Developing Local Order Parameters for Order–Disorder Transitions

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Developing Local Order Parameters for Order−Disorder Transitions From Particles to Block Copolymers: Application to Macromolecular Systems Ankita J. Mukhtyar and Fernando A. Escobedo* School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14850, United States

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S Supporting Information *

ABSTRACT: In part 1 of this two-paper series, a local order parameter framework was put forth that could track the changes occurring when block copolymer-like mesophases formed from a disordered state. The framework was developed using a two-particle model and involved identifying the local symmetries and geometric motifs that were unique to a given mesophase. In this paper, this framework is suitably modified to track the mesophase formation of standard coarse-grained bead−spring simulation models of polymers and oligomers. In particular, a mesoscale chain model typically employed with dissipative-particle dynamics is used to study the ordering transition of a linear symmetric diblock copolymer into a lamellar phase, and a more detailed bead−spring model of branched bolaamphiphile molecules is used to track the formation of a single diamond phase. These applications illustrate the robustness of the method in handling molecules with intramolecular degrees of freedom (including multiple chemical blocks and branched architectures), varying levels of coarsegraining, and rare mesophases with complex 3D order (like the single diamond phase). These features are suggestive of the potential suitability of the proposed framework as a tool to map transition pathways leading to complex macromolecular morphologies. describe ordering transitions with more “realistic” simulation models of BCPs. In this paper, we attempt to apply our OPs from Paper 1 to study the disorder-to-order transition of two polyphilic polymeric systems of practical importance: (i) the isotropicto-lamellar phase transition in a symmetric dBCP simulated via dissipative particle dynamics (DPD)10 and (ii) the isotropicto-single diamond (D1) phase transition in a bolaamphiphile system simulated via molecular dynamics. Unlike the systems studied in Paper 1 which have yet to be realized experimentally, the systems studied here have been well characterized in experiments11,12 and previous theoretical and simulation studies.13−17 Both applications entail distinct modifications to the OPs from Paper 1 and illustrate how such a framework can be extended to more complex systems. In particular, the lamellar dBCP system serves to illustrate how to “coarsen” the particle coordinates of multiatomic linear molecules into a minimalistic pseudocenter representation that can be mapped into a KM-like description. The bolaamphiphile system further illustrates how to extend the methodology developed for the gyroid network in Paper 1 to a relatively novel network phase (D1), formed by a molecule containing three types of chemical blocks (i.e., a triblock co-oligomer) and a branched (a “T-shaped bola”) structure. By demonstrating

1. INTRODUCTION Block copolymers (BCP) exhibit a rich phase behavior and have been studied extensively over the years to understand the factors that control their self-assembly into mesophases.1−3 For linear diblock copolymers (dBCP), for example, the strength of the incompatibility between blocks and their relative volume fractions are key characteristics that determine the tendency of these molecules to microphase segregate and undergo a “disorder-to-order” transition. In the disordered state the block domains lack any kind of discernible long-range structural order, such as layers (lamellar phase), cylinders (hexagonal phase), or interwoven networks (gyroid phase). Nanoscopic domains having certain short-range order or geometric motifs nucleate and grow as the system travels along the transition pathway toward the ordered state. Key to tracking this transition is the availability of a good “order parameter” (OP), a metric that quantifies the degree of order in the system.4,5 In part 1 of this two-part paper series, henceforth to be termed Paper 1, we discussed the need for a local OP based on geometric considerations that can be used in techniques like metadynamics6 or umbrella sampling7,8 to extract free energy barriers. We used the KM model9 to allow the simulation of ordered phases with a large number of repeating motifs (hence reducing finite size effects) and develop OPs for three of the archetypal BCP phases (lamellar, cylinder, and alternating gyroid). While the KM model served as a good platform to test our OPs, we had yet to confirm if these OPs are suitable to © XXXX American Chemical Society

Received: August 4, 2018 Revised: November 11, 2018

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DOI: 10.1021/acs.macromol.8b01683 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Snapshots of the lamellar phase obtained using the dBCP DPD model at f = 0.5, χN = 40, and Lbox = 21. (a) Original structure showing the multibead thick layered domains. (b) Top: replacing chains with pseudoblocks at block centers’ of mass (COM). Bottom: snapshot showing the original beads and the identified pseudoblocks. (c) Top: pairing nearest-neighbor pseudoblocks that do not share the same interface and identifying pseudocenters at the pair’s COM. Bottom: snapshot showing the pseudoblocks and pseudocenters. (d) Final reduced structure obtained with pseudocenters.

