CHEM465/865, 2006-3, Lecture 18, Oct. 16, 2006

Mass Transport

Tafel plot: reduction of Mn(IV) to Mn(III) at Pt in 7.5 M H2SO4 at 298 K, various concentrations.

E [V] vs. SHE

log 1

0[ j/

(A c

m-2

) ]

-η = E0 - E

E0

0

Corresponds to the following reaction scheme (cathodic

process, reduction), 0η < :

We have seen, how the rate of the reaction is influenced by E and by surface concentrations of involved species. Consider expression for the encountered current:

( ) ( )CCC C,0 s

ox

01exp

F E EIj nFk cA RT

α⎛ ⎞− −⎜ ⎟− = = − −⎜ ⎟⎝ ⎠

Note: sign of current and 0 0E E− <

Tafel-plot: electroanalytical tool to determine

(cathode) exchange current density C,0j

Tafel-slope ( )C

2.31

RTbFα

=−

(cathode) transfer coefficient Cα . Current depends exponentially on E. However, current cannot grow unlimitedly with E. Progress of a reaction is accompanied by concentration variations toward

interior of solution → affects a region that grows in

thickness with time!

Three forms of mass transport: diffusion: nonuniform concentrations + entropic “forces”, acting to smooth the uneven distributions

convection: action of a force on the solution (pump, gas flow in pressure gradient, hydraulic permeation, gravity, etc.). E.g. convection under laminar flow conditions:

ox oxx

c cvt x

∂ ∂= −

∂ ∂

migration: electrostatic effect, voltage variation in solution, difficult to calculate for real solutions due to ion solvation effects – try to avoid migration by adding supporting, inert electrolyte which levels potential variations in solution. Current density in solution with conductivity σ :

S

Ohmjxϕσ ∂

= −∂

Overall: Flux of electrical current at electrode involves electrode kinetics, and 3-dimensional diffusion, convection and migration – consistent treatment of all these effects within one system is impossible, in experiment as well as in theory!

⇒ understand conditions and electrode geometries under which mass transport limitations can be avoided Diffusion-limitations Transport of matter by random molecular motion (maximum entropy → striving for uniform distributions). Two limiting cases (as usual, realistic situations are between these limits):

rate of reaction >> rate of diffusion rate of reaction << rate of diffusion

Which case did we consider so far???

Now: Let’s focus on diffusion Diffusion normal to electrode surface (x-direction)

Fick’s first law (1855): ox

diffcJ Dx

∂= −

∂

Fick’s second law (1855): ox ox

2

2

c cDt x

∂ ∂=

∂ ∂

Fick’s second law: permits prediction of variation of concentration of different species as a function of position and time (need initial and boundary conditions) Important analogy: heat conduction Can be solved with all sorts of boundary and initial condition. Usually, the solution involves Laplace transforms. (for details: see J. Crank, The Mathematics

of Diffusion, 2nd Edition, 1975, Oxford University Press and H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press 1959).

Evaluate: How does diffusion affect current measured at an electrode?

Semiempirical treatment Comparison with full analytical solution Potential sweep experiment

Potential step experiment (using potentiostat)

E0

t0

E

1. Current limited entirely by diffusion

surface concentration: sox 0c ≈

oxc

x/µm10 20 30 Every reactant molecule that arrives at interface reacts immediately In which region of the Tafel-plot are we now? Region, where current does not depend on E any more!

oxc

x/µm10 20 30

δ

How deep does the disturbance penetrate into the solution? ⇒ mean free path or diffusion layer thickness:

Dtδ π≈

e.g. oxygen in solution: 2cm

s52.55 10D −≈ ⋅

⇒ µm90δ ≈ after 1s

What are the currents that diffusion can sustain?

s b box ox ox ox

diffc c c cj nFD nFD nFDx δ δ

∂ −= − ≈ − =

∂

Insert δ: b

diff oxDj nFctπ

= Cottrell-equation

boxc (no current)0E E≈

oxc

x/µm

δ10-3s

10-2s

10-1s

b (diffusion limited current)0E E−

10 20 30

0E E b− >>

Diffusion-limited current decreases with time! The longer a measurement takes, the more severe are the diffusion limitations. Reactant supplied from more distant regions.

