Ordering of block copolymer melts in confined geometry

Wrinkling pattern formation with periodic nematic orientation: From egg cartons to corrugated surfaces

Egg cartons, known as doubly sinusoidal surfaces, display a rich variety of saddles-cylinder-spherical patches organized with different spatial symmetries and connectivities. Egg carton surfaces, rich in functionalities, are observed in synthetic and biological materials, as well as across atomic and macroscopic scales. In this work we use the liquid crystal shape equation in the absence of elastic effects and normal stress jumps to predict and classify a family of uniaxial, equibiaxial, and biaxial egg cartons, according to the periodicities of the surface director field in nematic (N) and cholesteric (N*) liquid crystals under the presence of anisotropic surface tension (anchoring). Egg carton surface shape periodic solutions to the nonlinear and linearized liquid crystal shape equations predict that the mean curvature is a linear function of the orthogonal (along the surface normal) splay and bend contributions. Mixtures of egg carton surfaces (uniaxial, equibiaxial, and biaxial) emerge according to the symmetries of the nonsingular director field, and the spatial distributions of the director escape into the third dimension; pure uniaxial egg cartons emerge when the director escape has linelike geometries and mixtures of egg cartons arise under source or sink orientation lattices. Orientation symmetry and permutation analysis are incorporated into a full curvature (Casorati, shape parameter, mean curvature, and Gaussian curvature) characterization. Under a fixed anchoring parameter, conditions for maximal nanoscale curvedness and microscale maximal shape gradient diversity are identified. The present results contribute to various pathways to surface pattern formation using intrinsic anisotropic interfacial tension.

Analyzing spinodal decomposition of an anisotropic fluid mixture

Spinodal decomposition leads to spontaneous fluctuations of the local concentration. In the early stage, the resulting pattern provides explicit information about the material properties of the mixture. In the case of two isotropic fluids, the static structure factor shows the characteristic ring shape. If one component is a liquid crystal, the pattern is typically anisotropic and the structure factor is more complex. Using numerical methods, we investigate how structure factors can be used to extract information about material properties like the diffusion constant or the isotropic and the anisotropic contributions to the interfacial tension. The method is based on momenta taken from structure factors in the early stage of the spinodal demixing. We perform phase field calculations for an isotropic and an anisotropic spinodal decomposition. A comparison of the extracted results with analytic values is made. The calculations show that linear modes dominate in the beginning of the growth process, while non-linear modes grow monotonously in the region of the k-space for which damping is predicted by the linearized theory. As long as non-linear modes are small enough, linearized theory can be applied to extract material properties from the structure factor.

Influence of substrate interaction and confinement on electric-field-induced transition in symmetric block-copolymer thin films

In the present work, we study morphologies arising due to competing substrate interaction, electric field, and confinement effects on a symmetric diblock copolymer. We employ a coarse-grained nonlocal Cahn-Hilliard phenomenological model taking into account the appropriate contributions of substrate interaction and electrostatic field. The proposed model couples the Ohta-Kawasaki functional with Maxwell equation of electrostatics, thus alleviating the need for any approximate solution used in previous studies. We calculate the phase diagram in electric-field-substrate strength space for different film thicknesses. In addition to identifying the presence of parallel, perpendicular, and mixed lamellae phases similar to analytical calculations, we also find a region in the phase diagram where hybrid morphologies (combination of two phases) coexist. These hybrid morphologies arise either solely due to substrate affinity and confinement or are induced due to the applied electric field. The dependence of the critical fields for transition between the various phases on substrate strength, film thickness, and dielectric contrast is discussed. Some preliminary 3D results are also presented to corroborate the presence of hybrid morphologies.

Microphase separation patterns in diblock copolymers on curved surfaces using a nonlocal Cahn-Hilliard equation

We investigate microphase separation patterns on curved surfaces in three-dimensional space by numerically solving a nonlocal Cahn-Hilliard equation for diblock copolymers. In our model, a curved surface is implicitly represented as the zero level set of a signed distance function. We employ a discrete narrow band grid that neighbors the curved surface. Using the closest point method, we apply a pseudo-Neumann boundary at the boundary of the computational domain. The boundary treatment allows us to replace the Laplace-Beltrami operator by the standard Laplacian operator. In particular, we can apply standard finite difference schemes in order to approximate the nonlocal Cahn-Hilliard equation in the discrete narrow band domain. We employ a type of unconditionally stable scheme, which was introduced by Eyre, and use the Jacobi iterative to solve the resulting implicit discrete system of equations. In addition, we use the minimum number of grid points for the discrete narrow band domain. Therefore, the algorithm is simple and fast. Numerous computational experiments are provided to study microphase separation patterns for diblock copolymers on curved surfaces in three-dimensional space.

Simulated annealing study of asymmetric diblock copolymer thin films

We report a simulated annealing study of the morphology of asymmetric diblock copolymer thin films confined between two homogeneous and identical surfaces. We have focused on copolymers that form a gyroidal morphology in the bulk. The morphological dependence of the confined films on the film thickness and the surface-polymer interaction has been systematically investigated. From the simulations it is found that much richer morphologies can form for the gyroid-forming asymmetric diblock copolymer thin films, in contrast to the lamella-forming symmetric and cylinder-forming asymmetric diblock copolymer films. Multiple morphological transitions induced by changing the film thickness and polymer-surface interactions are observed.

