Krishnendu
Kundu
,
Marie Ramirez
Cohen
,
Akiva
Feintuch
,
Daniella
Goldfarb
and
Shimon
Vega
*
Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 7610001, Israel. E-mail: shimon.vega@weizmann.ac.il
First published on 28th November 2018
Dynamic Nuclear Polarization (DNP) is an efficient technique for enhancing NMR signals by utilizing the large polarization of electron spins to polarize nuclei. The mechanistic details of the polarization transfer process involve the depolarization of the electrons resulting from microwave (MW) irradiation (saturation), as well as electron–electron cross-relaxation occurring during the DNP experiment. Recently, electron–electron double resonance (ELDOR) experiments have been performed under DNP conditions to map the depolarization profile along the EPR spectrum as a consequence of spectral diffusion. A phenomenological model referred to as the eSD model was developed earlier to describe the spectral diffusion process and thus reproduce the experimental results of electron depolarization. This model has recently been supported by quantum mechanical calculations on a small dipolar coupled electron spin system, experiencing dipolar interaction based cross-relaxation. In the present study, we performed a series of ELDOR measurements on a solid glassy solution of TEMPOL radicals in an effort to substantiate the eSD model and test its predictability in terms of electron depolarization profiles, in the steady-state and under non-equilibrium conditions. The crucial empirical parameter in this model is ΛeSD, which reflects the polarization exchange rate among the electron spins. Here, we explore further the physical basis of this parameter by analyzing the ELDOR spectra measured in the temperature range of 3–20 K and radical concentrations of 20–40 mM. Simulations using the eSD model were carried out to determine the dependence of ΛeSD on temperature and concentration. We found that for the samples studied, ΛeSD is temperature independent. It, however, increases with a power of ∼2.6 of the concentration of TEMPOL, which is proportional to the average electron–electron dipolar interaction strength in the sample.
In solid solutions of free radicals, the DNP mechanism strongly depends on the EPR spectral width and concentration of the radicals used as polarizers. The commonly recognized DNP mechanisms, the solid effect6,7 (SE), the cross effect6,8–13 (CE) and thermal mixing14–18 (TM), all play a role in nuclear enhancement. The SE is responsible for the nuclear enhancement that is induced by MW excitation of electron-nuclear zero-quantum and double-quantum transitions. The CE also transfers electron polarization to nuclear polarization, but this time it is mediated by a dipolar interaction between neighboring electrons. In the TM description, the NMR enhancement is rationalized by a spin thermodynamic process14 between the spin heat baths19 of the multi-electron system and of the nuclei.20–28 When the polarizers are nitroxyl radicals with an in-homogeneously broadened EPR spectrum, the dominant DNP mechanisms are the SE and CE. Furthermore, the deviations from the TM predicted EPR line shape29,30 under DNP conditions at 95 GHz indicate that the contribution of the TM mechanism is negligible.31,32
Recently we made a distinction between the direct and the indirect CE mechanism. During a direct CE experiment one directly irradiates a pair of electron spins with a resonance frequency difference equal to the nuclear Larmor frequency (the CE conditions). During an indirect CE (iCE) enhancement process, the electron polarizations of a pair are affected by a redistribution mechanism of the electron depolarization throughout the full EPR spectrum, due to spectral diffusion.31 In solutions with radical concentrations above 10 mM, the MW irradiation initially burns a hole that then spreads throughout the EPR spectrum. Its lineshape reaches a steady state that is determined mainly by (i) the strength of the electron–electron dipolar interactions,33,34 (ii) the associated electron–electron cross relaxation, (iii) the electron spin–lattice relaxation and (iv) the strength of the applied MW amplitude.35–38 The polarization of the core and local nuclei39 coupled to the electrons is then enhanced by the iCE process that transfers the polarization difference of coupled electron spins fulfilling the CE conditions,40,41 to their nearby nuclei. In the next DNP stage, the enhanced polarizations of the local nuclei are transferred to the bulk nuclei by spin diffusion,42 governed by the nuclear dipolar interactions and longitudinal relaxation. Accordingly, once the depolarization profile within the EPR line shape is known and analyzed in terms of polarization gradients, it can be used to directly calculate the DNP lineshape, namely the DNP enhancement as a function of the MW irradiation frequency.
