590
Langmuir 2006, 22, 590-597
Structure of BRIJ-35 Nonionic Surfactant in Water: A Reverse Monte Carlo Study Gergely To´th* and A Ä da´m Madara´sz Institute of Chemistry, Eo¨tVo¨s UniVersity, P.O. Box 32, Budapest 1518, Hungary ReceiVed May 25, 2005. In Final Form: NoVember 7, 2005 There are some contradictions in the literature on the structure of micelles formed by the BRIJ-35 surfactant in water. One can find reasonable differences in the aggregation numbers and micellar sizes, but there is a lack of data on the intermicellar structure. In this study, we reevaluated the small-angle neutron scattering experiments performed previously on the BRIJ-35 surfactant in the concentration range of 5-200 g/dm3 at 20, 40, and 60 °C. The data were analyzed with a reverse Monte Carlo-type method developed recently for colloids. The micelles were modeled as spherical cores representing the hydrophobic parts and number of balls put on the cores to mimic the hydrated hydrophilic chains. The simulations provided data on the mean aggregation number and on the extent of hydration of the hydrophilic shell of the micelles. We obtained intermicellar pair-correlation functions indicating different micelle-micelle interactions from the usually assumed hard-sphere ones.
1. Introduction BRIJ-35 is the commercial name for CH3(CH2)11(OCH2CH2)23OH, a dodecyl-poly-ethylene-oxide-ether. The number 35 refers to the sum of the length of the alkyl chain and the ethylene-oxide groups. Its molecular weight is 1198 g/mol. Nonionic surfactants containing poly-ethylene-oxide chains as the hydrophilic part and n-alkyl chains as the hydrophobic part form micelles in water. If another solvent (e.g., an alcohol) is added to the binary BRIJ/water system, then microemulsions with many interesting properties are obtained.1-4 The systems have practical applications in cosmetics, pharmaceuticals, paints, cleaning, and biotechnology. BRIJ-35 is useful for the solubilization of membrane proteins and can also entrap hydrophobic reactants in liquidphase enzyme reactions.3 The physical chemistry and the structure of these micellar systems have been studied extensively, but one can find some contradictions in the results and some open questions. The critical concentration of BRIJ-35 needed to form micelles is small:5,6 0.04-0.10 g/dm3. The mean aggregation number, Nagg, of nonionic surfactants depends on the concentration and on the temperature. Becher7 proposed 40 in 1961, Tanford8 et al. found 40 in 1977, and Phillies et al.9 determined 37-47 in the temperature range of 10-40 °C for a concentration of 2-100 g/dm3 and 63 at 70 °C. Preu et al.2 found 41-49 at 25 °C in the concentration range of 1-150 g/dm3. In the same year (1999), Borbe´ly10 proposed it to be 34-64 at temperatures of 20-60 °C and concentrations of 5-200 g/dm3. Maire11,12 et al. published 40 in 2000. The geometry of the micelles is ambiguous. Rodlike micelles were proposed in the historical article of Becher7 in 1961. Tanford (1) Meziani, A.; Touraud, D.; Zradba, A.; Pulvin, S.; Pezron, I.; Clausse, M.; Kunz, W. J. Phys. Chem. B 1997, 101, 3620. (2) Preu, H.; Zradba, A.; Rast, S.; Kunz, W.; Hardy, E. H.; Zeidler, M. D. Phys. Chem. Chem. Phys. 1999, 1, 3321. (3) Schirmer, C.; Liu, Y.; Touraud, D.; Meziani, A.; Pulvin, S.; Kunz, W. J. Phys. Chem. B 2002, 106, 7414. (4) Tomsic, M.; Bester-Rogac, M.; Jamnik, A.; Kunz, W.; Touraud, D.; Bergmann, A.; Glatter, O. J. Colloid Interface Sci. 2006, 294, 194. (5) Sharma, B.; Rakshit, A. K. J. Colloid Interface Sci. 1988, 129, 139. (6) Sharma, K. S.; Patil, S. R.; Rakshit, A. K.; Glenn, K.; Doiron, M.; Palepu, R. M.; Hassan, P. A. J. Phys. Chem. B 2004, 108, 12804. (7) Becher, P. J. Colloid Sci. 1961, 16, 49. (8) Tanford, C.; Nozaki, Y.; Rohde, M. F. J. Phys. Chem. 1977, 81, 1555. (9) Phillies, G. G. J.; Hunt, R. K.; Strang, K.; Sushkin, N. Langmuir 1995, 11, 3408.
