Electrode Reactions in Slowly Relaxing Media
Standard models of reaction kinetics in condensed materials rely on the Boltzmann-Gibbs distribution for the population of reactants at the top of the free energy barrier separating them from the products. While energy dissipation and quantum effects at the barrier top can potentially affect the transmission coefficient entering the rate preexponential factor, much stronger dynamical effects on the reaction barrier are caused by the breakdown of ergodicity for populating the reaction barrier (violation of the Boltzmann-Gibbs statistics). When the spectrum of medium modes coupled to the reaction coordinate includes fluctuations slower than the reaction rate, such nuclear motions dynamically freeze on the reaction time-scale and do not contribute to the activation barrier. Here we consider the consequences of this scenario for electrode reactions in slowly relaxing media. Changing electrode overpotential speeds electrode electron transfer up, potentially cutting through the spectrum of nuclear modes coupled to the reaction coordinate. The reorganization energy of electrochemical electron transfer becomes a function of the electrode overpotential, switching between the thermodynamic value at low rates to the nonergodic limit at higher rates. The sharpness of this transition depends of the relaxation spectrum of the medium. The reorganization energy experiences a sudden drop with increasing overpotential for a medium with a Debye relaxation, but becomes a much shallower function of the overpotential for media with stretched exponential dynamics. The latter scenario characterizes electron transfer in ionic liquids. The analysis of electrode reactions in room-temperature ionic liquids shows that the magnitude of the free energy of nuclear solvation is significantly below its thermodynamic limit. This result applies to reaction times faster than microseconds and is currently limited by the available dielectric relaxation data.
The theory of nonadiabatic electron transfer at metal electrodes has been established in theoretical work by Marcus,Marcus (1965) Hush,Hush (1968, 1999) and LevichLevich (1965) and extensively tested by a number of seminal experimental studies by Chidsey,Chidsey (1991) Finklea,Finklea et al. (2001) and Savéant.Savéant (2002) Nonadiabatic rate is calculated from the Golden-Rule perturbation theory for individual electronic transitions between the localized electronic state in solution and delocalized conduction states in the metal electrode.Chidsey (1991); Hush (1999); Gosavi and Marcus (2000); Gosavi, Gao, and Marcus (2001); Migliore and Nitzan (2012) The Golden-Rule expression is limited by a low magnitude of the electrode-reactant electronic coupling. As the coupling increases, two effects become important. First, at still relatively small coupling magnitudes, the dynamical solvent effect alters the pre-exponential factor of the rate constant for homogeneous electron transfer in solutionZusman (1980); Sumi and Marcus (1986); Rips and Jortner (1987); Yan, Sparpaglione, and Mukamel (1988); Smith, Staib, and Hynes (1993) and for the electrode reaction.Morgan and Wolynes (1987); Chakravarti and Sebastian (1992); Matyushov (1994) At even higher electronic coupling, the barrier for electron transfer becomes affected in the regime known as adiabatic electrode reactions.Schmickler (1986); Gorodyskii, Karasevskii, and Matyushov (1991); Schmickler (1996); Hush (1999); Matyushov (2009a)
While the theory of the solvent effect on electron transfer accounts for friction (dissipation) at the top of the barrierKramers (1940); Smith, Staib, and Hynes (1993) and thus can be viewed as a nonequilibrium theory, it still assumes that the population of reactants near the barrier’s top is given by the Boltzmann-Gibbs probability distribution. Therefore, all traditional theories mentioned above anticipate that the statistical configurations of the reactants in the reactants’ well are assigned Boltzmann-Gibbs probabilities. The rate of an activated process is proportional to the equilibrium population of the activated state separated by the Gibbs energy barrier from the reactant state
where is the inverse temperature.
More detailed theories of the barrier crossing, following the original Kramers formalism,Kramers (1940) specifically consider the medium dynamics at the barrier top. Along these lines, the stable states picture proposed by Northrup and Hynes Northrup and Hynes (1980) subdivides the system phase space into reactant and product stable regions, to which equilibrium Boltzmann-Gibbs distribution applies, and the intermediate region near the barrier top where nuclear dynamics are treated separately. This approach leads to memory effects in barrier crossing related to the flux-flux correlation functions for the particles entering and leaving the intermediate region (I in Fig. 1). The separation into regions requires corresponding separation between the relaxation time inside each stable region and the reaction time : . Since relaxation within the reactant and product wells occurs fast on the reaction time-scale, one can use the Boltzmann-Gibbs statistics within the stable regions, and also apply it to calculate the flux-flux correlation functions according to standard recipes of the linear response approximation.Hansen and McDonald (2013) While the dynamics in the intermediate region are explicitly considered, the probability of reaching this region is still given by the Boltzmann-Gibbs distribution. This perspective is illustrated in Fig. 1, where the picture of the dividing surface (TS) of the transition-state theory is confronted with the intermediate region of Northrup and Hynes.
The nonergodic kinetics of chemical reactionsMatyushov (2007, 2009b) makes the next step in introducing the medium dynamics into the rate of activated barrier crossing. The main difference of this perspective compared to the classical theories discussed above is that the limitation of fast relaxation within the reactant and product phase sub-space, , is now lifted. The configurations allowed for sampling by the reactants and products are limited by the ability of the system to reach them on the time-scale of the reaction (closed areas in Fig. 1). If some parts of the phase space require longer time to reach, those configurations do not contribute to the statistical averages used to calculate the reaction rate. The nonergodic kinetics thus adopts an ensemble view of insufficient sampling of the reactant and product phase space, instead of attempting to solve the entire dynamic problem involving nonequilibrium relaxation.Northrup and Hynes (1980) The ensemble of stable states of Northrup and Hynes, from which the trajectories are pulled toward the barrier and which are assigned the Boltzmann-Gibbs weights, becomes depopulated, in addition to possible dissipative depopulation and recrossings at the top of the barrier considered in the Kramers and Northrup and Hynes models.
This simple reasoning has significant implications for rates determined through Eq. (1). Since the space of configurations is now restricted, the phase space contributing to the free energy profile along the reaction coordinate becomes restricted as well. The dynamics of the system now affect not only the barrier crossing event, but also the entire free energy profile of the reaction. The consequence of this perspective is best understood by considering the free energy profile of electron-transfer self exchange shown in Fig. 2. Restricting the available phase space lowers the reorganization energy of the reaction from its thermodynamic magnitude to a nonergodic value (see below). This alteration propagates into the corresponding alteration of the entire free energy surface and leads to a lower activation barrier .
The reorganization energy , and the barrier , are now functions of the rate constant. This is a general outcome of the nonergodic kinetics requiring that the free energy of activation is affected by the medium dynamics through relative magnitudes of the reaction time and . Since the rate itself is dependent on the barrier height, it needs to be calculated by solving a self-consistent equation
Equation 2 can be used directly to achieve a solution for , as we do below, or within a more complex formalism. Theories of barrier crossing,Northrup and Hynes (1980); Zusman (1980); Sumi and Marcus (1986); Rips and Jortner (1987); Yan, Sparpaglione, and Mukamel (1988); Smith, Staib, and Hynes (1993) can be applied to calculate the self-consistent activation dynamics, including cases of non-exponential population kinetics when the rate constant is not uniquely defined.LeBard, Kapko, and Matyushov (2008) We do not address these issues here and instead, for the sake of simplicity of formulation, focus on the free energy profile for electron-transfer reactions. The rates are calculated from the well-established Golden-Rule approach.Levich (1965) We address electrode reactions in this picture as an important class of reactions for which the electrode potential becomes a control parameter which can potentially drive the reaction out of ergodicity.
