Stability of a block-copolymer lamella in a strong electric field

Effects of Local Order Parameter Dependent Transport Coefficient in Diblock Copolymers Under Applied Electric Fields

We present an approach for constructing thermodynamically consistent time-dependent models relevant to thin films of diblock copolymers in applied electric fields. The approach is based on the principles of linear irreversible thermodynamics and in this work, it is applied to study the effects of electric  fields on thin  films of incompressible diblock copolymers. Enforcement of local incompressibility constraint at all times leads to a local order parameter dependent transport coefficient in the model for the diblock copolymers. The dependence of transport coefficient on the local order parameter is used to relate it with diffusion constant of Rouse chains and leads to sensitivity of the model to initial conditions. Also, transient behavior is found to be a affected when compared with an ad hoc model assuming a constant transport coefficient. Numerical results such as electric  field induced alignment of lamellae domains due to the  field are found to be in qualitative agreements with experiments.This approach opens up a systematic way of developing kinetic models for simulating effects of electrolytes added to thin  films containing diblock copolymers in the presence of applied electric  fields.

Nonequilibrium Processes in Polymer Membrane Formation: Theory and Experiment

Porous polymer and copolymer membranes are useful for ultrafiltration of functional macromolecules, colloids, and water purification. In particular, block copolymer membranes offer a bottom-up approach to form isoporous membranes. To optimize permeability, selectivity, longevity, and cost, and to rationally design fabrication processes, direct insights into the spatiotemporal structure evolution are necessary. Because of a multitude of nonequilibrium processes in polymer membrane formation, theoretical predictions via continuum models and particle simulations remain a challenge. We compiled experimental observations and theoretical approaches for homo- and block copolymer membranes prepared by nonsolvent-induced phase separation and highlight the interplay of multiple nonequilibrium processes─evaporation, solvent-nonsolvent exchange, diffusion, hydrodynamic flow, viscoelasticity, macro- and microphase separation, and dynamic arrest─that dictates the complex structure of the membrane on different scales.

Solvent Effect on the Self-Assembly of a Thin Film Consisting of Y-Shaped Copolymer

The self-assembly of an amphiphilic Y-shaped copolymer consisting of two hydrophilic branches and one hydrophobic branch in a thin film is investigated under different conditions by virtue of mesoscopic computer modelling, accompanied by doping with a single solvent, doping with a binary solvent, and those solvent environments together with the introduction of confinement defined by various acting distances and influencing regions. A cylindrical micellar structure is maintained, as it is in the thin film with the doping of either 10% hydrophobic solvent or 10% hydrophilic solvent, whose structure consists of the hydrophobic core and hydrophilic shell. Attributed to the hydrophobicity/hydrophilia nature of the solvents, different solvents play an obvious role on the self-assembled structure, i.e., the hydrophobic solvent presents as a swelling effect, conversely, the hydrophilic solvent presents as a shrinking effect. Further, the synergistic effect of the binary solvents on the self-assembly produces the lowest values in both the average volumetric size and free energy density when the quantity of hydrophobic solvent and hydrophilic solvent is equivalent. Interestingly, the solvent effect becomes more pronounced under the existent of a confinement. When a lateral-oriented confinement is introduced, a periodically fluctuating change in the cylindrical size occurs in two near-wall regions, but the further addition of either hydrophobic or hydrophilic solvent can effectively eliminate such resulting hierarchical-sized cylinders and generate uniform small-sized cylinders. However, with the introduction of a horizontal-orientated confinement, the copolymers self-assemble into the spherical micellar structure. Moreover, the further addition of hydrophobic solvent leads to a decrease in the average size of micelles via coalescence mechanism, in contrast, the further addition of hydrophilic solvent causes an increase in the average size of micelles via splitting mechanism. These findings enrich our knowledge of the potential for the solvent effect on the self-assembly of amphiphilic copolymer system, and then provide theoretical supports on improving and regulating the mesoscopic structure of nanomaterials.