Reentrant ordering transition of asymmetric copolymer solution film confined between polymer-grafted surfaces

We study the equilibrium morphology of an asymmetric A-B diblock copolymer solution film confined between homopolymer-grafted substrates by using self-consistent-field calculations. We find that on decreasing the copolymer concentration, a reentrant structural transformation between hexagonal --> lamellar --> hexagonal phases occurs as a result of the competition between the wetting effect of the brush surface and the bulk phase behavior of the asymmetric copolymer driven by the A-B interfacial tension.

Phase behavior in thin films of cylinder-forming ABA block copolymers: Mesoscale modeling

The phase behavior of cylinder-forming ABA block copolymers in thin films is modeled in detail using dynamic density functional theory and compared with recent experiments on polystyrene-block-polybutadiene-block-polystyrene triblock copolymers. Deviations from the bulk structure, such as wetting layer, perforated lamella, and lamella, are identified as surface reconstructions. Their stability regions are determined by an interplay between surface fields and confinement effects. Our results give evidence for a general mechanism governing the phase behavior in thin films of modulated phases.

Role of dissimilar interfaces in thin films of cylinder-forming block copolymers

We study the effect of dissimilar interfaces on the phase behavior of cylinder forming block copolymers in thin films by means of dynamic density-functional theory. In this article, we show that dissimilarity of the interfaces induces hybrid structures. These structures appear when the surface fields at the two interfaces stabilize different surface structures and/or reconstructions. We propose a general classification of hybrid structures and give an unifying description of phase behavior of cylinder forming block copolymer films. Our results are consistent with experimental observations.

Lamellar phases confined in quasicylindrical pores: lattice model results

A two-dimensional (2D) vector lattice model of microemulsions is applied to study the structure of lamellar phases confined in long rectangular pores. One-point distribution functions are calculated within mean field approximation. The effects of pore geometry and surface fields are considered. A 2D analog of an onion phase is favored by a pore with strongly hydrophilic walls. For neutral walls, far from the phase boundaries, the lamellar phase is stable inside the pore. By contrast, close to the lamellar-tubular phase boundary a pore with neutral walls favors a 2D tubular phase. This is the analog of capillary condensation. In all cases the excess pressure is calculated as a function of the pore geometry.

Influence of confinement on the orientational phase transitions in the lamellar phase of a block-copolymer melt under shear flow

In this paper, we incorporate some real-system effects into the theory of orientational phase transitions under shear flow [M. E. Cates and S. T. Milner, Phys. Rev. Lett. 62 1856 (1989) and G. H. Fredrickson, J. Rheol. 38, 1045 (1994)]. In particular, we study the influence of the shear-cell boundaries on the orientation of the lamellar phase. We predict that at low shear rates, the parallel orientation appears to be stable. We show that there is a critical value of the shear rate at which the parallel orientation loses its stability and the perpendicular one appears immediately below the spinodal. We associate this transition with a crossover from the fluctuation to the mean-field behavior. At lower temperatures, the stability of the parallel orientation is restored. We find that the region of stability of the perpendicular orientation rapidly decreases as shear rate increases. This behavior might be misinterpreted as an additional perpendicular to parallel transition recently discussed in literature.

Polymer adsorption-driven self-assembly of nanostructures

Driven by prospective applications, there is much interest in developing materials that can perform specific functions in response to external conditions. One way to design such materials is to create systems which, in response to external inputs, can self-assemble to form structures that are functionally useful. This review focuses on the principles that can be employed to design macromolecules that when presented with an appropriate two-dimensional surface, will self-assemble to form nanostructures that may be functionally useful. We discuss three specific examples: (a) biomimetic recognition between polymers and patterned surfaces. (b) control and manipulation of nanomechanical motion generated by biopolymer adsorption and binding, and (c) creation of patterned nanostructuctures by exposing molten diblock copolymers to patterned surfaces. The discussion serves to illustrate how polymer sequence can be manipulated to affect self-assembly characteristics near adsorbing surfaces. The focus of this review is on theoretical and computational work aimed toward elucidating the principles underlying the phenomena pertinent to the three topics noted above. However, synergistic experiments are also described in the appropriate context.

Density mismatch in thin diblock copolymer films

Thin films of diblock copolymer subject to gravitational field are simulated by means of a cell dynamical system model. The difference in density of the two sides of the molecule and the presence of the field causes the formation of lamellar patterns with orientation parallel to the confining walls even when they are neutral. The concentration profile of those films is analyzed in the weak segregation regime and a functional form for the profile is proposed.

Thin diblock copolymer films on patterned surfaces: computer simulations and the Frenkel-Kontorowa model

We study by direct numerical integration of the dynamical evolution equation the equilibrium configuration of a diblock copolymer thin film melt on a patterned surface. The surface has a large number of stripes and the mismatch between the bulk diblock spacing and the stripe width is small. We investigate primarily the formation of small discommensurations in the incommensurate phase and compare the results with the predictions of an analogous model of solid-state physics, the Frenkel-Kontorowa model.