ISO 13099-1:2012

INTERNATIONAL ISO
STANDARD 13099-1
First edition
2012-06-15
Colloidal systems — Methods for zeta-
potential determination —
Part 1:
Electroacoustic and electrokinetic
phenomena
Systèmes colloïdaux — Méthodes de détermination du potentiel zêta —
Partie 1: Phénomènes électroacoustiques et électrocinétiques
Reference number
©
ISO 2012
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Published in Switzerland
ii © ISO 2012 – All rights reserved

Contents Page
Foreword .iv
Introduction . v
1 Scope . 1
2 Terms and definitions . 1
2.1 Electric double layer . 1
2.2 Electrokinetic phenomena . 2
2.3 Electroacoustic phenomena . 4
3 Symbols . 5
4 Theory: general comments . 6
5 Elementary theories, Smoluchowski’s limit for electrokinetics . 7
5.1 General . 7
5.2 Electrophoresis . 7
5.3 Electroosmosis . 8
5.4 Streaming current or potential . 8
5.5 Sedimentation potential or current . 8
6 Elementary theories, Smoluchowski’s limit for electroacoustics . 8
6.1 General . 8
6.2 O’Brien’s theory for dynamic electrophoretic mobility . 9
6.3 Smoluchowski limit theory for dynamic electrophoretic mobility . 9
7 Advanced theories .10
8 Equilibrium dilution and other sample modifications .10
Annex A (informative) Electric double layer models .12
Annex B (informative) Surface conductivity.18
Annex C (informative) Debye length .20
Annex D (informative) Advanced electrophoretic theories .21
Annex E (informative) Advanced electroacoustic theories .24
Bibliography .26
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International
Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 13099 was prepared by Technical Committee ISO/TC 24, Particle characterization including sieving,
Subcommittee SC 4, Particle characterization.
ISO 13099 consists of the following parts, under the general title Colloidal systems — Methods for zeta-
potential determination:
— Part 1: Electroacoustic and electrokinetic phenomena
— Part 2: Optical methods
The following part is under preparation
— Part 3: Acoustic methods
iv © ISO 2012 – All rights reserved

