Field theoretic approach for block polymer melts: SCFT and FTS

Bicontinuous microemulsion in binary blends of complementary diblock copolymers

The phase behavior of binary blends of AB diblock copolymers of compositions f and 1 - f is examined using field-theoretic simulations. Highly asymmetric compositions (i.e., f ≈ 0) behave like homopolymer blends macrophase separating into coexisting A- and B-rich phases as the segregation is increased, whereas more symmetric diblocks (i.e., f ≈ 0.5) microphase separate into an ordered lamellar phase. In self-consistent field theory, these behaviors are separated by a Lifshitz critical point at f = 0.2113. However, its lower critical dimension is believed to be four, which implies that the Lifshitz point should be destroyed by fluctuations. Consistent with this, it is found to transform into a tricritical point. Furthermore, the highly swollen lamellar phase near the mean-field Lifshitz point disorders into a bicontinuous microemulsion (BμE), consisting of large interpenetrating A- and B-rich microdomains. BμE has been previously reported in ternary blends of AB diblock copolymer with its parent A- and B-type homopolymers, but in that system the homopolymers have a tendency to macrophase separate. Our alternative system for creating BμE is free of this macrophase separation.

Reaction-induced morphological transitions in a blend of diblock copolymers and reactive monomers: dissipative particle dynamics simulation

The dissipative particle dynamics (DPD) method is applied to the morphological transitions of microphase-separated domains in a mixture of symmetric AB-diblock copolymers and reactive C-monomers, where polymerization and cross-linking reactions take place among C-monomers. The initial structure for the DPD simulation is an equilibrated cylindrical domain structure prepared by the density-biased Monte Carlo method with density profiles obtained from the self-consistent field theory. By introducing a cross-linking reaction among reactive C-monomers, we confirmed that the DPD simulation reproduces the morphological transitions observed in experiments, where the domain morphology changes due to segregation between A-blocks of diblock copolymers and cross-linking networks of C-monomers. When the cross-linking reaction of C-monomers is sufficiently fast compared to the deformation of the domains, the initial cylindrical domains are preserved, while the distance between the domains increases. On the other hand, when the formation of the cross-linking network is slow, the domains can deform and reconnect with each other in the developing cross-linking network. In this case, we observe morphological transitions from the initial domain morphology with a large-curvature interface to another domain morphology with a smaller-curvature interface, such as the transition from the cylindrical phase to the lamellar phase. We calculated the spatial correlations in the microphase-separated domains and found that such correlations are affected by the speed of the formation of the cross-linking network depending on whether the bridging between microphase-separated domains occurs in a nucleation and growth process or in a spinodal decomposition process.

Gaming self-consistent field theory: Generative block polymer phase discovery

Block polymers are an attractive platform for uncovering the factors that give rise to self-assembly in soft matter owing to their relatively simple thermodynamic description, as captured in self-consistent field theory (SCFT). SCFT historically has found great success explaining experimental data, allowing one to construct phase diagrams from a set of candidate phases, and there is now strong interest in deploying SCFT as a screening tool to guide experimental design. However, using SCFT for phase discovery leads to a conundrum: How does one discover a new morphology if the set of candidate phases needs to be specified in advance? This long-standing challenge was surmounted by training a deep convolutional generative adversarial network (GAN) with trajectories from converged SCFT solutions, and then deploying the GAN to generate input fields for subsequent SCFT calculations. The power of this approach is demonstrated for network phase formation in neat diblock copolymer melts via SCFT. A training set of only five networks produced 349 candidate phases spanning known and previously unexplored morphologies, including a chiral network. This computational pipeline, constructed here entirely from open-source codes, should find widespread application in block polymer phase discovery and other forms of soft matter.

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.

Combining particle and field-theoretic polymer models with multi-representation simulations

Particle-based and field-theoretic simulations are both widely used methods to predict the properties of polymeric materials. In general, the advantages of each method are complementary. Field-theoretic simulations are preferred for polymers with high molecular weights and can provide direct access to chemical potentials and free energies, which makes them the method-of-choice for calculating phase diagrams. The trade-off is that field-theoretic simulations sacrifice the molecular details present in particle-based simulations, such as the configurations of individual molecules and their dynamics. In this work, we describe a new approach to conduct "multi-representation" simulations that efficiently map between particle-based and field-theoretic simulations. Our approach involves the construction of formally equivalent particle-based and field-based models, which are then simulated subject to the constraint that their spatial density profiles are equal. This constraint provides the ability to directly link particle-based and field-based simulations and enables calculations that can switch between one representation to the other. By switching between particle/field representations during a simulation, we demonstrate that our approach can leverage many of the advantages of each representation while avoiding their respective limitations. Although our method is illustrated in the context of complex sphere phases in linear diblock copolymers, we anticipate that it will be useful whenever free energies, rapid equilibration, molecular configurations, and dynamic information are all simultaneously desired.

