Advances in Colloid and Interface Science 211 (2014) 77–92
Contents lists available at ScienceDirect
Advances in Colloid and Interface Science
journal homepage: www.elsevier.com/locate/cis
Electrophoresis and stability of nano-colloids: History, theory and
experimental examples
C. Felix a, A. Yaroshchuk b, S. Pasupathi a, B.G. Pollet a, M.P. Bondarenko c, V.I. Kovalchuk c,⁎, E.K. Zholkovskiy c
a
b
c
South African Institute for Advanced Materials Chemistry (SAIAMC), University of the Western Cape, Modderdam Road, Bellville 7535, Cape Town, South Africa
ICREA and Department d'Enginyeria Química (EQ) Universitat Politècnica de Catalunya Av. Diagonal, 647, Edifici H, 4a planta, 08028, Barcelona, Spain
Institute of Biocolloid Chemistry of Ukrainian Academy of Sciences, Vernadskogo, 42, 03142 Kiev, Ukraine
a r t i c l e
i n f o
Available online 20 June 2014
Keywords:
Electrophoresis
Coagulation dynamics
Nano-suspensions
Standard Electrokinetic Model
DLVO theory
a b s t r a c t
The paper contains an extended historical overview of research activities focused on determining interfacial
potential and charge of dispersed particles from electrophoretic and coagulation dynamic measurements. Particular
attention is paid to nano-suspensions for which application of Standard Electrokinetic Model (SEM) to analysis of
experimental data encounters difficulties, especially, when the solutions contain more than two ions, the particle
charge depends on the solution composition and zeta-potentials are high. Detailed statements of Standard Electrokinetic and DLVO Models are given in the forms that are capable of addressing electrophoresis and interaction of
particles for arbitrary ratios of the particle to Debye radius, interfacial potentials and electrolyte compositions.
The experimental part of the study consists of two groups of measurements conducted for Pt/C nanosuspensions, namely, the electrophoretic and coagulation dynamic studies, with various electrolyte compositions.
The obtained experimental data are processed by using numerical algorithms based on the formulated models
for obtaining interfacial potential and charge. While analyzing the dependencies of interfacial potential and charge
on the electrolyte compositions, conclusions are made regarding the mechanisms of charge formation. It is
established that the behavior of system stability is in a qualitative agreement with the results computed from the
electrophoretic data. The verification of quantitative applicability of the employed models is conducted by calculating the Hamaker constant from experimental data. It is proposed how to explain the observed variations of
predicted Hamaker constant and its unusually high value.
© 2014 Elsevier B.V. All rights reserved.
Contents
1.
2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standard electrokinetic model and coagulation theory. Historical overview . . . . . . . . . .
2.1.
Relationships between interfacial potential and electrophoretic velocity . . . . . . . .
2.2.
Interactions of particles and dynamics of coagulation . . . . . . . . . . . . . . . . .
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.
Extracting interfacial potential from electrophoretic measurements . . . . . . . . . .
3.1.1.
Standard Electrokinetic Model . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.
Scalarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3.
Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.
Coagulation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1.
Electrostatic repulsion: general problem formulation
. . . . . . . . . . . .
3.2.2.
Numerical computation of electric field distribution . . . . . . . . . . . . .
3.2.3.
Obtaining Hamaker constant from electrophoretic and coagulation dynamic data
Experimental example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.
Pt/C nano-catalytic dispersions and their practical importance
. . . . . . . . . . . .
4.2.
Materials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
⁎ Corresponding author. Tel./fax: +380 44 424 8078.
E-mail address: [email protected] (V.I. Kovalchuk).
http://dx.doi.org/10.1016/j.cis.2014.06.005
0001-8686/© 2014 Elsevier B.V. All rights reserved.
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C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
5.
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . .
5.1.
Correlation between electrophoretic and stability data . . . . . .
5.2.
Influence of electrolyte composition on surface potential and charge
5.3.
Applicability of Standard Electrokinetic Model . . . . . . . . . .
6.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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87
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92
Nomenclature
1. Introduction
Latin letters
a
radius of particle;
Ck
concentration of the kth ion in the solution bulk;
ck
local concentration of the kth ion;
Dk
the kth ion diffusion coefficient;
E
external electric field strength;
en
unity vector of Cartesian coordinate system;
er, eθ, eφ unit vectors of spherical coordinate system;
eβ and eν unit vectors of bispherical coordinate system;
F
Faraday constant;
Gel(h)
electrostatic free energy;
GW(h)
van der Waals free energy;
H
Hamaker constant;
h
minimum distance between particle surfaces;
I
ionic strength;
I
unit tensor;
n
outward normal vector to a closed surface;
p
local pressure;
p∞
pressure in the solution bulk;
q
surface charge density;
R
gas constant;
r
vector coordinate;
rBA
vector whose origin and end coincide with the centers of
particles B and A, respectively;
r
polar radius in spherical coordinate system;
T
absolute temperature;
u
local velocity of liquid;
Ueph
electrophoretic velocity;
Udph
diffusiophoretic velocity;
X(h)
interaction force;
xn
Cartesian coordinate;
Y(r)
radial part of the streaming function;
Zk
electric charge of the kth ion in Faraday units;
z
axial coordinate;
Nano-particle systems have become one of the most important
objects in modern science and technology because of highly specific and
versatile properties of nano-particles, which are determined by their
small, close to molecular and atomic, size. Moreover, their properties
can be precisely tuned and functionalized by changing their size and composition and modifying their surface. In the recent less than two decades
innumerable amount of various types of nano-particles were obtained
having specific electronic, magnetic, optic, catalytic, biological and other
properties [1–7]. This opens many opportunities for their use in numerous important applications in technology and biomedicine. The studies
in this area are focused not only on the synthesis and tuning of nanoparticles properties but also on their interactions and behavior in various
media.
Liquid systems with dispersed nano-particles are particularly important. Nano-particle dispersions are widely used in technological
processes, e.g. for obtaining substrates covered by nano-particles or
nano-porous media. They are also very important for various biomedical applications. For most applications it is necessary to have
stable nano-dispersions, which do not change their properties with
time due to particle aggregation or chemical processes. The problem
of aggregative stability of solid-in-liquid dispersions is widely studied in
colloid science where it was shown that the stability is controlled by
surface forces acting between the particles [8,9]. In particular, attractive
(e.g. van der Waals) interactions facilitate aggregation of particles,
whereas repulsive (e.g. electrostatic or steric) forces tend to prevent
them from aggregation. Quantitative description of particle interactions
and aggregation in liquids is usually based on the approach, pioneered
by Derjaguin, Landau, Verwey and Overbeek (DLFO theory) [10,11].
After their foundational studies many efforts were devoted to the
derivation of equations describing the dependence of attractive and
repulsive forces on the distance between the particles under various
conditions. However, the application of these equations to the case of
nano-particle systems is not that straightforward. This is a consequence
of commensurability of the particles size and the characteristic distances where the surface forces are acting, which makes inapplicable
some common approximations.
The most important forces, stabilizing the dispersions, are repulsive
electrostatic forces that arise because the particle surfaces are usually
charged. The particle charge depends on the ionic composition of solution
surrounding them, and near the so-called iso-electric point the dispersion
loses its stability. Therefore, obtaining information on the particle charge
under various conditions is very important for understanding and
controlling the dispersion stability. A common approach to the relevant
information about the particle charge is via electrokinetic measurements,
in particular via the measurements of their electrophoretic mobility in
solutions. However, the interpretation of such measurements for nanosized particles is especially complicated.
Electrophoretic transport is driven by the electric forces acting on
the interfacial charges. The role of interfacial charges in the suspension
aggregative stability amounts to the electrostatic repulsion between the
similarly charged particles, which decreases the frequency of “successful”
collisions of particles participating in Brownian motion and thus the rate
of coagulation. Thus, both electrophoretic transport and aggregative stability of suspensions are controlled by the interfacial electric charge.
Greek letters
(β, ν)
bispherical coordinates;
ξk ¼ C ∞k zk =∑ z2n C ∞n dimensionless coefficient;
n
η
viscosity;
θ
spherical polar coordinate;
ε
dielectric permittivity;
κ
Debye parameter;
μk
the kth ion electrochemical potential;
Μk(r)
the function describing radial dependency of perturbation of kth ion electrochemical potential;
Π
effective pressure;
σ
stress tensor;
τ
coagulation time;
τSm
Smoluchowski time scale parameter;
Ψ
equilibrium electric potential;
ψ = ΨF/RT normalized equilibrium electric potential;
ζ
electric potential at the interface in equilibrium state
(zeta potential);
ζexp
measured zeta potential;
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
This charge arises at liquid/solid interfaces in solutions because of dissociation of interfacial acid or basic groups or preferential adsorption of
ions.
Both electrophoresis and electrostatic interactions are essentially
controlled by the properties of screening layers that surround charged
particles in electrolyte solutions. Having opposite charge signs, the
interfacial and screening (diffuse) layer charges form the interfacial
Electric Double Layer (EDL). Due to the thermal motion, the screening
layer has a non-zero thickness that can be estimated as the inverse
value, κ−1, of the Debye parameter, κ, given as [12]
rffiffiffiffiffiffiffiffi
2I
κ¼F
:
εRT
ð1Þ
In Eq. (1), F is the Faraday constant; R is the gas constant; T is the absolute temperature; ε is the electrolyte solution dielectric permittivity;
and I is the electrolyte ionic strength
I¼
1X
2
C z
2 k k k
ð2Þ
where zk and Ck are, respectively, the kth ion valence and bulk (far away
from the interface) concentration.
As is clear from Eq. (1), the EDL thickness decreases with increasing
ionic strength, I, and thus can be regulated by adding electrolyte to the
system. The latter is important since the thickness of diffuse part of
EDL strongly affects both electrophoresis and electrostatic interactions.
Since the thickness of EDL and the interfacial charge depend on the
electrolyte composition one can control electrophoretic transport and
aggregative stability by varying it. At the same time, parallel investigation of electrophoresis and coagulation as functions of solution compositions and establishing correlation between them can yield important
information for optimizing processes where particles simultaneously
experience electrophoresis and coagulation (for example, electrophoretic deposition, see below). Such integrated approaches have been
employed for obtaining information about the interfacial properties in
a number of studies [13–23].
The aim of the present study is to critically evaluate the existing
approaches to the investigations of electrophoresis and coagulation
with a particular emphasis on their applicability to the case of nanoparticle systems and the problems arising in this case. Based on the
statements of Standard Electrokinetic and DLVO Models we propose a
strategy that allows addressing electrophoresis and interaction of particles for arbitrary ratios of particles sizes to corresponding Debye radii.
As a practical example, this approach is illustrated by the application
of the strategy to the analysis of electrical interfacial properties of Pt/C
nano-particle suspension. Catalytic Pt/C nano-particles are used for
obtaining Membrane Electrode Assemblies (MEA) in Proton Exchange
Membrane Fuel Cells (PEMFC) [24–27].
