Dissipative Particle Dynamics Simulation of the Phase Behavior of T

Article pubs.acs.org/JPCB

Dissipative Particle Dynamics Simulation of the Phase Behavior of T‑Shaped Ternary Amphiphiles Possessing Rodlike Mesogens Xiaohan Liu, Keda Yang, and Hongxia Guo* Beijing National Laboratory for Molecular Sciences (BNLMS), State Key Laboratory of Polymer Physics and Chemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100080, China S Supporting Information *

ABSTRACT: We employed dissipative particle dynamics simulations to explore the phase behavior of T-shaped ternary amphiphiles composed of rodlike cores connected by two incompatible end chains and side grafted segments. By fine-tuning the number of terminal and lateral beads, three phase diagrams for the model systems with different terminal chain lengths are constructed in terms of temperature and lateral chain length, which have some common features and mostly compare favorably with experimental studies with the exception a couple of new phases. It is worthwhile to highlight that the mixed cylindrical phase and the perforated layer phase, as the experimentally observed mesophases exclusive for facial amphiphilies, are found in simulations for the first time. Also, a novel gyroid structure is observed in series of T-shaped ternary amphiphiles for the first time. Furthermore, by evaluating the effective volume fraction of lateral chains, the phase sequence spanning from conventional smectic layer phase via perforated layer structures and polygonal cylindrical arrays to novel lamellar mesophase is established, which is not just qualitatively consistent with the related experimental findings but even the stability windows of some mesophases quantitatively correspond well to experimental results. The success of reproducing the in-plane ordering of rods in the lamellar phase as well as the generic phase diagram of such T-shaped ternary amphiphiles in great detail implies that our genetic model qualitatively captures many of the characteristics of the phase behavior of real T-shaped molecules and could serve as a satisfactory basis for further exploration of self-organization in other related soft matter systems.

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stiff segments.6 As predicted by theories and simulations7,8 and confirmed by experiments,9,10 the asymmetrical amphiphiles/ copolymers possessing both chain flexibility and segmental rigidity can successfully self-organize into a complex variety of fascinating structures, such as arrowhead, zigzag, wavy lamellae, perforated lamellae with tetragonal or hexagonal lattices, hexagonal cylinders, and so on. Further, the nonlinear supramolecular topology, such as star,11 comb,12 cyclic copolymers,13 and dendrimers,14 has been demonstrated to be a viable parameter to trigger the formation of novel mesophases. Specifically, incorporating the rigid building block into the nonlinear supramolecule not only provides an efficient access to create diverse architectures of the individual amphiphiles but also offers a more remarkable control over the cooperative self-assembly process on the nanoscale. Nowadays, depending on the position and number of substituent groups grafted to the rigid backbones, a wide range of functional nonlinear supramolecules such as T-shaped,15,16 H-shaped,17 π-shaped,18 and X-shaped19 are synthesized, and considerable efforts are being made to study their assembly behavior.

INTRODUCTION For recent decades, the self-assembly of the amphiphilic molecules has received considerable attention because of its many applications in nanotechnology1,2 and exciting potential for exploring the design principles for advanced materials.3 To fabricate the complex and highly ordered nanoscale or mesoscale structures, the practical strategy is to build molecules with the appropriate building blocks so that they can organize in a preferred route that the specific block favors. This assembly behavior is often reckoned as “designer assemblies”4 or “bottomup” design.5 A prime task of realizing this goal is to understand the fundamental mechanism that motivates the spontaneous formation of various ordered morphologies. A complete description of the phase behavior for amphiphiles will depend on the thermodynamic condition and a large set of molecular variables, including chemical composition, molecular rigidity/ flexibility and supramolecular architecture and so on. Accordingly, it is crucial to systematically study the influence of various parameters on the phase behavior, and to determine the relevant mechanisms for the self-organization. Among various amphiphilic molecules, rod−coil amphiphiles are receiving special attention because their assembly is then governed not just by the microphase separation of the rod and coil components but also by the orientational alignment of the © 2013 American Chemical Society

Received: June 8, 2013 Revised: June 27, 2013 Published: June 28, 2013 9106

dx.doi.org/10.1021/jp405677u | J. Phys. Chem. B 2013, 117, 9106−9120

The Journal of Physical Chemistry B

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interesting simulations have been carried out to predict the complex ordered structures of bolaamphiphiles26−29 while the phase behavior of facial liquid crystals has remained less explored, even though quite exciting and distinguished mesophases are found experimentally for facial subtype of TLCs. For instance, Crane et al.26 performed coarse-grained (CG) molecular dynamics (MD) simulations to study the spontaneous formation of microstructures in the bolaamphiphile bulk. They established the global phase diagram, which qualitatively reproduces many experimental observations. At a given rod length, series of honeycomb cylinders and novel lamellar phases with the possible inplane long-range positional ordering are formed as a function of the side chain length. However, a full understanding of the principles underlying TLCs’ assembly is not achieved. Later, Bates and Walker27 developed a comparable CG model with softrepulsive nonbonded interactions and applied it to examine the phase diagram of T-shaped bolaamphiphiles. Also, their dissipative particle dynamics simulations reproduced many results found experimentally, i.e., the sequence of square, hexagonal columnar phases and a novel lamellar structure with increasing lateral length. Notably, Bates set a rather small repulsion parameter between terminal beads. While little is known of whether or not this small parameter value is the essential aspect to achieve the above-mentioned mesophases, and how the repulsion parameters between terminal-terminal beads affect the topology of mesophases is also not clear. Nevertheless, this rather small repulsion between terminal beads weakens the excluded volume interactions of the terminal beads, which may distort the excluded volume “shape” of the rod block buried between the two terminals. Very recently, Glozter et al.30 studied the importance of the rod block length for the topology change in self-assembled microstructures. Even though the common feature like the relatively weak repulsions between terminal beads in the above-discussed CG models is not included in their studied laterally tethered nanorods, the same qualitative behavior such as sequence of square and hexagonal honeycomb-like cylindrical mesophases and a novel lamellar phase with rod moieties adopting in-plane smectic order, is reproduced through changing the rod length. Consequently, a comparison with Bates’s observations clearly reveals that the composition of the TLC molecule is a dominant factor to modulate the complex variety of mesophases, and the relatively weak repulsions between terminal beads, which Glozter’s model ignores, are not absolutely essential for the topology changes in self-assembled microstructures. Note that, although changing different building block lengths can broaden the windows to tune the morphologies and may lead to diverse new structures, preliminary simulations have only explored the phase behavior of bolaamphiphiles via varying the number of lateral chain beads with invariable terminal bead number and varying end chain length is only utilized in the laboratories to study the polymorphism of facial amphiphiles.31 In order to fine-tune complex self-assembled structures in TLCs, in this paper we focus our attention to understand how controlling the sizes of both terminal and lateral substituents affects the complex interplay of the entropic and enthalpic incompatibility of the system and in turn tunes the morphology formation of complex phases. By setting up a “truly” minimal model universal for TLCs, we systematically study for the first time the influences from the above two effect factors on the final morphologies, which not only deepens our understanding on the physics underlying the self-assembly process but also provides valuable guidance to achieve the morphology control of the

