Kinetics of Hexagonal Cylinders to Face-centered Cubic Spheres Transition of Triblock Copolymer in Selective Solvent: Brownian Dynamics Simulation

Kinetics of Hexagonal Cylinders to Face-centered Cubic Spheres Transition of Triblock Copolymer in Selective Solvent: Brownian Dynamics Simulation*

Minghai Li†, Yongsheng Liu, Rama Bansil‡ Department of Physics, Boston University, Boston, MA 02215, USA Abstract The kinetics of the transformation from the hexagonal packed cylinder (HEX) phase to the face-centered-cubic (FCC) phase was simulated using Brownian Dynamics for an ABA triblock copolymer in a selective solvent for the A block. The kinetics was obtained by instantaneously changing either the temperature of the system or the well-depth of the Lennard-Jones potential. Detailed analysis showed that the transformation occurred via a rippling mechanism. The simulation results indicated that the order-order transformation (OOT) was a nucleation and growth process when the temperature of the system instantly jumped from 0.8 to 0.5. The time evolution of the structure factor obtained by Fourier Transformation showed that the peak intensities of the HEX and FCC phases could be fit well by an Avrami equation. Introduction It is well known that block copolymers exhibit a rich phase diagram with different ordered phases, such as 3 dimensional (3D) body center cubic (BCC)/

• This work is a part of Ph.D dissertation of Minghai Li, Boston University, 2008. † Current address: Department of Mechanical Engineering and Division of Material Science and Engineering, Boston University, Boston, MA, 02215, USA. ‡ Author to whom correspondence should be addressed. Email: rb@bu.edu 2

face center cubic (FCC) and the more complicated Gyroid and other bicontinuous phases, 2D hexagonal packed cylinder (HEX), and 1D lamellar (LAM) phases.1-4 A variety of self assembled micellar domain shapes (spherical, cylindrical or planar sheets) can be obtained from a block copolymer by varying composition of and number of blocks, or by varying the polymer concentration, temperature and solvent selectivity in a block copolymer of fixed composition. Block copolymers, like lyotropic liquid crystals, offer a unique system to investigate transformations that simultaneously involve a change in the shape of the micellar domains and the symmetry of the underlying lattice, for example from HEX cylinder to BCC. While there are many studies of the equilibrium phase diagrams and thermodynamics of solvent mediated interactions in block copolymer systems, the kinetics is not so well understood. A few studies have been reported on the kinetics of the HEX cylinder to BCC sphere transition5-8 but to the best of our knowledge, there is no published report on the kinetics on order-order transformation (OOT) of HEX cylinders to FCC spheres. Computational simulation methods can provide the microscopic structural changes involved in the transformation between different phases. Several computational simulations using molecular dynamics (MD),9-12 discrete MD,13 Brownian Dynamics (BD),14-19 Monte Carlo,20-22 dissipative particle dynamics,23 and time-dependent Ginsburg-Landau.24 With this view we have undertaken a computational study on the kinetics of cylinders to spheres transition using Brownian Dynamics. Brownian Dynamics is particularly suited for simulating polymer solutions because it correctly models the Langevin dynamics for 3

describing diffusion.15 The solvent is treated implicitly. In the simulation, the system is coarse-grained such that the elemental unit is not a single molecule or even a single monomer of the polymer, but rather a sphere representing the center of the mass of a cluster of many molecules. This sphere (denoted as monomer or bead in the later text) moves according to Newton laws of motion. There are two time scales in the polymer solution system: the short time scale of the motion of the solvent molecules whose mass is much less than that of the coarse-grained polymeric monomer, and the long time scale of the motion of the polymeric monomers. Brownian dynamics only simulates the longer time scale of polymeric monomers and not the short time scale of the solvent motion. Thus compared to all atom molecular dynamics, BD is more efficient and saves computational time in simulating the polymer solution system. For example, BD methods have been used for simulating polymer flow,16 phase diagram in surfactants modeled as sphere tethered to a chain15,17 and in block copolymer melts,14 solution,18 and polymer brushes systems.11,19 To the best of our knowledge, BD simulation has not been reported to study the kinetics of the HEX cylinders to FCC or BCC spheres transition for block copolymer in a selective solvent system.
Solvent selectivity further enriches the phase map and behavior of the block copolymers.25 It is well known that in tri-blocks it is possible to obtain either isolated or bridged micelles depending on whether the solvent prefers the outer A block or th

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