Self-Assembly of Janus Ellipsoids II: Janus Prolate Spheroids

Article pubs.acs.org/Langmuir

Self-Assembly of Janus Ellipsoids II: Janus Prolate Spheroids Wei Li*,† and James D. Gunton‡ †

Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, United States

‡

ABSTRACT: In self-assembly, the anisotropy of the building blocks and their formation of complex structures have been the subject of considerable recent research. Extending recent research on Janus particles and completing the study of Janus spheroids, we conduct a systematic investigation on the selfassembly of Janus prolate spheroids based on a primitive model that we proposed. Janus prolate spheroids are particles that have a prolate spheroidal body and two hemi-surfaces along the major axis coded with different chemical properties. Using Monte Carlo simulations, we investigate the effects of the aspect ratio on the self-assembly process. In contrast to the vesicle-like aggregates for Janus oblate spheroids, we obtain various ordered cluster structures for Janus prolate spheroids through selfassembly. With an increasing aspect ratio, we find a transition of cluster morphology, from vesicles to tubular micelles and micelles. In particular, a relatively small change in the aspect ratio leads to a rather significant change in morphology. We apply a cluster analysis to understand the mechanism associated with such a transition.

■

INTRODUCTION Self-assembly is the term used to describe the autonomous organization of pre-existing components from a disordered system into ordered structures or patterns. Because it provides a bottom-up approach to the design and preparation of functional materials, self-assembly has become one of the more important concepts in this century.1,2 It is well-known that isotropic particles can form into amorphous and crystalline structures.3 Furthermore, with the advance in synthesis technology, the fundamental building blocks of the selfassembling systems have become more complex, including irregular geometrical shapes and directional interactions. This consequently increases the complexity of the structures and enriches the properties that new materials can have.4−8 As a result, considerable experimental and simulation work has been devoted to study the effect of so-called “anisotropy dimensions”,4 such as surface coverage (patchiness), aspect ratio, etc.9−21 In this paper, we present a study of the self-assembly of Janus prolate spheroids by means of Monte Carlo simulation. Janus particles are particles with two hemi-surfaces coded with different chemical properties, such as hydrophobicity and hydrophilicity.14,22−27 Because of the recent success in synthesizing different kinds of Janus particles (colloidal or polymeric), the self-assembly of Janus particles and their potential application have received a significant amount of attention. For instance, considerable experimental and numerical work on Janus spheres has been reported.11,15,28−35 Moreover, the anisotropy dimensions of Janus particles have also drawn extensive attention. Studies have been conducted on the synthesis of non-spherical Janus particles36−38 as well as investigating new properties that the anisotropy dimensions bring.38−41 Recently, a study of synthesis and assembly of one © 2013 American Chemical Society

kind of Janus spheroid particle has been reported, which extends the search in a specific anisotropy dimension, namely, the aspect ratio of ellipsoidal Janus particles.42 Thus, to further explore this topic as well as to complete the “blueprint” of Janus spheroids, here, we study the self-assembly of Janus prolate spheroids with different aspect ratios systematically, using a primitive model proposed earlier.40 We find that an increase in the aspect ratio produces a transition in cluster morphology. We also provide a cluster analysis to gain a better understanding of this transition.

■

MODEL AND SIMULATION METHOD

A spheroid is a type of ellipsoid with the lengths of the semi-principal axes denoted by c ≠ a = b. The aspect ratio is defined as ε = c/a, with ε > 1.0 for a prolate spheroid and ε < 1.0 for an oblate spheroid. In our study, a Janus prolate spheroid is defined as a hemi-spheroidal patchy model, with its patch orientation along the c axis; the length of the prolate spheroid is set as c = σ/2. Analogous to the Kern−Frenkel patchy model,9 we choose the spheroids to interact through a pair potential that depends upon their separation and orientation.

