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J. Phys. Chem. B 2008, 112, 13842–13848
Ordered Packing of Soft Discoidal System Zhan-Wei Li, Li-Jun Chen, Ying Zhao, and Zhong-Yuan Lu* Institute of Theoretical Chemistry, State Key Laboratory of Theoretical and Computational Chemistry, Jilin UniVersity, Changchun 130023, China ReceiVed: May 17, 2008; ReVised Manuscript ReceiVed: August 21, 2008
A novel mesoscopic simulation method is adopted to study the ordered packing of the anisotropic disklike particles with a soft repulsive interaction, which possesses a modified anisotropic conservative force type used in dissipative particle dynamics. We examine the influence of the shape of the particles, the angular width of the repulsion, and the strength of the repulsion on the packing structures. Specifically, an ordered hexagonal columnar structure is obtained in our simulations. Our study demonstrates that an anisotropic repulsive potential between soft discoidal particles is sufficient to produce a relatively ordered hexagonal columnar structure. 1. Introduction Self-assembly is a governing principle by which materials form. The patterns arising from self-assembly are ubiquitous in nature, from the opalescent inner surface of the abalone shell to the internal compartments of a living cell.1,2 Great progresses had been made in studying the self-assembly of particles. The self-assembly of patchy particles is a representative and important example in this aspect, especially due to their complex assembly structures via anisotropic interactions between the patchy particles. For example, Glotzer and co-workers studied the self-assembly of patchy particles through molecular simulation and obtained chains, sheets, rings, icosahedra, square pyramids, tetrahedra, and twisted and staircase structures.2 Furthermore, they showed that patchy particles are capable of assembling from an initially disordered state into a diamond structure.3 Sciortino and co-workers studied the self-assembly of patchy particles into polymer chains4 and reported theoretical and numerical evaluations of the phase diagram for patchy colloidal particles of new generation.5 A recent report suggested that soft patchy particles can be prepared in solution via self-assembly of diblock copolymers.6 It implies that the self-assembly of patchy particles is actually related to the hierarchical assembly of block copolymers, which have been the focus of a lot of research.7-32 In experiments, various micelles and micelle complex structures had been obtained, for example, spheres, cylinders, bilayers, rodlike micelles, threadlike micelles, wormlike micelles, multilayer core-shell-corona spheres, and so on. Specifically, several studies reported the formation of core-corona disklike micelles and multilayer core-shell-corona disklike micelles,7,10-19,30,32 which are actually special examples of patchy particles due to their anisotropic interactions and shapes. These disklike micelles may be formed from ABC miktoarm star block terpolymers or flexible coil polymers as a consequence of the superstrong segregation regime,12-14 from diblock copolymers as a result of core block crystallization,16 from rod-coil copolymers,17,32 or from fluorination of the butadiene block copolymers.10,11 Moreover, Whitesides and Boncheva suggested that selfassembly is not limited to molecules; nonmolecular self* To whom correspondence should be addressed. E-mail: luzhy@ mail.jlu.edu.cn. Phone: 0086 431 88498017. Fax: 0086 431 88498026.
assembly offers a very promising route for making crystals of nanometer- and micrometer-scale components, and it may provide a way of assembling electrically or optically functional components.33 Nevertheless, to date, only a limited number of papers are available regarding further packing or self-assembly of disklike micelles, for example, into more advanced structures (called the secondary self-assembly).7,13,18 Further understanding of the self-assembly of the disklike micelles may benefit from the successful experiences in liquid crystals,34-49 dipolar particles,50-56 and disklike particles.57-63 For example, the Corner potential34,35 and the Gay and Berne (GB) potential36-39 have proved to be valuable for computer simulations of the anisotropic interactions in liquid crystal systems. A modified Lennard-Jones potential has also been adopted to model the phase behavior of reversibly assembled polymers.40 Atomistic molecular dynamics simulations were successfully employed to study the columnar phase formation by the disk-shaped molecules.58-60 Similarly, Wales and coworkers reported a helical columnar phase formed in a system of disklike ellipsoidal particles.62,63 However, Kremer et al. suggested that complete and quantitative description of the columnar mesophases is not really provided in atomistic molecular dynamics simulations because of the very small time and length scales they can reach.59 Therefore, a great deal of works were to serve as an input to coarse-graining techniques.64-66 It should be noted that computer simulations were also adopted to study other types of micellar packing into crystals, for example, by Travesset and co-workers, who provided a detailed investigation of self-assembled micellar crystals using molecular dynamics simulations.67 Their results led to a number of theoretical considerations and suggested a range of implications to experimental systems. The disklike micelle actually possesses the structure with a relatively hard core surrounded by a soft repulsive corona; thus the potential types originating from Lennard-Jones may not be suitable to describe the interactions between them. Especially, the attractive part of the Lennard-Jones potential is obtained via quantum chemistry to describe the interactions between two atoms with permanent dipoles.68 Therefore, a purely anisotropic soft potential is needed to model the self-assembly of these disklike micelles which consist of many atoms.