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.

Predicting surfactant phase behavior with a molecularly informed field theory

HYPOTHESIS

The computational study of surfactants and self-assembly is challenging because 1) models need to reflect chemistry-specific interactions, and 2) self-assembled structures are difficult to equilibrate with conventional molecular dynamics. We propose to overcome these challenges with a multiscale simulation approach where relative entropy minimization transfers chemically-detailed information from all-atom (AA) simulations to coarse-grained (CG) models that can be simulated using field-theoretic methods. Field-theoretic simulations are not limited by intrinsic physical time scales like diffusion and allow for rigorous equilibration via free energy minimization. This approach should enable the study of properties that are difficult to obtain by particle-based simulations.

SIMULATION WORK

We apply this workflow to sodium dodecylsulfate. To ensure chemical fidelity we present an AA force field calibrated against interfacial tension experiments. We generate CG models from AA simulation trajectories and show that particle-based and field-theoretic simulations of the CG model reproduce AA simulations and experimental measurements.

FINDINGS

The workflow captures the complex balance of interactions in a multicomponent system ultimately described by an atomistic model. The resulting CG models can study complex 3D phases like double or alternating gyroids, and reproduce salt effects on properties like aggregation number and shape transitions.

Inverse Design of Complex Block Copolymers for Exotic Self-Assembled Structures Based on Bayesian Optimization

Variable chain topologies of multiblock copolymers provide great opportunities for the formation of numerous self-assembled nanostructures with promising potential applications. However, the consequent large parameter space poses new challenges for searching the stable parameter region of desired novel structures. In this Letter, by combining Bayesian optimization (BO), fast Fourier transform-assisted 3D convolutional neural network (FFT-3DCNN), and self-consistent field theory (SCFT), we develop a data-driven and fully automated inverse design framework to search for the desired novel structures self-assembled by ABC-type multiblock copolymers. Stable phase regions of three exotic target structures are efficiently identified in high-dimensional parameter space. Our work advances the new research paradigm of inverse design in the field of block copolymers.

Accounting for the ultraviolet divergence in field-theoretic simulations of block copolymer melts

This study examines the ultraviolet (UV) divergence in field-theoretic simulations (FTSs) of block copolymer melts, which causes an unphysical dependence on the grid resolution, Δ, used to represent the fields. Our FTSs use the discrete Gaussian-chain model and a partial saddle-point approximation to enforce incompressibility. Previous work has demonstrated that the UV divergence can be accounted for by defining an effective interaction parameter, χ=z_∞χ_b+c₂χ_b ²+c₃χ_b ³+⋯, in terms of the bare interaction parameter, χ_b, used in the FTSs, where the coefficients of the expansion are determined by a Morse calibration. However, the need to use different grid resolutions for different ordered phases generally restricts the calibration to the linear approximation, χ ≈ z_∞χ_b, and prevents the calculation of order-order transitions. Here, we resolve these two issues by showing how the nonlinear calibration can be translated between different grids and how the UV divergence can be removed from free energy calculations. By doing so, we confirm previous observations from particle-based simulations. In particular, we show that the free energy closely matches self-consistent field theory (SCFT) predictions, even in the region where fluctuations disorder the periodic morphologies, and similarly, the periods of the ordered phases match SCFT predictions, provided the SCFT is evaluated with the nonlinear χ.

Excluded-Volume Interactions in Field-Theoretic Simulations: Multiconvolutions and Model Equivalence

To deal with divergences of functional integrals in field-theoretic simulations (FTS) of complex fluids, the microscopic density is often smeared by being replaced by a convoluted one, typically using a Gaussian masking function. The smearing changes radically the nature of nonbonded interactions of the original microscopic density and results in a regularized model that is free of ultraviolet (UV) divergences. In this work, we first resolve a few fundamental issues related with the use of masking functions for δ-interactions in FTS and then we detail a new methodology that builds on the concept of multiconvoluted inverse potentials and a principle of model equivalence for statistical weights to accommodate more physically relevant interactions in FTS. The capabilities of the new approach are highlighted by examining the Gaussian-regularized Edwards model (GREM) and the Yukawa potential. A successful test calculation of the excess chemical potential of a polymer chain in a good solvent with the GREM illustrates the power of the new theoretical framework.