Fijhar = −khrijriĵ

the use of our OPs in two different simulation models (of polymers and bolaamphiphiles), we hope to illustrate their potential usefulness as reaction coordinates for studies aimed at understanding order−disorder kinetic pathways in BCP or similar systems.

where kh = 4 is the spring constant used. The system is simulated at a number density = 3 and box length Lbox = 21 to represent melt conditions. Each chain consists of N = 8 beads, four from each block (f = 0.5), and χN is chosen to be 40 so that the system is in the intermediate segregation regime according with the known DPD phase diagram20 (noting that at the order−disorder transition or ODT, χNODT ≈ 28 for f = 0.5). The integration time step used is Δt* = 0.05, and simulations are run for 5 × 106 steps to ensure equilibration. 2.2. Mapping Structure into Coarse-Grained Pseudocenter Model. The lamellar phase shown in Figure 1a consists of approximately 5.0σ thick layers of alternating A and B domains, where σ is a characteristic length scale of the model roughly equal to the diameter of a bead in the dBCP molecule. This is in contrast to the one-particle thick lamellar domains obtained in the KM model, which could be thought of as representing the centers of a thicker domain. Because the OP we developed in Paper 1 essentially tracks cluster growth based on these domain “centers”, one approach to map the current DPD lamellar structure into a reduced, but equivalent KM-like lamellar structure is to identify surrogate bead “centers” in each layer. In this way, a DPD lamella is converted into a form that can be “directly” used by our OP. We do this in a multistep process: Step 1. Each block is replaced with a pseudoblock point located at the blocks’ center of mass (Figure 1b). At this point each lamella layer is roughly two pseudoblocks thick. Step 2. To try to reduce the layer thickness even more, each pseudoblock is paired to the closest pseudoblock of the same block type, but one that does not share the same interface; i.e., the distance between their respective interface beads (first bead of the other block) should be greater than a critical value lc. This value is taken to be lc = 5.0, considering that the distance between interface beads of neighboring chains that do not share the same interface would be on the order of the average lamellar thickness ≈5.5−6.0 (see Figure 2). These pairs are then replaced with a set of “pseudocenters” located at the pseudoblock pairs’ center of mass (Figure 1c). Iterating over all chains, we obtain the reduced form of the structure, as an attempt to approximately map the domain centers via pseudocenters (Figure 1d).

2. FORMATION OF THE LAMELLAR PHASE VIA DISSIPATIVE PARTICLE DYNAMICS One of the simplest morphologies formed by dBCPs is the lamellar phase consisting of alternating layers of two blocks (A and B). Several computational studies have used a highly coarse-grained soft-bead model, typically associated with dissipative particle dynamics (DPD), to describe the phase behavior of dBCPs and to analyze the disorder to order transition in the lamellar phase.15,16,18 Here we simulate the DPD model using the LAMMPS software19 to form the lamellar phase and apply our OP to track its transition from a disordered state. 2.1. Model Details. In the DPD model, polymers are represented as chains of soft beads and springs, where each bead represents a segment of a polymer chain that moves according to Newton’s equations of motion. The total force on a bead is given by the sum of the conservative FC, random FR, and frictional FD (dissipative) forces.10 Each bead interacts with all other beads through the conservative force: FijC

l a (1 − rij)riĵ (rij < 1) o o o ij =m o o (rij ≥ 1) o0 n

(2)

