Colloid and Polymer Sciencehttps://doi.org/10.1007/s00396-018-4360-5

ORIGINAL CONTRIBUTION

Template–assisted assembly of asymmetric colloidal dumbbellsinto desirable cluster structures

Hai Pham-Van1 ·Hoang Luc-Huy1 · Thuy Nguyen-Minh1

Received: 31 January 2018 / Revised: 16 June 2018 / Accepted: 20 June 2018© Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractIn this report, using Monte Carlo simulations, we investigate the cluster structure of colloidal dumbbells with asymmetricproperties in sizes via emulsion droplet evaporation. By tuning the diameter ratio of two spherical colloids, q = σl/σs withσl the diameter of the larger sphere and σs the diameter of the smaller sphere, we obtain clusters with both the compact andopen structures. For q < 1.2, we found a unique category of cluster isomers that minimize the second moment of the massdistribution. The uniqueness is lost when q ranges from 1.2 to 1.7. A further increase in the size ratio leads to a variety ofdifferent isomers with more complex configurations.

Keywords Asymmetric dumbbell · Pickering emulsion · Colloidal molecule · Droplet evaporation

Introduction

Assembly of anisotropic colloidal particles controlled viainterparticle interactions or geometric shapes has givenrise to a rich variety of novel structures of colloidalmaterials as well as unusual phase equilibria [1, 2]. Of themany available categories of anisotropic building blocks,colloidal dumbbells composed of two connected colloidalspheres have attracted considerable attention because theaspect ratio, size, interaction potential, and asymmetricfunctionalization properties of two constituent colloidalspheres can be easily manipulated by experiments [3–6].The self-assembly of colloidal dumbbells has been widelystudied in both computer simulations and experiments,revealing the spontaneous formation of crystal structures [7,8], micelles [9], vesicles [10], and bilayers [11, 12].Other investigations of the phase behavior of colloidaldumbbells have shown that even hard dumbbells exhibita complex phase diagram [7, 13–15]. In the case ofvibrating dumbbells, the structure and phase behaviorstrongly depend on the strength of attractive interactions,

� Hai Pham-Vanhaipv@hnue.edu.vn

1 Faculty of Physics, Hanoi National University of Education,136 Xuanthuy, Caugiay, Hanoi 10000, Vietnam

as well as their diameter ratio of the constituent spheresand their separation [11, 12, 16–18]. Vibrating dumbbellsnot only are served as a realistic model for diatomicmolecules [19] but also have potential applications inthe production of new types of colloidal crystals [20] orphotonic gap materials [21, 22].

In contrast to the large body of work aimed atunderstanding the crystal structure and the phase behaviorof colloidal dumbbells, little work has been done todescribe the process of cluster formation as well as clusterstructures. Clusters of colloidal dumbbells with controllablestructures are significant because they can be used as“colloidal molecules” for bottom-up fabrications of novelmaterials with unique magnetic, optical, and rheologicalproperties [23, 24]. Skelhon et al. [25] investigated thecluster formation of Janus dumbbells via the desorption ofa polymeric stabilizer. This procedure, which is supportedby simulations based on a constraint of surface areaminimization, leads to supracolloidal structures such asdimers, trimers, tetramers, and pentamers. However, theauthors have not found cluster configurations with a largernumber of constituent particles.

The evaporation technique of emulsion droplets pio-neered by Velev et al. [26–28] has been widely used to pre-pare clusters of monodispersed colloidal particles [29–34]and bidispersed colloidal particles [35]. These authors [35]found that the interparticle interaction and the wettability ofthe constituent spheres play an important role in the surface

Colloid Polym Sci

coverage of the smaller particles. In addition, the mini-mization of the second moment of the mass distribution,M2 = ∑nc

i=1 |ri − rcm|2 with ri being the position of theparticle i and rcm being only applies if the size ratio, is lessthan 3. Recently, Peng et al. [36] reported both experimentaland simulation work on the cluster formation of dumbbell-shaped colloids by using emulsions. The authors provedthat the minimization of the second moment of the massdistribution is applicable to symmetric colloidal dumbbellsbut not generally true for asymmetric colloidal dumbbell.However, they reported the assembly of cluster structuresof asymmetric colloidal dumbbell at only one size ratioq ≈ 1.36.

In previous work, we employed the model of Ref. [37] toinvestigate the role of the colloid–colloid interaction energyin competition with colloid–droplet interaction energyin the cluster formation of mostly symmetric colloidaldumbbells [38]. In the current paper, we consider in detailthe asymmetric size case to show that uniqueness of M2-minimal structures is lost when the size ratio between thelarge- and small-sized spheres is larger than 1.2. In addition,we analyze the cluster formation and size distributions ofnovel structures.