As a unique type of functional nonlinear amphiphilic supramolecules, T-shaped ternary liquid crystals (TLCs) have recently attracted more significant attention due to their novel self-assembled microstructures20 and diverse industrial applications, such as display device,21 tunable lasers, smart fluids, and semiconductors.22 This type of amphiphilic molecule contains a calamitic core, which is linearly connected by flexible chains at both ends, and an extra incompatible lateral chain is attached to the center of the rigid unit. There are two structural isomers denoted as bola- and facial amphiphiles. Bolaamphiphile, as shown in Figure 1a, is essentially a rigid triphenyl core with two

Figure 1. (a) A T-shaped bolaamphiphile based on a triphenylene core with two terminal polar 1,2-diol units and laterally attached semiperfluorinated chains reprinted from ref 24 and (b) a T-shaped facial amphiphile consisting of a rodlike p-terphenyl unit with two alkyl chains at both ends, and laterally attached an oligo(oxyethylene) spacer and a polar group (molecule 2 in ref 49). (c) The T-shaped coarse-graining model used in our simulation, where the rodlike core is consisted of three beads and the length of other segments is varying. Herein we only depict compound T2R3L3 with NT = 2 and NL = 3.

polar terminal groups and a lateral alkyl23 or semiperfluorinated chain.24 Akin to a bolaamphiphile, the facial molecule contains similar segments but only switches the positions of terminal and lateral segments,25 which is illustrated in Figure 1b. The phase behavior of TLCs is usually considered as a function of lateral chain length while keeping the lengths of other segments fixed. For a small lateral group, the TLCs exhibit the conventional smectic A (SmA) phase. Upon increasing the lateral chain length, series of cylinders with various cross-sectional shapes are obtained. With a further growth of the lateral chain length, a novel lamellar structure in which both rigid rods and flexible termini lie in the same stack units is likely to be formed. Although the experiments have revealed a rich variety of selfassembled superstructures for TLCs, direct observations of the self-organized morphologies on a molecular level in situ are very hard to realize with current experimental approaches. On the other hand, computer simulations have proved to be an excellent complementary tool for understanding the formation of selfassembled structures on the molecular scale and accessing the local conformation in more details. Very recently, several 9107



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However, the repulsion between rigid rodlike cores is preset to be relatively small as aRR = 15, which is representative of the effects of π−π stacking for triphenylene in the real T-shaped amphiphilic system. Additionally, the cutoff distance and bead mass were also set as the reduced units, that is, rc = m = 1. Then the reduced temperature unit was defined as ε/kB, and the scale of time was given by (mr2c /ε)1/2. Actually, in order to have insights into the interplay of aTT and the final microstructure, we designed comparative trials in which aTT is reduced from 20 to 5 with an interval of 5. As a result, a lamellar phase in which rigid rods are prone to lie in the sublayers in a smectic order (Lamsm phase) is observed when aTT = 20 and aTT = 15, while a lamellar phase in which rigid rods are prone to homogeneously lie in the sublayers (Lamiso phase) is observed at smaller aTT. Therefore we speculate that keeping an appropriate excluded volume for the CG terminal beads also plays a role, especially in lamellar phases with rods forming in-plane nematic-like or smectic-like arrangements. Additionally, we preset a relatively strong repulsive parameter between different segments in the backbone moieties to induce microphase separation. The complete list of repulsion parameters, aij, used are given in Table 1. In fact, the choice of

TLCs. To carry out this study, we use a mesoscopic simulation technique, dissipative particle dynamics (DPD). As DPD utilizes soft repulsive interactions and a momentum conserving thermostat, it has become a promising tool to study large-scale cooperative behavior, like self-assembly32 or phase transitions.33 The remainder of this paper is organized as follows. In section 2, we describe the simulation method, the generic CG model, as well as the important quantities to identify the mesophases and construct the phase diagrams. Then when presenting the simulation results in section 3, we begin with a phenomenological introduction, where the effects of lateral and terminal block lengths on the phase behavior of the model TLCs are described. Then quantitative analysis of the structure and/or dynamics of these phases are performed and comparisons with related experimental findings, and simulations are also made where is appropriate. In addition, we choose one compound and recheck the vicinity of phase transitions with thermodynamic quantities. In the last part of this section, by a further comparison between the three phase diagrams, we definitely show that much of the phase behavior in TLCs can be discussed by one of elementary properties of TLCs, the effective volume fraction of lateral components, whose value is undoubtedly determined by a complex interplay between enthalpy and entropy within the system and are affected by the molecular features such as terminal and lateral block lengths. Finally, the prime results of this paper are summarized in the last section.

Table 1. Repulsion Parameters for the CG Model

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METHODOLOGY Molecular Model and Method. DPD is a particle-based mesoscopic simulation technique that was first introduced by Hoogerbrugge and Koelman34 in 1992 and improved by Español and Warren35 in 1995. In DPD simulations, an individual particle represents a cluster of atoms or molecules, interacting via soft potentials. As such, the use of soft potential and a reduced number of interaction sites make DPD simulation a valuable approach over MD and Monte Carlo (MC) simulations to describe the phase morphology and dynamics of numerous soft matter systems at mesoscale while retaining the correct hydrodynamics. Full details of the implementation of the DPD method are given elsewhere.36,37 For a better understanding of how molecular composition influences the competition among various ordered mesophases as well as the basic physical principles that drive the self-assembly and the phase-behavior of T-shaped ternary amphiphiles, we construct a generic model based on the successful original CG model27 wherein two terminal blocks and one lateral block are attached to one rod block of a fixed length. As schematically illustrated in Figure 1c, in our model we have three types of beads, T, R, and L for terminal, rod, and lateral beads. Then depending on the number of terminal beads (NT) and the number of lateral beads (NL), each individual TLC molecule is represented by TNTR3LNL. Note that all beads have the same size and mass, and interact with the aforementioned soft repulsive potential. Hence the system is characterized by six repulsion parameters, which describe the mutual incompatibility of T, R, and L segments. We should point out that our model adopts a moderate repulsion between terminal-terminal beads with the repulsion strength of aTT = 20ε, which is expected to guarantee the correct excluded volume “shape” of rigid rods buried between the two terminals and the tendency to parallel alignment between rod blocks. Here ε = 1 is considered as the unit of energy. The lateral−lateral repulsive interaction is also relatively moderate as aLL = 20 in order to form lateral clusters.

