NMR diffusion-encoding with axial symmetry and variable anisotropy: Distinguishing between prolate and oblate microscopic diffusion tensors with unknown orientation distribution

Abstract

We introduce a nuclear magnetic resonance method for quantifying the shape of axially symmetric microscopic diffusion tensors in terms of a new diffusion anisotropy metric, D_Δ, which has unique values for oblate, spherical, and prolate tensor shapes. The pulse sequence includes a series of equal-amplitude magnetic field gradient pulse pairs, the directions of which are tailored to give an axially symmetric diffusion-encoding tensor b with variable anisotropy b_Δ. Averaging of data acquired for a range of orientations of the symmetry axis of the tensor b renders the method insensitive to the orientation distribution function of the microscopic diffusion tensors. Proof-of-principle experiments are performed on water in polydomain lyotropic liquid crystals with geometries that give rise to microscopic diffusion tensors with oblate, spherical, and prolate shapes. The method could be useful for characterizing the geometry of fluid-filled compartments in porous solids, soft matter, and biological tissues.

I. INTRODUCTION

Nuclear magnetic resonance (NMR) measurement of pore liquid translational diffusion is a powerful non-invasive method to investigate the structure of porous materials.^1,2 From a diffusion NMR perspective, it is fruitful to consider porous materials such as lyotropic liquid crystals,^3,4 paper,⁵ and brain tissue,⁶ as collections of microscopic “domains,” in which the water self-diffusion is approximately Gaussian and quantified by a diffusion tensor.⁷ Unfortunately, characterization of the material is hampered by the fact that the effects of diffusion tensor trace, anisotropy, and orientation are entangled when using conventional diffusion-encoding with the Stejskal-Tanner sequence,⁸ which is ubiquitous in both porous media research and clinical magnetic resonance imaging (MRI).

NMR pulse sequences employing diffusion-encoding in two sequential directions have been devised as a means of detecting microscopic diffusion anisotropy in materials that are isotropic on the macroscopic level due to an isotropic distribution of domain orientations.^9–14 Inspired by these older methods and the classical magic-angle spinning technique from solid-state NMR spectroscopy,¹⁵ we have recently introduced methods for detecting microscopic anisotropy using pulse sequences that can be switched between directional and isotropic diffusion-encoding.^16–18 Joint analysis of the directional and isotropic data yields quantitative estimates of both the microscopic anisotropy and the orientational order of the domains.^17,19 In common with other approaches for quantifying microscopic anisotropy,^20,21 our previous version relies on a single rotationally invariant metric that does not distinguish between prolate and oblate shapes of the underlying microscopic diffusion tensors. Since the tensor shape is directly related to the geometry of the compartments in which the studied pore liquid resides, measurements of the shape could be useful for distinguishing between planar and cylindrical pore geometries in materials ranging from liquid crystals³ and porous solids^22,23 to brain tumors.^24,25

Here, we suggest axially symmetric diffusion-encoding with varying anisotropy as a method to quantify the trace and shape of the diffusion tensors of an ensemble of identical microscopic domains having an unknown orientation distribution. Using a terminology based on one of the conventions for describing the chemical shift tensor in solid-state NMR,²⁶ we parameterize the diffusion-encoding tensor b with its trace b and anisotropy b_Δ, and, analogously, the diffusion tensor D with its isotropic value D_iso and anisotropy D_Δ. Positive and negative values of the anisotropies correspond to, respectively, prolate and oblate tensor shapes. We derive an analytical expression for the powder-averaged¹⁷ signal intensity as a function of the experimental variables (b, b_Δ) and the sought-for parameters (D_iso, D_Δ). This expression is a convenient fitting function when estimating (D_iso, D_Δ) from experimental data acquired as a function of (b, b_Δ). The experimental protocol is a modification of our recently introduced triple-stimulated echo pulse sequence for isotropic diffusion-encoding.¹⁸ We demonstrate the method by experiments on water in a series of polydomain lyotropic liquid crystals²⁷ with known geometries giving prolate, spherical, and oblate microscopic diffusion tensors.

II. THEORETICAL CONSIDERATIONS

A. Parameterization of the diffusion tensor

The directionality of a Gaussian diffusion process is captured in the diffusion tensor D.⁷ In its principal axis system (PAS), the tensor is diagonal with the elements D_xx^PAS, D_yy^PAS, and D_zz^PAS. In the context of this paper, it is convenient to parameterize the diffusion tensor with its isotropic value D_iso, anisotropy D_Δ, and asymmetry D_η, using a formalism reminiscent of the “Haeberlen notation” for the chemical shift tensor in solid-state NMR:^26,28

$${\displaystyle \begin{array}{l}{D}_{\mathrm{iso}}=\left(D_{\mathit{xx}}^{\mathrm{PAS}}+D_{\mathit{yy}}^{\mathrm{PAS}}+D_{zz}^{\mathrm{PAS}}\right)/3,\hfill \\ D_{\mathrm{\Delta }}=\frac{1}{3D_{\mathrm{iso}}}\left(D_{zz}^{\mathrm{PAS}}-\frac{D_{\mathit{yy}}^{\mathrm{PAS}}+D_{\mathit{xx}}^{\mathrm{PAS}}}{2}\right),\hfill \\ D_{\eta }=\frac{D_{\mathit{yy}}^{\mathrm{PAS}}-D_{\mathit{xx}}^{\mathrm{PAS}}}{2D_{\mathrm{iso}}{D}_{\mathrm{\Delta }}}.\hfill \end{array}}$$ (1)

The elements are ordered according to the convention |D_zz^PAS − D_iso| > |D_yy^PAS − D_iso| > |D_xx^PAS − D_iso|. The numerical factors in Eq. (1) are selected to get parameters in the ranges −1/2 ≤ D_Δ ≤ 1 and 0 ≤ D_η < 1. When D_iso, D_Δ, and D_η are given, the PAS elements of the diffusion tensor can be calculated from

$${\displaystyle \begin{array}{l}{D}_{\mathit{xx}}^{\mathrm{PAS}}=D_{\mathrm{iso}}\left[1-D_{\mathrm{\Delta }}\left(1+D_{\eta }\right)\right],\hfill \\ D_{\mathit{yy}}^{\mathrm{PAS}}=D_{\mathrm{iso}}\left[1-D_{\mathrm{\Delta }}\left(1-D_{\eta }\right)\right],\hfill \\ D_{zz}^{\mathrm{PAS}}=D_{\mathrm{iso}}\left(1+2D_{\mathrm{\Delta }}\right).\hfill \end{array}}$$ (2)

The tensor is axially symmetric when D_η = 0. As illustrated in Fig. 1, positive and negative values of D_Δ correspond to prolate and oblate tensor shapes, respectively. The convention for ordering the elements assures that the z-axis is the main axis of symmetry for both prolate and oblate tensors. In the axially symmetric case, it is convenient to define the radial and axial diffusivities, D_⊥ and D_||, through

$${\displaystyle \begin{array}{l}{D}_{\perp }=D_{\mathit{xx}}^{\mathrm{PAS}}=D_{\mathit{yy}}^{\mathrm{PAS}}=D_{\mathrm{iso}}\left(1-D_{\mathrm{\Delta }}\right),\hfill \\ D_{\left|\right|}=D_{zz}^{\mathrm{PAS}}=D_{\mathrm{iso}}\left(1+2D_{\mathrm{\Delta }}\right).\hfill \end{array}}$$ (3)

FIG. 1.

