Session A30: Focus Session: Block Copolymer Dynamics

A Renormalized Theory of Composition Fluctuations in Polymer Mixtures

Attempts to calculate corrections to Flory-Huggins theory using field theoretic methods have been plagued by an unwanted sensititivity to very short wavelength fluctuations, or to the value of an arbitrary cutoff length. This reflects the fact that the total free energy of a polymer mixture is dominated by effects of local fluid structure. Z.-G. Wang [1] has shown how the cutoff-dependence of a one-loop approximation for the free energy of a binary blend can be absorbed into an expression for the experimentally observed $\chi$ parameter, yielding a cutoff-independent renormalized theory of the contributions of mesoscopic fluctuations. We apply an analogous renormalization procedure to one-loop calculations of intramolecular and collective correlation functions at arbitrary wavenumbers in both homopolymer blends and copolymer melts, in which the dependence on local fluid structure is absorbed into the values of $\chi$ and of statistical segment lengths. We discuss predictions for homopolymer blends and diblock copolymer melts, and relationships to earlier work. [1] Z.-G. Wang, J. Chem. Phys. 117, 481 (2002). ~   

An efficient pseudo-spectral algorithm for the RPA response of ordered phases of block copolymer melts

We present a pseudo-spectral algorithm for calculating the linear response of monomer concentration field in ordered phases of block copolymer melts to small perturbations of the monomer chemical potential fields. The method is both considerably simpler and more efficient than the fully spectral perturbation theory presented by Laradji, Shi and coworkers.[1] The method will be used to re-examine the stability of the gyroid phase in diblock copolymer melts. \newline \newline [1] Laradji {\em et al.}, {\em Macromolecules}, {\bf 30}, 3242 (1997).   

Random isotropic structures and possible glass transitions in diblock copolymer melts

We study the microstructural glass transition in diblock-copolymer melts using a thermodynamic replica approach. Our approach performs an expansion in terms of the natural smallness parameter -- the inverse of the scaled degree of polymerization $\bar{N}$, which allows us to systematically study the approach to mean-field behavior as the degree of polymerization increases. We find that in the limit of infinite chain length, both the onset of glassiness and the vitrification transition (Kauzmann temperature) collapse to the mean-field spinodal, suggesting that the spinodal can be regarded as the mean-field signature for glass transitions in this class of microphase-separating system. We also study the order-disorder transitions (ODT) within the same theoretical framework; in particular, we include the leading-order fluctuation corrections due to the cubic interaction in the coarse-grained Hamiltonian, which has been ignored in previous studies on the ODT in block copolymers. We find that the cubic term stabilizes both the ordered (body-centered-cubic) phase and the glassy state relative to the disordered phase. In melts of symmetric copolymers the glass transition always occurs after the order-disorder transition (below the ODT temperature), but for asymmetric copolymers, it is possible for the glass transition to precede the ordering transition.   

Dynamics of PEO-PMMA diblock copolymers.

The structure and dynamics of a poly(ethylene oxide)--poly(methyl methacrylate) diblock copolymer (PEO-PMMA) are studied by molecular dynamics simulation using a united atom model, with emphasis on the junction point effect. These results are compared to those from a homopolymer blend. The intermolecular pair distribution function reveals that the internal packing of unlike segments is enhanced in the copolymer with respect to the blend. Thus the effective concentration of both PEO and PMMA decay towards the bulk faster as a function of local volume size in the copolymer than in the blend. The mean square displacement (MSD) illustrates differing mobilities of PEO and PMMA in both systems. In the copolymer, their mobilities are separated by less than in the blend. The junction point accelerates motion of PMMA up to a distance of 5A. In contrast, on both systems PEO mobility reach the average value close to the junction point. In spite of the different PMMA effective concentration on both systems copolymer and blend its dynamics are not affected by changes in local packing. In contrast PEO dynamics are strongly influenced by the differences on its effective concentration.   

Kinetics of Transition between HEX and Lamellar Phases in a triblock copolymer solution in a selective solvent.

Synchrotron based time-resolved small angle x-ray scattering (SAXS), was used to study the kinetics of ordering transition (OOT) between cylindrical micelles in HEX phase and lamellar (LAM) phase in a 0.4 (w/v) solution of a triblock of polystyrene (PS) and poly(ethylene-\textit{co}-butylene) (PEB), SEBS (PS-PEB-PS) copolymer in Dibutyl Phthalate (DBP), a selective solvent for the PS block. From a temperature ramp experiment the OOT was identified at about 137C and an ODT above 160 C. Several temperature jump experiments from HEX to LAM and the reverse were performed over the temperature range of 110-155C. Detailed analysis of the time evolution of the intensities of the Bragg peaks to follow the kinetics of the transition between HEX and LAM phases will be presented. A model to explain the transition mechanism will be discussed. This research was supported by NSF-DMR.   

