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IV. DISTRIBUTION OF ENERGY STATES FOR IONS OF REDOX SYSTEM IN SOLUTION

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,

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

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IV. DISTRIBUTION OF ENERGY STATES FOR IONS OF REDOX SYSTEM IN SOLUTION