Self-assembly mechanisms of triblock Janus particles - Journal of

Subscriber access provided by YORK UNIV

Nanochemistry

Self-assembly mechanisms of triblock Janus particles Hossein Eslami, Neda Khanjari, and Florian Müller-Plathe J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00713 • Publication Date (Web): 21 Dec 2018 Downloaded from http://pubs.acs.org on December 22, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 29 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

Journal of Chemical Theory and Computation

Self-assembly mechanisms of triblock Janus particles Hossein Eslami,1,2* Neda Khanjari,1 and Florian Müller-Plathe2 1Department

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

2Eduard-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]

1 ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 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

Abstract We present a detailed model to study the nucleation of triblock Janus particles from solution. The Janus particles are modeled as crosslinked polystyrene spheres whose poles are patched with sticky alkyl groups and their middle band is covered with negative charges. To mimic the experimental conditions, solvent, counterions and a substrate, on which the crystallization takes place, are included in the model. Many-body dissipative particle dynamics simulation technique is employed to include hydrodynamic and many-body interactions. Metadynamics simulations are done to explore the pathways for nucleation of Kagome and hexagonal lattices. In agreement with experiment, we found that nucleation of the Kagome lattice from solution follows a two-step mechanism. The connection of colloidal particles through their patches initially generates a disordered liquid network. Subsequently, orientational rearrangements in the liquid precursors lead to the formation of ordered nuclei. Biasing the potential energy of the largest crystal, a critical nucleus appears in the simulation box, whose further growth crystallizes the whole solution. The location of the phase transition point and its shift with patch width are in a very good agreement with experiment. The structure of the crystallized phase depends on the patch width; in the limit of very narrow patches strings are stable aggregates, intermediate patches stabilize the Kagome lattice, and wide patches nucleate the hexagonal phase. The scaling behavior of the calculated barrier heights confirms a first-order liquid-Kagome phase transition.


Page 2 of 29

Page 3 of 29 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


I. Introduction Recently, increasing attention has been paid to self-assembly of colloidal materials toward fabrication of nanocrystals, offering unique mechanical, electronic, and magnetic properties, essential for utility in magnetic, electronic, photovoltaic, sensing, biomedical, and catalytic applications.1 In the so-called bottom-up approach (self-assembly) information regarding the target structure is stored in the building blocks, to direct their assembly by physical/chemical interactions, without external control.2 In a new class of building blocks (anisotropic colloidal spheres) the directional interactions are encoded into the particles by covering some domains of their surfaces with patches. There exist synthetic challenges for producing such building blocks, as experimental decoration of the surface of nanoparticles with multiple well-defined sticky patches is not feasible. Therefore, current research in this direction is mainly oriented toward fabrication of simple patchy particles, capped on the poles with two patches (Janus particles). Experimental reports show that the Janus particles are capable to self-assemble into lowdimensional structures including linear chains,3 monolayers,4 helical clusters,5 as well as a twodimensional Kagome lattice.6 As complementary tools to experiment, molecular simulation methods are regarded as powerful tools for designing building block shapes and predictions of the self-assembled structures.7-12 Sophisticated computational techniques, based on anisotropy attributes of virtual hard-sphere building blocks, have been invented to simulate their self-assembly.13 Disk-like Janus particles have been modeled as a hard-sphere core capped with a soft coronal region.14 The Kern–Frenkel angle-dependent potential15 has been used to study the self-assembly in colloidal particles by several investigators.16-24 In this potential, soft patches interacting through a squarewell potential are introduced on the surface of a hard sphere core. While the above-cited models have successfully predicted diverse phases and morphologies of the finally organized lattices, two points are worth considering. 1) In all these simulations a very simplified model (mostly a hard-sphere model) of the nanoparticle is assumed. In fact, due to the computational cost, the role of factors such as solvent, surface charges, and supporting surface are mostly ignored. Experimental investigations, however, show that solvent, via its effect on the solubility and swelling of patchy particles tunes the magnitude of interactions between the particles, and hence, controls the reorganization 3 ACS Paragon Plus Environment


process. For charged colloids, surface charges and the spatial distribution of ions in solution also affect the nucleation mechanism.7,25 Another issue in the suspensions of charged colloids is many-body effects between the colloids and positional correlations of the counter-ions, whose importance in the prediction of the phase behavior of charged colloids has been noted by several authors.26 2) In such brute-force simulations attention is focused on the structure of finally organized pattern. However, many aspects of the self-assembly of even such simplified models, such as the microscopic picture of the crystal nucleation pathway is missing. This is of particular relevance for colloidal systems, for which the real time dynamical process are experimentally accessible,6 and hence, are the meeting point of experiment and simulation for nucleation studies. Although nucleation of colloidal particles is experimentally tractable, however, it is difficult to separate nucleation from the growth process in these experimental studies. Molecular simulations, on the other hand, allow exploration of cluster stability, structure, and the nucleation pathway, and captures unresolved aspects of the self-assembly. Recently, we have presented a detailed model of Janus particles and employed it to study the phase equilibria in solution of Janus particles.27 Here, we further study the mechanism of nucleation, and the effect of above-cited factors on the self-assembly pathway. Our simulations are designed to mimic the experimental conditions in self-assembly of triblock Janus particles into a Kagome lattice (an arrangement of interlaced triangles, as shown in Figure 1). To this end, we simulate a many-body dissipative particle dynamics (DPD)28 model of a real system of practical interest, namely a suspension of a charged colloid, polystyrene sulfonate (patched on opposite poles with sticky hydrocarbon patches)6 in water in contact with a coarse-grained model of charged graphene surface29 on one side and with vacuum on the other. Compared to the conventional DPD,30 in the many-body DPD method the conservative force is divided into two individual attractive and many-body repulsive parts. Although the conventional DPD is an adequate tool for studying systems with varieties of complexities, it is not a proper method for studying systems with free surfaces. Besides, in the conventional DPD method the effective force between particles is solely repulsive, and hence, the particles (patches) do not stick together. In our modeling, attractive patch-patch interactions are required to compete with the (screened) repulsive interactions between the particles. Moreover, the importance of many-body


Page 4 of 29

Page 5 of 29 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


interactions in the prediction of the solid-liquid phase behavior and clustering of charged colloidal particles is given addressed to by several authors.31-33 Taking into the effects of solvent, counterions, hydrodynamic and many-body interactions in our modeling, we have employed an advanced sampling scheme34 to simulate nucleation in two dimensions on a supporting surface in contact with the colloidal solution.

