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3.3
Electrokinetic phenomena
3.3.1 Introduction
In 1808, the Russian chemist Ferdinand Fiodorovich Reuss,
a colloid scientist, investigated the behaviour of wet clay. He
observed that the application of a potential difference not
only caused a flow of electric current, but also a remarkable
movement of water towards the negative pole. The transport
of a liquid through a porous medium soaked with the liquid
itself, with a potential difference applied to the boundaries
was subsequently called electroosmosis. In general, this term
indicates the movement of a liquid, with respect to a
stationary surface, that takes place inside porous media or
within capillaries as an effect of an applied electrical field.
The pressure necessary to counterbalance the osmotic flux is
referred to as electroosmotic.
In a second series of experiments, Reuss dipped two
tubes filled with water in a layer of wet clay and then
introduced two electrodes in the tubes. After having applied
a constant potential difference (and therefore an electrical
field), he observed a transport of clay particles towards the
positive pole in addition to electroosmosis. With this
experiment, transport phenomena of systems dispersed in a
fluid triggered by an electrical field were shown for the first
time. The phenomenon, called electrophoresis, would later
Fig. 1. A, schematic
illustration of an experiment
in which a streaming
potential is generated
as an effect of the flux
of a liquid through
a porous sect; B, schematic
illustration of Dorn’s
experiment demonstrating
the existence
of a sedimentation potential.
A
B
I
VOLUME V / INSTRUMENTS
see wide application in the selective separation of different
components present in a colloidal dispersion, as well as be
studied in all its different aspects.
Based on the consideration that a flux of water is usually
produced by a hydrostatic head, Reuss also performed an
ingenious experiment, ‘opposite’ of the first of the two
previously mentioned experiments. As illustrated in Fig. 1 A,
he measured the electrical potential difference displayed at
the boundary of a porous bed through which a fluid was
flowing. In this way, he discovered that a flux of water
though a porous membrane or a capillary generated a
potential difference called the streaming potential.
A fourth phenomenon, the opposite of electrophoresis,
was later discovered by Friedrich Ernst Dorn. If quartz
particles are permitted to fall in water, as shown in Fig. 1 B, it
is observed that an electrical potential difference called
sedimentation potential is located between two electrodes at
different heights.
The phenomena previously described are collectively
called electrokinetic phenomena, classified according to
Table 1.
A peculiar aspect of this phenomena is represented by
the fact that they emphasize coupled phenomena, where a
certain effect can be caused not only by the force directly
I
197
SURFACES AND DISPERSE SYSTEMS
Table 1. Classification of electrokinetic phenomena
Electrical forces
Mechanical forces
solid at rest
solid in motion
solid at rest
solid in motion
electroosmosis
electrophoresis
streaming potential
sedimentation potential
applied to it (for instance, the current generated by an
electrical field), but also by forces associated with different
effects (for instance, an electric current caused by a pressure
difference which typically generates a fluid flow). An
important confirmation of this aspect emerged 20 years after
Reuss experiments when another group of phenomena was
discovered presenting the same type of coupling:
thermoelectric phenomena. In particular, Thomas Johann
Seebek observed that by heating the ends of a bimetallic
couple at different temperatures, an electrical potential
difference was generated, whereas Jean-Charles-Athanase
Peltier noticed that, inversely, a current transport through the
couple caused a heat transfer from one junction to the other.
This group of phenomena found a descriptive frame
within the context of thermodynamics of irreversible
processes. In order to illustrate this aspect, we will refer to
the motion of a fluid (for instance, water) through a porous
sect or a membrane, expressing the fluxes of electrical
charges and of water molecules through the current intensity
I and the water volumetric flux JV . Their coupling can be
described by the following relationships:
[1]
I = L11∆ϕ + L12 ∆P
JV = L21∆ϕ + L22 ∆P
where Df indicates the electrical potential difference and
DP the hydrostatic potential difference, whereas parameters
L11, L22, L12, L21 are the phenomenological parameters. The
Onsager reciprocity relationship is applied to the mixed
terms Lij describing coupling by assuming L12⫽L21. From
these expressions, it is possible to notice that even if no
potential difference is applied (i.e. if Df⫽0), then simply
the presence of a pressure difference can produce an electric
current. On the other hand, if no pressure difference is
applied, the presence of an electromotive force can still
generate a water flux by electroosmosis. Moreover the
following relationships are valid:
[2]
J 
 I 
= L12 =  V 
= L21


