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EPR spectra of aqueous solutions of the singly-charged surfactant nitroxide 4-[N,N-dimethyl-N-(n-dodecyl)ammonium]-2,2,6,6-tetramethylpiperidinyl-N-oxy bromide-d16 (DCAT12) are studied as functions of the molar concentration, c = 0.1 – 8 mM, and the temperature from 273 to 353 K. This concentration range is below the critical micelle concentration, cmc, at which DCAT12 forms micelles. Spin-spin broadening of the EPR lines averaged over the three lines is separated into contributions due to spin exchange, Be, and dipolar, Bdip, interactions yielding values of the fractional broadening by spin exchange, Ω(T), that vary from near unity at 353 K to approximately 50% at 273 K. This compares with a variation from unity to approximately 77% for a neutral spin probe perdeuterated 2,2,6,6-tetramethyl-4-oxopiperidine-1-oxyl (PDT) over the same range. Unlike PDT and the Stokes-Einstein prediction, the broadening constant by spin exchange, d Be/dc, is not linear with T / η where η is the shear viscosity, instead following a quadratic dependence. Nevertheless, d Be/dc is remarkable close to a hydrodynamic prediction using the Stokes-Einstein equation modified to take the spin probe charge into account. Compared with PDT, values of d Be/dc are decreased and d Bdip/dc increased at all temperatures, while the values of the re-encounter rate, , deduced from line shifts, are reduced. Interestingly, values of d Bdip/dc, Ω(T), and are comparable for PDT and DCAT12 when compared at the same rotational diffusion rates.
Spin exchange between nitroxide spin probes is becoming a rich source of detailed information about molecular collisions in liquids.1–8 Spin exchange broadens and shifts the EPR lines while introducing a spin-exchange induced dispersion into the spectra.1–8 The broadening is a sum of the contributions from spin exchange, Be, and dipolar interactions, Bdip. The broadening due to spin exchange alone may be deduced from the spin exchange induced dispersion that may be subtracted from the total broadening to give the broadening by dipolar interactions. The spin exchange frequency, ωe, may be calculated from the broadening by spin exchange. The line shifts have turned out to be richer in content than previously thought being composed of two components, one varying linearly and the other quadratically with ωe. The quadratic term yields an independent measure of ωe. We model nitroxide-nitroxide collisions as first-time collisions, referred to as encounters, followed by a series of re-encounters prior to the diffusing pair’s escaping each other’s presence. The mean time between re-encounter collisions is denoted by τRE.
Particularly interesting was the discovery2 that re-encounter collisions can introduce a component into the line shifts that is linear in ωe for cases in which the spin exchange interaction is strong; i.e.,|Jτc|>> 1 where J and τc are the minus two times the exchange integral and the duration of the collision, respectively. We recently proposed that the experimentally observed linear term was indeed due to re-encounter collisions5 and have provided support from experiments involving both 14N and 15N nitroxides.6 Provided that this proves to be true after further testing, the linear term offers exciting insight into the behavior of two nitroxides while they are in close proximity colliding with one another within a solvent “cage.”
To summarize recent progress, we are able to independently measure the effects of (1) spin exchange interactions, (2) dipolar interactions, and (3) re-encounter collision rates at a single temperature with good precision. This work was motivated by asking what happens to these effects if the nitroxides are charged. There is some theoretical insight into the effect on spin exchange due to the work of Debye,9 but, to our knowledge, nothing is known on the other two. The reader might find it interesting to predict the qualitative effect of a charge on dipolar interactions and re-encounter collision rates.
