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Effective interactions in polydisperse systems of
penetrable macroions
a
Thiago Colla & Christos N. Likos
a
a
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
Published online: 01 Apr 2015.
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To cite this article: Thiago Colla & Christos N. Likos (2015): Effective interactions in polydisperse systems of penetrable
macroions, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, DOI:
10.1080/00268976.2015.1026295
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Molecular Physics, 2015
http://dx.doi.org/10.1080/00268976.2015.1026295
SPECIAL ISSUE IN HONOUR OF JEAN-PIERRE HANSEN
Effective interactions in polydisperse systems of penetrable macroions
Thiago Colla and Christos N. Likos∗
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
Downloaded by [Christos Likos] at 10:59 03 April 2015
(Received 29 January 2015; accepted 2 March 2015)
A coarse-graining approach based on the linear response theory is applied to deduce general expressions for the effective pair
potentials in a multi-component system of soft macroions. Within the underlying approximations, the effective pair potentials
can be written as simple convolutions between the intrinsic macroion charge distributions and a Yukawa-like potential which
effectively contains the averaged contributions from the small ions. Two different charge distributions are assigned to the
soft macroions: a Gaussian-like diffuse distribution and a uniform charge distribution inside the particle cores. The resulting
effective pair potentials are then used in an effective model based on the hyppernetted chain approximation to investigate
the structural properties of a two-component system of oppositely charged particles as a function of the various system
parameters. It is found that the condensation of counterions is much stronger in the case of particles with a Gaussian charge
distribution, leading to much weaker electrostatic interactions and less structured pair correlations in comparison with the
system of uniformly charged macroions.
Keywords: coarse-graining; microgels; charged colloids; statistical mechanics
1. Introduction
One of the most fundamental aspects of soft-matter systems is the intrinsic coexistence of many different types of
interacting components. While the presence of such large
amount of elements strongly increases the system complexity, it is also responsible for the rich phenomenology
inherent to these systems [1,2]. Another advantage resulting
from the rich variety of different species is the possibility
to explore many properties of these systems under a large
number of different conditions [3]. In many cases, the various components are characterised by quite different length
and time scales, opening the possibility to tune appropriate
effective interactions among the larger constituents by controlling specific properties on the smaller ones. Although
in many situations the smaller components have a little effect on the interactions among the mesoscopic ones, this
is not always the case. One example is the steric effects
induced by small sized species, which may strongly influence the effective interactions between colloidal particles
[4,5]. In the case of systems with charged particles, the
effects from the smaller components on the effective interactions between the bigger ones are even stronger due to
the long-range nature of the Coulombic interactions [3,6,7].
A classical example is the charge stabilisation of colloidal
particles against collapse driven by the short-range van der
Waals forces in polar solvents, which results from the dissociation of the small ionic groups that lie on the colloidal
surface [2,6,8].
∗
Corresponding author. Email: [email protected]
C 2015 Taylor & Francis
Even though the large asymmetry between the different components can be conveniently used to induce a
variety of effective interactions, it also renders the theoretical description enormously complex. Taking into account all the relevant interactions between components in
a large range of scales is an extremely demanding task,
and approximations prove to be essential in these circumstances. One convenient way of describing systems with
large asymmetries is to average out the contributions from
the smaller elements, resulting in an effective description
in which only the mesoscopic components are explicitly
considered [1,9]. This idea to formally describe the system through an average over their microscopic components
has been used in many different situations, and goes back
to the classical McMillan–Mayer theory of solutions [10].
Theoretically, the coarse-graining process can always be
accomplished by explicitly integrating out the degrees of
freedom from the components whose typical scales are
much smaller in comparison with the other components
[10]. In principle, this procedure leads to complex many
body, state-dependent effective interactions that implicitly
contain all the contributions from the small-scale particles
[9]. This averaging process represents however a powerful
tool that helps us to gain physical insight over the contributions from the microscopic components, allowing for further approximations that incorporate their mean effects on
the effective description. It is well established, for instance,
that addition of ionic components in systems of charged
Downloaded by [Christos Likos] at 10:59 03 April 2015
2
T. Colla and C.N. Likos
macromolecules leads to screened interactions among
them. On the other hand, effects from polar solvents can
be to a good extend described by a proper renormalisation
of the dielectric constant.
In practice, the averaging process over the microscopic
components can be performed through a number of different – sometimes equivalent – approaches. Effective interactions can be for instance defined in the context of density
functional theory, in which the density distributions of the
smaller components are calculated by a variational condition, assuming that they move under the influence of an effective field provided by the mesoscopic ones [1,9,11–14].
Alternatively, these averaged density profiles can be calculated as functional expansions over perturbations that
represent their interactions with the mesoscopic particles
[1,15–20]. Another way to define effective interactions in
mixtures is in the context of the Ornstein–Zernike (OZ)
equation, in which the effective pair interactions are defined by the requirement that they should provide the
same pair correlation functions as the ones in the original multi-component system [5,7,21,22]. For a complete
and recent review on the role of coarse-graining effective
interactions in soft-matter systems, we refer the reader to
Ref. [2].
In the case of polydisperse systems of charged
hard colloids, the classical theory of Derjaguin–Landau–
Verwey–Overbeek provides an excellent effective description [23,24], where the role of the small ions is essentially
to screen the Coulombic interactions beyond overlapping.
Similar Yukawa-like effective interactions also appear in
the description of soft charged molecules. In this situation, however, the ionic effects are not limited to screening
of the bare interactions. Due to the permeability of these
molecules to small ions, a large amount of counterions
will condense into the particle, attempting to neutralise
its bare charge [16,19,25]. As a consequence, the strength
of the effective interactions will be dramatically reduced
with respect to the ones from hard colloids – even in the
linear screening approximation. The effective pair potential in one-component systems of homogeneously charged
soft polyelectrolytes has been worked out by Denton in
the framework of a linear response approximation [16,26].
Apart from the pair interactions, the theory also allows
for the calculation of the so-called volume terms – zerothorder density-dependent contributions which may strongly
influence the system thermodynamics. The structure factors
resulting from these pair interactions were later on shown
to be in excellent agreement with experimental data, probing the validation of the underlying approximations [19].
Recently, Hanel et al. applied a similar approach in order
to extend the effective interactions to the case of layeredcharged microgels [20]. The linear response theory has also
been further extended by Chung and Denton to calculate effective interactions in a polydisperse system of hard colloids
with different sizes and charges [27]. Effects beyond the
Figure 1. Schematic representation of the system under consideration. The system is a multi-component mixture made up of
highly asymmetric spherical charged particles. Different colours
represent different charges assigned to the components. The big
spheres represent the mesoscopic components from group A, while
the small spheres belong to the group of microscopic particles B,
whose contributions we want to average out. All the soft particles
are in principle free to penetrate into one another.
linear ones on both ionic distributions [25,28–31] and particle swelling [25] in systems of penetrable macroions have
been recently investigated in the framework of the Poisson–
Boltzmann (PB) theory. The properties of soft unscreened
ionic systems, in which different charge distributions are
assigned to the penetrable ionic particles, have also been
extensively studied in the last years [32–35].
The aim of the present work is to study the effective interactions in the case of a mixture of penetrable macroions
in the presence of an arbitrary number of ionic species, in
the framework of the linear response approximation. Two
types of particle charge distributions for the mesoscopic
components will be considered: homogeneously charged
particles and a Gaussian-like charge distribution. Even
though the formalism will be developed for a quite general
situation of an arbitrary number of charged components,
emphasis will be later on given for the case of an effective
description in terms of a two-component solution made of
oppositely charged microgels.
