# Demixing by a Nematic Mean Field: Coarse-Grained Simulations of Liquid Crystalline Polymers

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Simulation Approach

## 3. Results

#### 3.1. Pure Systems

#### 3.2. Mixtures

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Whitesides, G.M.; Grzybowski, B. Self-assembly at all scales. Science
**2002**, 295, 2418–2421. [Google Scholar] [CrossRef] [PubMed] - Kato, T. Self-Assembly of Phase-Segregated Liquid Crystal Structures. Science
**2002**, 295, 2414–2418. [Google Scholar] [CrossRef] [PubMed] - Larson, R.G. The Structure and Rheology of Complex Fluids; Oxford University Press: Oxford, UK, 1999. [Google Scholar]
- Israelachvili, J.N. Intermolecular and Surface Forces, 3rd ed.; Academic Press: New York, NY, USA, 2011. [Google Scholar]
- Hamley, I.W. The Physics of Block Copolymers; Oxford University Press: Oxford, UK, 1998. [Google Scholar]
- Horsch, M.A.; Zhang, Z.; Glotzer, S.C. Simulation studies of self-assembly of end-tethered nanorods in solution and role of rod aspect ratio and tether length. J. Chem. Phys.
**2006**, 125, 184903. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tang, J.; Jiang, Y.; Zhang, X.; Yan, D.; Chen, J.Z.Y. Phase Diagram of Rod-Coil Diblock Copolymer Melts. Macromolecules
**2015**, 48, 9060. [Google Scholar] [CrossRef] - De Gennes, P.G.; Prost, J. The Physics of Liquid Crystals; Clarendon Press: Oxford, UK, 1993. [Google Scholar]
- Bai, P.; Kim, M.I.; Xu, T. Thermally Controlled Morphologies in a Block Copolymer Supramolecule via Nonreversible Order–Order Transitions. Macromolecules
**2013**, 46, 5531. [Google Scholar] [CrossRef] - Li, X.; Armas-Perez, J.C.; Martinez-Gonzalez, J.A.; Liu, X.; Xie, H.; Bishop, C.; Hernandez-Ortiz, J.P.; Zhang, R.; de Pablo, J.J.; Nealey, P.F. Directed self-assembly of nematic liquid crystals on chemically patterned surfaces: Morphological states and transitions. Soft Matter
**2016**, 12, 8595–8605. [Google Scholar] [CrossRef] [PubMed] - Choo, Y.; Mahajan, L.H.; Gopinadhan, M.; Ndaya, D.; Deshmukh, P.; Kasi, R.M.; Osuji, C.O. Phase Behavior of Polylactide-Based Liquid Crystalline Brushlike Block Copolymers. Macromolecules
**2015**, 48, 8315–8322. [Google Scholar] [CrossRef] - Dutta, D.; Fruitwala, H.; Kohli, A.; Weiss, R.A. Polymer Blends Containing Liquid Crystals: A Review. Polym. Eng. Sci.
**1990**, 30, 1005–1018. [Google Scholar] [CrossRef] - Gemünden, P.; Poelking, C.; Kremer, K.; Andrienko, D.; Ch. Daoulas, K. Nematic Ordering, Conjugation, and Density of States of Soluble Polymeric Semiconductors. Macromolecules
**2013**, 46, 5762–5774. [Google Scholar] [CrossRef] - Lee, Y.; Gomez, E. Challenges and Opportunities in the Development of Conjugated Block Copolymers for Photovoltaics. Macromolecules
**2015**, 48, 7385–7395. [Google Scholar] [CrossRef] - Kipp, D.; Wodo, O.; Ganapathysubramanian, B.; Ganesan, V. Achieving Bicontinuous Microemulsion Like Morphologies in Organic Photovoltaics. ACS Macro Lett.
