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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:lenz@thphy.uni-duesseldorf.de

† 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.

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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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This journal is ª The Royal Society of Chemistry 2009