Electrode kine-cs 1 Overview
•
•
•
•
•
•
•
•
•
•
Types of reac-ons Reac-on rates Arrhenius dependence Redox reac-ons Poten-al dependence of rate constants Butler-­‐Volmer theory Limi-ng cases Summary of Butler Volmer Transport processes Diffusion control 2 Types of reac-ons in electrochemical cells • A redox reac-on involves charge transfer across an electrode/electrolyte interface • The electrode conducts the electrons and the electrolyte (or membrane) conducts counterions (e.g., protons) • Three main types of electrode processes may be iden-fied: (a) ionic redox reac-ons, (b) gaseous redox reac-ons, and (c) phase change reac-ons. • In an ionic redox electrode process both the reactant and product are water-­‐soluble ionic species, although they have different charges. • In gaseous redox reac1ons a reactant or a product is a gas, e.g., oxygen, hydrogen, chlorine. If a product it will form bubbles if the electrolyte is a liquid 3 Types of reac-ons in electrochemical cells • Phase change electrode processes may involve phase forma-on or dissolu-on • In the case of electrodissolu1on, a surface metal ion detaches and enters the aqueous phase as a hydrated ion; the associated electron(s) is donated to the electrode • In electrodeposi1on, a hydrated metal ion receives electrons from the electrode and the resul-ng electroneutral metal becomes incorporated into the crystal structure of the metallic deposit • Faradaic reac-ons involve a transfer of charge directly by oxida-on and reduc-on (we will not look at non faradaic reac-ons) 4 Reac-on rates Consider a reac-on A↵B
(1) • The forward and backward rates of the reac-on are given by: rf = k f C A
(2) rb = k b C B
• kf and kb are rate constants (note that they are not constant!) • The net rate is (3) r = rf r b = k f C A k b C B
5 Reac-on rates • At equilibrium rf = rb so that kf
CB
=
=K
kb
CA
• K is called the equilibrium constant • Every kine-c theory requires that the kine-c equa-ons collapse to thermodynamic rela-ons (e.g. Nernst equa-on) for equilibrium condi-onds • Thermodynamics provides a descrip-on of the behaviour at equilibrium • Tells us whether a reac-on is feasible (spontaneous), • Informa-on about the reac-on rates is provided by kine-c theory 6 Arrhenius dependence • Rate constants vary with temperature exponen-ally (Arrhenius) k = A exp( Ea /(RT ))
(4) • Ea is the ac-va-on energy; R is the gas constant; A is called a pre-­‐exponen-al factor • Pre-­‐exponen-al factor usually depends algebraically on T
• The ac-va-on energy is an energy barrier that has to be surmounted before the reactants undergo reac-on 7 Arrhenius dependence • The exponen-al term has a probabilis-c interpreta-on (Boltzmann): probability of surmoun1ng the energy barrier. The higher the temperature the higher the probability • The pre-­‐exponen-al (aka frequency factor) is propor-onal to the number of a]empts at surmoun-ng the ac-va-on barrier before success 8 Redox reac-ons • Simplest redox reac-on is of the form (5) O + ne ↵ R
• O is the oxidized state and R is the reduced state. n is the number of electrons transferred • An oxida-on reac-on is one in which electrons are liberated. It occurs at an anode (nega-ve electrode) • A reduc-on reac-on is one in which electrons are taken up. It occurs at a cathode (posi-ve electrode) 9 Redox reac-ons • The equilibrium state is governed by Nernst’s equa-on Eeq = E 0