∆P ∆ϕ = 0
 ∆ϕ  ∆P = 0
Actually, when other information is unavailable, the
previous equations are also unable to provide the amount of
electric or volumetric flux, as thermodynamics alone is not
sufficient to calculate the values of the Lij coefficients. In
order to deal with this problem, it is necessary to extensively
investigate the influence that electric charges on the surfaces
have on the behaviour of fluids in contact with them, as
illustrated below.
solid and a liquid phase. As such, they apply to systems
where the ratio is high between the interphase area and the
volume, such as capillaries and porous materials soaked in
liquids or dispersions of solid particles in a liquid. In fact,
electrokinetic processes are due to the opposite charges on
the solid particles and in the liquid. On the solid, the charge
is generated by the presence of ions on the surface due to
either their selective adsorption from the solution or the
ionization of molecules present on the surface itself. These
phenomena do not arise in liquids characterized by small
values of the dielectric constant, such as chloroform,
diethylether and carbon disulphide. On the other hand, these
phenomena are observed in polar liquids such as acetone,
alcohols and especially water.
At the boundary, there is a segregation of positive or
negative electric charges perpendicular to the surface itself.
For example, on a silica surface in contact with an aqueous
solution, there are some hydroxylic groups, derived from
SiO2 hydration, that forms silic acid. This compound causes
the following ionic dissociation:
+
2−
H 2SiO3 ⫺
⫺SiO3 + 2H
䉴
䉳
producing negative charges on the surface that exert an
attractive action on hydrogen ions in the solution, forming an
electric double layer (Fig. 2). Another example is silver
H⫹
H⫹
SiO32⫺
198
SiO32⫺
SiO32⫺
A
H⫹
H⫹
H⫹
H⫹
SiO32⫺
SiO32⫺
H⫹
H⫹
H⫹
H⫹
SiO32⫺
SiO32⫺
H⫹
Electrokinetic phenomena arise from the polarization
process that takes place at the contact surface between a
H⫹
H⫹
SiO2
[3]
3.3.2 Formation and structure
of the electrical double layer
H⫹
H⫹
H⫹
SiO32⫺
SiO2
H⫹
SiO32⫺
SiO32⫺
SiO32⫺
H⫹
H⫹
H⫹
H⫹
H⫹
B
Fig. 2. Formation of an electric double layer on a silica surface:
A, plane surface; B, spherical surface.
ENCYCLOPAEDIA OF HYDROCARBONS
ELECTROKINETIC PHENOMENA
iodide particles suspended in a solution of potassium iodide,
the molecules of which are adsorbed on the surface.
The study of the characteristics of electrical double
layers was conducted by various authors who investigated
this problem at different levels. The first and most
significant studies, credited to G. Gouy and D.L. Chapman,
date back to the beginning of the Twentieth century. These
two authors described the surface as an infinite surface on
which a continuous electric charge is distributed in contact
with a solution containing point-like ions having opposite
charges. At an infinite distance from the surface, the
electrical potential identifies with that of the solution,
whereas when close to the surface, the potential gradually
varies until it assumes the values corresponding to the
surface itself. In this zone, two regions can be identified.
One region includes the ions adsorbed on the surface and the
other, called the diffuse region, encompasses the ions present
in the solution, whose distribution is determined by the
conflict between electrostatic interactions to which they are
subjected as well as random thermal movements. In general,
ion adsorption produces an electrostatic energy barrier that
hinders particle coagulation with the formation of a
precipitated phase which is more stable from a
thermodynamic point of view. In conclusion, at every
interphase there is an electrical double layer present,
originating from the asymmetry of the force field involved.
In order to describe the characteristics of the double
layer in quantitative terms, it is appropriate to refer to a
plane surface by simulating the layer of adsorbed ions with a
continuous charge distribution. This charge interacts with
ions of the opposite charge in the solution. This attraction
brings about their accumulation near the surface as opposed
to thermal agitation which promotes an even distribution in
the solution. Therefore, the distribution of i ions can be
expressed by the Boltzmann law:
 Z y( z ) e 
Ci ( z ) = Ci0 exp  − i

