# Identification of Some New Triply Periodic Mesophases from Molten Block Copolymers

## Abstract

**:**

## 1. Introduction

_{1}32 single gyroids [9,15,18], the centered rectangular C2mm cylinders [16], and a cubic bicontinuous P23 phase [16]. However, the morphologies reported so far are still limited for block copolymer systems. It seems that there are still many morphologies unexplored in block copolymers, as we recognize the emergence of the discontinuous mesophases with Pm3n (A15) and P4

_{2}/mnm (Frank–Kasper σ) symmetry in some copolymers via structural variations [19,20,21], and also the holey bicontinuous mesophases possessing I43d and Ia3d symmetry with high genera [22].

## 2. Theoretical Methods

_{0}+ W, where H

_{0}is given by Weiner measure of Gaussian chains as follows:

_{A}, ω

_{B}, and ω

_{C}in turn as the contour variable s passes through A, B, and then C blocks. It needs to be recalled that there is another function q

^{+}conjugate to q, so that q

^{+}starts reversely from the other chain end with ${q}^{+}(\overrightarrow{r},1)=1$.

_{p}are 12 and 4, respectively. This Bethe–Peierls mean-field energy is an improvement over Bragg–Williams (van der Waals) mean-field energy by taking the local packing of nearest neighbors into account [33]. The Cho–Sanchez model necessitates three homopolymer parameters such as ${\overline{\epsilon}}_{jj}$ for self-interactions, ${\sigma}_{j}$, and ${N}_{j}$. It is well known that a homopolymer with larger ${\overline{\epsilon}}_{jj}$ is denser and less compressible than one with smaller ${\overline{\epsilon}}_{jj}$. There is an additional parameter ${\overline{\epsilon}}_{ij}$ for cross i,j-interactions to describe mixture phase behaviors. Using the given EOS model, it was suggested that $\beta W\{{\overrightarrow{r}}_{j}\}={\displaystyle \int d\overrightarrow{r}}\beta {f}^{ni}({\widehat{\eta}}_{j}(\overrightarrow{r}))$, where ${f}^{ni}({\widehat{\eta}}_{j}(\overrightarrow{r}))$ is the localized non-Gaussian free energy (${f}^{ni}$ $\equiv \left({A}_{EV}+{A}_{nb}\right)/V$) per unit volume [25].

## 3. Results and Discussion

#### 3.1. Im3

^{3}lattice cells, and each chain contour was discretized into ${N}_{A}+{N}_{B}+{N}_{C}$ segments for A, B, and C blocks, respectively. The modified diffusion equation in Equation (3) along with Equations (7)–(9) were solved via the pseudospectral scheme [39]. The iteration at a given condition is continued until the variation of ${\omega}_{j}$’s is less than ~2.5 × 10

^{−6}. Using the single core of the Intel Xeon processor, it takes ~24 min per 1k iterations for the copolymer melts, and 40k iterations are necessary to reach the target tolerance. In the early stage, BCC stays, but eventually at the present compositions, there evolved a totally different and new morphology. Using Biovia Material Studio Mesodyn package, we visualized the 3-dimensional morphology of the copolymer melt, which is depicted in Figure 1. It is seen that the unit cell of the structure seems only 6 × 6 × 6${R}_{G}^{3}$ just as that of BCC. In our first look at the morphology, it possesses holey layers, not the dispersed micellar spheres.

_{1}3, Im3, I432, and I43m symmetry. For other body-centered I-type crystals suffer some systematic absence of various planes. In detail, Ia3 lacks 2, I4

_{1}32 does 4, I43d does 2 and 4, and Ia3d does 2, 4, 10, 12, etc. The periodicity, or equivalently the lateral unit cell length c is obtained as $c=2\pi /({q}_{1}/\sqrt{2})$ = $6{R}_{G}$, which is exactly identical to our visual inspection of the morphology.

^{3}lattice cells, and each chain contour was discretized into 40 + 60 segments for A and B blocks, respectively. The modified diffusion equation in Equation (3) along with Equations (4–6) were solved via the pseudospectral scheme [39]. The iteration at a given condition was continued until the incompressibility constraint (=$\sum {\varphi}_{i}}-1$) was less than $2.5\times {10}^{-7}$. In the same computational environment, it took ~13 min per 1k iterations for incompressible AB copolymer melts, and about 8.5k iterations were necessary in this case to reach the target tolerance. As was seen in Figure 4, the simulation was found to yield exactly the same morphology given in Figure 1 that we identify as Im3 symmetry for the ABC copolymer. The periodicity or the lateral unit cell length c was obtained as $c=2\pi /({q}_{1}/\sqrt{2})$ = $5{R}_{G}$, as it should. The correlation function ${S}_{AA}(q)$ for A block is given in Figure S1 as a Supplementary Material. It needs to be mentioned that our second trial with the initial density field generated using (130) reflection, whose contour plot is shown in Figure 3b, turns out that the evolved morphology was merely hexagonal P6/mm cylinders.

