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Structural properties of dendrimer–colloid mixtures
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 24 (2012) 284119 (12pp) doi:10.1088/0953-8984/24/28/284119
Structural properties ofdendrimer–colloid mixturesDominic A Lenz1,2, Ronald Blaak1 and Christos N Likos1
1 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria2 Institute of Theoretical Physics, Heinrich Heine University of Dusseldorf, Universitatsstraße 1,D-40225 Dusseldorf, Germany
Received 11 November 2011, in final form 2 February 2012Published 27 June 2012Online at stacks.iop.org/JPhysCM/24/284119
AbstractWe consider binary mixtures of colloidal particles and amphiphilic dendrimers of the secondgeneration by means of Monte Carlo simulations. By using the effective interactions betweenmonomer-resolved dendrimers and colloids, we compare the results of simulations of mixturesstemming from a full monomer-resolved description with the effective two-componentdescription at different densities, composition ratios, colloid diameters and interactionstrengths. Additionally, we map the two-component system onto an effective one-componentmodel for the colloids in the presence of the dendrimers. Simulations based on the resultingdepletion potentials allow us to extend the comparison to yet another level of coarse grainingand to examine under which conditions this two-step approach is valid. In addition, apreliminary outlook into the phase behavior of this system is given.
(Some figures may appear in colour only in the online journal)
1. Introduction
Typical soft matter systems include components with alarge separation of length scales and attributes, makingthe computational approach become a huge challenge tomaster. In particular, multicomponent mixtures, includingultrasoft particles like dendrimers or star polymers, presentenormous potentials and a wide range of possible set-upsand configurations. Therefore, several ways of computationalsimplifications of the full multicomponent mixture weredeveloped and successfully implemented [1–3], basedon effective descriptions on various levels and differentcomplexities. Despite the availability of fast computersystems, which allow us to model and simulate complexmolecules on the atomic level, it is still common practiceto attempt to simplify such systems because the time scalesand length scales required for effective descriptions areout of reach. A possible route to reduce the complexityof these systems is the choice of describing them at thelevel of effective interaction potentials. In its simplest form,this results in a situation in which all interactions betweenthe various species within the system area reduced toradially symmetric, effective pair potentials. This approach,although completely ignoring the usually complex and not
necessarily isotropic structure of the molecular species inthe system, allows for useful insights into the behavior ofsuch systems and has been applied successfully in varioussimulational studies [4–6]. A different approach, on whichmost of the theoretical investigations of binary mixtures arebased, is a one-component description, which employs asingle interaction potential that is obtained by integratingout the smaller component’s degrees of freedom. The bestknown example of this method is based on the so-calledAsakura–Oosawa (AO) model for a mixture of colloids andpolymers, in which interactions between colloids or withpolymers are modeled as hard-sphere-like, but interactionsbetween polymers are considered to be ideal. Consequently,the effect of the polymers on the colloids is fully determinedby their overall density and the available free volume in aconfiguration of colloidal spheres. Therefore, the mixture canbe described by an effective interaction between colloidalparticles only, which is the superposition of a directinteraction in combination with a polymer-induced depletionpotential.
In this work, we address the richness of possibilitiesof these different levels of description for binary mixturesystems, including the monomeric level, by performing acomparison of simulation results for a selection of different
J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
parameters, such as densities, composition ratios, particlesizes and interaction strengths. The mixture we consideris composed of soft colloidal particles and amphiphilicdendrimers, where particles from different species experiencea weak attraction with respect to each other. Dendrimers forma class of macromolecules providing enormous potentialsdue to their regularly branched architecture. They were firstsynthesized by Vogtle et al [7] and gradually attracted agreat deal of interest from various fields of scientific research.The development of new synthesis techniques has resulted ina variety of different dendrimeric systems. The control thatis possible in both architecture and the nature of functionalgroups of these macromolecules allow them to be applied incompletely different fields such as solubility enhancement,drug-delivery vectors or nanocarriers [8–11]. They combinecharacteristics of both colloidal and polymeric systems [12],and hence can encompass the range of soft particles withGaussian-shaped pair interactions [13, 14] when they areneutral, as well as homogeneously charged spheres whentheir end-groups carry charges [15]. A substantial amount ofresearch has been focused on the conformations, interactionsand phase behavior of neutral dendrimers in dilute andconcentrated solutions [16, 17]. This information is alsopartially accessible by small-angle neutron scattering in thecomparison of the scattering intensity of dendrimers withprotonated and deuterated end-groups [18]. It is therefore notsurprising that, due to their wide range of applications, theirability to induce new phenomena and challenges that arisein synthesis and experimental observations, dendrimers havebecome an important class of macromolecules that are widelystudied in soft matter research [7, 19–26].