**2015**, 4, 266–270. [Google Scholar] [CrossRef] - Vezie, M.S.; Few, S.; Meager, I.; Pieridou, G.; Dorling, B.; Ashraf, R.S.; Goñi, A.R.; Bronstein, H.; McCulloch, I.; Hayes, S.C.; et al. Exploring the origin of high optical absorption in conjugated polymers. Nat. Mater.
**2016**, 15, 746–753. [Google Scholar] [CrossRef] [PubMed] - Wang, S.; Fabiano, S.; Himmelberger, S.; Puzinas, S.; Crispin, X.; Salleo, A.; Berggren, M. Experimental evidence that short-range intermolecular aggregation is sufficient for efficient charge transport in conjugated polymers. Proc. Natl. Acad. Sci. USA
**2015**, 112, 10599–10604. [Google Scholar] [CrossRef] [PubMed] - Kipp, D.; Mok, J.; Strzalka, J.; Darling, S.B.; Ganesan, V.; Verduzco, R. Rational Design of Thermally Stable, Bicontinuous Donor/Acceptor Morphologies with Conjugated Block Copolymer Additives. ACS Macro Lett.
**2015**, 4, 867–871. [Google Scholar] [CrossRef] - Darling, S.B. Block copolymers for photovoltaics. Energy Environ. Sci.
**2009**, 2, 1266–1273. [Google Scholar] [CrossRef] - Müller, M.; de Pablo, J.J. Computational approaches for the dynamics of structure formation in self-assembling polymeric materials. Annu. Rev. Mater. Res.
**2013**, 43, 1–34. [Google Scholar] [CrossRef] - Müller, M.; Sun, D.W. Directing the self-assembly of block copolymers into a metastable complex network phase via a deep and rapid quench. Phys. Rev. Lett.
**2013**, 111, 267801. [Google Scholar] [CrossRef] [PubMed] - Ten Bosch, A.; Maissa, P.; Sixou, P. A Landau-de Gennes theory of nematic polymers. J. Phys.
**1983**, 44, 105–111. [Google Scholar] [CrossRef] - Wang, X.J.; Warner, M. Theory of nematic backbone polymer phases and conformations. J. Phys. A Math. Gen.
**1986**, 19, 2215–2227. [Google Scholar] [CrossRef] - Holyst, R.; Schick, M. Mixtures of rigid and flexible nematogenic polymers. J. Chem. Phys.
**1992**, 96, 721–729. [Google Scholar] [CrossRef] - Liu, A.J.; Fredrickson, G.H. Free Energy Functionals for Semiflexible Polymer Solutions and Blends. Macromolecules
**1993**, 26, 2817–2824. [Google Scholar] [CrossRef] - Liu, A.J.; Fredrickson, G.H. Phase Separation Kinetics of Rod/Coil Mixtures. Macromolecules
**1996**, 29, 8000–8009. [Google Scholar] [CrossRef] - Escobedo, F.; de Pablo, J.J. Monte Carlo simulation of athermal mesogenic chains: Pure systems, mixtures, and constrained environments. J. Chem. Phys.
**1997**, 106, 9858–9868. [Google Scholar] [CrossRef] - Fukuda, J. Phase separation kinetics of liquid crystalline polymers: Effect of orientational order. Phys. Rev. E
**1999**, 59, 3275–3288. [Google Scholar] [CrossRef] - Reenders, M.; ten Brinke, G. Compositional and Orientational Ordering in Rod-Coil Diblock Copolymer Melts. Macromolecules
**2002**, 35, 3266–3280. [Google Scholar] [CrossRef] - Pryamitsyn, V.; Ganesan, V. Self-assembly of rod–coil block copolymers. J. Chem. Phys.