#### 3.2. Metatron’s Cube with Pn3m Symmetry

^{3}lattice cells, and each chain contour was discretized into 40 + 60 segments. Finite compressibility was turned off. The modified diffusion equation was solved and the iteration at a given condition was continued until the target function (=$\sum {\varphi}_{i}}-1$) was less than $2.5\times {10}^{-7}$. In the same computational environment, less than 5k iterations were necessary to reach the target tolerance.

#### 3.3. P432 Symmetry

^{3}lattice cells, and each chain contour was discretized into 40 + 60 segments for A and B blocks, respectively. Finite compressibility was again turned off. The iteration at a given condition was continued until the target function (=$\sum {\varphi}_{i}}-1$) was less than $2.5\times {10}^{-7}$. In this case, 20k iterations were necessary to attain the target accuracy.

#### 3.4. Equilibrium Periodicity and Free Energies in the Incompressible Picture

_{1}32 symmetry are also included. It is well known that double gyroids are the stable morphology at the given segregation level and composition.

## 4. Conclusions

_{j}-mers with N being the overall size of the chosen copolymers, field-theoretic simulations based on Edwards Gaussian random-walk approach are performed for our purposes. Without finite compressibility, Helfand’s conventional self-consistent field analysis is undertaken to evaluate the canonical partition function at its saddle point while ensuring the incompressibility constraint. In case of compressible copolymers, the recently developed analysis is undertaken to combine Helfand’s theory with a molecular equation-of-state model.

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Three-dimensional morphology of the evolved nanostructure from compressible ABC copolymer melt (

**a**) in the simulation box of 12 × 12 × 12${R}_{G}^{3}$ and (

**b**) in the unit cell of 6 × 6 × 6${R}_{G}^{3}$ (1/8th of the simulation box). B and C domains are represented by green and blue color, respectively, and A domain as the matrix is erased intentionally.

**Figure 2.**Correlation function ${S}_{BB}(q)$ for B block plotted against the scattering vector q for the morphology given in Figure 1.

**Figure 3.**Contour plots for generic surface equations for Im3 mesophase using (

**a**) (132) reflection and (

**b**) (130) reflection along with their equivalent ones.

**Figure 4.**Three-dimensional morphology of the Im3 evolved from the generic test reflection given in Equation (8). Its unit cell morphology is given in plot (

**a**) and its expansion in a box of 10 × 10 × 10${R}_{G}^{3}$ is shown in plot (

**b**). A domain is represented by red color and B domain as the matrix is intentionally removed.

**Figure 6.**Correlation function ${S}_{AA}(q)$ for A block plotted against the scattering vector q for the morphology evolved from Equation (12).

**Figure 7.**Three-dimensional morphology of the mesophase evolved from the generic test reflection given in Equation (12) in a simulation box (unit cell) of 5 × 5 × 5${R}_{G}^{3}$. Its image is depicted by using Mesodyn (

**a**) and also by Matlab (

**b**) just for comparison purposes. A domain as the dispersed phase is only drawn here.

**Figure 8.**Correlation function ${S}_{AA}(q)$ for A block plotted against the scattering vector q for the morphology evolved from C(±Y) surface in a periodic box of 10 × 10 × 10${R}_{G}^{3}$.

**Figure 9.**Three-dimensional morphology of the network mesophase evolved from C(±Y) surface in a periodic box of 10 × 10 × 10${R}_{G}^{3}$. The unit cell morphology is depicted in plot (

**a**) by using Matlab in four different angles and its 1/8th piece is shown in plot (

**b**) by using Mesodyn just to reveal the tripod connections of the channels. A domain as the dispersed phase is only drawn here.

**Table 1.**Sets of molecular parameters of A/B/C constituents composing ABC triblock copolymers based on Cho–Sanchez model.

Parameter | ${\overline{\mathit{\epsilon}}}_{\mathit{j}\mathit{j}}/\mathit{k}$ (K) | ${\mathit{\sigma}}_{\mathit{j}}$ (Å) | ${\mathit{N}}_{\mathit{j}}/{\mathit{M}}_{\mathit{j}}\cdot \mathit{\pi}{\mathit{\sigma}}_{\mathit{j}}^{3}/6$ (cm^{3}/g) | |
---|---|---|---|---|

Polymer | ||||

A | 4107 | 4.04 | 0.41857 | |

B | 3000 | |||

C | 3000 |

Multiplicity | Position | Discretized Coordinates | ${\mathit{\eta}}_{\mathit{A}}(\overrightarrow{\mathit{r}})$ |
---|---|---|---|

