武汉大学 电化学课件 陈胜利 Chaper 3 -Mass transport

diffusion will be the only predominant mass transport mode within the thin layer near electrode surface ---- diffusion layer.
The changes on the local concentration of solution species due to mass transport
Ji1 Ji2
1 2 x
c 1 J i Di i x x x
c 2 J i Di i x x x dx
dx
(ci ) x x dx
c (ci ) x i dx x
c c 2 J i Di i i dx x x x x x
Navier – Stokes Equation ---- the newton’s first law (F = ma) for a fluid:
dv P 2 v f dt
ma
Force due to pressure gradient Force due to friction Natural convection due to density gradient
u vy
surf
x
u0 v y 0.33 x y
vx

The velocity perpendicular to solid surface
vx

surf 3
x2
x
The effective diffusion layer and its thickness

The convection-diffusion equation
D

Migration ------ Movement of a species driven by a gradient
of electric potential (i.e., electric field).
Ji

M
zi F ci Di RT
Convection ------ Movement of a species with the fluid flow.
ci cib
Convection is necessary for mass transport to reach steady-state
J i mi (cR cR )
b s
cis
Diffusion zone
0
ib
x
Steady-state mass transport in real electrochemical cells due to natural and forced convection
O ne R
x
0
s
0
s
mass transport coefficien t
diffusion coeffcient thickness of diffusion layer
The assumption of “Nernst boundary layer”
A stagnant layer of thickness exists in the vicinity the electrode surface, while the natural or forced convection maintains the concentration uniformly equal to its bulk value beyond x = id

In an electrochemical cell, various kinds of mass transport may function simultaneously, Ji = JiD + JiM + JiC . Usually, some measures may be taken so that only one kind of mass transport contribute predominantly and others are negligible.
Semi-empirical treatment of Steady-state mass transport

ci ci
x
Convection zone Diffusion zone
ci
l A
0
l
x
J i Di As steady - state is reached,
dci ci ci dx l

Steady-state process ---
ci 0 t Due to natural (or forced )
convection, the concentration gradient would not extend unlimitedly toward the bulk. A steady-state profile of concentration will established finally. ci 0 t
»Natural convection caused by density gradient »Forced convection caused by stirring or other hydrodynamic means
J i vci
C
The mass transport in electrochemical systems
x 0
JR
x 0
i nFvnet nFJO
x 0
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nFJ R
x 0
In the case that only diffusion is considered
dcO J O DO dx x 0
dcR J R DR dx x 0
DO 0 s 0 s JO (cO cO ) mO (cO cO ) l DR 0 s 0 s JR (cR cR ) mR (cR cR ) l
mi - - - mass transfer coefficien t of species i
i nFJO mO (cO cO ) nFJ R mR (cR cR )
Transient and Steady-state mass transport processes

Transient process --- The combined processes of the
development of concentration gradient due to interfacial reactions and the diffusion to eliminate the concentration would make the concentration under dynamic changing and a concentration profile gradually extends toward the bulk solution.
Rate of mass transport: the flux of species i, Ji (mol
s-1 cm-2) --- the amount of material passing through a unit area within an interval of unit time
The net rate of interfacial reaction equals to the flux of reactant O toward (or product R away from) the electrode surface at x = 0 O ne R
JO
Ob
Rb
x
JR
vnet J O
In a solution, the natural convection is the predominant mass transport mode and it maintains the concentrations of all species uniform. But the velocity of the fluid decreases sharply upon approaching a solid surface. Thus, within a very
The profiles of fluid velocity pf laminar flow near a solid surface

The velocity parallel to solid surface
surf ---- fluid surface layer, the distance beyond which the velocity of fluid is almost uniform, e.g., vy> 0.99 u0 ; Within surf ,
Fick’s second law of diffusion:
2ci ci J i1 J i 2 J i Di 2 t dx x
The relationship between the current density and the mass transport rate
Chapter 3
Mass Transport at Electrode/Electrolyte Interfaces
Mass transport


Mass transport : the movement of species in the media
(liquid, gas or solid).
x
The modes of mass transport

Diffusion ------ Movement of a species driven by a
gradient of chemical potential (i.e., concentration gradient).
J i Dici - - - - Fick' s first law of diffusion
thin layer near electrode surface, the convection can be neglected. The diffusion and migration become predominant. -------Nernst boundary layer
If excess of inert electrolyte (supporting electrolyte) is added to the solution and its concentration is much larger than the electro-active species, the electric field and the migration rate in the solution can be greatly reduced. Thus, the
0 s
ci 0 , i.e., J i is uniform within th e cappilary, thus t 0 s ci ci J i Di l --- the thickness of the diffusion layer l
dci dx
mass transport coefficient
The flux of speciesi :
J i J i J i J i Dici ci v
D
C
M
zi F Di ci RT
For steady - state convection - diffusion mass transfer :
ci J i Di 2ci vci 0 t