- Email: [email protected]
Behavior of charged species in supercritical heavy rare gas fluids
ARTICLE IN PRESS
Radiation Physics and Chemistry 76 (2007) 1225–1228 www.elsevier.com/locate/radphyschem
Behavior of charged species in supercritical heavy rare gas fluids Masaru Nishikawa Faculty of Engineering, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Atsugi 243-0292 Japan
Abstract Electron mobilities were measured in supercritical (SC) Xe and Kr as functions of pressure and temperature. Ion mobilities, reaction rates of electrons with C6F6, O2 and pyrazine were also measured in SC Xe. Electron mobility is strongly field-dependent and the lowfield mobility me exhibits a deep minimum at a density just below the critical density. A deformation potential model of the acoustic phonon scattering type can reproduce me change in SC Xe and Kr quantitatively. From positive ion mobility values, a hydrodynamic compressible continuum model indicates that the cluster radius goes to a maximum, 1 nm, at 291 K at the pressure of the isothermal compressibility, wT, peak. The rate constant for the reaction C6F6+e-C6F–6 sharply increases up to nearly 1015 molal s1 in this pressure region. Pyrazine was found to form neutral clusters in SC Xe prior to electron attachment. These and other data are compared with the observation made in SC ethane. r 2007 Elsevier Ltd. All rights reserved. Keyword: Supercritical fluids; Electron mobility; Ion mobility; Electron attachment reactions; Xenon
1. Introduction In the supercritical fluids (SCF) the density fluctuation is macroscopic near the critical point but it becomes microscopic when one moves away from the critical point. The SCF provides a unique stage for studying the interaction of macroscopic properties, and the variation in their scale, with microscopic charged species. Supercritical rare gas systems, in particular, are more suitable for the study to single out the effect, since they are composed of atoms. This overview is about our studies carried out in the last 10 years and focuses on transport phenomena of electrons and ions, and electron reactions in supercritical rare gas fluids.
2. Electron mobility 2.1. Xe. fast transients and hot electron effects The current due to the motion of electrons injected by pulse irradiation, drifting in the electric field E, is E-mail address: [email protected]. 0969-806X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2007.02.004
given by
i ¼ uD e=d
Z
d
ne ðx; tÞ dx,
(1)
0
where uD is the drift velocity, ne the concentration of electrons and d the electrode spacing. The Ramsauer–Townsend (R–T) minimum in momentum transfer crosssection is known to exist for heavy rare gas systems (McDaniel, 1989)even in the liquid phase (Gushchin et al., 1982). In Xe low-density gas the minimum is located at ca. 0.8 eV, but shifts to lower energy with increasing density in the liquid. The fast pulse conductivity method allows one to observe transient signal at a very short time domain. In supercritical xenon in the lower density region, very peculiar current traces were observed immediately after irradiation, i.e., first going positive and then negative (Holroyd et al., 2003). The negative current signal was predicted by McMahon and Shizgal (1985) and was first observed by in gaseous xenon (Warman et al., 1985). The initial positive signals were interpreted to be due to epithermal electrons with energies above the R–T minimum. Their uD is expected to behave normally with respect
ARTICLE IN PRESS M. Nishikawa / Radiation Physics and Chemistry 76 (2007) 1225–1228
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to the applied electric field. When the electrons are slowed down to the R–T minimum, the electrons moving along the direction of the electric field gain energy, while those moving against it continue to lose energy. Those electrons which have gained energy reach the steeply rising side of the cross-section curve and suffer more scattering events. Hence, they experience shorter free flight paths. Thus, those moving against the direction of the electric field contribute more to make uD negative and negative signal arises. Zero-field electron mobility. Electron mobility me in SC Xe was determined by using 60 ns X-ray pulse from a van de Graaff accelerator and from the straight portion of uD vs. E plot at the low field strength. (Fig. 1) A deformation potential model proposed by Basak and Cohen (BC) (Basak, Cohen, 1979) has been frequently used to interpret electron mobilities in rare gas systems. me ¼
pffiffiffiffiffiffi 5=2 2e_4 2p me =m h i , 5=2 3me ðkB T Þ1=2 N 2 V 0 20 kB TwT þ terms involving V 000 ; V 000 0
(2) where V0 is the energy of the bottom of the conduction band with respect to vacuum, primes are derivatives with respect to the density and m* is the electron effective mass. The BC model can predict the location of the maximum at a density where V00 ¼ 0 and the general trend in the mobility change with density near the maximum in liquid argon; however, the magnitude differs from experiments by several orders. In deformation potential theory, electrons are scattered by potential fluctuations at the bottom of the conduction band. The BC model considers potential fluctuations due solely to static density fluctuations, which are approximated by the long wavelength limit of structure factor S(0) ¼ NkBTwT,. Then, at the critical point where wT
Mobility/10-4m2V-1s-1
1000
100
10
1
0 0
2
4
6
N/1027 atoms
8
10
m-3
Fig. 1. Zero-field electron mobility in SC Xe. Experiment: J 291 K, K 293 K, n 303 K. Theory: Eq (2) – – –; 293 K, 303K; Eq (3) —— 293K, – – – – 303K. Data from (Holroyd et al., 2003).
