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Hydrogen ion chemisorption at electrodes
Surface Science @ North-Holland
101 (1980) 224-230 Publishing Company
HYDROGEN ION CHEMISORPTION AT ELECTRODES T.R. KNOWLES I. Instifute for Theoretica/ Physics. University of Hamburg. Jungiusstrasse 0. D-2MK) Hamburg 3h, F.R. Germany and GKSS Forschungszentrum,
Received
10 October
Postfach 160. D-2054 Geesthacht, F.R. Germany
1979: accepted
for publication
1X December
1979
The discharge of hydrogen ions at the metal+zlectrolyte interface is investigated with attention to both chemisorptive bonding interactions and hydration effects. Due to the strongness of the interactions, tunneling approximations are inadequate. In this work we describe the metal-electrolyte system in the framework of the Anderson-Newns model and obtain the Hartree-Fock solution for the adiabatic energy along the reaction path. The charging behavior and activation energy are studied as functions of the parameters describing the metal and electrolyte. The results show gradual bond formation, smooth reaction barriers. and smaller activation energies than obtained by tunneling estimates.
1. Introduction The elementary process of hydrogen ion discharge at electrodes is of interest both fundamentally and technologically [l]. Much attention has been devoted to understanding the factors determining the rate of the process Me- + H30+ + MH + HZ0
,
(1)
in which a hydrated proton is neutralized by a metal electron and becomes chemisorbed. The rate is known to be sensitive to material parameters and most importantly to the strength with which the metal chemisorbs H [2.3]. The current theories of change transfer reactions [4] are based on tunneling and, although appropriate for cases of weak coupling to the surface, the tunneling picture is at best a crude representation of the formation of chemisorption bonds. An approach to the study of reaction (1) which allows full consideration of the resonant electron hopping characterizing the MH bond may be based on the Anderson hamiltonian. In addition to providing a more accurate reaction barrier it may be used to derive explicitly various trends in the reaction rate as functions of the parameters defining the electrode and electrolyte. 224
T.R. Knowles
i Hydrogen
ion chemisorption
at electrodes
225
2. Reaction path and adiabatic energy
Data on the hydrogen evolution reaction show that (1) is a thermally activated process. The temperature dependence of the current (or rate) in an electrochemical cell at moderate (~50 mV) overpotential n is In i = -(E,,* + cq)/kBT
+ constant
,
where the transfer coefficient (Y= 0.5 and E,, is the activation energy. The activation barrier has its origin in the large proton hydration energy which places E,, the electron energy level on the H30C, above the Fermi level of most metal electrodes [5]. While E, = -I = -13.6 eV for a bare proton, the value in aqueous solution is G(Y)=
-I+Uy)+R(y)-eVH
,
(2)
where VH is the electrostatic potential at the H, L is the hydration energy and R is the repulsive potential energy of H-H20. Both these functions depend on an internal coordinate, y, describing the degree of excitation of the H30+. Usually, this reaction coordinate is described as the length of the H+-Hz0 bond [4]. In the equilibrium configuration y = y. the hydration energy is large: L(y,J = 11.6 eV and R(yo)= 1 eV. As shown in fig. 1, the placement of ea(yO) makes electron transfer energetically unfavored. In the usual description, the charge transfer process occurs only if ln(y) = lr, which is the condition for radiationless tunneling of an electron from the metal to the H30+. This situation requires a preliminary thermal excitation of the H30+, which lowers e,(y), and the energy necessary is the activation energy. It is helpful to picture the barrier in terms of the adiabatic potential energy of reactants R = {Me-, H30+} and products P” =
Fig. 1. The electron energy level of the HjO’ (e.) and the Fermi level of the electrode (EF); I is the H ionization energy, L and R are the hydration functions, C$ is the vacuum work function, and VH and VM are the shift in the electrostatic potential at the H+ and at the electrode from their vacuum values.
226
11k’. Knowit-s 1 Hydrogen ion chemisorption at electrodes
(M, H, HZO) or P = {MH, H20}, where in the latter case the H is chemisorbed on the metal electrode. The corresponding energies are l&(y)=-4
,
@a)
.
