Monte Carlo Study of Patchy Nanostructures Self-Assembled from a Single Multiblock Chain
Krajniak Jakub, Banaszak Michał *
Faculty of Physics, A. Mickiewicz University
ul. Umultowska 85, 61-614 Poznan, Poland
∗E-mail: mbanasz@amu.edu.pl
Received:
Received: 31 January 2013; revised: 09 May 2013; accepted: 16 May 2013; published online: 23 July 2013
DOI: 10.12921/cmst.2013.19.03.137-143
OAI: oai:lib.psnc.pl:449
Abstract:
We present a lattice Monte Carlo simulation for a multiblock copolymer chain of length N=240 and microarchitecture (10 − 10)12. The simulation was performed using the Monte Carlo method with the Metropolis algorithm. We measured average energy, heat capacity, the mean squared radius of gyration, and the histogram of cluster count distribution. Those quantities were investigated as a function of temperature and incompatibility between segments, quantified by parameter ω. We determined the temperature of the coil-globule transition and constructed the phase diagram exhibiting a variety of patchy nanostructures. The presented results yield a qualitative agreement with those of the off-lattice Monte Carlo method reported earlier, with a significant exception for small incompatibilities, ω, and low temperatures, where 3-cluster patchy nanostructures are observed in contrast to the 2-cluster structures observed for the off-lattice (10 − 10)12 chain. We attribute this difference to a considerable stiffness of lattice chains in comparison to that of the off-lattice chains.
References:
[1] I. W. Hamley, Developments in Block Copolymer Science and Technology, John Wiley & Sons, Berlin, 2004.
[2] L. Leibler, Theory of Microphase Separation in Block Copolymers, Macromolecules 13, 1602 (1980).
[3] M. W. Matsen and M. Schick, Lamellar phase of a symmetric triblock copolymer, Macromolecules 27, 187 (1994).
[4] K. Binder and W. Paul, Recent Developments in Monte Carlo Simulations of Lattice Models for Polymer Systems, Macromolecules 41, 4537 (2008).
[5] M. Banaszak, S. Wołoszczuk, T. Pakula, and S. Jurga, Computer simulation of structure and microphase separation in model A-B-A triblock copolymers, Phys. Rev. E. 66, 031804 (2002).
[6] M. Banaszak, S. Wołoszczuk, S. Jurga, and T. Pakula, Lamellar ordering in computer-simulated block copolymer melts by a variety of thermal treatments, J. Chem. Phys. 119, 11451 (2003).
[7] A. Knolla and R. Magerleb, Phase behavior in thin films of cylinder-forming ABA block copolymers: Experiments, J. Chem. Phys. 120 (2004).
[8] Z. Zhang and S.C. Glotzer, Self-assembly of patchy particles, Nano. Lett. 4, 1408 (2004).
[9] P. A. Rupar, L. Chabanne, M. A. Winnik, and I. Manners, Non-Centrosymmetric Cylindrical Micelles by Unidirectional Growth, Science 337(6094), 559 (2012).
[10] J. Zhang, Z.-Y. Lu, and Z.-Y. Sun, A possible route to fabricate patchy nanoparticles via self-assembly of a multiblock copolymer chain in one step, Soft Matter 7, 9944 (2011).
[11] K. Lewandowski and M. Banaszak, Intraglobular structures in multiblock copolymer chains from a Monte Carlo simulation, Phys. Rev. E. 84, 011806 (2011).
[12] D. F. Parsons and D. R. M. William, Single Chains of Block Copolymers in Poor Solvents: Handshake, Spiral, and Lamellar Globules Formed by Geometric Frustration, Phys. Rev. Lett. 99, 228302 (2007).
[13] H. G. Katzgraber, S. Trebst, D. A. Huse, and M. Troyer, Feedback-optimized parallel tempering Monte Carlo, J. Stat. Mech., 03018 (2006).
[14] A. Sikorski, Properties of star-branched polymer chains. Application of the replica exchange Monte Carlo method, Macromolecules 35(18), 7132–7137 (2002).
[15] D. Gront and A. Kolinski, Efficient scheme for optimization of parallel tempering Monte Carlo method, J. Phys. Condens. Mat. 19(3) (2007).
[16] K. Lewandowski, P. Knychala, and M. Banaszak, Parallel-Tempering Monte-Carlo Simulation with Feedback-Optimized Algorithm Applied to a Coil-to-Globule Transition of a Lattice Homopolymer, CMST 16(1), 29–35 (2010).
[17] T. Beardsley and M. Matsen, Monte Carlo phase diagram for diblock copolymer melts, Eur. Phys. J. E 32, 255 (2010).
[18] F. Wang and D. P. Landau, Efficient, multiple-range random walk algorithm to calculate the density of states, Phys. Rev. Lett. 86(10), 2050 (2001).
[19] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21, 1087 (1953).
[20] N. Lesh, M. Mitzenmacher, and S. Whitesides, A Complete and Effective Move Set for Simplified Protein Folding, RECOMB ’03, 188 (2003).
[21] K. Lewandowski, P. Knychala, and M. Banaszak, Protein-like behavior of multiblock copolymer chains in a selective solvent by a variety of lattice and off-lattice Monte Carlo simulations, Phys. Stat. Sol. (b) 245, 2524 (2008).
