . This value is within the error margins in accordance with values due to an approximation introduced by Tanford,8 V0 ) 0.54 nm3, and of measured values via a determination of the density of a micellar TDMAO solution,9 V0 ) 0.480 ( 0.003 nm3. The reason for the large quoted errors of V0(exptl) is related to the extremely small value of XMP itself. Therefore the absolute error quoted for XMP, which is not very different from errors found in other contrast variation experiments, in this case gives rise to an extremely large error in V0(exptl). 4.2. Micellar Shape at Different TDMAO Concentrations. In Figure 3 we present the scattering intensities I for different TDMAO concentrations as a function of Q. Again, ln(IQ) versus Q2 is shown. The solid lines are due to a fit described now. It is well-known that rodlike micelles show a wide distribution of lengths L. One of the models, the so-called ladder model, is used here to analyze the measured intensity distribution curves. According to this model10,11 the size distribution (mole fraction of a N-mer) XN is given by
XN ) βN/K
(3)
where N ) N0, N0 + 1, N0 + 2, ... . N0 and N are the aggregation numbers of the spherical or a rodlike micelle, respectively. The rodlike micelle has spherical
4392 J. Phys. Chem. B, Vol. 101, No. 22, 1997
Gorski and Kalus
TABLE 1a c, mM
〈L〉, nm
〈N〉
β
p
K × 10-8
L0
χ12
χ22
χ32
53.4 27.7 9.86 5.48 3.44 1.20 0.61
15 ( 2 11 ( 0.5 8.3 ( 0.6 6.7 ( 0.3 5.9 ( 0.2 5.0 ( 0.2
315 ( 35 245 ( 10 178 ( 12 144 ( 7 126 ( 4 108 ( 4
0.9961 ( 0.0004 0.9945 ( 0.0003 0.991 ( 0.001 0.988 ( 0.001 0.985 ( 0.001 0.978 ( 0.002
0.80 ( 0.02 0.75 ( 0.01 0.65 ( 0.03 0.57 ( 0.02 0.51 ( 0.02 0.42 ( 0.02
0.80 ( 0.16 0.81 ( 0.09 0.93 ( 0.21 1.00 ( 0.17 1.05 ( 0.15 1.48 ( 0.31
14.3 ( 0.3 12.0 ( 0.4 8.9 ( 0.3 7.9 ( 0.2 7.3 ( 0.2 6.2 ( 0.3 5.9 ( 0.4
2.9 3.2 4.0 2.2 1.8 0.9
8.7 10.2 12 5.8 2.9 0.95 0.9
8.3 2.9 1.2
a 〈L〉 is the mean length, 〈N〉 the mean aggregation number, p the polydispersity index, and β a parameter for rodlike micelles as explained in the text. c is the concentration of TDMAO in D2O. The length L0 for a model where the rodlike micelles have constant length is shown too. The fit qualities, described by the reduced χ2-value, for a model with and without length distribution of the cylindrical micelles, χ12 and χ22, are indicated as well. The temperature of the solution was (24 ( 0.5) °C. For the two lowest concentrations a model with a length distribution gives no improvement because the statistical errors of the scattering intensities are too large. For convenience we included reduced χ2-values, χ32, obtained by assuming that spherical micelles are present. K is a thermodynamic parameter described in the text and should be independent from c according to the ladder model.