Microscopic fractional anisotropy μFA vs. diffusion tensor anisotropy D_Δ for axially symmetric diffusion tensors. μFA and D_Δ are defined from the diffusion tensor eigenvalues in Eqs. (6) and (1), respectively, and the line is calculated with Eq. (8). Diffusion tensors are visualized as superquadric tensor glyphs⁵⁸ for the D_Δ-values labeled on the abscissa.

B. Ambiguity of previous microscopic diffusion anisotropy measures

Lasič et al.^17,29 and Jespersen et al.²¹ have independently introduced the measures μFA (microscopic fractional anisotropy) and FE (fractional eccentricity), respectively, as microscopic equivalents of the conventional FA (fractional anisotropy) parameter³⁰ in diffusion tensor imaging. When the microscopic tensors are perfectly aligned, the values of μFA, FE, and FA are equal, covering the range from 0 to 1. The MA (microscopic anisotropy) index of Lawrenz et al.²⁰ covers the same range but has a less straightforward relation to FA and is not considered further here. Orientational disorder leads to a reduction of FA according to¹⁷

$${\displaystyle \text{FA}=\text{OP}{\left[\mathrm{\mu }{\text{FA}}^{-2}+\frac{2}{3}\left({\text{OP}}^2-1\right)\right]}^{-1/2},}$$ (4)

where OP is the orientational order parameter defined by^31,32

$${\displaystyle \text{OP}=〈P_2\left(cos\theta \right)〉.}$$ (5)

In Eq. (5), P₂(x) = (3x² − 1)/2 is the second Legendre polynomial, θ is the angle between the main symmetry axes of the orientation distribution function and the microscopic diffusion tensor, and the angular brackets denote an ensemble average over the tensors in the investigated region of space.

A relation between the new and old metrics can be obtained by considering the definition of μFA from the diffusion tensor eigenvalues,¹⁷

$${\displaystyle \mathrm{\mu }\text{FA}=\sqrt{\frac{3}{2}}\sqrt{\frac{{\left(D_{\mathit{xx}}^{\mathrm{PAS}}-D_{\mathrm{iso}}\right)}^2+{\left(D_{\mathit{yy}}^{\mathrm{PAS}}-D_{\mathrm{iso}}\right)}^2+{\left(D_{zz}^{\mathrm{PAS}}-D_{\mathrm{iso}}\right)}^2}{{\left(D_{\mathit{xx}}^{\mathrm{PAS}}\right)}^2+{\left(D_{\mathit{yy}}^{\mathrm{PAS}}\right)}^2+{\left(D_{zz}^{\mathrm{PAS}}\right)}^2}}.}$$ (6)

Inserting the relations in Eq. (2) into Eq. (6) yields

$${\displaystyle \mathrm{\mu }\text{FA}=\sqrt{\frac{3\left(D_{\eta }^2+3\right)}{3D_{\mathrm{\Delta }}^{-2}+2D_{\eta }^2+6}}.}$$ (7)

For axially symmetric diffusion tensors, D_η = 0, the expression in Eq. (7) is simplified to

$${\displaystyle \mathrm{\mu }\text{FA}=\sqrt{\frac{3}{D_{\mathrm{\Delta }}^{-2}+2}}.}$$ (8)

The graph of Eq. (8) in Fig. 1 illustrates the limitations of the previous parameters μFA and FE for determining the diffusion tensor shapes. When μFA > 2^−1/2 ≈ 0.707, the tensor is unambiguously prolate, but when μFA is below this threshold value, the tensor could be either prolate or oblate. Conversely, the new measure D_Δ has a unique value for each of the axially symmetric shapes.

C. Orientational dependence of the diffusion tensor elements

Rotation of the PAS from the lab-frame using the Euler rotation matrices R_z(α), R_y(β), and R_z(γ) gives the following zz-element of the lab-frame diffusion tensor:

$${\displaystyle D_{zz}\left(\alpha ,\beta \right)=D_{zz}^{\mathrm{PAS}}{cos}^2\beta +\frac{D_{\mathit{yy}}^{\mathrm{PAS}}+D_{\mathit{xx}}^{\mathrm{PAS}}}{2}{sin}^2\beta -\frac{D_{\mathit{yy}}^{\mathrm{PAS}}-D_{\mathit{xx}}^{\mathrm{PAS}}}{2}{sin}^2\beta cos2\alpha .}$$ (9)

Using the relations in Eq. (1), this expression can be rearranged to

$${\displaystyle D_{zz}\left(\alpha ,\beta \right)=D_{\mathrm{iso}}\left[1+D_{\mathrm{\Delta }}\left(2P_2\left(cos\beta \right)-D_{\eta }{sin}^2\beta cos2\alpha \right)\right],}$$ (10)

which reduces to

$${\displaystyle D_{zz}\left(\beta \right)=D_{\mathrm{iso}}\left[1+2D_{\mathrm{\Delta }}{P}_2\left(cos\beta \right)\right],}$$ (11)

when D_η = 0. As a consistency check, we note that insertion of Eqs. (1) and (3) into Eq. (11) gives D_zz(0) = D_zz^PAS = D_|| and D_zz(π/2) = (D_xx^PAS + D_yy^PAS)/2 = D_⊥ as expected. Although the value of D_zz, as well as D_xx and D_yy, depend on the Euler angles, the trace of the diffusion tensor, given by

$${\displaystyle \text{tr}\left(\mathbf{D}\right)=D_{\mathit{xx}}+D_{\mathit{yy}}+D_{zz}=3D_{\mathrm{iso}},}$$ (12)

is rotationally invariant.

D. Parameterization of the diffusion-weighting tensor

The NMR signal is encoded with information about translational motion using a time-dependent magnetic field gradient G(t) = [G_x(t) G_y(t) G_z(t)]^T. The dephasing vector q(t) is given by the time integral,^1,2

$${\displaystyle \mathbf{q}\left(t\right)=\gamma \underset{0}{\overset{t}{\int }}\mathbf{G}\left(t^{\prime }\right)d{t}^{\prime },}$$ (13)

where γ is the magnetogyric ratio of the studied nucleus. The signal is recorded at the echo time t_E when the spin magnetization is rephased, i.e., q(t_E) = 0. Assuming anisotropic Gaussian diffusion, the signal amplitude S can be written as

$${\displaystyle S=S_0{e}^{-\mathbf{b}:\mathbf{D}},}$$ (14)

where S₀ is the signal intensity at zero gradient amplitude, and b : D denotes a generalized scalar product defined as

$${\displaystyle \mathbf{b}:\mathbf{D}=\sum _i\sum _j{b}_{ij}{D}_{ij}.}$$ (15)