diverges, m should become very small. Experimentally, however, no such m-values were observed (Kimura, Freeman, 1974; Huang, Freeman, 1981). Steinberger and Zeitak (1986) consider that the failure of the BC model near the critical point is due to the fact that the size of the density fluctuations represented by S(0) is too large compared with the de Broglie wavelength of electrons, and thus these fluctuations do not influence the motion of electrons. There are potential fluctuations arising from other mechanisms too, such as acoustical phonons. The BC model was modified to accommodate scattering by phonons by replacing wT with wS, as the mean square density fluctuation due to acoustic phonons is related to nkBTwS. (Ziman, 1972; Nishikawa, 1985; Ascarelli, 1986) This replacement dramatically improves the agreement between theory and experiment in the critical region and even around the maximum as well, although with the use of some adjustable parameters (Steinberger and Zeitak, 1986). When terms involving higher order derivatives of V0 are ignored, the equation is reduced to the expression used by Bardeen and Shockley (1950) for the acoustic phononlimited mobility. pffiffiffiffiffiffi 5=2 2e_4 2p me =m me ¼ (3) . 5=2 3me ðkB T Þ1=2 N 2 V 0 0 2 kB TwS In SC xenon in the lower density range, Eq. (3), using experimental values of V0’ (Altmann, Reininger, 1997) and values of electron effective mass m* (Plenkiewicz et al., 1992), was found to account for the density dependence remarkably well without having to use any adjustable parameter. However, the experimental me minimum lies lower than predicted by Eq (3). Huang-Freeman (1978) invoked quasilocalization of electrons on quasidroplets to explain unusually low values. The magnitude of the mean free path calculated from the me minimum of 4.25 cm2/Vs (Fig. 1) is about half the average atomic distance in the bulk. This comparison suggests there is indeed some kind of localization. Path integral Monte Carlo studies of Berne and co-workers showed that the electron in Xe exists in extended states at all densities, but prefers high density regions with lower V0 values in the fluid (Coker et al., 1987). Thus, when the spatial extent of the density fluctuations becomes comparable to the de Broglie wavelength of the thermal electrons, electrons are trapped by the clusters, i.e., quasidroplets by Freeman. Electrons are spread over the clusters, (Martyna and Berne, 1988) and low-density regions surrounding the clusters act as potential barriers for the electron transport from one cluster to another. It seems reasonable to attribute deviations of electron mobility from the theoretical curve to this effect. Kr (Nishikawa et al.) Fast transients and hot electron effects. Current traces obtained at short times after a 30 ps X-ray pulse at 2.5 MPa behave normally. At 1.25 MPa and below 30 V/cm, there is a small positive peak prior to major current signals, but in contrast to SC Xe no negative signal.