Figure 1. Cartoons of triblock Janus colloids with sticky patches on opposite poles, capable of formation of a two-dimensional Kagome lattice (left) and/or a hexagonal lattice (right), depending on the patch width. II. Method Direct simulation of nucleation faces the obstacle of high free energy barriers, which are unlikely to be crossed during the accessible time scales in brute force molecular simulations.34,35 Advanced sampling schemes, such as umbrella sampling36 and its extensions,37-39 the weighted histogram analysis,40 transition path sampling,41 replica exchange,42 metadynamics,34 and its extensions43 have been designed to accelerate sampling a representative portion of the phase space over an affordable time window. In such methods the free energy landscape is formulated in terms of a set of reaction coordinates (order parameters), q, along which different configurations of interest are distinguishable, i.e., F q   k BT ln Pq 

(1)

where kB is the Boltzmann constant, T is the temperature, F(q) is the Helmholtz free energy profile, and P(q) is the probability of observing the reaction coordinates q between q and q + dq. Augmenting the Hamiltonian of the system by an external bias term, the simulations are forced 5 ACS Paragon Plus Environment


Page 6 of 29

to explore regions of high free energy for obtaining reliable statistics of the reaction coordinates, and hence, the free energy. In the metadynamics method, which is employed here to surmount the free energy barrier, the bias term is introduced into the Hamiltonian in terms of a Gaussian. In this method, the biasing potential, U(q, t), is constructed as the sum of history-dependent Gaussians, centered at the visited points in the order parameter space at regular time intervals, i.e.,34   q  qt 2   U q, t   w exp   2   2 t t   

(2)

where, w and  are the height and the width of the added Gaussians, respectively, and t is the time. The order parameters must discriminate between different phases in the system, separated from each other by their corresponding free energy barriers. Historically, the bond (a vector connecting neighboring particles i and j) order parameters, introduced by Steinhardt et al.44, measure the order around a particle i. The Steinhardt et al.44 bond order parameters are defined as: qlm i  

Y  N i  1

b

m

j N b i

l

ij

, ij 

(3)

where θij and φij are the angles specifying the orientation of a bond between particles i and j, Nb(i) is the number of neighbors of particle i, and the Ylm are the spherical harmonics. In almost all simulations of nucleation and phase equilibria,45-47 the order is expressed in terms of a global order parameter, obtained by constructing rotational invariants of sum of qlm(i) vectors for all particles. Although such global order parameters are appropriate tools for simulations of phase equilibria, their limitations for simulation studies of nucleation have been warned in the literature.48-51 In this respect, we have recently developed a local order-parameterbased metadynamics method for nucleation studies.50 To distinguish between liquid, Kagome, and hexagonal phases (cf. Figure 1), here, we have examined a number of local order parameters. According to our calculations, we propose the following two order parameters to discriminate liquid, Kagome, and hexagonal phases: 1 i  

4  6  * ˆ ˆ     q i  q j  qˆ4 m i   qˆ4*m  j     6m 6m  N b i  jN b i   m  6 m  4 

1

and 6 ACS Paragon Plus Environment

(4)

Page 7 of 29 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


2 i  



1

    N i   b

j N b i

m  6 , 4

qˆ6 m i   qˆ6*m  j  

  qˆ i   qˆ  j 

m  2

4m

* 4m



(5)

where qˆlm i  

qlm i 

(6)

1/ 2

 l 2   qlm i    m l 

The order parameter 1 is  -0.1, 0.0, and 0.65 and 2 is  0.0, 0.8, and 0.7, for the particles involved in the liquid, hexagonal, and Kagome phases, respectively. In this work, the history-dependent biasing potentials in the space of 1 and 2, are progressively added to the system to fill the free energy basins of liquid/crystalline phases, and hence, to facilitate transition between different phases. A brief explanation of the many-body DPD method is given in SI, but for further details the reader is referred to reference.28 III. Simulations The colloidal nanoparticles are polystyrene (PS) sulfonate particles, whose self-assembly to the Kagome lattice has been reported by Chen et al.6 In experiment, the hydrophobic patches are made of monolayers of n-octadecanethiol, deposited on the thin gold films covering the surface of colloidal particles at opposite poles. Tuning the concentration of salt (Sodium sulfonate) in a 10 wt % solution of triblock Janus particles in water, strikes a balance between the Coulombic repulsions of the middle band and the patch-patch attractive interactions to direct the selfassembly toward formation of two-dimensional Kagome lattice on the surface of the container. Here a coarse-grained representation of colloidal particles, solvent, ions, and substrate has been invoked. The PS spherical nanoparticle is modeled as a highly crossliked PS chain, covered at the surface with PS beads. Each PS monomer in the nanoparticle (a sphere with a diameter of 5.0 nm) is regarded as a single DPD bead. The neighbouring beads on the surface of PS sphere are connected through a harmonic spring with an equilibrium bond length of ≈ 0.55 nm (tuned against the monomer-monomer bond lengths in coarse-grained models of PS).52 The structural anchor consists of three “beads” in a linear configuration at the centre of the hollow sphere. The nanoparticle surface beads are connected to the anchor beads as well. In this way, surface beads are unambiguously placed in one hemisphere, and anisotropy can be implemented. Single beads, each representing one hydrophobic n-hexyl group, are grafted to all PS beads on 7 ACS Paragon Plus Environment


