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PAPERMiha Ravnik and Slobodan ŽumerNematic braids : 2D entangled nematic liquid crystal colloids
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PAPERDominic Lenz et al.Colloid–dendrimer complexation
Soft Matter Volume 5 | Number 22 | 21 November 2009 | Pages 4341–4584
Modelling of soft matter
PAPER www.rsc.org/softmatter | Soft Matter
Colloid–dendrimer complexation†
Dominic A. Lenz,* Ronald Blaak and Christos N. Likos
Received 10th June 2009, Accepted 27th July 2009
First published as an Advance Article on the web 18th August 2009
DOI: 10.1039/b911357f
We consider amphiphilic dendrimers of the second generation by an appropriate modeling of the inter-
monomer interactions, introducing suitable attractive microscopic potentials between the inner,
solvophobic monomers and repulsive potentials between the outer, solvophilic ones. By employing
Monte Carlo simulations, we examine the selective adsorption of such dendrimers onto compact
colloidal particles that attract either the core or the shell of the dendrimer, while repelling the rest of the
molecule. We measure the effective colloidal–dendrimer interaction and we analyze in detail the
conformations of the dendrimer close to the colloid in dependence of the radius of the latter and the
microscopic energy parameters. A dendron-by-dendron adsorption of the dendrimer on the colloid
under suitable conditions is discovered.
I. Introduction
Dendrimers form a special class of macromolecules, which have
an enormous potential due to their regularly branched archi-
tecture. Although they were initially considered to be a curiosity,
shortly after their first synthesis by V€ogtle and coworkers,1 they
gradually drew vivid interest from various branches of science.
The building blocks of an idealized dendrimer consist of identical
monomeric units that have a functionality larger than two, i.e.,
they can chemically bind to three or more of the same type of
units. Starting with a single or a pair of these units, one can grow
in a controlled fashion successive generations by chemically
binding a new unit to all available binding sites. It is, however, by
no means necessary that all units are truly identical; for instance,
it is possible to tailor a dendrimer by using a different type of unit
for each generation and, in fact, even within a generation not all
units need be the same.2–4
The enormous flexibility to choose the building units of den-
drimers, in combination with a well-defined structure, makes
them an interesting object of research in physics, chemistry and
medicine, and has lead to applications in gene transduction,5
drug delivery,6,7 bio-sensors8 and contrast agents.2,9 Most
research on such systems to-date has been restricted to their
physical and chemical properties, and to their behavior in solu-
tion.10,11
The fact that by carefully choosing the different constituents
forming the dendrimers the overall behavior can be manipulated,
also makes them very interesting from a fundamental point of
view, because by tuning the generation number and/or bond
stiffness, they can be made to interact with each other as hard,
compact-sphere-like objects as well as very soft, blob-like
objects.12–14 This last property renders dendrimers into ideal
candidates for the study of cluster formation.15–19 Since dendritic
macromolecules of lower generation numbers have a rather open
Institute for Theoretical Physics II: Soft Matter, Heinrich-Heine-Universit€atD€usseldorf, Universit€atsstraße 1, D-40225 D€usseldorf, Germany. E-mail:[email protected]
† This paper is part of a Soft Matter themed issue on Modelling of softmatter. Guest editor: Mark Wilson.