In previous work [27, 28], we have demonstrated thediversity in adsorption behavior of amphiphilic dendrimersonto compact colloidal particles as well as on planarwalls. This includes phenomena like the gradual step-by-step adsorption of single branches of dendrimers andconformational characteristics like the living spider and deadspider alignment onto the surface [27, 28]. The extension tobinary mixtures at finite concentrations is the next logical stepin examining the properties of dendrimeric systems. The factthat the amphiphilic dendrimers used in this work are also ableto spontaneously form clusters [29, 30] only further enrichesthe possibilities offered by these particular systems.
The rest of this paper is organized as follows: afterintroducing the monomer-resolved description in section 2and the parameters of the composition in section 3, wepresent the modeling of the two-component level descriptionin section 4, the modeling of the one-component leveldescription using the depletion potentials in section 5, themapping between the different descriptions in section 6 andin section 7 we compare the results of all three descriptionlevels for a selected distribution of parameter sets. Finally insection 8 we draw our summary and concluding remarks.
2. Monomer-resolved description model
At the most basic level description of the current model,we find the full multicomponent mixture of dendrimers and
Table 1. Overview of the monomer interaction potential parametersused between core (c) and/or shell (s) monomers of the D2amphiphilic dendrimers.
Morse εµν αµνσ σµν/σ
cc 0.714 6.4 1cs 0.014 19.2 1.25ss 0.014 19.2 1.5
FENE Kµν σ 2 lµν/σ Rµν/σ
cc 40 1.875 0.375cs 30 3.75 0.75
colloids on the monomer level, in which the solvent degreesof freedom have been integrated out, thus forming in itselfalready a first step in the coarse-graining procedure. Withinthis framework, a dendrimer is described by a collectionof connected monomeric units, for which we employ themodel introduced by Mladek et al [29]. In particular, thismodel describes amphiphilic dendrimers of generation G =2 and functionality f = 3 starting from two central coremonomers. The resulting total of 14 monomers are dividedinto two classes of monomers. The eight monomers of theoutermost generation, g = 2, form the solvophilic shell of thedendrimer, denoted by the subscript s, and all monomers ofthe interior generations, g = 0 and 1, form the solvophobiccore, denoted by the subscript c. In so doing, one obtainsthe desired amphiphilic property for the dendrimers byintroducing suitable interactions between the various types ofmonomers. In this model, the interaction between any twomonomers, separated by a distance r, are modeled by theMorse potential [31], which is given by
β8Morseµν (r) = εµν{[e−αµν (r−σµν ) − 1]2 − 1},
µν = cc, cs, ss (1)
where σµν denotes the effective diameter between twomonomers of speciesµ and ν. This potential, and all followingpotentials, is expressed in units of the inverse temperatureβ = (kBT)−1, with kB denoting Boltzmann’s constant. TheMorse potential is characterized by a repulsive short-rangebehavior and an attractive part at long distances, whose depthand range are parameterized via εµν and αµν , respectively.The core monomer diameter σcc ≡ σ is chosen to be the unitof length. The bonds between adjacent monomers of relativedistance r are modeled by the finitely extensible nonlinearelastic (FENE) potential, given by
β8FENEµν (r) = −KµνR2
µν log
[1−
(r − lµν
Rµν
)2],
µν = cc, cs (2)
where the spring constant Kµν restricts the monomerseparation to be within the distance Rµν from the equilibriumbond length lµν . In order to stay within the limits ofexperimental feasibility, we use the parameter set (see table 1)from the D2 model from [29].
For modeling the interaction between the colloidalparticles and the monomers, we make use of the sameapproach as we have applied for investigating the adsorption
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
behavior of dendrimers onto colloids [28]. There, weassumed that a colloidal particle of radius RC consists ofa homogeneously distributed collection of Lennard-Jonesparticles with diameter σLJ and, without loss of generality, onecan assume a density ρσ 3
LJ = 1. For a simple power-like pairpotential acting between a volume element of the colloid anda test (i.e. monomer) particle of the form
βv(n)0 (r) = 4ε(σLJ
r
)n, (3)
and a dimensionless energy scale ε, the total interaction ona particle at position Er outside the spherical colloid can becalculated by simple integration over the whole colloid and issubsequently given by
β8(n)colloid(r) =
8περσ nLJ
r(n− 2)(n− 3)(n− 4)
×
[(n− 3)RC − r
(r − RC)n−3 +(n− 3)RC + r
(r + RC)n−3
]. (4)
This interaction potential diverges as a test particle approachesthe surface of the colloid from above, r ↓ RC, so that thecolloid is impenetrable to individual monomers. Additionally,since the potential depends only on the product of thedimensionless energy and density parameters ε and ρσ 3
LJ,a different choice for the value of ρσ 3
LJ can easily beincorporated by adjusting the energy parameter accordingly.