et al.8 found oblate ellipsoids in 1977, but in 1995, Phillies et al.9 proposed a spherical two-shell model with a hydrophobic core and a hydrated outer shell. Preu et al.2 used the spherical two-shell model, but they could reproduce the experimental data better with ellipsoids. Borbe´ly10 applied a hairy-sphere model.13,14 Here, the outer shell consists of partially entangled chains started at or near the surface of the hydrophobic core. The authors of the publications above came to the understanding that there is extensive hydration of the hydrophilic chains. That means that the aggregation number determines only the size of the hydrophobic core uniquely. In the case of the hydrophilic part, the so-called wetting factor is also important. Some results of the three most relevant papers on BRIJ-35/ water systems are recalled here. In these papers, the structure was investigated with small-angle scattering experiments. Ternary systems were investigated over the binary BRIJ-35/water systems in two of these papers. Preu et al.2 performed small-angle neutron scattering on binary BRIJ-35/D2O and ternary BRIJ-35/D2O/alcohol systems. The concentration of the surfactant was in the range of 1-150 g/dm3, and the temperature was 25 °C. The structure of the micelles was modeled with spherical and/or ellipsoidal core and shell models. The interparticle correlations were described by soft-sphere interactions and were calculated by the Percus-Yevick approximation. There were two parameters in the description of the interactions: the size and the softness of the micelles. The systems were supposed to be monodisperse. They got the following fitted parameters: Nagg ) 41-49, the radius of the micelles was 39.8-41.9 Å, and the radius of the core was 15.120.3 Å. In the case of the ellipsoids, the a minor semiaxis of the micelles was 33.0-36.0 Å with a b/a ratio of 1.4-1.8. The minor semiaxis of the core was 15.0-17.0 Å with an axis ratio of 1.2-1.7. Borbe´ly10 applied a more sophisticated model for the micelles. The hydrophobic part was modeled with a spherical core, and the hydrophilic part, with polymer chains tethered to the surface (10) Borbe´ly, S. Langmuir 2000, 16, 5540. (11) Maire, M.; Champeil, P.; Moller, J. V. Biochim. Biophys. Acta 2000, 86, 1508. (12) http://www.lipidat.chemistry.ohio-state.edu/cmc_4.html. (13) Svaneborg, C.; Pedersen, J. S. J. Chem. Phys. 2000, 112, 9661. (14) Pedersen, J. S. J. Chem. Phys. 2001, 114, 2839.
10.1021/la051380a CCC: $33.50 © 2006 American Chemical Society Published on Web 12/09/2005
Structure of BRIJ-35 Nonionic Surfactant in Water
of the core. The interparticle structure was modeled with hardsphere interaction solved by the Percus-Yevick approximation. The particle form factor and the calculation of the scattering intensity were developed by Pedersen et al.13,14 Polydisperse systems were assumed. They contained 5-200 g/dm3 surfactant in D2O, and the small-angle neutron scattering experiments were performed at 20, 40, and 60 °C. The equation of the scattering intensity contained seven parameters. The radii obtained for the micelles were 27.5-37.8 Å, and those obtained for the core were 15.8-19.6 Å. The hard-sphere interaction radii were 2.00-2.39 times the radii of the micelles, and the aggregation number was 34.4-64.1. Borbe´ly could not determine the hard sphere radii and the width of the polydispersity, σagg, at concentration lower than 25 g/dm3. σagg was between 9 and 42% of the aggregation number. The aggregation number, the radii of the micelles, and the core radii increased with increasing concentration, whereas the hard-sphere interaction radii and the standard deviation of the aggregation number decreased. Another parameter was introduced to estimate the starting point of the hydrophilic chains on the core. It may have accounted for the hydration omitted explicitly from the model. This parameter decreased for increasing concentration. Tomsic et al.15 investigated the binary BRIJ-35/water and the ternary BRIJ-35/water/alcohol systems by small-angle X-ray scattering and dynamic light scattering. The concentration of the surfactant was 5-250 g/dm3, and the temperature was 25 °C. The X-ray scattering data were evaluated using the generalized indirect Fourier transformation method.16-19 It is a quasi-model free method of determining the shape and size of the particles, and it is possible to use it simultaneously with the incorporation of interparticle interactions. For BRIJ-35, the interparticle structure was modeled as a mixture of polydisperse hard spheres solved by the Percus-Yevick approximation. The data evaluation provided intraparticle pair-distance distribution functions and electron-density contrast profiles. The radii of the core obtained were 16-18 Å, and the maximal intramicellar dimensions obtained were 90-105 Å. The latter can be considered to be the approximate diameter of the micelles. Spherical particles were proposed at moderate concentrations, but the shape of the intraparticle pair-distance distribution functions and the parameters of the modeled structure factor at high concentrations implied ellipsoidal distortions. The interaction radius of the interparticle hard-sphere contribution was between 39 and 46 Å. The higher values belonged to the low concentration. Similar micellar sizes were obtained using the dynamic light scattering data, and the diffusion coefficients were also determined. These coefficients were compared to those of monodisperse hard spheres. They showed a maximum at around 150 g/dm3 surfactant concentration and confirmed the elongation of the scattering particles also observed from the small-angle X-ray scattering results. One can find various interpretation methods for scattering data in the literature. Therefore, the structural results for the BRIJ-35/water mixtures may be divergent in some points. With our investigation, we would like to answer three questions: Is there any clear trend in the aggregation number with respect to the surfactant concentration? What is the extent of hydration of the hydrophilic chains? Is the intramicellar structure as simple (15) Tomsic, M.; Bester-Rogac, M.; Jamnik, A.; Kunz, W., Touraud, D.; Bergmann, A.; Glatter, O. J. Phys. Chem. B 2004, 108, 7021. (16) Brunner-Popela, J.; Glatter, O. J. Appl. Crystallogr. 1997, 30, 431. (17) Weyerich, B.; Brunner-Popela, J.; Glatter, O. J. Appl. Crystallogr. 1999, 32, 197. (18) Brunner-Popela, J.; Mittelbach, R.; Strey, R.; Schubert, K. V.; Kaler, E. W.; Glatter, O. J. Chem. Phys. 1999, 110, 10623. (19) Bergmann, A.; Fritz, G.; Glatter, O. J. Appl. Crystallogr. 2000, 33, 1212.