[22] E. Shakhnovich, Protein Folding Thermodynamics and Dynamics: Where Physics, Chemistry, and Biology Meet, Chem. Rev. 106, 1559–1588 (2006).
We present a lattice Monte Carlo simulation for a multiblock copolymer chain of length N=240 and microarchitecture (10 − 10)12. The simulation was performed using the Monte Carlo method with the Metropolis algorithm. We measured average energy, heat capacity, the mean squared radius of gyration, and the histogram of cluster count distribution. Those quantities were investigated as a function of temperature and incompatibility between segments, quantified by parameter ω. We determined the temperature of the coil-globule transition and constructed the phase diagram exhibiting a variety of patchy nanostructures. The presented results yield a qualitative agreement with those of the off-lattice Monte Carlo method reported earlier, with a significant exception for small incompatibilities, ω, and low temperatures, where 3-cluster patchy nanostructures are observed in contrast to the 2-cluster structures observed for the off-lattice (10 − 10)12 chain. We attribute this difference to a considerable stiffness of lattice chains in comparison to that of the off-lattice chains.
[1] I. W. Hamley, Developments in Block Copolymer Science and Technology, John Wiley & Sons, Berlin, 2004.
[2] L. Leibler, Theory of Microphase Separation in Block Copolymers, Macromolecules 13, 1602 (1980).
[3] M. W. Matsen and M. Schick, Lamellar phase of a symmetric triblock copolymer, Macromolecules 27, 187 (1994).
[4] K. Binder and W. Paul, Recent Developments in Monte Carlo Simulations of Lattice Models for Polymer Systems, Macromolecules 41, 4537 (2008).
[5] M. Banaszak, S. Wołoszczuk, T. Pakula, and S. Jurga, Computer simulation of structure and microphase separation in model A-B-A triblock copolymers, Phys. Rev. E. 66, 031804 (2002).
[6] M. Banaszak, S. Wołoszczuk, S. Jurga, and T. Pakula, Lamellar ordering in computer-simulated block copolymer melts by a variety of thermal treatments, J. Chem. Phys. 119, 11451 (2003).
[7] A. Knolla and R. Magerleb, Phase behavior in thin films of cylinder-forming ABA block copolymers: Experiments, J. Chem. Phys. 120 (2004).
[8] Z. Zhang and S.C. Glotzer, Self-assembly of patchy particles, Nano. Lett. 4, 1408 (2004).
[9] P. A. Rupar, L. Chabanne, M. A. Winnik, and I. Manners, Non-Centrosymmetric Cylindrical Micelles by Unidirectional Growth, Science 337(6094), 559 (2012).
[10] J. Zhang, Z.-Y. Lu, and Z.-Y. Sun, A possible route to fabricate patchy nanoparticles via self-assembly of a multiblock copolymer chain in one step, Soft Matter 7, 9944 (2011).
[11] K. Lewandowski and M. Banaszak, Intraglobular structures in multiblock copolymer chains from a Monte Carlo simulation, Phys. Rev. E. 84, 011806 (2011).
[12] D. F. Parsons and D. R. M. William, Single Chains of Block Copolymers in Poor Solvents: Handshake, Spiral, and Lamellar Globules Formed by Geometric Frustration, Phys. Rev. Lett. 99, 228302 (2007).
[13] H. G. Katzgraber, S. Trebst, D. A. Huse, and M. Troyer, Feedback-optimized parallel tempering Monte Carlo, J. Stat. Mech., 03018 (2006).
[14] A. Sikorski, Properties of star-branched polymer chains. Application of the replica exchange Monte Carlo method, Macromolecules 35(18), 7132–7137 (2002).
[15] D. Gront and A. Kolinski, Efficient scheme for optimization of parallel tempering Monte Carlo method, J. Phys. Condens. Mat. 19(3) (2007).
[16] K. Lewandowski, P. Knychala, and M. Banaszak, Parallel-Tempering Monte-Carlo Simulation with Feedback-Optimized Algorithm Applied to a Coil-to-Globule Transition of a Lattice Homopolymer, CMST 16(1), 29–35 (2010).
[17] T. Beardsley and M. Matsen, Monte Carlo phase diagram for diblock copolymer melts, Eur. Phys. J. E 32, 255 (2010).
[18] F. Wang and D. P. Landau, Efficient, multiple-range random walk algorithm to calculate the density of states, Phys. Rev. Lett. 86(10), 2050 (2001).
[19] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21, 1087 (1953).
[20] N. Lesh, M. Mitzenmacher, and S. Whitesides, A Complete and Effective Move Set for Simplified Protein Folding, RECOMB ’03, 188 (2003).
[21] K. Lewandowski, P. Knychala, and M. Banaszak, Protein-like behavior of multiblock copolymer chains in a selective solvent by a variety of lattice and off-lattice Monte Carlo simulations, Phys. Stat. Sol. (b) 245, 2524 (2008).
[22] E. Shakhnovich, Protein Folding Thermodynamics and Dynamics: Where Physics, Chemistry, and Biology Meet, Chem. Rev. 106, 1559–1588 (2006).