The conceptual framework of nonergodic kinetics clearly applies to electrode reactions in slowly relaxing media. Making the electrode overpotential more negative in the case of reduction considered below leads to an exponential increase of the electrode reaction rate constant, , according to the Tafel law.Tafel (1905); Bard and Faulkner (2001) With increasing reaction rate, the electrode reaction enters the window of nonergodic electrode kinetics when the time of electrode discharge and the characteristic relaxation time of the medium become approximately equal,
When this point is reached, the rate cannot be described by Eq. (1) any further and Eq. (2) should be used instead. Here, the overpotential is the deviation between the externally applied electrode potential and its equilibrium value : .Bard and Faulkner (2001)
The electrode reactions present us with a unique capability to move, through a potential scan of the electrode, from ergodic kinetics, following the basic prescriptions of the transition-state theory,Eyring, Lin, and Lin. (1980) to nonergodic kinetics with new rules for the reaction activation barrier affected by the medium dynamics. Here, we analyze the observable consequences of this general picture in application to electron transfer at the metal electrode. We start with a generic case of a solvent relaxing by a single-exponential Debye lawBöttcher (1973a) and then consider a more complex case when a continuous manifold of Debye processes, described by stretched exponential dynamics,Ediger, Angell, and Nagel (1996) is used for the solvent. For that purpose, we model electrode reactionsFietkau et al. (2006); Khoshtariya, Dolidze, and van Eldik (2009); Fawcett, Gaál, and Misicak (2011); Nikitina et al. (2014a) in room-temperature ionic liquids (RTILs).Arzhantsev et al. (2007); Stoppa et al. (2008); Hunger et al. (2009); Weingärtner (2014)
RTILs is a specific case of a broad list of materials characterized by stretched exponential dynamics and spanning several orders of magnitude in their relaxation times. Proteins is another example,Hong et al. (2011); Khodadadi and Sokolov (2015); Mondal, Mukherjee, and Bagchi (2017) along with a large number of glass formers.Ediger, Angell, and Nagel (1996) The dependence of the reorganization energy on the rate constant calculated here for RTILs is very shallow (Fig. 3). Importantly, the reorganization energy does not reach its thermodynamic value in the entire range of considered here. The accessible range of rate constants is limited by the current experimental window of dielectric spectroscopy.Stoppa et al. (2008) The dependence of calculated here turns out to be similar to that found previously for proteinsMartin and Matyushov (2015) (Fig. 3). This similarity suggests a common phenomenology for these types of media, which is quite distinct from a much sharper nonergodic crossover of the reorganization energy found for reactions in media with Debye relaxation.Ghorai and Matyushov (2006)
Ii Rate of electrode electron transfer
Essentially all complex media show complex dynamics, and hence deviations of their time-correlation functions from the simple one-exponential (Debye) form. This phenomenology is usually understood in two ways, which are not necessarily opposing each other. One can view the non-exponential kinetics as either a reflection of a large number of Debye decay relaxation processes with their characteristic time-scalesRichert (2002, 2015) or a continuous distribution of Debye decays resulting, mathematically, in complex dynamic susceptibilities and non-Debye functionalities for the dynamic response functions.Bagchi and Chandra (1991) The former mechanism is often linked to dynamic heterogeneityRichert (2002) and the latter mechanism can be related to intrinsically complex dynamics of condensed materials,Hansen and McDonald (2013) not necessarily involving spatial separation of dynamically distinct regions. The actual mechanism of appearance of many relaxation times is not central for the description of reactions in such complex media. The main feature from the viewpoint of the reaction dynamics is that nonequilibrium fluctuations of the nuclear modes coupled to the reaction need to be represented by several processes, some fast and some slow.
This perspective offers a possibility of dynamical freezing: that is, some of the modes cannot relax on the time-scale of the reaction.Chen and Meyer (1996); Görlach et al. (1995); Richert (2000); Goes et al. (2002); Matyushov (2007) The typical relaxation phenomenology of polar liquids and of many complex media (such as proteinsLakowicz (2000); Matyushov (2013); Qin et al. (2017)) involves two basic components: fast ballistic relaxationJimenez et al. (1994); Bagchi and Gayathri (1999) and much slower collective relaxation, which can be multiexponential or characterized by complex dynamics (such as stretched exponentialEdiger, Angell, and Nagel (1996); Ediger (2000); Richert (2002)). For charge-transfer transitions, the relevant dynamic property is the Stokes-shift time autocorrelation functionvan der Zwan and Hynes (1985); Carter and Hynes (1991); Smith, Staib, and Hynes (1993); Reynolds et al. (1996)
The correlation function projects the nuclear dynamics of the medium on the collective variable corresponding to the energy gap (as shown in Fig. 2) between electronic energies before and after the electronic transition, . For electrochemical reactions, the initial state of the cathodic process is the electron localized at the energy state in the conduction band of the metal and the reactant in the oxidized state with the energy . The initial energy is . The final state, after the electron transfer, has the energy of the reduced reactant . The energy gap is therefore . Since the energy level in the metal is screened from the medium fluctuations, one can put in Eq. (4). The time dependence of the energy gap fluctuations is caused by thermal agitation of the nuclear modes of the solvent medium, and the stationary value defines the equilibrium (Gibbs ensemble) reorganization energy Mukamel (1995)
In contrast to this thermodynamic reorganization energy, nonergodic reorganization energy arises in theories of activated transitions constraining the space of configurations available to the system. The constraint is dynamical, demanding that no relaxation process slower than the reaction time contributes to the statistical averages specifying the activation barrierMatyushov (2005, 2009b, 2015) (Fig. reffig:0). Such constraints are mathematically represented by the frequency filter when the nonergodic reorganization energy is expressed in terms of the time Fourier transform of the Stokes correlation function. One obtains
where is the imaginary part of the complex-valued Stokes-shift susceptibility following from the standard rules of the linear-response approximationHansen and McDonald (2013) formulated in terms of the collective variable . Its imaginary part, a loss function, determines the rate of loss of energy supplied to match a given value of (by say an optical excitation) into the thermal bath.
Here, is the reorganization energy of the fast modes, is the reorganization energy of the slow modes, and is the characteristic relaxation time of the slow component of .
Equation (7) is written in the form applicable to electrochemistry to stress the dependence of the reorganization energy on the electrode overpotential through the corresponding dependence of the the rate of electrode electron transfer . Note that is a thermodynamic free energy in the standard modelsMarcus and Sutin (1985) as given by Eq. (5). It does not depend of the electrode overpotential. However, when the reaction time becomes comparable in magnitude to the slow relaxation time , the reorganization energy gains a dependence on the overpotential. A representative calculation is shown in Fig. 4. When the reaction rate exceeds the rate of slow medium relaxation, the reorganization energy drops as a function of from its equilibrium value to the reorganization energy of the fast reaction
where the parameter
gives the ratio of the rate of activationless electrode reaction (the rate of electron tunneling) to the rate of medium relaxation. Here, ,Schmickler (1986); Gorodyskii, Karasevskii, and Matyushov (1991); Schmickler (1996); Gosavi and Marcus (2000); Matyushov (2009a) is the electron-transfer matrix element between the electronic state on the reactant and a single electronic state in the metal and is the density of states at the metal’s Fermi level.