Introduction
The basic theories and understanding of the electrokinetic and electroacoustic phenomena in a liquid
suspension, an emulsion, or a porous body are presented within this part of ISO 13099 as an introduction to
the subsequent parts, which are devoted to specific measurement techniques.
Many processes, from cleaning water, after either human or industrial fouling, to the creation of stable
pharmaceutical suspensions, benefit from an understanding of the charged surfaces of particles. Also, causing
the particles of a targeted mineral to have an affinity with respect to air bubbles, is a mechanism employed in
the recovery of some minerals.
It should be noted that there are a number of situations where electrokinetic and electroacoustic measurements,
without further interpretation, provide extremely useful and unequivocal information for technological purposes.
The most important of these situations are:
a) identification of the isoelectric point (or point of zero zeta-potential) by electrokinetic titrations with a
potential determining ion (e.g. pH titration);
b) identification of the isoelectric point by titrations with other reagents such as surfactants or polyelectrolytes;
c) identification of a saturation plateau in the adsorption indicating optimum dosage for a dispersing agent;
d) relative comparison of various systems with regard to their electric surface properties.
The determination of zeta-potential, which is not a directly measurable quantity, but one that is established by
the use of an appropriate theory, can be interpreted to establish the region of stability for some suspensions.
By determining the isoelectric point, conditions for the optimum coagulation of particles prior to either capture
in a filter bed or settling out in a lagoon can be set to facilitate the clean-up of fouled water.
This document follows the IUPAC Technical Report on measurement and interpretation of electrokinetic
phenomena (Reference [1]) and general References [2]–[5].
INTERNATIONAL STANDARD ISO 13099-1:2012(E)
Colloidal systems — Methods for zeta-potential
determination —
Part 1:
Electroacoustic and electrokinetic phenomena
1 Scope
This part of ISO 13099 describes methods of zeta-potential determination, both electric and acoustic, in
heterogeneous systems, such as dispersions, emulsions, porous bodies with liquid dispersion medium.
There is no restriction on the value of zeta-potential or the mass fraction of the dispersed phase; both diluted
and concentrated systems are included. Particle size and pore size is assumed to be on the micrometre scale
or smaller, without restriction on particle shape or pore geometry. The characterization of zeta-potential on flat
surfaces is discussed separately.
The liquid of the dispersion medium can be either aqueous or non-aqueous with any liquid conductivity, electric
permittivity or chemical composition. The material of particles can be electrically conducting or non-conducting.
Double layers can be either isolated or overlapped with any thickness or other properties.
This part of ISO 13099 is restricted to linear effects on electric field strength phenomena. Surface charge is
assumed to be homogeneously spread along the interfaces. Effects associated with the soft surface layers
containing space distributed surface charge are beyond the scope.
2 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
2.1 Electric double layer
NOTE The electric double layer (EDL) is a spatial distribution of electric charges that appears on and at the vicinity
of the surface of an object when it is placed in contact with a liquid.
2.1.1
Debye–Hückel approximation
model assuming small electric potentials in the electric double layer
2.1.2
Debye length
−1
κ
characteristic length of the electric double layer in an electrolyte solution
NOTE The Debye length is expressed in nanometres.
2.1.3
diffusion coefficient
D
mean squared displacement of a particle per unit time
2.1.4
Dukhin number
Du
dimensionless number which characterizes contribution of the surface conductivity in electrokinetic and
electroacoustic phenomena, as well as in conductivity and dielectric permittivity of heterogeneous systems
2.1.5
dynamic viscosity
η
ratio between the applied shear stress and the rate of shear of a liquid
NOTE 1 For the purposes of this part of ISO 13099, dynamic viscosity is used as a measure of the resistance of a fluid
which is being deformed by shear stress.
NOTE 2 Dynamic viscosity determines the dynamics of an incompressible newtonian fluid.
NOTE 3 Dynamic viscosity is expressed in pascal seconds.
2.1.6
electric surface charge density
σ
charges on an interface per area due to specific adsorption of ions from the liquid bulk, or due to dissociation
of the surface groups
NOTE Electric surface charge density is expressed in coulombs per square metre.
2.1.7
electric surface potential
s
ψ
difference in electric potential between the surface and the bulk liquid
NOTE Electric surface potential is expressed in volts.
2.1.8
electrokinetic potential
zeta-potential
ζ-potential
ζ
difference in electric potential between that at the slipping plane and that of the bulk liquid
NOTE Electrokinetic potential is expressed in volts.
2.1.9
Gouy–Chapman–Stern model
model describing the electric double layer
2.1.10
isoelectric point
condition of liquid medium, usually the value of pH, that corresponds to zero zeta-potential of dispersed particles
2.1.11
slipping plane
shear plane
abstract plane in the vicinity of the liquid/solid interface where liquid starts to slide relative to the surface under
influence of a shear stress
2.1.12
Stern potential
d
ψ
electric potential on the external boundary of the layer of specifically adsorbed ions
NOTE Stern potential is expressed in volts.
2.2 Electrokinetic phenomena
NOTE Electrokinetic phenomena are associated with tangential liquid motion adjacent to a charged surface.
2 © ISO 2012 – All rights reserved