(1)

where aij represents the strength of repulsion. Note that eq 1 implies that the “diameter” of a soft bead (where repulsion goes to zero) is unity and provides the reference length used in expressing various properties. All parameters are reported in reduced units. Beads of the same type interact with aii = 25 while for different-type beads aij = aii + 3.27χ, where χ is the Flory−Huggins interaction parameter used to quantify the degree of segregation between different blocks. It is this dependence on χ that allows a coarse-grained model like DPD to capture the main energetic disparity between blocks and to connect with experimentally determined values of χ. The beads are connected through a harmonic potential force: B

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pseudocenters (see Figure 3b) and a value large enough to get a good planar fit. The “neighbor” particles are used to fit a plane and identify the direction of the normal vector to the plane, denoted as the lamella-signature “LS” vector of particle i. The process is repeated until each particle has its own LS vector. To quantify the degree of alignment, the dot product between neighboring LS vectors is measured, and if its value exceeds 0.95, the corresponding two particles are considered connected. Figure 4 shows the histograms of the number of connections per particle for the disordered and ordered cases. From Figure 4 we define a threshold of ξ > 80 to determine if a particle is “lamellar-like”. This allows us to identify all the ordered particles and then cluster them based on their proximity using a cutoff of rc (see Figure 5). The growth of the ordered particles with time is shown in Figure 4. While the ordered particles do appear to form a single grain (which looks fractionated in Figure 5 due to the boundary conditions), the system size was not large enough to confirm the presence of a nucleation mechanism. Therefore, to minimize finite size effects, we tested the OP on a system consisting of around 220000 beads (8 times the previous system), keeping the same chain length and composition. The box length was doubled to 42 and the number density was kept at 3 to represent melt conditions. As before, the isotropic system was first allowed to equilibrate for 106 timesteps at χN = 25 and then subjected to an instantaneous jump to the desired χN of 40. The corresponding disorder-to-order transition can be seen in Figure 6, tracked by mapping the structure onto the coarse-grained pseudocenter model and using the OP as explained earlier. It can be seen that the qualitative features of the ordering process in the Lbox = 21 (Figure 5) and Lbox = 42 (Figure 6) systems are consistent, although the latter more clearly shows multiple lamellar grains forming and growing. Over a longer time scale (t* = 30000), the misaligned grains will become a single lamellar monodomain (not shown).

Figure 2. Criteria for selecting lc: distance between interface beads (denoted as Ix) of neighboring chains not sharing the same interface (6.04) is larger than that for those sharing the same interface (1.19). Pseudoblock i is hence paired to pseudoblock j and not k. Coloring scheme same as Figure 1.

Step 3. As an optional step, the closest pseudocenters from step 2 can be pairwise merged and the process iterated until the minimum distance between resulting merged sites is greater than some prescribed length scale lcg, i.e., a desired degree of length coarsening. While this step does not significantly change the effective (lamella) domain thickness attained in step 2, it can greatly reduce the total number of “sites” to be used for the evaluation of signature vectors, thus making the OP evaluation less expensive. Note also that this step could potentially replace step 2. For the ensuing analysis in this section, only the reduced structure obtained from step 2 will be used; sample calculations implementing step 3 are described in the Supporting Information. Figure 3a shows us how each step brings us closer to the domain center of the lamella layer. The results are only slightly sensitive to lc, and any value within the range of 4.5−6.0 would work sufficiently well to differentiate between the chains rooted to the same or different interfaces. 2.3. Growth of the Lamellar Phase. The system is first allowed to equilibrate for 106 timesteps at χN = 25 and f = 0.5, corresponding to the isotropic region of the phase diagram (but near the ODT). The final configuration obtained is then used as a starting point and subjected to an instantaneous jump to the desired χN of 40. The growth of the lamellar phase from disorder is tracked using a procedure similar to that described in section 2.3.1 of Paper 1 for the lamellar phase. Because the pseudocenters described here are the particles in Paper 1, henceforth we use these two terms interchangeably. For every particle i, a cutoff radius (rc) is defined within which all particles of the same type are identified as “nearest neighbors”. Here, rc is chosen to be 5.0 as this approximately corresponds to the first minimum in the pair radial distribution function of

