Long-Range Ordering in the Lyotropic Lamellar Phase Studied by

J. Phys. Chem. B 2010, 114, 165–173

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Long-Range Ordering in the Lyotropic Lamellar Phase Studied by High-Resolution Magnetic Resonance Diffusion-Weighted Imaging Kosma Szutkowski* and Stefan Jurga Department of Macromolecular Physics, Faculty of Physics, Adam Mickiewicz UniVersity, ul. Umultowska 85, PL61614 Poznan, Poland ReceiVed: July 28, 2009; ReVised Manuscript ReceiVed: NoVember 24, 2009

Diffusion-weighted magnetic resonance imaging (DW MRI) was applied to the lyotropic lamellar phase of the dodecylammonium chloride/water system (DDACl/H2O). In the course of employing a well-known medical imaging method, namely, diffusion tensor imaging (DTI), the system morphology was assessed accurately in the most straightforward way by two-dimensional visualization of eigenvectors associated with planar distribution of effective diffusion tensors throughout the whole slice with 40 µm in-plane resolution. Longrange order was observed in the studied lamellar phase, and morphology was best described by a combination of three- and one-dimensional diffusion. 1. Introduction

2. Theory

The matter of sampling morphologies over several orders of magnitude, e.g., from nano- to microscale, is important as long as desired properties of a complex lyotropic system are under consideration.1-4 Nowadays, a variety of “nano”applications have emerged due to numerous studies on diverse properties of amphiphile systems on a nanoscale. At the same time, a behavior on a microscale, such as long-range ordering, seems to attract less attention. Despite that, there are some interesting and appealing applications of such behavior. For example, optical properties are determined by a long-range orientational correlation, or simply long-range order, which renders them useful for novel applications such as optical display systems based on lyotropic chromonic liquid crystals (LCLCs).5 A quantitative characterization and monitoring of amphiphilic morphologies in a wide range of scales and various conditions are essential and central though. If we consider the morphology on a nanoscale, we think of methods like small-angle X-ray scattering (SAXS), transmission electron microscopy (TEM), or atomic force microscopy (AFM). However, considering structures on elevated scales, the so-called diffusion tensor imaging (DTI) coupled with high-resolution NMR microimaging can be exploited to assess long-correlation features of a diversity of amphiphilic systems.6-8 Long-range ordering and structural defects are in fact unambiguously analyzed by DTI, and in comparison with some classical “structural” methods, much more quantitative information is derived. One gets a precise spatial distribution of solvent diffusion anisotropy, related in a straightforward way to topological barriers such as smectic structures, double layers, platelates, micelles, and lamellas, thus affecting translational self-diffusion. Hence, DTI may fill a gap in the micro- and millimeter scale methods, especially when one is interested in the quantitative characterization of macroscopically ordered morphologies. As a model system for DTI investigation, a favorable sample was chosen. The sample was characterized by a relatively high water content, which assured sufficiently long transverse relaxation time T2 of water.

Calculation of Diffusion Tensors from Diffusion Maps. The chosen DTI scheme is based on prior calculation of apparent diffusion maps (ADC). Each diffusion map is calculated for a specific set of gradient vector directions. For the following directions, related to the principal axis system of the spectrometer, x, y, z, xy, xz, and yz, the elements of the diffusion tensor, Dxx, Dyy, Dzz, Dxy, Dxz, and Dyz, relate to ADCs as

* Corresponding author. E-mail: [email protected].