2.2.1
electroosmosis
motion of liquid through or past a charged surface, e.g. an immobilized set of particles, a porous plug, a
capillary or a membrane, in response to an applied electric field, which is the result of the force exerted by the
applied field on the countercharge ions in the liquid
2.2.2
electroosmotic counter-pressure
Dp
eo
pressure difference that is applied across the system to stop the electroosmotic flow
NOTE 1 The electroosmotic counter-pressure value is positive if the high pressure is on the higher electric potential side
NOTE 2 Electroosmotic counter-pressure is expressed in pascals.
2.2.3
electroosmotic velocity
v
eo
uniform velocity of the liquid far from the charged interface
NOTE Electroosmotic velocity is expressed in metres per second.
2.2.4
electrophoresis
movement of charged colloidal particles or polyelectrolytes, immersed in a liquid, under the influence of an
external electric field
2.2.5
electrophoretic mobility
µ
electrophoretic velocity per electric field strength
NOTE 1 Electrophoretic mobility is positive if the particles move toward lower potential (negative electrode) and
negative in the opposite case.
NOTE 2 Electrophoretic mobility is expressed in metres squared per volt second.
2.2.6
electrophoretic velocity
υ
e
particle velocity during electrophoresis
NOTE Electrophoretic velocity is expressed in metres per second.
2.2.7
sedimentation potential
U
sed
potential difference sensed by two electrodes placed some vertical distance apart in a suspension in which
particles are sedimenting under the effect of gravity
NOTE 1 When the sedimentation is produced by a centrifugal field, the phenomenon is called centrifugation potential.
NOTE 2 Sedimentation potential is expressed in volts.
2.2.8
streaming current
I
str
current through a porous body resulting from the motion of fluid under an applied pressure gradient
NOTE Streaming current is expressed in amperes.
2.2.9
streaming current density
J
str
streaming current per area
NOTE Streaming current density is expressed in coulombs per square metre.
2.2.10
streaming potential
U
str
potential difference at zero electric current, caused by the flow of liquid under a pressure gradient through a
capillary, plug, diaphragm or membrane
NOTE 1 Streaming potentials are created by charge accumulation caused by the flow of countercharges inside
capillaries or pores.
NOTE 2 Streaming potential is expressed in volts.
2.2.11
surface conductivity
σ
K
excess electrical conduction tangential to a charged surface
NOTE Surface conductivity is expressed in siemens.
2.3 Electroacoustic phenomena
NOTE Electroacoustic phenomena arise from the coupling between the ultrasound field and electric field in a liquid
that contains ions. Either of these fields can be primary driving force. Liquid might be a simple newtonian liquid or complex
heterogeneous dispersion, emulsion or even a porous body. There are several different electroacoustic effects, depending
on the nature of the liquid and type of the driving force.
2.3.1
colloid vibration current
CVI
I
CVI
a.c. current generated between two electrodes, placed in a dispersion, if the latter is subjected to an ultrasonic field
NOTE Colloid vibration current is expressed in amperes.
2.3.2
colloid vibration potential
CVU
a.c. potential difference generated between two electrodes, placed in a dispersion, if the latter is subjected to
an ultrasonic field
NOTE Colloid vibration potential is expressed in volts.
2.3.3
electrokinetic sonic amplitude
ESA
A
ESA
amplitude is created by an a.c. electric field in a dispersion with electric field strength, E; it is the counterpart of
the colloid vibration potential method
NOTE 1 See Reference [6].
NOTE 2 Electrokinetic sonic amplitude is expressed in pascals.
4 © ISO 2012 – All rights reserved

2.3.4
ion vibration current
IVI
a.c. electric current created from different displacement amplitudes in an ultrasound wave due to the difference
in the effective mass or friction coefficient between anion and cation
NOTE 1 See References [7][8].
NOTE 2 Ion vibration current is expressed in amperes.
2.3.5
streaming vibration current
SVI
streaming current that arises in a porous body when ultrasound wave propagates through it
NOTE 1 See References [9][10].
NOTE 2 A similar effect can be observed at a non-porous surface, when sound is bounced off at an oblique angle, see
Reference [11].
NOTE 3 Streaming vibration current is expressed in amperes.
3 Symbols
a particle radius
c
electrolyte concentration in the bulk
C double layer capacitance
dl
c
concentration of the ith ion species
i
D
diffusion coefficient of cations
+
D effective diffusion coefficient of the electrolyte
eff
Du
Dukhin number
D
diffusion coefficient of anions
−
e
elementary electric charge
F
Faraday constant
σ
K
surface conductivity
k Boltzmann constant
B
K conductivity of the dispersion medium
m
K conductivity of the dispersed particle
p
K conductivity of the dispersion
s
m parameter characterizing electroosmotic flow contribution to surface conductivity
N Avogadro’s number
A
p pressure
q electroosmotic flow rate per current
eo
R ideal gas constant
r radial distance from the particle centre
R radius of cell in electrokinetic cell model
c
T absolute temperature
U streaming potential
str
x distance from the particle surface
Z acoustic impedance
z valencies of the cations and anions
±
z valency of the ith ion species
i
ε vacuum permittivity
ε relative permittivity of the medium
m
ε relative permittivity of the particle
p
ζ electrokinetic potential, zeta-potential
η dynamic viscosity
κ reciprocal Debye length
µ electrophoretic mobility
µ dynamic electrophoretic mobility
d
ρ medium density
m
ρ particle density
p
ρ density of the dispersion
s
σ electric surface charge density
d
σ electric charge density of the diffuse layer
ϕ
volume fraction
ϕ critical volume fraction
over
d
ψ Stern potential
ψ(x) electric potential in the double layer
Ω
drag coefficient
ω
rotational frequency
ω critical frequency of hydrodynamic relaxation
hd
ω Maxwell–Wagner relaxation frequency
MW
4 Theory: general comments
Theory is an essential element in calculating zeta-potential from the measured data. However, there is no
theory which is valid for all real systems. Instead, there is a multitude of different theories that are each valid
for a certain subset of real dispersions and conditions. It is convenient to organize the theories into two groups:
elementary theories and advanced theories.
6 © ISO 2012 – All rights reserved

