Hiroki Takahashi

Carlos Fernandez De Alba

Daniel Lee

Vincent Maurel

Serge Gambarelli

Michel Bardet

Sabine Hediger

Anne-Laure Barra

Gaël De Paëpe
email: gael.depaepe@cea.fr

Optimization of an absolute sensitivity in a glassy matrix during DNP-enhanced multidimensional solid-state NMR experiments

Introduction

Dynamic nuclear polarization (DNP) is a hyperpolarization technique that transfers the larger polarization of electron spins to nuclear spins by microwave (w) irradiations near the electron spin resonances [START_REF] Overhauser | Polarization of nuclei in metals[END_REF][START_REF] Carver | Polarization of nuclear spins in metals[END_REF][START_REF] Carver | Experimental verification of the Overhauser nuclear polarization effect[END_REF][START_REF] Abragam | Principles of dynamic nuclear-polarization[END_REF][START_REF] Wind | Applications of dynamic nuclear polarization in C-13 NMR in solids[END_REF]. High-field DNP has been demonstrated to enhance sensitivity by one to two orders of magnitude for solid-state nuclear magnetic resonance (SSNMR) experiments. This high-field DNP work is indebted to the recent developments of high-power, high-frequency microwave sources (gyrotrons) [START_REF] Joye | Operational characteristics of a 14-W 140-GHz gyrotron for dynamic nuclear polarization[END_REF][START_REF] Bajaj | 250GHz CW gyrotron oscillator for dynamic nuclear polarization in biological solid state NMR[END_REF][START_REF] Idehara | Development of 394.6 GHz CW gyrotron (Gyrotron FUCWII) for DNP/proton-NMR at 600 MHz[END_REF][START_REF] Torrezan | Continuous-wave operation of a frequency-tunable 460-GHz second-harmonic gyrotron for enhanced nuclear magnetic resonance[END_REF][START_REF] Matsuki | Application of continuously frequency-tunable 0.4 THz gyrotron to dynamic nuclear polarization for 600 MHz solid-state NMR[END_REF][START_REF] Pike | A spectrometer designed for 6.7 and 14.1 T DNP-enhanced solid-state MAS NMR using quasi-optical microwave transmission[END_REF],

transmission lines [START_REF] Pike | A spectrometer designed for 6.7 and 14.1 T DNP-enhanced solid-state MAS NMR using quasi-optical microwave transmission[END_REF][START_REF] Woskov | Corrugated waveguide and directional coupler for CW 250-GHz gyrotron DNP experiments[END_REF][START_REF] Thurber | Low-temperature dynamic nuclear polarization at 9.4 T with a 30 mW microwave source[END_REF][START_REF] Bogdashov | Transmission line for 258 GHz gyrotron DNP spectrometry[END_REF], low-temperature magic-angle-spinning (MAS) probes [START_REF] Thurber | Low-temperature dynamic nuclear polarization at 9.4 T with a 30 mW microwave source[END_REF][START_REF] Barnes | Cryogenic sample exchange NMR probe for magic angle spinning dynamic nuclear polarization[END_REF][START_REF] Matsuki | Helium-cooling and -spinning dynamic nuclear polarization for sensitivity-enhanced solid-state NMR at 14 T and 30 K[END_REF] and efficient polarizing agents [START_REF] Hu | Dynamic nuclear polarization with biradicals[END_REF][START_REF] Ysacco | Properties of dinitroxides for use in dynamic nuclear polarization (DNP)[END_REF][START_REF] Hu | Polarizing agents and mechanisms for high-field dynamic nuclear polarization of frozen dielectric solids[END_REF] (TOTAPOL [START_REF] Song | TOTAPOL: a biradical polarizing agent for dynamic nuclear polarization experiments in aqueous media[END_REF], bTbK [START_REF] Matsuki | Dynamic nuclear polarization with a rigid biradical[END_REF], etc.) [START_REF] Barnes | High-field dynamic nuclear polarization for solid and solution biological NMR[END_REF][START_REF] Maly | Dynamic nuclear polarization at high magnetic fields[END_REF][START_REF] Hu | High-frequency dynamic nuclear polarization using biradicals: a multifrequency EPR lineshape analysis[END_REF][START_REF] Rosay | Solid-state dynamic nuclear polarization at 263 GHz: spectrometer design and experimental results[END_REF][START_REF] Nanni | THz dynamic nuclear polarization NMR[END_REF]. Griffin and co-workers demonstrated DNP experiments at 5 T [START_REF] Becerra | Dynamic nuclear-polarization with a cyclotron-resonance maser at 5-T[END_REF][START_REF] Becerra | A spectrometer for dynamic nuclear-polarization and electron-paramagnetic-resonance at high-frequencies[END_REF][START_REF] Gerfen | High-frequency (140 Ghz) dynamic nuclear-polarization -polarization transfer to a solute in frozen aqueous-solution[END_REF][START_REF] Hall | Polarization-enhanced NMR spectroscopy of biomolecules in frozen solution[END_REF] in the 1990's and the applicable magnetic fields have, since then, been extended to 9 [START_REF] Bajaj | Dynamic nuclear polarization at 9T using a novel 250GHz gyrotron microwave source[END_REF], 14.1 [START_REF] Matsuki | Dynamic nuclear polarization experiments at 14.1 T for solid-state NMR[END_REF] and recently 16.4 T [START_REF] Barnes | Dynamic nuclear polarization at 700 MHz/460 GHz[END_REF]. DNP enhancement factors vary depending on various factors such as the sample temperature and the magnetic field. Typical DNP enhancement factors in 1 H-C cross-polarization (CP) MAS experiments using the biradical TOTAPOL have been reported in the range of 20 -80 on small organic molecules [START_REF] Rosay | Solid-state dynamic nuclear polarization at 263 GHz: spectrometer design and experimental results[END_REF][START_REF] Mieville | A spinning thermometer to monitor microwave heating and glass transitions in dynamic nuclear polarization[END_REF][START_REF] Lange | The effect of biradical concentration on the performance of DNP-MAS-NMR[END_REF]] and 20 -30 on membrane proteins [START_REF] Linden | Neurotoxin II bound to acetylcholine receptors in native membranes studied by dynamic nuclear polarization NMR[END_REF][START_REF] Reggie | Dynamic nuclear polarization-enhanced solid-state NMR of a C-13-labeled signal peptide bound to lipid-reconstituted Sec translocon[END_REF][START_REF] Jacso | Characterization of membrane proteins in isolated native cellular membranes by dynamic nuclear polarization solid-state NMR spectroscopy without purification and reconstitution[END_REF] at ~100 K and 9.4 T.

The DNP enhancement factor (DNP) mentioned above is measured by comparing the NMR spectra acquired with and without w irradiation, usually in 1D experiments. However, for most of the applications of DNP, the only relevant factor to consider is a signal-to-noise ratio (S/N) per unit time available in multidimensional experiments. DNP indeed contributes to enhancement of S/N in DNP experiments, but other factors such as signal bleaching, DNP build-up time, spectral linewidths, etc. also contribute positively/negatively to S/N per unit time [START_REF] Vitzthum | Fractional spin-labeling of polymers for enhancing NMR sensitivity by solventfree dynamic nuclear polarization[END_REF][START_REF] Takahashi | Rapid natural-abundance 2D (13) C-(13) C correlation spectroscopy using dynamic nuclear polarization enhanced solid-state NMR and matrix-free sample preparation[END_REF][START_REF] Rossini | Dynamic nuclear polarization enhanced solid-state NMR spectroscopy of functionalized metal-organic frameworks[END_REF]. Thus, all of these factors need to be considered to evaluate effective DNP enhancement.

We have recently introduced a parameter dubbed the 'absolute sensitivity ratio (ASR)' to evaluate the effective DNP enhancement [START_REF] Takahashi | Rapid natural-abundance 2D (13) C-(13) C correlation spectroscopy using dynamic nuclear polarization enhanced solid-state NMR and matrix-free sample preparation[END_REF]. The ASR is experimentally obtained by comparing the S/N per unit time for a DNP experiment and a conventional SSNMR experiment. For conventional SSNMR, higher magnetic fields and different probes can easily be used to gain higher sensitivity. This fair comparison is extremely important to evaluate the actual usefulness of DNP and has already started to be used in the DNP community [START_REF] Lee | Enhanced solid-state NMR correlation spectroscopy of quadrupolar nuclei using dynamic nuclear polarization[END_REF][START_REF] Takahashi | Solid-state NMR on bacterial cells: selective cell wall signal enhancement and resolution improvement using dynamic nuclear polarization[END_REF][START_REF] Kobayashi | Analysis of sensitivity enhancement by dynamic nuclear polarization in solid-state NMR: a case study of functionalized mesoporous materials[END_REF]. The ASR can be rationalized by eight different factors that contribute to the effective DNP enhancement as follows:

ex seq weight LW bleach T1 T DNP ASR                 (1)
where T is the gain due to performing the measurement at a low temperature (increase of magnetization due to the Boltzmann distribution and the reduction of thermal noise), T1 takes into account the different repetition times for DNP and conventional NMR experiments, bleach is the factor accounting for signal bleaching, LW is the ratio of the linewidths, weight is the ratio of the effective sample weights, seq is the ratio of the effective magnetization after decays during the pulse sequence, and ex is the factor allowing extra effects (e.g. when using a different probe, magnetic field, etc.).