The diffusion-weighting matrix b is given by

$${\displaystyle \mathbf{b}=\underset{0}{\overset{t_{\mathrm{E}}}{\int }}\mathbf{q}\left(t\right){\mathbf{q}}^T\left(t\right)dt.}$$ (16)

In analogy with Eq. (1), the b-matrix can be parameterized with its trace b, anisotropy b_Δ, and asymmetry b_η:

$${\displaystyle \begin{array}{l}b=\text{tr}\left(\mathbf{b}\right)=b_{\mathit{xx}}^{\mathrm{PAS}}+b_{\mathit{yy}}^{\mathrm{PAS}}+b_{zz}^{\mathrm{PAS}},\hfill \\ b_{\mathrm{\Delta }}=\frac{1}{b}\left(b_{zz}^{\mathrm{PAS}}-\frac{b_{\mathit{yy}}^{\mathrm{PAS}}+b_{\mathit{xx}}^{\mathrm{PAS}}}{2}\right),\hfill \\ b_{\eta }=\frac{3}{2}\frac{b_{\mathit{yy}}^{\mathrm{PAS}}-b_{\mathit{xx}}^{\mathrm{PAS}}}{b{b}_{\mathrm{\Delta }}}.\hfill \end{array}}$$ (17)

The diagonal elements are ordered according to |b_zz^PAS − b/3| > |b_yy^PAS − b/3| > |b_xx^PAS − b/3|.

We want to emphasize that the definition of the b-matrix through Eqs. (13)–(16) is well established and can be found in standard text books on diffusion NMR and MRI, see, e.g., Chap. 4.4.1 in Price¹ or Chap. 9.7.2 in Callaghan,² but that our parameterization of the b-matrix using solid-state NMR conventions is novel and, as shown below, simplifies the notation and provides a framework for designing measurement protocols and analysis methods. Since the b-matrix is formed as the outer product of two vectors according to Eq. (16), it is by definition a second order tensor, and we will consequently denote it the b-tensor.

For completeness, we include the equations for calculating the PAS elements of the b-tensor when the trace, anisotropy, and asymmetry are known:

$${\displaystyle \begin{array}{l}{b}_{\mathit{xx}}^{\mathrm{PAS}}=\frac{b}{3}\left[1-b_{\mathrm{\Delta }}\left(1+b_{\eta }\right)\right],\hfill \\ b_{\mathit{yy}}^{\mathrm{PAS}}=\frac{b}{3}\left[1-b_{\mathrm{\Delta }}\left(1-b_{\eta }\right)\right],\hfill \\ b_{zz}^{\mathrm{PAS}}=\frac{b}{3}\left(1+2b_{\mathrm{\Delta }}\right).\hfill \end{array}}$$ (18)

Axially symmetric diffusion-encoding corresponds to b_η = 0, and in analogy with Eq. (3), the radial and axial b-values, b_⊥ and b_||, can be defined as

$${\displaystyle \begin{array}{l}{b}_{\perp }=b_{\mathit{xx}}^{\mathrm{PAS}}=b_{\mathit{yy}}^{\mathrm{PAS}}=\frac{b}{3}\left(1-b_{\mathrm{\Delta }}\right),\hfill \\ b_{\left|\right|}=b_{zz}^{\mathrm{PAS}}=\frac{b}{3}\left(1+2b_{\mathrm{\Delta }}\right).\hfill \end{array}}$$ (19)

Conventional directional diffusion-encoding is performed with b_Δ = 1, giving b_|| = b and b_⊥ = 0, while isotropic encod ing³³ is achieved when b_Δ = 0, resulting in b_|| = b_⊥ = b/3. A third special case is so-called “circular”³⁴ or “planar”³⁵ encoding, with b_Δ = − 1/2, corresponding to b_⊥ = b/2 and b_|| = 0.

E. Pulse sequence for axially symmetric diffusion-encoding

In order to study microscopic diffusion anisotropy in various porous materials, from detergent/water liquid crystals to brain tissue, we have recently proposed a series of pulse sequences that can be switched from isotropic to directional diffusion-encoding without affecting any of the timing variables of the sequence, the magnitude of q(t), or the b-value.^16–18 In the terminology of this paper, the previous pulse sequences allow for changing b_Δ between the discrete values 0 (isotropic) to 1 (directional) while b is constant and b_η = 0. Any of these sequences can be modified to cover the entire range of b_Δ, from −1/2 to 1, but for convenience, we limit the scope to our recently introduced triple-stimulated echo pulse sequence shown in Fig. 2.¹⁸ This sequence has superior performance when studying pore liquids with fast transverse relaxation in porous matrices with appreciable susceptibility-induced internal field gradients, but it is only applicable to hardware capable of producing short and intense gradient pulses. The bipolar gradient pulse pairs mitigate the effects of internal gradients and eddy currents, while the final spin-echo delay τ_e leaves sufficient time for complete settling of the eddy currents before starting the acquisition of an undistorted time-domain signal that gives a high-resolution NMR spectrum after Fourier transformation. Additionally, transverse relaxation editing of the acquired spectrum can be fine-tuned by adjusting the value of τ_e. In the absence of gradient pulses, extensive phase cycling would be required to select the desired coherence pathway. By carefully observing the lineshapes in the acquired ¹H spectra and performing reference experiments on isotropic liquids with known diffusion coefficients, we found that a simple two-step phase cycle, as described in the caption of Fig. 2, is sufficient provided that the three τ₂-delays include spoiler gradients in orthogonal directions, and the three sets of diffusion gradient are noncollinear.

FIG. 2.

Schematic of the triple-stimulated echo sequence with bipolar gradient pulse pairs and spin echo eddy current delay. The decaying sinusoidal symbolizes signal acquisition, while 90°_x and 180°_y RF pulses are shown as narrow and broad vertical lines, respectively. The magnetization is longitudinally stored during the delays with duration τ₂ and is only affected by transverse relaxation during the delays labeled with τ₁ and τ_e. Each stimulated echo block, indicated with colors (red, green, and blue) and numbered braces, contains two bipolar gradient pulse pairs with effective area Gδ and leading-edge separation Δ. The magnification of the first bipolar pulse pair (dashed box) shows the ramp time ε and the delay τ during which the 180°_y RF pulse is executed. The gradient directions for the three stimulated echo blocks are given by Eq. (20) and illustrated in Fig. 3. The trace b and anisotropy b_Δ of the diffusion weighting are given by Eqs. (23) and (24), respectively. A two-step phase cycle of ±x for both the initial 90° pulse and the receiver is sufficient if orthogonal spoiler gradients (not shown) are included during the τ₂ delays, and values of b and b_Δ close to 0 and 1, respectively, are avoided. (Adapted with permission from D. Topgaard, Microporous Mesoporous Mater. 205, 48 (2015). Copyright 2015 by Elsevier.)

