Self-consistent Field Theory Calculations Research Articles

Flow-driven translocation of polymer vesicle through a nanochannel

Flow-driven translocation of a polymer vesicle self-assembled from amphiphilic diblock copolymers through a nanochannel is studied using hybrid Lattice-Boltzmann Molecular Dynamics simulations. Three translocation modes are discovered: a single vesicle, small vesicles, and tiny micelles in sequence, respectively, as the width of the nanochannel is decreased, which is confirmed by the self-consistent field theory calculations. Accordingly, spherical vesicles, ellipsoidal vesicles, and worm-like micelles are formed sequentially after crossing the nanochannel. More importantly, we find that the critical flow rate monotonically decreases as the ratio of the polymer vesicle size and the width of the nanochannel decreases until it approaches 1. Our results provide a basis for characterizing the stability of vesicles or sorting vesicles with different sizes and also point to a new idea of regulating spherical vesicles to ellipsoidal vesicles or worm-like micelles.

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.

Chimney-Shaped Phase Diagram in a Polymer Blend Electrolyte.

The phase behavior of polymer blend electrolytes comprising poly(ethylene oxide) (PEO)/poly(methyl methacrylate) (PMMA)/lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) was determined using a combination of light and small angle neutron scattering (SANS) experiments. The results at a fixed temperature (110 °C) are presented on a PEO concentration versus salt (LiTFSI) concentration plot. The blends are miscible at all PEO concentrations in the absence of salt. With added salt, a region of immiscibility is obtained in PEO-lean polymer blend electrolytes; blends rich in PEO remain miscible at most salt concentrations. A narrow region of immiscibility juts into the miscible region, giving the phase diagram a chimney-like appearance. The data are qualitatively consistent with a simple extension of Flory-Huggins theory with a composition-dependent Flory-Huggins interaction parameter, χ, that was determined independently from SANS data from homogeneous blend electrolytes. Phase diagrams like the one we obtained were anticipated by self-consistent field theory calculations that account for correlations between ions. The relationship between these theories and measured χ remains to be established.

Reexamine the emergence and stability of the square cylinder phase in block copolymers.

Recently, a few AB-type multiblock copolymers have been successfully designed to form stable square cylinder phase based on self-consistent field theory (SCFT) calculations. The previous works only identify the stability region of the square phase but not analyzing its stability, which is closely related to the free-energy landscape. In this work, we have reexamined the stability of the square phase in $\mathrm{B_1 A_1 B_2 A_2 B_3}$ linear pentablock and $\mathrm{(B_1AB_2)_5}$ star triblock copolymers by drawing the free-energy landscape with respect to the two dimensions of a rectangular unit cell. Our results demonstrate that the square phase continuously transfers to the rectangular phase as the degree of packing frustration is gradually released. Moreover, the prolate contour lines of the free-energy landscape indicate the weak stability of the square phase in the $\mathrm{B_1 A_1 B_2 A_2 B_3}$ copolymer. In contrast, the stability of the square phase is notably improved in the $\mathrm{(B_1AB_2)_5}$ copolymer due to its enhanced concentration of bridging configurations. Our work sheds light on the understanding of the stability of the square cylinder phase in block copolymers. Accordingly, we propose some possible strategies for further designing new AB-type block copolymer systems to obtain more stable square phase.

Template Design for Complex Block Copolymer Patterns Using a Machine Learning Method.

This study represents the first attempt to address the inverse design problem of the guiding template for directed self-assembly (DSA) patterns using solely machine learning methods. By formulating the problem as a multi-label classification task, the study shows that it is possible to predict templates without requiring any forward simulations. A series of neural network (NN) models, ranging from the basic two-layer convolutional neural network (CNN) to the large NN models (32-layer CNN with 8 residual blocks), have been trained using simulated pattern samples generated by thousands of self-consistent field theory (SCFT) calculations; a number of augmentation techniques, especially suitable for predicting morphologies, have been also proposed to enhance the performance of the NN model. The exact match accuracy of the model in predicting the template of simulated patterns was significantly improved from 59.8% for the baseline model to 97.1% for the best model of this study. The best model also demonstrates an excellent generalization ability in predicting the template for human-designed DSA patterns, while the simplest baseline model is ineffective in this task.

