NMR can be used to measure self-diffusion constants with an accuracy approaching 1% for objects in solution from the size of molecules to micelles. Diffusion rates are measured by NMR using gradient pulses. There are three main methods used for such measurements: Pulsed Gradient Spin Echo (PGSE), Pulsed Field Gradient Stimulated Spin Echo (PFG-SSE) and bipolar pulse longitudinal eddy current delay (BPP-LED)

PGSE (fig. 1) is best suited to spectra consisting only of singlets where the transverse relaxation (T₂) is not much faster than the longitudinal relaxation (T₁).

Fig. 1. Pulse sequence for gradient PGSE

In most cases, T₂ is much shorter than T₁ and PFG-SSE (fig. 2) yields much more sensitivity than PGSE even though half the theoretically available signal is lost. When the spectrum contains multiplets, PGSE severely distorts them so PFG-SSE is a must.

Fig. 2. Pulse sequence for gradient PFG-SSE

For gradient systems such as ours (Bruker DRX 400 with BGU II gradient unit) that lack a B₀ compensation unit, bipolar pulses and eddy current reduction dramatically reduce the required phase cycling and improve the line-shape. Therefore we use BPP-LED (fig. 3). However, for very strong gradient pulses, artifacts such as waviness in the decay curves may occur and a fully phase-cycled PFG-SSE may be required to yield an accurate result.

Fig. 3. Pulse sequence for gradient BPP-LED

In all the experiments, the delays are kept constant and the experiment is repeated many (we usually use 32) times incrementing the gradient strength (fig. 4). A plot made of intensity against gradient strength (fig. 5). The intensity, I, is proportional to where γ is the gyromagnetic ratio (2.675 × 10^-8 rad s^-1 T^-1 for proton), g is the gradient strength, δ and Δ are delays and D is the diffusion constant. The diffusion constant can be extracted either using a non-linear curve fit to the resulting Gaussian decay or by a linear fit to ln(I) versus g² (fig. 6).

Fig. 4. Diffusion spectrum. The peak on the left decays faster and has a higher diffusion constant than the peaks on the right.

Fig. 5. Gaussian fit to diffusion peak intensity using a non-linear fit

Fig. 6. Linear fit to diffusion peak intensity

Often signals overlap or arise from multiple environments. In such cases a bigaussian or polygaussian decay results. This can be analyzed using a non-linear fit and does not lend itself to linear analysis. In fig. 7 the red and green Gaussian curves add to fit the experimental data.

Fig. 7. Bigaussian fit to diffusion peak intensity using a non-linear fit

Diffusion spectra are usually presented as a 2D plot with chemical shift on the horizontal axis and log(Diffusion constant) on the vertical axis (fig. 8). This representation is called a DOSY spectrum. The acquisition dimension is easily analyzed by a Fourier transform yielding high resolution in the frequency domain. However, analysis of the diffusion dimension involves an inversion of the Laplace transform (ILT). While this is quite accurate at up to 2% for a single decay, it has very low resolution when separating two or more overlapping signals with little chance of resolving diffusions of signals that have overlapping frequencies that differ by less than 30–50%.

Fig. 8. DOSY spectrum of Advil in DMSO-d₆ separating iboprofen, sucrose, water and DMSO

For best results care must be taken that the gradient pre-emphasis is correctly adjusted to yield a pulse of the expected shape and that thermal convection is not occurring. If either is the case, the decay curve will look decidedly unGaussian and even wavy. Sensitivity will be lost and the wrong value of the diffusion constant obtained. Convection in a regular 5 mm NMR tube is commonly observed at room temperature for low-viscosity solvents such as acetone and methanol or for other systems at elevated temperatures. Our solution is to use narrower (3 to 4 mm outer diameter, 0.85 to 2 mm inner diameter) regular Pyrex tubing containing the sample inserted into a 5 mm NMR tube. The narrower tube suppresses convection.

The self-diffusion constant is measured in m² s^-1 and is larger for smaller molecules and less viscous solvents. For example, at 25°C the self-diffusion constant is 2.299 × 10^-9 m² s^-1. For the less viscous acetone it is 4.57 × 10^-9 m² s^-1 while for the more viscous and larger octan-1-ol it is 1.4 × 10^-10 m² s^-1. Our equipment (Bruker DRX 400 with BGUII gradients) enables us to diffusion constants in the range 10^-7 to 10^-14 m² s^-1. The molecular size can be estimated from the Stokes-Einstein equation, where r is the van der Waals radius of the molecule in meters, k is the Bolztmann constant (1.380 × 10^-23 J K^-1), T is the temperature in Kelvin, η is the viscosity of the solution in Pascal seconds (Pa s = 1000 centipoises) and D is the self-diffusion constant. For example, the self-diffusion constant of 9,10-diphenylanthracene in THF-d₈ at 25°C is 1.04 × 10^-9 m² s^-1 and the viscosity is 0.501 mPa s. Applying the Stokes equation, the radius is calculated to be 0.42 nm, comparing well with the measured mean van der Waals radius of 0.41 nm.

However, the Stokes-Einstein equation assumes spherical molecules much larger than the solvent molecules. Small molecules diffuse faster than expected while large planar molecules diffuse slower than expected. In the above case, the small size of the molecule relative to THF-d₈ is counteracted by its planarity giving a near perfect result (fig. 9). In ionic solutions, the effective diffusion radius is extended by a solvent shell becoming significantly larger than the van der Waals radius.

Fig. 9. Comparison of the molecular size calculated from the Stokes-Einstein equation with the van der Waals radius

R. E. Hoffman, et al., J. Chem. Soc. Perkin 2, 1998, 1659-1664.

Stokes-Einstein equation can be more successfully applied to larger entities such as micelles that are usually spherical (fig. 10). However, if there is a significant difference in magnetic susceptibility inside and outside the droplets, the magnetic field of the spectrometer will distort the micelles and the diffusion rate will vary with gradient direction.

Fig. 10. Comparison of the emulsion droplet size calculated from the Stokes-Einstein equation with that determined by other methods

J.P.N. Duynhoven, et al., Magn. Reson. Chem., 2002, 40, S51-S59.

Another use for diffusion is the study of phase mobility in complex liquids. For example, an emulsion of oil and water can exist in three states: water in oil (w/o), bicontinuous or oil in water (o/w). A substrate that is hydrophilic (i. e., dissolves better in water) will diffuse much faster in an o/w or bicontinuous emulsion than in a w/o emulsion. Conversely, a substrate that is hydrophobic (i. e., dissolves better in oil) will diffuse much faster in an w/o or bicontinuous emulsion than in a o/w emulsion. A combination of the diffusion constants of both species indicates the state of the emulsion.

Fig. 11. Diffusion constants of substrates in water (circles) and oil (triangles) as a function of dilution used to differentiate three emulsion states: w/o, bicontinuous and o/w

A. Spernath, et al., J. Agric. Food Chem., 2003, 51, 2359-2364.

Diffusion spectra can be combined with any 2D technique in order to separate 2D spectra in the diffusion dimension. Fig. 12 shows an example of short-range ¹H-¹³C correlation combined with DOSY.

Fig. 12. DOSY-HSQC spectrum of a mixture of β-cyclodextrin, isopropyl-β-D-galactopyranoside and ethylbenzene in DMSO-d₆

To see this figure correctly place a red filter of the left eye and a cyan filter over the right eye.

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