Nanoscale Resolution of Electric-field Induced Motion in Ionic Diblock Copolymer Thin Films

Understanding the responses of ionic block copolymers to applied electric fields is crucial when targeting applications in areas such as energy storage, microelectronics, and transducers. This work shows that the identity of counterions in ionic diblock copolymers substantially affects their responses to electric fields, demonstrating the importance of ionic species for materials design. In situ neutron reflectometry measurements revealed that thin films containing imidazolium based cationic diblock copolymers, tetrafluoroborate counteranions led to film contraction under applied electric fields, while bromide counteranions drove expansion under similar field strengths. Coarse-grained molecular dynamics simulations were used to develop a fundamental understanding of these responses, uncovering a nonmonotonic trend in thickness change as a function of field strength. This behavior was attributed to elastic responses of microphase separated diblock copolymer chains resulting from variations in interfacial tension of polymer-polymer interfaces due to the redistribution of counteranions in the presence of electric fields.

Self-assembled morphologies of an amphiphilic Y-shaped weak polyelectrolyte in a thin film

Different from the self-assembly of neutral polymers, polyelectrolytes self-assemble into smaller aggregates with a more loosely assembled structure, which results from the repulsive forces acting between similar electrical compositions with the introduction of ions. The Y-shaped weak polyelectrolytes self-assemble into a core-shell type cylindrical structure with a hexagonal arrangement in a thin film, whose thickness is smaller than the gyration radius of the polymer chain. The corresponding formation mechanism consists of enrichment of the same components, adjustment of the shape of the aggregate, and the subsequent separation into individual aggregates. With the increase in the thickness of the thin film until it exceeds the gyration radius of the polymer chain, combined with the greater freedom of movement along the direction of thin film thickness, the self-assembled structure changes into a micellar structure. Under confinement, the repulsive force to the polymeric components is weakened by the repulsive forces among polyelectrolyte components with like charges, and this helps in generating aggregates with more uniform size and density distribution. In particular, when the repulsive force between the walls and the core forming components is greater than that between the walls and the shell forming components, such asymmetric confinement produces a crossed-cylindrical structure with nearly perpendicular arrangement of two cylinder arrays. Similarly, a novel three-crossed cylinder morphology is self-assembled upon removal of confinement.

One-dimensional Confinement Effect on the Self-assembly of Symmetric H-shaped Copolymers in a Thin Film

The self-assembly of a reformed symmetric H-shaped copolymer with four hydrophilic branches and one hydrophobic stem was systematically investigated. The existence of vacancies is vital to regulate the sizes of self-assembled cylinders to be able to form a hexagonal arrangement. With the introduction of horizontal-orientated confinement, a micellar structure is formed through a coalescence mechanism. The short acting distance and large influencing area of the confinement produces numerous small-sized micelles. Additionally, the cycled “contraction-expansion” change helps achieve hexagonal arrangement. In contrast, the introduction of lateral-oriented confinement with long acting distance and small influencing area cannot change the cylindrical structure. Under the fission mechanism, in which the larger cylinder splits into smaller ones, it is quite efficient to generate hierarchical-sized cylinders from larger-sized cylinders in the middle region and smaller-sized cylinders near both walls. The results indicate the possibility of regulating the characteristics of a nanomaterial by tuning the molecular structure of the copolymer and the parameters of the introduced confinement, which are closely related to the self-assembly structure.

Ordering block copolymers with structured electrodes

We study the kinetics of alignment and registration of block copolymers in an inhomogeneous electric field by computer simulations of a soft, coarse-grained model. The two blocks of the symmetric diblock copolymers are characterized by different dielectric constants. First, we demonstrate that a combination of graphoepitaxy and a homogeneous electric field extends the maximal distance between the topographical guiding patterns that result in defect-free ordering compared to graphoepitaxy alone. In a second study, the electric field in the thin block copolymer film is fabricated by spatially structured electrodes on an isolating substrate arranged in a one-dimensional periodic array; no additional topographical guiding patterns are applied. The dielectrophoretic effect induces long-range orientational order of the lamellae and, additionally, registers the lamellar structure with the electrodes due to the field inhomogeneities at the edges of the structured electrodes. Thus, orientational and translational order is established by the inhomogeneous electric field. The simulations identify a process protocol of time-dependent electric potentials that suppresses defect formation by initially forming a sandwich-like structure and subsequently reorienting these lying into standing lamellae that are registered with the structure of the electrodes. This process-directed self-assembly results in large defect-free arrays of aligned and registered lamellae using electrodes with a saw-tooth period of 4 lamellar periodicities, L₀, and a spacing of 10L₀.