The elementary theories encompass several theories for non-conducting solids having a common basis for all
electrokinetic and electroacoustic phenomena. These were originally established and applied to electrophoresis
in Reference [12] and subsequently developed for other electrokinetic and electroacoustic effects. These
theories have the remarkable features that they are valid for any shape, concentration of particles and any
geometry of pores in porous body.
An important feature of the elementary theories is that there is only one EDL parameter, the zeta-potential.
These theories allow direct calculation of zeta-potential from experimental data. All instruments that report
zeta-potential values apply one or another version of the elementary theories, which are appropriate for the
corresponding measurement.
Elementary theories have restricted validity range. Application of these theories beyond their validity range
leads to substantial errors in absolute values of the zeta-potential. These errors can be tolerated for purposes
of monitoring relative variations. However, there are many instances when accurate absolute values of zeta-
potential are required. There are more sophisticated theories for such cases.
These more specific theories, referred to here as advanced theories, are described in Annexes C and D.
These theories contain some other parameters of the EDL, such as Debye length, surface conductivity, Stern
potential (References [2]–[4]). Applications of the advanced theories to zeta-potential calculation are not that
direct. Additional assumptions or even measurement techniques are required.
5 Elementary theories, Smoluchowski’s limit for electrokinetics
5.1 General
There are three restrictions that define the range of validity of the Smoluchowski theory for any electrokinetic
and electroacoustic phenomena.
The first requirement is that the EDL be thin compared to the characteristic size of the heterogeneous
system (see A.4):
κa >> 1 (1)
Many aqueous dispersions satisfy this condition. This condition is not valid for nano-particles at low ionic
strength in aqueous solutions and for many organic liquids.
The second requirement is a negligible contribution of the surface conductivity (Annex B):
Du << 1 (2)
The third requirement is that the interface does not conduct normal electric current between phases. This
condition is valid for non-conducting particles, ideally polarized metal particles, and porous bodies with a non-
conducting matrix.
5.2 Electrophoresis
The Smoluchowski equation (Reference [12]) for the electrophoretic mobility, µ is
:
εε ζ
m0
μ = (3)
η
The derivation of this equation requires no EDL model.
5.3 Electroosmosis
The electroosmotic flow rate of liquid per unit current, q , can be derived from:
eo
εε ζ
m 0
q =− (4)
eo
ηK
m
where K is the conductivity, in siemens per metre, of the medium.
m
5.4 Streaming current or potential
In general, it is impossible to quantify the distribution of the electric field and the velocity in pores with unknown
or complex geometry. However, this fundamental difficulty is avoided at the Smoluchowski limit, when
hydrodynamic and electrodynamic fields have the same space distribution.
The value of the streaming potential U is obtained by the condition of equality of the conduction and streaming
str
currents (the net current vanishes). According to Kruyt (Reference [5]) this leads to the simple Formula (5):
U εε ζ
strm 0
= (5)
ΔpKη
m
Formula (5) does not contain geometric parameters, which makes it very convenient for determining zeta-potential.
5.5 Sedimentation potential or current
There is an analogue of the Smoluchowski equation [Formula (5)] for sedimentation potential, U :
sed
εε ζρ()−ρϕgd
mp0 m
U = (6)
sed
ηK
m
where d is the distance between the points where the potential is measured and g is the acceleration due to gravity.
6 Elementary theories, Smoluchowski’s limit for electroacoustics
6.1 General
Electroacoustic theory, following electrokinetic theory, operates with a notion that it is closely related to
electrophoretic mobility. This so-called dynamic electrophoretic mobility, µ , is the generalization of the
d
electrophoretic mobility concept for high-frequency oscillating particle motion.
The relationship between dynamic electrophoretic mobility and experimental electroacoustic data is not as
trivial as that in the case of electrokinetics. The additional theoretical step has been made through the O’Brien
electroacoustic theory (Reference [14]), which is valid for concentrated systems as well as dilute ones.
According to the O’Brien relationship, the average dynamic electrophoretic mobility, µ , is defined as:
d
ρ 1 
m
μ = A
d ESA 
ϕρ()−ρωAF() ()Z
pm

