# Phase Behavior of Gradient Copolymer Melts with Different Gradient Strengths Revealed by Mesoscale Simulations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mesoscopic Modelling

#### 2.1. Gradient Copolymer Melts with Polydisperse Monomer Sequence

#### 2.2. Dissipative Particle Dynamics and Gradient Copolymer Chain Model

#### 2.3. Simulation Details

## 3. Results and Discussion

## 4. Conclusions

## Supplementary Materials

**a**) Average composition of generated sequences $\gamma \left(i\right)$ compared with target average composition profile $f\left(i\right)$; (

**b**) block size distribution ${\varphi}_{b}\left(i\right)$ and (

**c**) compositional polydispersity $\sigma \left(i\right)$ for gradient copolymer melts with tan-h profile and $C=5$. All variables are functions of the position of segment $i$ in the chain. Melts with different chain length $N$ are compared. Figure S2: (

**a**) Average composition of generated sequences $\gamma \left(i\right)$ compared with the target average composition profile $f\left(i\right)$; (

**b**) block size distribution ${\varphi}_{b}\left(i\right)$ and (

**c**) compositional polydispersity $\sigma \left(i\right)$ for gradient copolymer melts with tanh profile and $C=1$. All variables are functions of the position of segment $i$ in the chain. Melts with different chain length $N$ are compared. Figure S3: Simulation flowchart where symbol ${a}_{AB}$ represents the repulsion between unlike species, ${P}_{OP}$ the order parameter and $S\left(\mathit{q}\right)$ the structure factor. Inset shows evolution of the order parameter and its first derivative as a function of ${a}_{AB}$. Part of the flowchart related to initial and production runs in LAMMPS is also shown in Scheme S1. Figure S4: (

**a**) Flowchart to find the equilibrium structure by means of structure factor $S\left(\mathit{q}\right)$ and scaling of the simulation box length ${L}_{box}$. Symbol ${a}_{AB}$ represents the repulsion between unlike species and ${P}_{OP}$ the order parameter. (

**b**) Flowchart related to the application of the reverse non-equilibrium molecular dynamics method. The symbol $\gamma $ stands for shear rate, and ${v}_{x}$ and $z$ stand for the $x$ component of velocity $v$ relative to $z$ direction, respectively. Part of the flowchart related to the application of shear in LAMMPS is also shown in Scheme S2. Figure S5: Examples of equilibrium structures obtained by means of structure factor $S\left(\mathit{q}\right)$ and unit cell box size ${L}_{eq}$, applying the flowchart in Figure S4a for gradient copolymers with a gradient strength $C=3$ and $\overline{f}=0.7$. (

**a**) Structure factor $S\left(\mathit{q}\right)$ of the configuration with initial box lengths $L=40{r}_{c}$, and (

**b**) structure factor $S\left(\mathit{q}\right)$ of the same configuration with new box dimensions. Related snapshots are shown on the right side. $A$ segments are omitted for clarity. Figure S6: Examples of equilibrium structures obtained by reverse non-equilibrium molecular dynamics following the flowchart in Figure S4b for diblock copolymers with $\overline{f}=0.7$. (

**a**) Structure factor $S\left(\mathit{q}\right)$ of initially twisted cylinders. (

**b**) Structure factor $S\left(\mathit{q}\right)$ of equilibrium configuration of hexagonally packed cylinders. The equilibrium configuration placed on the right side is obtained by first aligning the cylinders by shear flow. Then, the shear is turned off and the system equilibrated for a sufficient number of steps. (

**c**) Linear velocity profile maintained during shearing. Figure S7: Diblock copolymer phase diagram shown in ${\chi}_{AB}-\overline{f}$ plane, where ${\chi}_{AB}$ is the Flory-Huggins interaction parameter between unlike beads and ${f}_{A}$ the fraction of $A$ segments in the copolymer chain. Symbols represent simulation points, where red circles stand for lamellae, green for gyroid, blue for hexagonally packed cylinders, and pink for spherical nanostructures, respectively. Open circles represent the disordered phase. Black dashed lines denote approximate phase boundaries. Figure S8: Snapshots of lamellar configurations (left column) and front view of hexagonally packed cylinders (right column) obtained in our simulations. From top to bottom, we show snapshots for diblock copolymers and gradient melts $C=5$, $C=3$ and $C=2$. Figure S9: Lamellar configurations of gradient copolymers with weak gradient strength $C=1$ and (