The EPR lineshape, under conditions of MW irradiation typically used in DNP experiments, can be determined by a set of electron–electron double resonance (ELDOR) experiments.43,44 To analyze and extract the polarization gradients within the EPR spectrum, we introduced a mathematical model referred to as the electron spectral diffusion (eSD) model. This model is based on a set of coupled rate equations for the electron polarizations of the frequency bins composing the EPR spectrum. The rate constants, representing the electron spectral diffusion responsible for the electron depolarization during MW irradiation, are defined by a single eSD parameter ΛeSD. This single parameter, together with the spin relaxation rates, has been sufficient for reconstructing ELDOR frequency profiles31,45 and from them EPR and DNP spectra.46 In our recent theoretical study,47 we discussed the dependence of ΛeSD on the electron–electron dipolar interactions in the system, and in the present study, we experimentally address the question of how it depends in practice on the temperature and the radical concentration, which is directly related to the average dipolar interaction between the electron spins.
We report on ELDOR experiments carried out on samples of TEMPOL at different concentrations, where the CE is highly efficient, namely in the range of 20–40 mM in a 50:
50 (v/v%) DMSO/H2O amorphous glassy solvent at different temperatures. The eSD model is then used to analyze the ELDOR spectra and to extract the concentration and temperature dependence of ΛeSD. Additionally, we compare experimental temporal evolutions of the polarizations after MW irradiation with simulations relying on the values of ΛeSD. This comparison also reveals that to successfully reproduce the experimental data, it is necessary to introduce time-dependent SE parameters to the eSD model.
Measurements were carried out on three sets of samples referred to as I, II and III. Sample sets I and II were measured on our pulse EPR spectrometer (95 GHz).48 They were placed in a 0.9 mm outer diameter (OD) quartz capillary, containing about 2–4 μl, then degassed, and immediately frozen in liquid nitrogen. These samples were then introduced cold into the precooled cryostat of the spectrometer. In this spectrometer, cooling is done with a liquid helium flow cryostat. On sample set I, a series of ELDOR measurements was performed with different detection frequencies covering the full EPR line. From these ELDOR spectra, 2D-ELDOR maps were constructed. For these measurements, the superior SNR of the pulse EPR spectrometer was of importance and enabled obtaining reasonable ELDOR data, even when detecting at the edges of the EPR spectrum. Sample set II was used to test the reproducibility and estimate the uncertainties of the measurements. To test a broad range of temperatures and concentrations, we added the third set of samples, set III, which were measured using our hybrid pulsed-EPR-NMR spectrometer49 (henceforth called the DNP spectrometer). These measurements allowed experiments at 3 K, not possible for the EPR spectrometer. Synchronization between the EPR and NMR parts of this spectrometer enables the application of independent and simultaneous RF and MW irradiation pulse schemes. Similar to the EPR spectrometer, it operates at a magnetic field of 3.4 T, corresponding to a 1H-Larmor frequency of 144 MHz and an electron-Larmor frequency of approximately 95 GHz. Both spectrometers have a very similar design in terms of their microwave bridge, with two channels for applying two different MW frequencies during single measurements. For measurements on the DNP spectrometer, a series of samples at different concentrations were placed in a glass sample holder containing about 20 μl of the required solution, degassed, and sealed. The samples, placed in the probe head, were then cooled to cryogenic temperatures, by inserting the probe head into a cryogen-free variable temperature system from Cryogenic Limited, UK. The DNP setup does not support inserting precooled samples.