10.1021/jp804372s CCC: $40.75 2008 American Chemical Society Published on Web 10/11/2008
Ordered Packing of Soft Discoidal System
J. Phys. Chem. B, Vol. 112, No. 44, 2008 13843
Dissipative particle dynamics (DPD) is a successful technique to study the structure of soft condensed matters.69 Thus, in this work, we try to develop a modified anisotropic potential which is soft and repulsive in the form of DPD. With the modified potential, we perform mesoscopic simulations to study the ordered packing of soft discoidal systems, which represent the core-corona disklike micelles, the core-shell-corona disklike micelles, and so on. This mesoscopic simulation technique can be used to study the systems over greater length and time scales which are not accessible in traditional molecular dynamics (MD) simulations.27 We examine the influence of the shape of the particles, the angular width of the repulsion, and the strength of the repulsion on the packing structures. Interestingly, an almost defect-free hexagonal columnar structure is obtained in our simulations with such a purely repulsive soft potential. This structure is further analyzed in detail.
gi⊥ ) gi - (gi · ni)ni
(6)
with
gi )
Rij (nj · rij)rij (1 - rij)2µνf ν-1 2 r2
(7)
ij
The second term in eq 5 corresponds to the force λni along the vector ni, which constrains the vector length to be a constant of the motion. The quantity λ can be thought of as a Lagrange multiplier. The equations of both translational motion and rotational motion are integrated via a half-step leapfrog algorithm.40,70,71 In the rotational motion, a full step in the integration algorithm is advanced through the equation
2. Models and Simulation Details In the simulations, we adopt an anisotropic repulsive potential40 on the basis of the conservative potential in DPD.69 It is expressed as
Uij ) (1 - µf ν)
Rij (1 - rij)2 2
(1)
1 1 ui t + δt ) ui t - δt + δtgi⊥(t)/I 2 2
(
)
(
)
1 2 ui t - δt · ni(t) ni(t) (8) 2
[(
]
)
The step is completed using
1 ni(t + δt) ) ni(t) + δtui t + δt 2
(
where
f)
(ni · rij)(nj · rij)
(2)
rij2
and ni and nj are unit vectors assigning the orientation to particles i and j, respectively. The anisotropic factor f is unity if both ni and nj are parallel to rij (the vector connecting the particles i and j). If either ni or nj or both are perpendicular to rij, then f ) 0. If either ni or nj is antiparallel to rij, then fν ) -1 for ν odd, and fν ) 1 for ν even. Here, the magnitude of µ controls the shape of the particles, ν controls the angular width of the repulsion, and Rij controls the strength of the repulsion. The anisotropic conservative force between two particles is then given by
rij Rij + (1 - rij)2µνf ν-1 × rij 2rij (nj · rij)ni rij (ni · rij)nj + - 2f (3) rij rij rij
Fij ) Rij(1 - rij)(1 - µf ν)
[
]
The translational displacements of the particles follow the Newton’s equations of motion. The equations of rotational motion can be written as two first-order differential equations70
n˙i ) ui
(4)
u˙i ) gi⊥ /I + λn˙i
(5)
where I is the moment of inertia. Equation 4 simply defines ui as the time derivative of the orientation ni. Physically, the first term in eq 5 corresponds to the force g⊥i , the perpendicular component of gi, responsible for rotation of the particle, where
)
(9)
The simulations are performed in NVT ensemble. The weak coupling Berendsen thermostat, instead of the DPD thermostat, is used in the simulations to control the system temperature at the target value.71 We simulated systems of 24000 particles in a 20 × 20 × 20 cubic box with periodic boundary conditions. Here, we use the interaction cutoff radius as our unit of length, rc ) 1, and choose I ) m ) kBT ) 1. A time step of δt ) 0.002 is used. The total simulation steps are different, ranging from 4 × 106 to 8 × 106 depending on the reaching of the equilibrium. All calculations are performed with the in-housedeveloped parallel DPD code on an Intel Q6600 CPU. 3. Results and Discussion 3.1. Influence of µ and ν on the Packing Structures. For the anisotropic repulsive potential in eq 1, if we define that θi is the angle between the direction vector ni of particle i and the interparticle vector rij and θj is the angle between the direction vector nj of particle j and the interparticle vector rij, then eq 1 can be rewritten as
