1 Introduction

Heisenberg spin exchange (HSE) between paramagnetic particles is one of the few elementary bimolecular processes in solutions that is readily accessible to detailed experimental studies and to strict theoretical description. It arises when a colliding pair of radicals exchange electronic spin states at a rate proportional to their collision rate [1,2,3,4]. Because it depends on the mutual translational diffusion of spin probes, HSE allows one to study elementary molecular collisions in detail and has been widely used to address various problems of chemistry, physics, and molecular biology. One can obtain quantitative information on collision rates of molecules in solutions, including information about the structure of the colliding pair. Particularly valuable is the possibility of using spin exchange to study collisions in such complex systems as polymer solutions, multicomponent and heterogeneous mixtures, and especially biological systems.

In isotropic media, the rate of HSE between identical radicals is related to the radical concentration by the equation [1, 5]

$${\omega }_{\mathrm{ex}}\left(C\right)=p\cdot g\cdot 4\pi dDC,$$
(1)

where \(C\) is the concentration and \(d\) is the collision radius of the radical, \(D\) is the relative translational diffusion coefficient, \(g\) is a steric factor, and \(p\) is the probability of effective exchange per collision. This equation makes the approximation that spin exchange can only occur within the collision radius, whereas longer-range spin exchange interactions do occur in real systems.

Most detailed studies of the concentration dependence of HSE focus on the phenomenon in isotropic phases for which the dimensionality is well-defined, such as liquid solution. The Hubbell group has used the sensitivity of HSE to measure electrostatic interactions between charged nitroxide labels and probes to measure electrostatic potentials at biological interfaces. [6,7,8,9,10,11,12,13,14,15]. HSE has also been extensively used by the Hubbell group and by others to determine the solvent accessibility of spin labels attached to proteins by measuring broadening or spin relaxation due to HSE with paramagnetic species in solution [16,17,18] or in membranes [19].

Although HSE between paramagnetic probes has been studied in several complex and heterogeneous systems in which Eq. (1) may not apply, we are not aware of any systematic study of the detailed concentration dependence of HSE in such a system. One type of system that has been intensively studied is the percolation network [20, 21], which has applications in porous geological formations, aerogels, amorphous semiconductors, and biological membranes to name only a few. Specifically, the kinetics of elementary chemical reactions of a single species A, such as unimolecular decay or trapping \({\mathrm{A}}^{*}\to \mathrm{A}\), bimolecular coalescence \(\mathrm{A}+\mathrm{A}\to \mathrm{A}\), or annihilation \(\mathrm{A}+\mathrm{A}\to 0\), in such systems has been extensively characterized theoretically and verified experimentally [20,21,22,23]. In contrast to classical rate expressions for these reactions in isotropic homogeneous media, in percolation networks concentrations may appear with fractional exponents that depend upon the “fracton”, or “spectral” dimension of the network.

These considerations motivated us to identify a suitable system that might be expected to exhibit an effective dimensionality different from that of an isotropic solution and carry out a systematic study of spin probe concentration. A common and accessible example of such a network is a conductive ionic polymer such as Nafion, which has seen extensive use as conductive membranes in fuel cell applications. As revealed by cryo-electron tomography [24], the hydrated Nafion membrane consists of a network of aqueous channels lined with hydrophilic sulfonic acid groups interspersed with hydrophobic fluorocarbon backbone regions. The channels in the hydrated membrane are reported to have an average diameter 2.5 nm with a center-to-center spacing of 5.1 nm.

This system offers the additional advantage of an internal consistency check of the results. Most spin probe studies of ion exchange membranes are performed after exchange of the acidic proton on the internal sulfonate groups of the polymer with metal cations [25,26,27]. Without this pretreatment, the nitroxides are deactivated in the acidic environment. Thus, by initially loading the membrane with a high concentration of radical, it is possible to measure HSE over a wide range of concentrations without disturbing the sample, while at the same time measuring the kinetics of the radical disappearance.

This work presents a preliminary such study on a hydrated Nafion 117 membrane. We find that the rate of spin exchange as a function of probe concentration deviates from Eq. (1), and further that the decay kinetics of Tempone in the membrane does not follow classical kinetics. These results are discussed in the context of available percolation theories.

