Application of Nuclear Quadrupole Resonance Relaxometry to Study the Influence of the Environment on the Surface of the Crystallites of Powder

1 Introduction

The nuclear quadrupole resonance (NQR) relaxation times, spectra, and line shapes are very sensitive to the nuclei environment. In contrast to the nuclear magnetic resonance (NMR), the NQR frequencies are determined by the electric interactions, so the method allows us to directly investigate the electric fields at the atomic nuclei. NQR spectroscopy data are of fundamental importance for the understanding of the changes in the electron density distribution and for the observation of the physical phenomena occurring in crystals at the micro level. It can be applied to the studies of electron–nuclear interactions in a real crystalline structure and its fragments. It is important for investigating the internal local electric and magnetic fields, the effects of the defects and impurity nature on the properties of solids, lattice dynamics, especially in the field of the phase transitions and critical phenomena.

The NMR relaxometry is also of special importance due to a possibility of employing it for determining the surface area and pore volume of unknown structure [1]. The method is based on using an atomic nucleus of a fluid, imbibed into the pores, as a pore surface area, and it is widely used for characterising porous media, including a variety of rocks, zeolites, cements, etc. A process where the observed magnetisation exponentially decays to the equilibrium value characterises transverse relaxation time (T₂). The relaxation rate of the nuclear magnetisation of the atoms of liquid absorbed by a porous medium usually depends on the degree of their localisation caused by interactions with the pore walls [2] and the gradients of the internal fields.

It is well known that the longitudinal relaxation gives an information about the relatively fast motion, and the dispersion of the NMR relaxation can be studied by measuring the relaxation rate in a magnetic field of varying intensity. Slow dynamic processes can be investigated using the transverse relaxation time (T₂) or relaxation under the influence of a spin-locking pulse (T_1ρ), known as “a spin–lattice relaxation in the rotating frame”. The spin–lattice relaxation in the rotating frame T_1ρ is used, for example, in the magnetic resonance imaging for the purpose of medical diagnosis [3].

The spin interactions with various internal Hamiltonians taking place in the presence of the continuous radio frequency (RF) field of a spin-locking pulse are time dependent. Furthermore, it changes the direction of the spin quantisation and the other characteristics that determine the relaxation. The phenomenon of the spin locking in the presence of a time-dependent Hamiltonian was for the first time demonstrated by Redfield [4].

The dynamic information about the spin–lattice relaxation in the laboratory frame or in the rotating frame is contained in a spectral density function. One of the ways of studying the relaxation mechanism in the radiospectroscopy involves using the so-called “dispersion” and measurements of the relaxation times in magnetic fields of different intensities.

The previous studies showed that the relaxation times for the transverse component of the magnetisation decrease with decrease in the crystallite size. The effect of size of a powder granule on the shape and width of the NQR lines was investigated in our previous work [5]. The NQR linewidth is affected by many factors related to the crystalline type of the substance. These are the lattice defects such as dislocations, impurities, tensions, and other factors that cause the distortion of the electric field gradient near the quadrupole nuclei and lead to the broadening of the observed NQR line. Production process of the substances synthesis also affects the NQR line widths, so it can be difficult to predict which factor is of a greater importance in a particular case. Based on the experiments, we may conclude that unlike the atoms within the crystallite volume, the fluctuations of the surface atoms have a large amplitude directed particularly along the surface normal. The higher is the mobility of the atoms on the surface, the likely the relaxation time of surface spins to be shorter. Nevertheless, short relaxation times of the spins at the surface cannot be the sole factor shortening the relaxation time in the powdered samples. Even for very fine powders, the number of atoms on the surface is only a few percent of the number of atoms that in the bulk. This means that another factor is involved. Bloembergen [6] suggested that the heat transferred from the spins in the bulk to the paramagnetic impurities is due to the spin–spin diffusion. The paramagnetic impurities, in turn, transfer the heat to the lattice. The same mechanism is responsible for transfering heat to the surface spins. Since the spin–lattice relaxation time of the surface spins is very small, they can easily transfer the heat received from the bulk of the crystal to the lattice.

The main physical model of this process is the following: the rotating magnetisation diffuses to the surface where it splits in a very short time due to the strong coupling with the lattice (longitudinal magnetisation) and due to the internal field gradient (for the transverse component of the magnetisation). In contrast to the usual diffusion associated with a mass transfer, spin diffusion results only in a propagation of the spin excitation, whereas the carriers of the magnetic moments do not move. The measurements of the spins diffusion rates, which are dependent on the internuclear dipole interactions, give an information about the internuclear distances in the sample, which can be used to study the surface inhomogeneities.