Uij ) [1 - µ(cos θi cos θj)ν]
Rij (1 - rij)2 2
(10)
In order to fully understand the anisotropic repulsive potential presented in eq 10, we show the dependence of Uij on the interparticle distance for various particle configurations in Figure 1, keeping Rij ) 95, ν ) 6, and µ ) 0.7. If we choose that either θi or θj or both are equal to π/2, then f ) 0 and eq 10 becomes the standard DPD potential Uij ) (Rij/ 2)(1-rij)2. These particle packing configurations are always energetically unfavorable. If θi ) θj ) 0, that is, the particles are face-to-face packed, this particle packing configuration possesses a comparatively smaller potential energy at a given distance, as shown in Figure 1.
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Figure 1. Distance dependence of the anisotropic repulsive potential Uij with Rij ) 95, ν ) 6, and µ ) 0.7 for various configurations: θi ) 0, θj ) 0 (solid line) and θi ) 0, θj ) π/2; θi ) π/2, θj ) 0; or θi ) π/2, θj ) π/2 (dashed line).
Figure 3. Time evolution of the orientation order parameter Q with Rij ) 95 for (a) different µ while keeping ν ) 6 and for (b) different ν while keeping µ ) 0.7.
Figure 2. Angle dependence of the anisotropic repulsive potential Uij with Rij ) 95 for different µ and ν: µ ) 0.7, ν ) 6 (solid line); µ ) 0.7, ν ) 4 (dotted line); and µ ) 0.6, ν ) 6 (dashed line).
We then choose Rij ) 95, rij ) 0.4, θj ) 0, and vary θi to study the angle dependence of the anisotropic repulsive potential. The results are shown in Figure 2. It can be clearly seen that the particle packing configurations with θi ) π/2 should be unstable because they possess higher potential energies than the face-to-face packing configuration with θi ) 0 or π. µ actually controls the shape of the particles; the larger value of µ corresponds to the lower potential energy at θi ) 0 or π. The disklike particle is much flatter when the value of µ is larger. ν controls the angular width of the repulsion; the larger the value of ν, the wider the angular width of forming the unstable packing configurations. Thus, the larger value of ν favors the face-toface interparticle packing. In order to describe the influence of µ and ν on the packing structures of the disklike soft particles quantitatively, we calculate the orientation order parameter Q,40 expressed as
Q)
2 (N - 1)N
〈∑ ∑ i
j>i
〉
1 (3cos2 θij - 1) 2
(11)
Here, θij is the angle between ni and nj. Note that θij ) 0 corresponds to Q ) 1, whereas a random (isotropic) orientation of ni results in Q ) 0. First of all, we choose Rij ) 95 and ν ) 6 to analyze the influence of the parameter µ on the packing structure. Figure 3a shows the time evolution of the orientation order parameter
Q for different values of µ. For µ e 0.5, the packing structure is always isotropic, that is, Q ) 0. For µ ) 0.6 and 0.7, the particles begin to pack into ordered structures, and the equilibrium is reached after about 3 × 106 time steps. As shown in Figure 3a, the value of Q for µ ) 0.7 after equilibrium is larger than that for µ ) 0.6 since the potential energy for face-to-face particle packing is lower in the former case. However, when µ is even larger, the ordered packing structures cannot be found, as shown in Figure 3a. For µ ) 0.8, 0.9, and 1.0, Q equals 0. These larger values of µ correspond to very flat disklike particles, between which the repulsion is much soft, and the rotation of the disklike particle is easier. Furthermore, as common in standard DPD, the very soft interparticle repulsion may cause interpenetration between the particles. Thus, if the value of µ is too