ARTICLE IN PRESS M. Nishikawa / Radiation Physics and Chemistry 76 (2007) 1225–1228
The absence of negative transient current in SC Kr may be due to the fact that the wall of the R–T minimum is not steep enough to give rise to the predominant contribution of backward moving electrons as in Xe. Zero-field electron mobility. At 231 K me–value reaches a minimum of 25.3 cm2Vs-1 at 7.025 MPa (3.81 1027 atoms m-3). Similar to the case of Xe, the minimum lies at a density of approximately one-half of the critical density. Equation (3) with V0’-values and values of effective mass m* can also account for the density dependence around the minimum without any adjustable parameter. Applicability of the model. The magnitude of me in SCF around the critical density is much larger in ethane than in Xe, which is just the opposite of the trend in the liquid phase. The model can explain why the order in electron mobility in this density range is me (ethane)4 me (Kr)4 me (Xe). It is attributed to the difference in V00 2 wS in Eq. (3), the dominant factor being V00 which is largest for Xe, smaller for Kr and quite small for ethane. Ion mobility. Ion mobility, mi, values on all SCF measured are on the order of 10–3 cm2V1 s1 and decrease with increasing pressure (Itoh et al., 2001). In all cases when temperature is near the critical temperature, there is small but sharp minima at the pressure where wT has its peak. Ions are known to form large clusters in SCF by means of charge-induced dipole interactions, and their size strongly depends on the magnitude of wT (Itoh et al., 2001; Holroyd et al., 2005). The size of clusters drifting in SCF can be estimated from mi values by means of a hydrodynamic compressible continuum (HCC) model (Itoh et al., 2001). The HCC model is similar to the Stokes equation but uses the local viscosity estimated from local density buildup due to the ion electric field. The HCC model leads to the cluster radii rc maxima reaching 1 nm at mi minima in SC Xe and ethane. Electron attachment reaction. Reactions with pyrazine, pbenzoquinone, C2F4, C6F6, CCl4, CO2 and O2 were studied in SC Xe. In addition, reactions with NO, pyrimidine, methylpyrazine and styrene were studied in SC ethane. Rate constants for attachment, ka, in SC Xe to CO2, C2F4 and p-benzoquinone, were found to be o5 1011 m1 s1 (Holroyd et al., 2003). All small molecules, such as NO, CO2, O2 and C2F4., react slowly (o1 1012 m1s1) (Holroyd et al., 2005). The reactions with C6F6 and with CCl4 are extremely fast in SC Xe. The rate constant reaches almost 1 1015 m1s1 at pressures above 6.5 MPa at 293 K. Pyrimidine, pyrazine, and methylpyrazine react with an intermediate rate of around 1013 m1 s1. These reactions are reversible; detachment rate constants kd are below 108 s1 and decrease with pressure (density) increase. The attachment rate in SCF is characterized by a sharp increase at the pressure where wT reaches the peak. The magnitude of ka is not related to the electron affinity EA(eV) of respective solutes. The rate is to be understood in terms of matching between the V0-level of the fluid and energy required for the excess electron to hop vertically onto the unoccupied redox level of the solute molecule in
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the fluid (Henglein, 1977). For example, the level for small molecules like O2 is pulled down so deeply in the fluid by the polarization energy of the occupied level (negative ion state) from the gas phase value, there is no matching even for SC Xe with very low V0-level (-0.2–0.8 eV). Activation volume Va* and reaction volume change DVr for X þ e qf Ð X Va* and DVr are experimentally given by V a ¼ RTðq ln ka =qpÞT , DV r ¼ RTðq ln K=qpÞT ,
ð4Þ
where K is the equilibrium constant. They are the difference between the partial molar volumes V of products and reactants. V a ¼ V ðX Þ V ðX Þ V ðe qf Þ, DV r ¼ V ðX Þ V ðX Þ V ðe qf Þ.