In the next section we give a historical overview of the main statements of Standard Electrokinetic Model (SEM) and coagulation theory.
In Section 3.1 we present the fundamentals of SEM and a numerical
scheme of extracting the interfacial potential and charge from the electrophoretic mobility measured for various electrolyte compositions. The
numerical scheme of addressing coagulation dynamics is discussed in
Section 3.2. In this section, we also propose a method of verifying the
applicability of SEM. The proposed method consists in obtaining the
Hamaker constant, H, from the value of the Fuchs factor, W, which is
determined from the coagulation dynamics data. A short information
on Pt/C nano-catalytic dispersions and the details of experiments are
presented in Section 4. In Sections 5.1 and 5.2, we analyze the mechanisms of charge formation. Final discussion about the applicability of
SEM to our system is given in Section 5.3.
79
2. Standard electrokinetic model and coagulation theory.
Historical overview
2.1. Relationships between interfacial potential and electrophoretic velocity
The most widely used model for describing Electrokinetic Phenomena
is referred to as Standard Electrokinetic Model (SEM) [28,29]. According
to this model, one considers the electrolyte solution surrounding colloidal
particle as a continuous medium, which is described by applying approaches of electro- and hydrodynamics of continuous media combined
with the equations of physical macro-kinetics, which are mathematical
implementations of conservation laws for individual ionic species. The
fluxes of such species are expressed as superposition of convective, diffusion and electro-migration fluxes. Macroscopic equations are assumed to
be applicable up to the solid/liquid interface, which is considered a mathematical surface.
For these two limiting cases, κa → 0 and κa → ∞ (a is the particle
radius), the complex balance of forces acting on the charges in the
particle-electrolyte solution system was considered in classical papers
of, respectively, Debye and Hückel [12] and Smoluchowski [30,31].
These studies yielded simple results for electrophoretic velocity, Ueph,
in external electric field E
2εζ
E
3η
εζ
¼ E
η
Ueph ¼
Ueph
κa → 0
κa → ∞
ðaÞ
ðbÞ
ð3Þ
where ζ is the interfacial electric potential, which is defined in the
thermodynamic equilibrium state, with reference to the solution bulk; η
is the viscosity of electrolyte solution.
However, for intermediate values of κa, the relationship between
electrophoretic velocity and ζ-potential is much more complex. During
the 20th century, a number of theoretical approaches have been developed for addressing electrophoresis in various situations where the conditions of Eq. (3) are not satisfied [32–47]. For obtaining the surface
potential, one should use these theoretical results for extracting the
value of ζ from the experimental data.
Analysis of electrophoresis within the frameworks of SEM was
conducted by Henry [32] who determined the linear term in the expansion of electrophoretic velocity in the powers of ζ. The obtained
relationship enables one to address electrophoresis for low zeta potentials, ζe ¼ Fζ =RT≪1. In terms of apparent experimental quantity ζexp,
corresponding to a given ζ, the Henry result can be represented as
κa
ζ exp ¼ ζ 1−e ½5E7 ðκaÞ−2E5 ðκaÞ
ð4Þ
where E5(x) and E7(x) are the integral exponents of the fifth and
seventh order, respectively. Here ζexp is an apparent quantity, obtained
from the measured electrophoretic velocity, Ueph, by using the
Smoluchowski relationship, Eq. (3b). Note that, at given ζ, the electrophoretic velocity and thus the apparent value of zeta potential, ζexp,
depend on the parameter κa which, according to Eq. (1), depends on
the ionic strength, I. Thus, by using Eqs. (1) and (4), one can extract the
surface potential ζ from the experimentally measured quantity ζexp for
given particle radius, a, and ionic strength of solution, I. Importantly,
the electrolyte composition enters this relationship only via I (see
Eq. (2)). This simple scheme is applicable for ζe ¼ Fζ =RT≪1, only.
The case of higher zeta potentials was considered by various authors.
In particular, Overbeek [33,34] and Booth [35,36] determined several
leading terms in the expansion of electrophoretic velocity in the powers
of normalized zeta potential ζe ¼ Fζ =RT. In such an expansion, Overbeek
obtained the first, second and third order terms for a binary electrolyte
solution of general type. Booth considered symmetric electrolyte for
which he additionally obtained the fourth order term which turned
out to be non-zero for the case of ions of different mobilities.
80
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
For ζe≪1, the predictions of Henry [32] and Overbeek-Booth [33–36]
theories coincide. At higher ζ, the Overbeek–Booth results reveal a
slower increase of electrophoretic velocity with increasing ζ than that
defined by Eqs. (3) and (4). Moreover, at sufficiently large κa, the
electrophoretic mobility depends on ζ with a maximum that is reached
at ζeN3. Numerical results of computations based on SEM have been
reported by Wiersema et al. [37], and O'Brien and White [38]. These
results confirmed the behavior predicted by Booth and Overbeek, qualitatively, but yielded some important quantitative corrections. In particular, the authors of refs. [37,38] demonstrated that the expansions
deduced in refs. [33–36] underestimate both the electrophoretic velocity
and the value of ζ corresponding to the maximum which is observed at κa
N 3 and ζe≅5–7 for 1:1 electrolyte.
Approximate analytical results describing electrophoresis for
arbitrary zeta potentials were obtained for the case of κa N 1 and symmetric electrolyte solution in the pioneering study of Dukhin and Semenikhin
[39,40], and, later, in the papers of O'Brien and Hunter [41,42] and
Ohshima et al. [43]. Qualitatively, all the analytical expressions from
refs. [39–43] demonstrate similar behavior: at sufficiently high κa, the
electrophoretic mobility as a function of ζe passes through a maximum
and, for the limiting case when, simultaneously, κa → ∞ and ζe→∞,
approaches an asymptotic value. In terms of ζeexp ¼ Fζ exp =RT such a
common limit is given by ζeexp →2 ln ð2Þ. According to the comparative
analysis presented for κa = 50 in ref. [43], the analytical predictions
of refs. [39–43] nearly coincide for ζeb4. At higher potentials, the prediction by Ohshima et al. [43] reveals the highest accuracy, with deviation
from the numerical result by less than 1%.
To understand better the above discussed behavior, let us consider a
disperse system with κa ≫ 1 and a simple case of symmetric electrolyte
solution with equal diffusion coefficients of ions. In such a system,
applying an external electric field produces a steady state distribution
of electrolyte concentration in the electro-neutral zone (i.e. outside
EDL) around the particle. Importantly, in this concentration field,
always, the higher concentration zone is adjacent to the rear part
whereas the lower concentration zone is adjacent to the front part
(according to the direction of particle movement). There are two
reasons for such a structure of concentration distribution:
(i) The counterions and coions are transported by the electric field
through the EDL in, respectively, higher and lower proportions
than through the electrolyte solution bulk;
(ii) The coions and counterions are transported by the electric field,
respectively, in the same and opposite directions relative to the
electrophoresis of particle.
Consequently, the coions delivered by electric field to the rear zone
of moving particle are not withdrawn from there in equivalent amounts
through the EDL region. At the same time, the counterions being
transported against the particle motion are delivered in excess amounts
through the EDL to the rear zone, as well. Hence, both the co- and
counter-ion concentrations (i.e., the electrolyte concentration) become
higher than in the solution bulk, δc N 0. Similarly, coions and counterions
are withdrawn from the front zone toward the solution bulk and rear
zone, respectively. Accordingly, the electrolyte concentration in the
front part becomes lower than in the solution bulk, δc b 0 (Fig. 1).
Under the influence of external electrolyte concentration gradient,
a solid particle is involved into motion which is referred to as
diffusiophoresis [40,48–51]. The rate of diffusiophoresis, Udph, is proportional to the imposed concentration gradient and directed toward higher
electrolyte concentration. Hence, while applying electric field, additionally to pure electrophoresis, the particle is driven by diffusiophoresis in
the concentration field produced by the applied electric field. Since the
produced concentration gradient is proportional to applied electric
field, the diffusiophoresis of this type is also proportional to the electric
field. Being directed toward the larger concentration, which always
occurs close to rear part of the moving particle, such a field-induced
diffusiophoresis is always directed oppositely to the electrophoretic
Fig. 1. Illustration to the explanation of non-linear dependency of electrophoretic velocity
on zeta potential (see the text for more detail).
transport thereby decreasing the total electrophoretic velocity compared to the purely electrophoretic motion. At sufficiently high zeta
potentials diffusiophoresis results in decreasing the overall electrophoretic velocity with increasing ζ.
The asymptotic case κa → ∞ and ζe→∞, when the electrophoretic
velocity approaches a limiting value corresponding to ζeexp →2 ln ð2Þ,
corresponds to the situation when the particle is surrounded by a
vanishingly thin EDL that is infinitely permeable to the counterions
but absolutely impermeable to the coions. In such a case the electrochemical potential of counterions at the external surface of EDL is constant. By taking into account the approach developed in refs. [43–47],
electrophoresis can be considered a superposition of particle drifts due
to the gradients of electrochemical potentials of counterions and coions.
As the electrochemical potential of counterions is constant at the
surface, the particle is driven by the gradient of coion electrochemical
potential, alone. For κa → ∞, the coion electrochemical potential gradient
gives rise to the particle velocity equal to that given by the Smoluchowski
formula for the normalized zeta potential whose magnitude is equal to
2 ln(2).
By using the theoretical approaches of refs. [33–47], one can extract
from electrophoretic mobility the value of ζ for an arbitrary binary electrolyte solution of known concentration. As the zeta-potential is the interfacial electric potential in the thermodynamic equilibrium state, one
can use the equilibrium distribution of electric field and the electrostatic
boundary condition interrelating the interfacial charge density and the
normal electric displacement for determining the former one. Consequently, a set of zeta potentials obtained from electrophoretic measurements in solutions with different electrolyte compositions enable one to
obtain information on the isotherm of adsorption, which defines the
value of interfacial charge density.
In the cases when the electric charge is controlled by the adsorption of
H+ and OH− ions (or dissociation of interfacial basic or acidic groups),
one can study adsorption isotherm by using the above theoretical predictions for binary electrolyte solution [33–47] only within sufficiently
narrow ranges of pH values where one can neglect the presence of
H+ and OH− ions in the transport equations. For addressing the system
within a broad pH range one should consider at least ternary ionic
system. Such an analysis can be carried out numerically on the basis
of formalism developed in refs. [33–47].
For some systems, using the SEM model yields a perfect description.
A recent careful study [52] gives an example of that. While assuming the
surface charge density to be constant, the author demonstrates that the
maximum of the dependency of electrophoretic velocity on the zeta potential manifests itself as a minimum on the curve displaying the dependency of electrophoretic velocity on salt concentration. This conclusion
is supported by the perfect fitting of experimental data performed by
using the analytical result reported in [43]. However, in many cases
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
for properly addressing experimental data SEM requires modifications
[18,28,40].