aij

rod

coil

lateral

rod coil lateral

15 40 40

40 20 40

40 40 20

these parameters has been justified by the fact that our model can not only successfully reproduce phenomena like rod ordering in the lamellar phase but also reproduce generic phase diagrams of TLCs in great detail, and we will present them in the next section. Within the TLC chains, neighboring beads are bonded to each other by a harmonic spring force,27 FSij = −Crij, where C is the spring constant and C = 8.0 in this study. Additionally, to keep the rodlike core relatively straight and the global T-shaped molecular topology, a harmonic bending force is also applied to the molecule except for the flexible lateral chain, which takes the form Uθ = 1/2kθ(θ − θ0)2, where θ and θ0 are the current and specified angles formed by the adjacent bonds and kθ is a stiffness constant. All the values of kθ and θ0 empolyed in our current simulations are listed in Table 2 by taking the symmetric Table 2. Parameters for the Harmonic Bending Potential for the CG Model, the Numbers Listed Here Refer to Individual Bead of Model As Shown in Figure 1c angle

kθ

θ0/°

angle

kθ

θ0/°

1−2−3 2−3−4 3−4−5 4−5−6

4.0 4.0 4.0 4.0

180 180 180 180

5−6−7 3−4−8 5−4−8

4.0 4.0 4.0

180 90 90

T-shaped molecule T2R3L3 as an example. Finally, similar to that used in a pure rod system, an additional elastic force is applied between the first and the last beads of the rod block to produce extra rigidity as well as to save computational resource compared with those strategies to keep a complete stiff rod,38 which is given by Fl = −kex[rln − (n − 1)req]r̂ln = −Fn where kex = 500 is the elastic constant. According to Levine et al.,39 suitably semirigid mesogen chains are produced on taking kex = 500 in a wide range of temperatures. 9108



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size larger than 20rc when quenching from totally disordered configurations, even if we prolong the simulation time to 1 × 107 to 2 × 107 time steps, which are at least 1 order of magnitude of the generally used simulation time in amphiphilic systems.40,41 Clearly, this concerns the “supercooling” of the disordered state, and this supercooled state is not in equilibrium. Thereby, we did parallel simulations by taking different ordered equilibrium configurations obtained from higher temperatures as the starting states and then ran these simulations at T = 0.5 until equilibration is reached via checking the thermodynamic quantities as well as the structural properties. As a result, we observed identical welldefined morphologies at lower temperatures via cooling different ordered starting states. Furthermore, the lower bulk pressure (P) values of these ordered final structures compared with those of frozen-in disordered structures observed by quenching from an isotropic state demonstrate that the ordered mesophases reformed by self-organization via quenching from different ordered configurations at higher temperatures are thermodynamically stable. Additionally, we also found slow dynamics in the evolution process of some unusual mesophase structures such as the lamellar phase with rod in-plane order and the mixed columnar phase. In that case, to avoid metastable states, additional heating and cooling runs were performed by starting with different configurations at nearby state points, Then, by comparing the bulk pressure (P) values of the observed phases obtained from heating and cooling experiments as well as from the quenching experiment at the same state point, the thermodynamically stable phase is identified. Observables. Besides a direct view of various mesoscale selfassembled structures, a combination of various structural parameters including orientational or translational order parameters (S2 and τ1), structure factors SP(q) or angular distribution functions f(θ), and thermodynamic and dynamical properties such as bulk pressure P, mean square displacement (MSD)26 was used to characterize phases and determine phase boundaries. The orientational order parameter (S2) is used to examine the onset of the orientational order of molecular planes and rod blocks in respective columnar phase, SmA, and punched layer phase, which is commonly calculated as the largest eigenvalue through diagonalization of the ordering tensor Qαβ

In this study, to access a complex variety of mesophases and to calculate the phase diagrams in a wide parameter range at a finer resolution, we carried out a systematic study for a range of TNTR3LNL model molecules with NT = 1−3 and NL = 1−8. In our DPD simulations, we used the modified velocity−Verlet algorithm to integrate the equations of motion with a time step of Δt = 0.02. The noise amplitude was taken to be σ = 3.67. All simulations were performed in a NVT ensemble with periodical boundary conditions applied in all three dimensions and at a fixed system number density of 3.0, i.e., the system contains three beads per rc3. It is long known that in a NVT ensemble the size and the shape of the simulation box is not allowed to change; as a consequence, the size mismatch of mesophase periodicity and simulation box might lead to a tilted arrangement of directors (cylindrical axes/plane normals) with respect to the coordinate system and/or stretch or compress supermolecular structures in order to satisfy the fixed box dimensions. However, recent DPD simulations on the selfassembly of amphiphilic systems are widely performed in NVT ensembles by using a larger simulation box, and demonstrate that these distortions become small and the effect of the box’s shape and size on the phase behavior is minimized.40,41 These findings, together with the fact that the NPT ensemble is computationally more expensive and less easy to handle drive us to adopt a NVT ensemble in the present simulations. Considering the calculation economy, we selected the simulation box with different sizes for different molecular lengths. For example, for the short TLCs with a total bead number in each molecule less than 10, most runs involved 5000 molecules, in which the box size was significantly larger than quadruple the semirigid backbone length. While for the longer TLCs, most simulations were implemented in a box with a size of Lx × Ly × Lz (= 30rc), covering at least 4764 molecules, which was also sufficiently large to ensure that any molecule interacts with at most one periodic image of a neighbor molecule. Moreover, additional runs with larger simulation boxes (Lx = Ly = Lz = 40rc) were carried out to demonstrate that the phase diagrams and the final phase structures obtained from just mentioned simulation systems provide a good representation of those for a macroscopic sample. In the present work, we started simulations from isotropic configurations, which were constructed by equilibrating the systems under athermal conditions, i.e., by setting aij = 25 and T = 1.0 for all DPD beads. Then, those totally disordered configurations were quenched to any required temperature at desired aij values. Although this thermal treatment may prevent the system from being locked in one commensurate lamellar orientation and spacing,42 it cannot prevent trapping in metastable states, so we repeated the same quenching independently three times for each required temperature. The equilibration is considered to be reached when the thermodynamic quantities (such as the total energy E and bulk pressure P) as well as the structural properties (such as squared radius of gyration Rg2, squared end to end distance ⟨S2⟩, and structure factors Sp(q)) of the system do not change within the simulation time. The simulation time also depends on the molecular length and temperature. Typically, for those short TLCs containing less than 10 beads, we found that at least 106 steps were needed to obtain a well-equilibrated structure at each state point. While, for larger molecules containing more than 10 beads, equilibration runs of typically more than 2 × 106 steps were required to achieve reliable results. However, at lower temperatures, i.e., T = 0.5, because the relaxation time becomes extremely long, it is quite difficult to obtain well-defined microstructures in a box with a

Q αβ =

1 N

N

⎡3 1 ⎤ (û i)α (û i)β − δαβ ⎥ , α , β = x , y , z 2 2 ⎦

∑ ⎢⎣ i=1

(8)

where the unit vector ûi is defined differently in distinct microstructures. In a columnar phase, ûi represents the normal of the ith cross-sectional plane the rods embrace, which is calculated as the cross product of the unit vectors of a rod moiety and a lateral moiety. However, in the SmA and punched layer phase, ûi represents the long axis orientation of the ith rod and is defined as the end-to-end unit vector of the ith rods. In this context, the corresponding eigenvector is set as the orientation of the columnar axis and director for respective columnar phase and layered phase. The value of S2 in the isotropic phase will be close to zero, while it is nonzero in the nematic and smectic phases or when the rods are perpendicular to the columnar axis, and will tend to one in the limit of perfect orientational order. Moreover, to measure the positional order of rodlike units, the translational order parameter τ1 is calculated through the general function