In MAS-DNP experiments, samples of interest are usually dissolved [START_REF] Gerfen | High-frequency (140 Ghz) dynamic nuclear-polarization -polarization transfer to a solute in frozen aqueous-solution[END_REF][START_REF] Hall | Polarization-enhanced NMR spectroscopy of biomolecules in frozen solution[END_REF][START_REF] Rosay | Two-dimensional C-13-C-13 correlation spectroscopy with magic angle spinning and dynamic nuclear polarization[END_REF][START_REF] Rosay | Highfrequency dynamic nuclear polarization in MAS spectra of membrane and soluble proteins[END_REF][START_REF] Akbey | Dynamic nuclear polarization of deuterated proteins[END_REF] or suspended [START_REF] Linden | Neurotoxin II bound to acetylcholine receptors in native membranes studied by dynamic nuclear polarization NMR[END_REF][START_REF] Reggie | Dynamic nuclear polarization-enhanced solid-state NMR of a C-13-labeled signal peptide bound to lipid-reconstituted Sec translocon[END_REF][START_REF] Jacso | Characterization of membrane proteins in isolated native cellular membranes by dynamic nuclear polarization solid-state NMR spectroscopy without purification and reconstitution[END_REF][START_REF] Rosay | Sensitivity-enhanced NMR of biological solids: Dynamic nuclear polarization of Y21M fd bacteriophage and purple membrane[END_REF][START_REF] Van Der Wel | Dynamic nuclear polarization of amyloidogenic peptide nanocrystals: GNNQQNY, a core segment of the yeast prion protein Sup35p[END_REF][START_REF] Mak-Jurkauskas | Energy transformations early in the bacteriorhodopsin photocycle revealed by DNP-enhanced solid-state NMR[END_REF][START_REF] Bajaj | Functional and states of bacteriorhodopsin resolved by 250 GHz dynamic nuclear polarization-enhanced solid-state NMR[END_REF][START_REF] Bajaj | DNP enhanced frequencyselective TEDOR experiments in bacteriorhodopsin[END_REF][START_REF] Debelouchina | Dynamic nuclear polarization-enhanced solid-state NMR spectroscopy of GNNQQNY nanocrystals and amyloid fibrils[END_REF][START_REF] Barnes | Resolution and polarization distribution in cryogenic DNP/MAS experiments[END_REF][START_REF] Bayro | Intermolecular structure determination of amyloid fibrils with magic-angle spinning and dynamic nuclear polarization NMR[END_REF][START_REF] Sergeyev | Chemical shifts for the unusual DNA structure in Pf1 bacteriophage from dynamic-nuclear-polarization-enhanced solid-state NMR spectroscopy[END_REF][START_REF] Renault | Solid-state NMR spectroscopy on cellular preparations enhanced by dynamic nuclear polarization[END_REF] in glass-forming matrices (containing water and a cryo-protectant such as glycerol or DMSO) doped with polarizing agents and measured at low temperature (typically down to ~100 K). All the above factors are strongly dependent on the sample conditions (sample temperature, solvents and cryo-protectants, radical concentration, etc.). Therefore, the sample condition carefully needs to be adjusted to optimize these factors. It is important to note that seq can be a major factor for the reduction of an ASR when performing multidimensional experiments that contain refocused evolution (refocused INADEQUATE [START_REF] Lesage | Through-bond carbon-carbon connectivities in disordered solids by NMR[END_REF], SAR-COSY [START_REF] Lee | Sensitive absorptive refocused scalar correlation NMR spectroscopy in solids[END_REF], etc.) and/or spin-locking periods, since decay parameters such as refocusable transverse decay times (T2') [START_REF] Lesage | Through-bond carbon-carbon connectivities in disordered solids by NMR[END_REF][START_REF] De Paëpe | Transverse dephasing optimized solid-state NMR spectroscopy[END_REF] and spin-locking decay times (T1) are strongly affected by the sample condition [START_REF] Lange | The effect of biradical concentration on the performance of DNP-MAS-NMR[END_REF][START_REF] Potapov | Dynamic nuclear polarization-enhanced 1 H-13 C double resonance NMR in static samples below 20K[END_REF][START_REF] Rossini | One hundred fold overall sensitivity enhancements for Silicon-29 NMR spectroscopy of surfaces by dynamic nuclear polarization with CPMG acquisition[END_REF].

In this paper, we discuss various effects on the above parameters in order to optimize an ASR for DNP-enhanced multidimensional SSNMR experiments performed using a glassy-matrix. We will demonstrate that apparent linewidths, usually broader at low temperatures, can conceal a global shortening of other nuclear decay times, which can have deleterious effects on multidimensional experiments.

Application of these studies is demonstrated on natural-abundance cellulose using 2D refocused INADEQUATE. Moreover, we complement our 9.4 T DNP experiments by performing high-field EPR experiments at this magnetic field. Notably, it is shown that the rigidity of the frozen glassy matrices needs to be taken into account since microwave absorption is observed even below the glass transition temperature.

Softening of glass also results in a shortening of longitudinal relaxation times of electron spins (T1e) especially at high magnetic fields, which leads to a reduction of DNP efficiency.

Materials and methods

Sample preparation

The biradical TOTAPOL was synthesized according to the literature [START_REF] Song | TOTAPOL: a biradical polarizing agent for dynamic nuclear polarization experiments in aqueous media[END_REF]. 13 C-2-glycine, 13 C-urea, and natural abundance cellulose were purchased from Sigma-Aldrich (Lyon, France). Glycine and urea samples were dissolved in d6-DMSO/D2O/H2O (6:3:1 in volume ratio) solutions containing TOTAPOL at various concentrations specified in the text below or in the figure captions. The above solutions (40 L) were fully packed into 3.2-mm thin-wall zirconia rotors. 50 mg of the cellulose was suspended in the above DNP matrix along with 10 mM of TOTAPOL and centrifuged into a 3.2-mm thin-wall zirconia rotor. The desired room-temperature sample was then inserted and cooled to approximately 105 K directly in the precooled DNP probehead. When needed, the sample temperature was then increased in the probehead to the desired value.

DNP experiments

All experiments were performed on a Bruker AVANCE™ III 400 MHz wide-bore NMR system equipped with a 263 GHz gyrotron, a transmission line and a low temperature (~100 K) capable triple resonance 3.2-mm MAS probe [START_REF] Rosay | Solid-state dynamic nuclear polarization at 263 GHz: spectrometer design and experimental results[END_REF]. All data were recorded, processed and analyzed using Bruker Topspin 3.0. The sample temperature and MAS frequency were 105 K and 8 kHz, respectively, unless otherwise specified. The pulse sequences are shown in Fig. 1 and were used for T2' ( 13 C) measurements (a), T1 ( 1 H) measurements (b) and 2D refocused INADEQUATE experiments (c). 263-GHz microwaves (~5 W) were continuously irradiated throughout the DNP experiments. For all experiments, the 1 H RF amplitudes were 100 kHz for the π/2 pulses and the SPINAL-64 decoupling [START_REF] Fung | An improved broadband decoupling sequence for liquid crystals and solids[END_REF] during the evolution and detection periods unless otherwise stated. The 13 C CP pulse was applied at a RF field strength of 60 kHz and the 1 H RF field strength was adjusted to 80 kHz (at maximum) to match a Hartmann-Hahn condition using a linear ramp with a contact time of 1 -2 ms. The 13 C hard pulses were set to 80 kHz. Other NMR parameters are specified in the text below.

Precise measurement of the sample temperature relied on the protocol proposed by Thurber and Tycko, where the real sample temperature is shown to be phenomenologically related to the 79 Br T1 relaxation time constant in KBr [START_REF] Thurber | Measurement of sample temperatures under magic-angle spinning from the chemical shift and spin-lattice relaxation rate of Br-79 in KBr powder[END_REF]. A small quantity (5 mg) of KBr wrapped in Teflon was inserted at the bottom of the rotor, on the top of which the sample of interest was loaded as usual. The T1 relaxation time constant of 79 Br resonance was obtained by a standard saturation-recovery experiment.

High-field EPR experiments

CW High-field EPR spectra were recorded on a double-pass transmission EPR spectrometer operating in Faraday configuration at temperatures in the range of 100 K to 180 K. The DNP sample was placed in a Teflon cup and inserted at liquid N2 temperature in the variable temperature insert of the superconducting magnet. The Teflon cup is filled over a height of a few millimeters (typically ~8 mm), which corresponds to several wavelengths of the exciting radiation. The 285-GHz microwaves (λ~1.05 mm) were generated by a 95 GHz Gunn diode equipped with a tripler and were transmitted via a quasioptical set-up [START_REF] Barra | Broad-band quasi-optical HF-EPR spectroscopy: Application to the study of the ferrous iron center from a rubredoxin mutant[END_REF]. The detection of microwave absorption was performed with a hot electron InSb bolometer. The temperature was measured with the help of a Cernox sensor placed outside the modulation coil at the height of the sample. In this way, the thermometer is able to indicate the heating effect due to the modulation when it exists. It is to note that all high-field EPR experiments were performed in a regime below saturation of the signal.

Results and discussion

The eight parameters that compose the ASR in Eq. ( 1) are affected by many factors (see Table 1) including electron and nuclear relaxation times, microwave B1 irradiation amplitude, sample temperature, radical concentration, etc. Herein, we assume for simplicity that the magnetic field, the type of probe, etc. are the same and that the sample of interest is dissolved in a given DNP matrix. Most of parameters contributing to ASR (given in Table 1) are affected by radical concentration and sample temperature directly and/or indirectly. In this work, therefore, we will focus on the influence of these two experimental parameters on the measured ASR. a This becomes larger when using a suspension method.

Measurement of refocused transverse decay times in frozen solutions

The apparent transverse decay time, T2*, determines LW (see Table 1). Apparent spectral linewidths (1/T2*) for frozen solutions are typically much broader than for powdered solids. For instance, the 13 C linewidth of the -carbon peak of glycine in a frozen DMSO/water solution is 240 Hz in the absence of paramagnetic doping at ~100 K and 9.4 T (compare this to a linewidth of 49 Hz measured at 9.4 T and room temperature on powdered glycine). This broadening of the lines in frozen solutions corresponds to conformational distribution frozen at a low temperature [START_REF] Linden | Cryogenic temperature effects and resolution upon slow cooling of protein preparations in solid state NMR[END_REF]. This is one of the major limitations of the DNP approach, especially for biomolecular systems, which thus needs to be tackled by, for example, sparse isotopic labeling strategies. However, this inhomogeneous broadening has another potential risk as it can hide the influence of paramagnetic doping on the transverse relaxation times of nuclear spins. In such a situation, radical concentrations cannot be optimized by analysis of the apparent linewidths of the samples of interest.

