Effect of Materials Parameters on the Shape of Face-On Lamellae in Semi-Conducting Polymers: Insights From Qualitative Theory.

Polymer semiconductors frequently form crystals or mesophases with lamellae, that comprise alternating layers of stacked backbones and side chains. Controlling lamellar orientation in films is essential for obtaining efficient charge carrier transport. Herein, lamellar orientation is investigated in an application-relevant setup: lamellae assembled on a substrate that strongly favors face-on orientation, but exposed to a film surface that promotes orientation along an "easy" direction, other than face on. It is assumed that the face-on order propagates from the substrate, but the lamellae bend to reduce their surface energy. A qualitative free-energy model is developed. The deformation is investigated as a function of film thickness, effective Young modulus, anchoring coefficient, and easy direction at the free surface. The calculations highlight the importance of elastic constants - lamellae can substantially deform already when Young moduli are only an order of magnitude smaller than the values that are reported for crystals. Softer Young moduli are expected when lamellar assembly occurs in a non-solidified mesophase that can be an equilibrium or (more speculatively) a transient state prior to crystallization. The alternative scenario of a two-layered film is also evaluated, where edge-on and face-on grains form, respectively, at the free surface and substrate.

Hierarchical modelling of polystyrene melts: from soft blobs to atomistic resolution.

We demonstrate that hierarchical backmapping strategies incorporating generic blob-based models can equilibrate melts of high-molecular-weight polymers, described with chemically specific, atomistic models. The central idea is first to represent polymers by chains of large soft blobs (spheres) and efficiently equilibrate the melt on large scales. Then, the degrees of freedom of more detailed models are reinserted step by step. The procedure terminates when the atomistic description is reached. Reinsertions are feasible computationally because the fine-grained melt must be re-equilibrated only locally. We consider polystyrene (PS) which is sufficiently complex to serve method development because of stereo-chemistry and bulky side groups. Our backmapping strategy bridges mesoscopic and atomistic scales by incorporating a blob-based, a moderately coarse-grained (CG), and a united-atom model of PS. We demonstrate that the generic blob-based model can be parameterised to reproduce the mesoscale properties of a specific polymer - here PS. The moderately CG model captures stereo-chemistry. To perform backmapping we improve and adjust several fine-graining techniques. We prove equilibration of backmapped PS melts by comparing their structural and conformational properties with reference data from smaller systems, equilibrated with less efficient methods.

Splay-density coupling in semiflexible main-chain nematic polymers with hairpins.

A main-chain nematic polymer melt/solution exhibits macroscopic orientational order of main polymer chains, i.e., a preferred (nematic) direction. It has long been known that in such polymeric liquid crystals spatial density/concentration variations and distortions of the nematic direction are coupled, obeying a vectorial continuity constraint whose rigidity increases with chain length. Its vectorial nature precludes the application to flexible chains, where backfolds (hairpins) are present and apolar nematic symmetry is manifest, which has been its puzzling feature from the beginning. We now establish a description of the splay-density coupling in the case of arbitrary backfolding, devising a continuity constraint for the "recovered" polar order of the chain tangents and introducing hairpins as its new type of sources. Performing detailed Monte Carlo simulations of nematic monodomain melts of "soft" worm-like chains with variable length and flexibility, we show via their structure factors that the weakening of the coupling due to the backfolding can be consistently quantified on the macroscopic level.

Studying polymer solutions with particle-based models linked to classical density functionals: co-non-solvency.

