Field Theory Simulations Research Articles

The applications of the wormlike chain model on polymer physics

The continuum version of the wormlike chain model (WLC), which was initially developed by Saito, Takahashi and Yunoki in 1967, is particularly suitable for description of polymer conformational properties affected by the chain rigidity. The WLC model is capable of covering an extensive range of chain rigidity, from the flexible chains to the rigid chains, by tuning the persistence length directly. It is widely accepted as a coarse-grained model that can be used to capture the physical properties, such as conformation and structures, of a larger class of real polymers than the Gaussian chain (GSC). Recently, the WLC model attracts increasing interests because of its advantages in studying a variety of polymeric systems, including liquid crystalline polymers and conjugated polymers. This review article focuses on applications of the WLC model, incorporated in the framework of self-consistent field theory, which is an effective method in theoretical exploration of phase separation in polymer systems. The article also pays particular attention to the developments of the numerical schemes to solve the modified diffusion equation governing the probability distribution of polymers. In addition, we summarize recent applications of the self-consistent field theories based on WLC model in the following three areas: phase transitions in liquid-crystalline polymers; the influence of surface curvature on polymeric systems involving the chain orientation effects; self-assembly of wormlike block copolymers. These studies are beyond the scope of self-consistent field theories based on a GSC model, which have been utilized in a large number of theoretical studies in recent years. Finally, we propose the perspectives of theoretical developments in field-theory simulations based on the WLC model for future work. In the polymer literature, it is generally appreciated that chain-rigidity is an important factor that influences the properties of structural stabilities on the meso-scale. The theoretical studies indentify the key physical mechanisms that play crucial roles in many experimental systems with attractively promising applications in practice, for systems such as liquid crystalline polymers and organic solar cell based on the conjugated polymers.

Effect of 2-D Random Field Discretization on Failure Probability and Failure Mechanism in Probabilistic Slope Stability

This paper investigates, using the random field theory and Monte Carlo simulation, the effects of random field discretization on failure probability, pf, and failure mechanism of cohesive soil slope stability. The spatial sizes of the discretized elements in random field Δx, Δy in horizontal and vertical directions, respectively, are assigned a series of combinational values in order to model the discretization accuracy. The pf of deterministic critical slip surface (DCSS) and that of the slope system both are analyzed. The numerical simulation results have demonstrated that both the ratios of Δy/λy (λy = scale of fluctuation in vertical direction) and Δx/λx (λx = scale of fluctuation in horizontal direction) contribute in a similar manner to the accuracy of pf of DCSS. The effect of random field discretization on the pf can be negligible if both the ratios of Δx/λx and Δy/λy are no greater than 0.1. The normalized discrepancy tends to increase at a linear rate with Δy/λy when Δx/λx is larger than 0.1, and vice versa for pf of DCSS. The random field discretization tends to have more considerable influence on the pf of DCSS than on that of the slope system. The variation of pf versus λx and λy may exhibit opposite trends for the cases where the limit state functions of slope failure are defined on DCSS and on the slope system as well. Finally, the pf of slope system converges in a more rapid manner to that of DCSS than the failure mechanism does to DCSS as the spatial variability of soil property grows from significant to negligible.

Single Nanoparticle Localization in the Perforated Lamellar Phase of Self-Assembled Block Copolymer Driven by Entropy Minimization

Although precisely controlled microdomains of block copolymers (BCP) provide an excellent guiding matrix for multiple nanoparticles (NPs) to be controllably segregated into a desired polymer block, localization and positioning of individual NPs have not been demonstrated. Here, we report a unique one-to-one positioning phenomenon of guest Au NPs in the host BCP microdomains; each of polystyrene-functionalized Au NPs is embedded within the perforation domain of hexagonally perforated lamellar (HPL) morphology of poly(dimethylsiloxane-b-styrene) BCP. The local minimization of free energy achieved by the placement of Au NPs into the center of the perforation domain is theoretically supported by the self-consistent field theory (SCFT) simulation. We propose a novel design principle for more precisely controllable nanocomposites by developing a new route of NP arrangement within a polymer matrix.

Perpendicular Block Copolymer Microdomains in High Aspect Ratio Templates.