Alternating Gyroid Stabilized by Surfactant-like Triblock Terpolymers in IS/SO/ISO Ternary Blends

Block polymers self-assembled into chiral alternating-gyroid (GA) morphologies are desirable templates for fabricating photonic materials. Recent self-consistent field theory (SCFT) calculations on a model system predicted that adding a minuscule amount of ABC triblocks into a blend of thermodynamically symmetric double-gyroid-forming AB and BC diblocks with immiscible A and C blocks stabilizes GA. Here, we use SCFT to investigate the feasibility of this method in practice by computing the phase behaviors of ternary mixtures of poly(isoprene-b-styrene) (IS), poly(styrene-b-ethylene oxide) (SO), and poly(isoprene-b-styrene-b-ethylene oxide) (ISO). Addition of ISO indeed stabilizes GA; however, the phase window is relatively narrow, due to the packing frustration induced by the mismatch in preferred domain sizes between IS and SO diblock copolymers. Designing the system to have comparable diblock domain sizes significantly enhances the relative stability of GA against the competing alternating lamellae, and this stability is also sensitive to the degree of polymerization of the ISO midblock. Moreover, the stabilization of alternating morphologies is sensitive to temperature, and less ISO is needed at lower temperatures. The effect of some other design parameters, such as diblock volume fraction, is also investigated. These computational results provide insights into possible experimental strategies for producing GA and highlight the sensitivity of network phase formation to asymmetries in the thermodynamic parameters.

Patterning of Polymer-Functionalized Nanoparticles with Varied Surface Mobilities of Polymers.

The polymers can be either dynamically tethered to or permanently grafted to the nanoparticle to produce polymer-functionalized nanoparticles. The surface mobility of polymer ligands with one end anchored to the nanoparticle can affect the surface pattern, but the effect remains unclear. Here, we addressed the influence of lateral polymer mobility on surface patterns by performing self-consistent field theory calculations on a modeled polymer-functionalized nanoparticle consisting of immobile and mobile brushes. The results show that except for the radius of nanoparticles and grafting density, the fraction of mobile brushes substantially influences the surface patterning of polymer-functionalized nanoparticles, including striped patterns and patchy patterns with various patches. The number of patches on a nanoparticle increases as the fraction of mobile brushes decreases, favored by the entropy of immobile brushes. Critically, we found that broken symmetry usually occurs in patchy nanoparticles, associated with the balance of enthalpic and entropic effects. The present work provides a fundamental understanding of the dependence of surface patterning on lateral polymer mobility. The work could also guide the preparation of diversified nanopatterns, especially for the asymmetric patchy nanoparticles, enabling the fundamental investigation of the interaction between polymer-functionalized nanoparticles.

Orientation control of the hexagonal and lamellar phases in thin block copolymer films using in-plane AC electric field

Lamellar and hexagonal pattern rearrangements in thin films of polystyrene–block–poly(2-vinyl pyridine) and polystyrene–block–poly(4-vinyl pyridine) copolymers simultaneously exposed to an in-plane AC electric field and saturated chloroform vapor are studied with atomic force microscopy and analyzed via the numerical solution of the self-consistent field theory equations. It is demonstrated that the use of AC field with the root-mean-square strength higher than 8 V⋅μm−1 allows one to effectively orient microphase-separated domains along the field direction on the tens of microns scale. At the same time, the AC field considerably lowers the risk of breakdown and eliminates effects related to ionic transport in the presence of solvent vapor. The role of such factors as the exposure time, field strength and frequency is investigated. Theoretical considerations prove the equivalent effect of AC and DC fields on the structure of the copolymer film in a wide frequency range. Self-consistent field theory calculations of the free energy as a function of film thickness make it possible to exactly identify which phase dominates in the film. In particular, an apparent transformation of standing cylinders into long threads aligned in the field direction should be interpreted as the phase transition from the perpendicular to parallel hexagonal phase, which can take place in a certain range of film thicknesses, provided the electric field strength exceeds a certain threshold value. In the case of a perpendicular lamellar phase, the domain orientation along the direction of the electric field is always profitable. The observed morphological rearrangements under an in-plane field, which preserve connectivity between the film surfaces through the domains of the minor copolymer block, can be important for practical applications.