k BT 

where Ci (z) indicates their concentration at distance z from
the surface and Ci0 is the concentration in the bulk of the
solution; Zi represents the charge of ion i, y(z) its potential
energy, kB is the Boltzmann constant, T the temperature and
e the electric charge. By combining the previous equation
with the Poisson equation, one obtains:
infinite distance from the surface, the solution itself must be
electrically neutral, there is
[8]
∑C Z
0
i
i
=0
i
and it is possible to derive
[9]
y( z ) = y 0 e − χ z
where y0 is the value of the potential at the surface.
Parameter c is expressed by
[10]
χ2 =
8π e2
ε k BT
∑
Ci0 Z i2
i
Therefore, the potential decreases exponentially; the 1/c
term has the dimensions of a length and represents the width
where the surface double layer is basically located.
By applying [5] and using the Debye-Hückel
approximation, the following expression for the charge
density as a function of the coordinate z is obtained:
[11]
r=−
ε d 2y
ε 2 0 − χz
=−
χ ye
4π dz 2
4π
An important parameter in this analysis is the surface
electric charge density, referring to the unit area s0. Using
the previous equation, s0 is expressed by:
[12]
∞
∞
0
0
σ 0 = − ∫ r( z ) dz = ∫
ε d 2y
dz =
4π dz 2
=−
ε  dy
εχ y0
=
4π  dz  0
4π
It is possible to observe that the surface potential y0 is
related to both the surface charge density and the ionic
composition of the medium. For example, if c increases, the
double layer is compressed, and therefore either s0 increases
or y0 decreases.
[4]
d 2y
4π r
=−
dz 2
ε
which relates electrical potential to the volumetric charge
density r
 Z y( z ) e 
r=e
Ci0 Z i exp  − i

[6]
k BT 

i
and the following non-linear differential equation can be
derived:
 Z y( z ) e 
d 2y
4π
Z i eCi0 exp  − i
=−
[7]

