Screening effects and charge in
suspensions of colloidal spheres
Niels Boon
February 6, 2008
Master’s Thesis
Institute for Theoretical Physics
Utrecht University
Supervisor: dr. R. van Roij
Screening effects and charge in
suspensions of colloidal spheres
Niels Boon
February 6, 2008
Master’s Thesis in Theoretical Physics
Nicolaas Jacobus Henricus Boon
Institute for Theoretical Physics
Utrecht University
Supervisor: dr. R. van Roij
Contents
1 Introduction
1.1 Colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The system of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Attractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
7
8
2 Electrostatics and screening in suspensions of charged colloids
9
2.1 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 An alternative derivation of the Poisson-Boltzmann equation . . . . . . . . . 12
3 Wigner-Seitz cell theory
3.1 The model . . . . . . . .
3.2 The grand potential and
3.3 Charge renormalization
3.4 Gas-Liquid coexistence .
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15
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19
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21
21
21
24
24
26
26
29
5 Linear screening theory for the many-body problem.
5.1 The non-linear problem . . . . . . . . . . . . . . . . . .
5.2 Linearizing of the Poisson-Boltzmann equation . . . . .
5.3 Solving the Euler-Lagrange equations . . . . . . . . . .
5.4 Finding the effective interaction Hamiltonian . . . . . .
5.5 Calculating the free energy . . . . . . . . . . . . . . . .
5.6 Using the Gibbs-Bogoliubov inequality . . . . . . . . .
5.7 Gas-Liquid coexistence . . . . . . . . . . . . . . . . . . .
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31
31
32
34
36
38
39
39
. . . . . . .
free energy
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4 Porous spheres in oil
4.1 Introduction . . . . . . . . .
4.2 The model . . . . . . . . . .
4.3 Ions in oil and water . . . .
4.4 Charge regularization . . .
4.5 Calculations . . . . . . . . .
4.6 The zero-potential solution
4.7 Conclusion . . . . . . . . .
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6 The spurious charge in the colloids
43
6.1 The effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6
Contents
6.2
6.3
6.4
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Results for various particle distances . . . . . . . . . . . . . . . . . . . . . . . 45
Charge redefinition? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7 Combining linear multicentered and cell theory
7.1 The situation . . . . . . . . . . . . . . . . . . . .
7.2 Connecting the linear solution to the cell solution
7.3 A scheme to find the renormalized charge . . . .
7.4 An appropriate choice for the cell size . . . . . .
7.5 The Hamiltonian and the free energy . . . . . . .
7.6 Free energy and gas-liquid coexistence . . . . . .
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47
47
48
51
52
52
54
8 Various treatments of the correction term δΩ
55
8.1 Numerical accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.2 A new correction term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9 Linear theory using larger colloids
63
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9.2 Differences with earlier methods . . . . . . . . . . . . . . . . . . . . . . . . . 65
9.3 The grand potential and free energy . . . . . . . . . . . . . . . . . . . . . . . 65
10 Conclusion
69
11 Acknowledgments
71
A The correction term
73
A.1 Finding the cell-contribution from the linear screening theory . . . . . . . . . 73
A.2 Small cell diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Chapter 1
Introduction
1.1
Colloids
Although invisible to the bare eye, our every-day life is surrounded by giant amounts of
colloids. A few of the many examples in which colloids are present are butter, milk, paint
and fog. Colloids are typically small particles with sizes that vary between one millionth and
one thousands of a millimeter. This range in size is in between the typical size of molecules
and of macroscopic objects like grains of sand. Colloidal particles are large enough to be
essentially classical, non-quantummechanical objects and thus we can use classical (statistical) mechanics to describe them. For instance, effects like colloids tunneling through a
membrane do not occur. They can often be seen directly with a microscope. On the other
hand colloidal particles are small enough to perform Brownian motion in a solvent. Because
of this Brownian motion, colloidal systems can be described using statistical mechanics. In
order to make predictions about thermodynamic properties of these systems we have to calculate the probability of different configurations of colloidal systems, defined as a Boltzmann
weight. This is generally not straight-forward since colloidal systems may contain colloids
which vary in size, shape, composition and electric charge. Above that, there can be solvent molecules, salt ions or other molecules around. Since we have to account for all that,
calculations on the probabilities of different configurations become easily very complicated.
The standard way to still make sense out of such a system is by averaging out the effect
of many microscopic degrees of freedom. Then an effective system is obtained in which the
only remaining degrees of freedom are the colloidal positions.
1.2
The system of interest
The system which has our interest is a suspension consisting of a solvent(e.g. oil or water)
with colloids added. There are also ions present in this solvent, which can be released by the
colloids, but are often primarily due to added salt. In our case, the colloids are all spherical
and have the same size. We fix the electrostatic charge of the colloids and assume this charge
is uniformly distributed over the colloidal surfaces. This charge will be denoted by −Ze,
with −e the electron charge. Because of this charge, the ions in the suspension will change
a lot of the characteristics of the system. Therefore we assume a fixed salt concentration
8
Chapter 1. Introduction
inside the solvent.
1.3
Attractions
Experiments, like done by Tata, Mohanty and Valsakumar [1] clearly show that the colloids
inside the suspension can attract each other at small distances. How is this possible? Imagine
a system consisting of equally charged colloids, why would they want to be close to each other
while common sense and standard screening theory would suggest an electrostatic repulsion?
Figure 1.1: Obtained from [1]. Spherical charged colloids in a density-matched (H2 O − D2 O)
mixture. We clearly see evidence for attractions since the colloids (white dots) form clusters. Here
the colloidal volume fraction is η = 10−4 , the radius is a = 300 nm and the colloidal surface charge
is σe = 2.7 µC/cm. The ion concentration is estimated to be about 0.5 µM.
The classical Derjaguin, Landau, Verwey and Overbeek (DLVO) theory [2] [3] describes
the effective force between two charged colloids as a sum of repulsive screened-Coulomb and
attractive Van der Waals forces. The problem is that this Van der Waals component is only
present at very small particle distances while the observed effect is much longer ranged. We
must conclude that classical DLVO theory does not describe the phenomenon well. Other
experiments in which DLVO theory alone could not explain observed attractions in a colloidal suspenion are described in Refs.[4] [5].
There are other theories which describe the behavior these colloidal suspensions, but
in many of those no attractions are found. In others, attractions are found in parameter
regimes for which we should doubt if the theory is still valid as we will see in this thesis. We
go along the same way as Bas Zoetekouw did in his PhD thesis [6] and in Refs.[7] [8]. We
hope to further improve our understanding of attractions between charge-like colloids.
Chapter 2
Electrostatics and screening in
suspensions of charged colloids
2.1
The system
In this thesis, we are going to make some predictions about the equilibrium properties of
a special colloidal suspension consisting out of a solvent with dissolved colloids and monovalent co- and counterions. In our calculations we assume the ions to have a charge ±e
and to behave like point-like particles. Furthermore, we assume the colloids to be spherical
and of a fixed size and charge. We also describe the solvent as a dielectric continuum. The
parameters which characterize the suspension are
- V , the volume.
- N , the number of colloids, which are at positions Ri , (i = 1, .., N ).
- a, the radius of the colloids.
- −Ze, the charge of a colloid, with −e the electron charge.
- T , the temperature.
- , the dielectric constant of the solvent.
- N+ and N− , the number of positive resp. negative salt ions in the system.
Charge neutrality demands that N+ − N− = ZN . The extra amount of positive ions
comes from the colloids since they obtain their negative charge by releasing positive ions.
Furthermore, the charge on the colloids we assume to be smoothly distributed over the
surface of the colloid. Therefore this colloidal charge distribution, in units of e,is given by
q(r, {Ri }) =
N
X
−Z
δ(|r − Ri | − a).
2
4πa
i=1
The Hamiltonian which describes the system in general is written as
(2.1)
10
Chapter 2. Electrostatics and screening in suspensions of charged colloids
H = Hcc + Hcs + Hss ,
(2.2)
where the subscript c means colloid and s means salt. This Hamiltonian thus accounts for
colloid-colloid, colloid-salt and salt-salt interactions. One who knows the free energy of a
system knows all about it. Therefore our final goal will be to find the system’s Helmholtz
free energy, which is related to the Hamiltonian via
e−βF = trc tr+ tr− e−βH ,
(2.3)
1
,
where tr means taking the canonical trace of the degrees of freedom. As usual, β = kT
where k is Boltzmann’s constant. For the colloids, the trace inside Eq. (2.2) takes the form
Z
1
trc =
dRN ,
(2.4)
N !V N
where V is a volume which accounts for the internal degrees of freedom and kinetic energy of
the colloid. The value of V will not affect the characteristics of the suspension and therefore
it is irrelevant. For the salt ions the trace is defined as
Z
1
tr± =
drN± ,
(2.5)
N± Λ3±
where we integrate over all ion positions r± . It turns out to be convenient to view the
salt ions grand canonically. This means we are going to attach a chemical potential µ ± to
the ions instead of fixing their numbers. The physical picture which corresponds to this is
that we bring our system in diffusive contact with a large reservoir which contains the right
salt concentration. Later, we will see that if we choose µ± =
contains a salt concentration of cs .
ln(cs Λ3± )
,
β
then this reservoir
The so-called ’Donnan partition function’ looks like a Helmholtz free energy from the
colloids point of view, but is a grand potential in terms of the salt ions. This can been seen
from its relation to the free energy; F = F − µ+ N+ − µ− N− , where
e−βF =
∞
∞
X
X
e(βµ+ N+ +βµ− N− ) trc tr+ tr− e−βH .
(2.6)
N+ =0 N− =0
Since the colloidal configuration will be our main interest it is convenient to construct a
so-called ’effective Hamiltonian’, H, which only depends on the positions of the colloids and
in which the contributions of all possible configurations of the salt ions are included as an
effective Helmholtz free energy. The effective Hamiltonian is defined via
e−βH
∞
∞
X
X
= e−βHcc
e(βµ+ N+ +βµ− N− ) tr+ tr− e−βHcs +Hss
N+ =0 N− =0
= e
−βHcc −βΩ
e
.
(2.7)
Thus, H = Hcc + Ω, with Ω the ionic grand potential in the external field of the colloids.
Once we obtained H we can calculate F ’simply’ by taking the classical trace over all colloidal
positions,
2.2. The effective Hamiltonian
11
− βF = ln(trc e−βH ).
2.2
(2.8)
The effective Hamiltonian
We apply density functional theory (DFT) to find the effective Hamiltonian of our colloidal
system. DFT is a theoretical framework that relates (variational) density profiles of manybody systems to the thermodynamics of the system [10]. The thermodynamic average of the
positive and negative ion concentrations are described by ρ+ (r) and ρ− (r). In a mean-field
approximation, the effective Hamiltonian becomes the minimum of the following functional,
ρα (r)
drρα (r) ln
−1
cs
α=±
Z
1
dr [ρ+ (r) − ρ− (r) + q(r, {Ri })] φ(r, {Ri })
+
2
Z
X
drρα (r)βUαHC (r),
+
βH({Ri }) =
X
(2.9)
α=±
where φ(r) is the dimensionless Coulomb potential,
Z
ρ+ (r0 ) − ρ− (r0 ) + q(r, {Ri })
.
φ(r, {Ri }) = λB dr0
|r − r0 |
(2.10)
The colloidal charge distribution, q(r), was defined in Eq. (2.1). Furthermore, λB is the
2
so-called Bjerrum length and is related to the characteristics of the solvent, λB = βe . Note
that the differential form of Eq. (2.10) is the Poisson equation,
∇2 φ(r, {Ri }) = −4πλB [ρ+ (r) − ρ− (r) + q(r)]
(2.11)
The first line in the functional Eq. (2.9) is the entropy part of the ions. The second line
is the electrostatic part which accounts for Coulomb interactions between all the particles
in the system within a mean field approximation. The last line is the hard-sphere part of
the Hamiltonian and ensures that inside the colloidal cores the ion concentrations drop to
zero. This is done by assigning an infinite energy to ions at this position, so βUα (r) must
be defined such that it is infinite inside the colloidal cores, but vanishes outside.
By minimization of Eq. (2.9) with respect to the ionic densities, we obtain the EulerLagrange equation for the effective Hamiltonian. It is given by
ρ± (r)
HC
± φ(r) + βU±
= 0.
(2.12)
cs
Outside the colloidal spheres, the hard-core potential is zero. There we find the Boltzmann’s
equations in a natural form; ρ± (r) = cs exp[∓φ(r)]. If we insert the expressions for the ionic
densities into the Poisson equation, the Poisson-Boltzmann equation is found. Outside the
colloidal cores it reads
ln
∇2 φ(r) = κ2 sinh φ(r),
(2.13)
12
Chapter 2. Electrostatics and screening in suspensions of charged colloids
where we defined κ2 = 8πλB cs . At the surface of colloid i we find
ZλB
,
a2
where ni is the unit vector normal to the surface of colloid i.
ni · ∇φ(r) =
(2.14)
The problem of calculating H({Ri }) is now reduced to solving this multicentered nonlinear Poisson-Boltzmann equation (2.13) - (2.14) for a given colloidal configuration. Once
its solution φ(r, {Ri }) and hence ρ± (r), is known, it can be inserted into Eq. (2.9) to evaluate
H({Ri }).
2.3
An alternative derivation of the Poisson-Boltzmann
equation
We have just derived the Poisson-Boltzmann equation via the Euler-Lagrange equations
which stem from the minimum condition on the effective Hamiltonian. To get a bit more
feeling for what happens we shortly recapitulate what we have found, this time in a more
intuitive way.
The ion densities are determined as the Boltzmann factor associated with the potential
energy which a cation or anion needs to get from the reservoir into a certain position of
the suspension. Since the reservoir densities of both positive and negative species are c s the
expected densities becomes
ρ+ (r) = cs e−βE+ (r) ,
ρ− (r) = cs e−βE− (r) .
The energies E± (r) can be decomposed as the sum of the electric energy eψ(r) of a charge
±e at a position with a potential ψ(r) and the hard-core energy which vanishes outside the
colloidal cores and is infinite inside. Therefore, outside any colloid we find an ion distribution
which only depends on the electrostatic potential,
ρ± (r) = cs e∓βψ(r)e .
(2.15)
Now we consider the electric potential outside a colloidal particle. At the surface of the
particle the field is given by
Ze
,
a2
which follows from Gauss’ law. Outside the colloid we use the Poisson equation,
ni · ∇ψ(r) =
4π
e(ρ+ (r) − ρ− (r)).
Combining Eq. (2.16) and Eq. (2.17) yields
∇2 ψ(r) = −
∇2 ψ(r) =
8π
eρs sinh(βψ(r)e).
(2.16)
(2.17)
(2.18)
2.3. An alternative derivation of the Poisson-Boltzmann equation
Now using κ2 =
8βπe2 cs
13
and φ(r) = βψe, Eqs. (2.15) and (2.17) are rewritten as
ZλB
;
a2
∇2 φ(r) = κ2 sinh φ(r),
ni · ∇φ(r) =
(2.19)
(2.20)
The dimensionless potential φ(r) can be used to estimate wether or not a certain potential
difference is significant, since it should be of order 1 or larger then. The Debye or screening
length κ−1 is a measure for the distance in which the potential decays from its surface value
to the average potential, it is the length over which particles ’feel’ each other electrostatically.
The Poisson-Boltzmann equation (2.20) combined with boundary condtion (2.19) can
be solved numerically in the case that we consider a system constisting of one single colloid surrounded by an infinite volume of solvent. Since we might use spherical symmetry
together with an extra boundary condition that the potential should vanish in the large distance limit, lim|r|→∞ φ(r) = 0, we can calculate φ(r) in this case, which becomes a function
of the distance to the colloidal core.
Fig. 2.1 plots this potential for a small,κ−1 = 51 a, and a large,κ−1 = 5a, screening
length. It can be seen that the magnitude of the potential decays over a length scale which
corresponds to the screening length. In Fig. 2.2 we calculated the corresponding densities of
the cations, anions and the net charge density using Eq. (2.15). We see that the thickness
of the double-layer of positive ions is of the same order as the screening length. We also
see the densities of the cations and anions converge to their equilibrium value cs at large
distances again.