2 Materials and Methods

2.1 Sample Preparation and Measurement

Nafion 117 membranes (Ion Power, Inc., New Castle, DE) were first purified by heating to 75 °C for 1 h in 3% hydrogen peroxide followed by 1 h in deionized water, 1 h in 0.5 M sulfuric acid, and 1 h in deionized water. They were then cut to a size of 10 × 3 mm and equilibrated with a 118 mM aqueous solution of the Tempone probe (Sigma-Aldrich) that had been purged of oxygen by gently bubbling with saturated nitrogen gas. External liquid was removed from the membrane with a tissue and sealed in an EPR tube under a nitrogen atmosphere. The time evolution of the EPR spectrum was monitored in a Bruker EMX spectrometer at a microwave frequency of ~ 9.848 GHz. Spectra were acquired in a 2D field vs. time experiment at intervals of 90 s. The experimental parameters were constant for each field sweep, with a sweep width of 12.0 mT, an acquisition time of 81.92 ms, a time constant of 20.48 ms, field modulation amplitude of 0.05 mT and a microwave power of 2.0 mW. As noted above, Tempone was converted to an EPR-silent species over the course of the measurement. Spectra were acquired until the doubly integrated intensity of the spectrum was at most 1% of its initial value, over a period of at least 24 h. Two experimental runs were evaluated with 1831 and 2182 field scans, respectively.

2.2 Measurement of HSE

Heisenberg spin exchange was measured by least-squares fitting of the experimental lineshapes over the entire concentration range using the full lineshape calculation of Freed and coworkers [28, 29] based on the Stochastic Liouville Equation (SLE) [29, 30] as it is implemented in the “chili” function of the EasySpin package developed by Stoll [31, 32]. Because this program treats all transitions of the nitroxide at once, it will be referred to as the “full” SLE calculation. However, it does not include dipole–dipole interactions between radicals in an encounter pair.

Therefore, as a check of this method, the experimental spectra were also fitted using an analytical expression based on the SLE for pairs of interacting spins that are added to give the complete spectrum. The expression originally developed by Currin [2] has been recast by Salikhov in a paradigm that specifically identifies terms for transfer of spin coherence and spin decoherence (dephasing) and allows discrimination between the contributions of HSE and dipolar interactions to these terms [33,34,35,36,37]. This has been referred to as the “exact” lineshape calculation. Specifically,

$$J\left(B\right)=-I\mathrm{Re}\left\{\frac{G\left(B\right)}{1+VG\left(B\right)}\right\},$$
(2)

where I is the intensity of the spectrum in the linear response regime and

$$G\left(B\right)=\sum_{k}{G}_{k}(B)=\sum_{k}\frac{{g}_{k}}{-i\left({B}_{k}+{\delta }_{k}-B\right)-\left({\Gamma }_{0k}+W\right)}.$$
(3)

In Eq. (2) B is the spectrometer field and Bk, \({\Gamma }_{0k}\), and \({\delta }_{k}\) are, respectively, the resonance field, homogeneous linewidth in the absence of exchange, and exchange-induced shift of each hyperfine line k in the spectrum with statistical weight gk.[33] Also V and W, respectively, represent the rates of spin coherence transfer and spin dephasing due to electron spin–spin interactions at a given radical concentration. These quantities can be resolved into HSE and dipolar contributions as follows [33]:

$$\begin{gathered} V = \left( {K_{ex} - K_{dsct} } \right)C \hfill \\ W = \left( {K_{ex} - K_{dsct} } \right)C, \hfill \\ \end{gathered}$$
(4)

where Kex is the rate of HSE per unit concentration and Kdsct, and Kdsd are the rates of spin coherence transfer and spin dephasing (per unit concentration) due to dipole–dipole interactions between the colliding radicals.