2. Theory
We consider a mixture of several highly asymmetric charged
components, which for simplicity we will classify into two
distinct groups: NA mesoscopic components, each of which
carrying a total charge Zα and with radius Rα belonging
to group A, and a group B composed by NB microscopic
components of charge zi and radius ri (see Figure 1)1 . It
is assumed that both charge and size of the particles from
group A are in the orders of magnitude larger than the ones
belonging to group B, Zα zi and Rα ri . Overall charge
Molecular Physics
neutrality requires that
NA
HAB =
nα Zα +
Downloaded by [Christos Likos] at 10:59 03 April 2015
α=1
NB
ni zi = 0,
HBB = KB +
q2
σα (|r − r |)σβ (r ) dr dr
+
uαβ (r) =
ε
|r − r |
q 2 zi
σα (r ) dr
uαi (r) =
ε
|r − r |
q 2 zj zj
,
(2)
uij (r) =
ε r
where ε is the dielectric constant of the medium in which
the system is embedded on and u0αβ (r) is a short-range repulsive pair potential between the mesoscopic components.
Apart from these short-range interactions, the remaining
electrostatic contributions described above are written as
appropriate convolutions of the particle internal charge distributions and the usual Coulombic kernel (with the distributions from the ionic microscopic components represented
as point-like charges zi δ(r)). In an analogous fashion, the
total system Hamiltonian H can be split as
H = HAA + HAB + HBB .
(3)
nα
Defining the number density operator ˆ α (r) = k=1 δ(r −
r
kα ) for the mesoscopic A-component of type α and ρˆi (r) =
ni
m=1 δ(r − rmi ) for the microscopic B-component of type
i, the individual Hamiltonians in Equation (3) can be written
as
N
[ˆ α (r)ˆ β (r )
− δαβ δ(r − r )ˆ α (r)]uαβ (|r − r |)drdr
B B
1
2 i=1 j =1
N
[ρˆi (r)ρˆj (r )
− δij δ(r − r )ρˆi (r)]uij (|r − r |)drdr ,
(4)
where the second factor inside the integrals in HAA
and HBB has the role of suppressing the particles’ selfinteractions, and the factor 1/2 is introduced to avoid double
counting. The terms KA and KB represent the kinetic energy
from the components belonging to A and B, respectively:
KA =
KB =
p2
k
2m
k
k∈A
p2
k
,
2m
k
k∈B
(5)
where the summations are extended to all particles belonging to either groups A or B. The total system Helmholtz free
energy F can be formally obtained as
u0αβ (r)
N
ˆ α (r)ρˆi (r )uαi (|r − r |)drdr
N
(1)
i=1
A A
1
2 α=1 β=1
NA NB α=1 i=1
where nα and ni are the total number of particles belonging
to the components α and i, respectively.
Due to the strong size asymmetry, and since we focus in what follows mainly on electrostatic effects, we
will disregard the steric contributions from the smaller
B-components, and therefore from now on we approximate them as point-like charged particles (ri → 0). The
A-components are on the other hand characterised by a
continuous charge distribution qσα (r) (where q represents
the charge of a proton), which can in principle be arbitrary, apart from the requirement that its integral over the
whole space equals the total charge Zα q of the mesoscopic
component α. With these assumptions, the pair interactions
among A-components uαβ (r), the microscopic B pair potentials uij (r) and the crossed A–B interactions uαi (r) can
be written as
HAA = KA +
3
βF = − lne−βH ,
(6)
where β = 1/kB T and the symbol · represents the canonical trace over the momenta and position of all the system
particles. The effective interactions between particles from
group A can be formally obtained if one first trace out over
the components from group B, keeping both the momenta
and positions of the mesoscopic A-particles constant . Using the separation of the Hamiltonian described by Equation (3), and denoting by · A and · B the individual traces
over the components A and B, respectively, the total trace
in Equation (6) can be conveniently rewritten as
eff
e−βH = e−βHAA e−β(HAB +HBB ) B A ≡ e−βHAA ,
A
(7)
where in the last equality we have defined the effective
eff
interactions HAA
between the mesoscopic A-particles as
eff
ind
HAA
= HAA + HAA
(8)
ind
with the induced A interactions HAA
defined as
ind
HAA
= −kB T ln e−β(HAB +HBB ) B .
(9)
These induced interactions will explicitly depend only on
the collective positions from the A-particles, and represent
the trace over the space spanned by the degrees of freedom
of the microscopic B-particles for a fixed configuration of
the positions of all particles belonging to group A. It implicitly contains all the effects induced by the presence of
Downloaded by [Christos Likos] at 10:59 03 April 2015
4
T. Colla and C.N. Likos
the smaller B-components. Obviously, the effective A–A interactions defined in Equations (8) and (9) depend on the
specific configuration of all particles belonging to group
A, and have therefore an intrinsically many-body character.
The calculation of the induced interactions can be done in a
simple way if one relates them with equilibrium quantities
of an equivalent system composed by the B-components.
In the case of neutral systems, the induced interactions (9)
can be immediately identified with the Helmholtz free energy of a system of the B-particles under the influence of
an ‘external’ potential defined by their interactions with
the fixed A-components. In the present situation, however,
in which the B-components are charged, this equivalence
is not that straightforward, since the equivalent system of
B-particles does not obey charge neutrality. This can be
however easily resolved by replacing the Hamiltonians in
≡ HAB − E0 and
Equation (9) by the rescaled ones HAB
HBB ≡ HBB + E0 , where E0 is the interaction energy of
the ionic B-components
with a uniform neutralising background of charge Q = i ρ i zi , with ρ i being the number
density of the ionic component i. With this modification,
the induced Hamiltonian can be rewritten as:
ind
= −E0 − kB T lne−β(HAB +HBB ) B .
HAB
(10)
In this way, the induced interactions can be formally
identified with the free energy of an equivalent ionic system embedded on a neutralising background, and subjected
to an ‘external’ potential whose influence is equivalent to
the interactions in HAB . The induced interactions can now
be obtained by a process in which the mixed A–B interactions in Equation (10) are progressively ‘turned on’ from
zero to their final value. This is done by defining a control
parameter λ such that the pair potential uαi (r) is replaced
by λuαi (r) (with 0 ≤ λ ≤ 1) in Equations (4) and (10). The
ind
(λ) can be
resulting λ-dependent induced Hamiltonian HAA
written as
ind
(λ) = −E0 − kB T lne−β(λHAB +HBB ) B .
HAB
ind
= F0 − E0 +
HAA
1
H¯ AB (λ)dλ,
(13)
0
where F0 is, according to (11), the Helmholtz free energy
in a neutralof a NB -component system of ions
moving
ising background, F0 = −kB T ln e−βHBB B . The quantity
H¯ AB (λ) can be easily obtained from the second relation of
Equation (4), having in mind that the averaging process acts
on ρˆi (r) only:
H¯ AB (λ) =
NA NB ˆ α (r)ρiλ (r )uαi (|r − r |)drdr
α=1 i=1
≡
NB ρiλ (r)φi (r)dr,
(14)
i=1
where ρiλ (r) ≡ ρˆi (r)B is defined as the density distribution of the microscopic i component corresponding to the
fixed given strength parameter λ. In the last equality of (14),
we have defined an ‘external’ potential φi (r) which results
from the interactions between one particle of component i
and all the mesoscopic A-components, whose positions are
fixed in space:
NA φi (r) =
ˆ α (r )uαi (|r − r |)dr .