Chapter 2. Electrostatics and screening in suspensions of charged colloids
φ(r)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
φ(r)
14
1
2
3
4
5
6
1
2
3
r/a
4
5
6
r/a
(a)
(b)
Figure 2.1: The potential as a function of the distance to the colloid’s center, starting at the
colloidal surface. In Figures (a) and (b) we used screening lengths of κ−1 = 15 a and κ−1 = 5a,
respectively. Furthermore we used Z = 1000 and a/λB = 100.
10
10
ρ+(r)
ρ-(r)
ρ+(r) - ρ-(r)
8
6
ρ/cs
6
ρ/cs
ρ+(r)
ρ-(r)
ρ+(r) - ρ-(r)
8
4
2
4
2
0
0
1
2
3
4
r/a
(a)
5
6
1
2
3
4
5
6
r/a
(b)
Figure 2.2: The density of the cations, anions and net charge density as a function of the distance
to the colloid’s center, starting at the colloidal surface. Corresponding to the potential in Fig. 2.1.
Again, in (a) and (b) we used screening lengths of κ−1 = 51 a and κ−1 = 5a respectively. Similarly,
Z = 1000 and a/λB = 100.
Chapter 3
Wigner-Seitz cell theory
3.1
The model
The nonlinear character of Eq. (2.13) and the multicentered character of the boundary conditions (2.14) prevent an explicit analytic solution. We therefore resort to a cell approximation
in order to make predictions about the colloidal system.
Figure 3.1: Obtained from [1]. This figure schematically shows in different states how we aprroximate the system by using Wigner-Seitz cells.
The Wigner-Seitz cell theory was introduced by Alexander et al in 1984 [11]. The idea
is to split the colloidal system (Fig. 3.1 a) into N charge neutral cells, such that the total
ionic charge inside each of the cells exactly cancels the colloidal charge (Fig. 3.1 b). This
construction is generally possible for every colloidal decomposition but nevertheless the very
irregular geometry of the cells keep explicit calculations difficult.
Therefore we assume in addition that these cells are all spherical and of equal volume.
The colloids are located in the centers of these cells(see Fig. 3.1 c). By imposing this, the
picture which we get in mind looks like that every colloid is surrounded by a spherical cell
in which the electrostatic (screening-) charges are distributed spherically symmetric. There
is no difference between different colloids and cells. The volume of a cell is Vcell = 1/n, with
n = N/V the colloid density by the assumption that the cells fill up the whole volume. We
3
define the volume fraction of colloids as η = n 4πa
3 , the radius R of the cell is given explicitly
16
Chapter 3. Wigner-Seitz cell theory
3
by R = ( 4π
Vcell )1/3 =
a
.
η 1/3
Now Eqs. (2.13)-(2.14) reduce to an easier boundary-value problem for φc (r), the potential inside the cell. Since we obtained radial symmetry inside each cell we are left to solve
the spherically symmetric Poisson-Boltzmann equation,
1 ∂
2 ∂
r
φ
(r)
c
r2 ∂r
∂r
0
φc (a)
0
φc (R)
= κ2 sinh φc (r),
ZλB
,
a2
= 0.
(3.1)
(3.2)
=
(3.3)
By assumption we demand a vanishing electric field on the boundary of the cell, boundary
condition (3.3) signifies this cell’s neutrality.
3.2
The grand potential and free energy
We have already seen in Eq. (2.9) that the effective Hamiltonian is given by
XZ
ρα (r)
drρα (r) ln
βH({Ri }) =
−1
cs
Z
1
dr [ρ+ (r) − ρ− (r) + q(r)] φ(r)
+
2
Z
X
+
drρα (r)βUαHC (r),
(3.4)
α=±
where Uα is the hard-core potential energy. The Euler-Lagrange equations (2.12) which
stem from this Hamiltonian define the ionic densities. We see we get no contribution from
the colloidal cores, since these Euler-Lagrange equations makes that ρ± (r) vanish there. In
the cell model we may use that all cells are equal to each other and therefore the effective
Hamiltonian can been written as a sum of N equal grand potentials,
H({Ri }) = N ΩWSC ,
(3.5)
where ΩWSC is the contribution to the Hamiltonian of one single cell. This single cell grand
potential becomes:
βΩWSC
=
XZ
α=±
+
1
2
Z
ρα (r)
−1
drρα (r) ln
cs
cell
dr [ρ+ (r) − ρ− (r) + q(r)] φc (r),
(3.6)
cell
in which q(r) is defined as in Eq. (2.1). Of course, the only contribution from the colloidal
charge q(r) is that of the single colloid in the center of the cell. Now if we define the center
Z
of the cell to be positioned at r = 0, Eq. (2.1) simplifies to q(r) = −( 4πa
2 )δ(|r| − a), and
3.3. Charge renormalization
17
the integrals in Eq. (3.6) should be calculated over the interval a ≤ |r| ≤ R. If we apply
Boltzmann’s equations to the first line of Eq. (3.6) we get for this (entropy) part
ΩWSC,entropy
=
Z
dr
cell
= cs
Z
n
cs e−φc (r) [−φc (r) − 1] + cs eφc (r) [φc (r) − 1]
o
dr {φc (r) sinh φc (r) − 2 cosh φc (r)} ,
(3.7)
cell
and the second, electrostatic, part of Eq. (3.6) becomes also with help of the Boltzmann
equation
ΩWSC,electro
Z
1
=
dr [ρ+ (r) − ρ− (r) + q(r)] φ(r)
2 cell
Z
Z
cs
1
=
drφc (r)(e−φc (r) − eφc (r) ) +
drq(r)φc (r)
2 cell
2 cell
Z
Z
= − φc (a) − cs
drφc (r) sinh φc (r)
2
cell
(3.8)
Now we recombine the entropic and electrostatic part again. Since we have a radial symmetry
present we come to the following one-dimensional integral:
βΩWSC
Z
= − φ(a) + 4πcs
2
Z
R
drr2 {φc (r) sinh φc (r) − 2 cosh φc (r)}.
(3.9)
a
To obtain the Helmholtz free energy of the colloidal suspension we add the colloidal ideal
gas contribution to the grand potential. We get
F = Fid (N ) + N ΩWSC .
3.3
(3.10)
Charge renormalization
Since the solution to the Poisson-Boltzmann differential equation (2.13) is analytically unknown in this spherically symmetric cell system, we are forced to calculate solutions numerically. What if we would linearize the equation around some special potential? Then
we would find a differential equation which we can solve analytically. We use that to linear
order in (φc (r) − φ̃) yields
sinh φc (r) = sinh φ̃ + (cosh φ̃)(φc (r) − φ̃),
(3.11)
where φ̃ is the potential around which we linearize. Within this approximation the PoissonBoltzmann equation becomes
1 ∂
2 ∂
r
φc (r) = κ̃2 {tanh φ̃ − (φc (r) − φ̃)}.
(3.12)
r2 ∂r
∂r
In this equation κ̃ is defined via κ̃2 = κ2 cosh φ̃. We can solve this linearized equation. The
general solution may we written as
18
Chapter 3. Wigner-Seitz cell theory
λB −κ̃(r−R)
λB κ̃(r−R)
e
+ ν−
e
−1 ,
φc (r) = φ̃ + tanh[φ̃] ν+
r
r
(3.13)
with integration constants ν+ and ν− that are fixed by the boundary conditions. Suppose we
have found a solution for the non-linear Poisson-Boltzmann equation. We will now construct
a solution of the linearized equation (3.12) which matches it close to the cell boundary.
0
Therefore we choose the linearization point as φ̃ = φc (R) and demand that φc (R) = 0. We
see that
1
R
sinh(κ̃(r − R)) +
cosh(κ̃(r − R) − 1)
(3.14)
φL (r) = φ̃ + tanh[φ̃]
r
κ̃r
satisfies these boundary equations. In Fig. 3.2 we plot the nonlinear and linear solution
using various colloidal charges Z. The linear and nonlinear solutions deviate more from
each other at larger Z. This is what we expect since the overall variations in the potential
increase with the charge Z. The potential difference between the cell boundary and the
potential elsewhere in the cell reaches a maximum close to the colloidal surface. Therefore,
for large Z, we see that the linear theory describes the nonlinear potential less accurate there.
We may attach an effective charge Z ∗ to the linear solution like Alexander in Ref. [11].
This charge is equal to what a colloid would have in a system described by the linear
Poisson-Boltzmann equation (of course with the right linearization point chosen). We define
0
Z∗ ≡
a2 φL (a)
tanh[φ̃] 2
(κ̃ aR − 1) sinh[κ̃(R − a)] + κ̃(R − a) cosh[κ̃(R − a)] . (3.15)
=−
λB
κ̃λB
1
0
-1
φ(r)
-2
-3
-4
-5
Z=100
Z=1000
Z=10000
Z=100000
-6
-7
1
1.1
1.2
1.3
1.4
1.5
1.6
r/a
Figure 3.2: The potential φ(r) inside a cell for various Z. The full curves give the numerical
solutions. The dotted curves are the linear approximations. Note that these deviate more for larger
Z. Here we used η = 0.1, a/λB = 100 and κa = 10.
3.4. Gas-Liquid coexistence
19
Using this renormalized charge means that we use a lower charge, like we can see from
0
0
the derivatives of the potentials at r = a in Fig. (3.2). There we see that φ (a) ≥ φL (a),
and therefore Z > Z ∗ . Physically this can be understood by the fact that the non-linear
screening effect becomes more effective at higher potentials since the density of screening
ions grows exponentially instead of linearly with the potential. Therefore, close to the
colloidal surface the screening effect becomes stronger than what one would expect based on
the linearized Poisson-Boltzmann equation (3.12). In linear theory we then account for this
fact by introducing the renormalized charge (3.15) which is smaller than the bare charge,
Z ∗ < Z. In practice, having analytic expressions for the potential of this colloidal system
can for example be used to make predictions about electrostatic energies between colloids.
3.4
Gas-Liquid coexistence
Much research has confirmed that the Wigner-Seitz cell model does not predict a gas-liquid
coexistence for any set of parameters [12] [13] [14]. Apparently the assumption of spherical
charge-neutral cells simplifies too much to conserve this effect of attracting forces, or PoissonBoltzmann theory is just not capable of describing these effects, e.g. because of its mean-field
nature. Nevertheless the (nonlinearized) model can be used for the total range of system
parameters, since it keeps the correct expression for the Poisson-Boltzmann equation. In
Chapter 7 we therefore use this model in combination with a model which does predict a
gas-liquid coexistence to get benefit of both.
20
Chapter 3. Wigner-Seitz cell theory
Chapter 4
Porous spheres in oil
4.1
Introduction
In this chapter we will use an application of cell theory to find a theoretical support of an
experimentally observed phenomenon.
Micrometer sized silica spheres are often used in colloidal experiments. It is well known
that these spheres are very porous, empty volume fractions up to one fifth of the sphere’s
total volume are no exception. The typical linear dimension of the pore is of the order of
several nanometers. This porousity can be a serious problem to experimentalists, because
of the pores the spheres act like some kind of water absorber if they come into contact with
air. Therefore, before experiments in oil, most times these colloids are heated in an oven to
evaporate the collected water inside the pores. Nevertheless, this process is not easy since
the colloids first have to cool down a bit after their time in the oven. During this time they
start to collect moisture again. Also, if such particles are suspended in oil that contains trace
amounts of water, the water may be absorbed into the pores. We must conclude that it is
very hard to get the silica spheres water-free and thus we have to take this water into account.
4.2
The model
We are going to use a method based on Wigner-Seitz cell theory on a system which contains
N porous silica spheres per unit volume. The solvent is oil, but in the pores of the spheres
we assume there is water.
Let us consider a porous silica sphere which is placed in oil, such as in Fig. 4.1. We
consider the pores to be spherical, with a diameter D. All these Npores pores are assumed to
be filled with water such that water makes up for a volume fraction of χ in the total sphere,
given by
χ=
Npores D3
.
(2a)3
(4.1)
22
Chapter 4. Porous spheres in oil
Figure 4.1: A cross-section of the model-sphere which we have in mind. We imagine the pores
with diameter D are distributed homogeneously over the sphere’s core. We assume that these pores
are filled with water. Salt ions (blue and red) from the oil migrate into these pores. Therefore we
assume these pores to be connected (not drawn here).
This water inside the sphere might contain significant amounts both cations and anions at
densities ρ±water (r). These ions can migrate from the water in the pore to the oil and vice
versa, but every ion feels an energy-difference of ∆E± when migrating from the oil to the
water. This energy-difference will typically be negative and therefore the concentrations of
ions in the water will be higher than in the oil.
Since the pores are very narrow, the effective volume available for an ion in a pore is
3
π(D−2a± )3
, where a+ and a− are the radii of
not πD
6 , but is slightly less (see Fig.4.1) ,
6
cations and the anions, respectively. Therefore the available volume fractions are
(D − 2a± )3
χ.
(4.2)
D3
At a water-silica interface the silica contains negatively charged sites because these have lost
a positive ion to the water. It can easily be calculated that the interfacial area per unit
volume is 6χ
D , and therefore the density of negatively charged sites becomes:
χ± =
−6χσ(ψ(r))
,
(4.3)
D
where σ(ψ(r)) is the number of negatively charged sites per unit area (σ > 0), which may
depend on the local potential. We wish to solve the Poisson equation in this system. It
reads:
ρinterface(r) =
∇2 ψ(r) = −
4π
eρ(r),
sphere
(4.4)
where sphere is the effective dielectric constant inside the silica sphere, and ρ(r) is the charge
density. Now we make a simplification to our system: we assume that the charge density
everywhere in the sphere is given by
4.2. The model
23
ρ(r) = χ+ ρ+water (r) − χ− ρ−water (r) + ρinterface(r).
(4.5)
By doing this we have ’smeared out’ the ions and the charged interfaces over the inside of
the whole sphere. To be able to make this simplification we had to assume that the typical
screening length scale inside the sphere is much larger then the distance between the pores.
Namely, in this case ions from different pores are able to interact easily with each other
and thus the individual pores do not form isolated parts in which the value of the potential
differs strongly from its value in the surrounding silica. We thus assume that the diameter D
is much smaller than the local Debye length which signifies the distance over which charges
can feel each other.
Now we insert the Boltzmann distributions which relate the charge densities to the
potential into Poisson’s equation (4.4). We use that the oil-reservoir salt concentration is cs
to get
2
∇ ψ(r) = −
4πe
sphere
c s χ+ e
−β∆E+ −βψ(r)e
− c s χ− e
−β∆E− +βψ(r)e
6χσ(φ(r))
.
−
D
(4.6)
Here ψ(r)βe = φ(r) as before, and ∆E± denotes the self-energy differences of the ions in
2
cs
water and oil, see below. Using that κ2oil = 8βπe
we rewrite Eq. (4.6) as
oil
6χσ(φ(r))
κ2 oil
χ+ e−β∆E+−φ(r) − χ− e−β∆E−+φ(r) −
.
(4.7)
∇2 φ(r) = − oil
2sphere
Dcs
At the surface of the silica sphere the electric field will make a jump because of its charged
outer surface with negative surface charge density σ(φ(a)). Gauss’ law (which is the integral
form of Poisson’s equation) tells us that the change in the displacement field is proportional
to the free surface charge, such that
h
i
0
0
(4.8)
lim oil ψ (a + δ) − sphereψ (a − δ) = 4πeσ(φ(a)).
δ→0
By applying this in terms of φ(r) we get the following equations which describe the transition
from inside to outside the sphere. They are
0
0
φ (a+ )
= φ (a− )
φ(a+ )
= φ(a− ),
sphere
+ 4πλb,oil σ(φ(a)),
oil
(4.9)
(4.10)
2
+
−
where we used λb,oil = βe
oil the Bjerrum length in oil, and where a and a are shorthand
notations for a + δ and a − δ. The charge Z (in units of e) is defined using the derivative of
the potential in the oil as
Z=
a2
λb,oil
0
φ (a+ ).
(4.11)
To finish the description we only need a differential equation to describe the field outside
the charged sphere, r > a. We already found this equation in Chapter 2, Eq. (2.13). We get
24
Chapter 4. Porous spheres in oil
∇2 φ(r) = κ2oil sinh φ(r).
