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Physical and Interfacial
Electrochemistry 2013
Lecture 5
Electrode/solution
Interface
Solution
**+
+**
+ (^) -
**-
Excess positivecharge density**
Excess negativecharge density Electrode
xH
(^)
s
Distance
Potential HelmholtzLayer
C (^) DL ca. 0.6 nm
The electrified interface.
- The interface between two dissimilar interfaces is
electrified. Almost all surfaces carry an excess electric
charge.
- Hence when two dissimilar phases come into contact, charge
separation occurs in the interfacial region which results in
the generation of an interfacial potential difference or
electric field.
Excess negative charge density
Excess positive charge density
E field generated Bottom line : an electrical Double layer is set up at M/S interface.
Electro-neutrality is valid in bulk
solution.
- Consider a metal electrode in contact with an aqueous solution
containing salt (e.g KCl (aq)).
- The solution contains solvated charged ions and solvent dipoles.
- Forces experienced by ions and solvent molecules in bulk of
solution are isotropic : spherical symmetry operates. Ions and
water molecules (on a time average) experience forces which are
position and direction independent.
- There is no net alignment of solvent dipoles, and positive and
negative ions are equally distributed throughout any volume
element of the solution.
- Electroneutrality operates in bulk solution region very far from
electrode surface.
Electroneutrality breaks down
in surface region.
- What about the solution region next to the electrode
surface?
- In this region forces experienced by ions and solvent dipoles
are no longer isotropic and homogeneous. The forces are
anisotropic because of the presence of the electrode phase.
- New solvent structure, different from that of the bulk,
develops because of the phase boundary.
- Electro-neutrality breaks down on the solution side of the
interface.
- There will be a net orientation of solvent dipoles and a net
excess charge in any volume element of the solution
adjacent to the electrode surface.
- The solution side of the interface becomes electrified.
Solution
Excess positive charge density
Excess negative charge density
Electric field present at interface
Electrode
Solution
**+
+**
Excess positive charge density
Excess negative charge density Electrode
x (^) H
(^)
s
Distance
Potential
Helmholtz Layer
ca. 0.6 nm CDL
Simple representation of electrode/solution
interphase region : Helmholtz
compact layer model.
H
r DL H x
C C
0
C 20 60 Fcm ^2 Structure of thin double DL^ layer region modelled as a parallel plate capacitor with a plate separation of molecular dimension.
Solution
**+
+**
+ (^) -
**-
Excess positivecharge density**
Excess negativecharge density Electrode
xH
(^)
s
Distance
Potential
Helmholtz Layer
C ca. 0.6 nm DL
Numerical calculations
Using the Helmholtz
Model.
We need the following Relationships from basic Physics.
CH
Surface charge density on metal (Cm-2^ )
Helmholtz Capacitance (Fm-2^ )
Interfacial Potential difference (V)
H
r H x
C
0 = permittivity of vacuum = 8.854 x 10-12^ Fm - r = dielectric constant of solution.
Distance between plates of capacitor
- A fundamental problem is assigning a value for the dielectric constant of the solvent in the thin Helmholtz region.
- Solvent structure in this region differs considerably from that of the bulk solution.
- Have considerable dielectric saturation effects and so dielectric constant will be much lower than that associated with the bulk solution.
- The dielectric constant may also vary rapidly with distance in interface region.
( ) 5 6
( ) 78
Helmholtz
bulk r
r
x H
E
Electric field Strength (Vm-1)
Potential Distribution in Helmholtz Compact Layer.
( x )
x
xH
M
S
Potential distribution obtained using the Poisson-Boltzmann equation which relates charge density and electrostatic potential .
dx r
d
0
2
2
Ions treated as point charges. Hence can assume that excess charge density between electrode surface and OHP is zero, hence = 0.
x
dx
d
dx
d
2
M
H
M S
H S
M
x
x x
x
x x
x H
M S M
Linear potential profile in compact layer.
Radius of solvated ion = x (^) H
Simple models are not always
good ones.
- The simple Helmholtz picture is not complete since it predicts: - The double layer
capacitance is a constant
independent of ion
concentration and
electrode potential
Whereas:
- Experiment indicates that
the double layer
capacitance varies with
both of these quantities
in a definite manner.