The reorganization energy loses the status of the thermodynamic free energy and becomes a nonergodic energy parameter quantifying reorganization of the nuclear subsystem on the time-scales shorter than the electron tunneling rate (maximum (activationless) nonadibatic rateHush (1999); Hale (1968)). In the general case, the reorganization energy enters the rate of nonadiabatic electrode reaction, usually obtained by integrating individual nonadiabatic transitions over the Fermi-distributed conduction electrons in the metalLevich (1965); Chidsey (1991); Hush (1999); Migliore and Nitzan (2012); Oldham and Myland (2011); Newton and Smalley (2007)
where is the Fermi population function and the activation barrier is the standard result of the Marcus-Hush (MH) theory
If the Fermi population is taken at zero temperature the resulting relation isHale (1968)
where is the complementary error function.Abramowitz and Stegun (1972) From Eq. (12), at . This limit corresponds to an activationless electrode electron transfer, the rate of which is used in defining the parameter , as discussed above in connection with Eq. (9).
While Eq. (12) gives a reasonable approximation for the rate, a more accurate preexponential factor is calculated by integrating over the Fermi distribution at a finite temperature. The procedure suggested by Marcus and co-workersGosavi and Marcus (2000); Gosavi, Gao, and Marcus (2001) involves neglecting or Taylor expanding the Gaussian function in the -integral in Eq. (10). Merely dropping this term produces an analytical form
which is valid under the condition of . We have empirically introduced the factor 0.85 in the argument of to account for the terms dropped from the expansion of in Eq. (10). It brings Eq. (13) into good agreement with numerical integration. We note that the rate calculated here refers to a reactant placed at a specific distance from the electrode and has the units of inverse time. It can therefore be directly used for reactants adsorbed at the electrode. Alternatively, when mass transport becomes involved, the rate at a given distance is an input for the corresponding diffusional kinetics modelCompton and Banks (2009) producing the electrochemical rate in units of length/time.Matyushov (1994) Our examples below apply to adsorbed redox species to avoid complications from mass transport.
Iii Stationary electrode current
We want to establish how nonergodicity of the activation barrier for electrode electron transfer affects the observable electrode current. We will consider here the simplification of the electrochemical setup in which the reactants are assumed to be immobilized at the surface of the electrodeChidsey (1991); Yue et al. (2006); Newton and Smalley (2007) with the total surface concentration based on the surface concentrations of the oxidized, , and reduced, , forms of the reactant. This setup thus eliminates the need to consider the diffusive mass transport.Compton and Banks (2009)
In order to gain physical insights into how non-ergodicity modifies the electrode kinetics, we start with a simplified and somewhat unrealistic model in which we consider the cathodic process with a fixed concentration of the oxidized form . We thus consider the reduction process . Since the model is symmetric in respect to the overpotential, the oxidative branch follows from the sign flip, . We calculate the dimensionless current
from the total cathodic current passing through the electrode area .
Figure 5 shows from the rate constant determined by combining Eqs. (7) and (12). The main result from these calculations is the identification of the overpotential of the dynamical crossover determined by the condition (empirically found from Eqs. (8) and (12))
at which an approximately exponential (Tafel lawTafel (1905); Compton and Banks (2009)) dependence of the current on overptential switches to the saturation limit of activationless electron tunneling between the electrode and the reactant. The MH model for nonadiabatic electron transfer (dash-dotted line in Fig. 5) reaches the activationless regime at (as follows from the rate constant in Eq. (12) upon the substitution ). It requires a higher cathodic overpotential to reach the activationless reaction than in the nonergodic model.
More detailed information about the distribution of the reactant energy levels in the medium can be achieved by taking the derivative of the current over the overpotential.Becka and Miller (1992); Terrettaz, Cheng, and Miller (1996); Miller (1995) Within the MH model this observable directly probes the Gaussian distribution of the oxidized state. One obtains from Eq. (12)
is the Gaussian function (see Eq. (11) at ) of the driving force in the MH theory. In the nonergodic theory, one has to include the derivative of with respect to , which results in a slightly more complex relation.
The non-Gaussian shape of the nonergodic current derivative arises from the combination of two essentially Gaussian functions in one plot. The reorganization energy is high at low overpotentials and one sees the rising wing of the Gaussian probability function enroute to achieving the maximum at (cf. dashed and dash-dotted lines in Fig. 6). However, the reorganization energy drops due to nonergodic constraints before this limit is reached (Fig. 4). Correspondingly, further increasing (making it more negative) displays the decaying wing of the Gaussian function characterized by (Eq. (8)). What is seen in Fig. 6 is an overlap, in one plot, of these two Gaussian curves.
Iv Cyclic voltammetry
The calculations performed in the previous section represent the idealized situation when the surface concentration of the oxidized state is not affected by the passing of the cathodic current. Here, we turn to a more realistic description in terms of linear sweep cyclic voltammetry (CV), where the surface concentration changes when the overpotential is altered with the scan rate : . Thus the cathodic sweep runs from to with the scan rate .
The total electrode current is composed of the cathodic, , and anodic, , currents passing through the area under the applied overpotential. One can define, following Laviron,Laviron (1979) the scaled current
The equation for the scaled current is given in terms of the surface mole fractions of the oxidized, , and reduced, , adsorbates; is the total surface concentration. The equation for the current isHoneychurch (1999)
is the dimensionless scan rate. Further, the scaled rates for the oxidation and reduction reactions in Eq. (19) can be calculated either from integration over the Fermi distribution of electron states in Eq. (10) or from the step-wise approximation of the Fermi distribution, leading to Eq. (12). Perturbative models are also possible,Gosavi and Marcus (2000); Gosavi, Gao, and Marcus (2001); Migliore and Nitzan (2012) but they produce only minor corrections to the rate preexponential factor. In the case of assumed for the distribution of electrons in the metal, one gets
where are scaled rate constants, with “” and “” applied to “O” and “R”, respectively, and . Equation (13) is not very convenient for the modeling of cyclic voltammetry because of the limited range of overpotentials to which it applies.
The solution for is given asHoneychurch (1999)
where . We have additionally assumed that the sweep amplitude is large enough such that only the oxidized form is present at the electrode at : . With this initial condition, Eq. (22) is the solution of the following kinetic equation
The results of these calculations are shown in Fig. 7. The upper panel shows the cathodic CV peak at the scaled scanning rate in the MH and nonergodic kinetics. The nonergodic reorganization energy produces a lower barrier and a faster rate. The population of the oxidized state of the reactant is depleted faster, with the resulting shift of the peak potential to a lower value and a corresponding decrease in its amplitude. Large scanning rates eventually result in leveling off of the peak position as a function of (solid line in the lower panel in Fig. 7). However, reaching this regime requires very low relaxation times in Eq. (20). For most practical systems, only can be realized.
V Electrode reactions in ionic liquids
The standard Marcus model based on dipolar polarization of the bath predicts that the medium (solvent) reorganization energy is proportional to the Pekar factor, . While the optical dielectric constant is well defined for all media, the static dielectric constant is expected to diverge at for RTILs due to their intrinsic conductivity. This notion implies that a sample of conducting material placed in a constant external electric field must develop surface charges, and a corresponding macroscopic dipole moment, in order to render the field inside the conductor equal to zero.Landau and Lifshitz (1984) This perfect screening of the external field is what is meant by an infinite static dielectric constant,Lynden-Bell (2007a, b); Hansen and McDonald (2013) which cannot be “corrected” by the conductivity lossWeingärtner (2014) in the imaginary part of the frequency-dependent dielectric function reported by dielectric spectroscopy. The “static” dielectric constant of , often cited in the literature ( for protic RTILsSonnleitner et al. (2015); Weingärtner (2014)), mostly refers to the GHz domain of frequenciesStoppa et al. (2008); Hunger et al. (2009) (down to 0.2 GHz in Ref. Sonnleitner et al., 2015). Measurements in the kHz domain are dominated by electrode polarization effectsLeys et al. (2008); Serghei et al. (2009) and do not produce reliable data at high temperatures; dielectric data down to Hz are available near the glass transition.Ito and Richert (2007)
These comments are meant to emphasize that polar response in ionic liquids has to refer to a specific frequency window. Dielectric spectroscopy covers the time-scales relevant for solvation dynamics and for many redox reactions. However, since the long-time dynamics are dispersive and cover 2–3 orders of magnitude in time-scales,Arzhantsev et al. (2007); Stoppa et al. (2008) dynamic freezing of a part of the spectrum can always be an issue. Therefore, nonergodic effects are potentially important and RTILs are a primary target for the nonergodic theory of electron transfer. To understand consequences of nonergodic effects for electrode reactions, we perform here calculations modeling electrode reduction of ferrocene cationFietkau et al. (2006) in .