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as for hard or soft spheres? We started our study about 4 years ago with the simple idea to reevaluate the scattering data of Borbe´ly10 using a reverse Monte Carlo-type simulation20 to obtain a quasi-model free interparticle structure. Unfortunately, we could not get reliable results using a combination of Pedersen’s particle form factor and the reverse Monte Carlo method. Later, we tried other models (i.e., two-layer sphere and ellipsoidal models), but the results were not convincing. Recently, we developed a reverse Monte Carlo-type method where the scattering intensity was calculated on random scattering points confined in spatial solid bodies representing the mesoscopic particles. The method was tested and elaborated for a Ni-containing catalyst.21 In the present study, we applied this scheme to the BRIJ-35 micelle/water system to reevaluate the experimental data of Borbe´ly.
2. Model and Simulation Method Small-angle neutron or X-ray scattering and wide-angle light scattering are powerful tools for characterizing the materials in detail.22 The static part of the scattering data contains information on the structure of the samples on the nanometer scale. In most cases, there is a lack of macroscopic anisotropy in these systems, so the information is one-dimensional, and the scattering experiment results in the scattering intensity, I(q), as a function of the inverse variable q ) 4π sin(θ/2)/λ, a scalar connected to the scattering angle, θ, and the wavelength of the radiation, λ. The observed phenomena are rather complex in these experiments, but the scattering intensity is rather featureless. It seldom contains more than one or two peaks. The q range of the experiment is also limited, and sometimes it is impossible to determine the density of the scattering particles. It hinders the usual data evaluation from the inverse to real space with a direct Fourier transformation method of which the conditions are not fulfilled. Therefore, data processing may imply solely an analysis of the scattering intensity in q space, like the interpretation of the slope in the Porod region and also in the Guinier region, if the interparticle interactions are negligible. More information can be obtained only by the indirect Fourier transformation method or by modeling the system and varying the parameters of the model to reproduce the experimental data. Generally, the scattering intensity, I(q), depends on the atomic features and the position of the atomic scattering centers.22 If only one type of atom contributes to the scattering intensity in an isotropic system, then the equation is rather simple
I(q) ) N f 2(q) Sat(q)
(1)
where N is the number of the scattering centers and f(q) is the atomic scattering factor. Sat(q), the atomic structure factor, is connected to the real space pair-correlation function by a Fourier transformation
∫0∞ r2[gat(r) - 1]
Sat(q) ) 1 + 4πF0
sin(qr) dr qr
(2)
where r is the radial distance around an atom and F0 is the number density. The pair-correlation function, gat(r), represents the probability of finding an atom at a distance r around another one with respect to the average number density of the system. (20) To´th, G. Langmuir 1999, 15, 6718. (21) To´th, G.; Ko¨rmendi, K.; Vrabecz, A.; Bo´ta, A. J. Chem. Phys. 2004, 121, 3949. (22) Neutron, X-ray and Light Scattering: Introduction to an InVestigatiVe Tool for Colloidal and Polymeric Systems; Lindner, P., Zemb, Th., Eds.; NorthHolland Press: Amsterdam, 1991.
