Univerza v Ljubljani
Fakulteta za matematiko in fiziko
Electrical double layer
Seminar
Jasna Urbanija
Supervisor: prof. dr. Veronika Kralj-Iglič
Ljubljana, May 2007
1
Introduction
Within the theory of the electrical double layer, the electrostatic properties of a system
composed of charged surfaces and ionic solution are studied. Development of the electrical
double layer theory began at the beginnings of the 20th century, with the intention to
understand the stability of colloids and suspensions. Later on it became evident, that
the double layer theory is applicable to many different systems and is essential in understanding different chemical and biological processes. Therefore since the beginnings of
the theory development much effort has been put into improvement of the original GouyChapman model [2], [3], which is subjected to many simplifications and restrictions.
In this seminar I will present a possible extension of the simplest double layer theory.
First the interaction between two flat charged plates in the solution with point-like ions
will be presented, then, I will describe a system with divalent rod-like counterions where
internal charge distribution will be taken into account. The free energy of such a system
will reveal important consequences of taking into account the internal charge distribution
as well as the shape of particles. For completeness, an overview of the forces acting in
a double layer system will be presented, followed by a bit of historical overview. At the
end a possible application of the described theory in medical research will be presented
together with a brief discussion on the topic. We should bear in mind that in our case
this work is aimed towards the understanding of interactions between negatively charged
membranes and protein molecules with spatially distributed positive charge.
1
2
Forces between surfaces
Forces in a colloidal system are of different origin (electrostatic, solvation or structural,
undulation, steric, hydrophobic and hydrophilic,...). Taking into account all of them would
make computations of particle interactions too complicated. Therefore it is essential to
estimate the importance of each interaction force and take into the calculus just those that
are the most relevant for our system. When we study the long-range interactions, taking
into account the electrostatic forces would suffice, meanwhile at very short distances (few
nanometres) other forces become important and finally prevail.
Let us first take a look at the different electrostatic forces between molecules in the
gas phase. The strongest of all physical forces we shall be considering (stronger even
than most chemical binding forces) is the Coulomb ion-ion interaction (see table 1). The
potential energy of the Coulomb interaction falls with the first power of the distance
between two charges and thus the Coulomb force appears to be very long-ranged. But
as we shall see later in solutions the electric field can be shielded by the presence of
other charges and therefore decays more rapidly than in the case of isolated ions. At
large distances the decay can usually be described by an exponential function of the
distance, thus making Coulomb interactions effectively of much shorter range. This effect
of screening will become evident through the study of the double layer interaction in the
following sections. If the molecules have a multipole structure this should be taken into
account.
Table 1: Electrostatic interaction potentials between molecules in a gas phase.
Name
Coulomb ion − ion
Potential
q1 q2
4πε0 R
ion − dipole
q1 m2 Cosθ
4πε0 R2
dipole − dipole
m2 f(θφ)
− m14πε
3
0R
ion − induced dipole
− 2(4πε01)2 R4
αq 2
dipole − induced dipole
m2 α(1+3 Cos2 θ)
(Debye)
− 12(4πε0 )2 R6
Thermally averaged
m2 m22
dipole − dipole (Keesom) − 3(4πε01)2 kT
R6
Debye and Keesom interactions are both attractive and are two of three forces that
form van der Waals interaction (see table 1). A third one which is usually also predominant
(except for small highly polar molecules, such as water) is called dispersion force, and
2
could also be called spontaneous dipole - induced dipole interaction in consistence with
the notation of electrostatic forces in the table 1. Electrons by their motion around
the nucleus create instantaneous dipoles. These dipoles generate a field at the site of a
chosen molecule which responds immediately. Because Fritz London first analyzed this
mechanism in 1930 it is called the London dispersion interaction in his honor. It can be
described by quantum mechanical terms. Usually a Hamaker constant is connected with
a description of this force.