We demonstrate the potential of hybrid particle-based models, where interactions are introduced through functionals of local order parameters, in describing multicomponent polymer solutions. The link to a free-energy-like functional is advantageous for controlling the thermodynamics of the model. We focus on co-non-solvency - the collapse of polymer chains in dilute mixtures with two miscible good solvents, having different affinities towards the polymer. We employ a simple model where polymers and solvents are represented, respectively, by worm-like chains and single particles. Non-bonded interactions are captured by a polynomial which is third order in local densities and can, therefore, describe liquid-vapour coexistence. The parameterisation of the functional benefits from an elementary mean-field approximation to the statistical mechanics of the model. The model provides a framework for Monte Carlo simulations using a particle-to-mesh algorithm. Studies with conventional generic bead-spring and all-atom models have demonstrated that co-non-solvency is caused by preferential binding of the better solvent (termed cosolvent) with polymer. Hence, segmental loops bridged by cosolvent molecules are formed, initiating polymer collapse. The mesoscopic hybrid model differs conceptually from the conventional microscopic descriptions. Yet, it reproduces the same co-non-solvency mechanism supporting its universality. Films of adsorbed ternary solutions, showing co-non-solvency in the dilute regime, are considered at high concentrations. In this case, chains do not collapse. The properties of loops and tails of the adsorbed polymer agree with early theoretical predictions obtained for concentrated binary solutions.

Maier-Saupe model of polymer nematics: Comparing free energies calculated with Self Consistent Field theory and Monte Carlo simulations.

Self Consistent Field (SCF) theory serves as an efficient tool for studying mesoscale structure and thermodynamics of polymeric liquid crystals (LC). We investigate how some of the intrinsic approximations of SCF affect the description of the thermodynamics of polymeric LC, using a coarse-grained model. Polymer nematics are represented as discrete worm-like chains (WLC) where non-bonded interactions are defined combining an isotropic repulsive and an anisotropic attractive Maier-Saupe (MS) potential. The range of the potentials, σ, controls the strength of correlations due to non-bonded interactions. Increasing σ (which can be seen as an increase of coarse-graining) while preserving the integrated strength of the potentials reduces correlations. The model is studied with particle-based Monte Carlo (MC) simulations and SCF theory which uses partial enumeration to describe discrete WLC. In MC simulations the Helmholtz free energy is calculated as a function of strength of MS interactions to obtain reference thermodynamic data. To calculate the free energy of the nematic branch with respect to the disordered melt, we employ a special thermodynamic integration (TI) scheme invoking an external field to bypass the first-order isotropic-nematic transition. Methodological aspects which have not been discussed in earlier implementations of the TI to LC are considered. Special attention is given to the rotational Goldstone mode. The free-energy landscape in MC and SCF is directly compared. For moderate σ the differences highlight the importance of local non-bonded orientation correlations between segments, which SCF neglects. Simple renormalization of parameters in SCF cannot compensate the missing correlations. Increasing σ reduces correlations and SCF reproduces well the free energy in MC simulations.

The Cassie-Wenzel transition of fluids on nanostructured substrates: Macroscopic force balance versus microscopic density-functional theory.

Classical density functional theory is applied to investigate the validity of a phenomenological force-balance description of the stability of the Cassie state of liquids on substrates with nanoscale corrugation. A bulk free-energy functional of third order in local density is combined with a square-gradient term, describing the liquid-vapor interface. The bulk free energy is parameterized to reproduce the liquid density and the compressibility of water. The square-gradient term is adjusted to model the width of the water-vapor interface. The substrate is modeled by an external potential, based upon the Lennard-Jones interactions. The three-dimensional calculation focuses on substrates patterned with nanostripes and square-shaped nanopillars. Using both the force-balance relation and density-functional theory, we locate the Cassie-to-Wenzel transition as a function of the corrugation parameters. We demonstrate that the force-balance relation gives a qualitatively reasonable description of the transition even on the nanoscale. The force balance utilizes an effective contact angle between the fluid and the vertical wall of the corrugation to parameterize the impalement pressure. This effective angle is found to have values smaller than the Young contact angle. This observation corresponds to an impalement pressure that is smaller than the value predicted by macroscopic theory. Therefore, this effective angle embodies effects specific to nanoscopically corrugated surfaces, including the finite range of the liquid-solid potential (which has both repulsive and attractive parts), line tension, and the finite interface thickness. Consistently with this picture, both patterns (stripes and pillars) yield the same effective contact angles for large periods of corrugation.