592 Langmuir, Vol. 22, No. 2, 2006
To´ th and Madara´ sz
In the case of colloids, the scattering pattern contains information both about intraparticle and interparticle structures. The contribution of the scattering centers inside one object is usually collected into the particle form factor, P(q). For simple cases, the scattering intensity is a product of the intra- and interparticle structures, but for most real systems, the situation is more complicated. It is possible to derive complicated and approximate analytical expressions for the calculation of the intraparticle part of the scattering intensity for many systems.13,14,22 In the case of monodisperse spheres with a constant density of scattering atoms inside the spheres,
I(q) ) Nspn2 f(q)2 Psp(q) Ssp(q)
(3)
where Nsp denotes the number of particles here, n is the number of scattering centers in one object, and Ssp(q) is the structure factor of the spheres. Equation 3 is approximate for real systems. The homogeneous scattering density and the uniform spherical shape and size of the colloid particles are seldom fulfilled. There is an improved description if we use multicomponent-like equations incorporating the size and shape distribution of the particles both in P(q) and S(q).23 Otherwise, averaged functions can be used.16-18,24 It was shown that the averaged structure factors are approximately good descriptions for some simple systems. In the case of complex systems such as micelles of BRIJ-35, the particle form factor can be more sophisticated,13,14 and the simplification of the scattering intensity to a product of the intra and interparticle contribution is already questionable. Recently, we developed a method to avoid the separation of the intra- and interparticle contributions in the scattering intensity.21 We developed a reverse Monte Carlo-type simulation method25 where the scattering intensity was calculated according to eq 1 but the enormous number of primary scattering centers was reduced by using only representatives of them and geometrical constraints played the role of the shape of the colloid objects. The method was feasible on a Ni-containing catalyst, where the size, the shape, and the interparticle distributions of the assumed catalytic Ni cylinders were determined. Our method has some relationship to the DAMMIN technique.26,27 In the present study, we applied and developed our method to be suitable for the micelles of BRIJ-35. In the case of the Ni catalyst, we used only one type of scattering center. Here we used two types: one for the hydrophobic core of the micelles and one for the hydrophilic chains. The geometrical constraints, playing the role of the shape of the colloid particles, were also different here. The scattering intensity of a two-component system is
I(q) ) N[c1c2{f1(q) - f2(q)}2 + 2
2
∑R ∑β cRcβfR(q) fβ(q) sRβ(q)]
(4)
according to Faber and Ziman,28 where ∞ r2[gRβ(r) - 1] ∫r)0
sRβ(q) ) 1 + 4πF0
sin(qr) dr qr
(5)
is the partial structure factor, the number of scattering centers (23) Kotlarchyk, M.; Chen, S.-W. J. Chem. Phys. 1983, 79, 2461. (24) Pedersen, J. S. J. Appl. Cryst. 1994, 27, 595. (25) McGreevy, R. L.; Pusztai, L. Mol. Simul. 1988, 1, 359. (26) Svergun, D. I. Biophys. J. 1999, 76, 2879. (27) http://www.embl-hamburg.de/ExternalInfo/Research/Sax/ dammin.html. (28) Faber, T. E.; Ziman, J. M. Philos. Mag. 1965, 11, 153.
Figure 1. Schematic view of the model.
is N1 and N2, N ) N1 + N2, c1 ) N1/N, and c2 ) N2/N. f1(q) and f2(q) are the atomic scattering factors, and gRβ denotes the partial pair-correlation function. A schematic view of the model can be seen in Figure 1. A micelle was modeled as scattering centers placed inside a sphere (called the core), representing the hydrophobic core, and another type of scattering center inside small spheres (called balls) was placed on the surface of the large sphere. A ball represented the entangled hydrophilic chain of a BRIJ-35 molecule. The number of balls is equal to the aggregation number of the given micelle. The volume of the core was determined by the aggregation number multiplied by the partial volume of the hydrophobic chain (322 Å3, ref 10). The volume of one ball was calculated according to the volume of the hydrophilic chain of one BRIJ-35 molecule multiplied by W, the wetting factor (W × 1432 Å3, ref 10). The balls were placed randomly on the surface of the core. We let the balls overlap with each other slightly because it was impossible to put as many nonoverlapping balls on the surface of the core as there were monomers forming one micelle. N ) 200 micelles were put into a cube in the simulations (Figure 1). The cube was surrounded by its periodic images. The sizes of the micelles were different. Their distribution corresponded to a mean aggregation number, Nagg, with a given width of the Gaussian distribution, σagg. The volume of the cube was calculated according to the concentration of BRIJ-35. We used the experimental data of Borbe´ly at the concentrations of 5-200 g/dm3. The points representing the scattering centers were put randomly into the cores or the balls. A goal of our method developed for the Ni catalyst was that we were able to use significantly fewer points than the real number of primary scattering atoms. The number of atoms is around 106-107 in a system consisting of a few hundred BRIJ-35 micelles. Here, we used as many scattering points in the cores as the number of monomers that we had, and from the other type of scattering centers, we put about 2 to 3 times more into the balls. This means 35 000-50 000 points altogether, which is an acceptable number of points for a simulation. The theory behind the reduction of the points is detailed here for a two-component system.