τ1 = |⟨exp(2πiz /d)⟩| 9109

(9)



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respectively, and the lateral block length NL varied from 1 to 8, and examine the influence of both terminal and lateral blocks on the morphology formation of complex phases. Global Self-Assembling Structures. After a systematic exploration of parameter space of NL and T at constant rod length, we construct the two-dimensional phase diagrams for these three different model systems with NT =1, 2, and 3 in Figure 2. Here, we begin with the phase behavior of the model system T1R3L1−8. As illustrated in Figure 2a, the low temperature portion of the phase diagram shows that a sequence of smectic phase, polygonal LC cylinder structures, and lamellar phases are reproduced. More interestingly, as the size of the lateral chains is increased, the cross-sectional shape of the cylinders changes from tetragonal via pentagonal, hexagonal, to giant hexagonal lattice, which are labeled as Colsqu/p4mm, Colsqu1/p4gm, Colhex/p6mm, and Colhex/c2 mm. Note that the plane group symmetry behind the slash is added to distinguish the mesophases in which polygons orthogonal to the long axis of rods are generated with the same number of sides, but with a distinct cross-sectional shape. Indeed that precise sequence of columnar LC phases in the present simulation has been observed in experiments.24 Thereby, these resulting distinct arrangements are collectively called polygonal honeycomb,20 which can faithfully describe a wide range of two-dimensional (2D) periodic packing of parallel aligned generalized cylinders with polygonal cross section. As indicated in Figure 2a, with a further extending of the lateral chain length, a lamellar phase in which both rigid rods and flexible termini lie in the same stack units is formed. In particular, the in-plane arrangement of calamitic elements is manipulated by temperature. Upon cooling, the disorganized distribution of rodlike units is replaced by a smectic-like ordering, which is also found experimentally.44 Additionally, the two lamellar phases are labeled here as Lamiso and Lamsm, respectively. In sum, most of the ordered microstructures associated with bolaamphiphilies are observed in our phase diagram of Figure 2a, demonstrating that our generic model contains the essential features of TLCs, which can precisely control the LC phase behavior, although it lacks the complicating details of real systems. We then address the question of how terminal block length influences the competition among ordered phases. Figure 2b,c shows the phase diagrams for the case of model systems with NT = 2 and 3. We note that T2R3L1−8 and T3R3L1−8 model systems display many self-assembled microstructures that are qualitatively similar for the T1R3L1−8 system. For instance, usual smectic phase, square, and hexagonal columnar phases are seen in all three model systems. Also, the sequence of these ordered phases as a function of the lateral chain length is essentially similar to that of T1R3L1−8. In particular, by increasing the size of terminal blocks, one can carry out a systematic study of the TLC phase diagram over a range of rather narrowly spaced values of lateral chain volume fraction, so that some novel phases are likely to be found within the current narrow lateral chain volume fraction intervals studied. As illustrated in Figure 2b, in the T2R3L1−8 model system, a new assembled structure, a perforated layer (PL) phase with in-plane hexagonal perforations, which is labeled as ChLhex-like, is encountered between SmA and square columnar phase. In addition, the ChLhex-like phase has some structural features analogous to the experimental hexagonal channeled-layer phase (ChLhex), where the lateral aggregations coalesce into infinite cylinders. The main difference is that, instead of forming the interconnected channel structure, the lateral segments in the ChLhex-like phase aggregate to form discrete

where z is the coordinate of the mass center of the rod along the director, and d is the layer spacing. As d can be pre-estimated by the density profile, the maximum value of |⟨exp(2πiz/d)⟩| is given by slight adjustment of d, which determines the eventual value of τ1. τ1 vanishes in the isotropic and nematic phases, is nonzero in the smectic phase, and is unity in the limit of complete one-dimensional translational order.43 In addition, the planar orientational order parameter ψk is used to quantify the packing order of aggregations formed by lateral segments, which is obtained in the form of26 ψk =

1 N

N

∑

exp(ikθj) (10)

j=1

where θj is the angle between the projection of the vector terminated by the mass centers of two adjacent lateral clusters onto the cross-sectional plane and a fixed arbitrary axis on the same plane. When k = 6, ψ6 is planar hexatic order parameter; for k = 4, ψ4 gives the planar tetragonal order parameter. It is designed so that a value of ψk close to zero indicates that the lattice is strongly deformed, while that closer to unity indicates a highly perfect lattice. Another to fully characterize the phases, two correlation functions are monitored. One is the correlation between different ψk, which is calculated to evaluate the long-range order, and follows the general function as gk (r ) = ⟨∑ ψkiψkj*⟩

(11)

i≠j

Another is position-correlation SP(q), which is used to characterize the periodicity of layered structures and calculated by using the coordinates of junction points of T-shaped molecules as follows: N

SP(q) =

m 1 ⟨|∑ exp( −iq·ri)|2 ⟩ Nm i = 1

(12)

where ri is the position vector of joint point of the ith molecule, and Nm is the number of TLCs. The volume fraction of lateral chains in TLCs is an important variable to influence the liquid crystalline behavior. Following the algorithm proposed by Crane et al.,26 we calculate the volume fraction of side chains in a more realistic way: the first peak position of the radial distribution function g(r) of the beadlike pair is considered as the most probable bead diameter, which is cubed and multiplied by the corresponding bead number giving the volume of one species. Then the volume fraction can be obtained straightforwardly as follows: fL =

(dLprob)3 × NL (dRprob)3

× NR + (d Tprob)3 × NT + (dLprob)3 × NL (13)

prob prob where dprob are most probable bead diameter of L , dT , and dR lateral, terminal chains and rod units; NL, NT, and NR are bead number of lateral, terminal chains and rod units in TLCs.

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RESULTS AND DISCUSSION As the main objective of this work is to understand the interplay of the lengths of different building blocks in TLCs to create the desired ordered structures and expand the repertoire of the available morphologies, we focus on the three different model TLCs with the terminal block size fixed at NT = 1, 2, and 3, 9110



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Figure 2. Graphical representation of the mesophases obtained upon varying the reduced temperature for the different molecular conformations studied (varying the number of grafted side beads and terminal beads). Panels a−c refer to mesophases encountered by varying the number of terminal beads from 1 to 3. Structures 1−12 give the cartoon that captures the main structural features of each LC mesophase.

here at the composition of NL/(NL + NR + NT) = 2/7, close to the composition range of experimental PL phases. Therefore the occurrence of the ChLhex-like phase, closely related to the experimental PL phases, is not entirely unexpected. We can qualitatively

spheroidic clusters. Meanwhile, in this respect, the ChLhex-like phase has great similarity with another PL phase named Rho, where the discrete lateral clusters are self-organized to exhibit 3D rhombohedral symmetry.31 Also, the ChLhex-like phase is found 9111