Electric-field-induced lamellar to hexagonally perforated lamellar transition in diblock copolymer thin films: kinetic pathways

Symmetric block-copolymers, hitherto, are well known to evolve into parallel, perpendicular and mixed lamellar morphologies under the concomitant influence of an electric field and substrate affinity. In the present work, we show that an additional imposed confinement can effectuate a novel parallel lamellar to hexagonally perforated lamellar (HPL) transition in monolayer and bilayer films. Three dimensional numerical studies are performed using the Ohta-Kawasaki functional, complemented with an exact solution of Maxwell's equation. HPL is shown to stabilize at large substrate affinity in a narrow region of the phase diagram between parallel and perpendicular lamellar transitions in ultra-thin films. Additionally, we also identify perforated lamellae as intermediate structures during parallel-to-perpendicular lamellar transition. A systematic analysis using Minkowski functionals yields deeper insights into the associated kinetic pathways.

Instability and reorientation of block copolymer microstructure by imposed electric fields

The influence of electric fields on lamellar block copolymer microstructure is studied in the context of a density functional model and its sharp interface limit. A free boundary problem for domain interfaces of strongly segregated polymers is derived, which includes coupling of interface and electric field orientation. The linearized dynamics of lamellar configurations is computed in this context, leading to quantitative criteria for instability as a function of pattern wavelength, field magnitude, and orientation. Numerical simulations of the full model in two and three dimensions are used to study the nonlinear development of instabilities. In three dimensions, sufficiently large electric field magnitude always leads to instability. In two dimensions, the field has either stabilizing or destabilizing effects depending on the misorientation of the field and pattern. Even when linear instabilities are present, the dynamics can lead to stable corrugated domain interfaces which do not align with the electric field. Sufficiently high field strengths, on the other hand, produce topological rearrangement which may lead to alignment.

Phase diagrams of diblock copolymers in electric fields: a self-consistent field theory study

We investigated the phase diagrams of diblock copolymers in external electrostatic fields by using real-space self-consistent field theory. The lamella, cylinder, sphere, and ellipsoid structures were observed and analyzed by their segment distributions, which were arranged to two types of phase diagrams to examine the phase behavior in weak and strong electric fields. One type was constructed on the basis of Flory-Huggins interaction parameter and volume fraction. We identified an ellipsoid structure with a body-centered cuboid arrangement as a stable phase and discussed the shift of phase boundaries in the electric fields. The other type of phase diagrams was established on the basis of the dielectric constants of two blocks in the electric fields. We then determined the regions of ellipsoid phase in the phase diagrams to examine the influence of dielectric constants on the phase transition between ellipsoidal and hexagonally packed cylinder phases. A general agreement was obtained by comparing our results with those described in previous experimental and theoretical studies.

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.

Directed self-assembly of cylinder-forming diblock copolymers on sparse chemical patterns

Using both theory and experiment, we investigate the possibility of creating perfectly ordered block copolymer nanostructures on sparsely patterned substrates. Our study focuses on scrutinizing the appropriate pattern conditions to avoid undesired morphologies or defects when depositing cylinder-forming AB diblock copolymer thin films on the substrates which are mostly neutral with periodic stripe regions preferring the minority domain. By systematically exploring the parameter space using self-consistent field theory (SCFT), the optimal conditions for target phases are determined, and the effects of the chemical pattern period and the block copolymer film thickness on the target phase stability are also studied. Furthermore, as a sample experimental system, almost perfectly aligned polystyrene-block-poly(methyl methacrylate) (PS-b-PMMA) diblock copolymers are demonstrated. After the pattern transfer process, highly ordered Al nanodot arrays following the initial vertically aligned cylinder pattern are created. This systematic study demonstrates the ability to control the structure and the position of nanopatterns on sparse chemical patterns.

Electric field induced selective disordering in lamellar block copolymers

External electric fields align nanostructured block copolymers by either rotation of grains or nucleation and growth depending on how strongly the chemically distinct block copolymer components are segregated. In close vicinity to the order-disorder transition, theory and simulations suggest a third mechanism: selective disordering. We present a time-resolved small-angle X-ray scattering study that demonstrates how an electric field can indeed selectively disintegrate ill-aligned lamellae in a lyotropic block copolymer solution, while lamellae with interfaces oriented parallel to the applied field prevail. The present study adds an additional mechanism to the experimentally corroborated suite of mechanistic pathways, by which nanostructured block copolymers can align with an electric field. Our results further unveil the benefit of electric field assisted annealing for mitigating orientational disorder and topological defects in block copolymer mesophases, both in close vicinity to the order-disorder transition and well below it.