Perpendicular orientation of lamellar microdomains in a high interaction parameter block copolymer was obtained within high aspect ratio gratings functionalized with a preferential sidewall brush. The experiments used polystyrene-block-polydimethylsiloxane (PS-b-PDMS) with molecular weight 43 kg/mol within trenches made using interference lithography. The perpendicular alignment was obtained for both thermal and solvent annealing, using three different solvent vapors, for a range of film thicknesses and trench widths. A platinum (Pt) layer at the base of the trenches avoided the formation of a wetting layer, giving perpendicular orientation at the substrate surface. The results are interpreted using self-consistent field theory simulation and a Ginzburg-Landau analytic model to map the energies of lamellae of different orientations as a function of the grating aspect ratio and the surface energies of the sidewalls and top and bottom surfaces. The model results agree with the experiment and provide a set of guidelines for obtaining perpendicular microdomains within topographic features.

Scaling of polymer dynamics at an oil-water interface in regimes dominated by viscous drag and desorption-mediated flights.

Polymers are found near surfaces and interfaces in a wide range of chemical and biological systems, and the structure and dynamics of adsorbed polymer chains have been the subject of intense interest for decades. While polymer structure is often inferred from dynamic measurements in bulk solution, this approach has proven difficult to implement at interfaces, and the understanding of interfacial polymer conformation remains elusive. Here we used single-molecule tracking to study the interfacial diffusion of isolated poly(ethylene glycol) molecules at oil-water interfaces. Compared to diffusion in dilute aqueous solution, which exhibited the expected dependence of the diffusion coefficient (D) upon molecular weight (M) of D ∼ M(-1/2) for a Gaussian chain, the behavior at the interface was approximately D ∼ M(-2/3), suggesting a significantly more expanded polymer conformation, despite the fact that the oil was a poor solvent for the polymer. Interestingly, this scaling remained virtually unchanged over a wide range of oil viscosity, despite the fact that at low viscosities the magnitude of the diffusion coefficient was consistent with expectations based on viscous drag (i.e., Stokes-Einstein diffusion), and for high viscosity oil, the interfacial mobility was much faster than expected and consistent with the type of intermittent hopping transport observed at the solid-liquid interface. The dependence on molecular weight, in both regimes, was consistent with results from both self-consistent field theory and previous Monte Carlo simulations, suggesting that an adsorbed polymer chain adopted a partially swollen (loop-train-tail) interfacial conformation.

Supersymmetry in quantum optics and in spin-orbit coupled systems.

Light-matter interaction is naturally described by coupled bosonic and fermionic subsystems. This suggests that a certain Bose-Fermi duality is naturally present in the fundamental quantum mechanical description of photons interacting with atoms. We reveal submanifolds in parameter space of a basic light-matter interacting system where this duality is promoted to a supersymmetry (SUSY) which remains unbroken. We show that SUSY is robust with respect to decoherence and dissipation. In particular, the stationary density matrix at the supersymmetric lines in parameter space has a degenerate subspace. The dimension of this subspace is given by the Witten index and thus is topologically protected. As a consequence, the dissipative dynamics is constrained by a robust additional conserved quantity which translates information about an initial state into the stationary state. In addition, we demonstrate that the same SUSY structures are present in condensed matter systems with spin-orbit couplings of Rashba and Dresselhaus types, and therefore spin-orbit coupled systems at the SUSY lines should be robust with respect to various types of disorder. Our findings suggest that optical and condensed matter systems at the SUSY points can be used for quantum information technology and can open an avenue for quantum simulation of SUSY field theories.

The density of states approach for the simulation of finite density quantum field theories

Finite density quantum field theories have evaded first principle Monte-Carlo simulations due to the notorious sign-problem. The partition function of such theories appears as the Fourier transform of the generalised density-of-states, which is the probability distribution of the imaginary part of the action. With the advent of Wang-Landau type simulation techniques and recent advances [1], the density-of-states can be calculated over many hundreds of orders of magnitude. Current research addresses the question whether the achieved precision is high enough to reliably extract the finite density partition function, which is exponentially suppressed with the volume. In my talk, I review the state-of-play for the high precision calculations of the density-of-states as well as the recent progress for obtaining reliable results from highly oscillating integrals. I will review recent progress for the Z3 quantum field theory for which results can be obtained from the simulation of the dual theory, which appears to free of a sign problem.

