DOSY: Diffusion-Ordered NMR

Introduction

When a sample is a mixture, a proton spectrum becomes a superposition of several compounds, and deciding which peaks belong together can be very difficult. DOSY adds a second dimension that is not a frequency at all but a physical property: how fast each molecule moves through the solution. Large molecules diffuse slowly and small ones quickly, so every signal from one compound lines up at the same diffusion coefficient. DOSY has been called "NMR chromatography", because it separates components without physically separating them.

Core explanation

DOSY relies on pulsed field gradients . A gradient pulse makes the magnetic field vary linearly along the sample tube's z axis for a brief period δ, typically 1–5 ms. Nuclei at different heights then precess at slightly different frequencies and acquire a phase that depends on their position. This is the encoding step: each molecule's magnetisation is labelled with a helix of phase along the tube.

After a diffusion delay Δ, commonly 50–200 ms, an equal gradient is applied with the opposite effect, which unwinds the phase helix. If a molecule has not moved, its phase returns exactly to its starting value and it contributes fully to the signal. If it has diffused to a new height during Δ, the second gradient does not exactly cancel the first, leaving a residual phase. Averaged over many molecules moving randomly, these residual phases partially cancel, and the signal is attenuated. The faster the molecules diffuse, the greater the attenuation.

The attenuation follows the Stejskal–Tanner equation :

I = I₀ exp[−D γ² g² δ² (Δ − δ/3)]

where I₀ is the intensity with no gradient, γ is the magnetogyric ratio, g is the gradient strength, and D the diffusion coefficient. The experiment is repeated with typically 16–32 gradient strengths, increasing g stepwise. Each peak decays in a Gaussian-like fashion with g, and fitting its decay yields D. The processed result is plotted with chemical shift on the horizontal axis and D (usually on a logarithmic scale) on the vertical axis. All peaks of one compound fall on a horizontal line at their common D.

The Stokes–Einstein equation , D = kT ÷ (6πηr), links D to the hydrodynamic radius r of a roughly spherical molecule, the temperature T and the solvent viscosity η. Typical values in water at 25 °C are about 2.3 × 10⁻⁹ m² s⁻¹ for water itself, around 5 × 10⁻¹⁰ m² s⁻¹ for a sugar such as sucrose, and near 1 × 10⁻¹⁰ m² s⁻¹ for a small protein. Because D scales only approximately with the inverse cube root of mass, doubling mass reduces D by about 20%. Resolving two components therefore generally needs a difference in D of at least roughly 10–30%, depending on signal-to-noise and overlap.

Practical DOSY uses stimulated-echo sequences with bipolar gradients to reduce T₂ losses and eddy currents. Two hazards are important. Convection , caused by small temperature gradients in the tube, moves molecules systematically and makes D appear too large; convection-compensated sequences or narrower tubes reduce it. Overlapping peaks from two compounds decay as a sum of two exponentials, and a simple single-exponential fit gives an intermediate, misleading D that places the peak between the two compounds.

Formulae

I = I₀ exp[−D γ² g² δ² (Δ − δ/3)]; D = kT ÷ (6πηr). Units: D in m² s⁻¹, g in T m⁻¹, δ and Δ in s.

Step-by-step reasoning

1. Record spectra at increasing gradient strength with fixed δ and Δ. 2. Fit each peak's intensity decay to the Stejskal–Tanner equation. 3. Plot each peak at its chemical shift and fitted D. 4. Group peaks lying on one horizontal line as one component. 5. Check suspect intermediate D values for overlap.

Visual explanation

Picture a DOSY plot of a reaction mixture: along the bottom, a crowded proton spectrum; above it, three horizontal rows of spots. The top row, fastest diffusing, holds the solvent; the middle row a small starting material; the lowest row a larger product.

Real-world analogy

Imagine releasing marbles, footballs and beach balls onto a busy dance floor and photographing them a minute later. Small, light objects will have been kicked much further from their start than large ones. Sorting objects by how far they have wandered sorts them by size, without needing to pick any up.

Real-world example

DOSY is used to analyse polymer samples, detect adulterants in herbal products and foods, identify aggregation of drug molecules, and study host–guest binding: a small guest that binds a large host diffuses more slowly, and the change in D gives an estimate of the bound fraction.

Why?

Why does a larger molecule diffuse more slowly? It experiences greater viscous drag from the surrounding solvent. The Stokes–Einstein equation shows D is inversely proportional to hydrodynamic radius.

Common misconception

"DOSY separates molecules by mass." It separates by translational diffusion, which depends on hydrodynamic size and shape, solvent viscosity, temperature and aggregation. Two isomers of similar shape and mass usually cannot be distinguished.

Worked example

Question: Using the Stokes–Einstein equation, estimate the hydrodynamic radius of a molecule with D = 5.0 × 10⁻¹⁰ m² s⁻¹ in water at 298 K (η = 8.9 × 10⁻⁴ Pa s, k = 1.38 × 10⁻²³ J K⁻¹).

Reasoning: r = kT ÷ (6πηD) = (1.38 × 10⁻²³ × 298) ÷ (6π × 8.9 × 10⁻⁴ × 5.0 × 10⁻¹⁰) = 4.11 × 10⁻²¹ ÷ 8.39 × 10⁻¹² ≈ 4.9 × 10⁻¹⁰ m.

Answer: About 0.49 nm, consistent with a disaccharide-sized molecule.

Quick check

1. In a DOSY plot, why do all the peaks from a single compound appear at the same diffusion coefficient? Answer: They belong to one molecule that moves as a unit, so they share one D.

Exam focus

Explain gradient encoding and decoding and why diffusion causes attenuation. Quote the Stejskal–Tanner and Stokes–Einstein equations. Interpret DOSY plots and name convection and overlap as sources of error.

Advanced insight

Signals that decay as the sum of several exponentials can be analysed by multi-exponential fitting or by inverse-Laplace methods, but these are ill-conditioned and sensitive to noise. Adding DOSY encoding to 2D experiments such as HSQC-DOSY spreads overlapping signals first by carbon shift, allowing cleaner diffusion analysis in complex mixtures.

Summary

DOSY uses pulsed field gradients to label nuclei by position and measures how diffusion during Δ attenuates signals. Fitting decays with the Stejskal–Tanner equation gives D for each peak, and a DOSY plot groups signals by component. D reflects hydrodynamic size via the Stokes–Einstein equation. Convection and overlapping peaks distort the analysis.

Practice questions

1. What effect does increasing the gradient strength have on the signal of a fast-diffusing molecule? Answer: It attenuates it strongly, because the residual phase after decoding grows with gradient strength and diffusion. 2. A peak in a mixture appears at a D value between the lines of two components. Suggest why. Answer: It is probably an overlap of signals from both components, and a single-exponential fit gives an average D. 3. How does increasing solvent viscosity affect D? Answer: D decreases, because it is inversely proportional to viscosity in the Stokes–Einstein equation. 4. Why can a small guest molecule show a lower D when a large host is added? Answer: When bound, it moves with the large host, so its averaged diffusion coefficient falls.