Isotropic diffusion weighting using a triple-stimulated echo pulse sequence with bipolar gradient pulse pairs

Microporous and Mesoporous Materials xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Microporous and Mesoporous Materials journal homepage: www.elsevier.com/locate/micromeso

Isotropic diffusion weighting using a triple-stimulated echo pulse sequence with bipolar gradient pulse pairs Daniel Topgaard Division of Physical Chemistry, Department of Chemistry, Lund University, P.O.B. 124, SE-22100 Lund, Sweden

a r t i c l e

i n f o

Article history: Received 3 May 2014 Accepted 12 August 2014 Available online xxxx Keywords: Magnetic resonance Pulsed field gradient q-Vector Eddy current Aerosol-OT

a b s t r a c t Microscopic diffusion anisotropy in porous materials can be quantified from diffusion NMR data acquired with a combination of directional and isotropic diffusion encoding. A drawback with current pulses sequences for isotropic encoding is that they all rely on spin echo sequences, which are only applicable to pore liquids with long transverse relaxation times and porous materials with negligible internal magnetic field gradients. To mitigate these problems, we introduce a pulse sequence based on consecutive stimulated echo blocks with bipolar gradient pulse pairs giving equal diffusion encoding in three successive directions. By varying the angles between these directions, the pulse sequence can be tuned to give either directional or isotropic diffusion encoding. We demonstrate the new pulse sequence by experiments on detergent/water liquid crystals with lamellar, bicontinuous cubic, and reverse 2D hexagonal structures. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Pore space anisotropy can be observed through its effect on the translational diffusion of the pore liquids as measured with nuclear magnetic resonance (NMR). When the porous material is anisotropic on the macroscopic scale, the anisotropy is readily detected by diffusion NMR measurements with diffusion encoding in multiple directions [1,2]. Although the pore space is anisotropic, a wide range of materials exhibit limited diffusion anisotropy on the macroscopic scale if the anisotropic microcrystallites or ‘‘domains’’ lack orientational order. Examples of such materials are lyotropic liquid crystals [3], paper [4], inorganic porous solids [5,6], and brain tissue [7]. We have recently shown that isotropic diffusion weighting in conjunction with conventional directional measurements permits quantification of both the microscopic diffusion anisotropy and the orientational order of the microcrystallites [8]. Several pulse sequences giving isotropic diffusion weighting have been suggested [9–14], all of them relying on lengthy spin echo sequences that are sensitive to the effects of transverse relaxation and internal magnetic field gradients caused by magnetic susceptibility differences between the porous matrix and the pore liquid. Consequently, our previous experimental protocols are of limited use for

E-mail address: [email protected]

studies of inorganic solids such as mesoporous silica or zeolites. In order to extend the range of applicability, we here propose a pulse sequence using stimulated echoes to reduce the influence of transverse relaxation [15], and bipolar gradient pulse pairs to mitigate the effects of internal gradients and eddy currents [16]. The same combination of remedies has previously been used for both conventional directional diffusion weighting [17] and double pulsed field gradient experiments [18]. 2. Directional and isotropic diffusion weighting Anisotropic Gaussian diffusion can be described with the diffusion tensor D having the elements [19]

0

Dxx

B D ¼ @ Dyx Dzx

Dxy

Dxz

1

Dyy

C Dyz A:

Dzy

Dzz

ð1Þ

The off-diagonal elements vanish in the principal axis frame. The isotropic diffusivity Diso is obtained from the diagonal elements according to

Diso ¼ ðDxx þ Dyy þ Dzz Þ=3:

ð2Þ

Diso is independent of the orientation of the principal axis frame with respect to the lab frame.

http://dx.doi.org/10.1016/j.micromeso.2014.08.023 1387-1811/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: D. Topgaard, Micropor. Mesopor. Mater. (2014), http://dx.doi.org/10.1016/j.micromeso.2014.08.023

2

D. Topgaard / Microporous and Mesoporous Materials xxx (2014) xxx–xxx

In diffusion NMR and MRI, the signal is encoded for translational motion by application of a time-dependent magnetic field gradient G(t), from which the dephasing vector q(t) and the diffusion weighting matrix b can be defined through [20,21]

qðtÞ ¼ c

Z

t

Gðt 0 Þdt

0

(a)

z

n2

(b)

y n2

ζ

y

ψ

n1

n3

x

x

ð3Þ

n1

0

and

b¼

Z

n3 tE

T

qðtÞq ðtÞdt:

