Solid-Liquid and Solid-Solid Phase Diagrams of Self-Assembled

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C: Physical Processes in Nanomaterials and Nanostructures

Solid-Liquid and Solid-Solid Phase Diagrams of SelfAssembled Triblock Janus Nanoparticles from Solution Hossein Eslami, Kheiri Bahri, and Florian Müller-Plathe J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02043 • Publication Date (Web): 03 Apr 2018 Downloaded from http://pubs.acs.org on April 4, 2018

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The Journal of Physical Chemistry

Solid-Liquid and Solid-Solid Phase Diagrams of Self-Assembled Triblock Janus Nanoparticles from Solution Hossein Eslami,1,* Kheiri Bahri,1 and Florian Müller-Plathe2 1

Department of Chemistry, College of Sciences, Persian Gulf University, Boushehr 75168, Iran

2

Eduard-Zintl-Institut für Anorganische und Physikalische Chemie and Profile Area ThermoFluids & Interfaces, Technische Universität Darmstadt, Alarich-Weiss-Str. 8, 64287 Darmstadt, Germany

*

Corresponding author

Email: [email protected]

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Abstract A realistic model of triblock Janus particles, in which a cross-linked polystyrene sphere capped at the poles with hydrophobic n-hexyl groups and in the equatorial region with charges, is used to study the phase equilibrium boundaries for stabilities of quasi-two-dimensional liquid, Kagome, and hexagonal phases. The pole patches provide interparticle attraction, and the equatorial patches interparticle repulsion. The self-assembly has been studied in the presence of solvent, charges, and a supporting surface. An advanced sampling many-body dissipative particle dynamics simulation scheme, with the inclusion of many-body and hydrodynamic interactions, has been employed to drive the system from liquid to solid phases and vice versa. Our calculated phase diagrams indicate that in the limit of narrow pole patch widths (opening angle ≈ 65°), the Janus particles self-assemble to the more stable Kagome phase. The entropy-stabilized Kagome lattice is more stable than the hexagonal phase at higher temperatures. Increasing the pressure stabilizes the denser hexagonal vs. the Kagome lattice. Enlarging the pole patch width (varying the opening angle from 65° to 120°), promotes the bonding area, and hence, energetically stabilizes the close-packed hexagonal vs. the open Kagome lattice. A comparison with previous calculations, using the Kern-Frenkel potential, has been done and discussed.


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1. INTRODUCTION Microscale colloidal particles with anisotropic interactions (patchy particles) are counted among the building blocks, which are capable of forming the intended superstructures with a wide range of novel optical and mechanical properties. Although there exist synthetic challenges for fabrication of anisotropic hybrid building blocks, the main challenge is to exploit available patchy colloidal particles to self-organize them into intended ordered structures, applicable for advanced and functional materials and devices. In this respect, a fundamental understanding of the relation between the structure and property of the building blocks plays an important role. Molecular simulation tools offer the ability to track individual molecules in space and time and to predict the structure and phase behavior of the finally self-assembled structure. To date, a wide range of behaviours of patchy-particle models, including self-assembly and crystallization, have been investigated using computer simulation.1-8 A natural starting point to study the self-assembled structures of these colloidal building blocks is to view them as hard particles. In this respect, Monte Carlo (MC) simulation schemes have been developed to include special moves, allowing faster equilibration of patchy particles.9,10 Simplified hard-sphere models11 as well as the effective potentials, such as the Kern– Frenkel potential,12 in which the surface of a hard sphere is decorated with attractive patches, are used to simulate the self-organization.6,13,14 In these models a patch is described by a solid angle and the angular modulation of the interaction depends on the particle orientations. Self assembly of Janus ellipsoids is simulated using a model, similar to the Kern and Frenkel model, in which the ellipsoids interact through a pair-potential depending on their separation and orientation.15 Further studies involve modeling patchy particles using a simple angular modulation of the attractive part of the Lennard-Jones potential.16-20 Owing to the simplicity of the model, in most simulation studies, patchy anisotropic particles are modeled as decorated hard spheres. While these methods have been successful to provide valuable insights in predicting the structures of the finally organized phases, a more realistic methodology is needed to simulate Janus particles under experimental conditions.21,22 Although decoration of the surface of Junus particles with patches of varieties of sizes, shapes, and orientations is shown to direct the selfassembly toward formation of crystals of a wide range of symmetries, not all such computational designs are experimentally achievable. Therefore, inspired by experimental decorations of the



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Janus particles, rational designs of colloid particles, patch arrangements, and tuning the interaction parameters are desirable. Very recently we have developed such a model of triblock Janus particles in the presence of solvent and a supporting surface, on which the particles crystallize in two dimensions (unpublished results). As an extension of the previous work on the mechanism of nucleation, here we study the phase behavior of self-assembled structures.

2. METHOD On cooling a liquid to its freezing point, or below, usually a first-order liquid-solid phase transition happens. Often the crystallization process starts with a nucleation step, whereby an ordered nucleus is formed. Upon its growth to a critical size, the whole liquid crystallizes. The stable liquid and crystalline phases are minima in the free energy landscape. Once the system is trapped in a local minimum, it faces an obstacle of surmounting a high free energy barrier in order to cross into a neighbouring free energy minimum of a stable phase. This means that the brute force molecular simulation are likely to leave many relevant regions of configuration space unexplored. In order to direct a simulation along the liquid-solid transition, advanced sampling schemes have been developed.23-25 In most of the methods invented for sampling the whole configuration space within the accessible simulation time, one needs to identify a set of appropriate order parameters that quantify the degree of crystallization by discriminating between liquid and crystalline phases.26,27 Historically, orientation order parameters based on spherical harmonics have been employed for studying solid-liquid phase equilibria. In this scheme, a quantitative measure of structure around a particle i is characterized by the bond (a vector connecting neighboring particles i and j) order parameter, originally introduced by Steinhardt et al.28 It is defined as qlm (i ) =