Model and simulationmethod

Colloid–colloid pair interaction

A sketch of the model for two colloidal dumbbells isillustrated in Fig. 1a. Each dumbbell is composed of twospherical colloids of diameter σl (large sphere) and σs (smallsphere), respectively. The colloids in each dumbbell areseparated from each other by a distance that vibrates in the

range of λ ≤ l ≤ λ + �, where λ = (σl + σs) /2, �

is a parameter to determine the width of the short-rangedattractive square well.

The colloid-colloid pair interaction includes a short-ranged attraction and a Yukawa electrostatic repulsion. Forthe colloidal dumbbells, two colloidal species may havedifferent sizes. Thus, the colloid-colloid pair interaction canbe expressed by three equations

φll (r) =

⎧⎪⎪⎨

⎪⎪⎩

∞ r < σl

−εSW σl < r < σl + �

εYσl

e−κ(r−σl)

rotherwise,

(1)

φss (r) =

⎧⎪⎪⎨

⎪⎪⎩

∞ r < σs

−εSW σs < r < σs + �

εYσs

e−κ(r−σs)

rotherwise,

(2)

and

φls (r) =

⎧⎪⎪⎨

⎪⎪⎩

∞ r < λ

−εSW λ < r < λ + �

εYλe−κ(r−λ)

rotherwise,

(3)

where φll , φls , and φss are the large colloid–large colloid,large colloid–small colloid, and small colloid–small colloidpair interactions, respectively; εSW and � are the depth andwidth of the short-ranged attractive square well; εY andκ are the parameters to control the strength and range ofthe Yukawa repulsion, respectively; r is the center–centerdistance of particles.

As illustrated in Fig. 2a, the colloid–colloid pair potentialis plotted for a typical set of parameters given in Table 1.The potential above formulated is very similar to thepotential shape depicted in Ref. [39] to study the stability of

(a) (b)

Fig. 1 a Sketch of the model of colloidal dumbbells composed of twospecies (large and small spheres). Shown are the diameters of largecolloid, σl , and small colloid, σs . In the initial stage of the simula-tion, the colloid l-colloid s distance in the same dumbbell is l, whereasthe initial distance between any two colloids belonging to differentdumbbells is set to larger than σl + �. b Representative diagram of

a spherical colloidal particle located at an oil-water interface with theinterfacial tension γow. Also shown is the colloid-oil interfacial ten-sion γco and the colloid-water interfacial tension γcw, the height of thespherical cap h, and the contact angle θow. The Young equation canbe interpreted as a force balance of interfacial tensions (marked byarrows)

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(a)

(b)

Fig. 2 Illustration of the pair interactions (a) between two sphericalcolloids at size ratio q = σl/σs = 1.2 (b) colloid-droplet interaction atspontaneous droplet diameter σd(t) = 4σs , and with parameters givenin Table 1

nanoparticle shells. Experimentally, these parameters can betuned via the surface potential, concentration of electrolyte,and charges of particles [39].

Droplet-droplet pair interaction

The droplet–droplet pair interaction, dd , is a purely hard-sphere potential,

dd (r) ={ ∞ r < σd + σl

0 otherwise,(4)

where the droplet diameter σd is added to the larger colloiddiameter σl to ensure that no two droplets can share the samecolloid.

Table 1 Parameters of the pair interaction potentials used in thecomputer simulation

Physical quantity Description

εSW = 9 kBT Square-well depth

� = 0.09 σs Square-well width

k = 10 σ−1s Inverse Debye length

γ = 100 kBT

σ 2s

Oil–water interfacial tension

Colloid–droplet pair interaction

Figure 1b shows a schematic diagram of a colloidal particleat an oil–water (droplet–solvent) interface. The contactangle (measured in the water phase), θow, is related to thethree interfacial tensions by the Young equation

cos θow = γco − γcw

γow, (5)

where γco, γcw, and γow represent the colloid–oil, colloid–water, and oil–water interfacial tensions. The free energy−�Gint needed to detach colloid of diameter σc from theinterface due to the Pickering effect [40] is

− �Gint = π

4σ 2

c γow (1 ± cos θow)2 , (6)

where the minus and plus signs inside the bracketcorrespond to the removal of the colloid into the water phaseand into the oil phase, respectively. The binding energy isof the order of 103 − 105kBT , where kB is Boltzmann’sconstant, and T is the temperature, for nano-sized sphericalsilica, γow = 0.036 Nm−1 and θow = 90◦ [41]. However,Eq. 6 is only valid when the contact angle γow is formedby the planar interface. In the case of a particle located at aspherically curved water–oil interface, as shown in Fig. 1b,the free energy is a more complicated function of the oildroplet diameter [41].