(3b)
-L(y)-eV,
E!(y)=-I+R(y)-eVH &(y)=~Xy)-&h
where the zero (M. e . H, HZO}. potential of the R(y) have been
.
of energy
UC)
is taken
to be that of the separated system energy (Xl) and VM is the shift in the from the vacuum value. The functions L(y) and
E,,, is the adsorption
electrode
variously estimated [4. S] as a Morse molecular potential and an exponential core repulsion, respectively. In fig. 2, the energy curves (3) are sketched along with a typical energy curve obtained by the method of section 3. As the reaction coordinate y increases. the system proceeds from the reactant well along ER. At the intersection point G. where E,, = E?-. an electron may tunnel with a probability as estimated by Gurney [S]. and the system then evolves along the products curve as the neutral H-H,0 separates. If the H chemisorbs as it discharges then the tunneling condition is satisfied at the intersection of EK and Ep. The activation energy in both cases is given by the height of the corresponding intersection point as seen Reactton Coordinate
y
energy curves for the reactants (EK). the products with and without chemFig. 2. Adiabatic isorption (Ep and EF) and the adiabatic energy of the interacting system (E). as functions of the H”-&O stretching coordinate. The arrows show where the charge on the H’ is 10% and 90% of the full chemisorption value.
T.R. Knowles / Hydrogen ion chemisorption at electrodes
221
from the reactant well and is clearly lower for the chemisorption case. The charge on the ion is n = 0 on the ER curve and n = 1 on the EF or Ep curves. Although this tunneling picture may give a qualitative understanding of the effect of chemisorption on the activation energy, it is an unphysical description of the bond formation and charging behavior. Typical chemisorption energies of 2 eV correspond to electron hopping rates -lOI ss’ which is fast on the time scale of molecular vibrations with frequencies -lO’“s-‘. It becomes necessary to account for the return tunneling of the electron back into the electrode.
3. Model Hamiltonian
and Hartree-Fock
energy
To investigate the bonding of the hydrogen ion to the metal substrate it is natural to apply the Anderson-Newns model [6] of chemisorption theory. Here it must be modified to include hydration effects and thermal excitation of the H30+ during the charge transfer reaction. The metal/electrolyte system is represented by the Hamiltonian X= c W!L& 0
+x
D
4YkLGC7
+ P’ c
(c!kxr
.(T
+ Ck7Ckrr) 3
(4)
where CL and cf, are creation operators for electrons of spin u in the band state k and on the H,O+, respectively. The first term describes the metal electrode, which in the calculations below is specialized to a single, elliptical tight-binding d-band of width 4p and Fermi level +. The second term describes the absorbate and includes electron-electron correlation on the proton within the Hartree-Fock approximation:
Here E,(Y) is the electron energy level on the ion, U is the electron correlation strength and n, = (cLc,,), the expectation value of charge of spin CTon the adsorbate. The last term of (5) is the coupling, with strength p’ taken to be a constant. Note that in the present context the adsorbate level c,(y) contains hydration terms and is a function of y. as given in (2). The model may be solved [6] to determine the chemisorption energy, AE, defined as the change in energy resulting from coupling the (neutral) H-H?0 to the metal. It is a function of E,(Y), hence y, and may be expressed AE(y) = c AE’” - Un,n_, v
-E,(Y)
.
Measuring energy in units of 2/? and letting B = 2/312,AE’” is given by the
228
T.R. Knowles i Hydrogen ion chemisorption at electrodes
following
integrals: > 0 ,
for E,, + 1 -B F
l
AE’”
zc ,f’ I
tan:),
for E, + 1 -B
C de
;
< 0
c,;
where _B(l
_
E?)l/l
C=(B-l)E+E,, E/c =
’
(l-B)E,+B(2B++l)“? l-2B
-EF
,
and 0 < tan;’ < r, -r < tan:), < 0. The charge self-consistency condition
n, must be determined
n, = i3E ‘“Iat-, .