Diffusion-encoding is performed in a sequence of three directions n₁, n₂, and n₃ given by¹⁸

$${\displaystyle {\mathbf{n}}_i=\left[\begin{array}{l}{x}_i\hfill \\ y_i\hfill \\ z_i\hfill \end{array}\right]=\left[\begin{array}{c}\hfill cos{\psi }_{i}sin\zeta \hfill \\ \hfill sin{\psi }_{i}sin\zeta \hfill \\ \hfill cos\zeta \hfill \end{array}\right],}$$ (20)

where the azimuthal angles ψ_i are

$${\displaystyle {\psi }_i=\frac{2\pi }{3}\left(i-1\right),i=1,2,3,}$$ (21)

and the inclination ζ is constant. The directions are visualized in Fig. 3 for a series of values of ζ. Insertion of the gradient waveform into Eqs. (13) and (16) gives the b-tensor elements,¹⁸

$${\displaystyle \begin{array}{rcl}\hfill b_{\mathit{xx}}& \hfill =\hfill & b_{\mathit{yy}}=b\frac{1-{cos}^2\zeta }{2},\hfill \\ \hfill b_{zz}& \hfill =\hfill & b{cos}^2\zeta ,\hfill \\ \hfill b_{xy}& \hfill =\hfill & b_{yx}=b_{xz}=b_{zx}=b_{yz}=b_{zy}=0,\hfill \end{array}}$$ (22)

where

$${\displaystyle b=3{\left(\gamma G\delta \right)}^2\left(\mathrm{\Delta }-\frac{\delta }{3}-\frac{\tau }{2}-\frac{\epsilon }{2}-\frac{{\epsilon }^2}{6\delta }+\frac{{\epsilon }^3}{15{\delta }^2}\right),}$$ (23)

and the variables G, τ, δ, ε, and Δ are defined in Fig. 2. Since b_xx = b_yy and all off-diagonal elements are zero, the x, y, z-axes of the frame defined by the gradients can be identified as the PAS of the b-tensor. Inserting the diagonal elements into Eq. (17) yields

$${\displaystyle \begin{array}{l}{b}_{\mathrm{\Delta }}=\frac{3{cos}^2\zeta -1}{2}=P_2\left(cos\zeta \right),\hfill \\ b_{\eta }=0.\hfill \end{array}}$$ (24)

As shown in Fig. 3, the value of b_Δ can be varied through its dependence on the angle ζ; from conventional single-directional diffusion-weighting at ζ = 0°, via isotropic diffusion-weighting at the magic-angle ζ = acos(1/3^1/2) ≈ 54.7°,¹⁶ to planar encoding at ζ = 90°.³⁴ If the variables G, τ, δ, ε, and Δ remain constant, adjustment of ζ only affects b_Δ without influencing the values of b_η or b = tr(b).

FIG. 3.

Axially symmetric diffusion-encoding with variable anisotropy b_Δ. (Top) Gradient unit vectors n₁ (red), n₂ (green), and n₃ (blue) corresponding to the colors and numbered braces of the pulse sequence in Fig. 2. The vectors are located with three-fold symmetry about the z-axis according to Eqs. (20) and (21). The inclination ζ gives the value of b_Δ through Eq. (24). (Middle) checkerboard plot of the b-tensor elements calculated with Eq. (22). The gray scale of element ij is given by b_ij/b. (Bottom) visualization of the b-tensors as superquadrics.

F. Effective diffusion coefficient

In the b-tensor PAS, Eq. (15) can be written as

$${\displaystyle \mathbf{b}:\mathbf{D}=b_{\mathit{xx}}^{\mathrm{PAS}}{D}_{\mathit{xx}}+b_{\mathit{yy}}^{\mathrm{PAS}}{D}_{\mathit{yy}}+b_{zz}^{\mathrm{PAS}}{D}_{zz},}$$ (25)

which yields

$${\displaystyle \mathbf{b}:\mathbf{D}=\frac{b}{3}\left\{\left[1-b_{\mathrm{\Delta }}\left(1+b_{\eta }\right)\right]D_{\mathit{xx}}\right.\left.+\left[1-b_{\mathrm{\Delta }}\left(1-b_{\eta }\right)\right]D_{\mathit{yy}}+\left(1+2b_{\mathrm{\Delta }}\right)D_{zz}\right\}}$$ (26)

upon insertion of Eq. (18). For axially symmetric diffusion-encoding, b_η = 0, the expression is simplified to

$${\displaystyle \mathbf{b}:\mathbf{D}=\frac{b}{3}\left[\left(1-b_{\mathrm{\Delta }}\right)\left(D_{\mathit{xx}}+D_{\mathit{yy}}\right)+\left(1+2b_{\mathrm{\Delta }}\right)D_{zz}\right],}$$ (27)

which can be rewritten as

$${\displaystyle \mathbf{b}:\mathbf{D}=b\left[D_{\mathrm{iso}}+b_{\mathrm{\Delta }}\left(D_{zz}-D_{\mathrm{iso}}\right)\right]}$$ (28)

using Eq. (12). The D_zz element depends on the orientation of the diffusion tensor with respect to the b-tensor PAS according to Eq. (10), in the general case, or Eq. (11) when the diffusion tensor has axial symmetry. Inserting Eq. (11) into Eq. (28) yields

$${\displaystyle \mathbf{b}:\mathbf{D}=b{D}_{\mathrm{iso}}\left[1+2b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}{P}_2\left(cos\beta \right)\right].}$$ (29)

The factors following the b-value can be interpreted as an effective diffusion coefficient D_zz^eff(β, b_Δ) given by

$${\displaystyle D_{zz}^{\mathrm{eff}}\left(\beta ,b_{\mathrm{\Delta }}\right)=D_{\mathrm{iso}}\left[1+2b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}{P}_2\left(cos\beta \right)\right].}$$ (30)

Comparing Eqs. (11) and (30) shows that the effect of the inherent diffusion anisotropy is scaled by the value of b_Δ, which according to Eq. (24) can be tuned using the angle ζ illustrated in Fig. 3. The values of D_zz^eff are in the range between D_zz^eff(0, b_Δ) = D_iso(1 + 2b_ΔD_Δ) and D_zz^eff(π/2, b_Δ) = D_iso(1 − b_ΔD_Δ). When b_Δ = 0, the second term within the brackets in Eq. (30) vanishes and D_zz^eff = D_iso for all orientations of the diffusion tensor.