For amphiphilic dendrimers, the interactions with col-loidal particles depend on the nature of the monomer, i.e.whether it is a core or shell monomer. Depending on thechoice of which type plays the role of the solvophilic orsolvophobic units, this interaction will have an attractiverange or will be purely repulsive, respectively. For the scopeof this work we limit ourselves to the choice of colloidsthat are attractive for core monomers and repulsive for shellmonomers, which were previously denoted by CA-SR [27,28]. To model the attractive interactions of the colloid with thecore monomers, we use the standard Lennard-Jones interac-tion with the point particles forming the colloid, as mentionedbefore. The resulting interaction can then easily be expressedin terms of the derived interaction (4) and is given by
8corecolloid(r) = 8
(12)colloid(r)−8
(6)colloid(r) (5)
for a monomer at a distance r from the colloid center. Therepulsive interaction of the colloid with the shell monomersis obtained by employing a procedure similar to the one usedfor obtaining the WCA potential. Hereto, the distance r∗ islocated at which the interaction (5) attains its minimum value.The interaction is shifted to zero at this location and truncatedbeyond. Hence, the repulsive interaction between a colloidand a shell monomer is given by
8shellcolloid(r) ={
8corecolloid(r)−8
corecolloid(r
∗) r ≤ r∗
0 r > r∗.(6)
For the colloid–colloid interaction the WCA potential isused as well:
8CC(r) =4ε[(σC
r
)12−
(σC
r
)6+
14
]for r ≤ 21/6σC
0 otherwise,(7)
with energy scale ε and the colloidal diameter σC = 2RC. Inorder to simplify the parameters the value σLJ = σ is chosenfor the interactions between the colloid and both types ofmonomer species.
3. Composition
We consider binary mixture systems with NC colloidalparticles of radius RC and ND dendrimers, consisting of 14monomers each. This results in a total number of particlesN = NC + ND. Using the value RD = 3.36σ of the radius ofgyration of an isolated dendrimer as in [29], we define the sizeor asymmetry ratio between the dendrimer and colloid by q =RD/RC. The number densities for the colloids and dendrimersare given by ρC and ρD, respectively. This enables us tointroduce the (effective) volume fractions for both species by
ηC =4π3ρCR3
C (8)
for the colloids and
ηD =4π3ρDR3
D (9)
for the dendrimers. In addition, the composition ratio x isgiven by
x =ρD
(ρD + ρC). (10)
The simulation box of volume V is chosen to be cubical inshape to which the normal periodic boundary conditions areapplied. The sides of the simulation box under the varioussimulation conditions lie in the range between 40σ and 80σ .Three different colloid radii are considered, i.e. RC = 4σ , 8σand 12σ . For the interaction between the colloidal particlesand the dendrimers, we limit ourselves to two interactionstrengths characterized by ε = 2 and 0.5. In either case, onlythe core monomers experience an attractive force, whereas theinteraction with shell monomers is purely repulsive.
4. Effective two-component description
For the first level of coarse graining of the system at hand, i.e.in order to describe it as an effective simple binary mixture,we need to eliminate the internal degrees of freedom of thedendrimers. It was shown by Mladek et al [29] that theeffective interaction between two dendrimers separated by adistance r can readily be approximated by a double-Gaussiancore model (DGCM) potential, given by
8effDD(r) = ae−(
rb )
2− ce−(
rd )
2, (11)
where the parameters a, b, c and d are fitted to the measuredeffective interaction between two isolated dendrimers and areshown in table 2.
The effective interaction between colloids and den-drimers, 8eff
CD, can be obtained in a similar fashion. This was
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
Figure 1. The effective interactions 8effCD(r) between colloids and dendrimers for (a) ε = 0.5 and (b) ε = 2 for the three different colloid
radii RC = 4σ , 8σ and 12σ .
Table 2. Parameter values for the double-Gaussian core model(DGCM) potential for modeling the effective interaction betweendendrimers.
DGCM a/kBT b/RD c/kBT d/RD
DD 27.77 1.098 26.693 1.049
already explicitly demonstrated for the case ε = 0.5 [28].Figure 1 summarizes these results for both interactionstrengths ε = 0.5 and 2 and the three different colloid radiiRC = 4σ , 8σ and 12σ that we consider here. Whereas thecolloid–dendrimer interactions for ε = 0.5 remain purelyrepulsive, an increase in the monomer–colloid energy scaleenhances the importance of the core monomers for the totalinteraction. This results not only in the expected short- andlong-range repulsive interaction, but also in an intermediateattractive range of interactions for the choice ε = 2. It shouldalso be noted that, although the individual monomers of thedendrimer are not allowed to penetrate the colloidal particle,the center of mass of the same can be found, to some extent,in the interior of the colloid. This is due to the possibilityfor dendrimers to fold partially around the curved surface ofthe colloidal particles and results in a small range of finiteinteraction potentials with r < RC + RD shown in figure 1.
