J. Phys. Chem. B 2004, 108, 10177-10184
10177
Negative Charge Transport in Gaseous, Supercritical, and Liquid Carbon Dioxide Kengo Itoh,* Azusa Muraoka, Kazuo Watanabe, and Takashi Nagata Department of Basic Science, Graduate School of Arts and Sciences, UniVersity of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
Masaru Nishikawa Faculty of Engineering, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Astugi 243-0292, Japan
Richard A. Holroyd Chemistry Department, BrookhaVen National Laboratory, Upton, Long Island, New York 11973-5000 ReceiVed: March 1, 2004; In Final Form: April 26, 2004
The mobility of one of two types of fast moving negative charge carriers was measured by a time-of-flight method over a wide range of the solvent density F in gaseous, supercritical, and liquid carbon dioxide. Charge carriers are formed by laser photoinjection into neat CO2. At low densities below 2 mol/L, the mobility decreases inversely proportional to F6.3. The mobility becomes minimum near the critical density and increases at higher densities. The location of the mobility minimum approximately coincides with that of the isothermal compressibility maximum. The mobility increase at higher densities is understood as the result of the onset of the hopping electron transport or of the formation of the conduction band. Both of the hopping and the equilibrium (the conduction band) models are consistent with the mobility behavior observed.
1. Introduction Carbon dioxide is known to have a negative electron affinity (EA) -0.6 ( 0.2 eV1 and does not attach thermal electrons in low density gases. Yet in the studies of the gas density dependence of electron transport, the electron drift mobility was found to decrease abruptly at higher densities,2-5 and the involvement of cluster ions (CO2)n- in the charge transport was suggested.2,4,5 Warman et al.4 also reported the current spike right after the irradiation by 5 ns X-ray pulses, which can be interpreted as the result of an electron attachment-detachment reaction
e- + (CO2)n a (CO2)n-
(1)
Jacobsen and Freeman5 made further detailed transient conductivity studies of high-pressure gaseous CO2 over a wide range of density and temperature and deduced that an average of about 6 molecules are involved in the electron attachment process (1). Cluster anions (CO2)n- can be prepared via electron attachment in supersonic molecular beam experiments and are stable against autodetachment or dissociation at least for 2 ms for n ) 2 to 6.6,7 DeLuca et al.8 reported the photoelectron spectroscopy of (CO2)n- clusters for n ) 2-13 and found out that the vertical detachment energy (VDE) shows a sharp discontinuity at n ) 6. Tsukuda et al.9 extended the study to n ) 16 and found another discontinuity from n ) 13 to 14. These changes in VDE were interpreted as due to the core switching. The core of the cluster ions for n ) 2-5 and for n > 13 is the dimer (CO2)2-, whereas that for n ) 7-13 is the monomer CO2-. The cluster where n ) 6 is a special case where both conformations coexist. Saeki et al.10 later confirmed this interpretation by ab initio calculations; the most stable form of the isomers for n ) 3-6 was determined to be C2O4-(CO2)n-2. Recently, Shkrob and Sauer11 reported interesting findings in their transient photoconductivity experiments of solutions of * To whom correspondence should be addressed. E-mail: ckengo@ mail.ecc.u-tokyo.ac.jp.
photoionizable molecules (benzene, anthracene, and methylcyclohexane) in supercritical (sc) CO2. They found two different kinds of fast moving negative charge carriers. One is the quasifree electron, which is extremely mobile with mobilities over 10 cm2/Vs and short-lived; this is converted to the second type, the solvent anion, within 200 ps. The reported values of the mobility of the solvent anion, 0.008- 0.05 cm2/Vs under their experimental conditions (F ) 8.6-19.3 mol/L and T ) 41-65 °C) are 2∼10 times higher than those of other ions in sc CO2. Interestingly, the mobility increases with the solvent density, whereas other ions have the opposite tendency. The electron photodetachment spectrum of the solvent anion showed similar features to those of (CO2)n- clusters for n ) 6∼9, with the onset at 1.76 eV and a shoulder at 2.82 eV. Shkrob et al.12 more recently showed the solvent anion has the same optical absorption spectrum at 400∼500 nm in the pulse radiolysis experiments. The suggested mechanism of charge transport is hopping between neighboring clusters. Shkrob and Sauer11 reported the solvent anion mobility only in a limited density range where CO2 is strictly supercritical (P > Pc ) 7.38 MPa and T > Tc ) 31.1 °C). Considering the fact that cluster ions play an important role in charge transport in the gas phase at high pressures, there is a need to extend the study to lower densities continuously. Also, there is a good reason to confirm their mobility results by different measurements, as their method involved an extensive parameter fitting and was indirect in that the mobilities of ions originated from electronegative molecules (SF6, CCl4, and O2) added along with photoionizable molecules were required in order to deduce the solvent anion mobility. The reason they failed to determine the mobility at lower densities is that impurity anions had close mobility values and it was extremely difficult to evaluate all of the parameters. We report here the measurement of the mobility of negative charge carries by a time-of-flight method over a wide range of density in gaseous, supercritical, and liquid carbon
10.1021/jp049073l CCC: $27.50 © 2004 American Chemical Society Published on Web 06/22/2004
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Figure 1. Photoconductivity cell.