by the
(6)
Eqs. (5) and (6) are solved by numerical iteration. The energy, measured relative to the energy of separated reactants,
total is
system
E(y)=E%y)+AE(y)
A typical example of the energy curve calculated by this procedure is shown as the solid line in fig. 2. The charge on the adsorbate gradually increases from n = 0 to the full chemisorption value as y increases, and the arrows in fig. 2 show where the 10% and 90% values are attained. The rounding of the reaction barrier due to the coupling is a pronounced feature of these calculations and is discussed below. The height of the reaction barrier depends on the various parameters entering the model. For given 4, L, R, V, and V,, the point G of fig. 2 is fixed and defines the Gurney value of the activation energy E&,. The coupling causes the actual barrier to be lower, and the degree of lowering, 6E,,, = Et,, - E,,,
,
depends on B, /3’, lF and U. The dependence of 6E,,, on these parameters will now be discussed. Near the point G the curves L and R may be linearly approximated, and following Bockris [l, 41 we shall assume the slopes to be equal. From (2) ER and E$ are then linear functions of eF - E, near EF - E, = 0. In fig. 3 is shown
T.R. Knowles
/ Hydrogen
ion chemisorprion
229
at electrodes
Fig. 3. Adiabatic energy E for Ni near the top of the reaction barrier, angular energy barrier of the tunneling theory. Energy is in units of 2/3.
compared
with the
Fig. 4. The decrease in the activation energy due to chemisorption as a function (a) of B = 2p” for EF = 0, U = 3, (b) of EF for B = 3, CJ = 3, (c) of U for B = 3, EF = 0. Energy is in units of 2p.
the function E(y) near the barrier for Ni, using parameters given by Newns. The barrier is several volts below the Gurney prediction and even 3.36 eV below the value that would be obtained from the intersection of ER and Ep. The charge on the ion at the top of the barrier is 0.5, which may be shown to always be the case when L and R have equal slopes near G. Fig. 4 shows the dependence of SE,,, on the parameters j3, /3’, lF and U.
4. Discussion The model Hamiltonian approach provides a simple phenomenological description of the coupling of the H30+ to the electrode during neutralization. Parameters relevant to a particular metal/electrolyte system can at present only be obtained from the corresponding vacuum chemisorption data. The influence of solvent molecules on the process enter this study solely through the hydration energy contribution to E,(Y). Other dielectric effects which could alter the vacuum chemisorption values of p’ and U are expected to be less important, and are presently being investigated.
T.R. Knowles I Hydrogen ion chemisorption at electrodes
230
In general, /3’ and U are themselves dependent on both y and X, the M-H separation. To calculate reaction entropy, the dependence on x would be necessary, but for our present concern with activation energy it is irrelevant. Elementary considerations of molecular orbital overlap suggest that /3’ may be lowered by at most a factor d3 compared to the vacuum case, however the effect on bonding energy is partially counteracted by a similar lowering of U. The coupling in any case remains strong and the contention holds that a non-perturbative approach is necessary. The striking feature of the above numerical results is that E,,, is reduced by typically 2 eV below the predictions of the tunneling theories. The mathematical reason for this is clear. The coupling term
i?r= p’
c
(CL&, + Cb,,Ckrr)
kn
applied to the non-interacting perturbation correction W:
ground
state
with
E, < lF gives
a leading
case where, E,, is This value is larger for E,, = lF than for the chemisorption several volts below cr. Physically, the resonant exchange of the electron is more effective in lowering the system energy when E, = EF. This method allows one to quantify the barrier rounding that occurs in the chemisorptive neutralization process, and can form a basis for more quantitative studies using more refined estimates of the coupling parameters and hydration functions in the metal/electrolyte environment.
Acknowledgements The author is grateful helpful discussions.
to J. Appel,
P. Hertel,
H. Mertins,
and M. Pfuff for
References J.O’M. Bockris and A.K.N. Reddy. Modern Electrochemistry (Plenum. New York. lY70). A.N. Frumkin, Advan. Electrochem. Electrochem. Eng. 3 (lY6X) 2X7. L.I. Kristalik. Advan. Electrochem. Electrochem. Eng. 7 (1970) 2X3. A comprehensive review article is D.B. Matthews and J.O’M. Bockris, Mod. Aspects Electrochem. 6 (1971) 232. R.W. Gurney. Proc. Roy. Sot. (London) A134 (lY31) 137. D.M. Newns. Phys. Rev. 17X (1069) 1123.