ð4Þ

0

In Eqs. (3) and (4), c is the magnetogyric ratio and tE is the time of echo formation, i.e. where q(tE) = 0. The effects of RF pulses are included in G(t). The diffusion weighting b, the ‘‘b-value’’, is given by

b¼

Z

tE

qðtÞ2 dt

ð5Þ

Fig. 2. Gradient directions for the pulse sequence in Fig. 1 shown as (a) 3D view and (b) projection onto the xy-plane. The unit vectors n1 (red), n2 (green), and, n3 (blue) are given by Eqs. (10) and (11). In spherical coordinates, the three vectors have the polar angle f and the azimuthal angles w = 0°, 120°, and 240°. Directional and isotropic diffusion weighting is obtained for the angles f = 0° and 54.7°, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0

labeled with index i, encode the signal for diffusion in three consecutive directions ni given by

or, equivalently,

b ¼ bxx þ byy þ bzz :

ð6Þ

The signal intensity I is related to b and D through

I ¼ I0 expðb : DÞ;

ð7Þ

where I0 is the signal when b = 0 and b:D denotes a generalized scalar product defined by

XX b:D¼ bij Dij : i

ð8Þ

j

In general, b:D depends on the orientation of the diffusion tensor with respect to the lab frame. A special case is so-called ‘‘isotropic’’ or ‘‘trace’’ diffusion weighting [9], where the diagonal elements of b are equal and all off-diagonal elements are zero. In this case, Eq. (5) reduces to

I ¼ I0 expðbDiso Þ;

ð9Þ

which is invariant upon rotation of the diffusion tensor.

2

xi

3

2

cos wi sin f

3

6 7 6 7 ni ¼ 4 yi 5 ¼ 4 sin wi sin f 5; zi where

wi ¼

2p ði  1Þ; 3

i ¼ 1; 2; 3:

2

bxx ¼ byy ¼ b

sin f 2

2

3. Pulse sequence

bxy ¼ byx ¼ bxz ¼ bzx ¼ bxz ¼ bzx ¼ 0;

The triple-stimulated echo sequence for isotropic diffusion encoding is shown in Fig. 1. The three stimulated echo blocks,

where

τ2

2τ1

τ2

2τ1

τ2

τ1

RF

τe

Δ

gradient

1

ε

2

3

G

δ/2

τ

Fig. 1. Triple-stimulated echo pulse sequence for isotropic diffusion encoding of pore liquids with short transverse relaxation time. Narrow and broad vertical lines   symbolize 90x and 180y RF pulses, respectively. The sequence comprises three stimulated echo blocks, indicated with numbered braces, and a spin echo block with duration se before signal acquisition. The magnetization is longitudinally  stored during the s2 delays. The 180y pulses are located at the centers of the s1 and se delays. Each stimulated echo block contains two bipolar gradient pulse pairs, with D denoting the time between their leading edges. The magnification shows the shape of one bipolar pulse pair with amplitude G, ramp time e, inter-pulse spacing s, and effective area Gd. Each s2 delay includes a spoiler gradient (not shown). The phases of the first 90° pulse and the receiver are cycled in two steps ±x. The three gradient directions, symbolized with the colors red, green, and blue, are shown in Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

ð11Þ

As shown in Fig. 2, the direction vectors are distributed around the z-axis with threefold symmetry. Evaluation of the b-matrix elements through Eqs. (3) and (4) yields

bzz ¼ b cos f

τ1

ð10Þ

cos f

  d s e e2 e3 b ¼ 3ðcGdÞ2 D     þ 3 2 2 6d 15d2

ð12Þ

ð13Þ

and the variables G, d, D, s, and e are defined in Fig. 1. Isotropic diffusion encoding, i.e. bxx = byy = bzz = b/3, is achieved for the ‘‘magicangle’’ f = acos(1/31/2)  54.7°, while standard directional encoding corresponds to f = 0. In practice, it is beneficial to use a small, but non-zero, value of f also for directional encoding in order to reduce the need for RF and receiver phase cycling. The rationale for selecting the gradient directions in Eqs. (10) and (11) is that the b-value is independent of the value of f as long as the timing parameters and the gradient amplitudes remain constant. Any dependence of the detected signal on the value of f can then be attributed to the presence of microscopic diffusion anisotropy. Assuming exponential relaxation with the longitudinal and transverse relaxation times T1 and T2, the value of I0 is given by