**2004**, 120, 5824–5838. [Google Scholar] [CrossRef] [PubMed] - Kipp, D.; Ganesan, V. Influence of Block Copolymer Compatibilizers on the Morphologies of Semiflexible Polymer/Solvent Blend. J. Phys. Chem. B
**2014**, 118, 4425–4441. [Google Scholar] [CrossRef] [PubMed] - Yang, S.; Liang, B. Simulation of Phase-Separated Structures of Liquid- Crystalline Polymer/Flexible Polymer Blends. J. Polym. Phys. B Polym. Phys.
**2001**, 39, 2915–2921. [Google Scholar] [CrossRef] - Hamm, M.; Goldbeck-Wood, G.; Zvelindovsky, A.V.; Sevink, G.J.A.; Fraaije, J.G.E.M. Structure formation in liquid crystalline polymers. J. Chem. Phys.
**2002**, 116, 3152–3161. [Google Scholar] [CrossRef] - Doi, M. Introduction to Polymer Physics; Clarendon Press: Oxford, UK, 1996. [Google Scholar]
- Detcheverry, F.A.; Pike, D.Q.; Nealey, P.F.; Müller, M.; de Pablo, J.J. Monte Carlo simulation of coarse grain polymeric systems. Phys. Rev. Lett.
**2009**, 102, 197801. [Google Scholar] [CrossRef] [PubMed] - Helfand, E. Theory of inhomogeneous polymers: Fundamentals of the Gaussian random-walk model. J. Chem. Phys.
**1975**, 62, 999–1005. [Google Scholar] [CrossRef] - Ch. Daoulas, K.; Rühle, V.; Kremer, K. Simulations of nematic homopolymer melts using particle-based models with interactions expressed through collective variables. J. Phys. Condens. Matter
**2012**, 24, 284121. [Google Scholar] [CrossRef] [PubMed] - Gemünden, P.; Ch. Daoulas, K. Fluctuation spectra in polymer nematics and Frank elastic constants: A coarse-grained modelling study. Soft Matter
**2015**, 11, 532–544. [Google Scholar] [CrossRef] [PubMed] - Frenkel, D.; Smit, B. Understanding Molecular Simulations, 2nd ed.; Academic Press: New York, NY, USA, 2002. [Google Scholar]
- Sigaud, G.; Yoon, D.Y.; Griffin, A.C. Order in Nematic Phase of Semiflexible Polymers. Macromolecules
**1983**, 16, 875–880. [Google Scholar] [CrossRef] - Zhang, W.; Gomez, E.D.; Milner, S.T. Predicting Nematic Phases of Semiflexible Polymers. Macromolecules
**2015**, 48, 1454–1462. [Google Scholar] [CrossRef] - Zhang, W.; Gomez, E.D.; Milner, S.T. Using surface-induced ordering to probe the isotropic-to-nematic transition for semiflexible polymers. Soft Matter
**2016**, 12, 6141–6147. [Google Scholar] [CrossRef] [PubMed] - Nakai, A.; Shiwaku, T.; Wang, W.; Hasegawa, H.; Hashimoto, T. Process and Mechanism of Phase Separation in Polymer Mixtures with a Thermotropic Liquid Crystalline Copolyester as One Component. Macromolecules
**1996**, 29, 5990–6001. [Google Scholar] [CrossRef] - Bates, C.M.; Bates, F.S. Block Polymers—Pure Potential. Macromolecules
**2017**, 50, 3–22. [Google Scholar] [CrossRef] - Humphrey, W.; Dalke, A.; Schulten, K.J. VMD—Visual Molecular Dynamics. Mol. Graphics
**1996**, 14, 33–38. [Google Scholar] [CrossRef] - Images Were Created Using Paraview, an Open-Source Scientific Visualization Software. Available online: http://www.paraview.org/ (accessed on 3 March 2017).