RT
ln
nF
✓
CR
CO
◆
(6) • Relates the equilibrium poten-al of the electrode to the concentra-ons of the reactants and products • Kine-c theory must reduce to the Nernst equa-on at equilibrium • More generally, for a reac-on νOO+ne ⇔ νRR
Eeq = E 0
RT
ln
nF
✓
⌫R
CR
⌫O
CO
◆
νR
⎡
⎛
⎞⎤
RT
α
0
R
ln ⎜ νO ⎟ ⎥
⎢ Eeq = E −
nF ⎝ α O ⎠ ⎦
⎣
(6b) • Formally the concentra-ons should be ac-vi-es as shown in brackets 10 Redox reac-ons • Need to know the dependence of current on electrode poten-al • When a current is passed a change occurs at the electrode • Passage of a current incurs a penalty in terms of the electrode poten-al called overvoltage or overpoten-al – due to irreversibili-es ⌘ = Eeq E
(7) • The actual poten-al is E and the overpoten-al measures the departure of the actual poten-al from the equilibrium value • We will come back to the overpoten-al later 11 Redox reac-ons • At low currents, it is experimentally observed that the current is exponen-ally related to overpoten-al (8) ⌘ = a + b ln i
• i = current density, a and b are constants • Current density is per unit area • This equa-ons is called the Tafel equa-on • It is the basis of several techniques to measure electrochemical kine-c parameters 12 Redox reac-ons • For the redox reac-on (5), we can write the forward rate as Reac-on rate [mol m s ] × nF = charge transferred per second per rf = kf CO (0, t) = ic /nF
unit area = current density • C O (0,
t) is the surface concentra-on of O (at the cathode) and ic is the cathodic current density • Rate of the backward reac-on is: rb = kb CR (0, t) = ia /nF
-­‐2 -­‐1
• C
R (0,
t) is the surface concentra-on of R (at the cathode) and ia is the anodic current 13 Redox reac-ons • Net rate of reac-on is rnet = rf
i
ic ia
rb =
=
= kf CO (0, t)
nF
nF
kb CR (0, t)
• The reac-on constants are poten-al dependent • The forward reac-on (a reduc-on) is an electron accep-ng process -­‐ the rate increases as the electrode poten1al becomes more posi1ve. • For the backward (oxida-on) reac-on, the rate increases as the electrode poten1al becomes more nega1ve 14 Poten-al dependence of k
• Consider the point at which the net reac-on rate is zero (equilibrium): (9) kf CO (0, t) = kb CR (0, t)
• This occurs at zero net current because no net electrons transferred • Taking logs in (9): CO (0, t)
ln(kb ) ln(kf ) = ln
(10) C (0, t)
ic = ia
or
rf = rb
R
15 Poten-al dependence of k
• Since this is at equilibrium, we can use Nernst’s equa-on (6) in (10) to obtain (for n = 1) nF
(11) ln(kb ) ln(kf ) =
(E E 0 )
RT
• Differen-a-ng both sides w.r.t. E: RT
nF
✓
d
d
1
ln kb +
ln
dE
dE kf
◆
=1
16 Poten-al dependence of k
• This says that for a 1 electron transfer there are two quan--es: RT d
↵b =
ln kb
F dE
RT d
1
↵f =
ln
F dE kf
that sum to 1, i.e., ↵
b +
↵
f =
1 . Let ↵
f =
↵ . Then ↵b = 1 ↵
• These 2 quan--es are called symmetry factors 17 Poten-al dependence of k
• The reduc-on (or cathodic) symmetry factor (associated with the forward reac-on) is RT d
1
↵=
ln
(12) F dE kf
• The oxida-on (anodic) symmetry factor is 1
RT d
↵=
ln kb
F dE
(13) • α is a measure of the symmetry of the energy barrier to charge transfer. 0.5 means perfect symmetry 18 Poten-al dependence of k
• If the effect of changing the poten-al is iden-cal on both sides of the barrier (on the forward and backward rates), then α=0.5=1-­‐α. • Integra-ng (12): 1
↵F E
ln
=
+ const.