2
k BT 
ε

dz
i
where e is the dielectric constant of the liquid. An
approximated solution of [7], that attributes the dependency
of the potential on the z coordinate, can be easily obtained by
considering that Ziey(z)ⲐkBT⬍⬍1 (the Debye-Hückel
approximation), and using exp(⫺x)⬇1⫺x for x⬍⬍1.
Moreover, considering that in the bulk of the solution, i.e. at
[5]
∑
∑
VOLUME V / INSTRUMENTS
3.3.3 The Stern layer.
Electrokinetic potential
The approach described up to here, where ions are depicted
as charged points resulting in a homogeneous distribution of
surface charge, was improved by the theory proposed by
Otto Stern, who assigned a defined volume to ions, so that
the distance of their centres from the surface cannot be less
than the radius, as illustrated in Fig. 3. Furthermore, this
theory accounts for the fact that at short distances from the
surface, there can be chemical interactions between the ions
and the atoms of the surface itself, associated with the
adsorption phenomena, which occur when the ions reach a
distance from the surface comparable to the bond lengths.
Essentially, the double layer is divided in two parts separated
by a plane (the Stern surface) located at a distance
approximately equal to the radius of the hydrated ions. The
potential ranges from the value y0 at the surface to the value
yd at the Stern surface, and then decays to zero in the
diffused layer, in agreement with [9].
If the Stern layer is described as a condenser of width d,
the surface charge density can be expressed by the following
relationship:
[13]
ε
σ 0 = (y 0 − yδ )
δ
199
SURFACES AND DISPERSE SYSTEMS
performing measurements of an electrokinetic nature that
involve the relative motion of the solid surface with respect
to the liquid. Taking a plane surface encountered by a
current of fluid in laminar motion, it is possible to define an
ideal plane parallel to it where the shear stress is located and
where a rapid viscosity variation occurs. Actually, the real
position of this plane is unknown, since it is necessary to add
solvent molecules to surface ions. However, it is reasonable
to assume that this plane is located very close to the Stern
plane and for this reason, the z potential is only marginally
smaller than yd (see again Fig. 3). It is often assumed that
the values of z and yd are the same, a small error that can,
however, become significant for high ionic concentrations.
surface
Stern plane
surface of shear
3.3.4 Theory of electrokinetic
phenomena
The theory of electrokinetic phenomena involves both the
description of the electrical double layer and the fluid
dynamics description of motion, and it is therefore relatively
complicated. In order to formulate a model exemplifying the
situation, it is necessary to refer to a liquid layer of length l
containing electrolytes in contact with a plane surface under
the influence of an electrical field parallel to the surface
itself. Single ions will tend to move dragging the solvent
under the influence of an electrical force X, defined as
X⫽Df/l. This force is balanced by the friction force present
in the liquid. Therefore, in case of a stationary laminar
motion, each liquid layer of thickness dz (Fig. 4) moves
parallel to the surface with a velocity u that depends on z but
remains constant over time. An expression showing the
balance between the electrical force acting on the volume
and the friction force due to viscous forces is as follows:
diffuse layer
Stern layer
potential
y0
yd
z
0
d
distance
z
Fig. 3. Schematic representation of the formation
of an electric double layer according to Stern
with the corresponding graph of the electrostatic potential.
[14]
where m is the viscosity. Introducing the Poisson equation [5]
in this expression, it is possible to derive:
[15]
Electrokinetic phenomena originate from the fact that a
liquid moving tangentially to a surface does not drag along
the whole double layer. Only a portion is free to move with it
while another part remains anchored to the solid. In this way,
a charge separation parallel to the interface is created
generating a potential difference. If, conversely, an electrical
field is applied, the positive or negative charges generated in
the double layer tend to migrate toward the electrodes with
the opposite sign. If the solid is at rest, a movement of the
liquid phase takes place, as occurs in electroosmosis. If the
solid, however, is composed of a dispersion of particles,
these tend to move as in electrophoresis.
The double layer theory described up to here and widely
used to interpret surface phenomena, refers in any case to a
static equilibrium condition. Unfortunately it cannot be
directly verified by experiments. In fact, there are no
methods capable of measuring yd as they would require
placing an electrode on the plane which passes through the
centre of the first layer of adsorbed atoms. It is possible,
however, to determine another quantity, close to yd, called
the electrokinetic potential z or simply the zeta potential, by
200
d 2u
 du 
 du 
− µ   = µ 2 dz
X rdz = µ  
 dz  z
 dz  z + dz
dz
−
X ε d 2y
d 2u
=µ 2
2
4π dz
dz
y0
u(z)
z
y(z)
d
dz
z
Fig. 4. Graph showing electrostatic potential
and fluid velocity for a liquid in contact with a plane surface
and in tangential motion with respect to it, as a function
of the coordinate z perpendicular to the wall.
ENCYCLOPAEDIA OF HYDROCARBONS
ELECTROKINETIC PHENOMENA
Boundary conditions reflect the values that the potential
and the velocity of the fluid must have in the bulk of the fluid
at distance d from the surface which limits the zone in which
the liquid is stationary. Therefore, they can be written as:
y ⫽0
u ⫽ue
y ⫽z
u ⫽0
du/dz ⫽0
for z⫺⬁
䉴
for z ⫽d
where ue is the velocity in the bulk of the fluid.
Integrating [15] is relatively easy and, accounting for
boundary conditions, it permits the following relationships:
ε Xζ
[16] ue =
4πµ
4πµue
[17] ζ =
εX
The liquid flow is given by the product ueA⫽JV, where A
is the area of the layer. Keeping in mind that the field
intensity X is given by the ratio between the potential
difference and the length l of the medium where this
difference is applied, and using the Ohm relationship to
express the current intensity where the electric conductivity
k of the medium is introduced, then:
∆ϕ
l
and equation [17] becomes:
[18]
[19]
I = kA
ζ=
4πµkJV
εI
This relationship, known as the HelmholtzSmoluchowski equation, gives a linear relationship
between the liquid flow rate and the zeta potential and,
together with [16], plays an important role in the study of
electrokinetic phenomena. Since it does not contain the
characteristic geometric parameters of the system under
investigation, this expression offers a way to derive the
value of the zeta potential directly from the measured
values of JV and I. Its validity was confirmed by
experimental results, showing that the current intensity is
proportional to the volumetric flow.
Actually, Ohm’s law is not strictly valid for the systems
under investigation since the larger concentration of ions in
the diffused part of the double layer results in higher electric
conductivity as compared to the bulk of the fluid.
Accounting for both effects and in the case of a pipe, it is
possible to define an effective conductivity keff through this
relationship
[20]
keff = k +
b
k
A s
where k is the conductivity in the bulk of the fluid and ks
the conductivity on the surface, A the pipe area and b its
perimeter. Based on this last relationship, previous
equations should be appropriately modified, even if for
capillaries with a relatively large radius, the correction
needed due to the effect of surface conductivity is
basically negligible.
The approach just described refers to a simplified
geometrical situation where no effect is associated with
the surface curvature. In order to examine this aspect more
in depth, it is convenient to refer to a capillary with a
radius equal to r0 and a length equal to l, containing a
liquid to which a potential difference Df is applied, acting
VOLUME V / INSTRUMENTS
along its z axis. In this situation, the Poisson equation [5]
and the force balance [14] should be modified
accordingly. If r indicates the radial coordinate
perpendicular to the capillary axis with its origin on it,
considering the cylindrical symmetry of the system, the
two equations take this form:
[21]
1 ⭸  ⭸y 
4π r
r
 =−
r ⭸r  ⭸r 
ε
[22]
−∇Pz =
µ ⭸  ⭸u z 
r
 − Xr
r ⭸r  ⭸r 
Generally speaking, in [22], which corresponds to [14], a
pressure gradient along the cylinder axis was also
introduced. This approach allows for a unified vision for the
complementary phenomena: electroosmosis and an electric
current generated by a liquid flow.
Previous equations should satisfy the above mentioned
boundary conditions for r⫽r0⫺d, whereas for r⫽0, i.e. on
the cylinder axis, uz and y are both finite, with (⭸yⲐ⭸r)0⫽0.
Although the integration is relatively complex, by applying
the Debye-Hückel equation, it can be performed analytically,
permitting the following expression for the current intensity
and liquid flow rate, respectively:
[23]
[24]
2
 σ δ
k
X
=
+