It has long been appreciated that HSE is manifested as qualitative changes in the EPR spectrum as it increased with radical concentration. The changes theoretically predicted by Salikhov and coworkers [5, 33, 36,37,38,39] for fast-motional spectra of 14N nitroxides undergoing slow-to-intermediate exchange include (i) a broadening of the lines, (ii) a shifting of the positions of the \({m}_{\mathrm{I}}=\pm 1\) lines towards the center, and (iii) the development of asymmetry in the \({m}_{\mathrm{I}}=\pm 1\) lines that can be represented as an admixture of the Lorentzian dispersion lineshape. These effects were experimentally verified and phenomenologically quantified by Bales and coworkers using least-squares fitting of three Lorentzian lines to the spectrum [40, 41]. The fitting function for experimental first-derivative spectra is obtained by taking the derivative of Eq. (2) and adding in the dispersion line shape scaled by a factor ζ. Thus the absorption spectrum may be expressed as

$$G^{\prime}\left(B\right)={\sum }_{{m}_{I}=-1.0,+1}I{g}_{{m}_{\mathrm{I}}}\left\{\frac{2{\Gamma }_{{m}_{I}}\left({B}_{{m}_{I}}+{\delta }_{{m}_{I}}-B\right)}{{\left[{\left({B}_{{m}_{I}}+{\delta }_{{m}_{\mathrm{I}}}-B\right)}^{2}+{\Gamma }_{{m}_{I}}^{2}\right]}^{2}}+\zeta {m}_{I}\frac{{\left({B}_{{m}_{I}}+{\delta }_{{m}_{\mathrm{I}}}-B\right)}^{2}-{\Gamma }_{{m}_{I}}^{2}}{{\left[{\left(B_{{m}_{I}}+{\delta }_{{m}_{\mathrm{I}}}-{B}\right)}^{2}+{\Gamma }_{{m}_{I}}^{2}\right]}^{2}}\right\}$$
(5)

retaining the definition of terms in Eq. (2) and additionally defining \({\Gamma }_{{m}_{I}}={\Gamma }_{{m}_{I},0}+{\omega }_{\mathrm{ex}}\). The resonance fields \({B}_{{m}_{I}}\) also include the isotropic 14N hyperfine splitting \({a}_{\mathrm{N},0}\).

Bales and Peric identified a further effect of HSE on the spectrum, namely a change in the relative intensities of the \({m}_{I}=\pm 1\) and the \({m}_{I}=0\) lines resulting from magnetization transfer via HSE [41], which can be modeled by including a scaling factor \(d={g}_{\pm 1}/{g}_{0}\).

Least-squares fitting for the full 2-D time series of experimental data was automated in Matlab [42] using a modified Nelder-Mead simplex search algorithm (fminsearch). Equations (1–3) were programmed in Matlab macros, and the SLE calculations were performed using the chili function of the EasySpin Matlab package [31, 32]. Before fitting the series of spectra at different concentrations, Eq. (3) was fitted to a low-concentration spectrum with negligible HSE (i.e. assuming \({\delta }_{{m}_{I}}={\omega }_{\mathrm{ex}}=\zeta =0;d=1\)) including a Gaussian inhomogeneous broadening 2 \(\sigma\) (derivative peak-to peak). The parameters \({B}_{{m}_{I}}. {\Gamma }_{{m}_{I},0}\), and \(\sigma\) were fixed and the remaining parameters in Eqs. (1–3) varied as appropriate. For the fitting of the SLE lineshape calculations, only \({\omega }_{\mathrm{ex}}\) was varied, but all fits included a small correction for the instrumental microwave phase. Matlab functions were also programmed for the least-squares fits to \(C(t)\) and \({\omega }_{\mathrm{ex}}(C)\).

3 Results and Discussion

The kinetics of Tempone deactivation in the acidic Nafion membrane were investigated as an internal check of the fractal properties of the Nafion environment. Nitroxides undergo an acid-catalyzed redox disproportionation reaction as shown in Fig. 1 [43].Footnote 1 Kinetic studies of this reaction for a variety of 5- and 6-membered nitroxide rings including Tempone have shown that it obeys second-order kinetics in sulfuric acid solution over a wide range of acid and nitroxide concentrations [43].

Fig. 1
figure 1

Acid-catalyzed disproportionation reaction of nitroxides

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To test whether the disappearance of Tempone radical follows second-order kinetics in Nafion, a linearized kinetic plot was prepared of the inverse concentration of Tempone, as measured by the doubly integrated intensity of its EPR signal C vs. time, shown in Fig. 2A. A straight line fitted to the early part of the kinetic curve is included to emphasize the marked nonlinearity of the data, a clear indication that the disappearance of the Tempone radical departs significantly from classical second-order kinetics. A similar test was performed for first-order kinetics by plotting \(\mathrm{ln}C/{C}_{0}\) vs. time, shown in Fig. 2B. The nonlinearity is even more pronounced in this plot, suggesting that the effective order of this reaction may be between 1 and 2. Figure 2C shows a plot of C vs. time superimposed with the best-fit second-order (solid line) and first-order (dashed line) integrated rate laws. The residuals plotted at the bottom of this figure emphasize the deviation of the classical models from the observed kinetics.