Structure of BRIJ-35 Nonionic Surfactant in Water
Langmuir, Vol. 22, No. 2, 2006 593
We used fewer points than there are in reality. Therefore, our partial functions differ from the real ones. They are denoted with a prime. The partial structure factor at density F′ is
s′Rβ(q) ) 1 + 4πF′
∞ r2[g′Rβ(r) - 1] ∫r)0
sin(qr) dr qr
(6)
R and β refer to the different types of points. Introducing
sin(qr) dr qr bRβ ) sin(qr) ∞ F′ r)0 r2[g′Rβ(r) - 1] dr qr ∞ r2[gRβ(r) - 1] ∫r)0
F0
∫
(7)
m
χ ) 2
notation, the real partial structure factor is
sRβ(q) ) (s′Rβ(q) - 1)bRβ + 1
(8)
and the scattering intensity is equal to
I(q) ) a1s′11(q) + a2s′22(q) + a3s′12(q) + a4
(9)
where
a1 ) Nf12(q)b11 a2 ) Nf22(q)b22 a3 ) Nf1(q) f2(q)b12 a4 ) N[c1c2{f1(q) - f2(q)}2 + c12f12(q)(b11 - 1) + c22f22(q)(b22 - 1) + 2c1c2 f1(q) f2(q)(b12 - 1)] (10) The partial pair-correlation functions are normalized quantities. If the number of points is large enough (104 was a satisfactory choice here), then
gRβ(r) ≈ g′Rβ(r)
(11)
Therefore
bRβ ≈
F0 F′
enough free space for each micelle. The aggregation number of the micelles was determined according to an initial Nagg and σagg. The balls were placed randomly on the surface of the cores, taking care to minimize their overlap. The two types of scattering centers were positioned randomly in the cores and the balls. We used random numbers everywhere because we would like to avoid the artifact of predefined regular arrangements. The partial pair-correlation functions and the partial structure factors (eq 6) of the scattering points were calculated. Thereafter, the a1-a4 scalars of eq 9 were determined in such a way as to minimize χ2
(12)
In the case of neutron scattering or small-angle X-ray scattering, f(q) does not depend on q or the dependence is negligible. Practically, a1-a4 do not depend on q in these cases. If the experimental I(q) is not normalized, then the normalization can also be incorporated into a1-a4. We used eq 9 to calculate the scattering intensity of our model. We had the following parameters (or variables) in the simulation: the mean aggregation number (Nagg), the width of the Gaussian distribution of the aggregation number (σagg), the wetting factor of the hydrophilic chains (W), the a1-a4 scalars of eq 9, and the position of each micelle (three Descartes coordinates and three Euler angles for each micelle). According to our previous experience,20,21 the position of the micelles and some of the parameters (Nagg, σagg, and W) were determined by the reverse Monte Carlo method, whereas the a1-a4 scalars of eq 9 were determined with a deterministic minimum search in each reverse Monte Carlo trial. The algorithm was similar to the one used in the simulation of the Ni catalyst. At first, the initial configuration was set up. The coordinates of the centers of the micelles were determined. We started with random positions in the cubic simulation cell and we used a hard-sphere Monte Carlo simulation to ensure
∑ i)1
[IC(qi) - IE(qi)]2 σ2(qi)(m - 7)
(13)
where IC(qi) is the calculated scattering intensity (eq 9), IE(qi) is the experimental one with σ(qi) assumed error, and m is the number of experimental q points. The degrees of freedom were reduced with the number of parameters (7), omitting the position of the micelles, as is usual in the reverse Monte Carlo simulations. The a1-a4 scalars were determined in a cyclic 1D search with the method of the golden section.29 Five types of Monte Carlo moves were performed. The translation and rotation concerned one randomly selected micelle. The maximal move was 5 Å in each direction and 0.5 rad in each Euler angle. The other three moves all referred to micelles. The maximal change was 1% in Nagg, σagg, or W. If the change in the average aggregation number or the width of its distribution changed, then the aggregation number of a micelle, the balls, and the scattering points were reset for the micelle. In the case of a change in W, it was performed for all of the micelles. The shift in Nagg and σagg changed the number of BRIJ-35 molecules in the system because the number of micelles was fixed (N ) 200) in the simulations. In these cases, the side length of the simulation cell was rescaled to get the correct density. The global Monte Carlo changes were performed 50 times less than the translations or rotations of the micelles. The χ2 of the new configuration was calculated as detailed for the initial configuration, and it was compared to the χ2 of the old configuration. If it was smaller than the old one, then the new configuration was accepted. If it was larger than the old one, then it was accepted with a probability of P ) exp[-(χnew2 - χold2)/(2sv)]. The sv acceptance parameters were specific for each type of move. They were set to get reasonable acceptance ratios for the moves: 0.5-0.8 for the translations, 0.9 for the rotations, and 0.25 for the global moves. If a configuration was accepted, then the next Monte Carlo move was tried for this configuration. If a configuration was rejected, then the next trial originated from the old configuration. The overlap of the micelles was minimized by the method described in ref 21. Briefly, translations and rotations were not accepted if the overlap of the micelles increased. The overlap was not checked in global moves. The densities of the BRIJ-35 surfactant corresponded to the condition of Borbe´ly’s experiments: 5, 10, 25, 50, 100, 150, and 200 g/dm3. He measured the systems at three temperatures: 20, 40, and 60 °C. This gives a total of 21 simulations. 100 000300 000 translations and rotations were performed for each system. Depending on the density and the starting parameters (Nagg ) 50-80, σagg ) 10-20, and W ) 1-4), the equilibration took 30 000-50 000 moves. The data shown were averaged over 50 000 configurations. A calculation took a few days on a PC (29) Press, W. H.; Teukolsky, S. A.; Vetterling W. T.; Flannery, B. P. Numerical Recipes in Fortran; Cambridge University Press: Cambridge, U.K., 1992.
