5174
J. Phys. Chem. B 2007, 111, 5174-5179
Hexagonal Close Packing of Nonionic Surfactant Micelles in Water Xiangbing Zeng,*,† Yongsong Liu,‡ and Marianne Impe´ ror-Clerc§ Department of Engineering Materials, UniVersity of Sheffield, Sheffield S1 3JD, United Kingdom, Structural Genomics Consortium, UniVersity of Toronto, 100 College Street, Toronto, Ontario, Canada, M5G 1L5, and Laboratoire de Physique des Solides, Baˆ timent 510, UniVersite´ Paris-Sud, 91405 Orsay Cedex, France ReceiVed: December 21, 2006; In Final Form: February 15, 2007
The knowledge of the exact shapes of micelles in various micellar phases found in both lyotropic and thermotropic liquid crystals is very important to our understanding of the underlying principles of molecular self-assembly. In the current paper we present a detailed structural study of the hexagonal close packed (hcp, space group P63/mmc) micellar phase, observed in the binary mixtures of nonionic surfactant C12EO8 and water. The reconstructed electron density map of the phase shows perfectly spherical micelles. A spherical core/shell model of micelles, which fits the observed X-ray diffraction pattern satisfactorily, is subsequently constructed. The results confirm the previous assumption that the hcp phase consists of spherical close contacting micelles, each of which contains a low-density core of aliphatic parts and a high-density shell of hydrated ethylene oxide segments, with the gaps between the micelles filled by pure water.
Introduction The self-assembly of soft spherical objects, like micelles of nonionic copolymers in water,1 or in thermotropic phases of dendrimers, leads to many types of different 3D ordered packings. The simplest ones include the body-centered cubic (bcc, space group Im3hm) structure, and the two generic closepacked structures, the face centered cubic packing (fcc, space group Fm3hm) and the hexagonal close-packing (hcp, space group P63/mmc) (see Figure S1, Supporting Information). There are also different tetrahedrally close-packed (tcp) structures, where all interstices are tetrahedral. The simplest and the most widespread tcp structure is the cubic Pm3hn phase, with eight micelles in the unit cell. Several examples of another cubic phase of space group Fd3hm (24 micelles in the unit cell) are encountered in inverted micellar phases of surfactants and lipids.2,3 A tetragonal P42/mnm phase with 30 micelles in the unit cell has been observed only in dendrimeric thermotropic liquid crystals,4 along with a related quasicrystalline phase.5-7 For the first three packings, bcc, fcc, and hcp, all the micelles in the unit cell are crystallographically equivalent. Consequently one expects that all the micelles have the same shape and are probably nearly spherical. On the other hand, the tcp structures contain different types of micelles inside the unit cell, which are likely to have slightly different sizes and shapes. The fcc and hcp structures are the densest packings of hard spheres in the space, but are not observed in thermotropic systems. One hypothesis is that the formation of the bcc and the tcp phases is based on the minimization of the total contact area between the micelles, which can be achieved by slight deformations of the micelles. More precisely, bcc and the tcp structures (including the quasicrystalline phase) are better candidates to solve the mathematical Kelvin’s problem.8 This problem is to find the partition of space in cells of equal volumes that * Address correspondence to this author. E-mail:
[email protected]. † University of Sheffield. ‡ University of Toronto. § Universite ´ Paris-Sud.