large, the ordered packing structures will not be obtained in the simulations due to the soft nature of the repulsion. We also examine the influence of the parameter ν on the packing structures of the soft disklike particles, keeping Rij ) 95 and µ ) 0.7. As shown in Figure 3b, if ν is odd, the orientation parameter Q is 0 and the system is isotropic. As expected, for ν ) 2, 4, and 6, the larger ν is, the higher the degree of ordered packing is. However, when ν equals 8, Q is abnormally small. This is because the large ν leads to the narrow angular width of packing into ordered structure, as can be seen in Figure 2. As a consequence, in the simulations, the particles with ν ) 8 begin to pack in local domains rapidly and finally form two blocks of hexagonal columnar structures which possess unidentical domain orientation orders between each other. These two blocks of hexagonal columnar structures are kinetically trapped and cannot develop into a global hexagonal columnar structure in our simulations. Figure 4 shows the snapshots of the packing structures with different values of ν. These snapshots are generated by Qt-based Molecular Graphics Application (QMGA).72 Figure 4a and b shows the snapshots for ν ) 2 and 6 after reaching the
Ordered Packing of Soft Discoidal System
J. Phys. Chem. B, Vol. 112, No. 44, 2008 13845
Figure 5. (a) Time evolution of the orientation order parameter Q for different Rij while keeping µ ) 0.7 and ν ) 6. (b) The average orientation order parameter 〈Q〉 as a function of Rij.
Figure 4. Snapshots of the simulated systems in equilibrium with Rij ) 95 and µ ) 0.7 but with different ν: (a) ν ) 2, (b) ν ) 6, and (c) ν ) 8. For the sake of clarity, we only show a quarter of the particles in the systems.
equilibrium, respectively. For ν ) 2, only some of the particles pack into ordered structure, whereas for ν ) 6, almost all of the particles hexagonally pack into ordered structure. The snapshot for ν ) 8 is shown in Figure 4c. We can clearly observe two blocks of hexagonal columnar structures with different orientation directors. Such a trapped structure cannot further develop into a better hexagonal columnar structure in our simulations. Since for ν ) 6 we can obtain an ordered hexagonal columnar structure easily, we choose µ ) 0.7 and ν ) 6 in the following simulations unless otherwise indicated. 3.2. Influence of rij on the Packing Structures. In the simulations, we find that a proper choice of the repulsion strength Rij is very important to observe the highly ordered hexagonal columnar structures. To analyze the influence of Rij on the packing structures, we focus on the time evolution of the orientation order parameter Q as well as its average in equilibrium for different Rij, as shown in Figure 5. Figure 5a shows the time evolution of Q for different Rij; each of the
Figure 6. (a) Time evolution of the hexagonal order parameter Ψ6 for different Rij while keeping µ ) 0.7 and ν ) 6. (b) The average hexagonal order parameter 〈Ψ6〉 as a function of Rij.
simulations are started from the same initially isotropic configuration. Normally, 4.0 × 106 time steps are enough to reach equilibrium. However, for some higher values of Rij, such as Rij ) 105, 110, and 115, longer time steps of simulations are needed to reach equilibrium, as can be seen in Figure 5a. In Figure 5b, we present the average orientation order parameter 〈Q〉 after reaching equilibrium as a function of Rij. For Rij e 70, the orientation distributions of the disklike particles are isotropic. It can be attributed to the fact that the particle is very soft when Rij is small; as a consequence, it can rotate easily, and no ordered packing is preferred. For Rij ) 75 and 85, we find that hexagonal columnar structures coexist with some relatively large defects. Their typical snapshots obtained after equilibrium are similar to the one shown in Figure 4a. It implies that with increasing Rij, the disklike particles are not easy to rotate anymore and the locally ordered packing structures can be found. However, since the driving force of packing is relatively small when Rij ) 75 and 85, we still can observe a lot of defects in the systems. For further increasing Rij, the
