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The knot-linear polymeric ratios as a function of m polymer length L for chains having fixed chain rigidity (l p = 50.2 nm) and diameter (d = 2.8 nm). The ratios decrease with knot complexity and increase with the knotted ring length.
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The Coulomb energy E C is defined by the energy required to charge a conductive object and scales inversely to the self–capacity C, a basic measure of object size and shape. It is known that C is minimized for a sphere for all objects having the same volume, and that C increases as the symmetry of an object is reduced at fixed volume. Mathematicall...
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... A similar effect has been recently discussed in connection with the knotted state of ring polymers being related to the solvent quality conditions at which the chain ends are closed. 113 Natural rubber materials and their synthetic counterparts are nearly incompressible materials in both their dry and solution states, provided that these materials are well above their glass transition temperatures where the materials can be subjected to large deformation without fracture or failure by some other instability. The small compressibility of these materials has sometimes been considered, but in this work, we adopted the conventional model where compressibility is neglected. ...
We systematically examine the influence of varying temperature (T) over a large range in model poly(vinyl acetate) gels swollen in isopropyl alcohol. The theta temperature Θ, at which the second virial coefficient A2 vanishes, is found to be equal to within numerical uncertainty to the corresponding high molecular mass polymer solution value without cross-links, and we quantify the swelling and deswelling of our model gels relative to their size at T = Θ, as customary for individual flexible polymer chains in solutions. We also quantify the "solvent quality" dependence of the shear modulus G relative to G(T = Θ) and compare to the hydrogel swelling factor, α. We find that all our network swelling and deswelling data can be reduced to a scaling equation of the same general form as derived from renormalization group theory for flexible linear polymer chains in solutions so that it is not necessary to invoke either the Flory-Huggins mean field theory or the Flory-Rehner hypothesis that the elastic and mixing contributions to the free energy of network swelling are separable to describe our data. We also find that changes of G relative to G(T = Θ) are directly related to α. At the same time, we find that classical rubber elasticity theory describes many aspects of these semi-dilute solution cross-linked networks, regardless of the solvent quality, although the prefactor clearly reflects the existence of network defects whose concentration depends on the initial polymer concentration of the polymer solution from which the networks were synthesized.
... Polymer branching by itself can evidently lead to an approach to effective hyperuniformity. The specific form of topology and the form of loops and other topological constraints can be expected to modulate these topological correlations, as investigated in some detail in the case of knotting in the case of the average dimensions of isolated knotted ring polymers in solution [115,116] and the density of knotted ring polymers in the melt [117], and in the effect of branching density on the dimensions of isolated randomly branched or "network" polymers [118,119]. It has also been observed that when the topological complexity of branched polymer structures, defined appropriately for the particular type of branched polymer under discussion, e.g., average crossing number in knotted ring polymers, star arms in star polymers, etc., becomes large, then the branched polymers become more rigid, and this topological rigidification [115,116] can lead to a diminished packing efficiency as defined by the density. ...
... The specific form of topology and the form of loops and other topological constraints can be expected to modulate these topological correlations, as investigated in some detail in the case of knotting in the case of the average dimensions of isolated knotted ring polymers in solution [115,116] and the density of knotted ring polymers in the melt [117], and in the effect of branching density on the dimensions of isolated randomly branched or "network" polymers [118,119]. It has also been observed that when the topological complexity of branched polymer structures, defined appropriately for the particular type of branched polymer under discussion, e.g., average crossing number in knotted ring polymers, star arms in star polymers, etc., becomes large, then the branched polymers become more rigid, and this topological rigidification [115,116] can lead to a diminished packing efficiency as defined by the density. There are thus competing effects arising from the topological structures of the self-assembled molecules that can influence the ultimate thermodynamic and dynamic properties of the material in general. ...
We investigate a metallic glass-forming (GF) material (Al 90 Sm 10 ) exhibiting a fragile-strong (FS) glass-formation by molecular dynamics simulation to better understand this highly distinctive pattern of glass-formation in which many of the usual phenomenological relations describing relaxation times and diffusion of ordinary GF liquids no longer apply, and where instead genuine thermodynamic features are observed in response functions and little thermodynamic signature is exhibited at the glass transition temperature, T g . Given the many unexpected similarities between the thermodynamics and dynamics of this metallic GF material with water, we first focus on the anomalous static scattering in this liquid, following recent studies on water, silicon and other FS GF liquids. We quantify the “hyperuniformity index” H of our liquid, which provides a quantitative measure of molecular “jamming”. To gain insight into the T -dependence and magnitude of H , we also estimate another more familiar measure of particle localization, the Debye–Waller parameter 〈 u 2 〉 describing the mean-square particle displacement on a timescale on the order of the fast relaxation time, and we also calculate H and 〈 u 2 〉 for heated crystalline Cu. This comparative analysis between H and 〈 u 2 〉 for crystalline and metallic glass materials allows us to understand the critical value of H on the order of 10 –3 as being analogous to the Lindemann criterion for both the melting of crystals and the “softening” of glasses. We further interpret the emergence of FS GF and liquid–liquid phase separation in this class of liquids to arise from a cooperative self-assembly process in the GF liquid.