Until now we only considered the interactions of molecules in a vacuum or air. The
problem arises when molecules interact via a medium. In this case the many-body character of interaction also should not be neglected. Also the van der Waals forces are not
pairwise additive. In the case of electrostatic interactions between particles in electrolyte
solutions the medium is described as a dielectric, by introducing the dielectric constant
into the electrostatic equations. In the case of van der Waals forces between colloidal particles Lifshitz introduced a unified theory in 1950 which takes into account the many-body
effect and the retardation effect due to finite velocity of propagation of electromagnetic
waves. The non-aditive property of van der Waals forces is particulary important in the
interactions between large particles and surfaces in a medium. The van der Waals pair
potential falls with the sixth power of distance between interacting particles, but due
to retardation effects, with increasing separation, the dispersion energy of the interaction between two atoms begins to decay even faster, approaching −1/r7 dependence at
r > 100nm. The major limitation of the Lifshitz theory is that it treats both the surface
and the intervening solvent medium as structureless continuums and consequently does
not encompass molecular effects such as solvation forces and surface structural effects.
When two surfaces or particles approach closer than a few nanometres, continuum
theories of attractive van der Waals and repulsive double-layer forces often fail to describe
their interaction. Other forces come into play, that can be either attractive, repulsive or
oscillatory. Oscillatory solvation forces arise whenever liquid molecules are ordered into
quasi-discrete layers between two surfaces. In addition, surface-solvent interactions can
induce positional or orientational order in the adjacent liquid and give rise to a monotonic
solvation force which can be attractive or repulsive. Additional forces may arise from
electrostatic ion-binding and ion-correlation effects and from molecular ’bridging’ effects.
Solvation forces such as the hydrophobic force depend on the properties of the intervening
medium and also on the chemical and physical properties of the surfaces (whether they
are hydrophilic or hydrophobic). This is especially important in the study of interactions
of cell membranes. The cell membranes are basically composed of phospholipids, which
have a hydrophobic and a hydrophilic part.
The hydrophobic interaction is a strong attraction between hydrophobic molecules in
water. It is often stronger than attraction between the hydrophobic molecules in free
space. Because of its strength it was originally believed that some sort of hydrophobic
bond was responsible for this interaction. It is now acknowledged that no bond is associated with this mainly entropic phenomenon, which arises primarily from the rearrangements of H-bond configurations in the overlapping solvation zones as two hydrophobic
species approach each other, and could be of much longer range than any typical bond.
Hydrophilic effects can be recognized in propensity of certain molecules and groups
to be water soluble and to repel each other strongly in water, in contrast to the strong
3
attraction exhibited by hydrophobic groups. Strongly hydrated ions and zwitterions are
hydrophilic, but also some uncharged and even non-polar molecules can be hydrophilic if
they have an appropriate geometry and if they contain electronegative atoms capable of
associating with the H-bond network in water.
Thermal fluctuation or protrusion forces occur in the case when surfaces have thermally
mobile surface groups as is the case with the cell membranes. When two such surfaces
approach each other their protrusions become increasingly confined into a smaller region of
space, resulting in a repulsive force. Steric forces occur when the molecules that compose
the surface (polymers) detach from a surface and dangle out into the solution where they
are thermally mobile. On approach of another surface the entropy of confining these
dangling chains again results in a repulsive force. Such forces are essentially entropic or
osmotic.
In this paper we shall concentrate on the Coulomb forces between two charged surfaces
in ionic solution. Even this simple model is not yet completely revealed.
4
3
A bit of double layer history
The first approximative theory for the electrical double layer was given by Gouy [2] and
Chapman [3]. Within this theory an interaction between infinite homogenously charged
wall (described by surface charge density σ) and infinite electrolyte solution extending
on one side of the wall is discussed. Ions of the electrolyte are described by point-like
charges of both signs in water medium (which is defined by the permittivity constant
ε). According to this model the electric potential function and the corresponding average
charge distribution are computed in the neighborhood of the charged surface. We begin
with the Poisson equation for a planar surface:
ρ
d2 ψ(x)
=−
,
2
dx
εε0
(3.1)
where ρ(x) represents the charge density in dependence of distance x from the wall and
ψ(x) electric potential. We assume that charges are distributed according to the Boltzmann law:
X
ρ(x) = −NA e
Zi ci e−Zi eψ(x)/kT ,
(3.2)
i
NA is Avogadro’s number, ci is the bulk ionic concentration, Zi corresponding ionic valency, in our case, for 1:1 electrolyte (Z+ = 1, Z− = −1, c+ = c− = c(∞)) we arrive at
the Poisson-Boltzmann equation, for a symmetrical ionic solution:
d2 ψ(x)
2NA ec
eψ(x)
=
sinh(
),
dx2
εε0
kT
(3.3)
The solution of above equation gives the shape of the electric potential near the charged
surface (see [1] for the step by step integration):
"
y = 2 ln
ey0 /2 + 1 + ( ey0 /2 − 1) e−ξ
ey0 /2 + 1 − ( ey0 /2 − 1) e−ξ
#
(3.4)
where:
2ne2 Z 2
(3.5)
εε0 kT
and y0 is the value of y on the wall. The equation 3.4 seems to be complicated, but in
fact it is similar to an exponential decay as could be seen from the figure (Fig. 1 left).