4542 | Soft Matter, 2009, 5, 4542–4548
structure, they can easily overlap with each other and their
centers-of-mass can come arbitrary close to one another. Such
a soft interaction potential is required (but not sufficient) for the
formation of cluster solids. The crystal lattice of such a solid will
not change its lattice constant under compression but will rather
incorporate more particles per lattice site. Mladek et al.18 pre-
dicted that amphiphilic dendrimers are suitable candidates to
exhibit this behavior, which could possibly be enhanced even
more by the presence of nearby surfaces.19
In a previous study,20 the effect of planar walls on the behavior
of amphiphilic dendrimers of a second generation has been
analyzed. Since these molecules are formed by a solvophobic core
and a solvophilic shell, they exhibit a ‘‘dense-shell’’ structure,18
rather than the usual ‘‘dense core’’-forming dendrimers.10,21–23
The presence of an interacting surface evidently affects the
conformations of the dendrimers and, depending on whether the
core or shell monomers are attracted by the surface, two arche-
types of conformations have been found,20 which were named
‘‘dead spider’’ and ‘‘living spider’’ respectively. Here we concen-
trate on the effect that the curvature of a wall has on the
adsorption behavior of these amphiphilic dendrimers by making
use of Monte Carlo simulations.24 The rest of the paper is
organized as follows. In Section II we highlight the main features
of the dendrimer model18 and introduce the curved surface by
means of an idealized spherical colloid. By simulating a single
dendrimer and a colloidal sphere, we obtain the effective den-
drimer–colloid interaction in Section III. In Section IV we
analyze the change in conformation of the dendrimers due to the
adsorption. Since the adsorption behavior is also affected by the
dendrimer concentration, we examine the influence of the latter
in Section V and we finish in Section VI with a summary and
concluding remarks.
II. Model
To simulate amphiphilic dendrimers, we adopt the model intro-
duced by Mladek et al.18 We consider dendrimers of generation
G¼ 2 with two central core particles and functionality f¼ 3. The
resulting total of 14 monomers are divided in two classes, which
This journal is ª The Royal Society of Chemistry 2009
are required to obtain the amphiphilic property. The two central
monomers and the four monomers of the first generation form
the solvophobic core particles, which we label by the subscript C.
The eight monomers of the second generation form the sol-
vophilic shell of the dendrimer and are labeled by the subscript S.
Interactions between any two monomers (bonded or not),
separated by a distance r, are modeled by the Morse potential25
given by:
bFMorsemn (r) ¼ 3mn{[e�amn(r�smn) � 1]2 � 1}, mn ¼ CC, CS, SS (1)
where smn is the effective diameter between two monomers of
species m and n and b ¼ (kBT)�1 the inverse temperature, kB
denoting Boltzmann’s constant. This potential is characterized
by a repulsive short-range behavior and an attractive part at long
distances, whose depth and range are parametrized via 3mn and
amn, respectively. The diameter sc of the core monomers is chosen
to be our unit of length.
Bonding between two neighboring monomers within the
dendrimer is modeled by the finite extensible nonlinear elastic
(FENE) potential, which is given by:
bFFENEmn ðrÞ ¼ �KmnR
2mnlog
"1�
�r� lmn
Rmn
�2#; mn ¼ CC; CS (2)
where a spring constant Kmn restricts the monomer separation to
be within a distance Rmn from the equilibrium bond length lmn.
In order to achieve the intended amphiphilic behavior of the
dendrimers, while staying within the limits of what can be real-
ized in experiments, we use the set of parameters (see Table 1)
from the D2-model dendrimer,18 which also enables us to
compare the results with those stemming from the simulation of
dendrimers in the vicinity of a planar wall.20
To model the colloidal particle on which the dendrimers
adsorb, we make use of the same ideas as those employed for
modeling a planar wall.20 We assume that the colloidal particle of
radius Rs consists of a homogeneously distributed collection of
Lennard-Jones type particles with diameter sLJ and, without loss
of generality, density rs3LJ¼ 1. Consider first a simple, power-law
pair potential acting between a volume element of the colloid and
a hypothetical test (monomer) particle:
bvðnÞ0 ðrÞ ¼ 43
�s
r
�n
(3)
with some length scale s and dimensionless energy scale 3. Then,
the total interaction felt by a particle at position r outside the
spherical colloid is given by
Table 1 Overview of the monomer interaction potential parametersused between core (C) and/or shell (S) monomers
Morse 3 asc s/sc
CC 0.714 6.4 1CS 0.014 19.2 1.25SS 0.014 19.2 1.5
FENE K/sc2 l/sc R/sc
CC 40 1.875 0.375CS 30 3.75 0.75
This journal is ª The Royal Society of Chemistry 2009
bF(n)sphere(r) ¼ rb
Ðd3r0v(n)
0 (|r0 � r|)Q(Rs � |r0|) (4)
where r h |r| and Q(z) is the Heaviside step function. This
integral can readily be carried out to yield:
bFðnÞsphereðrÞ ¼
8p3rs3
xðn� 2Þðn� 3Þðn� 4Þ
"ðn� 3ÞXs � x
ðx� XsÞn�3þðn� 3ÞXsþx
ðxþ XsÞn�3
#
(5)
where x h r/s and Xs h Rs/s. The above potential diverges as the
test particle approaches the surface of the sphere from above,
r / Rs+, so that the colloidal particle is indeed impenetrable to
individual monomers. Further, it manifestly depends on the
dimensionless energy and density parameters 3 and rs3 only
through their product, so that a change in the internal density of
the colloid can be re-adsorbed into a modification of 3. Thus, we
vary here only the latter quantity, fixing the former at the value
rs3LJ ¼ 1 quoted above.