Uij = Uf (riĵ , nî , nĵ )

(1)

Considering the computational expense in determining the spatial relation between two spheroids, as illustrated in ref 40, the radial dependence is implemented by a so-called “quasi-square-well” potential

⎧ if particles overlap ⎪∞ U=⎨ ⎪ ⎩− u0H(σij + 0.5σ − rij) otherwise

(2)

Received: May 2, 2013 Revised: June 5, 2013 Published: June 6, 2013 8517

dx.doi.org/10.1021/la4016614 | Langmuir 2013, 29, 8517−8523

Langmuir

Article

Figure 1. Plot of second virial coefficient (B2) versus ellipsoid aspect ratio at three temperatures, T = 0.25, 0.28, and 0.33. (A) B2 of Janus ellipsoid− ellipsoid interaction, BJE 2 . Gray circles indicate the B2 of pure hardcore prolate spheroids. The dashed line is for the guide of B2 = 1.0. (B) (Upper IE IE panel) Scaled B2 of Janus ellipsoid−ellipsoid interaction (BJE 2 ) on isotropic ellipsoid−ellipsoid interaction (B2 ) and (lower panel) B2 dependence upon the aspect ratio. Color stars indicate the values from analytical calculation. temperature T is also expressed in the unit of u0 (with the Boltzmann constant kB = 1).

and the angular modulation is

⎧1 if nî riĵ ≤ 0, nĵ rjî ≤ 0 f (riĵ , nî , nĵ ) = ⎨ ⎩ 0 otherwise

■

⎪

⎪

RESULTS AND DISCUSSION We use the second virial coefficient B2 as a measure of the effective, temperature-dependent interaction between particles.9

(3)

where u0 is the well-depth, H(x) denotes the Heaviside step function, rij is the center−center distance between particles, r̂ij is the direction of the particle separation vector r⃗ij = r⃗i − r⃗j, and n̂i and n̂j are the patch orientation vectors. σij, the orientation-dependent range derived from a Gaussian overlap model, is given by −1/2 2 ⎡ ⎛ (riĵ nî − riĵ nĵ )2 ⎞⎤ χ ⎜ (riĵ nî + riĵ nĵ ) σ⎢ ⎟⎥ 1− ⎜ σij = + 2 ⎝ 1 + χnî nĵ 1 − χnî nĵ ⎟⎠⎥⎦ ε ⎢⎣

B2 =

1 2V

∫ dri ∫ drj ∫ dnî ∫ dnĵ [1 − e−βU ] ij

(5)

In particular, we use a scaled B2; i.e., B*2 = in our 3 discussion. Bhs = 2πσ /3 stands for the second viral coefficient 2 of a hard sphere, whose diameter is σ. For simplicity, we omit the “∗” in the notation for all of the B2 values in the following discussion. In our study, the phase diagram of Janus prolate spheroids is not yet known. Thus, we start our investigation by comparing the B2 of Janus ellipsoids to that of Janus spheres, whose phase diagram has been studied earlier.30 We show the results in Figure 1. We first give the analytic values of B2 for spherical particles (ε = 1.0) interacting through a square-well 3 potential; specifically, for isotropic spheres, BIS 2 = 1 − (δ − 9 βu0 3 βu0 JS 1)(e − 1), and for Janus spheres, B2 = 1 − (δ − 1)(e − 1)/4,40 where the interaction range δ has been chosen as δ = 1.5σ. Next, we give the numerical results for B2 of Janus ellipsoids, BJE 2 , obtained from Monte Carlo integration, for our “quasisquare-well” model. We plot the BJE 2 versus the aspect ratio ε at three temperatures, T = 0.25, 0.28, and 0.33 in Figure 1A. The gray circles represent the numerical results for the B2 of pure hardcore prolate spheroids (i.e., the “hydrophilic” parts cover the whole surfaces of the spheroids). We find that (a) the magnitude of BJE 2 decreases as the temperature and aspect ratio increase. This suggests that particles combine more “strongly” at lower temperatures. In addition, the larger the aspect ratio, the weaker the attraction between particles (for the same interaction cutoff, the interaction range depends upon the geometric shape of individual particles and the volume integral in eq 5 is reduced by a factor of the aspect ratio, i.e., ε−1). (b) The interaction between particles approaches a pure hardcore repulsion as the aspect ratio increases, because the probability of two hydrophobic parts finding each other becomes smaller when the particles are slimmer. B2/Bhs 2

