# SECULAR AND SEMI-NONSECULAR MODELS OF CROSS-POLARIZATION KINETICS FOR REMOTE SPINS: AN APPLICATION FOR NANO-STRUCTURED CALCIUM HYDROXYAPATITE

V. Klimavičius ab, F. Kuliesius a, E. Orentas b, and V. Balevičius a

a Institute of Chemical Physics, Vilnius University, Saulėtekio 3, 10257 Vilnius, Lithuania

b Department of Organic Chemistry, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania

Received 15 January 2021; revised 18 February 2021; accepted 19 February 2021

The 1H → 31P cross-polarization (CP) kinetics in the nanostructured calcium hydroxyapatite (nano-CaHA) was measured under moderate (5 kHz) magic-angle spinning (MAS) rate. This material was chosen as it contains the distanced 1H-31P spin pairs and the interactions between them are characterized by a relatively low dipolar coupling (b) that could be comparable with the spin-diffusion rates (R). Therefore, the physical legitimacy to use the secular solution of the quantum Liouville-von Neumann equation is doubtful. The semi-nonsecular model of spin dynamics was applied, and the results were compared with those obtained by the secular approach. The comparable results obtained by both models show that the secular model is applicable, with certain reservation, also in the case of |b| ≈ R. The extremely high anisotropy of spin diffusion in the nano-CaHA was deduced. This can be a matter of the applied approach, as the interactions of the 31P spins with the proton bath were neglected in both models. The high anisotropy could also be caused by the physical reasons that stem from the structural and proton diffusion features of CaHA. This material belongs to low-dimensional proton conductors possessing a large motional freedom for protons along OH- chains.

Keywords: solid-state NMR, cross-polarization, spin diffusion, magic-angle spinning, calcium hydroxyapatite

## 1. Introduction

Bone tissue consists of about 70% of calcium phosphates (CaPs) that make CaPs materials of choice for potential treating of bone diseases by repairing damaged bone tissues. For the  successful application in medicine, the process of CaP biomin-eralization and the interaction between CaPs and biological environment needs to be understood [1]. Hydroxyapatites (HAs) are thermodynamically the most stable form of CaPs, and therefore HAs are widely applied in implantology, orthopedic and periodontal surgery [2, 3]. Each particular application depends on HA structure, crystallinity, particles size and morphology  [4]. Calcium hy-droxyapatite (Ca10(PO4)6(OH)2, CaHA) represents a privileged member for the above applications due to its close resemblance to the mineral of hard tissues (bone, enamel, dentin, etc.) resulting in high biocompatibility [5].

CaHA is an attractive object for investigation from the  physical point of view. It belongs to the  class of low-dimensional proton-conducting materials possessing a  motional freedom for protons along OH- chains  [6, 7]. An interplay of the surface and bulk effects may further result in other interesting features of the  nanostructured CaHA (nano-CaHA) [8].

Cross-polarization (CP), combined with magic-angle spinning (MAS), is one of the 'classical' and most widely used methods in solid-state NMR spectroscopy  [9-11]. CP is a  powerful tool for studying fine structural details and dynamics in complex materials [12-14]. The processing of CP kinetic data, i.e. the consideration of the evolution of interactions between spins in time (contact time) provides the rates of spin diffusion and spin-lattice relaxation, the  profiles of distribution of dipolar coupling and some other parameters characterizing the effective sizes of spin clusters [15-17].

In the  present work, the  comparative study of application of secular and semi-nonsecular spin dynamics models for experimental CP MAS kinetics was conducted. To this end, the 1H → 31P CP MAS kinetics under moderate MAS rate (5  kHz) was measured. Nano-CaHA contains the distanced 1H-31P spin pairs and the  interactions between them are characterized by the relatively low dipolar coupling constant values that could be compared with the spin-diffusion rates. The earlier studies [15, 16] have revealed that the nano-CaHA is a slowly relaxing spin system. This makes the nano-CaHA very suitable for testing various microscopic quantum models of spin dynamics without taking into account the spin-lattice relaxation effects.

## 2. Experiment

The nano-CaHA was obtained from Aldrich (synthetic, 99.999%, from metal basis). The  material was characterized by scanning electron microscopy (SEM) and energy-dispersive X-ray analysis (EDX), for details see [16]. In order to remove the adsorbed water, the sample was vacuum-dried at 373 K for four days.