Stabilizing a Double Gyroid Network Phase with 2 nm Feature Size by Blending of Lamellar and Cylindrical Forming Block Oligomers

Molecular dynamics simulations are used to study binary blends of an AB-type diblock and an AB2-type miktoarm triblock amphiphiles (also known as high-χ block oligomers) consisting of sugar-based (A) and hydrocarbon (B) blocks. In their pure form, the AB diblock and AB2 triblock amphiphiles self-assemble into ordered lamellar (LAM) and cylindrical (CYL) structures, respectively. At intermediate compositions, however, the AB2-rich blend (0.2 ≤ xAB ≤ 0.4) forms a double gyroid (DG) network, whereas perforated lamellae (PL) are observed in the AB-rich blend (0.5 ≤ xAB ≤ 0.8). All of the ordered mesophases present domain pitches under 3 nm, with 1 nm feature sizes for the polar domains. Structural analyses reveal that the nonuniform interfacial curvatures of DG and PL structures are supported by local composition variations of the LAM- and CYL-forming amphiphiles. Self-consistent mean field theory calculations for blends of related AB and AB2 block polymers also show the DG network at intermediate compositions, when A is the minority block, but PL is not stable. This work provides molecular-level insights into how blending of shape-filling molecular architectures enables network phase formation with extremely small feature sizes over a wide composition range.

Order-to-Disorder Transition of Cylinder-Forming Block Copolymer Films Confined within Neutral Interfaces

We present the phase behavior of cylinder-forming block copolymer (BCP) films confined within neutral interfaces through theory and experiment. The Langevin field-theoretic simulation (L-FTS) is utilized to account for the compositional fluctuation effect on the order-to-disorder transition (ODT), and the ultraviolet divergence in the L-FTS result is eliminated by renormalization of the Flory–Huggins parameter χ, allowing accurate comparison of thickness-dependent (χN)ODT, where N is the polymerization index. The L-FTS is accelerated by utilizing a deep learning method, and we managed to achieve a 50–60% reduction in simulation time. The L-FTS results show that (χN)ODT decreases as the film thickness decreases at a very large value of the invariant polymerization index N̅, and BCPs undergo a phase transition from disordered to cylindrical phases through a spherical phase as predicted by the self-consistent field theory calculation. By performing the L-FTS at experimentally relevant low N̅, we show that the BCPs undergo a direct transition from disordered to cylindrical phases, and the (χN)ODT value of cylinder-forming BCPs confined within neutral interfaces increases as the film thickness decreases. To verify and complement the theoretical results, we investigate the ODT temperature (TODT) of cylinder-forming polystyrene-b-poly(2-vinylpyridine) (PS-b-P2VP) films supported on a neutral mat of P(S-r-2VP) random copolymers using grazing-incidence small-angle X-ray scattering. Our film experiments with low-N̅ PS-b-P2VP confirm the direct transition from disordered to cylindrical phases and the decrease in TODT with decreasing thickness below the onset thickness above which the TODT levels off.

Open-source platform for block polymer formulation design using particle swarm optimization.

Facile exploration of large design spaces is critical to the development of new functional soft materials, including self-assembling block polymers, and computational inverse design methodologies are a promising route to initialize this task. We present here an open-source software package coupling particle swarm optimization (PSO) with an existing open-source self-consistent field theory (SCFT) software for the inverse design of self-assembling block polymers to target bulk morphologies. To lower the barrier to use of the software and facilitate exploration of novel design spaces, the underlying SCFT calculations are seeded with algorithmically generated initial fields for four typical morphologies: lamellae, network phases, cylindrical phases, and spherical phases. In addition to its utility within PSO, the initial guess tool also finds generic applicability for stand-alone SCFT calculations. The robustness of the software is demonstrated with two searches for classical phases in the conformationally symmetric diblock system, as well as one search for the Frank-Kasper [Formula: see text] phase in conformationally asymmetric diblocks. The source code for both the initial guess generation and the PSO wrapper is publicly available.

Emergence of Hexagonally Close-Packed Spheres in Linear Block Copolymer Melts.