dioxide, in the hope of arriving at a more comprehensive understanding of the role of cluster ions in charge transport in this system. 2. Experimental Section The stainless steel conductivity cell sketched in Figure 1 is designed to withstand up to 20 MPa as well as to be vacuum tight. The cell body is a cubic shaped block with two holes drilled perpendicularly. Four flanges are sealed with C-seals obtained from Eagle Industry. The fourth harmonic 266 nm light pulses from the Nd:YAG laser (Spectra Physics, GCR-12S, 4∼5 ns fwhm, operated at 10 Hz) are introduced through the 10 mm thick quartz window. The photo cathode mounted on an Al holder is the thin (≈20 nm) Al film evaporated onto a quartz disk (1 mm thick, 15 mm in diameter). The area hit by the laser pulses is limited to 6 mm in diameter by the opening of the flange in order to avoid the effect of fringing field. The anode is also made of Al to ensure the uniformity in temperature. Although not shown in the figure, the electrodes are separated by d ) 1.90 mm with Macor spacers. Temperature was monitored by a sheath type Pt thermometer right behind the anode. Electrical contact is made through the Ceramaseal feedthroughs welded on the bottom flange. The 1/4 in. o.d. stainless steel tube welded onto the top flange is used for the gas inlet/ outlet and is connected to the main shut-off diaphragm valve (Nupro, SS-DLV51-VCR4) as well as to the pressure transducer (Kyowa Dengyo, PHS-200KA) and to a 75 cm3 stainless sidearm cylinder through two diaphragm valves in series. The signal from the pressure transducer was fed to an instrumentation amplifier (Kyowa, WGA-710A), and the pressure was read out to 0.01 MPa. The sidearm cylinder and the two diaphragm valves were conveniently used to condense the sample gas into the cell and to adjust the amount of the gas in the cell body. The portion of the cell including the cell body, the main shutoff valve, the pressure transducer, and one of the two diaphragm valves were placed in an Al thermostat oven and the temperature was controlled within 0.1 °C. The sidearm cylinder and the other diaphragm valve were placed outside of the oven. The cell was pumped to 1 mPa at 150 °C prior to filling with the sample. In our time-of-flight measurements, a negative voltage (V) is applied to the photo cathode from a high voltage power supply (Fluke 412B) and the cathode is hit by light pulses from behind to inject photoelectrons into neat CO2. The laser power was kept below 0.6 mJ/pulse to avoid the space charge effect. The electrons are converted to solvent radical anions in 200 ps in
Figure 2. Typical current traces observed: (a) 34.6 °C, 5.23 MPa, 2.88 mol/L, 2000 V, td ) 1.32 ms, µ ) 0.0137 cm2/Vs, average ) 1000 times, laser power ) 0.3 mJ/pulse, peak current ) 3.73 nA total charge ) 4.92 pC, (b) 34.6 °C, 8.52 MPa, 14.26 mol/L, 2000 V, td ) 2.31 ms, µ ) 0.00783 cm2/Vs, average ) 1000 times, laser power ) 0.3 mJ/pulse, peak current ) 2.38 nA, total charge ) 5.50 pC.
sc CO2. The negative charge carriers, anions or electrons, drift to the anode under the electric field E ) V/d. The transient current was averaged (typically 1000 times) and recorded with a digital oscilloscope (LeCroy 9310) after the amplification by a handmade current sensitive amplifier, which has a sensitivity of 108 V/A and a rise time of 10 µs. Ideally the current trace observed should be of a square shape with the height i ) n0eVµ/ d2 and the duration td ) d2/µV. Here n0 and µ are the total number of electrons injected and the mobility, respectively. From the drift time td, the drift mobility can be directly determined as µ ) d2/Vtd. The CO2 (99.9990%) obtained from Showa Tansan was further purified in a stainless steel vacuum/high-pressure line by degassing and storage over silica gel (Merck, extra pure) and sodium metal. The gas was first degassed by a freezepump-thaw method with liquid nitrogen several times and then condensed into a stainless steel cylinder containing silica gel which was pumped to 1 mPa at 380 °C prior to use. After being stored for 2∼3 days, the gas was transferred into another stainless steel cylinder and stored for several weeks over sodium metal. This purification procedure was repeated several times as required. After the purification, the gas was taken into the sidearm cylinder of the conductivity cell and was further degassed several times. The cell body was cooled to 5∼10 °C to condense the sample. Drift times as long as 2.5 ms were obtained in sc CO2, which is better than the lifetime of 11 µs reported by Shkrob and Sauer11 by more than 200 times. Thermophysical properties of CO2 were calculated based on an equation of state given by Ely.13 The critical constants (Tc ) 304.21 K, Pc ) 7.384325 MPa, Fc ) 10.6 mol/L) were also taken from ref 13. The dielectric constant is from ref 14. The viscosity was estimated by the Chung method.15 3. Results At low densities, observed current traces are almost square shaped (trace a in Figure 2) and drift times were easily determined. The short lasting tail is understood as being due to the electron attachment reactions to impurities. At high densities, the effect of impurities is more prominent (trace b) but the breaks
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Figure 3. Electric field dependence of drift velocity; open circle: 34.6 °C, 14.62 MPa, 18.49 mol/L, µ ) 0.0147 cm2/Vs; open square: 34.6 °C, 5.59 MPa, 3.20 mol/L, µ ) 0.0118 cm2/Vs; open triangle: 34.6 °C, 7.92 MPa, 9.42 mol/l, µ ) 0.00403 cm2/Vs.