G. Powder-averaged signal attenuation

Consider a macroscopic sample consisting of an ensemble of randomly oriented anisotropic domains having the same values of D_iso and D_Δ. In cases where there is some preferential alignment of domain orientations, it is possible to mimic the effects of random orientations by “powder-averaging” the data, i.e., recording data for a series of directions of the symmetry axis of the b-tensor and subsequently averaging the results over the various directions.¹⁷

Each domain gives rise to a signal that is obtained by inserting Eq. (29) into Eq. (14),

$${\displaystyle S\left(b,b_{\mathrm{\Delta }}\right)=S_0{e}^{-b{D}_{\mathrm{iso}}\left[1+2b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}{P}_2\left(cos\beta \right)\right]}.}$$ (31)

Integrating the contributions from all the domains gives

$${\displaystyle S\left(b,b_{\mathrm{\Delta }}\right)=S_0{\int }_0^{\pi /2}{P}_{\beta }\left(\beta \right)e^{-b{D}_{\mathrm{iso}}\left[1+2b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}{P}_2\left(cos\beta \right)\right]}\text{d}\beta ,}$$ (32)

where P_β(β) is the angular distribution function, normalized in the interval 0 ≤ β ≤ π/2. A random distribution of domain orientations corresponds to the distribution,³⁶

$${\displaystyle P_{\beta }\left(\beta \right)=sin\beta ,}$$ (33)

which upon evaluation of the integral in Eq. (32) yields

$${\displaystyle S\left(b,b_{\mathrm{\Delta }}\right)=S_0{e}^{-b{D}_{\mathrm{iso}}}\cdot \frac{\sqrt{\pi }}{2}\frac{e^{A/3}}{\sqrt{A}}\text{erf}\left(\sqrt{A}\right),}$$ (34)

where

$${\displaystyle A=3b{D}_{\mathrm{iso}}{b}_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }},}$$ (35)

and erf(x) is the error function. The expression in Eq. (34) is the key result of this paper and the basis for analysis of experimental data acquired with axially symmetric diffusion-encoding, giving estimates of D_iso and D_Δ, or, alternatively, D_|| and D_⊥.

Fig. 4(a) shows graphs of Eq. (34) for a range of values of D_Δ. With isotropic diffusion-encoding, i.e., b_Δ = 0, Eq. (34) reduces to

$${\displaystyle S\left(b,b_{\mathrm{\Delta }}=0\right)=S_0{e}^{-b{D}_{\mathrm{iso}}}.}$$ (36)

Consequently, for isotropic encoding, all values of D_Δ give the same single-exponential decay from which D_iso can be estimated without the confounding effects of anisotropy.¹⁶ When b is held constant at some finite value above approximately 2/D_iso, the magnitude and the sign of D_Δ can be determined from the characteristic variation of S as a function of b_Δ. With increasing values of |D_Δ|, the “edges” at b_Δ = − 0.5 and 1 become more pronounced by gaining amplitude with respect to the “valley” at b_Δ = 0. Prolate (D_Δ > 0) and oblate (D_Δ < 0) diffusion tensors can be distinguished by visual inspection of the S(b, b_Δ) data. As highlighted for the case |D_Δ| = 0.5 in Fig. 4(b), for prolates, the edges at b_Δ = − 0.5 and 1 have similar amplitudes, while for oblates, the edge at b_Δ = − 0.5 is considerably lower than the one at b_Δ = 1. The differences between the prolate and oblate cases can be discerned already at signal attenuation levels S/S₀ around 0.2.

FIG. 4.

Theoretical powder-averaged signal S(b, b_Δ) vs. the trace b and anisotropy b_Δ of the b-tensor. (a) Surface plots calculated with Eq. (34) for a series of diffusion tensor anisotropies D_Δ, defined in Eq. (1), corresponding to the diffusion tensors shown as superquadrics. The absolute value of D_Δ is increasing from left to right, while positive and negative values are in the top and bottom rows, respectively. Colored lines highlight S(b, b_Δ) for the values b_Δ = − 0.5 (red), 0 (blue), and 1 (green), while black lines indicate S(b, b_Δ) for integer values of bD_iso. (b) Selected slices, b_Δ = − 0.5 (red), 0 (blue), and 1 (green), from the surfaces in (a) for D_Δ = − 0.5 (solid) and 0.5 (dashed). The right-hand panel shows the corresponding effective diffusivity distributions P(D_zz^eff, b_Δ) calculated with Eq. (39) and subjected to minor Gaussian broadening. The distributions are scaled to the same maximum amplitude and vertically displaced.

H. Effective diffusivity distribution

Deeper insight into the dependence of S(b, b_Δ) on D_Δ can be obtained by considering S(b, b_Δ) as the Laplace transformation of a distribution of effective diffusivities P(D_zz^eff, b_Δ) according to^5,16

$${\displaystyle S\left(b,b_{\mathrm{\Delta }}\right)=S_0{\int }_0^{\infty }P\left(D_{zz}^{\mathrm{eff}},b_{\mathrm{\Delta }}\right)e^{-b{D}_{zz}^{\mathrm{eff}}}\hspace{0.17em}\text{d}{D}_{zz}^{\mathrm{eff}}.}$$ (37)

The distributions P(D_zz^eff, b_Δ) and P_β(β) are related through³⁶

$${\displaystyle P\left(D_{zz}^{\mathrm{eff}},b_{\mathrm{\Delta }}\right)=P_{\beta }\left(\beta \right)\left|\frac{\text{d}\beta }{\text{d}{D}_{zz}^{\mathrm{eff}}}\right|.}$$ (38)

Inserting P_β(β) from Eq. (33) and evaluating the derivative dβ/dD_zz^eff using Eq. (30) gives

$${\displaystyle P\left(D_{zz}^{\mathrm{eff}},b_{\mathrm{\Delta }}\right)=\frac{1}{2\sqrt{3D_{\mathrm{iso}}{b}_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\left[D_{zz}^{\mathrm{eff}}-D_{\mathrm{iso}}\left(1-b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\right)\right]}}}$$ (39)

in the range

$${\displaystyle min\left\{D_{\mathrm{iso}}\left(1-b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\right),D_{\mathrm{iso}}\left(1+2b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\right)\right\}<D_{zz}^{\mathrm{eff}}<max\left\{D_{\mathrm{iso}}\left(1-b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\right),D_{\mathrm{iso}}\left(1+2b_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\right)\right\},}$$ (40)

and P(D_zz^eff, b_Δ) = 0 otherwise. The distribution has a singularity at D_zz^eff = D_iso(1 − b_ΔD_Δ), corresponding to the domain orientation β = 90°. Equation (39) is analogous to the “powder-pattern” NMR spectrum obtained for an axially symmetric chemical shift anisotropy tensor.³⁷

The mean value of the distribution is D_iso, while the 2nd and 3rd central moments,¹⁷ μ₂ and μ₃, are

$${\displaystyle {\mu }_2=\frac{4}{5}{\left(D_{\mathrm{iso}}{b}_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\right)}^2}$$ (41)

and

$${\displaystyle {\mu }_3=\frac{16}{35}{\left(D_{\mathrm{iso}}{b}_{\mathrm{\Delta }}{D}_{\mathrm{\Delta }}\right)}^3.}$$ (42)

The moments μ₂ and μ₃ are measures of the width and asymmetry, respectively, of the distribution P(D_zz^eff, b_Δ). It is worth noting that the square in Eq. (41) renders μ₂ insensitive to the sign of D_Δ.