(4.12)
0
The boundary conditions on the whole system are φ (0) = 0 (because there is no point
0
charge at the origin) and φ (R) = 0 since we are using Wigner-Seitz cells, which must be
overall neutral.
4.3
Ions in oil and water
The smaller the radius of an ion, the larger the energy-difference it feels when migrating
between volumes with different dielectric constants. It can be derived from elementary
electrostatics that the potential energy of an ion is given by
E(a± , (r)) =
e2 1
.
2a± (r)
Therefore we find for the oil to water energy-difference
1
e2
1
∆E± =
−
,
2a± water
oil
(4.13)
(4.14)
in which we used water because we presume that the ions reside in water. Earlier, in
Eq. (4.7), we used an effective dielectric constant sphere inside the spheres because there
we described the screening of charges. We assumed this to take place over distances much
larger than the pore diameters and therefore had to use an average dielectric constant over
the water and silica. On the other hand, the oil-to-water energy-difference is much more a
local effect. This is why we here do not account for the fact that silica is around and choose
the dielectric constant of water in Eq. (4.14).
4.4
Charge regularization
The ionizable surface groups at the water-silica interface make that the porous silica sphere
will gain negative charge since these surface groups can loose a positive ion to the water. We
denote the ionized surface groups by A− , the cations that can bind to the surface by H + and
the bound,charge neutral, groups by AH. Chemically this reaction reads AH A− + H + .
The associated dissociation constant describes the equilibrium concentrations of the three
species. It is defined as
[A− ][H + ]
,
(4.15)
K=
[AH]
where the square brackets mean that we use the concentration of the associated species at
the silica-water interface. Earlier we stated that the number of ionized sites per unit surface
of interface depends on the potential. Now we see why, at high potentials there are much
fewer positive ions around to create charge neutral groups. Therefore we end up with a
higher ionized fraction. A little calculation which involves Boltzmann’s expression for the
positive ion density gives us now that
σ(φ(r)) = σ
K
,
cs e−β∆E+−φ(r) + K
(4.16)
4.4. Charge regularization
25
where σ is the total (maximum) number of ionizable sites per unit surface of interface. Note
that yields σ(∞) = σ and that we assumed here that all positive ions can attach to the
ionized surface groups, A− . In practice this is not always a very reasonable approximation
since cations may come from added salt and are chemically different than those released
by the interface and thus may not be able to bind to the surface groups. For simplicity,
however, we presume that all cations are H + .
26
4.5
Chapter 4. Porous spheres in oil
Calculations
We did numerical calculations where we solved the boundary value problem Eqs. (4.7) (4.12) on a grid. We chose fairly-large ion diameters which result in not too high energy
differences between water and oil: so a+ = 3.0 Å and a− = 3.6 Å. Since typically water = 80
and oil = 8 we find ∆E+ = −10.4 kT and ∆E− = −8.7 kT .
We also picked parameters for the sphere which we think correspond reasonably to
those in experiments. We choose 2a = 1000 nm, D = 10 nm and χ = 0.20. With these
we find using Eq. (4.2) that χ+ = 0.183 and χ− = 0.179. The dielectric constant which
we think describes the sphere well is sphere = 40. How the characteristic properties exactly
depend on this value should be investigated later.
The used dissociation constant of the silica in Ref. [6] is K = 0.32 nM. We will vary it
between this value and K = 3200 nM. Furthermore, the number of ionizable sites per unit
surface we choose to vary between σ = 36 µm−2 and σ = 0.36 · 104 µm−2 and is expected
to cover a reasonable range in σ. The smallest values corresponds to a few ionizable sites
on the colloidal outer surface and the largest value for σ corresponds to the situation where
sites are being as close to each other as one Bjerrum length (in water). This is extremely
close since it implies significant electrostatic energies between ionized sites.
In Fig. 4.2 we plot the potential at a low and a higher salt concentration respectively.
At the low salt concentration, which correspond to small κoil a, the potential is a negative
function which increases with the distance to the cell’s center. The discontinuity in its
derivative is due to crossing the silica-oil interface at r = a, where apart from a layer of free
surface charge, also a change in the dielectric constant is present. For the parameters used
in Fig. 4.2 (a) we conclude that the sphere becomes negatively charged since the derivative
of the potential is positive close to the colloidal surface. The same reasoning yields that we
must conclude for the parameters used in Fig. 4.2 (b) the sphere becomes positively charged.
The mechanism behind this sudden of charge sign comes from the observation that
positive ions gain more energy than the negative ions by moving from the oil resevoir to
the water inside the pores. Therefore, more positive than negative ions will gather inside
these pores. At low salt concentrations this effect is small since there are few ions present.
In this situation the sphere charges up negatively because of the dominating negative interfacial charge of the silica. At higher salt concentrations the positive ion charge inside the
pores ultimately becomes larger than the interfacial charge, which makes the total sphere
positively charged.
In Fig. 4.3 this effect can be seen at a high and a low colloidal packing fraction. Furthermore, by concentrating on the high packing fraction we see that the positive charge at high
salt concentrations can be much larger than the negative charge at low salt concentrations.
4.6
The zero-potential solution
As we saw, the charge of a porous sphere can turn out to be positive or negative. At low salt
concentrations the negative charging of the interface surface is the most dominant effect,
27
-9
0.8
-9.2
0.7
-9.4
0.6
-9.6
0.5
-9.8
0.4
φ(r)
φ(r)
4.6. The zero-potential solution
-10
0.3
-10.2
0.2
-10.4
0.1
-10.6
0
2
4
6
r/a
8
10
0
12
0
2
4
(a)
6
r/a
8
10
12
(b)
Figure 4.2: The potential inside a cell as a function of the distance r for κoil a = 10−3 in (a), and
κoil a = 10−1 in (b). Here we took η = 10−3 and K = 32 nM, σ = 3.6 · 1016 m−2 .
250
20
200
150
0
100
-40
-60
-80
-100
-3
10
13 -2
K=0.32 nM, σ=3.6 1013
m-2
K=32 nM, σ=3.6 1013 m-2
K=3200 nM, σ=3.6 10 m
15 -2
K=32 nM, σ=3.6 1017 m-2
K=32 nM, σ=3.6 10 m
-2
10
0
-50
-100
-150
-1
κoila
50
Z
Z
-20
10
0
10
-200
-3
10
(a)
13 -2
K=0.32 nM, σ=3.6 1013
m-2
K=32 nM, σ=3.6 10 m
K=3200 nM, σ=3.6 1013 m-2
15 -2
K=32 nM, σ=3.6 1017 m-2
K=32 nM, σ=3.6 10 m
-2
10
-1
κoila
10
0
10
(b)
Figure 4.3: The (negative) charge Z of the oil-dispersed silica spheres as a function of the screening
constant κoil a in the oil reservoir. The packing fractions are (a) η = 0.1 and (b) η = 10−3 .
whereas at high salt the charge becomes positive due to the massive pore occupation of
mainly the positive ions.
In the limiting case where all interface surface is inside the pores, the interface at
the outside can be ignored. The surface charge in Eq. (4.8) will merely contribute to the
characteristics of the potential. This limit is reached for high pore volume fractions and a
very small pore diameter. Since the pore density is given by npores = χ/( π6 D3 ), the total
3
2
3χ
2
pore interfacial area is 4πa
3 πD npores = 8πa D . The outer surface of the sphere is 4πa
and thus this last surface is negligible if
D 2χa.
(4.17)
In this limit we can analytically calculate if the spheres become positively or negatively
charged. The reason is that the potential is a positive function for positively charged spheres
and a negative function for negatively charged spheres. At the transition between a positive
to a negative charge we will find a vanishing potential inside the entire cell because this
φ(r) = 0 solution is supported by the boundary conditions and yields Z = 0.
For the set of parameters used in Fig. 4.2 we see that the potential is positive for positive values of Z and negative for negative values of Z. At the transition between positive
and negative Z, the potential is expected to vanish if condition (4.17) is satisfied.
28
Chapter 4. Porous spheres in oil
Now let us take a look at the transition between positively and negatively charged
spheres. At this point φ(r) = 0 for all r inside our cell. Using Eq. (4.7) we find that Z = 0
provided
χσ(0)
= χ+ e−β∆E+ − χ− e−β∆E− .
3Dcs
(4.18)
We also conclude that
χσ(0)
3Dcs
χσ(0)
3Dcs
< χ+ e−β∆E+ − χ− e−β∆E−
Positively charged,
(4.19)
> χ+ e−β∆E+ − χ− e−β∆E−
Negatively charged.
(4.20)
χ = 0.2
χ = 0.02
χ = 0.002
χ = 0.0002
140
120
100
Z
80
60
40
20
0
-20
10-3
10-2
10-1
100
Κa
Figure 4.4: Z as calculated for various pore diameters. η = 10−3 , K = 32 nM and σ = 3.6 ·
1016 m−2 .
For a particular set of parameters we now calculate the expected screening parameters
at which the charge Z vanishes via Eq. (4.18). Note that the value of the pore volume
fractions χ will not alter the value of the expected screening parameter since the left- and
the right-hand side of Eq. (4.18) scale linearly with χ. For η = 10−3 , K = 32 nM. and
σ = 3.6 · 1016 m−2 we find κa = 0.18 as the screening parameter for which the charge is
expected to vanish. Since for χ = 0.20 condition (4.17) is satisfied better than at extremely
low χ, e.g. χ = 0.00020 we find the calculated analytical value for κa the closest to the
value found numerically when using χ = 0.20. In Fig. 4.4 we can see this; the better
condition (4.17) is satisfied, the closer the analytically found value, κa = 0.18, is to the
numerically found value. Because of the charge on the outer surface the sphere’s total
charge stays negative (Z > 0) at larger salt concentration than what one would estimate
based on Eqs. (4.19) and (4.20), therefore for parameters which not satisfy Eq. (4.17) the
numerical value for κa is a bit shifted upwards.
4.7. Conclusion
4.7
29
Conclusion
We find a change in the sign of the colloidal charge at reasonable salt concentrations of
the oil. We also observe that close to the salt concentration where this change of sign
occurs, the colloidal charge depends strongly on the value of the salt concentration. At high
packing fractions, η ' 0.1, it is very well possible that the positive charge after adding salt
is far larger than the negative charge before the adding of salt (see Fig. 4.3 for example).
Furthermore, the zero-potential solution seems to give a reasonable first estimate about the
sign of colloid’s total charge in the regime where the pores have a large interfacial area
compared to the outer surface of the sphere.
30
Chapter 4. Porous spheres in oil
Chapter 5
Linear screening theory for the
many-body problem.
5.1
The non-linear problem
Suppose we have a suspension of N colloids at fixed posistions Ri (i = 1, .., N ) in the suspension. Like before we denote the colloidal radius by a, the charge by −Ze and the density
n = N/V .
We are interested in the ion distribution around the colloids, since this distribution is
needed to determine the grand potential of the ionic part of the system. The cations and
anions experience some electrostatic potential due to the charged colloids. We may view the
potentials which lead to this force as being external ones. They are given by:
U± (r) =
N
X
Vc± (|Ri − r|) ,
(5.1)
i=1
where the potential Vc± (r) is the colloid-ion pair potentials and it consists of a hard-core
part for small r and an Coulomb part for large r. It is given by
(
∞
r < a;
(5.2)
βVc± (r) =
∓ Zλr B r > a,
where λB is the Bjerrum length of the solvent. We extract the colloid-ion and ion-ion part of
the effective Hamiltonian as it was given in Chapter 2 and keep the mean-field approximation
in this equation for the grand potential. By doing this we find the following functional:
Z
e2
ρ(r)ρ(r0 )
drdr0
Ω[ρ+ , ρ− ] = Ωid [ρ+ ] + Ωid [ρ− ] +
2
|r − r0 |
Z
+
dr (ρ+ (r)U+ (r) + ρ− (r)U− (r)) ,
where the ideal-gas grand potential is given by
(5.3)
32
Chapter 5. Linear screening theory for the many-body problem.
Ωid [ρ± ] =
Z
1
3
ln ρ± (r)Λ± − 1 .
drρ± (r) µ± +
β
(5.4)
The Euler-Lagrange equations for ρ± (r) which follow from the last equation are βµ± −
ln(ρ± (r)Λ3± ) = 0. Far away from all colloids (in the reservoir), yields ρ± (r) = cs . Therefore
we see that the chemical potential should obey
µ± =
1
ln(cs Λ3± ),
β
(5.5)
and because of this Eq. (5.4) becomes
Ωid [ρ± ] =
1
β
Z
ρ± (r)
−1 .
drρ± (r) ln
cs
(5.6)
Now we return to the grand potential in the suspension. If we take the functional
derivative with respect to ρ± (r) we immediately see that ρ± (r) = 0 inside a colloid. Also
we easily find that outside any colloid the Euler-Lagrange equations become the well-known
Boltzmann equations, ρ± (r) = cs exp[∓φ(r)]. We recall
Z
ρ+ (r0 ) − ρ− (r0 ) + q(r0 )
φ(r) = λB dr0
,
(5.7)
|r − r0 |
where q(r) is the colloidal charge distribution, given by
q(r) =
N
X
−Z
δ(|r − Ri | − a).
2
4πa
i=1
(5.8)
In differential form, Eq. (5.7) becomes
∇2 φ(r) = H̄c (r, RN )κ2 sinh φ(r) −
N
ZλB X
δ(|r − Ri | − a),
a i=1
(5.9)
where H̄c (r, RN ) is a Heaviside-like function which (as a function of r) becomes 0 inside all
colloids and is 1 everywhere else.
Now solving this equation in our case is analytically impossible. Numerically it is also
terrible because of the multi-centered character. Therefore we are forced to make some
simplifications to it.
5.2
Linearizing of the Poisson-Boltzmann equation
The differential equation which defines φ(r) is impossible to solve analytically for this problem and therefore in this form it is useless for most purposes. If we want to make sense
out of it, we have more chances if the ion densities become linear functions of the potential.
Therefore we return to the grand potential and expand the ideal-gas part around yet unknown ion densities ρ̄± . Since we prefer to obtain Euler-Lagrange equations which are linear
5.2. Linearizing of the Poisson-Boltzmann equation
33
with respect to ρ± we better expand the grand potential to second order in (ρ± (r) − ρ̄± ).
We find for the ideal gas part
0
ρ̄±
− 1)V
c
Zs
Z
ρ̄±
1
+ ln
dr(ρ± (r) − ρ̄± ) +
dr(ρ± (r) − ρ̄± )2 ,
cs
2ρ̄±
βΩid [ρ± ] = ρ̄± (ln
(5.10)
such that Eq. (5.3) becomes
Z
0
0
0
ρ(r)ρ(r0 )
e2
drdr0
Ω [ρ+ , ρ− ] = Ωid [ρ+ ] + Ωid [ρ− ] +
2
|r − r0 |
Z
+
dr (ρ+ (r)U+ (r) + ρ− (r)U− (r)) .
(5.11)
As we will see below, we can not keep the infinite hard-core potential inside the colloids
since the ion densities become linear functions of the potential,φ(r). The Euler-Lagrange
equation namely reads
Z
ρ̄±
ρ± (r)
ρ+ (r0 ) − ρ− (r0 )
= 1 + U± (r) − ln
+ λB dr0
.
(5.12)
ρ̄±
cs
|r − r0 |
We see why infinite values for φ(r) will not hold. The calculated ion densities would then
become infinitely negative inside the colloidal cores. This is a main problem in linearizing.
To get the ion distributions exactly 0 inside the colloids we have to tune the values for the
hard-core potential specifically.
It is useful to rewrite the external potentials U± (r) as a sum of a charge dependent and
a charge independent part: U± (r) = ±V (r) + W (r). The charge dependent
P part is the part
which arises due to electrostatic contributions and is denoted by V (r) = i v(|r − Ri |). The
charge independent
part is due to hard-core effects and we denote it by
P
W (r) = i w(|r − Ri |). The single colloid functions v(r) and w(r) are given by
(
βv0
r < a;
βv(r) = −ZλB
(5.13)
r > a,
r
(
βw0 r < a;
βw(r) =
(5.14)
0
r > a,
with ”hard-core” potentials v0 and w0 to be determined. Using the linearized grand potential, Eq. (5.10), the linearized Euler-Lagrange equations become
ρ̄+
ρ+ (r) − ρ̄+
+
+ φL (r) + βW (r), = 0
cs
ρ̄+
ρ̄−
ρ− (r) − ρ̄−
ln
+
− φL (r) + βW (r) = 0.
cs
ρ̄−
ln
(5.15)
(5.16)
Whereas φ(r) did not include a hard-core part (5.7), note that now φL (r) also includes the
hard-core part of v(r). It is given by
34
Chapter 5. Linear screening theory for the many-body problem.