∑ Yl m (θij , ϕij ) N b (i ) j∈N b (i ) 1

(1)

where Nb(i) is the number of neighbors of particle i, θij and φij specify the orientation of the bond between particles i and j, and the Ylm are the spherical harmonics. Although the Steinhardt order parameters have been used extensively to study the phase equilibria in three-dimensional systems, they are not appropriate for characterizing liquid and crystalline phases in twodimensions. In our recent study on nucleation of quasi-two-dimensional Kagome and hexagonal 4 ACS Paragon Plus Environment

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phases (see Figure 1) we proposed the following two order parameters to characterize the free energy landscape for liquid, Kagome, and hexagonal phases: λ1 (i ) =

  qˆ (i ) ⋅ qˆ ( j ) − ∑ qˆ (i ) ⋅ qˆ ( j ) ∑ ∑  (i ) ( )   6

1 Nb

4

6m

j∈N b i

m = −6

* 6m

4m

* 4m

(2)

m = −4

and λ2 (i ) =



1 Nb

(i ) ∑( )  ∑ j∈ N b i

m = ±6 , ±4

qˆ6 m (i ) ⋅ qˆ6*m ( j ) −

 ∑ qˆ (i ) ⋅ qˆ ( j ) 4m

* 4m

m = ±4

(3)



where qˆlm (i ) =

qlm (i )

(4)

1/ 2

 l 2  ∑ qlm (i )   m=− l 

The liquid, Kagome, and hexagonal phases are identified by values of (-0.1, 0.0), (0.65, 0.7), (0.0, 0.8), respectively, in the order parameter space (λ1 , λ2).



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Figure 1. Pictorial representations of self-assembled triblock Janus particles into the Kagome (top panel) and hexagonal (bottom panel) crystals. The poles of Janus particles are capped with hydrophobic patches and the equatorial region is charged. The two staggered triangles show the triangular contacts of the particles, via patches, in the Kagome lattice. Having a set of proper order parameters in hand, we have employed our recently developed adaptive-numerical-bias metadynamics29 to bias the potential energy of the system. Metadynamics30 is a method rooted in the original idea of local elevation,31 in which the potential is biased progressively by adding Gaussians of definite height and width to it, i.e.,

(

)

  λ − λ (t ′) 2   U λ , t = w∑ exp −  2   2 σ   t ′≤t 

( )

(5)


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where, w and σ are the height and the width of the added Gaussians, respectively, and t is the time. In this work, the average order parameters (introduced in eqs. (2) and (3)) are employed to drive simulations from liquid to solid phases and vice versa. The average order parameters are defined as:

λ=

1 ∑ λ( j) N j

(6)

In extended versions of metadynamics,32,33 the shapes of Gaussian are tuned on the fly. In adaptive-numerical-bias metadynamics,29 the free energy surface is filled by progressively adding history-dependent numerical adaptive biasing potentials (instead of Gaussians), obtained by sampling the distributions along the order parameters. In order to sample the high-free-energy configurations, the biasing potentials are tabulated, at regular intervals, and their derivatives (for the sake of force calculation) are calculated numerically. Addition of such repulsive biasing potentials to the Hamiltonian of the system forces the collective variables away from their previously visited regions. The method is shown29 to overcome the free energy barriers without a priori knowledge of the barrier height and the shape of the free energy profile. Filling the free energy basins, a flat free energy landscape along the collective variable is obtained, over which the system performs an unrestricted random walk. Because of the large size of Janus particles simulated in this work and dealing with a huge number of solvent molecules, we have employed a many-body dissipative particle dynamics (DPD) simulation approach34 to conduct simulations from the liquid to crystalline phases and vice versa. In the DPD much longer time scales than those of atomistic simulations are achievable and the hydrodynamic interactions within the system are captured. Moreover, in the present many-body DPD approach, the effect of many-body interactions (emphasized in the literature for colloidal systems)35,36 are taken into account. In this work, during the course of many-body DPD simulations, we have included biasing potentials in the space of the proposed order parameters to drive simulations from liquid to Kagome, and/or hexagonal phases and vice-versa. The particle interactions are formulated in terms of three types of forces. The conservative force, FC, is defined as:



  rij   Aij 1 −  + B ρi + ρ j   rC     r  FijC =  Aij 1 − ij  eij     rC    0

(

)1 − αrr ij



C

  eij 

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rij ≤ αrC

αrC ≤ rij < rC

(7)

rij ≥ rC

where Aij (Aij < 0) and B (B > 0) are the strengths of attractive and repulsive forces, respectively, rC is the cutoff distance, α (α < 1) determines the range of attractive forces, eij=rij /rij, and ρ̅i is the average local density around particle i. In eq. (7) ρ̅i is defined as: r   1 − ij  ρi = ∑ 3 3 αrC  j ≠i 2πα rC  15

2

(8)

The contribution of repulsive forces to FijC is zero at r > αrC. A combination of DPD random and dissipative forces constructs the DPD thermostat. Here, this method is employed to sample the free energy surface for the phase equilibria between liquid and quasi-two-dimensional Kagome and hexagonal lattices.

3. SIMULATIONS In the present work we have tried to mimic the experiment by Chen et al.37 on the self-assembly of triblock Janus particles, namely polystyrene (PS) sulfonate nanoparticles, capped at the poles with the hydrophobic patches (monolayers of n-octadecanethiol) at an opening angle of 65°. The equatorial region is negatively charged and the particles are dissolved in water (at a concentration of 10 wt %). Addition of salt (NaCl) to the solution of triblock Janus particles selfassembles them into a quasi-two-dimensional Kagome lattice. In our many-body DPD modeling, the spherical PS nanoparticle is modeled as a hollow PS sphere with a diameter of 5.0 nm. Each DPD bead, covering the surface of the sphere, is representative of a styrene monomer and the distance to its neighbouring beads is ≈ 0.55 nm (corresponding to the size of PS beads reported in a recent coarse-grained model).38 To keep the PS beads at the surface, a linear “triatomic molecule” (which acts as a virtual molecule), cocenter with the PS sphere, encircled by the surface beads is used. The surface PS beads are linked to their neighbours and to the central atom of the central molecule as well as to the closer of its two terminal atoms in each hemisphere. Hydrophobic n-hexyl groups, represented by single beads, were grafted to all PS beads at the poles with an opening angle, δ , ranging from 65° to 120°. The 8 ACS Paragon Plus Environment

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equatorial PS beads were grafted by sulfonate beads (covering about 10 % of the nanoparticle (equatorial belt) surface) and Na+ counterions were added to balance the surface charges. A snapshot of the structure of the PS nanoparticles simulated in this work is shown in Figure 2.