Phase segregation of a symmetric diblock copolymer in constrained space with a square-pillar array

In this study, we apply a self-consistent field theory of polymers to study the structures of a symmetric diblock copolymer in parallel substrates filled with square-pillar arrays in which the substrates and pillars exhibit a weak preference for one block of the copolymer. Three classes of structures, i.e., lamellae, perpendicular cylinders, and bicontinuous structures, are achieved by varying the polymer film thickness, the pillar pitch (the distance between two centers of the nearest neighboring pillars), the gap and rotation of the pillars. Because of the confinement along horizontal directions imposed by the pillar array, eight novel types of perpendicular lamellar structures and eight novel types of cylindrical structures with various shapes and distributions occur. In the hybridization states of the parallel and perpendicular lamellar structures, several novel bicontinuous structures such as the double-cylinder network, pseudo-lamellae, and perforated lamellar structure are also found. By comparing the free energies of the various possible structures, the antisymmetric parallel lamellae are observed to be stable with the larger pillar gap at a certain film thickness. The structural transformations between the alternating cylindrical structures (alternating cross-shaped, square-shaped, and octagonal perpendicular cylinders) and parallel lamellae with increasing film thickness or pillar gap are well explained by the modified strong separation theory. Our results indicate that array confinement can be an effective method to prepare novel polymeric nanopattern structures.

Domain patterns in a diblock copolymer-diblock copolymer mixture with oscillatory particles

We investigate the orientational order transition of striped patterns in microphase structures of diblock copolymer-diblock copolymer mixtures in the presence of periodic oscillatory particles. Under certain conditions, although the macrophase separation of a system is almost isotropic, microphase separation of one diblock copolymer takes place and becomes anisotropic gradually. By changing the oscillatory frequency and amplitude, the orientational order transition of a striped microphase structure from the state parallel to the oscillatory direction to the state perpendicular to the oscillatory direction is observed. We also find that the order transition occurs when we change the initial composition ratio. Furthermore, we examine the domain size and the orientational order parameter of microstructure in the process of orientational order transition. The results may provide guidance for experimentalists. This model system can also give a simple way to realize orientational order transition of soft materials by changing the oscillatory field.

Phase-field modeling of the formation of lamellar nanostructures in diblock copolymer thin films under inplanar electric fields

Recent experiments show that external inplanar electric field can be employed to guide the molecular self-assembly in diblock copolymer (BCP) thin films to form lamellar nanostructures with potential applications in nanotechnology. We study this self-assembly process through a detailed coarse-grained phase-separation modeling. During the process, the free energy of the BCP films is modeled as the Ginzburg-Landau free energy with nonlocal interaction and electrostatic coupling. The resulting Cahn-Hilliard (CH) equation is solved using an efficient semi-implicit Fourier-spectral algorithm. Numerical results show that the morphology of order parameter formed in either symmetric or asymmetric BCP thin films is strongly influenced by the electric field. For symmetrical BCPs, highly ordered lamellar nanostructures evolved along the direction of the electric field. Phase nucleation and dislocation climbing in the BCP films predicted by the numerical simulation are in a good agreement with those observed in recent BCP electronanolithography. For asymmetrical BCPs, numerical simulation shows that nanodots are guided to align to the electric field. Furthermore, in the case of high electric field, nanodots formed in asymmetrical BCPs may further convert into highly ordered lamellar nanostructures (sphere-to-cylinder transition) parallel to the electric field. Effects of the magnitude of electric field, BCP asymmetry, and molecular interaction of BCPs on the self-assembly process are examined in detail using the numerical scheme developed in this study. The present study can be used for the prediction of the formation of nanostructures in BCP thin films and the quality control of BCP-based nanomanufacturing through optimizing the external electric fields.