φL (r) = λB
5.3
Z
0
ρ(r )
+ βV (r).
dr
|r − r0 |
0
(5.17)
Solving the Euler-Lagrange equations
We are going to choose φ̄(r) and ρ̄± (r) such that they are the average potential and ion
distributions respectively in the whole volume. Charge neutrality imposes directly that the
average ion charge density should cancel the colloidal charge density, thus
ρ̄ = ρ̄+ − ρ̄− = Zn.
(5.18)
After integrating the Euler-Lagrange equations over the whole volume we find
ρ̄± = cs exp ∓φ̄ − ηβw0 .
(5.19)
Now adding the equations for the positive and negative ion species with each other gives us
φ̄ = − sinh−1
Zn
exp [ηβw0 ] .
2cs
(5.20)
The average ion densities (5.19) are given in terms of φ̄, which is not a input parameter for
the system. Therefore we prefer to rearrange and combine Eqs. (5.19) and (5.20) to write
ρ̄± =
1 p
(Zn)2 + (2cs )2 exp(−2ηβw0 ) ± Zn .
2
(5.21)
Now we have equations for the average potential and ion distributions only depending on
system parameters. The only yet undefined variables are w0 and v0 . The way we are going
to determine them is by imposing that ρ+ (|r − Ri |) = ρ− (|r − Ri |) = 0 for any r inside a
colloid, so |r − Ri | < a for some i = 1, .., N . We take two specific linear combinations of the
Euler-Lagrange equations. Adding Eqs. (5.15) and (5.16) gives
ρ+ (r) ρ− (r)
+
− 2 = 2(βW (r) − ηβw0 ),
ρ̄+
ρ̄−
(5.22)
and (5.15) ·ρ̄+ −(5.16)·ρ̄− yields:
ρ(r) − ρ̄ = −(ρ̄+ + ρ̄− )(φL (r) − φ̄) − ρ̄(βW (r) − ηβw0 ).
(5.23)
No ions inside any colloid means that every linear combination of ρ+ (r) and ρ− (r) should
vanish. In especially, for any i < N ,
1
ρ+ (Ri ) ρ− (Ri )
+
= 0 ⇒ βw0 =
.
ρ̄+
ρ̄−
1−η
(5.24)
Eq. (5.23) is a defines ρ(r). It turns out to be useful
to take the equation into Fourier space.
R
We denote Rthe Fourier transform as f (k) = drf (r) exp(−ik · r) and its inverse follows
1
as f (r) = dk (2π)
3 f (k) exp(ik · r). The Fourier transform of the hard-core potential in
Eq. (5.23) becomes
5.3. Solving the Euler-Lagrange equations
4πaw0
W (k) =
k2
35
X
N
sin(ka)
− cos(ka)
eik·Ri ,
ka
i=1
(5.25)
where k = |k|. When taking the Fourier transform of Eqs. (5.17) and (5.23) we find
N
ρ(k) 4πa X ik·Ri
e
×
φL (k) = 4πλB 2 − 2
k
k i=1
λB
βv0 + Z
a
sin ka
cos ka − βv0
ka
.
(5.26)
As will become clear soon we now fix the yet unspecified parameter v0 as
βv0 = −Z
κ̄b
ρ̄+ − ρ̄−
+ βw0
.
1 + κ̄a
ρ̄+ + ρ̄−
(5.27)
We find that the Fourier transform of the charge density from Eq. (5.24) is given by
ρ(k) = (2π)3 ρ̄ + (ρ̄+ + ρ̄− )φ̄ + ρ̄ηβw0 δ 3 (k) − (ρ̄+ + ρ̄− )φL (k) − ρ̄W (k)
k2
ρ̄
3
+ (ρ̄+ + ρ̄− )φ̄
δ 3 (k)
= (2π)
1−η
k 2 + κ̄2
N
+
Z cos ka + κ̄k sin ka X ik·Ri
e
.
2
1 + κ̄a
1 + κ̄k 2
i=1
(5.28)
For this it was required to define a modified Debye screening parameter, κ̄, defined as
p
4πλB (ρ̄+ + ρ̄− )
1/4
p
−η
)2
,
= 4πλB (Zn)2 + (2cs exp
1−η
κ̄ ≡
(5.29)
where we used Eq. (5.21). This effective screening parameter does not deviate much from
the original one if the colloid density is low. It accounts for the extra screening due to the
counter ions and the reduced screening because other colloids hindering the forming of a
double-layer around the colloidal charges.
Now we calculate the Fourier inverse of the charge density (5.28) and the potential
(5.26), we find that both are a sum of N one-particle contributions. The charge distribution
P
is found to be ρ(r) = N
i=1 ρ1 (|r − R|), where

0
r < a;
(5.30)
ρ1 (r) = −Z κ̄2 exp(κ̄a) exp(−κ̄r)

r > a.
4π 1 + κ̄a
r
We see the screening charge distribution of a colloid perfectly vanishes inside its core. This
is due to our specific choice for v0 , Eq. (5.27). Nevertheless, note that the charge distributed
around colloid i does not vanish inside other colloids j 6= i.
36
Chapter 5. Linear screening theory for the many-body problem.
From Eq. (5.26) we find a correspondence between the charge distribution and the
potential, which yields with Eq. (5.28)
φL (r) = φ̄ − tanh φ̄ +
N
X
φ1 (|r − Ri |).
(5.31)
i=1
where the one-particle potential is given by
φ1 (r) = λB
5.4
Zeκ̄a e−κ̄r
.
1 + κ̄a r
(5.32)
Finding the effective interaction Hamiltonian
Now that we have found solutions of the linearized Euler-Lagrange equations of Eq. (5.11),
we can insert the equilibrium distributions again in order to evaluate its minimum. We
obtain
ρ̄±
Zn
ρ̄± ln
φ̄
1 +
cs
2
α=±
Z
1
ρ̄+ + ρ̄−
+
drρ(r)βV (r)
+ ηβw0
2
2V
Z
1
+
drβW (r) (ρ+ (r) + ρ− (r)) ,
2V
0
βΩ [ρ+ , ρ− ]
V
=
X
(5.33)
where V (r) and W (r) are defined in Eqs. (5.13) and (5.14) and where ρ(r) = ρ+ (r) − ρ− (r)
and ρ± (r) are the solutions found in Eq. (5.30).We again go to Fourier space, and hence the
Fourier transform of the charge dependent potential V (r) is needed. It can be calculated
that
βV (k) = −
4π
k3
βv0 + Z
λB
a
X
N
eik·Ri .
(5.34)
X λB
e−κ̄Rij
− Z2
+ βΦion ,
Rij
Rij
i<j
(5.35)
ka cos ka − βv0 sin ka
i=1
By using this and after tedious integrations we find that
0
βΩ [ρ+ , ρ− ] = (1 + Γ)
Zeκ̄a
1 + κ̄a
2 X
i<j
λB
where
X
ρ̄±
βΦion
1 (Zn)2
η
n Z κ̄λB
2ρ̄+ ρ̄−
ρ̄± ln
=
−1 −
+
−
.
V
cs
2 ρ̄+ + ρ̄−
1 − η ρ̄+ + ρ̄−
2 1 + κ̄a
±
(5.36)
The parameter Γ, which characterizes the deviation from a purely DLVO [2] [3] pair potential,
can be calculated to be
5.4. Finding the effective interaction Hamiltonian
Γ=
37
4π n
(1 + κ̄a)2 e−2κ̄a + (κ̄a)2 − 1 .
3
1 − η κ̄
(5.37)
In Fig. 5.1 we plot Γ as a function of κa for packing fractions η = 10−1 ,10−2 and 10−3 . As
can be seen Γ is typically much smaller than unity, and hence we ignore it in calculations.
10
0
η=0.1
η=0.01
η=0.001
Γ(κa)
10-1
10
-2
10-3
10-4
0
2
4
6
8
10
κa
Figure 5.1: The factor Γ defined in Eq. (5.37) as a function of η for various κa.
Now that we have found the grand potential of the ions interacting with the colloids
and with each other, all that remains is to add the electrostatic colloid-colloid interactions
to the effective Hamiltonian. It follows from elementary electrostatics that
Z Z
λB
q(r)q(r0 )
drdr0
2
|r − r0 |
X λB Z 2
λB Z 2
=
+N
,
Rij
2a
i<j
Vcc =
(5.38)
where we used q(r) as defined in Eq. (5.8). Note that the last term is a self-energy term
of the N colloids. Since this term is a constant as a function of the density as long as Z is
fixed, it can be neglected in phase-diagram calculations. Nevertheless, later we will describe
a theory where charges can change, then this term can become important. Therefore, we
keep it.
Now we are in a position where we can write down the effective interaction Hamiltonian
between the colloids. Using density functional theory we have accounted for the ion distributions around the colloids. The remaining interaction Hamiltonian therefore only depends
on the positions of the colloids. It reads
38
Chapter 5. Linear screening theory for the many-body problem.
0
H = ({R}, N, V, cs ) = Ω [ρ+ , ρ− ] + Vcc
= ΦD (V, n, cs ) + Φ0 (V, n, cs ) +
N
X
V2 (Rij ; n, cs ),
(5.39)
i<j
here, the Donnan-term, which accounts for the ideal-gas part of the average ion distribution,
is given by
X
βΦD
ρ̄±
=
−1 ,
(5.40)
ρ̄± ln
V
cs
±
and the other term includes self energies and also includes contributions of the average ion
distribution. It is
βΦ0
1 (Zn)2
η 2ρ̄+ ρ̄−
n ZλB
=−
+
+
.
V
2 ρ̄+ + ρ̄−
1 − η ρ̄+ + ρ̄−
2a 1 + κ̄a
(5.41)
This term is different from what is found in [6] since we included the (yet constant) selfenergy term of the colloids. The pair potential is found to be
(
∞
r < 2a;
V2 (r) =
(5.42)
−κ̄a
2
r > 2a,
(1 + Γ)Z>
λB e r
κ̄a
e
. The new screening parameter κ̄
where the so called DLVO charge is given by Z> = Z κ̄a+1
is larger then the original screening parameter κ.
5.5
Calculating the free energy
In the previous section we have calculated the effective Hamiltonian of the colloidal system.
We saw that it consists on a part which is dependent on the positions of the colloids but
there is also a part which is not, which are ΦD and Φ0 . Therefore the calculated free energy
is of the form
F = ΦD + Φ0 +
N
(ln(nV) − 1) + Fexc ,
β
(5.43)
where the third term is the colloidal ideal-gas free energy, and where Fexc is the excess free
energy given by
*N
+
X 0
Fexc =
V2 (Rij ) ,
(5.44)
i<j
where the average is to taken with Boltzmann weight e−βH . How can we calculate this
thermodynamic average? We do this by comparing the effective colloidal system to a similar
system from which we know the excess free energy accurately. Thanks to the CarnahanStarling approximation this is the hard-sphere fluid. The excess-free energy in this very
accurate approximation is
5.6. Using the Gibbs-Bogoliubov inequality
39
βFexc,HS
ηd (4 − 3ηd )
=
.
N
(1 − ηd )2
(5.45)
3
In this last equation ηd is the packing fraction of spheres with a radius d, ηd = n 4πd
3 . This
excess free energy is of course due to the decreased number of possible colloidal configurations
due to hard-core repulsions. More or less the same happens in our colloidal system. Since
the particles repel each other, the effective hard-sphere diameter will be larger then the
colloidal diameter, therefore ηd > η. The comparison with a fluid reference is only a good
one if our system itself is also in a fluid phase. At high densities, when a solid phase is
to be expected, we should for example use the free energy of a frozen hard-sphere system
as a reference. Since the final goal of our research is to check whether or not we can find
theoretical support for the existence of a gas-liquid phase coexistence we stick to the fluid
expression for the moment. This will apply for the parameters of interest.
5.6
Using the Gibbs-Bogoliubov inequality
In this section we will discuss very brief how the Gibbs-Bogoliubov inequality might be used
to calculate a good estimate for the excess free energy. A more detailed discussion can be
found in [6].
The Gibbs-Bogoliubov
inequality
uses the fact that if two positive functions are equally
R
R
normalized, dRN A({Ri }) = dRN B({Ri }), then
Z
Z
N
dR A(Ri ) ln A(Ri ) ≤ dRN A(Ri ) ln B(Ri ).
(5.46)
We use A({Ri }) = exp [β{Fexc,HS − HHS ({Ri })}] and
B({Ri }) = exp [β{Fexc,CS − HCS ({Ri })}], in which the subscript CS points to the colloidal
system and HS to the hard-sphere system. Then we find
*N
+
X
Fexc ≤ Fexc,HS +
(V (|Ri − Rj |) − VHS (|Ri − Rj |)
,
(5.47)
i<j
HS
where now the average is taken with a Boltzmann weight corresponding to the hard-sphere
Hamiltonian HHS . This equation is valid for any chosen hard-sphere diameter. So if we
minimize the right hand-size for d > 2a we find a very good approximation for the free
energy. To calculate all this, we need the radial distribution function g2HS (r) for a hardsphere fluid. In Ref. [9] the two dimensional Laplace-transform of this function, GHS (x), is
given on basis of the Padé approximation. After a tedious calculation one finds
ηd (4 − 3ηd )
ηd GHS (κd)
βFexc
2
≈ min
+ Z> κλB
.
(5.48)
d
N
(1 − ηd )2
κd
5.7
Gas-Liquid coexistence
The method described in Section 5.6 allows us to calculate the free energy of the colloidal
system. Actually we called it earlier the Donnan-ensemble. This is because the salt ions are
40
Chapter 5. Linear screening theory for the many-body problem.
described by a chemical potential instead of a fixed number and thus we are in between a
grand potential and a free energy. Nevertheless, the colloids are described by a fixed density N/V and therefore, from their point of view we are dealing with a Helmholtz free energy.
We can predict gas-liquid phase coexistence using this equation by making common
tangent constructions, as will be described in Chapter 7. Since the compressibility of this
system can become negative for certain parameters, these constructions are possible and
thus a gas-liquid coexistence is predicted. Fig.5.2 shows a calculated phase diagram in the
(cs , η) plane. Note that the solid phase is also taken in account.
Figure 5.2: Obtained from [6]. Phase diagram for a colloidal suspension as a function of colloidal
packing fraction η and reservoir salt concentration cs . Here a = 326 nm, Z = 7300. The solvent
is water at room temperature; λB = 0.72 nm. The solid lines denote fluid-solid and solid-solid
binodals. The dotted line is the underlying metastable gas-liquid binodal, which would be the
gas-liquid binodal if the solid-phase would not exist. The fluid-solid-solid triple point are denoted
by ×’s, the solid-solid critical point by a square.
We see in Fig. 5.2 that for salinities below cs = 20µM a large density gap between
the liquid and gas phase opens up. Though, as mentioned in Ref. [6] and as we will see in
Chapter 7, the colloidal charge is very high compared to a/λB , this means that the potential
at the surface will probably drop deeply below −kT /e and thus the linear theory will be
used outside its regime of validity. We should have a bit of caution while using these results.
Now we are at the point that we are familiar with cell theory and with linear theory.
Though cell theory can be used for all parameter regimes it will not predict attractions
between colloids, and thus no gas-liquid phase coexistence. The linear theory, on the other
hand, does predict phase coexistence. The problem is that we are using it outside its regime
of validity, the screening becomes non-linear. Therefore in Chapter 7 we alter the linear
theory to combine it will cell-theory and find a method for which the linearized potential
stays reasonably flat. But first, in Chapter 6, we investigate the effect of the linear-screening
artifact which is the presence of ions inside the colloidal hard cores.