594 Langmuir, Vol. 22, No. 2, 2006
Figure 2. Results of Nagg and σagg.
To´ th and Madara´ sz
Figure 3. Different volume ratios. The data are averaged over the temperature.
with a 2.4 GHz CPU. The code was written in C language, and it is available from the authors on request.
3. Results and Discussions The results on Nagg and σagg are shown in Figure 2. A decreasing trend was found for both Nagg and σagg at all three temperatures except at the two smallest concentrations, 5 and 10 g/dm3. The reliability of the data at small concentrations will be discussed later. For medium and high densities (50-200 g/dm3), Nagg is between 56 and 84. σagg was found to be around 20-25% of Nagg for all of the simulations. The temperature dependence is weaker than the dependence on the concentration, and the effect of the temperature is comparable in size to the estimated error of our calculations. Higher temperature seemed to increase Nagg and σagg. The statistical uncertainty is omitted from the graph to enhance its lucidity. It was calculated for each data point with the method of block averaging21,30 over 100 global reverse Monte Carlo trials. The uncertainty was between 0.5 and 4.1 for Nagg and between 0.5 and 2.7 for σagg at a significance of R ) 0.05. The wetting factor was between 1.7 and 5.4. W is a multiplication factor of the molar volume of the poly-ethyleneoxide chains in the calculation of the volume of the balls. We obtained a reasonable decrease in W with increasing concentration. At 200 g/dm3, W was 1.7-2, and at medium concentrations, it was 2-3. We mentioned in the previous section that a slight overlap of the balls was allowed to make possible the placing of all balls on the cores. Therefore, the water content of a micelle was calculated with a simple spherical core-shell model. Rmic, the radius of the micelles, was defined as the sum of the hydrophobic core radius (Rcore) and the diameter of the hydrophilic balls. Three different percentages are shown in Figure 3. The volume of the micelles versus the total volume was calculated as 100Vmic/V, where Vmic ) N(4/3)πRmic3 and V denotes the volume of the simulation cell. The percentage of water in the micelles with respect to the total amount of water was defined as 100(Vmic - VBRIJ)/(V - VBRIJ), where VBRIJ was calculated according to the molar volume of pure BRIJ-35. The volume percent of water in the hydrophilic shell equaled 100(Vmic VBRIJ)/Vmic. At small concentrations, the volume percent within the hydrophilic shell seemed to be large (up to 55%), but the reliability of these data were questionable. At medium and high concentrations, the ratio slightly decreased and tended to 40%. The temperature dependence was in the range of the estimated uncertainty ((1 to 6%) of the calculations. Therefore, the data shown in Figure 3 are averaged over the temperature. The mean radii of the hydrophobic cores and the micelles are presented in (30) Allen, M. P.; Tildesley D. J. Computer Simulations of Liquids; Clarendon Press: Oxford, U.K., 1987.
Figure 4. Core and micellar radii.
Figure 5. Intramicellar distance distribution functions.
Figure 4. For medium and high concentrations, the radii of the core were 16-19 Å, and the radii of the micelles were 33-39 Å with a decreasing trend for increasing concentrations. We compared our data to that of Tomsic et al. One of their main results was the intraparticle pair-distance distribution function because they did not use any geometric model for the intraparticle structure. They derived their distributions from smallangle X-ray data. The different contrast of the core and shell parts and the shape of the micelles together determined the distributions. We calculated some kind of pair-distance distribution function for the core-core, core-chain, and chainchain parts of the micelles where we assumed a unique positive contrast in both parts. Of course, our distributions were different from those of Tomsic et al. and from pair distributions containing small-angle neutron contrasts. One can see in Figure 5 that the maximal intramicellar distances were 67-86 Å. They were between 67 and 78 Å if the 5 and 10 g/dm3 concentrations were omitted.