Figure 5. Bottom: mobility of negative charge carriers as a function of the solvent density at intermediate and high densities. Open reversed triangle, 25.0 °C; open diamond, 30.0 °C; open circle, 34.6 °C; open square, 40.0 °C; open triangle, 45.0 °C. Error bars are shown for the points where errors of 3 to 5% are involved. Filled points were from ref 11. Filled square, 41 °C; filled triangle, 48.5 °C. Solid line: anion (Cl-, F-, and CO4-) mobilities from ref 11 at 41 °C. Dotted line: anion mobilities at 40 °C from ref 16. Dashed line: cation mobilities at 40 °C from ref 16. Top: isothermal compressibility.
Figure 4. Mobility of negative charge carriers as a function of the solvent density at low densities. Open circle, 34.6 °C; open square, 40.0 °C; and open triangle, 45.0 °C. Errors involved are less than 1% and the error bars are not shown because they are smaller than the size of the symbols. Filled points were calculated based on the equation given by Jacobsen and Freeman (ref 5) for respective temperatures.
are still clear enough to deduce drift times. In the intermediate density region, two drift times, one for electrons or solvent anions and the other for impurity anions, are too close to make out the difference, and some errors, maximum of 5%, are involved in the mobility values reported here. The amount of charge injected was on the order of a pC. All measurements were done below 11 kV/cm. The drift velocity is proportional to the electric field strength in the whole density range studied (Figure 3). Mobility values were determined at least at three different voltages and they generally agreed within 1%. A prompt signal corresponding to quasifree electrons was not observed in our measurements. Our amplifier has the rise time of 10 µs. In Figures 4 and 5, the solvent density dependence of the mobility of negative charge carriers is shown. All points are averages of data taken at several different voltages. Several
points are plotted in both figures. The agreement with the data reported by Shkrob and Sauer11 is reasonable. At 34.6, 40.0, and 45.0 °C, the pressure is lower than Pc below 6.0, 5.1, and 4.6 mol/L, respectively, and data taken there are for the gas phase. At 25.0 and 30.0 °C, measurements were done in the liquid phase at pressures higher than Pc (F > 17.2 and 14.9 mol/L, respectively). At densities below 2 mol/:, the mobility decreases with density abruptly (Figure 4). The dependence can be described by the equation
ln µ ) 34.7 - 6.3 ln F -
10 400 T
(2)
where units of cm2/Vs, mol/L, and K were used for µ, F, and T, respectively. This indicates that the apparent isochoric activation energy is almost constant in this density range at 0.89 eV. At higher densities, the mobility decreases somewhat gradually, goes through minimum near the critical density, and then increases (Figure 5). The mobility minimum shifts toward lower densities and becomes shallower as the temperature increases. The locations approximately coincide with those of isothermal compressibility maxima. The magnitudes correlate with the compressibility maxima as well. The higher the temperature, the shallower the mobility minima and the larger the compressibility maxima. At 25.0, 30.0, and 34.6 °C, the mobility tends to level off at high densities as pointed out by Shkrob and Sauer.11 Although the measurement at higher densities was not carried out at 40.0 and 45.0 °C, there seems to be a similar tendency.
10180 J. Phys. Chem. B, Vol. 108, No. 28, 2004
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4 TABLE 1: Parameter Values for µ ) ∑i)0 biGi
T [°C]
25
Fmin Fmax b0 b1 b2 b3 b4
30
34.6
40
45
17.3 19.8
15.2 19.6
11.5 19.3
9.9 17.6
7.7 16.9
1.692101 × 100 -3.558121 × 10-1 2.780959 × 10-2 -9.525218 × 10-4 1.205647 × 10-5
5.744395 × 10-1 -1.226065 × 10-1 9.512449 × 10-3 -3.109094 × 10-4 3.568992 × 10-6
-1.738471 × 10-1 5.322271 × 10-2 -6.032781 × 10-3 3.033051 × 10-4 -5.564735 × 10-6
-7.833847 × 10-2 2.996782 × 10-2 -3.975611 × 10-3 2.266147 × 10-4 -4.522753 × 10-6
-3.765230 × 10-2 1.872442 × 10-2 -2.863058 × 10-3 1.816355 × 10-4 -3.848675 × 10-6
TABLE 2: Apparent Isochoric Activation Energies and Pre-exponential Factors for µ ) D exp(-Ea/kT), µ - µa ) D exp(-Ea/ kT) (eq 10) and (µ - µa)T ) D exp(-Ea/kT) (eq 11) at High Densitiesa µ ) D exp(-Ea/T) F [mol/L] 13 14 15 16 17 18 19 a
eq 10
D ) ν0λ2/6k
eq 11
Ea [eV]
D [cm /Vs]
Ea [eV]
D [K cm /Vs]
Ea [eV]
ν0 [s ]
ν [s-1] at 40°C
µa [cm2/Vs]
1.01 × 106 2.11 × 106 2.97 × 106 2.05 × 106 1.19 × 106 2.53 × 105 1.27 × 105
-0.50 -0.52 -0.52 -0.50 -0.49 -0.44 -0.42
3.58 × 1010 2.37 × 109 4.38 × 108 1.64 × 108 5.56 × 107 1.49 × 107 2.78 × 106
-0.80 -0.71 -0.66 -0.63 -0.59 -0.55 -0.51
2.58 × 109 2.26 × 108 4.07 × 107 1.49 × 107 4.47 × 106 1.14 × 106 2.35 × 105
-0.82 -0.74 -0.69 -0.66 -0.62 -0.58 -0.54
1.51 × 1024 1.32 × 1023 2.38 × 1022 8.69 × 1021 2.62 × 1021 6.67 × 1020 1.38 × 1020
9.48 × 1010 1.34 × 1011 1.81 × 1011 2.32 × 1011 2.83 × 1011 3.18 × 1011 3.28 × 1011
0.0029 0.0027 0.0026 0.0025 0.0023 0.0021 0.0019
2
2
-1
ν ) ν0eEa/T
/Vs]
D
[cm2
Cation mobilities from ref 18 were used for µa. Values of ν0 are for λ ) 0.94 nm.