The experimental results of the ³⁵Cl and ¹⁴N NQR in composite and porous materials, an influence of the external environment and the crystallite size of the powder on the width of the NQR lines, as well as on the spin–spin and spin–lattice relaxation times are described in Ref. [7]. An idealised distribution function of the relaxation times is obtained assuming that the relaxation rate and concentration of the inhomogeneity decrease exponentially with increasing the distance from the surface of a crystallite deeper inside the bulk. It is shown that the model based on the spin diffusion explains the shortening of the signal decay time with the decrease in the material grain size.

The solution of the relaxation–diffusion equation for the nuclear magnetisation for the exponential change of relaxation rate in the surface layer of microcrystals [8] showed that the modality of the relaxation times distribution is determined by the value of the diffusion coefficient of the nuclear magnetisation and by the distribution of the local inhomogeneity near the surface.

The study of the phase transitions in molecular crystals of paradichlorobenzene by NQR relaxometry with the inverse Laplace transform (ILT) [9] allowed establishing the features of the β → α phase transition in a porous material. The crystallisation of potassium chlorate from an aqueous solution by means of ³⁵Cl NQR was studied in Ref. [10]. The effect of the crystal growth kinetics on the linewidths and the spin–spin relaxation time distributions was also demonstrated.

The effect of the surface crystallites of a powder on the relaxation time T_1ρ distribution was studied in the experiment where the potassium atoms on the surface of the KClO₃ crystallites were substituted by the sodium atoms by putting the sample in the aqueous solution of NaCl [11]. It is shown that the low-frequency component of a multimodal distribution in this case is shifted to the higher frequencies. This indicates that this component is caused by the surface molecules, which are replaced.

The aim of this work is an experimental study of the possibility of applying the NQR relaxometry with the ILT to the study of the surface state of the powder crystallites and the effects of the crystallite environment, in addition, the distribution of relaxation times T₁, T₂, and T_1ρ and the determination of the correlation between spin–spin and spin–lattice relaxation times.

2 Materials and Methods

The experimental studies were performed on the Tecmag Apollo NQR spectrometer powered by the TNMR software. To measure the spin–lattice relaxation time, we used an inversion-recovery pulse sequence 180° – τ – 90. A Carr–Purcell–Meiboom–Gill (CPMG) pulse sequence was used for measuring the spin–spin relaxation times T₂. The sequences with the spin-locking pulse with variable lengths were used to measure the spin–lattice relaxation in the rotating frame T_1ρ. When changing duration of the spin-locking pulse the oscillations of the signal intensity with a nutation frequency damped [12]. The spin–spin relaxation time in the rotating frame T_2ρ can be estimated from the decay of these oscillations and used further to analyse the parameters of the molecular dynamics. The methods mentioned earlier are well known and do not require detailed description. The TNMR “Kinetics Exp” script was used for studying how the processes evolve in time. All measurements were performed at a temperature T of 297 K.

We studied commercial samples of the chemical purity, and the granule size was reduced by grinding them mechanically in a mortar. The granules of the different size were obtained by sieving them through a mesh of a proper size. The granule size was then checked by a Horiba LabRAM HR Raman microscope.

Based on these measurements, we concluded that the average size of crystallites in our potassium chlorate powder was about 150 microns. The sample was studied in a number of different environments such as air, water, ethanol, soap solution, engine oil, epoxy resin, and a compound of the match heads. The match head compound contains 46.5% of KClO₃ as a solidified suspension with an admixture of K₂Cr₂O₇, S, Pb₃O₄, ZnO, ground glass, and the bone glue.

The ILT was performed using the regularised inverse Laplace transform (RILT) software by Iari-Gabriel Marino [13]. Main advantage of the RILT compared to other algorithms is the absence of the negative values and artifacts in the form of “wiggles” on the resulting curve of the relaxation times distribution. To obtain T₁ – T₂ and T_1ρ – T₂ correlations, we modified the RILT program for the case of the 2D inverse Laplace transformation.

The pulse sequence for T₁ – T₂ correlation is well described in the literature. The sequence for T_1ρ – T₂ correlation was composed by a sequence with a spin-locking pulse of variable duration and with an addition of a CPMG sequence (Fig. 1). After a π/2 pulse, which rotates the nuclear magnetisation in the transverse plane, the magnetisation relaxes in the presence of the B₁ field directed along the magnetisation in the rotating frame similar to the longitudinal magnetisation in a B₀ field for the case of NMR. This spin-locked magnetisation will relax with a time constant of the spin–lattice relaxation T_1ρ in the rotating frame and fade within duration of the spin-locking pulse with the B₁ field. After, a spin-locking pulse is followed by a CPMG sequence. The experimental data represent the dependence:

Figure 1:

The pulse sequence for study of the T_1ρ – T₂ correlation.