The solid-state NMR experiments were performed using an 600 MHz Bruker AVANCE NEO NMR spectrometer equipped with a 2.5 mm Bruker TriGamma triple resonance MAS probe. The  experiments were performed in 14.095 T magnetic field using an Ascend 54 mm standard-bore superconducting magnet. The resonance frequencies of 1H and 31P nuclei were 600.3 and 243.0 MHz, respectively. The 1H → 31P CP MAS experiments were performed for the  spinning sample at 5  kHz at the n = +1 Hartmann-Hahn (HH) matching condition. The CP contact was achieved with rectangular 71 and 76 kHz RF pulses for 31P and 1H, respectively. The sample temperature was set to 298 K and controlled by a  Bruker BCU II temperature regulation system. Spectra consisted of 7142 real data points and were registered using a single scan, the repetition delay was set to 125 s, that is equal to 5∙T1 (the  spin-lattice relaxation time). The  CP MAS kinetics were registered by varying the contact times from 50  μs to 10  ms in increments of 10 μs. Processing of CP MAS kinetics was carried out using the Microcal Origin 9 and MathCad 15.0 packages.

## 3. Theoretical models of CP kinetics

The most widely used theoretical model that exhibits the coherent oscillatory behaviour of CP intensity originates from the work of Müller et al. [18], the so-called I-I*-S model [11, 12, 19]. The system is treated as a strongly coupled I*-S spin pair (I = 1H and S = 31P spins in the present work) immersed in a spin bath consisting of the remaining I spins. The model assumes that only one spin I* interacts with the I-spin bath or infinite energy reservoir of I spins, that is described in a phenomenological way. The kinetics of the CP signal intensity I(t) is then expressed as

$I\left(t\right)={I}_{0}\left[1-\frac{1}{2}{\text{e}}^{-{k}_{2}t}-\frac{1}{2}{\text{e}}^{-{k}_{1}t}\mathrm{cos}\left(\frac{b}{2}t\right)\right],\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(1\right)$

where the parameters k1 and k2 are the spin-diffusion rate constants. The cosine-oscillation frequency is b/2, i.e. 1/2 of the dipolar splitting, that depends on the gyromagnetic ratios (γI, γS) of two interacting nuclei (I and S), the distance r between them and the angle θ between the r vector and the magnetic field:

$b=\frac{{\gamma }_{\text{I}}{\gamma }_{\text{s}}\hslash }{{r}^{3}}\frac{\left(3{\mathrm{cos}}^{2}\theta -1\right)}{2}.\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(2\right)$

Later on, the model was modified by Naito and McDowell [20] introducing the spin-lattice relaxation rate of spin I in the rotating frame (1/T) and the anisotropy of spin diffusion. Recently, the I-I*-S model and spin-diffusion properties were revised very thoroughly by Hirschinger and Raya by solving the quantum mechanical master equation using various approaches and the  formalism of spin-diffusion superoperators  [12, 19, 21]. For a  fast fluctuating I-spin bath, the  spin-diffusion super-operator for the reduced density operator $\stackrel{^}{\sigma }$ can be written as

$\begin{array}{rl}& \stackrel{^}{\stackrel{^}{\mathrm{\Gamma }}}\left(\stackrel{^}{\sigma }\right)={R}_{\mathrm{d}\mathrm{p}}^{\mathrm{I}}\left[{\stackrel{^}{\mathrm{I}}}_{z},\left[{\stackrel{^}{\mathrm{I}}}_{z},\stackrel{^}{\sigma }\right]\right]+\\ & +{R}_{\mathrm{d}\mathrm{f}}^{\mathrm{I}}\left\{\left[{\stackrel{^}{\mathrm{I}}}_{x},\left[{\stackrel{^}{\mathrm{I}}}_{x},\stackrel{^}{\sigma }\right]\right]+\left[{\stackrel{^}{\mathrm{I}}}_{y},\left[{\stackrel{^}{\mathrm{I}}}_{y},\stackrel{^}{\sigma }\right]\right]\right\}+\\ & +{R}_{\mathrm{d}\mathrm{f}}^{\mathrm{S}}\left\{\left[{\stackrel{^}{\mathrm{S}}}_{x},\left[{\stackrel{^}{\mathrm{S}}}_{x},\stackrel{^}{\sigma }\right]\right]+\left[{\stackrel{^}{\mathrm{S}}}_{y},\left[{\stackrel{^}{\mathrm{S}}}_{y},\stackrel{^}{\sigma }\right]\right]\right\},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(3\right)\end{array}$