ADCxx ) Dxx

(1)

ADCyy ) Dyy

(2)

ADCzz ) Dzz

(3)

1 1 ADCxy ) Dxx + Dyy + Dxy 2 2 1 1 ADCxz ) Dxx + Dzz + Dxz 2 2 1 1 ADCyz ) Dyy + Dzz + Dyz 2 2

(4) (5) (6)

By using eqs 1-6, the diffusion tensors are calculated in a much more convenient way than in typical medical applications, where tensors due to time limitations are often estimated from the minimum possible number of scans. The other advantageous feature of this approach is the possibility of analysis of multicomponent diffusion, for example, due to two or more diffusion coefficients. Characterization of Diffusion Tensors. One of the possible ways of representation of diffusion tensors is an ellipsoidal approximation. A diffusion ellipsoid shows a distance in three dimensions which is likely to be covered by diffusing molecules.9-11 The shape and orientation of the ellipsoid are thus crucial, especially for medical applications such as white matter tractography used for the early stage diagnosis of neurological diseases.11 On the other hand, in nonmedical applications, it is not always necessary to visualize ellipsoids. Instead, ellipsoids are characterized by specific shape measures, and directional anisotropy can be determined through eigenvectors e. The linear, planar, and spherical measures Cl, Cp, and Cs are calculated from resorted eigenvalues such that λ1 > λ2 > λ3.12

10.1021/jp9072087  2010 American Chemical Society Published on Web 12/17/2009

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J. Phys. Chem. B, Vol. 114, No. 1, 2010

Szutkowski and Jurga

Cl )

λ1 - λ2 λ1

(7)

Cp )

λ2 - λ3 λ1

(8)

λ3 λ1

(9)

Cs )

Ca ) 1 - Cs

(10)

Cl + Cp + Cs ) 1

(11)

where the value Ca determines a general deviation from the spherical case. Some examples of ellipsoidal approximations of diffusion tensors are shown in Figure 1. Eigenvectors corresponding to the biggest eigenvalues (the fastest diffusion) are parallel to the longest semiaxis of ellipsoids. The diffusion process which occurs in one dimension should manifest itself in high values of the linear measures Cl (Figure 1a). By analogy, high values of the planar case (Figure 1b) should reflect two-dimensional configuration of obstructions. The spherical case (Figure 1c) refers in general to isotropic diffusion. The other combinations of ellipsoids are also possible. Figure 1d shows a simple example of what the diffusion ellipsoid looks like for a combination of planar and linear diffusion. The combination of spherical and linear diffusion would result in a typical elongated ellipsoid. 3. Materials and Methods The lamellar phase of dodecylammonium chloride/water (DDACl/H2O) is easily obtained at room temperature for weight

concentrations ranging from 20 to 50%.13,14 The concentration of 33% by weight was selected. Dodecylammonium chloride was synthesized by the procedure given by Kertes15 from dodecylamine obtained from SIGMA. The sample was prepared by adding doubly distilled water to dry DDACl and sealed in an 8 mm NMR glass tube. The homogeneity was achieved throughout long time heating in the bath at a temperature around 90 °C. NMR Experiments. The measurements were performed using a Bruker Avance DMX 9.4 T NMR spectrometer and Micro2.5 probe generating gradient pulses up to 1 T/m in x-y-z directions; 15 mm r.f. birdcage was used. The experiments were controlled with Bruker Medical Paravision 2.1.1 and Bruker XWinNMR 2.6 PL6 operated on an SGI O2 workstation. Basic experimental parameters were as follows: field of view (FoV) 10 mm, matrix size 512 × 512 (T2 weighted image) and 256 × 256 (diffusion-weighted image), slice thickness 1 mm, and sinc5 shaped r.f. pulse for an accurate slice selection. A routine Multi Slice Multi Echo (MSME) pulse sequence was used for T2 weighted images and, respectively, Spin Echo Diffusion pulse sequence SE_DIFF for diffusion maps. The other parameters for SE_DIFF experiments were as follows: the repetition time 2.5 s, spin echo time 23 ms, number of accumulations 16, amplitudes of diffusion gradients 0, 100, 200, and 500 mT/m. The total experimental time for the acquisition of diffusion-weighted images was around 68 h. The T2 weighted experiment was performed at 28 °C, while the diffusion maps were acquired at room temperature to avoid sample heating and influence of temperature gradients. Also, transverse relaxation time is longer at lower temperatures due to the chemical exchange effect occurring in this sample.16 Furthermore, the

Figure 1. Examples of various diffusion tensors: (a) linear case for high Cl close to 1, (b) planar case for Cp close to 1, (c) spherical case for Cs close to 1, (d) mixed case.