101 (1980) 224-230 Publishing Company
HYDROGEN ION CHEMISORPTION AT ELECTRODES T.R. KNOWLES I. Instifute for Theoretica/ Physics. University of Hamburg. Jungiusstrasse 0. D-2MK) Hamburg 3h, F.R. Germany and GKSS Forschungszentrum,
Received
10 October
Postfach 160. D-2054 Geesthacht, F.R. Germany
1979: accepted
for publication
1X December
1979
The discharge of hydrogen ions at the metal+zlectrolyte interface is investigated with attention to both chemisorptive bonding interactions and hydration effects. Due to the strongness of the interactions, tunneling approximations are inadequate. In this work we describe the metal-electrolyte system in the framework of the Anderson-Newns model and obtain the Hartree-Fock solution for the adiabatic energy along the reaction path. The charging behavior and activation energy are studied as functions of the parameters describing the metal and electrolyte. The results show gradual bond formation, smooth reaction barriers. and smaller activation energies than obtained by tunneling estimates.
1. Introduction The elementary process of hydrogen ion discharge at electrodes is of interest both fundamentally and technologically [l]. Much attention has been devoted to understanding the factors determining the rate of the process Me- + H30+ + MH + HZ0
,
(1)
in which a hydrated proton is neutralized by a metal electron and becomes chemisorbed. The rate is known to be sensitive to material parameters and most importantly to the strength with which the metal chemisorbs H [2.3]. The current theories of change transfer reactions [4] are based on tunneling and, although appropriate for cases of weak coupling to the surface, the tunneling picture is at best a crude representation of the formation of chemisorption bonds. An approach to the study of reaction (1) which allows full consideration of the resonant electron hopping characterizing the MH bond may be based on the Anderson hamiltonian. In addition to providing a more accurate reaction barrier it may be used to derive explicitly various trends in the reaction rate as functions of the parameters defining the electrode and electrolyte. 224
T.R. Knowles
i Hydrogen
ion chemisorption
at electrodes
225
2. Reaction path and adiabatic energy
Data on the hydrogen evolution reaction show that (1) is a thermally activated process. The temperature dependence of the current (or rate) in an electrochemical cell at moderate (~50 mV) overpotential n is In i = -(E,,* + cq)/kBT
+ constant
,
where the transfer coefficient (Y= 0.5 and E,, is the activation energy. The activation barrier has its origin in the large proton hydration energy which places E,, the electron energy level on the H30C, above the Fermi level of most metal electrodes [5]. While E, = -I = -13.6 eV for a bare proton, the value in aqueous solution is G(Y)=
-I+Uy)+R(y)-eVH
,
(2)
where VH is the electrostatic potential at the H, L is the hydration energy and R is the repulsive potential energy of H-H20. Both these functions depend on an internal coordinate, y, describing the degree of excitation of the H30+. Usually, this reaction coordinate is described as the length of the H+-Hz0 bond [4]. In the equilibrium configuration y = y. the hydration energy is large: L(y,J = 11.6 eV and R(yo)= 1 eV. As shown in fig. 1, the placement of ea(yO) makes electron transfer energetically unfavored. In the usual description, the charge transfer process occurs only if ln(y) = lr, which is the condition for radiationless tunneling of an electron from the metal to the H30+. This situation requires a preliminary thermal excitation of the H30+, which lowers e,(y), and the energy necessary is the activation energy. It is helpful to picture the barrier in terms of the adiabatic potential energy of reactants R = {Me-, H30+} and products P” =
Fig. 1. The electron energy level of the HjO’ (e.) and the Fermi level of the electrode (EF); I is the H ionization energy, L and R are the hydration functions, C$ is the vacuum work function, and VH and VM are the shift in the electrostatic potential at the H+ and at the electrode from their vacuum values.
226
11k’. Knowit-s 1 Hydrogen ion chemisorption at electrodes
(M, H, HZO) or P = {MH, H20}, where in the latter case the H is chemisorbed on the metal electrode. The corresponding energies are l&(y)=-4
,
@a)
.
