# 快速时时彩: The electrode–electrolyte interface.pdf

The electrode–electrolyte interface A. Hamnett University of Strathclyde, Glasgow, UK 1 THE ELECTRIFIED DOUBLE LAYER Once an electrode, which for our purposes may initially be treated as a conducting plane, is introduced into an electrolyte solution, several things change. There is a sub- stantial loss of symmetry, the potential experienced by an ion will now be not only the screened potential of the other ions but will contain a term arising from the field due to the electrode and a term due to the image charge in the electrode. The structure of the solvent is also perturbed: next to the electrode, the orientation of the molecules of solvent will be affected by the elec- tric field at the electrode surface, and the net orientation will derive from both the interaction with the electrode and with neighboring molecules and ions. Finally, there may be a sufficiently strong interaction between ions and the electrode surface that the ions lose at least some of their inner solvation sheath and adsorb on the electrode surface. The classical model of the electrified interface is shown in Figure 1, [1] and the following features are apparent: 1. There is an ordered layer of solvent dipoles next to the electrode surface, the extent of whose orientation is expected to depend on the charge on the electrode. 2. There is, or may be, an inner layer of specifically adsorbed anions on the surface; these anions have displaced one or more solvent molecules and have lost part of their inner solvation sheath. An imaginary plane can be drawn through the centers of these anions to form the inner Helmholtz plane (IHP). 3. The layer of solvent molecules not directly adja- cent to the metal is the closest distance of approach of solvated cations. Since the enthalpy of solvation of cations is usually substantially larger than that of anions, it is normally expected that there will be insufficient energy to strip the cations of their inner solvation sheaths, and a second imaginary plane can be drawn through the centers of the solvated cations: this second plane is termed the outer Helmholtz plane (OHP). 4. Outside the OHP, there may still be an electric field and hence an imbalance of anions and cations extending in the form of a diffuse layer into the solution. 5. The potential distribution in this model obviously con- sists of two parts: a quasi-linear potential drop between the metal electrode and the IHP or OHP depending on the charge on the electrode surface and the corre- sponding planar ionic density, and a second part cor- responding to the diffuse layer; as we shall see below, in this part, the potential decays roughly exponen- tially through screening. However, there are subtleties about what can actually be measured that need some attention. 2 THE ELECTRODE POTENTIAL Any measurement of potential must describe a reference point, and we will take as this point the potential of an electron well separated from the metal and at rest in vacuo. By reference to Figure 2, [2] we can define the following quantities: Handbook of Fuel Cells – Fundamentals, Technology and Applications ? 2010 John Wiley it is positive and in the simple Sommerfeld theory of metals, [3] ε F = h 2 k 2 F /2m = h 2 (3πn e ) 2/3 /2m, where n e is the number density of electrons. 2. The work function Phi1 M which is the energy required to remove an electron from the inner Fermi level to vacuum. 3. The surface potential of the electrode, χ M , due to the presence of surface dipoles. At the metal–vacuum interface, these dipoles arise from the fact that the V b V Vacuum level Fermi level ε F Φ M μ M e V s eχ M Figure 2. Potential energy profile at the metal–vacuum boundary. Bulk and surface contributions to V are separately shown. (Repro- duced from Trassatti (1980) [2] with permission from Kluwer Academic/Plenum Publishers.) electrons in the metal can relax at the surface to some degree, extending outwards by a distance of the order of 1 ? A, and giving rise to a spatial imbalance of charge at the surface. 4. The chemical potential of the electrons in the metal, μ M e , a negative quantity. 5. The electrochemical potential ?μ M e of the electrons in metal M is defined as μ M e ? e 0 ψ M ? e 0 χ M ,whereψ M is the mean electrostatic potential just outside the metal, which will tend to zero as the free-charge density, σ, on the metal tends to zero; as σ → 0, then ?μ M e → μ M e ? e 0 χ M ≡?Phi1 M from Figure 2. Hence, for σ negationslash= 0, ?μ M e =?Phi1 M ? e 0 ψ M . 