In this work, we begin a series of studies of nitroxides bearing a single charge. In our first effort, we study the surfactant nitroxide 4-[N,N-dimethyl-N-(n-dodecyl)ammonium]-2,2,6,6-tetramethylpiperidinyl-N-oxy bromide-d16 (DCAT12) because above the critical micelle concentration (cmc) the presence of micelles10,11 presents an interesting system for those interested both in micelles and molecular collisions in liquids. Protons are replaced by deuterons at all positions in the ring except at the attachment point of the tail. Nitroxides similar to DCAT12, although not deuterated, have found use in biological studies.12–16
In this work, we restrict our attention to concentrations below the cmc and compare the results with those taken from the literature for a neutral nitroxide perdeuterated 2,2,6,6-tetramethyl-4-oxopiperidine-1-oxyl (PDT). There is no experimental information in the literature about the effect of a charge on re-encounter frequencies or the importance of dipolar versus spin exchange interactions. We were particularly curious about the effect of a single charge on the rate of re-encounter collisions, , because the doubly-charged peroxylamine disulfonate (Fremy’s salt) is the only case found so far in which the line shifts are well describe by the quadratic term only. We suggested two possibilities: (1) re-encounters are prevented by the electrostatic repulsion of the doubly-charged molecules or (2) strong spin exchange does not prevail.7 We planned this work thinking that we would gain insight into these possibilities; however, we find that strong spin exchange does prevail and is even faster than predicted by hydrodynamic theory provide that the charge is taken into account. Thus, we still don’t know if (1) or (2) or either explains the results in Fremy’s salt.
The only published work on spin-spin interactions of singly-charged nitroxides was by Martini and Bindi17 who measured the concentration broadening of the sodium salt of 2,2,5,5-tetramethylprrrolidin-1-oxyl-3-carboxolate in water as a function of temperature. Those authors claim to separate the effects of dipolar and spin exchange interactions; however, unfortunately, the spectra were incorrectly analyzed because inhomogeneous broadening due to unresolved hyperfine structure was not properly treated. Furthermore, Martini and Bindi17 measured only the line widths, from which reliable separation of the two interactions cannot be obtained.7
Eastman et al.18 have discussed the qualitative features of the behavior of exchange broadening based on standard Debye theory9,19 which breaks down at rather low concentrations. Others20 have tried to construct empirical descriptions of spin exchange between paramagnetic ions and spin probes.
Unfortunately, the field is at an awkward moment in which we are able to study molecular collisions in liquids in more detail than we can describe them theoretically: thus, the present paper offers a modest beginning to seek a better understanding of the collision process from the experimental side, hopefully offering some insights that will be useful in formulating better empirical understanding.
The theory has been presented in detail in this series and references therein1–8 which may be consulted for a detailed, conventional presentation. Here, we summarize the theory associated with the analysis of the EPR spectra in Table 1 and Table 2. All EPR spectral lines are least-squares fit to a model of the sum of the contributions by a Voigt absorption and Lorentzian dispersion. The quantities resulting from the fits are summarized in Table 1 and Table 2 summarizes eqs 1 – 11. The mixing parameter η(MI) permits the separation of the homogeneous and inhomogeneous contributions to the observed line width and is used to correct the parameters that are affected by the inhomogeneous contribution.21
Hydrodynamic estimates of the quantities of interest in this work are available22 employing the Stokes-Einstein equation as follows:
where η is the shear viscositiy, a is the hydrodynamic radius controlling the translational diffusion, and 2rex is the distance between the two nitroxides where spin exchange is effective. In eq 12, is a factor that is unity for neutral nitroxides and takes into account electrostatic interactions between charged particles. In 1942, Debye9 derived an expression for valid in the limit of small concentration. For singly-charged particles of the same charge sign, in the absence of added electrolyte, ; see, for example p. 538 of ref 23. We call eq 12 with , the hydrodynamic dilute (HD) limit. In the presence of added electrolyte, Eastman et al.18 have approximated , but point out that the results are qualitative. Note that the commonly quoted second-order spin exchange rate constant1 may be computed as follows: .
The hydrodynamic prediction for the dipolar broadening constant is given by
where 2 rc is the distance of closest approach between the two spin dipoles.22 The factor is equal to unity for uncharged nitroxides; however, to our knowledge, is unknown for charge particles.
For nitroxides, Ce = 0.485 cP · G / M deg and Cdip = 763 deg· G / cP · M. In the limit .22
The Stokes-Einstein prediction for the mean time between re-encounters2 is given by
where b is the distance between the spin probes at collision. The factor is equal to unity for uncharged nitroxides. One might think that because both re-encounters and encounters involve the collision of the two spin probes; nevertheless, a moment’s thought about Debye’s derivation9 of the HD brings into doubt this assumption. Debye followed Smoluchowski’s24 original approach and calculated the probability that two charged particles will collide under the influence of a screened electrostatic potential. The existence of a diffuse ionic cloud reduces the long-range electrostatic potential leading to in the HD limit; however, it is not obvious how or if the cloud would intervene between two nitroxides while diffusing in close proximity within a cage. Without the influence of the diffuse cloud, one might expect to be even less than 0.45.