where RIdp and RIdf are the (homonuclear) spin-diffusion rate constants of the I spin and RSdf is that (heteronuclear) of the S spin. The rate constants RIdf and RSdf are associated with the flip-flop terms of the homonuclear (I-I) and heteronuclear (I-S) di-polar Hamiltonians, respectively, and allow the complete thermal equilibration with the bath, whereas RIdp acts on the damping of the coherence driving the system to the internal quasi-equilibrium [21].

The usual approximation considers that the I-S interaction with environment is neglected, and thus the rate constant of the heteronuclear spin diffusion of the S spin is set as RSdf = 0. In this case, the spin-diffusion rate constants k1 and k2 are related with RIdp and RIdf as k1 = RIdf + RIdp/2 and k2 = RIdf, and Eq. (1) can be rewritten as

$I\left(t\right)={I}_{0}\left[1-\frac{1}{2}{\text{e}}^{-{R}_{\text{df}}^{\text{I}}}-\frac{1}{2}{\text{e}}^{-\left({R}_{\text{df}}^{\text{I}}+{R}_{\text{dp}}^{\text{I}}/2\right)t}\mathrm{cos}\left(\frac{b}{2}t\right)\right].\text{ }\text{ }\text{ }\left(4\right)$

Equations (1) and (4) are valid at the Hartmann- Hahn condition only if two secular approximations are satisfied: (i) the  applied RF fields are much stronger than the  I*-S coupling (ω1I, ω1S (ii) the I*-S coupling constant is much larger than the spin-diffusion rate constants (|b| ≫ RIdf, RIdp). Therefore, the  physical legitimacy to use Eqs. (1) and (4) for describing the  distanced, and thus weakly interacting spins (b ≈ RIdf, RIdp or even less) is in a certain doubt. Note that Alvarez et al. [22] have obtained the  analytical nonsecular solution of the master (Liouville-von Neumann) equation for arbitrary values of the  homonuclear spin-diffusion rate constants, however, for a static sample (no MAS) and neglecting the I-S interaction with environment (RSdf = 0). Nevertheless, the complete expression of the  general nonsecular equation is rather cumbersome to implement. In Ref.  [21] it was shown that the generalized Liouville-von Neu-mann quantum mechanical equation has a  semi-nonsecular analytical solution when |b| ≫ |RIdf-RSdf|

where

$\phi =\sqrt{{\left({R}_{\text{dp}}^{I}/2\right)}^{2}-{b}^{2}}.\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(6\right)$

From Eqs. (5) and (6) it is easy to see that transient oscillations of CP intensity will appear when φ becomes imaginary, i.e. b2 > (RIdp/2)2. If (RIdp/2)2 > b2, oscillations convert to an overdamped regime.

Both the secular (Eq. (4)) and semi-nonsecular (Eq.  (5)) models were applied processing the  experimental CP MAS kinetics data and the results were compared.

## 4. Results and discussion

As the  dipolar splitting b is an angular function, the proper angular averaging has to be carried out in order to apply Eqs. (4) and (5) to powder samples. For MAS experiments when the HH matching conditions ωII - ωIS = MAS are fulfilled for n = ±1 (used in the present work)

${b}_{±1}=\frac{{D}_{\text{IS}}}{2\sqrt{2}}\mathrm{sin}\left(2\beta \right)\text{,}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(7\right)$

where DIS is the heteronuclear I-S dipolar coupling constant (DIS = (1/2π) (μ0/4π) γIγS (h/2π)/r3, in Hz), β is the polar angle between r vector and the MAS rotor axis [23, 24], following [23], the angular averaging (AA) for Eq. (4) is carried out as

$\mathrm{cos}{\left(bt\right)}_{\text{AA}}=\frac{1}{2}{\int }_{0}^{\pi }\mathrm{cos}\left(b\left(\beta \right)t\right)\mathrm{sin}\left(\beta \right)\text{d}\beta ,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(8\right)$

and correspondingly it can be written for the oscillating part  [cosh(φt)  +  ...sinh(φt)] in Eq.  (5). The averaging for Eq. (4) was carried out analytically using the series of Bessel Jn functions [24, 25] as well as numerically:

