Fluctuation-Corrected Phase Diagrams for Diblock Copolymer Melts.

New developments in field-theoretic simulations (FTSs) are used to evaluate fluctuation corrections to the self-consistent field theory of diblock copolymer melts. Conventional simulations have been limited to the order-disorder transition (ODT), whereas FTSs allow us to evaluate complete phase diagrams for a series of invariant polymerization indices. The fluctuations stabilize the disordered phase, which shifts the ODT to higher segregation. Furthermore, they stabilize the network phases at the expense of the lamellar phase, which accounts for the presence of the Fddd phase in experiments. We hypothesize that this is due to an undulation entropy that favors curved interfaces.

Well-tempered metadynamics applied to field-theoretic simulations of diblock copolymer melts.

Well-tempered metadynamics (WTMD) is applied to field-theoretic simulations (FTS) to locate the order-disorder transition (ODT) in incompressible melts of diblock copolymer with an invariant polymerization index of N̄=104. The polymers are modeled as discrete Gaussian chains with N = 90 monomers, and the incompressibility is treated by a partial saddle-point approximation. Our implementation of WTMD proves effective at locating the ODT of the lamellar and cylindrical regions, but it has difficulty with that of the spherical and gyroid regions. In the latter two cases, our choice of order parameter cannot sufficiently distinguish the ordered and disordered states because of the similarity in microstructures. The gyroid phase has the added complication that it competes with a number of other morphologies, and thus, it might be beneficial to extend the WTMD to multiple order parameters. Nevertheless, when the method works, the ODT can be located with impressive accuracy (e.g., ΔχN ∼ 0.01).

Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation.

Field-theoretic simulations (FTS) provide an efficient technique for investigating fluctuation effects in block copolymer melts with numerous advantages over traditional particle-based simulations. For systems involving two components (i.e., A and B), the field-based Hamiltonian, Hf[W-,W+], depends on a composition field, W-(r), that controls the segregation of the unlike components and a pressure field, W+(r), that enforces incompressibility. This review introduces researchers to a promising variant of FTS, in which W-(r) fluctuates while W+(r) tracks its mean-field value. The method is described in detail for melts of AB diblock copolymer, covering its theoretical foundation through to its numerical implementation. We then illustrate its application for neat AB diblock copolymer melts, as well as ternary blends of AB diblock copolymer with its A- and B-type parent homopolymers. The review concludes by discussing the future outlook. To help researchers adopt the method, open-source code is provided that can be run on either central processing units (CPUs) or graphics processing units (GPUs).

Instability of the Microemulsion Channel in Block Copolymer-Homopolymer Blends.

Field theoretic simulations are used to predict the equilibrium phase diagram of symmetric blends of AB diblock copolymer with A- and B-type homopolymers. Experiments generally observe a channel of bicontinuous microemulsion (BµE) separating the ordered lamellar (LAM) phase from coexisting homopolymer-rich (A+B) phases. However, our simulations find that the channel is unstable with respect to macrophase separation, in particular, A+B+BµE coexistence at high T and A+B+LAM coexistence at low T. The preference for three-phase coexistence is attributed to a weak attractive interaction between diblock monolayers.

Field-theoretic simulations of bottlebrush copolymers.

Traditional particle-based simulations struggle with large bottlebrush copolymers, consisting of many side chains grafted to a backbone. Field-theoretical simulations (FTS) allow us to overcome the computational demands in order to calculate their equilibrium behavior. We consider bottlebrushes where all grafts are symmetric diblock copolymers, focusing on the order-disorder transition (ODT) and the size of ordered domains. Increasing the number of grafts and decreasing the spacing between them both raise the transition temperature. The ODT and lamellar period asymptotically approach constants as the number of grafts increases. As the spacing between grafts becomes large, the bottlebrushes behave like diblock copolymers, and as it becomes small, they behave like starblock copolymers. In between, the period increases, reaching a maximum when the spacing is approximately 0.35 times the length of the grafts. A comparison of FTS with mean-field calculations allows us to assess the effect of compositional fluctuations. Fluctuations suppress ordering, while having little effect on the period, as is the case for diblock copolymers.

Fluctuation effects in blends of A + B homopolymers with AB diblock copolymer.