The temporal behavior of the electron depolarization during MW irradiation was measured by recording EPR echo signals after various values of tMW. We also recorded the time-domain behavior of the electron polarization, following a short MW pulse of 100 μs, for monitoring the influence of the spectral diffusion process during the return of the EPR spectrum to its thermal equilibrium line shape.
![]() | (1) |
The spin–lattice relaxation of the jth bin is described by the equation:
![]() | (2a) |
The non-zero elements of the electron spin–lattice relaxation rate matrix, Rb1e, are therefore given by:
![]() | (2b) |
![]() | (3) |
The electron–electron cross relaxation mechanism for any two bins, j and j′, with polarizations Pe,j(t) and Pe,j′(t), respectively, is described by the equation:
![]() | (4a) |
![]() | (4b) |
ηj,j′ = exp(−(ωj − ωj′)/kBT) | (4c) |
The last rate matrix WbSE in eqn (1) represents electron depolarization due to the SE-DNP process. This is represented by an effective MW irradiation in the ZQ or DQ transitions, and again has only diagonal elements, given by:
![]() | (5) |
Solving eqn (1) for a set of parameters results in a polarization profile determining the values of Pe,j as a function of their bin resonance frequencies ωj. These profiles enable us to calculate the EPR spectra during MW irradiation, by taking the values of fj into account:
EEPRsim(ωj,t;ωMW) = fjPe,j(t) | (6) |
For consistency, from here on, we will present all frequencies in units of Hertz as Δνj = ωj/2π − νe and ΔνMW = ωMW/2π − νe, with νe = 94.8 GHz, and νn = ωn/2π. The parameters necessary for fitting the data are: T1e the electron spin–lattice relaxation, ΛeSD the spectral diffusion parameter, ν1 the MW irradiation intensity, νn the nuclear Larmor frequency, ASE the SE depolarization parameter and T2e the electron spin–spin relaxation. Before the data analysis we determined experimentally the shape of the EPR spectrum of the TEMPOL radical and its fj values, the values of T1e and ν1, set νn equal to the proton Larmor frequency and chose a value for ΔνMW. We also measured phase memory times, TM, which represent only a lower limit for T2e. Thus using these values, the remaining parameters ΛeSD, T2e and ASE are determined by a fitting procedure. The first parameter determines the overall depolarization profiles of the ELDOR spectra, the second defines the off-resonance excitation efficiency of the MW irradiation and the third is responsible for the spectral features induced by the SE-DNP process.
Using this rate equation we can now simulate ELDOR spectra with the following procedure: at first, after fixing the value for the widths of the frequency bins we define all bin frequencies ωj in units of Hertz and with respect to νe = 94.8 GHz: Δνj = (ωj − 2πνe)/2π. For each chosen set of parameters {ΛeSD,ASE(tMW),T2e} and MW irradiation at a specific bin frequency, Δνexc,j, we can calculate a normalized steady state EPR spectrum of the form
![]() | (7) |
In all calculations discussed here, the bin-width was set equal to Δνb = 2 MHz. This value was chosen because it is of the order of the average (e–e) dipolar interaction strength in the samples with the highest radical concentration. From our earlier calculations (not shown here), it follows that varying Δνb between 2 and 0.5 MHz does not affect the best-fit values for ΛeSD. Thus, to minimize the dimension of e(t) in eqn (1), we choose a bin size of 2 MHz. During the calculations, the Δνexc,j and Δνdet,j frequencies were always set at the center frequencies of one of the bins which, in turn, were fixed with respect to the EPR spectrum. This restriction is of course not valid for the experimental Δνexc and Δνdet values. To compare the simulated and experimental ELDOR results, the detection frequencies Δνdet,j during the calculations were set to be as close as possible to the experimental Δνdet values, by choosing Δνdet − Δνb/2 < Δνdet,j < Δνdet + Δνb/2, and the same rule follows for Δνexc.