**a**) $\overline{f}=0.5$, (

**b**) $\overline{f}=0.55$, (

**c**) $\overline{f}=0.6,$ and (

**d**) $\overline{f}=0.65$ in their overall composition. Scheme S1: Simplified LAMMPS simulation scheme for equilibration and production runs of gradient copolymer melts. LAMMPS keywords are highlighted in bold, variables are displayed in blue, comments in green and other text, like names of input and output files, are slanted. Scheme S2: Simplified LAMMPS simulation scheme for application of shear flow on gradient copolymer melts. LAMMPS keywords are highlighted in bold, variables are displayed in blue, comments in green and other text, like names of input and output files, are slanted. References [31,32,34,35,39,40,41,42,43,47,48] are cited in the supplementary materials.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bates, F.S.; Fredrickson, G.H. Block copolymers—Designer soft materials. Phys. Today
**1999**, 52, 32–38. [Google Scholar] [CrossRef] - Feng, H.B.; Lu, X.Y.; Wang, W.Y.; Kang, N.G.; Mays, J.W. Block Copolymers: Synthesis, Self-Assembly, and Applications. Polymers
**2017**, 9, 494. [Google Scholar] [CrossRef] [PubMed] - Yin, J.; Chen, Y.; Zhang, Z.H.; Han, X. Stimuli-Responsive Block Copolymer-Based Assemblies for Cargo Delivery and Theranostic Applications. Polymers
**2016**, 8, 268. [Google Scholar] [CrossRef] [PubMed][Green Version] - Eggers, S.; Eckert, T.; Abetz, V. Double thermoresponsive block-random copolymers with adjustable phase transition temperatures: From block-like to gradient-like behavior. J. Polym. Sci. Pol. Chem.
**2018**, 56, 399–411. [Google Scholar] [CrossRef] - Rabyk, M.; Destephen, A.; Lapp, A.; King, S.; Noirez, L.; Billon, L.; Hruby, M.; Borisov, O.; Stepanek, P.; Deniau, E. Interplay of Thermosensitivity and pH Sensitivity of Amphiphilic Block-Gradient Copolymers of Dimethylaminoethyl Acrylate and Styrene. Macromolecules
**2018**, 51, 5219–5233. [Google Scholar] [CrossRef] - Cui, J.; Ma, Z.; Pan, L.; An, C.H.; Liu, J.; Zhou, Y.F.; Li, Y.S. Self-healable gradient copolymers. Mater. Chem. Front.
**2019**, 3, 464–471. [Google Scholar] [CrossRef] - Zhao, R.B.; Shea, K.J. Gradient Methylidene-Ethylidene Copolymer via C1 Polymerization: An Ersatz Gradient Ethylene-Propylene Copolymer. Acs Macro Lett.
**2015**, 4, 584–587. [Google Scholar] [CrossRef] - Matyjaszewski, K. Advanced Materials by Atom Transfer Radical Polymerization. Adv. Mater.
**2018**, 30, 1706441–1706462. [Google Scholar] [CrossRef] - Ogura, Y.; Takenaka, M.; Sawamoto, M.; Terashima, T. Fluorous Gradient Copolymers via in-Situ Transesterification of a Perfluoromethacrylate in Tandem Living Radical Polymerization: Precision Synthesis and Physical Properties. Macromolecules
**2018**, 51, 864–871. [Google Scholar] [CrossRef] - Posel, Z.; Svoboda, M.; Limpouchova, Z.; Lisal, M.; Prochazka, K. Adsorption of amphiphilic graft copolymers in solvents selective for the grafts on a lyophobic surface: A coarse-grained simulation study. Phys. Chem. Chem. Phys.
**2018**, 20, 6533–6547. [Google Scholar] - Wang, W.Y.; Lu, W.; Goodwin, A.; Wang, H.Q.; Yin, P.C.; Kang, N.G.; Hong, K.L.; Mays, J.W. Recent advances in thermoplastic elastomers from living polymerizations: Macromolecular architectures and supramolecular chemistry. Prog. Polym. Sci.
**2019**, 95, 1–31. [Google Scholar] [CrossRef] - Sigle, J.L.; Clough, A.; Zhou, J.; White, J.L. Controlling Macroscopic Properties by Tailoring Nanoscopic Interfaces in Tapered Copolymers. Macromolecules
**2015**, 48, 5714–5722. [Google Scholar] [CrossRef] - Hodrokoukes, P.; Floudas, G.; Pispas, S.; Hadjichristidis, N. Microphase separation in normal and inverse tapered block copolymers of polystyrene and polyisoprene. 1. Phase state. Macromolecules
**2001**, 34, 650–657. [Google Scholar] [CrossRef] - Alam, M.M.; Jack, K.S.; Hill, D.J.T.; Whittaker, A.K.; Peng, H. Gradient copolymers—Preparation, properties and practice. Eur. Polym. J.