2a | 0,0,0 | 1,1,1 | 0.012331 |

1/2,1/2,1/2 | 9,9,9 | 0.012329 | |

6b | 0,1/2,1/2 | 1,9,9 | 0.34353 |

1/2,0,1/2 | 9,1,9 | 0.343527 | |

1/2,1/2,0 | 9,9,1 | 0.329676 | |

1/2,0,0 | 9,1,1 | 0.343811 | |

0,1/2,0 | 1,91, | 0.343831 | |

0,0,1/2 | 1,1,9 | 0.329386 | |

8c | 1/4,1/4,1/4 | 5,5,5 | 0.029919 |

1/4,3/4,3/4 | 5,13,13 | 0.344344 | |

3/4,1/4,3/4 | 13,5,13 | 0.344256 | |

3/4,3/4,1/4 | 13,13,5 | 0.030894 | |

3/4,3/4,3/4 | 13,13,13 | 0.029839 | |

3/4,1/4,1/4 | 13,5,5 | 0.344491 | |

1/4,3/4,1/4 | 5,13,5 | 0.344419 | |

1/4,1/4,3/4 | 5,5,13 | 0.030805 |

Multiplicity | Position | Discretized Coordinates | ${\mathit{\varphi}}_{\mathit{A}}(\overrightarrow{\mathit{r}})$ |
---|---|---|---|

2a | 0,0,0 | 9,9,9 | 0.924754 |

1/2,1/2,1/2 | 25,25,25 | 0.912959 | |

4b | 1/4,1/4,1/4 | 1,1,1 | 0.184598 |

1/4,3/4,3/4 | 17,17,17 | 0.109947 | |

3/4,1/4,3/4 | 17,1,17 | 0.109947 | |

3/4,3/4,1/4 | 1,17,17 | 0.109947 | |

4c | 3/4,3/4,3/4 | 17,17,17 | 0.890423 |

3/4,1/4,1/4 | 1,1,17 | 0.819492 | |

1/4,3/4,1/4 | 1,17,1 | 0.819492 | |

1/4,1/4,3/4 | 17,1,1 | 0.819492 |

Multiplicity | Position | Discretized Coordinates | ${\mathit{\varphi}}_{\mathit{A}}(\overrightarrow{\mathit{r}})$ |
---|---|---|---|

1a | 0,0,0 | 1,1,1 | 0.905336 |

1b | 1/2,1/2,1/2 | 17,17,17 | 0.890529 |

3c | 0,1/2,1/2 | 1,17,17 | 0.921184 |

1/2,0,1/2 | 17,1,17 | 0.86979 | |

1/2,1/2,0 | 17,17,1 | 0.903127 | |

3d | 1/2,0,0 | 17,1,1 | 0.882721 |

0,1/2,0 | 1,17,1 | 0.911596 | |

0,0,1/2 | 1,1,17 | 0.91643 |

**Table 5.**Comparison of the free energies for various morphologies of AB diblock copolymer melt at $N\chi $ = 14 and at ${\varphi}_{A}=0.4$.

Types of Copolymers | Morphology (Symmetry Group) | $\mathit{c}/{\mathit{R}}_{\mathit{G}}$ | $\mathit{\beta}\mathit{A}/\mathit{n}$ |
---|---|---|---|

AB | Double gyroids (Ia3d) | 8.727 | 3.2334 |

Fddd | 4.055 × 8.136 × 14.404 | 3.2358 | |

Im3 | 5.007 | 3.2364 | |

P6/mm (HEX) | 4.064 × 7.040 | 3.2370 | |

LAM | 3.553 | 3.2377 | |

P432 | 10.131 | 3.2408 | |

Metatron’s cube (Pn3m) | 5.304 | 3.2442 | |

BCC (Im3m) | 5.303 | 3.2442 | |

Double diamonds (Pn3m) | 5.475 | 3.2451 | |

Single gyroid (I4_{1}32) | 5.012 | 3.2461 | |

I43d | 8.800 | 3.2470 | |

Ia3d of g = 25 ^{b} | 8.910 | 3.2532 | |

Disorder | - | 3.3600 |

^{a}Fddd and P6/mm (HEX) morphologies require more than one lattice constant. Therefore, we included the optimized box dimensions in full for them.

^{b}g indicates the genus, which implies the number of independent holes on the dividing surface. It needs to be mentioned that double gyroids with the same Ia3d symmetry possess g = 5.

^{c}The target function for the incompressibility constraint ($\left|{\displaystyle \sum {\varphi}_{i}}-1\right|$) is less than 2.5 × 10

^{−7}for all the morphologies given in this table.

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**MDPI and ACS Style**

Cho, J.
Identification of Some New Triply Periodic Mesophases from Molten Block Copolymers. *Polymers* **2019**, *11*, 1081.
https://doi.org/10.3390/polym11061081

**AMA Style**

Cho J.
Identification of Some New Triply Periodic Mesophases from Molten Block Copolymers. *Polymers*. 2019; 11(6):1081.
https://doi.org/10.3390/polym11061081

**Chicago/Turabian Style**

Cho, Junhan.
2019. "Identification of Some New Triply Periodic Mesophases from Molten Block Copolymers" *Polymers* 11, no. 6: 1081.
https://doi.org/10.3390/polym11061081