µ
π r02

Is
( ) ( )
( )
 I 0 R0 I 2 R0  
 +
1 −
I12 R0
 

εζ I 2 R0
+
∇P
4πµ I1 R0
( )
( )
JV
X εζ I 2 ( R0 ) r02
−
∇P
=
2
π r0 4πµ I1 ( R0 ) 8π
I0, I1 and I2 are the modified Bessel functions of the first
type of the zeroeth, first and second order, R0⫽r0 c, whereas
sd is the surface charge calculated through equation [12],
using yd rather than y0.
The previous equations offer a general solution to the
problem, and specifically refer to electrophoresis if ⵜP⫽0,
and to a streaming potential if X⫽0.
3.3.5 Electroosmosis
As seen above, electroosmosis appears as a flux of a liquid
containing electrolytes when a potential difference Df is
applied to it. Ignoring the surface curvature, fluid motion
can be described by equation [16], and the liquid flow can be
expressed as:
[25]
JV = ueo A =
A∆ϕεζ
4πµl
where ueo is now defined as electroosmotic velocity. This
equation can be obtained with [24] if ⵜP⫽0 and if a high
value is assigned to the duct radius so that the radius
I2(R0)ⲐI1(R0) tends to unity. Expression [25] is also applied
for the calculation of fluxes through porous materials. In the
case of a circular duct with a radius equal to r0, and
accounting for the fact that there is a conductivity difference
between the surface layer and the liquid bulk, taking into
account equation [20], the [25] should be modified as
follows:
201
SURFACES AND DISPERSE SYSTEMS
[26]
JV =
Aεζ I

