Addressing the challenges of modeling the scattering from bottlebrush polymers in solution

Small‐angle scattering measurements of complex macromolecules in solution are used to establish relationships between chemical structure and conformational properties. Interpretation of the scattering data requires an inverse approach where a model is chosen and the simulated scattering intensity from that model is iterated to match the experimental scattering intensity. This raises challenges in the case where the model is an imperfect approximation of the underlying structure, or where there are significant correlations between model parameters. We examine three bottlebrush polymers (consisting of polynorbornene backbone and polystyrene side chains) in a good solvent using a model commonly applied to this class of polymers: the flexible cylinder model. Applying a series of constrained Monte‐Carlo Markov Chain analyses demonstrates the severity of the correlations between key parameters and the presence of multiple close minima in the goodness of fit space. We demonstrate that a shape‐agnostic model can fit the scattering with significantly reduced parameter correlations and less potential for complex, multimodal parameter spaces. We provide recommendations to improve the analysis of complex macromolecules in solution, highlighting the value of Bayesian methods. This approach provides richer information for understanding parameter sensitivity compared to methods which produce a single, best fit

Synthesis and Self-Assembly of Bottlebrush Block Polymers: Molecular Architecture and Materials Design

Bottlebrush polymers represent a unique molecular architecture and a modular platform for materials design. However, the properties and self-assembly of bottlebrush polymers remain relatively unexplored, in large part due to the synthetic challenges imposed by the sterically demanding architecture. This thesis describes our work to close this gap, connecting (1) the synthesis of polymers with precisely tailored molecular architectures, (2) the study of fundamental structure-property relationships, and (3) the design of functional materials. Chapter 1 introduces key concepts related to polymer architecture and block polymer phase behavior. Recent developments in the synthesis and self-assembly of bottlebrush block polymers are highlighted in order to frame the work presented in Chapters 2–6. Chapter 2 introduces a versatile strategy to design polymer architectures with arbitrary side chain chemistry and connectivity. Simultaneous control over the molecular weight, grafting density, and graft distribution can be achieved via living ring-opening metathesis polymerization (ROMP). Copolymerizing a macromonomer and a small-molecule co-monomer provides access to well-defined polymers spanning the linear, comb, and bottlebrush regimes. This design strategy creates new opportunities for molecular and materials design. Chapter 3 explores the physical consequences of varying the grafting density and graft distribution in two contexts: block polymer self-assembly and linear rheological properties. The molecular architecture strongly influences packing demands and therefore the conformations of the backbone and side chains. Collectively, these studies represent progress toward a universal model connecting the chemistry and conformations of graft polymers. Chapter 4 discusses the phase behavior of ABA' and ABC bottlebrush triblock terpolymers. Low-&#967; interactions between the end blocks promote organization into a unique mixed-domain lamellar morphology, LAMP. X-ray scattering experiments reveal an unusual trend: the domain spacing strongly decreases with increasing total molecular weight. Insights into this behavior provide new opportunities for block polymer design with potential consequences spanning all self-assembling soft materials. Chapter 5 describes other physical consequences of low-&#967; block polymer design. The ternary phase diagrams for ABC, ACB, and BAC bottlebrush triblock terpolymers reveal the influences of low-&#967; A/C interactions, frustration, and the molecular architecture. Potential non-equilibrium effects and crystallization in these bottlebrush polymers will also be discussed. Chapter 6 describes applications of bottlebrush polymers as functional materials. Self-assembly enables mesoscale structural control over many materials properties, such as reflectivity, conductivity, and modulus. The synthetic methods (Chapter 2) and physical insights (Chapters 3−5) provided in previous chapters illustrate opportunities for materials design. We will discuss AB brush diblock polymers that self-assemble to photonic crystals and ABA brush triblock copolymers in solid polymer electrolytes.</p

Addressing the challenges of modeling the scattering from bottlebrush polymers in solution

Small‐angle scattering measurements of complex macromolecules in solution are used to establish relationships between chemical structure and conformational properties. Interpretation of the scattering data requires an inverse approach where a model is chosen and the simulated scattering intensity from that model is iterated to match the experimental scattering intensity. This raises challenges in the case where the model is an imperfect approximation of the underlying structure, or where there are significant correlations between model parameters. We examine three bottlebrush polymers (consisting of polynorbornene backbone and polystyrene side chains) in a good solvent using a model commonly applied to this class of polymers: the flexible cylinder model. Applying a series of constrained Monte‐Carlo Markov Chain analyses demonstrates the severity of the correlations between key parameters and the presence of multiple close minima in the goodness of fit space. We demonstrate that a shape‐agnostic model can fit the scattering with significantly reduced parameter correlations and less potential for complex, multimodal parameter spaces. We provide recommendations to improve the analysis of complex macromolecules in solution, highlighting the value of Bayesian methods. This approach provides richer information for understanding parameter sensitivity compared to methods which produce a single, best fit