The transverse NMR signal decay resulting from conformational distribution, which leads to an inhomogeneous line-broadening, can be 'refocused' by introducing an echo period into the pulse sequence as illustrated in Fig. 1(a) for the 1D 13 C-CPMAS pulse sequence [START_REF] Lesage | Through-bond carbon-carbon connectivities in disordered solids by NMR[END_REF]. This sequence can be used to indirectly measure T2'. T2' is not affected by inhomogeneous (refocusable) interactions such as the chemical shift distribution but is affected by heteronuclear dipolar couplings (and thus the [in]efficiency of 1 H decoupling) and paramagnetic doping. Consequently, T2' is typically much longer than T2*. In order to truly observe the influence of paramagnetic doping, the efficiency of 1 H decoupling first needs to be analyzed.

Figure 2 shows the dependence of the peak height on the 1 H decoupling during the echo period using the pulse sequence in Fig. 1(a) on 2M 13 C-urea and 0.1 M 13 C-2-glycine with 20 mM TOTAPOL in d6-DMSO/D2O/H2O (6:3:1 by volume). Constant durations for the echo periods of 9.0 and 5.4 ms were chosen for urea and glycine, respectively. Two types of 1 H decoupling, CW decoupling (open symbols) and SPINAL-64 (filled symbols), were used during the echo period. As expected, SPINAL-64 is a more efficient decoupling technique. For the carbonyl group of urea (circles), where there are no 1 H spins directly bonded, relatively low power decoupling (90 kHz) is required to effectively suppress 13 C -1 H dipolar couplings, whereas over 120 kHz of decoupling strength is needed for the C  group of glycine (triangles) where two 1 H spins are covalently bonded. Therefore, with sufficient 1 H decoupling, the peak height reaches a plateau where the T2' of 13 C is mainly affected by paramagnetic doping. Herein, we use 100 (for urea) and 120 kHz (for glycine) of SPINAL-64 1 H decoupling during the echo period for T2' measurements, used to discuss the effects of radical concentration and sample temperature. For multidimensional NMR experiments, seq in Eq. ( 1) can be a major factor in the reduction of an ASR, particularly when the pulse sequence contains refocused evolution and/or spin-locking periods during which specific time constants have to be considered. In this case, we need to consider the cumulative effect of the DNP enhancement and the decays as a function of radical concentration. In order to evaluate such effects, it is convenient to use the reduced form of the ASR (ASR r ), described in our previous paper [START_REF] Takahashi | Rapid natural-abundance 2D (13) C-(13) C correlation spectroscopy using dynamic nuclear polarization enhanced solid-state NMR and matrix-free sample preparation[END_REF].

This ASR r keeps certain factors constant and thus permits the detailed analysis of the effects of the other parameters.

To observe the radical concentration dependence for a glassy DNP matrix (by comparing samples with and without polarizing agents at the same temperature), a ASR r can be written as: ). Thus, Eq. ( 2) can be rewritten as H)

seq radical LW bleach T1 DNP ex weight T LW T r weight,ex T, ASR ASR                   (2)
T 1 ( 1 H) T 2 ' ( 13 C) T 2 * ( 13 C) T 1 ( 1 H) / s TOTAPOL concentration / mM T 1 ( 1 H), T 2 ' ( 13 C), T 2 * ( 13 C) / ms (a) (b) (c)
Taking into account the discussions above, a 2D refocused INADEQUATE experiment was demonstrated on natural-abundance semi-crystalline cellulose in a glass-forming solution (the same DNP solution as above experiments). Semi-crystalline cellulose was suspended in the DNP matrix at 10 mM TOTAPOL. T2' was 26 ms that is long enough to perform NMR experiments that employ refocused evolutions (the total echo period used for the 2D experiment was 16 ms). The spectrum in Fig. 4 periods at a MAS frequency of 8 kHz. 64 scans were accumulated for each t1 point (the number of complex t1 points is 24 and the total t1 evolution time is 1.5 ms). The experimental data were apodized with 100 and 200 Hz of exponential line-broadening in t2 and t1 dimensions, respectively. The projection is shown on top.

Effects of sample temperature in glassy solutions

Temperature also changes sample conditions and influences DNP, T, T1 and seq in Eq. ( 1). bleach could potentially be affected by sample temperature but there was no clear influence of the temperature for the experiments reported here. Figure 5 (a) reports the effect of the sample temperature on the same set of parameters as the radical concentration dependence above (i.e. DNP enhancement factor and nuclear time decays T1, T1, T2', T2*). All parameters except for T2* appear sensitive to temperature variation and decay when the sample temperature increases from 100 to 160 K. The change in slope of the T1 and T2' curves around 140 K most probably reflects the approach of the glass transition temperature, in agreement with literature [START_REF] Mala | Temperature of glass transitions and the glass-forming region in the system lithium nitrate -dimethyl sulfoxide -water[END_REF].

This tendency is similar to that seen on the radical concentration where all the parameters are dominated by paramagnetic effects. For the temperature dependence, however, glass condition additionally needs to be considered since glass becomes softer as the temperature increases even below the glass transition temperature (around 140 K in our case). Softening of glass causes shortening of nuclear and electron relaxation times by molecular motion as well as microwave absorption by the glassy solution. Both phenomena will result in severe reduction of DNP efficiency and are specifically discussed in the section 3.4.

Similarly to the section 3.2, ASR r for sample-temperature dependence in a glassy DNP matrix (by comparing to the without-TOTAPOL sample at 105 K) can be expressed as

seq bleach T1 T(105K) T DNP ex weight T(105K) r weight,ex T(105K), ASR ASR                  . ( 4
)
Here again, LW  is neglected since T2* does not change in this temperature range in a frozen solution as seen in Fig. 5(a). T(105K) is the gain from Boltzmann distribution at 105 K (reduction of thermal noise is neglected herein for convenience.). It is very important to stress again that DNP does not represent a "real gain" of a DNP experiment especially when performing multidimensional experiments. Sample conditions such as radical concentration, sample temperature and DNP matrices need to be optimized by measuring an ASR. Lines are used as guides only.

Impact of glass softening probed by EPR

In this study, we have used the biradical TOTAPOL that provides inhomogeneous broadening of the EPR spectrum, which then facilitates one of the DNP mechanisms of cross effect (CE) [START_REF] Maly | Dynamic nuclear polarization at high magnetic fields[END_REF][START_REF] Hu | High-frequency dynamic nuclear polarization using biradicals: a multifrequency EPR lineshape analysis[END_REF][START_REF] Kessenikh | On explanation of experimtnat data on dynamic polarization of protons in irradiated polyethylenes[END_REF][START_REF] Hwang | Phenomenological model for the new effect in dynamic polarization[END_REF][START_REF] Wollan | Dynamic nuclear-polarization with an inhomogeneously broadened ESR line. 1. Theory[END_REF][START_REF] Hu | Quantum mechanical theory of dynamic nuclear polarization in solid dielectrics[END_REF]. The field sweep profile of a DNP experiment reaches a maximum when the number of pairs of radicals which more or less experience the CE matching condition is maximized (Fig. 6a). DNP (the maximum intensity of the filed sweep profile) based on this mechanism follows B0 -1 , since the number of electron-spin pairs experiencing the CE matching condition decreases as B0 increases, which results from inhomogeneous broadening caused by g-anisotropy (Fig. 6b). This field dependence (B0 -1 ) of DNP enhancements was theoretically and experimentally demonstrated at 5 and 9 T [START_REF] Hu | Polarizing agents and mechanisms for high-field dynamic nuclear polarization of frozen dielectric solids[END_REF][START_REF] Hu | Quantum mechanical theory of dynamic nuclear polarization in solid dielectrics[END_REF].

A recent theoretical study by Thurber et al. shows that in addition to the B0 -1 dependence above, a polarization transfer time constant (from electron spins to nuclear spins) is dependent on the magnetic field as B0 3 [72]. This is based on the level crossing model for MAS-DNP experiments [START_REF] Thurber | Theory for cross effect dynamic nuclear polarization under magic-angle spinning in solid state nuclear magnetic resonance: The importance of level crossings[END_REF][START_REF] Mentink-Vigier | Fast passage dynamic nuclear polarization on rotating solids[END_REF]. A longer time constant at a higher magnetic field will lead to smaller DNP enhancement provided that the other parameters (T1e, T1n, etc.) are the same. However, such parameters also depend on the magnetic field and on the sample condition (radical concentration, sample temperature, etc.) and are thus complicated to predict by theory.

In the following, we will discuss other contributions that strongly affect the DNP efficiency but are not explicitly included in the theory. Measurement of relaxation times of electron spins (T1e/T2e) is essential to improve our understanding of the DNP mechanism. However, such measurements are not straightforward to collect at high magnetic field (9.4 T in this case). In addition, DNP experiments are typically performed using radical concentrations one to two orders of magnitude larger than those used in EPR studies, which can significantly complicate data analysis. Nevertheless, even in the absence of direct measurements of electron spin relaxation times, their variations can be indirectly probed by measuring nuclear relaxation time decays (as long as paramagnetic effects significantly contribute to these nuclear time decays). In fact, nuclear time decays at 105 K and 20 mM TOTAPOL concentration are mainly dominated by the presence of radicals as seen in As seen in Fig. 5(a), all parameters except for T2* drop drastically above around 140 K in a d6-DMSO/D2O/H2O (6:3:1 by volume) solution, which corresponds to the glass transition temperature found in the literature [START_REF] Mala | Temperature of glass transitions and the glass-forming region in the system lithium nitrate -dimethyl sulfoxide -water[END_REF]. This suggests that another relaxation mechanism appears above that temperature yielding significantly shorter T1e. T1n was also measured at various temperatures on the same sample as used for Fig. 5(a) in absence of radical doping (Fig. 7). The same tendency as in Fig. 5 The composition of the glassy solution is the same as for Fig. 5.

As seen in Fig. 5(a), DNP drops drastically as the sample temperature increases. This may be related to relaxation times of electron spins. In fact, Sato et al. reported the temperature dependence of electron spin-lattice relaxation time (T1e) of nitroxyl radicals in glassy solutions at X-band [START_REF] Sato | Electron spin-lattice relaxation of nitroxyl radicals in temperature ranges that span glassy solutions to low-viscosity liquids[END_REF]. T1e shortens rapidly above glass-softening temperature since the motion-dependent process (relaxation due to spin rotation and modulation of g and A anisotropies) becomes dominant in the relaxation mechanism.