- A more elaborate model is required.
**-
-**
+ (^) -
**-
-**
Excess positive charge density
Electrode
**-
-**
s
x
ca. 1-10 nm
LD
Excess negative charge density
Solution
Gouy-Chapman model of diffuse double layer.
D
D r
D D
r D
r
D B
DL D r
L
C
C C
L
kT
ze L
C C
, 0 0
, 0 0
(^0000)
0 0
cosh
cosh cosh
2
cosh
Valid when 0 is small.
d 4 zec LD sinh 0
Charge density in diffuse layer
L (^) D = Debye Length. Measures diffuse layer thickness.
cosh 1
0
0
At potential of Zero charge:
0 0
D
r D L
C
0 , 0
(^) M D
Diffuse layer thickness.
The diffuse layer thickness is called the Debye Length and is given the symbol L (^) D. In many books this is denoted as 1/. For a z,z electrolyte the Debye length is given by the expression across. Evaluation of the constants gives a useful expression for computation.
1 / 2 2 2
0
1 / 2
(^2 2 22)
(^) zFc
RT
zec
L kBT r D
^
c
T z
LD r
- 3 1011
m (^) mol m- or mM
c/mol m-
1 10 100 1000
LD
/m
1e-
1e-
1e-
Note that the Debye Length increases as the ionic concentration decreases. The diffuse layer thickness will be greatest for the most dilute solutions.
9 1 /^2
LD c
(1,1) electrolyte, water r = 78, T = 298K
Stern model of the
interface region.
- Neither the Helmholtz compact layer model nor the Gouy- Chapman diffuse layer model is totally satisfactory.
- In the GC model the solvated ions are modelled as point charges. This neglect of ion size is unrealistic. In reality the solvated ion can only approach the electrode surface to a distance equal to its solvated radius a.
- Hence a more logical approach is to combine the features of the Helmholtz and Gouy- Chapman models. This was done by Stern. - The Stern model is as follows. Next to the electrode we have a region of high electric field and low dielectric constant (r value ca. 6) with a row of firmly held counter ions. Beyond that there is an ionic atmosphere (the diffuse layer) where there is a balance between the ordering electrostatic force and disordering thermal motions. The dielectric constant increases rapidly with distance in this region. - The electrical potential varies linearly with distance (ca. hydrated ion radius) within the inner compact layer and decreases in an approximate exponential manner with distance within the diffuse layer, decaying to zero in the bulk solution.
CDL CH C D
1 1 1
Stern model of solid/solution interphase region.
CDL CH CD
Series arrangement of capacitors.
The smaller of the two capacitances will determine the overall capacitance. If CH and CD are of very different size then the term containing the larger one may be neglected.
The diffuse layer capacitance will predominate when the solution concentration is low.
The Poisson-Boltzmann equation (II).
We now need to evaluate the charge density . The volume density of charge is obtained by adding together the product of the charge qj and concentration c (^) j of each ionic species j in the solution next to the electrode surface.
j
j j j
qj cj zec
Ion valence
fundamental charge We use the Boltzmann equation of statistical mechanics to obtain a relationship between the local counterion concentration c (^) j and the bulk concentration cj. To do this we need to evaluate the work w (^) j done in bringing the ion from a reference point at infinity , up to a point distance r from the electrode surface. We assume that this work is purely electrical in nature.
k T
w c c B
j j j exp
w (^) j ( r ) qj r zje ( r )
k T
ze r c c B
j j j
( ) exp
The Poisson-Boltzmann equation (III).
We are now in a position to write down the PB equation. This is a fairly complicated equation to solve from first principles. The exact form of the differential equation depends on the geometry. We shall assume a z,z electrolyte such as KCl or NaCl. The geometry determines the form that the ^2 operator takes. A planar geometry is used for macroelectrodes, whereas a spherical geometry is adopted for ultramicroelectrodes.
j (^) B
j j j
j
j j
kT
ze r
zec
zec r
r
exp
2 ()^1
dr
d
r
dr
d
r
dx
d
2 2
2
2
2 2
Planar geometry
Spherical geometry
c c c
z z z z,z valent electrolyte
k T
zec ze
kT
ze kT
zec ze dx
d x
B
B B
(^2) sinh
2 exp exp
2
k T
zec ze
kT
ze kT
zec ze dr
rd dr
d r
B
B B
(^2) sinh
(^12 2) exp exp The PB equation is solved for .
c
kT
ze
L D B
1 1 Debye Length, z,z electrolyte
The Poisson Boltzmann equation (IV).