Inserting the given potential 3.4 into the density equation 3.2 we get the corresponding charge distribution near the charged surface (Fig. 1 right). We see that near the
surface the concentration of counterions is much higher than its bulk value, meanwhile
the concentration of co-ions is diminished. An electric double layer is created, one layer
is represented by the charged diffuse surface and the other by the excess of the opposite
charge extending into the solution.
The whole system should be electrically neutral, which is described by the electroneutrality condition:
y = Zeψ/kT
σ=−
Z ∞
0
ρ dx = εε0
ξ = κx
Z ∞ 2
dψ
0
dx2
5
κ2 =
Ã
dx = −εε0
dψ
dx
!
.
x=0
(3.6)
Ψ HxL @mVD
ΡHxL
8
60
Ρ+ HxL
6
40
4
20
2
1
2
3
4
Ρ- HxL
Κx
1
2
3
4
Κx
Figure 1: Potential drop and distribution of the density of charges near a flat negatively
charged surface. The Debye length is 1/κ = 0.3nm, corresponding the bulk concentration
of both ions (c = 1M) and the surface charge σ = −0.16 As/m2 . Density distribution
approaches its bulk value (set to ρ = 1) with increasing distance.
Eq. 3.6 gives the relation between the surface charge density and potential ψ0 (or y0 ) on
the surface.
For small values of ψ, equation 3.3 can be linearized:
d2 ψ(x)
= κ2 ψ
2
dx
(3.7)
This fact was used by Debye and Hueckel [4] for the calculation of the electric double
layer around a spherical surface.
Within the Gouy-Chapman model the finite dimensions of the ions are neglected. In
diluted solutions this neglection is in some degree permissible, but in more concentrated
electrolyte solutions theory leads to unrealistically high concentrations of counterions near
the surface. Therefore Stern [5] tried to alter the model with the division of ions into two
populations. One population is considered as a layer of ions adsorbed on the surface and
resides close to the charged surface (Stern-layer), while the other population is described
as in Gouy-Chapman model.
Further, the theory was applied to the interaction of two flat double layers. Principles
of potential and ion density evaluation are the same as in the case of one layer, however,
the boundary conditions are different. We shall return to this problem in the following
section where a detailed study of the interaction of two flat charged surfaces in contact
with the solution of counterions is presented.
The next challenge in connection with the double layer interaction is to compute the
free energy of the system. This is actually the purpose of all double layer studies, since
we want to explain the stability conditions of different colloidal systems. Besides, the
minimization of the free energy gives the corresponding equilibrium density function for
ions. For example, the Boltzmann distribution functions are obtained by the minimization of the free energy consisting of electrostatic and entropic contributions where the
electrostatic energy is:
1Z d
εε0 E 2 A dx
(3.8)
W el =
2 0
6
and the entropic contribution is:
W
ent
= AkT
Z d
0
µ
(n+ ln
¶
µ
¶
Z d
n+
n−
−(n+ −n)) dx+AkT
(n− ln
−(n− −n)) dx , (3.9)
n
n
0
where n+ and n− are concentrations of positive and negative ions and n is a bulk concentration of both ions.
Until now only one charged and infinite surface in electrolyte solution was under consideration. Now we try to describe the interaction of two such equally charged surfaces
a distance D apart. The electrolyte solution is now filling te space between the two surfaces. Within the Gouy-Chapman model the free energy of the system of two flat surfaces
in contact with the electrolyte solution would always result in the repulsive interaction.