At X-band and 100 K on glycerol/H2O mixture, the dominant relaxation mechanism of electron spins is Raman process (

2 1e   T T
) and modulation of g-anisotropy does not contribute to it. However, the effect of modulation of g-anisotropy possesses B0 -2 dependence on T1e according to the above literature [START_REF] Sato | Electron spin-lattice relaxation of nitroxyl radicals in temperature ranges that span glassy solutions to low-viscosity liquids[END_REF], thus such modulation can be a dominant mechanism at 9.4 T or especially above even in the case of slow motion. Therefore, rigidity of glass, which depends on the temperature and the composition of the DNP solution, has to be considered for high-field DNP experiments to obtain a long T1e at high fields.

To further investigate the temperature dependence of DNP, EPR spectra of the same sample as in Fig. 5(a), 20 mM TOTAPOL with 2M 13 C-urea in d6-DMSO/D2O/H2O (6:3:1 by volume), was measured at 10 T in the temperature range between 100 and 175 K (Fig. 8a). Note that microwave absorption by TOTAPOL radical is detected and monitored without a cavity and that the absorption occurs over several wavelengths on this EPR setup (Fig. 8b). It is very important to understand that this is somewhat similar to the setup used in MAS-DNP experiments. As the temperature increases, the signal intensity of EPR spectra decreases much more rapidly than the Curie Law. This is an interesting phenomenon that requires further investigations.

Firstly the same experiments were repeated at X-band using a cavity where the sample was placed (in order to minimize E-field). In this case, the EPR signal intensities exactly followed Curie's law (data not shown). At this point, we clearly wonder whether the glass softening phenomenon (discussed above) can explain alone the drastic decay in EPR signals intensity observed at high magnetic field in absence of cavity. Since the frozen solutions are slightly "softened" when approaching the glass transition they might lead to increased non-resonant microwave absorption and thus to a decay in CW EPR intensity.

To prove this hypothesis, we first tried to indirectly detect the microwave absorption through the precise measurement of the sample-temperature increase arising when the microwave irradiation is applied.

This temperature increase is expected to become larger when the microwave absorption is higher, e.g. at higher absolute temperature. The experiments performed on the urea sample were, however, not conclusive, as the temperature difference of the sample with and without microwave irradiation did not show any temperature dependence in the 100 K-160 K range. Due to this protocol being not suitable to detect variations in microwave absorption by the DNP matrix, under our experimental conditions, it was therefore further investigated using high-field EPR at 10 T (285 GHz). The DNP matrix (same as above but without TOTAPOL) was placed on top of a MnO/MgO (0.01% MnO) mixture and microwaves were irradiated from the top (Fig. 8c). Figure 8 sample. This microwave absorption by glassy matrices at higher field has not been investigated due to the unavailability of an EPR setup at such high field. It is important to note that this phenomenon could become more serious at higher field.

Finally, DNP and the calibrated EPR signal intensities of TOTAPOL (the contribution from Curie's law is removed from Fig. 8(a) and therefore the main contribution is microwave absorption of the DNP matrix) are compared at various temperatures (Fig. 8e). They show the similar tendency as a function of temperature, which suggests that DNP efficiency is largely affected by microwave absorption of the DNP matrix. Further drop on DNP can be explained by a shortening of relaxation times of electron spins as discussed earlier.

Both microwave absorption by DNP matrices and shortening of relaxation times are affected by the quality of the glass formed by DNP matrices. Glass rigidity is a very important factor to be considered to obtain large DNP. One way to improve it is obviously lowering sample temperatures, which then lengthens the relaxation times of electron spins and avoids microwave absorption by DNP matrices. Another possible way (which does not require new instrumentation developments) is matrix-free DNP (MF-DNP) [START_REF] Takahashi | Rapid natural-abundance 2D (13) C-(13) C correlation spectroscopy using dynamic nuclear polarization enhanced solid-state NMR and matrix-free sample preparation[END_REF][START_REF] Takahashi | Towards structure determination of selfassembled peptides using dynamic nuclear polarization enhanced solid-state NMR spectroscopy[END_REF][START_REF] Takahashi | Matrix-free dynamic nuclear polarization enables solid-state NMR C-13-C-13 correlation spectroscopy of proteins at natural isotopic abundance[END_REF] where the motion-dependent process should not be a dominant mechanism for relaxation due to the absence of solvents. Furthermore, microwave absorption by DNP matrices is avoided in this case. MF-DNP has not yet been extensively studied in these aspects but it might be a favorable choice for sample preparation at a higher magnetic field.

Conclusions

We have demonstrated TOTAPOL concentration and sample temperature dependence of MAS-DNP experiments at 9.4 T. T1, T1 and T2' are severely affected by paramagnetic effects and glass conditions, whereas T2* is insensitive since it is mostly dominated by chemical shift distribution in a frozen solution.

All parameters that affect an effective DNP enhancement in multidimensional SSNMR experiments were taken into account. By choosing an adequate concentration of polarizing agents and optimizing sample conditions, a DNP-enhanced 13 C-13 C 2D refocused INADEQUATE spectrum of natural-abundance semicrystalline cellulose suspended in a typical DNP matrix doped with TOTAPOL was obtained in a reasonable experimental time.

Furthermore, the temperature dependence of DNP experiments was rationalized with high-field EPR experiments on the same sample at the same magnetic field. Shortening of electron spin relaxation times at high field and microwave absorption by DNP matrices depend on the glass condition, which then affect the DNP enhancement and nuclear spin relaxation times. At higher field, these contributions might become larger and bring additional reduction of DNP along with the B0 -1 dependence of DNP and the B0 3 dependence of the polarization transfer time constant.

  Optimization of an absolute sensitivity in a glassy matrix during DNP-enhanced multidimensional solid-state NMR experiments Hiroki Takahashi, Carlos Fernandez-De-Alba, Daniel Lee, Vincent Maurel, Serge Gambarelli, Michel Bardet, Sabine Hediger, Anne-Laure Barra, Gaël de Paëpe To cite this version: Hiroki Takahashi, Carlos Fernandez-De-Alba, Daniel Lee, Vincent Maurel, Serge Gambarelli, et al.. Optimization of an absolute sensitivity in a glassy matrix during DNP-enhanced multidimensional solid-state NMR experiments. Journal of Magnetic Resonance, 2014, 239, pp.91-99. 10.1016/j.jmr.2013.12.005. hal-02043248

Figure 1 .

 1 Figure 1. Pulse sequences for T2' ( 13 C) measurements (a), T1 ( 1 H) measurements (b) and 2D refocused

Table 1 .

 1 Parameters directly affecting the various ASR contributions.Relevant parametersDNPElectron and nuclear relaxation times, w B1e,

Figure 2 .

 2 Figure2. Influence of 1 H decoupling strength during an echo period (Fig.1a). The samples are 2M13 Curea (circles) and 0.1M 13 C-2-glycine (triangles) with 20 mM TOTAPOL in d6-DMSO/D2O/H2O (6:3:1 by



  are the parameters accounting for line-broadening caused by low temperature and radical doping, respectively. As seen above, radical LW  is negligible for frozen solution samples at a low concentration of polarizing agents (below 60 mM in this case). In a refocused INADEQUATE experiment, the signal-decay factor during the pulse sequence (seq) is mainly composed of factors caused by T1 and T2'

Figure 3

 3 Figure 3(c) shows the cumulative effect of the factors in Eq. (3) which were obtained

Figure 3 .

 3 Figure 3. (a,b) Concentration dependence of relaxation parameters on 2M 13 C-urea in d6-DMSO/D2O/H2O

  was obtained in 7 hours and all one-bond correlations of cellulose are seen and indicated by solid lines. Though signals from amorphous regions are broad and less intense, some of correlations of amorphous cellulose are also observed in the same spectrum indicated by dotted lines.

Figure 4 .

 4 Figure 4. A DNP-enhanced 13 C-13 C 2D refocused INADEQUATE spectrum of natural-abundance semi-

Figure 5 (

 5 b) shows the cumulative effect of the factors in Eq. (4) in the same sense as above for the concentration dependence. The effective DNP enhancement (compared to the without-TOTAPOL sample at 105 K) is represented by open circles. The same symbols are used as Fig.3(c) and open squares are added to represent DNP•(T/T(105K)).

Figure 5 .

 5 Figure 5. (a) Temperature dependence of relaxation parameters on 2M 13 C-urea in d6-DMSO/D2O/H2O

Figure 6 .

 6 Figure 6. (a) DNP field profile on 2M 13 C-urea with 15 mM TOTAPOL in d8-glycerol/D2O/H2O (6:3:1 by

  (a) is observed (with longer T1n relaxation times). These results presumably arise from molecular motion caused by softening of the glass. Impact of such glass softening is discussed in detail below both in terms of DNP enhancements (owing to shortening of T1e) and possible microwave absorption.

Figure 7 .

 7 Figure 7. Temperature dependence of proton T1 relaxation time constant in the absence of polarizing agents.

  (d) shows the experimentally obtained EPR signal intensities of MnO (filled circles). We then checked for a few temperature points that in absence of DNP matrix the MnO signals were very close to a Curie law (open circles and solid line). The discrepancy between both tendencies clearly indicates a variation in microwave absorption by the DNP matrix placed on the top on the MnO/MgO

Figure 8 .

 8 Figure 8. (a) 285 GHz CW EPR spectra on 20 mM TOTAPOL with 2M 13 C-urea in d6-DMSO/D2O/H2O

Acknowledgments

This work was partially supported by the French National Research Agency through the "programme blanc" (ANR-12-BS08-0016-01), the Labex ARCANE (ANR-11-LABX-0003-01) and funding from the RTB.


Frédéric Mentink-Vigier

Shimon Vega

Gaël De Paëpe
email: gael.depaepe@cea.fr

Fast and accurate MAS-DNP simulations of large spin ensembles

A deeper understanding of parameters affecting Magic Angle Spinning Dynamic Nuclear Polarization (MAS-DNP), an emerging nuclear magnetic resonance hyperpolarization method, is crucial for the development of new polarizing agents and the successful implementation of the technique at higher magnetic fields (> 10 T). Such progress is currently impeded by computational limitation which prevents the simulation of large spin ensembles (electron as well as nuclear spins) and to accurately describe the interplay between all the multiple key parameters at play.