Ordering in the Lyotropic Lamellar Phase

J. Phys. Chem. B, Vol. 114, No. 1, 2010 167

sample was heated to the isotropic phase and then slowly cooled to the lamellar phase to amplify the ordering effect. Apparent diffusion coefficient maps ADC were calculated using the “Image Fit Package” available in Paravision software. A single diffusion component was fit for six gradient directions, (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), and (0,1,1). The same spectrometer setup was used for the Fourier transform pulsed field gradient spin echo (FT-PGSE) experiments.17 DOSY displays were calculated with Bruker XWinNMR 3.5 software using the exponential fitting algorithm. Calculation of Diffusion Tensors. The elements of diffusion tensor for each voxel were determined directly from six highresolution diffusion maps. The procedure of conversion of ADCs to diffusion tensors is derived in Appendix A. Mathcad 11 software was used for initial calculation of diffusion tensors and then for diagonalization so that pairs of eigenvectors ei and eigenvalues λi (i ) 1,2,3) were determined for each voxel. The pairs of eigenvalues and eigenvectors were sorted by the magnitude λ1 > λ2 > λ3 for the ease of further calculations of specific anisotropy parameters Dl, Dp, Ds, and Da. The diffusion tensor imaging method was calibrated by performing the experiments on a water sample. The obtained spherical shape measure Ds for water was close to 1; however, the eigenvalues were slightly decreased in comparison with bulk water diffusivity obtained by PGSE. POM Textures. Polarizing optical microscopy (POM) textures were obtained by using a Leica DMLP microscope. The microscope was equipped with a heating table. 4. Results and Discussion A T2 weighted image recorded for the lamellar phase at 25 °C is shown in Figure 2a. Figure 2b shows corresponding DOSY displays calculated from FT-PGSE data as obtained for major gradient directions: Z(gx), Y(gy), Z(gz). Figure 2c shows corresponding spin echo attenuation curves together with fitted exponential decays (linear in log scale). As one may conclude from DOSY displays (Figure 2b), which also contain Fourier transformed spin echo signals (see the bottom-most part), the NMR signal comprises mainly water proton contribution at ca. 5 ppm. At the same time, there is no noticeable trace in the 1H spectra of alkyl chain protons, usually visible in NMR spectra at around 1-3 ppm. In the DDACl lamellar phase, due to strong dipole-dipole coupling, the transverse relaxation time T2 of alkyl chain protons is greatly reduced, which is evidenced by the broadening of the spectrum up to 10 kHz and the disappearance of a separate resonance from CH2 groups.16 It is clear, though, that since the alkyl proton resonances are distributed in a broad spectrum of NMR frequencies they are not detectible for sufficient spin echo times. Hereafter, what is seen afterward in PGSE and MR imaging data comes only from water protons. The evident inhomogeneities in the spin echo image (Figure 2a) reflect either changes in water density or differences in T2 contrast. Apart from variable density, the contrast due to spin-spin relaxation might be dependent on different angular orientations of lamellar domains with respect to the external magnetic field; the so-called anisotropy of dipole-dipole coupling should be concerned.18 The obtained T2 map of the slice shows almost uniform distribution of T2 relaxation time in the range between 70 and 90 ms (data not shown). However, most likely the contrast variations originate from both effects, for example, some differences in water distribution and T2 effect induced by the morphology of the lamellar phase. In particular, the latter conclusion is justified by direct comparison of bright

Figure 2. (a) T2 weighted image obtained for the lamellar phase of DDACl + H2O 33% wt at 25 °C. Matrix size 512 × 512; field of view 10 mm; repetition time 4.3 s; echo time 17.9 ms. (b) DOSY displays obtained from PGSTE for three directions at 28 °C and (c) corresponding spin echo attenuations (the same data set).