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).

Molecular Bonding in an Orbital-Free-Related Density Functional Theory

A density functional theory based on polymer self-consistent field theory is applied to systems of two atoms in order to show that this approach is capable of predicted molecular bonding. Periodic table elements from hydrogen up to neon are examined and homonuclear diatomic molecules are found to form for H₂, N₂, O₂, and F₂, in agreement with known results. The heteronuclear molecules CO and HF, which are known to exist under ambient conditions, are also found to be stable. Bond lengths for most of these molecules agree with experimental results to within less than 8%, with the exception of O₂ and F₂ which deviate more significantly. The bonding energy for H₂ is given and is within 16% of the known value, but fundamental vibrational frequencies do not agree well with experiment. The main approximations of the theory are very simple and include a Fermi-Amaldi correction to the electron-electron interaction to account for self-interactions and a basic expression for the Pauli potential to account for the exclusion principle. The self-consistent equations are solved in terms of basis functions that encode the cylindrical symmetry of diatomic molecules. Since orbitals are not used, the approach is related to orbital-free density functional theory.

Polymerization and Structure of Opposing Polymer Brushes Studied by Computer Simulations

A model of the polymerization process during the formation of a pair of polymer brushes was designed and investigated. The obtained system consisted of two impenetrable parallel surfaces with the same number of chains grafted on both surfaces. Coarse-grained chains embedded in nodes of a face-centered cubic lattice with excluded volume interactions were obtained by a ‘grafted from’ procedure. The structure of synthesized macromolecular systems was also studied. Monte Carlo simulations using the dynamic lattice liquid model were employed using dedicated parallel machine ARUZ in a large size and time scale. The parameters of the polymerization process were found to be crucial for the proper structure of the brush. It was found that for high grafting densities, chains were increasingly compressed, and there is surprisingly little interpenetration of chains from opposite surfaces. It was predicted and confirmed that in a polydisperse sample, the longer chains have unique configurations consisting of a stretched stem and a coiled crown.

The structure of polymer brushes: the transition from dilute to dense systems: a computer simulation study

Monodisperse polymer brushes were studied by means of Monte Carlo simulations. A coarse-grained model of a polymer brush was designed and the Cooperative Motion Algorithm was employed to model the polymerization process 'grafted from' and to study the structure of a brush immersed in a good solvent. The structure of brushes was determined as a function of the chain length and the grafting density. The influence of these parameters on the scaling properties of the brush was presented and discussed. A thorough analysis of the distribution of concentrations of the polymer segments and the distribution of chain free ends was also carried out. The analysis of the depth of penetration of the low molecular weight solvent into the brush area showed that the main factor determining the penetration is the grafting density. Good agreement between the simulation results and theoretical predictions is observed, especially for longer chains and higher grafting density. The origin of small quantitative differences between the simulation and theoretical results is discussed.

Sub-3-Nanometer Domain Spacings of Ultrahigh-χ Multiblock Copolymers with Pendant Ionic Groups

We investigated the temperature-dependent phase behavior and interaction parameter of polyethylene-based multiblock copolymers with pendant ionic groups. These step-growth polymers contain short polyester blocks with a single Li⁺SO₃^- group strictly alternating with polyethylene blocks of x-carbons (PESxLi, x = 12, 18, 23). At room temperature, these polymers exhibit layered morphologies with semicrystalline polyethylene blocks. Upon heating above the melting point (∼130 °C), PES18Li shows two order-to-order transitions involving Ia3̅d gyroid and hexagonal morphologies. For PES12Li, an order-to-disorder transition accompanies the melting of the polyethylene blocks. Notably, a Flory-Huggins interaction parameter was determined from the disordered morphologies of PES12Li using mean-field theory: χ(T) = 77.4/T + 2.95 (T in Kelvin) and χ(25 °C) ≈ 3.21. This ultrahigh χ indicates that the polar ionic and nonpolar polyethylene segments are highly incompatible and affords well-ordered morphologies even when the combined length of the alternating blocks is just 18-29 backbone atoms. This combination of ultrahigh χ and short multiblocks produces sub-3-nm domain spacings that facilitate the control of block copolymer self-assembly for various fields of study, including nanopatterning.