(15)
α=1
As the parameter λ grows from zero to one in Equation
(11), the strength of the ‘external’ potential provided by
the A-particles changes continuously from zero to its final
value, given by Equation (15). The corresponding mean
particle distributions ρiλ (r) can be conveniently written as
functional expansions with respect to the external stimuli
φi (r). Neglecting non-linear contributions on this expansion, the density profiles become
(11)
ρiλ (r)
Taking the derivative of this relation with respect to λ
provides
ind
HAB e−β(λHAB +HBB ) B
dHAB
(λ)
≡ H¯ AB (λ),
= −β(λH +H ) AB
BB
dλ
e
B
sides of (12) up to λ = 1:
=
(λ=0)
ρi
= ρi +
+
ρ˜iλ
ρ˜iλ
NB δρi (r)
δφj (r )dr
+
)
δφ
(r
j
j =1
+λ
NB χij (|r − r |)φj (r )dr ,
(16)
j =1
(12)
where H¯ AB (λ) indicates the mixed A–B interactions averaged over all possible phase-space configurations of particles belonging to B, keeping fixed the positions of the
A-particles, as well as the controlling parameter λ. The induced A–A interactions follow from the integration of both
where ρ i is the homogeneous density distribution for component i in the absence of external forces (λ = 0), and
χij (|r − r |) ≡ δρi (r)/δφj (r ) is the so-called linear response function, evaluated at λ = 0. In the last term of the
right-hand side, we have used the fact that δφi (r) = λφi (r).
The concentrations ρ˜iλ have been introduced in order to
keep fixed the overall particle concentrations on the distributions ρiλ (r) necessary to maintain charge neutrality as the
Molecular Physics
5
strength parameter λ changes
A A
1
2 α=1 β=1
N
+
(ρiλ (r) − ρi )dr = 0.
N
ˆ β (r )drdr .
ˆ α (r)uind
αβ (|r − r |)
(17)
(21)
The presence of the terms ρ˜iλ in (16) can be viewed
as the effect of an extra (λ-dependent) potential known
as Donnan potential, which has to be introduced in charged
systems in order to satisfy the constrain of electroneutrality.
Integrating both sides of (16), one easily find
Clearly, the second term of this expression is the interaction energy of the charged macroions with a uniform ionic
background. Since the system of A-particles is homogeneous (no external potential), this contribution is actually
independent of the particle coordinates. The induced pair
interactions uind
αβ (r) defined in (21) can be explicitly written
in terms of the linear response function χij (|r − r |) as
ρ˜iλ = −
NB λ χij (|r − r |)φj (r )drdr ,
V j =1
(18)
uind
αβ (r) =
NB NB uαi (|r − r |)χij (|r − r |)ujβ (r )dr dr .
Downloaded by [Christos Likos] at 10:59 03 April 2015
i=1 j =1
where V is the system volume. Using the relation ρi (r) =
ρˆi (r)B , the linear response function χij (|r − r |) can be
written in terms of the so-called total correlation function
hij (|r − r |) of the homogeneous system of B-particles:
χij (|r − r |) = −βρi (ρj hij (|r − r |) + δij δ(r − r )).
(19)
(22)
Finally, substitution of Equation (21) into (8) provides the
effective interactions among the mesoscopic A-particles,
which now contains the averaged contributions from the
microscopic B components:
Within this linear response approximation, the induced inind
can be easily obtained by inserting the denteractions HAB
sity profiles ρiλ (r) resulting from Equations (19), (16) and
(18) into Equations (14) and (13), and by performing the
(trivial) integration over the strength parameter. The result
is
ind
HAB
= F0 − E0 +
NB
i=1
1
+
2
NB NB (1)
ρ˜
ρi + i
2
φi (r)χij (|r − r |)φj (r )drdr , (20)
are the functions defined in (18) evaluated at
where
λ = 1. The first and second terms of this expression are
the corrections from the ionic system with its neutralising background. The third term on the other hand can be
interpreted as a first-order correction for the bare AA interactions, while the last term contains the second-order pair
interaction corrections. This becomes clear upon substitution of the ‘external’ potentials defined in Equation (15)
into the above expression, which then can be rewritten in
terms of the mesoscopic particle coordinates:
ind
HAB
= F0 − E0 +
×
NA NB
α=1 i=1
(1)
ρ˜
ρi + i
2
ˆ α (r)uαi (|r − r |))drdr
[ˆ α (r)ˆ β (r )
ind
ueff
αβ (r) = uαβ (r) + uαβ (r).
i=1 j =1
(1)
ρ˜i
where the effective pair potential ueff
αβ (r) is the sum of the
bare pair interactions uαβ (r) from Equation (2) and the
induced ones from Equation (22):
φi (r)dr
N
− δαβ δ(r − r )ˆ α (r)]ueff
αβ (|r − r |)drdr , (23)
A A
1
2 α=1 β=1
N
eff
HAA
= H0 +
(24)
The term H0 in Equation (23) is the collection of all the
remaining contributions that do not depend on the particle
coordinates:
NA NB 1
ˆ α (r)uαi (|r − r |)drdr
H0 = F0 − E0 +
2 α=1 i=1
⎤
⎡
NB 1
× ⎣ρi −
χij (|r − r |)φj (r )drdr ⎦
2V j =1
+
NA
nα uind
αα (r = 0),
(25)
α=1
where nα is the total number of particles of component
α. The last term of H0 represents the self-energy of the
macroions which is induced by their interactions with the
ionic system. Since the system as a whole is homogeneous,
the energy H0 provides a set of terms which do not depend
on the position of the macroions. These terms are therefore called the volume terms, since they do depend on the
6
T. Colla and C.N. Likos
particular macroscopic thermodynamic state of the system.
Although these terms do not influence the structural properties of the system, they are relevant in what concerns the
thermodynamic calculations.
Due to the large number of convolutions involved in
the calculation of both bare and induced interactions, it becomes convenient to rewrite the effective pair potential in
terms of its Fourier components. Taking the Fourier transforms of Equations (2), (22) and (24), the effective pair
interactions can be written in a compact form in Fourier
space as
Downloaded by [Christos Likos] at 10:59 03 April 2015
4π q 2
ˆ 0αβ (k) +
σˆ α (k)σˆ β (k)
uˆ eff
αβ (k) = u
εk 2
⎤
⎡
2 4π
q
zi zj χˆ ij (k)⎦ ,
× ⎣1 +
εk 2 ij
(26)
where uˆ 0αβ (k) are the Fourier components of the short-range
potential u0αβ (r). Once the response functions χij (r) for
the system of unperturbed B-components are known, the
coarse-grained, ionic averaged effective pair interactions
for the system of macroions with arbitrary internal charge
distributions σα (r) can be calculated by simply taking the
inverse Fourier transform of Equation (26). Similarly, the
volume terms (25) are also simplified in terms of Fourier
components of the induced and bare interactions:
A
1
nα uind
αα (r = 0)
2 α=1
N
H0 = F0 − E0 +
⎡
⎤
NB
NA
NA NA 1
⎦
+ lim ⎣
nα ρi uˆ αi (k) −
nα nβ uˆ ind
αβ (k) .
k→0
2V
α=1 i=1
α=1 β=1
(27)
In Fourier space, the linear response functions for the
ionic system χˆ ij (k) are directly related with the corresponding structure factors Sij (k), defined as 2
Sij (k) = δij + ρj hˆ ij (k).