(1)S(t1, 2nτ) = ∬f(T1ρ, T2)exp(−t1T1ρ)exp(−2nτT2)dT1ρdT2 + err (1)

which was analysed using a 2D ILT. Here, f(T_1ρ, T₂) is the desired distribution function, n is the number of echoes in a CPMG sequence, and err is an unknown summand of the error.

The spin–lattice relaxation in the rotating frame (T_1ρ) is a mechanism by which the excited magnetisation vector decays under the influence of a spin-locking radio-frequency excitation, i.e. via a weak effective magnetic field rotating in the x–y plane with the same frequency as the magnetisation vector. In this case, the spin–lattice relaxation is measured at the signal decay under the spin-locking conditions, which creates a rotating magnetic field at the resonant frequency [12]. T_1ρ filtering is used in NMR for separating the signals from a mixture of amorphous and crystalline substances.

3 Results and Discussion

The resulting distributions of the ³⁵Cl NQR relaxation times T₂, T₁, and T_1ρ for different environments of the KClO₃ crystallites are shown in Figures 2–4. Figure 2 shows that all samples have two characteristic components of spin–spin relaxation, which differ from each other by a factor of 2 or 3. The exception is the unimodal distribution in the match heads where the environment is not homogeneous. Both short and long spin–spin relaxation times depend on the material surrounding the potassium chlorate granules. For the interpretation of the bimodal distribution of T₂, a simple model proposed in paper [7] can be used. It is reasonable to assume that the concentration of the local inhomogeneity in a surface layer decreases exponentially with increasing the distance from the surface of the crystallite. Then, for an individual spherical crystallite of a radius b, we have

Figure 2:

The distribution of the T₂ relaxation times: w – water, et – ethanol, c – control, s – soap solution, o – engine oil, ep – epoxy resin, and m – match heads.

Figure 3:

The distribution of the T₁ relaxation times: w – water, et – ethanol, c – control sample, s – soap solution, o – engine oil, ep – epoxy resin, and m – match heads.

Figure 4:

The distribution of the T_1ρ relaxation times: w – water, et – ethanol, c – control, s – soap solution, o – engine oil, ep – epoxy resin, and m – match heads.

(2)n(r) = nsexp(−b − rr0) (2)

where r is the distance from the center, r₀ is the effective penetration depth of the inhomogeneity, and n_s is the surface concentration of the local inhomogeneity. It can be assumed that the signal decay 1/T₂ (relaxation rate) is proportional to the concentration of the local inhomogeneity n(r):

(3)1T2 = 1T2i + (1T2s − 1T2i)exp(−b − rr0) (3)

where T_2i is the relaxation time in the center of a crystallite with b/r₀→∞ and T_2s is the relaxation time on the surface.

The transverse relaxation of the nuclear magnetisation depends not only on the spin–spin interactions but also on the spin–lattice interaction, i.e., an energy transfer from the nuclear spins to the lattice can occur with a decrease in the transverse magnetisation. A variation of the long-time components of T₂ in Figure 2 with the change of environment is caused by the changes in the crystal lattice mobility. However, the short-time component of T₂ is also changed considerably because of the different character of lattice inhomogeneity near the crystallite surface.

Unlike the bimodal distribution function f(T₂), distribution function f(T₁) is unimodal practically for all samples. The spin–lattice relaxation time T₁ for the KClO₃ powder at room temperature is practically the same for all materials used as the sample environment. The only exception is the T₁ distribution for the match heads (Fig. 3), where the distribution is multimodal cause of the complex composition of the crystallite environment. In the first case, it is caused by the fact that the mobility of the crystalline lattice with relatively large crystallite size remains practically unaffected by the change of the environment. The crystallite size of the potassium chlorate in a solidified suspension of the bone glue (match heads) is significantly smaller, and the mobility of the lattice is more sensitive to the character of the environment. Thus, the relaxation time T₂* for a reference KClO₃ sample consisted of 150 μm crystallites was determined from the Lorentzian linewidth, which provided a value of 0.4 ms. For the KClO₃ microcrystals in the composition of a match head, the T₂* time is equal to 0.1 ms. The T₂* time can also be calculated for the spherical granules of an equivalent radius r(T₂*) in a way it was proposed in Ref. [14]. This model also enables an opportunity for us to estimate the characteristic granule size in the match heads we studied. The numerical calculation gives a value for r of 20 μm.

Wetting or non-wetting of solid surface by a liquid appears depending on the strength of interactions between the molecules of the surface layer. The greater is the surface roughness, the stronger are the properties of the surface, stimulating either attraction or repulsion of the liquid from the solid surface.