Supersymmetric quantum mechanics on the lattice: II. Exact results

Simulations of supersymmetric field theories with spontaneously broken supersymmetry require in addition to the ultraviolet regularisation also an infrared one, due to the emergence of the massless Goldstino. The intricate interplay between ultraviolet and infrared effects towards the continuum and infinite volume limit demands careful investigations to avoid potential problems. In this paper – the second in a series of three – we present such an investigation for N=2 supersymmetric quantum mechanics formulated on the lattice in terms of bosonic and fermionic bonds. In one dimension, the bond formulation allows to solve the system exactly, even at finite lattice spacing, through the construction and analysis of transfer matrices. In the present paper we elaborate on this approach and discuss a range of exact results for observables such as the Witten index, the mass spectra and Ward identities.

Evidence for spin-triplet odd-parity superconductivity close to type-II van Hove singularities

Searching for unconventional Cooper pairing states has been at the heart of superconductivity research since the discovery of BCS superconductors. In particular, spin-triplet odd-parity pairing states were recently revisited due to the possibility of tuning towards topological superconductors. In this context, it is interesting to note a recent proposal that such a spin-triplet pairing instability occurs when the band filling is near van Hove singularities (vHS) associated with momenta away from time-reversal invariant momenta named type-II vHS. However, this result was obtained within a weak coupling renormalization group with Fermi surface patch approximation. To explore superconducting instabilities beyond this weak coupling Fermi surface patch approximation, we perform systematic study on Hubbard model in a two-dimensional square lattice using three different methods: random phase approximation, large-scale dynamical mean field theory simulations with continuous time quantum Monte Carlo (CTQMC) impurity solver, and large-scale dynamical cluster simulations with CTQMC cluster solver. We find, in a wide doping range centered around the type-II van Hove filling, a two-fold degenerate, spin-triplet, odd-parity $p$-wave pairing state emerges due to repulsive interaction, when the Fermi surface is not sufficiently nested. Possible relevance of our findings to the recently discovered superconductors LaO$_{1-x}$F$_{x}$BiS$_{2}$, Ir$_{1-x}$Pt$_{x}$Te$_{2}$ and proposed doped BC$_{3}$ are also discussed.

Phases and phase transitions of a perturbed Kekulé-Kitaev model

We study the quantum-spin-liquid phase in a variant of the Kitaev model where the bonds of the honeycomb lattice are distributed in a Kekul\'e pattern. The system supports gapped and gapless ${Z}_{2}$ quantum spin liquids with interesting differences from the original Kitaev model, the most notable being a gapped ${Z}_{2}$ spin liquid on a kagome lattice. Perturbing the exactly solvable model with antiferromagnetic Heisenberg perturbations, we find a magnetically ordered phase stabilized by a quantum ``order by disorder'' mechanism, as well as an exotic continuous quantum phase transition between the topological spin liquid and this magnetically ordered phase. Using a combination of field theory and Monte Carlo simulations, we find that the transition likely belongs to the $3D\text{\ensuremath{-}}XY\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ universality class.

Supersymmetric quantum mechanics on the lattice: I. Loop formulation

Simulations of supersymmetric field theories on the lattice with (spontaneously) broken supersymmetry suffer from a fermion sign problem related to the vanishing of the Witten index. We propose a novel approach which solves this problem in low dimensions by formulating the path integral on the lattice in terms of fermion loops. For N=2 supersymmetric quantum mechanics the loop formulation becomes particularly simple and in this paper – the first in a series of three – we discuss in detail the reformulation of this model in terms of fermionic and bosonic bonds for various lattice discretisations including one which is Q-exact.

Calibrating Analytical Models for Semilocal Strings

In this work we calibrate two different analytic models of semilocal strings by constraining the values of their free parameters. In order to do so, we use data obtained from the largest and most accurate field theory simulations of semilocal strings to date, and compare several key properties with the predictions of the models. As this is still work in progress, we present some preliminary results together with descriptions of the methodology we are using in the characterisation of semilocal string networks.