Similarly, hydrodynamic estimates for rotational correlation times25 are available from the Stokes-Einstein-Debye equation yielding
where we have tacitly assumed that the hydrodynamic radius for translation is the same as for rotation.
The broadening constants d Be/dc and d Bdip/dc are rather independent of the probe size because a, rc, and re must be very similar.1,22 The rate does depend on the distance of closest approach b. Employing the reasonable assumption that this distance is equal to the diameter of the probes, 2a = b, we may determine a from eq 15 and use it to compute hydrodynamic predictions of the values of .
For uncharged molecules, the rotational rate is predicted to be 9 times that of the re-encounter rate independent of size, temperature, or viscosity, a result that ought to be straightforward to test.
and from the resulting values of κ, values of the re-encounter time, τRE, are computed using2
where γ is the gyromagnetic ratio of the electron. The determination of τRE becomes more difficult as dipolar interactions become more significant because of the error in separating Be from Btot.
Liquid trideuterioammonia (1.2 mL, 1.0 g) was allowed to vaporize into a vigorously stirred suspension of powdered anhydrous calcium chloride (3 g) in hexadeuterioacetone (10 g) at room temperature. The suspension was then stirred for 4 days. The orange reaction mixture was cooled in ice and quenched by slow addition of 5 mL of a solution of approx. 40% KOD in deuterium oxide with vigorous stirring. The supernatant was decanted and the yellow paste residue was extracted repeatedly with diethyl ether. The combined organic layer was dried over magnesium sulfate and the solvent was removed at reduced pressure to leave an orange residue (1.67 g) that crystallized when chilled. The sample was purified by flash chromatography on silica with diethyl ether eluent to provide 970 mg of a light orange oil which crystallized as long needles. This material was used without further purification.
To a solution of dimethylamine hydrochloride (2.26 g, 21.7 mmol, 5 eq) in methanol-O-d (5 ml) was added KOD (0.47 g) and the mixture was stirred until the base dissolved and KCl precipitated. I (953 mg, 5.54 mmol) in methanol-O-d (2.5 mL) was added and suspension was heated at 50 °C for 45 min. Then a solution of sodium cyanoborohydride (1.74 g, 5 eq “borane”) was added in methanol-O-d solution (5 ml) over 45 min. The reaction mixture was then stirred for an additional hour at 50 °C, then cooled. KOH (4 g) was added and the suspension was stirred until the pellets dissolved. The mixture was filtered by suction and the salts were washed with diethyl ether. The methanol layer was concentrated and extracted with diethyl ether. The combined ether solution was washed twice with a small amount of brine and dried over potassium carbonate. Solvent was removed at reduced pressure at no higher than 45 °C to leave an orange oil (972 mg, 88% crude yield). This material was purified by flash chromatography on silica, first with diethyl ether to eliminate a yellow impurity, then with 1 % butylamine in diethyl ether to elute the major component as a yellow oil, 868 mg. This was further purified by Kugelrohr distillation (60 deg, 0.1 mm) to provide a colorless oil, 659 mg (59%).
The tertiary amine II (471 mg, 2.36 mmol) and n-bromododecane (0.65 mL, 676 mg, 1.15 eq, 2.71 mmol) were heated together under a nitrogen atmosphere for 43 hr at 80 °C. The product after cooling was an orange glass which flaked upon grinding with a spatula. This was triturated several times with petroleum ether (30–60 °C); the first-formed sticky ball became a tractable solid after the third wash. This was dried with a pump to leave an off-white powder, 933 mg, 78% yield. This material was used without further purification.