(28)
By taking the Fourier transform in both sides of Equation
(19), one can easily find χˆ ij (k) = −βρi Sij (k). The structure factors can be in principle evaluated by means of the
traditional OZ equation for the unperturbed ionic system:
hij (r) = cij (r) +
NB
ρl
hil (r )clj (|r − r |)dr ,
(29)
l=1
where cij (r) are the so-called direct pair correlation functions. Considering the Fourier transformed version of this
equation, and defining the NB -dimensional square matrices
S of elements Sij (k), Cij of elements cˆij (k) and the diagonal matrix R of elements Rij = ρ i δ ij , the OZ Equation (29)
can be translated into the following matrix relation:
S = I + CRS,
(30)
where I is the NB -dimensional identity matrix. In order to
fully determine the elements of S, a second set of relations between the pair correlation functions – known as the
closure relations – is necessary. Assuming that the electrostatic correlations between the ionic components are weak,
we adopt here a leading order approximation known as the
mean spherical approximation (MSA), in which the direct
pair correlation functions for the point-like ions are approximated by their asymptotic limit, cij (r) = −βuij (r) =
−zi zj λB /r, where λB = βq2 /ε is the well-known Bjerrum
length. Within this approximation, the elements of the matrix C become cˆij (k) = −4π λB zi zj /k 2 , and Equation (30)
can be rewritten as
S=I−
4π λB
ZZT RS,
k2
(31)
where we have defined the NB -dimensional vector represented by the column matrix Z of components zi , i = 1,
. . ., NB . This equation can be easily solved for S by multiplying both sides of it (from the left) by the matrix ZZT R
of elements ρ j zi zj , and further noting that, according to
(31), ZZT RS = (I − S)k 2 /4π λB . The result can be explicitly written in terms of the corresponding matrix elements
of S as
Sij (k) = δij − 4π λB
zi zj ρj
,
κ 2 + k2
(32)
where κ 2 ≡ 4π λB ZT RZ = 4π λB i ρi zi2 defines the inverse of the so-called Debye screening length for the ionic
B-component system. Now, substitution of the corresponding linear response functions χˆ ij (k) = −βρi Sij (k) obtained
in this MSA into the expression for the Fourier components
of the effective interactions in Equation (26) provides a
very simple expression for the Fourier transformed coarsegrained pair interactions between the macroscopic charged
components:
ˆ 0αβ (k) + 4π λB
β uˆ eff
αβ (k) = β u
σˆ α (k)σˆ β (k)
.
k2 + κ 2
(33)
This expression for the effective pair potential is quite general, holding for arbitrary macroion charge distributions, as
long as the linear approximation for the ionic response and
correlations applies. This is usually the case at low electrostatic couplings, where the ionic correlations are weak. For
an aqueous solution at room temperature, the linear MSA
approximation is known to be only accurate in the situation
Molecular Physics
monovalent ions (|zi | = 1). In this case, addition of multivalent ions requires the use of more accurate approximations
to correctly account for the strong ionic correlations [36].
It is now an easy task to perform the inverse Fourier
transform of Equation (33), from which the following formal expression for the effective pair interactions in real
space is obtained
βuαβ (r) = βu0αβ (r) + λB
Downloaded by [Christos Likos] at 10:59 03 April 2015
×
e
σβ (r )dr dr .
|r − r |
(34)
e−κ|r−r | dr ,
|r − r |
(35)
the effective pair interactions (34) can be rewritten as
0
elec
ueff
αβ (r) = uαβ (r) + uαβ (r),
(36)
where the electrostatic interactions induced by the cloud of
small ions uelec
αβ (r) are
uelec
αβ (r) =
∇ 2 α (r) = −4π λB σα (r) +
σβ (r )α (|r − r |)dr .
NB
ρi zi e−zi α (r) − Q ,
(38)
σα (r )
i=1
Comparing this equation with the corresponding bare pair
interactions among the mesoscopic particles (first relation
in Equation (2)), we see that only difference is in the kernel of the convolution integrals, which changes from the
bare Coulombic interaction to a Yukawa-like screened potential. We, therefore, conclude that the major effects of the
ionic B-species on the effective pair interactions between
the mesoscopic charged components is – at least in the context of the linear approach – to screen the bare Coulombic
interactions among them, independently on the number of
both ionic and mesoscopic components. It is important to
note that the short-range repulse interactions are unaffected
at this level of approximation, in which the steric effects
from the small ions are neglected. Defining the screened
potential energy α (r) originated on the (screened) ionic
system by a particle of component α:
surrounding this macroion follows the Boltzmann distribution, ρi (r) = ρi e−βqzi α (r) . The corresponding mean electrostatic potential α (r) around the macroion can be obtained through the application of the Poisson equation,
which results in the traditional PB equation:
σα (|r − r |)
−κ|r −r |
βα (r) = λB
7
(37)
The above equations can be interpreted as expressing the
work that has to be done in order to bring two particles of
components α and β a distance r apart from each other,
when these are surrounded by an infinitely large electric
cloud that effectively screens the bare interactions. An
equivalent result can be easily obtained in the context of
the Debye–H¨uckel (DH) theory of ionic solutions. In fact,
the mean (bare) electrostatic potential around a particle
of component α immersed in an electrolyte composed by
the B-components and the corresponding neutralising background can be obtained by assuming that the ionic cloud
where α (r) ≡ βα (r), Q = i ρ i zi is the ratio between
the charge density of the neutralising background and the
proton charge q. In the present case of an open system,
linearisation of the ionic profiles in Equation (38) can be
made with respect to a point where r → ∞ (vanishing electrostatic potential), from which the following Helmholtz
equation for the electrostatic potential in the open system
is obtained
(∇ 2 − κ 2 )α (r) = −4π λB σα (r).
(39)
The Green function for the Helmholtz operator on the left
hand side is G(|r − r |) = e−κ|r−r | /4π |r − r |. The formal
solution for this equation can then be written as a convolution between its right-hand side and the corresponding
Green function G(|r − r |), and is therefore the same as
given by Equation (35). Now, application of the linear superposition principle (which holds in the present DH approximation) allows us to write down the pair interactions
between macroions of components α and β – which can be
identified as the work necessary to bring them at a distance
r from each other – as a convolution between the resulting electrostatic potential α (r) and the charge distribution
σβ (r), which provides exactly the electrostatic pair interactions described in Equation (37). This result applies not
only at infinite dilution, since the effects from the remaining
macroions are implicitly accounted for through the homogeneous background in Equation (38) (Jellium approximation). It is important to emphasise, however, that this DH
approach only provides the calculation of the effective pair
interactions, and the volume zeroth-order terms coming
from the ionic coarse-graining have to be introduced in an
ad hoc fashion at this level of approximation. Calculations
based on the PB equation are however extremely powerful
to provide physical insight on how the linear DH approach
can be extended to incorporate non-linear effects.
3. Effective pair potentials
Having established the theoretical concepts of the coarsegraining procedure on a general basis, we now proceed
to the calculation of the effective interactions for the particular case of multi-component systems composed of soft
macroions with their monovalent counterions, together with
8
T. Colla and C.N. Likos
added 1:1 electrolyte. Since we are mostly concerned on the
structural properties, we will focus only on the calculation
of the effective pair interactions, Equation (36), since the
volume contributions have no influence on these properties.