In this work, we present an alternative approach to existing Cross-Effect and Solid-Effect MAS-DNP codes that yields fast and accurate simulations. More specifically we describe the model, the associated Liouville-based formalism (Bloch-type derivation and/or Landau-Zener approximations) and the linear time algorithm that allows computing MAS-DNP mechanisms with unprecedented time savings. As a result, one can easily scan through multiple parameters and disentangle their mutual influences. In addition, the simulation code is able to handle multiple electrons and protons, which allows probing the effect of (hyper)polarizing agents concentration, as well as fully revealing the interplay between the polarizing agent structure and the hyperfine couplings, nuclear dipolar couplings, nuclear relaxation times, both in terms of depolarization effect, but also of polarization gain and buildup times. 56 experimental side by reporting DNP studies related to 57 the effect of polarizing agents' concentration, sample 58 spinning frequency, temperature, partial deuteration, 59

. Despite these attempts to 61 experimentally rationalize the performance of nitroxide 62 biradicals, a clear picture has still not emerged, mostly 63 because the DNP enhancement factor (defined as the 64 ratio of signal with and without µw, on/off = S on /S off ) 65 can lead to a misleading picture of the DNP perfor-66 mance while overestimating the true DNP gain signif-67 icantly [50]. 68 Previously, the most efficient and commonly used 69 DNP mechanism, the Cross Effect (CE) was crudely 70 described using arguments based on static (i.e. with-71 out sample spinning) experiments, which fails to pro-72 vide valuable insights such as the dependency with re-73 spect to the magnetic field, the polarizing agent geom-74 etry, the µw power, electron/nuclei relaxation times, 75 etc. Things have improved with the recent theoretical 76 developments by Tycko and Thurber, as well as Vega et 77 al. [50-54]. These contributions have not only brought 78 valuable insights into the DNP mechanism, but also il-79 lustrated the complexity of high magnetic field exper-80 iments and proven the necessity to rely on numerical 81 simulations to understand precisely the mechanisms at 82 play in MAS-DNP experiments. The mechanism has 83 been understood through numerical simulation tools 84 able to describe a series of discrete events that occur 85 periodically within one rotor period (rotor events) [50-2 53]. In the case of binitroxides, modeled by 2 electrons 87 and a nucleus, four types of rotor events need to be 88 considered: 89 • the µw rotor events, that induce a change in the 90 electron polarization, 91 • the electron Dipolar -J (exchange) rotor events 92 that tend to swap the electron polarization, 93 • the Cross-Effect rotor event (CE) that exchanges 94 part of the electron polarization difference to the 95 nuclear polarization of the hyperfine coupled nu-96 clei, 97 • and the Solid-Effect rotor event (SE), that ex-98 changes part of the electron polarization to the 99 nuclear polarization of the hyperfine coupled nu-100 clei. 101 The nuclear polarization under MAS-DNP conditions 102 is typically computed through the use of advanced 103 quantum mechanical simulations able to account for 104 relaxation. More specifically the evolution operator 105 was computed either in the Hilbert space by Thurber 106 and Tycko [51] or in the Liouville space in the work 107 of Mentink-Vigier et al. [50, 52, 54] and Mance et al. 108 [55]. These computations are largely in agreement and 109 were successfully used to simulate hyperpolarization as 110 well as depolarization effects for various systems and 111 experimental conditions, [50-55] and clearly highlight 112 the complexity of this multi-parameter problem. The 113 order and duration of the rotor events are directly re-114 lated to the structure of the polarizing agent but also to 115 the position and strength of the µw irradiation. For a 116 given crystallite orientation, each rotor event type can 117 typically occur 0, 2 or 4 per rotor period. When the 118 nuclear polarization has reached a steady-state value, 119 the electrons and nuclei still see periodic changes in 120 their polarization due to the rotor events and the elec-121 tron and nuclear relaxation (T e 1 , T n 1 ). Ultimately it 122 is important to note that the nuclear polarization can 123 only be equal to or lower than the maximum electron 124 polarization difference observed during the course of a 125 rotor period at steady state [50, 54]. 126 Based on these initial results, we now seek to further 127 develop this computational approach towards its usage 128 as a predictive tool. Challenges along this direction 129 are numerous but offer the perspectives to perform in 130 silico rational design of polarizing agents. This consti-131 tutes an important research direction since polarizing 132 agent design has so far relied on empirical approach.

133

The main hindrance in this approach is associated with 134 the duration of MAS-DNP simulations, especially for 135 large spin systems (> 4 spins) and the number of in-136 put parameters that needs to be accounted for such as 137 the spin relaxation times, the biradical geometry, the 138 µw power, the nuclear Larmor frequency, etc. So far

The combination of Dynamic Nuclear Polarization (DNP) and Magic Angle Spinning (MAS) is currently deeply impacting high-field solid-state Nuclear Magnetic Resonance (NMR). Although DNP is not a recent phenomenon, [START_REF] Overhauser | [END_REF]2] the potential of the technique for high magnetic fields was only recently demonstrated thanks to the pioneering work carried out in the Griffin laboratory [3][4][5][6]. Following this effort and the introduction of commercial systems [7], DNP experiments are currently carried out in an increasing number of laboratories, continuously expanding the scope of the technique. [8][9][10][11][12][13] The interest stems from the huge gain in NMR sensitivity obtainable though DNP, where the large intrinsic polarization of electrons is transferred to nuclei, using appropriate microwave (µw) irradiation. So far, most DNP experiments are conducted successfully at 10-20 T at around 100 K using high power continuous wave (CW) µw sources often combined with the use of binitroxide radicals as sources of free electrons, as introduced in the Griffin group [3,14,15]. Current on-going effort include improved polarizing agents and optimized sample preparation protocols [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], while challenging instrumental developments are also under progress in several laboratories, such as the design of more flexible high power µw sources, probe-heads with smaller diameter sample holders (rotors) and optimized µw coupling, the access to helium sample spinning and ultra-low temperature MAS-DNP experiments [34][35][36][37][38][39][40][41][42].

In addition, the field of MAS-DNP has also strongly benefited from recent improvements in theoretical un- [56,57] while Thurber and Tycko [53] have recently re-145 ported a MAS-DNP simulation code able to compute 146 the population changes of a 1000 × 3 spins system (2 147 electrons and 1 nucleus) randomly dispersed in a box 148 using the Landau-Zener (LZ) approximation. [58] This 149 last model allows describing the effect of biradical con- and that the superposition principle can be applied.

193

Importantly, we also show that the number of electrons and nuclei can easily be increased so as to account for multi electron effects as well as nuclear spin diffusion. This new feature can be used to compute DNP enhancement factors and polarization build-up times for the various protons present in large spin assemblies of tens to thousands of spins. This provides great insight into many key parameters, such as the electron concentration, the magnetic field dependency, polarizing agent geometries, and nuclear relaxation times of bulk versus local protons, explaining physically observed phenomena.

I. MATERIALS AND METHODS

The simulation codes have been written in Matlab (MATLAB and Statistics Toolbox Release 2013a,

The MathWorks, Inc., Natick, Massachusetts, United

States.) and optimized to minimize computational time. In particular we made use of the Suite Sparse Matlab toolbox when considering large spin system [59]. The simulations have been run on a Dell Precision T5500, using 2 Intel Xeon(R) CPU X5650 @ 2.67GHz (24 logical cores), using Ubuntu 15.10 as the OS.

Except when specified otherwise we performed calculations assuming a temperature of 100 K, either on a biradical geometry close to bTbK [60] using [g z , g y , g x ] = [2.0024, 2.0063, 2.0097] for the g-tensor values, or close to TOTAPOL using [g z , g y , g x ] = [2.0094, 2.006, 2.0017] as used in previous work [52,54,61]. The nitrogen hyperfine couplings, when included, had the following principal values [98, 16,17] MHz expressed in the principal axis system (PAS) of the closeby electron g-tensor.

As introduced in Mentink-Vigier et al. [50,54], we define the polarization gain B as the ratio between the nuclear polarization with µw (µw) irradiation and the nuclear Boltzmann Polarization

H(t) = H 0 (t) + H µw H 0 (t) = H Z (t) + H HF (t) + H J + H Dip (t) + H d (t), (1)
with

H Z (t) = i (g i (t)β e B 0 -ω µw ) S i,z - n ω n I n,z H HF (t) = i,n A z i,n (t) S i,z I n,z + 1 2 (A + i,n (t) S i,z I + n + A - i,n (t) S i,z I - n ) H J = i<i -2J i,i ( S i,z S i ,z + 1 2 ( S + i S - i + S - i S + i )) H Dip (t) = i<i D i,i (t)(2 S i,z S i ,z - 1 2 ( S + i S - i + S - i S + i )) H d (t) = n<n d n,n (t)(2 I n,z I n ,z - 1 2 ( I + n I - n + I - n I + n )) H µw = ω 1 i S i,x . (2)
where H Z (t) stands for the Zeeman interaction of the • the µw rotor events, active when ω i -ω µw = 0, 261 which induce changes in the electron polarization 262 through the effect of H µw ,

263

• the electron or nuclear dipolar rotor events, ac-

264

tive when ω i/n -ω i /n = 0, which induce exchanges of electron or nuclear polarizations through the action of the flip-flop of the dipolar coupling and exchange interaction when present,

• the CE rotor events, active when ω i -ω i ≈ ±ω n , which induce a exchange between the electron polarization difference and the proton polarization, through the combined effect of the flip-flop dipolar and pseudo-secular hyperfine couplings.

• the SE rotor events, active when ω i -ω µw ≈ ±ω n , which induce a transfer of polarization between the electron and the proton, through the combined effect of the pseudo-secular hyperfine interaction and µw irradiation.

For a simplified three spin system e ae bn, equation 2 can be rewritten as:

H Z (t) = (ω a (t) -ω µw ) S a,z + (ω b (t) -ω µw ) S b,z -ω n I n,z H HF (t) = A z (t) S a,z I n,z , + 1 2 (A + (t) S a,z I + n + A -(t) S a,z I - n ) H µw = ω 1 ( S x,a + S x,b ) H J = -2J a,b ( S a,z S b,z + 1 2 ( S + a S - b + S - a S + b )) H Dip (t) = D a,b (t)(2 S a,z S b,z - 1 2 ( S + a S - b + S - a S + b )) (3)
Note that in this simplified case, the nucleus is only coupled to electron a.