Ionic liquids are obviously different from dipolar molecular solvents,Wang et al. (2007); Fayer (2014) but still match in many respects dielectric properties of molecular liquids.Lynden-Bell (2007a, b); Shim and Kim (2009); Mladenova et al. (2016) Polar response of an ionic liquid is mostly due to translational motions of the ions. Rotations of molecular dipoles (mostly belonging to cationsKashyap and Biswas (2008)) were considered as an alternative mechanism of polar response, but are now viewed as a minor contribution to solvation as a result of recent computer simulations.Roy and Maroncelli (2012); Terranova and Corcelli (2013) Our goal here is to establish a mapping of charge density fluctuations onto fluctuations of a dipolar polarization field, thus connecting calculations of rates to dielectric measurements.
A structural fluctuation of an ionic liquid results in a fluctuation of charge density, which can be mapped on a dipolar polarization field. This can be illustrated by assigning dipoles to ionic translations (Fig. 8). The resulting dipolar field represents a fluctuating charge density by its divergence, . The charge density in reciprocal space relates to the longitudinal projection of the polarization field, . Therefore, charge density fluctuations produced by ionic translations can be mapped on the longitudinal polarization field, and the longitudinal dipolar response can be used to model solvation.
If the solute interacts with the charges of the ionic liquid via the electrostatic potential , the Stokes shift susceptibility can be calculated from the linear response approximation.Hansen and McDonald (2013) The result is conventionally written in reciprocal space
Here, a factor , which does not appear for the response of the bulk,Matyushov (1993); Dinpajooh, Newton, and Matyushov (2017) accounts for the fact that real-space integration is performed over half of the entire space since the space occupied by the polar liquid is restricted by the electrode. The tilde for in Eq. (24) and for the related functions below specifies the Laplace-Fourier transformHansen and McDonald (2013) (Laplace transform with the imaginary variable) of the corresponding time-dependent function. Both and are therefore complex-valued functions.
The longitudinal susceptibility can be expressed in terms of the longitudinal nonlocal dielectric function of the medium
The factor containing the high-frequency dielectric constant in this relation accounts for the screening of the permanent charges by the induced dipoles described by the Lorentz local field.Dinpajooh, Newton, and Matyushov (2017)
The longitudinal susceptibility function follows from the standard relations of the linear response theoryHansen and McDonald (2013)
Here, is the Laplace-Fourier transform of the normalized correlation function of the longitudinal (nuclear) polarization field representing electrostatic fluctuations of the solvent
Equation (28) can be viewed as a definition of the memory function, which generally is not directly related to observable properties. However, for the problem of charge fluctuations, the memory function is related to the observable property of longitudinal conductivity
Longitudinal conductivity connects the longitudinal electric current to the longitudinal Maxwell field as follows: , where both the current and the field oscillate with the frequency .
A general functionality for is unknown and requires extensive calculations even for model systems.Wilke, Chen, and Bosse (1999) The simplest approximation is to neglect the frequency dependence altogether. This representation conveniently connects the general formalism with the limiting case of Debye relaxation.Böttcher (1973a) One obtains from Eqs. (6) and (28)
Instead of a single Debye relaxation time in Eq. (7), a continuous distribution of relaxation times enters the polar response with the weights specified by the static longitudinal susceptibility . The contribution of the slow relaxation modes to the integral is reduced by the function , which tends to zero when the reaction rate is faster than the corresponding relaxation time.
The next level of approximation for the memory function allows one to circumvent some modeling difficulties by connecting to dielectric experiment incorporating the complex dynamics. It is achieved by factorizing into functions of and Munakata (1975); Fried and Mukamel (1990)
This factorization allows one to determine by satisfying the connection to the limit in Eq. (25). The final result is
where is the static dielectric constant. We put in the calculations presented below. This limit, taken in Eq. (32), implies that the frequency dependence of the longitudinal response is determined by the dielectric modulusIto and Richert (2007)
The memory function in Eq. (28) is therefore given by the relation
The static -dependent susceptibility function in Eqs. (26) and (32) presents the main challenge for the modeling of ionic liquids. In order to understand the difficulties involved, one can consider the case of a non-polarizable () 1:1 ionic liquid carrying unit ionic charges. The static susceptibility can be related to the structure factor describing charge translationsHansen and McDonald (2013)
Here, is the Debye-Hückel length, and the structure factor of the charge density fluctuations is the sum of the component density structure factors weighted with the component charges
Here, is the total number of RTIL ions, is the reciprocal-space density field of component , and the average is the density structure factor describing self and cross-correlations of the density fluctuations of the and components of the mixture. The structure factor describes fluctuations of the charge density in a homogeneous liquid caused by translations (density fluctuations) of the particles carrying molecular charges.
The charge structure factor satisfies the long wavelength limit
which guaranties .Hansen and McDonald (2013) This condition implies strong inhibition of the charge density fluctuations at compared to the density and rotational fluctuations. Density fluctuations at produce a nonzero compressibility, and rotational fluctuations at are related to the dielectric constant. In contrast, there are no macroscopic charge density fluctuations.
The requirement to satisfy the asymptote given by Eq. (37) makes direct modeling of polar response by dense ionic liquid a challenging task. It is often circumvented by models employing dipolar response for the longitudinal susceptibility. This is the route also adopted here. Our approach is to model through an effective dipolar field. The limit is set up through Eq. (37), which demands for nonolarizable liquids and a corresponding polarizability correction according to Eq. (25) for polarizable liquids. The limit of a dipolar function is , where is the standard one-particle parameter of the dielectric theories.Böttcher (1973b); Madden and Kivelson (1984) The effective squared molecular dipole here is not well defined in the case of ionic liquids. If one accounts for ionic translations to produce one-particle dipolar fluctuations (Fig. 8), the effective dipole should accommodate mean-square ionic displacements
where the sum runs over the charges and mean-square displacements of the ionic components with mole fractions . Further, is the molecular dipole introduced above, with the molar fraction accounting for the fact that molecular dipole moment is often associated with one of the ions, which is the cation for ().