Fig. 2
figure 2

Kinetic analysis of measured Tempone disappearance in acidified Nafion 117 membranes. A Linearized plot of 1/[Tempone] vs. time showing a significant deviation from the straight line expected for second-order kinetics. B Linearized plot of ln [Tempone]/[Tempone]0 vs time, showing a significant deviation from the straight line expected for first-order kinetics. C Plot of [Tempone] vs. time showing the best fits of integrated first order (dashed line) and second-order (solid line) rate laws. Residuals from the fits are show at the bottom

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The experimental design of monitoring the EPR spectrum as the spin probe decays in situ essentially eliminates the possibility that the observed deviation from Eq. (1) arises from nonlinearity in the probe concentration. This design follows a strategy employed in the initial HSE study of Bales and Peric with the spin probe PADS, which is deactivated at higher temperatures [40]. Because successive spectra are obtained under identical conditions, the intensity of the doubly integrated signal affords a very accurate measurement of the relative concentration of active spin probes; any nonideality in the partitioning of Tempone between the membrane and equilibration solution is immaterial. Finally, since the total number of probe molecules (both EPR-active and inactive) remains constant, any nonlinear effects of the probe concentration on the local viscosity, as reported by Eastman et al. [44, 45] are controlled for.

Figure 3 shows least-squares fits of the full SLE lineshape calculation to selected spectra over the time course of the Tempone disproportionation reaction. The parameters for the HSE-free spectrum obtained at long times were as follows: principal g-values of 2.00794, 2.00594, and 2.00342, principal 14N hyperfine values of 15.8, 15.8, and 103 MHz, and a peak-to-peak Gaussian inhomogeneous broadening of \(\sigma =0.06~\mathrm{ mT}\). For an isotropic rotational correlation time \(\tau =5.0\times {10}^{-9}\mathrm{ s}\), the homogeneous linewidths \({\Gamma }_{+\mathrm{1,0}},{\Gamma }_{\mathrm{0,0}}, {\Gamma }_{-\mathrm{1,0}}\) were 0.027, 0.028, and 0.031 mT, respectively. Only \({\omega }_{\mathrm{ex}}\) and the spectrum phase were varied in the fitting procedure. The fitted lineshapes (dashed lines) were essentially indistinguishable from the experimental ones up to a concentration of about 90 mM; at and above this concentration, small deviations began to appear as can be seen in the top spectrum of Fig. 3.

Fig. 3
figure 3

Nonlinear least-squares fits of the lineshape calculated by the “full” SLE equation (dashed lines) to experimental data (solid lines) at the indicated Tempone concentrations in the Nafion 117 membrane

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The different contributions to \({\omega }_{\mathrm{ex}}\) were identified using an approach suggested by Salikhov [39]: first, the linewidth, shift of the \({m}_{\pm 1}\) lines, and the magnitude of the dispersion components were determined by fitting Eq. (4) to the experimental spectra over the full concentration range. The fits to the experimental lineshapes for selected concentrations are shown in Fig. 4A. The values obtained for \({\omega }_{\mathrm{ex}}\), \({\delta }_{\pm 1}\), and ζ were then used to determine effective exchange rates as follows [5, 33]Footnote 2:

Fig. 4
figure 4

Nonlinear least-squares fits of A Eq. (4) and B the “exact” Eqs. (2) and (3) to experimental data at different concentrations in the Nafion 117 membrane. Dashed lines show calculated lineshapes and solid lines show experimental data