**2019**, 116, 394–414. [Google Scholar] [CrossRef] - Kim, J.; Gray, M.K.; Zhou, H.Y.; Nguyen, S.T.; Torkelson, J.M. Polymer blend compatibilization by gradient copolymer addition during melt processing: Stabilization of dispersed phase to static coarsening. Macromolecules
**2005**, 38, 1037–1040. [Google Scholar] [CrossRef] - Tao, Y.; Kim, J.; Torkelson, J.M. Achievement of quasi-nano structured polymer blends by solid-state shear pulverization and compatibilization by gradient copolymer addition. Polymer
**2006**, 47, 6773–6781. [Google Scholar] [CrossRef] - Mok, M.M.; Kim, J.; Torkelson, J.M. Gradient copolymers with broad glass transition temperature regions: Design of purely interphase compositions for damping applications. J. Polym. Sci. Pol. Phys.
**2008**, 46, 48–58. [Google Scholar] [CrossRef] - Aksimentiev, A.; Holyst, R. Phase behavior of gradient copolymers. J. Chem. Phys.
**1999**, 111, 2329–2339. [Google Scholar] [CrossRef] - Lefebvre, M.D.; de la Cruz, M.O.; Shull, K.R. Phase segregation in gradient copolymer melts. Macromolecules
**2004**, 37, 1118–1123. [Google Scholar] [CrossRef] - Jiang, R.; Jin, Q.H.; Li, B.H.; Ding, D.T.; Wickham, R.A.; Shi, A.C. Phase behavior of gradient copolymers. Macromolecules
**2008**, 41, 5457–5465. [Google Scholar] [CrossRef] - Tito, N.B.; Milner, S.T.; Lipson, J.E.G. Self-Assembly of Lamellar Microphases in Linear Gradient Copolymer Melts. Macromolecules
**2010**, 43, 10612–10620. [Google Scholar] [CrossRef] - Mok, M.M.; Pujari, S.; Burghardt, W.R.; Dettmer, C.M.; Nguyen, S.T.; Ellison, C.J.; Torkelson, J.M. Microphase separation and shear alignment of gradient copolymers: Melt rheology and small-angle X-ray scattering analysis. Macromolecules
**2008**, 41, 5818–5829. [Google Scholar] [CrossRef] - Mok, M.M.; Kim, J.; Wong, C.L.H.; Marrou, S.R.; Woo, D.J.; Dettmer, C.M.; Nguyen, S.T.; Ellison, C.J.; Shull, K.R.; Torkelson, J.M. Glass Transition Breadths and Composition Profiles of Weakly, Moderately, and Strongly Segregating Gradient Copolymers: Experimental Results and Calculations from Self-Consistent Mean-Field Theory. Macromolecules
**2009**, 42, 7863–7876. [Google Scholar] [CrossRef] - Ganesan, V.; Kumar, N.A.; Pryamitsyn, V. Blockiness and Sequence Polydispersity Effects on the Phase Behavior and Interfacial Properties of Gradient Copolymers. Macromolecules
**2012**, 45, 6281–6297. [Google Scholar] [CrossRef] - Jiang, R.; Wang, Z.; Yin, Y.H.; Li, B.H.; Shi, A.C. Effects of compositional polydispersity on gradient copolymer melts. J. Chem. Phys.
**2013**, 138, 074906. [Google Scholar] [CrossRef] [PubMed] - Pandav, G.; Pryamitsyn, V.; Gallow, K.C.; Loo, Y.L.; Genzer, J.; Ganesan, V. Phase behavior of gradient copolymer solutions: A Monte Carlo simulation study. Soft Matter
**2012**, 8, 6471–6482. [Google Scholar] [CrossRef] - Pakula, T.; Matyjaszewski, K. Copolymers with controlled distribution of comonomers along the chain. 1. Structure, thermodynamics and dynamic properties of gradient copolymers. Computer simulation. Macromol. Theor. Simul.
**1996**, 5, 987–1006. [Google Scholar] [CrossRef] - Sun, D.C.; Guo, H.X. Monte Carlo Studies on the Interfacial Properties and Interfacial Structures of Ternary Symmetric Blends with Gradient Copolymers. J. Phys. Chem. B
**2012**, 116, 9512–9522. [Google Scholar] [CrossRef] [PubMed] - Sun, D.C.; Guo, H.X. Influence of compositional gradient on the phase behavior of ternary symmetric homopolymer-copolymer blends: A Monte Carlo study. Polymer
**2011**, 52, 5922–5932. [Google Scholar] [CrossRef] - Fredrickson, G.H.; Milner, S.T.; Leibler, L. Multicritical Phenomena and Microphase Ordering in Random Block Copolymer Melts. Macromolecules
**1992**, 25, 6341–6354. [Google Scholar] [CrossRef] - Skvor, J.; Posel, Z. Simulation Aspects of Lamellar Morphology: Incommensurability Effect. Macromol. Theor. Simul.
**2015**, 24, 141–151. [Google Scholar] [CrossRef] - Posel, Z.; Rousseau, B.; Lisal, M. Scaling behaviour of different polymer models in dissipative particle dynamics of unentangled melts. Mol. Simulat.