Figure 2. Snapshot of a structure of a triblock Janus particle simulated in this work. The green, purple, and yellow sphere represent styrene, n-hexyl, and chloride beads (the counterions are not shown). The black spheres stand for the central triatomic molecule, which is cocenter with the spherical nanoparticle. Here, the opening angle δ = 120°. Similar to experiment,37 the triblock Janus particles (168 particles) were dissolved in water (at a concentration of ≈15 wt%). A combination of 5 water molecules was modeled as one DPD bead. The supporting substrate is a recently developed coarse-grained model for graphene,39,40 in which groups of eight carbon atoms are modeled as single beads. To model the silica substrate, employed in experiment, the graphene substrate was uniformly negatively charged (to a charge density of -2×10-3 nm-2). The many-body DPD parameters for water were obtained from simulation of pure water and those of the ions were taken from the reports of Ghoufi and Malfreyt.41 The DPD parameters for PS beads were parameterized according to a recent coarsegrained model.38 Compared to the PS beads, the graphene substrate beads were made more attractive, so that the particles get bound to the substrate. The patch-patch interactions were tuned more attractive than water-water or ion-ion interactions, to help the particles stick together via the patches. In order to settle the Janus particles, floating in water, on the substrate, a gravitational force normal to the substrate direction, is added according to the method of Yang and Yin.42 The many-body DPD parameters are reported elsewhere (unpublished results). All 9 ACS Paragon Plus Environment


simulations were performed with our simulation package, YASP.43 The time step was 0.1 ps and simulations were performed up to 3 µs. The electrostatic interactions were calculated using the Ewald summation method with charges distributed over the particles according to the method of González-Melchor et al.44 Adjusting the magnitude of the patch-patch (hexyl-hexyl) interaction parameters as well as the gravitational force, keeps the particles near the substrate. Additional electrolyte (NaCl) ion pairs were added to adjust the salt concentration at 5.0 mM. When all the Janus particles were within a distance 5 nm from the substrate, we started coupling the simulation box with a barostat.45 During the course of NpT ensemble simulations the sizes of the simulation box in the x and y directions (Lx and Ly, respectively) were reduced. As the substrate was not thermalized, the positions of substrate beads in the xy plane were scaled during the course of NpT ensemble simulations. However, this scaling changes the density of substrate beads, with the only effect of change in the substrate-solution interaction. To prevent this, whenever a change in the surface area of substrate (5 %) was observed, the compressed/expanded substrate was replaced with a new one, with the same size as the box size in xy plane.

4. RESULTS AND DISCUSSIONS 4.1. Specification of the Crystalline Structure The probability distribution of the average order parameters (averaged over all particles of the system) have been shown in Figure 3 for a system of patchy particles with an opening angle δ = 120° at p =101.3 kPa and T= 325 K (corresponding to 10 % supercooling). Here, the Janus particles are considered to be coordinating if their center-of-mass distances are less than 7.0 nm. The results show that the sole use of eitherλ1 or λ2 is not sufficient to discriminate between liquid, Kagome, and hexagonal phases. However in the (λ1 ,λ2 ) space, marked difference in shapes and positions of distributions for stable phases are seen. Both order parameters are large for both crystal structures and are invariant under rotations of the crystallite. This means that the configuration space can be biased by adding potentials to the potential energy of the system, as explained, to transform one phase into the other.


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Figure 3. Normalized distribution function for order parameters λ1 andλ2 (averaged over all particles) in the liquid, Kagome, and hexagonal phases. The patch width δ = 120°, p = 101.3 kPa, and T = 325 K (corresponding to 10 % supercooling).

4.2. Solid-Liquid Phase Equilibria The distributions of order parameters for a finite system near coexistence exhibit double maxima structures, located near the characteristic maxima of both phases. They correspond to minima in the free energy. If the two free-energy basins are equally deep, the phases are in equilibrium. Starting from a stable phase, we bias the potential energy by adding metadynamics potentials in the (λ1 ,λ2 ) space to fill the corresponding free energy basins. In our previous work (unpublished results) on nucleation, we employed local order parameters (limited to the particles forming the largest nucleus in the simulation box). However, in the present phase equilibrium study one needs to fill both basins of equal depth to perform several round trips between the coexisting phases. In other words, we can average the local order parameters over all particles in the system, as indicated in eq. (6). All simulations were started from a disordered (liquid) configuration, obtained by sedimenting Janus particles on the top of the supporting surface. In the initial liquid configuration the centers of mass of all Janus particles were within a distance of 5.0 nm from the surface. Metadynamics potentials were successively deposited in the order parameter space to 11 ACS Paragon Plus Environment


