Polymer Self-consistent Field Research Articles

Machine learning and polymer self-consistent field theory in two spatial dimensions.

A computational framework that leverages data from self-consistent field theory simulations with deep learning to accelerate the exploration of parameter space for block copolymers is presented. This is a substantial two-dimensional extension of the framework introduced in the work of Xuan et al. [J. Comput. Phys. 443, 110519 (2021)]. Several innovations and improvements are proposed. (1) A Sobolev space-trained, convolutional neural network is employed to handle the exponential dimension increase of the discretized, local average monomer density fields and to strongly enforce both spatial translation and rotation invariance of the predicted, field-theoretic intensive Hamiltonian. (2) A generative adversarial network (GAN) is introduced to efficiently and accurately predict saddle point, local average monomer density fields without resorting to gradient descent methods that employ the training set. This GAN approach yields important savings of both memory and computational cost. (3) The proposed machine learning framework is successfully applied to 2D cell size optimization as a clear illustration of its broad potential to accelerate the exploration of parameter space for discovering polymer nanostructures. Extensions to three-dimensional phase discovery appear to be feasible.

On the origins of spontaneous spherical symmetry-breaking in open-shell atoms through polymer self-consistent field theory.

An alternative approach to density functional theory based on self-consistent field theory for ring polymers is applied to neutral atoms hydrogen to neon in their ground-states. The spontaneous emergence of an atomic shell structure and spherical symmetry-breaking of the total electron density are predicted by the model using the ideas of polymer excluded-volume between pairs of electrons to enforce the Pauli-exclusion principle and an exact electron self-interaction correction. The Pauli potential is approximated by neglecting inter-atomic correlations along with other types of correlations, and comparisons to Hartree-Fock theory are made, which also ignores correlations. The model shows excellent agreement with Hartree-Fock theory to within the standards of orbital-free density functional theory for the atomic binding energies and density profiles of the first six elements, providing exact matches for the elements hydrogen and helium. The predicted shell structure starts to deviate significantly past the element neon, and spherical symmetry-breaking is first predicted to occur at carbon instead of boron. The self-consistent field theory energy functional that describes the model is decomposed into thermodynamic components to trace the origin of spherical symmetry-breaking. It is found to arise from the electron density approaching closer to the nucleus in non-spherical distributions, which lowers the energy despite resulting in frustration between the quantum kinetic energy, electron-electron interaction, and the Pauli exclusion interaction. The symmetry-breaking effect is found to have a minimal impact on the binding energies, which suggests that the spherical-averaging approximation used in previous work is physically reasonable when investigating atomic systems. The pair density contour plots display behavior similar to polymer macro-phase separation, where individual electron pairs occupy single lobe structures that together form a dumbbell shape analogous to the 2p orbital shape. It is further shown that the predicted densities satisfy known constraints and produce the same total electronic density profile that is predicted by other formulations of quantum mechanics.

Deep learning and inverse discovery of polymer self-consistent field theory inspired by physics-informed neural networks.

We devise a deep learning solver inspired by physics-informed neural networks (PINNs) to tackle the polymer self-consistent field theory (SCFT) equationsfor one-dimensional AB-diblock copolymers. The PINNs framework comprises two parallel feedforward neural networks that separately represent the segmental partition functions and self-consistent chemical potential fields. The two networks are coupled through a loss function incorporating the governing equation, initial and boundary conditions, and the incompressibility constraint. To avoid the metastable homogeneous solution, the network parameters are initialized based on known self-consistent fields obtained from the numerical pseudospectral method. For copolymers of length N at a given volume fraction of A block (f) and the reduced Flory-Huggins interaction parameter (χN), the minimization of the loss function leads to the converged network parameters that successfully capture the stable lamellar phase. The periodicity of the lamellar structure is correctly reproduced for the explored sets of [f,χN], irrespective of the presumed computational domain size for initialization. Moreover, the proposed PINNs are applicable to the inverse discovery of the interaction parameter and the embedded chemical potential fields for an observed structure. This capability of solving the inverse SCFT problem demonstrates the potential of using PINNs to accelerate the exploration of new polymeric materials.

An interpretation of quantum foundations based on density functional theory and polymer self-consistent field theory

The Feynman quantum-classical isomorphism between classical statistical mechanics in 3+1 dimensions and quantum statistical mechanics in 3 dimensions is used to connect classical polymer self-consistent field theory with quantum time-dependent density functional theory. This allows the theorems of density functional theory to relate non-relativistic quantum mechanics to a classical statistical mechanical derivation of polymer self-consistent field theory for ring polymers in a 4 dimensional thermal-space. One dynamic postulate is added to two static postulates which allows for a description of quantum physics from a 5 dimensional thermal-space-time ensemble perspective. A connection with aspects of classical field theory can be made in the classical limit.