(7)

ρ
m

μ = I
dCVI

ϕρ()−ρωAF() ()Z
pm

where A or I are normalized with respect to corresponding driving forces. A(ω) is an instrument constant
ESA CVI
found by calibration, and F(Z) is a function of the acoustic impedances of the transducer and the dispersion
under investigation. The densities of medium and particles, ρ and ρ , are required, as well as the volume
m p
fraction of particles ϕ.
According to O’Brien, a complete functional dependence of electroacoustics on key parameters, such as zeta-
potential, particle size and frequency, is incorporated into the dynamic electrophoretic mobility. For all cases
8 © ISO 2012 – All rights reserved

considered, the coefficient of proportionality between electroacoustic signal and dynamic electrophoretic
mobility is independent of particle size and zeta-potential. This feature makes the dynamic electrophoretic
mobility an important and central parameter of the electroacoustic theory.
There are two versions of the dynamic electrophoretic mobility theory at the Smoluchowski limit: when EDL
is thin and when surface conductivity is negligible. This is expressed with the same two conditions as in the
electrokinetic theory [Formulae (1) and (2)].
The first one is for dilute systems with negligible particle–particle interaction (Reference [14]), valid for
spherical particles only, but with no restriction on the particle size except that the size be small compared to
the sound wavelength. This is called “O’Brien theory for dynamic electrophoretic mobility”. It is more limited
than Smoluchowski theory for electrophoresis, which is valid for any particle shape and concentration.
The other version of the electroacoustic theory is almost as general as Smoluchowski theory for electrophoresis.
This can be achieved by restricting the frequency range. If the frequency is sufficiently low, dynamic electrophoretic
mobility becomes identical to the static electrophoretic mobility. Currently, this theory is available for CVI only
(References [15][16]). It is referred to as “Smoluchowski limit theory for dynamic electrophoretic mobility”.
6.2 O’Brien’s theory for dynamic electrophoretic mobility
This theory does not take into account particle–particle interactions, and is thus valid only in dilute dispersions.
2εε ζ
m 0
μ =+Gs()1 F('ω ) (8)
d  
3η
where
11++()j s
Gs = (9)
()
 
 
11++()jjss+ 29 32+−ρρ ρ
() ()
{}pm m
 
 
 
 
11+−jωε' ε
()
pm
 
F('ω ) = (10)
 