4. Discussion In dense gases, van der Waals clusters are formed
n(CO2) a (CO2)n
(3)
and the electron attachment-detachment reaction 1 occurs. The equilibrium is established in a short time compared with drift times.4,5 The reactions 1 and 2 can be thought of as an equilibrium between free and attached electrons5
e- a ea-
(4)
The Gibbs free energy, the enthalpy, and the entropy changes of this reaction (∆G, ∆H, and ∆S) are for the overall reaction. Both the cluster formation (3) and the electron attachment (1) contribute. The observed mobility is the time average of those of free electrons µf and of cluster ions, µa
1 K µ ) µf + µa 1+K 1+K Figure 6. Plots of µ versus 1000/T. Open circle, 13 mol/L; open triangle, 15 mol/L, open square, 17 mol/L; and open reversed triangle, 19 mol/l.
We previously reported transient conductivity measurements of supercritical ethane, xenon, and carbon dioxide irradiated by pulsed X-rays.16 In sc CO2, two types of ions with different mobilities were found, and we assigned the faster species to be negative ions. The results at 40 °C are shown in Figure 5 by a dotted line. They rather agree with mobilities of other negative ions.11 Considering the fact that an extensive effort in the purification was required in order to observe current traces as shown in Figure 2, it is concluded that what we reported is the mobility of impurity anions as pointed out by Shkrob and Sauer.11 Mobility values at high densities were fitted to and interpolated by fourth order polynomials. Only the data with errors less than 1% were used. The values of the parameters used are given in Table 1. The apparent isochoric activation energy Ea is in the range 0.42-0.52 eV (Table 2 and Figure 6) in agreement with 0.46 eV obtained by Shkrob and Sauer.11
(
K ) exp -
(5)
∆S ∆H ∆G ) exp kT k kT
)
(
)
Jacobsen and Freeman5 succeeded in reproducing the temperature dependence of the mobility, which changes nearly 3 orders of magnitude by using temperature independent parameter values for µf, ∆H, and ∆S at 0.52, 0.93, and 2.11 mol/L. Cation mobilities were used for µa. The values of µaF, µfF, ∆H, and ∆S were only weakly dependent on density. For example, µf [cm2/Vs] × F [mol/L] ) 28, 25, and 22 at 0.52, 0.93, and 2.11 mol/L, respectively. Filled points in Figure 4 were calculated from their data using their parameter values. At 40.0 °C, K is 0.0067, 2.3, and 310 at 0.52, 0.93, and 2.11 mol/L, respectively. If the first term is much greater than the second and if K is large, eq 5 can be approximated as
µ≈
µf ∆S ∆H ∆G ) µf exp ) µf exp + K kT k kT
( )
(
)
(5a)
Our isochoric activation energy of 0.89 eV is in agreement with -∆H (0.84, 0.91, and 0.99 eV at 0.52, 0.93, and 2.11 mol/L,
Negative Charge Transport in CO2
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respectively) used by Jacobsen and Freeman.5 The F-6.3 dependence of the mobility indicates an average of 5∼6 molecules are involved in the electron capture process. Note that in dilute gases µ is inversely proportional to F. The equilibrium (4) is shifted to the right as the density increases mainly due to the increase of van der Waals clusters. The core of cluster ions would be dimers. The linear relation shown in Figure 4 does not extend to lower densities where K is small, and the approximation (5a) is not valid. It is possible to evaluate the EA of clusters by using the data reported by Warman et al.,4 because the decay of the current spikes they observed is a measure of the rate (ka) of electron attachment (1). The lifetime of free electrons (τe) is about 10 ns at 16.1 atm and 19 °C (F ) 0.74 mol/L) from their Figure 1. Thus, ka[(CO2)n] ≈ 108 s-1. They also reported both µobs and µf (5.34 and 23.7 cm2/Vs, respectively) at the same pressure and temperature. The diffusion controlled rate 4πRµf/kT is 4.5 × 1014 M-1 s-1 if a reaction radius of 1 nm is assumed. Combining these two numbers yields [(CO2)n] equal to 2 × 10-7 M at 16.1 atm and 19 °C. The free energy is calculated by eq 5a from µobs and µf as -0.037 eV for 19 °C. The Gibbs free energy change associated with the electron localization process (4) may be given by other quantities as17
Nf Nt
∆G ) -EA + P- - V0 + kTln
(6)