Chapter 6
The spurious charge in the
colloids
6.1
The effect
In Chapter 5 the one-colloid charge distribution was chosen such that the double layer
around colloid i does not penetrate the hard core of colloid i itself, see Eq. (5.30). However,
this charge distribution will induce a charge inside other colloid cores j 6= i. The closer the
other colloids are, the larger this ’ghost charge’ becomes. Of course, the accumulation of
this charge is physically impossible. This is why we call it ’ghost-charge’ and it is an artifact
of using linear theory. In this chapter we are going to calculate the average amount of ghostcharge inside a colloid’s core. In Fig. 6.1 the effect is schematically drawn. Looking to the
picture, we expect that the effect becomes important at high densities and long screening
lengths. This will turn out to be the case.
Figure 6.1: Schematical illustration of two colloids and the graphs of the one-particle screening
ion densities which surrounds them. The upper part of this figure shows two colloids far apart from
each other. The lower part shows the two colloids at a close distance, such that the double layer of
the left one penetrates the hard core the right one and vice versa.
42
6.2
Chapter 6. The spurious charge in the colloids
Calculations
Let us choose the origin of the colloidal system at the center of colloid 1, which has a radius
a. We want to find the induced amount of charge inside this colloid due to the presence of
N − 1 other colloids at positions Ri (i = 2, .., N ), each surrounded by an ionic charge cloud
ρ1 (|r − Rij |). The equation which gives the induced charge is
hQind i =
Z
dr
|r|<a
N
X
ρ1 (|r − Rij |),
(6.1)
i=2
where ρ1 (r) is the one-colloid charge distribution given in Eq. (5.30). Now we suppose that
the average positions of the other colloid are described by the simplified pair correlation
function
(
0 r < c;
g2 (r) =
(6.2)
1 r > c.
We use the explicit equation for the charge distribution (5.31) and we find for the average
induced charge in a given colloid
Z
Z
dr nρ1 (|r − R|).
(6.3)
dR
hQind i =
|r|<a
|R|>c
The representation of ρ1 (r) as integral over its Fourier components is
Z
1
nZ κ̄2 e−ik·R eκ̄a 4π
ρ1 (|r − R|) =
eik·r ,
dk
(2π)3
4π
1 + κ̄a k 2 + κ̄2
(6.4)
and therefore hQind i becomes
hQind i = B
Z
dR
|R|>c
Z
dr
|r|<a
2
Z
dk e−ik·R
eik·r
,
+ κ̄2
k2
(6.5)
κ̄a
κ̄ e
where the constant B is defined as B = 4πnZ
(2π)3 1+κ̄a . In Appendix A the last two integrals
of Eq. (6.5) are integrated by contour integration, see Eq. (A.17). We find
hQind i = B
Z
|R|>c
−κ̄c
dR
e−κ̄R
a3
(2π)3
(κ̄a cosh κ̄a − sinh κ̄a)
R
(κ̄a)3
(1 + κ̄c)
a3
B(2π)3
(κ̄a cosh κ̄a − sinh κ̄a)
2
κ̄
(κ̄a)3
eκ̄a κ̄a cosh κ̄a − sinh κ̄a
= 3ηZe−κ̄c (1 + κ̄c)
1 + κ̄a
(κ̄a)3
= ηZξ(κ̄a, κ̄c),
=
e
(6.6)
where ξ(x, y) is given by
ξ(x, y) = 3e−y (1 + y)
ex x cosh x − sinh x
1+x
x3
(6.7)
We see that hQind i is linear in η and Z, as expected, and depends nontrivially only on κ̄a
and κ̄c.
6.3. Results for various particle distances
6.3
43
Results for various particle distances
We can select a minimal particle distance by altering the value of c. Using the condition
c = 2a the colloids are allowed to touch each other, this is the smallest value which we can
assign to c. At larger values for c we assume there is some distance between the colloids.
Fig. 6.2 shows ξ(κ̄a, κ̄c) as a function of κ̄a for several κ̄c.
1.2
c=2a
c=3a
c=4a
1
ξ(Κa,Κc)
0.8
0.6
0.4
0.2
0
0.01
0.1
1
10
κa
Figure 6.2: ξ(κa, κc) as defined in Eq. (6.7) for three values of c. Using ξ, the value Zηξ gives
the charge collected in a colloid due the double layers around other colloids.
From Fig. 6.2 we conclude that ghost charge begins to build up for κa < 1. We also
see that this point gets shifted by the value of c. This shifting effect is small since going
from c = 2a to c = 4a completely changes the characteristics of the pair correlation function
g2 (r). Therefore we are safe to assume that this point will be around κa = 1 for most two
pair correlation functions. This is also what we expect since the range of the one-particle
potential becomes relatively long for this value of κa.
6.4
Charge redefinition?
One could consider the idea to redefine the charge such that we compensate for the fact that
ionic charge builds up inside other colloids. Since the effective charge of a colloid drops to
Zeff = Z(1 − ξ(κa, κc)η), it is tempting to introduce a compensated colloid charge,
Zcomp =
Z
.
(1 − ξ(κa, κc)η)
(6.8)
Although this method might look promising at first sight, there is a problem with this
which cannot be neglected. By introducing a larger charge Zcomp we also introduce more
counter-ions and the total charge of the double layer increases by the same magnitude as
the colloid charge. Although we might get to a better description of the potential outside
the colloids, inside the colloid, the ghost charge even increases more. We saw that the grand
44
Chapter 6. The spurious charge in the colloids
potential of the linearized system is calculated by an integral over the whole volume. This
means we also take in account for (the degrees of freedom of) the ghost charge inside the
colloids. Therefore, redefining the charge does not solve the problem instantly and more
work is required to test ideas along these lines.
Chapter 7
Combining linear multicentered
and cell theory
7.1
The situation
We are concerned with a system of N colloids of radius a and charge −Ze at positions R i ,
in a dielectric continuum solvent in contact with a reservoir of pointlike cations and anions
−1/2
2
at a concentration 2cs , such that the reservoir’s screening length is κ−1 = 8πβe
.
The ion densities outside the colloidal hard cores are given by ρ± = cs e∓φ(r) , where
the dimensionless electrostatic potential satisfies the nonlinear Poisson-Boltzmann equation
(2.13) with the multicentered boundary conditions of Eq. (2.14). Whereas we linearised the
Poisson-Boltzmann equation in chapter 5 and assumed a cell geometry in chapter 3, we combine these two approaches here: we introduce a distance b > a, and describe the potential
φ(r) for |r − Ri | < b as φcell (|r − R|), with φcell (r) a solution of a cell, while in between the
colloids, |r − Ri | > b for all i, we linearize the Poisson-Bolltzmann equation. In contrast to
the original Wigner-Seitz cell theory the cells of radius b are not neutral since they are not
assumed to fill the whole system volume. We call their net charge −Qe (with Q < Z). In
other words, we do not demand that the electric field vanishes on the cell’s boundary, but
instead takes the value corresponding to the electric field of a point charge −Qe at distance b.
The size of the grid where we work with should be large enough to preserve the characteristics of the system. The problem to solve is one dimensional:
1 d
2 d
φcell (r)
= κ2 sinh φcell (r);
r
r2 dr
dr
0
ZλB
φcell (a) = 2 ;
a
0
QλB
φcell (b) = 2 .
b
(7.1)
(7.2)
(7.3)
One way to solve this equation for fixed Z and Q is to set φcell (a) at a trial value and impose
46
Chapter 7. Combining linear multicentered and cell theory
0
the value of φcell (a) given by the boundary equations. Then we solve Eq. (7.1) by ”walking”
0
B
from a to b over the grid. If φcell (a) was taken too small then we find φcell (b) < Qλ
b2 . For
φcell (a) choosen too large we find the opposite. Then, by adjusting this value of φcell (a) we
ultimately find an acceptable solution which matches condition (7.3).
7.2
Connecting the linear solution to the cell solution
In Chapter 5 we derived equations to the describe the potentials of colloids within a linear
theory. We are going to use these to describe the potential outside the cells. We derived
that
φL (r) = φ̄ − tanh φ̄ +
N
X
φ1 (|r − Ri |).
(7.4)
i=1
where φ1 (r) is the one-colloid potential. It has a characteristic DLVO like form and is given
by
Z ∗ eκ̄a e−κ̄r
,
(7.5)
1 + κ̄a r
in which Z ∗ is the effective charge of the colloids, which is yet unknown. Furthermore, like
in Chapter 3 we define:
∗
Z n
η
−1
φ̄ = − sinh
exp
,
(7.6)
2cs
1−η
1/4
p
−η
κ̄ =
4πλB (Z ∗ n)2 + (2cs exp
)2
.
(7.7)
1−η
φ1 (r) = λB
We use Z ∗ instead of Z because inside the cell charge renormalization takes place. The
charge gets screened more efficiently compared to what one would expect based on linear
theory. To calculate Z ∗ , we match the potential of the cell solution to the linearized solution.
We also match the electric fields of both solutions, since a discontinuous electric field would
mean the presence of a charged surface outside the colloids. Since the value of the linearized
potential depends on the positions of all colloids, we calculate its thermodynamic average.
If we set the origin at the center of colloid 1 we find
φcell (b)
0
φcell (b)
= hφL (br̂)i
(7.8)
= hr̂ · ∇φL (br̂)i ,
(7.9)
where the vector r̂ = r/r is a unit-length vector in the radial direction. Using Eq. (5.31) we
may expand the pieces between the expectation brackets to
φL (br̂) = φ̄ − tanh φ̄ + φ1 (b) +
N
X
φ1 (|br̂ − Ri |),
(7.10)
i=2
0
∇φL (br̂) = φ1 (b)r̂ +
N
X
i=2
∇φ1 (|br̂ − Ri |).
(7.11)
7.2. Connecting the linear solution to the cell solution
47
To calculate the thermodynamic average we have to use a colloidal pair-distribution function.
Getting this function would demand us to have equations about the electrostatic energies,
which is just what we are looking for. To keep things simple we use the mean-field pairdistribution function for hard-spheres of radius b, denoted by g2 (r) here, given by 0 for
r < 2b and 1 for r > 2b. We find
Z
φcell (b) = φ̄ − tanh φ̄ + φ1 (b) + n dRg2 (|R|)φ1 (|r − R|),
(7.12)
and for the field
0
0
φcell (b) = φ1 (b) + n
Z
0
dRg2 (|R|)φ1 (|r − R|)
(r − R) · r̂
.
|r − R|
(7.13)
By carrying out long integrations similar to those in Chapter 6 we find the following continuity equation for the potential:
φcell (b) = φ̄ − tanh φ̄ −
λB Z ∗ eκ̄a −κ̄b
e
[1 + ηc ζ(κ̄b)] ,
b 1 + κ̄a
(7.14)
while the continuity equation for the electric field becomes
0
0
λB Z ∗ eκ̄a 1 + κ̄b
λB Q
=
φ
(b)
=
φ
(b)
=
[1 − ηc ξ(κ̄b)] .
cell
cell
b2
b2 1 + κ̄a eκ̄b
(7.15)
In both equations ηc = (b/a)3 η is the packing fraction of the cells, and the functions ζ(x)
and ξ(x) are given by:
1 − e−2x
3
(1 + 2x)
,
2
x3
3 1 + 2x (1 + x)e−2x + x − 1
ξ(x) =
.
2 1+x
x3
ζ(x) =
(7.16)
(7.17)
Note the term linear in ηc ξ(κ̄b) in Eq. (7.15). This term is due to the electric field of the
colloids which are clouded around the centered one (colloid 1). We would like to see that in
the ηc = 1 limit the theory reduces to the Wigner-Seitz cell theory because then the cells
fill up the total volume. This only happens for ξ(κ̄b) = 1, which means low κb.
So, for κ̄b 1 this high ηc Wigner-Seitz limit is reached nicely, but for all other κ̄b this
is not the case. It would have been nice if ξ(κ̄b) = 1 for all κ̄b. The reason why this does not
happen in our theory is because of the particular choice we made for g2 (r) and the shape
of the cells. By assuming that the cells cannot come closer to each other then this distance
of 2b, we are contradicting the assumption that this high cell packing fraction, ηc → 1 limit
can be reached. It thus probably is a consequence of the spherical cell assumption.
Suppose that we would redefine ξ(x), such that ξ(κ̄b) = 1 for all κ̄b. Then, for all
screening parameters κ̄ we find the Wigner-Seitz cell model back in the limit ηc → 1. In this
case the electric field due to the other colloids would still scale linearly with η (or η c if one
0
likes) and φcell (b) → 0 for ηc → 1, so this choice seems to be all right. Furthermore, an on
48
Chapter 7. Combining linear multicentered and cell theory
0.6
1
0.5
0.8
ξ(κa)
ζ(κa)
0.4
0.3
0.2
0.4
0.2
0.1
0
10-2
0.6
10-1
100
κa
101
102
0
10-2
(a)
10-1
100
κa
101
102
(b)
Figure 7.1: The functions ζ(κa) and ξ(κa) as defined in Eqs.(7.16) and (7.17) respectively.
κ̄b depending choice for g2 (r) would make this possible though this would also change the
definition of ζ(x) (7.16). Nevertheless, since Winger-Seitz cell theory was an approximation,
we have no evidence that incorporating the high packing fraction Wigner-Seitz limit will
give us better results. Therefore we keep Eq.(7.17).
Now since we obtained Eqs. (7.14) and (7.15), i.e. a relation between the net cell charge
Q and the renormalised charge Z ∗ , we can write
eκ̄a 1 + κ̄b
Q
=
[1 − ηc ξ(κ̄b)] .
(7.18)
∗
Z
1 + κ̄a eκ̄b
For small η all contributions to the electric field are dominated by that of colloid 1
and then we find what we expect based on the point-charge potential from DLVO theory.
The net charge inside the sphere of radius a, Z ∗ , is related to the charge inside a larger
surrounding sphere of radius b, Q, via
Z∗
eκ̄b (1 + κ̄a)
= κ̄a
.
Q
e (1 + κ̄b)
(7.19)
Thus we obtained equations for the linearized charge Z ∗ which we might use to calculate the
one-particle potential outside the cells. Next we need Q and Z ∗ explicitly from Eqs. (7.14)
and (7.15).
7.3. A scheme to find the renormalized charge
7.3
49
A scheme to find the renormalized charge
The scheme we used to solve Z ∗ out of the Eqs. (7.14) and (7.15) differs slightly from the
one what was used in Ref. [6]. For fixed b it is done in the following way:
1 Pick a trial value for Z ∗ , for example Z ∗ = Z.
2 Calculate φ̄ and κ̄ from Eqs. (5.20) and (5.29).
3 Calculate Q from Eq. (7.15).
4 Solve the Poisson-Boltzmann inside the cell, using Q in the boundary equation.
This will give a value for φcell (b).
5 Using φcell (b) we find a new value for Z ∗ using Eq. (7.14).
6 If the new value for Z ∗ deviates only little from the original one in step 1 we
have found an acceptable solution, if not we take a new Z ∗ and go back to (2).
We found that Z ∗ is an unstable equilibrium point. It is very well possible to find this
equilibrium point but we should be aware that an iterative process in Z ∗ will not converge.
We find Z ∗ by decreasing the interval in which it can be found. Suppose we choose the
trial value for Z ∗ below its actual value, then the new value (as in step 4) will found to be
larger than the initial value, therefore we then conclude that the minimum of the interval is
larger then the initial value. On the other hand, if the trial value was choosen larger then
the actual value of Z ∗ then the new value is found to be smaller and we conclude that the
maximum of the interval is smaller than the initial value.
Figure 7.2: An example of the scheme to find the renormalized charge Z ∗ , which is drawn here
along the vertical axis. The grey boxes characterize the interval in which Z ∗ can be found. Along
the horizontal axis we see the result of the scheme described above after several cycles. By choosing
a trial value of Z ∗ , denoted by a blue cross, we find a new value for it by applying the scheme
above, this new value is denoted by a red cross. If the new value is larger then the trial value (as
in (1)) then we change the bottom of the interval to the trial value. If the new value is smaller (as
in (2)&(3)) then we change the top value of the interval to the trial value. We see on each step the
interval decreases by a factor 2.