kf
RT
• Use k f =
k f0 when E = E
0 (standard poten-al) ✓
◆
↵F
0
0
kf = kf exp
(E E )
(14) RT
19 Poten-al dependence of k
• Similarly ✓
kb = kb0 exp (1
F
↵)
(E
RT
E0)
◆
(15) • kf0 and kb0 are called standard (or condi-onal) rate constants • If the concentra-ons of O and R are equal and the poten-al is maintained at E0 to prevent current flow, equa-on (9) says that we must have kf0 =
kb0 • We can replace both with a single symbol k0. 20 Poten-al dependence of k
• k0 is called the standard rate constant • The larger the value of k0, the faster (Nernst) equilibrium will be a]ained. • Systems with small standard rate constants have slow kine-cs • k0 therefore a very important parameter in electrode kine-cs. • For simple redox reac-ons k0 is typically large, e.g., hydrogen oxida-on H2 ↵ 2H+ + 2e
21 Poten-al dependence of k
• For complex reac-ons it can be small, e.g., oxygen reduc-on O2 + 4H+ + 4e ↵ 2H2 O
• This reac-on is a mix of a direct 4 electron and a sequence of 2 electron transfers • To have a high rate constant k, need to have a high overpoten-al to counterbalance the low k0. This will reduce the cell voltage 22 Butler Volmer kine-c theory
• Recall that rnet = rf
i
ic ia
rb =
=
= kf CO (0, t)
nF
nF
• Or i = nF (kf CO (0, t)
• Using (14) and (15) ⇣
i = nF k 0 CO (0, t)e
↵nF
RT
kb CR (0, t)
kb CR (0, t))
0
(E E )
CR (0, t)e
(1
↵)nF
RT
0
(E E )
• Note that this is generalised for general n
23 ⌘
Butler Volmer kine-c theory
• At equilibrium (no current flowing) i = ic
ia = 0
0
ic = nF k CO e
E = Eeq = E0
↵nF
RT
(E E 0 )
= ia = nF k CR e
RT /(nF ) ln(CR /CO )
↵ 1
ic = ia = nF k 0 CR
CO
↵
0
(1
↵)nF
RT
(E E 0 )
(N ernst)
= i0
• i0 is called the exchange current density • Although there is no current flowing the forward and backward rates produce (equal) current = i0 24 Butler Volmer kine-c theory
• Recall defini-on of overpoten-al ⌘ = Eeq
E
RT
• Then E = Eeq − η = E −
ln(CR / CO ) − η
nF
0
• A more familiar form of BV is i >0
ia>0 c
⇣ ↵nF ⌘
⌘
(1 ↵)nF ⌘
↵ 1 ↵
(16) i = nF k 0 CR
CO
e RT
e RT
25 Graphical view (constant CO and CR)
Plot of j against η:
ij
ic = nF k
0
Oxidation
Reduc-on (cathode) ⌘
↵ 1 ↵ ↵nF
CR CO e RT
ηη
ic−ia
ia =
nF k
0
↵ 1 ↵
CR
CO e
(1
↵)nF ⌘
RT
Reduction
Oxida-on (anode) 26 Graphical view
i
Current increases slowly
η
Current increases quickly
Increasing α favours cathodic reaction (asymmetry)
27 Graphical view
ther factors also need to be taken into account
he magnitude of the exchange current density (jo) also determi
ential required for a significant current to flow:
Large i value incurs small overpoten-al (electrode poten-al stays near maximum value (magnitude) Decreasingi j
Decreasing
0
ij
o
For high jo:
observable curre
η
η
“Activation” loss
i0 /Am-2
jo /A m-2:
10
102
103
Large i value incurs large overpoten-al (electrode poten-al movers far from maximum value (magnitude) 28 Mass transport effects
Predicted by the
Butler-Volmer equation
i
j
Oxidation:
Red → Ox + ne-
ηη
Reduction:
Ox + ne- → Red
Further increase in |η| does not lead to higher current (magnitude) There is also a limit to how fast reactants can reach
reaction site: leads to plateau in anodic/cathodic branches
29 Limi-ng cases for BV
• For small overpoten-als i⇡
nF k
0
↵ 1 ↵
CR
CO
nF ⌘
=
RT
nF ⌘
i0
RT
⌘
RT
=
= Rct (Ohm’s law) i
nF i0
• Rct is called the charge transfer resistance • For large values of η, one of the anodic/
cathodic branches dominates. End up with Tafel rela-onship ⌘ = a + b ln i
30 Summary of BV
↵ 1
i = nF k 0 CR
CO
•
•
•
•
•
•
↵
⇣
e
↵nF ⌘
RT
e
(1
↵)nF ⌘
RT
⌘
For η = 0, i = 0
For large η, one branch dominates For small η, there is a linear rela-onship between overpoten-al and i
For fixed i, low concentra-on demands high η (polariza-on) Reactant supply limits the current (cannot grow indefinitely with η Can also write in the form (using defini-on of i0) ✓
◆↵ ✓
◆1 ↵ ⇣
⌘
(1 ↵)nF ⌘
↵nF ⌘
CR
CO
RT
i = i0
e RT
e
(17) CR,0
CO,0
• where the subscript ‘0’ indicates standard Nernst equilibrium condi-ons, i.e. ia=ic=i0
• To be more precise we would replace the concentra-on ra-os with ac-vi-es 31 Summary of BV