where Nf ) 2(2πmkT/h2)3/2 is the effective density of states of free electrons, Nt is the number density of electron traps and equal to [(CO2)n] in this case, P- is the polarization energy, and V0 is the conduction band energy. According to Shkrob and Sauer,11 the solvent anion in sc CO2 does not react with N2O which has an EA of 0.22 eV. As cluster ions have larger radii than solute molecules, they are less stabilized than molecular ions in sc CO2. A lower limit of EA is thus 0.22 eV. The isochoric activation energy of 0.89 eV contains both contributions from the electron attachment (1) and the cluster formation (3). The latter must be exothermic. Thus, 0.89 eV is an upper limit of EA. Equation 6 allows us to estimate EA if both P- and V0 are known. The polarization energy can be estimated with the Born equation. If a radius of 0.47 nm used by Shkrob and Sauer11 is assumed, P- ) -0.025 eV for F ) 0.74 mol/L. For many hydrocarbons, V0 was found to decrease linearly with density at low densities,18-22 V0 ) -BF. The slope (B) is 0.047 eV L/mol for ethane20 and 0.089 eV L/mol for n-pentane.23 We tentatively use 0.047 eV L/mol for ethane. Then V0 ) -0.035 eV at F ) 0.74 mol/L. kT ln (Nf/Nt) is calculated to be 0.307 eV for Nt ) [(CO2)n] ) 2 × 10-7 M at 16.1 atm and 19 °C. EA is thus estimated to be 0.35 eV. Considering the rough approximations and estimates made in the process of derivation, an error of 0.1 eV is possible. The values of ∆G can be also obtained from our data (eq 2) by using µf given by Jacobsen and Freeman5
ln K ) -31.5 ( 0.1 + 5.3 ln F +
10 400 ) -∆G/kT T
Equation 6 allows us, in turn, to estimate Nt(F,T). The results are shown in Table 3. Although these estimates of Nt can be off by orders, both the magnitude and the tendency that Nt is larger at higher densities or at lower temperatures seem to be reasonable. The concentration of electron traps is very low at these densities. At higher densities above 2 mol/L, the equilibrium (4) is further shifted to the right and electrons are attached most of
TABLE 3: Estimation of the Number Density of Electron Attaching Clusters at Low Densitiesa F T [mol/l] [°C] 1 1 1 1.5 1.5 1.5 2 2 2 a
34.6 40 45 34.6 40 45 34.6 40 45
K
∆G [eV]
P[eV]
V0 [eV]
Nf [M]
Nt [M]
2.29 1.71 1.19 4.44 3.86 3.34 5.97 5.38 4.86
-0.061 -0.046 -0.033 -0.118 -0.104 -0.091 -0.158 -0.145 -0.133
-0.034 -0.034 -0.034 -0.051 -0.051 -0.051 -0.067 -0.067 -0.067
-0.047 -0.047 -0.047 -0.071 -0.071 -0.071 -0.094 -0.094 -0.094
0.043 0.044 0.045 0.043 0.044 0.045 0.043 0.044 0.045
1.3 × 10-6 9.2 × 10-7 6.8 × 10-7 1.4 × 10-5 1.0 × 10-5 7.4 × 10-6 8.6 × 10-5 6.1 × 10-5 4.5 × 10-5
EA of 0.35 eV was assumed.
the time. The second term in eq 5 is not negligible any more. The mobility becomes comparable with that of impurity anions, which we believe to be a reasonable estimate of µa. At 3.73 mol/L and 45.0 °C, for example, µobs ) 0.0124 cm2/Vs, whereas µimp ) 0.0099 cm2/Vs. If µf ) 10 cm2/Vs is assumed and if µimp is used for µa, K becomes 4000 from eq 5. The mobility further decreases with density toward the minimum, closely following those of other ions (Figure 5). Although we currently do not have any direct information on µa, it would be natural to expect that µa behaves similarly to µimp. At densities above 12 mol/L, the mobility increases, in contrast to that of other ions for which the mobility decreases corresponding to the increase in viscosity. The locations of the mobility minima approximately coincide with those of the isothermal compressibility maxima (Figure 5). It was also repeatedly pointed out in our studies of electron reactions in supercritical fluids23-26 that the volume change associated with electron attachment reactions is negative and becomes minimum where the isothermal compressibility is maximum. The isothermal compressibility is related to the density fluctuation by27
〈∆N2〉 ) NkTχT N where N is the number of molecules in a unit volume. The dominant factor here is χT. The density fluctuation experimentally determined from small-angle X-ray scattering experiments by Nishikawa and co-workers28-32 behaves similarly to the isothermal compressibility as expected. It is maximum at densities slightly lower than the critical density and the location shifts toward lower densities as the temperature increases (Figure 4 of ref. 31). The magnitude is less at higher temperatures as well. The same correlation with the mobility minima is present in the density fluctuation as in the isothermal compressibility. The density fluctuation forms a ridge in the critical region.28-32 Nishikawa and co-workers31,32 pointed out that the ridge divides the supercritical state into two regions, liquidlike and gaslike; the behavior of many properties changes depending on which side of the ridge the system is in, including partial molar volumes, sound velocity, thermal conductivity, solubilities, and rates or equilibrium constants of various chemical reactions. Where the isothermal compressibility is maximum is understood as the point where the repulsive and attractive interactions between solvent molecules are balanced. It will be easier for molecules to reorganize local structures without paying much cost in energy. At lower densities, the attractive force prevails and molecules will tend to aggregate. Although the detailed mechanism is not clear at this point, it is likely that as the density of CO2 is increased and the system becomes liquidlike, a new channel of charge transport opens up.
