Multiple quantum filtered nuclear magnetic resonance of 23Na+ in uniformly stretched and compressed hydrogels

Stretching or compressing hydrogels creates anisotropic environments that lead to motionally averaged alignment of embedded guest quadrupolar nuclear spins such as 23Na+. These distorted hydrogels can elicit a residual quadrupolar coupling that gives an oscillation in the trajectories of single quantum coherences (SQCs) as a function of the evolution time during a spin-echo experiment. We present solutions to equations of motion derived with a Liouvillian superoperator approach, which encompass the coherent quadrupolar interaction in conjunction with relaxation, to give a full analytical description of the evolution trajectories of rank-1 (T^1±1), rank-2 (T^2±1), and rank-3 (T^3±1) SQCs. We performed simultaneous numerical fitting of the experimental 23Na nuclear magnetic resonance (NMR) spectra and rank-2 (T^2±1) and rank-3 (T^3±1) SQC evolution trajectories measured in double and triple quantum filtered experiments, respectively. We estimated values of the quadrupolar coupling constant CQ, rotational correlation time τC, and 3 × 3 Saupe order matrix. We performed simultaneous fitting of the analytical expressions to the experimental data to estimate values of the quadrupolar coupling frequency ωQ/2π, residual quadrupolar coupling ωQ/2π, and corresponding spherical order parameter S0*, which showed a linear dependence on the extent of uniform hydrogel stretching and compression. The analytical expressions were completely concordant with the numerical approach. The insights gained here can be extended to more complicated (biological) systems such as 23Na+ bound to proteins or located inside and outside living cells in high-field NMR experiments and, by extension, to the anisotropic environments found in vivo with 23Na magnetic resonance imaging.