5. Effective one-component description
In order to adequately map the two-component descriptiononto a one-component description, that is, onto a systemwith only colloids, one needs to integrate out all thedegrees of freedom of the dendrimers in an Asakura–Oosawamodel fashion [32]. The resulting effective interactionbetween colloids after this procedure contains two differentcontributions. The first component is the original directinteraction between colloids 8CC(r), as given in equation (7).The second contribution is a depletion potential 8depl
CC (r)originating from the presence of the dendrimers in the vicinityof the colloidal particles [33]. Hence, formally, the fulleffective interaction between two colloids on this level ofdescription is given by
8effCC(r) = 8CC(r)+8
deplCC (r). (12)
Implicitly it is assumed here that the depletion interactionis spherically symmetric. It should also be noted that the
depletion potential depends on the density of the surroundingdendrimers.
A relatively straightforward procedure for obtaining theunknown depletion potential runs via the depletion force,which can readily be measured in computer simulations.To this end, standard NVT Monte Carlo simulations areperformed for two colloidal particles at a fixed mutualdistance R12 = |ER12| = |ER2 − ER1|, where both colloids arelocated at the positions ER1 and ER2, respectively. Depending onthe nature of the colloid–dendrimer interaction, i.e. whetherit is attractive or repulsive, an aggregation or depletion,respectively, of dendrimers between the colloids arises. Thepresence of the second colloid results in a non-sphericallysymmetric dendrimer-density profile around either colloid,which causes a non-vanishing force that, for reasons ofsymmetry, is directed along the connection vector ER12.This depletion force EFdepl for a fixed distance R12 can bemeasured via
EFdepl(R12) =
⟨ND∑i=1
∇ER18CD(|ER1 − Eri|)
⟩R12
R12, (13)
where 〈·〉R12is an ensemble average over the positions Eri of
the ND dendrimers under the constraint of a fixed distance R12between both colloids and R12 denotes the unit vector in thedirection ER12. By measuring this depletion force for variousdistances, the full depletion force curve can be obtained,which upon integration will yield the depletion potential8
deplCC (r), which vanishes in the limit r→∞ by means of an
appropriate integration constant. Since the direct interactionbetween the colloids (7) allows for a small overlap betweenthe colloidal particles, the depletion potential needs to beevaluated also for some distances r < σC.
Figures 2(a), (c) and (e) show the results for the purelyrepulsive interaction between colloids and dendrimers (seefigure 1(a)), given by the energy parameter ε = 0.5, for thethree different colloid radii and for four different dendrimerdensities. Despite the absence of attractive interactionsbetween both components, the depletion potential is attractiveat short distances, the strength of which is growing onincreasing the dendrimer densities. Since it is the size ratioRD/RC between the dendrimers and the colloidal particles thatdetermines the width of the depletion zone in this represen-tation, the attractive range becomes narrower and deeper forincreasing colloid size. This is qualitatively in full agreement
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Figure 2. The depletion potentials 8deplCC (r) for the colloid–colloid interaction for different volume fractions ηD of dendrimers for each of
the interaction strengths ε and colloid radii RC. The distances are normalized by colloid diameter σC = 2RC.
with the behavior of the AO model. In addition, a repulsiverange in the measured depletion potential is visible when asingle layer of dendrimers is found between both surfaces.
Figures 2(b), (d) and (f) show the results for thesame parameter range of densities and colloidal sizes, butnow using the effective potential with attractive range (seefigure 1(b)) between dendrimers and colloids, given bythe energy parameter ε = 2. For this interaction it wasshown [27, 28] that, due to their adsorption behavior,dendrimers will form a complex molecular layer on thesurfaces of the colloidal particles. Within this layer, theusual conformation of dendrimers, in which the solvophobiccore monomers are found close to the center of mass andsurrounded by the solvophilic shell monomers, is almostinverted. The dendrimers in this layer adopt a ‘dead spider’conformation [28], which causes part of the core monomersto be closer to the colloid surface than those monomersbelonging to the shell. The same can be observed in the spacebetween two colloids. Upon approaching of the colloidalsurface, the cores of the dendrimers are adsorbed mainlyonto one of either surface, and hence repel the other surfaceby means of their shell monomers. This results in a weak,effective repulsive barrier between the colloids. For smallerdistances, the dendrimers are squeezed between both surfaces,exposing their core to either surface and act as a naturalglue and causing an additional, effective attraction betweenthe colloids. Naturally, this effect increases upon increasing
dendrimer density and, as visible in figures 2(d) and (e),forms an energetic minimum at roughly r = σC when reachingsufficiently high dendrimer density. The size of the colloidsreflects itself in the available surface for dendrimers to adsorband hence the amount of shared dendrimers between bothcolloidal surfaces. Consequently, the strength of the attractionalso increases with the size of the colloids. In the case ofthe smallest colloid with radius RC = 4σ in figure 2(b), theavailable surface is too small to fully exploit this behavior. Itsonset is, however, already visible.