50
7.4
Chapter 7. Combining linear multicentered and cell theory
An appropriate choice for the cell size
What should we choose for the cell radius b? We recall that b can be seen as the distance
from the center of a colloid (and closer) at which the potential becomes so large that linear
theory does not describe it well anymore. It marks the end of linear theory and begin of cell
theory. Typically linear theory is valid for |φL (r) − φ̄| < 1 so we demand that b should obey
∆φ ≡ φ̄ − φcell (b) = 1,
(7.20)
where we have skipped the absolute bars since we know that close to negatively charged colloid the potential becomes more negative. To find b we take a starting value for it, calculate
∆φ by finding the right value for Z ∗ . We raise b if ∆φ is too large and lower it if ∆φ is too
small. Ultimately we will find an acceptable value for b.
In Fig. 7.3 we show that the calculated value of b. It varies between b ' 1a and b ' 10a
in the (κa,η) plane covering several decades for Z = 1200 and a/λB = 92.6 . For ZλB /a > 1
at low densities and screening, the cells are significantly larger than the bare colloids. In
this high Z regime we see the same effect as in Chapter 3, the effective charge Z ∗ becomes
significantly smaller than Z. On the other hand, for ZλB /a < 1, the cells shrink to the
colloidal size for all densities and screening lengths. In this regime yields Z ∗ ∼
= Z.
Figure 7.3: Obtained from [6]. The calculated cell radius b for the parameters Z = 1200, a/λB =
92.6. lines of equal cell size are drawn black. Lines of equal cell packing fractions ηc = η(b/a)3 are
drawn red.
7.5
The Hamiltonian and the free energy
After a right value for b is picked the system can be properly divided into the interior of
cells of radius b and their outside. The effective Hamiltonian therefore, can also be split
in a non-linearized and a linearized part. In Chapter 5 we derived equations for the grand
potential of the whole volume of a linearized system. Since we want to use these again we
7.5. The Hamiltonian and the free energy
51
have to extract the contribution from inside the cells. Formally, the effective Hamiltonian
can be written as
H = Ωcells + ΩL − δΩL .
(7.21)
The first term is defined as
βΩcells
XZ
=
α=±
1
2
+
Z
ρα (r)
drρα (r) ln
−1
cs
cells
dr [ρ(r) + q(r)] φ(r).
(7.22)
cells
The second and third describe the Hamiltonian in the linearized part of the system, they
are:
XZ
(ρL,α (r) − ρ̄α )2
ρ̄α
−1 +
βΩL =
dr ρ̄α ln
cs
2ρ̄α
α=±
Z
1
+
dr [ρL (r) + q ∗ (r)] φL (r),
(7.23)
2
and
βδΩ =
XZ
α=±
+
1
2
Z
ρ̄α
(ρL,α (r) − ρ̄α )2
−1 +
dr ρ̄α ln
cs
2ρ̄α
cells
dr [ρL (r) + q ∗ (r)] φL (r).
(7.24)
cells
In these last equations ρL,α (r) denotes the density of ions of type α in our linear system,
PN
Z∗
ρL (r) = ρL,+ (r) − ρL,− (r) is the net charge distribution and q ∗ (r) = − 4πa
2
i=1 δ(|r − Ri | −
a) denotes the charge distribution due to the charged colloids. The first equation contains
integrals over the whole volume of the system, while the second (the correction part) only
integrates over the cell interiors.
Let us start with the cell part of the effective Hamiltonian (7.21). In Chapter 3 we
arrived at the grand potential for a Wigner-Seitz cell system (3.6). Although we do not have
charge neutral cells, it can easily be seen that the same equation still holds. We have
βΩcells
Z
= − φcell (a) + 4πcs
N
2
Z
b
dr r2 [φcell (r) sinh φcell (r) − 2 cosh φcell (r)] .
(7.25)
a
In Eq. (5.39) we defined the effective Hamiltonian for the linear system. Now we use this to
find the grand potential of the linear part ΩL of Eq. (7.21),
ΩL = ΦD (Z ∗ , κ̄, η) + Φ0 (Z ∗ , κ̄, η) +
N
X
i<j
where
V2 (Rij ; Z ∗ , κ̄),
(7.26)
52
Chapter 7. Combining linear multicentered and cell theory
X
ρ̄α
βΦD
=
−1 ,
ρ̄α ln
V
cs
α=±
(
∞
r < 2a;
V2 (r) =
−κ̄a
eκ̄a 2
(1 + Γ)(Z ∗ κ̄a+1
) λB e r
r > 2a,
(7.27)
(7.28)
and
1 (Z ∗ n)2
η
n Z ∗ λB
2ρ̄+ ρ̄−
βΦ0
=−
+
+
.
V
2 ρ̄+ + ρ̄−
1 − η ρ̄+ + ρ̄− 2a 1 + κ̄a
(7.29)
It will turn out that the correction term δΩ will alter the Donnan and the self-energy terms
as well as the pre-factor of the colloid-colloid pair potential. Chapter 8 is devoted to the
calculation of this correction term. For any given correction term we can start to make some
calculations about the free energy of the colloidal suspension.
7.6
Free energy and gas-liquid coexistence
Since we are interested in possible gas-liquid phase separations we calculate the free energy
per unit volume inside the system as defined in Eqs. (2.6) and (2.8). By extensivity we write
F (N, V, cs ) = V f (n, V, cs ), and we use the Gibbs-Bogoliubov inequality for the colloid-colloid
pair interaction term, using a hard-sphere system as referenceto calculate f . The chemical
potential and (osmotic) pressure follow then as
∂f
,
∂n
= nµ − f.
µ =
(7.30)
P
(7.31)
By calculating the compressibility, χ, defined by χ−1 = ∂P
∂n , we can calculate if there are
any unstable states in the system. Namely, if χ < 0 then a phase separation into a highand low-density state exists. In this case, chemical and mechanical equilibrium demand
that the high- and low-density states are of equal chemical potential and pressure. Such
calculations of phase coexistence are somewhat involved, but the calculation of χ −1 is more
straightforward. Below we focus therefore on χ−1 .
Chapter 8
Various treatments of the
correction term δΩ
Figure 8.1: Obtained from [6]. The blue part of each of the subfigures shows for different Z and
a/λB the predicted regions of negative compressibility. Along the long axis of the subfigures the
density changes over decades, along the short axis the salt concentration changes over a wide range.
The red areas were calculated using a different equation for the correction term δΩ, that will not
be discussed here.
54
8.1
Chapter 8. Various treatments of the correction term δΩ
Numerical accuracy
In Chapter 7 we described how to combine the Wigner-Seitz cell theory with linear theory
to obtain a new theory in which the linearized potentials do not deviate much from some
average potential in the linearized part of the system. We saw that to make computations
on the free energy, we more or less add the cell part to the linear part to the grand potential
and we extract the linear contribution in parts of the system where cell theory describes
the system. This correction term alters the equation which describes the grand potential
of the linear part. In Ref. [6] it was calculated. This correction term alters the effective
Hamiltonian of the system (7.21). To account for this we may as well alter the linear part
of the effective Hamiltonian to incorporate the correction term. We write
H = Ωcells + Ω˘L ,
(8.1)
where Ωcells is defined in equation 7.25 and in which the linear part Ω˘L becomes different
from its definition in equations (7.26)-(7.29). It was found that
Ω̆L
≡ ΩL − δΩ
=
∗
∗
Φ̆D (Z , κ̄, η) + Φ̆0 (Z , κ̄, η) + (1 − Υ(κ̄a))
N
X
V2 (Rij ; Z ∗ , κ̄),
i<j
where the volume term becomes
β Φ̆0
=
V
1−η
b3 − a 3
a3
2ρ̄+ ρ̄−
η
Z∗
φ̄ +
,
2
1 − η ρ̄+ + ρ̄−
(8.2)
and the Donnan term gets a bit reduced to
X
ρ̄α
β Φ̆D
= (1 − η)
ρ̄α ln
−1 .
V
cs
α=±
(8.3)
An important change is the reduced colloid-colloid interaction by a factor (1−Υ(κ̄a)), where
Υ(x) =
1 + x 1 − e−2x
.
x
2
(8.4)
In Figure 8.2 the factor (1 − Υ(κ̄a)) is plotted, showing that (1 − Υ(κ̄a)) 1 for κ̄a 1
while (1 − Υ(κ̄a)) = 21 for κ̄a 1.
Since we now have a full expression (8.1) for the effective Hamiltonian of the system
and may apply the Gibbs-Bogoliubov inequality for the remaining colloid-colloid interaction
term, we are able to calculate phase diagrams as was described in section 7.6. We calculated
regions of negative compressibility for various densities, salt concentrations, solvents and
colloidal charges.
By doing this we noticed that the value of the compressibility highly depends on the
number of grid points used to solve the Poisson-Boltzmann equation inside the cells. This
is remarkable because based on earlier experiences we would expect 1000 equally-sized grid
points would be enough for typical cells with b ' 1 - 10a. The more so since we saw no
8.1. Numerical accuracy
55
0.6
0.5
1 - Υ(κa)
0.4
0.3
0.2
0.1
0
10-2
10-1
100
κa
101
102
Figure 8.2: The prefactor for the colloid-colloid pair potential, 1 − Υ(κ̄a). We see that for small
κa the colloid-colloid pair potential does not contribute to the free energy at all.
significant change in the calculated value of b and Z ∗ for grid sizes larger than 1000.
The value of the compressibility χ seems to be a different case. Especially for small
densities we get an extreme sensitivity to the number of grid points. This seems to be
caused by the cell part of the grand potential, in especially in the parameter regime where
b/a becomes large and the potential at the colloidal surface becomes very negative. At very
negative values of the potential, the Poisson-Boltzmann equation becomes sensitive to the
value of the potential, since sinh(φ+∆φ)−sinh φ ≈ ∆φ cosh φ. On top of that, the integrand
of Eq. (7.25) becomes also sensitive to slight variations in the potential. Fig. 8.3 shows this
effect clearly. This plot was calculated for low colloidal volume fractions, η = 0.001 and a
low screening parameter, κa = 0.1 as well. The low volume fraction typically implies large
cells and the low screening parameter implies a highly negative potential on the colloidal
surface.
(r/a) (φ(r) sinh(φ(r)) - 2 cosh(φ(r)) + 2)
0
-2
-4
φ(r)
-6
-8
-10
-14
-16
1
1.5
2
2.5
3
100 points
200 points
500 points
1000 points
10000 points
1.5e+07
1e+07
5e+06
2
100 points
200 points
500 points
1000 points
10000 points
-12
2e+07
3.5
4
0
1
1.01
1.02
1.03
r/a
(a)
1.04
1.05
1.06
1.07
r/a
(b)
Figure 8.3: The potential (a) and the (dimensionless) integrand (b) of 7.25 as calculated inside a
cell using various grid sizes for Z = 1000, a/λB = 100, η = 0.001 and κa = 0.1. The horizontal axis
represents the distance from the center of the cell. The vertical axis represents the dimensionless
potential and the integrand in (a) and (b) respectively. Note that (invisible) tiny variations in the
potential may result in significant differences in the integrand and thus in the free energy.
On the left side of Fig. 8.3 the potential is plotted as a function of the distance to
the center of the cell. We can hardly distinguish the calculated potentials at various grid
56
Chapter 8. Various treatments of the correction term δΩ
sizes. Nevertheless, tiny variations in the potential may result in significant differences in
the integrand, which can be seen on the right side (b) of Fig. 8.3, which plots the integrand
as a function of the distance. This integrand differs strongly if one varies the number of grid
points. Therefore the associated values of integral (7.22) will differ as well, which introduces
an error in the calculated value for the free energy. The size of this error depends on the
number of grid points used.
In Figs. 8.4 and 8.5 we used the same conditions as in a specific sub figure of Fig. 8.1
with Z = 1000 and a/λB = 100. The figure spans many decades of the (η, κa) plane. We
see the phenomena which we described has giant impact on the regions of expected negative
compressibility as these region shrinks and in fact almost vanishes entirely when increasing
the number of grid points to 104 .
If we take a grid size between 500 and 1000 points it seems that we get a good agreement with Fig. 8.1. Also for many other values of Z and a/λB we saw the same phenomena
occur, the calculated negative compressibility seems to disappear for grid sizes large enough.
57
100
100
10
10
1
1
Κa
Κa
8.1. Numerical accuracy
0.1
0.01
-5
10
0.1
-4
10
-3
-2
10
10
-1
10
0.01
-5
10
0
10
-4
10
-3
(a) 500 points.
10
-1
10
0
(b) 2500 points.
100
10
10
1
1
Κa
Κa
10
η
100
0.1
0.01
10-5
-2
10
η
0.1
10-4
10-3
10-2
10-1
0.01
10-5
100
10-4
10-3
η
10-2
10-1
100
η
(c) 5000 points.
(d) 10000 points.
Figure 8.4: The regions of calculated negative compressibility for Z = 1000 and a/λB = 100.
It seems that these regions decrease if larger grids are used. The graphed lines corresponds to
Zn = 2cs , the line consisting of dots is probably an artifact of calculations.
gridpoints
100000
10000
1000
10-5
10-4
10-3
10-2
10-1
100
η
Figure 8.5: Horizontal cross sections out of Figure 8.4, various grid sizes are used. The filled
dots correspond to calculated negative compressibility. Here we fixed Z = 1000, a/λB = 100 and
κa = 0.1.
58
8.2
Chapter 8. Various treatments of the correction term δΩ
A new correction term
In Appendix A the correction term is recalculated via its definition in Eq. (7.24). We will
incorporate the found expression, Eq. (A.28), into existing equations. Similar to Eq. (8.1) but
b L,
using a different correction term, we now write the effective Hamiltonian as H = Ω cells + Ω
where Ωcells is given by Eq. (7.22) and
b L ≡ ΩL − δΩL = Ω
b D (Z ∗ , κ̄, η) + Ω
b 0 (Z ∗ , κ̄, η) +
Ω
N
X
i<j
c2 (Rij ; Z ∗ , κ̄).
V
(8.5)
Then we find the ideal gas contribution of the average salt density is given by
X
b D (Z ∗ , κ̄, η)
ρ̄α
βΩ
= (1 − ηc )
− 1),
ρ̄α (ln
V
cs
α=±
(8.6)
and also it can be shown that the self energy contribution is
b 0 (Z ∗ , κ̄, η)
ηc (Z ∗ n)2
η
n
2ρ̄+ ρ̄−
βΩ
=−
+
+ φ̄Z ∗
V
2(ρ̄+ + ρ̄− ) 1 − η ρ̄+ + ρ̄−
2
∗
Z nη
1
1
+
+
1 − η sinh 2φ̄ (1 − η) tanh φ̄
κ̄a
e
Z ∗ nλB 4π
1 + κ̄b
b
−1 .
−C
κ̄2
1 + κ̄a eκ̄b
(8.7)
b and G
b are given in Eqs.(A.12) and (A.13). Fortunately the pair potential
The values of C
stays DLVO-like but we need to redefine its weight by altering the effective charge. It reads
where
2
c2 (Rij ; Z ∗ , κ̄) = τbZ>
βV
λB
exp(−κ̄Rij )
,
Rij
b
b
τb = 1 + Γ − Υ(κ̄a) − 2CΘ(κ̄,
b) − 2GΘ(κ̄,
a)
(8.8)
(8.9)
which is of course only valid for r > 2a and in which Θ(κ̄, b) is defined in Eq. (A.20).
We have calculated the factor τb numerically at various salt concentrations κa and two
specific relative Bjerrum lengths via a/λB . In Fig. 8.6 the results are plotted. From this
figure we see that the calculated effective pair potential is predicted to become attractive at
low salinities and high η since τ is negative there. Furthermore we also see that this effect
becomes larger at small values of a/λB .
So what would this mean if true? This result is calculated using a simplified pair
distribution g2 (r). So suppose we would distribute the colloids randomly inside a solvent to
mimic this random pair distribution as well as possible. Then due to the interactions with
the salt ions, on average an effective attracting pair potential would show up between the
particles. If we now suppose that the system does not (yet) phase separate then Eqs. (8.6)
and (8.7) will probably still be valid since the local density does not change. The only way
for the system to minimize its free energy then is by forming small clusters of colloids. It
8.2. A new correction term
59
2
2
Κa=0.01
Κa=0.1
Κa=1
Κa=10
1
Κa=0.01
Κa=0.1
Κa=1
1
τ
0
τ
0
-1
-1
-2
-2
-3
10-5
10-4
10-3
10-2
η
(a)a/λB = 92.6.