ð5Þ
Excess electrons in SC ethane and Xe are quasifree, so that V ðe qf Þ ’ 0. Generally, both Va* and DVr show sharp minima at the pressure of wT peak. For X ¼ C6F6 in SC Xe, the minimum Va* value is -28 dm3/mol at 293 K (Holroyd et al., 2003). Very large negative DVr, –45 dm3/mole, was observed for the equilibrium reaction with pyrazine in SC ethane (Holroyd et al., 2000). These minima indicate that the volume changes are associated with the electrostriction volume Vel. Values of Vel calculated by the compressible continuum (CC) model that uses the local wT in the cluster (Nishikawa et al., 1998) can predict DVr for the electron attachment to pyrazine quite well (Fig. 2), and the predicted value indicates that the partial molar volume of neutral pyrazine in sc ethane is nearly zero. In contrast, DVr for the same reaction in SC xenon is small throughout, negative in the lower pressure region and become positive in the vicinity of wT maximum (Holroyd et al., 2007). Above the maximum they are practically zero, which means no further shift in K with pressure. Eq. (5) suggests that V for pyrazine ions and molecules in SC Xe are nearly equal and negative, indicating the formation of neutral clusters. Vel calculated by the CC model was found to agree with Va* for C6F6 reaction in SC Xe including the minimum above the pressure of wT peak. The agreement indicates that the transition state is close to the final ionic state, which is reasonable for molecules such as C6F6. The CC model can also explain the free energy change DGr for the equilibrium reactions. From thermodynamic cycles, DGr ¼ -EA+Pcc—V0. DGr for pyrazine in SC ethane is approximately given by DGrI Pcc—V0 , as EA is small, where Pcc is the polarization energy calculated by the model using local dielectric constants (Holroyd et al., 2000). It is concluded that the compressibility, both isothermal and adiabatic, thus density fluctuations, plays profoundly important roles in determining the behavior of charged species in the SCF.
ARTICLE IN PRESS M. Nishikawa / Radiation Physics and Chemistry 76 (2007) 1225–1228
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109
K/m-1
108
107
106 0
Volume/L mole-1
-10 -20 -30 -40 -50 -60 4
5
6
7
Pressure/MPa Fig. 2. Equilibrium constants and reaction volume changes DVr for e+pyrazine!pyrazine in SC ethane. Experiment: m 306K, J 310K, K 318K. Theory; CC model; —— 306K, - - - - - - 310K, – – – – 318K, Drude-Nernst equation (Drude, Nernst, 1894), – – – 310K. Data from (Holroyd et al., 2000).
Acknowledgments These researches are borne out of our (M.N. and R.A. Holroyd) long-standing project. Most part of these studies have been carried out at Brookhaven National Laboratory and supported under contract DE-AC02-98-CH10886 with US Department of Energy. We thank all the scientists at Brookhaven and University of Tokyo who participated in the project during the course of study, particularly, Kengo Itoh for his contribution to the theoretical part. We also appreciate financial support from Ministry of Education, Culture, Sports, Science and Technology and from Japan Society for Promotion of Science. References Altmann, K.N., Reininger, R., 1997. Density dependence of the conduction band minimum in fluid krypton and xenon from field ionization of (CH3)2S. J. Chem. Phys. 107, 1759–1764.
Ascarelli, G., 1986. Calculation of the mobility of electrons injected in liquid xenon. Phys. Rev. B 34, 4278–4288. Basak, S., Cohen, M.H., 1979. Deformation-potential theory for the mobility of excess electrons in liquid argon. Phys. Rev. B 20, 3404–3414. Coker, D.F., Berne, B.J., Thirumalai, D., 1987. Path integral Monte Carlo studies of the behavior of excess electrons in simple fluids. J. Chem. Phys. 86, 5689–5702. Drude, P., Nernst, W., 1894. Ueber Elektrostriktion durch freie Ionen. Z. Phys. Chem. 15, 79–85. Gushchin, E.M., Kruglov, A.A., Obodovskii, I.M., 1982. Electron dynamics in condensed argon and xenon. Sov. Phys. JETP 55, 650–655. Henglein, A., 1977. Electron reactivity as a function of phase. Can. J. Chem. 55, 2112–2123. Holroyd, R.A., Itoh, K., Nishikawa, M., 2003. Density inhomogeneities and electron mobility in supercritical xenon. J. Chem. Phys. 118, 706–710. Holroyd, R.A., Nishikawa, M., Itoh, K., 2000. Solvent clustering around