Figure 3 shows an example of the resulting composed ef-fective interaction between colloids as given by equation (12)for colloid radius RC = 8σ and both interaction strengths ε =0.5 and 2 for the dendrimer volume fraction of ηD = 0.06. Forthe case ε = 0.5 with the purely repulsive colloid–dendrimerinteraction, only a weak effective attractive range from thedepletion interaction remains. However, in the case ε = 2the direct colloid–colloid repulsive interaction can no longerfully compensate the depletion potential. The strong attractiveinteraction that remains between colloids already suggests thespontaneous aggregation of these colloids at small volumefractions of the dendrimer additives, as will be shown in thenext sections.
6. Mapping of the different descriptions
In order to compare the behavior of binary mixture systems asdescribed above, we implemented Monte Carlo simulations
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
Figure 3. The total effective interaction between colloids 8effCC(r),
as given by equation (12), for a colloid radius RC = 8σ , bothinteraction strengths ε = 0.5 and 2, and a dendrimer volumefraction ηD = 0.06.
in the three aforementioned levels of coarse graining forcomparable parameter sets. The correspondence betweenthe monomer-resolved description and the effective two-component description is straightforward, since obviouslyone only needs to choose the same densities for bothspecies, i.e. dendrimers and colloids, as well as identicalcolloidal sizes. It should be noted, however, that the fitteddendrimer–dendrimer interaction (11) and the measuredcolloid–dendrimer interactions as depicted in figure 1are idealized. Both interactions are based on two-particleinteractions only and hence neglect all many-particle effects.In general, either interaction will be distorted by the presenceof an additional particle, be it dendrimer or colloid. Hence, thevalidity of this approach is limited by the overall density. Itwas shown, however, that this approach provides surprisinglygood results even up to moderate densities [30].
Establishing the correspondence between these systemsand the one-component description is more troublesome.The depletion interaction is obtained in a system with onlytwo colloids, i.e. the zero density limit for colloids. In themixture, however, the dendrimer densities in the vicinity ofthe colloids is inhomogeneous. Hence, merely equating theoverall densities of either component will not be sufficient.Instead, we need to make the correspondence via the chemicalpotential of the dendrimers, i.e. the component that isintegrated out of the system. Hereto, the chemical potentialof the dendrimers is measured in the mixture by the Widomparticle-insertion method [34, 35]. The appropriate dendrimerdensity that needs to be assigned to the mixture is thenobtained by finding that particular density at which a puredendrimer system results in the same chemical potential thatis determined via the same Widom insertion method.
7. Results and comparison
For both interaction strengths and the various colloidalsizes that are considered here, a series of Monte Carlosimulations in the monomer-resolved level description havebeen performed. These systems, characterized by the colloidand dendrimer volume fractions (ηC, ηD), are also simulated
under the same conditions in the two-component effectivelevel description, using the same parameters. During thesecoarse-grained simulations, the chemical potential of thedendrimers was measured via the Widom insertion method.A separate series of simulations at this level was performed,containing only two colloidal particles immersed in a bathof dendrimers at various densities, from which the dendrimerdensity was chosen that results in the same chemical potentialfor the dendrimers as the full mixture. The depletion potentialsbetween the colloids obtained from this particular densitywere used in a simulation at the effective one-component leveldescription.
Figure 4 shows simulation snapshots for three exemplaryconfigurations in the ε = 0.5 case for both the monomer-resolved description and the two-component description.Upon visual inspection, the results between both types ofdescription look very similar. The clear separation betweencolloids and dendrimers already suggests the interactionbetween colloids and dendrimers to be repulsive, as wasalready shown in figure 1(a). It can also easily be observed thatboth components for the various densities and compositionsare not homogeneously distributed, but exhibit a weakaggregation. Hence, a weak effective attractive interactioncan be inferred between the colloids, which is confirmed infigure 3, and for the dendrimers as given by their effectiveinteraction (11). From our present work, it is not yet clearwhether this system will simply restrict itself to forming smalldomains where either of the components is dominant, or thatit will phase-separate on a macroscopic scale. The form ofthe colloid–colloid depletion potential strongly suggests that ademixing transition will take place at sufficiently high overalldensities, but calculating the phase diagram of the mixture isbeyond the scope of this work.