Figure 3. Lamellar texture obtained by polarizing optical microscopy for DDACl + H2O 33% at T ) 25 °C; magnification 100×.

structures (brighter voxels from Figure 2) with the lamellar texture obtained by polarizing optical microscopy (Figure 3). Regarding PGSE studies of macroscopic diffusion anisotropy, DOSY displays do not reveal any relevant macroscopic anisotropy of water diffusion. However, as we may derive from raw

168



Figure 4. Apparent diffusion coefficient maps (ADC) obtained for the partially aligned lamellar phase of DDACl + H2O 33% wt for T ) 21 °C (also see Materials and Methods section). Gradient directions are shown in brackets, matrix dimensions 256 × 256, FOV 10 mm.

Figure 5. Raw values of eigenvalues λ1, λ2, and λ3 and sorted according to the relevance λ1s > λ2s > λ3s (data calculated from previously obtained ADCs shown in Figure 4).

FT-PGSE data, a small anisotropy of water diffusion does exist at this stage (Figure 2c). The average diffusion of water is slightly faster in the Z direction which confirms that the sample is likely to be partially aligned in average due to the external magnetic field of a superconducting magnet.19,20 In normal conditions, where the lamellar phase is formed in the absence of the external magnetic field, the isotropic distribution of

lamellar directors is obtained, giving a powder pattern (Pake’s pattern) on the deuterium NMR spectrum. Accordingly, formation of the lamellar phase upon the presence of a magnetic field results in a partially oriented lamellar phase.20 The diffusion maps are shown in Figure 4. The gradient vector directions were varied as (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), and (0,1,1). At this stage, there are noticeable differences



Figure 6. Calculated maps of shape measures. Planar, Cp; overall anisotropy, Ca; linear, Cl; spherical, Cs (data calculated from sorted eigenvalues λ1s > λ2s > λ3s shown in Figure 5).

between water diffusivities measured along selected gradient directions. The brighter the point is, the faster the diffusion of water in a specific direction. The best image contrast can be perceived from a comparison of ADC maps gathered in directions (1,0,0) and (0,1,0). This comparison already suggests that in some areas of the image a high degree of anisotropy along with a long-range order does exist. Interestingly, the contrast in the (0 0 1) direction was much less pronounced. A plausible explanation is that the size of the voxel in the slice selection direction is bigger than the size in the planar direction. As a result, information about anisotropy in the Z direction is inevitably averaged out, and if high-resolution NMR microimaging is considered, it is hardly possible to equalize voxel dimensions. Apparent diffusion maps do provide some valuable information about diffusion anisotropy, and at least the size of the lamellar domain can be estimated. Nevertheless, there is one important drawback of such ADC maps. The analysis is dependent on the sample positioning and is especially sensitive to rotations around any gradient direction. To make the result independent of the laboratory frame, one has to calculate the elements of the effective diffusion tensor Def according to eqs 1-6 and then perform diagonalization resulting in a pair of eigenvalues λ and eigenvectors e. Figure 5a shows eigenvalues λ calculated from ADCs presented in Figure 4. Figure 5b shows eigenvalues sorted according to amplitude, which is denoted by the index “s”. The linear, planar, spherical, and anisotropy measures Cl, Cp, Cs,