Structure of BRIJ-35 Nonionic Surfactant in Water
The results were compared to the other previous investigations. Larger mean aggregation numbers than previously reported were obtained, especially at low concentrations. Only Phillies et al. suggested Nagg ) 63 at 70 °C, and Borbe´ly obtained data of 55 < Nagg for 50-200 g/dm3 at 40 and 60 °C. In the other studies, the average aggregation number lies between 40 and 50. Our data did not confirm the strong temperature dependence of the aggregation number suggested by some studies. Furthermore, we got decreasing aggregation numbers with respect to increasing concentration. If we compare the core radii, then our data corresponds well to that of other investigations. Most of the proposed radii are no more than 5 Å longer than the theoretical length of a fully stretched dodecyl chain,15 which is equal to 15.4 Å. According to the variables in our reverse Monte Carlo-type technique, our core radii had to show similar trends with respect to the concentration as the mean aggregation number. Our mean micelle radii are slightly (2-4 Å) larger than obtained previously with geometrical methods. If we compare the maximal extent of our pair-distance distribution functions to those of Tomsic et al., then our data suggested at least 2-6 Å smaller radii than their dynamic light scattering data or small-angle data analyzed with the generalized inverse Fourier transformation technique. This means that our data are within the minima and the maxima of the previous investigations. The studies took into account the hydration of the hydrophilic chains in different ways. One method was to optimize the mean aggregation number and the different radii2,10 independently or to fit an independent volume fraction parameter.15 The hydration emerged somehow in the intermicellar potentials (e.g., in the hypothetic hard-sphere or soft-sphere radii2,10,15). Other parameters are also connected to hydration (e.g., the chain-shift parameter of Borbe´ly). These approaches did not allow us to find direct evidence of the extent of hydration. In most cases, they showed a decrease in these parameters with increasing concentration or their relative decrease versus concentration; therefore, they showed that there is decreasing hydration with increasing concentration. As shown in Figure 3, we found slightly decreased hydration at medium and high concentrations. At the uncertain low concentrations, we obtained reasonable stronger hydration in the hydrophilic core. The dehydration was caused by the decrease in free water. The percentage of water molecules inside the micelles versus the total amount of water in the systems is shown together with the percentage of the volume of micelles with respect to the total volume in Figure 3. One can see that the micelles occupied 26% of the volume at the largest concentration and 10% of the water was in the micelles. The volume fraction parameters of Tomsic et al. were interpreted similarly: there is extensive hydration at low concentrations and a relatively less enhanced hydration at high concentration. How can one interpret our data at the molecular level? The core consists of stretched alkyl chains, and the surface of the core seems to be rough because its radius is larger than the length of the fully stretched hydrophobic chain. In the case of dilute systems, the large amount of excess water enhances the hydration of the hydrophilic chains. A fully hydrated chain is rather flexible, and it prefers not to touch the surface of the hydrophobic core. It tries to stick out from the surface. In the case of dense systems, there is a lack of water to hydrate the hydrophilic chains. The ethylene-oxide groups of the chains try to touch each other. An entangled structure takes shape, and the chains are less stretched. Here, the area on the core shielded by one chain is larger than in the fully hydrated case that spatially favors for the formation of smaller aggregates. We think that the competition of this and the trivial effect of the surfactant concentration determines the
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Figure 6. Reproduction of the scattering intensities at 293 K. Symbols, experimental data; lines, simulation.
mean aggregation number. In the case of BRIJ-35, the effect of dehydration seems to be strong, causing a decrease in the mean aggregation number at increasing surfactant concentrations. The length of the fully stretched polyoxiethylene chain (about 83 Å) is a few times longer than the observed thickness of the hydrophilic shell. This means that the chains were generally in an entangled structure here. The thickness of the hydrophilic shell was about 23.5 Å at low concentrations, where the average aggregation number was over 80 and extensive hydration was found. At high concentrations, where the aggregation number was around 60 and 40% of the outer shell was water, it was about 17.5 Å. The available surface of one hydrophilic chain on the hydrophobic core increased from 0.0011 to 0.0016 Å2 with increasing concentration. This means that at high concentrations a larger part of the hydrophilic chain was close to the hydrophobic core, which caused a decrease in hydration. Generally, a discussion of data evaluation starts with the quantification of the reproduction of the experimental data. We mention it only in this part of the section because data reproduction cannot be discussed separately from the intermicellar structure. The χ2’s defined in eq 13 were between 0.23 and 3.1. They were averaged values during the simulations, and they were some extent smaller at particular phases of the calculations. They increased linearly within the statistical error if the concentration increased. The overall weaker reproduction of the data of the dense systems can be interpreted as the limitation of our spherical (balls on a core) model because an elongation of the particles was proposed for this regime.2,15 Theoretically, low χ2 means an excellent fit, but one must be careful in taking that too literally. The experimental small-angle neutron scattering and the reproduced data are shown at 20 °C in Figure 6. One can see that the last 10 points of each set could not be correctly reproduced. These experimental points are weighted on a visible error. In the case of dilute systems, our reverse Monte Carlo method gave the opportunity to fit to the error of the points because there is the large possibility of spatial arrangement of the particles. If the model of the micelles is not perfect, then unphysical intermicellar arrangements also might improve the fit. This possibility is reduced for dense systems. The pair-correlation functions of the micellar centers are shown for two “overfitted” simulations in Figure 7. If we fitted our simulation without the last 10 experimental points, then the unphysical oscillations remained for these low-density systems. Furthermore, the oscillations did not vary if we changed the length of the g(r) functions of eq 5. The diffraction experiments were performed at two sampledetector distances.10 The frequency of the oscillations in r space was close to the corresponding q space intersect of the two experimental curves. We tried to rescale the two parts of the curves separately to ensure a smooth junction. Neither this nor
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Figure 7. Demonstration of overfitted intramicellar pair-correlation functions for the dilute (5 and 10 g/dm3) systems at 293 K.
Figure 8. Intramicellar pair-correlation functions for 150 and 200 g/dm3 concentrations at 293 K. The HB curves are the pair-correlation functions of the corresponding hard-body systems.
