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J. Phys. Chem. B 2005, 109, 2478-2486
REVIEW ARTICLES Properties and Reactions of Charged Species in Nonpolar Supercritical Fluids Richard Holroyd* Chemistry Department, BrookhaVen National Laboratory, Upton, New York 11973
Masaru Nishikawa Faculty of Engineering, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Astugi 243-0292, Japan
Kengo Itoh Department of Pure and Applied Science, UniVersity of Tokyo, Tokyo 153-8902, Japan ReceiVed: June 30, 2004; In Final Form: NoVember 22, 2004
This review focuses on the properties and reactions of charged species in supercritical fluids. The techniques of pulse conductivity and transient absorption are used to follow the behavior of charged species. We begin with a discussion of the mobilities, yields, and energy levels of electrons. Studies of the pressure dependence of electron attachment reactions lead to information on the magnitude of activation volumes. This as well as diffusion and energetics are factors that influence the rates of electron attachment. The free energy changes in electron attachment-detachment equilibrium reactions decrease rapidly at pressures where the compressibility maximizes. The transport properties of ions in supercritical fluids are also discussed, as these studies provide a straightforward indication of the degree of interaction between ions and the medium. We conclude with a review of electron transfer reactions in supercritical fluids.
Introduction In the last fifteen years, the uses of supercritical fluids in the chemical and semiconductor industries have burgeoned because supercritical fluids can be used as media for chemical synthesis, extraction, separation, surface cleaning, and processing.1-3 The useful characteristic of supercritical fluids common to all these applications is their ability to act as powerful solvents, a property that can be attributed to the accumulation of solvent density around solute molecules and the ability to exploit variation of supercritical conditions using pressure and temperature to “tune” conditions to favor particular reactions. Both features are closely related to the higher compressibility of supercritical fluids. As a result of the density augmentation that occurs, the partial molar volume of solutes is often large and negative. The role of clustering around neutral solutes in chemical reactions has been reviewed recently.4 The behavior of charged species in nonpolar supercritical fluids is of particular interest because solute-solvent interactions can be described in terms of theoretically tractable strong attractive charge-induced dipole interactions. Rearrangement of solvent molecules around the charges leads to alteration of the energy levels of charged species in solution by polarization. This “polarization energy” largely determines the reactions of charged species in solution. Moreover, polarization energy can be readily calculated using estimates of local dielectric constants. Nonpolar solvents exhibit low dielectric constants, which implies that the influence of electrostatic interactions can be relatively longranged and that density inhomogeneities spanning distances representing many molecules occur. Therefore, the utilization of supercritical fluids in chemical processing that involves charged species requires knowledge about the physical properties and reactivity of such species in the fluid, information that is largely
unknown at the present time. Hence, basic studies of the behavior of charged species in nonpolar supercritical solvents are important. Our interest in supercritical fluids grew out of studies of charged species in nonpolar liquids under high pressure that revealed the importance of the isothermal compressibility, χT, in determining the transport and reactivity of these species.5 χT is defined as
χT ) -
1 ∂V V ∂P T
( )
Studies of the pressure dependence of the electron mobility in liquids provided information about the volume change involved in electron trapping.5,6 This volume change can be negative or positive depending on the relative importance of the electrostriction and cavity volume terms. The former term dominates in the highly compressible liquids. Also, high pressure studies showed that large volume changes, between -80 and -300 cm3/mol, are associated with electron attachment reactions in liquids.5 These volume changes were too large to be explained by classical continuum models. To explain this large volume change within the framework of classical models, solidification of a shell of solvent molecules around the ion by electrostriction had to be invoked.7 In supercritical fluids, where χT is much larger than in ordinary liquids, more pronounced effects were expected and in fact extensive density augmentation around ions is found in the studies of the transport and reactions of charged species. Studies of these phenomena in supercritical fluids have led to information about the size of clusters around ions. For example the mobility of an ion, given by µion ) VD/E, where VD is the drift velocity in an electric field, E, is related to the size of the cluster moving with the ion in the fluid. The cluster radius, rs, can be estimated by means of the Stokes equation rs
10.1021/jp0471296 CCC: $30.25 © 2005 American Chemical Society Published on Web 01/22/2005
Review Article
Figure 1. Plot of k3 for electron attachment to NO versus pressure at (b) 33 °C, (O) 37 °C, and (2) 47 °C (left-hand scale) and compressibility of ethane versus pressure at (s) 33 °C, (- - -) 37 °C, and (- ‚ - ‚ -) 47 °C (right-hand scale).