(4)

with χ = (ε2 − 1)/(ε2 + 1). σij is introduced as an approximation to characterize the center−center distance between two prolate spheroids according to their orientations, such that there is no overlapping if rij ≥ σij.43 Therefore, H(σij + 0.5σ − rij) represents a “quasi-square-well” potential with an attractive interaction range equal to 0.5σ.40 It is to be noted that, for the case of ε = 1.0 (sphere), σij = σ; thus, the radial dependence part becomes a standard square-well potential with a cutoff range of 1.5σ. To summarize, the model describes the interaction between two Janus prolate spheroidal particles; i.e., the Janus prolate spheroid has two dissimilar hemi-surfaces (along the long axis), which interact differently with other particles. Describing an amphiphilic Janus prolate spheroid with this model for example, (a) the interactions between two hydrophobic hemi-surfaces of Janus spheroids are “quasisquare-well” potentials and (b) the interaction between a hydrophilic hemi-surface and other surfaces is just a hardcore repulsion. Our choice for the hydrophilic interactions corresponds to an experimental situation in which the salt concentration is sufficiently large as to screen the Coulomb interactions. We use a standard Monte Carlo simulation in the NVT ensemble to study the self-assembly of the Janus prolate spheroids. We choose three-dimensional systems of size L = 30σ, with a number density ρ = 0.02. We pick the low-density value to study the fundamental structures formed from direct self-assembly. Periodic boundary conditions are enforced to minimize wall effects. The simulations start from a random initial monomer conformation and run for 107 Monte Carlo steps (MCSs) (the position and orientation of each particle inside the system are tested for renewing at each MCS). We then analyze the data at the end of the simulation, with our final statistical results averaged over tens of runs. In the following, σ is taken as the unit of length and u0 is taken as the unit of energy. The 8518


Langmuir

Article

To understand the B2 dependence upon the hemi-surface patch and aspect ratio, we show the ratio between the B2 of Janus spheroids and isotropic spheroids (i.e., the “hydrophobic” IE parts cover the entire surfaces of the spheroids), BJE 2 /B2 , in the upper panel of Figure 1B and, for comparison, BIE in the lower 2 panel for reference. The results show that the aspect ratio has a significant impact on the reduction in the magnitude of B2, while the patch brings an extra decrease, and such a effect becomes more distinct for a larger aspect ratio. This may reduce the complexity of the structures formed in the selfassembling process. Thus, we arbitrarily split the map of Figure 1A into two regions: region 1, ε = 2.0−10.0, “structure-poor” region, and region 2, ε = 1.0−2.0, “structure-rich” region. From the discussion and similarity among the B2 curves for different temperatures, we set the system temperature at T = 0.25, which is believed to be far below the critical temperature, for all of the later simulations. Nevertheless, the results and structures that we find at this temperature are quite general for the gas-phase regime, which is similar to the case of Janus spheres.30 The simulation results also verify the simplicity in the cluster structures of region 1. We study the self-assembly for ε from 2.0 to 10.0 (step size of 1.0) and show typical system configurations and cluster structures in Figure 2. From the