The existing evidence indicates that ionic liquids are densely packed materials, in which ions are mostly involved in glassy cage rattlingDel Pópolo and Voth (2004) on a broad range of time-scales from hundreds of picoseconds to a few nanoseconds.Arzhantsev et al. (2007); Kashyap and Biswas (2010) This physical picture implies that mean-square displacements are mostly limited by the ion’s cage, , where is the hard-sphere diameter of the component. More precise assignment is difficult to make without explicit simulations of the longitudinal structure factors of an ionic liquid. However, the value assigned to and, correspondingly, to does not strongly affect the results of calculations. The reason is the damping effect of the solute electrostatic potential on the result of -integration in Eq. (24). More details on the parameterization of can be found in the supplementary material.sup ()
We consider a solute carrying the unit charge, with the radius placed at the distance from the electrode. The solute’s electrostatic potential in reciprocal space becomes
Here, is the radius of the solvent-accessible sphere, which is offset from the solute radius by an effective solvent radius , which needs to be determined (see below). The distance is the separation between the ion and its image in the metal electrode and is the spherical Bessel function of zeroth order.Abramowitz and Stegun (1972)
The image effects are always present in the dielectric calculations of the electrode reorganization energy,Liu and Newton (1994) and result in a linear dependence of the reorganization energy on the inverse distance to a metal electrode. The appearance of this dependence was questioned in the past based on the perceived screening of the image forces by molecular dipoles of a polar solvent.Phelps, Kornyshev, and Weaver (1990); Krishtalik, Alpatova, and Ovsyannikova (1991); Baranski, Winkler, and Fawcett (1991) The early calculations supporting these claims employed incorrect dipolar structure factors and were not supported by subsequent calculations.Fonseca and Ladanyi (1990); Raineri, Resat, and Friedman (1992); Matyushov (1993); Perng et al. (1996) The image effects are also found to be consistent with the standard expectations in more recent simulations of electrode reactions in ionic melts.Reed, Madden, and Papadopoulos (2008) The appearance of a substantial dependence of the thermodynamic on is also supported by our present calculations (Fig. S1 in the supplementary materialsup ()).
The ions making RTILs are typically large and have their size comparable to the size of redox pairs of the electrochemical experiment. For instance, the size of the cation in is estimated as Å from the van der Waals volume.Arzhantsev et al. (2007) An even larger value for an effective diameter of the ionic mixture, Å, is estimated here from the compressibility of ( Å is listed in Ref. Kashyap and Biswas, 2008). The approach based on compressibility typically provides a good estimate of the effective hard-sphere diameter of a liquid.Ben-Amotz and Herschbach (1990); Schmid and Matyushov (1995) As mentioned above, fluctuations of charge density are suppressed at and do not affect the macroscopic compressibility.Hansen and McDonald (2013) An effective hard-sphere diameter somewhat larger than the size of individual ions can arise from strong Coulomb correlations in ionic pairs which act as single entities from the perspective of compressibility (a typical size of the ionic pair in RTIL is ÅMezger et al. (2008)). The diameter of the ferrocene ion,Sikes (2001); Fietkau et al. (2006) which we use here for modeling, Å, is smaller than the effective solvent diameter . The result makes the application of continuum solvation models, which can be justified only in the limit , unreliable, and nonlocal susceptibility functions, depending on the reciprocal space -vector,Fried and Mukamel (1990); Bagchi and Chandra (1991); Matyushov (1993); Perng et al. (1996) are required.
We model the RTIL by using the following input parameters: the dielectric functionStoppa et al. (2008) , hard-sphere diameter Å, density g/cm, dipole moment of the cation D,Kashyap and Biswas (2008) and the high-frequency dielectric constant . The modeling of the limit of the dipolar susceptibility mapping the charge density fluctuations of the RTIL requires specifying the dipolar density parameter . The use of the effective dipole moment as given by Eq. (38) applies only to intermediate values of since related to ions’ translations decays to zero as , as follows from Eqs. (35) and (36). In the limit only the molecular point dipoles produce a non-zero-value of . The approximation of a point molecular dipole eventually breaks down at comparable to the intramolecular separation of charge.Raineri and Friedman (1999) However, that happens at in the range of wavevectors which mostly does not contribute to the solvation free energy. We therefore have chosen based on the density of molecular dipoles in the RTIL, which mostly comes from the D dipole of the cation. By adopting this approximation, we likely miss some excess polarity produced by ion translations at intermediate -values and resulting in the effective dipole in Eq. (38). We want to stress that is merely an effective polarity parameter characterizing the dipolar field mapping the charge fluctuations in RTIL. As mentioned above, the limit of large does not substantially affect the Stokes-shift susceptibility determined by -integration in Eq. (24).
RTILs in contact with solid substrates form layered structures.Mezger et al. (2008) The first layer of equal-sign ions forms a plane of charge with the surface charge density determined by the size and packing of the ions at the interface. The electrostatic potential drops approximately linearly in this layer,Reed, Madden, and Papadopoulos (2008); Vatamanu, Borodin, and Smith (2010) but, in contrast to conventional solution electrolytes,Bard and Faulkner (2001) it can drop below zero (overscreening) and then approach zero with increasing distance in oscillations reflecting layered structure of the RTIL in the interfaceRovere and Tosi (1986); Bazant, Storey, and Kornyshev (2011); Fedorov and Kornyshev (2014) The oscillations are low in amplitude at the equilibrium electrode potential, but increase in amplitude when electrostatic potential is applied to the electrode.Reed, Madden, and Papadopoulos (2008) Given this interfacial structure, it seems reasonable to assign the effective radius of the ions in the RTIL to the thickness of the Stern layer (defined as the layer where potential drops linearly). We place a spherical solute with the diameter Å corresponding to the ferrocene cation, at the distance from the electrode (see Fig. S2 in the supplementary materialsup ()). This placement allows us to assume that the reactant is outside the Stern layer and is in the part of the RTIL interface where oscillations of the potential decay to zero. Correspondingly, the Frumkin correctionsFedorov and Kornyshev (2014) for the electrode kinetics are minimized. The Frumkin corrections are generally viewed to be insignificant for RTILs.Reed, Madden, and Papadopoulos (2008); Fedorov and Kornyshev (2014)
Figure 9 shows the Stokes-shift loss function calculated for the ferrocene cation in . Because of the screening factor by the induced dipoles in Eq. (25), our calculations account for the effect of the solvent polarizabilityDinpajooh, Newton, and Matyushov (2017) often neglected in computer simulations based on non-polarizable force fields. The dependence of the loss function on the solute size is illustrated by additionally considering the cation of ferrocene carboxaldehydrateFietkau et al. (2006) with Å (dashed line).
The loss function is broadly distributed, in accord with the highly stretched dielectric relaxation of RTILs.Stoppa et al. (2008); Sonnleitner et al. (2015) Two characteristic peaks, at sub-picosecond and nanosecond range of frequencies, mirror the corresponding peaks in the dielectric loss functions. Integration of in Eq. (6) yields the nonergodic reorganization energy (Fig. 3). It slowly increases, with lowering , over a very broad range of rate values to a saturation magnitude resulting from the cutoff of the frequency domain in the dielectric experiment.Stoppa et al. (2008) Since low-frequency processes, not resolved by the dielectric experiment, are not included in our model of the dielectric constant, the dielectric function remains constant at low frequencies, thus producing a nearly constant value of the reorganization energy. Its value is below the thermodynamic reorganization energy,Nikitina et al. (2014b); com () eV, estimated here by putting in the susceptibility functions in Eqs. (32) and (33).
A shallow dependence of the reorganization energy on the rate constant, much different from a sharp change found for the Debye process (Fig. 4), is the result of the stretched exponential dynamics specific to the RTILs. A very similar slow variation of the reorganization energy with the changing observation window was found in extensive computer simulations of proteins extending to the microsecond length of trajectories.Martin and Matyushov (2015) The underlying phenomenology is likely common to both cases given a highly stretched dielectric relaxation of proteins.Hong et al. (2011); Nakanishi and Sokolov (2014) Consistent with the slow alteration in the reorganization energy, there are no discontinuities in the cathodic current curves (Fig. 10). The function is shown in the inset in Fig. 10. The reorganization energy is still changing with the overpotential due to nonergotic restrictions on the available phase space, but no discontinuities are seen in this case, in contrast to the Debye relaxation scenario shown in Fig. 4.
