Universal Dependence of Nuclear Spin Relaxation on the Concentration of Paramagnetic Centers in Nano- and Microdiamonds

An analysis of our data on 1H and 13C spin–lattice and spin–spin relaxation times and rates in aqueous suspensions of purified nanodiamonds produced by detonation technique (DNDs), DNDs with grafted paramagnetic ions, and micro- and nanodiamonds produced by milling bulk high-temperature high-pressure diamonds is presented. It has been established that in all the studied materials, the relaxation rates depend linearly on the concentration of diamond particles in suspensions, the concentration of grafted paramagnetic ions, and surface paramagnetic defects produced by milling, while the relaxation times exhibit a hyperbolic dependence on the concentration of paramagnetic centers. This is a universal law that is valid for suspensions, gels, and solids. The results obtained will expand the understanding of the properties of nano- and microdiamonds and will be useful for their application in quantum computing, spintronics, nanophotonics, and biomedicine.

In this paper, we analyze the results of measuring proton and carbon nuclear spinlattice and spin-spin relaxation times and rates in (i) aqueous suspensions of highly purified DNDs, (ii) aqueous suspensions of DNDs with grafted paramagnetic ions, (iii) powdered DNDs grafted with paramagnetic ions, and (iv) powdered micro-and nanodiamonds produced by milling bulk diamonds prepared by the high-temperature high-pressure (HTHP) method. We established that in all the studied materials the relaxation rates depend linearly on the concentration of nanodiamonds in suspensions, the concentration of grafted paramagnetic ions in suspensions and in powder samples, and surface paramagnetic defects produced by milling, while the relaxation times exhibit a hyperbolic dependence on the concentration of paramagnetic centers. This is a universal law that is valid for suspensions, gels, and solids. The results obtained will expand the understanding of the behavior of nano-and microdiamonds and will be useful for their applications in quantum computing, spintronics, nanophotonics, and particularly in biomedical applications.
Submicron diamond powders of Syndia SYP series, manufactured by L.M. Van Moppes & Sons SA, Geneva, Switzerland, and hereafter identified by the denomination SYP, were produced by milling initial CDFS HPHT microdiamond crystallites with an average particle size of 100 µm, which resulted in several fractions with average particle sizes of 18,30,86,130,208, and 386 nm [27]. An additional laboratory purification stage was carried out to exclude ferro-and paramagnetic impurities from the commercial SYP samples.
It is well established that the surface of DND particles is terminated by hydrogen atoms forming hydrocarbon, hydroxyl, and carboxyl groups [30,31]. They are the source of 1 H nuclear spins. 1 H and 13 C NMR measurements of powder samples were carried out at room temperature (T = 295 K) using a Tecmag (Houston, TX, USA) pulse solid-state NMR spectrometer and an Oxford superconducting magnet in an external magnetic field B 0 = 8.0 T, corresponding to the 1 H and 13 C resonance frequencies of 340.52 and 85.62 MHz, correspondingly. 13 C spin-lattice relaxation times T 1 were measured using a saturation comb pulse sequence ( π 2 pulse train) [34]. Magnetization recovery in measuring T 1 was fitted by a stretched exponential , which is characteristic of the spin-lattice relaxation through paramagnetic defects [12][13][14][15]19,20,[26][27][28][29][30][31]35]. Here, M ∞ is the equilibrium magnetization, and the parameter α varies in the range of 0.5 < α < 1. 13 C spin-spin relaxation times T 2 were measured using the Hahn echo method. 1 H NMR measurements of nanodiamond suspensions were carried out at a temperature of 310.1 K (37 • C). The 1 H spin-lattice relaxation times T 1 were measured using an inversion recovery pulse sequence [34], while the spin-spin relaxation times T 2 were measured using a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence [36].