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.

Quantitative Comparison of Field-Update Algorithms for Polymer SCFT and FTS

A wide range of field-update algorithms for polymer self-consistent field theory (SCFT) and field theoretic simulations (FTSs) are analyzed. We provide the first direct comparison between Anderson ...

Nonequilibrium Molecular Conformations in Polymer Self-Consistent Field Theory

The morphology of a multicomponent polymer melt within self-consistent field theory (SCFT) is completely characterized by the spatial density distribution of the components. SCFT therefore assumes ...

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.

Atomic shell structure from an orbital-free-related density-functional-theory Pauli potential

Polymer self-consistent field theory techniques are used to find radial electron densities and total binding energies for isolated atoms. Quantum particles are modelled as Gaussian threads with ring-polymer architecture in a four dimensional thermal-space, and a Pauli potential is postulated based on classical excluded volume implemented in the thermal-space using Edwards/Flory-Huggins interactions in a mean-field approximation. Other approximations include a Fermi-Amaldi correction for electron-electron self-interactions, a spherical averaging approximation to reduce the dimensionality of the problem, and the neglect of correlations. Polymer scaling theory is used to show that the excluded volume form of Pauli potential reduces to the known Thomas-Fermi energy density in the uniform limit. Self-consistent equations are solved using a bilinear Fourier expansion, with radial basis functions, for the first eighteen elements of the periodic table. Radial electron densities show correct shell structure, and the errors on the total binding energies compared to known binding energies are less than 9% for the lightest elements and drop to 3% or less for atoms heavier than nitrogen. More generally, it is suggested that only two postulates are needed within classical statistical mechanics to achieve equivalency of predictions with static, non-relativistic quantum mechanics: First, quantum particles are modelled as Gaussian threads in four dimensional thermal-space and, second, pairs of threads (allowing for spin) are subject to classical excluded volume in the thermal-space. It is shown that these two postulates in thermal-space become the same as the Heisenberg uncertainty principle and the Pauli exclusion principle in three dimensional space.

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.

Linear Scaling Self-Consistent Field Theory with Spectral Contour Accuracy.

We present a new methodology for polymer self-consistent field theory (SCFT) that has spectral accuracy in the contour dimension while retaining linear scaling of computational effort with system size. In contrast, traditional linear-scaling algorithms only have polynomial order accuracy. The improved accuracy allows for faster simulations and lower memory costs compared to traditional algorithms. The new spectral methods are enabled by converting from an auxiliary field representation to a recently developed "polymer coherent states" framework.

An alternative derivation of orbital-free density functional theory.

Polymer self-consistent field theory techniques are used to derive quantum density functional theory without the use of the theorems of density functional theory. Instead, a free energy is obtained from a partition function that is constructed directly from a Hamiltonian so that the results are, in principle, valid at finite temperatures. The main governing equations are found to be a set of modified diffusion equations, and the set of self-consistent equations are essentially identical to those of a ring polymer system. The equations are shown to be equivalent to Kohn-Sham density functional theory and to reduce to classical density functional theory, each under appropriate conditions. The obtained noninteracting kinetic energy functional is, in principle, exact but suffers from the usual orbital-free approximation of the Pauli exclusion principle in addition to the exchange-correlation approximation. The equations are solved using the spectral method of polymer self-consistent field theory, which allows the set of modified diffusion equations to be evaluated for the same computational cost as solving a single diffusion equation. A simple exchange-correlation functional is chosen, together with a shell-structure-based Pauli potential, in order to compare the ensemble average electron densities of several isolated atom systems to known literature results. The agreement is excellent, justifying the alternative formalism and numerical method. Some speculation is provided on considering the timelike parameter in the diffusion equations, which is related to temperature, as having dimensional significance, and thus picturing pointlike quantum particles instead as nonlocal, polymerlike, threads in a higher dimensional thermal-space. A consideration of the double-slit experiment from this point of view is speculated to provide results equivalent to the Copenhagen interpretation. Thus, the present formalism may be considered as a type of "pilot-wave," realist, perspective on density functional theory.