Using the cumulant expansion^17,38 of the signal,

$${\displaystyle ln\left[\frac{S\left(b,b_{\mathrm{\Delta }}\right)}{S_0}\right]=-b{D}_{\mathrm{iso}}+\frac{{\mu }_2}{2}{b}^2-\frac{{\mu }_3}{6}{b}^3+\cdots ,}$$ (43)

we see that the moments affect the signal in different ranges of b. When b → 0, the linear term dominates and the signal decay is determined by D_iso. Consequently, all graphs in Fig. 4(a) have the same initial decay rate which is independent of both b_Δ and D_Δ. The influence of diffusion anisotropy is first noticed at higher b where the quadratic term in Eq. (43) gains importance. This deviation from single-exponential decay can be used to estimate the magnitude of D_Δ via its relation to μ₂ according to Eq. (41). The cubic term in Eq. (43) starts to come into play at even higher b, and it is only in this range that the sign of D_Δ has any effect on the signal. These facts are illustrated in Fig. 4(b), showing plots of S(b, b_Δ) and P(D_zz^eff, b_Δ) for D_Δ = ±0.5 and three selected values of b_Δ. The initial decay of S(b, b_Δ) and the mean value of P(D_zz^eff, b_Δ) remain independent of both b_Δ and D_Δ. The initial curvature of logS vs. b and the width of P(D_zz^eff, b_Δ) depends on the magnitude of b_ΔD_Δ. The sign of D_Δ determines the symmetry of P(D_zz^eff, b_Δ), and has a visible influence on S(b, b_Δ) at values of b above approximately 2/D_iso for the case |D_Δ| = 0.5.

The behavior of S(b, b_Δ) at large b can be understood from the low-D_zz^eff limits of P(D_zz^eff, b_Δ). The most slowly decaying exponential component in the S(b, b_Δ) data is given by the lowest value of D_zz^eff that has non-zero amplitude. When D_Δ > 0, both b_Δ = − 0.5 and 1 have the limiting decay rate D_iso(1 − D_Δ). In a graph of logS vs. b, the b_Δ = − 0.5 and 1 lines consequently tend to become parallel as b → ∞. Analogous reasoning for the case where D_Δ < 0 shows that the limiting slopes are D_iso(1 + D_Δ/2) and D_iso(1 + 2D_Δ) for b_Δ = − 0.5 and 1, respectively, resulting in divergence of these two lines in the b → ∞ limit of a plot of logS vs. b.

From the reasoning above, we can formulate the following rule of thumb for distinguishing between prolate and oblate diffusion tensors by visual inspection of S(b, b_Δ) data. If the b_Δ = − 0.5 and 1 slices of the 2D data are approaching the same decay rate in the b → ∞ limit, the diffusion tensor is prolate (D_Δ > 0). Conversely, if these slices have different decay rates in the b → ∞ limit, the diffusion tensor is oblate (D_Δ < 0). Of course, these observations should be confirmed by regressing Eq. (34) onto the data.

III. MATERIALS AND METHODS

Experiments were carried out on hydrotropic liquid crystals of the detergent sodium 1,4-bis(2-ethylhexoxy)-1,4-dioxobutane-2-sulfonate (analytical grade, Sigma-Aldrich, Sweden) with the trade name Aerosol OT or AOT. Based on the equilibrium phase diagram of AOT/water,²⁷ the detergent concentration was chosen to give three different liquid crystalline phases: lamellar (70 wt. %), bicontinuous cubic (80 wt. %), and reverse hexagonal (85 wt. %). In order to allow for NMR observation of both ¹H and ²H nuclei, the water phase was an equal-weight mixture of ¹H₂O (Milli-Q quality) and ²H₂O (99.8% Armar Chemicals, Switzerland). The samples were weighed into 10 ml vials, which were sealed with screw caps and then centrifuged with alternating direction (cap-up/cap-down) until the samples turned homogenous. Subsequently, 400 μ l was transferred to 5 mm disposable NMR tubes.

The structures of the AOT/water liquid crystals were verified by recording small-angle X-ray scattering (SAXS)³⁹ data on samples sealed between two mica windows in a metallic block. The measurements were performed at 21 ± 1 °C with a SAXSLab instrument (JJ X-ray, Denmark), a pinhole-collimated system, equipped with a 100XL+ microfocus sealed X-ray tube (Rigaku, Texas) with Cu Kα radiation of wavelength 1.542 Å. The instrument was equipped with a Pilatus detector (Dectris Ltd., Switzerland), positioned to yield a scattering vector range of 0.027-0.73 Å⁻¹. The lattice spacings a of the liquid crystalline phases were calculated from the scattering vector Q₁ of the first Bragg peak in the SAXS data using the relations a = 2π/Q₁ (lamellar), a = 6^1/2 ⋅ 2π/Q₁ (cubic), and a = (2/3^1/2) ⋅ 2π/Q₁ (hexagonal).

NMR experiments were performed on a Bruker Avance-II 500 spectrometer operating at ¹H and ²H resonance frequencies of 500.13 and 76.77 MHz, respectively. The 11.7 T magnet was equipped with a Bruker MIC-5 microimaging probe giving maximum magnetic field gradients of 3 T/m in three orthogonal directions (Bruker, Germany). The probe was fitted with a 5 mm ²H/¹H RF insert, allowing for independent determination of the liquid crystalline phase structure by recording ²H NMR spectra⁴⁰ using 10 kHz spectral width, 25 ms acquisition time, 8 μ s 90° pulse length, and accumulation of a single transient. The ²H data were processed using the standard spectrometer software TopSpin 2.1 (Bruker, Germany).

The samples were investigated with the triple-stimulated echo sequence in Fig. 2 using ε = 0.1 ms, τ = 0.2 ms, τ₁ = 2.0 ms, δ = 1.2 ms and Δ = 106.6 ms. An 8 × 13 rectangular grid of the (b, b_Δ)-space was sampled by varying G and ζ. The maximum value of G was on the order of 0.5 T/m and adjusted for the different samples to reach approximately the same signal attenuation S/S₀ = 0.01. In order to reduce the need for RF and receiver phase cycling, the values of b_Δ were limited to the range from −0.48 to 0.96, rather than the full range from −0.5 to 1. The data were “powder-averaged” by repeated acquisitions for 39 different orientations of the b-tensor symmetry axis. These directions were chosen according to the electrostatic repulsion scheme.^41,42 An entire data set with 4056 data points, i.e., 8 b-values, 13 b_Δ-values, and 39 directions, was acquired in 10 h using a repetition time of 4 s and accumulation of two transients using the two-step phase cycle given in the caption of Fig. 2.

The time-domain signals were converted to high-resolution spectra through Fourier transformation, automatic phase correction,⁴³ and baseline correction using in-house Matlab scripts based on matNMR.⁴⁴ The AOT resonance lines were sufficiently narrow to be detectable with the triple-stimulated sequence for both the cubic and reverse hexagonal samples. Empirically, it was found that the water and AOT peaks overlap at 25 °C, but that the overlap is rendered insignificant by increasing the temperature to 70 °C. For consistency, all samples were studied at 70 °C. Although a τ_e-value of 2 ms is sufficient for the eddy currents to settle, longer values were used for the reverse hexagonal (20 ms) and cubic (320 ms) samples to reduce the amplitude of the AOT peaks located close to the water line. Using the lowest τ_e-value of 2 ms was essential for studying the lamellar sample, having a water T₂-value of only 5 ms.