Sample set | Temp. (K) | Conc. (mM) | T 1e (ms) | Sample set | Temp. (K) | Conc. (mM) | T 1e (ms) | Sample set | Temp. (K) | Conc. (mM) | T 1e (ms) |
---|---|---|---|---|---|---|---|---|---|---|---|
I | 20 | 20 | 5.3 | III | 5 | 25 | 5.5 | III | 10 | 35 | 48 |
I | 20 | 40 | 5.3 | III | 5 | 30 | 5.6 | III | 10 | 40 | 47 |
II | 10 | 20 | 41.3 | III | 5 | 35 | 5.3 | III | 20 | 20 | 280 |
II | 10 | 40 | 28.3 | III | 5 | 40 | 5.1 | III | 20 | 25 | 290 |
II | 20 | 20 | 5.4 | III | 10 | 20 | 63 | III | 20 | 30 | 278 |
II | 20 | 40 | 5.3 | III | 10 | 25 | 57 | III | 20 | 35 | 280 |
III | 5 | 20 | 5.5 | III | 10 | 30 | 56 | III | 20 | 40 | 260 |
Temperature | 5 K | 10 K | 20 K |
---|---|---|---|
T m (20 mM) | 0.53 μs | 0.49 μs | 0.46 μs |
T m (40 mM) | 0.35 ![]() |
0.33 μs | 0.28 μs |
![]() | (8) |
To test the reproducibility of these ELDOR profiles, we prepared an additional pair of samples with 40 mM and 20 mM radical concentrations, samples II40 and II20 respectively, and measured their ELDOR spectra. For these samples, we chose only three detection frequencies, Δνdet = −100, 0,
and 100
MHz. A comparison of the detection frequency Δνdet = 0
MHz is shown in the ESI,† Fig. S2. While samples I40 and II40 give very similar ELDOR spectra, some differences are noted between samples I20 and II20.
All the steady-state ELDOR data collected were analyzed by the simulation procedure described in Section 2.3, resulting in the best fit values for ASE and ΛeSD. In practice, we first determined the values of the proton ASE by reproducing the ELDOR spectra at frequencies outside the EPR spectrum. Manual eye inspection resulted in ASE = 4 MHz for all samples and at both temperatures. During the determination of ASE, the value for T2e was set at 10 μs for all cases. This T2e value determines also the off-resonance excitation efficiency of the MW irradiation and in particular, the shape of the ELDOR profiles after a short irradiation time, and is not necessarily equal to the phase-memory time Tm. The values of all ΛeSD extracted from these best-fit simulations are given in Table 3, showing that the ΛeSD values are basically the same for both sample sets within the uncertainty in the fitting parameters, and are generally temperature independent.
I20, 20 K | I40, 20 K | II20, 10 K | II20, 20 K | II40, 10 K | II40, 20 K | |
---|---|---|---|---|---|---|
Λ eSD (μs−3) | 500 | 2500 | 400 ± 50 | 400 ± 50 | 2500 ± 250 | 2500 ± 250 |
Fig. 2 shows the fitting of the ELDOR spectra of the II40 (a–c) and II20 (d–f) samples at 20 K (black) and 10 K (red) and at three detection frequencies, Δνdet = −100, 0,
and 100
MHz. We would like to emphasize that for all the fittings we have done in this work, the ΛeSD values were independent of the detection frequency, i.e. we assume ΛeSD is isotropic. This assumption is based on the understanding that the spectral diffusion is a result of a cross-relaxation process which we assume is isotropic. We notice that in particular at 20 K, we observe in the ELDOR spectra a broadening around Δνdet that originates from the 14N-SE depolarization. In the present simulations, these depolarization effects are not taken into account during the analysis and therefore the fit was based on reproducing the total width of the spectrum, relying less on the central part. The positions of the 14N-SE depolarizations and their effect on the ELDOR profiles determined by the hyperfine interaction with 14N will be treated in a separate publication.