2.2. Interactions of particles and dynamics of coagulation
According to the classical Smoluchowski theory [53], the dynamics
of coagulation process, which results in changes of size distribution of
particles, in particular, can be characterized by a time scale-parameter,
τ, given by
τ ¼ τSm W
ð5Þ
where τSm = 3η/4kBTn (kB ≈ 1.4 ⋅ 10− 23J/K is the Boltzmann constant;
n is the initial concentration of particles) is the time-scale parameter
corresponding to the so-called rapid or Smoluchowski coagulation,
which takes place when 100% of particle collisions lead to the formation
of doublets. In Eq. (5), the Fuchs factor, W, [54] describes the increase in
τ when, due to the repulsive forces between the particles, efficiency of
particle collisions becomes smaller than 100%
Z∞
W¼
0
exp½GðhÞ=kB T dh:
ð1 þ hÞ2
ð6Þ
In Eq. (6), h = (rAB − 2a)/2a, where rBA is the distance between the
particle centers (Fig. 2); G(h) is the free energy of a system of two
interacting particles defined with reference to the state where the
particles are separated by an infinitely large distance. The function
G(h) can be obtained by calculating the mechanical work performed
by interaction forces while particles are displaced from the state when
the distance between their centers is rAB = 2a(h + 1) to infinitely
large distance [8,55,56].
We consider interactions due to the electrostatic and van der Waals
forces whose contributions, Gel(h) and GW(h), to G(h) are assumed to be
additive
GðhÞ ¼ Gel ðhÞ þ GW ðhÞ:
ð7Þ
Study of interaction forces via coagulation dynamics consists in
measuring the time scale parameter τ and determining from it the
Fuchs factor, W, which, according to Eq. (6), is defined by the interaction
free energy as a function of distance separating the particles, G(h). By
using models for obtaining each of the contributions into the interaction
free energy, Gel(h) and GW(h), one can determine the unknown parameters of these models. These parameters should be obtained by fitting
the experimental dependencies of the measured parameter, W, on the
electrolyte composition. The fitting is conducted with the help of theoretical dependencies that are deduced by substituting the predicted
G(h) into Eq. (6).
Obtaining the electrostatic contribution, Gel(h), is based on the theory
independently developed by Derjaguin and Landau [10] and Verwey and
Overbeek [11]. According to their approach, the force acting on either of
the particles is determined via integration of the Maxwell stress tensor
over arbitrary closed surfaces enveloping the respective particle. The distributions of electric field and pressure, which are required for obtaining
the Maxwell tensor, are determined by considering thermodynamic and
mechanic equilibrium in the system within the frameworks of Poisson–
Boltzmann (P–B) equation, which is subject to the electrostatic boundary
conditions at the particle surfaces, and the hydrostatic momentum
balance conditions (also referred to as stress balance). The obtained
force is a function of the distance between the particles and is employed
for computing the minimum work which is expended while displacing
either of the particles far away from another. The latter work yields the
required function, Gel(h).
An equivalent method of obtaining Gel(h) consists in the calculation
of interface charge of particles at imposed surface potential by using the
solution of the P–B equation. Next, the charge obtained as a function of
surface potential should be integrated over the potential from the surface
potential corresponding to a single particle to the potential, which occurs
for a given distance between the particles [57].
Several approximate theoretical approaches have been used while
implementing the aforementioned schemes analytically. According to
the Derjaguin approximation [55,58,59], each of the particle surfaces is
represented as a set of quasi flat segments whose interaction is described by making use of the results obtained for two parallel infinite
planes. Ultimately, the contributions from separate couples of the
segments are added up. The Derjaguin approximation is valid for the
case of κa ≫ 1.
Another approximation, which is valid for arbitrary κa, is based on
the linearization of electric field distribution in terms of normalized surface potential [60]. Consequently, these results are valid for sufficiently
low surface potentials. One should also mention an approach based on
the assumption that the electric field in the system of two particles
with overlapped EDLs is a superposition of fields within the EDL of either
single particle [61]. The latter approach is valid when, simultaneously,
κa b 1 and κh N 1.
All the existing analytical results are inapplicable to the system of interest of this study because they are unable to address the case of κa ≅ 1
and high surface potentials. Thus, interpretation of our experimental
data requires numerical approaches that have been employed for addressing electrostatic interactions between two spheres [62–66].
There are some difficulties in addressing the contribution into the interaction free energy due to the van der Waals forces, GW(h). Rigorous
theoretical results for the interaction of two spheres in electrolyte solution still have not been obtained. Therefore, the most popular approach
described in the literature is based on the superposition approximation.
In this approximation, each of two interacting bodies is represented as a
set of infinitely small elements. Consequently, the interaction energy is
computed by adding up the energies of interaction between each of the
element pairs. Such a calculation is based on the assumptions that the
contributions of interaction of various pairs are additive. For the system
of two spheres, the above calculation scheme yields the classical result
of Hamaker [67]
GW
Fig. 2. Two interacting spherical particles (see the text for more detail).
81
h
i
8
"
#9
2
=
4 ð2 þ h=aÞ −2
H <
4
þ ln 1−
¼− ð8Þ
2 ;
6 :ð2 þ h=aÞ2 ð2 þ h=aÞ2 −4
ð2 þ h=aÞ
where H is the Hamaker constant, which depends on the materials of
both the bodies and the surrounding medium. The negative sign of
energy in the right hand side of Eq. (8) corresponds to the attraction.
The result given by Eq. (8) does not take into account the complex cooperative interactions within each of the bodies and the retardation effect,
which is a result of finite length of electromagnetic waves [8,55,56].
82
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
The flux-continuity equation can be represented in the following
form
Studies of surface forces by means of measuring the Fuchs factor, W,
have a long history that has been started by the work of Reerlink and
Overbeek [68], who determined both the surface potential and the
Hamaker constant by analyzing the dependency of experimentally measured W on the concentration of symmetrical electrolyte. The analyzed
system and the theoretical relationships employed for the interpretation are valid in the case of thin EDL. In the numerous studies, which
followed [68], the authors used optical methods, in particular, Light
Scattering for the determination of time evolution of particle-size distribution in the suspension and, having obtained the kinetic constant for
coagulation, determined W and extracted information on the surface
forces by using various modifications of the DLVO theory. A review of
these studies can be found in [18,69–71].
In Eq. (13), dμk = zkFdΦ + dμch
k is the differential of electrochemical
potential of kth ion; Φ is the local electric potential and μch
k is the chemical
potential of kth ion. Generally, μch
k depends on all the ion concentrations.
However, for ideal electrolyte solutions usually postulated in electrokinetic studies, dμch
k = RTdck/ck where ck = ck(r). When r → ∞, ck(r) → Ck.
The Stokes equation accounting for the electric body force, after
some transformations, can be represented, as [72]
3. Theory
X
η∇ ∇ u ¼ −∇Π−
C k ½ expð−ΨFzk =RT Þ−1∇μ k
∇ ∇μ k −
zk F
z F
∇μ k ∇Ψ ¼ − k u ∇Ψ:
RT
Dk
Extraction of the surface potential, ζ, from the apparent value ζexp, is
conducted by means of SEM according to the most consistent and
convenient formalism developed in refs. [39–43]. Below, we follow
the version of this formalism stated in review [72].
where the effective pressure, Π ¼ p−RT∑ C k ½ expð−ΨFzk =RT Þ−1, is
the deviation of local pressure, p, from its kvalue in the thermodynamic
equilibrium state relative to the solution bulk (the second term in the
right hand side of latter equality).
The continuity equation for liquid velocity is written in its usual form
∇ u ¼ 0:
3.1.1. Standard Electrokinetic Model
To determine the electrophoretic velocity, one should solve two
boundary value problems consequently. The first of these problems
describes the distribution of electric potential around a particle in the
thermodynamic equilibrium state, Ψ = Ψ(r), i.e., in the absence of externally applied electric field. This problem includes the Poisson–
Boltzmann (P–B) equation
FX
C z expð−Ψzk F=RT Þ
ε k k k
ð9Þ
where ∇ = en∂/∂xn; en and xn are the unit vector and the coordinate of a
Cartesian coordinate system. The P–B equation, Eq. (9), is subject to
these boundary conditions
Ψ¼ζ
Ψ→0
ð14Þ
k
3.1. Extracting interfacial potential from electrophoretic measurements
∇ ∇Ψ ¼ −
ð13Þ
at the particle surfaces
at infinity :
at the particle surface
The governing equations, Eqs. (13)–(15), are subject to boundary
conditions at the particle surface and infinity. The particle surface is
assumed to be impermeable for ions. This condition looks obvious for
the case of dielectric particles. It is also true for conducting but ideally
polarizable particles. As for small metallic particles, according to ref.
[73], one can approximately consider them as ideally polarizable since
the effective resistance of electrochemical reactions is much larger
than the resistance of electrolyte solution even in the case of catalytic
platinum. The latter enables one to set the kth normal flux to be zero
at the particle surface. We consider the problem in the reference system
linked to the particle. Consequently, the respective conditions take
these forms
n ∇μ k ¼ 0
at the particle surfaces
ð16Þ
ð10Þ
u ¼ 0 at the particle surfaces :
ð11Þ
At infinity, we impose a uniform external field strength, E, and zero
concentration gradients. These two physical conditions are expressed
this way
Thus, by solving the boundary-value problem given by Eqs. (9)–(11),
one obtains the spatial distribution of electric potential, Ψ(r), in the
thermodynamic-equilibrium state. By using the obtained function
Ψ(r, ζ), the interfacial charge density, q, is determined from the electrostatic condition written in the form
q ¼ −εn ∇Ψ
ð15Þ
ð12Þ
where n is the unit outward vector normal to the particle surface. The
latter equation yields the required relationship between the surface potential and charge density.
The second of the aforementioned problems is formulated for the
non-equilibrium mode when an external electric field is applied. The
problem formulation involves the function, Ψ(r), which is supposed to
be known from the solution of the first problem. The set of governing
equations of the second problem includes the continuity equations for
individual ionic fluxes and liquid flow and a version of Stokes equation,
which accounts for the presence of electric force acting on the EDL space
charge.
∇μ k ¼ −Fzk E
at infinity :
ð17Þ
ð18Þ
One more boundary condition should be set to impose zero total force
exerted on the particle. Such a force is a sum of electrical and mechanical
ones and is obtained via integration of the sum of the Maxwell and
viscous-stress tensors over any closed surface surrounding the particle.