We will consider two different cases for the internal charge
distributions of the macroions – a homogeneous charge distribution and a diffuse Gaussian-like charge distribution:
3Zα
(Rα − r)
4π Rα3
Zα
−r 2 /2Rα2
σα (r) = √
3 e
2π Rα
σα (r) =
Homogeneous,
Gaussian,
Downloaded by [Christos Likos] at 10:59 03 April 2015
(40)
where (x) is the usual Heaviside step function. Since both
distributions are isotropic, the effective electrostatic interactions in Equation (37) are in this case simplified to
r+r 2π ∞
uelec
(r)
=
σ
(r
)r
dr
α (R)RdR. (41)
β
αβ
r 0
|r−r |
Similarly, the linear electrostatic potential energy α
around the particle of component α given by Equation (35),
can be rewritten for the case of an isotropic charge distribution σ α (r) as
βα (r) =
4π λB −κr
e
κr
+ sinh(κr)
r
0
∞
σα (r )r sinh(κr )dr −κr σα (r )r e
dr . (42)
r
Inserting the charge distribution of a homogeneously
charged macroion (first relation in Equation (40)) into this
expression results in the following averaged electrostatic
potential:
3Zβ λB
[κr − (κRα + 1) sinh(κr)] r ≤ Rα ,
κ 3 Rα3 r
e−κr
3Zα λB
r > Rα ,
= 3 3 F (κRα )
κ Rα
r
(43)
The strong electrostatic correlations among the
macroions and their counterions, together with the possibility of the small ions to penetrate in the interior of
the macroions, will obviously result in a strong counterion adsorption inside those particles. As a consequence,
the macroions net charge will be considerably smaller
then its actual ‘bare’ charge. This quantity, known for soft
macroions as the particle effective charge, can be obtained
through the ionic density distributions by calculating the
total amount of ions that lies inside the macroion core. Alternatively, the effective charge can be computed through
the application of the Gauss law at the macroion surface.
This is not to be confused with the concept of the renormalised macroion charge, which is usually introduced to
account for non-linear effects in the field far away from the
macroion particle [25,29–31]. For the particle of component
α, the corresponding effective charge Zαeff can therefore be
written as
Zαeff = Zα + 4π
=−
Rα2
λB
NB
i=1
Rα
zi
ρi (r)r 2 dr
0
dα (r) ,
dr r=Rα
(45)
where ρ i (r) represents the ionic distribution around the
macroion of component α. In this relation, we have assumed
that the electric field remains finite at macroion centre. For
the case of a macroion described by the uniform charge distribution in Equation (40), substitution of the electrostatic
potential (43) into the above relation results in the following
expression for the effective charge as a function of the bare
one Zα in the linear approximation:
Zαeff =
3Zα e−κRα
(κRα + 1)F (κRα ).
κ 3 Rα3
(46)
βα (r) =
In a similar way, the effective charge corresponding to a
particle with the Gaussian charge distribution is obtained by
the substitution of Equation (44) into (45), which provides
the following result:
where we have defined the function F(x) ≡ xcosh (x) −
sinh (x). It is important to note that, contrary to the bare
Yukawa potential for point-like particles, the screened electrostatic potential for uniformly charged macroions remains
finite when r → 0. In the case of particles with the Gaussianlike charge distribution (second relation from Equation
(40)), the resulting averaged electrostatic potential is
Zαeff
2 2 κ Rα
r − κRα2
λB Zα
−κr
exp
e
1 + erf
βα (r) =
√
2r
2
2Rα
2
r
+
κR
α
− eκr 1 − erf
(44)
.
√
2Rα
1 − κRα
Zα e−1/2
2
=
(κRα + 1)e(κRα −1) /2 1 + erf
√
2
2
(47)
2
κRα + 1
2
−2
+ (κRα − 1)e(κRα +1) /2 1 − erf
.
√
π
2
(48)
The comparison between the effective charges for the
two charge distributions at different concentrations of
monovalent salt can be seen in Figure 2. There is a clear
difference between the counterion penetration in the two
Molecular Physics
The above integral can be formally evaluated by means
of a proper analytical extension in the complex k-plane,
followed by an analysis of the poles over this plane. Since
the range of the charge distributions is of the order of the
particle sizes, the large-distance decay of the electrostatic
interactions can be obtained by closing the integral over the
upper half of the complex plane. Application of the residue
theorem – together with the fact that the leading behaviour
is dominated by the pole at iκ – provides
cs = 0 mM
3
-3
4×10
cs = 10 mM
Zeff
-2
cs = 10 mM
3
2×10
0
0
3
2×10
3
Z
4×10
3
6×10
Downloaded by [Christos Likos] at 10:59 03 April 2015
Figure 2. Effective charges for penetrable soft macroions in the
linear approximation, corresponding to uniformly charged particles (solid curves) and particles with a Gaussian-like charge
distributions (dashed curves). The salt concentrations are cs = 0
mM (black curves), cs = 0.001 mM (red curves) and cs = 0.01
mM (blue curves). In all the cases, the particle radius is fixed at R
= 0.5 µm and the reduced concentration is ρR3 = 0.024.
cases, from which one can conclude that the effective charge
is strongly dependent on the macroion conformation. The
effective charges of a uniformly charged macroion are significantly larger than the ones from the Gaussian charge
distribution, corresponding to a much weaker ionic condensation. This is a consequence of the stronger electric
field inside the Gaussian particle resulting from its charge
inhomogeneity, leading to the trapping of a larger number
of counterions in this region. In both cases, the increase in
salt concentration leads to a stronger ionic condensation,
as expected. At zero salt concentration – where the particle is surrounded only by its counterions – the effective
charge in the case of a uniform distribution grows linearly
at small bare charges. A similar behaviour has been observed in the framework of the PB theory [25], probing the
validity of the linear approach in this limit. As the bare
charge becomes larger, the linear approximation for the
ionic response to its interaction with the strongly charged
macroions Equation (16) breaks down, and deviations are
expected in the calculated effective charges [25]. Furthermore, ionic correlations beyond the linear MSA approximation might play an important role on the resulting effective
charges, as recently demonstrated by Moncho-Jord´a in the
context of the primitive model hyppernetted-chain (HNC)
approximation [36].
Once the effective electrostatic potentials are calculated,
the corresponding electrostatic pair interactions uelec
αβ (r) can
be easily obtained by inserting them – together with the
corresponding charge distributions – into Equation (41).
Alternatively, these effective interactions can be obtained
by taking the inverse Fourier transform of the second term
of the right-hand side of Equation (33), which in this case
of isotropic charge distributions reduces to
βuelec
αβ (r) =
λB
π ir
∞
−∞
σˆ α (k)σˆ β (k) ikr
ke dk.
k2 + κ 2
9
(49)
βuαβ (r) ∼ λB σˆ α (iκ)σˆ β (iκ)
e−κr
r
(50)
showing that large-distance pair interactions in the linear
approach will be of Yukawa type, independently of both the
number of ionic (point-like) components and the charge distributions assigned to the macroions, as long as the charge
distribution functions σ α (r) decay sufficiently fast. The coefficients σˆ α (iκ) in (50) are given by
σˆ α (iκ) =
4π
κ
∞
σα (r)r sinh(κr)dr.
(51)
0
A similar result can be obtained regarding the electrostatic potentials α (r) in Equation (42). In particular, if
the functions σ α (r) vanish at distances beyond the particle radius r > Rα , the second integral in Equation
(42) will be zero in this region, and therefore the potential becomes βα (r) = λB σˆ α (iκ)e−κr /r. In the absence
of screening, this relation recovers the well-known result
βα (r) = λB σˆ α (0)/r = λB Zα /r for charged spherically
symmetric objects interacting through the bare Coulombic
potential.
