Notación y acrónimos
1 Capa de salida
3.2 Modelos empleados
In general for an elementary chemical reaction step to occur an activation barrier has to be overcome. [37]
A kf
kb
B (1.23)
The rate of this reaction step is then correlated with this activation barrier and is given by following relationship which was first introduced by Arrhenius and then extended in the transition state theory [37]:
where k is the respective rate constant for the forward (kf) and backward (kb) reaction respectively. When at equilibrium, kf = kb and is called k0. ΔG‡ (kJ mol-1) is the free
energy of activation that needs to be overcome in order to reach the transition state. R is the gas constant, T the temperature and A a pre-exponential factor which includes the statistical probability of collisions that lead to a reaction. [37] Thus, if the activation barrier is lowered, k is increased and the reaction proceeds more facile. This is the task of every catalyst. Generally it is achieved by a favourable catalyst-reactant interaction that stabilises the activated complex or transition state. [37] An electrochemical reaction is given by [37]:
O + ne− kf
kb
R (1.25)
when at equilibrium, the reaction will have the reversible potential according to the Nernst equation [37]: E = E0∗+RT nFln Cox∗ Cred∗ (1.26)
where E (V) is the equilibrium potential, E∗0 (V) is the formal standard potential, C∗oxand C∗red (in mol cm-3) the bulk concentrations of the oxidised and reduced species an n the number of electrons transferred. The potential current relationship was first empirically found by Tafel to be [37]:
η = a + blogj (1.27)
where a and b are constants. η (V) is the so called overpotential and describes the devia- tion form the respective equilibrium potential (Esystem−EOCV). For anodic processes, i.e. oxidations, η is positive and for cathodic processes, i.e. reductions, η is negative. Based on this observation the Butler-Volmer model of electrode kinetics was developed [37]. It depends upon the relative modulation of the energies and thus activation barrier upon change of potential. If the potential is increased relative to the OCP, the reduced species becomes less stable, which increases its relative free energy and hence the activation bar- rier towards oxidation becomes smaller as compared to the opposite process and vice versa. The imbalance in activation energies results in a net reaction which means a current. [37]
The following expression results form the derivation and describes the current-potential characteristic of a one electron elementary electrochemical reaction [37]:
j = j0 exp αnF η RT − exp −(1− α)nF η RT (1.28)
where j0 is the exchange current density and α is the transfer coefficient. The Tafel
relationship is a limiting case of the Butler-Volmer model, where the overpotential is sufficiently large, so the reverse reactions contribute less than 1% and can be neglected. [37] For a simple electrochemical step kinetic parameters can be extracted from a Tafel plot, as the constants a and b are then [37]:
a = 2.3RT
αF logj0 and b =
−2.3RT
αF (1.29)
Thus the exchange current density and the transfer coefficient can be extracted. [37] The exchange current density is correlated with the rate constant of an electrochemical step by the following relation [37]:
j0 = nF k0CoxαC (1−α)
red (1.30)
where Cox and Cred (in mol cm-3) are the concentrations of oxidised and reduced species
at the surface, repectively. This means ultimately j0, so k0 need to be increased and
therefore the principle of an electrocatalyst is similar to the catalyst; it has to favourably interact with the substrate and lower the activation barrier ΔG‡ for the reaction to be kinetically more facile. The additional requirement for an electrocatalyst as compared to a catalyst is that is has to be electrically conductive in order to minimise ohmic losses. Most technologically important reactions, including the ORR, are not simple elementary charge transfer steps as shown in equation 1.25. These usually involve a complex series of steps, including (i) diffusion of species to the catalyst surface (ii) adsorption and desorption of the unreacted substrate (iii) a series of electron and/or proton transfer steps (iv) desorption of intermediates (v) competitive blocking of sites (vi) desorption of the product and (vii) diffusion of the product away from the catalyst surface. [37] These steps can either occur successively or in parallel, so competitively. Every step has a rate associated with it and the complex interplay of these rate constants then defines the exact current potential
behaviour. [37, 39] These steps can be modelled and give mathematical expressions which can support proposed reaction mechanisms upon fitting with experimental data. [39]
G‡ Gn G0 rxn transition state Reaction coordinate Fr ee ener gy / a.u. n 4e-+4H++O 2+A 3e-+4H++O 2-A 3e-+3H++HOO-A 2e-+2H++H 2O+O-A e-+H++H 2O+HO-A 2H2O + A
Figure 1.8: Diagram showing the free energy of different intermediates for the ORR
along a reaction coordinate. The values for the activation barrier of a particular reaction
step ΔG‡n, the free energy of a particular reaction step ΔGn, the free energy of the
complete reaction ΔG0rxn and the energetic location of the respective transition state
are indicated. In this case the first electron transfer is the RDS, as it has the highest
activation barrier.4
For many reactions it is assumed that one step is significantly slower, i.e has a significantly higher activation barrier ΔG‡, than all other steps and is the RDS. [37] Figure 1.8 shows the reaction pathway described in Equations 1.7 to 1.12 (without 1.10).4 [36, 40] Even complex reactions can show some sort of Tafel behaviour. However, extracting the kinetic parameters as is done for the elemental step is not possible. [37] This is because the contributions from the other reaction steps cannot be neglected. [37] Nevertheless, Tafel slopes and apparent exchange current densities can be used to compare the kinetic facility of a reaction on similar materials to each other and get some insight.