Exploring Microstructures and Interphase Properties of Surface- Grafted Diblock Copolymers in a Homopolymer Melt by Self-Consistent Field Theory Simulations

Polymeric self-consistent field theory is used to investigate microstructures and interphase properties of diblock copolymers grafted onto solid surfaces in a homopolymer melt. The calculations show that the grafted diblock copolymers can self-assemble into hemispherical microstructures at low grafting densities of the diblock copolymers. The morphology transforms into hemicylinder-like and sandwich-like lamellar microstructures with an increase in the chain-grafting density. The effective thickness of the grafted block layer and the interphase width between the homopolymer melt and the grafted copolymers strongly depend on the physicochemical parameters of the system, such as the composition of the grafted copolymer, the chemical incompatibility between the different components, the length ratio of grafted copolymer to homopolymer, and the grafting density of the diblock copolymers. In addition, the above computational results of microphase-separated structures and interphase properties are qualitatively compared with our previous experimental observations. The comparison indicates that our theoretical results not only reproduce the general feature of the experimental observations, but also elucidate the internal structural information and complement the findings in the region of high grafting densities of diblock copolymers.

Effects of thermal fluctuations on directed self-assembly in cylindrical confinement

We investigate the directed self-assembly (DSA) of cylinder-forming block copolymers inside cylindrical guiding templates. To complement and corroborate our experimental investigations, we use field-theoretic simulations to examine the fluctuation-induced variations in the size and position of the cylindrical microdomain that forms in the middle of the guiding hole. Our study goes beyond the usual mean-field approximation and self-consistent field theory simulations (SCFT) and incorporates the effects of thermal fluctuations in the description of the self-assembly process using complex Langevin (CL) dynamics. In addition to CL simulations, we present an efficient SCFT-based approach that can inform about the positional error of the formed cylinders. In this new scheme, an external chemical-potential field is applied to displace the inner cylinder away from its centered, lowest energy configuration. In both our experimental and modeling efforts, we focus on two wall-wetting conditions: (1) minor-block-attractive sidewalls and bottom substrates and neutral top surfaces and (2) neutral sidewalls, substrates, and top surfaces. For both cases, we explore the properties of the formed cylinders, including fluctuations in the center position and the size of the domain, for various prepattern conditions. Our results indicate robust critical dimensions (CDs) of the DSA cylinders relative to the prepattern CD, with a standard deviation <0.9 nm. Likewise, we find that the DSA cylinders are accurately registered in the center of the guiding hole, with deviations in the hole-in-hole distance on the order of ∼0.7 to 1.4 nm, translating to errors in the hole-to-hole distance of ∼1 to 2 nm.

Phase behavior of electrostatically complexed polyelectrolyte gels using an embedded fluctuation model.

Nanostructured, responsive hydrogels formed due to electrostatic interactions have promise for applications such as drug delivery and tissue mimics. These physically cross-linked hydrogels are composed of an aqueous solution of oppositely charged triblocks with charged end-blocks and neutral, hydrophilic mid-blocks. Due to their electrostatic interactions, the end-blocks microphase separate and form physical cross-links that are bridged by the mid-blocks. The structure of this system was determined using a new, efficient embedded fluctuation (EF) model in conjunction with self-consistent field theory. The calculations using the EF model were validated against unapproximated field-theoretic simulations with complex Langevin sampling and were found consistent with small angle X-ray scattering (SAXS) measurements on an experimental system. Using both the EF model and SAXS, phase diagrams were generated as a function of end-block fraction and polymer concentration. Several structures were observed including a body-centered cubic sphere phase, a hexagonally packed cylinder phase, and a lamellar phase. Finally, the EF model was used to explore how parameters that directly relate to polymer chemistry can be tuned to modify the resulting phase diagram, which is of practical interest for the development of new hydrogels.

Directed self-assembly of linear arrays of block copolymer cylinders

We investigate the directed self-assembly (DSA) of cylinder-forming block copolymers inside extended guiding templates of different sizes and shapes. We focus in particular on templates that promote the formation of linear arrays of two or three cylinders for contact hole multiplication. Using self-consistent field theory (SCFT) simulations, we uncover the existence windows for two- and three-cylinder arrays and assess their stability relative to other defective morphologies present in the templates. From SCFT simulations, templates in which three DSA cylinders are the lowest free energy configuration are more prone to defective morphologies in comparison to narrower templates with a two cylinder ground state. The size, placement and pitch of the defect-free structures are estimated for different template sizes and shapes. Beyond the thermodynamic description of the self-assembly process and the ensuing equilibrium properties, we also consider the equally important issue of the kinetics of melting of defects into perfect structures. Using the string method, we compute the most probable kinetic pathways that anneal elementary defects into a desired perfect array of cylinders. Similar to our findings on defectivity levels, defects in three-cylinder-friendly templates are associated with high kinetic barriers > 8 kT while their counterparts in narrower templates more readily dissipate into two-cylinder states after crossing barriers < 2 kT. © 2014 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys. 2015, 53, 317–326