Field-theoretic simulations (FTSs) are performed on ternary blends of A- and B-type homopolymers of polymerization Nh and symmetric AB diblock copolymers of polymerization Nc. Unlike previous studies, our FTSs are conducted in three-dimensional space, with the help of two new semi-grand canonical ensembles. Motivated by the first experiment to discover bicontinuous microemulsion (BµE) in the polyethylene-polyethylene propylene system, we consider molecules of high molecular weight with size ratios of α ≡ Nh/Nc = 0.1, 0.2, and 0.4. Our focus is on the A + B coexistence between the two homopolymer-rich phases in the low-copolymer region of the phase diagram. The Scott line, at which the A + B phases mix to form a disordered melt with increasing temperature (or decreasing χ), is accurately determined using finite-size scaling techniques. We also examine how the copolymer affects the interface between the A + B phases, reducing the interfacial tension toward zero. Although comparisons with self-consistent field theory (SCFT) illustrate that fluctuation effects are relatively small, fluctuations do nevertheless produce the observed BµE that is absent in the SCFT phase diagram. Furthermore, we find evidence of three-phase A + B + BµE coexistence, which may have been missed in the original as well as subsequent experiments.

Self-Assembly of ABC Bottlebrush Triblock Terpolymers with Evidence for Looped Backbone Conformations.

Bottlebrush block copolymers offer rich opportunities for the design of complex hierarchical materials. As consequences of the densely grafted molecular architecture, bottlebrush polymers can adopt highly extended backbone conformations and exhibit unique physical properties. A recent report has described the unusual phase behavior of ABC bottlebrush triblock terpolymers bearing grafted poly(D,L-lactide) (PLA), polystyrene (PS), and poly(ethylene oxide) (PEO) blocks (LSO). In this work, a combination of resonant soft X-ray reflectivity (RSoXR), near edge X-ray absorption fine structure spectroscopy (NEXAFS), and self-consistent field theory (SCFT) was used to provide insight into the phase behavior of LSO and underlying backbone chain conformations. Consistent with SCFT calculations, RSoXR measurements confirm a unique mesoscopic ACBC domain connectivity and decreasing lamellar periods (d 0) with increasing backbone length of the PEO block. RSoXR and NEXAFS demonstrate an additional unusual feature of brush LSO thin films: when the overall film thickness is ~3.25d 0, the film-air interface is majority PS (>80%). Since PS is the midblock, the triblocks must adopt looping configurations at the surface, despite the preference for the backbone to be extended. This result is supported by backbone concentrations calculated through SCFT, which suggest that looping midblocks are present throughout the film. Collectively, this work provides evidence for the flexibility of the bottlebrush backbone and the consequences of low-χ block copolymer design. We propose that PEO blocks localize at the PS/PLA domain interfaces in order to screen the highest-χ contacts in the system, driving the formation of loops. These insights introduce a potential route to overcome the intrinsic penalties to interfacial curvature imposed by the bottlebrush architecture, enabling the design of unique self-assembled materials.

Detection of Surface Enrichment Driven by Molecular Weight Disparity in Virtually Monodisperse Polymers.

The preference for a shorter chain component at a polymer blend surface impacts surface properties key to application-specific performance. While such segregation is known for blends containing low molecular weight additives or systems with large polydispersity, it has not been reported for anionically polymerized polymers that are viewed, in practice, as monodisperse. Observations with surface layer matrix-assisted laser desorption ionization time-of-flight mass spectrometry (SL-MALDI-ToF-MS), which distinguishes surface species without labeling and provides the entire molecular weight distribution, demonstrate that entropically driven surface enrichment of shorter chains occurs even in low polydispersity materials. For 6 kDa polystyrene the number-average molecular weight (Mn) at the surface is ca. 300 Da (5%) lower than that in the bulk, and for 7 kDa poly(methyl methacryalate) the shift is ca. 500 Da. These observations are in qualitative agreement with results from a mean-field theory that considers a homopolymer melt with a molecular-weight distribution matched to the experiments.

Fluctuation correction for the critical transition of symmetric homopolymer blends.