I0 ¼

    1 6s1 þ se 3s2 exp  ; I90 exp  8 T2 T1

ð14Þ

where I90 is the signal after a single 90° pulse and the delays s1, s2, and se are defined in Fig. 1. The factor 1/8 = 1/23 originates from the fact that, for each of the three stimulated blocks, only half of the magnetization is stored in the longitudinal direction.

Please cite this article in press as: D. Topgaard, Micropor. Mesopor. Mater. (2014), http://dx.doi.org/10.1016/j.micromeso.2014.08.023

3

D. Topgaard / Microporous and Mesoporous Materials xxx (2014) xxx–xxx

5. Results and discussion Fig. 3 shows experimental signal attenuation data obtained with the new pulse sequence on a series of lyotropic liquid crystals with different nanoscale structures. For the lamellar and reverse hexagonal liquid crystals, the data acquired with directional diffusion encoding is markedly multi-exponential. This multi-exponentiality results from a combination of the locally anisotropic water diffusion within one microcrystallite, and the distribution of microcrystallite orientations [3]. The cubic liquid crystal gives single-exponential signal attenuation in agreement with the locally isotropic diffusion on the micrometer length scale of diffusional displacement during the ~100 ms observational time scale [25]. For the anisotropic phases, the multi-exponentiality disappears

D⊥

1

I / I0

D|| 0.1

0.01

(b)

1

I / I0

Liquid crystalline samples were prepared from 1H2O (Milli-Q quality), 2H2O (99.8% Armar Chemicals, Switzerland), and sodium 1,4-bis(2-ethylhexoxy)-1,4-dioxobutane-2-sulfonate (analytical grade, Sigma–Aldrich, Sweden) having the trade name AOT or Aerosol-OT. An equal-weight solution of 1H2O and 2H2O was mixed with AOT in 10 mL vials, giving AOT concentrations of 70, 80, and 85 wt.%. The samples were thoroughly mixed by repeated centrifugation with the vial cap alternatingly up or down. For each of the mixed solutions, a volume of 400 lL was transferred to a 5 mm disposable NMR tube, which was subsequently centrifuged to remove air bubbles. The samples were studied with NMR at 80 °C where they according to the equilibrium phase diagram have lamellar (70 wt.%), bicontinuous cubic (80 wt.%), and reverse 2D hexagonal (85 wt.%) structures [22]. NMR experiments were carried out on a Bruker AVII-500 spectrometer equipped with an 11.74 T standard bore magnet and a MIC-5 microimaging probe giving 3 T/m gradients in three orthogonal directions at a current of 60 A. The probe was fitted with a 5 mm 2H/1H RF insert allowing for facile 2H NMR controls of the liquid crystalline phase structure [23] during diffusion measurements with observation of the 1H signal. The microscopic diffusion anisotropy was investigated with the pulse sequence in Fig. 1 using d = 1.2 ms, e = 0.1 ms, s = 0.2 ms, D = 106.6 ms, and s1 = 2 ms, giving 12 ms total time of transverse relaxation during the three stimulated echo blocks. The b-value was incremented in an 8-step linear sequence by varying G up to a maximum value of approximately 0.5 T/m. This maximum value was adjusted to reach a signal attenuation of 0.01 for all samples. Directional and isotropic diffusion weighting was performed with gradient directions according to Eq. (10) using f = 9° and 54.7°, respectively. The non-zero value of f also for the directional encoding reduces the need for RF phase cycling to two steps, with only marginal effect on the observed values of D. In order to assure that the data corresponds to a random distribution of microcrystallite orientations, it was ‘‘powder-averaged’’ by repeating the acquisition and averaging the signal for 39 different orientations of the gradient frame with respect to the lab frame. These directions were distributed according to the electrostatic repulsion scheme [8,24]. The powder averaging is not necessary when using isotropic encoding, but was performed anyway in order to get the same signal-to-noise ratio as for the directional data. While a se value of 2 ms was sufficient to reduce the effects of eddy currents, the values 320 and 20 ms were used for the 80 and 85 wt.% samples, respectively, to attenuate the AOT resonance lines through T2-relaxation and J-couplings. The signal was recorded spectroscopically with 5 kHz spectral width and 1 s acquisition time. After Fourier transformation, phase correction, and baseline correction, the water peak was integrated and used for further analysis.