) e/6πηµion. A more accurate estimate can be obtained by taking the enhancement in viscosity around the ions into account.8 We find for example that positive ions move with at least one complete solvation shell, but larger radii are found in regions of higher compressibility. The pressure dependence of reaction rates and equilibrium constants of electron attachment reactions yields information on the reaction volume changes, ∆Vr, activation volumes, ∆Va, and partial molar volumes of the reactants and products involved, since ∆Vr ) (∂∆Gr/∂P)T and ∆Va ) -RT(∂ ln ka/∂P)T. In supercritical ethane, sharp increases in the rate constant for electron attachment to NO occur at the pressures where the isothermal compressibility maximizes (see Figure 1). The comparison of the volume information with various theoretical model calculations can be used to test the validity of such models. This review focuses on information about charged species, electrons, and ions, as studied by pulse conductivity and transient absorption. Since there are no free electrons in nonpolar media, the electrons must be introduced into the media; that is, they are excess electrons. This is accomplished in most of these studies by means of irradiation of the media with ionizing radiation. We begin with electron properties including the mobilities, yields, and energy levels of electrons in supercritical fluids. Then, we consider electron attachment reactions and the magnitude of activation volumes thereof. The factors that influence the rates of electron attachment are discussed including the role of diffusion and energetics. With some solutes, the reactions are reversible, and from the equilibrium studies, we obtain the free energy changes, which show sharp decreases at pressures where the compressibility maximizes. We explain this phenomenon in terms of the polarization energy change. The transport properties of ions and ion recombination rates are included, since these studies lead directly to information about the size of ionic species in the fluid. We conclude this review with a discussion of charge transfer reactions in terms of information obtained from the ion transfer studies. Electron Properties. Mobility. The electron mobility in three supercritical fluids is shown in Figure 2 for 0.2 < n/nc < 1.8, where n is the number density and nc the critical density. Surprisingly the mobility is higher in ethane and ethene than in xenon below the critical densities. The situation is totally reversed in the liquid state for these fluids where the mobility is very high for xenon, 8500 cm2 V-1 s-1,9 and extremely low for the hydrocarbons, for example, 0.0013 cm2 V-1 s-1 for ethane at 111 K.10 The electron mobility, µe, is considered to be determined by the strength of the scattering interaction
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Figure 2. Electron mobility versus density for (2) C2H6,68 (O) C2H4,41 and (b) Xe.19
between excess electrons and molecules of the medium. Thus, the scattering cross section, σ, is an important parameter. The classical formula, eq 1,11 shows the dependence of µe on the scattering cross section.