Monte Carlo simulations are performed on structurally symmetric binary homopolymer blends over a wide range of invariant polymerization indices, N¯. A finite-size scaling analysis reveals that certain critical exponents deviate from the expected 3D-Ising values as N¯ increases. However, the deviations are consistent with previous simulations and can be attributed to the fact that the system crosses over to mean-field behavior when the molecules become too large relative to the size of the simulation box. Nevertheless, the finite-size scaling techniques provide precise predictions for the position of the critical transition. Using a previous calibration of the Flory-Huggins interaction parameter, χ, we confirm that the critical point scales as (χN)c=2+cN¯-1∕2 for large N¯, and more importantly we are able to extract a reliable estimate, c≈1.5, for the universal constant.

Manipulating the ABCs of self-assembly via low-χ block polymer design.

Block polymer self-assembly typically translates molecular chain connectivity into mesoscale structure by exploiting incompatible blocks with large interaction parameters (χij). In this article, we demonstrate that the converse approach, encoding low-χ interactions in ABC bottlebrush triblock terpolymers (χAC [Formula: see text] 0), promotes organization into a unique mixed-domain lamellar morphology, which we designate LAMP Transmission electron microscopy indicates that LAMP exhibits ACBC domain connectivity, in contrast to conventional three-domain lamellae (LAM3) with ABCB periods. Complementary small-angle X-ray scattering experiments reveal a strongly decreasing domain spacing with increasing total molar mass. Self-consistent field theory reinforces these observations and predicts that LAMP is thermodynamically stable below a critical χAC, above which LAM3 emerges. Both experiments and theory expose close analogies to ABA' triblock copolymer phase behavior, collectively suggesting that low-χ interactions between chemically similar or distinct blocks intimately influence self-assembly. These conclusions provide fresh opportunities for block polymer design with potential consequences spanning all self-assembling soft materials.

Universality between Experiment and Simulation of a Diblock Copolymer Melt.

The equivalent behavior among analogous block copolymer systems involving chemically distinct molecules or mathematically different models has long hinted at an underlying universality, but only recently has it been rigorously demonstrated by matching results from different simulations. The profound implication of universality is that simple coarse-grained models can be calibrated so as to provide quantitatively accurate predictions to experiment. Here, we provide the first compelling demonstration of this by simulating a polyisoprene-polylactide diblock copolymer melt using a previously calibrated lattice model. The simulation successfully predicts the peak in the disordered-state structure function, the position of the order-disorder transition, and the latent heat of the transition in excellent quantitative agreement with experiment. This could mark a new era of precision in the field of block copolymer research.

Structure, Stability, and Reorganization of 0.5 L0 Topography in Block Copolymer Thin Films.

The structure, stability, and reorganization of lamella-forming block copolymer thin film surface topography ("islands" and "holes") were studied under boundary conditions driving the formation of 0.5 L0 thick structures at short thermal annealing times. Self-consistent field theory predicts that the presence of one perfectly neutral surface renders 0.5 L0 topography thermodynamically stable relative to 1 L0 thick features, in agreement with previous experimental observations. The calculated through-film structures match cross-sectional scanning electron micrographs, collectively demonstrating the pinning of edge dislocations at the neutral surface. Remarkably, near-neutral surface compositions exhibit 0.5 L0 topography metastability upon extended thermal treatment, slowly transitioning to 1 L0 islands or holes as evidenced by optical and atomic force microscopy. Surface restructuring is rationalized by invoking commensurability effects imposed by slightly preferential surfaces. The results described herein clarify the impact of interfacial interactions on block copolymer self-assembly and solidify an understanding of 0.5 L0 topography, which is frequently used to determine neutral surface compositions of considerable importance to contemporary technological applications.

Confinement effects on the miscibility of block copolymer blends.

Thin films of long and short symmetric AB diblock copolymers are examined using self-consistent field theory (SCFT). We focus on hard confining walls with a preference for the A component, such that the lamellar domains orient parallel to the film with an even number ν of monolayers. For neat melts, confinement causes the lamellar period, D, to deviate from its bulk value, Db, in order to be commensurate with the film thickness, i.e., L = νD/2. For blends, however, the melt also has the option of macrophase separating into ν(l) large and ν((s)) small monolayers so as to provide a better fit, where L = ν(l)D(l)/2 + ν(s)D((s))/2. In addition to performing full SCFT calculations of the entire film, we develop a semi-analytical calculation for the coexistence of thick and thin monolayers that helps explain the complicated interplay between miscibility and commensurability.