Fluctuation spectra in polymer nematics and Frank elastic constants: a coarse-grained modelling study.

Monte Carlo simulations of uniaxial nematic polymer melts are performed, based on a discrete worm-like chain model combined with soft, anisotropic non-bonded potentials. Different chain lengths are considered, the contour length of the longest being an order of magnitude larger than the persistence length. From equilibrated melt configurations, density and director fluctuation spectra are calculated and compared with analytical predictions available in literature. The latter typically correspond to hydrodynamic treatments of correlations and assume that there is no chain backfolding along the nematic director. Nevertheless, it is demonstrated that the analytical theories capture several features of the spectra obtained in the current simulations, where moderate backfolding of polymer chains is observed. Based on the available analytical expressions for density and director fluctuation spectra, material properties, such as Frank elastic constants, are extracted. Their dependence on polymerisation degree is studied and found to reproduce theoretically expected trends. For example, evidence is provided that the splay constant increases linearly with chain length, when effects of hairpins are negligible.

Communication: One size fits all: Equilibrating chemically different polymer liquids through universal long-wavelength description.

Mesoscale behavior of polymers is frequently described by universal laws. This physical property motivates us to propose a new modeling concept, grouping polymers into classes with a common long-wavelength representation. In the same class, samples of different materials can be generated from this representation, encoded in a single library system. We focus on homopolymer melts, grouped according to the invariant degree of polymerization. They are described with a bead-spring model, varying chain stiffness and density to mimic chemical diversity. In a renormalization group-like fashion, library samples provide a universal blob-based description, hierarchically backmapped to create configurations of other class-members. Thus, large systems with experimentally relevant invariant degree of polymerizations (so far accessible only on very coarse-grained level) can be microscopically described. Equilibration is verified comparing conformations and melt structure with smaller scale conventional simulations.

Can Polymer Helicity Affect Topological Chirality of Polymer Knots?

We investigate the effect of helicity in isolated polymers on the topological chirality of their knots with computer simulations. Polymers are described by generic worm-like chains (WLC), where helical conformations are promoted by chiral coupling between segments that are neighbors along the chain contour. The sign and magnitude of the coupling coefficient u determine the sense and strength of helicity. The corrugation of the helix is adjusted via the radius R of a spherical, hard excluded volume around each WLC segment. Open and compact helices are, respectively, obtained for R that is either zero or smaller than the length of the WLC bond, and R that is a few times larger than the bond length. We use a Monte Carlo algorithm to sample polymer conformations for different values of u, spanning the range from achiral polymers to chains with well-developed helices. Monitoring the average helix torsion and fluctuations of chiral order as a function of u, for two very different chain lengths, demonstrates that the coil-helix transition in this model is not a phase transition but a crossover. Statistical analysis of conformations forming the simplest chiral knots, 31, 51, and 52, demonstrates that topological mirror symmetry is broken─knots formed by helices with a given sense prefer one handedness over the other. For the 31 and 51 knots, positive helical sense favors positive handedness. Intriguingly, an opposite trend is observed for 52 knots, where positive helical sense promotes negative handedness. We argue that this special coupling between helicity and topological chirality stems from a generic mechanism: conformations where some of the knot crossings are found in "braids" formed by two tightly interwoven sections of the polymer.

Mesoscopic Modeling of a Highly-Ordered Sanidic Polymer Mesophase and Comparison With Experimental Data.