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average orientation order parameter 〈Q〉 becomes larger, as shown in Figure 5b. Their typical snapshots obtained after equilibrium are similar to the one shown in Figure 4b. It reflects that the driving force of ordered packing is enhanced when Rij is large enough; thus, most of the disklike particles are finally packed into ordered hexagonal columnar structures. It should be noted that, although 〈Q〉 is large when Rij g 105, much longer time steps of simulations are required to reach equilibrium, as can be seen in Figure 5a. This may be due to the kinetic slowing down effects of the high-energy barrier from large values of Rij between the isotropic and the ordered packing states. When the value of Rij is even larger, for example, Rij ) 150, the average orientation order parameter 〈Q〉 becomes small. At these large values of Rij, we cannot observe global ordered packing structures in the simulations. It reflects that the kinetic trapping is dominant when Rij is very large, although the hexagonal columnar structures are favored in equilibrium. To examine the influence of Rij on the hexagonal columnar structure, we also calculate the hexagonal order parameter Ψ6, expressed as34,40,42,58,60
Ψ6 )
|〈
1 N
〉|
∑ n1j ∑ wkl exp(6iθkl) b 〈kl〉
j
(12)
where nbj is the number of pairs of nearest neighbors of the jth particle, 〈kl〉 implies a sum over all possible pairs of neighbors, and θkl is the angle between the unit vectors along the projections of the interparticle vectors between particle j and its neighbors k and l onto a plane perpendicular to the director. wkl ) 1 if rjk and rjl lie within a cylinder of diameter 2 and thickness 2 centered at particle j, and it is zero otherwise. This range is chosen such that only particles in the first coordination shell contribute to the sum. Ψ6 takes unity for a phase with a perfect hexagonal order, zero for the isotropic phase, and a value between 0 and 1 for a phase with intermediate hexagonal order. The knowledge of the director orientation n is required to calculate the hexagonal order parameter Ψ6. However, the director is not known a priori in the computer simulations. Actually, the director fluctuates during the evolution of the system. The orientation of the director for a given configuration can be calculated from a second-rank tensor34,35,39,41,42,50,58-60 N
ΩRβ )
∑
1 i i 1 (3u u - δRβ) N i)1 2 R β
(13)
where R, β ) x, y, z and uRi is the direction cosine of the unit vector describing the orientation of particle i. The Ω tensor is calculated per 7 × 104 time steps, accumulated for 30 consecutive time steps, and then averaged and diagonalized. The eigenvectors of the diagonalized Ω tensor are used, and the one associated with the largest eigenvalue is identified as the director. The frequency of eigenvector determination depends on the frequency of sampling of the structural properties. Knowing the director orientation n, we can immediately calculate the hexagonal order parameter Ψ6 at a given time step. Figure 6a shows the time evolution of the hexagonal order parameter Ψ6 for different Rij, and Figure 6b shows its average in equilibrium 〈Ψ6〉 as a function of Rij. For Rij e 70, Ψ6 is nearly equal to 0, and we cannot observe ordered packing structures. For Rij ) 75 and 85, the hexagonal order parameter Ψ6 becomes larger, where we obtain hexagonal columnar structures with relatively large defects. When Rij g 85, the
Figure 7. Snapshots of the systems with Rij ) 95, µ ) 0.7, and ν ) 6, after (a) 0, (b) 1.4 × 106, and (c) 4 × 106 time steps of simulation. For the sake of clarity, we only show a quarter of the particles in the systems.