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$${\omega }_{\mathrm{ex},lw}={{\upgamma }_{\mathrm{e}}(\Gamma }_{{m}_{I}}-{\Gamma }_{{m}_{I},0})$$
$${\omega }_{\mathrm{ex},\zeta }={\upgamma }_{\mathrm{e}}{a}_{\mathrm{N},0}\frac{\zeta }{\sqrt{1+{\zeta }^{2}}}$$
(6)
$${\omega }_{\mathrm{ex},{\delta }_{\pm 1}}= {\gamma }_{\mathrm{e}}\sqrt{6{a}_{\mathrm{N},0}{\delta }_{\pm 1}}$$

The values of \({\omega }_{\mathrm{ex},lw}\) and \({\omega }_{\mathrm{ex},\zeta }\) were then used as initial estimates for the parameters W and V in the “exact” expression Eqs. (2) and (3), respectively. This expression was then fitted to the lineshapes varying these parameters and \({\delta }_{\pm 1}\). The fits of the exact expression to same set of experimental spectra are shown in Fig. 4B. The agreement is generally excellent for both Eq. (4) and the exact expression; slight deviations between the exact and experimental lineshapes appear at concentrations of ~ 90 mM and above, apparent in the top spectrum in Fig. 4B.

Figure 5 compares the effective exchange frequencies \({\omega }_{\mathrm{ex},lw}\), \({\omega }_{\mathrm{ex},\zeta }\), and \({\omega }_{\mathrm{ex},{\delta }_{\pm 1}}\) calculated according to Eqs. (5) with the exchange estimated from the full SLE calculation. It is evident from this plot that \({\omega }_{\mathrm{ex},lw}\) and \({\omega }_{\mathrm{ex},\zeta }\) both maintain a constant ratio with \({\omega }_{\mathrm{SLE}}\) over the concentration range shown: \({\omega }_{\mathrm{ex},lw}/{\omega }_{\mathrm{SLE}}\) = 0.61 ± 0.02 and \({\omega }_{\mathrm{ex},\zeta }/{\omega }_{\mathrm{SLE}}\) = 0.81 ± 0.02. More significantly, all these quantities exhibit a marked nonlinearity with concentration, as emphasized by the dashed lines in Fig. 5. Above ~ 30 mM, \({\omega }_{ex,{\delta }_{\pm 1}}\) is also proportional to \({\omega }_{\mathrm{SLE}}\) with a constant ratio \({\omega }_{\mathrm{ex},{\delta }_{\pm 1}}/{\omega }_{\mathrm{SLE}}\) = 1.33 ± 0.05 although it exhibits significant nonlinearity at lower concentrations.

Fig. 5
figure 5

Effective spin exchange rates \({\omega }_{\mathrm{ex},lw}\), \({\omega }_{ex,\zeta }\), and \({\omega }_{{\delta }_{\pm 1}}\) obtained from least-squares fits to the experimental data of Eq. (4) using Eqs. (5), and \({\omega }_{\mathrm{SLE}}\) obtained from least-squares fits of the full SLE lineshape calculation Each point represents a result from an individual fit to a spectrum at a given concentration. Dashed lines fitted to the low-concentration range are included to emphasize the nonlinearity of each quantity with concentration

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The ratios of the exchange rates \({\omega }_{\mathrm{ex},lw}\), and \({\omega }_{\mathrm{ex},\zeta }\) may be rationalized in terms of the new paradigm proposed by Salikhov [33, 34, 37], who noted that these quantities depend differently on the rates of spin coherence transfer and spin dephasing. The significantly higher values of \({\omega }_{\mathrm{ex},{\delta }_{\pm 1}}\) at concentrations above 30 mM most likely reflect re-encounters between pairs of radicals, an effect theoretically predicted by Salikhov [46] and later observed experimentally by Bales and coworkers in solvents exhibiting cage effects [47,48,49]. It is likely that re-encounters occur within the constrained environment of the aqueous phase in Nafion; the marked nonlinearity of \({\omega }_{\mathrm{ex},{\delta }_{\pm 1}}\) at low concentrations (long times) may also be a consequence of the structure of this phase as discussed below. It is not clear why \({\omega }_{\mathrm{SLE}}\) differs from the other effective exchange rates. As noted above, the SLE calculation of Freed and coworkers does not include dipolar terms in the encounter pair; if such interactions are affecting the EPR lineshape in the Nafion environment, the least-squares procedure may be finding an “effective” \({\omega }_{\mathrm{HE}}\) that accounts for such interactions.