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Furthermore, within each smectic layer, both rod blocks and terminal chains are well aligned along the layer normal. For instance, in the rod sublayer of compound T1R3L1 at T = 0.5, the layer thickness is 2.053, very close to the rod length (1.94), indicating a perfect parallel alignment of rodlike units. Meanwhile, the layer thickness of terminal sublayers is 1.16, which is almost twice the molecular length of the fully extended terminal segments (0.63) and gives the hint of a supermolecular end-to-end arrangement. A more detailed inspection of the smectic ordering is achieved by calculating the long-ranged orientational and positional order parameters of rods as well as their diffusion anisotropy. For compound T1R3L1 at T = 0.5, S2 = 0.758 and τ1 = 0.21 as well as D⊥/D|| ≥ 2 indicate unambiguously the high alignment of rodlike units. Structure of SmA/frm Phase. On reviewing Figure 2c, we found that only the compound T3R3L2 exhibits SmA/frm phase, which is a type of the experimentally reported PL phases with randomly distributed perforations, in the temperature range of T = 0.5−1.6. This naming of SmA/frm was defined by earlier experiments,31 wherein the abbreviation “frm” is short for filled random-mesh phase. Note, one important difference between this SmA/frm phase and the previously observed punched layer LC phases in the rod−coil molecules9 or lytropic system46 is that perforations in SmA/frm are filled with the third segment, while in the latter the holes are filled with the excessive component. Hereinafter, we exemplify the compound T3R3L2 at T = 1.0 to illustrate the structure of the SmA/frm phase. The T3R3L2 system exhibits a layered structure similar to the conventional SmA phase, as implied by direct visualization of Figure S1a as well as the regular oscillations of density profile in Figure S1b and the two equidistance peaks of SP(q) in Figure S1c. Figure S1 is given in the Supporting Information (SI). However, in contrast to the before-mentioned SmA phase, the orientation and position order of rodlike moieties in the SmA/frm phase are seriously reduced to S2 = 0.381 and τ1 = 0.0855, implying a probable nematic-like organization. Also, unlike the SmA phase, as clearly shown in Figure S1b, the rod density in terminal sublayers keeps at about 0.7 instead of zero, indicating a partial intercalation of the rigid segments into the neighboring sublayers and an imperfect ordering of rods within the rod layers. Furthermore, the interdigitation of sublayers is evidenced by the observation that layer thicknesses for rod-rich sublayers and terminal-rich sublayers are 1.442 and 3.199, respectively, smaller than the respective length of the fully extended rodlike (1.96) and the terminal (3.28) blocks. Indeed, all these data demonstrate again that in the SmA/frm phase, the rod blocks within each sublayer tend to have a nematic ordering. We note that within the same temperature range, the model system T3R3L1 exhibits SmA phase, but compound T3R3L2 forms SmA/frm phase. Apparently, the packing of bulky lateral blocks strongly disturbs the ordering of rods within rod sublayers and triggers the adjacent layers to interdigitate to some extent. In essence, the interplay between the steric effects of lateral chains and the parallel alignment tendency of rigid cores as well as microsegregation drives the formation of observed nematic-like layer in the packing of anisotropic elements, which is also believed to be the optimal arrangement to release the additional interfacial energy. Or speaking more accurately, the agglomeration of lateral groups as well as chain connectivity constraint weaken the microsegregation of immiscible segments in the backbone moieties as well as parallel alignment tendency of rigid cores; hence, the rodlike cores adopt a stochastic distribution in a restrained region and pack in a nematic-like arrangement.

rationalize the formation of the ChLhex-like phase as follows. Due to the different internal constraints employed on the lateral chain, the rigid rod core, and the terminal blocks in our simplified CG model, the true length ratio of lateral chain to rigid rod or effective volume fraction of the lateral chains is much lower than that conventionally calculated from the degrees of polymerization. Hence it may be due to the small realistic size of the lateral chains, which suppresses the lateral beads to assemble into honeycomblike arrays of cylinders along the plane normal, but instead triggers the formation of discrete spherical clusters within the rod layers and eventually makes a distinct ChLhex-like phase. When the terminal bead number further increases, corresponding to mapping out the phase diagram with more narrowly spaced values of lateral chain volume fraction, the T3R3L1−8 model systems exhibit an amazing richer phase behavior. As indicated in Figure 2c, not only are the PL phase with randomly distributed perforations (SmA/frm) and the ColhexΔ phase with triangular arrays of cylinders, which exist in typical real facial amphiphilic system,25 observed between SmA and square columnar phase, but even a new, unexpected network phase as gyroid is discovered when the lateral chain length is increased to NL = 7 or 8 at higher temperature. It is worth emphasizing that the bicontinuous network structure presented here is formed by lateral components while the other constituents form the matrix, which differs from the previously reported gyroid structure in the ABC triblock copolymers wherein the mutually interwoven tubes are formed by two different molecular segments and the third component comprises the matrix.45 However, herein the appearance of this new gyroid structure is not entirely surprising because at high temperatures the entropic contribution to the free energy dominates and possibly drives the system to a correlated unstructured state. At high temperatures, the strong thermal fluctuation effects cause undulating interfaces between nearby cylinders becoming more diffused, some intercylinder connections are formed and bring about the interweaving 4-fold coordinated lattices, which are structurally characteristic of gyroid morphology. Although these three phase diagrams in Figure 2 have many common features, further comparisons with experimental phase diagrams of real TLC systems indicate that there is a striking resemblance between the phase diagrams for model systems with longer terminal blocks and those for the facial TLC molecules, whereas for the model molecule with one distal bead at each end, the phase diagram shows a strong analogy to the experimental ones for bolaamphiphilies. Identification of Novel LC Phases. In the following, the quantitative analysis of the structure and/or dynamics of these novel liquid crystalline phases labeled on diagrams and identification of boundaries between them are described. Meanwhile, comparisons with related experimental findings and previous simulations are also made where is appropriate. Structure of SmA Phase. As indicated in Figure 2, these three different model systems with NT = 1, 2, and 3 exhibit the SmA phase when only one lateral bead is attached to the backbone. In the SmA phase, as shown in Figure 3a, incompatible segments of the backbones, that is, the rodlike core and the semiflexible terminal block, form periodic alternative layers with a long-range order, and the lateral chains are distributed in the corresponding rod layers. This arrangement is proved by the results of the density profile perpendicular to the director and structure factor SP(q). We take the T1R3L1 model at T = 0.5 as an example. As seen in Figure 3b,c, the density profile has the well-defined oscillatory form and the scattering peaks of SP(q) exhibit a position ratio of 1:2. 9112



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Figure 3. The quantitative characterization of the SmA phase for the NL = 1 and NT = 1 model at T = 0.5. (a) The snapshot of the equilibrium configuration, viewed perpendicular to the direction of the plane normal. Blue spheres represent the terminal chains, cyan spheres represent the rigid rodlike core, and red spheres represent the side grafted chains. (b) The density profile perpendicular to the plane normal. The different colors represent corresponding bead types. (c) The structure factor of the graft position of side chains.