Board-shaped polymers form sanidic mesophases: assemblies of parallel lamellae of stacked polymer backbones separated by disordered side chains. Sanidics vary significantly with respect to polymer order inside their lamellae, making them "stepping stones" toward the crystalline state. Therefore, they are potentially interesting for studying crystallization and technological applications. Building on earlier mesoscopic models of the most disordered sanidics Σd, we focus on the other extreme, near-crystalline order, and develop a generic model that captures a highly ordered Σr mesophase. Polymers are described by generic hindered-rotation chains. Anisotropic nonbonded potentials, with strengths comparable to the thermal energy, mimic board-like monomer shapes. Lamellae equilibrated with Monte Carlo simulations, for a broad range of model parameters, have intralamellar order typical for Σr mesophases: periodically stacked polymers that are mutually registered along their backbones. Our mesophase shows registration on both monomer and chain levels. We calculate scattering patterns and compare with data published for highly ordered sanidic mesophases of two different polymers: polyesters and polypeptoids. Most of the generic structural features that were identified in these experiments are present in our model. However, our mesophase has correlations between chains located in different lamellae and is therefore closer to the crystalline state than the experimental samples.

Speeding up intrinsically slow collective processes in particle simulations by concurrent coupling to a continuum description.

The difficulty to study intrinsically slow collective processes by computer simulation of particle models stems from multiple disparate time scales (e.g., stiff bonded interactions versus soft nonbonded interactions). Continuum models, which describe the system by collective variables rather than the coordinates of the individual molecular constituents, often do not suffer from this time-scale problem because the stiff microscopic degrees of freedom have been integrated out. We propose to concurrently couple these two descriptions by a heterogeneous multiscale method. We illustrate the technique by studying the Lifshitz-Slyozov coarsening mechanism in a binary polymer blend using a soft coarse-grained particle model and a Landau-Ginzburg-de Gennes free energy functional, respectively. A speedup of up to two orders of magnitudes is achieved.

Polymer-solid contacts described by soft, coarse-grained models.

The ability of soft, coarse-grained models to describe the narrow interface of a nearly incompressible polymer melt in contact with a solid is explored by numerical self-consistent field calculations and Monte-Carlo simulations. We investigate the effect of the discreteness of the bead-spring architecture by quantitatively comparing the results of a bead-spring model with different number of beads, N, but identical end-to-end distance, R(e), and a continuous Gaussian-thread model. If the width, ξ, of the narrow polymer-solid contact is smaller or comparable to the length of a statistical segment, b=R(e)/√N-1, strong differences in the interface tension and the density profiles between the two models are observed, and strategies for compensating the discrete nature of the bead-spring model are investigated. Compensating the discretization of the chain contour in the bead-spring model by applying an external segment-solid potential, we simultaneously adjust the interface tension and the density profile to the predictions of the Gaussian-thread model. We suggest that the geometry of the polymer-solid contact and the interface tension are relevant characteristics that a coarse-grained model of polymer-solid contacts must reproduce in order to establish a quantitative relationship to an experimental system.

Directed assembly of supramolecular copolymers in thin films: thermodynamic and kinetic advantages.

Using computer simulation of a coarse-grained model for supramolecular polymers, we investigate the potential of quasiblock copolymers (QBCPs) assembled on chemically patterned substrates for creating device-oriented nanostructures. QBCPs are comprised of AB diblock copolymers and supramolecular B segments that can reversibly bond to any available B terminus, on either the copolymers or the B oligomers, creating a polydisperse blend of B homopolymers, and AB and ABA copolymers. We demonstrate the defect-free replication of patterns with perpendicularly crossing, A-preferential lines, where the same QBCP can simultaneously replicate patterns differing by up to 50% in their length scales. We demonstrate how the pattern affects the distribution of molecular architectures and the key role of supramolecular associations for replicating patterns with different length scales.

Exploring Disordered Morphologies of Blends and Block Copolymers for Light-Emitting Diodes with Mesoscopic Simulations.