Efficient order-adaptive methods for polymer self-consistent field theory

A highly accurate and memory-efficient approach for the solution of polymer self-consistent field theory (SCFT) is proposed. The central idea is to combine spectral integration in the polymer chain contour variable with a spectral deferred correction technique to solve the SCFT modified diffusion equations with arbitrarily high order of accuracy. The result is a robust method that achieves high accuracy with a minimal number of discrete contour nodes, which translates into vastly reduced memory requirements and increased computational efficiency. In particular, this spectral deferred correction method enables the computation of strongly segregated systems with unprecedented accuracy. Moreover, the framework of deferred corrections allows us to adaptively increase the order of accuracy during the outer saddle point iteration to drastically reduce the cost of a SCFT computation.

Self-Consistent Field Lattice Model for Polymer Networks.

A lattice model based on polymer self-consistent field theory is developed to predict the equilibrium statistics of arbitrary polymer networks. For a given network topology, our approach uses moment propagators on a lattice to self-consistently construct the ensemble of polymer conformations and cross-link spatial probability distributions. Remarkably, the calculation can be performed “in the dark”, without any prior knowledge on preferred chain conformations or cross-link positions. Numerical results from the model for a test network exhibit close agreement with molecular dynamics simulations, including when the network is strongly sheared. Our model captures nonaffine deformation, mean-field monomer interactions, cross-link fluctuations, and finite extensibility of chains, yielding predictions that differ markedly from classical rubber elasticity theory for polymer networks. By examining polymer networks with different degrees of interconnectivity, we gain insight into cross-link entropy, an important quantity in the macroscopic behavior of gels and self-healing materials as they are deformed.

Broadly Accessible Self-Consistent Field Theory for Block Polymer Materials Discovery

Self-consistent field theory (SCFT) is a powerful tool for the design and interpretation of experiments on block polymer materials. In this Perspective, we lower the barrier to entry to the use of SCFT by experimental groups by two means. First, we present a pedagogical introduction to an improved version of the open-source Polymer Self-Consistent Field (PSCF) software package and of the underlying theory. Second, we discuss methods for generating robust initial guesses for the fields that are computed in SCFT. To demonstrate our approach, we present two case studies in which a typical desktop computer has been used to simulate the structure of (i) body-centered cubic, face-centered cubic, A15, and Frank–Kasper σ sphere-forming phases of a diblock copolymer melt and (ii) two core–shell morphologies of ABAC tetrablock terpolymers. A companion Web site provides all of the relevant software and detailed instructions for reproducing all results contained herein.

Multibody Interactions, Phase Behavior, and Clustering in Nanoparticle-Polyelectrolyte Mixtures.

We present the results of a computational study of the interactions, phase-behavior and aggregation characteristics of charged nanoparticles (CNPs) suspended in solution of oppositely charged polyelectrolytes (PEs). We used an extension of the mean-field polymer self-consistent field theory (SCFT) model presented in our earlier work ( Macromolecules , 2014 , 47 , 6095 - 6112 ) to explicitly characterize the multibody interactions in such systems. For dilute-moderate particle volume fractions, the magnitudes of three and higher multibody interactions were seen to be weak relative to the contributions from pair interactions. On the basis of such results, we embeded the pair-interaction potentials within a thermodynamic perturbation theory approach to identify the phase behavior of such systems. The results of such a framework suggested that the gas and FCC crystal phases were thermodynamically stable, whereas the fluidlike phase was metastable in such systems. To complement the parameters studied using SCFT, we used a recently developed multibody simulation approach to study the aggregation and cluster morphologies in CNP-PE mixtures. For low particle charges, such systems mainly exhibited clusters arising from direct contact aggregation between CNPs. However, for higher particle and PE charges and low PE concentrations, large regions of PE-bridged clusters were seen to form. We present a morphological phase diagram summarizing such results.

Pair interactions in polyelectrolyte-nanoparticle systems: Influence of dielectric inhomogeneities and the partial dissociation of polymers and nanoparticles.

We study the effective pair interactions between two charged spherical particles in polyelectrolyte solutions using polymer self-consistent field theory. In a recent study [V. Pryamitsyn and V. Ganesan, Macromolecules 47, 6095 (2015)], we considered a model in which the particles possess fixed charge density, the polymers contain a prespecified amount of dissociated charges and, the dielectric constant of the solution was assumed to be homogeneous in space and independent of the polymer concentration. In this article, we present results extending our earlier model to study situations in which either or both the particle and the polymers possess partially dissociable groups. Additionally, we also consider the case when the dielectric constant of the solution depends on the local concentration of the polymers and when the particle's dielectric constant is lower than that of the solvent. For each case, we quantify the polymer-mediated interactions between the particles as a function of the polymer concentrations and the degree of dissociation of the polymer and particles. Consistent with the results of our previous study, we observe that the polymer-mediated interparticle interactions consist of a short-range attraction and a long-range repulsion. The partial dissociablity of the polymer and particles was seen to have a strong influence on the strength of the repulsive portion of the interactions. Rendering the dielectric permittivity to be inhomogeneous has an even stronger effect on the repulsive interactions and results in changes to the qualitative nature of interactions in some parametric ranges.