How do concentration profiles and ( )diffj t look like?

box

2

0 2

,C mol,

mol l, cm s,

, A cm

5

5

1964851

2.55 100.510

nFc

D

jα

−

−

==

=

= ⋅=

=

0.0 0.5 1.00

1

2

3

j diff /

A c

m-2

t/s

bdiff ox

Dj nFctπ

=

2. Kinetic limitations and mass transport: MIXED KINETICS!

Finite surface concentration 0c >sox and balance of fluxes

⇒ boundary condition at electrode surface:

bs soxox ox and

00

00

exp

1 exp

E Ecc j nFk c

bE EtkD bπ

⎛ ⎞−⎜ ⎟= =⎜ ⎟⎛ ⎞− ⎝ ⎠⎜ ⎟+

⎜ ⎟⎝ ⎠

( )

Cdiff

Csox

rate of consumption (kinetics) = rate of supply (transport)

=

0

01

exp

j j

F E EnFk c

RTα⎛ ⎞− −

⎜ ⎟⎜ ⎟⎝ ⎠

ox

b ss ox oxox

sox

=

=

again use simple semiempirical treatment

insert

0

00

0

-

exp

exp

x

cnFD

x

E E c ck c D

b

Ek c

δ

δ

=

∂∂

⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠

− ( )b sox ox

sox

=

and solve f

or

0E Dc cb t

c

π

⎛ ⎞⎜ ⎟ −⎜ ⎟⎝ ⎠

and, thus

C box, wher e

0

0* 0* 00

0

exp

1 exp

E Eb

j j j nFk cE Etk

D bπ

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠= =

⎛ ⎞−⎜ ⎟+⎜ ⎟⎝ ⎠

With definitions of diffj and 0*j this can be rewritten

C bdiff ox

*diff

with (Cottrell)

E Eb Dj j j nFc

tE Ejj b

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠= =

⎛ ⎞−⎜ ⎟+⎜ ⎟⎝ ⎠

0

0* *00*

exp

1 ex

,

pπ

Be aware that diffj is a function of time!

Two important limiting cases are reproduced

Kinetic limitations more severe, diffj j<<0* * :

C 0

0* expE E

j jb

⎛ ⎞−⎜ ⎟=⎜ ⎟⎝ ⎠

Tafel-equation

Diffusion limitations more severe, diffj j>>0* * :

C box Dj nFc

tπ= Cottrell-equation

Two handles to steer between the limiting cases: t and E

Show results for

box

2

0* 2

0

, C mol,

mol l, cm s,

, A cm cm s

mV (298 K)

5

5

7

1964851

2.55 100.51010 ,

25.7

nFc

D

jk

RTbF

α

−

−

−

==

=

= ⋅=

=

=

⎛ ⎞= =⎜ ⎟⎝ ⎠

Reactant distribution and ( )Cj t for fixed V0 0.7E E− = .

0.000 0.002 0.0040.0000

0.0005

0.001010-1s10-3s 10-2s10-4s

c

x/cm 0.00 0.05 0.100

2

4

6

8 jt=0

j / A

cm

-2

t/s

η = 0.7 V

Consider: measurement of current-voltage relationship (potential-step experiment). How do you perform the measurement?

|E-E0|

0

t/s0

1

2

3

4

5

t3t2t1

After which time do you record the current? Assume: each point a new step from equilibrium. The measurement time is a property of the equipment (How many charges do you have to collect to reach appropriate accuracy current measurement).

