1. Introduction

Cross-polarization (CP), combined with magicangle spinning (MAS), is one of the ‘classical’ and most widely used methods in solid-state NMR spectroscopy [1–3]. CP proved to be a powerful tool studying fine structural details and dynamics in advanced complex materials  [4–6]. Naturally, carbons and protons are the main available dipolar coupling partners that decide the CP kinetics in most organic molecular systems. The processing of CP kinetic data, i.e. the  consideration of the evolution of communication 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 accounting for the  effective sizes of spin clusters [5–8].

The purpose of the present work was to extend the  CP MAS kinetic studies to a  very important class of matter  –  to soft low-dimensional proton-conducting materials, which possess high degrees of internal freedom for proton motion and proton disorder along the H-bonded chains. It is expected that the subtle coherent CP effects of quantum nature can be quenched upon increasing the incoherent intramolecular motion and local disorder. This could impede the  analysis of experimental data forcing one to create novel models and concepts of processing. The presence of remote spins (POO¹H in the present case) can also influence the extracted ¹H–¹³C dipolar coupling values that can be misinterpreted as due to a motional effect [9]. Poly(vinyl phosphonic acid) (pVPA) was chosen for these studies for several reasons. pVPA is the phosphonic acid tethered polymer that contains a high concentration of acid groups and adopts a  relatively simple structure. Furthermore, phosphonic acid is considered to form a strong hydrogen-bonding network involving both P=O and P–OH as proton acceptor and proton donor groups. The mobility of the phosphonic acid POOH groups and their acidic protons in pVPA was studied applying various solid-state NMR techniques  [10, 11]. Regarding industrial and other practical aspects, a considerable interest in recent years has arisen in polymer membranes containing phosphonic acid groups as these materials are very promising for fuel cell technologies [11].

Impedance spectroscopy was applied to study the activation energy of mobility of protons along H-bond chains using the experimentally measured temperature and frequency dependences of complex dielectric permittivity.

2. Experiment

Poly(vinyl phosphonic acid) (pVPA) was purchased from Sigma-Aldrich and used as received. The chemical structure of pVPA including the possible hydrogen-bond chain is shown in Fig. 1.

The solid state NMR experiments were performed using a  Bruker AVANCE III HD spectrometer with a double resonance CP MAS probe. The experiments were performed in the 9.4 T magnetic field using an Ascend wide bore superconducting magnet.

The ¹H→¹³C cross-polarization contact for ¹³C was achieved with a rectangular 71 kHz RF pulses at the n = ±1 Hartmann–Hahn (HH) matching condition for the spinning sample. The sample was spun at a magic angle at 10 kHz rate using a 4 mm zirconia rotor at room temperature. Spectra consisted of 3020 real data points and were accumulated by 16 scans with a repetition delay of 2.7 s. The  CP MAS kinetics were registered by varying the  contact times from 10  μs to 10  ms in increments of 10 μs. NMR spectra were processed using the Topspin 3.2 software. Some additional processing was carried out using the Microcal Origin 9 and MathCad 15.0 packages.

Impedance spectroscopy experiments were performed using a HP 4284A Precision LCR-meter. Dielectric measurements were carried out at 120–300 K temperature and 10²–10⁶ Hz frequency ranges using the custom-made chamber for powder measurement. The temperature in the sample was controlled using a  Keithley 2700 multimeter with a T-type thermocouple. The experimental data were collected during the cooling cycle at a cooling rate of 1 K/min. Low temperatures were achieved using the liquid nitrogen vapour.

3. Theoretical models of CP kinetics

The topic of CP kinetics, i.e. the dependence of CP signal intensity I(t) on the contact time (t), was reviewed in more detail in Refs. [1, 4]. The most widely used kinetic models that exhibit the coherent oscillatory behaviour of CP intensity originate from the work of Müller et al. [12]. The strongly coupled spin pair I–S (I  =  ¹H and S  =  ¹³C in the  present work) was considered there as an open quantum system interacting with remote I spins. This effect was represented by an isotropic spin-diffusion superoperator [1, 12, 13]. The kinetics of the CP signal intensity I(t) is then described by

where 1/T_1ρ is the spin-lattice relaxation rate of spin I in the rotating frame, which was added by Naito and McDowell [13]. Parameters k₁ and k₂ are first-order rate constants that describe the  decay of the  transient oscillation and the increase in the net magnetization, respectively. In the case of isotropic spin diffusion the constraining relation is k₁ = 3k₂/2 [12, 14]. The parameter b is dipolar splitting, which depends on the distance r between the two nearest interacting nuclei (I and S) and on the angle θ between the r vector and magnetic field direction:

