Architectural effect on the surface tension of an ABA triblock copolymer melt

The excess surface energy of lamellae formed by an ABA triblock copolymer melt oriented parallel to a neutral surface is evaluated using self-consistent field theory (SCFT). Consistent with experiments and previous SCFT calculations, we find a preference for the A-rich domains at the surface, which can only be attributed to the architectural asymmetry between the A and B blocks. The behavior was previously attributed to a loss of bridging configurations that occurs when the B-domain resides at the surface. Here we demonstrate that it is actually the presence of chain ends that reduces the excess surface energy of an A-rich domain relative that of a B-rich domain.

Inverse problems in dynamic cognitive modeling

Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov-Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations.

Fast and accurate SCFT calculations for periodic block-copolymer morphologies using the spectral method with Anderson mixing

We study the numerical efficiency of solving the self-consistent field theory (SCFT) for periodic block-copolymer morphologies by combining the spectral method with Anderson mixing. Using AB diblock-copolymer melts as an example, we demonstrate that this approach can be orders of magnitude faster than competing methods, permitting precise calculations with relatively little computational cost. Moreover, our results raise significant doubts that the gyroid (G) phase extends to infinite $\chi N$. With the increased precision, we are also able to resolve subtle free-energy differences, allowing us to investigate the layer stacking in the perforated-lamellar (PL) phase and the lattice arrangement of the close-packed spherical (S$_{cp}$) phase. Furthermore, our study sheds light on the existence of the newly discovered Fddd (O$^{70}$) morphology, showing that conformational asymmetry has a significant effect on its stability.

Melt brushes of diblock copolymer

Using self-consistent field theory (SCFT), we investigate the morphologies formed by a melt brush of AB diblock copolymers grafted to a flat substrate by their B ends. In addition to a laterally uniform morphology, SCFT predicts three ordered morphologies exhibiting different periodic patterns at the air surface: a hexagonal array of A-rich dots, an alternating sequence of A- and B-rich stripes, and a hexagonal pattern of B-rich dots. When the phase diagram of the tethered film is plotted as a function of A/B incompatibility, $\chi N$, and diblock composition, $f$, it resembles the bulk phase diagram with the periodic phases converging to a mean-field critical point at weak segregation. The periodic-phase region in the phase diagram shrinks with increasing grafting density and expands when the air surface acquires an affinity for the grafted B blocks.

Droplets of structured fluid on a flat substrate

We study the equilibrium morphology of droplets of symmetric AB diblock copolymer on a flat substrate. Using self-consistent field theory (SCFT), we provide the first predictions for the equilibrium droplet shape and its internal structure. When the sustrate affinity for the A component, $\eta_A$, is small, the droplet adopts a nearly spherical shape much like that of simple fluids. Inside the spherical droplet, however, concentric circular lamellar layers stack on top of each other; hence the thickness of the droplet is effectively quantized by a half-integer or integer number of layers. At larger $\eta_A$ and smaller contact angle, the area of the upper-most layer becomes relatively large, resulting in a nearly flat, faceted top surface, followed by a semi-spherical slope. This geometry is remarkably reminiscent of the droplet shapes observed with smetic liquid crystals.

Positioning Janus nanoparticles in block copolymer scaffolds

The periodic domains formed by block copolymer melts have been heralded as potential scaffolds for arranging nanoparticles in 3d space, provided we can control the positioning of the particles. Recent experiments have located particles at the domain interfaces by grafting mixed brushes to their surfaces. Here the underlying mechanism, which involves the transformation into Janus particles, is investigated with self-consistent field theory using a new multi-coordinate-system algorithm.

Unit-cell approximation for diblock−copolymer brushes grafted to spherical particles

This paper examines the equilibrium phase behavior of thin diblock-copolymer films tethered to a spherical core, using numerical self-consistent field theory (SCFT). The computational cost of the calculation is greatly reduced by implementing the unit-cell approximation (UCA) routinely used in the study of bulk systems. This provides a tremendous reduction in computational time, permitting us to map out the phase behavior more extensively and allowing us to consider far larger particles. The main consequence of the UCA is that it omits packing frustration, but evidently the effect is minor for large particles. On the other hand, when the particles are small, the UCA calculation can be readily followed up with the full SCFT, the comparison to which conveniently allows one to quantitatively assess the effect of packing frustration.

Compression of polyelectrolyte brushes in a salt-free theta solvent

This paper examines the normal force between two opposing polyelectrolyte brushes and the interpenetration of their chains that is responsible for sliding friction. It focuses on the special case of semi-dilute brushes in a salt-free theta solvent, for which Zhulina and Borisov [J. Chem. Phys., {\bf 107}, 5952, (1997)] have derived analytical predictions using the classical strong-stretching theory (SST) introduced by Semenov and developed by Milner, Witten and Cates. Interestingly, the SST predicts that the brushes contract maintaining a polymer-free gap as they are compressed together, which provides an explanation for the ultra-low frictional forces observed in experiment. We examine the degree to which the SST predictions are affected by chain fluctuations by employing self-consistent field theory (SCFT). While the normal force is relatively unaffected, fluctuations are found to have a strong impact on brush interpenetration. Even still, the contraction of the brushes does significantly prolong the onset of interpenetration, implying that a sizeable normal force can be achieved before the sliding friction becomes significant.