the poles as patches with an opening angle  = 65 (see Figure 2). Because of the smaller size of our particles, compared to experiment, here a smaller-chain alkyl group (n-hexyl), modeled as one DPD bead, is adopted to patch the poles. Sulfonate beads (nearly equally spaced, covering about 10 % of the nanoparticles’ surface) were grafted to the PS beads in the middle band of the nanoparticle and Na+ counterions were added to preserve electroneutrality. For solvent (water), a DPD bead represents 5 water molecules. For ion-ion and ion-water interactions, in previous reports on the DPD simulations of electrolyte solutions, the same set of interaction parameters as those of water-water interactions have been used.53-55 There are, however, reports in the literature where the many-body DPD parameters for ion-ion and water-water interactions are the same, but water-ion attractive interactions are stronger,56 leading to the stronger hydration of ions in water. In this work, we performed many-body DPD simulations on water to tune the water-water interactions parameters. In our modeling we mainly need to balance electrostatic interactions (surface charge repulsions and the screening of electrostatic interactions with tuning the ionic strength of the solution) against patch-patch attractions. Therefore, similar to the previous reports in the literature, the same set of many-body DPD interaction parameters for ion-ion, ion-water, water-water interactions were chosen.53-55 Like the parameterization of Groot,54 the charge structure function in our parameterization obeys the Stillinger–Lovett moment conditions.57 Our recent coarse-grained model of graphene,29 in which a combination of 8 C atoms are regarded as a single bead, is adopted here to model the substrate. The substrate beads were negatively charged (charge density = -2×10-3 nm-2) to mimic the silica substrate, employed in experiment.6 To facilitates patch-patch sticking, we chose more attractive patch-patch parameters. A gravitational force normal to the surface direction58 was employed to sediment the floating particles on the supporting surface. The many-body DPD parameters for water, PS beads, ions, and surface are tabulated in Table S1. The concentration of PS colloidal particles is adjusted close to the experimental value6 (15 wt%). Additional electrolyte, Sodium sulfonate, ion pairs (3.5 mM) were added to screen the Coulombic repulsions between the particles.


Page 8 of 29

Page 9 of 29 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


Figure 2. Snapshot of a configuration of a triblock Janus particle modeled in this work. The surface (orange) spheres are polystyrene beads, crosslinked to their nearest neighbours and connected to anchor beads (grey spheres). The green spheres represent the sticky hydrophobic patches covering the poles of the particle with an opening angles  = 65. The surface of the nanoparticle in the equatorial region is covered with negatively charged sulfonate beads (blue spheres). The yellow spheres indicate the counterions.

Four systems, in which the number of Janus particles ranges from 96 to 216, were simulated in this work. The number of DPD beads in these systems ranged from 322000 to 422000. An atomistic model representation of the largest system would include more than 6×106 atoms, which is almost impossible to simulate. The many-body DPD simulations were started with a random configuration of colloid and water beads in a large simulation box. All simulations were performed with our simulation package, YASP.59 The electrostatic interactions were calculated using the Ewald summation method, with the charge distributions according to the method explained by Ibergay et al.54 The time step for integration of equations of motion was 0.1 ps. Simulations were performed over a time window around 4 µs. IV. Results and Discussions A. Crystallization from Solution



Nucleation and subsequent crystal growth of Kagome and hexagonal phases have been examined by performing test metadynamics simulations at temperatures varying from 270 K to 320 K. The Janus particles on the surface freeze to the Kagome lattice (at p=101.3 kPa) at 308 K. We have shown distributions of the average local bond order parameters  1 and  2, for liquid, Kagome, and hexagonal phases at p = 101.3 kPa and T = 280 K ( 10 % supercooling) in Figure S1. Sharp well-resolved peaks at ( 1  -0.1 and  2  0.0), ( 1  0.65 and  2  0.7), and ( 1  0.0 and

2  0.8) are representatives of liquid, Kagome, and hexagonal phases, respectively. We have done long biased simulations at T = 308 K to study the pathways for nucleation of Kagome and hexagonal phases from solution. As in our previous local-order-parameter-based metadynamics method,50 the local order parameters, 1 and 2, are used to find the largest crystalline cluster. Clusters are determined as aggregates of particles having at least 4 connections with their neighbors. A pair of particles are considered to be connected if their 2 exceed a threshold value of 0.25. The average local order parameter is determined by averaging the local order parameter for the particles involved in the largest cluster. Figure 3 shows free energy contours, for a system of 168 triblock patchy particles at 308 K and 101.3 kPa, in the space of the average local order parameters,  1 and  2 (averaged over particles involved in the largest cluster). During the course of metadynamics simulations, the system performs several round trips between the two coexisting phases, allowing accurate calculation of the free-energies. According to the results in Figure 3, the disordered (liquid) phase is in equilibrium with Kagome; both stable phases are characterized by contours of zero free energy with free energy basins at  1  -0.1 and  2  0.05 and  1  0.6 and  2  0.63, respectively. Slight deviations in the values of order parameters for Kagome lattice from those reported in Figure S1 are due to the existence of defects in the Kagome nucleated from solution. The saddle point, which separates liquid from Kagome locates (23.6 ± 1.1) kJ mol-1 higher than both stable phases. This barrier height ( 9kBT) is comparable to that calculated in a previous simulation by Romano and Sciortino16 ( 5-10 kBT). The hexagonal phase is, however, unstable with respect to both (the Kagome and the liquid) phases; the free energy basin for the hexagonal phase, characterizes by 1  0.0 and 2  0.75, locates  8 kJ mol-1 higher than those for the other two phases. The barrier height for liquid to hexagonal phase conversion is (35.7 ± 1.2) kJ mol-1. The pathway for liquid to Kagome conversion includes variations in both order 10 ACS Paragon Plus Environment