Later [14] Eq. (1) was developed for the extended spin-systems S–I_N and under MAS,

where ω_0I is the Larmor frequency of I spins, and 〈S〉_qe is the quasi-equilibrium polarization of S spin. In the thermal equilibrium 〈S〉_qe has the following form:

The function g_n(t) in Eq. (3) incorporates the angular averaging as well as MAS effects on the cos-function (Eq. (1)) with n depending on which the HH sideband matching condition ω_II–ω_IS = nω_MAS was fulfilled [14]. In the case when the b distribution and its Fourier image can be approached by Gaussian functions, the function g_n(t) in Eq. (3) can be simply replaced by the Gaussian decay [5–7, 15]

where the  time constant T₂ is assumed to be approximately equal to the spin–spin relaxation time, at least at long contact times. Inserting Eqs. (4, 5) into Eq. (3), one obtains

However, the  coherent oscillatory behaviour, which is experimentally often observed in the initial stage of CP, is faded in the  calculated curves when a  Gaussian decay approach for g_n(t) is applied. To refine the coherent oscillatory behaviour, a more precise averaging of cos(2πbt/2) has to be carried out. The  digital averaging has been performed summing its values weighted by the fraction of spin pairs with a set of spatial parameters that corresponds to the oscillation frequency b_i/2,

where the  normalized spin-coupling distribution P(b) is introduced. Its profile is usually unknown and can be complex for soft disordered solids. Various shapes of P(b) can be tested for the practical use in the processing of the experimental CP kinetic curves and the most proper one chosen [5, 15].

4. Results and discussion

Despite the  fact that pVPA contains two carbon sites (–CH– and –CH₂–), due to a very small difference in the chemical shifts only the single signal at ~31 ppm was registered in the ¹³C NMR spectrum (Fig. 1). The ¹H → ¹³C CP MAS kinetics were measured and processed for pVPA using a  high density experimental data set (Fig. 2). As shown in the earlier works [5–7], this allows one to reduce the excessive freedom in the 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 a series of multi-parametrical models.

Table 1. The fit parameters of ¹H → ¹³C CP MAS kinetics of pVPA (Fig. 2) obtained using the models of isotropic spin diffusion (k₁/k₂ = 3/2) with T₂-averaging and cosine-averaging with the Lorentz-shaped spin-coupling distribution profile.

^* The dipolar splitting values are taken for comparison from Fig. 3.

The experimental ¹H → ¹³C CP MAS kinetic data (Fig. 2) was processed applying the isotropic spin-diffusion model with T₂-averaging with the  constraint k₁/k₂ = 3/2, i.e. using Eq. (6), and the cosine-averaging routine (Eqs. (3, 4, 7)). We suggested to use the Lorentz function L(b) ~ 1/(1 + ((b–b₀)/∆)²) to approach the spin-coupling distribution profile P(b) = L(b). In this case T₂ can be evaluated from the width of the Lorentz function (∆) as T₂ = 1/π∆. Smoothing of the  coherent oscillatory behaviour due to a Gaussian decay approach for g_n(t) (Eqs. (5, 6)) is demonstrated in detail (Fig. 2) at short contact time (<0.8 ms, see in the inset). The results of fitting are presented in Table 1.

A significant advantage of solid-state NMR methods is the access to static structural details as well as the site-resolved dynamic order parameters associated with the mobility of different functional groups of complex molecules, like liquid crystals, polymers, proteins, etc., in the  crystalline and powder samples [16, 17]. The obtained results on the CP MAS kinetics lead to certain conclusions on the fine dynamic features of pVPA as well.

It is gratifying to state that similar values of the  ¹³C  –  (¹H)_N spin-cluster sizes (N) have been obtained using both T₂- and cosine-averaging models without any constraint on the  flow of the  nonlinear curve fitting (χ²  →  minimum). Namely, the deduced values N ≈ (50–60)±40 (Table 1) can be recognized as the ‘effective’ number of spins I (protons) interacting with the carbons. Note the huge margins of the N error that appear to be significantly larger than in the cases of other spin-systems in solids that are more rigid and thus more ordered [6, 8]. Despite large errors, the finite value of N means that the spin-coupling and the CP transfer cover several tens of neighbouring C–H sites within pVPA.

Monitoring of the changes in the dipole–dipole coupling allows the identification of dynamic processes undergoing in the sample. The local dynamic order parameter S is defined as the ratio of the experimental dipolar coupling constant and the calculated static dipolar coupling constant [17–20]. In order to determine the experimental dipolar coupling constants the Fourier transform was carried out over the  measured CP intensity kinetics I(t) (Fig.  5). The  obtained dipolar splitting values (b) are almost identical with the maximum positions of the spin coupling distribution profile (b₀), which were determined by fitting the experimental kinetic curves with the cosine-averaging model (Table 1).