µe )
4e 1 Λ, where Λ ) nσ 3(2πmekBT)1/2
(1)
At very low densities, the mean free path, Λ, is greater than the de Broglie wavelength, which is 7.7 nm for an electron at room temperature. Electrons are simply scattered by individual molecules (i.e., single scattering), µe is relatively high, and µen is constant, indicating σ is constant. The mean free path becomes comparable to the de Broglie wavelength as n approaches the critical density and electrons begin to interact with many molecules simultaneously (multiple scattering); then, µen is no longer constant. As the density increases further above the critical density, the mobility increases in xenon, reaching a maximum comparable to that in semiconductors.12 These high mobility values in condensed systems indicate that the electrons are quasifree and in conduction bands. However, in hydrocarbons such as ethane, the mobility decreases as the density increases, and the concept of a mean free path loses its meaning as it is much shorter than molecular distances at such low mobility values. This strongly suggests electrons are trapped in cavities,13 and transport occurs by thermal activation from traps. In supercritical xenon at 293 K, the mobility goes through an unusually low minimum of ∼5 cm2 V-1 s-1 at a density just below the critical density. The deformation potential model of Basak and Cohen14 is often invoked to interpret such experimental results when electrons are in a quasifree state. In this model, the electron is scattered as a result of potential fluctuations due to density fluctuations.14,15 The scattering potential is related to density fluctuations by introducing a deformation potential, ∆V0 ) (dV0/dn)∆n + (1/2)(d2V0/dn2)∆n2 + ..., where V0 is the energy of the bottom of the conduction band with respect to vacuum. The density fluctuations, ∆n, are approximated by the long wavelength limit of the structure factor S(0) ) nkBTχT.
µe ) 2ep4x2π 3m*5/2(kBT)1/2n2[V0′2kBTχT + (terms involving V0′′, V0′′′)] (2)
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Figure 3. Free ion yield versus reduced density for (s) C2H6,33,34 (- - -) C2H4,33,34 and (- ‚ - ‚ -) Xe.32,35
In eq 2, m* is the reduced mass of the electron and the primes indicate derivatives. The density fluctuations of this type become quite large near critical densities as the isothermal compressibility, χT, diverges16 and are considered to be static relative to the electron motion. The model predicts a mobility maximum at the density where V0′ is zero and near zero mobility at the critical density, nc, where χT diverges. The loci of the maxima were verified experimentally in several fluids such as rare gas liquids17 and low molecular weight alkanes,18 but the mobility actually never goes to zero at nc. The magnitude of the maximum could not be estimated a priori and always needs to be normalized to the experiment. Near nc, eq 2 tends to exaggerate the effect of the fluctuations and the predicted values are smaller by several orders of magnitude than the experimental values.19 The use of χT was criticized, as the de Broglie wavelength of electrons is much shorter than the size of the fluctuations and electrons do not interact with these fluctuations.20,21 The substitution of the adiabatic compressibility, χS ) 1/Fω2, where F is the density and ω the speed of sound, for χT changes eq 2 into that for the acoustic phonon-limited mobility.15,22 When terms involving higher-order derivatives of V0 are ignored,21 it reduces exactly into the expression used by Bardeen and Shockley.22 In supercritical xenon in the density range below the critical density, where only V0′ is important, this acoustic phonon-limited mobility equation21 accounts for the density dependence going through the minimum remarkably well without having to use any adjustable parameter.19 Experimental values of V0′23 and calculated values of the electron effective mass, m*,24 were used in the calculation. Near the minimum, the experimental values are slightly below theoretical predictions, but this can be explained as being due to electron trapping onto clusters (large density fluctuations related to S(0) predominant near the critical density). This is because electrons tend to prefer higher density regions, where V0 values are lower and the void spaces between clusters act as barriers for electron transport.25 With further increase in density, the clusters merge together, facilitating transport. This model also explains the order of magnitude of mobilities in this density range in ethane, ethene, and xenon, as shown in Figure 2, as due to the difference in the values of V0′2 and χS for these fluids. The dominant factor is V0′2 which is largest for Xe and quite small for the hydrocarbons, as is clear from an examination of Figure 4. More detail can be found in recent reviews.26,27
