Accessing the long-time limit in diffusion NMR: the case of singlet assisted diffusive diffraction q -space

The latest developments in the field of long-lived spin states are merged with pulsed-field gradient techniques to extend the diffusion time beyond what is currently achievable in standard q- space diffusive-diffraction studies. The method uses nearly-equivalent spin-1/2 pairs that let diffusion times of the order of many minutes to be measured allowing access to the long-time limit in cavities of macroscopic size (millimeters). A pulse sequence suitable to exploit this regime has been developed and validated with the use of numerical simulations and experiments.


Introduction
Molecular translational motions such as flow, free and restricted diffusion, convection etc. can be studied with nuclear magnetic resonance techniques. [1][2][3] Generally speaking, all methods rely on preparing some form of spin order (at a given position and time), then following its fate as the molecules undergo the translational motion of interest. Clearly, spin order must survive the duration for which the motion is to be studied and, consequently, all these techniques are limited by the characteristic decay rate of the spin order prepared. The two main pulse sequences in this area are the pulsed gradient spin echo 4 (PGSE) and the pulsed gradient stimulated echo 5 (PGSTE). Both these sequences use pulsed magnetic field gradients to encode and, later on, decode the molecular position. The PGSE method creates transverse spin order whose characteristic decay time is conventionally indicated as T 2 whereas the PGSTE method creates longitudinal spin order whose characteristic decay time is indicated as T 1 . For small molecules in low-viscosity solvents and at high magnetic field strength T 1 ~ T 2 and the PGSE experiment is preferred because it can deliver twice as much signal as the PGSTE. Outside of this condition typically T 2 < T 1 and therefore the PGSTE is preferred.
In the most common practice, the intensity of the signal acquired at the end of these sequences is plotted as a function of the gradient strength or measurement time. The decaying signal can be fit to theoretical or empirical models (see Theory section below) to yield information on the diffusion coefficient, the pore size, porosity and many other structural parameters of the material in which the molecules are imbibed. 6 When dealing with molecules confined in a restricted space, the measured diffusion coefficient appears smaller than when molecules diffuse in an unrestricted environment. 3 In the event of relatively mono-disperse compartment sizes, strong spatial coherence between the diffusing spins causes minima in the intensity of the signal at selected and equally spaced values of the gradient strength. These can be correlated with the characteristic size of the restricting geometry. Such experiments have been termed as diffusive diffraction in analogy with X-ray diffraction. In the NMR field, the diffusive diffraction technique is also known as q-space diffraction for the reasons that will be clear below. Given the ability of q-space diffraction to obtain information on geometry, the technique has been widely used to characterize sizes and shapes in porous media relevant to various disciplines ranging from material sciences 3,[7][8][9][10][11][12][13][14][15][16] to medicine [17][18][19][20] , including a form of NMR-based cytology 21-22 . However, the largest measurable size of pores and compartments is limited to about 50-100 μm due to the "limited" time that longitudinal order survives while molecules diffuse in a medium.
More recently, long-lived spin order was used in NMR diffusion experiments [23][24][25][26][27] and, more relevant to this paper, in diffusive diffraction q-space measurements [28][29] . Thanks to the fact that long-lived spin order has a characteristic decay time, T S , which is typically one order of magnitude bigger than T 1 , these singlet-enhanced q-space experiments have extended the scope of q-space spectroscopy to cavity sizes of up to ~400 μm. 29 However, those singletenhanced diffusion methods are no longer up-to-date with the developments in the field of long-lived spin order and can therefore be improved. The methodology used in those pioneering papers [28][29] requires the use of a spin-locking radiofrequency field turned on for the length of the diffusion period. This constitutes a fundamental limit in the sense that, if the required radiofrequency field is "too strong" and/or the diffusion time "too long" one can incur sample heating (with connected convection issues) and, in the worst case, damage to the probe. Because of this, the diffusion time must be kept within specific hardware limits.
Furthermore, singlet-bearing molecules suitable for these spin-locking methods, have singlet order lifetimes which are longer than T 1 but, typically, not very long in absolute terms. For these reasons, the benefits of using spin-locking singlet diffusing methods instead of more conventional techniques (not based on singlet order) is not spectacular.
In this paper, we consolidate the latest developments in the field of long-lived spin order and merge them with diffusion NMR standard techniques to obtain a method that is able to exploit diffusion times of the order of tens of minutes, and possibly much longer depending on the molecular spy. We demonstrate the use of this technique in the context of q-space diffraction to measure compartment sizes of the order of a few millimeters.