The dependence of the reorganization energy on overpotential should cause a generally non-parabolic form for the activation barrier , as was already shown for a solvent with Debye relaxation in Fig. 6. Since the dependence is significantly smoothed out in the case of the RTILs, one does not observe sharp features in the derivative of the current with respect to the overpotential and, instead, a smooth curve follows (Fig. 11). It is shallower on its left wing, reflecting a higher reorganization energy at less negative overpotentials (Fig. 10 inset).
The thermodynamic reorganization energy shown in Fig. 3 is close in magnitude to what one anticipates for molecular polar solvents.Lynden-Bell (2007a, b); Shim and Kim (2009) However, since the relaxation of RTILs is much slower, fast electrode reactions in RTILs have puzzled many researchers.Nikitina et al. (2014b); Mladenova et al. (2016) Our resolution of this puzzle is in terms of nonergodic effects on the solvent reorganization energy: the nonergodic character of the electrode reaction, due to complex dynamics of RTILs, leads to a significantly lower reorganization energy (Fig. 3) and a correspondingly lower activation barrier. We note that our results generally apply to nuclear solvation produced by RTILs. A corresponding reduction of both the reorganization energy and of the driving force is expected for homogeneous electron-transfer reactions between the reactants dissolved in RTILs.
Vi Discussion and conclusions
The model presented here deals with the consequences of reaction nonergodicity for electrode reactions in media with sufficiently slow relaxation such that the relaxation time and the reaction time are comparable in magnitude. This condition leads to breaking of ergodicity for the statistics of configurations in the reactant and product wells (Fig. 1). Ergodicity breaking, and the related difficulties in formulating the Gibbs distribution,Palmer (1982) is a general phenomenon most commonly encountered in phase transitions and relaxation of glassy materials.Crisanti and Ritort (2000) It reflects the inability of a statistical ensemble to completely sample all its allowed phase space within the time of observation. The problem is circumvented by defining Gibbs-weighted averages on a restricted sub-space of the entire phase space of the system.Palmer (1982) In other words, the statistical averages are calculated with the Boltzmann-Gibbs weights within a subspace
This formalism assumes that only configurational fluctuations can be slow and momentum transfer is always fast, making the kinetic temperature ( in Eq. (40)) well-defined.
The definition of the restricted phase space ( in Eq. (40)) is less straightforward for activated transitions, when the activation barrier can be altered by either changing the thermodynamic state or some external conditions. This difficulty becomes particularly severe for chemical reactions. In this case, the rate of the reaction itself defines the observation window, which therefore has to be adjusted self-continuously with the account for the dependence of the activation barrier on the rate (Eq. (2)). The restricted ensemble becomes dynamically restricted, which is achieved by specifying the phase space region in Eq. (40)
The result of this definition of statistical averages is the appearance of the dependence on the rate constant in the free energy assigned to reactants and products. When applied to the rate of activated transitions, this notion leads to the formulation of the nonergodic chemical kinetics.Matyushov (2005, 2015) Precursors of this concept can be found in the isomerization theory of van der Zwan and Hynesvan der Zwan and Hynes (1985) and in the Sumi-Marcus theory of the dynamic solvent effect on electron transfer.Sumi and Marcus (1986) In both cases, solutions for the two-dimensional barrier crossing problem allow the dynamics to drive the reaction along a path deviating from the lowest barrier height. Nonergodic kinetics replaces the dynamic solution with an ensemble perspective (similar to the replacement of the Newtonian dynamics with the Gibbs canonical ensemble).Matyushov (2009b) The advantage of this approach is that it is not limited to a small number of dynamical coordinates and can be applied to systems with many stochastic modes coupled to the reaction coordinate and producing complex dynamics not allowed by simple dynamic models.
Electrode reactions are a particularly important class of activated proccesses because of the freedom to alter the reaction free energy and the activation barrier through the electrode overpotential. One can therefore drive the reaction to the nonergodic regime by merely sweeping the electrode potential. Here, we formulated a theory of electrode reactions in slowly relaxing media and have shown that switching to nonergodic kinetics is indeed possible by adjusting the electrode overpotential.
The transition from the kinetics based on the free energy of activation following from the Gibbs ensemble (thermodynamics) to the nonergodic regime requiring account of the relaxation times is rather abrupt with the sweep of the electrode potential when the relaxation of the medium is characterized by a single Debye process. The transition occurs when the equality condition (Eqs. (3) and (15)) between the rate and the Debye relaxation time is met. Allowing a broad distribution of relaxation times, which is the situation realized for RTILs, broadens the transition and leads to a relatively shallow dependence of the activation parameters (reorganization energy here) on the external tuning of the activation barrier. However, the dependence of the reorganization energy on the electrode overpotential is preserved in this case as well (Fig. 10 inset). It leads to a generally non-parabolic dependence of the activation barrier on the electrode overpotential. The nonergodic reorganization energy of electron transfer in RTILs depends on the observation window specified by the reaction rate. It slowly changes when the reaction rate is altered over several orders of magnitude (Fig. 3), similarly to electron transfer in proteins.Martin and Matyushov (2015) The phenomenology found here seems to be quite general and there are significant reasons to believe that this picture should be common to a number of materials with stretched exponential dynamics.
See supplementary material for the parametrization of the nonlocal susceptibilities of RTILs.
Acknowledgements.This research was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Energy Biosciences, Department of Energy (DE-SC0015641).
- Marcus (1965) R. A. Marcus, J. Chem. Phys. 43, 679 (1965).
- Hush (1968) N. S. Hush, Electrochim. Acta 13, 1005 (1968).
- Hush (1999) N. S. Hush, J. Electroanal. Chem. 470, 170 (1999).
- Levich (1965) V. G. Levich, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 4, edited by P. Delahay (Interscience, New York, 1965) pp. 1–124.
- Chidsey (1991) C. E. D. Chidsey, Science 251, 919 (1991).
- Finklea et al. (2001) H. O. Finklea, K. Yoon, E. Chamberlain, J. Allen, and R. Haddox, J. Phys. Chem. B 105, 3088 (2001).
- Savéant (2002) J.-M. Savéant, J. Phys. Chem. B 106, 9387 (2002).
- Gosavi and Marcus (2000) S. Gosavi and R. A. Marcus, J. Phys. Chem. B 104, 2067 (2000).
- Gosavi, Gao, and Marcus (2001) S. Gosavi, Y. Q. Gao, and R. A. Marcus, J. Electroanal. Chem. 500, 71 (2001).
- Migliore and Nitzan (2012) A. Migliore and A. Nitzan, J. Electroanal. Chem. 671, 99 (2012).
- Zusman (1980) L. D. Zusman, Chem. Phys. 49, 295 (1980).
- Sumi and Marcus (1986) H. Sumi and R. A. Marcus, J. Chem. Phys 84, 4894 (1986).
- Rips and Jortner (1987) I. Rips and J. Jortner, J. Chem. Phys. 87, 2090 (1987).
- Yan, Sparpaglione, and Mukamel (1988) Y. J. Yan, M. Sparpaglione, and S. Mukamel, J. Phys. Chem. 92, 4842 (1988).
- Smith, Staib, and Hynes (1993) B. B. Smith, A. Staib, and J. T. Hynes, Chem. Phys. 176, 521 (1993).
- Morgan and Wolynes (1987) J. D. Morgan and P. G. Wolynes, J. Phys. Chem. 91, 874 (1987).
- Chakravarti and Sebastian (1992) N. Chakravarti and K. L. Sebastian, Chem. Phys. Lett. 193, 456 (1992).
- Matyushov (1994) D. Matyushov, J. Electroanal. Chem. 367, 1 (1994).
- Schmickler (1986) W. Schmickler, J. Electroanal. Chem. 204, 31 (1986).