- The PB equation fully describes the pertinent electrical properties of the diffuse layer. However it can only be solved analytically for a few special situations. For the most general cases a numerical solution has to be adopted.
- The PB equation for flat planar surfaces can be rigorously solved for z,z electrolytes.
- This cannot be done rigorously for spherical surfaces. Approximate solutions of varying degrees of accuracy have been produced. •A reasonable approach valid both for planar and spherical interfaces involves the Debye-Huckel approximation , which results in the transformation of the PB equation into a linear form as indicated across. This approximation will be valid provided that the potential at the surface of the electrode is not too large.
2 2
2
2 2
dx
d
Planar geometry
(^)
L D
x
ze kT ^0 exp^ x^^ ^0 exp
B
DH approximation
Approximate form Of potential distribution
sinh 2 0 2
1 2
2
d
d
2 tanh^1 tanh 02 exp
0 exp
2 tanh^1 exp
Variation of electrostatic potential with distance in the diffuse layer region. The potential is effectively exponentially decaying with distance from solid surface.
Poisson/Boltzmann equation : planar surface.
k T
ze
kT
ze
B
B
2
2 0 ^0
x L
x D
Thickness of Diffuse layer
Large surface potential
Small surface potential Debye-Huckel approximation
Normalised potential Normalised distance
Full solution of PB equation
Neglect compact layer
Variation of diffuse layer
capacitance with
potential.
How good is the diffuse layer
Theory in practice?
Diffuse layer model also applies for colloidal particle/solution interface. Double layer modelling still being Performed at research level to various degrees of sophistication.
0
2
0
2
coth
tanh
f c
c
f c
c
0 lntanh
0 lntanh
0 0 0
0 0 0
f
f
Electroneutrality breakdown in diffuse layer region : Planar surface.
Counterion concentration increases close to charged solid surface and co-ion concentration decreases close to charged surface.
Co-ion depletion
Counter ion excess
Typical variation of CDL with applied potential. Hg/aqueous KCl interface.
Constant capacity Region.
Capacitance minimum
Capacitance maximum
Explained by Helmholtz model
Explained by Gouy-Chapman model
Modern models incorporating specific adsorption of ions in the inner compact layer, allied with a model for the water structure in the inner layer explain the capacitance maximum
CDL
RCT Electrode/solution interface
Electrical equivalent circuit
Ideally Polarizable Interface : (^) RCT
Ideally non-polarizable Interface: RCT 0
Polarizable and non-polarizable interfaces.
No leakage of Charge across M/S interface
Charge transfer occurs across M/S interface
Measures ET Across interface
Interfacial structure
CDL
RCT
i
i (^) C
i (^) F
RS
Electrode
Solution
Simple equivalent circuit representation of electrode/solution interface region.
Faradaic current
DL charging current
i iC i F
Resistance of solution
Double layer charging current always present in addition to Faradaic current in electrochemical measurements.
Evaluation of CDL (and hence ic ) always necessary when making kinetic measurements at short timescales.
Experimental interrogation of
electrode/solution interfaces.
- Conventional electrochemical techniques.
- Based on measurement of current, potential and charge.
- CV, RDV, RRDV, PSCA, CIS etc.
- Applied both to macrosized and microelectrodes.
- Theory, instrumentation , and practice well developed.
- No direct information on microscopic structure of electrode/solution interface.
- Spectroscopic techniques.
- Provides useful chemical information anout species at interfaces.
- FTIR, Raman, UV/VIS, XPS, EXAFS, Ellipsometry, EC/NMR (new technique, very specialised, limited application at present).
- Scanning probe microscopy.
- High resolution topographical imaging of electrode surfaces with atomic resolution. Surface reactivity also probed with high spatial resolution.
- STM, AFM, SECM.
Refer to: P.A. Christensen, A. Hamnett, Techniques and Mechanisms in Electrochemistry, Chapman and Hall, UK, 1994 for details concerning Spectroscopic techniques.