end caps on each side. The radius of the cylindrical section of the micelles is assumed to be equal to the radius of the spherical micelle and depends not on the lengths L of the micelles. β describes the width of the length distribution function, the number-averaged aggregation number 〈N〉, and the numberaveraged squared aggregation number 〈N2〉:
〈N〉 ) N0 + β/(1 - β)
(
〈N2〉 ) 〈N〉 N0 +
(
β β 1+ 1-β N0(1 - β) + β
(4a)
))
(4b)
K ) exp((∆ - N0δ)/kBT) is a Boltzmann factor corresponding to a free energy gain of a surfactant molecule to be situated in the cylindrical section of the micelle, δ, compared to the free energy of the molecules in the spherical end caps, ∆. Both, δ and ∆, have their origin in a substantial lowering of the free energy of the aggregates compared to having all the monomers dispersed in water. Notice that ∆ and δ have negative values and that |δ| > ∆/N0. Otherwise no rodlike micelles can be present according to this model. The parameter β is related to the mole fraction of the monomers in the solution X1 and to the δ according to β ) X1 exp(-δ/kBT). The width of the size distribution function can be characterized by a polydispersity index
p ) (〈N2〉 - 〈N〉2)1/2/〈N〉
(5)
This factor, as well as 〈N〉, β, and 〈L〉, the mean length of the micelles, is shown for convenience in Table 1. 〈L〉 was calculated using the somewhat simplified formula 〈N〉V0 ) πR2〈L〉, where V0 is the volume of a monomer in the micelle. We assumed that the volume of a monomer is V0 ) 0.480 nm3. This value was obtained from the measured density of a solution of TDMAO in water. From this measurement a micellar density of 0.851 g/cm3 was extracted.9 With the molar weight of TDMAO of 257.4 g/mol we obtained the quoted value of V0. We mention here that this value is not in accordance with an approximation due to Tanford.8 According to this approximation we should expect a value of around 0.54 nm3. At the moment we cannot give an explanation for this discrepancy. As mentioned above, the radius of the cylindrical micelles is R ) 1.85 ( 0.02 nm. It turned out that the introduction of a length distribution improved the fit to the measured scattering intensities substantially. This is shown in Figure 4, where for the sample with a concentration of 53.4 mM results of fits are presented for the length distribution model according to eq 3 as well as for rods with a constant length. The size distribution according to eq 3 fits our results quite well. It turns out that, for low concentrations, fits with a model where the length L0
Figure 4. Scattering intensities for the 27.7 mM solution. Notice the ln(IQ) versus Q2 plot. The solid line for the upper curve is due to a model with a length distribution. The solid line for the lower curve (which for convenience is shifted downward) is due to the best fit with a constant length. The reduced χ2-values, giving a measure for the quality of the fits, are 3.2 and 10.2, respectively.
of the rodlike micelle is monodisperse slowly join the fit quality of the size distribution model. In principle this is in accordance with the decrease of the polydispersity index p found within the framework of the ladder model. Therefore we present in Table 1 the value of L0 and the number characterizing the goodness of both fits, χ12 for the length distribution and χ22 for the monodisperse model, as well. χ2 is the usual reduced χ2value. We mention that other broad functions describing the length distribution of the micelles can fit the results nearly equally well. For TDMAO concentrations c below ∼1 mM, the length of the spherocylinders is not very different from the diameter of the end cap. For convenience we present in Table 1 χ32-values we got for purely spherical micelles. It is evident that at least for c ) 0.61 mM, the difference between the quality of fits assuming rods or spheres is not very pronounced. Furthermore, for c ) 0.61 mM the χ2-value χ12 of the ladder model did not give any improvement compared to χ22 for the model without length distribution. Therefore we were not able to extract any reliable β-values for this concentration. Compared to the results of light measurements2 the length of the rods in that publication turns out to be larger by a factor of about 1.5. We mention here that with SANS no assumptions concerning radius and other parameters, like rotational diffusion parameters, have to be made for an evaluation of the shape of the micelle. The surface area A of a TDMAO monomer bound in a micelle can be extracted and is given approximately by A ≈ 2π(R〈L〉 + R2)/〈N〉. We found that A is around 0.6 nm2 with an estimated
Structure Determination of TDMAO Micelles in Water
J. Phys. Chem. B, Vol. 101, No. 22, 1997 4393 was observed by Sheu et al.13 in a solution of charged micelles. They claimed that δ, which is the energy advantage of inserting an additional monomer into a micelle of size equal to or greater than N0, may depend on concentration c in reality. If we adopt the mean value of K, which is given by 〈K〉 ) (0.91 ( 0.07) × 108, we can conclude that (∆ - N0δ)/kBT ) 1.8.3 ( 0.1. The deduced magnitude of 〈K〉 is not unexpected. Missel et al.11 found for example, for sodium dodecyl sulfate micelles where the charges were screened by a 0.8 M NaCl solution, K-values of such a magnitude, too. The question is how large ∆ and δ are themselves. From the definition of β ) X1 exp(-δ/kBT) and by setting X1 ≈ cmc, which is an approximation, we get δ/kBT ≈ -13 and finally ∆/(N0 kBT) ≈ -12·65.
Figure 5. The mean aggregation number of the spherocylindrical micelles as a function of the TDMAO concentrations. The aggregation number N0 ) 55 for spherical micelles is indicated by a solid dot. The dashed curve is due to a theory valid for large concentrations, c. More details are described in the text.