A schematic illustration of the rotor events is provided in figure 1. We can observe the evolution of the electron and proton polarizations for the three main rotor events (µw, dipolar and CE). The effect of the µw and electron dipolar rotor events can be seen on the electron polarization curves whereas the effect of the CE rotor event appears too small to be seen on that scale. On the other hand the presence of CE rotor events is observed on the proton polarization curve.

Note that the SE rotor events can not be observed in this simulation.

As described in previous work, the calculation of the spin evolution involves solving the time dependent (due to the sample spinning) master equation, including all necessary interactions and relaxation processes. This requires the computation of the propagator superoperator of a rotor period, using small stepwise integration and calculating this propagator superoperator U κ for every (κ th ) rotor-step [52,54]. For the three-spin system of two electrons and one nucleus the spin density matrix, representing the state of the system at any time ρ(t) has a dimension of 2

327

In this work, we achieve a drastic reduction of the 328 vector size σ to shorten significantly the computational 329 time. To that purpose, we assume that the spin dy-330 namics are well described by a series of successive rotor events, with each corresponding to a two-level anticrossing problem,along with spin-lattice relaxation towards Boltzmann equilibrium. As discussed in the following section, each rotor event is treated/computed using either Bloch-type derivation in a reduced Liouvillian subspace for which relaxation can be simply introduced or with the LZ formalism (accounting for the variation in polarization operator). In between rotorsevents, the spin dynamics are well described by the secular part of the full Hamiltonian (see Eq. 3) and correspond to a return to Boltzmann equilibrium. In the following, section A describes the general approach to treat and approximate the spin dynamics in between rotor events and during each individual rotor event.

Section B and C describe the computation of a propagator superoperator over a rotor period when all rotor events are treated with the LZ formalism or using a combination of the LZ and Bloch-type formalisms, respectively. Finally section D describes the extension to large spin assemblies (multiple electrons and protons).

A. Independent diabatic rotor events -The Bloch-type approach and rotor synchronized propagation Every rotor event previously described involves only two energy levels. Therefore in the following we assume that rotor events, which correspond to diabatic passages, are well separated and independent. For a given rotor event involving the two levels |1 and |2 , one can derive an effective Hamiltonian of the form:

H 12 (t) =
B (t) =     0 0 0 0 s 1,2,eq z /T 1 -1/T 1 -ξ 12 x (t) ξ 12 y (t) 0 ξ 12 x (t) -1/T 2 -∆ω 12 (t) 0 -ξ 12 y (t) ∆ω 12 (t) -1/T 2     (

401

As described previously, [52,54] the rotor synchronized

402
propagator is simply obtained by step integration over 403 one rotor period:

404 U rotor = Ns κ=1 U κ (10)
where N s stands for the number of integration points, As a result the T 1 and T 2 relaxation values were not constant during sample rotation because of the strong state mixing that occurs during the D-J and CE rotor events. On the contrary, with the Bloch-type approach presented in this work, we assume such relaxation times to be constant in order to optimize the computational performance. Such an approximation can be justified by the fact that the duration of state mixing is shorter than the electron T 2 in most cases and that the coherences created during the events decay after the mixing period. As shown below, this is further validated by the very good agreement with full Liouville calculations. More details can be found in the SI.

Accounting for relaxation effects is especially important when probing cases with large ω 1 and/or short electronic T 2 . This is notably also the case for strong µw irradiation strength, large dipolar and/or J interactions, radicals with narrow EPR lines such as Trityl, and short electron relaxation times. In all other cases, the Bloch type treatment can be simplified using the LZ approximation.

B. Independent diabatic rotor events -Combining Landau-Zener approximation and rotor synchronized propagation

Coming back to the two level system as defined by the Hamiltonian in eq. 5 we can follow Vitanov [64] and define a scaled dimensionless coupling parame- of the the S 1,2 z ) operator in the spin density operator 451 expansion in eq. 6. This translates onto s 1,2 z as:

ter
452 s 1,2 z (t + × ) = 1 -2 12 s 1,2 z (t × ) (11)
with

453 12 = 1 -exp -π|ξ 12 (t × )| 2 2| d dt ∆ω 12 | t× . ( 12
)
Here

U k = exp( R 1 δt) × U LZ,κ = 0 0 [s 12,eq z (1 -e -δt/T1 )] [e -δt/T1 (1 -2 12 κ )] (13)
If there is no crossing during this interval, then U LZ,κ is identity.

On the three-spin system the rotor events usually occur at different time-steps, and in the following we assume that they can be successively treated. This allows to rely on a LZ formulae for each rotor event separately. For each type of event one can identify the two levels involved in the diabatic passage and define the coefficient s 477

U κ = exp( R 1 (κδt) × δt) U µw LZ,κ U D-J LZ,κ U CE LZ,κ U SE LZ,κ (14)
where R 1 represents the longitudinal relaxation dur-478 ing the time interval [t κ , t κ + δt] and its 4×4 evolution 479 operator has the form:

480 R 1 =    1 0 0 0 s eq a,z/T e 1,a -1 /T e 1,a 0 0 s eq b,z/T e 1,b 0 -1 /T e 1,b 0
s eq n,z/T n 1 0 0 -1 /T n 1    .
The explicit forms of the Liouvillians in Eq. 14 are

481
given in the SI. As above, the rotor synchronized prop-482 agator is obtained by step integration over one rotor 483 period:

484 U rotor = Ns κ=1 U κ (15)
The novelty here relies on the fact that the LZ for- and the µw and D-J rotor events using "Bloch-type" 532 derivations. For a two electrons and one nucleus spin

L B (t) =                0 0 0 0 0 0 0 0 0 0 s eq a,z (t)/T e 1 -1/T e 1 ω1 0 0 0 0 DJ ab (t) 0 0 0 -ω1 -1/T e 2 -∆ωa(t) 0 0 0 0 0 0 0 0 -∆ωa(t) -1/T e 2 0 0 0 0 0 0 s eq b,z (t)/T e 1 0 0 0 -1/T e 1 ω1 0 -DJ ab (t) 0 0 0 0 0 0 -ω1 -1/T e 2 -∆ω b (t) 0 0 0 0 0 0 0 0 -∆ω b (t) -1/T e 2 0 0 0 0 -DJ ab (t) /2 0 0 DJ ab (t) /2 0 0 -1/T e 2,ZQ -∆ωD(t) 0 0 0 0 0 0 0 0 ∆ωD(t) 0 0 s eq n,z /T n 1 0 0 0 0 0 0 0 0 -1/T n 1               
where we defined a transverse relaxation time for the

U κ = U B,κ U LZ,κ (17)
where respectively. These additional interactions induce intermolecular rotor events (dipolar, CE, SE) that can efficiently be computed using the LZ computational approach. More precisely, the code generates N copies of a 3 spin system (2 electrons and one nucleus) with a fixed configuration, which are randomly dispersed and oriented in a box.

U B,κ = exp( L B (κδt)×δt),
In order to allow the use of the LZ approach, some constraints must be applied:

• Two identical crystallite orientations cannot coexist within the same box (in order to avoid frequency degeneracy).

• The intermolecular electron-electron distance (d min ) is always larger than 1.7 nm (∼ 10 MHz dipolar coupling) so that LZ can be applied ac-600 curately.

601

• For simplicity, the effects on the spin system of 602 the dipolar rotor events between electrons that 603 are more than d max = 6.4 nm (∼ 0.2 MHz dipolar 604 coupling) are ignored.

605

The calculation is then modified in order to account 606 for intermolecular rotor events. For a given integration 607 step, the propagator is now written:

608 U κ = U B,κ × U LZ,κ (18)
where where U B,κ is obtained from the previous sec-

Including additional nuclei

The introduction of additional (hyperfine coupled) nuclei leads to more CE/SE rotor events but also to a new type of event, the nuclear-nuclear dipolar rotor event, [51,54] which occurs when two nuclei with different hyperfine couplings have the same instantaneous resonant frequency. These rotor events allow hyperpolarized nuclei close to the unpaired electrons to propagate their polarizations to more distant spins. This MAS-dependent effect has been accounted for theoretically [see ref [54] for details] to predict a reduction/removal of the so-called diffusion barrier present in the static case [67][68][69][70][71]. As in the electron spins' case, the nuclear dipolar rotor events induce a partial exchange of the nuclear polarization which can be accurately computed using the LZ derivation. Such a treatment is applied to "local nuclei" (also called closeby or ENDOR nuclei) for which hyperfine couplings are larger than the mean nuclei-nuclei dipolar interaction.

In addition we can refine our model and add additional "bulk" nuclei, which are not directly coupled to the electrons but are in contact with some of the "local nuclei". The nuclear spin diffusion among the bulk nuclei is simulated using rate equations that equilibrates the polarization between two connected nuclei [66,72,73] and for nucleus j is given by ds z,j dt = j -r SD j,j (s z,j -s z,j ) + s eq z,j

T 1,j

where j corresponds to the index of the neighboring nucleus, and r SD j,j = d 2 j,j T n 2 /4. Hence the assumption used to build our model containing two electrons (e a , e b ) and N n nuclei can be listed as follows:

• CE rotor events involving "local" nuclei occur when |ω a -ω b | ≈ ω n . Each CE rotor event, involving a given nucleus, can be a priori treated using the derivation provided in the SI for a 3 spin case. In addition this CE rotor event condition is a priori also influenced by the presence of other hyperfine coupled nuclei which induce splittings of the effective electron resonance, leading to a quasi-continuum of CE sub-conditions.

Overall these additional splittings can be safely ignored since they all contribute identically to the CE polarization transfer and it simplifies the treatment.

•

U κ = U B,κ × U LZ,κ (19)
where

685 U LZ,κ = N L j=1 U CE,j LZ,κ × N L j=1 U SE,j LZ,κ × N L j=1 N L j =n U dip,j,j ZL,κ
and where the Bloch part accounts for electrons' dy- In figure 4 Remarkably, the ratio between the positive and negative part of the spectrum is also ω 1 dependent. Note that simulations performed with multiple biradicals in a box gave the same normalized profile (not shown). were easily computed with this reduced 3 spin system.