pyrazine ions in the high-compressibility region of supercritical ethane. J. Phys. Chem. B 104, 11585–11590. Holroyd, R.A., Nishikawa, M., Itoh, K., 2005. Properties and reactions of charged species in nonpolar supercritical fluids. J. Phys. Chem. B 109, 2478–2486. Holroyd, R. A., Presses, J., Nishikawa, M., Itoh, K., 2007. Energetics and volume changes in electron attachment to pyrazine in supercritical xenon. J. Phys. Chem. B., in press (Published on Web 01/11/2007 ASAP Article, jp065922o). Holroyd, R.A., Wishart, J.F., Nishikawa, M., Itoh, K., 2003. Reactions of charged species in supercritical xenon studied by pulse radiolysis. J. Phys. Chem. B 107, 7281–7287. Huang, S.S.-S., Freeman, G.R., 1981. Electron transport in gaseous and liquid argon: effect of density and temperature. Phys. Rev. A. 24, 714–724. Itoh, K., Holroyd, R.A., Nishikawa, M., 2001. Ion mobilities in supercritical ethane, xenon, and carbon dioxide. J. Phys. Chem. A 105, 703–709. Kimura, T., Freeman, G.R., 1974. Mobility of electrons in xenon in the critical region. J. Chem. Phys. 60, 4081–4082. Martyna, G.J., Berne, B.J., 1988. Structure and energetics of Xe-n. J. Chem. Phys. 88, 4516–4525. McDaniel, E.W., 1989. Atomic Collisions—Electron and Photon Projectiles. Wiley Interscience, New York. Nishikawa, M., 1985. Electron mobility in fluid argon: application of a deformation potential theory. Chem. Phys. Lett. 114, 271–273. Nishikawa, M., Holroyd, R., Itoh, K., 1998. Electron attachment to NO in supercritical ethane. J. Phys. Chem. B 102, 4189–4192. Nishikawa, M., Holroyd, R. A., Presses, J., Mobility of electrons in supercritical Kr—role of density fluctuaions. in preparation. Plenkiewicz, B., Frongillo, Y., Plenkiewicz, P., Jay-Gerin, J.-P., 1992. Calculation of the effective mass of excess electrons in fluid Ar, Kr, Xe. J. Chem. Phys. 94, 6132–6134. Steinberger, I.T., Zeitak, R., 1986. Estimation of electron mobilities near the critical point in simple nonpolar fluids. Phys. Rev. B 34, 3471–3474. Warman, J.M., Sowada, U., De Haas, M.P., 1985. Transient negative mobilities of hot electrons in gaseous xenon. Phys. Rev. A 31, 1974–1976. Ziman, J.M., 1972. Principles of the Theory of Solids, second ed. Cambridge University Press.
Radiation Physics and Chemistry 76 (2007) 1225–1228 www.elsevier.com/locate/radphyschem
Behavior of charged species in supercritical heavy rare gas fluids Masaru Nishikawa Faculty of Engineering, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Atsugi 243-0292 Japan
Abstract Electron mobilities were measured in supercritical (SC) Xe and Kr as functions of pressure and temperature. Ion mobilities, reaction rates of electrons with C6F6, O2 and pyrazine were also measured in SC Xe. Electron mobility is strongly field-dependent and the lowfield mobility me exhibits a deep minimum at a density just below the critical density. A deformation potential model of the acoustic phonon scattering type can reproduce me change in SC Xe and Kr quantitatively. From positive ion mobility values, a hydrodynamic compressible continuum model indicates that the cluster radius goes to a maximum, 1 nm, at 291 K at the pressure of the isothermal compressibility, wT, peak. The rate constant for the reaction C6F6+e-C6F–6 sharply increases up to nearly 1015 molal s1 in this pressure region. Pyrazine was found to form neutral clusters in SC Xe prior to electron attachment. These and other data are compared with the observation made in SC ethane. r 2007 Elsevier Ltd. All rights reserved. Keyword: Supercritical fluids; Electron mobility; Ion mobility; Electron attachment reactions; Xenon
1. Introduction In the supercritical fluids (SCF) the density fluctuation is macroscopic near the critical point but it becomes microscopic when one moves away from the critical point. The SCF provides a unique stage for studying the interaction of macroscopic properties, and the variation in their scale, with microscopic charged species. Supercritical rare gas systems, in particular, are more suitable for the study to single out the effect, since they are composed of atoms. This overview is about our studies carried out in the last 10 years and focuses on transport phenomena of electrons and ions, and electron reactions in supercritical rare gas fluids.
