CHEM465/865, 2006-3, Lecture 10, 27nd Sep.
, 2006
Gouy-Chapman model (1910)
The double layer is not as compact as in Helmholtz rigid layer.
Consider thermal motions of ions:
Tendency to increase the entropy and make the layer diffuse
(while electrostatic attraction tends to keep ions close to
interface and make the layer more rigid).
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0 0 0 0 0 0 0 0 0 0 Metal 00000000000000000000000
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Distribution of ions: determined by
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 electrostatics and stat. mechanics
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 Assumptions:
000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 Ions: point ions, highly mobile
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 Solvent: dielectric continuum
000000000000000000000000000000000
Metal: perfect conductor
σM
0 x
Distribution of ions with distance x from interface:
apply Boltzmann statistics
ni = n 0
zF
exp −
i (ϕ ( x ) − ϕ )
S
i
RT
ni: particle density (#particles/cm3)
Regions of large ϕ ( x ) : reduced density of positive ions,
enhanced density of anions (compared to bulk).
Charge density involving all ionic species:
ρ ( x ) = ∑ ni zie0 with e0 = 1.602 ⋅ 10-19 C
i
=∑
ni zie0 exp −
0 (
zi F ϕ ( x ) − ϕ S )
i RT
Galvani potential in solution obeys Poisson’s equation:
d2ϕ ( x ) ρ ( x)
=−
dx 2 εε 0
Resulting expressions is the Poisson-Boltzmann equation:
∑ ni0 ziq exp −
d2ϕ ∆ 1 zi F ϕ ∆ ( x )
=− , where ϕ ∆ ( x ) = ϕ ( x ) − ϕ
S
dx 2
εε 0 i RT
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Note: This equation is precisely equal to the Debye-Hückel
theory of ionic interactions in dilute solutions, which determines
the charge distribution around a central ion.
Next: Let’s consider a z-z-electrolyte
zFϕ ∆ ( x ) zFϕ ∆ ( x )
(*) ρ ( x ) = n0 zq exp − − exp
RT RT
Debye-Hückel approximation
Let’s further assume that the potential variation is small,
zF ϕ ∆ ( x )
<< 1
RT
i.e. potential variations are ϕ ∆ << 25 mV at room-T. This well-
known approximation corresponds to the so-called
Debye-Hückel approximation.
If we use this approximation in the charge distribution (*) of the
z-z electrolyte and then insert this charge distribution into the
Poisson-Boltzmann equation, then we arrive at the linear
Poisson-Boltzmann equation:
d2ϕ ∆
2
= κ 2
ϕ∆
dx
Here, κ is the inverse Debye-length,
1/ 2
( )
2 zF 2 c 0 1
κ = =
εε 0 RT LD ,
c 0 : bulk electrolyte concentration (far away from interface)
κ or its inverse LD are important characteristics of the
electrolyte.
Debye-length LD : quantitative measure of width of space charge
region within which ϕ ( x ) decreases from ϕ to ϕ
M S
–
tremendous importance in electrochemistry and biology.
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The following effects are important to remember:
Larger electrolyte concentration c
0
→ smaller L
D
i.a.w: the double layer becomes less diffuse
Higher temperature → larger L D
i.a.w.: the double layer becomes more diffuse
Potential and charge distribution in the electrolyte solution are
given in Debye-Hückel approximation by:
σM
ϕ∆ ( x ) = exp ( −κ x )
εε 0κ
ρ ( x ) = −σ Mκ exp ( −κ x )
i.e. they are exponential functions of the distance from the
interface, i.e. the excess surface charge density σ M on the
metal is balanced by an exponentially decaying space-charge
layer in solution.
The double layer capacity in this approximation is given by
εε 0
Cd,DH = εε 0κ =
LD
i.e. it is given by the plate capacitor formula with the Debye
length as the effective plate separation.
Table: Debye length at various electrolyte concentrations
(1-1 electrolyte)
c0 / mol l-1 10-4 10-3 10-2 10-1
LD / Å 304 96 30.4 9.6
Higher concentrations:
steeper potential drop in solution
narrower space charge region, smaller LD
higher double layer capacities, C d,DH
In practice: The Debye-Hückel (DH) )approximation is not valid at
large electrolyte concentration. It works well as long as the
potential variation does not exceed ϕ ∆ ≈ 80 mV .
General Case: Nonlinear Poisson-Boltzmann-equation
For a z-z-electrolyte with concentration c0 the Poisson-
Boltzmann equation has an explicit solution. This will be added
as an appendix. Here only the main results will be given.