Page 10 of 29

Page 11 of 29 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


parameters, quantifying the Kagome-like and hexagonal-like orders for crystallites extending to their second coordination shells. The higher stability of the Kagome with respect to the hexagonal phase and lower barrier height for conversion of liquid to Kagome than that for liquid to hexagonal phase, facilitating liquid to Kagome conversion, is in agreement with experiment.6 The location of calculated liquid-Kagome phase equilibrium point (p = 101.3 kPa and T = 308 K) is close to the experimentally reported phase equilibrium point6,25 (ambient conditions). The highest temperature for liquid-Kagome phase equilibrium in the simulations of Romano and Sciortino,16 however, ranges from ≈ 125 K to ≈ 180 K (calculated based on their reported T* = 0.15 and experimental6 and calculated60 potential well-depths, ranging from ≈ 7 kBT to ≈10 kBT).



Figure 3. Free energy contours for equilibrium between liquid ( 1  -0.1 and  2  0.05) and Kagome ( 1  0.6 and  2  0.63) phases in a system containing 168 Janus particles at p=101.3 kPa and T = 308 K. The basin for the metastable hexagonal phase ( 1  0.0 and  2  0.75) locates ( 8 ± 0.8) kJ mol-1 higher than those for liquid and Kagome phases. The barrier heights for nucleation of Kagome and hexagonal lattices from solution at this temperature are (23.6 ± 1.1) kJ mol-1 and (35.7 ± 1.2) kJ mol-1, respectively. Contours are 1.5kBT apart.

C. Mechanism of Nucleation Our procedure for studying the nucleation mechanism was to form a two-dimensional weakly interacting (unconnected) solution of Janus particles, brought to the surface as explained in Sec. III. The Janus particles are bound to the surface, by tuning the surface interaction parameters (see Table S1) and the gravitational force as explained in Sec. III, but they are laterally mobile on the top of surface. We define the zero of time as the moment when all the Janus particles are brought 12 ACS Paragon Plus Environment

Page 12 of 29

Page 13 of 29 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


to a distance less than 5 nm from the substrate. With the shrinkage of sizes of simulation box along x, Lx, and y, Ly, directions in the NpT ensemble simulations, the Janus particles get close together and neighboring particles start linking via the sticky patches, but avoid the unfavorable contacts between their charged equatorial regions. Snapshots of the simulation box are shown in Figure 4. At low density, fusion of Janus particles forms dimers and subsequently strings with dangling bonds (see Figure 4 panels a-c). Further reduction of the box size and rotation of Janus particles, introduces triangular aggregates (Figure 4 panel d), and subsequently, disordered ring structures (Figure 4 panels e and f). The structure of the sample at this stage is a disordered network of strings and rings. The number of contacts per particle depends on the patch size. The patch size examined here (corresponding to the experimental value)6 is just large enough to allow triangular contacts, but is too small to allow more than two contacts per patch. The NpT metadynamics driven crystallization (Figure 4 panels d-i) will be discussed further bellow.



Figure 4. Snapshots of simulation box along the crystallization path of Kagome from a solution of 168 Janus particles at p =101.3 kPa and T=308 K. Setting t = 0 as the moment the Janus particles are within a distance 5 nm from the substrate, snapshots (a)-(i) belong to simulation times 0.04, 0.12, 0.18, 0.25, 0.45, 0.7, 1.0, 1.05, and 1.1 µs, respectively. The dimensions of simulation box (Lx=Ly) in snapshots (a)-(c) are 150, 110, 99, respectively. All other snapshots are taken from the NpT simulations along the crystallization path. In snapshots (e)-(g) the largest nucleus is shown in yellow. The nucleus shown in snapshot (g) corresponds to the critical nucleus. Snapshot (h) represent the growth process and snapshot (i) is a defected (see the top left portion) Kagome lattice. In snapshot (i) Lx=Ly =82.3 nm.


Page 14 of 29

Page 15 of 29 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


We have shown in Figure 5 a quantitative analysis of the number of isolated particles, particles with two contacts (strings), particles involved in triangular contacts, and those involved in the Kagome lattice, with the passage of time. The number of particles with no contacts decreases steadily as the box size is reduced. When the particles get close enough together, they start forming dimers and larger strings. The concentration of strings passes through a maximum and reduces at longer times. Reduction in the fraction of strings is accompanied by the formation of triangular contacts, facilitating formation of disordered rings. The occurrence of a maximum in the fraction of strings shows that they are kinetically favored intermediates during the crystallization pathway. The chain lengths in strings, depending on the time, vary from 2 to  20 particles. With the increase in the chain length, the strings adopt varieties of conformations, leading to further intra- and inter-chain contacts, and hence, formation of ring structures. Therefore, as the number of Janus particles involved in strings reduces, the number of triangular contacts increases sharply. These findings are in complete agreement with experimental observations of Chen et al.6 The tendency of strings being substituted by triangular contacts (connected ring structures) depends on the patch width; strings are the stable aggregates for small patch widths. Wider patches destabilize the strings in favor of the formation of ring structures. Therefore, as the number of Janus particles involved in strings reduces, the number of triangular contacts increases sharply.