minimize the total area of these cells. Kelvin proposed the truncated octahedra with slightly curved faces (bcc structure) to be the best solution in 1887. Weaire and Phelan9 in 1994, however, have shown that the Pm3hn phase with cells of equal volume gives a slightly better solution. In the case of the micellar Pm3hn phase of thermotropic dendrimers, Ziherl and Kamien10 have shown that the minimization of the total area of the cells could be related to the maximization of the orientational entropy of the aliphatic chains forming the soft corona of the micelles. In the Pm3hn structure there are two different types of micelles. The unit cell contains eight micelles, two of them are expected to be more spherical than the other six which are expected to have a slightly ellipsoidal shape (see Figure S1, Supporting Information). In the cubic Fd3hm phase there are also two types of micelles, and there is strong evidence, both experimental and theoretical, suggesting that the two micelles are significantly different in size.2,3 However, there have been only a few experimental efforts in determining the exact shape of the micelles in micellar phases by TEM11 or SAXS.12 We present here such an analysis of the hcp micellar phase by reconstruction of the 3d electron density from the SAXS data. Only very few examples of this hcp packing are known. It has been observed in the binary phase diagram of nonionic surfactant C12EO8/water13,14 (see Figure S3, Supporting Information) and very recently in the P123/water/ethanol15 system. Other examples are encountered in mesoporous materials, which are related hybrid organic/inorganic materials also based on the self-assembly of soft micelles.16 The reason why an hcp packing is formed instead of an fcc one is not obvious, even for hard spheres. These two types of packing differ only in the stacking sequence of the planes (ABC for fcc and AB for hcp) and in the local environment of the micelles (Figure S2, Supporting Information). Calculations have shown that the stacking entropy of the fcc phase is only very slightly greater than that of the hcp phase.17 This difference is very weak, and nucleation and growth kinetics play a role in the relative stability of these two phases. Experimental results confirm that these two structures
10.1021/jp0687955 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/13/2007
Close Packing of Nonionic Surfactant Micelles
J. Phys. Chem. B, Vol. 111, No. 19, 2007 5175 TABLE 1: Experimental and Calculated d-Spacings, Relative Intensities, and Phases of Diffraction Peaks for 34.3 wt % C12EO8 in Water at 10 °Ca
Figure 1. Powder X-ray diffraction pattern of the hcp phase (top curve), with diffraction peaks resolved. I(q)q2 is plotted to take directly into account the Lorentz correction for a powder sample. The deviation of the fitted curve from the experimental one is shown at the bottom.
are indeed closely related. In the CnEOm/water phase diagrams, a small change of the number of ethyloxy group modifies the stability of the various micellar phases: the C12EO12/water phase diagram18 exhibits a fcc phase and not an hcp phase as in the C12EO8/water phase diagram (see Figure S3, Supporting Information). In the P123/water/ethanol phase diagram,15 depending on the amount of ethanol, a mixture of fcc/hcp or the hcp phase only is observed. Experimental Section C12EO8 (99% purity, Nikkol) solution in distilled water (34.3 wt %) was allowed to equilibrate at room temperature for several days, then transferred into a capillary for SAXS (small-angle X-ray scattering) experiments at station 8.2 of the Daresbury Synchrotron Radiation Source, UK. A fixed wavelength of 1.54 Å was used, and a quadrant detector was used for collection of SAXS data. The sample-detector distance was about 1.5 m, which was calibrated by using long chain paraffins with known spacings. Sample was cooled with liquid nitrogen and the temperature was controlled to within a degree with a Linkam hot-stage. Intensities of the diffraction peaks were measured for reconstruction of the 3-D electron density map. To minimize the inaccuracy in measuring the relative intensities due to the possible existence of large domains in the sample, the capillary was rotated in the collection of the diffractograms. Peaksolve was used for intensity integration and resolving overlapping peaks. Results and Discussion The powder diffraction pattern of the hcp phase of the 34.3 wt % mixture of C12EO8 in water (H2O) was recorded at 10 °C and is shown in Figure 1. The lattice parameters are a ) 7.02 ( 0.05 nm and c ) 11.46 ( 0.05 nm. These values are in good accordance with those from the two previous studies13,14 performed in heavy water (D2O) instead of H2O. The c/a ratio (1.632) is very close to the theoretical value (x8/3 = 1.633) for a hcp phase formed by close contacting hard spheres. Ten diffraction orders were observed, and their index and relative intensities (Lorentz corrected) are listed in Table 1. The intensities are corrected by a factor of q2 (Lorentz correction) to take into account the fact that the SAXS experiment was carried out on a powder sample with a linear detector.19 The extinction conditions agree with those of space group P63/mmc, as expected for the hcp phase.
index (hkl)
exptl d-spacing (nm)
calcd d-spacing (nm)
intensity I(q)q2
phase
100 002 101 102 110 103 200 112 201 004
6.01 5.65 5.32 4.17 3.53 3.26 3.08 3.02 2.97 2.91
6.08 5.73 5.37 4.17 3.51 3.23 3.04 2.99 2.94 2.87
11.3 16.6 100.0 29.4 25.5 18.7 2.4 9.6 4.4 1.3
+ + + + + + -
a
a ) 7.02 ( 0.05 nm, c ) 11.46 ( 0.05 nm.