and Ca are shown in Figure 6. The lamellar morphology is usually described as two-dimensional, which means that free diffusion occurs freely in two dimensions while molecular translations along the third direction are hindered. In average, as derived from calculations, the values of spherical shape measure Cs are bigger than those of planar shape Cp and linear Cl cases. At the same time, the linear case is much bigger than planar diffusion. This implies that the diffusion of the water lamellar phase is a combination of three-dimensional, spherical, and linear diffusion. This is not the result one may expect since the geometry of the lamellar phase is usually described as twodimensional and should be characterized by high values of a planar shape Cp. Instead, the diffusion of water is characterized by a combination of spherical and linear tensors resulting in slightly flattened elongated ellipsoids. A possible explanation is that lamellas are permeable to water diffusion, for istance, due to the presence of defects such as perforations. This result is in fact confirmed by our other results of high-concentration lamellar phases of DDACl/H2O (above 40% wt) where PGSE data were analyzed by restricted diffusion models.21 The spin echo decays in 40-60% DDACl lamellar phases fit 3D restricted diffusion models rather than 2D diffusion (to be published). Figures 7 and 8 show a series of 2D vector field plots generated for different projections of eigenvectors e into three major planes XY-XZ-YZ. Eigenvector e1 is related to the fastest diffusion characterized by eigenvalue λ1s; eigenvector e2 is related to eigenvalue λ2s; and, respectively, e3 is related to the slowest diffusivity and eigenvalue λ3s (Figure 5b). One can make

170



Figure 7. Projection of the eigenvectors onto XY, XZ, and YZ planes. The rectangular region is magnified and then shown in Figure 8. Each visualized eigenvector corresponds to a single voxel of dimension 39 × 39 × 1000 µm (X-Y-Z) (data calculated from previously obtained ADCs shown in Figure 4).

a plausible assumption that the direction of the fastest diffusion will indicate the way of alignment within the lamellar domain. For further analysis, the rectangular region of interest (ROI) was selected so that it is placed on the border between two lamellar domains. Figure 8 shows a magnification of ROI from Figure 7. There are two regions indicated in Figure 8, region (1) and region (2). Region (1) is one lamellar domain, characterized by a long-range order, and one direction of alignment, and region (2) denotes the second domain of a different direction of alignment. There is a pronounced border between those two regions. In the following, we will analyze the directions of eigenvectors to reveal in a quantitative way the local order within lamellar

domains. If one wishes to analyze the direction of eigenvectors, the best way is to find a viewpoint direction that is parallel to the vector of interest so that the value of the projection is small and the vector itself resembles a dot. The best example of this approach is shown in Figure 8g, where the slowest diffusion λ3s is seen from the XY plane so that most of the projections of eigenvectors e3 into the XY plane are pointlike. This is just an indication that eigenvectors e3 in Figure 8g are more parallel to the XY plane than to the other planes. Now, if we analyze the fastest diffusion in region (1), we see that it occurs in the XZ plane, while the fastest diffusion at region (2) occurs in the YZ plane. The average angles between directors can be calculated. It can be derived from the vector field plot that the domains in



Figure 8. Projection of the eigenvectors onto XY, XZ, and YZ planes in the region of interest from Figure 7. The region of interest for which corresponding anisotropy parameters are shown in Figure 9 is depicted in (a). Each visualized eigenvector corresponds to a single voxel of dimensions 39 × 39 × 1000 µm (X-Y-Z).

regions (1) and (2) are almost perpendicular. Furthermore, the lamellar director b n is parallel to the slowest diffusion eigenvalue λ3s and eigenvector e3 (Figure 8g). This means that in this particular slice the lamellar surfaces are perpendicular to the external magnetic field. The changes in morphology can be traced throughout the transition between regions (1) and (2). The rectangular region (three voxels wide) shown in Figure 8a is further analyzed in Figure 9. The region was chosen so that a domain transition region is tracked. There is a slight increase in Cs and Cp values and a small decrease in Cl in the region where the joint zones

between lamellas of different orientations are present. The increase in the spherical tensor shape measure Cs in the transition region is reasonable due to the summation of two perpendicular diffusion tensors in one region. The degree of anisotropy is decreased in the transition region.