Recently, disordered blends of semiconducting and insulating polymers have been used to prepare light-emitting diodes with increased luminous efficiency. Because the thermodynamic stability of the disordered phase in blends is limited, equivalent diblock copolymers (BCPs) could be an alternative. However, the choice between disordered blends and BCPs requires understanding structural differences and their effect on charge carrier transport. Using a hybrid mesoscopic model, we simulate blends and equivalent BCPs of two representative semiconducting and insulating polymers: poly(p-phenylene vinylene) (PPV) and polyacrylate. The immiscibility is varied to mimic annealing at different temperatures. We find stable or metastable disordered morphologies until we reach the mean-field (MF) spinodal. Disordered morphologies are heterogeneous because of thermal fluctuations and local segregation. Near the MF spinodal, segregation is stronger in BCPs than in the blends, even though the immiscibility, normalized by the MF spinodal, is the same. We link the spatial distribution of PPV with electric conductance. We predict that the immiscibility (temperature at which the layer is annealed) affects electrical percolation much stronger in BCPs than in blends. Differences in the local structure and percolation between blends and BCPs are enhanced at a high insulator content.

Can Soft Models Describe Polymer Knots?

Similar to macroscopic ropes and cables, long polymers create knots. We address the fundamental question whether and under which conditions it is possible to describe these intriguing objects with crude models that capture only mesoscale polymer properties. We focus on melts of long polymers which we describe by a model typical for mesoscopic simulations. A worm-like chain model defines the polymer architecture. To describe nonbonded interactions, we deliberately choose a generic "soft" repulsive potential that leads to strongly overlapping monomers and coarse local liquid structure. The soft model is parametrized to accurately reproduce mesoscopic structure and conformations of reference polymer melts described by a microscopic model. The microscopically resolved samples retain all generic features affecting polymer topology and provide, therefore, reliable reference data on knots. We compare characteristic knotting properties in mesoscopic and microscopically resolved melts for different cases of chain stiffness. We conclude that mesoscopic models can reliably describe knots in those melts, where the length scale characterizing polymer stiffness is substantially larger than the size of monomer-monomer excluded volume. In this case, simplified local liquid structure influences knotting properties only marginally. In contrast, mesoscopic models perform poorly in melts with flexible chains. We qualitatively explain our findings through a free energy model of simple knots available in the literature.

Generic Model for Lamellar Self-Assembly in Conjugated Polymers: Linking Mesoscopic Morphology and Charge Transport in P3HT.

We develop a generic coarse-grained model of soluble conjugated polymers, capable of describing their self-assembly into a lamellar mesophase. Polymer chains are described by a hindered-rotation model, where interaction centers represent entire repeat units, including side chains. We introduce soft anisotropic nonbonded interactions to mimic the potential of mean force between atomistic repeat units. The functional form of this potential reflects the symmetry of the molecular order in a lamellar mesophase. The model can generate both nematic and lamellar (sanidic smectic) molecular arrangements. We parametrize this model for a soluble conjugated polymer poly(3-hexylthiophene) (P3HT) and demonstrate that the simulated lamellar mesophase matches morphologies of low molecular weight P3HT, experimentally observed at elevated temperatures. A qualitative charge-transport model allows us to link local chain conformations and mesoscale order to charge transport. In particular, it shows how coarsening of lamellar domains and chain extension increase the charge carrier mobility. By modeling large systems and long chains, we can capture transport between lamellar layers, which is due to rare, but thermodynamically allowed, backbone bridges between neighboring layers.

An efficient Monte Carlo algorithm for the fast equilibration and atomistic simulation of alkanethiol self-assembled monolayers on a Au(111) substrate.