The powder-averaged water signal S(b, b_Δ) was analyzed by least-squares fitting of Eq. (34) to the experimental data using the “lsqcurvefit” Matlab routine with three adjustable parameters: the initial signal intensity S₀, the isotropic diffusivity D_iso, and the diffusion tensor anisotropy D_Δ. The fitting procedure was initiated with D_Δ-values of both −0.4 and 0.9, and the best fit was chosen according to a least-squares criterion. The axial and radial diffusivities, D_|| and D_⊥, were calculated from D_iso and D_Δ using Eq. (3). The confidence intervals of the fit parameters were estimated by least-squares analysis of 10⁴ data sets obtained by bootstrap resampling⁴⁵ of the in total 4056 acquired data points.

IV. RESULTS AND DISCUSSION

A. Characterization of the liquid crystalline phases

The phase structure of detergent/water liquid crystals is sensitive to parameters such as water concentration, temperature, and the presence of impurities.²⁷ In order to verify that the AOT/water samples have the expected structures, they were investigated with ²H NMR and SAXS as shown in Fig. 5. While ²H NMR can distinguish between anisotropic and isotropic phases using the presence or absence, respectively, of ²H quadrupolar splittings,⁴⁰ SAXS gives more detailed information about the crystallographic space group and lattice spacings from the positions of the Bragg peaks.³⁹

FIG. 5.

²H NMR spectra and SAXS data for the AOT/water liquid crystalline phases. Characteristic Bragg peaks are labeled in the magnifications of the SAXS data. (a) 70 wt. % AOT: lamellar phase giving ²H quadrupolar splitting and SAXS data with Bragg peaks at the relative positions 1:2. (b) 80 wt. % AOT: cubic phase with Ia3d space group resulting in a ²H singlet and Bragg peaks at 3^1/2:2:7^1/2:8^1/2:10^1/2:11^1/2:12^1/2:13^1/2 relative positions. The peaks marked with a star (*) originate from a reverse hexagonal phase. (c) 85 wt. % AOT: reverse hexagonal phase with ²H quadrupolar splitting and Bragg peaks at the relative positions 1:3^1/2:2. The rightmost column shows schematic illustrations of the water compartment geometries, with arrows indicating the lattice spacing calculated from the SAXS data.

The experimental data are consistent with the literature phase diagram of AOT/water:²⁷ the 70 wt. % sample gives ²H quadrupolar splitting and SAXS Bragg peaks at the relative positions 1:2, verifying an anisotropic phase with lamellar symmetry; the 80 wt. % sample yields a ²H singlet and a multitude of Bragg peaks at positions indicating Ia3d cubic symmetry;⁴⁶ and the 85 wt. % sample shows ²H quadrupolar splitting and Bragg peaks at the relative positions 1:3^1/2:2, consistent with 2D hexagonal symmetry. From previous studies, it is known that the cubic phase is bicontinuous⁴⁶ and that the 2D hexagonal phase is of the reverse type, i.e., with water rods in a continuous detergent matrix.²⁷ The values of the lattice spacing are on the nanometer length-scale as reported in Fig. 5.

The SAXS data for the 80 wt. % sample also contain peaks at 1:3^1/2:2 relative positions, indicating the presence of a reverse hexagonal phase in addition to the cubic one. It should be noted that NMR and SAXS experiments were performed at 70 and 21 °C, respectively, on different samples taken from the same equilibrated detergent/water mixtures, and that both diffusion and ²H NMR experiments were performed on the very same sample under exactly the same conditions in terms of temperature and hardware. Since the ²H data indicate that the NMR sample is completely isotropic, we suggest that the partial conversion of the SAXS sample to reverse hexagonal phase took place when it was smeared onto the mica plate of the sample holder. This part of the SAXS sample preparation is susceptible to water evaporation, leaving areas of higher AOT concentration than in the original mixture.

During the ∼10 ms acquisition time of the ²H NMR experiment, the water molecules diffuse on the order of 10 μ m. Well-defined quadrupolar splittings of the type shown in Figs. 5(a) and 5(c) occur only if the water molecules during the acquisition time do not exchange between crystallites having significantly different orientations, thus indicating that the crystallites are at least tens of micrometers in size. Assuming that the water concentration is uniform and that there is only a single liquid crystalline phase in the sample, the shape of the ²H spectrum reflects the distribution of anisotropic domain orientations with respect to the main magnetic field.^40,47 The spectrum in Fig. 5(a) features a Pake doublet,⁴⁸ which is a signature of randomly oriented domains, while the pronounced “shoulders” of the distorted Pake doublet in Fig. 5(c) show that the water channels in the reverse hexagonal phase have a slight preference for orientations in parallel with the magnetic field.⁴⁹ This partial alignment of the anisotropic domains necessitates the use of powder-averaging when acquiring the diffusion data.

To summarize, the ²H NMR and SAXS characterization of the liquid crystalline phases verify that the NMR samples have the required structures for testing our new approach.

B. Axially symmetric diffusion-encoding applied to liquid crystals

Fig. 6 displays experimental data obtained on the liquid crystalline phases shown in Fig. 5 using the triple-stimulated echo pulse sequence in Fig. 2. Visual inspection of the graphs reveals that the acquired data are consistent with the known microstructures; the anisotropic phases (lamellar and reverse hexagonal) feature a marked dependence of S(b, b_Δ) on the b-tensor anisotropy b_Δ, while there is no such dependence for the isotropic phase (cubic). When b_Δ = 0, all phases give single-exponential signal decays from which the isotropic diffusivity D_iso can be determined according to Eq. (36).

FIG. 6.

Experimental data obtained with the pulse sequence in Fig. 2 for AOT/water liquid crystals of the (a) lamellar, (b) cubic, and (c) reverse hexagonal types as described in Fig. 5. (Left) powder-averaged water signal S(b, b_Δ) vs. the trace b and anisotropy b_Δ of the b-tensor. Circles: experimental data points sampled on a rectangular grid in the (b, b_Δ)-space; lines: fit of Eq. (34) to the experimental data using the initial signal intensity S₀, the isotropic diffusivity D_iso, and the diffusion anisotropy D_Δ as adjustable parameters. Colors emphasize S(b, b_Δ) for the values Δ_b = − 0.48 (red), 0 (blue), and 0.96 (green). (Middle) selected 1D slices along the b-dimension of the 2D S(b, b_Δ) data. Symbols and colors have the same meanings as in the left panel. (Right) effective diffusivity distributions P(D_zz^eff, b_Δ) calculated with Eq. (39) using the values of D_iso and D_Δ obtained from the fit and b_Δ = − 0.48 (red), 0 (blue), and 0.96 (green). The distributions have been subjected to minor Gaussian broadening, scaled to the same maximum amplitude, and vertically displaced.