From the measured intensities I(h,k,l), the 3-D electron density map E(x,y,z) can be reconstructed by Fourier transform by using the general formula
E(x,y,z) )
∑ xI(h,k,l) exp[i2π(hx + ky + lz) + φh,k,l]
(1)
h,k,l
If the origin of the structure is also an inverse center, i.e., E(x,y,z) ) E(-x,-y,-z), the formula can be simplified to
E(x,y,z) )
∑ xI(h,k,l) cos[2π(hx + ky + lz) + φh,k,l]
(2)
h,k,l
and the phase angle φh,k,l for each diffraction order is either 0 or π. However, as the phase angle cannot be directly determined from the X-ray experiment, the choice of the phase combination needs to be made on the basis of the physical merits of the electron density map, with the help of existing knowledge about the system. For liquid crystals the right phase combination should normally be the one to give well-defined low- and highdensity regions,5 and can be decided by studying the electron density maps and histograms obtained. In the current case this was done in two steps. First electron density maps were reconstructed by using only the first six diffraction orders with all the possible phase combinations (64 in total), and the best phase combination is chosen on the merit of the reconstructed maps and histograms. The phases of the remaining four diffraction orders were determined in a similar way by adding them to the reconstruction, while the phases of the first six diffraction orders were fixed. As the phase of each diffraction order is either 0 or π, the changes to the electron density map upon changing the phase of even just one diffraction order are always nontrivial. This effect is clearly shown in Figure S4 (Supporting Information), where the best reconstructed electron density map (using the first six diffraction orders only) is compared to those whose choice of phases are only different for one of the six diffraction orders. Consequently the best phase combination can be distinguished from others fairly easily. The best reconstructed electron density map is shown in Figure 2, and the phases of the diffraction orders used are listed in Table 1. In parts a and b of Figure 2, the perfectly spherical iso-electron surfaces enclose the low-density regions, i.e., where the aliphatic chains are, in the structure. This shows that the hcp phase is indeed formed by packing of spherical micelles self-assembled from the molecules. The micelles form layers perpendicular to the c direction, and in each layer the micelles are packed on the same 2-D hexagonal lattice (the densest packing in a plane). For two consecutive layers the origins of the 2-D lattices are shifted in the plane toward each other, so
5176 J. Phys. Chem. B, Vol. 111, No. 19, 2007
Figure 2. Reconstructed electron density map of the hcp phase. Spherical isoelectron surfaces (yellow and cyan) enclose the low-density regions. The level is chosen so that the radius of the spheres is 2.0 nm (the theoretical value for the low density aliphatic core) in parts a and b. A higher density level is chosen for the isoelectron surfaces in parts c and d, which show regions of intermediate densities around the octahedral interstices. False color electron density maps in the plane of yellow spheres are also shown; the palette used is shown at the bottom. The self-assembly of C12EO8 molecules into micelles is schematically drawn on top of part b, the aliphatic parts are shown as black curvy lines, and the polar parts are white.
Figure 3. Relative electron density as a function of distance from the center of nearest micelle in the reconstructed electron density map of the HCP phase. Scattered black dots represent points sampled in the whole unit cell, and curves A, B, C, and D are sampled along different directions from the center of a micelle (shown in the inset as purple arrows). For comparison the density-distance curve of the best-fit model is also shown (broken line). The vertical dotted line indicates the maximum radius of hard spheres in the hcp structure (3.51 nm). Inset: One micelle and its twelve nearest neighbors in the HCP phase.
that the position of each micelle in one layer is always in the center of three micelles in the other layer when viewed from the top. Such ...ABABAB... repetition in the c-direction leads to the hcp structure in the end. To make this point clearer, in
Zeng et al.
Figure 4. (a) Best fit of the experimental form factor [Fcs(q)]2 with a core-shell model without a Debye-Waller term (eq 3). The paramters are Rcore ) 2.0 nm, Rtot ) 3.43 nm, Fcore:Fshellfit:Fwater ) 0.826:1.063:1. (b) Best fit of the experimental form factor [Fcs(q)]2 with a core-shell model and a Debye-Waller term with x〈u2〉 ) 0.6 nm (eq 6). The parameters are Rcore ) 1.9 nm, Rtot ) 3.42 nm, Fcore:Fshellfit:Fwater ) 0.826: 1.056:1.