The least-squares values of W and V obtained by fitting the exact lineshape expression are plotted in Fig. 6 together with the initial estimates \({\omega }_{\mathrm{ex},lw}\) and \({\omega }_{\mathrm{ex},\zeta }\) for the low-concentration region in which these parameters maintain constant ratios. Also plotted is the sum V + W for each fit. Whereas the least-squares values of W are reasonably close to the initial guess of \({\omega }_{\mathrm{ex},lw}\) with a ratio \(W/{\omega }_{\mathrm{ex},lw}=0.852\pm 0.004\). Those of V are significantly different from the initial guess, with a ratio \(V/{\omega }_{\mathrm{ex},\zeta }=0.26\pm\) 0.01. However, the sum V + W is close to \({\omega }_{\mathrm{ex},\zeta }\) over this concentration range, with a ratio \((W+V)/{\omega }_{\mathrm{ex},\zeta }=0.94\pm 0.02\).

Figure. 6
figure 6

Comparison of fitting parameters \({\omega }_{\mathrm{ex},lw}\) (circles) and \({\omega }_{\mathrm{ex},\zeta }\) (squares) obtained from fitting the three Lorentzian lines (Eq. (4)) and the parameters W (triangles) and V (inverted triangles) from fitting exact lineshape expression (Eqs. (2) and (3)) to the experimental spectra of Tempone in Nafion 117. The sum of W and V (plus signs) is also plotted for comparison to \({\omega }_{\mathrm{ex},\zeta }\)

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Equations (3) in principle allows an estimate of the contribution of dipolar interactions to the exchange interaction from the data in Fig. 6. Subtracting these two equations gives \(\frac{1}{2}\left({K}_{dsct}+{K}_{dsd}\right)=(W-V)/C\), or an average contribution of dipolar interactions to spin coherence transfer and decoherence of about 0.12 ± 0.01 MHz mM−1. Thus, in contrast to low-viscosity isotropic solvents in which dipolar interactions are negligible, they appear to be quite significant in the Nafion environment.

The consistent behavior of \({\omega }_{\mathrm{ex},\upzeta }\), \({\omega }_{\mathrm{ex},lw}\) and \({\omega }_{\mathrm{SLE}}\) as a function of concentration increases confidence that the observed nonlinearity is not an artifact of the different lineshape fitting procedures used, but rather an intrinsic property of the Nafion matrix. These deviations from the classical expression Eq. (1) as well as the departure of Tempone disproportionation kinetics from second order will now be considered in the context of percolation theory after a brief review of the relevant aspects of that theory. The reader is referred to the monographs references [20, 21] for greater detail.

A percolation system is modeled as a network of nodes or links that may be open with probability \(p\). At some critical threshold \({p}_{c}\), which depends on the network geometry, enough sites are open to form an “infinite cluster” that provides an open path through the network. Two important characteristics of the network are \({p}_{\infty }\), the probability that a node belongs to the infinite cluster, and \(\xi\), the characteristic length of finite clusters in the network. These are governed, respectively, by the critical exponents \(\beta\) and \(\nu\) as follows:

$$\begin{gathered} p_{\infty } \left( p \right) = \left( {p - p_{c} } \right)^{\beta } \hfill \\ \xi \left( p \right) = \left( {p - p_{c} } \right)^{ - \nu } . \hfill \\ \end{gathered}$$
(7)

Note that \({p}_{\infty }\to 1\) and \(\xi \to \infty\) as \(p\to {p}_{c}\). The fractal dimension of the network is then defined as \({d}_{f}=d-\beta /\nu\), where \(d\) is the Euclidean dimension in which the network is embedded. In Nafion, these properties are generally expressed in terms of water concentration [50] instead of occupation probability, but the principles are the same.

The disproportionation of the Tempone spin probe is an example of the “annihilation” reaction that has been extensively characterized in percolation networks, and for which theoretical rate expressions are available. It is represented by the reaction \(\mathrm{A}+\mathrm{A}\to 0\) where A is assumed to be diffusing within the network with concentration\({C}_{\mathrm{A}}\), and 0 represents the EPR-inactive products. This reaction has a time-dependent rate constant for which the expression near the percolation threshold is [51]

$$\frac{d\left[\mathrm{A}\right]}{dt}={-k}_{0}{t}^{1-{d}_{s}/2}{\left[\mathrm{A}\right]}^{2},$$
(8)

where \({k}_{0}\) is proportional to the collision frequency at \(t=0\) and \({d}_{s}\) is the so-called “spectral” or “fracton” dimension of the percolation network. The spectral dimension measures the interplay between the fractal geometry of the conductive phase of the membrane and the dynamics of the diffusion, and it plays a role equivalent to the dimensionality of standard Euclidean systems.