5. Conclusions The lamellar morphology is usually described as twodimensional, which means that diffusion should occur in two

172



Figure 9. Linear Cl, planar Cp, and spherical Cs tensor shape parameters and apparent diffusion tensor eigenvalues λ1s > λ2s > λ3s shown versus voxel positions in the region of interest depicted in Figure 7a.

dimensions, while molecular translations along the third direction should be hindered. As derived from our studies, the values of spherical shape measure Cs are bigger than those of planar shape Cp and linear Cl cases. This implies that the diffusion of water in the studied lamellar phase is a combination of threedimensional and linear diffusion. Most likely the lamellas in DDACl/H2O 33% wt are permeable to water diffusion, and the presence of defects and perforations should be taken under consideration. Acknowledgment. This work was partially supported by research grant No. N N202 128536 (Poland). We appreciate Dr. Zbigniew Fojud for synthesis of DDACl. Appendix A

) -bgˆ D gˆ [ M(TE) M(0) ]

[

3

]

Gz )

gˆTyy ) [0, 1, 0]

(A.1)

T ) [0, 0, 1] gˆzz

i)1

where M(TE) is the NMR signal amplitude for a given echo time TE and bij * 0; M(0) is the amplitude of the NMR signal for bij ) 0; bij are elements of the gradient matrix b; and Def ij are elements of the effective diffusion tensor Def. The relation between magnetic field gradient Gz of the Bz field and gradient vectors g is given by

∂Bz ∂Bz ∂Bz i+ j+ k ) gx + gy + gz ∂x ∂y ∂x

gˆTxy ) [√0.5, √0.5, 0] gˆTyz ) [0, √0.5, √0.5]

For this particular set of gradient vectors and bxy ) byx, bxz ) bzx, byz ) bzy, Dxy ) Dyx, Dxz ) Dzx, and Dyz ) Dzy, we get the following set of equations

gyz ) √0.5(gy + gz)

(A.4)

gzx ) √0.5(gz + gx)

(A.5)

Then eq A.1 may be rewritten in the following form

12

( ) ( ) ( )

Mxx ) -bxxgˆTxxDefgˆxx ) -bxxDxx M(0) Myy ln ) -byygˆTyyDefgˆyy ) -byyDyy M(0) Mzz T ef ln ) -bzzgˆzz D gˆzz ) -bzzDzz M(0) ln

where i, j, and k are unit vectors in the (xyz) laboratory frame. Gradient vectors of an equal amplitude in (xy), (yz), and (zx) directions may be written as

(A.3)

(A.7)

gˆTxz ) [√0.5, 0, √0.5]

(A.2)

gxy ) √0.5(gx + gy)

(A.6)

gˆTxx ) [1, 0, 0]

3

∑ ∑ bijDijef ) b · Def

M(TE) )M(0) j)1

ef

and gˆ ) g/|g|. The diffusion tensor is symmetric; hence, not all nine elements have to be calculated. Only six of them are relevant: D11 ) Dxx, D22 ) Dyy, D33 ) Dzz, D12 ) Dxy ) Dyx, D13 ) Dxz ) Dzx, D23 ) Dyz ) Dzy. If diffusion maps are not under consideration, the minimum number of scans needed for the calculation of six diffusion tensor elements is seven. One additional experiment has to be run without the weighting gradient for the calculation of spin density maps M(0). Once we operate on diffusion maps, we need to perform at least six scans in noncolinear gradient directions gˆ. The most basic set of gradient vectors is given by

The spin echo amplitude M as a function of b-values is given by8

ln

T

ln

( )

ln

(A.8) (A.9) (A.10)

Mxy 1 1 1 ) -bxygˆTxyDefgˆxy ) -bxy Dxx + Dyy + Dxy M(0) 2 2 2 (A.11)

(

)


( )

(

( )

(

ln

ln


Mxz 1 1 1 ) -bxzgˆTxzDefgˆxz ) -bxz Dxx + Dzz + Dxz M(0) 2 2 2 (A.12)

)

Myz 1 1 1 ) -byzgˆTyzDefgˆyz ) -byz Dyy + Dzz + Dyz M(0) 2 2 2 (A.13)

)

The solution yields in the elements of the effective diffusion tensor Deff. Then, if we substitute the left sides of these equations by apparent diffusion maps (ADCs), we obtain relations between diffusion maps ADCij and diffusion tensor elements Dij

ADCxx ) Dxx

(A.14)