In the case of homogeneously charged macroions, the
Fourier components of the charge distributions, σˆ α (k), are
given by
3Zα
sin(κRα )
,
σˆ α (k) = 2 2 cos(κRα ) −
κ Rα
κRα
(52)
while the ones corresponding to the diffuse Gaussian charge
distribution are
σˆ α (k) = Zα ek
2
Rα2 /2
.
(53)
The effective electrostatic pair interaction can now be
calculated either by the explicitly evaluation of Equation
(49) or by the substitution of the appropriate electrostatic
potentials (43) and (44) into Equation (41). The result for
the case of homogeneously charged macroions of components α and β can be divided into three regions. Assuming
without loss of generality that Rα ≥ Rβ , the potential in the
10
T. Colla and C.N. Likos
region where r ≤ Rα − Rβ is
9Zα Zβ λB 2 4 3
κ Rβ − 3F (κRβ )eκRα
βuαβ (r) =
2κ 6 Rα3 Rβ3 3
sinh(κr)
r ≤ Rα − Rβ ,
× (κRα + 1)
r
(54)
whereas in the region Rα − Rβ < r < Rα + Rβ the pair
interactions are
βuαβ (r) =
9Zα Zβ λB
[γαβ (r) + F (κRα )(κRβ + 1)e−κ(r+Rβ )
2κ 6 Rα3 Rβ3 r
− e−κRα (κRα + 1)(κRβ sinh[κ(r − Rβ )] + cosh[κ(r − Rβ )])],
Downloaded by [Christos Likos] at 10:59 03 April 2015
(55)
and finally, for the non-overlapping region r ≥ Rα + Rβ
the electrostatic pair potential for the uniformly charged
macroions are
βuαβ (r) =
9Zα Zβ λB
e−κr
,
F
(κR
)F
(κR
)
α
β
r
κ 6 Rα3 Rβ3
r ≥ Rα + Rβ . (56)
In Equation (55), we have defined the function γ αβ (r)
as
γαβ (r) =
κ4 1 2
Rα − r 2 Rβ2 − (r − Rα )2
2 2
Rβ4
(r − Rα )4
2r 3
3
R − (r − Rα ) −
+
+
3 β
4
4
2
κ
(57)
+ (r 2 − Rα2 − Rβ2 ) + 1.
2
It can be verified that the above expressions are reduced
to the effective pair potential in an one-component system
of uniformly charged particle described in Refs [16] and
[19] in the limit when Zα = Zβ ≡ Z and Rα = Rβ ≡ R.
In the case of a system of particles with Gaussian-like
charge distributions, the effective electrostatic pair interactions are
2
2
κ 2 Rαβ
r − κRαβ
Zβ Zβ λB
−κr
exp
βuαβ (r) =
1 + erf
e
√
2r
2
2Rαβ
2
r + κRαβ
− eκr 1 − erf
(58)
,
√
2Rαβ
where Rαβ ≡ Rα2 + Rβ2 . It is easy to verify that, in the
situation of an one-component system in the unscreened
limit when κ → 0, the above interactions correctly reproduces the limit of Gaussian charged particles interacting through the bare Coulombic potential [33,35], where
βu(r) = Z2 erf (r/2R)/r. It is also easy to check that the
effective potential asymptotically behaves like a screened
Yukawa potential.
The averaged electrostatic pair potentials for both the
cases of uniform and Gaussian-like distributions depend
on a large number of control parameters, such as the
particle concentrations, the size and charge asymmetries,
the screening parameter (or equivalently the amount of
added 1:1 electrolyte) and the solvent permissibility. A rich
variety of effective interactions can be therefore induced by
the proper adjustment of these quantities. In Figure 3, the
effective electrostatic potentials resulting from Equations
(54), (55), (56) and (58) are compared for the specific case
of an asymmetric two-component system with particles of
opposite charge Z + = −Z− = 500, same radius R + = R−
= 0.5μm and same concentrations. Clearly, the effective
electrostatic interactions at overlapping distances are quite
different for the situations of Gaussian and homogeneous
distributions. In the case of uniformly charged macroions,
the electrostatic forces when the particles overlap are much
stronger when compared with the ones from the Gaussian
system. This is consistent with the fact that the ionic condensation is much weaker in the case of uniformly charged
macroions, and therefore the strength of the electrostatic
interactions is enhanced with respect to the Gaussian
particles. In both cases, addition of salt leads to a decrease
of the electrostatic interactions, resulting from the stronger
screening effects. As the particle–particle distance grows
larger, the effective interactions are in both cases similar
to the ones of a Yukawa system of oppositely charged
particles.
It is important to emphasise that the effective pair interactions in Equations (54), (55), (56) and (58) have been
obtained in the framework of the linear approximation, and
therefore their validity breaks down when the macroion–
ion interactions are too strong – namely at large macroion
charges. For the case of uniformly charged macroions surrounded by monovalent ions in an aqueous solution at room
temperature, it has been recently shown [25,28] that the
linear approximation is sufficiently accurate for macroion
charges up to Zα ≈ 1500. It is important to note that the
results for both pair interactions and correlation functions
in this work are therefore within the range of validity of the
linear approximation.
4. Structure of the binary charged system
The investigation of the structural properties of the multicomponent effective system requires – apart from the already described averaged pair electrostatic interactions –
the further introduction of a short-range repulsive pair
potential in Equation (36), which is important to avoid the
unstable collapse of the oppositely charged soft particles on
the top of one another. Here, we model this short-range repulsions through the traditional Hertzian potential between
Molecular Physics
0
(a)
ρR = 0.024
3
ρR = 0.060
200
3
Z+ = - Z- = 500
100
0
1
-100
3
4
-300
Z+ = - Z- = 500
cs = 0 mM
0
0
(c)
1
Z+ = - Z- = 500
100
ρR = 0.024
3
ρR = 0.060
-100
Z+ = - Z- = 500
-4
-200
-4
0
-300
2
r [μm]
3
4
(d)
cs = 10 mM
cs = 10 mM
1
3
3
ρR = 0.024
3
ρR = 0.060
200
0
2
r [μm]
3
β u+-(r)
β u++(r)
300
2
r [μm]
ρR = 0.024
3
ρR = 0.060
-200
cs = 0 mM
0
(b)
3
β u+-(r)
β u++(r)
300
Downloaded by [Christos Likos] at 10:59 03 April 2015
11
4
0
1
2
r [μm]
3
4
Figure 3. Electrostatic pair interactions for a binary system of equally sized, oppositely charged macroions with charge Z + = −Z− =
500 and radius R = 0.5 µm. The concentrations of both components are the same, and are characterised by the reduced density ρ + R3 =
ρ − R3 ≡ ρR3 . The solid curves represent the interactions for the homogeneously charged particles, while the dashed ones correspond to the
interactions between particles with the Gaussian charge distribution. Black curves stands for ρR3 = 0.024, and the red ones correspond
to ρR3 = 0.060. (a) and (b) represent the system in the absence of added monovalent salt (cs = 0 mM), while the curves in (c) and (d)
correspond to the salt concentration cs = 10−4 mM.
deformable soft particles:
βu0αβ (r) = βuHαβ = αβ 1 −
=0
r
(Rα + Rβ )
5/2
r ≤ Rα + Rβ ,
r > Rα + Rβ ,
(59)
where the strength parameters αβ are related with the Poisson’s ratio and the Young’s modulus of the particle material
[19], and depend on the particle size of components α and β.