Since we consider amphiphilic dendrimers, where core and
shell particles interact differently, the interactions of the mono-
mers with the colloid will also differ and can either have an
attractive range or are purely repulsive. For those monomers that
experience an attractive interaction to the colloid, we employ the
usual Lennard-Jones interaction between the monomers and the
point particles forming our colloid. The resulting interaction
with a monomer at a distance r from the colloid center therefore
follows from eqn (5) and is given by
F(att)sphere(r) ¼ F(12)
sphere(r) � F(6)sphere(r) (6)
where s¼ (sLJ + sm)/2 and m corresponds to either a core or shell
monomer, depending on which of those is attracted to the sphere.
To model the repulsive interaction of the colloid with the
remaining species of monomers, we use the same method as for
the shifted Lennard-Jones type of interaction, i.e., we determine
the distance r* at which the minimum energy in the interaction (6)
is found, and apply the appropriate truncation and shift to the
attractive potential of eqn (6), rendering the total interaction
purely repulsive, viz.:
FðrepÞsphereðrÞ ¼
(FðattÞsphereðrÞ � F
ðattÞsphereðr*
Þ r # r*
0 r . r*
: (7)
In what follows, we will switch between the case in which the
core monomers are attracted to the colloid and the shell ones
repelled from it and the complementary one. To simplify the
parameters, we chose sLJ ¼ sc. Note that in the limit Rs / N
this reduces to the planar wall interaction.20
III. The effective sphere–dendrimer interaction
In order to investigate the effect colloidal particles have on
nearby dendrimers, we measure the effective dendrimer–colloid
potentials. Hereto, we perform Monte Carlo simulations in the
canonical ensemble, i.e., at constant particle number N, simula-
tion box volume V, and temperature T. In particular, we simulate
the interaction of a single dendrimer, represented in a coarse-
grained fashion by its center-of-mass, with a single colloidal
particle. One method to calculate the effective interaction is by
Soft Matter, 2009, 5, 4542–4548 | 4543
Fig. 2 Effective dendrimer–colloid interactions for the two different
dendrimer types, varying the energy parameter 3 and the colloidal radius
Rs, as indicated in the legends. (a) CA-SR with 3 ¼ 0.5, (b) CA-SR with
3 ¼ 1.0, (c) CR-SA with 3 ¼ 0.5, and (d) CR-SA with 3 ¼ 2.0. The
interaction is plotted against the distance h between the colloidal sphere
and the center-of-mass of the dendrimer. The insets in panels (b) and (c)
applying the umbrella sampling technique24 to constrain the
center-of-mass of the dendrimer within a certain range from the
spherical particle, measure the probability of finding the den-
drimer at a given distance, and combine such probability profiles
in order to obtain the full effective interaction potential.20
Here we adopt an alternative but equivalent method, by
keeping the center-of-mass of the dendrimer at a fixed distance
from the spherical particle. This can easily be achieved by using
a Monte Carlo type of move in which a random displacement of
one of the monomers is generated and combined with a correct-
ing translation of the full dendrimer in the opposite direction. A
summation of the individual forces on each of the monomers by
the colloid then yields the effective force of which the ensemble
average is calculated. For reasons of symmetry this average will
be directed along the vector connecting the centers-of-mass of the
sphere and the dendrimer. Finally, the effective dendrimer–
colloid potential can be obtained by integrating the force over the
distance.