22++jωε' ε
()
pm
 
a ωρ
ω
m
s = ; ω'=
2η ω
MW
The frequency dependence of dynamic electrophoretic mobility is determined by the two factors s and ω′’ and j is
the square root of −1. The factor G(s), reflects the frequency dependence related with the inertia effects, whereas
the factor F(ω′’), represents the influence of the Maxwell–Wagner polarization of the EDL (References [17]–[19]).
For aqueous colloids, the inertia factor, G(s), plays a more important role than the EDL polarization factor,
F(ω′’). The inertia factor dramatically reduces the magnitude of the dynamic mobility of larger particles at high
frequencies. In addition to reducing the amplitude, the inertia factor also causes a lag in the particle motion
relative to the external driving force, and this interposes a phase shift on the dynamic electrophoretic mobility.
This phase shift reaches a maximum value of 45° at the high-frequency limit.
Neither of these two factors is important at low frequency when the G(s) and F(ω′’) factors approach 1 and 0,5,
respectively. This means that at sufficiently low frequency, dynamic electrophoretic mobility depends only on
those factors that are taken into account by the Smoluchowski theory for electrophoresis.
6.3 Smoluchowski limit theory for dynamic electrophoretic mobility
This version of the theory (References [15][16]) is important because it is as general as Smoluchowski theory,
within its range of validity — sufficiently low frequency. It is adequate for particles of any size, any shape and,
most importantly, any concentration including highly concentrated systems.
There are two critical frequencies that determine the valid range of this theory, together with the Smoluchowski
restrictions, Formulae 1 and 2.
The critical frequency of hydrodynamic relaxation (ω ) determines the range of the particle inertia and that of
hd
the influence of the factor G(s). This factor becomes negligible when frequency ω satisfies the condition:
η
ωω<< = (11)
hd
ρ a
m
The critical frequency of EDL electric polarization ω presents a frequency range where the second factor,
MW
function F(ω′’), becomes negligible:
K
m
ωω<< = (12)
MW
εε
m 0
The expression for dynamic electrophoretic mobility at this limit was derived only for colloid vibration current
(References [15][16]):
()ρρ− ρ
εε ζ K
ps m
ms0
μ = (13)
d
η K ()ρρ− ρ
m pm s
7 Advanced theories
Application of the advanced theories is much more complicated, because other parameters of the EDL are
involved. On the other hand, application of these theories allows much more detailed description of the electric
surface properties. The two most important parameters are the Debye length and Dukhin number. There are
several ways of characterizing these parameters for the proper use of the advanced theories. They are described
in Annex C. Increasing complexity of the measuring procedure yields more detailed determination of these
additional parameters, which in turn leads to the more accurate zeta-potential value. There are several distinctive
steps in this balance between complexity of the measuring procedure and accuracy of the zeta-potential.
Estimation of the Debye length using conductivity is simpler than measuring Dukhin number. On the other
hand, information on the Debye length allows us only limited modelling of the surface conductivity (Annex B).
This approach requires the assumption that surface conductivity is only associated with the diffuse layer. It
is so-called “standard Debye length model” where according to Lyklema (Reference [2]) the Dukhin number
becomes dependent only upon Debye length and zeta-potential (Formula B.4). There are two analytical theories
of electrophoresis that can be applied in this case: simplified Dukhin–Semenikhin theory (Formula D.10) and
O’Brien theory (Formula D.11).
The next level of complexity requires direct measurement of the Dukhin number, which takes into account
surface conductivity not only in the diffuse layer, but under it and even in the Stern layer. There are a number of
studies indicating the importance of this stagnant layer surface conductivity (Reference [2]). Direct measurement
of the Du can be achieved using several different experimental procedures (Reference [2]). One of the most
suitable is based on the conductivity measurement, as presented in Annex C. Calculation of zeta-potential can
be based on the numerical procedure by O’Brien and White (D.2), or general Dukhin–Semenikhin analytical
theory (D.5).
The overlap of EDL is another factor that complicates theoretical interpretation. It becomes important for
concentrated nano-particles and non-polar dispersions. In the latter case, it can appear even at rather low
volume fractions, as discussed in A.4. There are two theories that take into account this factor for both
electrophoresis (D.6) and electroacoustics (E.4).
Each particular study of zeta-potential requires a certain level of accuracy. This accuracy determines the
complexity of the theoretical interpretation procedure and the choice between either elementary or appropriate
advanced theory.
8 Equilibrium dilution and other sample modifications
A comparison of different methods is complicated by the fact that the zeta-potential is not only a property of
particles alone, but also depends on the chemical equilibrium between particle surface and the liquid. Any variation
of the liquid chemical and ionic composition affects this equilibrium and, consequently, affects zeta-potential.
10 © ISO 2012 – All rights reserved