Potential distribution in solution
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00 00 00 00 00 00 00 00 00 00 00 00Metal 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 Solution
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ϕ0 (interface)
000000000000000000000000000000000000000 ϕ∆,0 = ϕ0 - ϕS
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000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ϕS (bulk)
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σM
0 x
The relation between potential and charge density of the diffuse
layer is given by
dϕ ∆ zF ϕ ∆ ,0
= ( 8 RT εε 0 c 0 )
1/ 2
σ M = εε 0 sinh
dx x = 0 2 RT
Differentiating this expression gives the differential capacity in
the Gouy-Chapman model:
∂σ M εε 0 zF ϕ ∆ ,0
Cd,GC = = cosh , where ϕ ∆ ,0 = E − Epzc
∂ϕ LD 2 RT
∆ ,0
The variation of this capacitance with electrode potential E is
thus given by
Cd,GC This model predicts correctly the
minimum in capacity at the pzc.
For large E, it predicts an
unlimited rise of the capacity.
0 E-Epzc
Why would the DL capacity rise unlimitedly in this approach?
Which assumption is responsible for this unphysical behaviour?
It is assumed that ions are point charges, which could approach
the electrode surface arbitrarily close with increasing E. This
leads to a very small charge separation (consider the plate
capacitor as an analogue!). The capacity rises without limit.
What is the solution out of this “capacity catastrophe”???
Ions have a finite size! Arbitrarily close approach is not possible.
Ions are stopped from approaching the electrode at distances
that correspond to their radii.
A further refinement of the double layer models takes this effect
into account.
The Stern Model (1924)
accounts for finite size of ions
compact diffuse
layer layer
combines the Helmholtz and Gouy- 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
Chapmann models 00 00 00 00 00 00 00 00 00 00Metal 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 Solution
00000000000000000000000000000000
two parts of double layer: 00000000000000000000000000000000
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(a) compact layer (“rigid layer”) of ions 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
at distance of closes approach
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00000000000000000000000000000000
00000000000000000000000000000000
(b) diffuse layer. 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000
The compact layer, x < xH, is charge- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
free (linear variation of potential)! σM
0
xH
Mathematically: two capacitors in series with total capacity
1 1 1 x LD
= + = H +
Cd Cdiff,H Cdiff,GC εε 0 zF ϕ ∆ ,0
εε 0 cosh
2 RT
Far from Epzc (large ϕ ∆ ,0 = E − Epzc ):
C diff,H << C diff,GC ⇒C d ≅ C diff,H Helmholtz, rigid
Close to Epzc (small ϕ ∆ ,0 = E − Epzc ):
C diff,H >> C diff,GC ⇒C d ≅ C diff,GC Gouy-Chapman, diffuse
Remember: the smaller guy always wins (well, not always, but
here)!
Effect of electrolyte concentration:
smaller electrolyte concentration
smaller κ (LD larger), double is more diffuse!
C becomes more important
diff,GC
The Stern model reproduces gross features of real systems.
Potential distribution in interfacial layer and capacity variation
with E are shown in the figures below.
What happens upon increasing the electrolyte concentration?
Potential distribution Double layer capacitance
charge mobile
free ions
Further refinement: Grahame model (1947)
Some ions (usually anions) loose hydration shell – smaller radii!
Ions of smaller radii could approach the electrode closer.
Distinguish three different regions!
Inner Helmholtz plane (IHP): through centers of small,
partially solvated ions
Outer Helmholtz plane (OHP): through centers of fully
solvated ions
Outside OHP: diffuse layer
This is the best model so far. It shows good correspondence to
the experimental data.
Why are the potential distributions (in (b)) different for positive
and negative E?
This is due to a difference between anions and cations. Anions
have a less rigid solvation shell. They become more easily
desolvated. Thus, desolvated anions could form the IHP. This
happens at positive E. Cations keep their solvation shell. They,
thus, cannot approach the electrode closer than to the OHP.
Appendix: Electrocapillary measurements
Only applicable to liquid electrodes (metals), based on
measurement of surface tension (Lippmann’s method). The
method was developed specifically for mercury electrodes.
This is a null-point technique, i.e. the measurement of a physical
property is performed when the system has reached an
equilibrium state. Such measurements are rather accurate.
Balanced forces: Surface tension of mercury in the capillary
counterbalances the force of gravity
2π rcγ cos θ = π rc 2 ρHg hg
rc is the capillary radius,
γ is the surface tension,
θ is the contact angle
h is the height of the capillary column of Hg.
In the measurement, the contact angle θ is measured with a
microscope. Above relation then provides values of the surface
tension γ. As the experimental parameter, the electrode potential
E can be varied.
What happens upon changing E?
When the system is balanced (reaching a new equilibrium –
minimum of Gibbs free energy!), the change in surface energy is
exactly balanced by the electrical work,
− Asdγ = QdE
where As is the surface area and Q is the excess charge on the
surface.
Therefore, the charge density on the Hg electrode is given by
Q dγ
σM = =−
As dE .
It is, thus, possible to measure the surface charge as a function
of E.