Figure 5. Time-dependence of the fraction of Janus particles, classified in terms of their number of contacts; particles with no contacts, with two contacts, those involved in triangular contacts, and particles involved in the largest Kagome nucleus. The sudden jump in the fraction of particles involved in the Kagome crystals (t > 1 µs) represents the growth process. In Figure 6 we plot a contour map of the number of nearest neighbors (particles whose center-ofmass distances is less than 7.0) in the ( 1 ,  2) plane for crystallization of particles to Kagome at p =101.3 kPa and T = 308 K. The crystallization is associated with a monotonic increase in both order parameters (see Figure 3) from the liquid basin to the Kagome basin. During the crystallization path, fluctuations in the number of neighbors (density) occur. Experimental reports of Chen et al.6 verify these findings. The crystallization pathway of Kagome, therefore, includes condensation of Janus particles to disordered liquid network in its early stages. Subsequently, structural rearrangements in these liquid domains form organized nuclei. In other words, the nucleation obeys a two-step mechanism, where initial liquid-like precursors are formed and subsequently particles in the dense liquid-like domains rearrange to construct solidlike nuclei. In fact, fluctuations in the density of the disordered phase results in the formation of disordered nucleation sites. Such disordered clusters, locating in the front side (toward Kagome) 16 ACS Paragon Plus Environment

Page 16 of 29

Page 17 of 29 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


of wide liquid basin, can undergo structural rearrangements to be converted to more organized crystallites.

Figure 6. Contour maps for the average number of neighbors of Janus particles (within a cutoff distance of 7.0 nm form their geometrical centers), along the crystallization path, at p =101.3 kPa and T=308 K. The snapshot of the simulation box in Figure 4 (panel d) shows a liquid configuration, with a density 15 % lower than that of the Kagome lattice. We start metadynamics simulations from such a liquid configuration, to fill the liquid phase free energy basin. Our analysis indicates that nucleation of Kagome starts from dense disordered liquid aggregates (Figure 4, panel d). This dense liquid domain, with triangular contacts, further undergoes structural rearrangements to form an organized Kagome nucleus (a typical small nucleus, in which the particles are shown in yellow, is seen in panel e of Figure 4). Our analysis shows that Kagome nucleates from a dense liquid droplet containing triangular contacts. This is also evident from the results in Figure 5, in which the number of particles involved in Kagome remains zero until the number of particles with triangular contacts reaches a threshold limit. The structural arrangement of a dense liquid droplet for formation of an ordered nucleus is an evidence for a two-step mechanism of nucleation of Kagome, confirmed in experiment.6 It is worth mentioning that this two-step 17 ACS Paragon Plus Environment


Page 18 of 29

mechanism of nucleation of Kagome from solution has not been detected in former simulations in the literature. Besides, the simulations of Romano and Sciortino16 show that crystallization of Kagome precedes by the formation of chains. However, according to our results, nucleation of Kagome does not start until a critical concentration of triangular contacts is reached (in agreement with experiment).6 Our metadynamics simulation scheme in the local-order-parameter space allows biasing the potential for the initial crystallites. Biasing the potential increases the number of Janus particles in the nucleated phase. At t = 0.7 µs a few nuclei exist in the simulation box; the structure of the largest nucleus is shown in Figure 4 (panel f). At t  1.0 µs, the critical nucleus forms (Figure 4, panel g), upon its growth the whole liquid crystallizes. We have given a quantitative order in the critical nucleus in SI. Our findings (see Figure S2) show that the core of the nucleated Kagome cluster is more Kagome-like ordered, while the particles at the surface of the largest clusters are less ordered. Further growth of the critical nucleus leads to the combination of the nucleated sites and a fast transition to the solid phase. A sudden jump in the fraction of particles involved in the largest Kagome nucleus (see Figure 5) at t > 1.0 µs indicates that at this time the grown nuclei in the box quickly merge (the crystal growth stage) toward formation of a single crystal phase. A snapshot of the simulation box after merging the critical nucleus with other crystalline regions in the box, in which more than 65 % of Janus particles belong to the Kagome lattice is shown in Figure 4 (panel h). The final Kagome lattice typically contains the defects (a typical configuration is shown in Figure 4, panel i). We have further examined, in Figure 7, the structural similarity between the Janus particles involved in the largest Kagome crystal and those attaching to its surface (the particles surrounding it) with the passage of time. This correlation is defined as: C t   1 i, t 1 i,0

(7)

where 1 (i, t) is the order parameter for particle i at time t. Here t=0 indicates the moment of attachment of a particle to the pre-existing nucleus. Here 1 is shifted so that it has always positive values, and the correlation function C(t) is normalized to 1 for the Kagome lattice. Our analysis is done for the system along the nucleation path from the initial stages a stable nucleus is formed to the critical nucleus. At t > 0 the attached Janus particles are included in the nucleated Kagome phase. Similar to the analysis of Keys and Glotzer,61 we count only particles attaching permanently to the surface. At t < 0, the attaching Janus particles are in the liquid region 18 ACS Paragon Plus Environment

Page 19 of 29 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


surrounding the largest nuclei. The results in Figure 7 show that the order in particles attached to the surface (t ≥ 0) of the Kagome nuclei largely correlates with the order in the preexisting nucleated Kagome phase. Also the correlation function for attaching atoms to the largest nuclei (t ≤ 0) decays much slower than that for particles in the liquid phase.