(3b)
-L(y)-eV,
E!(y)=-I+R(y)-eVH &(y)=~Xy)-&h
where the zero (M. e . H, HZO}. potential of the R(y) have been
.
of energy
UC)
is taken
to be that of the separated system energy (Xl) and VM is the shift in the from the vacuum value. The functions L(y) and
E,,, is the adsorption
electrode
variously estimated [4. S] as a Morse molecular potential and an exponential core repulsion, respectively. In fig. 2, the energy curves (3) are sketched along with a typical energy curve obtained by the method of section 3. As the reaction coordinate y increases. the system proceeds from the reactant well along ER. At the intersection point G. where E,, = E?-. an electron may tunnel with a probability as estimated by Gurney [S]. and the system then evolves along the products curve as the neutral H-H,0 separates. If the H chemisorbs as it discharges then the tunneling condition is satisfied at the intersection of EK and Ep. The activation energy in both cases is given by the height of the corresponding intersection point as seen Reactton Coordinate
y
energy curves for the reactants (EK). the products with and without chemFig. 2. Adiabatic isorption (Ep and EF) and the adiabatic energy of the interacting system (E). as functions of the H”-&O stretching coordinate. The arrows show where the charge on the H’ is 10% and 90% of the full chemisorption value.
T.R. Knowles / Hydrogen ion chemisorption at electrodes
221
from the reactant well and is clearly lower for the chemisorption case. The charge on the ion is n = 0 on the ER curve and n = 1 on the EF or Ep curves. Although this tunneling picture may give a qualitative understanding of the effect of chemisorption on the activation energy, it is an unphysical description of the bond formation and charging behavior. Typical chemisorption energies of 2 eV correspond to electron hopping rates -lOI ss’ which is fast on the time scale of molecular vibrations with frequencies -lO’“s-‘. It becomes necessary to account for the return tunneling of the electron back into the electrode.
3. Model Hamiltonian
and Hartree-Fock
energy
To investigate the bonding of the hydrogen ion to the metal substrate it is natural to apply the Anderson-Newns model [6] of chemisorption theory. Here it must be modified to include hydration effects and thermal excitation of the H30+ during the charge transfer reaction. The metal/electrolyte system is represented by the Hamiltonian X= c W!L& 0
+x
D
4YkLGC7
+ P’ c
(c!kxr
.(T
+ Ck7Ckrr) 3
(4)
where CL and cf, are creation operators for electrons of spin u in the band state k and on the H,O+, respectively. The first term describes the metal electrode, which in the calculations below is specialized to a single, elliptical tight-binding d-band of width 4p and Fermi level +. The second term describes the absorbate and includes electron-electron correlation on the proton within the Hartree-Fock approximation:
Here E,(Y) is the electron energy level on the ion, U is the electron correlation strength and n, = (cLc,,), the expectation value of charge of spin CTon the adsorbate. The last term of (5) is the coupling, with strength p’ taken to be a constant. Note that in the present context the adsorbate level c,(y) contains hydration terms and is a function of y. as given in (2). The model may be solved [6] to determine the chemisorption energy, AE, defined as the change in energy resulting from coupling the (neutral) H-H?0 to the metal. It is a function of E,(Y), hence y, and may be expressed AE(y) = c AE’” - Un,n_, v
-E,(Y)
.
Measuring energy in units of 2/? and letting B = 2/312,AE’” is given by the
228
T.R. Knowles i Hydrogen ion chemisorption at electrodes
following
integrals: > 0 ,
for E,, + 1 -B F
l
AE’”
zc ,f’ I
tan:),
for E, + 1 -B
C de
;
< 0
c,;
where _B(l
_
E?)l/l
C=(B-l)E+E,, E/c =
’
(l-B)E,+B(2B++l)“? l-2B
-EF
,
and 0 < tan;’ < r, -r < tan:), < 0. The charge self-consistency condition
n, must be determined
n, = i3E ‘“Iat-, .
