Principle of thickness measurement of inorganic particles in organic-inorganic nanocomposites based on dipolar couplings. The dephasing of X nuclei ( in this picture) in the inorganic particles by in the organic matrix depends strongly on their distance from the interface with the organic matrix. HARDSHIP NMR eliminates dephasing by protons in the inorganic phase (dashed lines); thus, fast dephasing proves a large fraction of X nuclei near the interface, i.e., thin particles , while slow but observable dephasing on a time scale indicates greater particle thickness (up to for phosphates).
Basic cycle of the proton pulses in the HARDSHIP pulse sequence. Heteronuclear recoupling by one 180° pulse every half rotation period , as in REDOR (Ref. 5), alternates with dephasing of transverse (I spin) coherence flanked by 90° pulses (filled rectangles) of opposite phases. Heteronuclear recoupling for rotation periods is shown; , i.e., with a single 180° pulse at , is also commonly used in HARDSHIP experiments.
Schematic representation of spin diffusion between nearby matrix protons coupled to a distant X nucleus in the inorganic phase. (a) When the two X–H vectors are similar in length and orientation , then the couplings and the corresponding phases do not differ much. (b) A significant change in the heteronuclear phase and resulting interference with heteronuclear recoupling occurs after spin diffusion to protons at distances approximately equal to the heteronuclear distance in question.
Effects of the basic double cycle of the HARDSHIP pulse sequence on (a) protons with long that are on resonance and (b) protons with short . The corresponding I-spin parts of the coherences are indicated pictorially above the pulses. The density operators at various times are given on the right for the simplest examples of these spin systems. The duration of the heteronuclear recoupling periods, which is in general, is shown as . The figure shows heteronuclear dephasing periods.
Four versions of the HARDSHIP sequence for different spinning frequencies. (a) Slow-spinning version . The heteronuclear recoupling and a short homonuclear dipolar-dephasing period are incorporated into one rotation period. (b) Standard version for intermediate spinning speeds . One rotation period with a recoupling pulse in its center for heteronuclear dephasing alternates with one rotation period for homonuclear dephasing. (c) Version for relatively fast spinning . This is desirable when the couplings in the inorganic particles are relatively strong, since faster spinning increases the relaxation time and shortens the dephasing period of duration in the pulse sequence. The heteronuclear recoupling period is kept significant by extending it to rotation periods, with one 180° pulse per . (d) Version for relatively fast spinning and a spread of frequencies in the inorganic phase (including a short ). Compared to (c), every odd-numbered homonuclear dephasing period is extended to two rotation periods with a 180° pulse at its center, which refocuses dephasing by isotropic-shift dispersion. Except for these refocusing 180° pulses, all pulses in (a)–(d) are composite, as shown in Fig. 6. In all the sequences, the 90° pulse immediately after a period labeled “no ,” where no coherence exists, is optional. The reference intensity is obtained by strong decoupling during the full time where the X-spin magnetization is transverse.
Schematic representation of the effects of composite pulses, which reduce the sensitivity to pulse-strength errors, when applied to on-resonance coherence during the HARDSHIP experiment. (a) Cycle of the simplest HARDSHIP pulse sequence with composite 180° (Ref. 21) and 90° pulses indicated. (b) Rotations of initial coherence, as produced by a composite 90° pulse that minimizes the residual component (inner pulse phases and arrows) and by the complementary composite pulse that restores the on-resonance coherence with long exactly to (outer pulse phases and arrows). (c) Composite-pulse phase sequence in the pulse scheme of Fig. 5(d). (d) Corresponding rotations, which return on-resonance coherence to regardless of pulse length.
