^{1}, John Christodoulou

^{1}, Paul D. Barker

^{1}, Christopher M. Dobson

^{1,a)}and Guy Lippens

^{2,b)}

### Abstract

NMRdiffusion experiments employing pulsed field gradients are well established as sensitive probes of the displacement of individual nuclear spins in a sample. Conventionally such measurements are used as a measure of translational diffusion, but here we demonstrate that under certain conditions rotational motion will contribute very significantly to the experimental data. This situation occurs when at least one spatial dimension of the species under study exceeds the root mean square displacement associated with translational diffusion, and leads to anomalously large apparent diffusion coefficients when conventional analytical procedures are employed. We show that in such a situation the effective diffusion coefficient is a function of the duration of the diffusion delay used, and that this dependence provides a means of characterizing the dimensions of the species under investigation.

We thank Giorgio Favrin and Shang-Te Danny Hsu for helpful discussions and the MRC, the Wellcome and Leverhulme Trusts, and the Cambridge Nanoscience Centre for financial support of this work.

I. INTRODUCTION

II. RESULTS

A. Calculation of for rotating systems

1. The effect of restricted diffusion on

B. Derivation of NMR observables from rotational functions

1. Application to a point sphere

2. Application to a rigid rod

C. Calculation of NMRdiffusion data

III. DISCUSSION

IV. METHODS

A. Calculations

B. Monte Carlo simulations

### Key Topics

- Diffusion
- 60.0
- Nuclear magnetic resonance
- 22.0
- Rotation measurement
- 8.0
- Cumulative distribution functions
- 6.0
- Amyloids
- 5.0

## Figures

(Color online) (a) Illustration of how rotational motion contributes to the displacement function of a point at distance from the center of a sphere. (b) When this single rotation is considered in addition to a translational diffusion, it can be seen that it acts to shift the Gaussian translational displacement distribution by a constant distance . (c) for spheres of radii (red), (green), and (blue) for using the explicit form of Eq. (5) given in Appendix A. For each sphere, the distribution in the absence of rotation is shown, together with the distribution obtained in the limit of unrestricted diffusion. Remarkably, the displacement function for the rotating sphere closely resembles that of the sphere. The inset shows the difference between the displacement distribution functions in the static and rotating cases. For spheres of radii , there is no difference between these two cases. For larger spheres, rotational motion promotes an increase in longer displacements at the expense of shorter displacements. (d) Inset: simulated restricted distribution functions for rods of length and radius , calculated as described in Sec. IV. Distribution curves are drawn for from (black) and compared to an unrestricted diffusion simulation (red). These values are fitted to Eq. (3) to obtain an effective radius, and plotted against in the main figure (black points). These values are compared with the theoretical estimate for from Eq. (7) (red) and that for a rod of length (green) which reaches its unrestricted diffusion limit much more rapidly than a rod.

(Color online) (a) Illustration of how rotational motion contributes to the displacement function of a point at distance from the center of a sphere. (b) When this single rotation is considered in addition to a translational diffusion, it can be seen that it acts to shift the Gaussian translational displacement distribution by a constant distance . (c) for spheres of radii (red), (green), and (blue) for using the explicit form of Eq. (5) given in Appendix A. For each sphere, the distribution in the absence of rotation is shown, together with the distribution obtained in the limit of unrestricted diffusion. Remarkably, the displacement function for the rotating sphere closely resembles that of the sphere. The inset shows the difference between the displacement distribution functions in the static and rotating cases. For spheres of radii , there is no difference between these two cases. For larger spheres, rotational motion promotes an increase in longer displacements at the expense of shorter displacements. (d) Inset: simulated restricted distribution functions for rods of length and radius , calculated as described in Sec. IV. Distribution curves are drawn for from (black) and compared to an unrestricted diffusion simulation (red). These values are fitted to Eq. (3) to obtain an effective radius, and plotted against in the main figure (black points). These values are compared with the theoretical estimate for from Eq. (7) (red) and that for a rod of length (green) which reaches its unrestricted diffusion limit much more rapidly than a rod.

(Color online) Rotational diffusion factors for spheres (black surfaces) and rods (white surfaces). (a) The restricted diffusion factor from Eq. (7). When this factor equals unity, rotations are unrestricted and is independent of . For short diffusion delays and larger particles, the factor is less than unity, and rotations are restricted. (b) The behavior of the preexponential factor . When equals unity, the ST equation is being obeyed, and rotational motion will not contribute to NMR diffusion measurements. The surfaces with red lines are calculated in the freely rotating limit, and the surfaces with blue lines are calculated for restricted diffusion, with for both spheres and rods. Increasing causes the surface in the restricted case to tend towards that of the freely rotating limit, and decreasing causes it to tend to the static limit of unity. When , i.e., where species are larger than , rotational motion will significantly contribute to the observed signal attenuation measured in a PFGSE experiment.

(Color online) Rotational diffusion factors for spheres (black surfaces) and rods (white surfaces). (a) The restricted diffusion factor from Eq. (7). When this factor equals unity, rotations are unrestricted and is independent of . For short diffusion delays and larger particles, the factor is less than unity, and rotations are restricted. (b) The behavior of the preexponential factor . When equals unity, the ST equation is being obeyed, and rotational motion will not contribute to NMR diffusion measurements. The surfaces with red lines are calculated in the freely rotating limit, and the surfaces with blue lines are calculated for restricted diffusion, with for both spheres and rods. Increasing causes the surface in the restricted case to tend towards that of the freely rotating limit, and decreasing causes it to tend to the static limit of unity. When , i.e., where species are larger than , rotational motion will significantly contribute to the observed signal attenuation measured in a PFGSE experiment.

(Color online) Simulated NMR intensity data and for rigid rods. (a) Intensity data as a function of for rods of length , and ranging from in the freely rotating limit (blue lines), under restricted rotation (red lines), and in the static limit (green lines). As plotted, the gradient of the intensity data is independent of in the static regime. (b) against obtained from taking the gradients of the plots in (a). is a function of the experimental diffusion delay. (c) Simulated data for of increasing length in the freely rotating limit (blue surface) and static limit (green surface). (d) Variation of effective diffusion coefficients with rod length and . As the rod length exceeds , in the freely rotating limit is significantly larger than that expected from the effects of translational diffusion alone.

(Color online) Simulated NMR intensity data and for rigid rods. (a) Intensity data as a function of for rods of length , and ranging from in the freely rotating limit (blue lines), under restricted rotation (red lines), and in the static limit (green lines). As plotted, the gradient of the intensity data is independent of in the static regime. (b) against obtained from taking the gradients of the plots in (a). is a function of the experimental diffusion delay. (c) Simulated data for of increasing length in the freely rotating limit (blue surface) and static limit (green surface). (d) Variation of effective diffusion coefficients with rod length and . As the rod length exceeds , in the freely rotating limit is significantly larger than that expected from the effects of translational diffusion alone.

The vectors that define the rotation of a point on the surface of a sphere.

The vectors that define the rotation of a point on the surface of a sphere.

The effect of sets of equivalent rotations on for ellipsoids.

The effect of sets of equivalent rotations on for ellipsoids.

## Tables

Translational, , and rotational, , friction factors, preexponential factors , and limiting preexponential functions for idealized geometries. is the viscosity, is the radius, and is the rod length.

Translational, , and rotational, , friction factors, preexponential factors , and limiting preexponential functions for idealized geometries. is the viscosity, is the radius, and is the rod length.

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