6. The potential energy of the electrons, V ,whichisa negative quantity that can be partitioned into bulk and surface contributions as shown. Clearly, from Figure 2, μ M e = ε F + V b . Of the quantities shown in Figure 2, Phi1 M is measurable, as is ε F , but the remainder are not, and must be calculated. Values of 1–2 eV have been obtained for χ M , though smaller values are found for the alkali metals. If two metals with different work functions are placed in contact, there will be a flow of electrons from the metal with the lower work function to one with the higher work function; this will continue until the electrochemical potentials of the electrons in the two phases are equal. This change gives rise to a measurable potential difference between the two metals termed the contact potential or Volta potential difference. Clearly Delta1 M 1 M 2 Phi1 = e 0 Delta1 M 2 M 1 ψ,where Delta1 M 2 M 1 ψ is the Volta potential difference between a point close to the surface of M 1 and one close to the surface of M 2 , both points being in the vacuum phase; this is an experimentally measurable quantity. The actual number of electrons transferred is very small, so that the electron 2 The electrode–electrolyte interface densities of the two phases will be unaltered, and only the value of the potential V will have changed. If we assume that, on putting the metals together, the χ M vanish, and we define the potential inside the metal as φ, then the equality of electrochemical potentials also leads to ?μ M 1 e + e 0 Delta1 M 1 M 2 φ + μ M 2 e = 0 (1) This internal potential, φ, is not directly measurable; it is termed the Galvani potential, and is the target of most of the modeling discussed below. Clearly, if the electrons are transferred across the free surfaces and vacuum between the two metals, we have Delta1 M 1 M 2 Phi1 = Delta1 M 1 M 2 φ + Delta1 M 2 M 1 ψ. Once a metal is immersed in a solvent, a second dipolar layer will form at the metal surface due to the alignment of the solvent dipoles. Again, this contribution to the poten- tial is not directly measurable, and, in addition, the metal dipole contribution itself will change since the distribution of the electron cloud will be modified by the presence of the solvent. Finally, there will be a contribution from free charges both on the metal and in the electrolyte. The over- all contribution to the Galvani potential difference between metal and solution then consists of these four quantities, as shown in Figure 3. [2] If the potential due to dipoles at the metal–vacuum interface for the metal is χ M and for the solvent–vacuum interface is χ S , then the Galvani poten- tial difference between metal and solvent can be written either as Delta1 M S φ = (χ M + δχ M ) + (χ S + δχ S )(2) or as Delta1 M S φ = Delta1 M S χ + Delta1 M S ψ (3) where δχ M , δχ S , are the changes in surface dipole for metal and solvent on forming the interface. In equation (2) we pass across the interface, and in equation (3) we pass into the vacuum from both metal and solvent. As before, the value of Delta1 M S ψ, the Volta potential difference, is measurable experimentally, but it is evident that we cannot associate this potential difference with that due to free charges at the interface, since there are changes in dipole contribution on both sides as well. Even if there are no free charges at the interface (at the point of zero charge (PZC)), the Volta potential difference is not zero unless δχ M = δχ S i.e., the free surfaces of the two phases will still be charged unless the changes in surface dipole of solvent and metal balance exactly. In practice, this is not the case: careful measurements [4] show that Delta1 Hg H 2 O ψ =?0.26V at the PZC, showing that the dipole changes do not, in fact, compensate. Historically, this discussion is of considerable interest, since ? M S φ g M S (ion) g S (dip) OHP M D (e) g M (dip) S Figure 3. Components of the Galvani potential difference at a metal–solution interface. (Reproduced from Trassatti (1980) [2] with permission from Kluwer Academic/Plenum Publishers.) a bitter dispute between Galvani and Volta over the origin of the electromotive force (EMF) when two different metals are immersed in the same solution could, in principle, be due just to the Volta potential difference between the metals. In fact, it is easy to see that if conditions are such that there are no free charges on either metal, the difference in potential between them, again a measurable quantity, is given by Delta1E σ=0 = Delta1Phi1 + (Delta1 M S ψ) σ=0 (4) showing that the difference in work functions would only account for the difference in electrode potentials if the two Volta terms were actually zero. 3 INTERFACIAL STATISTICAL THERMODYNAMICS OF THE DIFFUSE LAYER Development of a self-consistent theory for the double layer has proven extremely difficult, since the presence of an electrode introduces an essentially non-isotropic element into the equations. This manifests itself in the need for 3 Thermodynamics and kinetics of fuel cell reactions atermw(x k ) in the potential energy of the ion deriving from the electrode itself, where x k is the distance between the kth particle and the electrode surface. It is possible to work within the McMillan–Mayer theory of solutions, and a complete account of w must include the following contributions: 1. A short-range contribution, w s (x k ), which takes into account the nearest distance of approach of the ion to the electrode surface. For ions that do not specifically adsorb, this will be the OHP, distance h from the electrode. For ions that do specifically adsorb, w s (x k ) will be more complex, having contributions both from short-range attractive forces and from the energy of de-solvation. 