those self-assembly systems are close to (or in) the equilibrium states. The differences between the plateaus indicate the differences in cluster sizes and structures. It is to be noted that because of the thermal fluctuation and cluster−cluster aggregation, there are tiny variations in the cluster morphology along with the simulation time, which result in the slow evolution of the potential energy curves. However, such changes would not affect the fundamental structures formed from the self-assembly in the gas-phase region and the dependence of the cluster morphology upon the aspect ratio, which is the main focus of this study. Thus, we believe it is sufficient to study the character of the self-assembly systems in this window (∼107 MCSs) and will comment on such later stage evolutions in the following results. It is also interesting to notice that all of the potential curves scaled by the aspect ratio, i.e., ⟨Uij⟩/ε, seem to collapse onto a master curve as a function of the simulation time (whereas the deviation of the ε = 2.0 curve might be caused by the difference in the equilibration “time” because of the different overall particle volume fraction), which is reminiscent of the law of “corresponding states” and the possible extension of B2 scaling in calculating the phase diagram.44,45 The typical morphologies for different aspect ratios are shown in Figure 4. From the snapshots, we find that, in the upper panel, the final configurations of ε = 1.0, 1.1, and 1.2 (at 107 MCSs) are quite similar; i.e., the systems consist of vesicles (double-layer shells), micelles (single-layer shells), and small oligomers. This also explains why the potential curves of these three aspect ratios are close to each other in Figure 3. The small differences in energy for ε = 1.1 and 1.2 in comparison to ε = 1.0 are due to the change in monomer geometry, which leads to small changes in the cluster sizes and, hence, energy. As ε reaches 1.3 (Figure 4D), the morphology changes, the vesicle structure becomes unstable, and a tubular (cylindrical) micelle structure dominates the configuration. Interestingly, the tubular micelles are also found for the larger aspect ratios (ε = 1.4 and 1.7); however, the longitudinal length decreases dramatically as the aspect ratio increases beyond 1.3. Thus, as shown in Figure 4F, the micelle structures become the dominant part of the configuration for ε = 1.7. This change in morphology in the neighborhood of ε = 1.3 makes this narrow region rather interesting. Note that a relatively small change in ε results in such a significant change in morphology. This sensitive dependence upon a control parameter in anisotropy dimensions could also be true in other self-assembly systems and, thus, potentially be of importance in the design of new materials. To gain a better qualitative understanding of the properties of the clusters formed in those systems, we plot the statistical results for the cluster distribution with a bin size ΔNp = 5 in Figure 5. Then, we give a rough interpretation of the dominant peaks in the cluster distribution. For ε = 1.0, the dominant peak at Np ≅ 40 corresponds to a vesicle and is consistent with earlier results for Janus spheres.13,30 The region around the secondary peak at Np ≅ 20 corresponds to micelle aggregates. For ε = 1.2, vesicles of sizes Np ≅ 20−60 dominate in the cluster distribution. The more or less flat portion shows a rather wide range of vesicle size, corresponding to a slimmer particle than for smaller aspect ratios. The curve for ε = 1.3 shows an oscillatory, “periodic” behavior in size, because the dominant peaks occur at about Np = 5, 15, 25, 35, etc., which suggests that the chain structure forms from the assembly of “sub” building blocks (Np ≅ 5−9) rather than from monomers. In other

Figure 2. Configuration of the entire system (T = 0.25 and ρ = 0.02) after 107 MCSs for ellipsoid aspect ratios (A) ε = 3.0 and (B) ε = 10.0. (C and D) Largest clusters in those two configurations with cluster size Np = 42 and 26, respectively. The “hydrophobic” (hydrophilic) hemi-surfaces are coded in red (blue).

figure, we see that most clusters in the two simulation boxes are micelles with (complete or incomplete) “star” structures. One would expect that the size of the stable cluster (the number of particles in the cluster) depends upon the geometric shape of the monomer (because of the possible difference in packing fraction). However, their properties, such as volumes and chemical characteristics (e.g., hydrophobic inside and hydrophilic outside), are quite similar and rather simple. Now, we focus on region 2; we simulate the systems with ε from 1.0 to 2.0 and step size of 0.1. We show the evolution of the ensemble average potential in the upper panel of Figure 3A and a “zoom-in” section of the later stage in Figure 3B. The plateaus of the potential curves at the later stage suggest that 8519


Langmuir

Article

Figure 3. System potential evolution along with MCS (T = 0.25 and ρ = 0.02) for aspect ratios from 1.0 to 2.0. (A) (Upper panel) Ensemble average of system potential ⟨Uij⟩ and (lower panel) scaled system potential ⟨Uij⟩/ε. (B) “Zoom-in” section at the later stage of the system potential evolution in the semi-log plot.