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Two possible transport mechanisms discussed by Shkrob and Sauer11 in sc CO2 are thermal activation to the conduction band and hopping between neighboring clusters. In the former, the equilibrium similar to (1)
eqf- + (CO2)n a (CO2)nis assumed and the transport is via thermal activation to the conduction band. Here eqf denotes quasifree electrons in the conduction band. The mobility is given by
1 K + µa µ ) µqf 1+K 1+K where K is the equilibrium constant of the reaction
eqf- a eaThese equilibria and the mobility expression are formally the same as (1), (4), and (5) except that free electrons are replaced by quasi-free electrons. We refer to (1), (4), (5), (5a), and (6) for the equilibrium model without notice hereafter. When K is large but the second term is not negligible, the mobility expression (5) is approximated as
µqf 1 K + µa ≈ + µa ) µ ) µqf 1+K 1+K K
(
µqf exp -
∆S ∆H + + µa (7) k kT
)
In the latter hopping model, electrons hop between neighboring clusters without going through the conduction band
(CO2)n- + (CO2)n f (CO2)n + (CO2)n-
(8)
The mobility is given by33,34
µ)
eλ2ν + µa 6kT
( )
ν ) ν0 exp -
Ea kT
(9)
where the mobility of cluster ions µa is added in order to take into account the drift while electrons are attached on clusters. Here λ and ν are the hopping distance and the hopping frequency, respectively, ν0 is the attempt frequency, and Ea is the barrier height. In both the equilibrium and the hopping models µa, which is not known at present, is required for analysis. We shall use cation mobility values at 40 °C we previously reported16 for µa regardless of temperature. According to Shkrob and Sauer,11 the activation energy for mobilities of ions is less than 20 mV. Plots of µ-µa or (µ-µa)T versus 1/T give straight lines (Figure 7). Parameter values for
( ) Ea kT
(10)
( )
(11)
µ - µa ) D exp and
(µ - µa)T ) D exp -
Ea kT
are given in Table 2. These activation energies are slightly higher than those without the correction term µa.
Figure 7. Plots of (µ-µa) versus 1000/T (bottom) and (µ-µa)T versus 1000/T (top) at open circle, 13 mol/L; open triangle, 15 mol/L; open square, 17 mol/L; and open reversed triangle, 19 mol/L. Cation mobilities at 40 °C from ref 16 were used for µa.
The hopping model qualitatively explains the increase of the mobility with density. As the density is increased, the average distance between clusters become shorter, and it is easier for electrons to hop. In fact, the hopping frequency, which was calculated assuming the constant hopping distance of 0.94 nm ) 0.47 nm × 2, increases and the activation barrier decreases with density as shown in Table 2. Hopping transport was also reported for benzene, toluene,35 and o- and m-xylenes36 under high pressures as well as for liquid C6F637 and CS238 (a close analogue of CO2) at normal pressure. The magnitudes of the hopping mobility, 0.08 (benzene), 0.06 (toluene), 0.04 (oxylene), 0.06 (m-xylene), 0.018 (C6F6), and 0.0078 cm2/Vs (CS2) at room temperature are quite comparable with the mobility values obtained here. The hopping frequencies, 2.4 × 1012 (benzene, λ ) 0.66 nm), 2.0 × 1012 (toluene, λ ) 0.70 nm), 1.3 × 1012 (o-xylene, λ ) 0.70 nm) and 2.1 × 1012 s-1 (mxylene, λ ) 0.71 nm) at room temperature are similar, too. However, there is a distinct difference in the activation barrier height. It is 0.11 eV for C6F6, 0.12 eV for benzene, 0.13 eV for toluene, 0.15 eV for o-xylene, 0.14 eV for m-xylene and 0.06 eV for CS2, whereas it is more than 0.54 eV for sc CO2. The attempt frequency is also higher than 2.6 × 1014 (benzene), 3.1 × 1014 (toluene), 5.1 × 1014 (o-xylene), and 5.4 × 1014 s-1 (m-xylene), corresponding to the higher activation energies. We speculated the hopping process in liquid benzene under high pressures involves some kind of molecular reorientation based on the similarity of the activation energy to that for the viscosity (0.12 eV). In the case of sc CO2, the activation energy for the viscosity is almost zero. For example, the viscosity is calculated to be 785 µP at 20 °C and 786 µP at 65 °C at 19 mol/L. According to Shkrob and Sauer,11 their activation energy of 0.46 eV for the mobility is equivalent to the loss or exchange of just two CO2 molecules. They also pointed out that the hopping process may involve core switching. In the case of aromatics, an electron is trapped on a single molecule, whereas in sc CO2, it is an aggregate of molecules, which captures the electron.