Interplay between Depletion and Electrostatic Interactions in Polyelectrolyte–Nanoparticle Systems

We use a numerical implementation of polymer self-consistent field theory to study the effective interactions between two spherical particles in polyelectrolyte solutions. We consider a model in which the particles possess fixed charge density and the polymers contain a prespecified amount of dissociated charges. We quantify the polymer-mediated interactions between the particles as a function of the particle charge, polymer concentrations and particle sizes. We study the interplay between depletion interactions, which arise as a consequence of polymer exclusion from the particle interiors, and the electrostatic forces which result from the presence of charges on the polymers and particles. Our results indicate that for weakly charged and uncharged particles, the polymer-mediated interactions predominantly consist of a short-range attraction and a long-range repulsion. When the particle charge is increased, the interactions become purely repulsive. A longer range, albeit weaker, bridging attraction was also evident for some parametric regimes. We demonstrate that the short-range attraction and the longer-range repulsion can be modeled as a sum of a depletion-like attraction and an electrostatic Debye–Huckel like repulsion. However, the amplitude and range underlying the depletion and electrostatic interactions are shown to possess a complex relationship to the parameters of our system. We present scaling arguments and analytical theory to rationalize some of the dependencies underlying the parameters governing the interaction potentials.

SCFT Study of Microphase Separation in Mixed Polymer Brushes Grafted on Cylindrical Surface

We report on microphase separation behaviors of mixed polymer brushes grafted onto an infinitely long cylindri- cal rod by performing polymer self-consistent field theory (SCFT) calculation with "masking" technique. The "masking" technique is especially suitable to deal with systems of confined polymers grafted onto curved surfaces. We have developed a method to solve the morphology of block copolymers confined into complicated topographic surfaces with SCFT. In this paper, this unique technique is extended to solve the SCFT for nanorod grafted by mixed polymer brushes. Furthermore, the use of simple Cartesian grids in a cubic computational cell with periodic boundary conditions makes it possible to solve dif- fusion equations in SCFT by utilizing an efficient and highly accurate pseudo-spectral method involving fast Fourier trans- form. Both parallel rippled phase and ring-shaped phase are predicted. We have investigated the influences of the cylinder radius, grafting density and interaction between the two incompatible grafting polymers on the stability of the two typical phases. Our results show that the system prefers the ring-shaped phase with the increase of the cylinder radius, grafting den- sity and interaction between the two grafting polymers. Phase diagrams involving these parameters are constructed, and we explain the reason of the transition between the parallel rippled phase and ring-shaped phase in terms of the degree of phase segregation. Again, the degree of phase segregation is higher with larger cylinder radius, grafting density and interaction between the two grafting polymers. By comparing the degree of phase segregation and free energy of the parallel rippled and ring-shaped phases at the same condition, we found that the ring-shaped phase favors the entropic part of the free energy while the parallel rippled phase significantly reduces the enthalpy. Therefore, when the degree of phase segregation is low, the free energy of the system is dominated by the enthalpy, leading to the parallel rippled phase; when the degree of phase segregation is high, the free energy of the system is dominated by the entropic part and the ring-shaped phase is stable. We also found that the domain numbers of parallel rippled phase and the period of alternating ring-shaped phase vary with the radius of cylinder. These predictions are expected to be helpful in rational design and fabrication of such novel polymer brushes.

Comparison of Pseudospectral Algorithms for Field-Theoretic Simulations of Polymers

We compare three pseudospectral algorithms for mean-field polymer self-consistent field theory (SCFT) simulations and beyond mean-field field-theoretic simulations (FTS) using the complex Langevin (CL) sampling technique. In agreement with a study by Stasiak and Matsen, we find that for SCFT the fourth-order algorithm developed by Ranjan, Qin, and Morse usually outperforms the other pseudospectral algorithms. In contrast, for CL simulations we find that the second-order algorithm adapted to SCFT by Rasmussen and co-workers often outperforms the fourth-order methods not only in computational speed but also, surprisingly, in accuracy at a fixed contour resolution. This suggests an intricate coupling between pseudospectral schemes for chain contour integration and pseudotime stepping methods in CL simulations. Finally, the nominal stability of the algorithms is examined for the case of homogeneous fields.