Suspensions of Purified DNDs and DNDs with Grafted Paramagnetic Ions
As is known, DND particles exhibit intrinsic localized paramagnetic defects: (i) P1 nitrogen paramagnetic centers distributed over the entire diamond core and (ii) unpaired electron spins of dangling bonds positioned mainly in the near-surface layer [26,[28][29][30][31]. The total defect density in DND particles measured by the EPR method is around 6 × 10 19 spin/g [26,[28][29][30][31]. In DND suspensions, the relaxation of the proton nuclear spins of the solvent is accelerated, owing to the interaction of protons with unpaired electron spins of the aforementioned paramagnetic defects [16][17][18]. The contributions of the DND-inherent paramagnetic defects to the experimentally measured proton spin-lattice and spin-spin relaxation rates R exp 1 and R exp 2 in suspensions are described [16][17][18] by the second term of the equations where T solv on the DND content. This finding is in accordance with the fundamentals of the spin relaxation theory [22][23][24], which demonstrates a linear dependence of the relaxation rate on the concentration of paramagnetic centers/defects. Herewith, as it follows from Equations (1) and (2) and the experimental data presented in Figures 1 and 2, both proton spin-lattice and spinspin relaxation times demonstrate a hyperbolic dependence on the concentration C DND of nanodiamonds in suspension according to Equations (3) and (4): Materials 2022, 15, x FOR PEER REVIEW 4 of Figure 1. Dependence of the spin-lattice relaxation rate R1 (circles) [16] and the spin-lattice relax tion time T1 (triangles) of water protons in aqueous DND suspensions on the DND concentration   This experimental result contrasts with the linear concentration dependence of T1 and T2 reported by Sękowska et al., [21]. The latter is inconsistent with that published in the literature and the fundamentals of the relaxation phenomena in nuclear spin systems, which casts some doubt on the correctness of the measurements and conclusions made in Ref. [21]. Herewith, we note that the measurements of Sekowska et al., particularly those of T2, were carried out in a limited range of nanodiamond concentrations, which causes some difficulties in establishing the character of the concentration dependence measured by these authors.
Similar dependencies were obtained for suspensions of the gadolinium-grafted DND (Gd-DND), which are shown in Figures 3 and 4. Contributions of paramagnetic gadolinium ions grafted to the DND surface to the spin-lattice and spin-spin relaxations of water protons in this case are: Figure 2. Dependence of the spin-spin relaxation rate R 2 (circles) [16] and the spin-spin relaxation time T 2 (triangles) of water protons in aqueous DND suspensions on the DND concentration.
This experimental result contrasts with the linear concentration dependence of T 1 and T 2 reported by Sękowska et al., [21]. The latter is inconsistent with that published in the literature and the fundamentals of the relaxation phenomena in nuclear spin systems, which casts some doubt on the correctness of the measurements and conclusions made in Ref. [21]. Herewith, we note that the measurements of Sekowska et al., particularly those of T 2 , were carried out in a limited range of nanodiamond concentrations, which causes some difficulties in establishing the character of the concentration dependence measured by these authors.
Similar dependencies were obtained for suspensions of the gadolinium-grafted DND (Gd-DND), which are shown in Figures 3 and 4. Contributions of paramagnetic gadolinium ions grafted to the DND surface to the spin-lattice and spin-spin relaxations of water protons in this case are: where C Gd is the Gd(III) ions concentration in suspension.
where Gd C is the Gd(III) ions concentration in suspension.   . Figure 4. Dependence of the spin-spin relaxation rate R2 (circles) [16] and the spin-spin relaxation time T2 (triangles) of water protons in aqueous Gd-DND suspensions on the Gd 3+ ion concentration.
Therefore, the proton spin-lattice and spin-spin relaxation times reveal a hyperbolic dependence on Gd C : Figure 4. Dependence of the spin-spin relaxation rate R 2 (circles) [16] and the spin-spin relaxation time T 2 (triangles) of water protons in aqueous Gd-DND suspensions on the Gd 3+ ion concentration.
Therefore, the proton spin-lattice and spin-spin relaxation times reveal a hyperbolic dependence on C Gd : We note that Gd(III) ions have a large unpaired electron spin of S = 7/2 and a large magnetic moment of 7.9 µ B (here, µ B is the Bohr magneton), thus their contribution to relaxation exceeds the DND contribution by more than an order of magnitude [16][17][18].
In addition to our data, we mention the measurements of an aqueous solution of the nanodiamond-polyglycerol-gadolinium(III) conjugate DND-PG-Gd(III) [37]. The relaxation rates R 1 of water protons in this material show a linear dependence on the Gd concentration in magnetic fields of 1.5 T, 3.0 T, and 7.0 T.