The local order parameter S was calculated in the well-known way [17–21] as

where α is the angle of instantaneous orientation of the dipole–dipole coupling tensor with respect to the ‘symmetry axis of fast motion’ [17] or the polar angle between the  internuclear vector r_IS and the end-to-end vector of the polymer chain [21], 〈…〉 represents the dynamic average over the molecular reorientations. The  static constant D_stat for the  ¹³C–¹H dipolar coupling was taken as 23.0±0.3 kHz for the ‘frozen’ C–H bond that corresponds to r_C–H ~ 1.09–1.10 Å (Eq. (2)).

The  coupling constants D_CH were determined from the b values, rescaling them by a factor of $\sqrt{2}$ because the HH matching for n = ±1 was fulfilled in the present experiments. Finally, the order parameter was calculated using the D_CH value as the average of those obtained from the dipolar splitting (Fig. 3) and the b₀ values from the curve fitting routine using a  cosine-averaging model (Table 1). The  obtained value S = 0.63±0.02 points toward the significant local dynamic and disorder in pVPA.

However, the  abovementioned definitions of the angle α in Eq. (9), given in Refs. [17] and [21] (which are often identical), make the visualization of the  local disorder in polymers and other complex molecules not obvious or even sophisticated. The simplified visualization can be done if the internal motion of vector r_IS(t) is modelled as a restricted movement in a cone with the semi-angle θ₀ (Fig. 4). The order parameter S is then given by [16, 19, 20]

The amplitude of this motion (bending, rocking, twisting, etc.) is qualitatively visualized by the cone semi-angle θ₀. The S and θ₀ values for carbon sites in pVPA are given in Fig. 4. These values significantly differ from those obtained for proteins, other biological macromolecules and polymers [16, 18]. For instance, the S values for CH, CH₂ and CH₃ groups in poly(3-hydroxybutyrate) were found to be 0.81, 0.80 and 0.43, respectively [18]. Much higher S values within 0.98–1.0 were determined in more rigid and thus more ordered systems, like alanine and glycine [8, 19, 20]. Hence, the S value obtained for pVPA is an intermediate case. Namely, according to the obtained S value (Fig. 4), the local disorder of the C–H bonds in pVBA is between those for rather rigid C–H bond configurations having S = 0.8–1.0 and highly disordered –CH₃ groups (S ~ 0.4).

The proton mobility along the  H-bond chain (Fig.  4) was studied by impedance spectroscopy. Dielectric spectroscopy experiments for pVPA were performed in the  120–300  K temperature range. The  temperature dependences of real (εʹ) and imaginary (εʺ) parts of the complex dielectric permittivity ε* = εʹ–iεʺ of pVPA powder are shown in Fig. 5. At low temperatures, the experimental real part of dielectric permittivity shows a stable value of 2.8. A sharp increase in complex dielectric permittivity values is observed at high temperatures (above 225 K) that is typical for relaxations due to mobile charge carriers. In the further processing of experimental data the real part of conductivity σ› was calculated as

where ω is the angular frequency and ɛ₀ is the dielectric permittivity of vacuum. According to the Jonscher power law, σʹ is related to the direct current conductivity σ_DC:

Here A and S are the constants. The σ_DC values were estimated using Eq. (12) and the experimental set of the frequency dependences of conductivity at various temperatures. The dependence of σ_DC on temperature is presented as the Arrhenius plot in Fig. 6.

The determined activation energy E_a = 59±7 kJ/mol in pVPA powder (Fig. 6) is comparable with the values 45 and 65 kJ/mol obtained for as-prepared and annealed pVPA, respectively [10]. However, it is lower than in the  pVPA/poly(ethylene oxide) (PEO) blends (from 95 to 78 kJ/mol, depending on the PEO content in the blend) [11]. As there are no other mobile species than protons, the proton conductivity in pVPA can be realized via H-bond breaking and forming processes only. The above E_a values fit well into a general vista of H-bond energies and point to a two-step Grotthuss mechanism for proton migration: (1) displacement of a proton along a hydrogen bond; (2) transfer of the proton to another oxygen with formation of a new H-bond [10, 11]. The  Arrhenius type behaviour of the  proton conductivity means that the polymer chain motion has no significant effect on the  proton dynamics  [11]. On the other hand, it can be supposed that the presence of the proton-motion ‘highway’ (Fig. 4) significantly accelerates the spin-lattice relaxation process in pVPA. The spin-lattice relaxation rate of protons in the rotating frame 1/T_1ρ is almost one order faster (Table 1) than those observed for other polymers without proton conductivity paths [22]. In relation to this, it is interesting to note the presence of narrow peaks at multiples of the MAS frequency in the dipole coupling distribution profile (10 kHz, Fig. 3). The presence of such peaks is considered as the confirmation of the existence of a periodic quasi-equilibrium state having a lifetime depending on T_1ρ [23]. Such peaks have been observed in other systems: glycine [8], ferrocene and alanine [23], as well as in some polymers – poly(3-hydroxybutyrate) [18] and poly(2-hydroxyethyl methacrylate) (pHEMA) [22], where T_1ρ is definitely longer than the spin-diffusion time, i.e. T_1ρ  >>  1/k₂. Therefore the  fact itself that the peaks in the pVPA dipole coupling distribution profile are discerned at 10 kHz MAS (Fig. 3) even in such limiting case T_1ρ ~ T_dif = 1/k₂ (Table 1) is quite remarkable. For rigorous confirmation the advanced CP MAS experiment has to be carried out achieving a better signal-to-noise ratio.