![]() | ||
Fig. 2 The experimental (dotted lines) and simulated (solid lines) steady state ELDOR spectra EELDORexp/sim(Δνexc,[Δνdet]) are shown for sample II40 in (a–c), and for sample II20 in (d–f), for two temperatures, 10 K (red) and 20 K (black). The detection frequencies are Δνdet = −100 MHz in (a and d); Δνdet = 0 in (b and e); and Δνdet = 100 MHz in (c and f). In each case, the solid lines are the simulated spectra EELDORsim(Δνexc,[Δνdet]) using the eSD model with the parameters listed in Table 1. The thermal equilibrium EPR spectrum is shown in green. |
The estimated error of the best-fit ΛeSD parameters was determined on the basis of the least squares fitting while keeping the parameters T2e and ASE fixed. To show this, we plot in Fig. S3 in the ESI† the sums of the squares of the differences for all Δνexc,j, between the simulated and experimental ELDOR profiles of the II20 and II40 samples at 20 K, as a function of ΛeSD for the fixed values Δνdet = −100, 0, and 100
MHz. A similar procedure was performed for these samples at 10 K. The ΛeSD values for each sample and temperature were set equal to the average of the best fit values for each Δνdet value, plus/minus their maximum deviations from this average value. These deviations are added to the ΛeSD values in Table 3.
In Fig. 3a and b, we show a reconstruction of the steady-state EPR spectra of sample I40 and sample I20, respectively, as a function of Δνdet and for the fixed values Δνexc = −200, −100,
0,
and 100
MHz, derived from the 2D plot in Fig. 1. These spectra are obtained by extracting slices of the 2D plots, EELDORexp(Δνexc,[Δνdet]), of constant excitation frequency and varying detection frequency, EEPRexp(Δνdet,[Δνexc]), and then multiplying them with the EPR lineshape function, EEPReq(Δνdet), of TEMPOL recorded with low concentration at thermal equilibrium.55 Finally, these EPR spectra are presented by setting the maxima of EEPReq(Δνdet) to unity. This way of presenting the data gives a better perspective on the effect of spectral diffusion when burning a hole. We can, of course, look also at the simulated EPR spectra, EEPRsim(Δνdet,[Δνexc]), of the samples using the ΛeSD parameters given in Table 3, as shown in Fig. 3c and d. As expected, we see in the simulated ELDOR spectra that the eSD model shows narrower MW excitation profiles around Δνdet than their corresponding experimental profiles, because 14N-SE effects are not taken into account.
![]() | ||
Fig. 3 Experimental and simulated steady-state EPR spectra EEPRexp/sim(Δνdet,[Δνexc]) derived from EELDORexp/sim(Δνexc,[Δνdet]) × EEPReq(Δνdet) obtained after a MW irradiation period tMW = 20 ms of sample I40 in (a and c), and of sample I20 in (b and d). The experimental spectra in (a and b) are extracted from the 2D-ELDOR spectra shown in Fig. 1 at Δνexc = −200 MHz (black), Δνexc = −100 MHz (red), Δνexc = 0 MHz (blue) and Δνexc = 100 MHz (magenta). The spectra in (c and d) are colored accordingly. The EPR spectrum at thermal equilibrium, EEPReq(Δdet), is shown in green. The simulated EPR spectra obtained using the eSD model with parameters as in Table 3 are shown in (c and d). |
Looking at the results in Table 3, we see that an increase of the radical concentration by a factor of 2 causes an increase of ΛeSD by a factor of 6. This increase manifests itself as a profound expansion of the depolarization area when comparing the EPR spectra of I20 and I40. Another observation is that the ΛeSD parameter stays constant when the temperature of the samples is varied between 10 and 20 K, while T1e varies by a factor of 6–8 (see Table 1).
