Electrochemistry MAE-295
Dr. Marc Madou, UCI, Winter 2012
Class V Transport in Electrochemistry (II)
Table of Content
 Reynolds Numbers
 Low Reynolds Numbers
 OHP, Diffusion Layer Thickness, Hydrodynamic Boundary
Layer Thickness
 Mixing in low Reynolds number fluids to enhance
electrochemical reactions
Reynolds Numbers
 The dimensionless Reynolds
number is given by:
where v is the mean velocity of
an object relative to the fluid (SI
units: m/s), L is a characteristic
linear dimension (SI: m),μ is the
dynamic viscosity of the fluid
[SI: Pa·s or N·s/m² or kg/(m·s)]
and ν is the kinematic viscosity
(ν: μ / ρ) (m²/s) and r is the
density of the fluid (SI: kg/m³)
 Note that multiplying the
Reynolds number
yields:
which is the ratio of:
 Or also:
by
Reynolds Numbers
• Small systems are less turbulent than large ones (e.g., flow in very thin pipes is laminar).
• Slow flows are laminar, while fast flows are turbulent.
• More viscous materials are less turbulent (e.g., oil in a pipeline is less turbulent than water in the
same pipeline).
Low Reynolds Numbers
 Creeping flow also known as “Stokes Flow” or “Low Reynolds
number flow”
 Occurs when Re << 1
 r,v (often U is used), or L are very small, e.g., micro-organisms,
MEMS, nano-tech, particles, bubbles
  is very large, e.g., honey, lava
Low Reynolds Numbers (Stokes flow)
 In micro-fluidics, Re<1
 In Laminar flow the viscous force is dominant over the inertial force
 Inertial forces are pretty much irrelevant
Purcell 1977
http://www.youtube.com/user/Swimmers1
Low Reynolds Numbers
• Micro and nano
1mm
Typical size of a chip
technology enabled
100m
Extended lenght of DNA
Micro-channel
10m
Microstructure and micro-drops
Cellular scale
1m
100nm
Radius of Gyration of DNA
Colloid and polymer molecular size
10nm
Re D,max 
UDh


10 mms 500 m
1
mm
2
s
5
Low Reynolds Numbers
 Fluids in micro-channels and nano-channels
 Here we are specifically interested in working with
electrodes in confined spaces: so back to
electrochemistry !!
 To maximize the supply of electro-active species to
electrodes in such confined spaces one relies very
often on forced convection.
 If migration is suppressed the mass transport will be
under diffusion-convection control.
Micro-channels
Nano-tubes (some of the smallest
channels).
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 The outer Helmholtz plane (OHP) is considered to be the approximate
site for electron transfer.
 Nonspecifically adsorbed ions also reside in the diffuse layer (Nernst
layer) extending some distance from the electrode surface. The thickness
of the diffuse layer is dependent upon ionic strength of the buffer, and for
stirred aqueous solutions the thickness of the diffuse layer varies between
0.01 and 0.001 mm. And has been found to be 0.05 cm in many cases of
unstirred aqueous electrolytes. The nature of the diffuse layer can have a
significant impact on the rate of electron-transfer since the actual
potential felt by a reactant close to the electrode is dependent upon it.
 From the above the thickness of the Nernst layer is strongly dependent
on the condition of the hydrodynamic flow due to say stirring or other
convective effects. The double layer on the other hand is typically less
than 1 nm and is not influenced by stirring.
 So how does the hydrodynamic boundary layer influence the diffusion
layer? First what is the hydrodynamic boundary layer?
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 In physics and fluid mechanics, a boundary layer is the layer of fluid
in the immediate vicinity of a bounding surface where the effects of
viscosity are significant.
 When fluid flows past an immersed body, a thin boundary layer will
be developed near the solid body due to the no-slip condition (i.e.,
fluid is stuck to the solid boundary). The flow can be treated as
inviscid flow outside of this boundary layer, while viscous effects are
important inside of this boundary layer.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 The characteristics of flow past a flat plate with
finite length L subject to different Reynolds
numbers (Re = ρUL/μ) are shown in the figures
on the right. At a low Reynolds number (Re =
0.1), the presence of the flat plate is felt in a
relatively large area where the viscous effects are
important.
 At a moderate Reynolds number (Re = 10), the
viscous layer region becomes smaller. Viscous
effects are only important inside of this region,
and streamlines are deflected as fluid enters it.
 As the Reynolds number is increased further (Re
= 107), only a thin boundary layer develops near
the flat plate, and the fluid forms a narrow wake
region behind the flat plate. (a) Re = 0.1, (b) Re
= 10 and (c) Re = 107
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 Consider now a flow past an infinite
long flat plate. Also define the
Reynolds number using the local
distance x (i.e., the distance from the
leading edge along the flat plate as the
characteristic length).
 The local Reynolds number is then
given by: Rex = ρUxX/μ
 The flow becomes turbulent at a
critical distance xcr downstream from
the leading edge. The transition from
laminar to turbulent begins when the
critical Reynolds number (Rexcr)
reaches 5×105. The boundary layer
changes from laminar to turbulent at
this point.
 The concept of a boundary layer was
introduced by Prandtl (1904) for
steady, two-dimensional laminar flow
past a flat plate using the Navier-Stokes
(NS) equations. Prandtl's student,
Blasius, was able to solve these
equations analytically for large
Reynolds number flows.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 Based on Blasius' analytical solutions, the boundary layer
thickness (δ) for the laminar region is given by :
where δ is defined as the boundary layer thickness in which
the velocity is 99% of the free stream velocity (i.e., y = δ, u
= 0.99U).
 To compare the thickness of the Nernst Layer and the Prandtl
layer we need to introduce a few more dimensionless
numbers.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 (1) velocity profile; (2) Prandtl
boundary layer; (3) Nernst
boundary layer.
 Prandtl boundary layer thickness
(hydrodynamic):δp ≈
5(νx/U)1/2,
 Nernst boundary layer thickness
(diffusion)δN ≈ D 1/3ν
1/6(x/U)1/2,
with U, fluid velocity; ν,
kinematic viscosity; and D,
diffusion constant.
 The Nernst diffusion layer is a
sublayer of the Prandtl layer.
 As in the Prandtl layer there is no
motion of the solution in the Nernst
layer.
 The Nernst and the Prandtl layers are
the regions where the concentration
and the tangential velocity gradients
are at a maximum.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 The Schmidt Number, Sc, is a
 Prandtl Number: describes the
thickness of the hydrodynamic
boundary layer compared with the
thermal boundary layer. It is the
ratio between the molecular
diffusivity of momentum to the
molecular diffusivity of heat.
 Small values of the Prandtl number
with kinematic viscosity,, and mass
(< 1) in a given fluid indicates that
thermal diffusion occurs at a greater
diffusivity Dc.
rate than momentum diffusion and
 Small values of the Schmidt number
therefore heat conduction is more
(<1) diffusion dominates over
effective than convection.
convection. It physically relates the
Conversely if the Prandtl number is
large (greater than 1), momentum
relative thickness of the
hydrodynamic layer and mass-transfer diffuses at a greater rate than heat
and convection is more effective than
boundary layer.
conduction.
dimensionless parameter representing
the ratio between momentum
transport and mass transport by
diffusion. It is defined as:
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 The Schmidt number is the mass
transfer equivalent of the Prandtl
Number. For gases, Sc and Pr have
similar values (≈0.7) and this is used as
the basis for simple heat and mass
transfer analogies.
 A quantitative treatment of the relative
extension of dp vs dN shows that it is
governed by Sc. For aqueous
electrolytes  = 10-2 cm2s-1, the
Schmidt number is about 1000.
 Based on
1