Since from now on we consider only equally sized particles,
the strength parameter will be same for all the pair interactions, αβ ≡ . The Hertz potential (59) is the characteristic
short-range repulsion in the systems of microgel particles,
where the repulsion for overlapping distances comes from
the mutual deformation of their internal polymer chains.
The parameter controls the strength of this short-range
repulsion. For small values of this parameter, the oppositely charged particles will be able to penetrate themselves
in order to minimise their electrostatic energy. As grows
larger, the particles can no longer penetrate each other, and
in the limit of very large strength parameters we expect the
system to behave similarly to a system of hard charged particles. Obviously, the repulsive Hertzian interactions will
play a paramount role on the system structure, dictating the
degree in which oppositely charged particles are allowed to
overlap each other. This is to be contrasted with the situation of oppositely charged hard colloidal particles, where
the volume exclusion strongly constrains the resulting system structure, especially at high packing fractions.
Figure 4 shows the effective pair interactions in a binary system of uniformly charged particles with Z + = 420
and Z− = −Z + /2 = −210. In the case of positive–negative
interactions, there is a clear competition between the shortrange repulsion and the electrostatic attraction. At small
, particle interpenetration is favourable, as showed by the
minimum in u± (r) at r ≈ R for = 100 (Figure 4(a)).
As the strength of the short-range repulsion increases, they
rapidly become dominant over the electrostatic attraction at
short distances, and the minimum in the positive–negative
potential is shifted to larger values r ≈ 2R. The minimum
also becomes sharper, meaning that the particles have less
freedom to oscillate around their equilibrium positions. In
the case of positive–positive repulsion, the increase of the
short-range repulsion has a weaker effect, since the electrostatic repulsion at contact is already very large, making particle penetration energetically unfavourable even at
small (Figure 4(b)). A similar behaviour is observed in the
negative–negative repulsions, where then the effects from
T. Colla and C.N. Likos
75
(a)
25
0
ε = 100
ε = 750
ε = 2000
(b)
75
βu++(r)
50
βu+-(r)
100
ε = 100
ε = 750
ε = 2000
100
50
75
βu--(r)
12
50
25
25
0
-25
0
1
2
3
r [μm]
0
0
0.5
1
1.5
r [μm]
2
0
2.5
1
2
3
r [μm]
increasing are a bit more pronounced due to the weaker
electrostatic repulsion (inset in Figure 4(b)).
Once the effective interactions are properly defined in
the multi-component system, the corresponding structure
can be analysed by means of the OZ equation, Equation
(29). For the case of an isotropic effective two component
system made up of oppositely charged particles, this equation in Fourier space can be translated as
In order to establish a full set of relations, the above
OZ equation is here supplemented with the HNC relation
among all the components:
cαβ (r) = hαβ (r) − βuαβ (r) − ln[hαβ (r) + 1].
1 2
(1 − ρ− cˆ−− (k))cˆ++ (k) + ρ− cˆ+−
(k)
D(k)
1 2
(1 − ρ+ cˆ++ (k))cˆ−− (k) + ρ+ cˆ+−
hˆ ++ (k) =
(k)
D(k)
cˆ+− (k)
hˆ +− (k) = hˆ −+ (k) =
,
(60)
D(k)
where the function D(k) is given by
2
D(k) = (1 − ρ+ cˆ++ (k)) (1 − ρ− cˆ−− (k)) − ρ+ ρ− cˆ+−
(k).
(61)
(a)
g++(r)
g+-(r)
g--(r)
Z+ = 420
Z- = 210
ε = 100
g(r)
3
2
10
(b)
6
g++(r)
g+-(r)
g--(r)
Z+ = 420
Z- = 210
ε = 750
4
(c)
g++(r)
g+-(r)
g--(r)
Z+ = 420
Z- = 210
ε = 2000
8
g(r)
4
(62)
The above HNC relation is known to be very accurate for
systems interacting with Yukawa-like potentials [37], as
well as for systems of soft particles [38]. For the calculated effective pair potentials βuαβ (r) (with α, β = ±), the
above set of equations is evaluated as follows. First, a set of
guess functions for the direct correlation functions cαβ (r)
are estimated. Taking the Fourier transform of these functions and inserting them into the OZ equation, Equations
(60) and (61), allows for the calculation of the corresponding Fourier components of the total correlation functions
hˆ αβ (k), from which the functions hαβ (r) are evaluated. Inserting the total correlation functions in the HNC relation,
Equation (62), provides new estimations for the total correlation functions cαβ (r). The whole process is then repeated
until numerical convergence is achieved.
hˆ ++ (k) =
g(r)
Downloaded by [Christos Likos] at 10:59 03 April 2015
Figure 4. Effective pair interactions for a binary system of equally sized, oppositely charged macroions with asymmetric charge Z + =
420 and Z− = −Z + /2 = −210. All particles are uniformly charged, and both components possesses the same radius R = 0.5 µm and
reduced concentration ρR3 = 0.06. The strength parameters of the Hertz potential are = 100 (black curves), = 750 (red curves) and
= 2000 (blue curves). The inset in (b) shows the repulsive negative–negative interactions.
6
4
2
1
0
0
2
1
2
r [μm]
3
4
0
0
1
2
r [μm]
3
4
0
0
1
2
r [μm]
3
4
Figure 5. Pair correlation functions between the positively (black curves) and negatively (blue curves) charged components, as well as
the positive–negative pair correlations (red curves) for a system of oppositely charged macroions with homogeneous charge distributions,
in the absence of added salt. The particle charges are Z + = 420 and |Z− | = 210. Both components have the same size R = 0.5 µm
and concentrations ρR3 = 0.06, corresponding to a reduced screening length of κR = 0.824. The strength parameters of the short-range
Hertzian potential are = 300 (a), = 750 (b) and = 2000 (c).
Molecular Physics
5
(a)
g++(r)
g+-(r)
g--(r)
Z+ = 420
Z- = 210
ε = 100
g(r)
g++(r)
g+-(r)
g--(r)
Z+ = 420
Z- = 210
ε = 750
4
g(r)
2
6
(b)
3
g++(r)
g+-(r)
g--(r)
Z+ = 420
Z- = 210
ε = 2000
4
2
1
(c)
g(r)
3
13
2
1
0
0
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Figure 6.
1
2
r [μm]
3
4
0
0
1
2
r [μm]
3
4
0
0
1
2
r [μm]
3
4
Same as in Figure 5, but for a system of macroions with the Gaussian charge distribution from Equation (40).
In Figure 5, the calculated pair correlation functions for
the system of uniformly charged macroions in the absence
of added salt are shown for increasing values of . All the
particles are equally sized (R + = R− = 0.5 µm), but the
positive particles are twice as charged as the negative ones
(Z + = 420 and Z− = −210), corresponding to the pair
interactions depicted in Figure 4. In this case, the system
structure is strongly influenced by the charge asymmetry.
At the lowest value of , the competition between electrostatic attraction and the short-range repulsion leads to a
significant particle penetration, as clearly suggested by the
first peak of g + − (r) in (Figure 5(a)). Interesting enough, the
position of this peak is shifted with respect to the minimum
of u + − (r) in Figure 4(a). This is because the interactions
in this charge asymmetric case are strongly dominated by
the positive–positive repulsion: as two positively charged
particles try to penetrate into a negative one, they experience a mutual repulsion which will effectively reduce particle penetration. The strong oscillations in the correlation
functions in this case suggest a scenario in which particles
are assembled to form small chains (most likely with four
particles each). Again, the chain structure is dictated by
the stronger repulsion between positive particles. While the
positive particles are constrained to be as far as possible
from one another, the weaker repulsion between negative
particles means that they have more freedom to oscillate
around their equilibrium positions, resulting to the broader
first peak of g−− (r) as compared with the on from g + + (r).