In order to be able to compare the various descriptionlevels, we need to make a more quantitative analysis ratherthan a mere visual inspection. The most relevant propertiesof the systems are found in the structure, which can becharacterized by the pair correlation functions. To this end,the three different radial distribution functions gαβ(r) canbe measured, where the subscripts α, β are representingthe colloidal (C) or dendrimer (D) particle species, andin either case the center of mass is used to measure therelative distances. A selection of these correlation functionsfrom simulations performed at the monomer-resolved levelare presented in figure 5, for a few different densitiesand compositions in the case of ε = 0.5 and RC = 8σ .Although the structures become more pronounced uponincreasing the overall volume fraction of the system, theyremain qualitatively the same. From the distribution functiongCC(r) in figure 5(a) it can be seen that the softness of thecolloid–colloid interaction only results in a small range ofoverlap. The effective repulsive colloid–dendrimer interaction(see figure 1(a)), based on a two-particle calculation, remainsvalid for these densities as can be seen from figure 5(b),because the centers of mass of the dendrimers stay welloutside the colloidal particle. The most fascinating property,however, is found in the radial distribution of the dendrimersin figure 5(c), where a non-zero correlation is found for zero
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
Figure 4. Typical simulation snapshots for simulations performed for the repulsive colloid–dendrimer system with interaction strengthε = 0.5. The snapshots on the left (a), (c) and (e) correspond to monomer-resolved simulations, and the ones on the right (b), (d) and (f) tosimulations performed at the two-component level description. The systems are characterized by (a) and (b): RC = 8σ ,(ηC = 0.157, ηD = 0.047), x = 0.2; (c) and (d): RC = 4σ , (ηC = 0.019, ηD = 0.027), x = 0.3; (e) and (f): RC = 8σ ,(ηC = 0.275, ηD = 0.115), x = 0.15. Color code: core monomers black (generation 0) and light blue (generation 1), shell monomers red(generation 2), colloids yellow, coarse-grained dendrimers green. The sphere diameters correspond to the respective actual sizes.
distance between dendrimers. This apparent counterintuitivebehavior of the dendrimers displays their ability to formclusters of overlapping particles [30]. It should be stressedthat this is not an artifact, but describes a true, existingphenomenon. It is caused by their amphiphilic nature,which enables the centers of mass of two dendrimers tobecome arbitrarily close, but at the same time disallowsthis behavior for individual monomers from which theyare built. Although it is known that this cluster behaviorexists for these types of dendrimers [27, 28], what makesit surprising here is the fact that the clustering takes place
at a much lower density than is found for pure dendrimersystems, i.e. the clustering is enhanced by the presence ofcolloidal particles.
The three different levels of description are comparedin figure 6, where the three pair correlation functions areshown for two selected densities/composition ratios forcolloids of radius RC = 8σ . Obviously, for the one-componentdescription only the colloid–colloid correlation gCC(r) can beshown. The plots on the left column side compare the threemeasured correlation functions for a relatively dilute system,where the agreement between all three levels of description
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
a
b
c
Figure 5. The radial distribution functions measured at themonomer-resolved level description for colloids with radiusRC = 8σ and interaction strength ε = 0.5 at four different volumefractions (ηC, ηD) and composition ratios x. (a) Colloid–colloidgCC(r), (b) colloid–dendrimer gCD(r) and (c) dendrimer–dendrimergDD(r). The inset additionally provides the estimated error bars ingDD(r) for the system with the highest particle density.
is almost perfect. It is not surprising that some discrepancyarises for the much higher density on the right column sideof figure 6, but the overall agreement is still very good. Boththe colloid–colloid and the dendrimer–dendrimer correlationare overestimated by the higher levels of coarse graining,whereas the colloid–dendrimer correlation is underestimated.The explanation is found in having neglected the many-bodyinteractions that are made in both coarse-graining steps. Sincemost of the dendrimers stay at distances larger than their ownradius of gyration from the colloidal surfaces, the main sourcewill be that of the overestimation of the dendrimer–dendrimer
interaction by the DGCM interaction (11). The results for thelarger colloidal particles with RC = 12σ are very similar.
Figure 7 shows monomer-resolved results, analogouslyto figure 5, but for smaller colloids with RC = 4σ . Tworepresentative density combinations have been chosen, adilute system and a system with higher density. The resultsare very similar to those for the larger colloids. Since themechanisms responsible for the structural characteristics areessentially the same at a lower colloidal scale, caused bythe smaller size of the colloids, this is not surprising. Itis also evident that on a smaller colloid less space fordendrimers to adsorb is available, leading to a less distinctshape of the results compared to the ones shown in figure 5.Figure 7(a) additionally includes the direct comparison of themonomer-resolved level description and the one-componentdepletion description. The agreement of the two descriptionsis clearly visible. The results for the case of colloids withradius RC = 4σ suggests that the overall structural and particlecharacteristics do not differ very much compared with the caseof colloids with radius RC = 8σ .
Figure 8 shows simulation snapshots for two differentvolume fractions/composition ratios in the case of an effectiveattractive interaction between colloids and dendrimerscharacterized by ε = 2. Also here, a close resemblance isfound between the monomer-resolved simulations and thosethat have been performed at the level of the two-componentmixture. For a low density system presented in figures 8(a)and (b), the dendrimers again show a tendency to aggregateand, in addition, to weakly adsorb at the colloids, thus creatinglarger spaces void of any particles. For the higher densitiesand larger colloid size shown in figures 8(c) and (d), theadsorption has increased substantially and the dendrimersappear to be found mainly in between pairs of colloids,thus binding actively to both and gluing them together. Onvisual inspection, the resulting structure is best described asan interconnected, gel-like network including voids. Noticethat, in strong contrast to figures 4(e) and (f), colloids anddendrimers remain well mixed with one another down tothe scale of the individual particle size. Here, dendrimersact as the ‘glue’ between the colloids, whereas in the caseε = 0.5 they act as depletants. Thus the physical mechanismfor the colloidal attractions induced by the dendrimers is verydifferent in the two cases.