- Gorodyskii, Karasevskii, and Matyushov (1991) A. V. Gorodyskii, A. I. Karasevskii, and D. V. Matyushov, J. Electroanal. Chem. 315, 9 (1991).
- Schmickler (1996) W. Schmickler, Interfacial Electrochemistry (Oxford University Press, New York, 1996).
- Matyushov (2009a) D. V. Matyushov, J. Chem. Phys. 130, 234704 (2009a).
- Kramers (1940) H. Kramers, Physica 7, 284 (1940).
- Northrup and Hynes (1980) S. H. Northrup and J. T. Hynes, J. Chem. Phys. 73, 2700 (1980).
- Hansen and McDonald (2013) J.-P. Hansen and I. R. McDonald, Theory of simple liquids, 4th ed. (Academic Press, Amsterdam, 2013).
- Matyushov (2007) D. V. Matyushov, Acc. Chem. Res. 40, 294 (2007).
- Matyushov (2009b) D. V. Matyushov, J. Chem. Phys. 130, 164522 (2009b).
- LeBard and Matyushov (2008) D. N. LeBard and D. V. Matyushov, J. Phys. Chem. B 112, 5218 (2008).
- LeBard, Kapko, and Matyushov (2008) D. N. LeBard, V. Kapko, and D. V. Matyushov, J. Phys. Chem. B 112, 10322 (2008).
- Tafel (1905) J. Tafel, Z. Phys. Chem. 50, 641 (1905).
- Bard and Faulkner (2001) A. J. Bard and L. R. Faulkner, Electrochemical Methods. Fundamentals and Applications, 2nd ed. (Wiley, New York, 2001).
- Eyring, Lin, and Lin. (1980) H. Eyring, S. H. Lin, and S. M. Lin., Basic Chemical Kinetics (Wiley-Interscience, New York, 1980).
- Böttcher (1973a) C. J. F. Böttcher, Theory of electric polarization, Vol. 2 (Elsevier, 1973).
- Ediger, Angell, and Nagel (1996) M. D. Ediger, C. A. Angell, and S. R. Nagel, J. Phys. Chem. 100, 13200 (1996).
- Fietkau et al. (2006) N. Fietkau, A. D. Clegg, R. G. Evans, C. Villagr n, C. Hardacre, and R. G. Compton, ChemPhysChem 7, 1041 (2006).
- Khoshtariya, Dolidze, and van Eldik (2009) D. E. Khoshtariya, T. D. Dolidze, and R. van Eldik, Chemistry –A European Journal 15, 5254 (2009).
- Fawcett, Gaál, and Misicak (2011) W. R. Fawcett, A. Gaál, and D. Misicak, J. Electroanal. Chem. 660, 230 (2011).
- Nikitina et al. (2014a) V. A. Nikitina, A. V. Rudnev, G. A. Tsirlina, and T. Wandlowski, J. Phys. Chem. C 118, 15970 (2014a).
- Arzhantsev et al. (2007) S. Arzhantsev, H. Jin, G. A. Baker, and M. Maroncelli, Journal of Physical Chemistry B 111, 4978 (2007).
- Stoppa et al. (2008) A. Stoppa, J. Hunger, R. Buchner, G. Hefter, A. Thoman, and H. Helm, J. Phys. Chem. B 112, 4854 (2008).
- Hunger et al. (2009) J. Hunger, A. Stoppa, S. Schr dle, G. Hefter, and R. Buchner, ChemPhysChem 10, 723 (2009).
- Weingärtner (2014) H. Weingärtner, J. Mol. Liq. 192, 185 (2014).
- Hong et al. (2011) L. Hong, N. Smolin, B. Lindner, A. P. Sokolov, and J. C. Smith, Phys. Rev. Lett. 107, 148102 (2011).
- Khodadadi and Sokolov (2015) S. Khodadadi and A. P. Sokolov, Soft Matter 11, 4984 (2015).
- Mondal, Mukherjee, and Bagchi (2017) S. Mondal, S. Mukherjee, and B. Bagchi, Chem. Phys. Lett. , 1 (2017).
- Martin and Matyushov (2015) D. R. Martin and D. V. Matyushov, J. Chem. Phys. 142, 161101 (2015).
- Ghorai and Matyushov (2006) P. K. Ghorai and D. V. Matyushov, J. Phys. Chem. B 110, 1866 (2006).
- Richert (2002) R. Richert, J. Phys.: Condens. Matter 14, R703 (2002).
- Richert (2015) R. Richert, Adv. Chem. Phys. 156, 101 (2015).
- Bagchi and Chandra (1991) B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991).
- Chen and Meyer (1996) P. Chen and T. J. Meyer, Inorg. Chem. 35, 5520 (1996).
- Görlach et al. (1995) E. Görlach, H. Gydax, P. Lubini, and U. P. Wild, Chem. Phys. 194, 185 (1995).
- Richert (2000) R. Richert, J. Chem. Phys. 113, 8404 (2000).
- Goes et al. (2002) M. Goes, M. de Groot, M. Koeberg, J. W. Verhoeven, N. R. Lokan, M. J. Shephard, and M. N. Paddon-Row, J. Phys. Chem. A 106, 2129 (2002).
- Lakowicz (2000) J. R. Lakowicz, Photochemistry and Photobiology 72, 421 (2000).
- Matyushov (2013) D. V. Matyushov, J. Chem. Phys. 139, 025102 (2013).
- Qin et al. (2017) Y. Qin, L. Zhang, L. Wang, and D. Zhong, J. Phys. Chem. Lett. 8, 1124 (2017).
- Jimenez et al. (1994) R. Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, Nature 369, 471 (1994).
- Bagchi and Gayathri (1999) B. Bagchi and N. Gayathri, Adv. Chem. Phys. 107, 1 (1999).
- Ediger (2000) M. D. Ediger, Annu. Rev. Phys. Chem. 51, 99 (2000).
- van der Zwan and Hynes (1985) G. van der Zwan and J. T. Hynes, J. Phys. Chem. 89, 4181 (1985).
- Carter and Hynes (1991) E. A. Carter and J. T. Hynes, J. Chem. Phys. 94, 5961 (1991).
- Reynolds et al. (1996) L. Reynolds, J. A. Gardecki, S. J. V. Frankland, and M. Maroncelli, J. Phys. Chem. 100, 10337 (1996).
- Mukamel (1995) S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995).
- Matyushov (2005) D. V. Matyushov, J. Chem. Phys. 122, 084507 (2005).
- Matyushov (2015) D. V. Matyushov, J. Phys.: Condens. Matter 27, 473001 (2015).
- Marcus and Sutin (1985) R. A. Marcus and N. Sutin, Biochim. Biophys. Acta 811, 265 (1985).
- Hale (1968) J. M. Hale, J. Electroanal. Chem. 19, 315 (1968).
- Oldham and Myland (2011) K. B. Oldham and J. C. Myland, Journal of Electroanalytical Chemistry 655, 65 (2011).
- Newton and Smalley (2007) M. D. Newton and J. F. Smalley, Phys. Chem. Chem. Phys. 9, 555 (2007).
- Abramowitz and Stegun (1972) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
- Compton and Banks (2009) R. G. Compton and C. E. Banks, Understanding Voltammetry (World Scientific, Singapore, 2009).
- Yue et al. (2006) H. Yue, D. Khoshtariya, D. H. Waldeck, J. Grochol, P. Hildebrandt, and D. H. Murgida, The Journal of Physical Chemistry B, J. Phys. Chem. B 110, 19906 (2006).
- Becka and Miller (1992) A. M. Becka and C. J. Miller, J. Phys. Chem. 96, 2657 (1992).
