The Current State of Measuring Bimolecular Spin Exchange Rates by the EPR Spectral Manifestations of the Exchange and Dipole–Dipole Interactions in Dilute Solutions of Nitroxide Free Radicals with Proton Hyperfine Structure

Abstract

Experimental studies of 4-hydroxy-2,2,6,6-tetramethylpiperidine 1-oxyl (Tempol) in 60 wt% aqueous glycerol were carried out for temperatures from 273 to 340 K. Selective isotope substitution allowed comparisons between the experimental spectral manifestations of spin exchange and dipole–dipole interactions for protonated, deuterated, ¹⁵N, and ¹⁴N Tempol. Theoretical spectra were computed from a rigorous theory specifically formulated to include proton hyperfine interactions over a wide range of spin exchange and dipole–dipole interactions to compare with the experimental data. For spin exchange and dipole–dipole interactions small compared with the proton hyperfine coupling constant, spectra were calculated with perturbation theory to gain insight into the behavior of individual proton lines. The theoretical and experimental spectra were analyzed by least-squares fitting to Voigt shapes or by a new two-point method. For most accessible experimental designs, the comparisons are rather good; however, for an experiment constrained to low concentrations and high viscosities, the methods are less accurate.

Appendices

Appendix 1: Exaggerated Values of $\langle V_{{{\text{disp}}M}} /V_{{{\text{pp}}M}} \rangle$

Figure 16 shows thirteen Lorentzian absorptions (a) and inter-manifold dispersions (b) of binomial relative intensities, light lines, and the sums, bold lines. (c) The sum of the absorption and dispersion manifolds. On this scale, there appear to be 9 lines because the outer 4 are not observable. Compare with Fig. 1, except the intra-manifold dispersions are not shown and a greatly exaggerated value of $\langle V_{{{\text{disp}}M}} /V_{{{\text{pp}}M}} \rangle$ = 0.3 is employed to emphasize the asymmetry of the spectrum. The lf- and hf-manifolds are spaced by the ¹⁵N hyperfine coupling constant, $A_{0}$ = 22.0 G; however, the central 12 G of each trace has been removed to emphasize the structure; thus, the nitrogen spacing is about twice as large as it appears. The intrinsic Lorentzian line widths for the two manifolds are $\Delta H_{{{\text{pp}} + 1}}^{\text{L}}$ = 0.4680 G and $\Delta H_{{{\text{pp}} - 1}}^{\text{L}}$ = 0.6521 G, respectively, and $a$ = 0.26 G. The absorption–dispersion admixtures (c) and (f) are asymmetric, reminiscent of line shapes that are now familiar for nitroxides undergoing HSE; e.g., Figure 9a of Ref. [23, #3822]. Of course, in this latter case, the IHB will have collapsed, so the manifolds are effectively Lorentzian, in contrast with Fig. 16. The quantities $\Delta \varOmega_{{{\text{pp}}M}}^{\text{man}}$, $Y_{\hbox{max} M}^{\text{man'}}$, and $Y_{\hbox{min} M}^{\text{man'}}$ are defined.

Fig. 16

a Thirteen Lorentzian absorptions and b inter-manifold dispersions of binomial relative intensities, light lines, and the sums, bold lines. c The sum of the absorption and dispersion manifolds. On this scale, there appear to be 9 lines because the outer 4 are not observable

The resulting manifolds have $V_{\text{disp}}^{\text{man}} /V_{\text{pp}}^{\text{man}}$ = 0.2675 (lf) and 0.2708 (lf), 11 and 10% smaller than $V_{{{\text{disp}}M}} /V_{{{\text{pp}}M}}$, respectively. When transformed by Eq. (49), $\langle V_{\text{disp}}^{\text{man}} /V_{\text{pp}}^{\text{man}} \rangle$ = 0.300 for both manifolds.