It is convenient to choose such a surface as a sphere with infinitely
large radius. In this case, the electrical force acting on the totally
electro-neutral volume inside the surface is zero. Accordingly, the integral
of Maxwell stress tensor over the chosen surface, S∞ turns out to be zero,
too. Consequently, the required boundary condition takes this form
∮ ΠIþη ∇uþð∇uÞ n∞ dS ¼ 0
ð19Þ
S∞
where I = enen, n∞ is the unit vector normal to the surface S∞. Since the
function Ψ(r) is known as a solution of the first boundary-value problem
given by Eqs. (9)–(11) and (13)–(15) subject to the boundary conditions
of Eqs. (16)–(19) yield a closed problem formulation enabling one to find
the unknown functions μk(r), u(r) and Π(r). By considering the limit of
r → ∞ for the velocity field, u(r), and transforming to the reference
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
83
system linked to the liquid at infinity, one obtains the electrophoretic
velocity, as
In Eqs. (25) and (26), we also suggest a convenient normalization of
unknown functions. By combining Eqs. (13), (25) and (26), we obtain
Ueph ¼ −U ¼ − lim uðrÞ:
d2 Μ k 2 dΜ k
2
− 2 Μk ¼
þ
r dr
dr2
r
ð20Þ
r→∞
dΜ k 3
dψ
− mk ur
zk
2
dr
dr
ð27Þ
Thus, by solving consequently the boundary value problems given
by Eqs. (9)–(11) and (13)–(19) and by using the limiting transition of
Eq. (20), we can interrelate the electrophoretic velocity Ueph and surface
potential ζ for any electrolyte solution used in experiment.
where the electrokinetic parameter, mk, is given by
3.1.2. Scalarization
We assume that the particle is a sphere having radius a.
While using the spherical coordinate system shown in Fig. 3 and
taking into account that the system has spherical symmetry at equilibrium, Eqs. (9)–(12) are rewritten as
By using Eqs. (25) and (26), the boundary conditions of Eqs. (16)
and (18) take these forms
X
1 d
2 dψ
2
r
¼ −ðκaÞ
ξk expð−ψzk Þ
2 dr
dr
r
k
ð21Þ
ψðaÞ ¼ ζe
ψð∞Þ ¼ 0
q¼−
ð22Þ
εRT dψ
ð1Þ
Fa ∂r
ð23Þ
mk ¼
2
2 ε
RT
:
3 ηDk F
ð28Þ
dΜ k
ð1Þ ¼ 0
dr
ð29Þ
dΜ k
ð∞Þ ¼ −1:
dr
ð30Þ
To derive a convenient form of the Stokes equation, Eq. (14), one
should apply operator ∇ × to both sides of Eq. (14) and substitute the
electrochemical potential, μk(r, θ), and velocity, u(r, θ) in the forms
given by Eqs. (25) and (26). While making use of such a substitution,
the functions ur(r) and uθ(r) can be represented in a form that follows
from Eq. (15)
ur ðr Þ ¼ −
2
Y
r2
ð31Þ
where
2
ξk ¼ C k zk =C k zk
ðaÞ
ψ ¼ ΨF=RT
ðbÞ
ζe ¼ ζF=RT
ðcÞ :
ð24Þ
uθ ðr Þ ¼
1 dY
:
r dr
ð32Þ
In the presence of a uniform electric field E at infinity the system has
an axial symmetry. We choose the spherical coordinate system with
unit vectors er, eθ, eφ and the polar axis directed along the vector E
(Fig. 3). The problem symmetry dictates the following angular dependencies
The transformation scheme described above leads to the following
form of Eq. (14)
μ k ðr; θÞ ¼ zk FEaΜk ðr Þ cosðθÞ
Further, Eq. (33) is subject to boundary conditions that are obtained
by combining the boundary conditions of Eqs. (17) and (19) with
Eqs. (26), (31) and (32). Vector boundary condition Eq. (17) transforms
into two scalar conditions
uðr; θÞ ¼ Eε
RT
½u ðr Þ cosðθÞer þ uθ ðr Þ sinðθÞ eθ :
ηF r
ð25Þ
ð26Þ
d2
2
−
dr 2 r 2
!2
2
Y ¼ −ðκaÞ
X
k
ξk
d expð−zk ψÞ
Μk :
dr
Y ð1Þ ¼ 0
dY
ð1Þ ¼ 0:
dr
ð33Þ
ð34Þ
For specifying the boundary condition of Eq. (19), which imposes
zero value of the total force exerted on the particle, we will use the results of refs. [72,74] where this condition was obtained at the spherical
cell border. By considering the limiting case of infinitely large cell radius
that corresponds to a single particle or very dilute suspension, the
boundary condition of Eq. (19) is rewritten in the form
!
d2
2
2 d
r
þ 2r
−
→0:
Yj
dr
dr 2 r 2
r→∞
Fig. 3. Particle in external electric field. Spherical coordinate system (see the text for more
detail).
ð35Þ
Thus, the governing Eqs. (21), (27) and (33) subject to the boundary
conditions (22), (23), (29)–(32), (34) and (35) make up a closed problem formulation that enables one to determine N + 2 (N is the number
of ions) unknown functions ψ(r), Μk(r), and Y(r).
By using the function Y(r) to be obtained and combining Eqs. (20),
(26) and (31), one can determine the electrophoretic velocity, Ueph.
84
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
Then the measured value of zeta potential, ζexp, normalized by RT/F is
represented as
Y
ζeexp ¼ 2 lim 2 :
r→∞ r
ð36Þ
Since Y ¼ Y r; ζe; mk ; κa; zk ; ξk , Eq. (36) yields the required relationship between the measured and actual surface potentials, ζeexp and ζe for
any given composition of electrolyte. Next, we will discuss numerical
algorithm of determining ζe from a given value of ζeexp by using the
boundary value problem stated above.
3.1.3. Numerical analysis
For given functions ψ(r) and Μk(r), the distribution Y(r) is obtained
by solving the boundary value problem Eqs. (33)–(35). It can be shown
that the solution of this problem can be represented in the following
form
"Zr
!
2ðκaÞ2
3 2 3
3x4
3r 2
f ðxÞdx
ur ¼ −
− x þ xþ
−
2r
2
9
10r 3 10x
1
#
3
1
3 2 1
1
−1− 3 A þ
r − þ 3 B
þ
2r
10
2 5r
2r
ð37Þ
While addressing either Smoluchowski or Henry case, the right hand
side of Eq. (27) can be omitted. For the Smoluchowski case, it differs
from zero within the vanishingly thin (κa → ∞) EDL region, only. The
latter is a result of the presence of dψ/dr in the right hand side of
Eq. (27). For the Henry case [31], which yields the linear term in the
expansion of electrophoretic mobility in the powers of ζe, one should
substitute into Eq. (41) the function corresponding to ζe ≡ 0, Μk(r, 0),
since Eq. (27) already contains factors proportional to ζe besides Μk(r).
Hence, for both the limiting cases, Eq. (21)
into a homoge transforms
neous equation whose solution, Μ k ¼ Μk r; ζe; κa , satisfying boundary
conditions (29) and (30) has this form
1
Μk r; ζe; ∞ ¼ Μ k ðr; 0; κaÞ ¼ Μ ðr Þ ¼ −r 1 þ 3 :
2r
Thus, for both the Smoluchowski and Henry cases, the function
Μk(r) is independent of ζe and κa and turns out to be the same for all
the ions.
Now, we substitute Eqs. (42) into (41) and combine the derived
equation with Eq. (11). The integral obtained in this manner should
be taken by parts three times in series while accounting for Eq. (22).
Finally, we arrive at the following expression
ζeexp ¼ ζe−
where
Z∞ 1
f ðr Þ ¼
X
k
Z∞
A¼
ξk
d exp½−zk ψðrÞ
Μ k ðr Þ
dr
ð38Þ
2
x f ðxÞdx
ð39Þ
f ðxÞ
dx:
x
ð40Þ
1
Z∞
B¼
1
While using Eqs. (31), (36) and (38)–(40), the expression for the
apparent zeta potential, ζeexp , takes the form
2
ðκaÞ
ζeexp ¼ −
9
X
k
Z∞ 1
2
þ 2r −3r
ξk
r
1
h
i
∂ exp −z ψ r; ζe
k
∂r
Μk r; ζe dr:
ð41Þ
Thus, Eq. (41) yields the interrelation between the apparent zeta potential, ζeexp , and the surface potential, ζe. The dependency on the latter
value is contained
in the normalized
of equilibrium poten
distributions
tial, ψ r; ζe , and the function Μ k r; ζe attributed to the kth ion with the
help of Eq. (25). In ref. [45], an integral relationship similar to Eq. (41)
was derived for z:z electrolyte solution. Hence, Eq. (41) yields a generalization of the result of ref [45] for the case of mixed electrolyte solution.
e
e
As stated above, for interrelating
ζexp and ζ with
the help of Eq.(41),
one should know the functionsψ r; ζe andΜ k r; ζe . The functionψ r; ζe
is determined separately by solving
the
boundary value problem given by
e requires a more complex scheme
r;
ζ
Eqs. (21) and (22). Obtaining
Μ
k
since the functions Μk r; ζe appear in
Eq.(27) together with ur which, in
turn, depends on all the functions Μ k r; ζe via the integral relationship of
Eq. (37). However, in two limiting cases, Eq. (27)does
not contain ur that
allows obtaining the required set of functions Μ k r; ζe by solving Eq. (27)
subject to boundary conditions (29) and (30), separately. These two
limiting cases corresponding to κa → ∞ (Smoluchowski limit [30,31])
and ζe≪1 (Henry case [32]) were mentioned in Section 2.
ð42Þ
5
2
− 4 ψdr:
6
r
r
ð43Þ
For the Smoluchowski limiting case, κa → ∞, the integral in the right
hand side of Eq. (43) approaches zero. This can be understood by considering that |ψ|/rn ≤ |ζ| exp[−(r − 1)κa]. Consequently, substituting
the right hand side of the later inequality into the integral of Eq. (37)
one can see that the integral approaches zero when κa → ∞. Hence,
Eq. (43) and, thus, Eq. (41) lead to the expected result for the
Smoluchowski limit, ζeexp ¼ ζe.
While dealing with the Henry case, from the boundary value problem
given by Eqs. (21) and (22), one obtains:
exp½−ðr−1Þκa
2
þO ζ :
ψ ¼ ζe
r
ð44Þ
By substituting Eqs. (44) into (43), we arrive at Eq. (4) which is
equivalent to the Henry's result [32]. When the parameter κa is
known, Eq. (4) allows determining ζe for any measured value ζeexp .
For a given value of κa, the relationship between ζeexp and ζe is independent of electrolyte composition for each of the two limiting cases
discussed above. Such an independency takes place since the distributions Μk(r) given by Eq. (42) become the same for all the ions. Generally,
the functions Μk(r) corresponding to various ions differ from each other
and can be determined with the help of computational scheme described
below.
Let us solve Eq. (27) with respect to Μk(r) by considering the right
hand side of Eq. (27) as a known function. After satisfying the boundary
conditions (29) and (30), the obtained solution can be represented in
this form
Zr
1
1
Μk ¼ −ð1 þ Hk Þ 1 þ 3 r þ
3
2r
r−
x4
r
!
zk
dΜ k 3
dψ
dx
− mk ur
2
dx
dx
1
ð45Þ
where the integration constants Hk are given by
Hk ¼
1
3
Z∞ dΜk 3
dψ
dr:
− mk ur
zk
2
dr
dr
1
ð46Þ
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
Now, we consider the set of integral relationships given by Eqs. (37)
and (45). According to these relationships, at the point with coordinate
r = r∗, the functions ur(r) and Μk(r) take values, ur(r∗) and Μk(r∗), that
are expressed through the distributions Μk(r) for 1 b r b r∗. Consequently,
both ur(r) and Μk(r) can be determined numerically by gradually
increasing r in the integrals on the right hand sides of Eqs. (31) and
(41). Recall that, when r → ∞, the asymptotic value approached by
ur(r) is (−ζeexp ).