Figure 4. V0 versus density for (- - -) C2H6,39 V0 for (‚‚‚‚) C2H4sthis curve was calculated using the WS method38 based on a value of V0 of -0.044 at a density of 0.075 g/cm3, estimated from the optical model40 and a value of the electron mobility at the same density,41 and V0 for (s) Xe.23
Hot Electrons. A feature specific to xenon and other rare gases is that the mobility of electrons depends on their kinetic energy. In radiolysis, electrons emerge from the initial ionization processes with excess energy, taking many nanoseconds to thermalize,19 since the only thermalization mechanism available is elastic collisions, once the electron kinetic energy falls below that of the lowest electronic excited state, which for xenon is 8.4 eV.28 The mobility of hot electrons is significantly different from that of thermal electrons. The drift velocity and the mobility are also field-dependent because electrons become “hot” by gaining energy from the applied electric field and the energy is not readily dissipated. In ethane and CO2, thermalization presumably occurs in picoseconds, as is the case for heavier hydrocarbons,29 where there are mechanisms, other than momentum transfer, that can take up electron energy efficiently. Electrons in fluid xenon can be thermalized faster by adding a small amount of a polyatomic nonreacting solute such as ethane.19 Like argon and krypton, the Ramsauer-Townsend (R-T) minimum in collision cross sections exists in gaseous xenon at around 0.6 eV, and it persists in dense fluids, shifting to lower energy values.9,19 The influence of the R-T minimum is pronounced in low density fluids below 1.5 × 1021 cm-3, causing transient electron signals to go negative at times shorter than 40 ns.19,30 This peculiar phenomenon was explained by the heating of electrons moving along the field to the high energy side of the R-T minimum, thereby decreasing their contribution to current signals.31 At a higher density, electron signals behave normally with respect to the applied field. Free Ion Yields. Following the initial ionization events, electrons travel some distance from their geminate ions before thermalizing. The electrons and ions can then either undergo geminate recombination or escape to diffuse freely. The yield of electrons that escape this recombination per 100 eV absorbed is termed the free ion yield, Gfi. High ion yields facilitate the study of ionic reactions in these fluids. The thermalization distance depends inversely on density. Thus, free ion yields in the molecular fluids ethane and ethene decrease with density,
Review Article as shown in Figure 3.32-35 The yields are around 1-2 ion pairs per 100 eV near critical densities which is an order of magnitude larger than the yields in the liquid state, where for liquid ethane and ethene, Gfi ) 0.16 and 0.01, respectively.36 The lower density of supercritical fluids means that electrons can travel further while losing energy; that is, the electron ranges are longer than they are in liquids. An estimate of the ranges, R, may be obtained from Gfi values and the following expression for the probability of escape for an ion pair: Pesc ) exp(-e2/ RkBT). The ranges calculated assuming Pesc ) Gfi/Gtotal ) Gfi/4 are ∼80 nm for ethane and ∼30 nm for ethene near critical densities. The low density also means that the dielectric constant, , is less ( for ethane is 1.25 at the critical density). The Onsager escape distance37 is the distance at which the thermal kinetic energy of the electron, kBT, is balanced by the potential energy of the electron in the Coulombic field of the ion, and it is longer here than in liquids. This distance is 44 nm at the critical density in ethane. The situation in xenon is quite different; the free ion yield remains high, actually increasing slightly as the density increases.32 The Onsager escape distance in xenon is similar to that in ethane; however, the electron travels well beyond this distance in the time it takes to thermalize, which is several nanoseconds.19 Thus, the escape probability is close to unity. This feature makes xenon a good choice for the study of reactions of electrons and ions. Conduction Band Energy. The most fundamental parameter of excess electrons in dense media is V0, the energy of the bottom of the conduction band measured from the vacuum level. V0 is usually treated by the Springett-Jortner-Cohen (SJC) model38 and is considered to consist of two parts, namely, the attractive polarization energy taking into account the screening effect and the repulsive kinetic energy arising from the impenetrable hard core. V0 can be positive or negative depending on the type of interaction between the electrons and the molecules of the media. If the interaction is predominantly repulsive, V0 is positive, and if attractive, V0 is negative. As V0 is the ground