(a) Outline of the simulation procedure for obtaining HARDSHIP curves. (b) Angles relevant in the calculation of the heteronuclear coupling frequencies and powder averaging (see also the text). The nanoparticle is shaded light gray, the plane containing the rotor axis and the field darker gray. [(c) and (d)] Signal intensity as a function of dephasing time for (c) laterally wide -thick phosphate platelets and (d) -diameter spherical phosphate nanoparticles, assuming recoupling periods of . Curves for various depths , in steps of , from the interface ( distance) are shown, as well as the overall dephasing curve (dashed line). The proton density in the matrix was . (e) Parameters for biexponential fits of the data in (c) and of similar data for phosphate sheets of 1, 2, and thickness. Decay constants (open symbols) and (filled symbols), on a logarithmic scale, and the weighting factor (see inset) are plotted as a function of depth . For , the decay is single exponential on the time scale. The dashed line is a curve . (f) Same as (e) for spherical phosphate nanoparticles of 3, 6, 9, and diameter. These parameters can be used to produce HARDSHIP fits without explicit orientational averaging.
dephasing in hectorite clay. (a) Schematic representations of the structure of pure hectorite clay. Oxygen atoms are indicated by open circles, except in OH groups (filled circles). The lines connecting oxygen atoms are not chemical bonds. Silicon and hydrogen atoms are denoted by Si and H, respectively. The closest Si–H distance is . (b) Schematic of hectorite clay exfoliated in a hydrophilic polymer. (c) HARDSHIP data (squares) and REDOR data (triangles) of pure hectorite clay. (d) HARDSHIP data of hectorite clay dispersed in poly(vinyl alcohol) (circles), with a fit curve based on the structural parameters shown in (b) (see the text for details). All data were acquired at MAS after direct polarization ( 90°-pulse excitation) with recycle delays and refocused detection for signal enhancement (Ref. 20). The relaxation time of the OH protons is in pure hectorite clay and in the presence of polymer (Ref. 20). The pulse sequence shown in Fig. 5(b) was used.
dephasing in nanodiamond ( diameter of crystalline core) with mostly protonated surface (the interior is free of protons). (a) HARDSHIP (triangles) vs REDOR (squares) dephasing. The HARDSHIP pulse sequence shown in Fig. 5(b) was used, but with of homonuclear dipolar dephasing between the 90° pulses. Filled symbols and scale at the bottom: MAS. Open triangles and scale at the top: MAS. In the plot, the time values of the MAS data have been scaled by a factor of 0.5 (see scale at the top of the plot), in order to confirm the proportionality of the HARDSHIP dephasing rate with . The full line is a fit curve for spherical nanoparticles of diameter with 60% of the surface carbons directly bonded to . (b) Direct-polarization NMR spectrum of nanodiamond, obtained at MAS. The downfield shoulder is highlighted in gray. (c) decay ( relaxation) for two different regions in the NMR spectrum. Top curve: ; bottom curve: . Recycle delay: ; spinning frequency: . The dashed lines are guides to the eye.
dephasing in apatites (calcium phosphates). (a) HARDSHIP (filled circles: peak height; open circles: peak area) and REDOR (triangles) dephasing of pure NIST hydroxyapatite. Recycle delay: ; spinning frequency: , except for the REDOR data . The HARDSHIP pulse sequence of Fig. 5(d) was used with and . The inset shows the decay under decoupling (filled inverted triangles: peak height; open squares: peak area) and the decay of the hydroxide protons (filled diamonds), again at MAS. The dashed lines are guides to the eye. (b) , , and scaled-up spectra of NIST hydroxyapatite obtained in HARDSHIP with . Preferential dephasing of a broad component is clearly seen. (c) HARDSHIP dephasing of bioapatite in native mouse bone (matrix: collagen). Recycle delay: ; spinning frequency: . The full curve shown was simulated assuming a -thick (P-to-P) platelet. The inset shows the decay under decoupling.
Simulated HARDSHIP dephasing curves for various nanoparticle thicknesses (defined here as the distance between X-nucleus surface layers, which is in hectorite clay, in nanodiamond, and in the bone-apatite simulation). Dephasing times for are shown on the bottom scale, those for on the top scale, for . The distance between X-nucleus layers, as well as the closest H–X distance, was in the simulations; the density of the matrix was . (a) Dephasing for wide platelets of thickness. The curve is shown dashed because for such thin particles, details of the X-nucleus layering may significantly change the dephasing curve; for instance, the X-nucleus layer distance in hectorite clay (see Fig. 8) is , not as assumed in the present simulations. (b) Dephasing for spheres of thickness.
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