• An even more general form is for a reac-on νOO+ne ⇔ νRR
α nFη
−(1−α )nFη
⎞
ν (1−α )
ν α ⎛
RT
RT
i = i0 (CO CO,0 )
(CR CR,0 ) ⎜⎝ e − e
⎟⎠
• The deriva-on is the same except that throughout we must take care of the stoichiometric coefficients (including in the Nernst equa-on) – law of mass ac-on • The defini-on of i0 will also be modified accordingly 32 O
R
Transport processes
•
•
•
•
•
•
•
•
•
•
In ba]eries and fuel cells, reactants are transported to the reac-on sites by various mechanisms, depending on the reactant and the system Charge is also transported by conduc-on and by migra-on For non-­‐charged reactants, the transport mechanisms and diffusion and convec-on In a H2 fuel cell, H2 and O2 are in gaseous form and diffuse through the layers to reach the reac-on zone There may be addi-onal resistances such as a diffusion boundary layer In a flow ba]ery, transport of non charged reactants is primarily by convec-on. There may be diffusion control near the reac-on surfaces due to boundary layer effects For charged species, an addi-onal mode of transport is migra-on Under the influence of a poten-al field, charged ions will move at a rate propor-onal to the poten-al gradient This is how counterions (posi-ve or nega-ve) move through the electrolyte, across which an ionic poten-al gradient exists The importance of ion transport is that the main ohmic limita-ons are due to the movement of ions (which is required for electroneutrality and/or for complementary reac-ons 33 Transport processes
• There are four basic factors which control the rate of charge transfer at the electrodes of a fuel cell or ba]ery o Mass transfer (movement of reactants from the bulk solu-on to the electrode surface) o Electron transfer at the electrode-­‐solu-on interface (e-­‐ transfer kine-cs) o Coupled chemical reac-ons, preceding or following electron transfer (e.g., protona-on o Surface reac-ons (adsorp-on, desorp-on, of intermediates) • As we have seen, the poten-al beyond the equilibrium poten-al that is required to draw a current is called overpoten-al η = Eeq − E
• Overpoten-al is a sum of terms that account for the total electrochemical process at the electrode, including o mass transfer o electron-­‐transfer 34 Diffuse double layers
• An electric double layer is formed on surfaces that come into contact with an ionic aqueous solu-on • Consider an anode (+ve electrode). Ca-ons (-­‐ve charge) migrate and adsorb to the electrode surface due to electrosta-c forces • The Stern layer is defined as the compact layer of these immobile ions strongly adsorbed to the electrode surface • Beyond the Stern layer is the so-­‐called diffuse layer where ions are mobile under the coupled influence of electrosta-c forces and diffusion • The two layers together are called an electric double layer (EDL) • There are different models with different levels of complexity for an EDL • The liquid in the EDL is immobile or stagnant, meaning no movement by convec-on (bulk mo-on) • Outside this EDL is the bulk fluid, where electroneutrality holds • The Stern layer has a capacitance effect (charge is separated) but in most cases rela-ng to ba]eries and fuel cells the effect is small and can be ignored • The mass transport in the EDL is however significant 35 Diffuse double layers
Anion (-­‐ve charge ion) Ca-on (+ve charge ion) • Bulk solu-on (electroneutrality) outside the diffuse layer • Length of the diffuse layer is called the Debye length • Oten the concern is only for the mass transport limita-ons in Stern layer Diffuse layer the diffuse layer, where Figure 2: Based on figure 5.28 in [2, p. 170]. Legend: a: charged surface, b: Stern layer, c: shear
Posi-vely charged only o-on by plane, d: diffusive layer, e: surface potential
, f: Stern potential
, g:m
zeta
potential
⇣ diffusion surface (anode) or migra-on is possible S
d
to the Brownian motion and the electrostatic forces. Differently signed charges repel each other,
whereas equally signed charges attract each other.