A new Monte Carlo algorithm is presented for the simulation of atomistically detailed alkanethiol self-assembled monolayers (R-SH) on a Au(111) surface. Built on a set of simpler but also more complex (sometimes nonphysical) moves, the new algorithm is capable of efficiently driving all alkanethiol molecules to the Au(111) surface, thereby leading to full surface coverage, irrespective of the initial setup of the system. This circumvents a significant limitation of previous methods in which the simulations typically started from optimally packed structures on the substrate close to thermal equilibrium. Further, by considering an extended ensemble of configurations each one of which corresponds to a different value of the sulfur-sulfur repulsive core potential, sigmass, and by allowing for configurations to swap between systems characterized by different sigmass values, the new algorithm can adequately simulate model R-SH/Au(111) systems for values of sigmass ranging from 4.25 A corresponding to the Hautman-Klein molecular model (J. Chem. Phys. 1989, 91, 4994; 1990, 93, 7483) to 4.97 A corresponding to the Siepmann-McDonald model (Langmuir 1993, 9, 2351), and practically any chain length. Detailed results are presented quantifying the efficiency and robustness of the new method. Representative simulation data for the dependence of the structural and conformational properties of the formed monolayer on the details of the employed molecular model are reported and discussed; an investigation of the variation of molecular organization and ordering on the Au(111) substrate for three CH3-(CH2)n-SH/Au(111) systems with n=9, 15, and 21 is also included.

Single-chain dynamics in a homogeneous melt and a lamellar microphase: a comparison between Smart Monte Carlo dynamics, slithering-snake dynamics, and slip-link dynamics.

We investigate the ability of Monte-Carlo algorithms to describe the single-chain dynamics in a dense homogeneous melt and a lamellar phase of a symmetric diblock copolymer. A minimal, coarse-grained model is employed that describes connectivity of effective segments by harmonic springs and where segments interact via soft potentials, which do not enforce noncrossability of the chain molecules. Studying the mean-square displacements, the dynamic structure factor, and the stress relaxation, we show that local, unconstraint displacements of segments via a Smart Monte Carlo algorithm give rise to Rouse dynamics for all but the first Monte Carlo steps. Using the slithering-snake algorithm, we observe a dynamics that is compatible with the predictions of the tube model of entangled melts for long times, but the dynamics inside the tube cannot be resolved. Using a slip-link model, we can describe the effect of entanglements and follow the different regimes of the single-chain dynamics over seven decades in time. Applications of this simulation scheme to spatially inhomogeneous systems are illustrated by studying the lamellar phase of a symmetric diblock copolymer. For the local, unconstraint dynamics, the single-chain motions parallel and perpendicular to the interfaces decouples; the perpendicular dynamics is slowed down but the parallel dynamics is identical to that in a homogeneous melt. Both the slithering-snake dynamics and the slip-link dynamics give rise to a coupling of parallel and perpendicular directions and a significant slowing down of the dynamics in the lamellar phase.

Thermodynamics of a Compressible Maier-Saupe Model Based on the Self-Consistent Field Theory of Wormlike Polymer.

This paper presents a theoretical formalism for describing systems of semiflexible polymers, which can have density variations due to finite compressibility and exhibit an isotropic-nematic transition. The molecular architecture of the semiflexible polymers is described by a continuum wormlike-chain model. The non-bonded interactions are described through a functional of two collective variables, the local density and local segmental orientation tensor. In particular, the functional depends quadratically on local density-variations and includes a Maier⁻Saupe-type term to deal with the orientational ordering. The specified density-dependence stems from a free energy expansion, where the free energy of an isotropic and homogeneous homopolymer melt at some fixed density serves as a reference state. Using this framework, a self-consistent field theory is developed, which produces a Helmholtz free energy that can be used for the calculation of the thermodynamics of the system. The thermodynamic properties are analysed as functions of the compressibility of the model, for values of the compressibility realizable in mesoscopic simulations with soft interactions and in actual polymeric materials.

Morphology of multi-component polymer systems: single chain in mean field simulation studies.

Recent work exploring phase separation and self-assembly in multicomponent polymer fluids using a particle-based self-consistent field simulation method is reviewed. The computational method is placed in the context of classical molecular dynamics and Monte Carlo simulations as well as field-theoretic approaches. Its potential is illustrated by applications ranging from spinodal decomposition in symmetric polymer blends and the ordering of diblock copolymers in the bulk to more complex phenomena such as solvent evaporation from thin polymer films and the fabrication of three-dimensional bicontinuous diblock copolymer morphologies reconstruction on patterned substrates.