Multiple quantum filtered nuclear magnetic resonance of (23)Na(+) 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 (23)Na(+). 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 ([Formula: see text]), rank-2 ([Formula: see text]), and rank-3 ([Formula: see text]) SQCs. We performed simultaneous numerical fitting of the experimental (23)Na nuclear magnetic resonance (NMR) spectra and rank-2 ([Formula: see text]) and rank-3 ([Formula: see text]) SQC evolution trajectories measured in double and triple quantum filtered experiments, respectively. We estimated values of the quadrupolar coupling constant C(Q), rotational correlation time τ(C), 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 ω(Q)/2π, residual quadrupolar coupling [Formula: see text] , and corresponding spherical order parameter [Formula: see text] , 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 (23)Na(+) 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 (23)Na magnetic resonance imaging.