![]() | ||
Fig. 4 Experimental (dotted lines) and simulated (solid lines) ELDOR spectra, EELDORexp/sim(Δνexc,[tMW,Δνdet]), of sample II40 at 10 K, are shown for 5 different values of tMW with detection frequencies Δνdet = 0 MHz in (a), Δνdet = 100 MHz in (b), and of sample II20 also for 5 values of tMW, with detection frequencies Δνdet = 0 MHz in (d) and Δνdet = 100 MHz in (e). In (a and b) spectra are plotted for tMW = 0.15, 0.6, 1, 4, 20 ms and in (d and e) for tMW = 0.4, 1, 2, 8, 20 ms, in black, red, blue, green and magenta, respectively. The simulated solid lines are obtained using the eSD model with parameters listed in Table 1. The tMW dependent values of ASE used during the simulations are plotted for sample II40 in (c) and for sample II20 in (f). The frequency positions of the 1H-SE features in the spectra are marked with arrows. |
The τdep values are a complex function of the eSD parameter ΛeSD, the MW amplitude ω1 and the T1e value and of course, the values of Δνdet and Δνexc. If the electron depolarization would have been determined solely by the spectral diffusion mechanism, we would have expected the τdep times to be about inversely proportional to ΛeSD. Inspection of the τdep values of sample II40 and sample II20, with the ratio between their ΛeSD values equal to ∼6, reveals that the ratio between the τdep values at equal Δνdet and Δνexc with different T1e values varies between 3.5 and 5.5.
To verify further the validity of the eSD model, we also followed the temporal evolution of the electron polarizations back to equilibrium after the MW irradiation, on sample II20 at 10 K. The pulse sequence used for the evaluation consists of MW irradiation of fixed duration tMW, followed by a varying time delay trelax before the echo detection pulses. Here we chose a short MW irradiation time tMW = 100 μs, preventing any significant cross-relaxation induced spectral diffusion effect during this pulse. Normalized EPR spectra EEPRexp(trelax,Δνdet,[Δνexc]) were constructed by performing a large set of ELDOR experiments within the range of −200 MHz < Δνdet < 100 MHz as a function of trelax. The results are shown in Fig. 6a together with the thermal equilibrium EPR spectrum.
![]() | ||
Fig. 6 Experimental (a) and simulated (b) EPR spectra, EEPRexp/sim(Δνdet,[trelax,Δνexc]), of sample II20 at 10 K, obtained after a short MW pulse of tMW = 100 μs and an increasing delay time, trelax, shown in black (0.01 ms), red (0.2 ms), cyan (0.7 ms), green (5 ms), magenta (20 ms) and violet (200 ms). The simulated spectra in (b) are derived using the eSD model with parameters tabulated in Table 3. In (a) the 14N-SE depleted signals are marked with arrows. |
The narrow hole in the spectrum created by the short MW irradiation broadens due to the spectral diffusion, and after 5 ms the normalized EPR spectrum reaches a flat profile that shows that the unnormalized EPR spectrum itself has reached the same shape as the thermal equilibrium EPR spectrum, but with a smaller overall intensity. This intensity grows to its equilibrium value with a time constant T1e, that is the same for all Δνdet values. In addition to the hole at Δνexc = 0, we observe the effect of SE processes of 1H and 14N. In Fig. 6b we show the simulated normalized EPR spectra using again the value of ΛeSD derived from the ELDOR analysis. The broadening of the initial hole and the equilibration of the intensity over the whole spectrum are clearly reproduced by the simulation in Fig. 6b. However, a comparison between the experimental and simulated spectra shows a significant deviation. The sharp features in the experiment are missing in the simulation and the level of equilibration is much closer to 1 than in Fig. 6a. The reason for this difference is the fact that the 14N-SE depolarization process is not taken into account in the calculations. A preliminary simulation taking the 14N-SE process into account is shown in Fig. S4 (ESI†). The evolution of EEPRexp/sim(trelax,[ΔνdetΔνexc]) at detection frequencies that are not in the initial hole, burnt by the MW irradiation, during the initial 5 ms is shown in Fig. S4 (ESI†) for both samples. The initial decay can be fitted by single exponentials with a decay constant, τdecay. The time constants of these fitting procedures are summarized in Tables S3 and S4 (ESI†), which show that their experimental and simulated values agree within 10%. As this decay process is a direct consequence of spectral diffusion effects, this agreement of experiments and simulations is another indication that the model describes properly the spectral diffusion process.