magnitude, i.e., ψ6(r⊥) = 0.33 and ψ4(r||) = 0.28. Moreover, by calculating the long-rang correlation between positions of lateral clusters within the layer, as typically plotted in Figure 4, one clearly see that the g6(r⊥) curve drawn in open circles exhibits regular oscillations with the five peak values larger than 0.12, implying that in ChLhex-like phase lateral clusters pack into a long-ranged 2D hexagonal lattice. In contrast, for the SmA/frm phase, g6(r⊥) as presented by open squares in Figure 4 is near-zero, indicating a random arrangement of lateral clusters within the layer. As mentioned earlier, the tendency of the side chains to agglomerate into bigger clusters seriously disturbs the parallel packing of rods, which makes PL phases quite different from the SmA phase. In this context, we expect that the motion of rod segments may be severely inhibited by the growing lateral clusters as compared to that in SmA phase. To illustrate this, we calculated the distribution of lateral cluster size in terms of bead number and the mean-square-distance of rodlike cores in three typical systems of T1R3L1 at T = 0.5, T3R3L2 at T = 1.0, and T2R3L2 at T = 0.9 corresponding to SmA, SmA/frm, and ChLhex-like phases, respectively, and the results are summarized in Figure 5. For the lateral beads in model T1R3L1, as reflected by the corresponding probability distribution function in Figure 5a, the formation of dimers, trimers, or larger aggregates is very limited, and the lateral beads are predominantly in a

Structure of ChLhex-like Phase. Upon visual inspection of Figure 2b, it is clear that only compound T2R3L2 exhibits the ChLhex-like phase, a new type of PL phases, at the temperature range of T = 0.5−1.0. As mentioned earlier, this ChLhex-like phase has some structural features quite similar to those experimentally reported PL phases. For instance, as typically illustrated in Figure S2a, the compound T2R3L2 at T = 0.9 shows a similar tendency of the side chains to agglomerate into bigger clusters as the SmA/frm phase. Also, as we can see in Figure S2b,c, the density profile and structure factor derived from this phase are qualitatively in agreement with those from the SmA/frm phase. Herein Figure S2 is provided in the SI. However, apart from these similarities, ChLhex-like phase has some unique and intriguing structural features, such as the formation of channellike perforations and 3D ordering of spheroidic lateral clusters, as indicated by Figure 4a and Figure S2a. Here we used inplane hexatic order parameter ψ6(r⊥) and interlayer tretagonal order parameter ψ4(r||) to characterize the three-dimensional (3D) ordering of lateral clusters, where r⊥ and r|| represent respective directions of perpendicular to and parallel to the plane normal. We take a typical system of T2R3L2 at T = 0.9 as an example. Interestingly, we find that the ChLhex-like phase possesses the in-plane hexatic order and interplane tretagonal order, as the corresponding order parameters are of moderate 9113



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Figure 4. The snapshot (a) and in-plane hexagonal order correlation function g6(r⊥) (b) of lateral clusters within the rod-rich planes for the ChLhex-like phase for the NL = 2 and NT = 2 model at T = 0.9. In panel a, the terminal beads are deleted for visual convenience. In panel b, the open circles are related to ChLhex-like phase, while the open squares are related to SmA/frm phase as a reference.

Figure 5. (a) The distribution of lateral cluster size in terms of bead number in layered phases where the inset gives the corresponding distribution when cluster size is in the range of 12−27. The blue cylinders symbolize lateral clusters for SmA phase of T1R3L1 at T = 0.5, the red ones symbolize lateral clusters for SmA/frm phase of T3R3L2 at T = 1.0, and the dark cyan ones symbolize lateral clusters for ChLhex-like phase of T2R3L2 at T = 0.9. (b) The mean square distance of mass centers of rodlike cores as a function of time where the color code is the same with that in panel a.

rods f(θ) for model system T3R3L3 at T = 1.2 and depicted it in Figure 6b. One can find three peaks located approximately at 60°, 120°, and 180°, which is consistent with the microstrcuture derived from the visual inspection of Figure 6a. In fact, the tiling pattern of regular triangular columns is a general mode of selfassembly in soft-matter.20 For example, a related structure has also been observed on a larger length scale of hundreds of nanometers in morphologies of ABC star triblock copolymers.47 Further, such a tessellation of triangles was recently reported as a novel six-point star motif assembled by DNA single strands on the length scale of tens of nanometers.48 Structure of Colsqu2/p4gm and Colsqu1/p4gm Phases. According to Figure 2, as a novel type of numerous variations of the basic cylinder phase, the Colsqu1/p4gm phase is observed at a reduced temperature range of 0.5−1.1 for compound T1R3L3 and 0.5−1.4 for compound T2R3L6. On the other hand, another variant, Colsqu2/p4gm, is observed for model T3R3L4 at the reduced temperature range of 0.5−1.3. Note that these two ordered structures are regarded as topological dual, in which Colsqu1/p4gm is a periodic organization of deformed pentagonal cylinders, as typically shown in Figure 6e, while Colsqu2/p4gm

unibead distribution, suggesting that in the SmA phase the lateral chains stay well dispersed within rod layers. By contrast, in both SmA/frm and ChLhex-like phases, the lateral beads are in an inhomogeneous distribution. As indicated in Figure 5a, the lateral chains are associated into highly polydisperse clusters with the cluster size in the scope of 1−27, which may result from the frustration between the alignment of rods and the spatial demands of lateral chains. Last but not least, the significant agglomerating tendency of lateral chains in the ChLhex-like phase is further detected by the large proportion as twice as that in the SmA/frm phase when the cluster size exceeds 10 beads. More importantly, the growth of clusters is accompanied by the suppressed movement of rodlike units with the expansion of lateral moieties shown in Figure 5b, which reaches a consensus with the experimental declaration of a higher viscosity of ChLhex-like phase than the other layered mesophases.31 Structure of ColhexΔ Phase. As illustrated in Figure 2c, the experimental triangular arrays of cylindrical phase, that is, the ColhexΔ phase, is detected for compound T3R3L3 at reduced temperature of 0.5−1.3 . To study this structure in details, we measured the distribution of the angle formed by the neighboring 9114



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Figure 6. Snapshots of equilibrium configurations viewed along the direction of the columnar axis and the distribution function f(θ) of the angles formed at the vertices for liquid crystalline columnar phases. (a,b) ColhexΔ for the NL = 3 and NT = 3 model at T = 1.2; (c,d) Colsqu2/p4gm for the NL = 4 and NT = 3 model at T = 1.2; (e,f) Colsqu1/p4gm for NL = 3 and NT = 1 at T = 0.8. The white lines in panels a, c, and e are depicted to highlight unit cells with polygonal shapes.