2k 
4πµ  k + s 
r 

particle moves at constant velocity. In developing this
analysis, Erich Armand Hückel derived the following
expression of mobility:
0
Electroosmosis can be adopted for the solution of
practical problems, related in particular to the dehydration
of porous materials. To this purpose, a humid mass is
located between two electrodes connected to the opposite
poles of an external source of electric current. The current
causes a transfer of water towards the cathode where it is
removed by pumping. In the meantime, the anodic mass is
compressed. The electroosmotic phenomenon is ideal for
reconditioning humid rooms, particularly in restoration
operations. In practice, two electrodes, positive and
negative, are installed in the walls and in the floor
connected with an electric generator that delivers a
continuous low tension electrical current. These
applications, however, are rare since this is a slow and
expensive process. Techniques involving the use of
electroosmosis to remove water from oil were also studied.
3.3.6 Electrophoresis
Particles representing a dispersed phase in a liquid whose
charge is due to the formation of a double layer on the
surface, show the behaviour described in Fig. 5 when
subjected to the action of an electric field. Under the
influence of this field, each particle moves together with the
compact layer of ions present in the double layer, whereas
the diffuse ionic atmosphere tends to move in the opposite
direction. Choosing a system of coordinates fixed on the
particle, and neglecting the geometrical factors related to its
shape, the situation is identical in principle to that previously
seen when describing electroosmosis and therefore, in first
approximation, it is legitimate to apply equation [16]. This is
confirmed by many experiences revealing the existence of a
proportionality between the electrical force and particle
velocity, which in this case is indicated by uep, called the
electrophoretic velocity. Another parameter is also used
called the electrophoretic mobility (v), equal to the particle
velocity under the influence of a unit force:
uep
εζ
=
[27] v =
X 4πµ
An alternative way to tackle the problem consists in
focusing attention on a spherical charged particle, which
moves in a fluid as an effect of an electrical field. In
stationary conditions, the force of electrical nature acting
on the particle is balanced by viscous friction forces,
which can be expressed by Stokes’ law, where the
Fig. 5. Variation of ion distribution as an effect of the motion
of a charged particle in a liquid.
202
[28]
v=
εζ
6πµ
which, as seen, differs from [27] for a numerical factor equal
to 2/3. The previous equation was derived by assuming that
the electrical conductivity of the particle is equal to that of
the medium and that its dimensions are small with respect to
the width of the double layer. It is included in the description
of electrophoretic phenomena occurring in non-aqueous
solutions.
In a more detailed analysis, D.C. Henry explicitly
accounted for the geometrical shape of the particle in the
configuration of the electric field that forms around it. In a
quantitative approach, Henry suggested modifying equation
[16] as follows:
[29]
uep =
ε Xζ  r 
f 
πµ  δ 
where a complicated function ( f ) of the r/d ratio between
the particle radius and the width of the diffuse layer appears.
If a constant value equal to 0.25 is attributed to it, [29]
identifies with [16].
Another situation, called the relaxation effect, is
associated with the disturbance of the symmetry of the
electrical field in the diffuse layer. The cause is due to
polarization decreasing the effective value of the electric
force which, in turn, influences the parameters
determined by the electric force. This results in a
retardation due to the opposite flux of counterions which
causes additional friction. Actually, thanks to the
electrical conductivity increase in the double layer and
the diffusive processes in it, the system tends to recover
the symmetrical configuration, requiring a relaxation
time which depends on the electrokinetic potential, the
product of the electric conductivity multiplied by the ion
dimensions, and the valence of the electrolytes present in
the system.
An important field of application of electrophoresis is
the separation of high molecular weight complex organic
compounds in a solution. The difference between their
migration velocities is due to their charges and dimensions.
This approach proved very useful in biochemistry,
particularly for the classification of proteins in blood
plasma. There are several different analytical chemistry
techniques that perform this operation.
3.3.7 Streaming potential
When an electrolytic solution flows through a capillary due
to pressure difference, the presence of an electrical potential
difference is measured between two electrodes located at the
ends of the duct. This streaming potential occurs when the
fluid flow transports the ions present in the mobile part of
the electric double layer that forms close to the capillary
surface.
The potential difference Df along the capillary axis
therefore generates a streaming current Is. For a cylindrical
capillary with a radius r0 and a length l, this current has the
following form:
ENCYCLOPAEDIA OF HYDROCARBONS
ELECTROKINETIC PHENOMENA
r0
[30]
I s = 2π ∫ u(r ) r(r )dr
3.3.8 Sedimentation potential
0
where r(r) usually expresses the charge density as a function
of the radial coordinate r, while u(r) represents the liquid
velocity as a function of r, which can be expressed by the
Poiseuille equation, by integrating the Navier-Stokes
equation for the stationary laminar motion in a circular duct:
[31]
[37]
∆P 2
u (r ) =
(r0 − r 2 )
4 µl
By neglecting the effect of curvature, r(r) can be
expressed by the equation [11]. In this case, the integration
of [30] is relatively easy, resulting in the following
expression:
εζ∆PA
Is =
[32]
4πµl
By expressing the current intensity using Ohm’s law, it is
possible to derive the following equation for the difference in
potential measured at the ends of the capillary:
[33]
∆ϕ =
εζ∆P
4πµk
In stationary conditions, current Is must be balanced by a
current I which returns the charge to the system through the
liquid bulk and its surface, shown by
∆ϕ
π r 2 k + 2π r0 ks
l
Since Is⫹I⫽0 in a stationary condition, equation [33] is
more appropriately written as:
εζ∆P
[35] ∆ϕ =