hexagonal order parameter Ψ6 becomes larger, and the defects of the hexagonal columnar structure become less and less. However, with further increasing of Rij when Rij g 115, the hexagonal order parameter Ψ6 turns to be smaller. In the case of Rij ) 150, the system is always isotropic during the simulation. The results shown in Figure 6 are consistent with those in Figure 5. The Rij dependence of the hexagonal order parameter Ψ6 can be also explained with similar reasons as those used for that of the orientation order parameter Q. 3.3. Analysis of the Hexagonal Columnar Structure in Detail. In the case of Rij ) 95, shorter time steps of simulations are needed to equilibrate the system, and a relatively ordered hexagonal columnar structure can be obtained in equilibrium. Thus, in the following, we will study the hexagonal columnar structure in detail while keeping Rij ) 95. Typical snapshots during the evolution of the hexagonal columnar structure starting from an initially isotropic phase
Ordered Packing of Soft Discoidal System
J. Phys. Chem. B, Vol. 112, No. 44, 2008 13847 and perfectly ordered systems. On our simulation time scales, the relatively ordered hexagonal columnar structure with defects cannot relax to the perfect structure due to kinetic trapping. In order to characterize the structure of the hexagonal columnar phase, we calculate three positional correlation functions, g(r), g⊥(r⊥), and g|(r|).34,35,39,41,42,50,52,58-60 The first one is the orientationally averaged pair distribution function g(r), which provides information about the probability of finding a particle at a distance r from the one at the origin
g(r) )
1 4πNFr2
〈
N
N
i
j*i
〉
∑ ∑ δ(r - rij)
(14)
It enables us to distinguish between the ordered and the disordered phases easily. The second distribution function is the perpendicular radial distribution function g⊥(r⊥), which gives information about the arrangement of particles in planes perpendicular to the director n
g⊥(r⊥) )
1 2πltNFr⊥
〈
N
N
i
j*i
〉
∑ ∑ δ(r⊥ - rij,⊥)
(15)
where |n · rij| < lt/2, lt ) 2 is the slice thickness of the particles considered, and rij,⊥ ) (|rij|2 - |n · rij|2)1/2. We have also calculated the distribution function parallel to the director g|(r|), which is sensitive to the arrangement of the disklike particles in layers
g|(r|) ) Figure 8. The radial distribution functions: (a) g(r), (b) g⊥(r⊥), (c) g|(r|).
are shown in Figure 7, which is also generated by QMGA.72 Figure 7a shows the isotropic phase in the beginning of the simulation. Figure 7b shows the packing structure of the disklike particles after a 1.4 × 106 time steps simulation. We can clearly observe the typical hexagonal columnar structure, but with a lot of defects. After about 4.0 × 106 time steps, a relatively ordered hexagonal columnar structure is obtained, as shown in Figure 7c. The defects in Figure 7c may be mainly dominated by the kinetic trapping due to the presence of energy barriers. Here, to determine the average energy per particle of a perfectly ordered system, we construct a perfect packing by growing columns along the 〈111〉 direction in the simulation box and adjusting the spacing between the particles to arrive at the box size of 20 × 20 × 20. After 1.8 × 106 time steps simulation, a well-ordered hexagonal columnar structure is obtained (as shown in Supporting Information Figure S1), for which the average energy per particle is 13.525, in contrast to the average energy per particle of 13.805 for the structure as shown in Figure 7c. The perfect hexagonal columnar structure is indeed lower in energy. Moreover, we can see that the initially disorderd system does not relax to perfect order, and the initially perfectly ordered system does not sample a relatively ordered state in the allowed simulation time. Therefore, we speculate that a energy barrier exists between the relatively ordered
1 lxlyNF
〈∑ ∑ N
N
i
j*i
〉
δ(r| - n · rij)
(16)
where lx and ly are the box dimensions along X and Y axes, respectively. The director n is calculated from eq 13. For all of these distribution functions, δ is the Dirac delta function, and a histogram with a width of 0.01 is constructed of all pair separations falling within such a range. The calculated radial distribution functions are shown in Figure 8 for the system with Rij ) 95 in equilibrium. Typically, a doublet in the pair distribution functions g(r) and g⊥(r⊥) can be taken as the signature of a hexagonal arrangement of the particles.34,42 As can be seen in Figure 8a and b, a clear doublet between 1.4 and 2 appears, which is the evidence of the hexagonal columnar structure. The perpendicular radial distribution function g⊥(r⊥), as shown in Figure 8b, possesses a large peak at around r ) 0, which is attributed to the packing of the particles in the same column.52,58,59 Moreover, we can approximately estimate the aspect ratio (L/D) of the disklike particles in the system with Rij ) 95 using the pair distribution functions g(r) and g⊥(r⊥). The first peak located at 0.36 as shown in Figure 8a is due to the face-to-face packing of the particles along the director. Thus, the thickness of the disklike particle L is roughly taken as 0.36. Similarly, we can obtain the diameter of the disklike particle D through the second peak (at 0.95) in Figure 8a and through Figure 8b, which corresponds to the sideby-side packing configurations of disklike particles. Thus, the aspect ratio (L/D) of the disklike particles is about 0.38. Figure 8c shows the radial distribution function parallel to the director g|(r|), which indicates a uniform mass distribution along the director. Therefore, there is not layered structure in the system.