10-1
100
-3
10-5
10-4
10-3
10-2
10-1
100
η
(b) a/λB = 200.
Figure 8.6: The calculated factor τb as a function of the packing fraction at different (relative)
Bjerrum lengths. Different salt concentration are taken into account by the spanning a wide range
in κa. At low salt and high densities, τb has the tendency to become negative. Here Z = 1200.
might even be possible that a full phase separation into two clearly distinct layers is favorable
if a negative compressibility is found. We did some calculations and indeed found a negative
compressibility for the parameters as used in Fig. 8.6. This negative compressibility we
found at sufficient high packing fractions. Nevertheless, this result should be doubted since
the compressibility in many cases does not become positive again in the dense limit, η → 1.
Maybe this is due to the fact that in this limit we the difference between the hard-sphere
pair potential which is used in the Gibbs-Bogoliubov approximation and the simplified pair
potential g2 (r) induces an over- or undercounting in some parts of the correction term.
Further research could probably point out what exactly causes this problem.
60
Chapter 8. Various treatments of the correction term δΩ
Chapter 9
Linear theory using larger
colloids
9.1
Introduction
In the previous section we fitted the more or less exact spherical solution from cell theory
to the linearized one using colloids with charge Z ∗ and radius a such that the potential
is described well both inside and outside the cells with radius b. Nevertheless, to be able
to do this we had to use that the cells cannot overlap since this would certainly destroy
the spherically symmetric potential. Therefore we are allowed to view the suspension of
cells as a new suspension of larger colloids. These new colloids have a yet undetermined
radius b and charge Q0 . The no-overlap condition between cells then reduces to applying
a simple hard-sphere repulsion between the new colloids. A disadvantage of this shows up
if the packing fraction of these new colloids becomes large. Then, in real suspensions, the
cells will start to overlap. This would correspond to overlapping colloidal cores in this new
view which is absolutely impossible. We conclude that we do not account for all possible colloidal configurations and therefore we should have doubts about the validity in the
high packing fraction regime. On the other hand, overlapping cells were also not allowed in
the previous method as in Chapter 7. Maybe both are equally bad at high packing fractions.
The main advantage of this view to the problem is that it simplifies the calculation of
the effective Hamiltonian. Namely, we can easily combine the free energy of the linear part
in this colloidal system with the free energy of the cells. Whereas in the theory described
in Chapter 7 a complicated correction term had to be extracted to compensate for an overcounting in the effective Hamiltonian we expect this correction term to simplify in this case.
Let us take a look at the equation for the one-colloid potential for these new colloids.
It is given by
φ1 (r) = λB
Q0 eκ̄b e−κ̄r
.
1 + κ̄b r
(9.1)
The average potential and the effective Debye screening parameter become using equations
62
Chapter 9. Linear theory using larger colloids
(a)
(b)
(c)
Figure 9.1: The colloids in suspension (a) are placed inside spherical cells (b) of radius b and
charge Q0 , both variables are yet undetermined. The system outside these cells can be viewed
as a new colloidal suspension containing colloids of radius b and charge Q0 (c). By choosing an
appropriate value for b we may describe this new colloidal suspension by the multi-centered linear
theory of Chapter 5. To describe the inside of this new colloids we use cell theory of Chapter 3.
(5.20) and (5.29)
−1
φ̄ = − sinh
κ̄ =
p
4πλB
ηc
Q0 n 1−η
e c ,
2cs
(9.2)
1/4
−ηc
)2
,
(Q0 n) + (2cs exp
1 − ηc
2
(9.3)
where ηc = (b/a)3 η is the packing fraction of the larger colloids, which is of course the same
as the packing fraction of the original cells. We now can apply the same reasoning again
to get continuity equations between the cell part and the linear part of our system like we
found in Eqs. (7.14) and (7.15). This time we may set a = b, we obtain for the potential
λB Q 0
[1 + ηc ζ(κ̄b)] ,
(9.4)
b 1 + κ̄b
where ζ(κ̄b) was defined in Eq.(7.16).The continuity equation for the electric field becomes
φcell (b) = φ̄ − tanh φ̄ −
0
λB Q
λB Q 0
= φcell (b) =
[1 − ηc ξ(κ̄b)] ,
(9.5)
2
b
b2
here, ξ(κ̄b) was defined in Eq.(7.17). Now we see why the term Q0 differs from Q; the
charge of the new colloids only reduces to Q0 in the low η limit. Since the used linear theory
does not incorporate a one-particle ion distribution Eq (5.30) which vanishes inside all other
colloids, also some net positive ion charge from the double-layers of other colloids entered
our colloid of interest. This is the effect which was described in Chapter 6 and makes that
the net charge of these new colloids is less then Q0 . We find Q = Q0 (1 − ηc ξ(κ̄b). Note
that in a non-linear theory we never would have a build up of ionic charge inside the new
colloids since then we can threat the colloidal cores as being hard-spheres, not allowing the
double-layer of colloids i to penetrate the colloidal core of colloid j, i 6= j.
The Wigner-Seitz cell limit for these larger new colloids is characterized by the condition that the electric field at the boundary of the larger colloids vanishes if ηc → 1. This
limit is properly reached at low screening lengths like we saw before in section 7.2 but we
9.2. Differences with earlier methods
63
see this limit is only reached because of the artifact in linear theory. Because Q → 0 in the
Wigner-Seitz limit, these new colloids(or cells) in systems which are close to this limit build
up a ghost-charge which cancels the negative charge Q0 . Therefore the net charge of the
new colloid vanishes and the electric field at the surface of the new colloid drops to 0.
What does this mean? In combination with ghost-charge we describe the potential and
ionic charge distributions outside the new colloids well since Eq. (9.5) just forces us to choose
a larger charge Q0 to match the electric field in the cell. Though, the grand potential in
the linear theory takes account for the degrees of freedom and electrostatic energy of all the
ions in the system, also those inside the colloidal cores. Now a problem arises since these
ghost-ions are actually not there of course. Nevertheless, in the effective Hamiltonian (5.39)
we still account for them. We conclude that we have to take care close to the Wigner-Seitz
limit. Contributions of the ghost-ions must then be taken into account.
9.2
Differences with earlier methods
The scheme to solve the continuity conditions (9.4) and (9.5) is similar to the one we used
before in Chapter 7. Now let us take a look at the predicted cell sizes using both the method
described here and the method used in Chapter 7.
10
10
κa=0.01
κa=0.1
κa=1
κa=10
8
κa=0.01
κa=0.1
κa=1
κa=10
8
b/a
6
b/a
6
4
4
2
2
0
10-6
10-5
10-4
10-3
η
(a)
10-2
10-1
100
0
10-6
10-5
10-4
10-3
η
10-2
10-1
100
(b)
Figure 9.2: The calculated cell radius b/a for a system with charge Z = 1000 and colloidal size
a/λB = 92.6. The results of the method used in Chapter 7 (a) and the method used here (b), are
(at least for this set of parameters), indistinguishable.
As we see, this method gives agreement on the values for b. This gives us a hint that
the calculated cell radius does not strongly depend on the method used. There might be
differences for different parameters, this could be investigated later.
9.3
The grand potential and free energy
The fact that we introduced new colloids with a radius b would suggest that the correction
term in the equation for the effective Hamiltonian, Eq. (7.21) will simplify since the cells
are now of equal size as the (large) colloids and we thus do not have a volume between the
colloidal surface and the cell’s boundary. The charges inside this colume were taken into
account by the grand potential Ωcell of the cell model as well as by the grand potential
64
Chapter 9. Linear theory using larger colloids
ΩL of the multi-centered linear model. Since this volume vanishes we are interested in the
remaining expression for the correction term.
It might seem that a correction term will not be necessary because of this vanishing
volume between the colloidal surface and the cells boundary. This is not entirely true since
the grand potential of the multi-centered linear theory accounts for electrostatic energy of
the larger colloids in the electrostatic potential field. The cell-model grand potential Ω cell
accounts for the electrostatic energy of all the charges inside the cells in a potential field,
effectively it thus also includes the electrostatic energy of the larger colloids. We conclude
that there is a double counting for the colloidal charge Q0 and may account for this by
introducing an intuitive correction term to the linear part of the grand potential
φcell (b)Q0
βδΩ
= −n
.
(9.6)
V
2
A more mathematical way to calculate the correction term would be to integrate the
linearized effective Hamiltonian, using the equilibrium ionic density profiles, inside the cells.
This is what we did to calculate the correction term in Appendix (A). It showed that
corrected Eq. (8.5) describes the grand potential in the linearized part after taking into
account for the correction term (A.28). Now we adjust the colloidal and cell parameters to
a → b, b → b, Z ∗ → Q0 to fit the new model, where the cell size is equal to the colloidal size
and the colloidal charge becomes Q0 . We obtain H = Ωcells + Ω̆L , where Ωcells is given by
Eq. (7.22) and by including the correction term into ΩL we find
Ω̆L ≡ ΩL − δΩ = Ω̆D (Q0 , κ̄, ηc ) + Ω̆0 (Q0 , κ̄, ηc ) +
N
X
V˘2 (Rij ; Q0 , κ̄),
(9.7)
i<j
where
X
β Ω̆D (Q0 , κ̄, ηc )
ρ̄α
= (1 − ηc )
− 1),
ρ̄α (ln
V
cs
α=±
β Ω̆0 (Q0 , κ̄, ηc )
−ηc (Q0 n)2
ηc
2ρ̄+ ρ̄−
n
=
+
+ φ̄Q0
V
2(ρ̄+ + ρ̄− ) 1 − ηc ρ̄+ + ρ̄−
2
1
1
Q0 nηc
+
+
,
1 − ηc sinh 2φ̄ (1 − ηc ) tanh φ̄
(9.8)
(9.9)
and
exp −κ̄r
2
.
β V˘2 (Rij ; Q0 , κ̄) = τ̆ Z>
λB
r
Now the DLVO factor becomes
τ̆ = 1 + Γ − Υ(κ̄b) − 2(C̆ + Ğ)Θ(κ̄, b).
(9.10)
(9.11)
The densities C̆ and Ğ are given by
Q0 n
φ̄
(1 +
);
2
tanh φ̄
Q0 n (1 + φ̄)
Ğ =
.
1 − ηc
2
C̆ =
(9.12)
(9.13)
9.3. The grand potential and free energy
65
So what makes Eq. (9.7) much more complicated than the grand potential which would
follow from the more or less intuitive correction term in Eq. (9.6)? By looking closely to
the definition of the correction term we can discover this. If calculating the correction term
we integrate the linearized effective Hamiltonian supplied with the densities ρ± (r) which
minimize it over the inside of the colloid. By doing this we automatically account for the
entropy and electrostatic energy of the ’ghost ions’ inside the colloids, which in a full theory
would never appear.
We might choose not to integrate over the colloids interiors since in the full multicentered theory of Chapter 5 we did not do this either. We may do this by altering the
definitions of integrals (A.3) and (A.4) such that the colloidal cores are excluded from
integrations. Then we find another expression for the correction term which not accounts
for ghost-charges in the colloids of radius b. Nevertheless, we incorporate this correction
term in ΩL to find the for (corrected) linear part of the effective Hamiltonian
Ω̈L = ΩD (Q0 , κ̄, ηc ) + Ω̈0 (Q0 , κ̄, ηc ) +
N
X
V2 (Rij ; Q0 , κ̄).
(9.14)
i<j
In this expression ΩD (Q0 , κ̄, ηc ) is just the original linear Donnan term of Eq. (7.27), and
also V2 (Rij ; Q0 , κ̄) is defined as in Eq. (7.28). The only term which alters is the self-energy
term, which is now defined via
2ρ̄+ ρ̄−
β Ω̈0 (Q0 , κ̄, ηc )
ηc
n
=
+ φ̄Q0 .
V
1 − ηc ρ̄+ + ρ̄−
2
(9.15)
Using these expressions we did not find any numerical evidence for the existence of large
scale attractions at typical parameters for which we would expect it, the compressibility was
found to be negative nowhere. A remark to be made is that due to problems described in
Section 8.1 we could not check extremely low density regimes.
The numerical results for τ̆ and the free energy using the more complex equations, 9.7
have the same characteristics as described in Section (8.2). We find negative compressibility
above certain densities but we do not see the compressibility reaching positive values again
at large packing fractions, (η ≈ 0.7). Further research might probably clarify this.
66
Chapter 9. Linear theory using larger colloids
Chapter 10
Conclusion
In this thesis we made predictions about colloidal suspensions by applying the PoissonBoltzmann equation. We used Wigner-Seitz cell theory and a multi-centered linear screening
theory to determine the characteristics of suspensions containing charged spherical colloids.
We discovered that the spherical symmetry obtained by the Wigner-Seitz model could well
be used to make some calculations on a special suspension where the porousity of the colloids
turned out to be of great importance. We found that the charge of these (silica) colloids,
under certain circumstances, varies heavily when changing the salt concentration of the solvent, which was oil. It is also possible to observe a change from negative to positive charge
at rising salt concentrations.
In later chapters, by combining the cell model with the linear screening model, we tried
to find evidence for long-range attractions between (non-porous) spherical colloids. We discovered that numerical computations can be extremely sensitive to the number of grid points
used to solve the Poisson-Boltzmann equation inside the cells. This complicates calculations
on the phase behaviour of these suspensions since they may take lots of computational time.
To combine the cell model and the linear model we had to take in account for correction
terms which excluded the doubly counted entropic and electrostatic contributions to the
effective colloidal Hamiltonian by both models. Although under certain circumstances the
effective pair potential between the colloids was found to become attractive we could not
explain a fluid-fluid phase seperation observed in for example Ref. [1]. The reason was that
while using these models the colloidal compressibillity was predicted to become negative
even in the high colloidal packing fraction limit which is unphysical. We have good hope
that future research can figure out what is precisely going on here.
68
Chapter 10. Conclusion
Chapter 11
Acknowledgments
Mijn dank gaat uit naar René van Roij voor zijn fijne begeleiding en naar anderen die een
prestatie- en/of sfeerbevorderende invloed op mij gehad hebben tijdens mijn afstuderen.
Toen ik in 2003 als tweedejaars student in Utrecht belandde trof ik tussen de colleges door
een aantal medestudenten aan die elkaar al wat beter kenden. Wat destijds begon als een
groep om de pauzes mee te vullen, kreeg vanaf een zeker sporttournooi de naam QSSP en
stond garant voor stapels afwas en de sterk daaraan gecorreleerde gezelligheid. Daarnaast,
tegen het einde van mijn studie, heb ik het geluk gehad dit onderzoek bij René van Roij te
kunnen doen. Alhoewel we gestart waren met een hele ruime onderzoeksvraag kristalliseerde
het geheel plotseling toch uit naar hele specifieke vraagstukken. Dit was een mooie ervaring.
Ik zie dankzij dit alles uit naar de tijd die komen gaat nu blijkt dat ik als promovendus deze
omgeving voorlopig nog niet hoef te verlaten.
70
Chapter 11. Acknowledgments
Appendix A
The correction term
A.1
Finding the cell-contribution from the linear screening theory
We combine cell-theory with the linearized theory to obtain an expression for the grand
potential of the whole system. By doing that we get an over counting in this potential
since the cells are doubly counted. At distances closer then b to the colloid cores the grand
potential can best be calculated by cell theory. We want to extract the contribution of the
linearized theory in this part of the system. The correction term we will call βδΩ. To be
able to calculate this term we will frequently use the following equations, which can be found
in Eqs. (5.15) and (5.16):
−η
ρ̄+ − ρ̄− = −2cs exp
sinh φ̄ = Z ∗ n
1−η
−η
−Z ∗ n
.
ρ̄+ + ρ̄− = 2cs exp
cosh φ̄ =
1−η
tanh φ̄
(A.1)
(A.2)
Furthermore, we define:
Z
Z
dr =
cells
i=1
dr =
colloids
N Z
X
N Z
X
i=1
dr,
(A.3)
dr.