is composed of tetragonal and triangular cylinders as seen in Figure 6c. Herein topological dual refers to a couple of columnar phases with similar cross-sectional tessellations,20 i.e., polygonal honeycombs with exchangeable nodes and tiles, wherein the tile gives the number of sides of the cross-sectional polygon, and the node is the valence of the vertex in the corner of a polygon. Additionally, it is worth mentioning that ColhexΔ and Colhex/ p6mm observed in our simulations is another pair of topologically identical structures. Interestingly, in order to deeply understand the complex planar packing mode of polygonal honeycomb, Tschierske et al.20 sorted the 2D periodic packing of

polygons into Laves tiling motif and its dual, the Archimedean tiling motif. From a topological perspective, the Colsqu1/p4gm and ColhexΔ phases belong to the classic Laves tilings composed of identical polygons incorporating different nodes, while the Colsqu2/p4gm and Colhex/p6mm phases fall into the category of Archimedean tilings. Here, to provide a much more detailed picture of the Colsqu2/p4gm and Colsqu1/p4gm phases, we will study the topology of these two mesophases in the following. Literature has reported49 that the typical packing pattern of two-dimensional pentagons with p4gm symmetry is 90°-turn herringbone-like. In this context, for the Colsqu1/p4gm phase, we 9115



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Figure 7. Snapshots of equilibrium configurations and the distribution function f(θ) of the angles formed at the vertices for liquid crystalline columnar phases. (a,b) Colsqu/p4mm for NL = 4 and NT = 2 at T = 1.0; (c,d) Colhex/p6mm for NL = 4 and NT = 1 at T = 1.0. The white lines in panels a and c are depicted to highlight unit cells with polygonal shapes.

175° shown in Figure 6d are attributed to defects arising from the natural structure of the Colsqu2/p4gm phase incommensurate with the exact dimensions of the simulation box. Nevertheless, it is worthwhile to address that this mixed trigonal and tetragonal cylindrical mesophase has not been reported in the preliminary simulations and is identified for the first time. Structure of Colsqu/p4mm Phase and Colhex/p6mm Phase. As seen in Figure 2, with the further increase of lateral block length, the conventional cylindrical phases, Colsqu/p4mm and Colhex/p6mm, are detected in all three phase diagrams. Similarly, the cross-sectional architecture of cylinders is firmly verified by the distribution function f(θ). As displayed in Figure 7 b,d, for Colsqu/p4mm the supposed peaks in f(θ) are at 90° and 180°, while for Colhex/p6mm f(θ) is supposed to display a single peak at 120°; all these agree well with the simulation data for compound T2R3L4 at T = 1.0 and compound T1R3L4 at T = 1.0, respectively. Structure of Gyroid and Colhex/c2 mm Phase. The gyroid structure is detected in Figure 2c as a high-temperature phase for the compound T3R3L7 at T = 2.6−3.7 and T3R3L8 at T = 2.1−3.8, while upon cooling, hexagonal cylinders are encountered. Similar results that gyroid structure occurs after heating hexagonal cylindrical phase have been predicted by the theoretical phase diagram of rod−coil diblock copolymers lately.50 Additionally, the mixed columnar phase Colhex/c2 mm

suppose all the angles formed at 4-fold vertices are 90°, then the four expected peaks at 90°, 114.3°, 131.4°, and 180° in f(θ)28 are supposed to be encountered, which are schematically shown in Figure 6f reproduced by the simulation data for the compound T1R3L3 at T = 0.8. Because of the incompatibility of the mesophase periodicity with the box dimensions, the crosssectional pattern of Colsqu1/p4gm phase is partially irregular as shown in Figure 6e. On the other hand, the broadened peaks as well as the extra shoulder peak at approximate 50° in Figure 6f indicate the diffuse boundaries between the microsegregated domains in the Colsqu1/p4gm phase, resembling a realistic core− shell-in-cylinder structure to some extent. The topological analogue of pentagonal cylindrical structure is the mixed triangular and square cylinders with 5-fold verties. As typically shown in Figure 6c, the packing arrangement of three molecular segments in the Colsqu2/p4gm phase can be described as a mixture of pentagons, trigons, and tetragons, wherein the lateral columns illustrated as the red columns stack to form pentagons and the terminal blue columns aggregate to form squares and triangles, which are regarded as the frameworks enclosing the packing unit cells. For Colsqu2/p4gm phase, the expected locations of peaks in f(θ) are at 60°, 90°, 120°, and 150°, which are consistent with the observation of Figure 6d produced by a typical Colsqu2/p4gm system of compound T3R3L4 at T = 1.2. The broad peaks and a shoulder peak at about 9116



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Figure 8. Snapshots of the (a) gyroid phase for compound C3R3L8 at T = 1.0 and (b) Colhex/c2 mm phase for compound C1R3L7 at T = 0.8.

Figure 9. Snapshots of the novel lamellar phases with bulky side grafted chains. (a) Snapshot of Lamsm phase for compound T1R3L8 at T = 0.8; (b) Snapshot of Lamiso phase for compound T1R3L8 at T = 2.0; Panels c and d are the face views of snapshots of Lamsm, Lamiso phases without lateral chains for clarification.

When the size of lateral chain grows, the additional interfacial energy is released by the formation of structures with anisotropic curvature, and then the Colhex/c2 mm mesophase is highly expected. Structure of Novel Lamellar Phases. According to Figure 2, compound T1R3L8 at T = 0.5−2.0 and T2R3L8 at T = 3.0−3.1 exhibit novel lamellar phases with rods oriented perpendicular to the layer normal. Additionally, the in-plane packing mode of rigid segments is significantly dependent on the temperature. For instance, at low temperatures of 0.5−0.8, the rigid anisotropic units of compound T1R3L8 adopt a side-by-side alignment and exhibit a long-range translational order within the rigid layer planes, as seen in Figure 9a,c, which is a characteristic property

with both elongated and regular hexagons is observed for compound T1R3L7 at the temperature range of 0.5−1.0. As shown in Figure 8b, within these elongated hexagons, two opposite sides are stretched, i.e., two sides of these hexagons are formed by an end-to-end dimer of rod blocks. In the earlier simulation study, Bates also found this elongated hexagonal cylindrical morphology between regular hexagons and lamellar structure, which agrees with our results.27 The formation of intermediate Colhex/c2 mm mesophase between columnar and lamellar phases can be tentatively rationalized by a nonuniform interfacial curvature. The microsegregation between the terminal groups and the remainder part of the backbone tends to impose a certain interfacial curvature. 9117