13848 J. Phys. Chem. B, Vol. 112, No. 44, 2008 Similar to the results of ref 42, the fall in g|(r|) at higher separations is due to the lack of particles at high separations when sampling in a plane perpendicular to the director since such a plane is not parallel to the XY surface. We have describe the hexagonal columnar structure quantitatively by the hexagonal order parameter Ψ6, as shown in Figure 6. With the help of Ψ6 and the distribution functions g(r), g⊥(r⊥), and g|(r|), we can easily identify the hexagonal columnar structure from the equilibrium packing of the soft discoidal particles. 4. Conclusions In summary, we have presented a modified anisotropic conservative potential which is soft and repulsive in the form of DPD, by which we can model the ordered packing of the soft discoidal systems. Actually, the anisotropic disklike particles with a soft repulsive interaction correspond to the systems such as the core-corona disklike micelles and the core-shell-corona disklike micelles. By examining the influence of the shape of the particles, the angular width of the repulsion, and the strength of the repulsion on the packing structures, we can obtain ordered hexagonal columnar structures in equilibrium in some specific conditions. Here, we have adopted a novel mesoscopic simulation method that can be used to study the soft anisotropic systems over greater length and time scales efficiently. Our study demonstrates that an anisotropic repulsive potential between soft discoidal particles is sufficient to produce a relatively ordered hexagonal columnar structure. Acknowledgment. This work is supported by NSFC (20490220, 20774036) and Fok Ying Tung Education Foundation. Supporting Information Available: The snapshot of the well-ordered hexagonal columnar structure. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Glotzer, S. C. Science 2004, 306, 419. (2) Zhang, Z.; Glotzer, S. C. Nano Lett. 2004, 4, 1407. (3) Zhang, Z.; Keys, A. S.; Chen, T.; Glotzer, S. C. Langmuir 2005, 21, 11547. (4) Sciortino, F.; Bianchi, E.; Douglas, J. F.; Tartaglia, P. J. Chem. Phys. 2007, 126, 194903. (5) Bianchi, E.; Largo, J.; Tartaglia, P.; Zaccarelli, E.; Sciortino, F. Phys. ReV. Lett. 2006, 97, 168301. (6) Srinivas, G.; Pitera, J. W. Nano Lett. 2008, 8, 611. (7) Cui, H.; Chen, Z.; Zhong, S.; Wooley, K. L.; Pochan, D. J. Science 2007, 317, 647. (8) Pochan, D. J.; Chen, Z.; Cui, H.; Hales, K.; Qi, K.; Wooley, K. L. Science 2004, 306, 94. (9) Lodge, T. P.; Bang, J.; Park, M. J.; Char, K. Phys. ReV. Lett. 2004, 92, 145501. (10) Ren, Y.; Lodge, T. P.; Hillmyer, M. A. Macromolecules 2001, 34, 4780. (11) Ren, Y.; Lodge, T. P.; Hillmyer, M. A. Macromolecules 2002, 35, 3889. (12) Zhou, Z.; Li, Z.; Ren, Y.; Hillmyer, M. A.; Lodge, T. P. J. Am. Chem. Soc. 2003, 125, 10182. (13) Li, Z.; Kesselman, E.; Talmon, Y.; Hillmyer, M. A.; Lodge, T. P. Science 2004, 306, 98. (14) Lodge, T. P.; Hillmyer, M. A.; Zhou, Z.; Talmon, Y. Macromolecules 2004, 37, 6680. (15) Li, Z.; Hillmyer, M. A.; Lodge, T. P. Macromolecules 2006, 39, 765. (16) Richter, D.; Schneiders, D.; Monkenbusch, M.; Willner, L.; Fetters, L. J.; Huang, J. S.; Lin, M.; Mortensen, K.; Farago, B. Macromolecules 1997, 30, 1053. (17) Wu, J.; Pearce, E. M.; Kwei, T. K.; Lefebvre, A. A.; Balsara, N. P. Macromolecules 2002, 35, 1791. (18) Zhu, J.; Jiang, W. Macromolecules 2005, 38, 9315. (19) Weiss, T. M.; Narayanan, T.; Wolf, C.; Gradzielski, M.; Panine, P.; Finet, S.; Helsby, W. I. Phys. ReV. Lett. 2005, 94, 038303.
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