Figure 4. Configuration of the entire system after 107 MCSs (T = 0.25 and ρ = 0.02) for ellipsoid aspect ratios (A) ε = 1.0, (B) ε = 1.1, (C) ε = 1.2, (D) ε = 1.3, (E) ε = 1.4, and (F) ε = 1.7.

words, the tubular micelle is formed from a hierarchical selfassembly. Such a periodicity dies out quickly as the aspect ratio increases, and the sub building blocks are fairly large (for instance, Np ≅ 20 for ε = 1.4 and 1.7). As a result, the “docking” points (exposed “hydrophobic” regions) on the surface of sub building blocks that attract each other can be easily eliminated in the equilibration process (the adjustment to lower the free energy because of surface tension effects). Such an effect is not helpful in generating larger ordered or higher hierarchical structures, because the repulsion of “hydrophilic” surfaces tend to keep the micelle-like structures in the clusters. We also note that, because of the late stage cluster−cluster aggregation process, there might be small changes in the cluster distribution function with time. With regard to the tubular micelle structure, it continues growing along with the simulation time until it exhausts the sub building blocks in its neighborhood and the regularity is transmitted along the whole tube. However, for the micelle and vesicle aggregates, once they

form the complete shells, their repulsive outer surfaces would slow the aggregation with other monomers and aggregates. In consequence, the micelles and vesicles are rather isolated and stable (in the gas micellar phase). We give one example of this in our discussion of Figure 8 below. Next, we evaluate the distribution of the scalar product n̂1n̂2 for all pairs of bonded ellipsoid particles for different ε and give the results in Figure 6. The two peaks that are near −1.0 (the patch orientation vectors of a pair of particles face toward each other) and 1.0 (the two patch orientation vectors are parallel) for ε = 1.0 (Janus sphere) are quite consistent with previous investigations.30 The similarity of the curves ε = 1.0, 1.1, and 1.2 is consistent with the cluster morphologies being similar in the three cases. Furthermore, new order (peaks) builds up within the intermediate region that reflects the typical close packing structures because of the constraint of geometrical shapes. This becomes more obvious for larger aspect ratios, which increase the system symmetry and reduce the complexity 8520


Langmuir

Article

Figure 7. Plot of the distance r(i) of the ith particles to the center of mass of the cluster, corresponding to the configuration (ε = 1.1 and Np = 49) shown in the inset.

Figure 5. Distribution of cluster size (Np) inside the system after 107 MCSs for different ellipsoid aspect ratios.

the moments of inertia of the two spherical shells, I = 2(m1r12 + m2r22)/3 = 85.07, where m1 = 9 and m2 = 40. We then calculate the principal moments of inertia for the cluster by diagonalizing the inertia tensor numerically and find the results I1 = 85.93, I2 = 85.47, I3 = 83.38, and the moment of inertia I = (I1 + I2 + I3)/ 3 = 84.93. I1, I2, and I3 are approximately the same and indicate the spherical symmetry of the cluster. The result for I is consistent with our prediction about the vesicle structure of the cluster. Finally, we show the typical clusters for ε = 1.2, 1.4, and 1.3 in Figure 8. These configurations are from the simulations run

Figure 6. Distribution of orientation correlation (n̂1n̂2) over all pairs of bound ellipsoid particles for different aspect ratios.

of cluster morphologies. However, in contrast to the other two groups of curves (ε = 1.0, 1.1, and 1.2 and ε = 1.4, 1.5, 1.7, and 2.0), ε = 1.3 is a distinct case, because the different configurations of pairs of particles have an equal probability of occurrence (the flat region of the distribution function). This reflects the cylindrical symmetry of the ordered tubes, in contrast to the spherical symmetry of micelles. In combination with the results of Janus oblate spheroids,40 we find that this type of Janus spheroid particle (Janus patch along the c axis) prefers to form bilayer or multilayer structures for ε = 0.1−1.2 (generally two peaks in the pair orientational correlation), tubular micelle structures around ε = 1.3, and mostly micelles for ε greater than 1.3. To study the cluster morphology in a more quantitative way, we apply a simple inertia tensor analysis for a typical example. As shown in Figure 7, we plot the distribution of the particle distance from the cluster center of mass, for the cluster (ε = 1.1 and Np = 49) in the inset. From the result, we find that such a cluster consists of two layers, with 9 ellipsoids in the first layer (inner shell) and 40 ellipsoids in the second layer, with two average radii r1 = 0.85 and r2 = 1.74, respectively. Thus, we give an estimate of the moment of inertia I by computing the sum of