All in all, these simulations indicate the interest of using rather low biradical concentrations (<10 mM) and relatively long electron relaxation times (up to T e 1 ≤ 0.5 ms). At this point, it is important to note that the results discussed here were obtained for a given biradical geometry (close to TOTAPOL) and a selected set of nuclear/electron relaxation times, and neglecting the role of the bulk nuclei. We believe that such simulations give a good qualitative picture of the CE MAS-DNP mechanism but that different results could be obtained with other input parameters, and when the bulk nuclei are taken into account the behavior of the concentration may be changed. relaxation mechanism [69] given by T n 1 :

Main magnetic field dependency

942 1 T n 1 ∝ |A ± a,n | 2 (S(S + 1)) τ 1 + ω 2 n τ 2
where τ is an electron spin-flip correlation time that can be close to either T e 1 or T e 2 depending on the concentration [69]. The nuclear relaxation is then proportional to the square of the hyperfine coupling (T n 1 ∝ |A a,n | -2 ∝ r 6 ). Nonetheless, the biradicals are not the only source of relaxation. Indeed, in an undoped frozen solution the relaxation time has still a finite value (of 30 -80 s [19,30,50]), which originates from other relaxation mechanisms such as proton-proton dipolar relaxation or dissolved paramagnetic oxygen. It is reasonable to assume that these additional mechanisms are responsible for the relaxation of the bulk nuclei, while the local protons relax under the influence of the biradicals. The nuclear relaxation rates can then be written as the sum of the two contributions

1 T n 1 = 1 T n 1,a-n + 1 T n 1,Bulk
As the exact values are not known, we assumed that 943 the relaxation time of nucleus i is given by ied and the simulations are reported in Figure 11. As 1054 in the 3 spins system case, the increase in the mag-

944 1 T n 1,i = 1 T n 1,1 A a,1 A a,i 2 + 1 T n 1,Bulk (20)

55 derstanding.

 55 Several groups have contributed from the full quantum calculations (Hilbert-or Liouville-based) 140 have been limited to 3 or 4 spins because extension to 141 larger spin system requires the introduction of approxi-142 mations. For instance, size restrictions have been used 143 and developed by Karbanov et al. in the static case 144

150

  centration and was used to simulate a depolarization 151 mechanism in the absence of µw irradiation. 152 In this work we present another approach based on 153 Liouville-type calculations that yields fast and accu-154 rate simulations compared to exact Liouville calcula-155 tions. Since the goal is to build a tool that can possibly 156 be used in a quasi-predictive fashion, the code needs: 157 • to be as fast as possible by building a periodic 158 propagator, 159 • to be able to increase both the number of electron 160 and nuclear spins while keeping a moderate sized 161 problem (linear scaling), 162 • to be accurate in predicting the electron polar-163 ization difference at steady-state, i.e. account 164 for electron T e 1 and T e 2 relaxation during rotor 165 events. 166 The code presented is fast and flexible (as detailed be-167 low) and constitutes the first step towards predictive 168 MAS-DNP simulations. For the 3 spins case (2 elec-169 trons and 1 nucleus), a 30-point CE field sweep profile 170 can be computed in only 20 to 30 minutes (on a desktop 171 computer) while keeping an excellent agreement with 172 the previous code (based on full-Liouville calculations 173 which take 6 -8 hours). This represents a significant 174 improvement in time-savings, about a factor 15 for a 175 3-spin system, which allows checking efficiently the ef-176 fects of changing different parameters, such as the po-177 larizing agent geometry, the external magnetic field B 0 , 178 the MAS rotation frequency, the relaxation rates and 179 the spin-spin interaction strengths. In this model, each 180 type of rotor event is treated separately using a simpli-181 fied subspace (spanned by reduced Liouvillian opera-182 tors). The relevant spin dynamics during a rotor event 183 are computed either by relying on Bloch-type equa-184 tions accounting for electronic and nuclear relaxation 185 or using the LZ formula. The former are particularly 186 important for rotor events that act on timescales of the 187 order of the relaxation times, which therefore cannot 188 be ignored. As demonstrated in the following, each 189 rotor event is thus described with a set of first-order 190 linear equations. The overall evolution is then com-191 puted assuming that every rotor event is independent 192

P

  n (µw on) P n (Boltzmann) the nuclear depolarization factor Depo as the ratio between the nuclear polarization without µw irradiation and the nuclear Boltzmann Polarization Depo = P n (µw off) P n (Boltzmann) and the classical "DNP enhancement factor" On/Off as the ratio of nuclear polarization with and without µw irradiation On/Off = P n (µw on) P n (µw off) One should note that On/Off does not represent the 237 true polarization gain induced by biradicals [53, 54], 238 which is given by B . 239 Finally we define the DNP build-up time T B as the 240 time-constant obtained by fitting the nuclear polariza-241 tion build-up curves with a single exponential function. 242 II. FAST SIMULATION CODE: 243 SUPERPOSITION OF REDUCED 244 LIOUVILLIAN SUBSPACE SPIN DYNAMICS 245 High magnetic field MAS-DNP simulations can be 246 efficiently computed in the µw rotating frame as de-247 scribed in refs [50-55]. The corresponding Hamilto-248 nian for N e electrons (with indices i, i = 1, N e ) and 249 N nuclei (with indices n, n = 1, N ) can be written: 250

12 B

 12 9) from which the propagator of a short time inter-382 val [κδt, (κ + 1)δt] can be obtained via U (κt)δt).384Following this approach, the complete derivation for 385 each type of rotor event can be obtained (see Support-386 ing Information). The full Liouvillian superoperator -387 accounting for all types of rotor events -can then be 388 obtained by applying the superposition principle since 389 the rotor events are assumed to be independent. In the 390 end, the spin dynamics can be described by the evolu-391 tion of a σ vector with dimension 18 and thus by a 18 392 × 18 Liouvillian superoperator. The elements of σ(t) 393 in this case become the 18 prefactors of the operators 394 { E, S a,z , S a,y , S a,x , S b,z , S b,y , S b,x , S DJ ZQ,y , S DJ ZQ,x I n,z , Information for the derivation details). 397 This constitute a drastic size reduction compared to 398 64 x 64 in the full Liouville approach that results in 399 massive time savings with minimal compromise on the 400 accuracy (as demonstrated in the rest of the paper).

405

  and U κ = exp( L κ × δt) with δt = 1/(N s ν r ) and L κ = 406 L B (κδt). 407 Note that in previous MAS-DNP calculations, the 408 relaxation times T 1 and T 2 were determined after diag-409 onalization of the Hamiltonian (without the µw term).

  410

  . As is shown in the SI, the changes in all of these coefficients can be transferred to changes in only the coefficients {s a,z , s b,z , s n,z } of the operators { S z,a , S z,b , I z,n }. Adding relaxation, the Liouville superoperator representing all events while relying on LZ formula, operates on the vector {1, s a,z , s b,z , s n,z } and the propagator at step κ can be written in the basis E, S z,a , S z,b , I z,n as follows:

  533 system, this can be done resulting in a Liouville super-534 operator of dimension 10 × 10 operating on the vec-535 tor {1, s z,a , s y,a , s x,a , s z,b , s y,b , s x,b ,, s D-J y,ZQ , s D-J z,ZQ , s z,n } 536 where the LZ part of the CE and SE events has matrix 537 elements only between {s z,a , s z,b , s z,n } and the Bloch 538 part of the µw and D-J events has the form 539

  540 D-J rotor events T e 2,ZQ = T e 2 /2 and DJ ab = (D a,b + 541 2J a,b ) (See SI for details). The time-step integration is 542 still used to obtain the periodic propagator 543 U rotor = Ns κ=1 U κ(16)and at each time-step κ, the propagator is the product 544 between the Bloch and LZ part 545

  551D. MAS-DNP simulation of large spin ensembles (tens to thousands of spins): extension to multiple electrons and nuclei Here we extend our three-spin system by adding many electrons and nuclei with the aim to generate more realistic simulations able to reproduce the electron concentration and nuclear spin diffusion effects found in contemporary experiments. The presence of intermolecular electron-electron dipolar interactions generates a MAS-induced spectral diffusion phenomenon which tends to equilibrate the polarization throughout the EPR line,[53] and directly impacts the intramolecular polarization difference and thus the overall nuclear polarization enhancement. Moreover, the presence of additional nuclei induces new CE rotor events and tends to equilibrate the polarization among the nuclei. In order to meet both challenges, two models and codes were developed and are described below.1. Increasing the number of electron spins: accounting for the electron concentration effect (the box model)We simulate N biradicals (each modeled by 2 electrons and 1 proton) randomly distributed in a box so as to meet a given biradical concentration. In this model, referred as the box model in this work, interactions between biradicals are restricted to nearest neighbors. This approximation allows keeping an efficient computational code while accounting for intermolecular dipolar rotor events, i.e. electron spectral spin diffusion, in a similar fashion as nuclear spin diffusion process[65,66]. A similar approach was previously described by Thurber et al.[53]. During standard DNP experiments, the biradical concentration is ∼ 5 -30 mM which translates to ∼ 0.3 -1.5 MHz and 0.45 -2 kHz of intermolecular dipolar and hyperfine couplings

. 5 FIG. 2 .

 52 FIG.2. Example of a spin system used in the box model corresponding here to a random distribution of N = 40 biradicals (15 mM concentration). In blue electron of type "a", yellow, electron of type "b", orange, nucleus. The black dotted lines correspond to the intermolecular electron-electron dipolar couplings active during the simulation.

Figure 2

 2 Figure2shows a typical random distribution gener-

accurate simulations but is beyond the scope of the 706 current article. 707 Figure 3 10 FIG. 3 .FIG. 4 .