Scaling behavior of the reorientation kinetics of block copolymers exposed to electric fields

We have followed the reorientation kinetics of various block copolymer solutions exposed to an external electric DC field. The characteristic time constants follow a power law indicating that the reorientation is driven by a decrease in electrostatic energy. Moreover, the observed exponent suggests an activated process in line with the expectations for a nucleation and growth process. When properly scaled, the data collapse onto a single master curve spanning several orders of magnitude both in reduced time and in reduced energy. The power law dependence of the rate of reorientation derived from computer simulations based on dynamic density functional theory agrees well with the experimental observations. First experiments in AC electric fields at sufficiently high frequencies confirm the notion that the reorientation process is dominated by differences in the dielectric constants rather than by mobile ions.

Undulation instability in block-copolymer lamellae subjected to a perpendicular electric field

We examine the stability of lamellar stacks in the presence of an electric field, , applied normal to the lamellae. Calculations are performed with self-consistent field theory (SCFT) supplemented by an exact treatment of the electrostatic energy for linear dielectric materials. The calculations identify a critical electric field, , beyond which the lamellar stack becomes unstable with respect to undulations. This rapidly decreases towards zero as the number of lamellae in the stack diverges. Our quantitative predictions for are consistent with previous experimental measurements by Xu and co-workers.

Electric field versus surface alignment in confined films of a diblock copolymer melt

The dynamics of alignment of microstructure in confined films of diblock copolymer melts in the presence of an external electric field was studied numerically. We consider in detail a symmetric diblock copolymer melt, exhibiting a lamellar morphology. The method used is a dynamic mean-field density functional method, derived from the generalized time-dependent Ginzburg-Landau theory. The time evolution of concentration variables and therefore the alignment kinetics of the morphologies are described by a set of stochastic equations of a diffusion form with Gaussian noise. We investigated the effect of an electric field on block copolymers under the assumption that the long-range dipolar interaction induced by the fluctuations of composition pattern is a dominant mechanism of electric-field-induced domain alignment. The interactions with bounding electrode surfaces were taken into account as short-range interactions resulting in an additional term in the free energy of the sample. This term contributes only in the vicinity of the surfaces. The surfaces and the electric field compete with each other and align the microstructure in perpendicular directions. Depending on the ratio between electric field and interfacial interactions, parallel or perpendicular lamellar orientations were observed. The time scale of the electric-field-induced alignment is much larger than the time scale of the surface-induced alignment and microphase separation.

Self-consistent field study of the alignment by an electric field of a cylindrical phase of block copolymer

Self-consistent field theory is applied to a film of cylindrical-forming block copolymer subject to a surface field which tends to align the cylinders parallel to electrical plates, and to an external electric field tending to align them perpendicular to the plates. The Maxwell equations and self-consistent field equations are solved exactly, numerically, in real space. By comparing the free energies of different configurations, we show that for weak surface fields, the phase of cylinders parallel to the plates makes a direct transition to a phase in which the cylinders are aligned with the field throughout the sample. For stronger surface fields, there is an intermediate phase in which cylinders in the interior of the film, aligned with the field, terminate near the plates. For surface fields which favor the minority block, there is a boundary layer of hexagonal symmetry at the plates in which the monomers favored by the surface field occupy a larger area than they would if the cylinders extended to the surface.

Ion complexation: a route to enhanced block copolymer alignment with electric fields

We investigate the enhanced alignment of lamellar microdomains under an electric field by addition of lithium chloride (LiCl) into polystyrene-block-poly(methyl methacrylate) (PS-b-PMMA) copolymers. A significant increase of dielectric contrast resulting from the formation of lithium-PMMA complexes markedly reduces the critical electric field strength required to overcome the preferential interactions of one block with the substrate, providing a route to achieve the complete alignment of microdomains in block copolymer thin films.

Converting the nanodomains of a diblock-copolymer thin film from spheres to cylinders with an external electric field

We investigate the ability of an applied electric field to convert the morphology of a diblock-copolymer thin film from a monolayer of spherical domains embedded in the matrix to cylindrical domains that penetrate through the matrix. As expected, the applied field increases the relative stability of cylindrical domains, while simultaneously reducing the energy barrier that impedes the transition to cylinders. The effectiveness of the field is enhanced by a large dielectric contrast between the two block-copolymer components, particularly when the low-dielectric contrast component forms the matrix. Furthermore, the energy barrier is minimized by selecting sphere-forming diblock copolymers that are as compositionally symmetric as possible. Our calculations, which are the most quantitatively reliable to date, are performed using a numerically precise spectral algorithm based on self-consistent-field theory supplemented with an exact treatment for linear dielectric materials.