Negative Charge Transport in CO2 The activation process for hopping in sc CO2 is apparently different from those in other systems. In the equilibrium model, the mobility depends on both K and µqf. Shkrob and Sauer11 estimated K is 700-7000 and the Gibbs free energy change of electron trapping is -0.14 to -0.21 eV at 18.9 mol/L and 41 °C from their mobility value assuming µqf is in the range 10 to 100 cm2/Vs. They further argued that the equilibrium (1) is not likely in sc CO2 because an unrealistically large negative entropy change less than -400 J/molK is required in order to reconcile the difference between the Gibbs free energy change of -0.14 to -0.21 eV and the heat of the reaction -1.76 eV estimated from the onset of the electron photodetachment spectrum. They referred to the gasphase value -274 J/mol K by Jacobsen and Freeman5 and -51 to -59 J/mol K in hydrocarbon solutions39 together with -80 to -90 J/mol K for gas-phase clustering (X-(CO2)n-1 + CO2 ) X-(CO2)n, X- ) halide anion, n e 5),40,41 claiming -400 J/mol K is equivalent to producing a monomer anion and binding four CO2 molecules to it and is an overestimate in sc CO2 where electron attaching clusters preexist. We would like to point out, however, that the enthalpy change associated with the reaction 4 is not -1.76 eV. The clusters must be vibrationally excited after electrons are photodetached. The energy left behind must be included in 1.76 eV as well. Thus, 1.76 eV is an upper limit of -∆E. In fact, they also reported that the electron photodetachment is possible even at 1.17 eV though the efficiency is much less. Further, an entropy change as large as -720 J/mol K is possible in supercritical fluids.26 This is because electron attachment reactions are followed by electrostriction. Solvent molecules are attracted by an ion as a result of charge-induced dipole interactions. A region of high density is formed around the ion. The effect is particularly large in highly compressible supercritical fluids. The P∆V work associated with the electrostriction is included in 1.76 eV as well. The claim they asserted does not apply. The possibility of the equilibrium model cannot be excluded. The mobility expression for the equilibrium model, eq 7, can be rewritten by using the electron lifetime τe and the rate of electron detachment kd as
µ - µa ≈
µqf ) (µqfτe)kd K
Shkrob and Sauer11 showed that the product uqf × τe increases by a factor of 3.3 from 6 × 10-10 cm2/V at 14 mol/L to 2 × 10-9 cm2/V at 18 mol/L at 41 °C (Figure 4 of ref 11). The ratio of µ-µa at respective densities at 40 °C is (0.0198-0.0021)/ (0.0101-0.0027) ≈ 2.4. Thus, the increase of µqfτe with density explains the increase of the mobility if kd decreases from 0.0077/6 × 10-10 ) 1.3 × 107 s-1 at 14 mol/L to 0.0177/2 × 10-9 ) 9 × 106 s-1 at 18 mol/L. The tendency of kd to decrease with density was also observed for electron attachmentdetachment reactions to solute molecules (CO2, pyrimidine, and pyrazine) in sc ethane.25,27 The magnitude of kd is comparable, too. Both the increase in µqfτe and the decrease in kd are responsible for the changes in the mobility with density. Only the former explains the mobility increase. On the other hand, µqfτe increases only by 33% for a 24 °C temperature increase from 1.1 × 10-9 at 41 °C to 1.5 × 10-9 cm2/V at 65 °C at 16 mol/L from Figure 4 of ref 11, while µ-µa nearly doubles over a 10.4 °C temperature increase from 0.011 to 0.0025 ) 0.0095 cm2/Vs at 34.6 °C to 0.0207-0.0025 ) 0.0182 cm2/Vs at 45.0 °C. Most of the temperature dependence is attributed to kd. The quantity µqfτe/µ shown by Shkrob and Sauer11 in their Figure 27S is understood as τd ) 1/kd obtained from the electron
J. Phys. Chem. B, Vol. 108, No. 28, 2004 10183 photodetachment experiments in the framework of the equilibrium model, as long as µa is negligibly small compared with µ. From the figure, it is seen that kd more than doubles from 9 × 106 at 41 °C to 2.4 × 107 s-1 at 65 °C at 16 mol/L. Thus, the density and temperature dependence of the mobility observed is consistent with the equilibrium model as well. The new channel of electron transport is the conduction band in this model. One of the important factors that determine which transport path, the hopping or the conduction band, is preferred is the energy level of the conduction band V0. If V0 is low enough, there is a fair chance for electrons to be thermally activated to the conduction band. If V0 is high, electrons would rather directly hop to the neighboring clusters without going through the conduction band. It is possible to make a rough estimate of an upper limit of V0 in the framework of the equilibrium model, because a large portion of the isochoric activation energy Ea for the mobility must be for kd and because the activation energy for kd may be given by EA - P- + V0 in the model. At 19 mol/L, Ea is 0.50 eV for the corrected mobility µ-µa and