Figure 8. Typical clusters of (A) ε = 1.2 and Np = 69, (B) ε = 1.4 and Np = 39, and (C) ε = 1.3 and Np = 88.

up to 4 × 107 MCSs, in contrast to all our other results. These runs were carried out to check the stability of the configurations found at the end of 107 MCSs and confirm our earlier conclusions. We see again the transition from vesicle to tubular micelle and micelle, as well as the regularity in the tube of ε = 1.3. Furthermore, Figure 8C also illustrates the formation of such a tubular structure by attaching sub building units one by one and reorganizing the particles into an ordered geometrical arrangement. For ε = 1.4, however, such an anisotropy modulation, namely, the change in length of semi-axes a = b = c/ε from 0.385 to 0.357 (in comparison to ε = 1.3), weakens the formation of the higher hierarchical order through such an attaching process and, hence, leads to the limitation in “chain” extension and reduces the total length. It is to be noted that, for 8521


Langmuir

Article

(5) Yang, S.-M.; Kim, S.-H.; Lim, J.-M.; Yi, G.-R. Synthesis and assembly of structured colloidal particles. J. Mater. Chem. 2008, 18, 2177. (6) Pawar, A. B.; Kretzschmar, I. Fabrication, assembly, and application of patchy particles. Macromol. Rapid Commun. 2010, 31, 150. (7) Du, J.; O’Reilly, R. K. Anisotropic particles with patchy, multicompartment and Janus architectures: Preparation and application. Chem. Soc. Rev. 2011, 40, 2402. (8) Sacanna, S.; Pine, D. J. Shape-anisotropic colloids: Building blocks for complex assemblies. Curr. Opin. Colloid Interface Sci. 2011, 16, 96. (9) Kern, N.; Frenkel, D. Fluid−fluid coexistence in colloidal systems with short-ranged strongly directional attraction. J. Chem. Phys. 2003, 118, 9882. (10) Bianchi, E.; Largo, J.; Tartaglia, P.; Zaccarelli, E.; Sciortino, F. Phase diagram of patchy colloids: Towards empty liquids. Phys. Rev. Lett. 2006, 97, 168301. (11) Hong, L.; Cacciuto, A.; Luijten, E.; Granick, S. Clusters of amphiphilic colloidal spheres. Langmuir 2008, 24, 621. (12) Giacometti, A.; Lado, F.; Largo, J.; Pastore, G.; Sciortino, F. Effects of patch size and number within a simple model of patchy colloids. J. Chem. Phys. 2010, 132, 174110. (13) Sciortino, F.; Giacometti, A.; Pastore, G. A numerical study of one-patch colloidal particles: From square-well to Janus. Phys. Chem. Chem. Phys. 2010, 12, 11869. (14) Jiang, S.; Chen, Q.; Tripathy, M.; Luijten, E.; Schweizer, K. S.; Granick, S. Janus particle synthesis and assembly. Adv. Mater. 2010, 22, 1060. (15) Chen, Q.; Whitmer, K.; Jiang, J.; Bae, S.; Luijten, S. C.; Granick, E.; Supracolloidal, S. Reaction kinetics of Janus spheres. Science 2011, 331, 199. (16) Chen, Q.; Bae, S. C.; Granick, S. Directed self-assembly of a colloidal kagome lattice. Nature 2011, 469, 381. (17) Singh, J. P.; Lele, P. P.; Nettesheim, F.; Wagner, N. J.; Furst, E. M. One- and two-dimensional assembly of colloidal ellipsoids in ac electric fields. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2009, 79, 050401. (18) Romano, F.; Sciortino, F. Two dimensional assembly of triblock Janus particles into crystal phases in the two bond per patch limit. Soft Matter 2011, 7, 5799. (19) Pons-Siepermann, I. C.; Glotzer, S. C. Design of patchy particles using quaternary self-assembled monolayers. ACS Nano 2012, 6, 3919. (20) Odriozola, G.; de J. Guevara-Rodriguez, F. Communication: Equation of state of hard oblate ellipsoids by replica exchange Monte Carlo. J. Chem. Phys. 2011, 134, 201103. (21) Odriozola, G. Revisiting the phase diagram of hard ellipsoids. J. Chem. Phys. 2012, 136, 134505. (22) Paunov, V. N.; Cayre, O. J. Supraparticles and “Janus” particles fabricated by replication of particle monolayers at liquid surfaces using a gel trapping technique. Adv. Mater. 2004, 16, 788. (23) Lattuada, M.; Hatton, T. A. Synthesis, properties and applications of Janus nanoparticles. Nano Today 2011, 6, 286. (24) Andala, D. M.; Shin, S. H. R.; Lee, H.-Y.; Bishop, K. J. M. Templated synthesis of amphiphilic nanoparticles at the liquid−liquid interface. ACS Nano 2012, 6, 1044. (25) Kaufmann, T.; Gokmen, M. T.; Rinnen, S.; Arlinghaus, H. F.; Prez, F. D.; Ravoo, B. J. Bifunctional Janus beads made by “sandwich” microcontact printing using click chemistry. J. Mater. Chem. 2012, 22, 6190. (26) Yang, S.; Guo, F.; Kiraly, B.; Mao, X.; Lu, M.; Leong, K. W.; Huang, T. J. Microfluidic synthesis of multifunctional Janus particles for biomedical applications. Lab Chip 2012, 12, 2097. (27) Kim, H.; Carney, R. P.; Reguera, J.; Ong, Q. K.; Liu, X.; Stellacci, F. Synthesis and characterization of Janus gold nanoparticles. Adv. Mater. 2012, 24, 3857. (28) Walther, A.; Müller, A. H. E. Janus particles: Synthesis, selfassembly, physical properties, and applications. Chem. Rev. 2013, DOI: 10.1021/cr300089t.