drive simulation from liquid to Kagome or hexagonal phases. We have shown in our previous study that, in the limit of very narrow pole patches, the Janus particles stick together via the patches (one contact per patch) to form stable strings. Strings are also shown to be intermediates for formation of two-dimensional aggregates. Increasing the patch width to δ ≈ 65°, as in experiment,34 the particles are able to establish two contacts per patch (cf. Figure 1) and stabilize the Kagome lattice. For the sake of comparison with experiment, the narrowest patch width examined here is that examined in experimental study of self-assembly to Kagome lattice.34 Our recent simulations20 aslo indiacte that this patch width is just wide enough to self-organize the triblock Janus particles to the Kagome lattice. It is worth mentioning that, similar to experiment,37,46 in all cases addition of salt, to screen the Coulombic repulsion of the equatorial region is requred to self assemble crystalline structures from the solution. In Figure 4 we have shown the solid-liquid phase diagram for Janus particles whose poles are patched at δ ≈ 65°. At low temperatures, enthalpy dominates the phase transition; because of the lower molar volume of hexagonal (compared to the Kagome and liquid), it is more stable phase. The Kagome phase, however, has a higher vibrational entropy than that of the hexagonal phase, because the latter phase has less space to vibrate. This means that at higher temperatures the entropy stabilizes the Kagome phase more, compared to the hexagonal phase. The higher stability of open colloidal structures with respect to close-packed lattices has also been discussed by Mao et al.46 Expectedly, compared to the hexagonal phase, the lower-density Kagome phase is the more stable phase at lower densities (pressures). A comparison of our calculated phase diagram with that reported by Romano and Sciortino5 indicate that in both cases the topologies of the phase diagrams are the same. However, due to the differences in the modeling schemes of Janus particles, and inclusion of charges, solvent, and hydrodynamic interactions in our model, the exact locations of phase equilibrium points are not the same. It is worth mentioning that a similar effect of change in force field and interaction range on the phase equilibria of quasicrystals has been reported by Pattabhiraman and Dijkstra.47 Investigating the effect of shape of the interaction potentials and interaction ranges of four potentials, Pattabhiraman and Dijkstra47 reported formation of (dodecagonal) quasicrystals, in two dimensions, in all cases. However, they found that depending on the shape of the potential, formation of dodecagonal quasicrystals took place at various temperatures. Lack of experimental data on the phase diagram of the present triblock patchy 12 ACS Paragon Plus Environment

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particles prevents us from direct quantitative comparisons against experiment. However, higher stability of Kagome phase with respect to the hexagonal phase at higher temperature (due to higher rotational and vibrational entropies of Kagome) is also reported experimentally.46 Another strength point of the present modeling is that the reported pressure-density-temperature range is comparable with experimental conditions.37,46 For example, for Janus particles (δ ≈ 65°), our results show that formation of Kagome lattice is in equilibrium with liquid solution at p = 101.3 kPa and T = 308 K, which is close to the ambient conditions at which experiment is done.37 On the other hand, the highest temperature at which Kagome is in equilibrium with liquid, as reported by Romano and Sciortino, is T* = 0.15, which corresponds to ≈ 125-180 K (adopting the potential well-depth of ≈ 7-10 KBT, reported in experiment,48 in previous simulations on similar systems,49 and compatible with the calculated free energy barrier height). Moreover, the stability range of Kagome lattice, reported by Romano and Sciortino5 extends to T* = 0. Considering the fact that entropy stabilizes Kagome over hexagonal phase, the hexagonal phase is expected to be the most stable phase at very low temperatures. Our calculated phase diagram also demonstrates the conditions of equilibrium between liquid, Kagome, and hexagonal phases. However, the reported phase diagram by Romano and Sciortino5 includes equilibrium between the Kagome, hexagonal, and fluid phases.



Figure 4. Temperature-density phase diagrams for triblock Janus particles. The top, middle, and bottom panels represent patch widths with opening angles δ = 65°, 90°, and 120°, respectively.


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Increasing the patch width increases the number of contacts per particle, and hence, stabilizes the hexagonal phase more. The phase diagrams for Janus particles covered with two wider patches (δ ≈ 90° and δ ≈ 120°) are also shown in Figure 4. As patches wider than δ ≈ 120° would allow three neighbours of a given particle in two dimensions and stabilize other lattices,46 we have not considered δ > 120°. The freezing temperature of our model of Janus particles increases with increasing the patch width (between δ ≈ 65° to δ ≈ 120°, the freezing temperature at p = 101.3 kPa increases from 308 K to 361 K). The patch size-dependence of the freezing temperature is in line with the calculations of Romano et al.50 A comparison of phase diagrams in Figures 4 indicates that in all cases the phase diagrams have the same topology. However, with increasing the patch width, the phase stability boundary of the Kagome shifts to higher densities and higher temperatures. Due to the softness of Janus particles, increasing the patch width enlarges the particle-particle contact area, and hence, increases the vibrational entropy and decreases the energy. The former factor is responsible for higher stability of Kagome at higher temperatures, but the latter factor stabilizes more the hexagonal phase at higher densities. Further increase in the temperature disorders the particle orientations to liquid phase. Increasing the pressure, introduces larger contact areas and more order in the particles to stabilize the hexagonal phase at higher densities. This is due to the higher compressibility of the Kagome vs. hexagonal phase. A more stabilization of the liquid phase with increasing the contact area has also been reported in a recent theoretical investigations of the patchy particles by Kalyuzhnyi et al.51 The stability of the Kagome phase over a range of conditions demonstrates that it is possible to set up a competition between close-packed and open lattices by suggesting the best choice of the external conditions (tuning the temperature, pressure, and patch width). An analogy between the present results for phase equilibria in triblock two-patch particles with the four-patch particles simulated by Romano et al.50,52 and by Smallenburg and Sciortino53 for self-assembly into 3-dimensional lattices, shows that in both cases there is a competition between an open (diamond in 3 dimensions and Kagome in 2 dimensions) and a denser (bcc in 3 dimensions and hexagonal 2 dimensions) solid phases. The patch size is reported51 to have a slight impact on the crystallization of 4-patch Kern-Frenkel model, containing a single bond per patch. An increase in the range of the attractive potential in 4-patch Janus particles is shown to favor a close-packed (bcc) vs. an open (diamond) lattice.51-55 Our results, on the other hand, show a higher sensitivity of the structure of the finally self-assembled phase on the patch size. This can 15 ACS Paragon Plus Environment