The hexagonally close-packed (HCP) sphere phase is predicted to be stable across a narrow region of linear block copolymer phase space, but the small free energy difference separating it from face-centered cubic spheres usually results in phase coexistence. Here, we report the discovery of pure HCP spheres in linear block copolymer melts with A = poly(2,2,2-trifluoroethyl acrylate) ("F") and B = poly(2-dodecyl acrylate) ("2D") or poly(4-dodecyl acrylate) ("4D"). In 4DF diblocks and F4DF triblocks, the HCP phase emerges across a substantial range of A-block volume fractions (circa fA = 0.25-0.30), and in F4DF, it forms reversibly when subjected to various processing conditions which suggests an equilibrium state. The time scale associated with forming pure HCP upon quenching from a disordered liquid is intermediate to the ordering kinetics of the Frank-Kasper σ and A15 phases. However, unlike σ and A15, HCP nucleates directly from a supercooled liquid or soft solid without proceeding through an intermediate quasicrystal. Self-consistent field theory calculations indicate the stability of HCP is intimately tied to small amounts of molar mass dispersity (Đ); for example, an HCP-forming F4DF sample with fA = 0.27 has an experimentally measured Đ = 1.04. These insights challenge the conventional wisdom that pure HCP is difficult to access in linear block copolymer melts without the use of blending or other complex processing techniques.

High-throughput prediction of stress\u2013strain curves of thermoplastic elastomer model block copolymers by combining hierarchical simulation and deep learning

We achieved high-throughput prediction of the stress–strain (S–S) curves of thermoplastic elastomers by combining hierarchical simulation and deep learning. ABA triblock copolymer with a phase-separated structure was used as a thermoplastic elastomer model. The S–S curves of the ABA triblock copolymers were calculated from the hierarchical simulation of self-consistent field theory calculations and coarse-grained molecular dynamics simulations. Because such hierarchical simulations require considerable computational resources, we applied a deep learning technique to accelerate the prediction. Sets of phase-separated structures and the S–S curves obtained from the hierarchical simulation were used to train a 3D convolutional neural network. Using the trained network, we confirmed that the predicted S–S curves of the untrained structures accurately reproduced the simulation results. These results will enable us to design novel polymers and phase-separated structures with desired S–S curves by high-throughput screening of a wide variety of structures.Graphic abstract

Data for Order and Disorder in ABCA′ Tetrablock Terpolymers

Self-consistent field theory calculations appearing in the paper, along with scripts for generating the plots of those data using Polymer Self Consistent Field (PSCF) code.

Order and Disorder in ABCA' Tetrablock Terpolymers.

Self-assembly of poly(styrene)-block-poly(isoprene)-block-poly(lactide)-block-poly(styrene) (PS-PI-PLA-PS' or SILS') tetrablock terpolymers, where the volume fractions of the first three blocks are nearly equivalent, was studied both experimentally and using the self-consistent field theory (SCFT). SCFT indicates that addition of the terminal PS' chain to a low-molecular-mass, hexagonally packed cylinders forming, SIL precursor can produce a disordered state due to preferential mixing of the polystyrene end-blocks with the PI and PLA midblocks in the SILS' tetrablock, alleviating the unfavorable contact between the highly incompatible PI and PLA segments. In contrast, SCFT predicts that higher-molar-mass triblock precursors will maintain an ordered morphology upon addition of the terminal PS' block due to stronger overall segregation strengths. These predictions were tested using three sets of SILS' polymers that were synthesized based on three precursor SIL triblock polymers differing in total molar mass (14, 30, and 47 kg mol-1) and varying the length of the terminal PS' chain. In the lowest-molar-mass set of tetrablock polymers, the shift from order to disorder was observed in the materials at ambient temperature as the molar mass of the terminal PS' block was increased, consistent with SCFT calculations. Disorder with longer S' chain lengths was not found in the two higher-molar-mass polymer sets; the medium-molar-mass set showed both microphase separation and long-range order based on transmission electron microscopy (TEM) and small-angle X-ray scattering (SAXS), while the largest of these block polymers microphase separated but showed limited long-range order. The combination of the experimental and theoretical results presented in this work provides insights into the self-assembly of ABCA'-type polymers and highlights potential complications that arise from frustration in accessing well-ordered materials.

The size and affinity effect of counterions on self-assembly of charged block copolymers.