Figure 7. Correlation in the order of attaching Janus particles from the solution to the surface of the largest Kagome nucleus. The start of time (t=0) corresponds to the moment the particles attach to the surface of the preexisting Kagome nucleus. The blue and red curves represent correlation of orders in the attaching particles to the surface of the largest Kagome nucleus after and before attachment, respectively. The black curve is correlation of order for the rest of the particles in the liquid phase. For definition of C(t) see eq. 10 in the text. D. System-Size Dependence of Barrier Heights We have done metadynamics simulations for liquid-Kagome equilibrium in 96, 144, 168, and 216 Janus particles in solution. The calculated barrier heights can be used to determine the type of the freezing transition in the present quasi-two-dimensional system. It is worth considering that because of the difference in the type of order, distinguishing solid from liquid, in two- and three-dimensional system, understanding the mechanism of two-dimensional melting/freezing is of theoretical importance. While experimental studies of phase ordering in confined quasi-twodimensional colloid suspension are supportive for a first-order transition,62 computer simulations



of melting in two-dimensional systems indicate that mechanism of melting depends on the nature of intermolecular interactions.63 We have examined the size-dependence of the free energy barrier heights for liquid to Kagome crystallization Figure 8. The scaling of barrier heights as the square-root of the number of particles (N1/2) satisfies the scaling characteristics of the first-order transition in quasi twodimensional systems.62,63 Due to the increase in the size of the interface with increasing the size of the system, the barrier height for nucleation indicates system-size dependency.

Figure 8. Size-dependence of the barrier height for nucleation of Kagome from solution at p =101.3 kPa and T=308 K. The scaling behavior of free energy as the square-root of the number of Janus particles, N1/2, is an indication of a first-order phase transition.

E. Effect of Patch Size The number of contacts per Janus particle is controlled by a balance between the Coulombic repulsions of the equatorial region and the attractive patch-patch interactions. In fact the patch size adopted by Chen et al.6 ( = 65) in Figure 2, is wide enough to allow two contacts per patch, favoring formation of Kagome lattice, but is narrow enough to disfavor formation of a hexagonal phase. In two dimensions, the particles can have at most 6 nearest neighbors. As the hydrophobic patches allow considerable orientational freedom within the initially formed disordered network, the Janus particles can form denser phases than Kagome upon an increase in 20 ACS Paragon Plus Environment

Page 20 of 29

Page 21 of 29 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


the patch width. To quantify the effect of patch width on the structure of the resulting crystal, here we have done simulations on triblock Janus particles whose poles are patched with an opening angle  = 90. The initial test simulations show that for this patch width the melting temperature increases to 345 K. The increase in the freezing temperature, at p = 101.3 kPa, with increasing the patch width is in agreement with the reports of Romano et al.64 Free energy contours for crystallization of Kagome and hexagonal phases from a solution of Janus particles ( = 90) are shown in Figure S3. Our findings show that in this case the hexagonal phase, which is more stable than Kagome, is in equilibrium with the liquid phase. Also in this case, nucleation obeys a two-step mechanism. Following the formation of disordered aggregates, a structural rearrangement in the aggregates leads to formation of initial hexagonal nuclei. V. Conclusions We have developed a sophisticated model to mimic self-assembly in triblock Janus particles. It contains many details of experimental relevance. The Janus particles are modeled as crosslinked polystyrene spheres whose poles are patched with sticky alkyl groups and their middle band is covered with negative charges. Besides, we have included the solvent and a substrate, on which the crystallization takes place. Moreover, hydrodynamic and many-body interactions are taken care of within the context of many-body DPD simulation approach. An advanced sampling technique, metadynamics, employing two efficient local order parameters, has been employed to accelerate sampling along the transition path. The local order parameters hold information regarding the order in the first- and the second-coordination shells of a central particle and show well-resolved peaks for liquid and the crystalline phases examined in this work. Unlike methods in which global order parameters (leading to multiple cluster generation, and hence, overestimating the nucleation free energy barriers) are used, in the present local order parameter approach50 the bias is sent to the largest crystal in the box. Our results on the nucleation pathway from liquid solution to Kagome reveal a sequence of the phase-ordering processes; isolated adsorbed spheres link together to form strings. Followed by an initial increase in the concentration of strings, a decrease in their concentration (due to the formation of triangular contacts and rings) occurs at longer times. Subsequently, structural rearrangements in the dense liquid domains form organized nuclei. In other words, our findings indicate that nucleation of both the Kagome and the hexagonal phase follows a two-step 21 ACS Paragon Plus Environment


mechanism; formation of metastable amorphous high-density liquid precursors followed by their reorganization into ordered structures. Biasing the potential energy of the initially formed nucleation site grows it to a critical nucleus. The core of the critical nucleus is Kagome-like ordered, while the particles at its surface are liquid-like ordered. The particles attaching to the surface of the growing Kagome nuclei have a much slower dynamics than those in the liquid, confirming the solid-like character of the nucleated phase. Depending on the patch width, the nucleation is characterized by a competition between an open (Kagome) and a dense (hexagonal) phase. In the limit of very narrow patches strings are stable aggregates. Wider patches (opening angle of 65), favor four contacts per particle and stabilize the Kagome lattice. Similar to the phase transition in confined quasi-two-dimensional colloid suspensions,63 here size-dependency of calculated barrier heights confirms a first-order liquid to Kagome phase transition. Our calculated barrier height for liquid-Kagome phase equibrium ( 10kBT), agree with the previously reported value16, a few kBT, employing a simple Kern-Frenkel potential. Furthermore, the calculated barrier height for the nucleation of the Kagome lattice from liquid solution is, at ambient conditions, lower than that of the hexagonal phase (in complete agreement with experiment).6 Also, the location of the phase transition point (p = 101.3 kPa and T = 308 K) agrees very good with experiment.6 The increase in the freezing temperature, at p = 101.3 kPa, with increasing the patch width is in agreement with the reports of Romano et al.64 Our results confirm that Kagome lattice is stable only over a narrow window of patch width, supporting difficulty in experimental design of accurate patch widths leading to formation of Kagome. The ability of the model presented here for quantitative prediction of the stable crystalline phase, depending on the patch width, provides a guide for designing new patchy particles for tailoring new nanostructured compounds. 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.