Compound III (933 mg, 2.08 mmol) was dissolved in methanol (5 mL) and the solution was added to a solution of sodium tungstate dihydrate (65 mg) and tetrasodium EDTAate trihydrate in water (4 mL). This mixture deposited some salts. Hydrogen peroxide (30%, 0.75 mL) was added dropwise and the cloudy mixture was stirred at room temperature for 3 days. Small portions of sodium tungstate dihydrate and tetrasodium EDTAate trihydrate were added, followed by methanol (2 ml), and hydrogen peroxide (30%, 0.3 mL), and the orange nonhomogeneous mixture was heated in an oil bath at 35 °C. After one more day, the mixture was cooled to room temperature and more hydrogen peroxide (30%, 0.3 mL) was added. After each of the following two days, further portions of sodium tungstate dihydrate, tetrasodium EDTAate trihydrate, and hydrogen peroxide (as just previously) were added and stirring was continued at room temperature. After a total of ten days the mixture was extracted with dichloromethane (2 × 20 mL). This extract was dried over magnesium sulfate and solvent was removed at reduced pressure to leave reddish-orange thick oil, 808 mg, which began to solidify upon standing. This was triturated twice with petroleum ether (30–60 °C) with centrifugation to provide an orange, soapy solid. Three crystallizations from a mixture of petroleum ether (30–60) and dichloromethane provided the desired cationic nitroxide, 60 mg. The mother liquors were combined and another crop of crystals was collected by a similar procedure, 90 mg.
A stock solution of 37 mM DCAT12 was prepared in Milli-Q water by weight and diluted by weight to form a series of molar concentrations c. EPR affords a convenient measure of radical concentrations, c(EPR), because they are proportional to the doubly-integrated spectral intensity. For DCAT12, values of c(EPR) were computed using eq 2 (Table 2). Factor in eq 2, which depends on the temperature, was determined using freshly prepared aqueous solutions of PDT employing the same sample preparation, placement, and experimental conditions. For DCAT12, averaged over c = 2 – 8 mM, c(EPR) = 0.95 ± 0.02 c. Curiously, for c = 0.1 mM, c(EPR) = 0.89 ± 0.01 c, thus there is a loss in detected signal of the surfactant nitroxide at low concentrations. The accuracy of c(EPR) is estimated to be ± 2%; the relative concentrations are accurate to ± 1%. Undegassed samples were sealed into 50-µL pipettes and placed into sample tubes housed in Bruker’s nitrogen gas-flow temperature control dewar. The temperature, accurate to ± 1 K, was measured with a thermocouple placed just above the active portion of the microwave cavity. During each spectrum, the temperature was stable to ± 0.1 K. Five EPR spectra were acquired, one after the other, with a Bruker 300 ESP X-band spectrometer interfaced with Bruker's computer employing a sweep time of 41 s; time constant, 10 ms; microwave power, 5 mW; and sweep width, 70 G. The modulation amplitude was set at 1.0 G, comparable to the minimum observed total line widths. This improves the signal-to-noise ratio of low concentration spectra and broadens the Gaussian component of the lines by the difference in columns 3 and 4 of Table 3, leaving the Lorentzian component unchanged. The magnetic field sweep width was measured with Bruker’s NMR Gaussmeter operating in the 1-mG resolution mode and was averaged over a day’s run.
EPR spectra of DCAT12 at concentrations below the cmc are similar to those published in recent papers.3,4,7,8 Fits of Gaussian-Lorentzian sum functions to these spectra yielded parameters summarized in Table 1 and were analyzed to give the quantities in Table 2. The results in the limit c → 0, where dipolar and spin exchange interactions are negligible, are tabulated in Table 3.
At low concentrations and temperatures, the ratio of the inhomogeneous to the intrinsic line width, , reaches values in excess of 3. Thus, even though this probe is deuterated, reducing significantly the effect of unresolved hyperfine structure, it is critical to correct for inhomogeneous broadening in order to obtain accurate values of the intrinsic line width, , from which the broadening is calculated. For example, at 298 K, the overall line width, ; thus, ignoring the correction leads to an error of 265%, and this percentage is a strong function of both temperature and concentration, so even the relative values of the broadening are not valid when computed from .
