IV. DISTRIBUTION OF ENERGY STATES FOR IONS OF REDOX SYSTEM IN SOLUTION
Tải bản đầy đủ - 0trang
24
Kohei Uosaki and Hideaki Kita
*VB = WB " iwB =
fcve4
JJo
N'£( W) W(E)DO(E) dE
- kWBc% I
Jo
Jo
N(E) W(E)DR(E) dE (43)
where fCB and fCB are the anodic and cathodic component of the
conduction band current, the rate constants of which are icCB and
fcCB, respectively. Similar notation is used for the valence band
current. N(E), N'(E), W(E), DO(E)9 and DR(E) represent the
number of occupied and unoccupied states in the semiconductor,
tunneling probability, and the distribution of occupied (reduced)
and of unoccupied (oxidized) state of redox couple, respectively,
between energy E and E + dE. E is taken as zero at the bottom
of the valence band, cR and c% are the surface concentrations of
reduced and oxidized species, respectively.
In Sections II and III we correlated the electronic energy level
of the solid phase, EF, and that of electrolyte, Vredox. This comparison is useful with regards to understanding the potential
difference between the two phases and the potential distribution
of this interface. However, to describe the kinetics of the electron
transfer reactions at the surface, it is necessary to know the energy
distributions of occupied and unoccupied levels of the semiconductor and those of the redox couple, as one can see from Eqs. (42)
and (43). It is well known87 that the energy distribution in a
semiconductor in the dark is described by Fermi statistics with a
function for density of states. The energy distributions of redox
couples are, however, still a very controversial issue.88 There are
essentially two major approaches to describe this problem. One
approach was originated by Gurney.86 In this model, the energy
distribution is considered to be due to the vibration-rotation interaction between the particular ion and solvent molecules. According
to the second model,61 the energy level of the ion fluctuates around
its most probable value due to the thermal fluctuation of solvent
dipoles.
2. Gurney's Model
Gurney86 developed a model for the neutralization of H+ and Cl~.
Let us first consider the H+ case. In vacuum the ionization potential,
Theoretical Aspects of Semiconductor Electrochemistry
I
I
-1- H
-L H
(a)
25
(b)
+
Figure 12. (a) Ionization of H to H in vacuum and electron in vacuum at oo.
Energy required is the ionization potential, (b) Ionization of H to H + in water and
electron in vacuum at oo. Energy required is (/ — W).
/, is required to complete the following process (Fig. 12a):
H -> H+ + e~
(44)
This step is analogous to that of removing an electron from the
Fermi level of the solid phase to the vacuum, the energy of which
is given by the work function,
. Thus, the ionization potential
represents the energy level of electrons of H/H + (H as an occupied
state and H+ as an unoccupied state). In an aqueous solution, ions
are hydrated and the energy required for
H -» H:q + e"
(45)
is smaller than / by the hydration energy, W9 and the energy level
of electrons of H/H + in an aqueous solution is given by I - W
(Fig. 12b). In the above consideration, it was assumed that the
mutual potential energy of the ion and adjacent water molecule
before the neutralization was — W and that after the neutralization
it was zero. This is not quite correct. Figure 13 shows the potential
energy diagram for this process. The potential energy of the neutral
state ion, i.e., H, is represented by a curve such as abc. The force
between H and H2O is repulsive at all distances. The curve for the
stable vibrational levels of H3O+ is represented by defgh. The lowest
vibration-rotation level of H3O+ is ef and the energy difference
between ef and h is the hydration energy of positive ion, H+ in
this case, which is in the lowest vibration-rotation level, Wo. The
transition representing neutralization of an ion in its lowest vibration-rotation level is on this diagram fb.t In accordance with the
t The Gerischer model, which will be discussed later, allows the transition at much
smaller H + -H 2 O distances.
26
Kohei Uosaki and Hideaki Kita
H*+H20
Distance
Figure 13. Potential energy diagram of H + + H 2 O and H + H 2 O systems.86 See text
for details.