Without the charged plane, there is no preferred direction of motion. Brownian motion and
electrostatic forces negotiate a homogeneous equilibrium. Within this equilibrium there exists on
average no space charge and therefore no potential difference. The potential in this area, further
36 446
N
●
Nernst–Planck equation
Nernst Layers
Nernst layer — Figure
tained by extrapolation of the linear part of the concentration vs. distance curve to the bulk concentration, as
shown in the Figure. The Nernst-layer approximation
• Some-mes (e.g., in cyclic voltammetry) a boundary This force makes the ion move at veloc
layer or Nernst layer is is equal to the Stokes’ frictional force, 
assumed to by
exist J = −cv,
is given
• This is not the same as a double layer but is a .
Jdiffuse = −Dgradc
− cD(zF#RT)gradϕ
simplified model to account See for alsom!
Onsager
relations,
! migrati
ass transfer limita-ons and
subentry
convective
diffusion.
• This layer !
has no direct Ref.: [i] Atkins PW () Physical chemistry. Oxfo
physical interpreta-on (it is Oxford, pp –
phenomenological) • It is a stagnant region in which transport is governed Nernstian behavior ! Nernst equilibrium
by diffusion • The length of this boundary Nernstian
electrode
reaction
— An ! el
layer is u
but can !
be equilib
proceeding
atnknown electrochemical
the approximated ! Nernst equation is applicable. See
Nernstian reaction ! Nernst equilibrium
37 Nernstian response ! Nernst equilibriu
Transport processes
•
•
•
•
•
•
•
•
All modes of transport in ionic solu-ons are captured in the Nernst Planck equa-on (in 1D) ∂Ci (x) zi F
∂φ (x)
J i (x) = −Di
−
Di Ci
+ Ci v(x)
∂x
RT
∂x
J(x) = flux of species i towards an electrode at a distance x from the electrode (mole m2 s–1 ) Di = diffusion coefficient (m2 s–1) – in a liquid this is around 10–10 m2 s–1 Ci(x) = concentra-on gradient at distance x from electrode ϕ(x) = poten-al gradient at distance x from electrode v(x) = velocity at which species i moves (m s-­‐1) z i is the charge number on i (e.g., 2–, 2+ or 3+) +ve charged species move in the poten-al field from high to low poten-al just as they move from high to low concentra-on 38 Transport processes
• The current density generated by this movement of ions is directly propor-onal to the flux ii = zi FJ i
• The total current density generated by all ions ⎞ ∂φ
∂Ci ⎛ F 2
2
i = ∑ i ii = zi FJi = −F ∑ i zi Di
−⎜
z DiCi ⎟
∑
i i
∂x ⎝ RT
⎠ ∂x
• Note that the last term in the Nernst Planck equa-on disappears when summed because electroneutrality has to hold ( ∑
i z i C i = 0 ) • From this we can define an ionic conduc-vity F2
κ=
RT
∑ z DC
2
i i
i
i
• There are 2 contribu-ons to the current, the first being diffusion current and the second being migra-on current 39 •
•
•
Diffusion control
In many cases, mass transport is controlled by diffusion alone in the Nernst layer The experimental setup (e.g., cyclic voltammetry) can ensure this by the concentra-ons of species and suppor-ng electrolyte Consider the following situa-on z+
A
(s)
O(s)
N
NO
A
CAO
C