The cmc of CAT12 (not deuterated) was measured by Fox using conductivity with the following results: 8.7 ± 0.02, 9.3 ± 0.02, and 9.1 ± 0.02 mM at 298, 313, and 333 K, respectively.11 Except for Figure 1 – Figure 3 all of the results in this paper are derived from c(EPR) ≤ 8 mM, thus pertaining to spin-spin interactions in the absence of micelles. Figure 1 gives spectra of DCAT12 well above the cmc at three temperatures, showing the presence of the micelle line as a broad, underlying single line. The least-squares fits to the three narrow monomer lines and the micelle line are given in Figure 2. The smaller trace in each case in Figure 1 is the difference in the experimental spectrum and the fits. The three monomer lines and the micelle line in Figure 2 are sums of absorption and dispersion lines of Voigt shape. To save space we have not presented separate traces of the absorption and dispersion components; examples of these separated traces may be found.8
Figure 3 shows values of c(EPR) for DCAT12 derived from fits of the narrow lines versus c. The cmc = 8.0 mM ± 0.5 is clearly observable as the point where the monomer concentration levels. This break point is constant with temperature within experimental uncertainty; thus, the cmc determined from these data are somewhat smaller than those published by Fox.11 Only data in the range c < 8 mM are used in the rest of the present paper; however, it is interesting that employing data up to 12 mM, fitting just the narrow lines does not affect the results presented in Figure 4 – Figure 9. In fact, fitting the narrow spectrum and ignoring the micelle spectrum yields reasonable results up to approximately 14 mM at T > 313 K. Above this concentration, the fitting of the center narrow line begins to be affected by the presence of the micelle line; however, the outside lines continue to give results near those that properly include the micelle line. Below 283 K, the micelle line becomes so broad that it does not appear in the residuals of narrow-line only fitting.
Values of Btot vary linearly with c at all temperatures with coefficients of correlation larger than 0.999. For a typical plot of Btot versus c, see, for example, Figure 4 of ref7. In contemplating the challenge of studying DCAT12 as opposed to non-surfactant nitroxides, it is useful to note that the maximum broadening observed in the present work is approximately 0.25 G compared with, for example, 5 G observed in ref7. Line widths in the absence of spin-spin broadening are in the range 0.86 – 1.16 G.
Figure 4 shows values of the total broadening constant, d Btot/dc, as a function of T / η. Data for PDT taken from ref7 are included and also in Figure 6 – Figure 9 for comparison. The straight lines are least-squares fits constrained to the origin with slopes 0.30 and 0.24 G · cP / M · K, for PDT and DCAT12, respectively. Although d Btot/dc for DCAT12 is fit satisfactorily by a straight line, a slight curvature concave upward is apparent.
Figure 5 shows representative plots of averages of vs. <Btot/A0 > at 298 K (triangles, r = 0.991) and 353 K (squares, r = 0.999). For clarity, the ordinate of the 298 K data is displaced by + 0.025. The straight lines are linear fits with coefficients of correlation indicated in the parentheses. Individual results from 5 spectra are plotted to show the reproducibility; the differences in are smaller than the plot symbols.
The slopes of the linear least-squares fit lines in Figure 5 yield the percentage broadening by exchange, Ω(T), which are presented vs. T / η in Figure 6a and vs. in Figure 6b. Data for PDT in water (diamonds) and PDT in 70 wt% glycerol (circles) taken from the literature are included.7 Figure 6a shows significant differences between the neutral PDT in the two solvents and the singly-charged surfactant DCAT12 in water; however, these differences are smaller when viewed as functions of the rotational correlation rate, Figure 6b.
The broadening constant by spin exchange, d Be/dc, as a function of T / η is shown in Figure 7. The solid straight line is a least-squares fit with slope 0.35 ± 0.01 G · cP / M · K (r = 0.986) for PDT. The solid line through the DCAT12 data is a quadratic constrained to the origin as follows: d Be/dc = (0.114 ± 0.006)T / η + (1.26 ± 0.009) × 10−4 (T / η)2, r = 0.999 in G/M with T / η in K/cP. The error bars are propagated from the errors in Ω(T)and the total broadening constant. The dashed line is the Stokes-Einstein prediction for PDT, eq 12 with rex = a. The dotted line is the HD limit prediction for DCAT12 showing an remarkable agreement with the experimental data.