The effect of counterions' size and affinity on the microphase separated morphologies of neutral-charged diblock copolymers is investigated systematically using a random phase approximation (RPA) and self-consistent field theory (SCFT). The phase diagrams as a function of χAB and fA at different counterion sizes and different affinities to neutral blocks are constructed, respectively. Stability limits calculated using the RPA are in good agreement with the disorder-body-centered cubic phase boundaries from SCFT calculations. It was found that increasing the size of counterions causes the phase diagram to shift upward and leftward, which is attributed to electrostatic interactions and the intrinsic volume of counterions. The domain size of the ordered phase shows an unexpected tendency that it decreases with increasing counterions' size. The counterions' distributions in H and G phases demonstrate that it is electrostatic interaction, instead of packing frustration, that plays a leading role in such systems. For finite size counterions, with the increase in affinity between counterions and neutral blocks, the phase diagram shifts upward, indicating the improved compatibility between different blocks. Furthermore, the affinity effect between counterions and neutral blocks can be mapped into an effective Flory parameter χAB ' = χAB + 0.27χBC.

Open-source code for self-consistent field theory calculations of block polymer phase behavior on graphics processing units.

Self-consistent field theory (SCFT) is a powerful approach for computing the phase behavior of block polymers. We describe a fast version of the open-source Polymer Self-Consistent Field (PSCF) code that takes advantage of the massive parallelization provided by a graphical processing unit (GPU). Benchmarking double-precision calculations indicate up to 30× reduction in time to converge SCFT calculations of various diblock copolymer phases when compared to the Fortran CPU version of PSCF using the same algorithms, with the speed-up increasing with increasing unit cell size for the diblock polymer problems examined here. Where double-precision accuracy is not needed, single-precision calculations can provide speed-up of up to 60× in convergence time. These improvements in speed within an open-source format open up new vistas for SCFT-driven block polymer materials discovery by the community at large.

Controllable interfacial adhesion behaviors of polymer-on-polymer surfaces during fused deposition modeling 3D printing process

The controllable interfacial adhesion behaviors of polymer-on-polymer surfaces during fused deposition modeling (FDM) three-dimensional (3D) printing process were examined by combining the experimental tests and self-consistent field theory (SCFT) to further get insight into the interfacial bonding behaviors between currently- and previously-deposited layers. The results show that the interfacial bonding strength can be availably tailored by the filling density and cooling rate. An unusual and interesting phenomenon was also observed that the interfacial bonding strength firstly increases to an optimal value and then decreases as the filling density increases. Both the experimental observations and SCFT calculations revealed that the better strength of interfacial adhesion is mainly attributed to the larger width of wetted interface and degree of inter-molecular diffusion between two adjacent layers. The critical mechanism behind the controllable interfacial adhesion strength of polymer-on-polymer surfaces were precisely clarified. The present work may provide useful information for preparing lightweight FDM parts with high performance and a general understanding of the interfacial bonding behaviors of polymer-on-polymer surfaces.

Estimating the Segregation Strength of Microphase-Separated Diblock Copolymers from the Interfacial Width.

The ever-growing catalog of monomers being incorporated into block polymers affords exceptional control over phase behavior and nanoscale structure. The segregation strength, , is the fundamental link between the molecular-level detail and the thermodynamics. However, predicting phase behavior mandates at least one experimental measurement of for each pair of blocks. This typically requires access to the disordered state. We describe a method for estimating from small-angle X-ray scattering measurements of the interfacial width between lamellar microdomains, , in the microphase-separated melt. The segregation strength is determined by comparing to self-consistent field theory calculations of the intrinsic interfacial width, , as a function of the mean-field . The method is validated using a series of independent experimental measurements of and , measured via the order-disorder transition temperature, . The average absolute relative difference between calculated from and the value calculated from is a modest 11%. Corrections for nonplanarity of the interfaces are investigated but do not improve the agreement between the experiments and theory. Published 2019. This article is a U.S. Government work and is in the public domain in the USA.

Thermodynamic geometry of Ising ferromagnet in an external magnetic field: A self-consistent field theory calculation

We present a complete geometrical description for the ferromagnetic Ising model in the pair approximation as introduced by Balcerzak (2003) using self-consistent field theory. A metric is defined in a two-dimensional phase space of magnetization (M) and nearest-neighbour correlation function (C). Based on the metric elements an expression for the thermodynamic Ricci scalar (R) is derived in terms of the lattice coordination number q. We study R as the temperature (T), magnetic field (h) and exchange energy coupling (J) are varied and show that there are T and h dependent critical properties for q=6. By direct comparison, we demonstrate that the special case q=2 provides a consistent behaviour with the already known exact formula in Janyszek and Mrugała work (1989) for the one-dimensional Ising model.