Contrary to intuition it is not electrostatic part that plays a decisive role in repulsive
interaction between two charged surfaces in an electrolyte solution but entropic. To explain the stability of colloidal systems and processes as coagulation, the celebrated DLVO
theory (Derjaguin and Landau [6], Verwey and Overbeek [1]) was developed. In this theory both repulsive double layer forces and attractive van der Waals forces are taken into
account. Fig. 2 shows the free energy contributions of DLVO interactions of the system
versus distance between two charged surfaces. DLVO theory gave a possible explanation
of the stability of some colloidal systems but still contains a number of approximations:
ions are considered as point charges and it is assumed that the Poisson-Boltzmann equation remains valid also at fairly high concentrations, the influences of the solvent are not
well described and the solution is modeled as a homogeneous dielectric medium.
Figure 2: Energy contributions versus distance profiles of DLVO interaction (adapted
from [7]).
7
Various attempts have been made to incorporate excluded volume effects (finite size
of counterions) into the theory [8]. One of them is based on a lattice statistics model,
which will also be the basis for the calculations in the following sections.
8
4
Double layer free energy for counterions only
In this section, our aim is to compute the free energy of a system of two double layers.
The electrolyte solution contains counterions only (see figure 3).
Figure 3: Schematic illustration of two equally charged planar surfaces, interacting
through an electrolyte solution that contains monovalent counterions.
The lattice model has been chosen in our case to incorporate finite ion dimensions into
the calculations. The lattice model is any system of particles attached to a set of lattice
sites. In our case these particles are positive and negative ions. We restrict the discussion
here to the case in which the binding on any one site is independent of the binding on the
remaining sites. Actually we begin with a system of N molecules bound not more than
one per site to a set of M equivalent, distinguishable, and independent sites. The free
energy of a cell is obtained directly from the canonical partition function:
F = −kT lnQ .
(4.10)
And the cell partition function can be written as a product of the particles partition
Q
functions Q = N
m qm , with an additional factor due to the configurational degeneracy [9]:
Q
M! N
m qm
Q=
N !(M − N )!
(4.11)
The single particle canonical partition function qm is:
qm =
X
e−εmi /kT
(4.12)
i
where i runs through all possible energy states of the particle. With the use of the
Stirling approximation for large n (lnn! ' n lnn − n), by supposing that the number
of the attached molecules is much smaller than the number of sites (N/M ¿ 1), and
summing up the contributions of the whole system composed of charged surfaces and the
electrolyte solution, the free energy is obtained (see Appendix A):
Z
F = kT
Z
[n ln(nv0 ) − n] dV − kT
N
Y
ln(
m
9
qm )
dV
,
M v0
(4.13)
where v0 is a volume of one particle. The first integral on the right side of Eq. (4.13)
represents the entropic contribution to the total free energy meanwhile the last term is
electrostatic term and can be rewritten as presented in Eq. (3.8). Finally, the free energy
F per area A of the system is:
F/A = kT
Z D
0
Ã
1Z D
dψ
[n(x) ln(n(x)v0 ) − n(x)] dx +
εε0
2 0
dx
!2
dx
(4.14)
The electric potential ψ(x) and the density of the number of counterions n(x) are computed by using the Poisson-Boltzmann equation to obtain the desired free energy.
The Poisson-Boltzmann equation (Eq. (3.1)) is solved by taking into account the
relevant boundary conditions. The electroneutrality condition for the system reads:
¯
σe0
dy ¯¯
¯
=
,
dx ¯x=0 εε0 kT
(4.15)
where sigma is charge density of the surface and function y is defined as given in Eq.
(3.5). The function y reaches an extreme in the middle between both charged surface
because of the symmetry of the problem, giving us the condition:
¯
dy ¯¯
¯
=0
dx ¯x=D/2
(4.16)
Because we now treat the solution that contains counterions only, the Poisson-Boltzmann
equation reads:
d2 y
κ2 −y
=
−
e
(4.17)
dx2
2
3
2.5
2
F/kT A
1.5
1
0.5
0
−0.5
−1
0
2
4
6
8
10
D [nm]
Figure 4: The free energy (full line), electrostatic energy (dashed line) and entropic part
of the free energy (dotted line) as functions of the distance between the equally charged
surfaces. The model parameters are σ = 0.1As/m2 and v0 = 5nm3 .