This presents a problem for methods that require extreme dilution of the sample. Dilution can change the
chemical composition of the liquid, if special measures are not taken. The sample preparation shall follow a
procedure so that zeta-potential is not changed from the original system to the diluted sample.
This procedure requires, upon dilution, not only that particles and their surfaces remain identical between the
original and diluted systems, but also that liquids remain identical. This condition is not easy to satisfy if both
dilution and surfactant stabilization of the sample are involved. The sample preparation procedures can affect
liquid composition tremendously.
The sample preparation shall use the so-called equilibrium dilution procedure, which employs the same
liquid as in the original system as a diluent. After dilution, the only parameter that has changed is the particle
concentration. Only sample preparation based on equilibrium dilution provides zeta-potential values that are
identical between the original system and the diluted sample.
There are two approaches to the collection of the liquid used for dilution. The first consists of extracting a
supernatant using sedimentation or centrifugation. This supernatant can be used for diluting the initial sample
to the degree that is optimal for the particular measuring technique. This method is suitable for large particles
with sufficient density contrast. It is not very convenient for nano-particles and soft biological systems.
The other approach, more suitable for nano- and bio-colloids is to employ dialysis. Dialysis membranes are
required that are penetrable for ions and molecules, but not for colloidal particles (Reference [15]).
In some rare cases, there can be a need to prepare more concentrated samples from the dilute ones. This can
be achieved by initially separating particles from the liquid and the re-dispersing them in the same liquid but at
a higher volume fraction.
Annex A
(informative)
Electric double layer models
A.1 General
Electric double layer (EDL) is a spatial distribution of electric charges that appears on and at the vicinity of the
surface of an object when it is placed in contact with a liquid. This object might be a solid particle, gas bubble,
liquid droplet, or porous body. The structure consists of two parallel layers of electric charges. One layer (either
positive or negative) coincides with the surface of the object. It is the electric surface charge. The other layer
is in the fluid. It electrically screens the first one. It is diffuse, because it forms under the influence of electric
attraction and thermal motion of free ions in fluid. It is called the diffuse layer.
Interfacial EDL is usually most apparent for dispersions and emulsions with sizes on the micrometre or even
nanometre scale or porous bodies with pore sizes on the same scale.
EDL plays a very important role in real world systems. For instance, milk exists only because fat droplets are
covered with an EDL that prevents their coagulation into butter. EDLs exist in practically all heterogeneous
liquid based systems, such as blood, paint, inks, ceramic slurries, cement slurries, etc.
The earliest model of the electrical EDL is usually attributed to Helmholtz (Reference [13]) who treated the EDL
mathematically as a simple capacitor, based on a physical model in which a single layer of ions is adsorbed at
the surface with compensating countercharges in the solution. Later Gouy and Chapman (References [20][21])
made significant improvements by introducing a diffuse model of the EDL, in which the electric potential decays
exponentially away from the surface into the bulk liquid. The Gouy–Chapman model fails for highly charged
EDLs. In order to resolve this problem, Stern (Reference [22]) suggested the introduction of an additional layer
adjacent to the surface that is now called the Stern layer. The combined Gouy–Chapman–Stern model is the
one most commonly used.
There are some limitations in the Gouy–Chapman–Stern model:
— ions are effectively point charges;
— the only significant interactions in the diffuse layer are coulombic;
— electric permittivity is constant throughout the electric double layer;
— dynamic viscosity of fluid is constant beyond the slipping plane.
There are more recent theoretical developments studying these limitations of the Gouy–Chapman–Stern
model, which have been reviewed in References [1]–[4].
Figure A.1 illustrates an interfacial EDL in more detail (Reference [2]). The reason for the formation of a relaxed
(equilibrium) double layer is the non-electric affinity of charge-determining ions for a surface. This process
leads to the build-up of an electric surface charge. This surface charge creates an electrostatic field that
then affects the ions in the bulk of the liquid. This electrostatic field, in combination with the thermal motion
of the ions, creates a countercharge, and thus screens the electric surface charge. The net electric charge in
this screening diffuse layer is equal in magnitude to the net surface charge, but has the opposite sign. As a
result, the complete structure is electrically neutral. Some of the counterions might specifically adsorb near the
surface and contribute to the Stern layer. The outer part of the screening layer is usually called the diffuse layer.
The diffuse layer, or at least part of it, can move under the influence of tangential stress. There is a conventionally
introduced slipping plane that separates mobile fluid from fluid that remains attached to the surface. The
electric potential at this plane is called the electrokinetic potential or zeta-potential.
12 © ISO 2012 – All rights reserved

