Fluctuations, Phase Transitions, and Latent Heat in Short Diblock Copolymers: Comparison of Experiment, Simulation, and Theory

Assessment of the partial saddle point approximation in field-theoretic polymer simulations

Field-theoretic simulations are numerical treatments of polymer field theory models that go beyond the mean-field self-consistent field theory level and have successfully captured a range of mesoscopic phenomena. Inherent in molecularly-based field theories is a "sign problem" associated with complex-valued Hamiltonian functionals. One route to field-theoretic simulations utilizes the complex Langevin (CL) method to importance sample complex-valued field configurations to bypass the sign problem. Although CL is exact in principle, it can be difficult to stabilize in strongly fluctuating systems. An alternate approach for blends or block copolymers with two segment species is to make a "partial saddle point approximation" (PSPA) in which the stiff pressure-like field is constrained to its mean-field value, eliminating the sign problem in the remaining field theory, allowing for traditional (real) sampling methods. The consequences of the PSPA are relatively unknown, and direct comparisons between the two methods are limited. Here, we quantitatively compare thermodynamic observables, order-disorder transitions, and periodic domain sizes predicted by the two approaches for a weakly compressible model of AB diblock copolymers. Using Gaussian fluctuation analysis, we validate our simulation observations, finding that the PSPA incorrectly captures trends in fluctuation corrections to certain thermodynamic observables, microdomain spacing, and location of order-disorder transitions. For incompressible models with contact interactions, we find similar discrepancies between the predictions of CL and PSPA, but these can be minimized by regularization procedures such as Morse calibration. These findings mandate caution in applying the PSPA to broader classes of soft-matter models and systems.

Field theoretic approach for block polymer melts: SCFT and FTS

This perspective addresses the development of polymer field theory for predicting the equilibrium phase behavior of block polymer melts. The approach is tailored to the high-molecular-weight limit, where universality reduces all systems to the standard Gaussian chain model, an incompressible melt of elastic threads interacting by contact forces. Using mathematical identities, this particle-based version of the model is converted to an equivalent field-based version that depends on fields rather than particle coordinates. The statistical mechanics of the field-based model is typically solved using the saddle-point approximation of self-consistent field theory (SCFT), which equates to mean field theory, but it can also be evaluated using field theoretic simulations (FTS). While SCFT has matured into one of the most successful theories in soft condensed matter, FTS are still in its infancy. The two main obstacles of FTS are the high computational cost and the occurrence of an ultraviolet divergence, but fortunately there has been recent groundbreaking progress on both fronts. As such, FTS are now well poised to become the method of choice for predicting fluctuation corrections to mean field theory.

Network Model of the Disordered Phase in Symmetric Diblock Copolymer Melts

We present a model for the order-disorder transition of symmetric A-B diblock copolymer melts in which the disordered phase is treated as a bicontinuous network, and in which self-consistent field predictions of properties of an analogous ordered network are used to estimate some properties. Such a model is shown to accurately predict the latent heat of this transition. The dependence of the location of the transition upon the invariant degree of polymerization N[over ¯] is shown to be consistent with a simple hypothesis that the disordered bicontinuous structure is stabilized relative to an analogous ordered network by a nearly constant entropy per network junction.

Why Is Gyroid More Difficult to Nucleate from Disordered Liquids than Lamellar and Hexagonal Mesophases?