dN
( )  Sc 2
d Pr
the Prandtl
layer is 10 to 30 times

thicker than the Nernst layer.
 Voltammograms we saw exhibit a
sigmoidal (wave) shape. If the stirring
rate (U) is increased, the diffusion
layer thickness becomes thinner,
according to:
dN 
B
Ua
where B and a are constants for a
given system.
As a result, the

concentration gradient becomes
steeper, thereby increasing the limiting
current. Similar considerations apply
to other forced convection systems,
e.g., those relying on solution flow or
electrode rotation. For all of these
hydrodynamic systems, the sensitivity
of the measurement can be enhanced
by increasing the convection rate.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 In aqueous the Prandtl layer is
10-fold larger than the
Nernst layer, indicating
negligible convection within
the diffusion layer .
 Additional means for
enhancing the mass transport
and thinning the diffusion
layer, include the use of
ultrasound, heated electrodes.
Mixing in low Reynolds number fluids to
enhance electrochemical reactions
 The question we address now is
how to mix reactants in a small
microreactor in the absence of
turbulence? The primary resistance
to mixing by convection, we saw
earlier, is controlled by a thin layer
of stagnant fluid adjacent to a solid
surface.
 In this hydrodynamic boundary
layer, the flow velocity, V, varies
from zero at the surface, that is, the
no-slip condition, to the value in
the bulk of the fluid, that is, V∞.
Laminar flow around an object of
length L and boundary layer δ is
given by:
 At the microfluidic level, mixing is like
trying to stir syrup into honey, and two
liquids, traveling side-by-side through a
narrow channel, only become mixed after
several centimeters because mass transport
in the microdomain is traditionally limited
to simple diffusion . To decide which
transport type dominates, diffusion or
convection, one must inspect the Péclet
number.
Mixing in low Reynolds number fluids to
enhance electrochemical reactions
 The Péclet number represents the ratio of
mass transport by convection to mass
transport by diffusion
 The higher the Péclet number, the more
the influence of flow dominates over
molecular diffusion. In liquids, the
diffusion coefficient of a small molecule
typically is about 10−5 cm2/s. With a
velocity of 1 mm/s, in a channel of 100μm height, the Péclet number is on the
order of 100. This elevated value suggests
that the diffusion forces are acting more
slowly than the hydrodynamic transport
phenomena: for mixing by diffusional
forces one must have Pe < 1.
At low Reynolds numbers, stirring is like
kneading dough for making bread, with
stretching of fluid elements to increase the
diffusional interface and folding to decrease
distance over which species have to diffuse.
Creating chaotic pathlines for dispersing fluid
species effectively in smooth and regular flow
fields is called “chaotic advection.”
 Chaotic advection results in rapid distortion and
elongation of the fluid/fluid interface, increasing
the interfacial area across which diffusion occurs,
which increases the mean values of the gradients
driving diffusion, leading to more rapid mixing.

Mixing in low Reynolds number fluids to
enhance electrochemical reactions
 Mixing can also be improved on by
using a fractal approach, a design akin
to how nature uses passive mixing . In
nature, only the smallest animals rely
on diffusion for transport; animals
made up of more than a few cells
cannot rely on diffusion anymore to
move materials within themselves.
They augment transport with hearts,
blood vessels, pumped lungs, digestive
tubes, etc. These distribution networks
typically constitute fractals. Fractals
are an optimal geometry for
minimizing the work lost as a result of
the transfer network while maximizing
the effective surface area