partly be attributed to the longer range of patch-patch attractions in our models of Janus particles compared to the Kern-Frenkel model (in the present many-body DPD approach, the interactions are taken into account up to the second shell neighbours of DPD beads). Another reason is the softness of particles and patches in our model, which allows a larger surface area of contacts per patch. Both factors stabilize the close-packed (hexagonal) vs. open (Kagome) lattices. The hydrodynamic interactions are reported to facilitate formation of chain-like and open colloidal structures.56 Therefore, it is reasonable to assume that the inclusion of hydrodynamics interactions in the present DPD scheme, with inclusion of solvent molecules, facilitates formation of Kagome lattice. Compared to the phase diagrams reported by Romano and Sciortino,5 our phase diagrams are restricted to a narrower range of pressures/densities. This is due to the fact that in our simulations liquid-solid phase boundaries are calculated, while in their calculations fluid-solid phase boundaries are shown. Moreover, in our simulations the Janus particles are allowed to move in three dimensions. Therefore, an increase in the pressure moves the particles away from the surface into the solution, which is also the case in experiments. To give an estimate of the free energy difference between different phases and the transition paths for inter-conversion of phases, we have plotted in Figure 5 the free energy contours for liquid, Kagome, and hexagonal phases for Janus particles patched up to an opening angle δ = 120° at T = 361 K and p =101.3 kPa. Our calculations show that for this patch size, the liquid and hexagonal phases are in equilibrium and the Kagome phase is unstable with respect to the others. The liquid phase has to surmount a free energy barrier ∆G = (19 ± 2.1) kJ mol-1, i.e. ≈ 5 kBT to convert to the hexagonal phase. The free energy barrier for conversion of liquid to Kagome phase is, however, larger (≈ 42 ± 2.8) kJ mol-1, i.e. ≈ 15 kBT. This finding is in agreement with experimental report of Chen et al.,37 where a higher stability for the Kagome phase is found by tuning the patch width to an opening angle δ = 65°. The calculated free energy contours show how the crystalline phases (Kagome and hexagonal) transform in response to the pressure/density and temperature changes. It is worth mentioning that calculations by Romano and Sciortino,5 in which one order parameter to distinguish fluid and solid phases is employed, no observable path for Kagome-hexagonal phase transition is seen.


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Figure 5. Free energy contours (kJ mol-1) for equilibrium between liquid (λ1 ≈ -0.1 andλ2 ≈ 0.0) and hexagonal (λ1 ≈ 0.0 andλ2 ≈ 0.8) phases in a system containing 168 Janus particles patched with hydrophobic patch up to an opening angle (δ = 120°) at T = 361 K and p =101.3 kPa. The basin for the metastable Kagome phase (λ1 ≈ 0.6 andλ2 ≈ 0.7) locates (19 ± 2.1) kJ mol-1 higher than those for liquid and hexagonal phases. Free energy barriers for nucleation of Kagome and hexagonal lattices from solution at this temperature are (≈15 ± 2.3) kJ mol-1 and (≈ 42 ± 2.8) kJ mol-1, respectively. Contours are 1.5 kBT apart.



We have also shown, in Figure 6, typical snapshots of simulation boxes for self-assembly of 168 triblock Janus particles, patched with hydrophobic patch up to an opening angle (δ = 120°), at T = 361 K and p =101.3 kPa. The snapshots shown in Figure 6 indicate liquid, Kagome, and hexagonal phases as well as configurations indicating equilibrium between liquid-Kagome, liquid-hexagonal, and Kagome-hexagonal phases. Both solid phases contain defects. In the liquid phase, the particles stick together via the hydrophobic patches to form disordered rings and chains. To self-assemble into an ordered solid phase, the particles in the dense liquid domains arrange their orientations to form organized nuclei. This is a typical behavior for a two-step mechanism of nucleation. Further growths of crystalline domains lead to the formation of crystalline (Kagome and/or hexagonal) phases. In agreement with experimental findings of Chen,57 the snapshot for Kagome-hexagonal phase equilibria contains elongated Kagome rings with lattice mismatch between the highdensity and low-density domains. The collapse of the open Kagome lattice to hexagonal phase has also been reported by Mao et al.,46 in which the lateral compression (due to gravitational force in a tilted sample) was applied on heavier and larger silica spheres (compared to PS spheres36).


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Figure 6. Typical snapshots of simulation box for self-assembly of 168 triblock Janus particles, patched with hydrophobic patches up to an opening angle (δ = 120°), at T = 361 K and p =101.3 kPa. The first row, from left to right, shows configurations for liquid, Kagome, and hexagonal phases. The second-row snapshots (from left to right) belong to configurations identifying liquid to Kagome, liquid to hexagonal, and hexagonal to Kagome (close to the top of free energy barrier) phase transitions.



We have shown in Figure 7 the time evolution of the fraction of Janus particles (δ = 90°), classified as isolated particles (particles with no contact), strings, particles with triangular contacts, and particles involved in the Kagome lattice. The number of isolated spheres decreases as the lateral dimensions of the simulation box reduces. The Janus particles then stick together via the sticky patches and form disordered chains and rings, whose concentration passes through a maximum when the size of the simulation box is further reduced. A further lateral compression of the simulation box and the increased chain length of the strings, lead to formation of ring structures. At this stage Janus particles form triangular contacts, which are known experimentally37 as initial precursors for formation of Kagome lattice. An increase in the concentration of Janus particles with triangular contacts leads to a further orientation of particles toward formation of Kagome nuclei. Growing the size of initial Kagome nuclei in the simulation box, the critical nucleus is formed. At this stage a sudden crystallization of the liquid phase as a result of interconnections between different nucleated domains in the box (growth) happens (see Figure 7). Time evolution of the number of isolated spheres, strings, particles involved in triangular contacts, and the correlation between formation of triangular contacts and the Kagome lattice is in complete agreement with experimental observation of Chen et al.37 While in the simulations of Romano and Sciortino,5 the crystallization of Kagome is reported to preceded by the formation of chains, our results indicate that formation of triangular contacts and a critical concentration of triangular contacts are needed to nucleate the Kagome phase. Formation of triangular contacts and disordered rings structures are also seen in the snapshot representing liquid structure in Figure 6. Furthermore, a two-step mechanism of nucleation of Kagome form solution, in agreement with experiment,37 is reported in our previous study (unpublished results). These findings are a step forward toward elucidation of pathway, not detected in previous MC simulations of phase equilibria.