Franck-Condon principle, bq represents the positive mutual potential energy of the molecular components before they have moved
apart. If Rn is the value of this repulsive potential energy resulting
from neutralization of an ion in its nth vibration-rotation level, the
neutralization energy of H 3 O + in its nth vibration-rotation level,
E+n , is given by
E+n = I - Wn - Rn
(46)
Thus, this energy corresponds to the following process:
(H - H 2 O) + + e~ -» H - H2O
(47)
This means that a free electron from infinity is introduced into the
solution to occupy the electron state in H 3 O + in its nth vibrationrotation level without change of the solvation structure (FranckCondon principle). Therefore, this energy corresponds to the energy
of unoccupied states, i.e., H 3 O + .
Similar arguments can be applied to Cl~ with minor
modification (Fig. 14). In this case the potential energy curve for
C1~-H2O (klmnp) lies below the axis and px is the ionization
potential of the unhydrated negative ion, Cl~. Curve a'b'c' represents the potential energy curve for H2O-C1. The lowest vibration
rotation level of C1~-H2O is Im and the energy difference between
Im and p is the hydration energy of negative ion, Cl~ in this case,
Theoretical Aspects of Semiconductor Electrochemistry
27
Figure 14. Potential energy diagram of C P + H 2 O and Cl + H 2 O systems.86 See text
for details.
which is in the lowest vibration-rotation level, Wo. From this figure
it is clear that the neutralization energy of C r - H 2 O in its nth
vibration-rotation level, E~ , is given by
(48)
En =
In both cases, if the vibration-rotation energy is given by (7,
the distribution is given by the Boltzmann distribution:
N(U) = Noeiu°-u)/kT
(49)
where Uo is the vibration-rotation energy of an ion in its ground
level and is represented by the energy level of ef in Fig. 13 and by
Im in Fig. 14. If E and Eo are the neutralization energies of an ion,
the corresponding vibration-rotation energies of which are U and
UOi respectively, it is clear from Fig. 13 that (E - Eo) is greater
than (U - L/Q); and, between E and Eo the same number of levels
exists as those between U and Uo, but they are distributed over a
larger range of energy. Thus, for H +
N(E) =
Noe(E°-E)/ykT
(50)
where y = \E0- E\/(U0- U) > 1. Similarly, the distribution
function for Cl~ is given by
N(E) = Noe(E-E°)/ykT
(51)
The distribution given by Gurney is shown in Fig. 15, where £ $
Kohei Uotaki aod HiOeOi Ktta
28
Eo
E5
~E
Figure 15. Distribution of density of states as a function of energy (Gurney*s
model). 86 EQ and EQ are the neutralization energy of H + - H 2 O and Cl" - H 2 O,
in its ground level, respectively.
and EQ represent the neutralization energy of H3O+ and Cl~,
respectively, in their ground levels. These distributions allow only
the energy levels of E > Eo for positive ions and E < Eo for negative
ions. This limitation originates from the fact that Gurney did not
allow states in which the ion-water distance is shorter than its
ground state. In Fig. 13, only the states at which the H+-H2O
distance is larger than/ are allowed and in Fig. 14, only the states
at which the Cr~H 2 O distance is larger than m are allowed. It
seems, however, that there is no reason to inhibit the states in which
the ion-water distance is shorter than its ground level. Thus, there
should be energy levels with E < Eo for a positive ion and with
E > Eo for a negative ion. Equation (50) should be used if E > Eo
and Eq. (51) should be used if E < Eo. The value of y depends
on the shape of the potential energy curve of an ion before and
after neutralization. As far as H+ is concerned, y < 1 for E < Eo,
from Fig. 13, which means that the number of states decreases
more quickly when E < Eo than when E > Eo. For Cl", it is not
possible to determine y from Fig. 14 and the exact shape of the
potential energy diagram H2O-C1 and H2O-C1" must be known to
calculate y.