Theory
When molecules diffuse in a medium, the total probability density that a particle is found at position r at time t is given by 3 : (1) with being the particle density and the conditional probability density that a particle is found at position r, at time t after having been at position at time .
The conditional probability density above fulfills Fick's second law: (2) with D indicating the diffusion coefficient. Eq. 2 can be solved analytically only in a few regimes. In the case of freely-diffusing particles in one dimension, the solution can be derived using the method of the Fourier transform to obtain: If diffusion happens in a restricted geometry, then the solution of Eq. 2 becomes more complicated. However, for a few simple geometries an analytical solution is available. One such example, relevant to this work, is the case of molecules diffusing between perfectly reflecting walls separated by a distance l. In this case (still in one dimension for the sake of simplicity), the conditional probability of Eq. 2 assumes the form 30 : The most basic NMR diffusion pulse sequence (PGSE) is shown in Fig. 1a. If one assumes that the duration of the gradient is negligible (i.e. δ ≪ Δ, narrow-pulse approximation) then the effect of the field gradient is to impart a phase shift to spins located at position .
After the application of both gradients, and neglecting spin relaxation, the intensity of the signal at the echo point is given by: Attenuation of the signal in Eq. 5 is the result of the net phase shift that a spin accumulates when it reaches position z at the time of the second field gradient pulse (at t = ∆) having been at position at the time of the first field gradient pulse (t = 0). The central quantity: has the dimension of m -1 and gives the name to the q-space technique.
Assuming that molecules are equally distributed in space at time t = 0, the NMR signal at the echo point of a PGSE experiment (for freely diffusing molecules in one dimension) is found by inserting Eq. 3 into Eq. 5 to obtain: A plot of (f indicates free diffusion) versus q and for ∆ = 10 s and D = 2.0 × 10 -9 m 2 s -1 is shown in Fig. 2 (grey line). In the case of diffusion between parallel planes, where the propagator is given by Eq. 4, the signal at the echo point of a PGSE experiment has the analytical form: Note the features (diffraction minima) that appear at q × l = k (with the integer k = 1, 2, …).
The position of these minima gives direct information about the geometrical parameters of the space the molecules are diffusing within (in this simple case the wall distance l). A plot of ( indicates diffusion restricted by parallel planes) versus q and for ∆ = 10 s, D = 2.0 × 10 -9 m 2 s -1 and l = 100 μm is also shown in Fig. 2 (black line). Two limiting behaviours are worth noticing: i) the short time-limit (short), defined by the condition , for which Eq. 8 reduces to: (9) Note that this is the same as the expression obtained in the free diffusion case (see Eq. 7); and ii) the long-time limit (long), defined by the condition , for which Eq. 8 reduces to: In this regime the NMR signal becomes independent of both the diffusion time Δ and the diffusion coefficient D. Similar equations, derived for the more general case in which the walls are not perfectly reflecting (i.e. including a term to account for spin relaxation occurring at the walls), are available in literature [31][32][33] .
Spin relaxation during the experiment (transverse for PGSE and longitudinal for PGSTE) also contributes to signal decay and poses an upper limit to Δ, which in turn limits the largest value of l measurable via a PGSE or PGSTE technique. Consequently, the long- S (q,Δ) time diffusion limit cannot be reached in geometries whose characteristic sizes are above a threshold that is defined by the absolute value of T 1 . This limit is of the order of 100 μm.

The PGM2S2M pulse sequence
In order to access the long-time limit for geometries whose characteristic dimensions exceed this limit we have developed, implemented and tested a pulsed-gradient magnetisation-to-singlet-to-magnetisation pulse sequences (PGM2S2M) that uses long-lived singlet spin order to store molecules' positional information for a time exceeding T 1 (typically an order of magnitude or more). The PGM2S2M pulse sequence is sketched in Fig. 3.

Numerical Simulations
To validate the proposed methodology, numerical spin-dynamics simulations of the pulse sequences in Fig. 1 and 3 have been run using a custom-made code (available upon request) programmed in Wolfram Mathematica 11. Molecular diffusion was handled using the following strategy: and PGSTE, in the case of unrestricted diffusion ( Fig. 1a and b). These simulations assume N = 5000 molecular positions distributed along the z-direction and within parallel walls separated by l = 0.5 mm. The diffusion coefficient was assumed D = 1.0 × 10 -9 m 2 s -1 , δ = 0.5 ms, τ = 5 ms and Δ = 1 s, to mimic unrestricted diffusion conditions. The value of g was arrayed in 16 equally spaced steps between 0 and 1.5 T m -1 . The simulated system is made by two spin-1/2 nuclei with a difference in chemical shift frequencies and a scalar coupling , chosen to mimic the experimental case discussed below (see Scheme 1). Fig. 1 and 3 done using custom made routines that use the SpinDynamica package 38  The simulated curves have the expected shape and were fitted to retrieve the correct diffusion coefficient. The reduction in intensity by half when using PGSTE instead of PGSE (compare Fig. 1a and b) is well-known and due to the fact that only the y-component of the transverse magnetisation generated after the very first 90x pulse can be converted into longitudinal magnetisation by a second 90x pulse. 3 The simulation results in Fig. 1c and d compare PGSTE and PGM2S2M in the case of restricted diffusion. These simulations use the same parameters as Fig. 1a and b but with Δ set to 300 s to allow molecules to reach, and be reflected by, the walls (relaxation at the walls was neglected for the sake of simplicity). The simulated curves display diffraction minima at q = 2000 and 4000 m -1 consistent with the theory. The intensity of the first point at q = 0 m -1 (i.e. g = 0 T m -1 ) in Fig. 1d is very close to 2/3 (slightly lower due to rounding the values of n 1 to the nearest even integer), which is the maximum theoretical amplitude transfer for the transformation of magnetization-to-singlet-to- This is evident in the experiments discussed below.