Dynamics of Opposing Polymer Brushes: A Computer Simulation Study

Opposing polymer brush systems were synthesized and investigated by molecular modeling. Chains were restricted to a face-centered cubic lattice with the excluded volume interactions only. The system was confined between two parallel impenetrable walls, with the same number of chains grafted to each surface. The dynamic properties of such systems were studied by Monte Carlo simulations based on the dynamic lattice liquid model and using a highly efficient parallel machine ARUZ, which enabled the study of large systems and long timescales. The influence of the surface density and mean polymer length on the system dynamic was discussed. The self-diffusion coefficient of the solvent depended strongly on the degree of polymerization and on the polymer concentration. It was also shown that it is possible to capture changes in solvent mobility that can be attributed to the regions of different polymer densities.

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).

Field-theoretic simulations beyond δ-interactions: Overcoming the inverse potential problem in auxiliary field models

Modern field-theoretic simulations of complex fluids and polymers are constructed around a particle-to-field transformation that brings an inverse potential u^-1 in the model equations. This has restricted the application of the framework to systems characterized by relatively simple pairwise interatomic interactions; for example, excluded volume effects are treated through the use of δ-function interactions. In this study, we first review available nonbonded pair interactions in field-theoretic models and propose a classification. Then, we outline the inverse potential problem and present an alternative approach on the basis of a saddle-point approximation, enabling the use of a richer set of pair interaction functions. We test our approach by using as an example the Morse potential, which finds extensive applications in particle-based simulations, and we calibrate u^-1 with results from a molecular dynamics simulation. The u^-1 thus obtained is consistent with the field-theoretic model equations, and when used in stand-alone self-consistent field simulations, it produces the correct fluid structure starting from a random initial state of the density field.

Fluctuation correction for the order-disorder transition of diblock copolymer melts

The order-disorder transition (ODT) of diblock copolymer melts is evaluated for an invariant polymerization index of N¯=10⁴, using field-theoretic simulations (FTS) supplemented by a partial saddle-point approximation for incompressibility. For computational efficiency, the FTS are performed using the discrete Gaussian-chain model, and results are then mapped onto the continuous model using a linear approximation for the Flory-Huggins χ parameter. Particular attention is paid to the complex phase window. Results are found to be consistent with the well-established understanding that the gyroid phase extends down to the ODT. Furthermore, our simulations are the first to predict that the Fddd phase survives fluctuation effects, consistent with experiments.

Temperature dependence of the conductivity of filled diblock copolymers

We demonstrate that an insulating diblock copolymer system (DBC) filled with conductive fillers can be used as an electrically responsive soft composite that changes its conductivity in response to temperature-induced changes in its morphology. By combining the phase field model describing the morphology of the DBC system with the Monte Carlo simulations and the resistor network model describing electrical properties of the filler network, we calculate the conductivity of this composite. The calculated conductivity is found to essentially depend, in particular, on the temperature of the composite. Changing the temperature is shown to result in morphological changes in the DBC system causing the structural changes in the filler network. In particular, the order-disorder transition in the host DBC system is found to be accompanied by the conductor-insulator transition in the filler network. The effect of the difference between the affinities of the fillers for dissimilar copolymer blocks on the composite conductivity, as well as the effect of the repulsive and attractive interaction between fillers, is considered in detail.

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.

Single chain in mean field simulation of flexible and semiflexible polymers: comparison with discrete chain self-consistent field theory

Single chain in mean field (SCMF) simulation is a theoretical framework performing Monte Carlo moves of explicit polymer chains under quasi-instantaneously updated external fields which were originally imported from the self-consistent field theory (SCFT). Even though functional-based hybrid simulations are often used to compare the results of SCFT and MC simulation, the adoption of a finite number of coarse-grained segments makes direct comparison rather difficult. In this study, we perform SCMF simulation of block copolymers using various chain models and quantitatively compare it with discrete chain SCFT (DCSCFT) which finds the mean field solution of polymers with a finite number of segments. By comparing free energy and natural period of the symmetric block copolymer lamellar phase, we systematically show that DCSCFT serves as an intermediate step between SCMF simulation and SCFT. In addition, by adopting angle dependent bond potential, we perform SCMF simulation of semiflexible polymers using bead-spring and freely jointed chain models. As the chain stiffness increases, the lamellar phase tends to align perpendicular to the surfaces when confined between two neutral walls. We also investigate the effects of fluctuation and chain stiffness on the distribution of chain ends. The tendency of chain end segregation towards the surfaces turns out to increase as the chain stiffness increases for both homopolymer and block copolymer systems.