Figure 2 micelles in the two layers are shown in yellow and cyan, respectively. When the level of the iso-electron surface is increased, in addition to the original spherical objects shown in Figure 2a,b, regions of intermediate density start to appear around the octahedral interstices in the structure. The octahedral interstices are the points in the structure which are furthest away from any micelles. This is again in line with the micellar model of the structure, where the gaps between the micelles are filled by water, which have a density in between those of the aliphatic part and the polar part of the C12EO8 molecule. Figure 3 shows in more detail how the electron density is distributed around each micelle in the reconstructed electron map. In the hcp phase, each micelle is surrounded by 12 nearest neighbors: 6 of them are in the same plane, 3 on top, and 3 below (Figure 4). The curves show how the electron density changes with distance from the center of the micelle along four different directions: A to the center of closest micelle in the same plane, B to the center of closest micelle above or below, C to the closest tetragonal interstices, and D to the closest octagonal interstices. For all curves the electron density is minimum in the center of the micelle. Within 2 nm from the center, the electron density always increases continuously with increasing distance from the center, and the four curves are almost exactly the same. The curves only start to fork out gradually when the distances are increased further. Curves in directions A and B are almost the same: the electron density continues to increase with the distance from the center, it
Close Packing of Nonionic Surfactant Micelles
J. Phys. Chem. B, Vol. 111, No. 19, 2007 5177
TABLE 2: Core/Shell Model of the Form Factor of the Micelles form factor [Fcs(q)]2 hcp structure factor Shcp(q)
peaks
intensities (a)
intensities (b)
x〈u2〉 ) 0 (eq 3)
x〈u2〉 ) 0.6 nm (eq 6)
∆ ) 8.9c
∆ ) 8.0c
(hkl)
M multiplicity
phase
intensity
exptl
calcd
exptl
calcd
100 002 101 102 110 103 200 112 201 004
6 2 12 12 6 12 6 12 12 2
+ + + + + + -
1 4 3 1 4 3 1 4 3 4
67.8 74.7 100 88.2 38.25 18.7 14.4 7.2 4.4 5.85
68.4 81.1 91.5 89.4 42.4 19.3 7.5 4.6 2.8 1.2
77.3 86.6 118.2 115.8 55.9 29.2 23.7 12.1 7.5 10.2
77.8 94.4 108.7 116.0 61.5 31.6 14.8 10.3 7.2 4.2
Rcore ) 2 nm; Rtot ) 3.43 nm; Fcore:Fshellfit:Fwater ) 0.826:1.063:1. 1.056:1. c ∆ ) 100[∑i |Iexpi - Icali|]/[∑i Iexpi]. a
b
gradually levels off at around 3 nm, and it reaches the maximum in the middle of the two micelles (3.51 nm from both centers). Meanwhile, for curves C and D, with increasing distance the electron density first increases quickly and arrives at the maximum at around 2.7 nm, but then decreases slowly and ends with an intermediate value. Furthermore, points in the unit cell are sampled on a dense grid (35 × 35 × 57 ) 69 825 in the unit cell), for each point its electron density and the distance from the nearest micelle is calculated. These electron densitydistance relationships were presented as scattered dots in the background in Figure 3. These results agree at least qualitatively with the micellar model where each micelle should have a lowdensity core (aliphatic chains) and a high-density shell (EO chains and water) and is surrounded by intermediate density waters. Because the shape of the micelles appears to be spherical, we can directly model the diffraction peak intensities by the product of the hcp structure factor Shcp(q) by the form factor of the micelles, [Fcs(q)]2, taking a core/shell model to describe the micelles. The expression of the intensity of a diffraction peak after Lorentz correction is given in eq 3, where M is the multiplicity of a Bragg peak.