3. FORMATION OF THE “SINGLE” DIAMOND IN BOLAAMPHIPHILES Bolaamphiphiles are a class of molecules that consist of a hydrophobic backbone (e.g., a polyphenyl core) with two hydrophilic end groups (e.g., polar diol groups). T-shaped bolaamphiphiles (TBA) also include a third “block” consisting

Figure 3. (a) Schematic demonstrating how a single lamella block-domain is reduced to fewer beads lying closer to the center plane (black dashed line). It shows: I = original beads, II = pseudoblocks, and III = pseudocenters. Coloring scheme same as Figure 1. (b) Pair radial distribution function of the reduced structure containing the pseudocenters. C

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Figure 4. Left: distribution function of the number of connections per particle for the lamellar phase (DPD model). Two particles are connected if the dot product of their LS vectors is >0.95. Right: plot showing the growth of the ordered phase with time.

Figure 5. Lamellar phase formation from a disordered state for f = 0.5, χN = 40, and Lbox = 21. Red and blue correspond to the ordered A and B particles (or pseudocenters) of the diblock copolymer that grows from the isotropic liquid (colored in gray). Time is reported in reduced units.

Figure 6. Lamellar phase formation from a disordered state for f = 0.5, χN = 40, and Lbox = 42. Multiple ordered domains detected using our OP. Coloring scheme same as Figure 5. Time is reported in reduced units.

of a laterally attached flexible chain (e.g., alkyl or fluoroalkyl chains).21,22 The incompatibility between segments allows these molecules to self-assemble into a variety of different morphologies that can be tuned through changes in temperature and lateral chain length and design. A recent simulation study by Sun et al. confirmed the rich phase behavior exhibited by TBA, where they observed the formation of the rare “single” diamond (D1) and the “single” plumber’s nightmare (P1) phases, along with other more commonly observed morphologies.17 Here, we attempt to use the principles that govern our OPs to detect the growth of the D1 phase from disorder. This D1 phase, as originally predicted by the Sun et al. simulation study, has also been experimentally realized.23 We adapt the methodology used to track the formation of the alternating gyroid network (that lies on a BCC lattice) in Paper 1 to the D1 phase, given that the latter is also a 3D network that lies on a cubic lattice but involves a different lattice symmetry (FCC). 3.1. Model Details. The model implemented by Sun et al. in ref 17 was adopted to simulate a system of 2700 TBA molecules, with a chain length of 6 (Nrigid) and 11 (Nflx) beads for the rigid backbone and flexible lateral chain respectively

(see Figure 7). Unlike the soft beads used in the DPD model, here beads interact via a Lennard-Jones type of potential and

Figure 7. Schematic of the TBA molecule for Nflx = 11 and Nrigid = 6. Red beads correspond to the hydrophilic end groups, blue beads correspond to the linear aromatic core, and green beads correspond to the hydrophobic flexible lateral chain.

hence represent a more detailed (less coarse-grained) representation of the intra- and intermolecular interactions. All parameters are reported in reduced units. MD simulations were performed using the LAMMPS software,19 and forces were integrated using the velocity-Verlet algorithm with an integration step size of Δt* = 0.005. The simulations were conducted in an NVT ensemble at a constant density η = 0.45 D

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Figure 8. Left: simulated snapshot of the “single” diamond (D1) phase consisting of 2700 TBA molecules, obtained at T* = 1.05, η = 0.45, Nflx = 11, and Nrigid = 6. The red and blue correspond to the rigid backbone of the TBA that forms the D1 phase, with the red representing the hydrophilic end groups (that constitute the nodes) and blue representing the rigid rods (that constitute the struts). Green represents the flexible lateral chain that forms the continuous phase. Right: equivalent skeleton model (consisting of pseudonodes and struts) obtained after clustering and reducing nodal clusters into single beads and drawing struts as single rods.