The discussion above defines the steps of numerical scheme to be
used: (i) the function ψ r; ζe; κa is determined by solving the P–B
boundary value problem given by Eqs. (21) and (22); (ii) certain initial
values of the integration constants A, B and Hk appearing in Eqs. (37)
and (45) are assumed; (iii) while using Eqs. (31), (32) and (41), the
functions ur(r) and Μk(r) are computed by gradually increasing r; (iv)
the obtained distributions are used for recalculation of A, B and Hk by
means of Eqs. (39), (40) and (46), respectively.
3.2. Coagulation dynamics
In this present section, we consider the interpretation scheme for
the experimental rate of disperse system coagulation. As it was stated
in Section 2.2, we will extract the Hamaker constant, H, from the
Fuchs factor, W, which is estimated from the coagulation dynamics.
Next, we consider the scheme of obtaining, Gel(h) for given zeta
potential and electrolyte composition. Since we deal with moderate
κa, ζeN1 and mixed electrolyte solution, the problem will be solved
numerically for a rather general case.
3.2.1. Electrostatic repulsion: general problem formulation
We consider two particles separated by a distance, l, and bearing
either constant surface potential, ζ, or constant surface charge, q, that
are determined from the electrophoretic mobility measurements following the scheme described in the previous sections. Both the particle
charge and potential are assumed to be the same for two particles.
The system containing two particles and the infinite volume of
surrounding electrolyte solution is considered to be in thermodynamic
and mechanic equilibrium. Consequently, the distribution of electric
potential, Ψ, is obtained as a solution of P–B problem given by
Eqs. (9)–(11) with a reservation that, in the limiting case of constant
surface potentials, the same potential ζ is set at the surfaces of each of
the particles. For analyzing the case of constant surface charge at the
surface of each of the particles, one should use Eq. (12) instead of
Eq. (10) for setting the electrostatic boundary condition.
By using the solution of P–B problem, Ψ(r), one can determine the
force, acting on either of two particles, X. To this end, the stress tensor,
σ, should be integrated over the particle surface Sp, as
X ¼ ∮ σ n dS
ð47Þ
Sp
ε∇Ψ∇Ψ− 2ε Ι∇Ψ
∇Ψ−Ιp is the stress tensor. On the right
where σ ¼
hand side of the latter expression, the first two terms represent the
Maxwell tensor, and the third term gives the contribution of pressure,
p, into the total stresses. The local pressure can be interrelated with
the local value of the potential by means of mechanical-equilibrium
condition, which can be written in the form
∇ σ ¼ 0:
ð48Þ
By combining Eqs. (9), (47) and (48), after some transformations,
one obtains
p−p∞ ¼ RT
X
C k ½ expð−ΨFzk =RT Þ−1
ð49Þ
By using Eq. (48) and the tensor version of the Gauss theorem, one
can prove the following equality
∮ σ nA dS ¼ −
Sp
1
2að1 þ hÞ
Z
σ rBA dS
ð50Þ
Ssym
where Ssym is the symmetry plane; rBA is the vector whose origin and
end coincide with the centers of particles B and A, respectively.
Now, we consider the force XA and XB exerted on the particles A and
B respectively. This is obtained by combining Eqs. (46), (47), (49) and
(50) and by using symmetry considerations
XA ¼
rBA
rAB
X¼−
X ¼ −XB
2að1 þ hÞ
2að1 þ hÞ
ð51Þ
where the force magnitude, X, which is obviously the same for both
particles, is expressed as an integral over the symmetry plane,
which is perpendicular to the line connecting the sphere centers,
Ssym
X¼
#2
)
Z ( "
X
ε
r
∇Ψ− 2 BA 2 ðrBA ∇ΨÞ þ RT
C k ½ expð−zk ΨF=RT Þ−1 dS:
2
4a ð1 þ hÞ
k
Ssym
ð52Þ
Thus, Eqs. (51) and (52) enable one to compute the electrostatic interaction force exerted on the interacting particles when the equilibrium
electric potential distribution, Ψ(r), is known. At a given particle radius,
this force is a function of the distance between the particles, X = X(h).
Consequently, the contribution of electrostatic forces into the interaction
free energy, Gel(h), is determined as
Z∞
Gel ðhÞ ¼
XðhÞdh:
ð53Þ
h
In summary, the electrostatic contribution to the system free energy is
obtained by integral of Eq. (53) where the interaction force magnitude,
X(h), is computed by using Eq. (52), which depends on the potential
distribution, Ψ(r). The latter distribution is obtained as a solution of the
non-linear boundary value problem given by governing Eq. (9) subject
to the boundary condition (12) (for constant surface potential) or (10)
(for constant surface charge). Both the latter conditions are set at the surface of each of the particles. One more boundary condition is given by
Eq. (11). Numerical solution of this problem for two particle system is
considered next.
3.2.2. Numerical computation of electric field distribution
For κa ≫ 1, the above outlined scheme of obtaining Gel(h) is simplified by using the Derjaguin approximation [56,62,63] discussed in
Section 2.2. In this approximation, the first term in curly brackets in
Eq. (52) is omitted for being small at κa ≫ 1. For moderate κa, this
term yields a noticeable contribution. In such a case, one should solve
complete P–B problem and take into account all the terms represented
in Eq. (52). The analysis of this type is given in ref. [73], which will be
used in our calculations.
By following ref. [66], we will compute the distribution Ψ(r)
and the electrostatic interaction force magnitude in a bi-spherical
coordinate system. Taking into account the problem axial symmetry,
one can represent the ∇ operator in terms of new coordinates (β, ν)
as
k
where p∞ is the pressure far away from the particles.
85
∇¼
coshðν Þ− cosðβÞ
∂
∂
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eβ
þ eν
∂β
∂ν
a hðh þ 2Þ
ð54Þ
86
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
where eβ and eν are the unit vectors of the bi-spherical coordinate
system. Differentiating eβ and eν satisfies the following rules:
∂eβ
sinhðνÞ
¼ eν
coshðν Þ− cosðβÞ
∂β
∂eν
sinðβÞ
¼ eβ
coshðν Þ− cosðβÞ
∂ν
ðaÞ
ðcÞ
∂eβ
sinðβÞ
ðbÞ
¼ −eν
coshðν Þ− cosðβÞ
∂ν
:
∂eν
sinhðν Þ
ðdÞ
¼ −eβ
coshðν Þ− cosðβÞ
∂β
ð55Þ
By combining Eqs. (9), (24a,b), (54) and (55), we arrive at the
following dimensionless version of the P–B equation written in
bi-spherical coordinates
3
½ coshðν Þ− cosðβ Þ ∂
sinβ
∂ψ
∂
sinβ
∂ψ
þ
hðh þ 2Þ sinβ
∂β coshðν Þ− cosðβÞ ∂β
∂ν coshðν Þ− cosðβ Þ ∂ν
X
2
¼ −ðκaÞ
ξk expð−ψzk Þ:
k
ð56Þ
Boundary conditions at the particle surface are rewritten as
ψðβ; ν 0 Þ ¼ ζeðconstant potentialÞ
ð57Þ
or
coshðν0 Þ− cosðβÞ ∂ψ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðβ; ν0 Þ ¼ e
qðconstant chargeÞ
κa hðh þ 2Þ ∂ν
ð58Þ
where e
q ¼ qF=εκRT. The coordinate surface, ν = ν0, coincides with the
surface of one of the particles.
Instead of setting the same condition at the surface of another particle,
we will use the system symmetry which allows us to set the following
conditions at the symmetry plane ν = 0
∂ψ
ðβ; 0Þ ¼ 0
∂ν
ð59Þ
and at the axis β = 0, π
∂ψ
ð0; νÞ ¼ 0
∂β
∂ψ
ðπ; ν Þ ¼ 0:
∂β
ð60Þ
Thus, Eq. (56) subject to boundary conditions (57) (or (58)), (59) and
(60) forms a closed problem formulation for obtaining the function
ψ(β, ν). This problem is numerically solved by conducting a discretization
of the second order differential equation for obtaining equations to be
solved with the help of an iteration scheme. We used the iteration
method of Newton–Raphson that enabled us to reduce the non-linear
problem to several linear iterations. Finally the obtained function
ψ(β, 0) is substituted into integral (52) which is rewritten in the form
X ðhÞ ¼ πε
2 Zπ (
)
2
RT
ðκaÞ hðh þ 2Þ X ξk
∂ψ
½
ð
ð
Þ
Þ−1
ð
Þ
sinðβ Þdβ:
exp
−z
ψ
β;
0
þ
β;
0
k
F
∂β
½1− cosðβÞ2 k zk
0
ð61Þ
sum of electrostatic, Gel(h), and van der Waals, GW(h), parts, Eq. (7).
The function Gel(h) is obtained by using calculations presented in
Sections 3.2.1 and 3.2.2 for a surface potential ζ and electrolyte composition. By using numerical calculations based on the SEM, ζ is determined
from ζexp obtained from the electrophoretic measurements. The function,
GW(h), which contains, H, is substituted in the form given by Hamaker's
Eq. (8) [67]. For a set of electrolyte concentrations and zeta potentials,
the Hamaker constant is fitted to make the calculated Fuchs factor, W,
match its value estimated from the coagulation-dynamic experiments.
4. Experimental example
4.1. Pt/C nano-catalytic dispersions and their practical importance
The experimental part of this study deals with a system of considerable practical interest namely colloidal suspensions of composite Pt/C
nano-particles. The Pt/C composites are commonly employed as catalyst
materials in Proton Exchange Membrane Fuel Cells (PEMFC) for both anodic and cathodic catalytic reactions. Fuel cells are considered to be the
most technically viable solution for clean and sustainable future energy
scenarios. While consuming fuel (hydrogen or hydrogen rich substances)
and oxidant (oxygen or air), the fuel cell generates electrical energy and
produce water as the waste [75].
The PEMFC are an especially interesting type of fuel cells due to their
inherent advantages such as high power density, reduced system
weight, simplified construction and quick startup. PEMFC are suitable
for portable, transport and stationary applications [76,77]. The main
component of the PEMFC is the Membrane Electrode Assembly (MEA)
which consists of a proton exchange membrane located between two
porous electrodes (anode and cathode) [78]. Electrochemical reactions,
both anodic and cathodic, take place at the electrodes and are promoted
by the use of a catalyst. Pure Pt or Pt in combination with other Pt group
metals (PGM), either supported or unsupported, are most suitable for
electrochemical reactions in PEMFC. Because of the use of Pt and PGM,
the MEA represents the most expensive component of the PEMFC. Therefore active research is carried out for improving catalyst utilization.