5. Conclusions

It can be stated that the isotropic spin-diffusion approach well describes the ¹H → ¹³C CP kinetics in pVPA, i.e. in the spin-system containing the adjacent (or directly bonded) spins. The best fit between experimental data and theory was achieved applying the  averaging model with the  Lorentz spin-coupling distribution profile that reproduces coherent CP intensity oscillations in the initial stage of polarization transfer (at short contact times). The rates of spin-lattice relaxation of protons and spin diffusion are of the same order and both occur in the time scale of milliseconds.

The values of the spin-cluster size (N ~ 50±40) have been obtained for this spin-system without any constraint on the flow of the nonlinear curve fitting. Despite large errors, the  finite value of N means that the CP transfer process covers several tens of neighbouring C–H sites within pVPA.

According to the determined value of the order parameter (S  =  0.63±0.02), the  local disorder of the C–H bonds in pVPA is between those for rather rigid C–H bond configurations having S = 0.8–1.0 and for highly disordered –CH₃ groups when S ~ 0.4. Probably, this is due to the mutual influence of the C–H mobility and the proton transfer along the H-bond chain, which is characterized by relatively low activation energy (E_a = 59±7 kJ/mol).

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. One of us (V. Chizhik) thanks the Magnetic Resonance Group for the  hospitality during his research stay in the  frame of the  Agreement of Collaboration between Saint Petersburg and Vilnius Universities.

LOKALIOSIOS NETVARKOS ŽEMOS DIMENSIJOS H-RYŠIO PROTONŲ LAIDININKUOSE TYRIMAI CP MAS KINETIKOS IR IMPEDANSO SPEKTROMETRIJOS METODAIS

Santrauka

Ištirta ¹H–¹³C CP (kryžminės poliarizacijos) MAS (magiško kampo sukimo) kinetika, vykstanti poli(vinyl fosfoninėje rūgštyje) (pVPA), t.  y. medžiagoje, kuriai būdingas didelis protonų judėjimo išilgai vandenilinių ryšių (H-ryšių) grandinės laisvės laipsnių skaičius. Dėl šios savybės pVPA priskiriama medžiagų klasei, vadinamajai protonų laidininkei. Parodyta, kad šioje medžiagoje vykstančios CP MAS kinetikos eksperimentiniai duomenys gali būti aprašyti taikant izotropinės sukinių difuzijos modelį. Nustatytos sukinių difuzijos bei sukinių ir gardelės relaksacijų spartos, taip pat parametrai, nusakantys ¹H ir ¹³C sukinių sąveiką ir efektyviuosius sukinių spiečių matmenis. Lokaliosios tvarkos parametras S ≈ 0,63±0,02, apskaičiuotas remiantis eksperimentiškai išmatuotos dipolinės ¹H–¹³C sąveikos konstantos ir apskaičiuotosios statinės konstantos vertėmis, yra daug mažesnis už S vertes, kurios aptinkamos daugelyje giminingų molekuliniams fragmentams polimerų ir amino rūgščių serijose. Lokalioji C–H jungčių netvarka pVPA užima tarpinę padėtį tarp stingiųjų C–H konfigūracijų, kurioms yra būdingos S vertės, artimos 1,0 (S = 0,8–1,0), ir metilo (–CH₃) grupių, kurių didelė netvarka (S ∼ 0,4) atsiranda dėl lengvai aktyvuojamo šių grupių sukimosi. Galima teigti, kad nemažas C–H jungčių tvarkos parametro sumažėjimas yra nulemtas intensyvaus rūgštinių protonų judėjimo išilgai H-ryšių grandinės. Impedanso spektroskopijos metodu nustatyta šio judesio aktyvavimo energija yra E_a = 59±7 kJ/mol. Sukinių difuzijos bei sukinių ir gardelės relaksacijos spartos tirtajame polimere yra tos pačios eilės, ir šie vyksmai prateka milisekundžių skalėje.