Similarly to the previous study of [37, 38], we assumethat the colloid–water interfacial tension is equal to thecolloid–droplet interfacial tension, γco = γcw, so that θow =90◦. This assumption is reasonable since a change in thecontact angle seems to not have an influence on the finaloutcomes [42]. In addition, we neglect the influence of theadsorbed colloid on the oil–water interfacial curvature suchthat the droplet remains spherical.

The evaporation of the dispersed oil droplet implies thatthe droplet diameter is initially larger and eventually smallerthan the colloid diameter. In order to mimic this situation,the colloid–droplet potential φcd (r) is given as follows. Ifthe diameter of droplets σd is larger than that of the colloidsσi that is σd > σi , the colloid–droplet adsorption energy is

id ={

−γπσdhσd − σi

2< r <

σd + σi

20 otherwise,

(7)

and when σd < σi ,

id =

⎧⎪⎪⎨

⎪⎪⎩

−γπσ 2d r <

σi − σd

2−γπσdh

σi − σd

2< r <

σi + σd

20 otherwise,

(8)

Colloid Polym Sci

where i = l, s denotes the two colloidal species in thedumbbell, and h is the height of the spherical cap resultingfrom the colloid–droplet intersection (see Fig. 1b) given by

h = (σi/2 − σd/2 + r) (σi/2 + σd/2 − r)

2r. (9)

The parameter γ is the droplet–solvent interfacial tension inorder to control the colloid–droplet interaction strength. SeeFig. 2b for an illustration of the colloid–droplet pair potential.

Simulationmethod

We carry out Metropolis Monte Carlo (MC) simulation inthe canonical (NV T ) ensemble for a ternary mixture ofNd droplets (Nd = 11 − 44) and Nc colloidal dumbbells(Nc = 250) formed by two colloidal species. The dropletand colloid packing fractions are ηd = 0.15 and ηc =0.03−0.06, respectively. The initial droplet diameter is fixedat σd(0) = 8σs . The parameters of the pair interactionsare given in Table 1. We note that the colloid–colloidattractive potential is chosen in such a way that once thephysical bonds between the colloids have formed via dropletevaporation, they are irreversible; meanwhile, the repulsivebarrier is also high enough to hinder the spontaneousclustering, that is, clustering not mediated by droplets.Different from Ref. [38] where authors investigated the roleof the colloid–droplet adsorption interaction in competitionwith the Yukawa repulsion, we restrict ourselves to thecase of a strong colloid–droplet adsorption interaction.Therefore, we set the droplet–colloid adsorption energywith the interfacial tension γ = 100 kBT

σ 2s

, which is much

larger than the Yukawa electrostatic repulsion (≈ 9 kBT ).Simulations are performed at different size ratios q = σl/σs

between 1.0 and 3.0. For a fixed set of parameters, statisticaldata are collected by running 40 independent simulations.Each simulation consists of 106 MC sweeps that for smallmaximum displacement steps of colloids dc = 0.01σs

and droplets dd = dc

√σs/σd can reproduce the dynamics

of Brownian dynamics simulations [43]. In each sweep,all particles are attempted to be moved once on average,while the collective motion of particles in the cluster, i.e.,collective translational and rotational cluster moves, is notperformed because such collective motion only plays a rolein dense colloidal systems [44]. The droplet diameter isshrunk at a fixed rate so that the droplets vanish after 5×105

MC sweeps. This leaves the remaining 5 × 105 MC sweepsto equilibrate the simulation system.

The initial random configuration of non-overlappingcolloidal particles and droplets is prepared in a cubic boxwith the periodic boundary condition. The initial distancebetween the large colloid and small colloid in the samedumbbell (see Fig. 1a) is set smaller than λ+�, whereas theinitial distance between any colloidal species that belong todifferent dumbbells is set larger than σ1 + �. As a result,no two dumbbells bind together in the initial stage of thecomputer simulation.