Figure 9. Probability distributions of the correlation functions d6 and d4 for the ordered (single diamond) phase and disordered phase of TBAs at T* = 1.05, η = 0.45, Nflx = 11, and Nrigid = 6. (a) d6 distributions showing significant overlap. (b) d4 distributions showing significant separation, suggesting a discriminating threshold of d4 > 0.66. (c, d) Two-dimensional histograms of d6 with rij2 for disordered and ordered phases, respectively. (e, f) Two-dimensional histograms of d4 with rij2 for disordered and ordered phases, respectively.

and a temperature T* = 1.05 (using the Nosé−Hoover thermostat). η was defined in the same way as mentioned in ref 17 and corresponds to a dense (solvent-free) fluid. Because the system disorders for T* > 1.1, the chosen conditions are relatively close to the melting point, consistent with a small degree of supercooling. Under these conditions, we obtain the “single” diamond phase as shown in Figure 8. This structure was converted into its equivalent skeleton form using the network analysis mentioned in ref 17, where each strut is a

cluster of the blue beads (i.e., the hydrophobic backbone) and each node is a cluster of the red beads (i.e., the hydrophilic end groups). Note that the skeleton representation approach of ref 17 plays a similar role to the structure reduction approach we described to map the DPD lamella phase into pseudocenters in section 2.2. For example, in the D1 phase the roughly spherical symmetry of a node allows the use of the center of mass of all beads making up that node as the representative position of the node, to be henceforth denoted as a pseudonode. This gives us E

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Macromolecules an ordered state node size distribution that consists of an average of 20−30 beads per node. In the disordered state, where there is no coherent network structure, only those clusters that lie within the node size distribution are classified as being potential nodes. 3.2. Order Parameter for the Single Diamond Phase. In section 2.3.2 of Paper 1, it was shown that the symmetry of a cubic phase can be captured using vectors described by a set of spherical harmonic functions. Here we extend that idea to develop an OP for the D1 phase, whose nodes lie on a diamond cubic lattice having FCC symmetry. All further analysis involved only the position of the pseudonodes, which were obtained from the skeleton form shown in Figure 8. These coordinates were used to compute the bond OPs described by eq 3

Figure 10. Distribution function of the number of connections per pseudonode for the D1 phase. Two pseudonodes are considered connected if their correlation function d4 is >0.66.

N (i)

b 1 ql , m(i) = ∑ Yl ,m(θi ,j , ϕi ,j) Nb(i) j = 1

(3)

where Yl,m(θi,j,ϕi,j) are the spherical harmonics, m ∈ [−l, l], and θi,j and ϕi,j are the polar and azimuthal angles of ri,j (distance vector between pseudonodes i and j). Nb(i) denotes the total number of neighbors for every pseudonode, found within a cutoff distance rc (= the position of the first g(r) minimum). The symmetry index l is chosen to be 4, after comparing the probability distributions of the q6 and q4 correlation functions, computed using eq 4 for all pairs of neighboring pseudonodes, in both the ordered and disordered cases (see Figure 9a−f).

D1 phase, consisting of pseudonodes and struts, from a disordered cocontinuous network, shown by dotted lines, in Figure 11. This suggests that similar to what we observed in the dBCP-like phases in Paper 1, the bolaamphiphile molecules also appear to quickly nanophase separate, in this case into a disordered cocontinuous network, which only later begins to “align” or order into the ordered network. These results further support our hypothesis that disorder-to-order transitions in network morphologies can be mapped by identifying and tracking the pseudonode positions alone, provided the final structure lies on a cubic lattice with well-defined symmetry.

l

dl(i , j) =

∑m =−l ql , m(i)ql*, m(j) l

l

(∑m =−l |ql , m(i)|2 )1/2 (∑m =−l |ql , m(j)|2 )1/2

(4)