As explained in Sec. II above, the sign of the diffusion anisotropy D_Δ can be estimated visually by comparing the b_Δ = 0 data with the ones at the positive and negative extremes of b_Δ. In order to reduce the need for phase cycling, the data were acquired in the range of b_Δ from −0.48 to 0.96 rather than from −0.5 to 1. In practice, the visual comparison of the b_Δ = − 0.5 to 1 data described in Sec. II can be applied equally well to the b_Δ = − 0.48 to 0.96 data. From Eq. (40) follows that the prolate diffusion tensor (D_Δ > 0) for the reverse hexagonal phase gives b_Δ = − 0.48 and 0.96 data with the same decay rate D_zz^eff = D_iso(1 − 0.96D_Δ) when b → ∞. As shown in the graph of logS vs. b in Fig. 6(c), this prediction is experimentally confirmed as the b_Δ = − 0.48 and 0.96 data tend to become parallel at high b. Conversely, the oblate diffusion tensor (D_Δ < 0) for the lamellar phase gives the b → ∞ decay rates D_zz^eff = D_iso(1 − 0.48|D_Δ|) and D_iso(1 − 1.92|D_Δ|) for data acquired with b_Δ = − 0.48 and 0.96, respectively, having the result that these two values of b_Δ give noticeably different slopes in the high-b range of the logS vs. b plot in Fig. 6(a).

Regression of Eq. (34) onto the experimental data yields quantitative estimates of D_iso and D_Δ, which can also be expressed in terms of the axial and radial diffusivities, D_|| and D_⊥, using Eq. (3). Table I is a collection of these values, including the 95% confidence intervals as obtained with bootstrapping. The fit results are consistent with the visual inspection of the data and the phase structures described in Fig. 5: the nanometer-thick water sheets of the lamellar phase results in an oblate diffusion tensor with D_Δ = − 0.38; the three-dimensionally interconnected pore space of the bicontinuous cubic phase gives a spherical diffusion tensor with D_Δ = 0.00; and the nanometer-thick water rods in the reverse hexagonal phase yield a prolate diffusion tensor with D_Δ = 0.74.

TABLE I.

Microscopic diffusion anisotropy parameters estimated by regressing Eq. (34) onto the experimental data in Fig. 6. The axial and radial diffusivities, D_|| and D_⊥, are related to the isotropic diffusivity D_iso and diffusion tensor anisotropy D_Δ via Eq. (3). The uncertainties correspond to 95% confidence interval as determined with bootstrapping.

	Diso/10−10 m2 s−1	DΔ	D||/10−10 m2 s−1	D⊥/10−10 m2 s−1
Lamellar	8.74 ± 0.05	−0.38 ± 0.01	2.1 ± 0.2	12.1 ± 0.2
Cubic	5.82 ± 0.02	0.000 ± 0.002	5.82 ± 0.03	5.82 ± 0.02
Reverse hexagonal	2.81 ± 0.02	0.74 ± 0.02	7.0 ± 0.2	0.74 ± 0.05

The ideal lamellar and reverse hexagonal structures in Fig. 5 would give D_Δ-values of −0.5 and 1, respectively. The magnitudes of the experimentally observed values, −0.38 and 0.74, are smaller than the ideal ones, which is often taken as an indication of the presence of defects, “holes,” in the structure.⁴⁰ Such defects are expected when the liquid crystal is close to a phase transition. In the present case, both the lamellar (70 wt. %) and the reverse hexagonal (85 wt. %) phases are close to a transition into the cubic phase that is thermodynamically stable in a narrow concentration interval around 80 wt. %.²⁷

The effective diffusivity distributions P(D_zz^eff, b_Δ) calculated from the fitted and experimental parameters using Eq. (39) give further insight into the observed shapes of S(b, b_Δ) in line with the reasoning above. In particular, the narrow distributions at all values of b_Δ for the cubic phase in Fig. 6(b) correspond to the overlapping single-exponential decays of S(b, b_Δ). For the anisotropic phases in Figs. 6(a) and 6(c), finite values of b_Δ give broader distributions and curvature in the graphs of logS vs. b. For the reverse hexagonal phase in Fig. 6(c), the lowest values of D_zz^eff with non-zero amplitudes are the same when b_Δ = − 0.48 and 0.96, thus giving similar decay rates of S(b, b_Δ) at high b-values.

The narrow confidence intervals reported in Table I result from the large number of acquired data points, 4056 in total and 104 when powder-averaged, in relation to the number of fit parameters, which amounts to only three (S₀, D_iso, and D_Δ). Despite the somewhat exaggerated duration of the experiment, over 10 h, we chose to acquire such a large number of points in order to verify that Eq. (34) describes the data well over the entire 2D (b, b_Δ)-space. In future applications of our approach, the data acquisition protocol could be optimized to yield maximum precision of the fit parameters for given constraints on the available experiment time, signal-to-noise ratio, maximum gradient strength, and nuclear relaxation times.⁵⁰ Such optimization is, however, beyond the scope of this paper.

V. CONCLUSIONS AND OUTLOOK

Axially symmetric diffusion-encoding with varying anisotropy b_Δ permits accurate quantification of the microscopic diffusion tensor anisotropy D_Δ. Our approach for data acquisition and display allows prolate, spherical, and oblate diffusion tensors to be distinguished by simple visual inspection of the data. Quantitative analysis of the experimental data is based on the expression in Eq. (34), which is the key result of this paper. This expression could be modified to include multiple components with different diffusion properties, which could then be resolved in the experimental data if they have significantly different values of the isotropic diffusivity D_iso. Such a separation and correlation of the isotropic and anisotropic features are analogous to the variable-angle spinning⁵¹ and switched-angle spinning⁵² techniques in solid-state NMR spectroscopy.

Although we have here implemented the protocol in a type of pulse sequence suitable for classical porous media research using equipment capable of generating high-amplitude magnetic field gradients, we foresee that it could be adapted also to hardware with limited gradient capabilities, e.g., clinical MRI scanners, by using spin-echo sequences with numerically optimized gradient waveforms.^17,19,53 In favorable cases, prolate and oblate diffusion tensors give visibly different signals already at attenuation levels S/S₀ around 0.2, indicating that the signal-to-noise ratios observed in vivo with analogous pulse sequences¹⁹ could be sufficient for successful clinical implementation of the method. As for a clinical application, our method could differentiate tumors by their average cell shape, either prolate or oblate, which is today not possible if the tumor cells are randomly oriented, and the diffusion is macroscopically isotropic.¹⁹

We have here used powder-averaging to circumvent a lack of knowledge about the orientation distribution function of the anisotropic domains. Once the values of D_iso and D_Δ have been quantified, the directional dependence of the signal could be used to estimate the orientation distribution function by, e.g., spherical deconvolution.⁵⁴

In its current form, the method is ideal for accurate quantification of diffusion anisotropy in, e.g., lyotropic liquid crystals and porous solids without the need for tedious sample preparation procedures to either completely align the microscopic domains³² or to assure that they have an isotropic orientation distribution.^3,11 Consequently, we suggest that our method could be used for investigations of currently debated topics such as the existence of defects in nonionic surfactant lamellar phases,⁵⁵ the presence of “slow” water at distances exceeding hundreds of nanometers from biomembranes,⁵⁶ and the relations between diffusion and orientation heterogeneity of zeolite particles.⁵⁷