Page 22 of 29

Page 23 of 29 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


Supporting Information: Description of the many-body DPD method, probability distributions for the local order parameters, a quantitative measure of the order in the Kagome critical nucleus as a function of distance from its center, the effect of patch size on the nucleation pathway, and the many-body DPD parameters. This information is available free of charge via the Internet at http://pubs.acs.org



References (1) (2) (3) (4)

(5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)

Yan, L-T.; Popp, N.; Ghosh, S-K.; and Alexander Böker, A. Self-Assembly of Janus Nanoparticles in Diblock Copolymers. ACS Nano 2010, 4, 913–920. Jiang, Y.; Löbling, T. I.; Huang, C.; Sun, Z.; Müller, A. H. E.; Russell, T. P. Interfacial Assembly and Jamming Behavior of Polymeric Janus Particles at Liquid Interfaces. ACS Appl. Mater. Interfaces, 2017, 9, 33327–33332. 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, 4720–4728. Xiao, Q.; Rubien, J. D.; Wang, Z.; Reed, E. H.; Hammer, D. A.; Sahoo, D.; Heiney, P. A.; Yadavalli, S. S.; Goulian, M.; Wilner, S. E.; Baumgart, T.; Vinogradov, S. A.; Klein, M. L.; Percec, V. Self-Sorting and Coassembly of Fluorinated, Hydrogenated, and Hybrid Janus Dendrimers into Dendrimersomes. J. Am. Chem. Soc. 2016, 138, 12655– 12663. 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. Chen, Q.; Bae, S. C.; Granick, S. Directed Self-assembly of a Colloidal Kagome Lattice. Nature 2011, 469, 381–384. Chen, T.; Zhang, Z.; Glotzer, S. C. A Precise Packing Sequence for Self-assembled Convex Structures. Proc. Natl. Acad. Sci. USA 2007, 104, 717–722. Roussel, T. J.; Vega, L. F. Modeling the Self-assembly of Nano Objects: Applications to Supramolecular Organic Monolayers Adsorbed on Metal Surfaces. J. Chem. Theory Comput. 2013, 9, 2161–2169. 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. Arai, N.; Yausoka, K.; Zeng, X. C. Self-Assembly of Triblock Janus Nanoparticle in Nanotube. J. Chem. Theory Comput. 2013, 9, 179–187. 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. Lindgren, E. B.; Derbenev, I. N.; Khachatourian, A.; Chan, H-K.; Stace, A. J.; Besley, E. Electrostatic Self-Assembly: Understanding the Significance of the Solvent. J. Chem. Theory Comput. 2018, 14, 905–915. Lin, H.; Lee, S.; Sun, L.; Spellings, M.; Engel, M.; Glotzer, S. C.; Mirkin, C. A. Clathrate Colloidal Crystals. Science 2017, 355, 931–935. Vanakaras, A. G. Self-organization and Pattern Formation of Janus Particles in Two Dimensions by Computer Simulations. Langmuir 2006, 22, 88–93. Kern, N.; Frenkel, D. Fluid–fluid Coexistence in Colloidal Systems with Short-ranged Strongly Directional Attraction. J. Chem. Phys. 2003, 118, 9882–9889. 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. Romano F.; Sciortino, F. Patterning Symmetry in the Rational Design of Colloidal Crystals, Nature Commun. 2012, 3, 975.


Page 24 of 29

Page 25 of 29 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


(18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36)

Reinhardt, A.; Romano, F.; Doye, J. P. K. Computing Phase Diagrams for a Quasicrystalforming Patchy-particle System, Phys. Rev. Lett. 2013, 110, 255503. Mahynski, N. A.; Panagiotopoulos, A. Z.; Meng, D.; Kumar, S. K. Stabilizing Colloidal Crystals by Leveraging Void Distributions. Nature Commun. 2014, 5, 4472. Newton, A. C.; Groenewold, J.; Kegel, W. K.; Bolhuis, P. G. Rotational Diffusion Affects the Dynamical Self-assembly Pathways of Patchy Particles. Proc. Natl. Acad. Sci. USA 2015, 112, 15308–15313. Reinhart W. F.; Panagiotopoulos, A. Z. Equilibrium Crystal Phases of Triblock Janus Colloids. J. Chem. Phys. 2016, 145, 094505. 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. Reinhart W. F.; Panagiotopoulos, A. Z. Crystal Growth Kinetics of Triblock Janus Colloids, J. Chem. Phys. 2018, 148, 124506. 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. Mao, X.; Chen, Q.; Granick, S. Entropy Favours Open Colloidal Lattices, Nature Mater. 2013, 12, 217. 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. Eslami, H.; Bahri, K.; Müller-Plathe, F. Solid-Liquid and Solid-Solid Phase Diagrams of Self-Assembled Triblock Janus Nanoparticles from Solution. J. Phys. Chem. C 2018, 122, 9235–9244. Warren, P. B. Vapor-Liquid Coexistence in Many-Body Dissipative Particle Dynamics. Phys. Rev. E: 2003, 68, 066702. Eslami, H.; Karimi-Varzaneh, H. A.; Müller-Plathe, F. Coarse-Grained Computer Simulation of Nanoconfined Polyamide-6,6. Macromolecules 2011, 44, 3117–3128. Español, P.; Warren, P. B. Statistical Mechanics of Dissipative Particle Dynamics. Europhys. Lett. 1995, 30, 191−196. Dobnikar, J.; Y. Chen, Y.; Rzehak, R.; von Grünberg, H. H. Many-body Interactions and the Melting of Colloidal Crystals. J. Chem. Phys. 2003, 119, 4971. Driscoll, M.; Delmotte, B.; Youssef, M.; Sacanna, S.; Donev, A.; Chaikin, P. Unstable fronts and motile structures formed by microrollers. Nature Phys. 2017, 13, 375–379. Vanya, P.; Crout, P.; Sharman, J.; Elliott, J. A. Liquid-phase Parametrization and Solidification in Many-body Dissipative Particle Dynamics. Phys. Rev. E 2018, 98, 033310. Laio A.; Parrinello, M. Escaping Free Energy Minima. Proc. Natl. Acad. Sci. USA 2002, 99, 12562−2566. Hawk, A. T.; Konda, S. S.; Makarov, D. E. Computation of Transit Times using the Milestoning Method with Applications to Polymer Translocation. J. Chem. Phys. 2013, 139, 064101. 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.