s
y
d
y
z
2 3
4 5 6
Key
1 Stern layer
2 Positively charged particle
3 Negatively charged diffuse layer
4 Stern plane
5 Slipping plane
6 Debye length
Figure A.1 — Illustration of the double layer structure according to the Gouy–Chapman–Stern model
The electric potential on the external boundary of the Stern layer versus the bulk electrolyte is referred to as
d
the Stern potential, ψ . The electric potential difference between the bulk of the fluid and the surface is called
s
the electric surface potential, ψ .
General experience indicates that the plane of shear (slipping plane) is located very close to the outer Helmholtz
plane that determines the Stern potential. The layer between this plane and the interface is usually called the
“stagnant layer”. Both planes are abstractions of reality. This means that, in practice, the zeta-potential is equal
d
to or lower in magnitude than the Stern potential, ψ .
A characteristic value of the electric potentials in the EDL at the slipping plane is 25 mV with a maximum value
around 100 mV for aqueous systems. The chemical composition of the sample that brings zeta-potential to zero
is called the point of zero zeta-potential or isoelectric point. It is usually determined by the pH value of the solution.
There are useful mathematical relationships between EDL parameters that can be found in the literature.
Several geometric configurations of the EDL are presented here.
A.2 Flat surfaces
−1
The EDL thickness is characterized by the so-called Debye length, κ , defined by:
cz
22 ii
κ = F (A.1)
∑
εε RT
m 0
i
where
F is the Faraday constant;
R is the gas constant;
T is the absolute temperature;
ε is the dielectric permittivity of vacuum;
o
ε is the relative permittivity of the liquid;
m
c is the molar concentration of the ith ion species.
i
z is the absolute valency of the ith ion species.
i
For a symmetrical electrolyte z = −z = z.
+ −
For a flat surface and a symmetrical electrolyte of concentration c, there is a straightforward relationship
d d
between the electric charge density in the diffuse layer, σ , and the Stern potential, ψ , namely:
d
Fψ
d
σε=− 8 ε cRT sinh (A.2)
m 0
2RT
If the diffuse layer extends to the surface, Formula A.2 can then be used to relate the surface charge to the
surface potential.
Sometimes it is helpful to use the concept of a differential EDL capacitance. For a flat surface and a symmetrical
electrolyte, this capacitance per area is given by:
d
dσ Fψ
C ==εε κcosh (A.3)
dl m 0
d
dψ 2RT
ψψ=
For a symmetrical electrolyte, the electric potential, ψ, at a distance, x, from the flat surface into the EDL, is given by:
 
tanh zFψ xR4 T
()
 
exp()−κx = (A.4)
d
tanh zFψ 4RT
()
The relationship between the electric charge density and the potential over the diffuse layer for an asymmetric
electrolyte is given by:
112/

dd dd
 
σψ=− sgn 2εε cRTzνψexpe− +−νψxp z −−νν (A.5)
() m 0 ++() −−() +−
 
 
where ν is the number of cations and anions produced by dissociation of a single electrolyte molecule, and
±

d
ψ is a dimensionless potential given by:
d
Fψ

d
ψ = (A.6)
RT
For the general case of an arbitrary electrolyte mixture, there is no analytical solution.
14 © ISO 2012 – All rights reserved

A.3 Spherical isolated electric double layers
There is only one geometric parameter in the case of a flat EDL, namely the Debye length 1/κ. In the case
of a spherical EDL, there is an additional geometric parameter, namely the radius of the particle, a. The ratio
of these two parameters, κa, is a dimensionless parameter that plays an important role in colloid science.
Depending on the value of κa, two asymptotic models of the EDL exist.
The thin EDL model corresponds to colloids in which the EDL is much thinner than particle radius, or simply:
κa >>1 (A.7)
The vast majority of aqueous dispersions satisfy this condition, except for very small nano-particles in low
−3
ionic strength media. If it is assumed that for an ionic strength greater than 10 mol/l, corresponding to the
majority of natural aqueous systems, the condition κa >> 1 is satisfied for virtually all particles having a size
above 100 nm.
The opposite case of a thick EDL corresponds to systems where the EDL is much larger than the particle
radius, or simply:
κa << 1 (A.8)
Many aqueous nano-dispersions at low ionic strength and the vast majority of dispersions in hydrocarbon
media, having inherently very low ionic strength, satisfy this condition.
These two asymptotic cases allow one to picture, at least approximately, the EDL structure around spherical
particles as shown in Figure A.2:
-1
k
2a
a)  thin double layer b)  thick double layer
Key
−1
κ Debye length
2a particle diameter
Figure A.2 — Illustration of the “thin” and “thick” double layer models
A general analytical solution exists only for low potential:
RT
d
ψ << (A.9)
F
At 25 °C, the value for the right hand side is approximately 25,8 mV.
This so-called Debye–Hückel approximation yields the following expression for the electric potential in a
spherical EDL, ψ(r), at a distance, r,
...