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of a Lamsm phase. At elevated temperature of 0.9−2.0, a Lamiso phase where the anisotropic cores randomly distribute within the rigid subspace is observed as depicted in Figure 9b,d. Actually, these two novel lamellar structures are previously obtained in the laboratories for bolaamphiphilies.26 Moreover, two such novel lamellar phases, Lamsm and Lamiso, have been also found in the closely relevant systems of laterally mono- or multitethered nanorods.30,15 Aside from the direct observation, the distinct rod assignments are characterized with the help of the orientation order parameters within each rigid layer. These results are summarized in Figure S3 provided in the SI. As shown in Figure S3a, S2 = 0.4−0.5 at T = 0.7−0.8 and S2 = 0.1−0.2 at other temperatures imply that the orientational order of rods in a Lamsm phase is much higher than that in a Lamiso phase. Further quantitative test, such as measurement of the positional order of rods, is also used to distinguish between these two phases. As drawn in Figure S3b, the positional order parameter τ1 at T = 0.7 and T = 0.8 is relatively higher with τ1 = 0.1−0.2 for all five layers, whereas at higher temperatures the positional ordering of Lamiso phase vanishes as τ1 is close to zero. Besides, the rod orientations in different layers of Lamsm phase are strongly correlated. We found that the angle between the director in each individual rigid layer and the director of any selected layer at both temperatures of T = 0.7 and T = 0.8 is rather small, i.e., less than 10° as typically depicted in Figure S3c, which does not just mean that the rod orientation of different layers is consistent, but also that different layers are coherent in the Lamsm phase. Identification of Phase Boundaries. After demonstrating how to characterize the structure of the various liquid crystalline mesophases using combinations of results obtained from the structural properties and dynamic properties, in the following we shall present how we estimate the phase boundaries. The values of the thermodynamic properties, e.g., bulk pressure P and the nematic order parameters S2 were used to identify positions of phase transitions. For simplicity, we only plot the results of just one system. Figure S4 depicts the evolution of P and S2 of compound T1R3L1 as a function of temperature. Note that by monitoring the structural properties and snapshots of configurations for T1R3L1 in the previous section, the phase sequence is established to be SmA phase at T = 0.5−0.8 and isotropic state at higher temperatures. Correspondingly, we observed that both P and S2 undergo an obvious discontinuous change at approximate T = 0.83, which implies a first-order phase transition between isotropic and LC phases. In addition, at T = 0.5−0.8 the orientational ordering of rod segments in compound T1R3L1 is indicated by a higher value of S2 in the range of 0.4−0.77 as displayed in Figure S4a, whereas at T = 0.83, S2 suddenly falls to an extremely small value very close to zero. In the case of bulk pressure P in Figure S4b, despite the overall increasing trend as a function of temperature, clearly the vertical bar at T = 0.83 divides the figure into two regions where the P values in the right zone are obviously larger than 19.0, while at lower temperatures, the P values are relatively smaller indicating more condensed state, which is consistent with the characteristics of LC phases. The Phase Sequence of TLCs. Due to the nonequivalent effects of tuning terminal and lateral chain length, we need a generalized variant to identify the dominant reason for the assembly behavior in our model systems. In an alternative approach to compare the three model systems in a universal manner, we replot the phase diagrams presented in Figure 10 as a function of effective volume fraction of lateral chains f L. As mentioned in the Methodology section, the value of f L with

Figure 10. The sequence of phase transitions in T-shaped ternary amphiphiles as a function of f L, where green arrowheads indicate different structures manipulated by temperature.

respect to individual mesophase is obtained by considering the real packing distance of beads at the same type. Then, further comparisons between these three model systems in the phase space of f L indicate that there are striking similarities among these three phase diagrams, i.e., the stability windows of each mesophase in the three phase diagrams are almost matching each other quantitatively. As such, Figure 10 offers a unique view to understand how phase behavior is affected by the molecular composition. Our calculated correlation between the volume fraction of lateral chain f L and the observed mesophase type for TLC molecules can mostly be compared favorably with the experimental results of bolaamphiphilic systems, wherein f L is obtained based on the lateral and overall volume of a single molecule as calculated using the crystal volume increments reported by Immirzi.51 For instance, Figure 10 showcases that the SmA phase is observed when the lateral chain volume fraction is in the range of f L = 0.08−0.16, which is within the experimental reported range of f L ≤ 0.28.24 Also, the experiments indicate that the square columnar mesophase is supposed to be encountered when the volume fraction of lateral chain is approximately 0.36,24 which roughly falls into the range of 0.28−0.42 as sketched in Figure 10. Besides, the obtained volume fractions of lateral chains for pentagonal and hexagonal columnar as well as lamellar structure are all close to those in the experiments.24 However, the elongated hexagonal mesophases in our case exist in the slightly larger volume fractions, i.e., f L ≈ 0.60, when compared with the preliminary experimental results of f L = 0.53−0.56.24 As for the reminder microstructures, the SmA/frm, ChLhex-like, and mixed cylindrical structures shown in Figure 10 correspond well in qualitative terms to the facial analogical system. We remark that the sequence of these microstructures as a function of f L shown here was found experimentally, while the correlated specific scope of volume fraction of lateral chain has not been documented. On the other hand, we notice that the sequences of ColhexΔ and ChLhex-like phase as well as gyriod and Colhex/c2 mm phase have not been demonstrated both in the experiments and simulations so far and are claimed in our work for the first time. Overall, our TLC models display a universal liquid crystalline behavior embracing both bolaamphiphilies and facial amphiphilies. Insight into the design and control in new and complex mesophase by relatively simple amphiphilic molecules is further envisioned. 9118



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CONCLUSIONS We have used dissipative particle dynamics to study the phase behavior of T-shaped ternary amphiphiles composed of rodlike cores connected by two incompatible end chains and side grafted segments. We first adopt a relatively moderate repulsion at both ends to guarantee the correct excluded volume “shape” of rigid rods, which leads to the spontaneous formation of in-plane smectic ordering of rods in the lamellar phase. In order to expediently explore the organization of more complex welldefined morphologies, we fine-tune the number of terminal and lateral beads, and actually by this manipulation our model indeed successfully reproduces the preliminary observations as well as series of novel mesophases. Three phase diagrams for the model systems with different terminal chain lengths, which were constructed in terms of temperature and lateral chain length, contain mostly experimentally observed microstructures of T-shaped bolaamphiphiles, such as lamellae and cylinders with orthogonal polygonal tiling patterns. Also, we have successfully reproduced the experimental observations for facial liquid crystals, such as mixed triangular and square cross-sectional honeycombs, perforated layer structures that are observed in the simulations for the first time. Detailed quantitative characterization is given to elaborate the structure of individual liquid crystalline mesophase and identify the phase boundaries. Finally, we constructed a normalized phase diagram of T-shaped ternary amphiphiles as a function of f L, which not only can be quantitatively compared with the experimental results but also offer a unique view to understand the universal phase behavior in TLCs. The success of this minimalistic model indicates that the fundamental elements, i.e., the shape, rigidity, connectivity, short-range repulsion and segregation between different components, are essential to reproduce the generic properties of real TLCs systems. Therefore, we believe that this model could serve as a satisfactory basis for further exploration of self-organization in other related soft matter systems where more complex molecular topology and functional building blocks are implemented.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Quantitative characterization of the SmA/frm phase for the NL = 2 and NT = 3 model at T = 1.0, quantitative characterization of the ChLhex-like phase for the NL = 2 and NT = 2 model at T = 0.9, arrangement of rods in a single rigid layer of lamellar structures of compound T1R3L8 in the temperature range of 0.7−2.0, and the order parameter S2 of rod segments in compound T1R3L1 and bulk pressure of compound T1R3L1 as a function of temperature. This information is available free of charge via the Internet at http://pubs.acs.org

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; phone: 86-10-82618124; fax: 86-10-62559373. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful for the support of the NSF of China (21174154, 50930002 and 20874110). We also thank the supercomputing center of CAS for supporting computing resources. 9119



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Dissipative Particle Dynamics Simulation of the Phase Behavior of T

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