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Figure 7. Time evolution of the fraction of Janus particles (δ = 90°), classified by their surrounding. The legend shows fraction of isolated spheres, those involved in strings, those with triangular contacts, and particles involved in the largest Kagome nucleus.

Specification of the Crystalline Structures In order to quantify the structural differences for the liquid, Kagome, and hexagonal phases, we have shown in Figure 8 the orientational distribution function, φ (r), defined as follow. φ (r ) =

∑

m = ±6 , ±4

qˆ6 m (0 ) ⋅ qˆ6*m (r ) −

∑ qˆ (0) ⋅ qˆ (r ) 4m

(9)

* 4m

m = ±4

For both solid phases, the orientational distribution functions do not decay with distance, which implies that both configurations display behaviors characteristic to solid phases, exhibiting longrange orientational orders. The orientational distribution function for the disordered liquid phase, however, exhibits an exponential decay with distance. Larger scale fluctuations in the orientational distribution function for Kagome, compared to that of hexagonal phase, is due to larger scale fluctuations in the positions of particles in the former phase.



Figure 8. Orientational distribution function for hexagonal (full curve), Kagome (dashed curve), and liquid phases for Janus particles patched with hydrophobic patches up to an opening angle (δ = 90°), at T = 332 K and p =101.3 kPa.

5. CONCLUSIONS In this work a realistic model of triblock Janus particles, in which a spherically cross-linked polystyrene chain capped at the poles with hydrophobic n-hexyl groups and containing charges in the equatorial region is used to study the phase equilibrium boundaries for the stability of liquid, Kagome and hexagonal phases. The self-assembly has been studied in the presence of solvent, counterions, and a supporting surface. Many-body dissipative particle dynamics simulations, which permit long-time scale simulations with the inclusion of many-body and hydrodynamic interactions within the system, are performed. An advanced sampling scheme, adaptive-numerical-bias metadynamics,29 has been employed to drive simulations from liquid to solid phases and vice versa. We have simulated triblock Janus particles of varying hydrophobic pole patch sizes (the opening angle varying from 65° to 120°) over a wide range of temperatures and pressures.


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Our calculated phase boundaries indicate that in the limit of narrow patch widths (opening angle ≈ 65°), the Janus particles self-assemble to the more stable Kagome phase in ambient conditions. Increasing the patch width, however, shifts the self-assembly toward formation of denser hexagonal phase. Enlarging the patch width, promotes the bonding area, and hence, energetically stabilizes the close-packed hexagonal vs. the open Kagome lattice. At low temperatures, the increased particle-paticle attractions order the particle orientations and reduce the kinetic bottleneck toward formation of the hexagonal phase. Lowering the pressure, however, stabilizes the lower-density Kagome more (compared to the higher-density hexagonal) phase. Our results are in qualitative agreement with those reported by Romano and Sciortino6, in their simulation of 2-dimensional Janus particles, using the Kern-Frenkel potential.12 However, compared to the phase diagrams reported by Romano and Sciortino,5 our phase diagrams are restricted to a narrower range of pressures/densities. This is due to the fact that in our simulations liquid-solid phase boundaries are calculated, while in their calculations fluid-solid phase boundaries are shown. Moreover, in our simulations the Janus particles are allowed to move in three dimensions. Therefore, an increase in the pressure moves the particles away from the surface into the solution, which is also the case in experiments. A comparison between the phase boundaries in the present quasi-two-dimensional system with the three-dimensional structures formed by four-patched particles50,52-55 reveal that in both cases there is a competition between an open (diamond in 3 dimensions and Kagome in 2 dimensions) and a close-packed (bcc in 3 dimensions and hexagonal 2 dimensions) solid phases. Our calculated phase boundaries for quasi-two-dimensional open and close-packed phases are more sensitive to the patch width, compared to those of 4-patch particles,50 studied using the Kern-Frenkel model. We believe that the origin of this deviation is partly due to the longer range of patch-patch attractions and higher softness of particles and patches in our model of Janus particles, compared to the Kern-Frenkel model. These factors promote bonding and shifts the self-assembly toward the denser hexagonal (compared to Kagome) phase at higher pressures.

Acknowledgements We gratefully acknowledge the support of this work by the Deutsche Forschungsgemeinschaft (DFG), project MU1412/25-1. H. Eslami also acknowledges the research committee of Persian Gulf University. 23 ACS Paragon Plus Environment


References (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

(14) (15) (16) (17)

(18) (19) (20)

(21)