We define a bond between two colloidal spheres oftypes i and j when their distance is smaller than or equalto σi + σj/2, with i, j = l, s. A cluster is a group ofcolloidal particles connected with each other by a networkof bonds. We use the number of bonds nb as an indicatorof the compactness of the cluster. Hence, each cluster ischaracterized by both the number of bonds nb and thenumber of colloidal particles nc belonging to this cluster.For a cluster with the same number of constituent colloidsnc, a larger nb indicates a more closed structure as comparedto a cluster with less bonds. Clearly, a single dumbbell canbe regarded as a trivial cluster with nc = 2, nb = 1.These trivial clusters will not be considered in the followinganalysis.

Result and discussion

Figure 3 presents some typical snapshots of the simulatedsystem at four different stages of the time evolution at

Fig. 3 Snapshots of simulation system for a diameter ratio q = 1.2at different stages of the time evolution (a) initial configuration bafter 2.8 × 105 MC sweeps, c after 5 × 105 MC sweeps, and dafter 106 MC sweeps. Droplets are shown in dark blue spheres. Each

colloidal dumbbell is made of a large (red) sphere and a small (yellow)sphere. Colloidal particles that belong to clusters formed due to dropletevaporation are colored differently, that is, purple and green spheresrepresent large spheres and small spheres, respectively

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σl/σs = 1.5. In Fig. 3a, the initial configuration consistsof non-overlapping spheres of both the colloidal dumbbells(red sphere–yellow sphere) and the large (blue) droplets.

(a)

(b)

(c)

(d)

Fig. 4 Colloid–colloid radial distribution functions, gcc(r), as afunction of the scaled distance r/σs in the final stage of the simulation.Results are shown at four different values of the size ratio: a q = 1, bq = 1.2, c q = 1.5, and d q = 2.0

After 2.8 × 105 MC sweeps, the droplets have shrunk andsimultaneously trapped several colloidal dumbbells at theirsurface (Fig. 3b). After 5 × 105 MC sweeps, the dropletscompletely vanish and the colloidal dumbbells arrange intoclusters (Fig. 3c). The colloidal species that belong to acluster configuration are colored differently. At the end ofthe simulation, i.e., after 106 MC sweeps, the presence of allclusters induced by the droplets demonstrates the stabilityof the clusters against thermal fluctuations (Fig. 3d).

We investigate the assembly of the colloidal dumbbellsinto clusters by means of the colloid–colloid radialdistribution functions (RDFs), gcc(r). Figure 4 shows theresults for gcc(r) at different size ratios q after 106 MCsweeps. As shown in Fig. 4a, for the case of q = 1,gcc(r) has a pronounced peak at r = σs corresponding

(b)

(a)

(c)

Fig. 5 a Typical M2-minimal structures found in the final stage of thecomputer simulations at q = 1.2. Each column includes the number ofcolloids nc with a corresponding bond-number nc, the cluster structure,and its polyhedron and name just below. b Same as a but for q = 1.5.Only additional configurations not observed at q = 1 are shown. cSame as a–b but for q = 2.0. Only additional configurations notobserved at both q = 1 and q = 1.5 are shown

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to a large number of bonds between colloidal particles. Atdistances larger than σs , gcc(r) in the inset to the figureshows small peaks indicating that clusters have formed. Forq = 1.2, in Fig. 4b, gcc(r) exhibits a sharp peak and twoless well-defined side peaks, one on each side of the centralpeak. While the central peak at r = (σl + σs)/2 is mainlydue to the bond between the colloidal species in the samedumbbells, the left peak originates from the bond formationbetween small colloids, and the right peak at r = σl is dueto the bonds between large colloids. The width of all threepeaks is the same, consistent with the width of the short-ranged attractive square well �. Apparently, as q increases,the central and right peak shift to greater distances (Fig. 4c–d). In addition, an increase in the size ratio, or equivalentlycolloid packing fraction, leads to a larger probability oftrapping colloids in the initial stages of the simulations,signaled by a higher height of the central peak.

Figure 5a shows cluster structures obtained for the sizeratio q = 1.2. We observe stable clusters of uniquestructures that are identical to the cluster configurationsmade of symmetric size dumbbells [36, 38] or singlecolloids [37]. Such clusters include tetrahedron (nc =4), octahedron (nc = 6), snub disphenoid (nc =8), gyroelongated square dipyramid (nc = 10), andicosahedron (nc = 12). This result can be explained bythe following argument. Colloids trapped on the dropletsurface feel the adsorption energy with the interfacialtension γ = 100kBT/σ 2

s much larger than the Yukawarepulsive interaction and thermal energy. As a result, bothcolloidal species once trapped are strongly localized at thedroplet surface and brought closely together during dropletevaporation. Therefore, the final clusters possess a compact

structure satisfying the relation nb = 3nc−6, identical to thestructures of colloidal clusters obtained through template-assisted assembly [36, 45]. Our result indicates that theuniqueness of compact clusters made of colloidal dumbbellsis maintained until q = 1.2.

