Three-state analytic theory of two-dimensional nuclear magnetic resonance in systems with coupled macro- and micropores

Two-dimensional (2D) nuclear magnetic resonance (NMR) experiments involve a sequence of longitudinal ($T_1$) and transverse ($T_2$) measurements. In a previous paper we showed that if each of these 1D measurements can be represented by two exponential decays then there can be an accurate analytic solution for the 2D measurements with no additional information. In this paper we extend the theory to the case where there are three decay channels for the 1D measurements. The resulting analytic theory introduces a single free parameter, which is a rotation angle in the vector space spanned by the normal modes. Our predictions agree quite well with numerical results based on the microporous grain consolidation ($\mu \text{GC}$) model. The theory allows one to deduce information about decay modes in situations in which they may not be measurable in a conventional 1D measurement because the amplitude of that mode is too small.

Figure 1

A microporous version of the grain consolidation model in which the large grains are composed of smaller grains whose centers lie on a lattice constructed by subdividing the original unit cell.

Figure 2

One-dimensional lifetime distributions are shown for two values of the surface relaxation parameter. In both cases three modes are clearly evident.

Figure 3

Four 2D relaxation time cross-plots are shown for ${\rho }_1=2.5\mu \mathrm{m}/\mathrm{s}$ and ${\rho }_2=20\mu \mathrm{m}/\mathrm{s}$. The contour axis is logarithmic. The black lines represent the positions of the 1D peaks seen in Fig. 2 (2.011 s, 0.71 s, and 0.089 s).

Figure 4

Four 2D relaxation time cross-plots are shown for ${\rho }_1=0$ and ${\rho }_2=20\mu \mathrm{m}/\mathrm{s}$, with the same logarithmic axes for the contours as in Fig. 3. The black lines represent the positions of the 1D peaks seen in Fig. 2 (2.011 s, 0.71 s, and 0.089 s).

Figure 5

One-dimensional decay of $M\left(t\right)$ in the $\mu \text{GC}$ model with $\rho =20\mu \text{m}/\mathrm{s}$. The data and the three-exponent fit essentially overlie each other but when the slowest exponential contribution is subtracted from the data the remaining difference is well represented by two exponentials, except for deviations at the longer times. Each single exponential component is also plotted using a dotted line. It is clear that the simulation data is well represented by the sum of three single exponential decay functions.

Figure 6

One-dimensional decay of $M\left(t\right)$ in the $\mu \text{GC}$ model with $\rho =2.5\mu \text{m}/\mathrm{s}$, with the same conventions as in Fig. 5. It is difficult to deduce accurate values for the amplitude and the decay rate of the $j=2$ mode because the systematic noise at larger times is more dominant than it is in Fig. 5.

Figure 7

Decay rates ${\mu }_j\left(\rho \right)$ for the $\mu \text{GC}$ model. The open circles represent rescaled values of $\mu$ deduced for the smaller scale $\mu \text{GC}$ model reported in Ref. [9] as per Eq. (17). The value ${\mu }_2\left(0\right)=0.07{\text{s}}^{-1}$ was deduced from a special simulation as described in the text. The values ${\mu }_3\left(0\right)=0.67{\text{s}}^{-1}$ and ${\mu }_3\left(0.5\right)=0.75{\text{s}}^{-1}$ were deduced from fits to the 2D data shown in Figs. 12 and 14.

Figure 8

Amplitudes of each of the observed modes in the decay of an initially uniform magnetization, as functions of the assumed value of the surface relaxivity, $\rho$.

Figure 9

Storage matrix elements ${\mathcal{M}}_{ij}\left(t_s\right)$ for the $\mu \text{GC}$ model with ${\rho }_1=2.5\mu \text{m}/\mathrm{s}$ and ${\rho }_2=20\mu \text{m}/\mathrm{s}$. The analytic results are shown as continuous curves, the results of our simulations are shown with symbols. The off-diagonal elements are shown in both the middle panel and the lower panel so that one may see the behavior on different time scales. The data obey the exact sum rule, Eq. (14).

Figure 10

The angular dependence of the coefficients $C_{ij}^{\left(l\right)}$ as defined in Eq. (12) for the case ${\rho }_1=2.5\mu \text{m}/\mathrm{s}$ and ${\rho }_2=20\mu \text{m}/\mathrm{s}$.

Figure 11

Storage matrix elements ${\mathcal{M}}_{ij}\left(t_s\right)$ for the $\mu \mathrm{GC}$ model with ${\rho }_1=2.5\mu \mathrm{m}/\mathrm{s}$ and ${\rho }_2=20\mu \mathrm{m}/\mathrm{s}$, as in Fig. 9, but with a deliberately mischosen value for the rotation angle $\delta$. The resulting computed curves for $M_{ij}\left(t_s\right)$ are well off the mark. Same conventions as in Fig. 9.

Figure 12

Storage matrix elements ${\mathcal{M}}_{ij}\left(t_s\right)$ for the $\mu \text{GC}$ model with ${\rho }_1=0$ and ${\rho }_2=20\mu \text{m}/\mathrm{s}$, using the same conventions as in Fig. 9.

Figure 13

The angular dependence of the coefficients $C_{ij}^{\left(l\right)}$ as defined in Eq. (12) for the case ${\rho }_1=0$ and ${\rho }_2=20\mu \text{m}/\mathrm{s}$.

Figure 14

Storage matrix elements ${\mathcal{M}}_{ij}\left(t_s\right)$ for the $\mu \text{GC}$ model with ${\rho }_1=0.5 \mu \text{m}/\mathrm{s}$ and ${\rho }_2=20 \mu \text{m}/\mathrm{s}$, using the same conventions as in Fig. 9.