I(q) ) M(q)Shcp(q)[Fcs(q)]2
(3)
The form factor amplitude Fcs(q) for a core/shell model is recalled in eq 4 and the form factor Fsphere(q,R) of a sphere in eq 5.14
Fcs(q) ) (Fshell - Fcore)VcoreFsphere(q,Rcore) + (Fwater -Fshell)VtotFsphere(q,Rtot) (4) Fsphere(q,R) ) 3[sin(qR) - qR cos(qR)]/(qR)3
(5)
Here Fcore, Fshell, and Fwater are the electron density of the core, the shell, and water, respectively. A similar approach has been followed by Imai et al. to model the scattering curve for more diluted samples inside the isotropic region of the C12EO8/water phase diagram.14 In their case, the form factor of the micelles is the same, but to deduce the structure factor it is necessary to model the interaction potential between the micelles, except for very dilute samples. The advantage of examining the hcp phase is that the structure factor has a simple well-known expression, because the micelles are on an ordered 3D lattice and not in a
x〈u2〉 ) 0.6 nm; Rcore ) 1.9 nm; Rtot ) 3.42 nm; Fcore:Fshellfit:Fwater ) 0.826: liquid state as in the isotropic phase. The values of the structure factor Shcp(q) are given in Table 2. To take into account the thermal fluctuations of the positions of the micelles, a Debye-Waller factor can be added, where 〈u2〉 is the mean square displacement of the center of a micelle with respect to the node of the hcp lattice. The expression of the intensity is then
I(q) ) M(q)Shcp(q) exp(-〈u2〉q2/3)[Fcs(q)]2
(6)
Using the Lindemann criteria for the melting of a crystal, we can estimate the upper value of 〈u2〉. By analogy with a recent study of the Debye-Waller term in the solid helium fcc and hcp phases, we take the following upper limit for the Lindemann ratio: x〈u2〉 = 0.2(a/2) = 0.1a = 0.7 nm.20 In fitting the form factor, an average volume of 0.350 nm3 is assumed for one -C12H25 alkyl chain, corresponding to an average electron density for the core Fcore ) 276 e-/nm3. The electron density of water is Fwater ) 334 e-/nm3. Fits with values of Rtot greater than a/2 ) 3.51 nm are excluded, because they would correspond to an overlap of the micelles. A good fit of the data can be obtained without taking into account the DebyeWaller term and it is slightly improved with a Debye-Waller term with x〈u2〉 ) 0.6 nm, which is consistent with the Lindemann criteria. The two best fits, with and without the DW factor, have very similar parameters and are shown in Figure 4 and in Table 2. All the measured Bragg peaks are located within the second oscillation of the form factor curve, so that the phase contribution from the form factor is the same for all the peaks. The phases are then fixed by the hcp structure factor only (Table 2) and they are the same as the ones used for the electron density reconstruction (Table 1). This validates the criteria we adopted in choosing the phase combination during reconstruction of electron density. The best fit core radius Rcore without the DW factor is 2.0 nm, which is virtually the same as calculated from the composition of the sample (34.3 wt %) assuming that the core contains the aliphatic parts of the molecules. This is longer than the extended length of the aliphatic chain -C12H25, which can be calculated to be ∼1.6 nm. Thus the other best fit with the
DW factor, with a mean square displacement x〈u2〉 ) 0.6 nm, is more reasonable. The displacements of each micelle as a whole contribute to this rather large x〈u2〉 value. The effect of
5178 J. Phys. Chem. B, Vol. 111, No. 19, 2007 the Debye-Waller factor is to smear the boundaries between the core and shell regions, and between the shell and the water, making the density profile of the core/shell model more realistic. This smearing could lead to a small overlap of the shells for large x〈u2〉 values. However, no deep penetration of the shells is expected, because the hexagonal close packing results from a hard-sphere interaction. Taking the core radius to be 2.0 nm, the aggregation number of the micelles can be calculated to be 96. The shell density is deduced from the fits to be Fshell ) 355 e-/nm3. This corresponds to a shell of 64% hydration volume fraction, with about 30 water molecules for one EO8 group. The fits also indicate that Rtot is very close to a/2, in line with the fact that hcp is one of the densest packings for close contact hard spheres. Imai et al.14 have studied the C12EO8/water