10
The analitical solution of Eq. (4.17) is:
µ
·
y = y0 − ln 1 + tg2
κ − y0
e 2
2
µ
D
−x
2
¶¸¶
,
(4.18)
where y0 ≡ y(x = D/2). We can calculate the density distribution from the potential:
n(x) = n0 e−y ,
(4.19)
n0 being a bulk concentration of counterions. Inserting the electric potential (3.4) and
the density function (4.19) of the number of counterions into the Eq. (4.14) gives us the
desired free energy in dependence of the distance between the surfaces (Fig. 4).
The total free energy results in repulsive interaction. We can see (Fig. 4) that the entropic energy contribution at small distances by far exceeds the electrostatic contribution
which is attractive by itself. The main part of the free energy is thus of entropic origin.
11
5
Double layer free energy for divalent rod-like ions
In this chapter we derive the free energy of two interacting double layers with rod-like
counterions in the solution between the two plates (Fig. 5). Flat surfaces represent a
negatively charged biological membrane while rod-like particles are an approximation for
dimeric proteins with spatially distributed charge.
Figure 5: Schematic illustration of two equally charged planar surfaces (lipid membranes)
interacting in a solution that contains multivalent rod-like particles (proteins), with spatially distributed charge.
For simplicity we suppose that within an ion equal charges e = Ze0 are a distance l
apart, one at each end of the particle. We refer to one of the two charges as a reference
charge. The concentration of all the reference charges is n(x). The location of the second
charge of the particle is then given by a conditional probability p(s|x), denoting the
probability to find the second charge at position x + s if the reference charge is at x (Fig.
5). The total charge density is therefore:
ρ(x) = Ze0 n(x) + Ze0
1 Zl
n(x − s) p(s|x − s) ds
2l −l
(5.20)
With the introduction of the rotational degree of freedom a new term should be added to
the entropic part of the free energy. We add constraints and write the functional:
1Z D
L
=
εε0
A
2 0
Ã
dψ
dx
+kT
!2
dx + kT
Z D
0
Z D
0
[n(x) ln(n(x)v0 ) − n(x)] dx+
1 Z
dx n(x)
p(s|x) lnp(s|x) ds+
2l
12
(5.21)
1 Z
p(s|x) ds
2l
0
0
The first two terms are equal as in the case of electric double layers with the monovalent
counterions (see chapter 4). The third term is a contribution due to the rotational degree
of freedom. The last two terms originate from constraints requiring the ectroneutrality of
the system (Eq. 5.22) and the probability condition (Eq. 5.23):
+µkT
Z D
2Zn(x) dx + kT
Z D
0
Z D
n(x)λ(x)
Ze0 n(x) dx = σ
(5.22)
1 Z
p(s|x) ds = 1.
(5.23)
2l
We integrate Eq. ( 5.23) over all possible orientations. When integrating near the surfaces
all orientations are not possible due to steric effects. Variation of the functional (5.21)
gives the number density and probability distribution functions for the equilibrium case:
p(s|x) =
and
n(x) =
Ze0 ψ(x+s)
1
e− kT ,
q(x)
q(x) − Ze0 ψ(x+s) −2Zµ
kT
e
v0
where
(5.24)
(5.25)
1 Z l − Ze0 ψ(x)
q(x) =
e kT
ds.
(5.26)
2l −l
µ is obtained by inserting n(x) (Eq. 5.25) into the electroneutrality condition (Eq. 5.22).
n(x) is then inserted into the Eq. (3.1) from which potential can be obtained by solving
it numerically.
The free energy of the system is obtained by inserting the potential into the Eq. (5.21).
Figure 6 shows the free energy in dependence on the distance between the charged surfaces
for two different surface charge densities σ and for two different lengths of the counterions.
If σ is large enough we get a minimum of the free energy at the distance which is equal
to the length of the particles. The minimum free energy is more pronounced for longer
counterions.