by the
(6)
Eqs. (5) and (6) are solved by numerical iteration. The energy, measured relative to the energy of separated reactants,
total is
system
E(y)=E%y)+AE(y)
A typical example of the energy curve calculated by this procedure is shown as the solid line in fig. 2. The charge on the adsorbate gradually increases from n = 0 to the full chemisorption value as y increases, and the arrows in fig. 2 show where the 10% and 90% values are attained. The rounding of the reaction barrier due to the coupling is a pronounced feature of these calculations and is discussed below. The height of the reaction barrier depends on the various parameters entering the model. For given 4, L, R, V, and V,, the point G of fig. 2 is fixed and defines the Gurney value of the activation energy E&,. The coupling causes the actual barrier to be lower, and the degree of lowering, 6E,,, = Et,, - E,,,
,
depends on B, /3’, lF and U. The dependence of 6E,,, on these parameters will now be discussed. Near the point G the curves L and R may be linearly approximated, and following Bockris [l, 41 we shall assume the slopes to be equal. From (2) ER and E$ are then linear functions of eF - E, near EF - E, = 0. In fig. 3 is shown
T.R. Knowles
/ Hydrogen
ion chemisorprion
229
at electrodes
Fig. 3. Adiabatic energy E for Ni near the top of the reaction barrier, angular energy barrier of the tunneling theory. Energy is in units of 2/3.
compared
with the
Fig. 4. The decrease in the activation energy due to chemisorption as a function (a) of B = 2p” for EF = 0, U = 3, (b) of EF for B = 3, CJ = 3, (c) of U for B = 3, EF = 0. Energy is in units of 2p.
the function E(y) near the barrier for Ni, using parameters given by Newns. The barrier is several volts below the Gurney prediction and even 3.36 eV below the value that would be obtained from the intersection of ER and Ep. The charge on the ion at the top of the barrier is 0.5, which may be shown to always be the case when L and R have equal slopes near G. Fig. 4 shows the dependence of SE,,, on the parameters j3, /3’, lF and U.
4. Discussion The model Hamiltonian approach provides a simple phenomenological description of the coupling of the H30+ to the electrode during neutralization. Parameters relevant to a particular metal/electrolyte system can at present only be obtained from the corresponding vacuum chemisorption data. The influence of solvent molecules on the process enter this study solely through the hydration energy contribution to E,(Y). Other dielectric effects which could alter the vacuum chemisorption values of p’ and U are expected to be less important, and are presently being investigated.
T.R. Knowles I Hydrogen ion chemisorption at electrodes
230
In general, /3’ and U are themselves dependent on both y and X, the M-H separation. To calculate reaction entropy, the dependence on x would be necessary, but for our present concern with activation energy it is irrelevant. Elementary considerations of molecular orbital overlap suggest that /3’ may be lowered by at most a factor d3 compared to the vacuum case, however the effect on bonding energy is partially counteracted by a similar lowering of U. The coupling in any case remains strong and the contention holds that a non-perturbative approach is necessary. The striking feature of the above numerical results is that E,,, is reduced by typically 2 eV below the predictions of the tunneling theories. The mathematical reason for this is clear. The coupling term
i?r= p’
c
(CL&, + Cb,,Ckrr)
kn
applied to the non-interacting perturbation correction W:
ground
state
with
E, < lF gives
a leading
case where, E,, is This value is larger for E,, = lF than for the chemisorption several volts below cr. Physically, the resonant exchange of the electron is more effective in lowering the system energy when E, = EF. This method allows one to quantify the barrier rounding that occurs in the chemisorptive neutralization process, and can form a basis for more quantitative studies using more refined estimates of the coupling parameters and hydration functions in the metal/electrolyte environment.
Acknowledgements The author is grateful helpful discussions.
to J. Appel,
P. Hertel,
H. Mertins,
and M. Pfuff for
References J.O’M. Bockris and A.K.N. Reddy. Modern Electrochemistry (Plenum. New York. lY70). A.N. Frumkin, Advan. Electrochem. Electrochem. Eng. 3 (lY6X) 2X7. L.I. Kristalik. Advan. Electrochem. Electrochem. Eng. 7 (1970) 2X3. A comprehensive review article is D.B. Matthews and J.O’M. Bockris, Mod. Aspects Electrochem. 6 (1971) 232. R.W. Gurney. Proc. Roy. Sot. (London) A134 (lY31) 137. D.M. Newns. Phys. Rev. 17X (1069) 1123.