**2014**, 40, 1274–1289. [Google Scholar] [CrossRef] - Karatrantos, A.; Composto, R.J.; Winey, K.I.; Kroger, M.; Clarke, N. Modeling of Entangled Polymer Diffusion in Melts and Nanocomposites: A Review. Polymers
**2019**, 11, 876. [Google Scholar] [CrossRef][Green Version] - Posel, Z.; Posocco, P. Tuning the Properties of Nanogel Surfaces by Grafting Charged Alkylamine Brushes. Nanomaterials
**2019**, 9, 1514. [Google Scholar] [CrossRef] [PubMed][Green Version] - Guskova, O.A.; Seidel, C. Mesoscopic Simulations of Morphological Transitions of Stimuli-Responsive Diblock Copolymer Brushes. Macromolecules
**2011**, 44, 671–682. [Google Scholar] [CrossRef] - Posocco, P.; Hassan, Y.M.; Barandiaran, I.; Kortaberria, G.; Pricl, S.; Fermeglia, M. Combined Mesoscale/Experimental Study of Selective Placement of Magnetic Nanoparticles in Diblock Copolymer Films via Solvent Vapor Annealing. J. Phys. Chem. C
**2016**, 120, 7403–7411. [Google Scholar] [CrossRef] - Posel, Z.; Posocco, P.; Fermeglia, M.; Lisal, M.; Pricl, S. Modeling hierarchically structured nanoparticle/diblock copolymer systems. Soft Matter
**2013**, 9, 2936–2946. [Google Scholar] [CrossRef] - Posocco, P.; Posel, Z.; Fermeglia, M.; Lisal, M.; Pricl, S. A molecular simulation approach to the prediction of the morphology of self-assembled nanoparticles in diblock copolymers. J. Mater. Chem.
**2010**, 20, 10511–10520. [Google Scholar] [CrossRef] - Posel, Z.; Posocco, P.; Lisal, M.; Fermeglia, M.; Pricl, S. Highly grafted polystyrene/polyvinylpyridine polymer gold nanoparticles in a good solvent: Effects of chain length and composition. Soft Matter
**2016**, 12, 3600–3611. [Google Scholar] [CrossRef] - Karatrantos, A.; Clarke, N.; Kroger, M. Modeling of Polymer Structure and Conformations in Polymer Nanocomposites from Atomistic to Mesoscale: A Review. Polym. Rev.
**2016**, 56, 385–428. [Google Scholar] [CrossRef] - Posel, Z.; Svoboda, M.; Colina, C.M.; Lisal, M. Flow and aggregation of rod-like proteins in slit and cylindrical pores coated with polymer brushes: An insight from dissipative particle dynamics. Soft Matter
**2017**, 13, 1634–1645. [Google Scholar] [CrossRef] [PubMed][Green Version] - Espanol, P.; Warren, P.B. Perspective: Dissipative particle dynamics. J. Chem. Phys.
**2017**, 146, 150901. [Google Scholar] [CrossRef] - Groot, R.D.; Madden, T.J. Dynamic simulation of diblock copolymer microphase separation. J. Chem. Phys.
**1998**, 108, 8713–8724. [Google Scholar] [CrossRef] - Martinez-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. [Google Scholar] [CrossRef] - Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular-Dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef][Green Version] - Nguyen, T.D.; Plimpton, S.J. Accelerating dissipative particle dynamics simulations for soft matter systems. Comp. Mater. Sci
**2015**, 100, 173–180. [Google Scholar] [CrossRef][Green Version] - Beranek, P.; Posel, Z. Phase Behavior of Semiflexible-Flexible Diblock Copolymer Melt: Insight from Mesoscale Modeling. J. Nanosci. Nanotechno.
**2016**, 16, 7832–7835. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brown, J.R.; Sides, S.W.; Hall, L.M. Phase Behavior of Tapered Diblock Copolymers from Self-Consistent Field Theory. Acs Macro Lett.
**2013**, 2, 1105–1109. [Google Scholar] [CrossRef] - Brown, J.R.; Seo, Y.; Maula, T.A.D.; Hall, L.M. Fluids density functional theory and initializing molecular dynamics simulations of block copolymers. J. Chem. Phys.
**2016**, 144, 124904. [Google Scholar] [CrossRef] - Brown, J.R.; Seo, Y.M.; Sides, S.W.; Hall, L.M. Unique Phase Behavior of Inverse Tapered Block Copolymers: Self Consistent Field Theory and Molecular Dynamics Simulations. Macromolecules
**2017**, 50, 5619–5626. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Examples of copolymers with different monomer sequences along the chain. The left side shows bars with monomer density profiles and the right side displays possible monomer sequences in the chain. (