Block copolymers, surfactants, and biomolecules form lamellar, hexagonal, and gyroid mesophases. Across these systems, the nucleation of lamellar from the disordered liquid is the easiest and the nucleation of gyroid the most challenging. This poses the question of what are the factors that determine the rates of nucleation of the mesophases and whether they are controlled by the complexity of the structures or the thermodynamics of nucleation. Here, we use molecular simulations to investigate the nucleation and thermodynamics of lamellar, hexagonal, and gyroid in a binary mixture of particles that produces the same mesophases as those of surfactants and block copolymers. We demonstrate that a combination of averaged bond-order parameters q̅₂ and q̅₈ identifies and distinguishes the three mesophases. We use these parameters to track the microscopic process of nucleation of each mesophase and investigate the existence of heterogeneous nucleation (cross-nucleation) between mesophases. We estimate the surface tensions of the liquid/mesophase interfaces from nucleation rates using classical nucleation theory and find that they are comparable for the three mesophases with values that are about a third of those expected for liquid-crystal interfaces. The driving forces for nucleation, on the other hand, are quite different and increase in the order gyroid < hexagonal < lamellar at any temperature. We find that the nucleation rates of the mesophases follow the order of their driving forces. We conclude that the difficulty to nucleate the gyroid originates in its lower temperature of melting and extremely low entropy of melting compared to those of the hexagonal and lamellar mesophases.

Polymer Semiflexibility Induces Nonuniversal Phase Transitions in Diblock Copolymers

The order-disorder phase transition and the associated phase diagrams of semiflexible diblock copolymers are investigated using the wormlike chain model, incorporating concentration fluctuations. The free energy up to quartic order in concentration fluctuations is developed with chain-rigidity-dependent coefficients, evaluated using our exact results for the wormlike chain model, and a one-loop renormalization treatment is used to account for fluctuation effects. The chain length N and the monomer aspect ratio α directly control the strength of immiscibility (defined by the Flory-Huggins parameter χ) at the order-disorder transition and the resulting microstructures at different chemical compositions f_{A}. When monomers are infinitely thin (i.e., large aspect ratio α), the finite chain length N lowers the χN at the phase transition. However, fluctuation effects become important when chains have a finite radius, and a decrease in the chain length N elevates the χN at the phase transition. Phase diagrams of diblock copolymers over a wide range of N and α are calculated based on our fluctuation theory. We find that both finite N and α enhance the stability of the lamellar phase above the order-disorder transition. Our results demonstrate that polymer semiflexibility plays a dramatic role in the phase behavior, even for large chain lengths (e.g., N≈100).

Origins of low-symmetry phases in asymmetric diblock copolymer melts

Cooling disordered compositionally asymmetric diblock copolymers leads to the formation of nearly spherical particles, each containing hundreds of molecules, which crystallize upon cooling below the order-disorder transition temperature (TODT). Self-consistent field theory (SCFT) reveals that dispersity in the block degrees of polymerization stabilizes various Frank-Kasper phases, including the C14 and C15 Laves phases, which have been accessed experimentally in low-molar-mass poly(isoprene)-b-poly(lactide) (PI-PLA) diblock copolymers using thermal processing strategies. Heating and cooling a specimen containing 15% PLA above and below the TODT from the body-centered cubic (BCC) or C14 states regenerates the same crystalline order established at lower temperatures. This memory effect is also demonstrated with a specimen containing 20% PLA, which recrystallizes to either C15 or hexagonally ordered cylinders (HEXC) upon heating and cooling. The process-path-dependent formation of crystalline order shapes the number of particles per unit volume, n/V, which is retained in the highly structured disordered liquid as revealed by small-angle X-ray scattering (SAXS) experiments. We hypothesize that symmetry breaking during crystallization is governed by the particle number density imprinted in the liquid during ordering at lower temperature, and this metastable liquid is kinetically constrained from equilibrating due to prohibitively large free energy barriers for micelle fusion and fission. Ordering at fixed n/V is enabled by facile chain exchange, which redistributes mass as required to meet the multiple particle sizes and packing associated with specific low-symmetry Frank-Kasper phases. This discovery exposes universal concepts related to order and disorder in self-assembled soft materials.

