# Advances in the Interpretation of Frequency-Dependent Nuclear Magnetic Resonance Measurements from Porous Material

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## Abstract

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## 1. Introduction

## 2. The $\mathbf{3}\mathbf{\tau}$ Model

#### 2.1. The Physics behind the $3\tau $ Model

#### 2.2. Calculating the Spin–Lattice Relaxation Rate Dispersion ${R}_{1}\left(f\right)$

#### 2.3. Comparison of Models

## 3. Results

#### 3.1. Mortar

#### 3.2. Plaster

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NMR | nuclear magnetic resonance |

FFC-NMR | fast-field-cycling NMR |

BMSD | bulk-mediated surface diffusion |

## References

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**Figure 1.**The figure shows, at left, two water molecules at time $t=0$. A pair of intermolecular hydrogen spins are connected by a vector ${\mathbf{r}}_{0}$. The molecules diffuse within the porous material and the same two hydrogen atoms are separated by vector $\mathbf{r}$ after a time t. The frequency-dependent relaxation rate ${R}_{1}\left(f\right)$ is determined by the probability density function describing the probability that a pair of spins are separated by ${\mathbf{r}}_{0}$ at $t=0$ and by $\mathbf{r}$ at time t.

**Figure 2.**The arrows represent in-layer spin diffusion (red), desorption (green) and bulk diffusion (blue). In (

**a**), a measurement of the relaxation rate ${R}_{1}\left(f\right)$ is dominated by the interaction of fixed paramagnetic impurities with both the water in the surface layer and in bulk. The density of paramagnetic impurities is modelled by a single effective layer (dashed line) placed $2\delta $ below the surface. In (

**b**), ${R}_{1}\left(f\right)$ is due to layer–layer, bulk–bulk and bulk–layer interactions. The surface layer of water is assumed to be $\delta =0.27$ nm thick.

**Figure 3.**Fast-field-cycling nuclear magnetic resonance (FFC-NMR) spin–lattice relaxation rate data for a hydrated mortar by Barberon et al. [10], presented as a function of frequency. The solid curve represents the best fit obtained using the 3$\tau $ model and the 3TM fitting package. The open circles represent the residuals with the horizontal line indicating a residual of zero.

**Figure 4.**FFC-NMR spin–lattice relaxation rate data for a plaster paste by Korb [5], presented as a function of frequency. The solid curve represents the best fit obtained using the 3$\tau $ model and the 3TM fitting package. The open circles represent the residuals with the horizontal line indicating a residual of zero.

**Figure 5.**(

**top**) The flow diagram provides a simplistic illustration of the function of the 3TM fitting package used to interpret FFC-NMR dispersion data. (

**bottom**) A screen grab of the interface post-fitting is presented.

**Table 1.**List of physical quantities that may be provided by an FFC-NMR measurement from a porous material by the Korb, bulk-mediated surface diffusion (BMSD) and $3\tau $ model types.

Quantity | Korb | BMSD | $3\mathit{\tau}$ | Comments |
---|---|---|---|---|

${\tau}_{\ell}$ | ✓ | ✓ | ✓ | Diffusion correlation time for fluid at the pore surface. |

${\tau}_{d}$ | ✓ | ✓ | Desorption time constant for the surface fluid. | |

${\tau}_{b}$ | ✓ | Bulk fluid diffusion time constant related to the diffusion coefficient. | ||

${N}_{\sigma}$ | ✓ | Each model employs a scaling factor proportional to the number of paramagnetic ions per unit volume in the solid. 3$\tau $ provides the equivalent spin density for the effective paramagnetic layer. | ||

${N}_{\ell}$ | ✓ | For water, the surface spin density is normally set to 66.6 spins/nm${}^{3}$ as for the bulk. ${N}_{\ell}$ can act as an additional fit parameter within the $3\tau $ model. | ||

$x\approx S/V$ | ✓ | The dimensionless ratio of the volume of the pore surface (thickness assumed to be $\delta =$ 0.27 nm) to the pore volume. | ||

h | ✓ | The “planar-pore-equivalent" pore thickness is equal to $5.4\times {10}^{-4}/x$ in units of $\mathsf{\mu}$m. Useful characteristic pore dimension. | ||

$\alpha $ | ✓ | ✓ | The Lévy parameter is a measure of the departure from Fickian dynamics. $\alpha =1$ in most BMSD models. Lévy dynamics is trivially introduced into $3\tau $ but is not necessary to secure good fits to FFC-NMR data. | |

${\tau}_{d}/{\tau}_{\ell}$ | ✓ | ✓ | The ratio of time constants is approximately equal to the number of hops a spin makes at a surface before desorption. It is linked to surface affinity (see text). | |

${T}_{1}/{T}_{2}$ | ✓ | ✓ | ✓ | The ratio of spin–lattice relaxation time to the spin–spin relaxation time is sometimes available at a spot frequency from separate ${T}_{1}$–${T}_{2}$ correlation measurements. Easily estimated by any model for comparison. |

**Table 2.**Results of a fit of the $3\tau $ model to a FFC-NMR dispersion using 3TM for a hydrated mortar due to Barberon and co-workers [10].