2. Electron mobility 2.1. Xe. fast transients and hot electron effects The current due to the motion of electrons injected by pulse irradiation, drifting in the electric field E, is E-mail address: [email protected]. 0969-806X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2007.02.004
given by
i ¼ uD e=d
Z
d
ne ðx; tÞ dx,
(1)
0
where uD is the drift velocity, ne the concentration of electrons and d the electrode spacing. The Ramsauer–Townsend (R–T) minimum in momentum transfer crosssection is known to exist for heavy rare gas systems (McDaniel, 1989)even in the liquid phase (Gushchin et al., 1982). In Xe low-density gas the minimum is located at ca. 0.8 eV, but shifts to lower energy with increasing density in the liquid. The fast pulse conductivity method allows one to observe transient signal at a very short time domain. In supercritical xenon in the lower density region, very peculiar current traces were observed immediately after irradiation, i.e., first going positive and then negative (Holroyd et al., 2003). The negative current signal was predicted by McMahon and Shizgal (1985) and was first observed by in gaseous xenon (Warman et al., 1985). The initial positive signals were interpreted to be due to epithermal electrons with energies above the R–T minimum. Their uD is expected to behave normally with respect
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to the applied electric field. When the electrons are slowed down to the R–T minimum, the electrons moving along the direction of the electric field gain energy, while those moving against it continue to lose energy. Those electrons which have gained energy reach the steeply rising side of the cross-section curve and suffer more scattering events. Hence, they experience shorter free flight paths. Thus, those moving against the direction of the electric field contribute more to make uD negative and negative signal arises. Zero-field electron mobility. Electron mobility me in SC Xe was determined by using 60 ns X-ray pulse from a van de Graaff accelerator and from the straight portion of uD vs. E plot at the low field strength. (Fig. 1) A deformation potential model proposed by Basak and Cohen (BC) (Basak, Cohen, 1979) has been frequently used to interpret electron mobilities in rare gas systems. me ¼
pffiffiffiffiffiffi 5=2 2e_4 2p me =m h i , 5=2 3me ðkB T Þ1=2 N 2 V 0 20 kB TwT þ terms involving V 000 ; V 000 0
(2) where V0 is the energy of the bottom of the conduction band with respect to vacuum, primes are derivatives with respect to the density and m* is the electron effective mass. The BC model can predict the location of the maximum at a density where V00 ¼ 0 and the general trend in the mobility change with density near the maximum in liquid argon; however, the magnitude differs from experiments by several orders. In deformation potential theory, electrons are scattered by potential fluctuations at the bottom of the conduction band. The BC model considers potential fluctuations due solely to static density fluctuations, which are approximated by the long wavelength limit of structure factor S(0) ¼ NkBTwT,. Then, at the critical point where wT
Mobility/10-4m2V-1s-1
1000
100
10
1
0 0
2
4
6
N/1027 atoms
8
10
m-3
Fig. 1. Zero-field electron mobility in SC Xe. Experiment: J 291 K, K 293 K, n 303 K. Theory: Eq (2) – – –; 293 K, 303K; Eq (3) —— 293K, – – – – 303K. Data from (Holroyd et al., 2003).
diverges, m should become very small. Experimentally, however, no such m-values were observed (Kimura, Freeman, 1974; Huang, Freeman, 1981). Steinberger and Zeitak (1986) consider that the failure of the BC model near the critical point is due to the fact that the size of the density fluctuations represented by S(0) is too large compared with the de Broglie wavelength of electrons, and thus these fluctuations do not influence the motion of electrons. There are potential fluctuations arising from other mechanisms too, such as acoustical phonons. The BC model was modified to accommodate scattering by phonons by replacing wT with wS, as the mean square density fluctuation due to acoustic phonons is related to nkBTwS. (Ziman, 1972; Nishikawa, 1985; Ascarelli, 1986) This replacement dramatically improves the agreement between theory and experiment in the critical region and even around the maximum as well, although with the use of some adjustable parameters (Steinberger and Zeitak, 1986). When terms involving higher order derivatives of V0 are ignored, the equation is reduced to the expression used by Bardeen and Shockley (1950) for the acoustic phononlimited mobility. pffiffiffiffiffiffi 5=2 2e_4 2p me =m me ¼ (3) . 