Population genetics of the highly polymorphic RPP8 gene family

Plant nucleotide-binding domain and leucine-rich repeat containing (NLR) genes provide some of the most extreme examples of polymorphism in eukaryotic genomes, rivalling even the vertebrate major histocompatibility complex. Surprisingly, this is also true in Arabidopsis thaliana, a predominantly selfing species with low heterozygosity. Here, we investigate how gene duplication and intergenic exchange contribute to this extraordinary variation. RPP8 is a three-locus system that is configured chromosomally as either a direct-repeat tandem duplication or as a single copy locus, plus a locus 2 Mb distant. We sequenced 48 RPP8 alleles from 37 accessions of A. thaliana and 12 RPP8 alleles from Arabidopsis lyrata to investigate the patterns of interlocus shared variation. The tandem duplicates display fixed differences and share less variation with each other than either shares with the distant paralog. A high level of shared polymorphism among alleles at one of the tandem duplicates, the single-copy locus and the distal locus, must involve both classical crossing over and intergenic gene conversion. Despite these polymorphism-enhancing mechanisms, the observed nucleotide diversity could not be replicated under neutral forward-in-time simulations. Only by adding balancing selection to the simulations do they approach the level of polymorphism observed at RPP8. In this NLR gene triad, genetic architecture, gene function and selection all combine to generate diversity

Concentration Dependence of the Size and Symmetry of a Bottlebrush Polymer in a Good Solvent

Bottlebrush polymers consist of a linear backbone with densely grafted side chains which impact the rigidity of the molecule. The persistence length of the bottlebrush backbone in solution is influenced by both the intrinsic structure of the polymer and the local environment, such as the solvent quality and concentration. Increasing the concentration reduces the overall size of the molecule because of the reduction in backbone stiffness. In this study, we map out the size of a bottlebrush polymer as a function of concentration for a single backbone length. Small-angle neutron scattering measurements are conducted on a polynorbornene-based bottlebrush with polystyrene side chains in a good solvent. The data are fit using a model which provides both the long and short axis radius of gyration (R_(g,2) and R_(g,1), respectively), providing a measure for how the conformation changes as a function of concentration. At low concentrations, a highly anisotropic structure is observed (R_(g,2)/R_(g,1) ≈ 4), becoming more isotropic at higher concentrations (R_(g,2)/R_(g,1) ≈ 1.5). The concentration scaling for both R_(g,2) and the overall Rg is evaluated and compared with predictions in the literature. Coarse-grained molecular dynamics simulations were also conducted to probe the impact of concentration on bottlebrush conformation, showing qualitative agreement with the experimental results

Concentration Dependence of the Size and Symmetry of a Bottlebrush Polymer in a Good Solvent

Bottlebrush polymers consist of a linear backbone with densely grafted side chains which impact the rigidity of the molecule. The persistence length of the bottlebrush backbone in solution is influenced by both the intrinsic structure of the polymer and the local environment, such as the solvent quality and concentration. Increasing the concentration reduces the overall size of the molecule because of the reduction in backbone stiffness. In this study, we map out the size of a bottlebrush polymer as a function of concentration for a single backbone length. Small-angle neutron scattering measurements are conducted on a polynorbornene-based bottlebrush with polystyrene side chains in a good solvent. The data are fit using a model which provides both the long and short axis radius of gyration (R_(g,2) and R_(g,1), respectively), providing a measure for how the conformation changes as a function of concentration. At low concentrations, a highly anisotropic structure is observed (R_(g,2)/R_(g,1) ≈ 4), becoming more isotropic at higher concentrations (R_(g,2)/R_(g,1) ≈ 1.5). The concentration scaling for both R_(g,2) and the overall Rg is evaluated and compared with predictions in the literature. Coarse-grained molecular dynamics simulations were also conducted to probe the impact of concentration on bottlebrush conformation, showing qualitative agreement with the experimental results