Apart from the fact that the dendrimers cluster, which iseven more enhanced by the presence of the colloids in thesystem, the most interesting structural property they exhibitfor the interaction strength ε = 2 is found in their ability to actas a glue for the colloids. In order to examine this behavior inmore detail, the pair correlation functions between dendrimersas well as between dendrimers and colloids are shown infigure 9 for the intermediate sized colloids with RC = 8σ andvarious density/composition ratios. The colloid–dendrimercorrelation function gCD(r) is comparable with the results forthe ε = 0.5 case, except that the adsorption is much stronger,as was already expected from the effective colloid–dendrimerinteractions in figure 1(b). The radial distribution functionsgDD(r) for the dendrimers (see figure 9(b)) not only revealthe peak for small distances related to the clustering of
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
Figure 6. The pair correlation functions gαβ for systems with RC = 8σ and ε = 0.5 as obtained at the different levels of description. (a), (c)and (e) on the left correspond to a low density (ηC = 0.16, ηD = 0.05), and (b), (d) and (f) on the right correspond to a much denser systemat (ηC = 0.27, ηD = 0.12).
the dendrimers, but also indicate the presence of additionalsubstructure on distance beyond the first neighbor. The exactnature of this substructure cannot be determined on thebasis of the available simulation results, and a full analysisfalls outside the scope of the present work. It is clear,however, that it is connected to the network structure visiblein figures 8(c) and (d), which is not fully random. It is, forinstance, conceivable that chain-like structures are formedin the network, which suggest an increased probability infinding dendrimers being adsorbed on opposite sides of thecolloid, forming the bonds between the colloids within achain. Such correlations would indeed be visible in thedendrimer–dendrimer pair correlation function at compatibledistances as found here.
A comparison between results stemming from thetwo-component level description and those from monomer-resolved simulation for the attractive colloid–dendrimersystem ε = 2 is shown in figure 10 in terms of thecolloid–dendrimer pair correlation function. The position ofthe first peak in the distribution, at both low and high colloiddensities, is reproduced well by the coarse-grained approach.This is not too surprising, as it is mainly the strong effective
colloid–dendrimer that is responsible for this absorptionpeak. For larger distances, however, the agreement is lesssatisfactory and small discrepancies occur. Whereas for thelower density the monomer-resolved simulation shows moresubstructure, for the higher density the reverse is true, and itshows less substructure than the two-component descriptiondoes. The two seem to be somewhat in contradiction with eachother. However, the composition ratios are also different, andin combination with almost a factor 4 difference in overallvolume fraction, the systems correspond to different kinds ofstructures that cannot be inferred from this simple structuralanalysis. It does indicate that the many-body effects that areneglected by the coarse-graining procedure might be muchmore relevant here than in the repulsive system with ε = 0.5.This is also confirmed by figure 11 where the dendrimer radialdistribution function between both approaches is comparedfor the lower density case (corresponding to figure 11). Thefinite value attained by the correlation functions at smalldistances indicates the presence of a significant number ofoverlapping dendrimers, and hence that results based on onlytwo-particle interactions can seriously suffer from the lack ofmany-body interactions.
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a
b
c
Figure 7. The radial distribution functions measured at themonomer-resolved level description for colloids with radiusRC = 4σ and interaction strength ε = 0.5 at two different volumefractions (ηC, ηD) and composition ratios x = 0.3.(a) Colloid–colloid gCC(r), (b) colloid–dendrimer gCD(r) and(c) dendrimer–dendrimer gDD(r).
8. Conclusions
We have made a direct comparison between three differentlevels of description for a binary mixture of colloidal particlesand amphiphilic dendrimers. Starting from a fully detaileddescription that contains colloidal particles and dendrimersexplicitly formed by monomeric units with a solvophobicor solvophilic nature, we performed simulations to measurethe effective interactions between two dendrimers, as wellas between a dendrimer and a series of individual colloidalparticles with different radii and interaction strengths. Byusing the center of mass of both dendrimers and colloidalparticles, and assuming the validity of these effective
interactions at finite densities, a first coarse-grained leveldescription is constructed. In so doing the computationalefficiency is enhanced, at the cost of a simplificationin the form of neglecting many-body effects. On thistwo-component level description, a second step of coarsegraining has been performed by means of an AO-liketreatment. To this end, simulations of this level descriptionwere performed in order to measure the depletion potentialinduced by the dendrimers on the colloidal particles. Withthe aid of this depletion potential and the direct colloidalinteractions, a second level coarse model is obtained, inwhich only a single component, i.e. the colloidal particles, aresimulated in a bath of dendrimers that result in an additionalinduced colloid–colloid interaction. In fact, the starting modelitself has already been a coarse-grained model, becausethe solvent in which colloidal particles and dendrimers areimmersed is simulated implicitly and is only present in thepair interactions between the various components at that levelof description.