Fluctuation Effects in Semiflexible Diblock Copolymers

We present a simulation study of the equilibrium thermodynamic behavior of semiflexible diblock copolymer melts. Using discretized wormlike chains and field-theoretic Monte Carlo, we find that concentration fluctuations play a critical role in controlling phase transitions of semiflexible diblock copolymers. Polymer flexibility and aspect ratio control the order-disorder transition Flory-Huggins parameter χ_ODTN. For polymers with low aspect ratios, fluctuations strongly elevate the phase transition χ_ODTN at finite molecular weights. For high aspect-ratio polymers, chain semiflexibility decreases the phase transition χ_ODTN. We find that the simulated phase behavior agrees well with our recently developed fluctuation theory based on wormlike chain configurations and a one-loop treatment of concentration fluctuations.

Characterizing the Interface Scaling of High χ Block Copolymers near the Order-Disorder Transition

Advancements in the directed self-assembly of block copolymers (BCPs) have prompted the development of new materials with larger effective interaction parameters (χ_e). This enables BCP systems with phase separation at increasingly small degrees of polymerization (N). Very often these systems reside near the order-disorder transition and fit between the weak and strong segregation limits where the behavior of BCP systems is not as thoroughly understood. Utilizing resonant soft X-ray reflectivity (RSoXR) enables both the BCP pitch (L₀) and interface width (w_M) to be determined simultaneously, through a direct characterization of the composition profile of BCP lamellae oriented parallel to a substrate. A series of high χ_e BCPs with χ_e ranging from ≈0.04 to 0.25 and χ_eN from 19 to 70 have been investigated. The L₀/w_m ratio serves as an important metric for the feasibility of a material for nanopatterning applications; the results of the RSoXR measurement are used to establish a relationship between χ_e and L₀/w_m. The results of this analysis are correlated with experimentally established limits for the functionality of BCPs in nanopatterning applications. These results also provide guidance for the magnitude of χ_e needed to achieve small interface width for samples with sub-10 nm L₀.

Implications of Grain Size Variation in Magnetic Field Alignment of Block Copolymer Blends

Recent experiments have highlighted the intrinsic magnetic anisotropy in coil-coil diblock copolymers, specifically in poly(styrene-block-4-vinylpyridine) (PS-b-P4VP), that enables magnetic field alignment at field strengths of a few tesla. We consider here the alignment response of two low molecular weight (MW) lamallae-forming PS-b-P4VP systems. Cooling across the disorder-order transition temperature (T_odt) results in strong alignment for the higher MW sample (5.5K), whereas little alignment is discernible for the lower MW system (3.6K). This disparity under otherwise identical conditions of field strength and cooling rate suggests that different average grain sizes are produced during slow cooling of these materials, with larger grains formed in the higher MW material. Blending the block copolymers results in homogeneous samples which display T_odt, d-spacings, and grain sizes that are intermediate between the two neat diblocks. Similarly, the alignment quality displays a smooth variation with the concentration of the higher MW diblock in the blends, and the size of grains likewise interpolates between limits set by the neat diblocks, with a factor of 3.5× difference in the grain size observed in high vs low MW neat diblocks. These results highlight the importance of grain growth kinetics in dictating the field response in block copolymers and suggests an unconventional route for the manipulation of such kinetics.

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.

Poly(cyclohexylethylene)-block-poly(ethylene oxide) Block Polymers for Metal Oxide Templating

A series of poly(cyclohexylethylene)-block-poly(ethylene oxide) (CEO) diblock copolymers were synthesized through tandem anionic polymerizations and heterogeneous catalytic hydrogenation. Solvent-annealed CEO diblock films were used to template dense arrays of inorganic oxide nanodots via simple spin coating of an inorganic precursor solution atop the ordered film. The substantial chemical dissimilarity of the two blocks enables (i) selective inclusion of the inorganic precursor within the PEO domain and (ii) the formation of exceptionally small feature sizes due to a relatively large interaction parameter estimated from mean-field analysis of the order-disorder transition temperatures of compositionally symmetric samples. UV/ozone treatment following incorporation produces an ordered arrangement of oxide nanodots and simultaneously removes the block polymer template. Herein, we report the smallest particles (6 ± 1 nm) templated from a selective precursor insertion method to date using a block polymer scaffold.