We will distinguish two types of colloidal particles that differ
in their interaction with the core and shell monomers of the
dendrimers. The first type is characterized by an attractive
interaction with the core and a repulsive interaction with the shell
(CA-SR).20 Fig. 1(a) shows the effective interaction for a colloid
with a radius Rs ¼ 8sc and several different interaction strengths
3. In order to facilitate the comparison with results of different
sphere diameters and the limiting case of a planar wall, the
interaction is plotted as a function of the distance h between the
center-of-mass of the dendrimer and the surface of the spherical
colloid. Similar to the case of a planar wall, the interactions
remain purely repulsive for relatively weak values of 3 ( 1. For
intermediate interaction strengths, the dendrimer experiences
a repulsive barrier at a distance of roughly 4sc and an attractive
well at closer distances. For 3 T 3, the barrier disappears almost
completely.
The second type of spherical particle we consider, has the
opposite properties, i.e., it repels the core monomers of the
dendrimers and attracts the shell monomers (CR-SA). For the
same sphere radius Rs ¼ 8sc the resulting effective interaction
potential is shown in Fig. 1(b). Since in this case the shell
monomers that are close to the sphere are already attracted, the
dendrimer does not experience a barrier, but only an attractive
well, and only at small distances do the short-range repulsive
forces become dominant. A striking difference between the two
types of colloids can be observed for the large values of 3.
Fig. 1 Effective interactions between a colloidal particle with radius
Rs ¼ 8sc and a single dendrimer for different values of interaction
strength 3: (a) the CA-SR case; (b) the CR-SA case. The interaction is
plotted against the separation h between the dendrimer’s center-of-mass
and the colloidal surface.
4544 | Soft Matter, 2009, 5, 4542–4548
Whereas in the CA-SR type the effective interaction remains
rather smooth, it turns out that for a colloid of the type CR-SA
the interaction becomes more irregular. As will be explained in
the next section this is not an artifact but it is rather related to
structural changes in the dendrimer itself.
It is evident that the curvature of the colloidal sphere has
a large influence on the adsorption behavior of the dendrimer.
For large radii, the curvature is negligible and one could easily
describe the system by using a planar wall. For smaller radii,
especially those of the same order of magnitude as the dendrimer
itself, the dendrimer will behave very differently. In order to
examine this more closely, we compare in Fig. 2 the effective
dendrimer–colloid interaction obtained by Monte Carlo simu-
lations for different radii and for both types of spheres.
The fact that the interaction becomes weaker on decreasing the
sphere radius is not surprising. This is a direct consequence of the
fact that the total interaction of the colloidal particle scales with
the size of the sphere as follows from eqn (5). The observation that
the interaction strength remains finite at a distance h¼ 0 can also
easily be explained, by realizing that the center-of-mass of the
show details at zoomed-in regions of h.
Fig. 3 Simulation snapshots of the two archetypes of dendrimer
conformations near a colloidal sphere of radius Rs ¼ 8sc. (a) ‘‘Dead
spider’’-conformation for a CA-SR type of sphere. (b) ‘‘Living spider’’-
conformation for a CR-SA type of sphere.