ADCyy ) Dyy

(A.15)

ADCzz ) Dzz

(A.16)

1 1 ADCxy ) Dxx + Dyy + Dxy 2 2 1 1 ADCxz ) Dxx + Dzz + Dxz 2 2 1 1 ADCyz ) Dyy + Dzz + Dyz 2 2

(A.17) (A.18) (A.19)

It is important to note that for a different set of gradient vectors we get different relations. Gradient pulses applied in the main directions affect only the diagonal elements of the diffusion tensor Def (eqs A.14-A.16). Any other combination of gradient pulses, like two or three simultaneous gradient directions (eqs A.17-A.19), affects off-diagonal elements of the diffusion tensor. Since the diffusion tensor is symmetric, it can be diagonalized giving both eigenvalues λ and eigenvectors e of diffusion tensor.

References and Notes (1) Fo¨rster, S.; Konrad, M. J. Mater. Chem. 2003, 13, 2671. (2) Lauw, Y. J. Colloid Interface Sci. 2009, 332, 491. (3) Lu, G.; Zhang, X.; Cai, X.; Jiang, J. J. Mater. Chem. 2009. (4) Whitesides, G. M.; Grzybowski, B. Science 2002, 295, 2418. (5) Boiko, O.; Nastishin, Y. A.; Vasyuta, R.; Nazarenko, V. Lyotropic chromonic liquid crystals with negative biregringence for display applications. Proceedings of the 14th International Symposium: AdVanced Display Technologies, 2006. (6) Basser, P. J.; Pierpaoli, C. Magn. Res. Med. 1996, 111 (3), 209. (7) Basser, P. J.; Mattiello, J.; Lebihan, D. Biophys. J. 1994, 66 (1), 259. (8) Basser, P. J.; Mattiello, J.; Lebihan, D. J. Magn. Res. B 1994, 103 (3), 247. (9) Shimony, J. S.; McKinstry, R. C.; Akbudak, E.; Aronovitz, J. A.; Snyder, A. Z.; Lori, N. F.; Cull, T. S.; Conturo, T. E. Radiology 1999, 212, 770. (10) Le Bihan, D.; Mangin, J. F.; Poupon, C.; Clark, C. A.; Pappata, S.; Molko, N.; Chabriat, H. J. Magn. Res. Imaging 2001, 13, 534. (11) Masutani, Y.; Aoki, S.; Abe, O.; Hayashi, N.; Otomo, K. Eur. J. Radiol. 2003, 46, 53. (12) Westin, C. F.; Maier, S. E.; Mamata, H.; Nabavi, A.; Jolesz, F. A.; Kikinis, R. Med. Image Anal. 2002, 6, 93. (13) Gault, J. D.; Kavanagh, E.; Rodrigues, L. A.; Gallardo, H. J. Phys. Chem. 1986, 90, 1860. (14) Gault, J. D.; Leite, M. A.; Rizzatti, M. R.; Gallardo, H. J. Colloid Interface Sci. 1988, 122, 587. (15) Kertes, A. S. J. Inorg. Nucl. Chem. 1965, 27, 209. (16) Szutkowski, K.; Stilbs, P.; Jurga, S. J. Phys. Chem. C 2007, 111, 15613. (17) Stilbs, P. J. Colloid Interface Sci. 1982, 87, 385. (18) Yang, X.; Tony, F.; Nancy, B.-W.; George, L. J. Magn. Reson. Imaging 1997, 7, 887. (19) Fojud, Z.; Szczesniak, E.; Jurga, S.; Stapf, S.; Kimmich, R. Solid State Nucl. Magn. Reson. 2004, 25, 200. (20) Wachowicz, M.; Jurga, S.; Vilfan, M. Phys. ReV. E 2004, 70, 031701. (21) Callaghan, P. T.; Soderman, O. J. Phys. Chem. 1983, 87, 1737.

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Long-Range Ordering in the Lyotropic Lamellar Phase Studied by

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