Figure 17 shows the low-field line of the admixture, Fig. 1c, on an expanded scale showing the resonance field, $H_{\text{lf}}^{\text{abs}}$; i.e., where the absorption crosses the baseline (where the dispersion is maximum); and where the admixture crosses the baseline, $H_{\text{lf}}^{\text{obs}}$. In this case, in which the lf-manifold is positive because HSE dominates, $H_{\text{lf}}^{\text{obs}}$ is up field from $\langle H_{ - 1/2} \rangle$; the opposite is true at hf, so the separation between the two admixtures, $A_{\text{obs}}$, is less than between the resonance frequencies, $A_{\text{abs}}$. For DD, the reverse occurs. See Fig. 11a of Ref. [11] for an experimental example of an increasing $A_{\text{obs}}$ as $C$ increases.

Fig. 17

The lf-manifold of Fig. 16 showing the resonance field, $H_{ - 1/2}$ and where the admixture crosses the baseline

Figure 18 shows values of $V_{\text{disp}}^{\text{man}} /V_{\text{pp}}^{\text{man}}$ derived by the two methods as a function of $\chi_{M}$.

Fig. 18

Values of $V_{{{\text{disp}}M}}^{\text{man}} /V_{{{\text{pp}}M}}^{\text{man}}$ from fitting, open circles, and from the two-point method, open squares as functions of $\chi_{M}$. The closed circles are obtained from the open circles by applying the correction Eq. (49). For input values of $\langle V_{{{\text{disp}}M}} /V_{{{\text{pp}}M}} \rangle$ ranging from 0 to 0.4, the fitting method yields the same value, while the two-point method yields somewhat different values as shown by the scatter of the open squares

Appendix 2: IHB Spectra Remain Excellent Voigt Line Shapes Under HSE or DD

Spectra were generated for HSE and DD with Eq. (1) and for $C$ = 0 from Ref. [23, #3822]. The input parameters are given in the caption to Fig. 16. The spectra were fit with Eq. (44) and the differences between the fit and the spectra, the residuals, were computed. Figure 19 shows the lf-manifold of spectra and the resulting residuals for conditions near and including incipient resolution. The $C$ = 0 spectra were computed by holding the proton spacing constant at $a$ = 0.4 G and varying $\Delta H_{\text{pp}}^{\text{L}}$ such that the range of $\chi_{ - 1/2}$ overlapped that of the HSE and DD results.

Fig. 19

Spectra for the ${\text{lf}}$-manifold only computed from Eq. (1) for a ¹⁵N nitroxide, $A_{0}$ = 22 G with hyperfine coupling to 12 equivalent protons, $a$ = 0.4 G, with $K_{\text{ex}} C/\gamma$ = a 0.03 G, b 0.06 G, c 0.09 G; or with $W_{\text{dd}} C/\gamma$ = d 0.0525 G, e 0.1575 G, f 0.2625 G. $\Delta H_{\text{pplf}}^{\text{L}} (0)$ = 0.45 G. The spectra were fit with Eq. (44) and the residuals computed. The smaller trace overlying each spectrum shows the residual. The fit is shown only for a in order not to obscure the distortions in the other spectra that are evident near the peaks of b–e. A practiced eye will also detect distortions in f

Figure 20 shows the maximum value of the residuals, $R_{ \hbox{max} }$, as a fraction of $V_{\text{pplf}}^{\text{man}}$. This plot shows that for $\chi_{ - 1/2}$ < 2, the difference in the Voigt shape and the spectrum is 1% or less; thus, for all three cases, HSE, DD, and $C$ = 0, the spectra are accurately modelled as Voigt shapes. For the hf-manifolds of all spectra, with $\Delta H_{{{\text{pp}}M}}^{\text{L}} (0)$ = 0.69 G, $R_{ \hbox{max} } /V_{\text{lf}}^{\text{man}}$ < 0.004 for HSE, DD, and $C$ = 0, all having $\chi_{\text{lf}}$ < 1.9 and showing no hint of incipient resolution.