**Figure 1.**Global order parameter, S, as a function of the orientational coupling μ, obtained by Monte Carlo simulations (symbols). Polymer chains are composed of $N=8$ segments, and $\kappa =3.5$. Lines are only a guide to the eye. Insets are instantaneous polymer configurations in the isotropic and nematic phases, different chain colors are used to facilitate visualization.

**Figure 2.**Phase diagram in the parameter space (μ, N) for different degrees of flexibility, parametrized by κ, obtained by Monte Carlo simulations. Symbols indicate the transition values, ${\mu}_{t}$, at which the isotropic-nematic transition occurs. Lines are only a guide to the eye. The associated persistence lengths, from small to large κ values, are: ${\ell}_{p}/b\approx 1.6,2.5,3.0$ and $4.5$.

**Figure 3.**Relationship between the orientational coupling at the I-N transition, ${\mu}_{t}$, and the degree of flexibility, parametrized by κ, obtained by Monte Carlo simulations (symbols) at $N=24$. Green (dashed) line is the mean field prediction in the limit of very large N, ${\mu}_{t}\sim {\kappa}^{-1}$. The red (continuous) line is a fit to a power law ${\mu}_{t}\sim {\kappa}^{-\alpha}$, with exponent $\alpha \approx 0.84\pm 0.05$.

**Figure 4.**Phase diagrams in Figure 2 in terms of the renormalized parameters $\mu {\kappa}^{\alpha}$ and $Nb/{\ell}_{p}$.

**Figure 5.**Phase diagram in the parameter space (${f}_{\mathrm{coil}}$, ${N}_{\mathrm{coil}}$) for a mixture of rigid and flexible polymers, obtained by Monte Carlo simulations. Red circles indicate the points in the phase diagram that give rise to phase separated samples while black circles represent those conditions for which a single phase is obtained. Lines are only a guide to the eye and highlight the phase boundary, these lines were obtained by using splines on the simulation data. Symbols and continuous (blue) line are results for ${\kappa}_{\mathrm{coil}}=0.25$ (${\ell}_{p}\approx 0.44b$), ${\kappa}_{\mathrm{stiff}}=5.0$ (${\ell}_{p}\approx 4.5b$) and $\mu =4.0$. The dashed green line is the phase boundary obtained for a sample with less dissimilarity between polymers, characterized by ${\kappa}_{\mathrm{coil}}=2.0$ (${\ell}_{p}\approx 1.6b$), ${\kappa}_{\mathrm{stiff}}=5.0$ and $\mu =3.5$.

**Figure 6.**Three-dimensional cross-section of an instantaneous polymer configurations in the phase separated state, stiff polymer chains are shown in red and blue, segments of flexible chains are shown as yellow dots to facilitate visualization [45].

**Figure 7.**Iso-surface of the composition scalar parameter $\psi \left(\mathbf{r}\right)=0$ corresponding to the interface between the rigid and flexible domains (green surface). Red lines are the local nematic director $\widehat{\mathit{n}}\left(\mathbf{r}\right)$ where local ordering $S\left(\mathbf{r}\right)\ge 0.7$ [46].

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramírez-Hernández, A.; Hur, S.-M.; Armas-Pérez, J.C.; Cruz, M.O.d.l.; De Pablo, J.J.
Demixing by a Nematic Mean Field: Coarse-Grained Simulations of Liquid Crystalline Polymers. *Polymers* **2017**, *9*, 88.
https://doi.org/10.3390/polym9030088

**AMA Style**

Ramírez-Hernández A, Hur S-M, Armas-Pérez JC, Cruz MOdl, De Pablo JJ.
Demixing by a Nematic Mean Field: Coarse-Grained Simulations of Liquid Crystalline Polymers. *Polymers*. 2017; 9(3):88.
https://doi.org/10.3390/polym9030088

**Chicago/Turabian Style**

Ramírez-Hernández, Abelardo, Su-Mi Hur, Julio C. Armas-Pérez, Monica Olvera de la Cruz, and Juan J. De Pablo.
2017. "Demixing by a Nematic Mean Field: Coarse-Grained Simulations of Liquid Crystalline Polymers" *Polymers* 9, no. 3: 88.
https://doi.org/10.3390/polym9030088