error of about 10%. This value can be compared with a surface area of TDMAO against air, A ) 0.51 nm2, as quoted by Oetter et al.1 We can see that with increasing concentration the mean length or the mean aggregation number 〈N〉 of the micelles increases (see Figure 5). This effect was observed in ref 2 also and is predicted from a theoretical point of view, for example, in refs 8,11-14. An extrapolation of the mean aggregation number 〈N〉 of the mean length 〈L〉 or of L0 for the samples with c e 3.44 mM to c f 0 seems to give a value that is not near the value N0 ≈ 55 of a spherical micelle with radius R ) 1.85 nm. (V0 and therefore N0 were two of the ingredients of our fit program.) A spherical micelle is, according to the ladder model, the shape expected at a TDMAO concentration c ) cmc. It is claimed that for large concentrations the mean aggregation number 〈N〉 can be represented according to
〈N〉 - N0 ) prop c1/2
(6)
For convenience of the reader we represented such a curve in Figure 5 by a dashed line. Such a behavior is expected for the ladder model, where13
(
)
〈N2〉 - N0 /(X - X1)1/2 ) exp((∆ - N0δ)/2kBT)β1-N0/2 ‚ N (1 + 1/(β + N0(1 - β)))/(β + N0(1 - β))1/2 (7) ≈ 2xK for β f 1
(8)
For high concentrations X (mole fractions) (X - X1) can be approximated by (X - Xcmc). X1 is the mole fraction of the monomers in the solution. Furthermore, for high concentrations β f 1, and therefore 〈N2〉/〈N〉 f 2〈N〉 - N0. If ∆ and δ, which are ingredients of K, do not or only weakly depend on concentration X and X . Xcmc, we see that 〈N〉 - N0 becomes proportional to X1/2 or c1/2, which is indeed seen in Figure 5 by the dashed curve. Using now N0 and β and setting X1 ) cmc, we can extract from eq 7 the value of K ) exp((∆ - N0δ)/kBT) for all measured concentrations. These K-values are shown in Table 1 too and should be independent from the concentration c. Indeed, we observe a tendency that K increases slightly with decreasing c, but this tendency is not very pronounced when taking into account the error bars. We mention here that a reverse tendency
5. Conclusions By means of a contrast variation experiment it was shown that the scattering length density of the rodlike TDMAO micelles is homogeneously distributed. The radius of these micelles was determined to be 1.85 ( 0.03 nm. At room temperature (25 °C) we observed an increase of length of the rodlike micelles with increasing TDMAO concentration c. We have shown that the micelles have a pronounced length distribution which was analyzed with the help of the so-called ladder model. Basically two thermodynamic parameters are required to characterize the system: δ and ∆/N0. δ, the energy advantage of inserting an additional monomer into a micelle of size equal to or greater than a micelle having the minimum aggregation number N0 ) 55, turned out to be -13 in units of kBT. ∆/N0, where ∆ is the free energy of micellization of the spherical micelle with aggregation number N0, was found to be -12.65 in units of kBT. There might be a slight indication that ∆/N0 - δ depends on concentration c, increasing in fact a little bit with decreasing concentration. We mention that for c ) 53.4 mM, the mean length 〈L〉 ) 15 nm) of the micelles is less than the mean distance of 21 nm between the micelles, which is characteristic for a dilute solution. Acknowledgment. We thank the Joint Institute for Nuclear Research JINR, Dubna, for getting access to the MURN spectrometer and Professor Dr. Heinz Hoffmann for the gift of TDMAO. This research is supported by the BMBF of the Bundesrepublik Deutschland under Grant 03-DU4BAY-6. References and Notes (1) Oetter, G.; Hoffmann, H. J. Dispersion Sci. Technol. 1988-89, 9, 459. (2) Hoffmann, H.; Oetter, G.; Schwandner, B. Prog. Colloid Polym. Sci. 1987, 73, 95. (3) Gorski, N.; Gradzielski, M.; Hofmann, H. Langmuir 1994, 10, 2594. (4) Ostanevich, Yu. M. Makromol. Chem., Macromol. Symp. 1988, 15, 91. (5) Vagov, V.; Kunchenko, A. B.; Ostanevich, Yu. M.; Salamatin, I. M. JINR Report P14-83/898; Dubna, Russia, 1983. (6) Kostorz, G. Treatise Mater. Sci. Technol. 1979, 15, 277. (7) Herbst, L.; Kalus, J.; Schmelzer, U. J. Phys. Chem. 1993, 97, 7774. (8) Tanford, Ch. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; John Wiley: New York, 1973. (9) Gradzielksi, M. Diplomarbeit, Universita¨t Bayreuth, Bayreuth, Germany, 1989. (10) Chen, S.-H.; Sheu, E. Y.; Kalus, J.; Hoffmann, H. J. Appl. Crystallogr. 1988, 21, 751. (11) Missel, P. J.; Mazer, N. A.; Benedeck, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (12) Mukerjee, P. J. Phys. Chem. 1972, 76, 565. (13) Sheu, E. Y.; Chen, S.-H. J. Phys. Chem. 1988, 92, 4466. (14) Lin, T. L.; Chen, S.-H.; Gabriel, N. E.; Roberts, M. F. J. Phys. Chem. 1987, 91, 406.