The absorption of a substance from the volume of a liquid and its concentration on a surface of a solid can be explained by the presence of adsorption forces of a different nature, acting between the atoms and molecules on the surfaces. The layer is formed on the interfacial surface, the amount of which in disperse systems is very big. The surface tension σ is related to the concentration of adsorbed component C, its chemical potential μ, and excess Gibbs adsorption Γ [15].

The molecules of the surface layer have an excess of potential energy. The energy can be decreased spontaneously by reducing the surface tension in the system or diminishing the surface area. This is possible as a result of the reduction in the surface tension or the interfacial surface area. Speaking about the colloidal particles, their surface area can be decreased by a well-known process called coagulation. Here, the surface tension can be decreased by the attractive forces between the molecules forming an environment and that on the surface of the crystallites. Surface tension of KClO₃ relative to air at 20°°C is σ=81 mN/m [16]. Among the environments we studied, ethanol has the lowest surface tension relative to air (σ=22.39 mN/m at 20°C). The interfacial tension generally is lower than that for the free phases.

Thus, decreasing the surface energy of a solid is explained by adsorption of molecules of a liquid on the surface and by minimising the imbalance between the atoms forming a surface layer of the solid. Both have a significant effect on the distribution of the relaxation times.

In contrast to T₁, the spin–lattice relaxation time in the rotating frame T_1ρ is much more sensitive to the change of the environment of a crystallite (Fig. 4).

The B₁ field decreases the influence of dipole relaxation, static dipole interactions, chemical exchange, and of the field gradients on the signal. The T_1ρ time is always greater than the T₂ time. In a typical experiment for measuring the T_1ρ time, a spin-locking pulse duration increases while its amplitude remains constant. As an alternative, it is also possible to measure T_1ρ as a function of the amplitude of the spin-locking pulse B₁ with a constant pulse duration. “T_1ρ dispersion” curve illustrates a variation in the spin–lattice relaxation times in the rotating frame, as a function of γB₁ value.

The correlations of the T₁ – T₂ and T_1ρ – T₂ relaxation times in a reference sample of potassium chlorate are shown in Figures 5–7. The found T₁ – T₂ distribution shows that the spin–lattice and spin–spin relaxation times are strongly correlated. The same applies to the correlation between the T_1ρ and T₂ relaxation times.

Figure 5:

T₁ – T₂ correlation for KClO₃ at T=77 K.

Figure 6:

T₁ – T₂ correlation for KClO₃ at T=297 K.

Figure 7:

T_1ρ – T₂ correlation for KClO₃ at T=297 K.

The 2D distribution appears to be a more versatile for studying our system than a one-dimensional (1D) distribution, because it includes more experiments. The peaks of the various components in the 1D distribution of the relaxation times (T₁ or T₂) may overlap, but they are best resolved in the 2D correlation distribution of f (T₁, T₂). By means of 2D distribution, 1D distributions of T₁ and T₂ can be obtained by integration over the corresponding dimension.

The T₁/T₂ ratio can be obtained from the correlation spectrum, which is important for studying the molecular mechanisms of the surface relaxation. The T₁/T₂ ratio of the relaxation times depends on the crystallite size, crystallite shape, and on the properties of the environment. If we account longitudinal and transverse relaxation as the spin–lattice relaxation, then the T₁/T₂ ratio is 1. For the reference KClO₃ sample at T=77 K (Fig. 5), the ratio T₁/T₂=120 and 1200 for the various peaks of the bimodal T₂ distribution, respectively. At temperature T=297 K, these values of T₁/T₂ become 1.5 and 5.0, respectively (Fig. 6), which appears due to a decrease in the T₁ relaxation time with increasing temperature.

On 2D distribution of the relaxation times T_1ρ and T₂ (Fig. 7), only a single peak with a T_1ρ/T₂ ratio of 1.2 is confidently detected.

4 Conclusions

In conclusion, we would like to summarise the main results of our work; it was found that the distributions of the T₁ relaxation times for all environments of the potassium chlorate crystallites we studied are practically unimodal and the T₂ distributions are bimodal. A simple model based on the exponential distribution of the local inhomogeneity in the surface layer of microcrystallites was used for the interpretation of the bimodal character of the T₂ relaxation times distribution.

The changes in the spin–spin relaxation times and spin–lattice relaxation times in the rotating frame for the KClO₃ microcrystallites, placed in a various environments, can be explained by adsorption of molecules of the liquid onto a surface of the crystallites and by the change of the atomic mobility in the surface layer of a solid. Better understanding of the role of the impurities, dislocations, lattice defects, and how all the NQR parameters vary with temperature requires further investigation of the surface of the microcrystals by means of NQR relaxometry. This method can be successfully used for studying the surface phenomena in solids.