5=2 3me ðkB T Þ1=2 N 2 V 0 0 2 kB TwS In SC xenon in the lower density range, Eq. (3), using experimental values of V0’ (Altmann, Reininger, 1997) and values of electron effective mass m* (Plenkiewicz et al., 1992), was found to account for the density dependence remarkably well without having to use any adjustable parameter. However, the experimental me minimum lies lower than predicted by Eq (3). Huang-Freeman (1978) invoked quasilocalization of electrons on quasidroplets to explain unusually low values. The magnitude of the mean free path calculated from the me minimum of 4.25 cm2/Vs (Fig. 1) is about half the average atomic distance in the bulk. This comparison suggests there is indeed some kind of localization. Path integral Monte Carlo studies of Berne and co-workers showed that the electron in Xe exists in extended states at all densities, but prefers high density regions with lower V0 values in the fluid (Coker et al., 1987). Thus, when the spatial extent of the density fluctuations becomes comparable to the de Broglie wavelength of the thermal electrons, electrons are trapped by the clusters, i.e., quasidroplets by Freeman. Electrons are spread over the clusters, (Martyna and Berne, 1988) and low-density regions surrounding the clusters act as potential barriers for the electron transport from one cluster to another. It seems reasonable to attribute deviations of electron mobility from the theoretical curve to this effect. Kr (Nishikawa et al.) Fast transients and hot electron effects. Current traces obtained at short times after a 30 ps X-ray pulse at 2.5 MPa behave normally. At 1.25 MPa and below 30 V/cm, there is a small positive peak prior to major current signals, but in contrast to SC Xe no negative signal.
ARTICLE IN PRESS M. Nishikawa / Radiation Physics and Chemistry 76 (2007) 1225–1228
The absence of negative transient current in SC Kr may be due to the fact that the wall of the R–T minimum is not steep enough to give rise to the predominant contribution of backward moving electrons as in Xe. Zero-field electron mobility. At 231 K me–value reaches a minimum of 25.3 cm2Vs-1 at 7.025 MPa (3.81 1027 atoms m-3). Similar to the case of Xe, the minimum lies at a density of approximately one-half of the critical density. Equation (3) with V0’-values and values of effective mass m* can also account for the density dependence around the minimum without any adjustable parameter. Applicability of the model. The magnitude of me in SCF around the critical density is much larger in ethane than in Xe, which is just the opposite of the trend in the liquid phase. The model can explain why the order in electron mobility in this density range is me (ethane)4 me (Kr)4 me (Xe). It is attributed to the difference in V00 2 wS in Eq. (3), the dominant factor being V00 which is largest for Xe, smaller for Kr and quite small for ethane. Ion mobility. Ion mobility, mi, values on all SCF measured are on the order of 10–3 cm2V1 s1 and decrease with increasing pressure (Itoh et al., 2001). In all cases when temperature is near the critical temperature, there is small but sharp minima at the pressure where wT has its peak. Ions are known to form large clusters in SCF by means of charge-induced dipole interactions, and their size strongly depends on the magnitude of wT (Itoh et al., 2001; Holroyd et al., 2005). The size of clusters drifting in SCF can be estimated from mi values by means of a hydrodynamic compressible continuum (HCC) model (Itoh et al., 2001). The HCC model is similar to the Stokes equation but uses the local viscosity estimated from local density buildup due to the ion electric field. The HCC model leads to the cluster radii rc maxima reaching 1 nm at mi minima in SC Xe and ethane. Electron attachment reaction. Reactions with pyrazine, pbenzoquinone, C2F4, C6F6, CCl4, CO2 and O2 were studied in SC Xe. In addition, reactions with NO, pyrimidine, methylpyrazine and styrene were studied in SC ethane. Rate constants for attachment, ka, in SC Xe to CO2, C2F4 and p-benzoquinone, were found to be o5 1011 m1 s1 (Holroyd et al., 2003). All small molecules, such as NO, CO2, O2 and C2F4., react slowly (o1 1012 m1s1) (Holroyd et al., 2005). The reactions with C6F6 and with CCl4 are extremely fast in SC Xe. The rate constant reaches almost 1 1015 m1s1 at pressures above 6.5 MPa at 293 K. Pyrimidine, pyrazine, and methylpyrazine react with an intermediate rate of around 1013 m1 s1. These reactions are reversible; detachment rate constants kd are below 108 s1 and decrease with pressure (density) increase. The attachment rate in SCF is characterized by a sharp increase at the pressure where wT reaches the peak. The magnitude of ka is not related to the electron affinity EA(eV) of respective solutes. The rate is to be understood in terms of matching between the V0-level of the fluid and energy required for the excess electron to hop vertically onto the unoccupied redox level of the solute molecule in
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the fluid (Henglein, 1977). For example, the level for small molecules like O2 is pulled down so deeply in the fluid by the polarization energy of the occupied level (negative ion state) from the gas phase value, there is no matching even for SC Xe with very low V0-level (-0.2–0.8 eV). Activation volume Va* and reaction volume change DVr for X þ e qf Ð X Va* and DVr are experimentally given by V a ¼ RTðq ln ka =qpÞT , DV r ¼ RTðq ln K=qpÞT ,
ð4Þ
where K is the equilibrium constant. They are the difference between the partial molar volumes V of products and reactants. V a ¼ V ðX Þ V ðX Þ V ðe qf Þ, DV r ¼ V ðX Þ V ðX Þ V ðe qf Þ.