We are aware of no theoretical treatment that fully addresses the physics of our experimental observation, i.e. an expression for two separate processes that have different dependencies upon the rate of collisions between identical species A in a percolation network. The spin exchange case differs from the annihilation reaction in that the radicals remain active after a collision in which they undergo spin exchange. However, if the Tempone disproportionation is truly diffusion-limited, then both processes depend upon the rate of collision between Tempone molecules, and Eq. (7) might approximate the spin exchange case. By analogy, Eq. (1), then becomes

$${\upomega }_{\mathrm{ex}}\left(C,t\right)={\omega }_{0}{t}^{1-{d}_{s}/2}C\left(t\right),$$
(9)

where \({\omega }_{0}\) is the HSE rate at zero time and includes all the additional factors appearing in Eq. (1).

We now apply these results from percolation theory to the experimental data, starting with the Tempone decay curve. The solid line in Fig. 7A shows the least squares fit of the stretched-time second-order decay, Eq. (7), to the experimental decay curve. The functional form and least-squares parameters are given in Table 1. The fitted decay function overlaps quite well with the data as can be seen in the plot of the residuals (upper plot, Fig. 7B). The value obtained for the fracton dimension \({d}_{s}=1.42\) is also quite consistent with the ideal value of 4/3 expected for a percolation cluster embedded in three Euclidian dimensions [23, 51].

Fig. 7
figure 7

A Points show experimental measurement of Tempone concentration (normalized) in the Nafion membrane vs. time. Only every 10th point is shown to allow visualization of the least-squares stretched-time second-order expression, Eq. (8) (solid line) and a double exponential function. (dashed line) to the data. (B) Residuals for the two functions plotted in (A). (C) Resolution of the two components of the double-exponential decay function

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Table 1 Least-squares parametersa for different functions fitted to experimental Tempone decay C/C0
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Figure 8 shows a fit of Eq. (9) to the experimental \({\omega }_{\mathrm{ex},\upzeta }(C)\) curve. The fit overlaps with the experimental data reasonably well as can be seen from the residuals shown in the upper plot of Fig. 8B and the least-squares parameters given in Table 2. The fracton dimension obtained from this fit is \({d}_{s}=1.96\). The \({\omega }_{\mathrm{ex},lw}(C)\) and \({\omega }_{\mathrm{SLE}}(C)\) data were also fitted with this function (solid lines in Fig. 5) and gave a consensus value of \({d}_{s}=1.98\pm 0.03\).

Fig. 8
figure 8

A Exchange rate as a function of normalized concentration (points) fitted with a stretched time function (Eq. (8), solid line) and the double exponential scaling model described in the text described in the text (dashed line). The fits are practically indistinguishable in this plot. B Residuals for the stretched time and double exponential fits

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Table 2 Least-squares parametersa for different functions fitted to experimental \({\omega }_{\mathrm{ex}}(C)\)
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Although both spin exchange and reaction kinetics exhibit deviations from classical dependence on concentration in Nafion, the rather different values of \({d}_{s}\) obtained from these independent experiments lead us to consider alternative models. It is likely that the approximation of using Eq. (8) to describe \({\omega }_{\mathrm{ex}}(C)\) is not completely valid and a more rigorous theory must be derived for the spin exchange case. The stretched-time functions also do not explain the curious nonlinear behavior of \({\delta }_{\mathrm{ex},\pm 1}\) observed at low concentrations.

Another possibility is suggested by the observation that the decay of the Tempone radical is reasonably well-fit by a double exponential. Figure 6C shows a double-exponential function that fits closely to the data as indicated by the residuals in the lower plot of Fig. 6B. Least-squares parameters for this fit are given in Table 1. This plot also shows the two components of the fit: approximately 70% of the spin probe population decays relatively quickly with a rate of \(1.9\times {10}^{-4} {\mathrm{s}}^{-1}\), with the remainder decaying at about one fifth of that rate.