(A.4)
|r−Ri |<b
|r−Ri |<a
We recall that the correction term was defined in Eq. (7.24) as
βδΩ =
XZ
α=±
Z
1
ρ̄α
(ρl,α (r) − ρ̄α )2
+
dr ρ̄α (ln
dr [ρL (r) + q ∗ (r)] φL (r)
− 1) +
cs
2ρ̄α
2 cells
cells
72
Appendix A. The correction term
We use the fact that since we are in the linearized regime there yields
φ̄) − βW (r) to see
ρl,± (r)−ρ̄±
ρ̄±
= ∓(φL (r)−
X (ρl,α (r) − ρ̄α )2
1
1
= − (φL (r) − φ̄)(ρl,+ (r) − ρ̄+ ) + (φL (r) − φ̄)(ρl,− (r) − ρ̄− )
2ρ̄
2
2
α
α=±
1
− βW (r)((ρ+ (r) − ρ̄+ ) + (ρ− (r) − ρ̄− ))
2
(A.5)
and if we also include that ρ̄+ − ρ̄− = Z ∗ n (5.18) we find
(ρl,α (r) − ρ̄α )2
2ρ̄α
1
(φL (r) − φ̄)(−ρL (r) + Z ∗ n)
2
βW (r)
+
(ρ̄+ + ρ̄− − (ρ+ (r) + ρ− (r)))
2
1
1
1
= − (φL (r)ρL (r)) − φ̄(Z ∗ n − ρL (r)) + Z ∗ nφL (r)
2
2
2
βW (r) Z ∗ n
βW (r)
−
(ρ+ (r) + ρ− (r)).
−
2 tanh φ̄
2
=
(A.6)
Now we are at the point to make some serious simplifications to the correction term. We
find
XZ
ρ̄α
βδΩ =
dr ρ̄α (ln
− 1)
cs
α=± cells
Z
Z
Z
1
1
1
dr(Z ∗ n − ρL (r)) + Z ∗ n
drφL (r)
dr(φL (r)ρL (r)) − φ̄
−
2 cells
2 cells
2
cells
Z
1
+
dr [ρL (r) + q ∗ (r)] φL (r)
2 cells
Z
Z
1
1
Z ∗n
−
−
drW (r)
drW (r)(ρ+ (r) + ρ− (r)).
2 cells
tanh φ̄ 2 cells
We prefer to denote all integrals in terms of the potential instead of charge densities. Therefore we use
ρL (r) = ρ̄+ 1 − (φL (r) − φ̄) − βW (r) − ρ̄− 1 + (φL (r) − φ̄) − βW (r)
= (1 − βW (r))(ρ̄+ − ρ̄− ) + (ρ̄+ + ρ̄− )(φL (r) − φ̄)
N
X
−η
∗
cosh φ̄(− tanh φ̄ +
φ1 (r − Ri ))
= (1 − βW (r))Z n − 2cs exp
1−η
i=1
N
Z ∗n X
φ1 (r − Ri ),
= W (r)Z n +
tanh φ̄ i=1
∗
where we used Eq. (7.4) for φL (r). Similarly we find
ρ+ (r) + ρ− (r) =
N
X
Z ∗ n(W (r) − 1)
+ Z ∗ n tanh φ̄ − Z ∗ n
φ1 (r − Ri ).
tanh φ̄
i=1
(A.7)
A.1. Finding the cell-contribution from the linear screening theory
73
Since W (r) vanishes outside the colloids, some cell integrals become integrals over colloidal
cores. Some integrals can be calculated instantly since their integrand is a constant. It can
be shown that the correction term becomes:
Z
X
1
ρ̄α
− 1) +
βδΩ = N
dr [q ∗ (r)] φL (r)
Vcell ρ̄α (ln
c
2
s
cells
α=±
−
+
+
+
=
1
1
N Vcell Z ∗ nφ̄ + N Vcell Z ∗ n(φ̄ − tanh φ̄)
2
2
Z
N
∗
X
Z nφ̄
1
dr
φ1 (r − Ri )
)
( Z ∗n +
2
2 tanh φ̄ cells i=1
N Z ∗η
1
1
+
η − 1 sinh 2φ̄ (1 − η) tanh φ̄
Z
N
Z ∗n
(1 + φ̄) X
φ1 (r − Ri )
dr
1 − η colloids
2
i=1
X
ρ̄α
1
N Vcell ρ̄α (ln
− 1) − N Vcell Z ∗ n tanh φ̄
cs
2
α=±
+ I1 + I2 + I3 ,
where
I1 =
b
I2 = C
b
I3 = G
Z
1
2
Z
Z
drq ∗ (r)φL (r),
(A.8)
(A.9)
cells
dr
cells
N
X
dr
colloids
φ1 (r − Ri ),
(A.10)
i=1
N
X
φ1 (r − Ri ).
(A.11)
i=1
b and G
b are charge densities given by:
The constants C
∗
b = Z n (1 + φ̄ ),
C
2
tanh φ̄
∗
b = Z n (1 + φ̄) .
G
1−η
2
(A.12)
(A.13)
Let us focus on the integral in I2 . We may write
Z
dr
cells
N
X
i=1
φ1 (r − Ri ) = N
Z
drφ1 (r) + 2
r|<b
XZ
i<j
drφ1 (r − Rij ).
r|<b
Now we apply the explicit expression for the one-particle potential in our linearized system,
Eq. (7.4), and use that this potential vanishes at distances smaller then the colloidal radius.
74
Appendix A. The correction term
N
Z
Z
r=b
e−κ̄r
r
r=0
r=a
κ̄a e
−N Z ∗ λB
(1 + κ̄r)e−κ̄r 1 + κ̄a
r=b
κ̄a
N Z ∗ λB 4π
e
1 + κ̄b
−1
κ̄2
1 + κ̄a eκ̄b
drφ1 (r) = −N Z ∗ λB
r|<b
=
=
eκ̄a
1 + κ̄a
dr4πr2
(A.14)
It is relatively easy to prove that the Fourier transform of the one-particle potential is
φ1 (k) = −Z λB
1
φ1 (r − Rij ) =
(2π)3
Z
∗
eκ̄a
1 + κ̄a
κ̄2
4π
,
+ k2
(A.15)
and we may also use that
dkφ1 (k)e−ik·Rij eik·r
(A.16)
to obtain that
Z
drφ1 (r − Rij ) = A
r|<b
Z
dr
r|<b
Z
dke−ik·Rij
eik·r
,
κ̄2 + k2
(A.17)
κ̄a e
4π
∗
Z
λ
with A ≡ − (2π)
3
B 1+κ̄a . When we write k and r in their polar coördinates representation, we can perform the θ and φ integral in both integrals, leaving us thus a double
one-dimensional integral:
Z
Z b
eikRij − e−ikRij
k2
sin(kr)
drφ1 (r − Rij ) = 2πA
dk 2
dr4πr2
2
k
+
κ̄
ikR
kr
ij
r|<b
0
0
Z ∞
Z
b
k 2 eikRij
sin(kr)
= 2πA
dk 2
.
dr4πr2
k + κ̄2 ikRij a
kr
−∞
Z
∞
Since the integrand is even in k. We see the integrand has poles at k = ±iκ̄ and they are
of order 1. At k = 0 we do not have a pole since for small kr we know that sinkrkr = O((kr)2 ).
To calculate the integral we promote k to a complex number and make a contour
integration in the positive imaginary part of the k plane. In particular, for a regular function
χ(k) which decays exponentially with increasing ik we have
Z
so
Z
∞
dk
−∞
χ(k)
χ(κ̄i)
χ(k) = 2πi
=
2πiRes
,
κ̄2 + k 2
κ̄2 + k 2 k=iκ̄
2iκ̄
−κ̄2 e−κ̄Rij
drφ1 (r − Rij ) = 4π iA
−2iκ̄2 Rij
0|r|<b
2
Z
b
0
dr4πr2
sin(iκ̄r)
iκ̄r
(A.18)
A.1. Finding the cell-contribution from the linear screening theory
e−κ̄Rij
Rij
Z
b
75
sinh(κ̄r)
κ̄r
0
r=b
eκ̄a
4πr3
e−κ̄Rij ∗
Z λB
(κ̄r cosh(κ̄r) − sinh(κ̄r))
= −
3
Rij
1 + κ̄a (κ̄r)
r=0
∗ κ̄a 2
Z e
e−κ̄Rij
= Θ(κ̄, b)
λB
.
1 + κ̄a
Rij
= (2π)3 A
drr2
(A.19)
Here we used that
Θ(κ̄, b) = −
4π(1 + κ̄a) b3
(κ̄b cosh(κ̄b) − sinh(κ̄r))
Z ∗ eκ̄a (κ̄b)3
(A.20)
The result (A.19) of the integral we will later put back in Eq. (A.10) to obtain the full
expression for I2 . For I3 we repeat this method, but then we integrate over the radial
interval [0, a]. The expression for I3 therefore becomes linear in Θ(κ̄, a). We are left with
the integral in I1 . We write
I1
=
=
=
+
1
2
1
2
Z
Z
drq ∗ (r)φL (r)
cells
1
drq (r)(φ̄ − tanh φ̄) +
2
cells
∗
Z
drq ∗ (r)
cells
1
(tanh φ̄ − φ̄)N Z ∗
2
N N
1 X X −Z ∗
δ(|r − Rj | − a)φ1 (r − Ri ),
2 i=1 j=1 4πa2
N
X
φ1 (r − Ri )
i=1
(A.21)
where the case i = j is a special one, since we then integrate the potential of a colloid over
its own surface. If we split off this part of the double sum we can easily come to
I1
=
+
1
N (Z ∗ )2 λB 1
(tanh φ̄ − φ̄)N Z ∗ +
2
2a
1 + κ̄a
N Z
∗
X
−Z
dr
δ(|r| − a)φ1 (r − Rij ),
2
4πa
i<j cell i
(A.22)
R
where we now choose r = 0 at the center of each cell for each new i. Let us call I 1 the
second,integral, part of the last equation. To solve this part we go to Fourier space again.
Similar like before, we find
R
I1 =
−Z ∗ A X
4πa2 i<j
Z
dr
cell i
Z
dk
1
ik·Rij −ik·r
e
e
δ(|r|
−
a)
,
k2 + κ̄2
(A.23)
where we choose A to be the same constant as before. Now we again write both k and r in
76
Appendix A. The correction term
their polar coördinate representation and work out the angular integrals to find:
R
I1
N
=
−Z ∗ A X
4πa2 i<j
= −2πZ ∗ A
Z
∞
0
N Z
X
i<j
= −4πZ ∗ A
N Z
X
i<j
k 2 (eikRij − e−ikRij )
dk 2
k + κ̄2
2ikRij
∞
dr4πr2 δ(r − a)
dk
k 2 (eikRij − e−ikRij ) sin ka
k 2 + κ̄2
2ikRij
ka
dk
sin ka eikRij
,
k 2 + κ̄2 2iaRij
−∞
∞
−∞
Z
sin kr
kr
(A.24)
where we had to use the fact that this integral is even in k. Again we have poles at k = ±iκ̄.
We apply the same contour integration around k = iκ̄ to find
R
I1
2
∗
= −8π iZ A
N
X
sin(iκ̄a) e−κ̄Rij
i<j
2iκ̄
2iaRij
N
= −2π 2 Z ∗ A
= Υ(κ̄a)
in which
Υ(x) =
sinh(κ̄a) X e−κ̄Rij
κ̄a
Rij
i<j
Z ∗ eκ̄a
1 + κ̄a
2
λB
N
X
e−κ̄Rij
i<j
Rij
,
1 + x 1 − e−2x
.
x
2
(A.25)
(A.26)
This is exactly the same as what was found in [6] and [7]. We now have found all terms
which define the correction to the grand potential. Combining all the terms yields
βδΩ
N
=
+
+
+
+
+
X
ρ̄α
1
Vcell ρ̄α (ln
− 1) − Vcell Z ∗ n tanh φ̄
c
2
s
α=±
Z ∗η
1
1
+
η − 1 sinh 2φ̄ (1 − η) tanh φ̄
κ̄a
∗
1 + κ̄b
e
b Z λB 4π
C
−
1
κ̄2
1 + κ̄a eκ̄b
∗
eκ̄a 1 + κ̄a
b Z λB 4π
−
1
G
κ̄2
1 + κ̄a eκ̄a
1
(Z ∗ )2 λB 1
(tanh φ̄ − φ̄)Z ∗ +
2
2a
1 + κ̄a
N ∗ κ̄a 2
X
λB e−κ̄Rij
Z e
(Υ(κ̄a) + 2CΘ(κ̄, b) + 2GΘ(κ̄, a))
.
1 + κ̄a
N Rij
i<j
(A.27)
A.2. Small cell diameters
77
∗
3
We can simplify this equation a bit by using that tanh φ̄ = − ρ̄+Z+nρ̄− and recall that ηc = η ab 3
to find
X ηc
βδΩ
ρ̄α
ηc − 1 (Z ∗ )2 n
=
ρ̄α (ln
− 1) +
N
n
cs
2 ρ̄+ + ρ̄−
α=±
−
+
+
1
(Z ∗ )2 λB 1
(φ̄)Z ∗ +
2
2a
1 + κ̄a
∗
1
Z η
1
+
η − 1 sinh 2φ̄ (1 − η) tanh φ̄
κ̄a
∗
1 + κ̄b
e
b Z λB 4π
C
−
1
κ̄2
1 + κ̄a eκ̄b
+ (Υ(κ̄a) + 2CΘ(κ̄, b) + 2GΘ(κ̄, a))
A.2
N ∗ κ̄a 2
X
λB e−κ̄Rij
Z e
.
1 + κ̄a
N Rij
i<j
(A.28)
Small cell diameters
In the limit b → a the cells shrink to the size of the colloid. If we take a look at the correction
term, we see it does not vanish in this limit. This is due to various reasons. The correction
term includes the interactions between the colloidal charge and all the ions in the system.
These interactions are included because the cell theory also accounts for this. Furthermore,
as we see in Chapter 5 the used linear theory does not predict all ionic charge to be outside
the colloidal cores. This is an unwanted effect since we defined the colloidal cores to be hard
spheres. As we see there, this effect becomes important for low κa and high densities. By
definition, the correction term takes account for the ionic densities inside the spheres and
therefore subtracts contribution of these ’ghost’-ions to the grand potential.
We might therefore say that this correction term would also increase the correctness of
the grand potential in the case of a purely linear theory (so without cell parts).This might be
done by setting the cell radius equal to the colloid radius. Though, since the correction term
eliminates the grand potential-contribution of all charge inside the cells, also the contribution
of the negative charge on the colloidal surface gets eliminated. Therefore it is suggested to
pick a cell radius of a − δ, for positive δ, and consider the limit δ → 0.
78
Appendix A. The correction term
Bibliography
[1] B.V.R. Tata, P.S. Mohanty and M.C. Valsakumar, Direct Evidence for Long-range Attraction between Like-Charged Colloids.
[2] B. Derjaguin and L. Landau, Acta Physicochim., URSS 14, 633 (1941).
[3] J.W. Verwey and J.T.G. Overbeek, Theory of the stability of lyotropic colloids (Elsevier,
Amsterdam, 1948).
[4] K. Ito, H. Hiroshi, and N. Ise, Science 263, 66 (1994).
[5] B.V.R. Tata, E. Yamahara, P.V. Rajamani, and N. Ise, Phys. Rev. Lett. 78, 2660 (1997).
[6] B. Zoetekouw, Phase behavior of charged colloids. Utrecht 2006.
[7] B. Zoetekouw and R. van Roij, Volume terms for charged colloids: a grand-canonical
treatment, Phys. Rev. E 73, 021403 (2006); cond-mat 0510226.
[8] B. Zoetekouw and R. van Roij, Nonlinear screening and gas-liquid separation in suspensions of charged colloids, Phys. Rev. Lett 97, 258302 (2006); cond-mat 0611290.
[9] S. Bravo Yuste, M. López de Haro and A.Santos, Phys. Rev. A 43,5418 (1991).
[10] R. van Roij, Lecture notes on soft condensed matter theory (2007).
[11] S. Alexander, P.M. Chaikin, P. Grant, G. Morales, P. Pincus and D. Hone, J. Chem.
Phys. 80, 5776(1984).
[12] H.H. von Grünberg, R. van Roij, and G. Klein, Europhys. Lett. 55, 580 (2001).
[13] M.N. Tamashiro and H. Schiessel, J. Chem. Phys. 119, 1855 (2003).
[14] Y. Levin, M.C. Barbosa, and M.N. Tamashiro, Europhys. Lett. 41, 123 (1998).