**b**) Monomer composition profile $f\left(i\right)$ for the $A$ segment in a copolymer chain with total length $N$. Figure shows all tan-h composition profiles considered here, with different gradient strengths ($C=\left\{1,2,3,5\right\}$) and composition profiles of corresponding diblock copolymers (DBC).

**Figure 2.**Gradient copolymer phase diagrams. Melts with C = {(

**a**) 5, (

**b**) 3, (

**c**) 2, (

**d**) 1)} are shown in the ${\chi}_{AB}N-\overline{f}$ plane, where ${\chi}_{AB}$ is the Flory-Huggins interaction parameter between unlike beads and $\overline{f}$ is the fraction of $A$ segments in the copolymer chain. Symbols represent simulation points, where red circle stands for lamellae, green circle for gyroid, blue circles for hexagonally packed cylinders, and pink circles for spherical nanostructures, respectively. Open circles represent the disordered phase. Black dashed lines denote approximate phase boundaries of corresponding diblock copolymers.

**Figure 3.**Distribution of chains $N\left({f}_{A}\right)$ as a function of the fraction of $A$ segments that become ${f}_{A}$ in copolymer chains for (

**a**) $\overline{f}=0.5$, (

**b**) $\overline{f}=0.7$ and (

**c**) $\overline{f}=0.9$. Only gradient copolymer melts with $C=\left\{5,1\right\}$ are shown by black and red bars, respectively. Green bar denotes corresponding diblock copolymers with $N\left({f}_{A}\right)=1$.

**Figure 4.**(

**a**) Order parameter ${P}_{OP}$ and (

**b**) chain mean-squared radius-of-gyration ${R}_{g}^{2}$ for the lamellar phase (LAM, $\overline{f}=0.5$, open circle), hexagonally packed cylinders (CYL, $\overline{f}=0.8$, open square), and the gyroid phase (GYR, $\overline{f}=0.5$, open triangle) for all the different melts considered here. Error bars within size of the symbol are not shown. (

**c**) Configurational snapshots of lamellar (top) and front view of hexagonally packed cylinder (bottom) assemblies with $\left({a}_{AB}{r}_{c}\right)/\left({k}_{B}T\right)=40$. Snapshots of diblock copolymers and gradient copolymers with $C=2$ are shown in left and right column, respectively. For clarity, only $A$ segments are displayed. Additional snapshots are reported in the Supplementary Materials in Figures S8 and S9.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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**

Beránek, P.; Posocco, P.; Posel, Z. Phase Behavior of Gradient Copolymer Melts with Different Gradient Strengths Revealed by Mesoscale Simulations. *Polymers* **2020**, *12*, 2462.
https://doi.org/10.3390/polym12112462

**AMA Style**

Beránek P, Posocco P, Posel Z. Phase Behavior of Gradient Copolymer Melts with Different Gradient Strengths Revealed by Mesoscale Simulations. *Polymers*. 2020; 12(11):2462.
https://doi.org/10.3390/polym12112462

**Chicago/Turabian Style**

Beránek, Pavel, Paola Posocco, and Zbyšek Posel. 2020. "Phase Behavior of Gradient Copolymer Melts with Different Gradient Strengths Revealed by Mesoscale Simulations" *Polymers* 12, no. 11: 2462.
https://doi.org/10.3390/polym12112462