This journal is ª The Royal Society of Chemistry 2009
Fig. 4 Radial density profiles of the dendrimer’s monomers for a CA-SR
dendrimer close to a sphere with radius Rs ¼ 8sc and 3 ¼ 1.0. Two
different, fixed distances h between the dendrimer’s center-of-mass and
the colloidal surface are shown: (a) h ¼ 3sc and (b) h ¼ 6sc. The profiles
correspond to core monomers of generation 0 (Z), generation 1 (C0), and
shell monomers (S).
dendrimer can actually lie within the sphere, due to the curvature
of its surface. Whereas the individual interactions of the mono-
mers at this position would indeed diverge, a dendrimer that lies
against the surface will follow the curvature, thus adopting
a conformation for which the center-of-mass falls within the
colloidal particle. This is illustrated in Fig. 3(b) where a dendrimer
in the vicinity of a sphere of the CR-SA type with radius Rs¼ 8sc is
shown. Consequently, the effective interaction will still diverge,
but at distances h < 0 provided the sphere is large enough. For
small spherical particles the effective interaction remains finite,
because the dendrimer can fold itself around it.
For relatively weak interactions, 3 ( 1, in the case where the
colloidal particle attracts the core monomers (CA-SR–type), the
interaction remains purely repulsive for all radii, as can be seen in
Fig. 2(a) and (b). The plateaus found for the large spheres, see
insets of Fig. 2(b), correspond to structural rearrangements of the
dendrimer and disappear for smaller values of the sphere radius.
In the opposite case of a dendrimer of the CR-SA type, shown in
Fig. 2(c) and (d), we observe that the depth of the attractive range
decreases with decreasing sphere radius and the minimum shifts
towards the center, which signals that the center-of-mass of the
dendrimer can reach deeper in the interior of the sphere.
Fig. 5 Radial density profiles of the dendrimer’s monomers for CR-SA
dendrimers close to a sphere with radius Rs ¼ 8sc and 3 ¼ 3. Four
different center-of-mass to surface distances h are considered: (a) h ¼1.5sc, (b) h ¼ 3sc, (c) h ¼ 4sc, and (d) h ¼ 5sc. The insets show the
cumulative number of monomers; the horizontal axis there has the same
meaning as that of the main plots.
IV. Dendrimer conformations
The natural conformation for an isolated amphiphilic dendrimer
is to have the core monomers, in general, in close proximity to the
center-of-mass and that the distance of its monomers from the
center increases with the generation number. This is a direct
consequence of the bonds within the macromolecule. Although
for simple, athermal dendrimers one can observe that for
entropic reasons outer monomers can be found close to the
center, this is just a relatively small fraction of the overall
monomer population. This process of back-folding is easily
suppressed by the presence of additional interactions, e.g., in
charged dendrimers.26 Also in the present case of amphiphilic
dendrimers, the presence of solvophilic end groups in the shell
and a solvophobic units in the core will retain the natural den-
drimer structure in bulk.
The vicinity of nearby objects, however, will affect the internal
structure of the dendrimers. For planar walls, two archetypes of
configuration were found.20 In the case of a CA-SR wall, the core
monomers are found near the wall and the shell particles are
repelled by it. For the other type of wall, the CR-SA case, it is the
shell monomers that adsorb to the surface. These two different
dendrimer conformations, the ‘‘dead spider’’ and ‘‘living spider’’
respectively, are also found for colloidal spheres as illustrated in
Fig. 3. It is clear that the conformations become deformed due to
the curvature of the surface, an effect that increases with smaller
sphere radii.
To analyze the conformations in more detail, we consider the
radial density profiles of the monomers around a colloidal
sphere. In Fig. 4 we show radially-averaged monomer densities
around a colloid with radius Rs ¼ 8sc and interaction strength
3 ¼ 1.0 for the case of a colloidal particle of type CA-SR. These
profiles are resolved in the individual generation of the mono-
mers, i.e., the two central core particles of generation 0 (Z), the
four core particles of generation 1 (C0), and the eight monomers
of generation 2 that form the shell (S). For these profiles, the
This journal is ª The Royal Society of Chemistry 2009
center-of-mass of the dendrimer is kept fixed during the simu-
lation. Fig. 4(b) shows the profiles for a relatively large distance
h ¼ 6sc between the dendrimer and the surface of the spherical
particle. At this distance the conformation of the dendrimer is
very much like the one in bulk, and only a small asymmetry in the
distribution of the shell arises due to its repulsion by the colloid.