2k 
4πµ  k + s 
r 

[34]
I=
)
(
0
Finally, it is interesting to observe that by applying
thermodynamics of irreversible processes and accounting for
the Onsager reciprocity condition, on the basis of equation
[3], it is possible to obtain
[36]
I s ∆P
=
JV ∆ϕ
Since JV⫽Aue, if equation [16], obtained by applying the
Helmholtz-Smoluchowski model, is attributed to ue, it is
easy to verify that the expression of Is identifies with [32],
thereby confirming the compatibility of the two approaches.
VOLUME V / INSTRUMENTS
During the deposition of solid particles in a fluid, the
presence of a potential difference can be measured as shown
in Fig. 1 B. In this case, pressure is replaced by gravity, and
the force acting on a spherical particle of radius r is
therefore represented by the following relationship:
mg =
4 3
π r (d − d0 )g
3
where d and d0 are the density of the solid and the fluid,
respectively; g is the acceleration of gravity; and m is the
apparent mass of the particle.
By replacing the previous equation with [33] and in the
case of a swarm of particles with density n (number of
particles per unit volume), the following expression is
obtained for the electrical potential difference measured
between two electrodes located at the ends of a column of
height l:
[38]
∆y =
r 3 ( d − d0 ) gnεζ
l
3µk
Actually, the application of this relationship is not always
easy, as real systems are normally polydispersed, and
furthermore, particles are not spherical. An additional
complication is due to the difference of the velocities for the
various particles.
Even though they have not found specific industrial
applications, sedimentation potentials are of remarkable
interest as they take parte in atmospheric discharges.
Bibliography
Fridrikhsberg D.A. (1986) A course in colloid chemistry, Moscow,
Mir.
Newman J.S. (1973) Electrochemical systems, Englewood Cliffs (NJ),
Prentice-Hall.
Shaw D.J. (1970) Introduction to colloid chemistry and surface
chemistry, London, Butterworths.
Voyutsky S.S. (1978) Colloid chemistry, Moscow, Mir.
Sergio Carrà
Dipartimento di Chimica, Materiali
e Ingegneria Chimica ‘Giulio Natta’
Politecnico di Milano
Milano, Italy
203