The dipolar broadening constant, d Bdip/dc, is given vs. η/T in Figure 8a and vs. τrot in Figure 8b. The Stokes-Einstein prediction for PDT, eq 13, is shown as the dashed line in Figure 8a. Data for PDT in 70 wt% glycerol taken from the literature7 are off scale in Figure 8a but are included in Figure 8b. Note that there is a dramatic difference in the ranges of the abscissae of Figures 8a and 8b. The dipolar broadening constant for DCAT12 appears to reach a plateau of approximately 16 ± 3 G / M in water at approximately 283 ± 10 K. A similar plateau of 25 ± 1 G / M is observed for PDT in 70 wt% glycerol at approximately 273 K; however, at a value of η/T that is 30 – 50 times higher.7 The difference in the rotational mobilities of the two nitroxides corresponding to these two plateaus is about a factor of 4. It is clear that the SE prediction for dipolar broadening can only be useful very near the origin (high values of T) of Figure 5 where, unfortunately, the data are subject to large errors. When Ω approaches unity, the calculation of Bdip becomes the small difference in the two large quantities Be and Btot, eq 11. Furthermore, the computation of Be from the dispersion is subject to larger errors at high values leading to systematic errors that yield values of Ω slightly larger than unity, as is evident in Figure 6. These produce negative values of Bdip, a non-physical result. We hope to improve our determination of Ω when it approaches unity; however, in the meantime, the best we can do is to compare the initial slopes of the experimental data in Figure 8a with the SE prediction of . This results in for PDT in 70 wt% glycerol, 0.3 ± 0.02 for PDT in water, and 0.07 ± 0.01 for DCAT12 in water. Thus, there is an order of magnitude reduction in the value of (an order of magnitude increase in the dipolar broadening constant) for the charged DCAT12. How much of that is due to the charge and how much is due to other factors, for example the attractive hydrophobic forces expected from a surfactant, is not yet known.
Plots of Aabs / A0 vs < Be / A0 > (not shown, see, for example, Figure 10 of ref5) were fit to eq 17 and values of the re-encounter time, τRE, were computed using eq 18. The re-encounter rate is plotted as a function of T / η in Figure 9a and of in Figure 9b. The dashed lines in Figure 9a are eq 14, assuming that b = 2a where the hydrodynamic radii are determined as follows: a plot of (Table 3) versus T / η is linear (r = 0.998) with slope (0.0528 ± 0.0015) × 109 s · cP / K. From eq 15, we compute a = 4.17 Å for DCAT12. Using data from ref7 the same procedure produces a = 2.67 Å for PDT. The dotted line is the HD limit prediction for DCAT12 assuming that . Note that this hydrodynamic radius for PDT is considerably smaller than a = 3.5 Å computed from the method of Bondi and used in Figure 4 of ref8. The dashed line in Figure 9b is a plot of eq 16 that shows remarkable consistency with the experimental data except for DCAT12 at 273 and 283 K. The dashed lines in Figures 9a and 9b are essentially the same content; however, in 9a the rotational data are smoothed by fitting to eq 15, while in 9b they retain their individual measured values.
Our expectation was that the broadening constants due to spin exchange d Be/dc would be reduced due to coulomb repulsion and Figure 7 shows this to be so. The dashed line is the SE prediction for an uncharged molecule, eq 12 with a = rex and the dotted line is the HD limit for singly-charged molecules. The nearly quantitative agreement between the HD limit and the experimental results for DCAT12 shows that strong exchange very likely prevails. Equation 12 shows remarkable agreement with PDT in a series of alkanes,8 a series of aromatics,26 as well as in water, Figure 7;7 however, none of the experimental curves of d Be/dc vs. T / η is strictly linear. Note that the hydrodyamic radius does not come into the theory.
From values of , we find that the initial increase of d Bdip/dc is 14 ± 2 times faster than the hydrodynamic prediction for DCAT12 in water. For PDT, the initial increase is 3.3 ± 0.2 times faster in water and 1.0 ± 0.8 in 70% glycerol. The result in Figure 8b, where PDT in two solvents and DCAT12 in water show similar increases when plotted versus τrot, is very interesting. That the results for PDT in two solvents would form a common curve makes sense because the hydrodynamic radius is the same. However, that DCAT12 would also form a common curve is unanticipated. Note that the circles in Figure 8b representing data for PDT in 70% glycerol are completely off scale in Figure 8a where they ought to be compared with the dashed line. We resist making too much of this result until more data become available. Note that the very interesting effects of dipolar interaction on line spacing and the dispersion predicted by Galeev et al.27 are not observed in the present work because the low values of T / η where these effects may be manifest were avoided.