4. CONCLUSION In this paper, we have implemented the local OPs developed in Paper 1 toward more complex polymeric or oligomeric molecular models, namely, of linear diblock copolymers and of T-shaped bolaamphiphiles. The OPs in Paper 1 are based on the description of characteristic domain “centers” of the different phases, i.e., the centers of individual layers in the lamellar phase, the axis of cylinders in the hexagonal phase, or the centers of mass of nodes in the gyroid network. Hence, to be able to use those OPs for the lamellar phase consisting of multiparticle thick domains, we needed to convert it into a suitable single-particle thick representation. By doing so, we showed that we can effectively track the growth of the lamellar phase from a disordered state. We also extended the method developed for the alternating gyroid toward another network morphology, the “single” diamond phase obtained via the self-assembly of T-shaped bolaamphiphile molecules. In this case, each nodal cluster characteristic of this phase can be readily reduced to a single domain center or nodal point. Because the nodes of the diamond phase are arranged in an FCC cubic lattice, we showed that the symmetry of any FCC-like motif was readily identified with the help of a bond orientational OP. Both of these applications confirm the use of the OPs we proposed in Paper 1 as effective metrics to track the evolution of systems undergoing disorder-to-order transitions. The underlying principle is the same: Identify the characteristic symmetry of the phase by analysis of the “finite elements” making up the domain centers and represent it through a set of appropriate vectors. The correlation between vectors then gives us the (local) spatial extent of order in the system. An important consideration regarding the choice of neighbor

where ∗ indicates the complex conjugate. Indeed, the distributions for the two phases overlap more significantly for d6 (Figure 9a) than for d4 (Figure 9b), indicating that the latter possesses better discriminatory power. Analysis of the corresponding 2-D histograms over squared distances and d6/d4 correlations of particle pairs in the ordered phase reveals that the distribution based on d4 (Figure 9f) shows a stronger correlation than that based on d6 (Figure 9d), with the former being significantly less diffuse than the latter. This further confirms that the choice of symmetry index 4 is more discriminating of the order characteristic of the D1 phase. We can then define two neighboring pseudonodes to be connected if their d4 value is larger than a threshold value d4,c, i.e., d4 > d4,c = 0.66. Using this definition, we can effectively identify all the ordered pseudonodes, but we should also ascertain how many such connections per pseudonode are characteristic for both the ordered and disordered phases. Figure 10 shows the distribution function for the number of connections per pseudonode, denoted by ξ. From this, we can define another threshold ξc to unambiguously identify “diamond-like” pseudonodes so that if ξ(i) > ξc then pseudonode i is considered “diamond-like”, while if ξ(i) ≤ ξc it is considered to be “liquid-like”. Based on the distributions shown in Figure 10, ξc was set as 3 as this would minimize both false negatives and false positives (for comparison, see section S1 of Supporting Information for calculations with ξc = 2). Using this criterion, the selection process is performed until all the “diamond-like” pseudonodes have been identified. These pseudonodes are then clustered depending on their spatial proximity to one another, using the cutoff rc. This allowed us to track the nucleation and growth of the ordered F

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Figure 11. Growth of the D1 phase of TBAs at T* = 1.05, η = 0.45, Nflx = 11, and Nrigid = 6. The ordered domain is identified using the correlation function d4 for the pseudonodes. Red corresponds to the pseudonodes and blue to the struts of the D1 phase that grows from the isotropic liquid (colored in gray). For clarity, only the rigid rods are shown. Time is reported in reduced units (t*/1000).

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cutoff (rc) used for our OPs is that it should be of the order of the characteristic repeat-unit domain width or “period length” of the phase. For example, rc for the lamellar phase was approximately equal to the lamella spacing, i.e., the distance corresponding to the first g(r) minimum, irrespective of the model used (KM or DPD). For the gyroid phase in Paper 1, this translated into a longer distance to capture the proper “period length”. For the DPD model used in this paper where interaction beads strongly overlap with each other yielding a high number density, the proper choice of rc translated into a rather large threshold number of connections per particle (ξ ∼ 80) to discriminate between disordered and lamellar phases. For an asymmetric diblock copolymer leading to lamella layers with different thicknesses, a different cutoff may be needed for each of the block atom types, a choice that again would be based on the first minimum of the corresponding radial distribution function. We intend to extend and develop further the OP approach advocated here, toward realistic molecular models that selfassemble into the cylinder phase, spherical phase, and double gyroid phase seen in BCP, as well as employ them as reaction coordinates to extract free energy barriers and estimate nucleation rates using rare-event sampling techniques.6−8,24,25 Such calculation will also allow us to more quantitatively assess the quality of our OPs, e.g., via analysis of committor probabilities.4,26 We expect that by introducing suitable modifications and extensions, the proposed framework would also be effective in describing the ordering transition of BCP models with more complex architectures (e.g., having different bead diameters or branching)27−29 and in surfactant−solvent systems 30,31 or multicomponent BCP−homopolymer blends,32,33 all known to exhibit complex ordered phases.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01683.