- Terrettaz, Cheng, and Miller (1996) S. Terrettaz, J. Cheng, and C. J. Miller, J. Am. Chem. Soc. 118, 7857 (1996).
- Miller (1995) C. J. Miller, “Physical electrochemistry: Principles, methods, and applications,” (Marcell Dekker, 1995) Chap. Heterogeneous electron transfer kinetics at metallic electrodes, pp. 27–79.
- Laviron (1979) E. Laviron, J. Electroanal. Chem. 101, 19 (1979).
- Honeychurch (1999) M. J. Honeychurch, Langmuir, Langmuir 15, 5158 (1999).
- Landau and Lifshitz (1984) L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).
- Lynden-Bell (2007a) R. M. Lynden-Bell, The Journal of Physical Chemistry B, J. Phys. Chem. B 111, 10800 (2007a).
- Lynden-Bell (2007b) R. M. Lynden-Bell, Electrochemistry Communications 9, 1857 (2007b).
- Sonnleitner et al. (2015) T. Sonnleitner, D. A. Turton, G. Hefter, A. Ortner, S. Waselikowski, M. Walther, K. Wynne, and R. Buchner, J. Phys. Chem. B 119, 8826 (2015).
- Leys et al. (2008) J. Leys, M. W bbenhorst, C. Preethy Menon, R. Rajesh, J. Thoen, C. Glorieux, P. Nockemann, B. Thijs, K. Binnemans, and S. p. Longuemart, J. Chem. Phys. 128, 064509 (2008).
- Serghei et al. (2009) A. Serghei, M. Tress, J. R. Sangoro, and F. Kremer, Phys. Rev. B 80, 654 (2009).
- Ito and Richert (2007) N. Ito and R. Richert, J. Phys. Chem. B 111, 5016 (2007).
- Wang et al. (2007) Y. Wang, W. Jiang, T. Yan, and G. A. Voth, Acc. Chem. Res. 40, 1193 (2007).
- Fayer (2014) M. D. Fayer, Chem. Phys. Lett. 616-617, 259 (2014).
- Shim and Kim (2009) Y. Shim and H. J. Kim, J. Phys. Chem. B 113, 12964 (2009).
- Mladenova et al. (2016) B. Y. Mladenova, D. R. Kattnig, B. Sudy, P. Choto, and G. Grampp, Phys. Chem. Chem. Phys. 18, 14442 (2016).
- Kashyap and Biswas (2008) H. K. Kashyap and R. Biswas, J. Phys. Chem. B 112, 12431 (2008).
- Roy and Maroncelli (2012) D. Roy and M. Maroncelli, J. Phys. Chem. B 116, 5951 (2012).
- Terranova and Corcelli (2013) Z. L. Terranova and S. A. Corcelli, J. Phys. Chem. B 117, 15659 (2013).
- Matyushov (1993) D. V. Matyushov, Mol. Phys. 79, 795 (1993).
- Dinpajooh, Newton, and Matyushov (2017) M. Dinpajooh, M. D. Newton, and D. V. Matyushov, J. Chem. Phys. 145, 064504 (2017).
- Boon and Yip (1980) J. P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill Inc., 1980).
- Wilke, Chen, and Bosse (1999) S. D. Wilke, H. C. Chen, and J. Bosse, Phys. Rev. E 60, 3136 (1999).
- Munakata (1975) T. Munakata, Prog. Theor. Phys. 54, 1635 (1975).
- Fried and Mukamel (1990) L. E. Fried and S. Mukamel, J. Chem. Phys. 93, 932 (1990).
- Böttcher (1973b) C. J. F. Böttcher, Theory of Electric Polarization, Vol. 1 (Elsevier, Amsterdam, 1973).
- Madden and Kivelson (1984) P. Madden and D. Kivelson, Adv. Chem. Phys. 56, 467 (1984).
- Del Pópolo and Voth (2004) M. G. Del Pópolo and G. A. Voth, J. Phys. Chem. B 108, 1744 (2004).
- Kashyap and Biswas (2010) H. K. Kashyap and R. Biswas, J. Phys. Chem. B 114, 254 (2010).
- (103) See supplementary material at [URL will be inserted by AIP] for details of calculations and experimental procedures.
- Liu and Newton (1994) Y.-P. Liu and M. D. Newton, J. Phys. Chem. 98, 7162 (1994).
- Phelps, Kornyshev, and Weaver (1990) D. K. Phelps, A. A. Kornyshev, and M. J. Weaver, J. Phys. Chem. 94, 1454 (1990).
- Krishtalik, Alpatova, and Ovsyannikova (1991) L. I. Krishtalik, N. M. Alpatova, and E. V. Ovsyannikova, Electrochim. Acta 36, 435 (1991).
- Baranski, Winkler, and Fawcett (1991) A. S. Baranski, K. Winkler, and W. R. Fawcett, J. Electroanal. Chem. 313, 367 (1991).
- Fonseca and Ladanyi (1990) T. Fonseca and B. M. Ladanyi, J. Chem. Phys. 11, 8148 (1990).
- Raineri, Resat, and Friedman (1992) F. O. Raineri, H. Resat, and H. L. Friedman, J. Chem. Phys. 96, 3068 (1992).
- Perng et al. (1996) B.-C. Perng, M. D. Newton, F. O. Raineri, and H. L. Friedman, J. Chem. Phys. 104, 7153 (1996).
- Reed, Madden, and Papadopoulos (2008) S. K. Reed, P. A. Madden, and A. Papadopoulos, J. Chem. Phys. 128, 124701 (2008).
- Ben-Amotz and Herschbach (1990) D. Ben-Amotz and D. R. Herschbach, J. Phys. Chem. 94, 1038 (1990).
- Schmid and Matyushov (1995) R. Schmid and D. V. Matyushov, J. Phys. Chem. 99, 2393 (1995).
- Mezger et al. (2008) M. Mezger, H. Schroder, H. Reichert, S. Schramm, J. S. Okasinski, S. Schoder, V. Honkimaki, M. Deutsch, B. M. Ocko, J. Ralston, M. Rohwerder, M. Stratmann, and H. Dosch, Science 322, 424 (2008).
- Sikes (2001) H. D. Sikes, Science 291, 1519 (2001).
- Raineri and Friedman (1999) F. O. Raineri and H. L. Friedman, Adv. Chem. Phys. 107, 81 (1999).
- Vatamanu, Borodin, and Smith (2010) J. Vatamanu, O. Borodin, and G. D. Smith, Phys. Chem. Chem. Phys. 12, 170 (2010).
- Rovere and Tosi (1986) M. Rovere and M. P. Tosi, Rep. Prog. Phys. 49, 1001 (1986).
- Bazant, Storey, and Kornyshev (2011) M. Z. Bazant, B. D. Storey, and A. A. Kornyshev, Phys. Rev. Lett. 106, 1247 (2011).
- Fedorov and Kornyshev (2014) M. V. Fedorov and A. A. Kornyshev, Chem. Rev. 114, 2978 (2014).
- Nikitina et al. (2014b) V. A. Nikitina, S. A. Kislenko, R. R. Nazmutdinov, M. D. Bronshtein, and G. A. Tsirlina, J. Phys. Chem. C 118, 6151 (2014b).
- (122) The reorganization energies reported in Ref. Nikitina et al., 2014b are calculated by dividing the values from MD simulations by .
- Nakanishi and Sokolov (2014) M. Nakanishi and A. P. Sokolov, J. Non-Cryst. Solids 407, 478 (2014).
- Palmer (1982) R. G. Palmer, Adv. Phys. 31, 669 (1982).
- Crisanti and Ritort (2000) A. Crisanti and F. Ritort, Physica A 280, 155 (2000).