Fig. 20

Maximum value of the residuals, $R_{ \hbox{max} }$, as a fraction of $V_{ - 1/2}^{\text{man}}$ from fits to Eq. (44). Diamonds, DD; squares, HSE; circles $C$ = 0. Open symbols, $a$ = 0.2 G; closed, $a$ = 0.4 G. The vertical arrows indicate values of $\chi_{ - 1/2}$ corresponding to spectra in Fig. 19a–f, respectively

One question is how well does the Voigt model IHB spectra; another, just as important, is how well do the fit parameters compare with the known input parameters. This can only be done for $C$ = 0 spectra because we do not know the input values except there. In Fig. 21 this question is addressed over the range of $\chi_{ - 1/2}$ in Fig. 20 and higher, where input and fit values of line widths of $C$ = 0 spectra are displayed. The solid squares are input values of $\langle \Delta H_{{{\text{pp}}M}}^{\text{L}} \rangle$ and the open squares, the fit values of $\Delta H_{{{\text{pp}}M}}^{\text{L(Voigt)}}$. The solid circles show input values of $\Delta H_{{{\text{pp}}M}}^{\text{G(Voigt)}} = a\sqrt {\alpha N}$, from Eq. (8), and the open squares, the fit values of $\Delta H_{{{\text{pp}}M}}^{\text{G(Voigt)}}$. The vertical arrows indicate values of $\chi_{\text{lf}}$ corresponding to the spectra in Fig. 19a, b, respectively. The inset shows the spectrum and residuals for the $\chi_{M}$ = 2.85; compare with Fig. 19a, b, respectively. It is clear that excellent accuracy is obtained in support of Assumption 1, even for spectra that are clearly partially resolved. This interesting fact holds for spectra showing considerably more resolution that in the inset; however, detailed investigation is outside the scope of this paper.

Fig. 21

Input and fit values of line widths of $C$ = 0 spectra. Solid squares, input values of $\langle \Delta H_{{{\text{pp}}M}}^{\text{L}} \rangle$ and open squares, fit values of $\Delta H_{{{\text{pp}}M}}^{{{\text{L}}({\text{Voigt}})}}$. Solid circles, input values of $\Delta H_{{{\text{pp}}M}}^{{{\text{G}}({\text{Voigt}})}} = a\sqrt {\alpha N}$, from Eq. (8), and open squares, fit values of $\Delta H_{{{\text{pp}}M}}^{{{\text{G}}({\text{Voigt}})}}$. The vertical arrows indicate values of $\chi_{ - 1/2}$ corresponding to two of the spectra in Fig. 19a, b, respectively. The inset shows the spectrum and residuals for the $\chi_{M}$ = 2.85; compare with Fig. 19a, b, respectively

Appendix 3: The Reason that the p-Parameter is in Error for the Low-Field Line of 14 NH at 273 K

Figure 22 shows values of $- p_{\text{lf}} A_{0} /3$ and $+ p_{\text{hf}} A_{0} /3$ for 14 NH at 273 K taken from Fig. 11b on a larger scale. We observe that, while the hf results are reasonably linear, the lf results are not.

Fig. 22

Values of $- p({\text{lf}})A_{0} /3$ (squares) and $p({\text{hf}})A_{0} /3$ (circles) for 14NH at 273 K shown here on a larger scale than in Fig. 11b. The straight lines are linear fits, showing that the ${\text{hf}}$ results are reasonably linear; however, the ${\text{lf}}$ results are not

Figure 23 shows a spectrum at 273 K together with the residual, which shows an impurity line overlapping the main spectrum in the vicinity of the measurement of $p({\text{lf}})$. The impurity does not appear to amount to much on this scale, but because the asymmetry is so small, it makes a large difference in the $p$-parameter. Clearly there is no significant extraneous line for the high-field line so it’s not affected.

Fig. 23

EPR spectrum of 10-mM 14 NH at 273 K together with the residual where an impurity line is evident in the lf-manifold

Note also that for the cf and hf-manifolds where interference due to extraneous lines is not severe, the line shape is an excellent Voigt as shown by the small residuals.