In both cases the effects of series truncation (Σ, Bessel) and the increment of discretization (ΣN, numerical integration) were checked. The  results are shown in Fig. 1. The truncation of Bessel series has no significant influence on the precision of calculations due to a very steep suppression (~1/(2k)2) of the contributions from J2k(x) of higher order. A more delicate situation is encountered when applying the numeri cal method - certain computing artifacts may flash (see the range 8-10 ms in Fig. 1) if the increment of integration is set too rough. This can be checked and mended by optimizing the  discretization (e.g. increasing N in Eq.  (9) from 100 to 1000, Fig.  1). Generally speaking, the analytical and the numerical averaging provide practically identical results over the whole range of contact time often used in the kinetic CP MAS experiments. However, the numerical method is more universal, more convenient for programming and can be applied in the  cases where the oscillations are described by more complicated (not a single cosine) functions.

The 1H → 31P CP MAS kinetics in nano-CaHA were measured and processed using a high density experimental data set (1000 points, Fig. 2). As shown in the earlier works [15-17, 25], this allows one to reduce the excessive freedom in th e nonlinear curve fitting targeting its flow towards the 'true' minimum on the multi-parameter surface χ2, i.e. to the minimal sum of weighted squares of deviations of the chosen theoretical model curve from the experimental one. This enables one to test and verify a series of multi-parametrical models. The results of fitting are presented in Table 1.

A similar agreement between the  theory and the experiment was achieved over the whole contact time range using both models. It is reflected in the similar statistical parameters R2 (the correlation coeﬃcient) and χ2. Two most significant discrepancies are noticeable over the whole kinetic curve: (i)  high frequency (tens of  kHz) oscillations of weak intensity at very short contact times (<1 ms) and (ii)  the  non-random residuals close to 2  ms (Fig. 2). The physical origin of the first one is most likely related with the RF field values, that in principle might be revealed by some additional experiments varying the  MAS rate and the  settings of HH matching parameters. The second discrepancy might originate from structural features of the na-no-CaHA itself. It is known that in pure crystalline CaHA each P atom has two neighbouring protons distanced at 0.385 nm, further two at 0.42 nm, while others are 0.6 nm or more away [26]. The coupling constants DIS were determined from b values, rescaling them by a factor of √2 because the HH matching n = ±1 was fulfilled in the present experiments. DIS values of 2200-2500 Hz (Table 1) correspond to P-H distances of 0.267-0.277  nm. Distances of 0.21-0.25 nm are typical of the P-O-H structures that are found in some related systems, such as calcium phosphate gelatin nano-composites  [27] or sol-gel derived SnO2 nanoparticles capped by phos-phonic acids [28]. In CaHA, protons are not part of the phosphate group, and thus such short P...H contacts should not be observed. However, in nano-Ca-Ha the P-O-H structural motifs with P...H distances of 0.20-0.25 nm can be present on the surface layers [29]. Most importantly, the studied models deal with the  single dominant P-H couplings between the spins at the shortest distances only, whereas other couplings were not taken into account.

Table 1. The fit parameters of 1H → 31P CP MAS kinetics in nano-CaHA (Fig. 2) obtained using secular and semi-nonsecular models without and with angular averaging (AA).
Model ⇒ Secular, Eq. (4) Semi-nonsecular, Eq. (5)
No AA Numerical AA No AA Numerical AA
RIdp, s-1 3520 2030 4110 2030
RIdf, s-1 22 18 19 7
DIS, Hz 1620 2220 2200 2500
I0, a.u. 1.525 1.564 1.555 1.738
R2/χ2, % 0.976/2.6 0.988/1.8 0.989/1.7 0.986/2.0

Finally, the  extremely high anisotropy of spin diffusion in the nano-CaHA (RIdp/RIdf ~ 100-200), never seen for other studied spin systems, e.g. glycine [17] or some polymers [30, 31], is noteworthy. It can be a matter of the applied models, as the I-S interaction with environment is neglected in both models (RSdf = 0). On the other hand, the high anisotropy could be caused by the physical reason related with structural and proton diffusion features of CaHA. The  long-range proton diffusion pathway in CaHA was evidenced by high-temperature neutron diffraction technique and bond valence method [6]. The proton diffusion via reorientation of hydroxide ions (OH-) is a complex process resembling a sinusoidal pattern. It consists of one-dimensional proton diffusion pathways along the c axis in the hexagonal channel and two-dimensional proton migration pathway network on the ab planes.