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Effect of connectivity cutoff value on OP for single diamond phase, pseudocode for coarse-graining DPD lamellar model, sensitivity analysis of OP cutoffs in lamellar DPD model (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Fernando A. Escobedo: 0000-0002-4722-9836 Notes

The authors declare no competing financial interest. The preceding paper referred to as Paper 1 in the text can be found at DOI: 10.1021/acs.macromol.8b01682.

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ACKNOWLEDGMENTS Funding support from NSF Award DMR-1609997 is gratefully acknowledged. The authors are grateful to Yangyang Sun for valuable discussions and suggestions.

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REFERENCES

(1) Bates, F. S.; Fredrickson, G. H. Block Copolymers  Designer Soft Materials. Phys. Today 1999, 52 (2), 32−38. (2) Mai, Y.; Eisenberg, A. Self-assembly of block copolymers. Chem. Soc. Rev. 2012, 41, 5969−5985. (3) Matsen, M. W.; Bates, F. S. Origins of Complex Self-Assembly in Block Copolymers. Macromolecules 1996, 29, 7641−7644. (4) Peters, B. Recent advances in transition path sampling : accurate reaction coordinates, likelihood maximisation and diffusive barriercrossing dynamics. Mol. Simul. 2010, 36 (15), 1265−1281. G