2. A contribution from the charge on the surface, w Q e (x k ). If this charge density is written Q e , then elementary electrostatic theory shows that w Q e (x k ) will have the unscreened form w (Q e ) (x k ) = const. + z k e 0 Q e εε 0 x k (5) 3. An energy of attraction of the ion to its intrinsic image, w im (x k ), of unscreened form w (im) (x k ) = z 2 k e 2 0 16πεε 0 x k (6) In addition, the energy of interaction between any two ions will contain a contribution from the mirror potential of the second ion; u(r ii ) is now given by a short-range term and a term of the form u (el) (r ij ) = z i z j e 2 0 4πεε 0 parenleftBigg 1 r ij ? 1 r ? ij parenrightBigg (7) where r ? ij is the distance between ion i and the image of ion j. Note that there are several implicit approximations made in this model: the most important is that we have neglected the effects of the electrode on orienting the solvent mole- cules at the surface. This is highly significant: image forces arise whenever there is a discontinuity in dielectric function, and the simple model above would suggest that at the least, the layer of solvent next to the electrode should have a dielectric function rather different, especially at low frequencies, from the bulk dielectric function. Implicit in equations (5–7) is also the fact that the dielectric constant of water, ε W , is assumed independent of x, an assumption again at variance with the simple model presented in Figure 2. In principle, these deficiencies could be overcome by modifying the form of the short-range potentials, but it is not obvious that this will be satisfactory for the description of the image forces, which are intrinsically long-range. The zeroth-order solution is the Go¨uy–Chapman theory dating from the early part of the 20th century. [5, 6] In this solution, the ionic atmosphere is ignored, as is the mirror image potential for the ion. The variation of potential with distance from the electrode surface reduces to d 2 φ dx 2 =? summationdisplay ∝ z ∝ e 0 n 0 ∝ exp{?β[z ∝ e 0 φ(x)]} εε 0 (8) where we have built in the further assumption that w s (x) = 0forxhand w s (x) =∞for xh. This corresponds to the hard sphere model introduced above. Whilst equa- tion (8) can be solved for general electrolyte solutions, a solution in closed form can be most easily obtained for a 1–1 electrolyte with ionic charges ±z. Under these circum- stances, equation (8) reduces to d 2 φ dx 2 = 2ze 0 n 0 εε 0 sin h parenleftBig ze 0 kT φ parenrightBig (9) where we have assumed that n 0 + = n 0 ? = n 0 . Integration under the boundary conditions above gives: Q e = radicalBig (8kTεε 0 n 0 ) sinh parenleftbigg ze 0 φ(h) 2kT parenrightbigg (10) φ(h) = 2kT ze 0 sinh ?1 parenleftBigg Q e radicalbig (8kTεε 0 n 0 ) parenrightBigg (11) φ(x) = 4kT ze 0 tanh ?1 braceleftbigg e ?κ(x?h) tan h parenleftbigg ze 0 φ(h) 4kT parenrightbiggbracerightbigg (12) and κ is given by κ 2 = 2z 2 e 2 0 n 0 /εε 0 kT. These are the cen- tral results of the Go¨uy–Chapman theory, and clearly if ze 0 φ(h)/4kT is small, then φ(x) ～ φ(h)e ?κ(x?h) ,andthe potential decays exponentially in the bulk of the electrolyte. The basic physics is similar to the Debye–H¨uckel theory of inter-ionic interactions, in that the actual field due to the electrode becomes screened by the ionic charges in the electrolyte. Clearly κ is a measure of the screening length; for a 1 M electrolyte of unit 1 : 1 charge, the value of κ is 3.2 × 10 9 m ?1 , corresponding to a screening distance of just 3 ? A, as indicated above. Developments of the theory have been the subject of considerable analytical investigation, [1] but there has been relatively little progress in devising more accurate theories, since the approximations made even in these simple derivations are very difficult to correct accurately. 4 The electrode–electrolyte interface 3.1 Specific ionic adsorption and the inner layer Interaction of the water molecules with the electrode surface can be developed through simple statistical models. Clearly, for water molecules close to the electrode surface, there will be several opposing effects: the hydrogen bonding tending to align the water molecules with those in solution, the electric field tending to align the water molecules with dipole moments perpendicular to the electrode surface, and dipole–dipole interactions tending to orient the nearest neighbor dipoles in opposite directions. Simple estimates [7] basedon20kJmol ?1 for each H bond, suggest that the orientation energy pE,wherep is the dipole moment of water, becomes comparable to this for E ～ 5 × 10 9 Vm ?1 ; such field strengths will be associated with surface charges of the order of 0.2–0.3 C m ?2 or 20–30 μCcm ?2 assuming p =