 70731034 Figure3shows a typical cone distribution. The po-

  (a) and (b), solid lines are the complete 758 Liouville calculations, dashed lines are respectively the 759 Hybrid method in (a), and LZ method in (b). In the 760 full Liouville model B increases as the MAS frequency 761 is increased up to 2-3 kHz, then after a maximum, B 762 decreases highlighting a reduction of the DNP mech-763 anism efficiency. This loss is more drastic in the case 764 of weak dipolar couplings compared to larger ones, a behavior that was previously explained [54], and could be summarized as: the MAS frequency is increased, the dipolar rotor events do not succeed in maintaining a large polarization difference between the electrons |P a -P b | max , leading to lower B . The Hybrid method generates almost an identical outcome as compared to the Liouville one, with a surprisingly good numerical accuracy. Similarly, a good agreement is achieved using the LZ as it reproduces well at a high MAS frequency even if it remains a bit off in the slow MAS regime. For a 3 spin system problem, the two simplified methods capture the spin physics. It highlights the accuracy of the methods at a fraction of the time cost, as simulations are 15 times faster for the Hybrid, and 20 times for the LZ approach. B. Accurate DNP field-sweep profile: the bTbK example Thanks to the significant time-savings, one can now easily account for the presence of 14 N spins in the binitroxides, which induce hyperfine EPR lineshifts. The resonant frequency of electron i can be written as follows, assuming that the nuclear state of the 14 N is m i,I and that the secular hyperfine coupling is A I i,z : ω i (m I ) = g i βB 0 + m i,I A I i,z . Note that this approach has already been implemented by other groups [51, 53, 55] using other numerical codes with the goal to improve the field-sweep accuracy. The importance of this feature is illustrated in Figure 5 for the bTbK biradical case using a three spin system. Figure 5 (a) shows the simulated DNP field-sweep profile in the presence and absence of nitrogen hyperfine couplings. Their presences induce clear edges in the g z part of the profile, as well as a slight decrease of the positive maximum. In the end, this demonstrates that this new simulation tool is able to generate a field sweep profile in excellent agreement with previously published experimental data.[16]. This is of course of prerequisite for future work targeting in silico radical design. In addition, Figure 5 (b) shows the effect of an increase of the µw irradiation strength ω 1 on the DNP field-sweep profile: not only can it change the fieldsweep profile (e.g. features in the negative part of the field sweep) but it can also increase the enhancement factor at the optimal field position (about 4.696 T).

FIG. 5 .FIG. 6 .

 56 FIG. 5. (Three spins (e-e-n) simulations) Theoretical DNP Field sweep computed with the bTbK geometry in the three spin system case. In figure (a), with (red) or without accounting the 14 N hyperfine couplings. (b) Effect of the µw irradiation strength on the DNP Field sweep for 4 irradiation strengths (black circles ω1/2π = 0.1 MHz, blue squares ω1/2π = 0.4 MHz, green down-pointing triangles ω1/2π = 0.7 MHz, red diamonds ω1/2π = 1MHz. For all the simulations, T e 1 = 0.3 ms, T e 2 = 1 µs, T n 1 = 0.2 s, A1,a/2π = 1.5 MHz, νµw/2π = 131.725 GHz, νr = 5 kHz and ω1/2π = 0.7 MHz for figure (a) where the µw irradiation strength has been used to obtain a good agreement with experimental data published in [16].

Figure 6 (

 6 Figure 6 (a) and (d) show that the polarization gain B decreases with the concentration and that this effect is more pronounced for longer nuclear T n 1 (0.1 versus 4 s) and electronic (0.1 up to 1 ms) T e 1 relaxation times. Similarly Depo (Figure 6 (b) and (e)) decreases (which means a greater depolarization effect) with increased concentrations and/or longer electron T e 1 . At this point it is worth noting that intermolecular effects can account for up to a factor 4 difference in terms of Depo between isolated and coupled 3 spin systems at large biradical concentrations (> 10 mM). The presence of additional intermolecular electron-electron dipolar rotor events leads to a MAS-induced spectral diffusion, a mechanism that tends to equilibrate the electron polarization through the EPR line [53]. Such an effect is stronger at large electron concentration and for long T e 1 . Even with a simple "TOTAPOLlike" 3 spin system where only one nucleus collects the electron spin polarization difference, the MAS-induced

  Figure 7 presents similar simulations as in Figure 6 but exploring the effect of the magnetic field (corresponding to 200 to 800 MHz 1 H Larmor frequencies) on the CE MAS-DNP efficiency. The spin system is the same as in the previous section and the biradical concentration fixed to 15 mM. Overall the polarization gain B (panels (a) and (d)) decreases with increasing magnetic field and this effect is more pronounced for short nuclear relaxation times (0.1 versus 4 s). In-line with the previous section, we observe that the polarization gain B is hardly affected by the presence of MAS induced spectral diffusion at short electron relaxation times T e 1 = 0.1 ms (blue squares) but strongly decreased at longer times (T e 1 > 0.3 ms). For instance, for T e 1 = 1 ms, B is reduced from 380 to 200 when taking into account intermolecular interactions (15 mM concentration) at B 0 = 4.7 T. Figure 7 also illustrates that long T e 1 are especially preferred at high magnetic fields (> 15 -20 T) but not necessarily for lower field studies. Once more these simulations illustrate the importance of relying on the polarization gain B and not the On/Off enhancement factor since they clearly give two different qualitative pictures: longer T e 1 always provide higher On/Off values whereas the situation is more complex in terms of real polarization gain. Notably, long T e 1 (1 ms) at low field yield a lower polarization gain. The discrepancy between B and On/Off can be explained by looking at the depolarization factor Depo . It is worth noting that a significant part of the depolarization comes from intermolecular effect and that this contribution gets smaller at higher fields. This is actually consistent with recent experimental findings[50]. The depolarization effect (intra and intermolecular contribution) is larger at low magnetic fields and/or for long T e 1 values. The theoretical results presented here are inline with the trends previously observed experimentally and theoretically for On/Off , Depo , and B [50, 54]. We must of course emphasize that the results presented here were obtained for 40 biradicals with a TOTAPOL-like geometry in a box with a set of interaction and relax-920 ation parameters. Therefore these results should not 921 straightforwardly be compared with experimental ob-922 servations from samples that contain different types of 923 biradicals and differ in their nuclear conformations. 924 D. Accounting for multiple nuclear spins 925 In this section we investigate the polarization of a 926 large set of protons in a two-electron system as de-927 scribed in section II D 2. The spin system considered 928 in this section consists of isolated biradicals coupled 929 to a set of 400 protons. These protons are divided in 930 Local(or ENDOR) protons directly coupled to the elec-931 trons and bulk protons coupled to the Local/ENDOR 932 protons that exchange their polarization via spin diffu-933 sion. A key feature of the model is that one can easily 934 introduce non-uniform nuclear relaxation times among 935 the local protons in order to account for the fact that 936 they are not all located at the same distance from the 937 electrons. More precisely the model presented below 938 assumes that the electron spin-flips induce through the 939 pseudo-secular hyperfine coupling (A ± a,n ) a field fluc-940 tuation at the proton position that leads to a nuclear 941

where T n 1 , 1 FIG. 7 .FIG. 8 .Figure 9 FIG. 9 .

 117899 FIG. 7. (Box model simulation) Dependence of the polarization gain B (a and d), depolarization Depo (b and e) and On/Off (c and f) as a function of the main magnetic field and electron relaxation time , T e 1 = 0.1 ms (blue squares), 0.3 ms (green up-pointing triangles), 0.5 ms (red down pointing triangles), and 1 ms (purple diamonds). The biradical structure used corresponds to TOTAPOL, T e 2 = 1 µs, A1,a/2π = 1.5 MHz, ωµw/2π = 263.45 GHz, B0 = 9.394 T, νr = 8 kHz and ω1/2π = 0.7 MHz. Top figures (a-c) T n 1 = 0.1 s, and bottom figures (d-f) T n 1 = 4 s. Simulations with the Hybrid approach have been performed by averaging 10 randomly distributed boxes containing 40 biradicals orientations (picked up among 144 ZCW crystal orientations) with a concentration of 15 mM, and dmin = 2 nm. Dotted and solid lines represent respectively the isolated and interacting biradical case.

1055

  netic field leads to a significant decrease of the po-1056 larization gain B while increasing the build-up times 1057 T B . The calculations are inline with experimental ob-1058 servations reported for the two water soluble biradi-1059 cals TOTAPOL and AMUPol [21, 50]. The polariza-1060 tion build-up times increase with the magnetic field for 1061 both biradicals, whereas the higher dipolar coupling in 1062 AMUPol explains why it polarizes better and faster 1063 than TOTAPOL.

  1064

FIG. 10 .

 10 FIG. 10. (Bulk model simulation) Effect of the number of bulk nuclei N Bulk on the final B (black curve, right axis) and build-up time TB (black dashed curve, right axis) for T n 1,Bulk = 60 s, and B (red curve, left axis) and buildup time TB (red dashed curve, right axis) for T n 1,Bulk = 5 s. Simulations performed for TOTAPOL geometry with ω1/2π = 0.85 MHz, T e 1 = 0.3 ms, T e 2 = 1 µs, A1,a/2π = 3 MHz, ωµw/2π = 263.45 GHz, B0 = 9.394 T, νr = 8 kHz, the closest proton relaxation time was T n 1,1 = 0.1 s.

Figure 12

 12 Figure12shows the dependence of both the polariza-

  1120

  and U ZL,κ = U SE,κ ×

	546
	547	U CE,κ . This (Hybrid) approach ensures accurate sim-
		ulations even for short electron T e 2 (i.e < 2 µs for ni-

548 troxides at 9 T), for large ω RE interactions (e.g strong 549 µw fields) or narrow EPR line widths (e.g in the case 550 of Trityl).

  The local nuclei are connected among themselves

	675
	676	via nuclear dipolar rotor events. The LZ ap-
		proach can be used safely here to describe the
	680
	681	• The last shell of the Local nuclei are connected
	682	to bulk nuclei via nuclear dipolar rotor events.
	683	• The evolution operator at each step can thus be
	684	written as

677

energy crossing with ∆ω 0 = (A z,1 -A z,2 )/2.

678

• Couplings among "bulk" nuclei are introduced 679 through a semi-classical spin-diffusion treatment.

ACKNOWLEDGEMENTS 1121 Daniel Lee, Sabine Hediger and Yonatan Hovav