Samples and methods
To test the capabilities of the proposed method, singlet-enhanced q-space diffraction NMR experiments were used to measure the shortest dimension (defined as the depth below) of three different rectangular capillary tubes. The capillaries, sized A) 50×8×0.4 mm, B) 50×8×0.8 mm and C) 48×4×2 mm (L×W×D, internal dimensions, 10% precision), were purchased from Wale Apparatus (Hellertown, US) in the case of A and B and from CM Scientific (Silsden, UK) in the case of C. Each of these three capillaries has been fused to the J-Young valve of a 10 mm OD gas-tight LPV NMR tube for support and to allow degassing.
Each of these three resulting (capillary) tubes has been filled with a solution 0.9 M of 1-(ethyl-d5),4-(propyl-d7)(Z)-but-2-enedioate (Scheme 1, a custom-made singlet-bearing molecule that possesses a long singlet-order lifetime [39][40] and that is easily available in our laboratory) in deuterated methanol for the tube carrying capillary A and in deuterated acetone for those carrying capillaries B and C. The solutions were degassed through a three step pump-thaw cycle in order to remove dissolved oxygen.

Scheme 1: Molecular scheme of 1-(ethyl-d5) 4-(propyl-d7)(Z)-but-2-enedioate used in all experiments discussed in this paper
All experiments were run on a Bruker 11.7 T Avance III NMR instrument equipped with a 10 mm 1 H/ 13 C resonator and a gradient system able to deliver pulsed field gradients of up to 1.5 T m -1 . The capillaries were placed in the NMR probe and their alignment adjusted until their width (see specifications above) was perpendicular to the y-axis gradient as determined by the width of the y-profile obtained by running a 1D imaging experiment with field gradient pulses on that axis. All experiments have been run at room temperature (21 ºC) and without using the probe's temperature controller to achieve a more sample temperature and minimize convection flow.

Experiments
Sample A was used to optimize the basic parameters in the PGM2S2M sequence; these were found to be: τ = 20.8 ms, n 1 = 16 and n 2 = 8. A standard saturation recovery experiment measured a T 1 decay constant of 15.5 ± 0.1 s. The singlet order decay constant was measured using the M2S2M method 37 and was found to be T S = 250 ± 10 s. The unrestricted diffusion coefficient was measured using PGSTE (with bipolar gradients, Δ = 200 ms and δ = 3 ms) and was found to be D 0 = 7.7 × 10 -10 m 2 s -1 .
Sample A was then used to test the capabilities of the PGM2S2M pulse sequence ( compartments of such sizes and above (see Fig. 5a and b). Conversely, singlet-enhanced qspace experiments, obtained with the use of the PGM2S2M pulse sequence, can successfully detect the diffraction minima occurring at q × l = integer (only the first 2 shown, see Fig. 5c and d) as expected for the case of molecular diffusion restricted between parallel walls separated by l. In order to probe the new limits of the technique, now limited by T S rather than T 1 , we ran the PGM2S2M pulse sequence to measure the intensity of the NMR signal as a function of q for the rectangular capillaries B and C whose characteristic dimensions are l = 0.8 ± 0.08 and l = 2.0 ± 0.2 mm (see Fig. 6). The value of ∆ was 7 min for B and 10 min for C. The diffusion coefficient of the compound in Scheme 1 dissolved in acetone-d 6 was found to be  Fig. 6 (gray points). In the same figure the experimental data are fitted to Eq. 10 to yield l = 0.756 ± 0.002 and l = 2.06 ± 0.02 for B and C, respectively ( Fig. 6a and b). These values are well within the capillaries' tolerances given by This methodology is fundamentally limited by the fact that the singlet bearing molecules suited for this method, display lifetimes longer than T 1 but with an absolute magnitude that is often not very large (typically of the order or 10-100 s). As explained above, even in the case of such molecules having significantly longer singlet order lifetimes, hardware limitations will prevent access to significantly longer diffusion times. Conversely, the methodology proposed in this paper can be run at any magnetic field strength and does not require continuous radiofrequency irradiation. Furthermore, molecules suited for the method proposed in this work, often display singlet order lifetimes that are many minutes [42][43] up to hours 44 in length. This is the key to accessing larger compartment sizes.

Conclusions
We have introduced a method which extends the long-time diffusion limit in q-space diffusion diffraction experiments by an order of magnitude to measure compartment sizes of up to 2 mm. This is well beyond the limits of current q-space experiments based on PGSE or PGSTE pulse sequences, and previous q-space measurements that have used singlet-order.
The effect of wall relaxation has been deliberately omitted in this work and is currently under study. Singlet order is immune to internal gradients and, providing one can access singlet order in the presence of such gradients, this technique may be advantageous over traditional methods when working with real systems. Here, susceptibility inhomogeneities do cause large internal gradients that typically compromise, or at least complicate, NMR diffusion measurements. We are working on verifying this hypothesis and, in the case of large susceptibility differences, developing an apparatus to perform these experiments in low magnetic field, where internal gradients are largely minimized.