Malescio, G.; Pellicane, G. Stripe Phases from Isotropic Repulsive Interactions, Nat. Mater. 2003, 2, 97–100. Zhang, Z.; Glotzer, S. C. Self-assembly of Patchy Particles. Nano Lett., 2004, 4, 1407– 1413. Michele, C. De.; Gabrielli, S.; Tartaglia, P., Sciortino, F. Dynamics in the Presence of Attractive Patchy Interactions. J. Phys. Chem. B 2006, 110, 8064–8079. Sciortino, F.; Giacometti, A.; Pastore, G. Phase Diagram of Janus Particles. Phys. Rev. Lett. 2009, 103, 237801. 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, 5799–5804. Preisler, Z.; Vissers, T.; Smallenburg, F.; Sciortino, F. Crystals of Janus Colloids at Various Interaction Ranges. J. Chem. Phys. 2016, 145, 064513. Tang, X.; Rupp, B.; Yang, Y.; Edwards, T. D.; Grover, M. A.; Bevan, M. A. Optimal Feedback Controlled Assembly of Perfect Crystals. ACS Nano 2016, 10, 6791–6798. Pattabhiraman, H.; Dijkstra, M. On the Formation of Stripe, Sigma, and Honeycomb Phases in a Core–corona System. Soft Matter 2017, 13, 4418. Keys, A. S.; Chen, T.; Glotzer, S. C. Self-assembly of Patchy Particles into Diamond Structures Through Molecular Mimicry. Langmuir, 2005, 21, 11547–11551 Reinhardt, A.; Frenkel, D. DNA Brick Self-assembly with an Off-lattice Potential. Soft Matter 2016, 12, 6253–6260. Lin, H.; Lee, S.; Sun, L.; Spellings, M.; Engel, M.; Glotzer, S. C.; Mirkin, C. A. Clathrate Colloidal Crystals. Science 2017, 355, 931–935. Kern, N.; Frenkel, D. Fluid–fluid Coexistence in Colloidal Systems with Short-ranged Strongly Directional Attraction. J. Chem. Phys. 2003, 118, 9882–9889. Li, Z.-W.; Lu, Z.-Y.; Zhu, Y.-L.; Sun, Z.-Y. Soft Janus Particles: Ideal Building Blocks for Template-free Fabrication of Two-dimensional Exotic Nanostructures. Soft Matter 2014, 10, 5472–5477. Mahynski, N. A.; Lorenzo, R.; Likos, C. R. Panagiotopoulos, A. Z. Bottom-up Colloidal Crystal Assembly with a Twist. ACS Nano 2016, 10, 5459−5467. Liu, Y.; Li, W.; Perez, T.; Gunton, J. D.; Brett, G. Self Assembly of Janus Ellipsoids. Langmuir 2012, 28, 3−9. Romano, F.; Sanz, E.; Sciortino, F. Phase Diagram of a Tetrahedral Patchy Particle Model for Different Interaction Ranges. J. Chem. Phys. 2010, 132, 18450. Guo, R.; Liu, Z.; Xie, X-M.; Yan, L-T. Harnessing Dynamic Covalent Bonds in Patchy Nanoparticles: Creating Shape-shifting Building Blocks for Rational and Responsive Self-assembly. J. Phys. Chem. Lett. 2013, 4, 1221–1226. Reinhardt, A.; Romano, F.; Doye, J. P. K. Computing Phase Diagrams for a Quasicrystalforming Patchy-particle System Phys. Rev. Lett. 2013, 110, 255503. Guo, R.; Mao, J.; Xie, X-M.; Yan, L-T. Predictive Supracolloidal Helices from Patchy Particles. Sci. Rep. 2014, 4, 7021. Reinhardt, A.; Schreck, J. S.; Romano, F.; Doye, J. P. K. Self-assembly of Twodimensional Binary Quasicrystals: A Possible Route to a DNA Quasicrystal. J. Phys.: Condens. Matter 2017, 29, 014006. Ma, C.; Wu, H.; Huang, Z-H.; Guo, R-H.; Hu, M-B.; Kibel, C.; Yan, L-T.; Wang, W. A Filled-honeycomb-structured Crystal Formed by Self-assembly of a Janus 24 ACS Paragon Plus Environment

Page 24 of 28

Page 25 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(22)

(23)

(24)

(25)

(26) (27)

(28) (29) (30) (31)

(32)

(33) (34) (35) (36) (37) (38)

(39)

Polyoxometalate–silsesquioxane (POM–POSS) Co-cluster. Angew. Chem. Int. Ed. 2015, 54, 15699 –15704. Huang, Z.; Chen, P.; Yang, Y.; Yan, L-T. Shearing Janus Nanoparticles Confined in Two-dimensional Space: Reshaped Cluster Configurations and Defined Assembling Kinetics. J. Phys. Chem. Lett. 2016, 7, 1966−1971. Torrie, G. M.; Valleau, J. P. Monte Carlo Free Energy Estimates Using Non-Boltzmann Sampling: Application to the Sub-critical Lennard-Jones Fluid. Chem. Phys. Lett. 1974, 28, 578−581. Major, D. T.; Gao, J. An Integrated Path Integral and Free-energy Perturbation−umbrella Sampling Method for Computing Kinetic Isotope Effects of Chemical Reactions in Solution and in Enzymes. J. Chem. Theory Comput. 2007, 3, 949–960. Kumar, S.; Rosenberg, J. M.; Bouzida, D.; Swendsen, R. H.; Kollman, P. A. The Weighted Histogram Analysis Method for Free-energy Calculations on Biomolecules. I. The Method. J. Comput. Chem. 1992, 12, 1011−1021. Ten Wolde, P. R.; Ruiz-Monterob, M. J.; Frenkel, D. Simulation of Homogeneous Crystal Nucleation Close to Coexistence. Farady Disscuss. 1996, 104, 93−110. Eslami, H.; Khanjari, N.; Müller-Plathe, F. A Local Order Parameter-based Method for Simulation of Free Energy Barriers in Crystal Nucleation. J. Chem. Theory Comput. 2017, 13, 1307−1316. Steinhardt, P. J.; Nelson D. R.; Ronchetti, M. Bond-orientational Order in Liquids and Glasses. Phys. Rev. B 1983, 28, 784−805. Khanjari, N.; Eslami, H.; Müller-Plathe, F. Adaptive-numerical-bias Metadynamics. J. Comput. Chem. 2017, 38, 2721–2729. Laio, A.; Rodriguez-Fortea, A.; Gervasio, F. L.; Geccarelli, M.; Parrinello, M. Assessing the Accuracy of Metadynamics. J. Phys. Chem. B 2005, 109, 6714−6721. Huber, T.; Torda, A.; van Gunsteren, W. F. Local Elevation: A Method for Improving the Searching Properties of Molecular Dynamics Simulation. J. Comput.-Aided Mol. Des. 1994, 8, 695−708. Bulo, R. E.; Donadio, D.; Laio, A.; Molnar, F.; Riegerand, J.; Parrinello, M. Site Binding of Ca2+ Ions to Polyacrylates in Water: A Molecular Dynamics Study of Coiling and Aggregation. Macromolecules 2007, 40, 3437–3442. Branduardi, D.; Bussi, G.; Parrinello, M. Metadynamics with Adaptive Gaussians. J. Chem. Theory Comput. 2012, 8, 2247–2254. Warren, P. B. Vapor-liquid Coexistence in Many-body Dissipative Particle Dynamics. Phys. Rev. E: 2003, 68, 066702. Dobnikar, J.; Chen, Y.; Rzehak, R.; von Grünberg, H. H. Many-body Interactions and the Melting of Colloidal Crystals. J. Chem. Phys. 2003, 119, 4971–4985. Driscoll, M.; Delmotte, B.; Youssef , M.; Sacanna, S.; Donev, A.; Chaikin, P. Unstable Fronts and Motile Structures Formed by Microrollers. Nature Phys. 2017, 13, 375–380. Chen, Q.; Bae, S. C.; Granick, S. Directed Self-assembly of a Colloidal Kagome Lattice. Nature 2011, 469, 381–384. Qian, H-J.; Carbone, P.; Chen, X.; Karimi-Varzaneh, H. A.; Liew, C. C.; Müller-Plathe, F. Temperature-transferable Coarse-grained Potentials for Ethylbenzene, Polystyrene and Their Mixtures. Macromolecules 2008, 41, 9919–9929. Eslami, H.; Karimi-Varzaneh, H. A.; Müller-Plathe, F. Coarse-grained Computer Simulation of Nanoconfined Polyamide-6,6. Macromolecules 2011, 44, 3117–3128. 25 ACS Paragon Plus Environment