As the size ratio increases, e.g., q = 1.5, someadditional structures are found, as shown in Fig. 5b. Here,we observe that the nc-sphere cluster (cluster consists ofnc spheres), except for the cluster with nc = 12, canbe decomposed into one (nc − 1)-sphere clusters withthe polyhedral configuration that minimizes M2 and oneparticle outside this polyhedron. For example, the six-spherecluster (leftmost of Fig. 5b) is composed of one five-sphere cluster (triangular dipyramid) and one additionalparticle. The cluster structure with nc = 12 is identical tothe icosahedron except for one missing particle inside theicosahedron (marked by a red filled circle) but plus anotherone outside the polyhedron. Increasing the size ratio further,for example, to q = 2.0, generates more different isomersthat are similar to the less-compact structures (see Fig. 5c).

Figure 6 shows stacked histograms of the number ofclusters at different size ratios. The total height of thecolumns represents the number of cluster Nnc with nc

colloids and the height of each bar (colored differently)is the number of clusters with bond number nb. It can beseen that the number of clusters, especially small clusters,increases with increasing q. This can be explained by alarger probability available to capture colloids at the dropletsurface, in agreement with the result obtained by analysis ofcolloid–colloid radial distribution. Furthermore, for q ≤ 1.2(Fig. 6a and b), only a unique nc-isomer is found, except fornc = 4. For the size ratio q > 1.2, e.g., q = 1.5, Fig. 6c

Fig. 6 Distribution of thenumber of clusters Nnc as afunction of the number ofcolloids nc in the cluster in thefinal stage of simulation. Resultsare shown for different sizeratios a q = 1, b q = 1.2, cq = 1.5, and d q = 2.0. Thenumerical label in differentlycolored regions indicates thebond number nb

(a) (b)

(c) (d)

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shows two different structures: one belongs to M2-minimalisomers with nc colloids and the other isomers with (nc −1)colloids plus one particle (as above shown in Fig. 5b and c,respectively). Finally, when q ≥ 1.7, e.g., q = 2.0, we finda variety of different isomers of open and closed structures.We interpret the occurrence of these structures as a directresult of the spatial hindrance of large spherical colloidsagainst the bond formation between small spherical colloidsat large size ratios.

Conclusion

We have investigated the cluster assembly of colloidaldumbbells via emulsion droplet evaporation using kineticMonte Carlo simulations. Each colloidal dumbbell iscomposed of two colloidal spheres separated by a distancethat can fluctuate in a small range. Colloids interactvia a short-ranged attractive and longer-ranged repulsiveinteraction whose interaction strength is chosen to avoidthe spontaneous formation of clusters, and to ensurethat physical bonds between colloids are permanent. Thedroplet–droplet interaction is a hard-core repulsion withan effective hard-sphere diameter chosen so that anytwo droplets cannot merge due to a shared colloid.The colloid–droplet adsorption interaction is aimed atmodeling the Pickering effect, which has a minimum at thedroplet surface. To model the evaporation of droplets inexperiments, the droplet diameter shrinks at a fixed rate.

In the case where the adsorption energies of the dropletswith both types of colloids are much larger than theYukawa repulsion and thermal fluctuation, a unique M2-minimal isomer for size ratios less than 1.2 is found.In addition to these isomers, for size ratios in the range1.2 to 1.7, we found a particular category of clusterswith a M2-minimal compact core and one protruding arm.A further increase in the size ratio leads to a varietyof different isomers with more complex configurations.It should be noted that while the well-defined closedcluster structures produced by the template-assisted strategyhave been reported experimentally and simulation, theoccurrence of complex, less compact structures as wellas their stability in experiments is in general unknown.Moreover, the less compact clusters (low nb) can in turnused as building blocks for synthesis of syndiotactic,chiral [46], “crisscross” [47] structures rather than thecrystalline structures or aggregates traditionally observedin closed cluster structures. Therefore, our result could, inprinciple, be a promising way toward the preparation ofnovel and complex colloidal molecules.

Funding information This work was funded by Vietnamese NationalFoundation for Science and Technology Development (NAFOSTED)under grant number 103.02-2017.328.

Compliance with ethical standards

Conflict of interest The authors declare they have no conflicts ofinterest.

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