micellar system at different concentrations using SAXS. Fitting of the observed scattering profile for disordered micelles in a dilute sample (3 wt %) suggests the parameter of the micelles to be Rc ) 1.32 nm and Rtot ) 3.55 nm. While the overall radius of the micelle is similar to that of ours in the ordered hcp phase, the core radius is significantly smaller. This suggests that the degrees of hydration of the shells are significantly different for dilute and concentrated solutions (the hydration number per EO8 group is ∼120 for the 3 wt % sample14). At the same time, SANS experiments on the C12EO8/D2O system21 suggest very similar micelle parameters: Rcore ) 1.99 ( 0.01 nm and Rtot ) 3.39 ( 0.014 nm for a more concentrated solution (18 wt % at 10 °C). In the SAXS spectrum, the cutoff value in q is ∼2.5 nm-1 (Figure 1). As a consequence, only ten independent diffraction orders (and their symmetrical equivalents) were used in reconstructing the electron density map, which is expected to possess a limited spatial resolution of ∼1.25 nm. It would thus be useful to examine the cutoff effect on the reconstructed electron density map to establish how faithfully it preserves the features of the original map. For this purpose, a simulated reconstruction from the best-fit model (without the DW term) is carried out. Again, electron density-distance curves from the center of a micelle along four different directions are plotted along with that of the model (Figure S5, Supporting Information). The features of the curves from the simulated reconstruction are almost the same as the experimental one, suggesting a good quality fit to the experimental data. It can be clearly seen that in the reconstructed map the boundary between the core and the shell is largely smeared compared to that in the model map. The maximum electron density observed in the reconstructed map is also less prominent, i.e., the contrast of the reconstructed map is lower. The general initial up and then down trend of density with increasing distance from the micelle center in the curves is preserved. However, it is also clear in Figure 2 that a quantitative description of the structure based on reconstructed map must be done with extreme care, and overexplanation of the data, beyond the experimental capability, must be avoided. The size of the core and shell regions and the spherical symmetry of the micelles are well established from the electron reconstruction we have performed. Obviously, the core/shell model is only a simplification of the true density profile and more complicated models for the density distribution of the PEO chains inside the shell could be tested.22 This would improve the vision of the details of the density distribution inside the core and shell regions. It is also clear from discussions above that to discriminate different density models, improvements of the experimental data would be necessary. However, experi-
Zeng et al. mentally it is difficult to achieve much higher resolution as the higher the q value is, the weaker the diffraction intensities. At the same time, as diffraction peaks become denser with increasing q, resolving nonequivalent diffraction peaks becomes increasingly difficult, if not impossible, with powder diffraction. The problem can be solved by single-crystal X-ray diffraction experiments, which are considerably more difficult, in addition to the complication of preparing single-crystal samples. In addition to the hcp phase, CnEOm + water systems demonstrate a number of other micellar phases such as fcc, bcc, and Pm3hn. It will be our future work to examine the shapes of micelles in these other phases and compare them with each other. This would hopefully shed new light on how the shapes of micelles are linked to their modes of packing. Conclusions Quantitative analysis of the SAXS spectrum of the hcp phase found in nonionic surfactant C12EO8-water binary system has been carried out. The electron density map of the phase has been reconstructed from the measured diffraction intensities, and it confirms that the phase is formed by the packing of close contacting, spherical micelles. Experimental SAXS data are satisfactorily fitted by using a spherical core/shell model of micelles. The parameters of the model provide a more quantified description of such micelles, and fit nicely with the molecular dimensions. Acknowledgment. X.Z. thanks the University Paris-Sud (France) for the invitation and financial support for his visit to the Laboratoire de Physique des Solides in 2006. Supporting Information Available: Figures showing the direct micellar phases observed in nonionic surfactant/water phase diagrams (Figure S1), a comparison of the fcc and the hcp packings (Figure S2), the C12EO8 and C12EO12 binary phase diagrams in water (Figure S3), reconstructed electron density maps using the first six diffraction orders (Figure S4), and the density-distance curves for a simulated reconstruction from the best-fit model (Figure S5). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Hamley, I. W.; Daniel, C.; Mingvanish, W.; Mai, S. M.; Booth, C.; Messe, L.; Ryan, A. J. From hard spheres to soft spheres: The effect of copolymer composition on the structure of micellar cubic phases formed by diblock copolymers in aqueous solution. Langmuir 2000, 16 (6), 25082514. (2) Seddon, J. M.; Robins, J.; Gulik-Krzywicki, T.; Delacroix, H. Inverse micellar phases of phospholipids and glycolipids. Phys. Chem. Chem. Phys. 2000, 2 (20), 4485-4493. (3) Delacroix, H.; GulikKrzywicki, T.; Seddon, J. M. Freeze fracture electron microscopy of lyotropic lipid systems: Quantitative analysis of the inverse Micellar cubic phase of space group Fd3m (Q(227)). J. Mol. Biol. 1996, 258 (1), 88-103. (4) Ungar, G.; Liu, Y. S.; Zeng, X. B.; Percec, V.; Cho, W. D. Giant supramolecular liquid crystal lattice. Science 2003, 299 (5610), 1208-1211. (5) Ungar, G.; Zeng, X. B. Frank-Kasper, quasicrystalline and related phases in liquid crystals. Soft Matter 2005, 1 (2), 95-106. (6) Zeng, X. B. Liquid quasicrystals. Curr. Opin. Colloid Interface Sci. 2005, 9 (6), 384-389. (7) Zeng, X. B.; Ungar, G.; Liu, Y. S.; Percec, V.; Dulcey, S. E.; Hobbs, J. K. Supramolecular dendritic liquid quasicrystals. Nature 2004, 428 (6979), 157-160. (8) Rivier, N. Kelvins Conjecture on Minimal Froths and the Counterexample of Weaire and Phelan. Philos. Mag. Lett. 1994, 69 (5), 297-303. (9) Weaire, D.; Phelan, R. A. Counterexample to Kelvin Conjecture on Minimal-Surfaces. Philos. Mag. Lett. 1994, 69 (2), 107-110. (10) Ziherl, P.; Kamien, R. D. Maximizing entropy by minimizing area: Towards a new principle of self-organization. J. Phys. Chem. B 2001, 105 (42), 10147-10158.
Close Packing of Nonionic Surfactant Micelles (11) Mariani, P.; Luzzati, V.; Delacroix, H. The cubic phases of lipidcontaining systems. Structure analysis and biological implications. J. Mol. Biol. 1988, 204, 165-189. (12) Balagurusamy, V. S. K.; Ungar, G.; Percec, V.; Johansson, G. Rational design of the first spherical supramolecular dendrimers selforganized in a novel thermotropic cubic liquid-crystalline phase and the determination of their shape by X-ray analysis. J. Am. Chem. Soc. 1997, 119 (7), 1539-1555. (13) Clerc, M. A new symmetry for the packing of amphiphilic direct micelles. J. Phy. II 1996, 6 (7), 961-968. (14) Imai, M.; Yoshida, I.; Iwaki, T.; Nakaya, K. Static and dynamic structures of spherical nonionic surfactant micelles during the disorderorder transition. J. Chem. Phys. 2005, 122 (4). (15) Soni, S. S.; Brotons, G.; Bellour, M.; Narayanan, T.; Gibaud, A. Quantitative SAXS analysis of the P123/water/ethanol ternary phase diagram. J. Phys. Chem. B 2006, 110, 15157-15165.
J. Phys. Chem. B, Vol. 111, No. 19, 2007 5179 (16) Tolbert, S. H.; Schaeffer, T. E.; Feng, J.; Hansma, P. K.; Stucky, G. D. A new phase of oriented mesoporous silicate thin films. Chem. Mater. 1997, 9, 1962. (17) Mau, S. C.; Huse, D. A. Stacking entropy of hard-sphere crystals. Phys. ReV. E 1999, 59 (4), 4396-4401. (18) Sakya, P.; Seddon, J. M.; Templer, R. H.; Mirkin, R. J.; Tiddy, G. J. T. Micellar cubic phases and their structural relationships: The nonionic surfactant system C12EO12/water. Langmuir 1997, 13 (14), 3706-3714. (19) Janssen, T.; Janner, A.; Looijenga-Vos, A.; de Wolff, P. M. International Tables for Crystallography; Kluwer: London, UK, 1995. (20) Arms, D. A.; Shah, R. S.; Simmons, R. O. X-ray Debye-Waller factor measurements of solid He-3 and He-4. Phys. ReV. B 2003, 67 (9). (21) Constantin, D.; Freyssingeas, E. Unpublished results. (22) Pedersen, J. S.; Svaneborg, C. Scattering from block copolymer micelles. Curr. Opin. Colloid Interface Sci. 2002, 7 (3-4), 158-166.