DOI: 10.1021/acs.macromol.8b01683 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules (5) Wales, D. J. Perspective : Insight into reaction coordinates and dynamics from the potential energy landscape. J. Chem. Phys. 2015, 142, 130901. (6) Barducci, A.; Bonomi, M.; Parrinello, M. Metadynamics. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2011, 1, 826−843. (7) Auer, S.; Frenkel, D. Quantitative Prediction Of CrystalNucleation Rates For Spherical Colloids: A Computational Approach. Annu. Rev. Phys. Chem. 2004, 55, 333−361. (8) Auer, S.; Frenkel, D. Numerical prediction of absolute crystallization rates in hard-sphere colloids. J. Chem. Phys. 2004, 120, 3015−3029. (9) Kumar, A.; Molinero, V. Self - assembly of Mesophases from Nanoparticles. J. Phys. Chem. Lett. 2017, 8, 5053−5058. (10) Groot, R. D.; Warren, P. B. Dissipative Particle Dynamics: Bridging the Gap between Atomistic and Mesoscopic Simulation. J. Chem. Phys. 1997, 107, 4423. (11) Hashimoto, T.; Sakamoto, N. Nucleation and Anisotropic Growth of Lamellar Microdomains in Block Copolymers. Macromolecules 1995, 28, 4779−4781. (12) Hashimoto, T.; Sakamoto, N.; Koga, T. Nucleation and growth of anisotropic grain in block copolymers near order-disorder transition. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1996, 54, 5832−5835. (13) Carilli, M. F.; Delaney, K. T.; Fredrickson, G. H. Nucleation of the lamellar phase from the disordered phase of the renormalized Landau-Brazovskii model. J. Chem. Phys. 2018, 148, 054903. (14) Pike, D. Q.; Detcheverry, F. A.; Müller, M.; De Pablo, J. J. Theoretically informed coarse grain simulations of polymeric systems. J. Chem. Phys. 2009, 131, 084903. (15) Groot, R. D.; Madden, T. J. Dynamic simulation of diblock copolymer microphase separation. J. Chem. Phys. 1998, 108, 8713. (16) Gavrilov, A. A.; Kudryavtsev, Y. V.; Chertovich, A. V. Phase diagrams of block copolymer melts by dissipative particle dynamics simulations. J. Chem. Phys. 2013, 139, 224901. (17) Sun, Y.; Padmanabhan, P.; Misra, M.; Escobedo, F. A. Molecular dynamics simulation of thermotropic bolaamphiphiles with a swallow-tail lateral chain: formation of cubic network phases. Soft Matter 2017, 13, 8542−8555. (18) Sandhu, P.; Zong, J.; Yang, D.; Wang, Q. On the comparisons between dissipative particle dynamics simulations and self-consistent field calculations of diblock copolymer microphase separation. J. Chem. Phys. 2013, 138, 194904. (19) Plimpton, S. Fast Parallel Algorithms for Short − Range Molecular Dynamics. J. Comput. Phys. 1995, 117, 1−19. (20) Martínez-Veracoechea, F. J.; Escobedo, F. A. Simulation of the gyroid phase in off-lattice models of pure diblock copolymer melts. J. Chem. Phys. 2006, 125, 104907. (21) Cheng, X.; et al. Calamitic Bolaamphiphiles with (Semi) Perfluorinated Lateral Chains : Polyphilic Block Molecules with New Liquid Crystalline Phase Structures. J. Am. Chem. Soc. 2003, 125 (36), 10977−10996. (22) Tschierske, C. Liquid crystal engineering − new complex mesophase structures and their relations to polymer morphologies, nanoscale patterning and crystal engineering. Chem. Soc. Rev. 2007, 36 (12), 1930−1970. (23) Poppe, S.; Prehm, M.; Zeng, X.; Ungar, G.; Tschierske, C. Tshaped bolapolyphiles with cubic LC phases. In the 13th European Conference on Liquid Crystals, Manchester, UK, September 2015. (24) Allen, R. J.; Valeriani, C.; ten Wolde, P. R. Forward flux sampling for rare event simulations. J. Phys.: Condens. Matter 2009, 21 (46), 463102. (25) Borrero, E. E.; Escobedo, F. A. Reaction coordinates and transition pathways of rare events via forward flux sampling. J. Chem. Phys. 2007, 127 (16), 164101. (26) Bolhuis, P. G.; Chandler, D.; Dellago, C.; Geissler, P. L. Transition path sampling: throwing ropes over rough mountain passes, in the dark. Annu. Rev. Phys. Chem. 2002, 53, 291−318.

(27) An, T. H.; et al. Solution self-Assembly of block copolymers containing a branched hydrophilic block into inverse bicontinuous cubic mesophases. ACS Nano 2015, 9, 3084−3096. (28) Matsen, M. W. Effect of architecture on the phase behavior of AB-type block copolymer melts. Macromolecules 2012, 45, 2161− 2165. (29) Jiang, W.; Qiang, Y.; Li, W.; Qiu, F.; Shi, A. C. Effects of Chain Topology on the Self-Assembly of AB-Type Block Copolymers. Macromolecules 2018, 51, 1529−1538. (30) Mondal, J.; Mahanthappa, M.; Yethiraj, A. Self-assembly of gemini surfactants: A computer simulation study. J. Phys. Chem. B 2013, 117, 4254−4262. (31) Siperstein, F. R.; Gubbins, K. E. Phase separation and liquid crystal self-assembly in surfactant-inorganic-solvent systems. Langmuir 2003, 19, 2049−2057. (32) Martinez-Veracoechea, F. J.; Escobedo, F. A. The Plumber’s Nightmare Phase in Diblock Copolymer/Homopolymer Blends. A Self-Consistent Field Theory Study. Macromolecules 2009, 42, 9058− 9062. (33) Martínez-Veracoechea, F. J.; Escobedo, F. A. Bicontinuous Phases in Diblock Copolymer/Homopolymer Blends: Simulation and Self-Consistent Field Theory. Macromolecules 2009, 42, 1775−1784.

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DOI: 10.1021/acs.macromol.8b01683 Macromolecules XXXX, XXX, XXX−XXX