A
O
z+
C
CAS
f
O + ne − ←
⎯⎯
⎯
→
R
⎯
k
OS
(rA)
kb
CBSRS
C
R(s)
B(s)
Electrode
N
NBR
Solution
CBR
C
B
R
δ
40 Diffusion control
• Flux of O to the surface (in mol m-­‐2): N O = kO ( CO − COS )
(COS = CO − NO
kO )
(18) • Flux of R from the surface: N R = kR (CRS − CR )
(CRS = CR + N R
kR )
(19) • Rate of consump1on of O by the reac1on (in mol m-­‐2) – see slides 15/16: rO = (1 A ) dnO dt = k f COS − kbCRS = (ic − ia ) nF = inet nF
(20) • Under steady-­‐state condi1ons, we must have (21) N = r = i nF (i = i )
O
O
net
41 Diffusion control
• The mass transfer coefficients kR and kO can be related to the diffusion coefficients of O and R and the Nernst layer thickness kO = DO / δ
kR = DR / δ
• The rate of genera1on of R is: N R = rR = i nF = rO
(22) • Subs1tute (18)/(19) into (20): rO = k f ( CO − N O kO ) − kb ( C R + N R kR )
• According to (21) and (22) N
R =
N
O = r O so rO = k f (CO − rO kO ) − kb (CR + rO kR )
= k f CO − rO k f kO − kbCR − rO ( kb kR )
(
(23) )
42 Diffusion control
• Therefore rO =
(
k f CO − kbCR
1 + k f kO
)
i
=
+ ( kb kR ) nF
• As we have seen (equa1ons (14) and (15)), kf and kb are dependent on poten1al kf =
kf0
exp
✓
↵F
(E
RT
0
E )
◆
✓
kb = kb0 exp (1
↵)
F
(E
RT
• For small E k b 
k f i.e., the cathodic reac1on dominates (reduc1on). In this case 1
1
1
=
+
i nFk f CO nFkOCO
E0)
43 ◆
Diffusion control
• Depending on the rela-ve magnitudes of kf and kO, the electrode reac-on will be kine-c or mass transport controlled • When k O 
k f (kine-c control) i = nFk f CO
• When k f 
k O (mass transport control) i = nFkOCO = nFDOCO / δ
• This is the cathodic limi-ng current, when COS = 0, as seen from (18) and (21). This happens when the reactant is consumed as soon as it reaches the electrode surface and cannot be transferred faster 44 Quiz
Effect of transfer coefficient
10
= 0.5
5
j / A cm
= 0.9
0
-2
i
-5
= 0.1
-10
-0.2
-0.1
0.0
0.1
1. Iden-fy which of these curves corresponds to charge transfer coefficients of 0.1, 0.5 and 0.9 Explain your answer 0.2
η/ V
45 Quiz
2. Consider a reac-on in which the concentra-on of the reduced species is not limi-ng and can be ignored. The Butler-­‐
Volmer equa-on (17) simplifies to •
•
•
•
nFη
−(1−α )nFη
⎛ αRT
⎞
RT
i = i0 ( CO CO,0 ) ⎜ e
−e
⎟⎠
⎝
Iden-fy the cathodic and anodic currents For α = 0.5, write this expression in a simpler form using a hyperbolic func-on If the current of the reac-on is fixed, explain why the overpoten-al grows as the concentra-on of O decreases. What is the limi-ng behaviour What stops i from going to +∞ or −∞? α
46 Quiz
Consider again the equa-on 3. i = i0 (CO CO,0 )
1−α
nFη
−(1−α )nFη
⎛ αRT
⎞
RT
−e
⎜⎝ e
⎟⎠
with α = 0.5 • For a fixed current i = if write down an expression for the overpoten-al in terms of if , i0 , CO , CO,0, n, R, F, and T
• For fixed values of the oxidant concentra-on, explain the trend on slide 30 as the exchange current density is increased 47 Quiz
4. Perform the analysis on slides 46 and 47 for the case k f  kb
to determine the currents in the cases of kine-c and mass transfer control. 48