Another consequence is that negative particles will be in average closer to one another in comparison with the positive
ones.
By increasing , the strong energetic penalty for particle penetration (see Figure 4(a)) leads to the break of the
chain structure. Now, the energy gain corresponding to two
oppositely charged particles at contact is not enough to overcome the penalty in having two positively charged particles
attached to a single negative one (due to the strong positive–
positive repulsion), and the chain formation breaks down.
The strong peak in g + − (r) suggests in this case a strong
tendency of formation of single dipoles (Figure 5(b) and
5(c)). Another structure that might become favourable in
this case corresponds to the situation of two negative particles attached to a positive one, together with a free positive
particle. In this case, the weaker repulsion between the negative particles allows them to move around the periphery of
the positive one, leading to the wide spread peak between
r ≈ 2R and r ≈ 4R in Figure 5(b) and 5(c).
In the case of macroions with the Gaussian-like charge
distribution, the electrostatic effects are much weaker, the
pair correlations are less structured as compared with the
ones from the uniformly charged particles, and the effects
from the short-range repulsion clearly dominate. This is
shown in Figure 6, in which the same pair correlation functions as in Figure 5, but now for the case of Gaussian
particles, are displayed. Even for the smaller considered,
the degree of particle penetration is very small, and the
system structure is quite different from the one corresponding to uniformly charged particles. As a consequence of
the weaker electrostatic forces, there are very small differences between the positive–positive and negative–negative
pair correlations. Again, the strong peak in g + − (r) at
r ≈ 2R clearly indicates the formation of dipolar pairs.
The peaks in the positive and negative correlations at this
same point suggest that now the weak electrostatic forces
lead to structures in which positive and negative particles
are symmetrically attached around the axis of a dipolar pair.
Finally, the effects of increasing the screening parameter κ – or alternatively the concentration of monovalent
salt – over the underlying particle correlation functions are
shown in Figure 7. The particles are uniformly charged,
and the strength parameter is fixed in = 300. Again,
the two components have the same radius R = 0.5 µm, are
equally concentrated ρ + R3 = ρ − R3 = 0.095, and are now
also equally dissociated (Z + = −Z− = 550). Similarly to
the parameter , the screening length plays a key role in
controlling the relative strength between the short-range repulsion and the electrostatic effects. At the lowest possible
inverse Debye screening length κR = 1.378 (corresponding
to the absence of added salt), the electrostatic effects are
clearly the dominant ones, and the system is structured in
14
T. Colla and C.N. Likos
4
5
g++(r)
g+-(r)
3
g++(r)
g+-(r)
Z+ = 550
Z- = 550
cs = 0.1 mM
3
g(r)
2
2
2
1
0
Z+ = 550
Z- = 550
cs = 0.005 mM
4
g(r)
3
g(r)
g++(r)
g+-(r)
Z+ = 550
Z- = 550
cs = 0 mM
4
0
1
1
2
4
r [μm]
6
8
0
0
2
4
r [μm]
6
0
0
2
4
6
r [μm]
Downloaded by [Christos Likos] at 10:59 03 April 2015
Figure 7. Positive–positive (black curves) and positive–negative (red curves) pair correlation functions for a system of oppositely charged
macroions with uniform charge distributions at different salt concentrations. Both components have the same size R = 0.5 µm, same
concentration ρR3 = 0.095 and charges with same magnitude Z + = |Z− | = 550. The salt concentrations inside the system are cs = 0 mM
(a), cs = 0.005 mM (b) and cs = 0.1 mM (c), corresponding to reduced inverse Debye screening length of κR = 1.378, κR = 3.939 and
κR = 16.562, respectively.
a perfect periodic way, with alternating positive and negative particles typical of a chain-like formation. As the
salt concentration increases (stronger screening), the system clearly becomes less ordered. At the largest screening
considered κR = 16.562 (cs = 0.1 mM), the electrostatic
interactions are so screened that there is virtually no distinction between positive and negative charges anymore: the
system essentially behaves like a neutral system of particles
interacting through the short-distance Hertzian repulsion
(Figure 7(c)).
Although the OZ equation provides a very simple
and efficient way to extract information about the system structure, its application for strongly interacting multicomponent system is still very limited. This is because
these systems usually display complex self-assembly structural formations, from which a detailed analysis cannot be
performed through the pair correlations alone. In this situations, more elaborated quantities – such as for instance
the Euler characteristic [39,40] – have to be also considered
to fully describe the underlying system morphology. It is
however important to emphasise that the pair correlations
do help to get physical insight over the effects from the various system parameters on the local structure. Furthermore,
they are extremely useful to identify the regions in the large
parameter space that are interesting to look at closely using
more sophisticated approaches.
5. Conclusions
We have employed the formalism of the linear response
approximation to formally obtain the effective interactions
in charged systems with an arbitrary number of charged
components. To this end, the system was separate into two
classes – the mesoscopic particles and the microscopic
ones – whose typical length scales are order of magnitudes apart from one another. The degrees of freedom of
the smaller components were then effectively integrated in
the framework of the linear approximation, resulting in general expressions for the effective interactions between the
charged mesoscopic components. The formalism has been
then applied to study the effective interactions between soft
charged particles with two different intrinsic charge distributions: homogeneously charged particles and a diffuse,
Gaussian-like charge distribution. The short-range repulsion between those macroions was set to be the Hertzian
potential for deformable particles, typical in the description of microgel particles. The electrostatic interactions,
the effective charge and the pair correlation functions for
both internal charge distributions were compared at different system parameters. It was found that the counterion
penetration is much larger in the case of the Gaussian distribution, leading to a much weaker effective electrostatic
interaction among these particles.
In what concerns the structural properties, it was shown
that the balance between the short-range Hertzian potential and the electrostatic effects in a system of oppositely
charged macroions plays a crucial role in determining the
particle structure as represented by the pair correlation
functions. When the electrostatic effects dominate, the system structure at high concentrations resembles the one in
which positive and negative particles are periodic structured
in a chain-like formation. Further investigation of the particle conformation in these system requires, however, the
use of more advanced techniques which go beyond the calculation of pair correlation functions [39–41], since these
are not enough to describe the complex system morphology in a complete way. It would be also interesting to study
the underlying time-dependent system structure as it approaches equilibrium. For all this analysis, it is convenient
to study the calculated effective interactions by means of
computer simulation techniques – such as Brownian dynamics and molecular dynamics simulation – which allow
for a detailed investigation of the particle self-assembly.
Work along these lines is currently in progress.
Molecular Physics
Acknowledgements
It is a pleasure and a great honour to dedicate this work to Prof.
Jean-Pierre Hansen, whose enormous contributions to the theory
of liquids – and in particular of systems of charged particles –
have dramatically improved our knowledge about these systems.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was partially supported by the Conselho Nacional de
Pesquisa (CNPq) [grant number 236575/2012-0].
Notes
1.
Downloaded by [Christos Likos] at 10:59 03 April 2015
2.
Throughout this paper, we adopt the convention that the mesoscopic components are denoted by Greek indices.
We adopt here for calculational convenience an asymmetric
definition of the structure factor with respect to indices exchange. This plays however no role in what follows, since
the response function χˆ ij (k) = −βρi Sij (k) does recover exchange symmetry between indices i and j.
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