longer runs, there is a small change in the cluster distribution function for ε = 1.3 shown in Figure 5, resulting from further cluster−cluster aggregation that preserves the ordered structure of the chains and shifts the dominant peak to the next higher value of Np. Meanwhile, one would expect that the cluster distribution function for ε = 1.2 and 1.4 remains almost the same (i.e., the dominant peak positions remain unchanged, while the peak values might vary slightly), because micelle and vesicle aggregates are rather stable.

■

CONCLUSION In summary, we have studied a theoretical model for Janus prolate spheroids and have used Monte Carlo simulations to investigate its self-assembly for different aspect ratios. We have limited our investigation to a low-temperature, low-density regime that has been studied by others for the case of Janus spheres.11,13,14,30 In contrast to the vesicle aggregates obtained from the self-assembly of Janus oblate spheroids,40 we find more ordered cluster structures for Janus prolate spheroids, such as vesicles, tubular micelles, and micelles. Furthermore, we find that a relatively small increase in the aspect ratio leads to the transition of cluster morphology. In this manner, we have completed the study of the self-assembly of Janus spheroid particles, where the patch is along the c axis. If we compare our results to studies of surfactant aggregates in dilute aqueous solutions, the aspect ratio functions as a “packing parameter” for Janus spheroidal particles.46 Specifically, vesicle (bilayer) structures dominate the configuration for Janus spheroids with ε less than 1.2; tubular micelles are formed for ε around 1.3; as ε increases further, such a tubular order quickly disappears and leads to a reduction in the chain length; and then micelle structures dominate the configuration when ε exceeds 2.0. This transition in aggregate structures is similar to that of a surfactant as the role of its hydrophobic tail becomes weaker.47 Overall, the findings based on the anisotropy of geometric shape are quite general for the lowdensity regime that demonstrates the effect of anisotropy in directing the self-assembly. Thus, we believe that studies of different anisotropy dimensions of the basic building blocks could add further precise control in self-assembly and raise additional questions for future studies.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by a grant from the G. Harold and Leila Y. Mathers Foundation.