Effects of polydispersity on the order-disorder transition of diblock copolymer melts

The effect of polydispersity on an AB diblock copolymer melt is investigated using latticebased Monte Carlo simulations. We consider melts of symmetric composition, where the B blocks are monodisperse and the A blocks are polydisperse with a Schultz-Zimm distribution. In agreement with experiment and self-consistent field theory (SCFT), we find that polydispersity causes a significant increase in domain size. It also induces a transition from flat to curved interfaces, with the polydisperse blocks residing on the inside of the interfacial curvature. Most importantly, the simulations show a relatively small shift in the order-disorder transition (ODT) in agreement with experiment, whereas SCFT incorrectly predicts a sizable shift towards higher temperatures.

Kinetics of layer hopping in a diblock copolymer lamellar phase

In the ordered state, symmetric diblock copolymers self-assemble into an anisotropic lamellar morphology. The equilibrium thickness of the lamellae is the result of a delicate balance between enthalpic and entropic energies, which can be tuned by controlling the temperature. Here we devise a simple yet powerful method of detecting tiny changes in the lamellar thickness using optical microscopy. From such measurements we characterize the enthalpic interaction as well as the kinetics of molecules as they hop from one layer to the next in order to adjust the lamellar thickness in response to a temperature jump. The resolution of the measurements facilitate a direct comparison to predictions from self-consistent field theory.

Monte Carlo phase diagram for a polydisperse diblock copolymer melt

The phase diagram for an AB diblock copolymer melt with polydisperse A blocks and monodisperse B blocks is evaluated using lattice-based Monte Carlo simulations. Experiments on this system have shown that the A-block polydispersity shifts the order-order transitions (OOTs) towards higher A-monomer content, while the order-disorder transition (ODT) moves towards higher temperatures when the A blocks form the minority domains and lower temperatures when the A blocks form the matrix. Although self-consistent field theory (SCFT) correctly accounts for the change in the OOTs, it incorrectly predicts the ODT to shift towards higher temperatures at all diblock copolymer compositions. In contrast, our simulations predict the correct shifts for both the OOTs and the ODT. This implies that polydispersity amplifies the fluctuation-induced correction to the mean-field ODT, which we attribute to a reduction in packing frustration. Consistent with this explanation, polydispersity is found to enhance the stability of the perforated-lamellar phase.

New fast SCFT algorithm applied to binary diblock copolymer/homopolymer blends

We present an efficient strategy for mapping out the classical phase behavior of block copolymer systems using self-consistent field theory (SCFT). With our new algorithm, the complete solution of a classical block copolymer phase can be evaluated typically in a fraction of a second on a single-processor computer, even for highly segregated melts. This is accomplished by implementing the standard unit-cell approximation (UCA) for the cylindrical and spherical phases, and solving the resulting equations using a Bessel function expansion. Here the method is used to investigate blends of AB diblock copolymer and A homopolymer, concentrating on the situation where the two molecules are of similar size.

Structure variation and evolution in microphase-separated grafted diblock copolymer films

The phase behavior of grafted d-polystyrene-block-poly(methyl methacrylate) diblock copolymer films is examined, with particular focus on the effect of solvent and annealing time. It was observed that the films undergo a two-step transformation from an initially disordered state, through an ordered metastable state, to the final equilibrium configuration. It was also found that altering the solvent used to wash the films, or complete removal of the solvent prior to thermal annealing using supercritical CO2, could influence the structure of the films in the metastable state, though the final equilibrium state was unaffected. To aid in the understanding to these experimental results, a series of self-consistent field theory calculations were done on a model diblock copolymer brush containing solvent. Of the different models examined, those which contained a solvent selective for the grafted polymer block most accurately matched the observed experimental behavior. We hypothesize that the structure of the films in the metastable state results from solvent enrichment of the film near the film/substrate interface in the case of films washed with solvent or faster relaxation of the nongrafted block for supercritical CO2 treated (solvent free) films. The persistence of the metastable structures was attributed to the slow reorganization of the polymer chains in the absence of solvent.

Effect of salt on the compression of polyelectrolyte brushes in a theta solvent

Classical strong-stretching theory (SST) predicts that, as opposing polyelectrolyte brushes are compressed together in a salt-free theta solvent, they contract so as to maintain a finite polymer-free gap, which offers a potential explanation for the ultra-low frictional forces observed in experiments even with the application of large normal forces. However, the SST ignores chain fluctuations, which would tend to close the gap resulting in physical contact and in turn significant friction. In a preceding study, we examined the effect of fluctuations using self-consistent field theory (SCFT) and illustrated that high normal forces can still be applied before the gap is destroyed. We now look at the effect of adding salt. It is found to reduce the long-range interaction between the brushes but has little effect on the short-range part, provided the concentration does not enter the salted-brush regime. Consequently, the maximum normal force between two planar brushes at the point of contact is remarkably unaffected by salt. For the crossed-cylinder geometry commonly used in experiments, however, there is a gradual reduction because in this case the long-range part of the interaction contributes to the maximum normal force.

Effect of architecture on the phase behavior of AB-type block copolymer melts

Equilibrium phase diagrams are calculated for a selection of two-component block copolymer architectures using self-consistent field theory (SCFT). The topology of the phase diagrams is relatively unaffected by differences in architecture, but the phase boundaries shift significantly in composition. The shifts are consistent with the decomposition of architectures into constituent units as proposed by Gido and coworkers, but there are significant quantitative deviations from this principle in the intermediate-segregation regime. Although the complex phase windows continue to be dominated by the gyroid (G) phase, the regions of the newly discovered Fddd (O^70) phase become appreciable for certain architectures and the perforated-lamellar (PL) phase becomes stable when the complex phase windows shift towards high compositional asymmetry.