Bockris88"96 and his colleagues essentially followed Gurney's
model for the hydrogen evolution reaction, but they included the
interaction between the electrode and hydrogen. They usually used
y = 1 at E > Eo and neglected the distribution at E < Eoss~93 as
Gurney did or assumed a very sharp drop of the number of states
at E <
E0.27>94-96
Theoretical Aspects of Semiconductor Electrochemistry
29
3. Gerischer's Model
Gurney treated the neutralization of ions (H + + e~ -> H and Cl~ ->
e~ + Cl), which is a rather complicated process that involves the
strong interaction between an electrode and an reaction intermediate such as an adsorbed hydrogen for the hydrogen evolution
reaction. Gerischer extended Gurney's treatment to simple redox
reactions such as
(ox)s
(52)
(red) solv
In this case the potential energy diagram is something like the one
shown in Fig. 16, in which / is the ionization energy of the reduced
species in vacuum. The corresponding reaction can be written as
red -» ox + e^>
(53)
In Fig. 16, Orox and orred are the reaction coordinates corresponding
to the most stable states of the oxidized and reduced forms and
0 Uox and 0 Ured represent the energies corresponding to each form.
E is the energy change in the reaction shown in Eq. (52). In other
words, E is the energy required to introduce a free electron from
infinity into the solution and to occupy the electron state in an
oxidized form without changing the solvation structure (FranckCondon principle) and, thus, gives the energy of the unoccupied
states. Consequently, Eq. (52) actually has the same meaning as
Eq. (47). The reverse process gives the energy of the occupied states
UJ
O
CL
I
\
//
A
E
orox orR»d
/ O x -fsolv+ei
I
j
^ ^
Red+ solv.
Distance
Figure 16. Potential energy diagram of ox + solv. and red 4- solv. systems.24
30
Kohei Uosaki and Hideaki Kita
(reduced form). Gerischer neglected the zero-point energy and
represented the vibration-rotation level by a smooth continuous
curve, as shown in Fig. 16. Here, 0£ox and 0£red are the energies
of unoccupied and occupied states in the most stable state, respectively. He also used the Boltzmann distribution to describe the
energy distribution of both forms as Gurney did. The most significant difference between Gurney's treatment and Gerischer's is
that Gurney did not allow the configuration where the H+-H2O
distance is smaller than the ground level as shown before but
Gerischer did. The equation given by Gerischer is very similar to
that of Gurney:
kT
Lg(£)expl
(55)
where g{E) is a weighting factor and DOX(E) and D red (£) are the
density of states of oxidized and reduced forms, respectively.! To
use these equations, the values of exp[-( Uox - 0Uox)/kT] and
exp[-((7red - 0Ured)/kT] should be known as functions of E. Figure
17 shows Uox - o Uox and UTed - 0 ^red as functions of E. From these,
Gerischer found DOX(E) and D red (£) as shown in Fig. 18, which
appeared in his original publication.24 DOX(E) and Drcd(E) are
equivalent to N(E) of H+ and CP, respectively, of Gurney's
treatment. The curves are somewhat asymmetrical, but in his later
publications, the curves are more symmetrical and show Gaussian
distribution,5'8 which is also obtained by the continuum model
t Although A£ instead of E is used in his original paper, 24 AE actually means E
in Gurney's treatment and Gerischer used E later.5 Khan commented 88 ' 96 that
Gerischer plotted the density of states not as a function of the energy, £, with
respect to vacuum but as a function of the difference of energy, A£, of the electron
in the oxidized and reduced ion. It is, however, clear from the above argument
that these two terms have the same meaning.
Figure 17. Uox - 0UOX and UTcd - 0Urcd as a function of electron energy, E.24
shown below. Since y( = \0Eox - E\/(0UOX - Uox)) > 1 when E >
but < 1 when E < 0EOX, one cannot expect a symmetric distribution. Exact distribution curves can be calculated by knowing the
potential energy curves for particular redox couples.
The energy where
0EOX
is, in a sense, equivalent to the Fermi level of the solid state
phase at which the occupancy of an electron is 1/2, as in this
case. Therefore, Gerischer suggested that this energy be called
^F,redox-5'8'2326'54'55 The significance of ^F.redox was discussed in
Section II.
(0
c
Q
Figure 18. Density of states as a function of electron energy (Gerischer's model).24
32
Kohei Uosaki and Hideaki Kits
4. Continuum Solvent Polarization Fluctuation Model
The model of Gurney as well as those of Bockris and of Gerischer
assume that an energy level change is caused by the vibrationrotation interaction between the central ion and solvent molecules
and describe this change by using the Boltzmann distribution. In
the present model, the surrounding solvent is considered to be a
continuum dielectric. Marcus,97"99 Dogonadze,100'101 Christov,102
and Levich103104 presented the model in this category and the model
of Marcus is most often employed in the literature of semiconductor
electrochemistry. Here we describe the model presented by Morrison,61 which is similar to but less rigorous than that of Marcus.