There are several MEA preparation techniques which alter the way
that the catalyst layer is formed. Each technique is aimed at improving
MEA performance and reducing the catalyst loading and thereby overall
cost. The catalyst layer can be deposited either onto the gas diffusion
layer known as a catalyst coated substrate (CCS) or directly onto the
membrane known as a catalyst-coated membrane (CCM) [79]. Electrophoretic deposition (EPD) is a highly efficient process for the production
of films and coatings. EPD is easy to implement, low cost, fast and applicable to a wide variety of materials [80]. EPD has already been successfully demonstrated for the deposition of catalytic layers in MEAs [24–27].
Deposition occurs when the particles collect via coagulation at the electrode (or membrane) surface and form a relatively compact and homogeneous film [81,82]. The knowledge of both electrophoretic mobility
and stability of catalyst suspensions is of paramount importance for the
optimization of EPD catalyst-layer formation. Usually, EPD is accompanied by considerable pH changes close to the electrode (membrane)
surface. Therefore, the electrophoretic mobility and stability should be
understood within a broad pH range for various salt concentrations, i.e.
we face the necessity to extract the values of zeta potential for ternary
electrolyte system. This can be done by using the algorithms described
above in the theoretical sections.
The obtained function X(h) is substituted into the integral of Eq. (53)
for obtaining Gel(h).
4.2. Materials
3.2.3. Obtaining Hamaker constant from electrophoretic and coagulation
dynamic data
The above stated scheme allows one to determine the Fuchs factor,
W, as a function of ζ, the electrolyte composition, particle radius and
the Hamaker constant. To this end, we will use the integral expression
given by Eq. (6) to substitute there the interaction free energy as a
HiSpec 4000, 40 wt.% Pt/C (Johnson Matthey, United Kingdom) was
used as received as catalyst material for all experiments. Ultrapure H2O
(18.3 MΩ cm) was obtained via a Zeneer Power III water purification
system (Human Corporation, South Korea). Ionic strength of the suspensions was controlled by addition of NaCl (KIMIX, South Africa)
while the pH was adjusted by the addition of NaOH (KIMIX, South
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
87
Africa) and HClO4 (KIMIX, South Africa). Catalyst suspensions were
prepared by mixing the Pt/C powder (0.1 or 0.03 mg) with 10 ml NaCl
electrolyte solution (0.1 to 40 mM). The pH of the suspensions was
monitored using the Metrohm 827 pH lab (Metrohm, Switzerland)
equipped with a Primatrode pH electrode. Homogeneous suspensions
were obtained by means of ultrasonic dispersion for 5 min via the
Biologics 3000 ultrasonic homogenizer (Biologics, Inc., USA) fitted
with micro tip ultrasonic finger. The power of the homogenizer was
set at 40% with pulser set to the off position (0%). The initial diameter
of particles in suspension was about 280 nm.
4.3. Methods
Transmission Electron Micrograph of the dry Pt/C powder were
obtained using a Tecnai G2 F20 X-Twin Mat200 kV Field Emission
TEM, operating at 200 kV (Fig. 4).
Measurement of electrophoretic mobility and particle size of Pt/C
particles in aqueous solutions were obtained using the Zetasizer Nano
ZS (Malvern Instrument Ltd., United Kingdom) as shown in Fig. 5.
The instrument was fitted with a production standard 532 nm,
50 mW diode laser source. The Zetasizer instrument measures electrophoretic mobility via a 3 M-PALS technique which is a combination of
laser doppler velocimetry (LDV) and phase analysis light scattering
(PALS). Particle size was measured via Dynamic Light Scattering (DLS)
also known as Photon Correlation Spectroscopy (PCS). The instrument
is capable of measuring particle size between 0.6 nm–6 μm and the electrophoretic mobility of particles with a size range of 3 nm–10 μm [83,
84]. A syringe was used to fill a semi-disposable capillary cell with the
sample which was then immersed into a temperature controlled block
holder to avoid thermal gradients in the absence of the applied electric
field [85]. Electrophoretic mobility was measured by applying a fixed
voltage of 100 V and programming the instrument to record 3 electrophoretic mobility values for each sample. Particle size was obtained by
averaging 10 size values obtained over a 600 s time interval with a measurement recorded every 60 s. All measurements were performed at
25 °C.
While studying aggregation, we compare the coagulation rates for
systems with different salt concentrations and different pHs. To check the
possibility of addressing all the studied cases in terms of a single time
scale parameter, τ, given by Eq. (5), to which we will refer as the coagulation time, we analyze each of the time dependencies of the “particle
size” (as the device displays it) that have been obtained for various solution compositions. For each of the dependencies, dimension vs. time, we
determine its own τ by considering the initial stage of coagulation.
Fig. 5. Malvern Zetasizer Nano ZS instrument and semi-disposable capillary cell (inset).
Finally, we redraw all the experimental curves by representing the
“particle size” as a function of time normalized by the coagulation
time, τ, determined for each of the curves separately.
The results of implementing the above described scheme are
represented in Fig. 6, where all the experimental points could be collapsed to a single smooth curve. Such behavior reveals that obtaining τ
for each of the solution composition yields the required information on
the system aggregative behavior.
5. Results and discussion
As discussed above, for finite values of κa, the predictions of electrophoretic velocity can noticeably deviate from the results given by
Eq. (3). However, the devices measuring electrophoretic velocity often
display the experimental data in terms of zeta potential by assuming
that the Smoluchowski relationship given by Eq. (3b) is valid. The latter
means that the displayed quantity, ζexp, is obtained from the measured
electrophoretic velocity, Ueph, by using Eq. (3b), as
ζ exp ¼
U eph η
:
Eε
ð62Þ
1000
900
800
2a, nm
700
600
500
400
c=3 pH=10
c=3 pH=11
c=3 pH=11.5
c=10 pH=10
c=10 pH=11
c=10 pH=11,5
c=15 pH=10
c=15 pH=11
c=15 pH=11,5
c=20 pH=10
c=20 pH=11
c=20 pH=11,5
c=25 pH=10
c=25 pH=11
c=25 pH=11,5
c=40 pH=10
c=40 pH=11
c=40 pH=11,5
300
1E-3
0,01
0,1
1
t/τ
Fig. 4. TEM image of 40% Pt/C (JM HiSpec 4000).
Fig. 6. Size of particles vs. normalized time for various electrolyte solution compositions
(NaCl concentration 3, 10, 15, 20, 25 and 40 mM; pH 10, 11 and 11.5).
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
For sufficiently high κa, the displayed and actual values of zeta
potentials should coincide, ζexp = ζ. However, for finite κa, the latter
equality does not hold, and ζexp turns out to be a function of ζ, the
particle radius, a, and the electrolyte composition. Consequently, in the
general case, to determine the value of ζ from the measured ζexp one
should know the particle radius, a, the electrolyte solution composition
and the relationships which express ζexp through ζ, a and the parameters
describing the solution composition. Such a relationship should theoretically be established on the basis of a model, as discussed above. Below we
consider a particular example where the proposed methodology is
applied to the analysis of electrical interfacial properties of Pt/C nanoparticles suspension.
20
-6
Volume fraction 10
ζ exp,
ζ
10
-20
-30
-40
-50
-60
-70
-80
-90
-100
2
4
6
8
5.2. Influence of electrolyte composition on surface potential and charge
One can suggest two mechanisms whose simultaneous action can
lead to the behavior of zeta potential displayed by curves in Fig. 7 a and
b, namely, (i) changes of surface charge due to the binding-release of
H+ and OH− ions, and (ii) the decrease of potential due to the EDL
compressing, which occurs while increasing the ionic strength and thereby decreasing the Debye length. The latter mechanism manifests itself
when the base, NaOH, concentration becomes sufficiently high.
To understand the role of the first mechanism, we consider the
behavior of surface charge density as a function of pH at constant salt
concentrations, 10−4 M and 10−3 M (Fig. 8 a and b). At some quite
10
12
14
pH
5.1. Correlation between electrophoretic and stability data
(a)
-6
Volume fraction 10
ζ exp,
ζ , run 1
ζ exp,
ζ, run 2
10
0
-6
Volume fraction 4*10
ζ exp,
ζ, run 1
ζ exp,
ζ, run 2
ζ exp,
ζ, run 3
-10
-20
ζ , mV
The calculation scheme described in Section 3.1 enables us to obtain
the apparent value of zeta potential ζexp from its actual value ζ for arbitrary values of ζ and κa. By solving the inverse problem, one can estimate
both the surface potential, ζ, and charge density, q, from ζexp and κa that
are known from experiment. The parameter κa is obtained by determining particle radii and specifying Eq. (3) for ternary electrolytes employed
in experiments, namely, the mixtures of NaCl with either NaOH or HCl.
The calculation scheme of Section 3.1 is also specified for these electrolyte
solutions. Below, we present the results of such calculations for the experimental system described above.
The results of the first group were obtained while measuring the
electrophoretic mobility for sufficiently low concentrations of salt,
10− 4 M and 10− 3 M, within a wide pH range which includes both
the low and high pHs where the suspension becomes unstable.
The curves in Fig. 7a and b display the behavior of both the apparent
and actual surface potentials, ζexp and ζ, as functions of solution pH for the
10−4 M and 10−3 M concentrations of salt. The presented data were obtained for two solid-phase volume fractions, (10−6 and 4 ∗ 10−6), for
which the suspension can be considered infinitely diluted in terms of
electrophoresis (but not coagulation). Accordingly, the data, except for a
few points, are close to each other.
In both the graphs, while increasing pH within the acidic range, the
positive surface potential decreases and reaches zero at rather low pHs.
With the further increase of pH, the potential becomes negative and increases in absolute value until reaching a maximum magnitude within
the alkaline range but close to the neutral pH. The final decrease of potential is observed within the alkaline range. Importantly, ζexp and ζ
nearly coincide for acidic and alkaline pHs, but, within the neutral
range of pH, the actual potential magnitude, |ζ|, exceeds that of the
apparent value, |ζexp|, by a factor of about 2, for the salt concentration
10−4 M, and 1.2, for 10−3 M.
These results correlate with our observation of system stability, according to which the system remains relatively stable at neutral pHs,
but addition of acid significantly accelerate coagulation which reaches
the maximum rate at about pH ~ 3 ÷ 4. At such pHs, as it follows from
the electrophoretic mobility measurements, the particle charge dramatically decreases, and thus the electrostatic repulsion weakens, which
leads to the acceleration of coagulation.