Bottlebrush Block Polymers: Quantitative Theory and Experiments.

The self-assembly of bottlebrush block polymers into a lamellar phase was investigated using a combination of experiment and self-consistent field theory (SCFT). Nine diblock bottlebrush polymers were synthesized with atactic polypropylene side chains (block A) and polystyrene side chains (block B) attached to poly(norbornene) backbones of various contour lengths, L, and the resulting lamellar structures were analyzed using small-angle X-ray scattering. The scaling of the lamellar period, d0 ∼ L(γ), exhibited an increasing exponent from γ ≈ 0.3 at small L to γ ≈ 0.9 at large L. The small exponents occurred for starlike molecules where the size of the side chains is comparable to L, while the larger exponents occurred for the more brushlike molecules where the side chains extend radially outward from the backbone. The bottlebrushes were then modeled using flexible side chains of types A and B attached to a semiflexible backbone with an adjustable persistence length, ξb. The resulting SCFT predictions for d0 showed remarkable quantitative agreement with the experimental data, where ξb was similar to the radius of the bottlebrushes. The theory was then used to examine the joint-distribution functions for the position and orientation of different segments along the backbone. This revealed a bilayer arrangement of the bottlebrushes in the lamellar phase, with a high degree of backbone orientation at the A/B interfaces that almost completely vanished near the center of the domains. This finding clearly refutes the prevailing interpretation that the large scaling exponent γ is a result of highly extended backbone conformations.

Universality of block copolymer melts.

Simulations of five different coarse-grained models of symmetric diblock copolymers are compared to demonstrate a universal (i.e., model-independent) dependence of the free energy and order-disorder transition (ODT) on the invariant degree of polymerization N̄. The actual values of χN at the ODT approach predictions of the Fredrickson-Helfand (FH) theory for N̄ ≳ 10(4) but significantly exceed FH predictions at lower values characteristic of most experiments. The FH theory fails for modest N̄ because the competing phases become strongly segregated near the ODT, violating an underlying assumption of weak segregation.

Quantized contact angles in the dewetting of a structured liquid.

We investigate the dewetting of a disordered melt of diblock copolymer from an ordered residual wetting layer. In contrast to simple liquids where the wetting layer has a fixed thickness and the droplets exhibit a single unique contact angle with the substrate, we find that structured liquids of diblock copolymer exhibit a discrete series of wetting layer thicknesses each producing a different contact angle. These quantized contact angles arise because the substrate and air surfaces each induce a gradient of lamellar order in the wetting layer. The interaction between the two surface profiles creates an effective interface potential that oscillates with film thickness, thus, producing a sequence of local minimums. The wetting layer thicknesses and corresponding contact angles are a direct measure of the positions and depths of these minimums. Self-consistent field theory is shown to provide qualitative agreement with the experiment.

Morphology Induced Spinodal Decomposition at the Surface of Symmetric Diblock Copolymer Films.

Atomic force microscopy is used to study the ordering dynamics of symmetric diblock copolymer films. The films order to form a lamellar structure which results in a frustration when the film thickness is incommensurate with the lamellae. By probing the morphology of incommensurate films in the early ordering stages, we discover an intermediate phase of lamellae arranged perpendicular to the film surface. This morphology is accompanied by a continuous growth in amplitude of the film surface topography with a characteristic wavelength, indicative of a spinodal process. Using self-consistent field theory, we show that the observation of perpendicular lamellae suggests an intermediate state with parallel lamellae at the substrate and perpendicular lamellae at the free surface. The calculations confirm that the intermediate state is unstable to thickness fluctuations, thereby driving the spinodal growth of surface structures.

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.

Droplet shape of an anisotropic liquid.

We investigate how a droplet of a complex liquid is modified by its internal nanoscale structure. As the liquid passes from an isotropic disordered state to an anisotropic layered morphology, the droplet shape switches from a smooth spherical cap to a terraced hyperbolic profile, which can be modeled as a stack of thin concentric circular disks with a repulsion between adjacent disk edges. Our ability to resolve the detailed shape of these defect-free droplets offers a unique opportunity to explore the underlying physics.