state energy of the quasifree electrons, the level of V0 relative to the energy of the localized state determines whether the electrons can stay quasifree or not in the dense system. V0 values cannot easily be estimated a priori, since the hard-core radius is not known, and are usually determined experimentally. The change in work function of a metal when immersed in the dense fluid is the method often used, but other techniques are available. Recently, a method utilizing the field ionization of a Rydberg state of a solute23 has been devised, which seems to be more accurate than other methods. Values of V0 as a function of density are shown in Figure 4 for these fluids. This energy is important in many ways. As was shown above, the mobility of quasifree electrons depends on the derivative dV0/dn. The magnitude of V0 may also influence the energetics of electron attachment reactions (see later section). In ethane, V0 is slightly negative and changes little with density in the supercritical range.39 For ethene, V0 is similar and also shows little dependence on density in this range. This curve was calculated using the SJC method38 based on a value of V0 of -0.044 at a density of 0.075 g/cm3, estimated from the optical model 40 and a value of the electron mobility at the same density.41 A similar calculation for ethane gives reasonable agreement with the experimental curve shown. For xenon, V0 changes dramatically throughout the critical region, becoming quite low at high densities.23,42 Electron Attachment Rates. The volume changes in electron attachment reactions provide evidence for the large density
J. Phys. Chem. B, Vol. 109, No. 7, 2005 2481 augmentation around ions in nonpolar fluids. For the solutes (S) that have been studied so far, the rate of electron attachment, reaction 3,
e- + solute f solute-
(3)
generally increases as the pressure increases in supercritical fluids. More specifically, the rate constant for this reaction, k3, is much higher above nc than below nc. Plots of ln k3 versus pressure exhibit a sharp rise in the region of high compressibility, as shown in Figure 1 for nitric oxide in supercritical ethane.43 Similar results were obtained for C2F4 in ethane and C6F6 in xenon.44 The absolute values of the rates are spread over a wide range. For small molecules such as O2,45 NO, and C2F4, the rate constants are around 1011 m-1 s-1 at pressures above nc. Note that m stands for molal. The slow rate for O2 has been attributed to the large energy gap between the V0 level and the vertical attachment energy for this reaction.46 For the solutes pyrimidine,47 pyrazine, and methylpyrazine48 in ethane, k3 is around 1013 m-1 s-1, and for C6F6 in xenon, k3 is close to 1015 m-1 s-1 at pressures above 70 bar at 20 °C.44 The rate constant observed for attachment to C6F6 in Xe is one of the highest ever measured in a fluid. Even so, the rate is well below the diffusion limit at high pressures in Xe, given by the usual expression kD ) 4πrDe, using 1 nm for the reaction radius, r. The mobility of the electron, µe, and thus the diffusion coefficient, De, which is equal to µekBT/e increases rapidly with pressure above nc, yet the C6F6 rate remains fairly constant at high pressure. Electron attachment rates tend to plateau at high mobilities because the time spent by the electron near the acceptor becomes shorter, resulting in a maximum rate of ∼3 × 1014 M-1 s-1.49 M stands for molar in this case. In the same units, our rate constant for C6F6 is 5.6 × 1014 M-1 s-1, within a factor of 2 of the maximum rate. ActiVation Volumes. The activation volume for a reaction can be determined from the change in the attachment rate with pressure.
∆Va* ) -RT(∂ ln k3/∂P)T
(4)
This equation is valid provided that the rate constant is in pressure-independent concentration units such as molality, which is used in our studies. The sharp increases in rate, as shown in Figure 1, lead to minima in plots of ∆Va* versus pressure. For example, for NO in ethane, minimum values of ∆Va* ) -27 L/mol at 33 °C and -9.2 L/mol at 37 °C were reported.43 These very large volume changes, that occur where the isothermal compressibility, χT, is a maximum, signify a large role for electrostriction, that is, the response of molecules in the neighborhood of an ion to the electrostatic force of the charge on the ion. The activation volume is the difference in partial molar volumes at the activated state, νj(S-*), and the partial molar volumes of the reactants:
∆Va* ) νj(S-*) - νj(S) - νj(e-)
(5)
The latter term in eq 5 is the partial molar volume of the electron, which is small, at least near critical densities where the electron is quasifree and would therefore not perturb the solvent. At higher densities, the electron in ethane is trapped in a cavity. Nevertheless, estimates indicate that νj(e-) is
