Threshold Strain Value for Perpendicular Orientation in Dynamically Sheared Diblock Copolymers

The relationship between structure and rheology in a three-dimensional sheared lamellar mesophase

The evolution of a lamellar mesophase from an initially disordered state under shear is examined using simulations of a mesoscale model based on a concentration field ψ that distinguishes the hydrophilic and hydrophobic components. The Landau-Ginzburg free-energy functional is augmented by a term that is minimised for sinusoidal modulations in the concentration field with wavelength λ = (2π/k), and the dynamical equations are the model H equations. The structure and rheology are determined by the relative magnitudes of the diffusion time for coarsening, (λ²/D) and the inverse of the strain rate ^-1, and the Ericksen number, which is the ratio of the shear stress and the layer stiffness. When the diffusion time is small compared with the inverse of the strain rate, there is a local formation of misaligned layers, which are deformed by the imposed flow. There is near-perfect ordering with isolated defects at low values of the Ericksen number, but the defects result in a significant increase in viscosity due to the high layer stiffness. At high values of the Ericksen number, the concentration field is deformed by the mean shear before layers form via diffusion. Cylindrical structures aligned along the flow direction form after about 8-10 strain units, and these evolve into layers with disorder through diffusion perpendicular to the flow. The layers are not perfectly ordered, even after hundreds of strain units, due to the creation and destruction of defects via shear. The excess viscosity is low because the layer stiffness is small compared with the applied shear at a high Ericksen number. This study provides guidance on how the material parameters and imposed flow can be tailored to achieve the desired rheological behaviour.

Shear flow simulations of smectic liquid crystals based on the Gay–Berne fluid and the soft sphere string-fluid

Smectic liquid crystal undergoing shear flow.

Directed self-assembly of block copolymers: a tutorial review of strategies for enabling nanotechnology with soft matter

Self-assembly of soft materials is broadly considered an attractive means of generating nanoscale structures and patterns over large areas. However, the spontaneous formation of equilibrium nanostructures in response to temperature and concentration changes, for example, must be guided to yield the long-range order and orientation required for utility in a given scenario. In this review we examine directed self-assembly (DSA) of block copolymers (BCPs) as canonical examples of nanostructured soft matter systems which are additionally compelling for creating functional materials and devices. We survey well established and newly emerging DSA methods from a tutorial perspective. Special emphasis is given to exploring underlying physical phenomena, identifying prototypical BCPs that are compatible with different DSA techniques, describing experimental methods and highlighting the attractive functional properties of block copolymers overall. Finally we offer a brief perspective on some unresolved issues and future opportunities in this field.

Shear-induced topology changes in liquid crystals of the soybean lecithin/DDAB/water system

The viscoelastic behavior of the two different liquid crystalline lamellar phases and the liquid crystalline cubic phase of the mixed soybean lecithin/DDAB system in water was studied through rheology, with mechanical parameters studied as a function of composition. The swollen or diluted lamellar region is formed by vesicles, and its characteristic flow curve presents two-power law regions separated by a region where viscosity passes through a maximum. Yield stress and shear-dependent flow behavior were also observed. The microstructure suffers transformation under shear stress, and rheological response shifts from thixotropic to antithixotropic loops. Similar rheological behavior has been observed for samples in the collapsed or concentrated lamellar region, at the water-rich corner of the phase diagram. Vesicle formation may therefore occur by shearing the initial stacked and open bilayers. However, concentrated lamellar samples in the water-poor part of the phase diagram are less sensitive to shear effects and show plastic behavior and thixotropy. All lamellar samples manifest high elasticity. The dynamic responses of both lamellar topologies, i.e., vesicles and open bilayers, are comparable and exhibit an infinite relation time. The bicontinuous cubic, liquid crystalline phase is highly viscous. Its dynamic response cannot be modeled by a Maxwell model.

Orientation selection in lamellar phases by oscillatory shears

In order to address the selection mechanism that is responsible for the unique lamellar orientation observed in block copolymers under oscillatory shears, we use a constitutive law for the dissipative part of the stress tensor that respects the uniaxial symmetry of a lamellar phase. An interface separating two domains oriented parallel and perpendicular to the shear is shown to be hydrodynamically unstable, a situation analogous to the thin layer instability of stratified fluids under shears. The resulting secondary flows break the degeneracy between the parallel and perpendicular lamellar orientation, leading to a preferred perpendicular orientation in certain ranges of parameters of the polymer and of the shear.

Shear alignments in three-dimensional simulations of lamellar phases

Experiments studying layer orientation of sheared lamellar phases have consistently observed not only the seemingly obvious parallel orientation (with layers parallel to the shear plane), but also the so called perpendicular orientation (with layer normals along the vorticity direction) at the condition of higher shear frequencies and near the lamellar phase transition. We find that three-dimensional simulations of a deterministic mesoscopic dynamical equation (without thermal fluctuations) under the convection of simple shear flows have shown exactly such a dependence. The simulations show the important role played by the transverse orientation (layer normal along the velocity direction) and highlight the mechanism of the shear alignment being the competition between the shear frequency and the mesoscopic time scale of pattern organization.

Viscoelastic response of diblock copolymers to oscillatory shear

A mesoscopic model for diblock copolymers is combined with a simple rheological description of the viscoelastic contrast of its two phases. Under oscillatory shear the contrast generates secondary velocity fields and substantial deviations of volume-averaged flow parameters. The validity of our analytical findings is tested with three-dimensional numerical simulations. Furthermore, we consider the effect of advection by the generated flows on the stability of ordered lamellar states and propose a new criterion for the selection of orientations.

Shear-induced grain boundary motion for lamellar phases in the weakly nonlinear regime

We study the effect of an externally imposed oscillatory shear on the motion of a grain boundary that separates differently oriented domains of the lamellar phase of a diblock copolymer. A direct numerical solution of the Swift-Hohenberg equation in shear flow is used for the case of a transverse/parallel grain boundary in the limits of weak nonlinearity and low shear frequency. We focus on the region of parameters in which both transverse and parallel lamellae are linearly stable. Shearing leads to excess free energy in the transverse region relative to the parallel region, which is in turn dissipated by net motion of the boundary toward the transverse region. The observed boundary motion is a combination of rigid advection by the flow and order parameter diffusion. The latter includes break up and reconnection of lamellae, as well as a weak Eckhaus instability in the boundary region for sufficiently large strain amplitude that leads to slow wave number readjustment. The net average velocity is seen to increase with frequency and strain amplitude, and can be obtained by a multiple scale expansion of the governing equations.

Cell dynamics simulations of shear-induced alignment and defect annihilation in stripe patterns formed by block copolymers

The effect of large amplitude oscillatory shear on two-dimensional stripe patterns formed by block copolymers was investigated using cell dynamics simulations. A global orientational order parameter S and a correlation function for stripe normals G(r-r(')) were used to characterize the degree of stripe orientation under shear. The kinetics of stripe alignment, quantified by S, at various shear and quench conditions were studied as a function of strain amplitude, shear frequency, and temperature. The mechanisms of shear alignment and defect annihilation were investigated. A critical shear condition for complete alignment of stripes along the shear direction was also identified.