Figure 9. Integrated number of neighbors for different concentrations at 293 K.
the omission of the questioned q-space data stopped the oscillations in r space. We think that the data contained some experimental bias that seemed to be dominant at low concentrations and negligible at medium and high concentrations. Borbe´ly’s original data evaluation was not sensitive to it because he used an analytical intramicellar model where the hard-sphere model was solved by the Percus-Yevick integral equation. He could not overfit the interparticle structure. Therefore, we think that our results on the 5 and 10 and partly on the 25 g/dm3 systems should be interpreted very carefully. Some selected micelle-micelle pair-correlation functions are shown in Figure 8. There were unphysical oscillations if the concentration of the systems was lower than 50 g/dm3. The integrated number of neighbors is shown in Figure 9. The number of micelles in the first coordination shell was 2.5 to 11 in the density range of 50 to 200 g/dm3.
To´ th and Madara´ sz
Figure 10. Percolation data: the number of chains with respect to the length of the fictive bond for different concentrations at 293 K.
Simulations provide microscopic information on the possible existence of particle pairs, chains, and networks. It can be especially interesting for microemulsions where changes in the turbidity and the existence of cloud points are practically important. We were interested in these properties, and we performed a percolation analysis on the configurations. A fictive bond length was introduced, and the number of chains (or separated particles) was calculated as the variable of this bond length. The results are shown in Figure 10. In the case of highand medium-density systems, it consisted of one network if the fictive bond length was slightly larger than the maximum of the first peak in the g(r) function. In the case of low-density systems, a few percent of the micelles formed pairs around the fictive bond length of the first g(r) peak, but the percolation threshold of network formation was not sharp. We did not pay attention to these pairs because of the suspected experimental bias. In the evaluation of scattering data, the interparticle structure is often described with hard-body but mostly with hard-sphere interaction. The effective hard-sphere radius was a separated parameter in the data evaluation of Borbe´ly and Tomsic et al. One of the original aims of this study was to model the intermicellar structure relatively free from any model with the reverse Monte Carlo method. We performed hard-body Monte Carlo simulations to distinguish the effect of the excluded volume and the fit to the experimental data. These simulations were realized with fixed values of Nagg, σagg, and W averaged for a given concentration of the simulations detailed above. The hard bodies corresponded to the balls in the core model shown in Figure 1. One can see in Figure 8 that the reverse Monte Carlo results significantly differed from the hard-body ones. The data were also far from the experimental results, and the corresponding χ2’s were at least one magnitude larger than the reverse Monte Carlo ones. This means that the intermicellar structure of this surfactant cannot be related simply to the excluded volume of the micelles. Effective hard-sphere diameters are frequently used in the evaluation of small- angle scattering data and one can find a diameter that provides at least reasonable first-neighbor distances, but this does not give a correct description of the overall intermicellar structure. The nonnegligible limits of the one-component hard-sphere colloid model in comparison to asymmetric binary hard-sphere models are studied in detail in one of our other manuscripts in preparation.31
4. Conclusions We used a reverse Monte Carlo-type method to reevaluate the experimental data on BRIJ-35/water systems determined by Borbe´ly.10 The structure of a micelle was modeled as a spherical (31) Vrabecz, A.; To´th, G., submitted for publication, 2005.
Structure of BRIJ-35 Nonionic Surfactant in Water
core representing the hydrophobic part, and a number of balls were placed on the core to mimic the hydrated hydrophilic chains. The optimized parameters of the micelle were the mean aggregation number (Nagg), the width of the Gaussian distribution of the aggregation number (σagg), and the wetting factor of the hydrophilic chains (W). The intermicellar structure was determined quasi-model-free by the reverse Monte Carlo technique. We determined the structure for medium and high concentrations. Our approach did not work correctly at low concentrations, where we suspected a dominant experimental bias. Our model is sensitive to the experimental errors at low densities because an unphysical overfit to the error and compensation for the lack of a model are possible. We got a decreasing trend in the mean aggregation number with increasing concentrations, from 84 down to 56 in the reliable 50-200 g/dm3 concentration range. We think that this can be explained by the various extents of hydration at different concentrations. At low concentrations, the excess water caused fewer entangled hydrophilic chains, and the chains extend into
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the solvent. At high concentrations, the dehydration produced a compact and less-flexible micellar structure with more hydrophilic-hydrophobic chain contacts, which hinders the formation of large micelles. The systems seemed to be polydisperse, and the σagg parameter of the Gaussian distribution for the aggregation number was between 25 and 30% of the mean aggregation numbers. We determined the intermicellar structure for medium and high concentrations. We compared our structures to those of micelles interacting as hard bodies, and we got reasonable differences. The pair-correlation functions indicate stronger interactions than the hard-body one. Acknowledgment. Support from Hungarian Research Grant T43542 and the Be´ke´sy fellowship is gratefully acknowledged. We thank Dr. Sa´ndor Borbe´ly for the experimental data and the reviewers for carefully checking the manuscript. LA051380A