For a smaller distance of the dendrimer, as is shown in Fig. 4(a)
for h ¼ 3sc, roughly one of the core particles is found on the
surface of the sphere and the other core monomers are distrib-
uted over a wider range. The number of core monomers that get
adsorbed increases with the interaction strength.
Similar observations can be made in the case of a sphere of
type CR-SA. Results for a sphere with radius Rs¼ 8sc are shown
in Fig. 5 for four different distances of the dendrimer to the
colloidal surface, where we focus on a rather strong interaction
energy, 3 ¼ 3. The reason for this is to examine more closely the
irregular behavior in the effective interaction displayed in
Fig. 1(b). For a small dendrimer-center to colloid-surface
distance, h¼ 1.5sc, Fig. 5(a), the center-of-mass of the dendrimer
lies deep within the sphere (the individual monomers are always
Soft Matter, 2009, 5, 4542–4548 | 4545
Fig. 7 Typical dendrimer conformations with a small colloidal sphere of
radius Rs ¼ sc at different distances h of the colloidal surface from the
dendrimer center-of-mass. The values of h are expressed in units of sc.
found outside the sphere). The shell monomers are found in
a very narrow range close to the surface, which is due to the
attraction exerted on them. The core monomers, which are
repelled, are found somewhat further away. If the dendrimer
distance to the colloidal surface is increased to h ¼ 3sc, Fig. 5(b),
the single peak distribution of the shell monomers has been split
in a double peaked structure, whereas the core monomers
distribution has become more narrow. It should be noted,
however, that the central core monomers (Z) are on average
further away from the sphere center than the core monomers of
generation 1 (C0). The cumulative profiles shown in the inset,
reveal that exactly one of the shell monomers has been detached
from the surface of the sphere.
Fig. 5(c) shows the profiles for a distance h ¼ 4sc. At this
distance also the single-peaked distribution of the core mono-
mers of generation 1 has become bimodal. The cumulative
profiles indicate that two shell monomers and one core mono-
mer, i.e., one branch of the dendrimer, has been freed from the
sphere and extends into the implicit solvent. If the dendrimer-to-
colloidal surface distance is increased to h¼ 5sc, as can be seen in
Fig. 5(d), another shell monomer has left the surface of the
sphere. This behavior indicates that the adsorption of the den-
drimer on a sphere of the type CR-SA is a gradual process in
which on approaching the sphere the natural conformation of the
dendrimer is modified in a step-by-step fashion and explains the
irregular behavior in the effective interaction shown in Fig. 1(b).
It should be noted that a strong interaction was used to highlight
this behavior, but that this scenario also applies to weaker
interactions. For the case of a CA-SR sphere, a similar behavior
can be observed albeit less pronounced.
Fig. 6 shows the density profiles for the more extreme case of
a very small sphere with radius Rs ¼ sc of the type CA-SR. Since
the total interaction scales proportional to the volume of the
sphere, it is rather weak. In fact, although the effective interac-
tion in this case is repulsive, the colloid is small enough to be
placed inside the dendrimer. In other words, instead of the
dendrimer being adsorbed on the colloid, we find that the colloid
Fig. 6 Radial density profiles of the dendrimer’s monomers for a CA-SR
dendrimer close to a sphere with radius Rs¼ sc and 3¼ 0.5. The values of
the center-of-mass to surface distances h are: (a) h ¼ 0.1sc, (b) h ¼ 1.0sc,
(c) h ¼ 2.0sc, and (d) h ¼ 4.0sc.
4546 | Soft Matter, 2009, 5, 4542–4548
gets absorbed by the dendrimer. The density profiles for the
shortest distance between the center-of-mass and the colloid
indicate that the colloid sticks to typically two of the core
monomers, which is energetically the most favorable. If the
colloid is moved away from the center-of-mass, such a tight
binding to two core particles is no longer possible. For the larger
distance, the presence of the colloidal sphere only distorts the
natural conformation of the dendrimer to expel the shell
monomers from its direct vicinity. This is also illustrated in
Fig. 7, where typical configurations of a dendrimer that captures
a spherical particle are shown.