The conditional probability function p(s|x) as a function of the projection of counterions on the x axis is presented in Fig. (7). Calculations show that energetically the
most favorable distance between the surfaces corresponds to the length of the rod-like
particles (fig. 6). At the distance betweeb the surfaces that equals the length of the counterion two orientations of the counterions are preferred, parallel and perpendicular to the
surface. Particles that orient perpendicular to the surface act as bridges between two
surfaces. This bridging mechanism is responsible for the attractive interaction between
equally charged surfaces. That can not be obtained within the Gouy-Chapman model.
13
2.5
2
2
1.5
2
1.5
[1/nm2 ]
σ=0.033 As/m2
1
F/AkT
1
F/AkT
[1/nm2 ]
σ=0.1 As/m
0.5
σ=0.1 As/m2
σ=0.033 As/m2
0.5
0
0
−0.5
0
1
2
3
4
D[nm]
5
6
7
8
1
2
3
4
5
6
7
8
D[nm]
Figure 6: Electrostatic free energy as a function of the distance between two equally
charged plates for two different surface charge densities. Model parameters are l = 2nm
and v0 = 65 nm3 (left) and l = 5nm and v0 = 65 nm3 (right).
6
15
5
10
σ=0.033 As/m2
σ=0.1 As/m2
p(s|x = 0)
p(s|x = 0)
4
3
2
5
1
0
−5
0
s[nm]
0
−5
5
0
s[nm]
5
Figure 7: Conditional probability density as a function of the projection of the rod-like
particles on the x axis (perpendicular to the surfaces) for two different surface charge
densities. The equilibrium distance between the plates is D = 5nm and is equal to the
length of the particles l = 5nm.
14
6
Application of the electrical double layer theory in
the medicine
The above described theory is used to explain the adhesion of giant phospholipid vesicles
in the presence of certain proteins and/or antibodies (Fig. 8).
A
B
C
Figure 8: A sequence showing the process of vesicle adhesion in 30s intervals. The first
picture is taken 14 minutes after the addition of HCAL monoclonal anti-ß2GPI antibodies
to the charged vesicles (POPC:cholesterol:cardiolipin=2:2:3).
The experiments were made in order to study the mechanisms involved in the antiphospholipid syndrom (APS). APS is an autoimmune disease characterized by thrombotic events and/or pregnancy morbidity. The etiology and underlying mechanisms of
APS are not yet understood, however, activation of the coagulation system is evident.
Antiphospholipid antibodies are present in the sera of patients with APS. These antibodies were found to interact directly with phospholipids constituting cell membranes
(e.g. cardiolipin) or to bind to an antigen beta2-glycoproteinI (ß2GPI). The interactions
between phospholipid membranes, protein cofactors and antibodies can be studied in a
system of giant phospholipid vesicles. The change of vesicles shape and coalescence of
vesicles due to the presence of protein cofactors and/or antibodies in the solution can be
observed under phase contrast microscope. The giant phospholipid vesicles were prepared
(by electroformation method) with three different lipids: POPC (1-Palmitoyl-2-Oleoylsn-Glycero-3-Phosphocholine), cholesterol and cardiolipin in different proportions. With
such selection of lipids we obtain negatively charged surfaces in contact with electrolyte
solution. The vesicles are stable for days. For comparison we also prepared uncharged vesicles with POPC and cholesterol only (in proportions POPC:cholesterol=4:1). Interactions
15
between vesicles, antibodies and ß2GPI were studied under phase contrast microscope.
Figure 9: Adhesion of giant unilamellar vesicles 27 minutes after the insertion of HCAL
monoclonal anti-ß2GPI antibodies into the solution with negatively charged (A) and neutral (B)vesicles.
It turns out that antibodies alone cause the adhesion of charged vesicles, but not
uncharged vesicles (Fig. 9). Adhesion also does not occur if vesicles are prepared only
with very small amount of cardiolipin. Monoclonal antibodies (HCAL monoclonal antiß2GPI antibodies) were used in the experiments in order to avoid the variability in the case
of using different IgG fractions obtained from human sera. If we describe the antibodies
as rod-like molecules with spatially distributed charge the explanation of the adhesion is
possible.
