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.

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.

Dynamic coarse-graining of polymer systems using mobility functions.

We propose a dynamic coarse-graining (CG) scheme for mapping heterogeneous polymer fluids onto extremely CG models in a dynamically consistent manner. The idea is to use as target function for the mapping a wave-vector dependent mobility function derived from the single-chain dynamic structure factor, which is calculated in the microscopic reference system. In previous work, we have shown that dynamic density functional calculations based on this mobility function can accurately reproduce the order/disorder kinetics in polymer melts, thus it is a suitable starting point for dynamic mapping. To enable the mapping over a range of relevant wave vectors, we propose to modify the CG dynamics by introducing internal friction parameters that slow down the CG monomer dynamics on local scales, without affecting the static equilibrium structure of the system. We illustrate and discuss the method using the example of infinitely long linear Rouse polymers mapped onto ultrashort CG chains. We show that our method can be used to construct dynamically consistent CG models for homopolymers with CG chain lengthN= 4, whereas for copolymers, longer CG chain lengths are necessary.

Comparing equilibration schemes of high-molecular-weight polymer melts with topological indicators.

Recent theoretical studies have demonstrated that the behaviour of molecular knots is a sensitive indicator of polymer structure. Here, we use knots to verify the ability of two state-of-the-art algorithms-configuration assembly and hierarchical backmapping-to equilibrate high-molecular-weight (MW) polymer melts. Specifically, we consider melts with MWs equivalent to several tens of entanglement lengths and various chain flexibilities, generated with both strategies. We compare their unknotting probability, unknotting length, knot spectra, and knot length distributions. The excellent agreement between the two independent methods with respect to knotting properties provides an additional strong validation of their ability to equilibrate dense high-MW polymeric liquids. By demonstrating this consistency of knotting behaviour, our study opens the way for studying topological properties of polymer melts beyond time and length scales accessible to brute-force molecular dynamics simulations.

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.

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.

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.

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.

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.

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.

Equilibrating high-molecular-weight symmetric and miscible polymer blends with hierarchical back-mapping.

Understanding properties of polymer alloys with computer simulations frequently requires equilibration of samples comprised of microscopically described long molecules. We present the extension of an efficient hierarchical backmapping strategy, initially developed for homopolymer melts, to equilibrate high-molecular-weight binary blends. These mixtures present significant interest for practical applications and fundamental polymer physics. In our approach, the blend is coarse-grained into models representing polymers as chains of soft blobs. Each blob stands for a subchain with N b microscopic monomers. A hierarchy of blob-based models with different resolution is obtained by varying N b. First the model with the largest N b is used to obtain an equilibrated blend. This configuration is sequentially fine-grained, reinserting at each step the degrees of freedom of the next in the hierarchy blob-based model. Once the blob-based description is sufficiently detailed, the microscopic monomers are reinserted. The hard excluded volume is recovered through a push-off procedure and the sample is re-equilibrated with molecular dynamics (MD), requiring relaxation on the order of the entanglement time. For the initial method development we focus on miscible blends described on microscopic level through a generic bead-spring model, which reproduces hard excluded volume, strong covalent bonds, and realistic liquid density. The blended homopolymers are symmetric with respect to molecular architecture and liquid structure. To parameterize the blob-based models and validate equilibration of backmapped samples, we obtain reference data from independent hybrid simulations combining MD and identity exchange Monte Carlo moves, taking advantage of the symmetry of the blends. The potential of the backmapping strategy is demonstrated by equilibrating blend samples with different degree of miscibility, containing 500 chains with 1000 monomers each. Equilibration is verified by comparing chain conformations and liquid structure in backmapped blends with the reference data. Possible directions for further methodological developments are discussed.

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.

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.

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.

Effect of mesoscale ordering on the density of States of polymeric semiconductors.

A multiscale simulation scheme, which incorporates both long-range conformational disorder and local molecular ordering, is proposed for predicting large-scale morphologies and charge transport properties of polymeric semiconductors. Using poly(3-hexylthiophene) as an example, it is illustrated how the energy landscape and its spatial correlations evolve with increasing degree of structural order in mesophases with amorphous, uniaxial, and biaxial nematic ordering. It is shown that the formation of low-lying energy states in more ordered systems is mostly due to larger (on average) conjugation lengths and not due to electrostatic interactions. The proposed scheme is general and can be applied to a wide range of polymeric organic materials.

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.

Equilibration of High Molecular Weight Polymer Melts: A Hierarchical Strategy.

A strategy is developed for generating equilibrated high molecular weight polymer melts described with microscopic detail by sequentially backmapping coarse-grained (CG) configurations. The microscopic test model is generic but retains features like hard excluded volume interactions and realistic melt densities. The microscopic representation is mapped onto a model of soft spheres with fluctuating size, where each sphere represents a microscopic subchain with Nb monomers. By varying Nb, a hierarchy of CG representations at different resolutions is obtained. Within this hierarchy, CG configurations equilibrated with Monte Carlo at low resolution are sequentially fine-grained into CG melts described with higher resolution. A Molecular Dynamics scheme is employed to slowly introduce the microscopic details into the latter. All backmapping steps involve only local polymer relaxation; thus, the computational efficiency of the scheme is independent of molecular weight, being just proportional to system size. To demonstrate the robustness of the approach, microscopic configurations containing up to n = 1000 chains with polymerization degrees N = 2000 are generated and equilibration is confirmed by monitoring key structural and conformational properties. The extension to much longer chains or branched polymers is straightforward.

Simulations of nematic homopolymer melts using particle-based models with interactions expressed through collective variables.

We develop a hybrid Monte Carlo approach for modelling nematic liquid crystals of homopolymer melts. The polymer architecture is described with a discrete worm-like chain model. A quadratic density functional accounts for the limited compressibility of the liquid, while an additional quadratic functional of the local orientation tensor of the segments captures the nematic ordering. The approach can efficiently address large systems parametrized according to volumetric and conformational properties, representative of real polymeric materials. The results of the simulations regarding the influence of the molecular weight on the isotropic-nematic transition are compared to predictions from a Landau-de Gennes free energy expansion. The formation of the nematic phase is addressed within Rouse-like dynamics, realized using the current model.