This observation is consistent with so-called scaling models of the long-time behavior of macroscopic systems [52, 53] in which they exhibit a crossover time \({t}^{\times }\) at which the observed decay switches from a stretched exponential (fractal kinetics) to a simple exponential (classical kinetics). This behavior can be qualitatively understood in the following way. The self-diffusion coefficient of particles diffusing in a percolation network varies with time [54], approaching a constant value when the distance scale of the diffusion is larger than the characteristic length of the finite clusters in the network, which is the length scale at which the network can be considered a homogeneous medium. Thus, at long times, separate ensembles of diffusing particles may emerge [21], one consisting of particles in clusters of finite size, for which there is an upper limit on the particle’s displacement, and the other diffusing freely in the infinite cluster over long distance scales and exhibiting classical kinetics.

To test whether this two-ensemble model could explain the \({\omega }_{\mathrm{ex}}(C)\) data, different collision rates were assumed for each of the populations, producing different proportionality constants \({\omega }_{1}~ \mathrm{and }~{\upomega }_{2}\) between C and \({\omega }_{\mathrm{ex}}\) as follows:

$${\omega }_{\mathrm{ex}}\left(C(t\right))={\omega }_{1}{A}_{1}\mathrm{exp}\left(-{k}_{1}t\right)+{\omega }_{2}{A}_{2} \mathrm{exp}(-{k}_{2}t)$$
(10)

The calculated fit of Eq. (9) is also shown in Fig. 8A, and the least-squares values of \({\omega }_{1}~ \mathrm{and }~{\upomega }_{2}\) are given in Table 2. Although the \({\omega }_{ex}\left(C\left(t\right)\right)\) calculated from Eq. (10) is nearly indistinguishable from the fit of Eq. (9) as shown in Fig. 8A, differences are evident from the residuals that are compared in Fig. 8B. The RMSD for the fit of Eq. (10) is somewhat, but not significantly, lower than that for Eq. (8).

The relative collision rates found for the two components are consistent with the scaling model: at early times (corresponding to high concentrations), radicals diffusing locally in finite clusters undergo fractal diffusion so that the collision rate per unit concentration as reflected by \({\omega }_{1}\) is lower than the \({\omega }_{2}\) observed at longer times. This model may also explain the unusual nonlinear behavior of \({\omega }_{\mathrm{ex},{\delta }_{\pm 1}}\). At short times, the confined spaces of the finite clusters lead to a high rate of re-encounters per encounter in this ensemble, which is reflected in the higher relative \({\omega }_{\mathrm{ex}}\) that is derived from the shift of the outer 14N hyperfine lines. At the longest times, corresponding to the lowest concentrations, the ensemble of radicals in the infinite cluster appear to undergo classical bulk diffusion, reducing the relative number of re-encounters so that \({\omega }_{\mathrm{ex}.{\updelta }_{\pm 1}}\) approaches \({\omega }_{\mathrm{ex},\zeta }\) and \({\omega }_{\mathrm{ex},lw}\).

4 Conclusion

This detailed study of spin exchange over a wide range of spin probe concentrations in in a well-characterized percolation network demonstrates the possibility of using this method to obtain quantitative information about the nanoscale properties in media with fractal properties. In principle, the ability to characterize the separate effects noted above will require a more rigorous interpretation of the concentration dependence of SHE in this type of percolation network. As one Reviewer suggested, it will be useful to verify disproportionation is indeed responsible for the deactivation of the nitroxide by testing whether treating the membrane with alkali restores an EPR signal. More detailed interpretation of the results may also require development of a theory for the steady-state collision rate of probes in the presence of an annihilation reaction with a different rate. Such a treatment should also include an analysis of re-encounter rates according to the methods proposed in the extensive studies of Salikhov, Bales, and coworkers [38, 47, 55,56,57]. The influence of the different network parameters could be studied by varying the degree of membrane hydration, including values below the percolation threshold, and by varying the acidity of the aqueous phase by changing the degree of sulfonation in the polymer. The fracton dimension of the polymer could also be varied by using polymers with different constraints on their backbone conformations such as sulfonated polyaryl ketones.