In Figure 9, the dashed lines are the hydrodynamic predictions of , using the rotational motion to fix the hydrodynamic radii of PDT and DCAT12, respectively ignoring the charge of DCAT12. The dotted line represents the dilute limit of the expected rate for the charged DCAT12 employing . Unexpectedly, DCAT12 shows a re-encounter rate near that expected if it were not charged. Figure 9b showing that the universal prediction eq 16 holds rather well for these two nitroxides is a consequence of the good agreement of the experimental data with the SE result in Figure 9a. Equation 16 might not be expected to hold for DCAT12 for another reason: the value of hydrodynamic radius derived from rotational diffusion is expected to be less than that derived from translational diffusion because DCAT12 is a flexible probe.28 One observes, for example, that the hydrodynamic radii of a series of doxyl labeled stearic acid methyl esters determined from τrot depend on the point of attachment of the nitroxide moiety while their geometric radii are the same.28 Using a more realistic hydrodynamic radius for translation would reduce the theoretical prediction even more; i.e., the results would be even further from the theory. Finally, we’ve remarked that could be less than which would reduce the theoretical prediction even further. Therefore, we cannot assume general validity of eq 16 based on these limited results; nevertheless, if we are able to gain confidence in its generality from further work, a tremendous practical benefit will accrue because a measurement of τrot could be used to fix . Then, the line shifts could be used to provide an independent measurement of ωe that could be used to verify values derived from the dispersion. Therefore, the re-encounter rate for DCAT12 is higher than expected and we cannot think of a simple explanation except that while two nitroxides are in close proximity, translation of the unpaired electron localized on the 2p orbital on nitrogen is strongly coupled to the rotation motion. Are we perhaps measuring re-encounters with a contribution from rotational motion which may not be strongly affected by coulomb repulsion? The basic idea is shown in Figure 10 that shows that a rotation of two PDT molecules by π/2 produces an effective translation of the unpaired electrons. An even larger effect would be expected for a larger molecule like DCAT12. Perhaps the curious result in Figure 8b may be understood partially by the mechanism of Figure 10.
One other point is worth mentioning. The simplified SE equation, used to derive eq 12 – eq 14, is written using so-called stick boundary conditions for a spherical particle.29 The same is true for the Stokes-Einstein-Debye equation leading to eq 15. Many authors have considered modifications to account for different boundary conditions and non-spherical shapes for both rotation and translation; see, for example ref29 and references therein. Therefore one expects systematic errors to arise in eq 12 – eq 14 (translation) on the one hand and eq 15 (rotation) on the other. Assuming that these errors would be similar for rotation and translation, then determining a from τrot and using it in the Stokes-Einstein equation is expected to compensate for the errors somewhat. Kovarski et al.30 showed that this approach is sound in rather complex fluid systems. The idea of combining translational and rotational hydrodynamic equations has been used, for example, by Tarjus and Kivelson31 as the basis of their eq 3.
The net result of decreasing d Be/dc and increasing d Bdip/dc for DCAT12 relative to PDT is to reduce rather dramatically the total broadening constant as shown in Figure 6a. This means that we must be careful not to assume that spin exchange dominates for charged nitroxides even in water at room temperature, where the percentage broadening by exchange is less than 75 %. The intriguing correlation of values of Ω between PDT in two solvents and DCAT12 when plotted versus the rotational rate cannot be predicted from hydrodynamic theory.
It is useful to keep in mind that the dipolar broadening is rather small, reaching a maximum of 0.23 G for 8 mM DCAT12 at 273 K. Future studies of charged (non-surfactant) nitroxides will provide a wider range of data because one will not be limited to a maximum concentration of c < cmc.
In water, compared with the uncharged PDT, the singly-charged surfactant DCAT12 shows (1) decreased values of d Be/dc, (2) increased values of d Bdip/dc, and (3) decreased values of . Item (1) is remarkably well predicted by the HD limit of the SE equation, eq 12. Item (3) is well predicted by the SE result, eq 14, using the hydrodynamic radius estimated from the rotational correlation time, however, only if we ignore the charge. The SE prediction for d Bdip/dc is reasonably accurate only for PDT in 70 wt% glycerol near the origin. For PDT and DCAT12 in water, the increase of d Bdip/dc with η/T is more rapid than the hydrodynamic prediction. Values of Ω and d Bdip/dc for PDT in water and 70 wt% glycerol, as well as DCAT12 in water form approximately common curves when plotted against , or τrot, respectively, results that cannot be predicted by hydrodynamic theory.
The authors gratefully acknowledge support from NIH grant 2 S06 GM048680.