The simulations have been performed on amphiphilicdendrimers of the second generation, of which only thesolvophobic core particles are attracted by the colloidalparticles, and the solvophilic shell particles experiencea purely repulsive interaction. For these interactions twodifferent choices of interaction strength have been chosenthat result in a purely repulsive colloid–dendrimer interaction(ε = 0.5) and a partial attractive one (ε = 2). Furthermorethree different sizes of colloid radii, and various differentdensities and composition ratios, have been used. The twotypes of interactions lead to different behavior of the mixture,which is even more enriched by the capability of these typesof dendrimers to form clusters of fully overlapping particles.
For the repulsive colloid–dendrimer interaction, the threedifferent approaches result in an excellent agreement of thevarious pair correlation functions even up to an overall volumefraction of 0.40. The binary mixture itself is characterized byan effective weak attraction between the colloidal particlesinduced by the dendrimers and results in the formation ofsmaller, dendrimer-rich domains on the lower levels of coarsegraining. In addition, the presence of the colloidal particles,irrespective of size, enhances the potential to form clusters bythe dendrimers, compared to that of pure dendrimer systems.The fact that these clusters are formed also immediatelyimplies that many-particle effects do play a role, but theagreement between the radial distributions from the threelevels of model description clearly indicate the validity of theapproach.
In the case of the attractive colloid–dendrimer interaction,only the part of the features visible in the correlation functionsthat correspond to the adsorption of the dendrimers on thecolloids is reproduced well. Some discrepancies, however,do appear on the longer length scales. This is caused by anunexpected behavior of the system related to the adsorptionpropensity, during which the conformation of the dendrimersis inverted, i.e. the core monomers move more to the outsideof the dendrimers with respect to the center of mass, inorder to bind optimally to the surface of the colloids. Onthe approach of two colloids, however, they partially bind to
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
Figure 8. Simulation snapshots for the attractive colloid–dendrimer interaction ε = 2 for (a) and (c) the monomer-resolved representationand (b) and (d) the two-component level description. The system parameters are: (a) and (b): RC = 4σ , (ηC = 0.131, ηD = 0.031), x = 0.2;and (c) and (d): RC = 8σ , (ηC = 0.21, ηD = 0.14), x = 0.1. Color code as in figure 4.
either colloidal surface, thus gluing the colloids together andforming aggregates in a fashion different from the depletionmechanism. For higher densities, there appears to be agel-like network formed by colloidal particles and sustainedby dendrimers that bind them. Some of the correlations thusformed are not well produced by the second level description,and in combination with the clustering of dendrimers, indicatethat many-body interactions do play a more important role andbecome relevant for a more detailed structural analysis of thesystem.
On the whole, we have demonstrated that either ofthe three different levels of description of this mixture ofdendrimers and colloidal particles can be used to determinethe main characteristics of this system. To the best ofour knowledge, the comparison between monomer-resolvedsimulations with simulations based on depletion interactionshas not been performed on soft mixtures. Comparisonsbetween the depletion approach and experiments do exist,but even those are extremely rare for soft mixtures [36].By comparing the scattering intensity of dendrimers withprotonated and deuterated end-groups from small-angleneutron scattering [18], experiments might shed some light onthese issues, but both synthesis and measurements are far fromtrivial. The fact that these coarse-grained approaches are thissuccessful, taking into account the complications due to theamphiphilic nature of the dendrimers, shows that the type ofapproach used here could be employed for a variety of similar,
a
b
Figure 9. The pair correlation functions for simulations performedon the monomer-resolved level description for R = 8σ and ε = 2,using four different densities/composition ratios.
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J. Phys.: Condens. Matter 24 (2012) 284119 D A Lenz et al
a b
Figure 10. The colloid–dendrimer pair correlation function obtained by monomer-resolved and coarse-grained simulations for colloids ofradius RC = 8σ and ε = 2 with (a) ηC = 0.04, ηD = 0.055, x = 0.05; and (b) ηC = 0.21, ηD = 0.14, x = 0.1.
Figure 11. The dendrimer–dendrimer pair correlation functionobtained by monomer-resolved and coarse-grained simulations forcolloids of radius RC = 8σ and ε = 2 with ηC = 0.04, ηD = 0.055and x = 0.05.
and possibly simpler, systems. In addition, it was shownthat a binary mixture of colloidal particles and amphiphilicdendrimers can hide many surprises, and that a full analysisof the phase behavior is worthwhile but it falls outside thescope of the present work.
Acknowledgments
This work has been supported by the Marie Curie InitialTraining Network ITN-COMPLOIDS, FP7-PEOPLE-ITN-2008, project no. 234810, and by the Wolfgang Pauli Institute(Vienna).
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