Eliminating the Trade‐Off between the Throughput and Pattern Quality of Sub‐15 nm Directed Self‐Assembly via Warm Solvent Annealing

The directed self‐assembly (DSA) of block copolymers (BCPs) has been suggested as a promising nanofabrication solution. However, further improvements of both the pattern quality and manufacturability remain as critical challenges. Although the use of BCPs with a high Flory‐Huggins interaction parameter (χ) has been suggested as a potential solution, this practical self‐assembly route has yet to be developed due to their extremely slow self‐assembly kinetics. In this study, it is reported that warm solvent annealing (WSA) in a controlled environment can markedly improve both the self‐assembly kinetics and pattern quality. A means of avoiding the undesirable trade‐off between the quality and formation throughput of the self‐assembled patterns, which is a dilemma which arises when using the conventional solvent vapor treatment, is suggested. As a demonstration, the formation of well‐defined 13‐nm‐wide self‐assembled patterns (3σ line edge roughness of ≈2.50 nm) in treatment times of 0.5 min (for 360‐nm‐wide templates) is shown. Self‐consistent field theory (SCFT) simulation results are provided to elucidate the mechanism of the pattern quality improvement realized by WSA.

A multi-species exchange model for fully fluctuating polymer field theory simulations.

Field-theoretic models have been used extensively to study the phase behavior of inhomogeneous polymer melts and solutions, both in self-consistent mean-field calculations and in numerical simulations of the full theory capturing composition fluctuations. The models commonly used can be grouped into two categories, namely, species models and exchange models. Species models involve integrations of functionals that explicitly depend on fields originating both from species density operators and their conjugate chemical potential fields. In contrast, exchange models retain only linear combinations of the chemical potential fields. In the two-component case, development of exchange models has been instrumental in enabling stable complex Langevin (CL) simulations of the full complex-valued theory. No comparable stable CL approach has yet been established for field theories of the species type. Here, we introduce an extension of the exchange model to an arbitrary number of components, namely, the multi-species exchange (MSE) model, which greatly expands the classes of soft material systems that can be accessed by the complex Langevin simulation technique. We demonstrate the stability and accuracy of the MSE-CL sampling approach using numerical simulations of triblock and tetrablock terpolymer melts, and tetrablock quaterpolymer melts. This method should enable studies of a wide range of fluctuation phenomena in multiblock/multi-species polymer blends and composites.

Thin Film Morphologies of Bulk-Gyroid Polystyrene-block-polydimethylsiloxane under Solvent Vapor Annealing

Thin film morphologies of a 75.5 kg/mol polystyrene-block-polydimethylsiloxane (PS-b-PDMS) diblock copolymer subject to solvent vapor annealing are described. The PS-b-PDMS has a double-gyroid morphology in bulk, but as a thin film the morphology can form spheres, cylinders, perforated lamellae, or gyroids, depending on the film thickness, its commensurability with the microdomain period, and the ratio of toluene:heptane vapors used for the solvent annealing process. The morphologies are described by self-consistent field theory simulations. Thin film structures with excellent long-range order were produced, which are promising for nanopatterning applications.

Harnessing Anisotropic Nanoposts to Enhance Long-Range Orientation Order of Directed Self-Assembly Nanostructures via Large Cell Simulations.

Self-assembly behaviors of cylinder-forming diblock copolymers directed by an array of anisotropic nanoposts with elliptical shape are explored by large cell simulations of self-consistent field theory. The strategy using elliptical nanoposts allows us to generate long-range order cylinders with single orientation by suppressing other selections of cylinder alignment. The anisotropy of nanoposts plays a significant role in improving the tolerance of commensurability conditions between the dimensions of nanopost lattices and the period of cylinders. Moreover, the local defect structures could be regulated through varying the spacing and orientation of elliptical nanoposts. This work may provide useful guidelines for designing the topographical templates and lay the groundwork for fabricating well-ordered nanostructures of block copolymer lithography.