-6
Volume fraction 4*10
ζ exp,
ζ, run 1
ζ exp,
ζ, run 2
0
-10
ζ, mV
88
-30
-40
-50
-60
-70
2
4
6
8
10
12
14
pH
(b)
Fig. 7. Dependency of apparent (dashed lines) and actual (solid lines) zeta potentials on
pH for salt concentration 10−4 M (a) and 10−3 M (b); the volume fractions are 10−6
and 4 ∗ 10−6.
low pH value, the particle charge turns from positive to negative and,
then, increases in magnitude with increasing pH and reaches a value
of about 0.02 C/m2 in a concentrated base solution. In the alkaline
range, the curve behavior resembles a Langmuir isotherm.
At neutral and acidic pH, we do not observe such adsorption saturation. Instead, there is a slow linear dependency on pH, while the concentrations of hydrogen (and hydroxyl) ions change substantially. Note that
the pH axis in Fig. 8 is decimal logarithmic with respect to the OH− ion
concentration. Perhaps, such behavior is a result of existence of two
types of surface ionic groups with different properties. Saturation of
groups that belong to one of the types can coincide at, approximately,
pH = 7, with start of ion binding (or releasing) by groups of another type.
For better understanding which ions take part in forming the surface
charge, we determine all the ion concentrations in the immediate vicinity
of surface, CSk, by using the Boltzmann distribution
S
C k ¼ C k exp −ζezk :
ð63Þ
We plot these concentrations against the surface charge density, q,
which is calculated for each of the points by using the electrolyte composition and the calculated value of surface potential corresponding to
this point. Generally, at given ζ, any relation between CSk and q, should
also depend on the electrolyte composition. Therefore, the set of points
CSk vs. q, plotted for different electrolyte compositions in different experiments should be spread over a certain area in the graph. However,
while assuming that the surface charge is formed due to the interaction
(dissociation or adsorption) of surface groups with the kth sort of ions
only, the surface charge is completely defined by the concentration CSk
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
-6
Volume fraction 10
run 1
-6
Volume fraction 4*10
run 1
run 2
0,005
0,000
-0,005
q, C/m
2
-0,010
-0,015
-0,020
-0,025
-0,030
2
4
6
8
10
12
14
pH
(a)
-6
Volume fraction 10
, run 1
, run 2
-6
Volume fraction 4*10
, run 1
, run 2
, run 3
0,005
0,000
-0,005
q, C/m
2
-0,010
-0,015
-0,020
-0,025
-0,030
2
4
6
8
10
12
14
pH
(b)
Fig. 8. Dependency of surface charge density on pH at salt concentration (a) 10−4 M and
(b) 10−3 M; the volume fractions are 10−6 and 4 ∗ 10−6.
and the parameters of adsorption (dissociation-binding) isotherm and
is independent of electrolyte composition. In such a case, the corresponding points in the graph CSk(q) are expected to lie on a smooth
line, the isotherm, or to be close to it (taking into account the experimental error).
Let us now consider the positions of points plotted in the graph
of Fig. 9 for H+, Na+ and Cl− ions. As it is clear from Fig. 9, Na+ and
Cl− ions do not form the surface charge because the points corresponding
1000
89
to these ions approach to a smooth line at a relatively large charge,
|q| N 0, 005C/m2, only. Note that the points corresponding to the
H+ ions make up a set which can be approximated by a smooth line.
Consequently, we conclude that the charge is formed either by hydrogen
or by hydroxyl ions.
Importantly, by using the present approach, it is impossible to distinguish whether the charge is formed by binding of OH− ions or releasing
of H+ ions. The charge in this region strongly depends on pH (see Fig. 8 a
and b).
Near the maximum charge, |q| ≈ 0, 02C/m2, we have a set of points
plotted for the Na+ ions that nearly form a smooth line. However, it
does not mean that the Na+ ions form the charge. This can be understood by considering that the Na+ ion is the only counterion (cation)
in the solution. Consequently, under conditions of locally flat double
layer, which is satisfied when ζexp and ζ are close to each other, and sufficiently high ζ, one can establish this relationship between the charge q
and the concentration C sNaþ
e εRT ∂Ψ
q¼−
F ∂r pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffi
≈− 2εRT C sNaþ :
ð64Þ
r¼a
Thus, Eq. (64) strictly interrelates the charge density and the concentration C sNaþ , and, hence, is independent of electrolyte composition.
However, this interrelation is not an adsorption isotherm.
The second group of experiments has been conducted to elucidate the
mechanisms of decrease of the surface potential magnitude at high pHs,
as shown in Fig. 7 a and b.
The principal purpose of these experiments was to figure out whether
this decrease is due to the compression of EDL or there are changes in the
ion adsorption or binding. To answer this question we increased the ionic
strength by adding a salt, NaCl, instead of the base, NaOH, i.e., we maintained pH in each of experiments.
The curves of Fig. 10 show the dependency of surface potential on
the salt concentration. The difference between the behaviors of curves
plotted for ζ and ζexp is noticeable but smaller than in the case of low
salt concentrations. One can see that the absolute value of surface potential decreases except for two points in high concentration range
which deviate from the decreasing trend within the limits of experimental error.
Obtaining the surface charge density corresponding to the surface
potential, ζ, and the ion concentrations presented in Fig. 10 yields a remarkable result. Although ζ decreases in magnitude with increasing salt
concentration, the surface charge magnitude increases (Fig. 11).
The opposite behavior of potential and charge demonstrated by
curves in Figs. 10 and 11 is explained by the increase of concentration
of hydroxyl ions and decrease of that of hydrogen ions in the immediate
vicinity of particle surface, see Eq. (63). Accordingly, the adsorption of
100
-35
10
1
-40
0,1
-45
ζ, mV
c, mM
0,01
1E-3
1E-4
1E-5
+
H
+
Na
Cl
1E-6
1E-7
1E-8
1E-9
-0,030
ζ, pH=10
ζ, pH=11
ζ, pH=11.5
ζexp, pH=10
-50
-55
ζexp, pH=11
-60
ζexp, pH=11.5
-65
-0,025
-0,020
-0,015
-0,010
q, C/m
-0,005
0,000
0,005
2
Fig. 9. Concentrations of H+, Na+ and Cl− ions that correspond to given values of surface
charge in different experiments.
0
10
20
30
40
CNaCl, mM
Fig. 10. Dependency of surface potential on salt concentration for different pHs: ζ — solid
lines, ζexp — dashed lines.
90
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
pH=10
pH=11
pH=11.5
q, C/m
2
-0,010
-0,015
-0,020
-0,025
0
10
20
30
40
CNaCl, mM
Fig. 11. Dependencies of surface charge density on salt concentration for different fixed
pHs.
potential-defining OH−-ions (and/or desorption of H+ ions) increases.
Thus, the behavior of charge as a function of salt concentration additionally confirms that the surface charge is formed due to the adsorption or
desorption of OH− or H+ ions, respectively. Clearly, the charge magnitude is higher for higher pHs that are illustrated in Fig. 11 by the positions
of the respective curves.
It can be expected that, at a given ionic strength, the charge will be
higher for systems having higher pH. To verify that, in Fig. 12, we constructed a graph similar to plot of Fig. 11, but, as the horizontal axis, we
used the ionic strength of mixed electrolyte solution, NaCl and NaOH, instead of the salt concentration.
In Fig. 12, at the same ion strength, the points corresponding to
higher pHs, at the same time, correspond to larger charge magnitudes.
Accordingly, the lowest curve, which corresponds to the largest charge
magnitude, was plotted for the lowest salt concentration when almost
the whole solution ionic strength is due to the NaOH ions.
In summary, binding of OH− ions (and/or release of H+ ions) is the
principal mechanism of charging the particle surface. While increasing
the electrolyte solution pH, one observes an increase of the negative
surface charge by absolute value. Nevertheless, the negative surface
potential magnitude decreases due to the compression of EDL which
occurs when the ionic strength increases.
5.3. Applicability of Standard Electrokinetic Model
While considering coagulation, the free energy of particle interaction
should be compared with the energy of thermal motion kT, which, for
the room temperature, is about 4 ⋅ 10−21J. By using the Hamaker
constant of about 10−20J, the Van der Waals energy is estimated to be
pH=10
pH=11
pH=11.5
CNaCl=0.1 mM
Table 1
Measured quantities: zeta potential, ζexp, coagulation time, τ, ionic strength, I. Calculated
quantities: surface potential, ζ, surface charge density, q, and Hamaker constant, H.
ζ
q
H
mV
C/m^2
J
0.03 mg/ml Pt/C in 0.1 mM NaCl, τSm = 300 s
4.7
−23.7
1064.3
0.120
4.8
−24.7
2003
0.116
4.9
−24.4
12,801
0.113
5
−26.4
47,170
0.110
12
−54.3
39,018
10.100
12.1
−54.7
11,620
12.689
12.2
−49.9
2792.3
15.949
−32.00
−33.50
−33.10
−36.20
−60.85
−60.70
−54.30
−0.00101
−0.00104
−0.00102
−0.00111
−0.01768
−0.01972
−0.01894
1.03E-18
1.13E-18
1.03E-18
1.24E-18
1.66E-19
1.46E-19
1.07E-19
mM NaCl, τSm = 300 s
1171.85846
1.016
1792.53255
1.013
2777.72166
1.010
3742.66225
1.008
2458.15181
13.589
1831.41644
16.849
1387.91573
20.953
−27.85
−29.10
−30.30
−32.80
−52.90
−51.96
−52.50
−0.00216
−0.00237
−0.00247
−0.0027
−0.01691
−0.01836
−0.0207
1.46E-19
1.59E-19
1.71E-19
2.04E-19
1.12E-19
9.63E-20
8.72E-20
0.03 mg/ml Pt/C in 1 mM NaCl repeats, τSm = 300 s
5
−28.9
1134.12405
1.010
−34.45
5.1
−29.9
1665.31577
1.008
−35.85
−0.00285
−0.00298
2.36E-19
2.54E-19
0.1 mg/ml in 0.1 mM NaCl, τSm = 100
4.2
−14.6
388.909118
4.5
−18.0
943.820157
12.1
−51.7
2813.28198
12.2
−46.6
1019.51999
13.5
−15.2
545.963125
pH
ζ
Coagulation
Ionic
(exp)
time, s
strength
mV
0.03 mg/ml Pt/C in 1
4.8
−23.7
4.9
−24.65
5
−25.6
5.1
−27.55
12.1
−48.5
12.2
−48.1
12.3
−48.9
mM
s
0.163
0.132
12.689
15.949
316.328
−18.75
−23.70
−56.95
−50.35
−15.40
−0.00065
−0.00076
−0.01805
−0.01715
−0.02014
2.06E-19
4.23E-19
1.31E-19
9.30E-20
7.94E-22
-0,015
0.1 mg/ml in 0.1 mM NaCl repeats, τSm = 100 s
4.7
−29.1
6018.24953
0.120
4.8
−28.9
8911.66667
0.116
−40.30
−40.10
−0.0013
−0.00128
1.53E-18
1.54E-18
-0,020
0.1 mg/ml Pt/C in 1 mM NaCl, τSm = 100 s
6
−24.6
403.432586
1.001
12.3
−43.2
1283.4414
20.953
−29.00
−46.00
−0.00235
−0.0175
7.47E-21
6.68E-20
0.1 mg/ml in 1 mM NaCl repeats, τSm =
6
−32.8
894.712644
6.1
−32.5
601.771404
6.3
−32.4
1902.24475
6.4
−33.1
5206.99029
−39.70
−39.20
−39.10
−40.10
−0.00335
−0.0033
−0.00329
−0.00338
3.12E-19
3.07E-19
2.97E-19
3.06E-19
-0,005
2
-0,010
q, C/m
significant at distances of the order of magnitude of particle radius,
Eq. (8). Clearly, at such distances Derjaguin approximation is not applicable. As for the electrostatic repulsion, its contribution is important up
to several Debye lengths. When EDL is sufficiently thin, the rapid coagulation can occur in the secondary minimum.