Magnetic Photon Splitting: Computations of Proper-time Rates and Spectra

The splitting of photons in the presence of an intense magnetic field has recently found astrophysical applications in polar cap models of gamma-ray pulsars and in magnetar scenarios for soft gamma repeaters. Numerical computation of the polarization-dependent rates of this third order QED process for arbitrary field strengths and energies below pair creation threshold is difficult: thus early analyses focused on analytic developments and simpler asymptotic forms. The recent astrophysical interest spurred the use of the S-matrix approach by Mentzel, Berg and Wunner to determine splitting rates. In this paper, we present numerical computations of a full proper-time expression for the rate of splitting that was obtained by Stoneham, and is exact up to the pair creation threshold. While the numerical results derived here are in accord with the earlier asymptotic forms due to Adler, our computed rates still differ by as much as factors of 3 from the S-matrix re-evaluation of Wilke and Wunner, reflecting the extreme difficulty of generating accurate S-matrix numerics for fields below about \teq{4.4\times 10^{13}}Gauss. We find that our proper-time rates appear very accurate, and exceed Adler's asymptotic specializations significantly only for photon energies just below pair threshold and for supercritical fields, but always by less than a factor of around 2.6. We also provide a useful analytic series expansion for the scattering amplitude valid at low energies.Comment: 13 pages, AASTeX format, including 3 eps figures, ApJ in pres

ABA Triblock Brush Polymers: Synthesis, Self-Assembly, Conductivity, and Rheological Properties

The synthesis, self-assembly, conductivity, and rheological properties of ABA triblock brush polymers (BBCPs) with grafted polystyrene (A block, N_(PS) = 21) and poly(ethylene oxide) (B block, N_(PEO) = 45) side chains are reported. Two backbone molecular weights (N_A:N_B:N_A = 11:78:11 and 15:119:15) were investigated with lithium bis(trifluoromethane)sulfonimide (LiTFSI) doping ratios 2 ≤ [EO]:[Li+] ≤ 20. Blends with 2 ≤ [EO]:[Li+] ≤ 10 suppress PEO crystallization and self-assemble into hexagonally packed cylinders of the minority gPS component. Conductivity is on the order of 10^(–3) S/cm at 105 °C with a corresponding elastic modulus ca. 10^4 Pa. The optimum conductivity occurs at a blend ratio near 10:1 [EO]:[Li+], similar to that reported for linear block copolymer analogues

Confinement and Processing Can Alter the Morphology and Periodicity of Bottlebrush Block Copolymers in Thin Films

Bottlebrush block copolymers (BBCPs) are intriguing architectural variations on linear BCPs with highly tunable structure. Confinement can have a significant impact on polymer assembly, giving rise to changes in morphology, assembly kinetics, and properties like the glass transition. Given that confinement leads to significant changes in the persistence length of bottlebrush homopolymers, it is reasonable to expect that BBCPs will see significant changes in their structure and periodicity relative to the bulk morphology. Understanding how confinement influences assembly will be important for designing BBCPs for thin film applications including membranes, integrated photonic structures, and potentially BCP lithography. In order to study the effects of confinement on BBCP conformation and morphology, a blade coating was used to prepare films with continuous variation in film thickness. Unlike thin films of linear BCPs, islands/holes were not observed, and instead mixtures of parallel and perpendicular morphologies emerge after annealing. The lamellar periodicity (L₀) of the morphologies is found to be thickness dependent, increasing L₀ with decreasing film thickness for blade coated films. Films coated out of tetrahydrofuran (THF) resulted in a single well-defined lamellar periodicity, verified through atomic force microscopy (AFM) and grazing incidence small-angle X-ray scattering (GISAXS), which increases dramatically from the bulk value (30.6 nm) and continues to increase as the film thickness decreases. The largest observed L₀ was 65.5 nm, and this closely approaches the estimated upper limit of 67 nm corresponding to a fully extended backbone in a bilayer arrangement. Films coated out of propylene glycol methyl ether acetate (PGMEA) resulted in a mixture of perpendicular lamellae and a smaller, likely cylindrical morphology. The lamellar portion of the film shows the same thickness dependence as the lamellae observed in the THF coated films. The scaling of the lamellar L₀ with respect to film thickness follows predictions for confined semiflexible polymers with weak excluded volume interactions and can be related to models for confinement of DNA. Spin coated films shows the same reduction in periodicity, although at very different film thicknesses. This result suggests that the material has shallow free-energy barriers to transitioning between different L₀ and morphologies, a property that could be taken advantage of for patterning diverse structures with a single material