EA - P- + V0 < 0.50 eV At 19 mol/L from the Born equation for rcluster ) 0.47 nm, Pis -0.52 eV. If EA ) 0.35 eV determined above is used, V0 must be lower than -0.37 eV at this density. The value may be too low compared with those in other fluids.18-22 However, V0 is generally a minimum at around 2Fc, which is about 21 mol/L for CO2. V0 minima are really low in high mobility fluids where the conduction band is formed; -0.38 eV for methane,42 -0.39 eV for isobutene,20 -0.52 eV for neopentane,22 and -0.61 eV for tetramethylsilane,22 but are relatively high in low mobility fluids; -0.19 eV for ethane,18 -0.27 eV for propane,19 -0.19 eV for n-butane,20 and -0.28 eV for n-pentane.21 A value of -0.37 eV is not impossible if carbon dioxide belongs to the former group. It is tentatively concluded that, if the electron transport is through the conduction band, the V0 minimum will be as low as -0.37 eV. Yet we have to note that this estimate can have an error of 0.1 eV or more and that the structure and the size of cluster ions where electrons are trapped may be different in sc CO2 from that in low-density gases. The ab initio calculations by Saeki et al.10 showed n ) 6 cluster anions have a relatively flat structure with a D2h symmetry C2O4- core and four CO2 molecules placed in the same plane. Both sides of the plane are vacant. It will be difficult to find out such configurations in high-density sc CO2. Increases in density may cause Nt to decrease. This is consistent with the equilibrium model, because the equilibrium (4) will shift to the left. However, there will be a wide range of distributions in the size and the structure of the clusters in sc CO2 and they are continuously fluctuating. Larger clusters may be responsible for electron trapping. It is clear that further studies are needed to answer the question, hopping or equilibrium, which explains the actual charge transport in sc CO2 better, and to clarify the role of cluster ions. One conceivable way to attack this problem is to measure V0. Higher V0 values than -0.37 eV estimated above makes the transport through the conduction band unlikely. Another possibility, which can be made with conventional experimental techniques, is the measurement of the mobility in diluted sc CO2 with other solvents such as Xe or ethane. If hopping is the case, the mobility will decrease significantly as the hopping distance is increased. THz pulse probe is a recent technique that measures mobility at picosecond times.43 It would measure the quasi-free electron mobility. Although it would not tell us
10184 J. Phys. Chem. B, Vol. 108, No. 28, 2004 about the equilibrium at longer times directly, the information would be quite useful for detailed examination of the equilibrium model. In summary, the mobility of one of two types of fast moving negative charge carriers was measured by a time-of-flight method over a wide density range in fluids of neat carbon dioxide. In low-density CO2, the mobility decreases with density due to the increase of the number density of van der Waal clusters which trap electrons temporarily. The mobility reaches a minimum where the isothermal compressibility is maximum and increases at higher densities. This is understood as the result of either the onset of the hopping transport or of the formation of the conduction band. Both the hopping and the equilibrium models are consistent with the mobility data at high densities. Acknowledgment. This research was partially supported by a Grand-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. K.I., M.N., and R.H. are supported by a grant from the Japanese Society for the promotion of Sciences under Japan-U.S. Cooperative Science Program, Joint Research. R.H. is supported under contract DEAC02-98-CH10886 with U.S. Department of Energy by its Division of Chemical Sciences, Office of Basic Energy Sciences. References and Notes (1) Compton, R. N.; Reinhard, P. W.; Cooper, C. O. J. Chem. Phys. 1975, 63, 3821. (2) Lehning, H. Phys. Lett. 1968, 28A, 103. (3) Allen, N. L.; Prew, B. A. J. Phys. B. 1970, 3, 1113. (4) Warman, J. M.; Sowada, U.; Armstrong, D. A. Chem. Phys. Lett. 1981, 82, 458. (5) Jacobsen, F. M.; Freeman, G. R. J. Chem. Phys. 1986, 84, 3396. (6) Klots, C. E.; Compton, R. N. J. Chem. Phys. 1977, 67, 1779. (7) Klots, C. E.; Compton, R. N. J. Chem. Phys. 1978, 69, 1636. (8) DeLuca, M. J.; Niu, B.; Johnson, M. A. J. Chem. Phys. 1988, 88, 5857. (9) Tsukuda, T.; Johnson, M. A.; Nagata, T. Chem. Phys. Lett. 1997, 268, 429. (10) Saeki, M.; Tsukuda, T.; Nagata, T. Chem. Phys. Lett. 2001, 340, 376. (11) Shkrob, I. A.; Sauer, M. C. J. Phys. Chem. B 2001, 105, 4520. (12) Shkrob, I. A.; Sauer, M. C.; Jonah, C. D.; Takahashi, K. J. Phys. Chem. A 2002, 106, 11855. (13) Ely, J. F. Supercritical Fluid Properties, Final Report. NTIS report D985011203, 1984.
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