(a)

0.1

0

5

10

15

0 0.5 1

0

5

10

15

0 0.5 1

0.01

(c)

1

I / I0

4. Experimental

0.1

D⊥

dire

D||

ction

al

0.01 0

5 10 b / 109 sm-2

15

0 0.5 1 D / 10-9 m2s-1

Fig. 3. Experimental results obtained with the triple-stimulated echo pulse sequence in Fig. 1 for water in (a) lamellar, (b) bicontinuous cubic, and (c) reverse 2D hexagonal AOT/water liquid crystals. The left panels show normalized water signal intensity I/I0 (circles) vs. diffusion-weighting b for directional (blue) and isotropic (black) diffusion encoding. The insets are illustrations of the water porespace geometry on the length scale of 10 nm. Lines are least-squares fits of Eqs. (16) and (17) to the experimental data, with the fit results displayed in the right-hand panels as distributions of apparent diffusivities P(D) calculated with Eq. (15). The value of I0 used for normalizing the data is the same for both directional and isotropic encoding. The distributions are scaled to give the same maximum amplitude. The labels Djj and D? indicate the diffusivities parallel and perpendicular, respectively, to the main symmetry axis of the structure.

when applying isotropic diffusion encoding, while the initial slope remains unchanged [13]. For the cubic phase, there is no observable difference between directional and isotropic encoding, verifying that the directional and isotropic versions of the proposed pulse sequence provide the same b-value. From this simple visual inspection of the signal-vs-b data, we conclude that the pulse sequence efficiently removes the effects of microscopic diffusion anisotropy, and thus gives data that is appropriate for further analysis to quantify the microscopic anisotropy [8]. An isotropic distribution of microcrystallite orientations yields a distribution of apparent diffusivities P(D) given by [13]

1 PðDÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; for minðDjj ; D? Þ 6 D 6 maxðDjj ;D? Þ 2 ðD  D? ÞðDjj  D? Þ PðDÞ ¼ 0; otherwise: ð15Þ In Eq. (15), Djj and D? are the apparent diffusion coefficients parallel and perpendicular, respectively, to the main symmetry axis of the crystallite. Laplace transformation of Eq. (15) gives the signal attenuation function [13]

pffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I p ebD? pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi erf bðDjj  D? Þ ; ¼ 2 I0 bðDjj  D? Þ

Please cite this article in press as: D. Topgaard, Micropor. Mesopor. Mater. (2014), http://dx.doi.org/10.1016/j.micromeso.2014.08.023

ð16Þ

4

D. Topgaard / Microporous and Mesoporous Materials xxx (2014) xxx–xxx

which approaches

I ¼e I0

Djj þ2D? b 3

Acknowledgments

ð17Þ

in the limit jDjj  D? j ! 0, as well as when applying isotropic encoding. The directional and isotropic experimental data in Fig. 3 was jointly analyzed by least-squares fitting of Eqs. (16) and (17) using I0, Djj , and D? as adjustable parameters, and the fit results are displayed as plots of the corresponding distributions P(D) calculated with Eq. (15). The distributions were subjected to minor Gaussian smoothing to avoid the singularity at D ¼ D? . As shown in Fig. 3, the anisotropic phases give broad distributions with directional encoding. This broadening is eliminated when applying isotropic encoding. Data similar to Fig. 3 has previously been obtained with spin echo pulse sequences at echo times of approximately 50 and 100 ms [8,13]. Despite serious attempts, we have not been able to obtain artefact-free isotropic diffusion encoding at shorter echo times on the Bruker microimaging system that we have at our disposal. The reason for this failure remains unknown, but is most probably related to the gradient amplifiers not being able to accurately play out the rather complex gradient modulation functions when decreasing their duration and increasing their amplitude. With the herein proposed pulse sequence, we obtain sufficient diffusion weighting at only 14 ms of transverse relaxation, which is short enough to study the 70 wt.% AOT/water lamellar phase with a water T2-value of 5 ms. 6. Conclusions We have proposed and experimentally demonstrated isotropic diffusion weighting with a stimulated echo pulse sequence, thus extending the applicability of our approach for quantifying microscopic diffusion anisotropy to pore liquids experiencing background field gradients or having transverse relaxation times of only a few milliseconds.