(37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54)

Rajamani, R.; Naidoo, K. J.; Gao, J. Implementation of an Adaptive Umbrella Sampling Method for the Calculation of Multidimensional Potential of Mean Force for Chemical Reaction in Solution. J. Comput. Chem. 2003, 24, 1775−1781. Major, D. T.; Gao, J. Implementation of the Bisection Sampling Method in Path Integral Simulations. J. Mol. Graph. Model. 2005, 24, 121−127. 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. Dellago, C.; Bolhuis, P. G.; Csajka, F. S.; Chandler, D. Transition Path Sampling and the Calculation of Rate Constants. J. Chem. Phys. 1998, 108, 1964−1977. Cole, D. J.; Tirado-Rives, J.; Jorgensen, W. L. Enhanced Monte Carlo Sampling through Replica Exchange with Solute Tempering. J. Chem. Theory Comput. 2014, 10, 565–571. Khanjari, N.; Eslami, H.; Müller-Plathe, F.; Adaptive-numerical-bias Metadynamics. J Comput Chem. 2017, 38, 2721-2729. Steinhardt, P. J.; Nelson D. R.; Ronchetti, M. Bond-orientational Order in Liquids and Glasses. Phys. Rev. B 1983, 28,784−805. Auer, S.; Frenkel, D. Prediction of Absolute Crystal Nucleation Rate in Hard-Sphere Colloids. Nature 2001, 409, 1020–1023. Trudu, F.; Donadio, D.; Parrinello, M. Freezing of a Lennard-Jones Fluid: From Nucleation to Spinodal Regime. Phys. Rev. Lett. 2006, 97, 105701. Palmer, J. C.; Martelli, F.; Liu, Y.; Car, R.; Panagiotopoulos, A. Z.; Debenedetti, P. G. Metastable Liquid-Liquid Transition in a Molecular Model of Water. Nature 2014, 510, 385−388. Ten Wolde, P. R.; Ruiz-Monterob, M. J.; Frenkel, D. Simulation of Homogeneous Crystal Nucleation Close to Coexistence. Farady Disscuss. 1996, 104, 93−110. Leyssale, J.-M.; Delhommelle, J.; Millot, C. Atomistic Simulation of the Homogeneous Nucleation and of the Growth of N2 Crystallites. J. Chem. Phys. 2005, 122, 104510. 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. Eslami, H.; Sedaghat, P.; Müller-Plathe, F. Local bond order parameters for accurate determination of crystal structures in two and three dimensions. Phys. Chem. Chem. Phys. 2018, 20, 27059−27068. 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. Groot, R. D. Electrostatic Interactions in Dissipative Particle Dynamics Simulation of Polyelectrolytes and Anionic Surfactants. J. Chem. Phys. 2003, 118, 11265. Ibergay, C.; Malfreyt, P.; Tildesley, D. J. Electrostatic Interactions in Dissipative Particle Dynamics: Toward a Mesoscale Modeling of the Polyelectrolyte Brushes. J. Chem. Theory Comput. 2009, 5, 3245–3259.


Page 26 of 29

Page 27 of 29 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


(55) (56) (57) (58) (59) (60) (61) (62) (63) (64)

Vaiwala, R.; Jadhav, S.; Thaokar, R. Electrostatic Interactions in Dissipative Particle Dynamics-Ewald-like Formalism, Error Analysis, and Pressure Computation. J. Chem. Phys. 2017, 146, 124904. 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. Stillinger, F. H.; Lovett, R. Do Variational Formulations for Inhomogeneous Density Functions Lead to Unique Solutions? J. Chem. Phys. 1991, 94, 7353. 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. J. R. Wolters, G. Avvisati, F. Hagemans, T. Vissers, D. J. Kraft, M. Dijkstra, and W. K. Kegel, Self-assembly of “Mickey Mouse” Shaped Colloids into Tube-like Structures: Experiments and Simulations, Soft Matter 2015, 11, 1067−1077. Keys, A. S.; Glotzer, S. C.; How do Quasicrystals Grow? Phys. Rev. Lett. 2007, 99, 235503. Etienne P. Bernard, E. P.; Krauth, W. Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition. Phys. Rev. Lett. 2011, 107, 155704. Engel, M.; Joshua A. Anderson, J. A.; Glotzer, S. C.; Isobe, M.; Bernard, E. P.; Krauth, W. Hard-disk Equation of State: First-order Liquid-Hexatic Transition in Two Dimensions with Three Simulation Methods. Phys. Rev. E 2013, 87, 042134. Romano, F.; Russo, J.; Tanaka, H. Influence of Patch-size Variability on the Crystallization of Tetrahedral Patchy Particles. Phys. Rev. Lett. 2014, 113, 138303.



TOC Graphic


Page 28 of 29

Page 29 of 29 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


Free energy contours for conversion of liquid, Kagome, and hexagonal phases. 253x133mm (300 x 300 DPI)

ACS Paragon Plus Environment

Self-assembly mechanisms of triblock Janus particles - Journal of

Recommend Documents