Powder DNDs with Grafted Paramagnetic Ions
Similar dependences of the nuclear spin relaxation in nanodiamonds on the concentration of the paramagnetic ions were obtained in our measurements of powder samples. In this case, the spin-lattice relaxation rate R 1 = 1 T 1 of the nuclear spin I, which interacts with the unpaired electron spin S of the paramagnetic ion, is given by the expression [14,15,19,20,22,[29][30][31] Here, γ I and γ S are the nuclear and electron gyromagnetic factors, ω I is the nuclear Larmor angular frequency, r is the distance from the nucleus to the paramagnetic ion, τ e is the correlation time of the electron spin of the paramagnetic ion, and N S is the number of paramagnetic ions in the particle. The obtained dependences of the 1 H and 13 C spin-lattice relaxation times and rates on the concentration of paramagnetic Cu 2+ and Gd 3+ ions grafted to the DND surface are presented in Figures 5-8. All these data show a linear dependence of the spin-lattice and spin-spin relaxation rates R 1 and R 2 and a hyperbolic dependence of the relaxation times T 1 and T 2 on the paramagnetic ions concentration. This finding is consistent with the fundamentals of the spin relaxation theory [22][23][24], which demonstrates a linear dependence of the relaxation rate on the concentration of paramagnetic centers.

Powder HPHT Nanodiamonds
Let us move on to the powder nanodiamonds of the Syndia SYP series manufac tured by L.M. Van Moppes & Sons SA (Switzerland) by milling the initial high-pressure high-temperature (HPHT) microdiamond crystallites with an average particle size o ∼100 μm. According to the size distribution datasheets provided by the manufacturer this milling process provides several SYP fractions with average particle sizes of 18, 30 86, 132, 208, and 386 nm, respectively, which were used in our study along with the initia SYP micro CDFS of the size of 100 μm. In Figure 9, we present our NMR measurement of SYP nanodiamonds of various sizes, in which the main contribution to relaxation i

Powder HPHT Nanodiamonds
Let us move on to the powder nanodiamonds of the Syndia SYP series m tured by L.M. Van Moppes & Sons SA (Switzerland) by milling the initial high-p high-temperature (HPHT) microdiamond crystallites with an average particle ∼100 μm. According to the size distribution datasheets provided by the manu this milling process provides several SYP fractions with average particle sizes o 86, 132, 208, and 386 nm, respectively, which were used in our study along with th SYP micro CDFS of the size of 100 μm. In Figure 9, we present our NMR measu of SYP nanodiamonds of various sizes, in which the main contribution to relax Figure 8. Dependence of the 13 C spin-spin relaxation rate R 2 (circles) and the spin-spin relaxation time T 2 (triangles) in Gd-DND powders on the Gd 3+ ion concentration.

Powder HPHT Nanodiamonds
Let us move on to the powder nanodiamonds of the Syndia SYP series manufactured by L.M. Van Moppes & Sons SA (Switzerland) by milling the initial high-pressure hightemperature (HPHT) microdiamond crystallites with an average particle size of ∼100 µm. According to the size distribution datasheets provided by the manufacturer, this milling process provides several SYP fractions with average particle sizes of 18, 30, 86, 132, 208, and 386 nm, respectively, which were used in our study along with the initial SYP micro CDFS of the size of 100 µm. In Figure 9, we present our NMR measurements of SYP nanodiamonds of various sizes, in which the main contribution to relaxation is made by paramagnetic defects (mainly unpaired electron spins of broken bonds) associated with surface and subsurface defects that appear during the process of diamond milling [27]. Such paramagnetic centers produced by mechanical damage (e.g., milling) are often found in insulators and semiconductors, including diamonds, and are observed in EPR experiments [38][39][40][41]. On diminishing the average size of the ND fraction, the density of these defects increases from 7.6 × 10 18 spin/g in the fraction of the largest particle size to 3.3 × 10 19 spin/g in the fraction of the smallest particle size. Figure 9 clearly shows the linear dependence of the spin-lattice relaxation rate and the hyperbolic dependence of the spin-lattice relaxation time on the concentration of the paramagnetic defects in this series of materials. aterials 2022, 15, x FOR PEER REVIEW 9 milling [27]. Such paramagnetic centers produced by mechanical damage (e.g., mill are often found in insulators and semiconductors, including diamonds, and are obser in EPR experiments [38][39][40][41]. On diminishing the average size of the ND fraction, density of these defects increases from 7.6 × 10 18 spin/g in the fraction of the largest p cle size to 3.3 × 10 19 spin/g in the fraction of the smallest particle size. Figure 9 cle shows the linear dependence of the spin-lattice relaxation rate and the hyperbolic pendence of the spin-lattice relaxation time on the concentration of the paramagn defects in this series of materials. Hyperbolic-like concentration dependence of T1 was recently obtained in meas ments of the 1 H spin-lattice relaxation of aqueous solutions of nanodiamonds of 18 125 nm in diameter prepared by the HPHT technique [42].
The data obtained in our measurements demonstrate the universality of the pendence of the nuclear spin relaxation in nanodiamonds on the concentration of paramagnetic centers both in suspensions and in powder samples. This is a universal that is valid for solutions, suspensions, gels, and solids [12-16-18,22,23,27,29-31].