The work was financially supported by the Swedish Research Council (Grant Numbers 2009–6794 and 2011–4334). Stefanie Eriksson is acknowledged for skillful preparation of the liquid crystal samples. References [1] E.O. Stejskal, J. Chem. Phys. 43 (1965) 3597–3603. [2] M.E. Moseley, J. Kucharczyk, H.S. Asgari, D. Norman, Magn. Reson. Med. 19 (1991) 321–326. [3] P.T. Callaghan, O. Söderman, J. Phys. Chem. 87 (1983) 1737–1744. [4] D. Topgaard, O. Söderman, J. Phys. Chem. B 106 (2002) 11887–11892. [5] K.J. Packer, F.O. Zelaya, Colloids Surf. 36 (1989) 221–227. [6] F. Stallmach, J. Kärger, C. Krause, M. Jeschke, U. Oberhagemann, J. Am. Chem. Soc. 122 (2000) 9237–9242. [7] M. Lawrenz, J. Finsterbusch, Magn. Reson. Med. 69 (2013) 1072–1082. [8] S. Lasicˇ, F. Szczepankiewicz, S. Eriksson, M. Nilsson, D. Topgaard, Front. Phys. 2 (2014) 11. [9] S. Mori, P.C.M. van Zijl, Magn. Reson. Med. 33 (1995) 41–52. [10] E.C. Wong, R.W. Cox, A.W. Song, Magn. Reson. Med. 34 (1995) 139–143. [11] R.A. de Graaf, K.P.J. Braun, K. Nicolay, Magn. Reson. Med. 45 (2001) 741–748. [12] J. Valette, C. Giraudeau, C. Marchadour, B. Djemai, F. Geffroy, M.A. Ghaly, D. Le Bihan, P. Hantraye, V. Lebon, F. Lethimonnier, Magn. Reson. Med. 68 (2012) 1705–1712. [13] S. Eriksson, S. Lasicˇ, D. Topgaard, J. Magn. Reson. 226 (2013) 13–18. [14] D. Topgaard, Micropor. Mesopor. Mater. 178 (2013) 60–63. [15] J.E. Tanner, J. Chem. Phys. 52 (1970) 2523–2526. [16] R.F. Karlicek Jr., I.J. Lowe, J. Magn. Reson. 37 (1980) 75–91. [17] R.M. Cotts, M.J.R. Hoch, T. Sun, J.T. Marker, J. Magn. Reson. 83 (1989) 252–266. [18] N. Shemesh, Y. Cohen, J. Magn. Reson. 212 (2011) 362–369. [19] P.J. Basser, J. Mattiello, D. Le Bihan, Biophys. J. 66 (1994) 259–267. [20] W.S. Price, NMR Studies of Translational Motion, Cambridge University Press, Cambridge, 2009. [21] P.T. Callaghan, Translational Dynamics & Magnetic Resonance, Oxford University Press, Oxford, 2011. [22] B. Jönsson, B. Lindman, K. Holmberg, B. Kronberg, Surfactants and Polymers in Aqueous Solution, John Wiley & Sons Ltd., 1998. [23] T.M. Ferreira, D. Bernin, D. Topgaard, Annu. Rep. NMR Spectrosc. 79 (2013) 73–127. [24] M. Bak, N.C. Nielsen, J. Magn. Reson. 125 (1997) 132–139. [25] B. Balinov, U. Olsson, O. Söderman, J. Phys. Chem. 95 (1991) 5931–5936.

Please cite this article in press as: D. Topgaard, Micropor. Mesopor. Mater. (2014), http://dx.doi.org/10.1016/j.micromeso.2014.08.023