Quantity | Value | Comments |
---|---|---|

${\tau}_{\ell}$ | 0.27 $\mathsf{\mu}$s | Best fit result for the surface water diffusion correlation time. |

${\tau}_{d}$ | 1.8 $\mathsf{\mu}$s | Best fit desorption time constant. |

${\tau}_{b}$ | 21 ps | Best fit bulk water diffusion time constant equivalent to a diffusion coefficient of 0.6 × 10${}^{-9}$ m${}^{2}$s${}^{-1}$ |

${N}_{\sigma}$ | 0.028 ions/nm${}^{3}$ | The best fit paramagnetic ion number density is close to the measured value of 0.03 [10]. |

x | 0.00073 | The best fit surface-to-volume ratio. |

h | 0.74 $\mathsf{\mu}$m | The planar-pore-equivalent pore thickness. |

${\tau}_{d}/{\tau}_{\ell}$ | 6.5 | The number of surface hops of water before desorption. |

${\tau}_{\ell}$ | 0.15–0.42 $\mathsf{\mu}$s | Range of values obtained from good fits (see text). |

${\tau}_{d}$ | 1.2–2.1 $\mathsf{\mu}$s | Range of values obtained from good fits (see text). |

${\tau}_{b}$ | 21 ps | All good fits yielded the same value of the bulk diffusion time constant. |

$<\phantom{\rule{-0.166667em}{0ex}}{\tau}_{\ell}\phantom{\rule{-0.166667em}{0ex}}>$ | 0.24 $\mathsf{\mu}$s | Mean value of ${\tau}_{\ell}$ from the spread of good fits. |

$<\phantom{\rule{-0.166667em}{0ex}}{\tau}_{d}\phantom{\rule{-0.166667em}{0ex}}>$ | 1.5 $\mathsf{\mu}$s | Mean value of ${\tau}_{d}$ from the spread of good fits. |

${T}_{1}/{T}_{2}$ | 2.9–3.3 | Range of values obtained from the set of good fits. The experimental value is typically 4 for cement paste at $f=$ 20 MHz [15]. |

**Table 3.**Results of a fit of the $3\tau $ model to an FFC-NMR dispersion for a plaster paste from Korb [5] using 3TM.

Quantity | Value | Comments |
---|---|---|

${\tau}_{\ell}$ | 2.4 $\mathsf{\mu}$s | Best fit result for the surface water diffusion correlation time. |

${\tau}_{d}$ | 6.5 $\mathsf{\mu}$s | Best fit desorption time constant. |

${\tau}_{b}$ | 13 ps | Best fit bulk water diffusion time constant equivalent to a diffusion coefficient of 0.9 × 10${}^{-9}$ m${}^{2}$s${}^{-1}$ |

${N}_{\ell}$ | 73 spins/nm${}^{3}$ | The best fit ${}^{1}$H spin density for the surface layer is similar to the 66.6 spins/nm${}^{3}$ for bulk water. |

x | 0.00144 | The best fit surface-to-volume ratio. |

h | 0.38 $\mathsf{\mu}$m | The planar-pore-equivalent pore thickness. |

${\tau}_{d}/{\tau}_{\ell}$ | 2.7 | The number of surface hops of water before desorption. |

${T}_{1}/{T}_{2}$ | 3.9 | The experimental value is typically 4 $f=$ 20 MHz for cement paste [15]. |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Faux, D.; Kogon, R.; Bortolotti, V.; McDonald, P.
Advances in the Interpretation of Frequency-Dependent Nuclear Magnetic Resonance Measurements from Porous Material. *Molecules* **2019**, *24*, 3688.
https://doi.org/10.3390/molecules24203688

**AMA Style**

Faux D, Kogon R, Bortolotti V, McDonald P.
Advances in the Interpretation of Frequency-Dependent Nuclear Magnetic Resonance Measurements from Porous Material. *Molecules*. 2019; 24(20):3688.
https://doi.org/10.3390/molecules24203688

**Chicago/Turabian Style**

Faux, David, Rémi Kogon, Villiam Bortolotti, and Peter McDonald.
2019. "Advances in the Interpretation of Frequency-Dependent Nuclear Magnetic Resonance Measurements from Porous Material" *Molecules* 24, no. 20: 3688.
https://doi.org/10.3390/molecules24203688