■

REFERENCES

(1) Whitesides, G. M.; Grzybowski, B. Self-assembly at all scales. Science 2002, 295, 2418. (2) Palma, C.-A.; Cecchini, M.; Samori, P. Predicting self-assembly: From empirism to determinism. Chem. Soc. Rev. 2012, 41, 3713. (3) Whitesides, G. M.; Boncheva, M. Beyond molecules: Selfassembly of mesoscopic and macroscopic components. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 4769. (4) Glotzer, S. C.; Solomon, M. J. Anisotropy of building blocks and their assembly into complex structures. Nat. Mater. 2007, 6, 557. 8522


Langmuir

Article

(29) Hu, J.; Zhou, S.; Sun, Y.; Fanga, X.; Wu, L. Fabrication, properties and applications of Janus particles. Chem. Soc. Rev. 2012, 41, 4356. (30) Sciortino, F.; Giacometti, A.; Pastore, G. Phase diagram of Janus particles. Phys. Rev. Lett. 2009, 103, 237801. (31) Ren, B.; Ruditskiy, A.; Song, J. H. K.; Kretzschmar, I. Assembly behavior of iron oxide-capped Janus particles in a magnetic field. Langmuir 2012, 28, 1149. (32) Rosenthal, G.; Gubbins, K. E.; Klapp, S. H. L. Self-assembly of model amphiphilic Janus particles. J. Chem. Phys. 2012, 136, 174901. (33) Hagy, M. C.; Hernandez, R. Dynamical simulation of dipolar Janus colloids: Equilibrium structure and thermodynamics. J. Chem. Phys. 2012, 137, 044505. (34) Li, Z.-W.; Lu, Z.-Y.; Sun, Z.-Y.; An, L.-J. Model, self-assembly structures, and phase diagram of soft Janus particles. Soft Matter 2012, 8, 6693. (35) Fujii, S.; Yokoyama, Y.; Miyanari, Y.; Shiono, T.; Ito, M.; ichi Yusa, S.; Nakamura, Y. Micrometer-sized gold-silica Janus particles as particulate emulsifiers. Langmuir 2013, 29, 5457. (36) Liu, B.; Zhang, C.; Liu, J.; Qu, X.; Yang, Z. Janus non-spherical colloids by asymmetric wet-etching. Chem. Commun. 2009, 26, 3871. (37) Walther, A.; Drechsler, M.; Rosenfeldt, S.; Harnau, L.; Ballauff, M.; Abetz, V.; Müller, A. H. E. Self-assembly of Janus cylinders into hierarchical superstructures. J. Am. Chem. Soc. 2009, 131, 4720. (38) Güell, O.; Sagués, F.; Tierno, P. Magnetically driven Janus micro-ellipsoids realized via asymmetric gathering of the magnetic charge. Adv. Mater. 2011, 23, 3674. (39) Park, B. J.; Lee, D. Equilibrium orientation of nonspherical Janus particles at fluid−fluid interfaces. ACS Nano 2012, 6, 782. (40) Liu, Y.; Li, W.; Perez, T.; Gunton, J. D.; Brett, G. Self assembly of Janus ellipsoids. Langmuir 2012, 28, 3. (41) Li, W.; Liu, Y.; Brett, G.; Gunton, J. D. Encapsulation by Janus spheroids. Soft Matter 2012, 8, 6027. (42) Shah, A. A.; Schultz, B.; Kohlstedt, K. L.; Glotzer, S. C.; Solomon, M. J. Synthesis, assembly, and image analysis of spheroidal patchy particles. Langmuir 2013, 29, 4688. (43) Berne, B. J.; Pechukas, P. Gaussian model potentials for molecular interactions. J. Chem. Phys. 1972, 56, 4213. (44) Noro, M. G.; Frenkel, D. Extended corresponding-states behavior for particles with variable range attractions. J. Chem. Phys. 2000, 113, 2941. (45) Liu, H.; Kumar, S. K.; Sciortino, F.; Evans, G. T. Vapor−liquid coexistence of fluids with attractive patches: An application of Wertheim’s theory of association. J. Chem. Phys. 2009, 130, 044902. (46) Chu, Z.; Dreiss, C. A.; Feng, Y. Smart wormlike micelles. Chem. Soc. Rev. 2013, DOI: 10.1039/C3CS35490C. (47) Nagarajan, R. Molecular packing parameter and surfactant selfassembly: The neglected role of the surfactant tail. Langmuir 2002, 18, 31.

8523


Self-Assembly of Janus Ellipsoids II: Janus Prolate Spheroids

Recommend Documents