## 5. Conclusions

The semi-nonsecular model of spin dynamics has been applied for analysing 1H → 31P CP MAS kinetics obtained for the powdered sample under moderate MAS rate. The  results have been compared with those obtained by the secular approach.

The comparable results obtained by both models mean that the condition |b| ≫ RIdf and RIdp, stated as necessary to derive the secular solution of master equation, appeared to be not very crucial. The secular model can be used with certain reservations also in the case of |b| ≈ RIdp.

Extremely high anisotropy of spin diffusion (RIdp/RIdf ~ 100-200) in the nano-CaHA is deduced for the first time. This ratio is the same for secular and nonsecular approaches. Probably, it is caused by the spin dynamics in the low-dimensional proton bath in the nano-CaHA.

## Acknowledgements

The authors acknowledge the  Center of Spectroscopic Characterization of Materials and Electronic/ Molecular Processes (SPECTROVERSUM, www. spectroversum.ff.vu.lt) at the  Lithuanian National Center for Physical Sciences and Technology for the use of spectroscopic equipment. This research is funded by the  European Social Fund under Measure No.  09.3.3-LMT-K-712-19-0022 'Development of Competences of Scientists, Other Researchers and Students Through Practical Research Activities'. We thank Professor Jérôme Hirschinger (Strasbourg University) for helpful discussion.

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### SEKULIARUSIS IR PUSIAU NESEKULIARUSIS SUKINIŲ KINETIKOS MODELIAI NUTOLUSIEMS SUKINIAMS: TAIKYMAS NANOSTRUKTŪRIZUOTAM KALCIO HIDROKSIAPATITUI

V. Klimavičius ab, F. Kuliesius a, E. Orentas b, V. Balevičius a

a Vilniaus universiteto Cheminės fizikos institutas, Vilnius, Lietuva

b Vilniaus universiteto Organinės chemijos katedra, Vilnius, Lietuva

Santrauka

Istirta 1H  →  31P kryzminės poliarizacijos (CP) tai-kant magiskojo kampo sukimą (MAS) kinetika, vyks-tanti nanostruktūrizuotame kalcio hidroksiapatite (na-no-CaHA). Matavimai atlikti kambario temperatūroje (T = 298 K) sukant bandinį 5 kHz dazniu. Siai medziagai yra būdingas didelis protonų mobilumas isilgai OH-... OH-... grandinių. Dėl sios savybės CaHA priskiriamas medziagų klasei, vadinamajai protonų laidininkei. Eks-perimentiniai CP MAS duomenys buvo apdoroti taikant sekuliarųjį ir pusiau nesekuliarųjį sukinių kinetikos mo-delius. Sekuliarusis apibendrintos kvantinės mechani-nės Liouville'o - von Neumann'o lygties sprendinys yra isvedamas ispildant dvi asimptotines sąlygas: 1) pridėtų-jų radiolaukų dazniai ω1 yra daug didesni uz dominuo- jantį dipolinį I-S (siame darbe I = 1H ir S = 31P) sukinių sąveikos suskilimą, t. y. ω1I, ω1S ≫ |b|; 2) I-S sąveikos suskilimas yra daug didesnis uz sukinių difuzijos spartas (|b| ≫ RIdf, RIdp). Pastaroji sąlyga yra sunkiai ispildoma tokiai nutolusių sukinių sistemai, kokia yra nano-CaHa. Kyla abejonių dėl sekuliariojo modelio taikymo. Gautieji kokybiskai identiski rezultatai byloja, kad CP MAS ki-netikos gali būti aprasytos taikant tiek sekuliarųjį, tiek pusiau nesekuliarųjį sukinių kinetikos modelius, netgi tais atvejais, kai |b| ir RIdp yra tos pačios eilės dydziai. Aptikta anomaliai didelė sukinių difuzijos anizotropija, kuri nepriklauso nuo taikyto modelio. Tai gali būti sie-jama su zemos dimensijos 1H sukinių dinamika nano-CaHA protonų rezervuare.