Tafel-plots recorded with distinct measurement times:

0.0 0.5 1.0

-4

-2

0t = 1 st = 10-1 st = 10-2 st = 10-3 st = 10-4 s

2b

log 10

(j / A

cm-2)

|η|/V

kinetics diffusion

less sensitiveequipment

|E-E0| / V

Compare with full solution

ox ox

box ox

b s oxox ox ox

Diffusion equation:

Initial condition (t = 0): everywhere

Boundary conditi

ons:

a

nd

2

2

00

0

e: x0 : p -x

x x

c cD

t xc c

E E cc c k c D

b x=

∂ ∂=

∂ ∂=

⎛ ⎞− ∂⎜→ ∞ ⎟= =⎜

⎠=

⎟ ∂⎝

soxox

wh ere 00

0

- exp,x

E Ec kKc Kx D b

=

⎛ ⎞−∂ ⎜ ⎟⇒ = =⎜ ⎟∂ ⎝ ⎠

Solution: straightforward mathematics (use Laplace transform) can be found in H.S. Carslaw, J.C. Jaeger, Conduction

of Heat in Solids, Oxford University Press 1959.

( ) ( )

( ) ( ) ( )

( ) ( )

oxbox

oxbox

Variation of concentration with and

Concentration at x = 0

x t

erfc erfc

erfc

er

:

Current density:

fc

2

2

00* 2

,1 exp ,

4 4

0,exp

exp exp

c x t x xKx K Dt K Dtc Dt Dt

c x tK Dt K Dt

c

E Ej j K Dt K Dt

b

⎛ ⎞ ⎛ ⎞= − + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

==

⎛ ⎞−⎜ ⎟=⎜ ⎟⎝ ⎠

Compare the two solutions: good correspondence!

0.000 0.002 0.0040.0000

0.0005

0.001010-3s 10-1s10-2s10-4s

c / m

ol c

m-3

x / cm

Approximate solution is sufficient for a basic understanding. The semiempirical approximation works pretty well.

0.00 0.05 0.100

2

4

6

8

approximation

exact

jt=0j /

A c

m-2

t/s

Consider potential sweep experiment: 0E E At− =

0.0 0.5 1.00.00

0.05

0.10

onset ofreduction

diffusion

sweep rate

|E-E0| = 100 mV/s * t

j / A

cm-2

|E-E0| / V

How to confine concentration variations to a thin region and not let them become limiting?

Control transport: vigorous stirring (e.g. rotating disc electrode) and supporting electrolyte

⇒ good transport due to convection and migration

Control electrode geometry: microelectrodes

Diffusion Overpotential

In a voltammetric experiment, the electrode potential E is controlled with a potentiostat (chronoamperometric measurement). The current density j is determined by the following relation

C

*diff

( current through ce ) ll

E Eb

j j j jE Ej

j b

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠= ≡

⎛ ⎞−⎜ ⎟+⎜ ⎟⎝ ⎠

0

0*00*

,

exp

1 exp

.

This can be easily rewritten as (j-η relation)

*diff

bj j

jj b

⎛ ⎞⎜ ⎟⎝ ⎠=

⎛ ⎞+ ⎜ ⎟

⎝ ⎠

00

exp

1 exp

η

η .

Now assume that the experiment is performed under current control (chronopotentiometric measurement).

What is the value of η corresponding to a fixed value of j ?

Solve for |η| …

*diff

j jj j b b⎛ ⎞⎛ ⎞ ⎛ ⎞

+ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

0

0 1 exp expη η

*diff

j jj j b

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

0 1 expη

Take ln on both sides and collect all j-dependent terms on right hand side:

K diff*diff

j jb bj j

⎛ ⎞⎛ ⎞= − − = +⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠0ln ln 1η η η

First term: usual Tafel-equation in absence of mass

transport limitations – reaction overpotential: Kη

Second term: overpotential due to mass transport

limitations – diffusion overpotyential: diffη

0.0 0.2 0.4 0.6 0.8 1.00.7

0.8

0.9

Ecell =1

.23

V -

|ηk|-

|ηdi

ff| / V

j / Acm-2

0.0 0.5 1.00.0

0.2

0.4

jdiff = 1.1 Acm-2

j0 = 10-7 Acm-2

diffusion, ηdiff

kinetics, ηk

over

pote

ntia

ls

j / Acm-2