(40)

(41)

(42) (43) (44)

(45)

(46) (47)

(48) (49)

(50) (51)

(52) (53) (54)

(55) (56) (57)

Eslami, H.; Müller-Plathe, F. How thick is the Interphase in an Ultrathin Polymer Film? Coarse-grained Molecular Dynamics Simulations of Polyamide-6,6 on Graphene. J. Phys. Chem. C 2013, 117, 5249–5257. Ghoufi, A.; Malfreyt, P. Coarse Grained Simulations of the Electrolytes at the Water−air Interface from Many Body Dissipative Particle Dynamics. J. Chem. Theory Comput. 2012, 8, 787−791. Yang, L.; Yin, H. Parametric Study of Particle Sedimentation by Dissipative Particle Dynamics Simulation. Phys. Rev. E 2014, 90, 033311. Müller-Plathe, F. YASP: A Molecular Simulation Package. Comput. Phys. Commun. 1993, 78, 77−94. González-Melchor, M.; Mayoral, E.; Velázquez, M. E.; Alejandre, J. Electrostatic Interactions in Dissipative Particle Dynamics Using the Ewald Sums. J. Chem. Phys. 2006, 125, 224107. Mehdipour, N.; Faramarzi, G. Metal–nonmetal Transition in Fluid Cesium: A Many-body Dissipative Particle Dynamics Simulation Approach. J. Chem. Eng. Data 2017, 62, 3938–3945. Mao, X.; Chen, Q.; Granick, S. Entropy Favours Open Colloidal Lattices. Nature Mater. 2013, 12, 217–222. Pattabhiraman, H.; Dijkstra, M. The Effect of Temperature, Interaction Range, and Pair Potential on the Formation of Dodecagonal Quasicrystals in Core-corona Systems. J. Phys.: Condens. Matter 2017, 29, 094003. Chen, Q.; Whitmer, J. K.; Jiang, S.; Bae, S. C.; Luijten, E.; Granick, S. Supracolloidal Reaction Kinetics of Janus Spheres. Science 2011, 331, 199–202. Wolters, J. R.; Avvisati, G.; Hagemans, F.; Vissers, T.; Kraft, D. J.; Dijkstra, M.; Kegel, W. K. Self-assembly of “Mickey Mouse” Shaped Colloids into Tube-like Structures: experiments and simulations. Soft Matter 2015, 11, 1067–1077. Romano, F.; Russo, J.; Tanaka, H. Influence of Patch-size Variability on the Crystallization of Tetrahedral Patchy Particles. Phys. Rev. Lett. 2014, 113, 138303. Kalyuzhnyi, Y. V.; M. Holovko, M.; Patsahan, T.; Cummings, P. T. Phase Behavior and Percolation Properties of the Patchy Colloidal Fluids in the Random Porous Media. J. Phys. Chem. Lett. 2014, 5, 4260−4264. Romano, F.; Sanz, E.; Sciortino, F. Role of the Range in the Fluid-crystal Coexistence for a Patchy Particle Model. J. Phys. Chem. B 2009, 113, 15133–15136. Smallenburg, F.; Sciortino, F. Liquids More Stable Than Crystals in Particles with Limited Valence and Flexible Bonds. Nature Phys. 2013, 9, 554–558. Noya, E. G.; Vega, C.; Doye, J. P. K.; Louis, A. A. The Stability of a Crystal with Diamond Structure for Patchy Particles with Tetrahedral Symmetry. J. Chem. Phys. 2010, 132, 234511. Preisler, Z.; Vissers, T.; Smallenburg, F.; Sciortino, F. Crystals of Janus Colloids at Various Interaction Ranges. J. Chem. Phys. 2016, 145, 064513. Tanaka, H.; Araki, T. Simulation Method of Solloidal Suspensions with Hydrodynamic Interactions: Fluid Particle Dynamics. Phys. Rev. Lett. 2000, 85, 1338–1341. Chen, C. Synthesis and Self-assembly of Multiblock Janus Particles, PhD Dissertation, University of Illinois at Urbana-Champaign; 2012.


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TOC



Phase diagram for triblock Janus particles. 165x88mm (300 x 300 DPI)

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Solid-Liquid and Solid-Solid Phase Diagrams of Self-Assembled

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