V. Dendrimer–colloid mixture
In the preceding sections we analyzed the effective colloid–den-
drimer interaction and the change in conformation of a single
dendrimer due to the presence of a nearby colloidal sphere. In
this section we examine the adsorption behavior of dendrimers
with a finite density in the vicinity of a single, isolated colloidal
sphere. To this end, we have employed Monte Carlo simulations
to study the two types of colloidal spheres for various interaction
strengths, radii, and densities, where the position of the sphere is
kept fixed. The simulation box, for which periodic boundary
conditions are applied, is chosen to be cubic in shape and has
a size of 40 � 40 � 40 s3c. Given the finite range of the interac-
tions and the maximal extent a dendrimer can have, this guar-
antees that only the nearest images of dendrimers interact with
each other. Two typical snapshots from such simulations are
shown in Fig. 8. On the left panel, pertaining to CA-SR
Fig. 8 Snapshots from simulations of N ¼ 100 dendrimers with a sphere
of radius Rs ¼ 8sc. (a) The CA-SR case with 3 ¼ 0.5; (b) the CR-SA case
with 3 ¼ 2.
This journal is ª The Royal Society of Chemistry 2009
Fig. 9 Monomer radial density profiles from simulations of N ¼ 100
dendrimers around a sphere with radius Rs¼ 8sc for the CA-SR case [(a),
(c), (e)], and the CR-SA case [(b), (d), (f)]. The interaction strengths are
3 ¼ 0.5 [(a), (b)], 3 ¼ 1.0 [(c), (d)], and 3 ¼ 2.0 [(e), (f)]. The profiles are
shown for the center-of-mass, the core monomers, and the shell mono-
mers, as indicated in the legend of panel (a).
Fig. 10 Monomer radial density profiles from simulations of N ¼ 100
dendrimers around spheres of varying radii, as indicated in the legends of
panels (a) and (b). Different panels correspond to the CA-SR case with
3 ¼ 1.0 [(a), (c), (e)], and the CR-SA case with 3 ¼ 0.5 [(b), (d), (f)]. The
profiles are shown for the center-of-mass [(a), (b)], the core monomers
[(c), (d)], and the shell monomers [(e), (f)].
dendrimers, it can be seen that there is a zone around the colloid
that is depleted of the monomers. Quite the contrary, on the right
panel that refers to CR-SA dendrimers there is strong adsorption
of the latter on the colloidal surface.
In Fig. 9 we show the radial density profiles of the dendrimers
for a fixed sphere radius Rs ¼ 8sc and N ¼ 100 dendrimers for
various interaction strengths 3. In order to have a clearer
understanding of their adsorption behavior, not only the density
profiles of the centers-of-mass are shown, but also the monomer
density profiles of core and shell monomers independently.
In the case in which the core monomers are attracted by the
sphere (CA-SR-type) and 3¼ 0.5, the interaction is still too weak
to lead to an effective attraction. Due to the finite density
however, dendrimers are pushed against the sphere, leading to an
induced ordering. This brings the dendrimers close enough to the
sphere, so that their internal conformation is affected and some
core monomers attach to the surface, as indicated by the small
peak in Fig. 9(a). On increasing the interaction strength to 3 ¼ 1,
Fig. 9(c), the effective dendrimer–sphere interaction remains still
repulsive, but has formed a plateau. This enables the dendrimer
to approach the sphere closer and form an initial adsorption
layer. The density in this layer further grows for 3 ¼ 2, Fig. 9(e),
where the effective interaction now has an attractive range, so
that almost all dendrimers in the system have been adsorbed.
Whereas the core monomers form a clear structure, the shell
monomers are only at short ranges repelled by the sphere, but are
otherwise more or less homogeneously distributed, only
restricted by the internal bonds of the dendrimer.