Other Materials Containing Gadolinium Ions
The universality of the dependences of nuclear spin relaxation in nanodiamond the concentration of paramagnetic centers in both suspensions and powder samples tained in our measurements is supported by data on other non-diamond lantha complexes promising for NMR/MRI diagnostic probes [43]. For example, proton rel tion rates for Gd2O3 nanodisks of different diameters and Gd-doped iron oxide nano ticles of various sizes and shapes were measured in water after the nanoparticle sur functionalization with polyethylene glycol (PEG) dibasic acid. Both relaxation rate and R2 reveal a linear dependence as a function of the Gd and Gd-Fe concentrat [44,45].
The relaxation rates R1 and R2 of water protons taken at room temperatur aqueous solutions of SiO2-coated quantum dots with grafted Gd-DOTA complexe various concentrations ranging from 0.125 to 4 μM reveal a linear dependence on the Hyperbolic-like concentration dependence of T 1 was recently obtained in measurements of the 1 H spin-lattice relaxation of aqueous solutions of nanodiamonds of 18 and 125 nm in diameter prepared by the HPHT technique [42].
The data obtained in our measurements demonstrate the universality of the dependence of the nuclear spin relaxation in nanodiamonds on the concentration of the paramagnetic centers both in suspensions and in powder samples. This is a universal law that is valid for solutions, suspensions, gels, and solids [12][13][14][15][16][17][18]22,23,27,[29][30][31].

Other Materials Containing Gadolinium Ions
The universality of the dependences of nuclear spin relaxation in nanodiamonds on the concentration of paramagnetic centers in both suspensions and powder samples obtained in our measurements is supported by data on other non-diamond lanthanide complexes promising for NMR/MRI diagnostic probes [43]. For example, proton relaxation rates for Gd 2 O 3 nanodisks of different diameters and Gd-doped iron oxide nanoparticles of various sizes and shapes were measured in water after the nanoparticle surface functionalization with polyethylene glycol (PEG) dibasic acid. Both relaxation rates R 1 and R 2 reveal a linear dependence as a function of the Gd and Gd-Fe concentrations [44,45].
The relaxation rates R 1 and R 2 of water protons taken at room temperature in aqueous solutions of SiO 2 -coated quantum dots with grafted Gd-DOTA complexes at various concentrations ranging from 0.125 to 4 µM reveal a linear dependence on the Gd concentration [46].
Longitudinal relaxation rates and transverse relaxation rates as a function of concentration for aqueous solutions of gadolinium diethylenetriamine-pentaacetic acid (Gd-DTPA) Materials 2022, 15, 5774 9 of 11 and gadolinium DTPA-bis methylamide (Gd-DTPA BMA) at 23 • C represent a linear regression of the data, from which the relaxation rates R 1 and R 2 were determined [47].
These results support well the above findings about the universality of the dependence of nuclear spin relaxation on the concentration of the paramagnetic centers both in suspensions and powder materials.

Conclusions
It has been established that the dependences of the nuclear spin-lattice and spin-spin relaxation times and rates in nano-and microdiamonds on the concentration of intrinsic paramagnetic defects, surface-grafted ions, and milling-induced paramagnetic defects reveal a universal behavior for both suspensions and powder samples. The relaxation rates show linear concentration dependence, while the relaxation times exhibit hyperbolic dependence on the concentration of paramagnetic centers. This is a universal law that is valid for solutions, suspensions, gels, and solids. The data obtained will expand the understanding of the behavior of nanodiamonds and will be useful for their applications in quantum computing, spintronics, nanophotonics, and biomedicine. In our opinion, this is particularly important for the use of the nanodiamond suspensions as contrast agents and phantoms for MRI [16][17][18]20,49].
Funding: This research received no external funding.