Let us consider the following electron transfer reaction:
SOAZ f e~ -> SfKz~x
(57)
by dividing it into the following three steps:
SOAZ -> S,-Az
Z
(59)
2 1
(60)
S,A + e~ -> S.A "
2 1
(58)
2 1
S.A " -> SyA "
The first step represents the change in polarization or change in
dipole configuration around the ion A. The second step is the
electron transfer during which the polarization is frozen at the
configuration denoted as S, (Franck-Condon principle). The third
step represents the subsequent relaxation of the polarization of the
medium to its new equilibrium value. The energy required for the
first and third step can be obtained by considering the energy, AEp,
necessary to cause fluctuations from the equilibrium polarization
of the dielectric around an ion without changing the charge on the
ion. A parameter 8 is introduced such that the polarization of the
dielectric corresponds to eZ ± e5, where Z is the actual charge on
the ion. A polarization is defined to correspond to a central charge
differing from the central charge actually present, eZ, by ±e8. It is
necessary to know an expression for the energy increase in the
polarized dielectric medium when the central charge remains eZ,
but the polarization fluctuates from its equilibrium configuration
to a new configuration, which would normally correspond to a
charge e(Z ± 8) on the ion. Marcus97 and Dogonadze et al.100
Theoretical Aspects of Semiconductor Electrochemistry
33
derived the expression as
A£ p - S2A
(61)
where A is the reorganization energy and is given by
where a is the ionic radius and KOP and KS are the optical and the
static dielectric constant of the medium, respectively. One must
note that this is very similar to the Born expression for the solvation
energy per ion, W,105
A better expression for A including changes in the inner sphere is98
^
<64)
where the / ' s are force constants for the jth bond and Ax, is the
displacement in the bond length. Thus, the energy level shift from
its equilibrium value Eo (= Eox if the ion is the oxidized form) to
a value E in the first step is accomplished by a change in polarization
5i5 requiring an energy 52A. For the third step, the initial polarization
of the surrounding medium corresponds to an ionic charge e(Z 8i) but the central charge is now e(Z - 1). Thus, the change in
polarization required to bring the system to its new equilibrium
configuration corresponds to a change in central charge of 1 - 8t.
Therefore the energy released in the third step is (1 - ^) 2 A. The
energy change in the second step is given by Ecs - E, where Ecs is
the energy of the conduction band edge at the electrode surface.
The total energy change of the three steps which represents
the energy required to transfer an electron to an ion in solution
with an appropriate change of equilibrium polarization of the
medium, A, is given by
A = 8]\ - (1 - S^X + E - Ecs
(65)
The energy A should be independent of intermediate state and,
therefore, of E and 8t. Let us calculate A for the equilibrium
34
Kohei Uosaki and Hideaki Kit*
situation, i.e., 8t = 0 and E = Eo. In this case,
A = - A + Eo - Ecs
(66)
Equations (65) and (66) give
- A + Eo - Ecs = 8]X - (1 - S,)2A + E - Ecs
(67)
Thus,
By inserting Eq. (68) into Eq. (61), one obtains
A E , = ^ ^
(69)
This gives the thermal energy required to shift the energy level from
Eo to an arbitrary energy E. The energy distribution function D(E)
is given by
D(E) oc Qxp(-AEp/kT)
(70)
By using Eq. (63)
where the preexponential term is a normalizing constant. For the
oxidized and reduced forms, Eox and £ re d, respectively, should be
used for Eo. This function gives a symmetric curve (Gaussian) with
a central energy of Eo. The energy difference between Eox and E red
can be obtained by considering the following cycle:
SOAZ + e~ -> SoA2"1
(72)
2 1
2 1
(73)
2 1
2
(74)
2
2
(75)
SoA " -> Sy-A "
S/A " -> S/A + e~
S/A -> S0A
Step (72) represents the electron transfer step from the solid to the
unoccupied level of ion in its equilibrium state, 0EOX. The energy
change for this step is given by 0£Ox - E, where E is the initial