This journal is ª The Royal Society of Chemistry 2009
For the opposite, CR-SA spheres, we always have an attractive
range in the effective interaction, which enables the dendrimer to
adsorb on the colloidal particle. For the weak interaction 3¼ 0.5,
shown in Fig. 9(b), the dendrimers do not even modify their bulk
conformation, and bind with one or more of the shell monomers
directly to the sphere. On increasing the interaction strength to
3 ¼ 1 and 3 ¼ 2, more dendrimers are adsorbed on the sphere.
They attach to the sphere with an increasing number of shell
monomers per dendrimer, as indicated by the bimodal shell–
monomer density profiles, Fig. 9(d) and 9(f).
The effect of the radius of the sphere is examined in Fig. 10,
where three different radii are selected. In order to facilitate the
comparison between the different profiles, they are grouped in
density profiles of the centers-of-mass, the core monomers, and
the shell monomers. It can be seen that for all profiles an
increasing sphere radius results in a more pronounced structure.
This is a direct consequence from the fact that the interaction
between the colloidal sphere and a dendrimer, according to eqn
(5), scales with its size.
The fact that the same simulation box-size is used for each
sphere radius in Fig. 10 also results in a more structured profile,
because for the larger spheres an increased effective dendrimer
density is obtained. In Fig. 11 this is illustrated by comparing the
different radial profiles from a radius Rs ¼ 8sc sphere at various
densities. In the CA-SR case, the increased density, forces
a larger fraction of the dendrimers to be near the spherical
particle. Due to the repulsive force on the shell monomers, those
dendrimers need to adjust their conformations as can be seen
from the density profiles for the core monomers. Since in this
Soft Matter, 2009, 5, 4542–4548 | 4547
Fig. 11 Monomer radial density profiles for different numbers of den-
drimers N, as indicated in the legend of panel (a), and fixed colloidal
radius Rs ¼ 8sc. Different panels show the CA-SR case with 3 ¼ 1.0 [(a),
(c), (e)], and the CR-SA case with 3 ¼ 0.5 [(b), (d), (f)]. The profiles are
shown for the center-of-mass [(a), (b)], the core monomers [(c), (d)], and
the shell monomers [(e), (f)].
case 3 ¼ 1.0, the effective interaction still remains repulsive and
the sphere induces a layered structure of dendrimers. In the CR-
SA case the shape of the profiles is almost unaffected by the
increased density and mainly scales with the density. Only the
distance of the adsorbed layer with respect to the sphere is
slightly increased, which is necessary in order to create enough
space for those dendrimers to bind to the sphere.
VI. Conclusions
We have analyzed the behavior of amphiphilic dendrimers in the
vicinity of colloidal spheres. The curvature of the sphere has
important consequences for the adsorption behavior of the
dendrimers, specifically for the internal structure of the den-
drimers. The two archetypes of conformations found in an earlier
study,20 the ‘‘dead spider’’ in the case of a CA-SR planar wall and
the ‘‘living spider’’ for a CR-SA type of wall, are also present for
spherical colloidal particles with the same type of interaction.
It was shown that the adsorption on a sphere or planar wall is
a gradual process that depends on the strength of interaction and
the nature of the surface. In order to get adsorbed on a CA-SR
type of surface, the dendrimers need to overcome an initial
barrier due to the repulsive interaction of the shell. Thereafter,
4548 | Soft Matter, 2009, 5, 4542–4548
the conformation of the dendrimer needs to be inverted in order
to bring the core monomers close enough to the surface to get
bonded by it, which requires relatively strong interactions. It is
much easier for dendrimers to get adsorbed by a surface of the
type CR-SA, because the shell monomers that bind to the surface
are on average already at the outside of the dendrimer. But also
in this case the conformation of the dendrimer is affected in the
process. It was shown that in the latter case the adsorption
becomes a step-by-step process in which single shell monomers
get adsorbed one-by-one.
Acknowledgements
This work has been supported by the Deutsche For-
schungsgemeinschaft (DFG) and the Heinrich-Heine-Uni-
versit€at D€usseldorf, project number 01020027.
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