Kinetics of HEX-BCC Transition of Cylinders to Spheres: Comparison of Time-resolved SAXS data with a Model of Coupled Anisotropic Fluctuations

The kinetics of the transition of HEX cylinders to BCC spheres was studied by coupling anisotropic fluctuations on cylinders, similar to the pearling instability, according to which the amplitude of a transverse wave along the length of the cylinder grows causing the cylinder to break up into spheroidal droplets. We find that the sphere BCC phase arises with phase shifts of 0, \textbf{\textit{4$\pi $/3}} and \textbf{\textit{8$\pi $/3}} for the sinusoidal waves on 3 neighboring cylinders on the HEX lattice, which correspond to the minimum of overlap volume of rippled cylinders and wavelength \textit{$\lambda $ }related to the nearest neighbor distance of the rippled cylinders by $d=2\sqrt 2 \lambda /3$. The azimuthally averaged scattering function from an un-oriented system of cylinders, as well as the 2-dimensional scattering from an oriented system was calculated with varying amplitude of the fluctuation. The results are in excellent agreement with time-resolved SAXS measurements of the kinetics of this transition in a Styrene (S)-ethylene-co-butylene (EB)-Styrene (S)) triblock copolymer in mineral oil, a selective solvent for the EB block.   

Modeling of twist grain boundaries in block copolymers: structure, stability, and motion

Twist grain boundaries, which are widely observed in block copolymer samples of lamellar phase, have been investigated through both direct numerical solution and multiscale analysis of a coarse-grained mesoscopic model equation. We show that the twist boundary profile can be well described by two sets of appropriate amplitude equations characterizing the slow evolution of lamellae. Stability of the grain boundary configuration has been examined, and our results show that the boundary width, albeit varying with twist angle, is of order $\epsilon^{-1/4}$, with $\epsilon$ the measure of the distance from the order-disorder threshold. We also study the motion of twist grain boundaries subjected to slow transversal modulations of lamellae, and obtain both analytically and numerically the traveling velocity of the boundary as well as its dependence on modulation wave number.   

Removal of non-equilibrium microdomain defects in block copolymer thin film simulations

In recent years, there has been increased interest in using microphase-separated block copolymer thin films as sub-optical lithographic masks in next generation semiconductor and magnetic media fabrication. However, if such techniques are to evolve into a useful and commercially feasible lithographic tool, one must have control over, or at least an understanding of the in-plane ordering of the block copolymer microdomains. In the context of self-consistent field theory (SCFT), we introduce new simulation techniques intended to efficiently remove non-equilibrium microdomain defects in 2D block copolymer simulations, and thus allow further study of equilibrium defect populations associated with 2D systems (cf., KTHNY theory) and/or defects induced by confinement. The first technique, which we call \emph{spectral amplitude filtering}, is used in parallel with a saddle point relaxation algorithm. Spectral amplitude filtering zeros out all Fourier components with amplitude below some fraction of the maximum amplitude. This encourages symmetries associated with the dominant Fourier modes to rapidly set in. The other technique is a variation on a force-biased, Fourier-accelerated Monte Carlo algorithm, which is shown to be efficient in controlled removal of non-equilibrium microdomain defects.   

Mixed Lamellae in Symmetric Diblock Copolymer Thin Films

For symmetric diblock copolymers confined between two flat and homogeneous surfaces, three morphologies (parallel, perpendicular, and mixed lamellae) have been obtained in experiments. While the effects of surface preference (for one of the two blocks) and film thickness on the thin-film morphology are well understood, less studied is the influence of a hard (impenetrable) surface on the copolymer chain conformations, referred to as the ``hard-surface effect''. It is this effect that favors the perpendicular lamellae between two neutral surfaces over parallel lamellae, at all film thicknesses. It also leads to the formation of mixed lamellae between asymmetric surfaces. Here we use the self-consistent field (SCF) theory in continuum to study the morphology of mixed lamellae. The SCF equations are solved in real space with high accuracy. Results under different boundary conditions (zero-density vs. non-flux) are compared to examine their effects on the copolymer chain conformations near the surface. We also study in detail the chain conformations at the T-junction in the mixed lamellae. Conditions under which the mixed lamellae are a stable phase (over parallel and perpendicular lamellae) are determined, and compared with experiments and Monte Carlo simulations.   

Correlated defect dynamics in block copolymer melts

With in-situ scanning force microscopy we image the ordering of cylindrical microdomains in a thin film of a diblock copolymer melt. Tracking the evolution of individual defects reveals their annihilation pathways via interfacial undulations and formation of transient phases, such as spheres and lamella. Repetitive transitions between distinct defect configurations suggest a cooperative movement of chain clusters. The microdomain dynamics is correlated on a length scale of several domain spacings. Characteristic times of structural relaxations range from $\sim $1 to $\sim $100 min.   

Dislocation Density and Orientational Order of Spherical Microdomains in Shear-Aligned Block Copolymer Thin Films

Studies of annealed monolayers of cylindrical diblock copolymers (striped patterns) showed that the disclination density dictated the orientational correlation length. Here we test the role of dislocations in the orientational order of shear aligned hexagonal patterns. Shearing block copolymer thin films using a viscous fluid overlayer creates long-range orientational order of the microdomains, extending over a centimeter or more, by eliminating grain boundaries. However, some isolated dislocations remain that perturb the both translational and local orientational order of the hexagonal lattice formed by the spherical microdomains in these thin films. Atomic force microscopy was used to image the lattice, and image analysis software was written to determine dislocation density and alignment quality using an orientational order parameter. In such shear- aligned films, a linear relationship between the density of isolated dislocations and the orientational order parameter was found.