The nature and energetic shape of the activation barrier can be explained with micro- scopic theories of charge transfer, among which the Marcus theory is the most prominent one. [37] It holds for electrochemical charge transfer as well chemical charge transfer. [37] An extensive treatment is outside the scope of this work and the interested reader is re- ferred to the respective literature. [37, 41–43] A summary will be given for completeness. The theory distinguishes between outer sphere and inner sphere electron transfer. While in the outer sphere electron transfer the solvation shell of the reactant stays intact during the reaction step and the molecule does not interact with the electrode material directly,
the inner sphere mechanism involves a strong direct interaction of the substrate with the electrode, i.e. chemisorption. Almost all electrocatalytic reactions are inner sphere mechanisms, as this very catalyst-substrate interaction defines the role of most catalysts. The basis of the theory is the assumption that the electron transfer is a radiationless isoenergetic process. This means the electron has to move from the donating state, i.e. electrode or reduced reactant, to the same energy level of the receiving species, i.e. re- duced reactant or electrode, respectively. [37, 41–43] Expressed in a very simplified way, the theory considers the reaction coordinate in terms of bond lengths of initial and final state with contributions from solvent molecules and terms associated with the movement of charged species through an electrical field. An electron transfer can then only occur when the physical state of the initial and the final state are the same. This is then the transition state. [37, 41–43] The following relationship was derived for an electrochemical system [37]: ΔG‡f = λ 4 1 + F (E− E 0) λ 2 (1.31)
where ΔG‡f (kJ mol-1) is the activation energy of the forward step and λ (kJ mol-1) is the reorganisation energy that results from the derivation as the amount of work necessary to bring the configuration in terms parameters such as bond length, bond angle, solvation shell and spatial orientation of the initial state to that in the final state. At equilibrium ΔG‡≈ λ4. This means the reorganisation energy is smaller when the reduced and oxidised state are more similar to each other and k0 becomes larger hence the reaction more
facile. [37] It has to be mentioned that this relationship was derived for an outer sphere electron transfer and expressions for inner sphere electron transfers are more complex and subject to ongoing research. [35, 37]
From this consideration one can in a simplified way understand the principle of a catalyst to lower ΔG‡. This can be done by inducing a transition state which leads to a configura- tion in which the two states are arranged to be similar, e.g. in terms of bond length. For the first step of the ORR, the electron transfer to form O−2, a favourable interaction with a catalyst could be the donation of electron density from the catalyst into the antibond- ing orbital of the oxygen molecule, creating a state where this intermediate is similar to the initial as well as the final state, by increasing the O-O bond length. If the catalyst facilitates this with the RDS, the reaction is enhanced significantly. An ideal catalyst
however would facilitate this for every step of a multi step mechanism. To achieve this, the active site would have to be highly complex and be able to change during the course of the reaction. To date only enzymes might behave in such a way.
In catalyst development it is useful to find descriptors, e.g. material properties, which govern catalyst activity. [36, 40, 44] This potentially enables a more rational catalyst design and minimises the trial and error approach. The most basic activity descriptor for catalysts as well as electrocatalysts is the energy of adsorption Eads of a substrate with the catalyst surface. [36] The Sabatier principle states that the most active catalyst should bind the substrate strong enough to facilitate a good interaction. If it binds too strongly however the desorption is hindered and the catalyst becomes poisoned. [36] Plotting Eads or any other activity descriptor versus the activity for different materials usually results in an ascending branch and a descending branch, with the most active material on the top.
Figure 1.9: A plot of the electrochemical biding energy of O2on different metal surfaces
and the respective ORR activity. [Reprinted with permission from Ref. [36]. Copyright 2004 American Chemical Society.]
An example of such a plot for the ORR is shown in Figure 1.9, it is called a Volcano curve, owing to its shape. The activity is given as the logarithm of current density. It can be seen that Pt is the by far the most active pure metal for the ORR. [36] The same applies for the HOR. [8] In complex mechanisms as for the ORR, different descriptors have been used for metal surfaces, such as binding constants of certain intermediates or the d-band center of the metal. [40, 45]