16
7
Discussion
Mean field theory is used to describe double layer interactions in all previously described
models. Other basic approximations of Gouy-Chapman model are: homogenous surface
charge distribution described by surface charge density sigma, electrolyte is composed of
point-like ions and solution is described by dielectric constant which is independent of the
electric field variation and ion concentration, electric field behind the charged plates is not
taken into account. At short separations of the two charged surfaces further factors come
into play: ion correlation effects, image forces arising on the boundaries with different
dielectric constants, solvation forces and discreteness of surface charges. Because the ions
are taken to be infinitely small, the concentrations of counterions on the charged surfaces
can become unreasonably large in Gouy-Chapman model. Therefore the theory breaks
down at very small distances.
Regarding surfaces of spherical particles, surface curvature might also play an important role in interaction energies. But we do not expect this to play a major role when
using giant phospholipid vesicles because the membrane thickness and counterion sizes
(some nm) are much smaller than an average size of the vesicle (some µm)
The problem of finite ion size was partially resolved in our model. Ion size near the
charged surface and charge distribution within an ion are taken into account. Other
approximations remain unsolved. Different theories (hypernetted chain theory, modified
Poisson-Boltzmann etc.) were made trying to incorporate additional factors, but because
of their complexity in practice Gouy-Chapman model still remains the most useful tool
for fast and lucid estimation of the forces in the electrical double layer. From this point
of view our model is an upgrade of the Gouy-Chapman model, but remains transparent.
Excluded volume effect has been incorporated into the double layer interaction through
the lattice statistics model in our case. The Gouy-Chapman model predicts repulsion
between equally charged surfaces [1]. We have shown, that taking into account spatial
distribution of charges may lead to the attractive interaction due to the orientational
ordering of counterions with spatially distributed charge. However, only for large enough
surface charge densities the minimum of the free energy was obtained. This is in agreement
with computer simulations, where ion-ion correlation effects were taken into account for
systems with multivalent counterions. Experimental observations can be explained with
the results of developed theory.
Systems where interactions between charged membranes are mediated by particles of
different shapes and with internal charge distribution are common in different biological
and chemical environments and thus still represents a challenging problem in electrical
double layer studies. Many similar, at first glance simple, but actually complex and poorly
understood phenomena in the ”bio” field are still waiting to be explored in the future.
17
Appendix A: Derivation of the entropic part of free
energy from the partition function
We derive the entropic contribution to the free energy:
Z
F
entropic
= kT
from the partition function:
Q=
[n ln(nv0 ) − n] dV
(A.27)
M!
.
N !(M − N )!
(A.28)
Using the Stirling approximation for large n (lnn! ' n lnn − n) in Eq. (A.28) gives:
lnQ = M lnM − M − N lnN + N − (M − N ) ln(M − N ) + (M − N ) ,
(A.29)
Rewriting the last term we get:
lnQ = M lnM − M − N lnN + N − (M − N ) ln(M (1 −
N
)) =
M
= M lnM − N lnN + N − (M − N ) lnM − (M − N ) ln(1 −
(A.30)
N
)
M
Finally:
N
N
) − (M − N ) ln(1 − ).
(A.31)
M
M
Now we introduce new variables v0 and n. We define v0 as volume of one particle so that
M v0 = V cell , where V cell is the volume of the cell with M sites of volume v0 . n is the
number density of counterions defined as n = N/V cell . Total free energy is defined as:
lnQ = −N ln(
Z
F
and F
cell
tot
=
F
cell
dV
,
V cell
(A.32)
= −kT lnQ. Equation A.31 is inserted into the equation A.32:
F
Z
entropic
kT
=
·
µ
1
N
dV n ln(nv0 ) +
1−
v0
M
¶
µ
N
ln 1 −
M
¶¸
(A.33)
If we now assume that N/M ¿ 1, the second term can be approximated by ln(1+x) ≈ x,
giving us the final version of the entropic part of the free energy that we wanted to derive
(Eq. A.27).
18
References
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[3] Chapman D.L.: Philos. Mag., (6) 25, (1913) 475
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[5] Srtern O.: Z. Electrochem., 30, (1934) 508
[6] Derjaguin B.V. and Landau L.: Acta Physicochim. URSS, 14, (1941) 633
[7] Israelachvili J.N.: Intermolecular and surface forces, Academic press(1991)
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19