ð5Þ
Excess electrons in SC ethane and Xe are quasifree, so that V ðe qf Þ ’ 0. Generally, both Va* and DVr show sharp minima at the pressure of wT peak. For X ¼ C6F6 in SC Xe, the minimum Va* value is -28 dm3/mol at 293 K (Holroyd et al., 2003). Very large negative DVr, –45 dm3/mole, was observed for the equilibrium reaction with pyrazine in SC ethane (Holroyd et al., 2000). These minima indicate that the volume changes are associated with the electrostriction volume Vel. Values of Vel calculated by the compressible continuum (CC) model that uses the local wT in the cluster (Nishikawa et al., 1998) can predict DVr for the electron attachment to pyrazine quite well (Fig. 2), and the predicted value indicates that the partial molar volume of neutral pyrazine in sc ethane is nearly zero. In contrast, DVr for the same reaction in SC xenon is small throughout, negative in the lower pressure region and become positive in the vicinity of wT maximum (Holroyd et al., 2007). Above the maximum they are practically zero, which means no further shift in K with pressure. Eq. (5) suggests that V for pyrazine ions and molecules in SC Xe are nearly equal and negative, indicating the formation of neutral clusters. Vel calculated by the CC model was found to agree with Va* for C6F6 reaction in SC Xe including the minimum above the pressure of wT peak. The agreement indicates that the transition state is close to the final ionic state, which is reasonable for molecules such as C6F6. The CC model can also explain the free energy change DGr for the equilibrium reactions. From thermodynamic cycles, DGr ¼ -EA+Pcc—V0. DGr for pyrazine in SC ethane is approximately given by DGrI Pcc—V0 , as EA is small, where Pcc is the polarization energy calculated by the model using local dielectric constants (Holroyd et al., 2000). It is concluded that the compressibility, both isothermal and adiabatic, thus density fluctuations, plays profoundly important roles in determining the behavior of charged species in the SCF.
ARTICLE IN PRESS M. Nishikawa / Radiation Physics and Chemistry 76 (2007) 1225–1228
1228
109
K/m-1
108
107
106 0
Volume/L mole-1
-10 -20 -30 -40 -50 -60 4
5
6
7
Pressure/MPa Fig. 2. Equilibrium constants and reaction volume changes DVr for e+pyrazine!pyrazine in SC ethane. Experiment: m 306K, J 310K, K 318K. Theory; CC model; —— 306K, - - - - - - 310K, – – – – 318K, Drude-Nernst equation (Drude, Nernst, 1894), – – – 310K. Data from (Holroyd et al., 2000).
Acknowledgments These researches are borne out of our (M.N. and R.A. Holroyd) long-standing project. Most part of these studies have been carried out at Brookhaven National Laboratory and supported under contract DE-AC02-98-CH10886 with US Department of Energy. We thank all the scientists at Brookhaven and University of Tokyo who participated in the project during the course of study, particularly, Kengo Itoh for his contribution to the theoretical part. We also appreciate financial support from Ministry of Education, Culture, Sports, Science and Technology and from Japan Society for Promotion of Science. References Altmann, K.N., Reininger, R., 1997. Density dependence of the conduction band minimum in fluid krypton and xenon from field ionization of (CH3)2S. J. Chem. Phys. 107, 1759–1764.
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