The experiments and their analysis in this section have shown that, for each sample, a single ΛeSD value can explain the steady state ELDOR spectra, as well as the time behavior of the electron polarizations during and after MW irradiation. This shows that this empirical parameter indeed represents the dipolar interaction driven spectral diffusion mechanism. In the next section, we discuss in which manner ΛeSD depends on the average electron–electron dipolar interaction in our samples.
![]() | ||
Fig. 7 Experimental (dotted lines) and simulated (solid lines) steady-state ELDOR spectra EELDORexp/sim(Δνexc,[Δνdet]) of sample III40 in (a) and sample III20 in (b), with Δνdet = 0 MHz obtained at different temperatures are shown in magenta (3 K), blue (5 K), red (10 K) and black (20 K). The simulated spectra are derived using the eSD model with parameters tabulated in Table 3. |
Sample | III20 | III25 | III30 | III35 | III40 |
---|---|---|---|---|---|
A SE/2π (MHz) | 1.5 | 1.5 | 2 | 2 | 2 |
Λ eSD | 100 ± 20 μs−3 | 200 ± 20 μs−3 | 250 ± 25 μs−3 | 400 ± 25 μs−3 | 600 ± 50 μs−3 |
For all concentrations c, the value of ΛeSD was found to be temperature independent. The ASE parameter under the steady state conditions needed to be varied from 2 to 1.5 MHz to reproduce the depolarization for Δνexc values outside the EPR spectral range. The errors associated with ΛeSD were evaluated in the same way as described in the previous section. The relevant least square analysis for the data of the III20, III25 and III30 samples at 10 K is shown in Fig. S6 (ESI†). Another observation is the similarity between the T1e temperature dependence for all samples, as shown in Table 1. We can, therefore, conclude that changes in the ELDOR profiles as a function of temperature for a particular concentration are all a result of the temperature dependence of T1e. The dependence of ΛeSD in μs−3 units on concentration in mM units is plotted in Fig. 8. The points in this figure can be analyzed by fitting them to a simple exponential function:
ΛeSD(c) = λc![]() | (9) |
![]() | (10) |
The temperature independence of ΛeSD may be related to the weak temperature dependence of the phase memory time (Tm), here observed for samples III20 and III40 and presented in Table 2. While the contribution of the dipolar flip–flop process on Tm57–59 is expected to decrease its value at high concentration,38,60 its temperature dependent influence is negligible. Similarly, the dipolar induced cross relaxation increases ΛeSD with concentration and leaves it unchanged as a function of temperature. Naively, we can say that the weak temperature dependence of Tm is a consequence of the temperature-independent cross-relaxation process rate in this range, which also approximately justifies the temperature independent behavior of ΛeSD obtained by ELDOR simulations.
We are aware of the fact that the ΛeSD(c) values of the present samples, measured on the DNP spectrometer, are about four times smaller than what was obtained from the samples measured on the EPR spectrometer. Together with this, we observe that the T1e values for each concentration measured using the DNP spectrometer are about 1.2 times longer than those measured using the EPR spectrometer. Whether these two differences are correlated is not clear at the moment. Changes in the distribution of the radicals caused by the sample preparation methods for the two spectrometers could be an origin for the differences in ΛeSD and T1e values. The “EPR” samples are frozen quickly by insertion into liquid nitrogen where, in the case of “DNP”, they are rather cooled slowly. Other differences may arise from the different B1 distributions in the samples, knowing that the EPR spectrometer exhibits a better B1 homogeneity at the position of the sample than the DNP spectrometer. We expect, however, that all these effects will not affect the concentration and temperature dependence of ΛeSD.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp05930f |
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