In our experiments, for all samples, we always observed an initially
linear dependency of aggregate size on time. The latter allows us to
estimate the time-scale parameter of coagulation dynamics, τ. By
using the estimated value of τ and Eq. (5), we determined the
Fuchs factor, W, for each experiment. The Smoluchowski time, τSm,
has been estimated at τSm = 300s and τSm = 100s for the 10− 6 and
4 ∗ 10− 6 particle volume fractions, respectively. By using the numerical
scheme outlined in Section 3.2.3, the Hamaker constant, H, was determined for each of the employed electrolyte compositions and the surface
potential, ζ, extracted from the electrophoretic data. In the interpretation
this potential was considered independent of inter-particle distance. To
evaluate the deceleration of coagulation, from which the Hamaker constant is determined, we divide the coagulation time by τSm.
In Table 1, we present the results of such calculations conducted for
experiments whose results are shown in Figs. 7 and 8 in terms of surface
potential and charge as functions of pH. We do so for two salt concentrations, 10−3 M and 10−4 M, and two particle volume fractions, 10−6 and
4 ∗ 10−6. While preparing Table 1, we excluded the data with coagulation times shorter than the Smoluchowski value, τSm, given by Eq. (5).
-0,025
0
10
20
30
40
I, mM
Fig. 12. Dependency of surface charge density on ionic strength, I, at different pHs.
100 s
1.001
1.001
1.001
1.000
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
Besides, we excluded the cases where coagulation is so slow that the
changes in the aggregate size are comparable to the experimental error.
For the alkaline range, we obtain the Hamaker constant
1.28 ⋅ 10−19J(±0.28 ⋅ 10−19J) (in parentheses we present the standard
deviation) when the salt concentration is 10−4 M. A close value (within
the standard deviation) is obtained in the same alkaline range but at
higher salt concentration, 10−3 M: − 0.91 ⋅ 10−19J(± 0.19 ⋅ 10−19J).
In acidic media the average Hamaker constant turns out to be larger,
especially, for the case of lower concentration of salt, 10 − 4 M,
10.19 ⋅ 10− 19J(± 4.79 ⋅ 10− 19J), while at 10− 3 M, the deviation is
smaller, and, with account for standard deviation, lies near the
upper limit of the above result predicted for the base region,
2.18 ⋅ 10− 19J(± 0.93 ⋅ 10− 19J).
As possible reasons for the computed Hamaker constant variations
one can consider an error in the measured ζ-potential, that strongly influences the electrostatic barrier, and accumulation of particles simultaneously in the primary and secondary potential minimum, that is
accompanied by a slow particle transition between them [70]. However
sometimes the discrepancy between the theory and experiment is so
large that it cannot be explained by such reasons, especially for small
particles (of the order of 100 nm) [71]. Alternative explanation can relate to heterogeneity of the particle surface. A higher value of Hamaker
constant obtained for the acidic range is indicative of an additional
attraction that is not taken into account within the framework of SEM
and manifests itself as an apparent increase in the Hamaker constant.
Such an additional attraction can be associated with heterogeneity of
surfaces which often leads to the appearance of an attractive mean
force between the surfaces bearing mosaic charge. A stronger impact
of heterogeneity can be expected for higher electrolyte concentrations.
However, even at the lower ionic strengths, the EDL is relatively thin
compared with the particle size (κa ≈ 5). Hence, sufficiently large regions with different charge densities can manifest themselves. Recall
that we deal with carbon particles modified by metallic platinum.
Accordingly the surface heterogeneity is quite possible. The dynamics
of particles interaction can be also significant for the aggregation kinetics
[71,86].
In Table 2, we present the coagulation times estimated from the
second group of experiments whose results are shown in Figs. 10–12
and corresponding estimates of Hamaker constant. In these calculations,
we assumed the surface potential to be independent of the distance between the particles.
The electrolyte concentration is the major parameter defining the
coagulation rate since it defines the ionic strength and, thus, the EDL
91
thickness. When the salt concentration is lower than 10− 2 M, coagulation is nearly absent. At 1.5 · 10− 2 M, the coagulation occurs at a
noticeable rate that further increases at 2 · 10− 2 M. At concentrations of 2.5 · 10− 2 M and 4 · 10− 2 M, the electrostatic repulsion
does not affect the coagulation, which is characterized by the
Smoluchowski rate. Computing the mean Hamaker constant for the
concentration range 1.5 · 10−2 M–2 · 10−2 M (here the result is most
reliable), we obtain H ≈ 0.71(± 0.12) · 10− 19 J. This value is lower
than that estimated for the low salt concentration case which was presented before. The smaller Hamaker parameter means slower coagulation than that expected from the theory.
6. Conclusions
Parallel investigation of electrophoresis and coagulation, both
depending on ionic solution composition, can yield important information about the interfacial properties of particles and aggregative stability
of their dispersions. However, in the case of nano-particle systems interpretation of experimental results is not easy and requires significant
modifications of traditional approaches. An overview of academic literature has revealed that reasonable semi-quantitative correlations between electrophoretic mobility and stability of model nano-colloidal
systems could be established within the scope of Standard Electrokinetic
Model. At the same time, in many cases application of SEM to the analysis
of experimental data encounters difficulties, in particular, when the solutions contain more than two ions, the particle charge depends on the
solution composition and zeta-potentials are high. Such situations demand novel developments of SEM.
As an experimental example, we studied the influence of electrolyte
composition on the aggregative stability of a diluted suspension of
280 nm carbon particles modified by metallic platinum and experimentally established the following behavior of this disperse system. The system remains relatively stable at neutral pH but starts to coagulate when
an acid is added. When adding a base (NaOH), the system is stable until
pH reaches the value of 12.2. With further increase of pH, the system coagulates, and the coagulation is more rapid than at low pH. Importantly,
the coagulation threshold, observed at high pHs, is independent of the
salt concentration in contrast to that observed at low pHs.
We attempted to explain the above behavior through the variation of
electrostatic repulsion forces that are described by the DLVO theory [10,
11]. For example, the observed dependency of stability on pH can occur
due to the variation of surface charge density. Such a variation takes
place because the pH controls both the concentrations of hydrogen/
Table 2
Initial data and results of calculation of Hamaker constants for assumptions: Pt/C of 10−6 volume fraction at various salt concentrations and pHs, Smoluchowski coagulation time τ Sm ¼300 s.
C NaCl
mM/l
pH
3
3
3
10
10
10
15
15
15
20
20
20
25
25
25
40
40
40
10
11
11.5
10
11
11.5
10
11
11.5
10
11
11.5
10
11
11.5
10
11
11.5
ζ
Coagulation time, s
Ionic strength
mM
ζ
mV
q
C/m^2
H
ζ = const
J
22,066.4
38,687.7
27,237.4
11,371.7
30,236.4
25,450.2
1407.5
1770.6
857.2
352.2
1251.7
565.5
355.8
681.2
368.8
422.5
267.9
289.1
3.1
4
6.162278
10.1
11
13.16228
15.1
16
18.16228
20.1
21
23.16228
25.1
26
28.16228
40.1
41
43.16228
−53.8
−55.65
−55.65
−47.3
−51.2
−54
−41.9
−47
−49.6
−40.05
−44.25
−46.53
−39.55
−46.3
−47.85
−43.6
−42.5
−40.8
−0.0084
−0.0099
−0.0123
−0.0126
−0.0146
−0.0171
−0.0133
−0.0158
−0.0179
−0.0145
−0.0167
−0.0187
−0.0159
−0.0196
−0.0213
−0.0226
−0.0221
−0.0216
2.67E-19
2.43E-19
1.89E-19
1.05E-19
1.15E-19
1.15E-19
6.78E-20
8.20E-20
8.56E-20
5.50E-20
6.30E-20
6.68E-20
4.74E-20
6.18E-20
6.40E-20
4.40E-20
4.28E-20
3.81E-20
(exp)
mV
−45.8
−47.9
−49.0
−43.2
−46.7
−49.3
−39.1
−43.7
−46.1
−37.8
−41.6
−43.8
−37.5
−43.7
−45.2
−41.6
−40.7
−39.1
92
C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92
hydroxyl ions and their adsorption or dissociation of interfacial ionogenic
groups. Another mechanism manifests itself at sufficiently large deviations of pH from the neutral value. In this case, the pH changes affect
the ionic strength and thus the Debye screening length, thereby changing
the electrostatic interaction forces even at constant interfacial charge. The
mechanism associated with changing the Debye length is also expected
to be responsible for the decrease of electrostatic repulsion when increasing the salt concentration.
To verify if the changes in electrostatic interactions really control the
dependency of stability on pH, we used a theoretical model that
accounted for electrostatic and Van der Waals interactions. The model
enables one to predict the interaction energy as a function of distance
between the particles for arbitrary values of surface potentials and
ratio of Debye to particle radii. The computed energy was employed
for the prediction of coagulation time, which is usually being determined in coagulation experiments.
To obtain information on the dependencies of surface potential and
charge on electrolyte composition, we measured electrophoretic mobility
of the Pt/C particles and extracted the surface zeta-potential from it by
using the Standard Electrokinetic Model. This calculation has demonstrated that, within certain ranges of ion concentrations, the surface potential, which is employed in the stability model, has noticeably larger
absolute value than the apparent zeta potential estimated from the mobility by using Smoluchowski formula.
The self-consistency of our approach was checked via computing the
Hamaker constant for a set of experiments conducted at various electrolyte compositions. If the model is adequate such calculations should yield
the same value of this constant (within the experimental error). However, we obtained the Hamaker constants varying from one experiment to
another. In the case of high solution pHs, the variation is not very significant, and one can conclude that the model describes the system behavior, at least, semi-quantitatively. For low pHs, the computed Hamaker
constant varied substantially and sometimes took anomalously high
values. Supposedly, this behavior is a manifestation of the surface heterogeneity, which can be expected for the studied particles.
Acknowledgments
Financial support from European Commission within the scope of FP7
(project acronym “CoTraPhen”, Grant Agreement Number: PIRSES-GA2010-269135) is gratefully acknowledged. The work was supported by
COST action CM1101. The authors would like also to thank Prof. S.S.
Dukhin for valuable discussions.
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