Graphical abstract
... ZENO 1-5 is a software tool for characterizing polymers and nanoparticles, used toward the design of dispersions formed from them. ZENO has been applied to systems formed from synthetic polymers of different molecular topologies (linear, star, knotted and unknotted ring polymers, randomly branched polymers), 6-10 proteins and other biological macromolecules, [11][12][13][14] nanoparticles such as graphene and carbon nanotubes, [15][16][17] DNA orgami constructs, 18 and nanoparticles grafted with layers of DNA. 19 ZENO has also been utilized in the characterization of cell shapes. ...
We describe an extension of the ZENO program for polymer and nanoparticle characterization that allows for the precise calculation of the virial coefficients, with uncertainty estimates, of polymeric structures described by arbitrary rigid configurations of hard spheres. The probabilistic method of virial computation used for this extension employs a previously developed Mayer-sampling Monte Carlo method with overlap sampling that allows for a reduction of bias in the Monte Carlo averaging. This capability is an extension of ZENO in the sense that the existing program is also based on probabilistic sampling methods, and involves the same input file formats describing polymer and nanoparticle structures. We illustrate the capabilities, demonstrate the accuracy, and quantify the efficiency of this extension of ZENO by computing second, third and fourth virial coefficients and associated difficulty metrics for model polymeric structures having different shapes. We obtain good agreement with literature estimates available for some of the model structures considered.
... This general trend has been established by theoretical and experimental studies on bulk polymers by Dudowicz et al., 62 Stukalin et al., 64 and Kunal et al. 65 Of course, polymer molecular mass and topology (ring, star, bottlebrush, and structures, etc.) are also important, as these molecular parameters can also alter molecular packing efficiency. 54,55,66,67 The extension of these relationships to thin polymer films and polymer nanocomposites where there are large gradients in mobility can exist is not obvious. ...
We utilize recently introduced chemically specific but coarse-grained models of poly(ethylene oxide) (PEO) and poly(methyl methacrylate) (PMMA) to explore the influence of monomer architecture on the dynamics of supported thin polymer films based on molecular dynamics simulations. In particular, we contrast differences in the molecular packing and mobility gradients in these materials near the substrate and "free" interface regions. As expected, relaxation is generally enhanced in the free surface region relative to the film interior (and bulk), and the degree of enhancement is similar for both PEO and PMMA. However, the dynamical changes near the substrate are more sensitive to monomer structure, and are enhanced with increasing polymer−substrate interaction strength, ε. PMMA is relatively stiff compared to PEO and has a side group of appreciable size, and we find that the dynamics of PMMA near the substrate are slowed significantly more in comparison to PEO for the same substrate. Substrate interactions lead to a notable difference of local fragility near the substrate that appears to arise from a higher cohesive interaction strength of the PMMA chains in this region. Our data also reveal the inadequacy of the these coarse-grained polymer models to reproduce the experimentally known differences in the fragility of these materials. However, this technical shortcoming is not expected to alter our qualitative conclusions regarding the comparative effect of substrate interactions on relatively flexible polymers such as PEO versus a relatively stiff polymer such as PMMA. ■ INTRODUCTION It is generally appreciated that the molecular mobility of thin polymer films near interfaces can be significantly modified from the bulk. 1−7 Both experimental and computational studies have sought to quantify the effect of the interfacial mobility gradient near the interfaces through estimates of the local glass transition temperature T g in the interfacial region, 8−10 the film (shear) viscosity η, 11,12 segmental relaxation time 6,13−24 of the entire film, and average molecular diffusion coefficient in thin films. 25−27 Theoretical studies 28−33 have also shed light on the molecular origin of these changes in dynamics near interfaces. There is a general consensus that linear polymers near their "free" interfaces exhibit an enhanced mobility in the this interfacial region, 12,16,34−39 while the molecular mobility near the substrate can be either enhanced or diminished, depending on the interaction between supporting substrate and polymer. 34,40−44 The highly mobile layer at the polymer−air interface can be "liquid-like" in terms of mobility even when the temperature T is below the T g of polymer film as a whole, 4,45 while the mobility near a highly attractive substrate is normally diminished near the supporting substrate. As a result, the overall properties of the supported polymer film reflect the competition between the dynamics of the free and supporting substrate. 40,46−48 On the other hand, since polymers can be built from a myriad of monomer types, sequences of different monomers, and topologies (e.g., linear, ring, star, or comb), we expect the resulting interfacial mobility to depend on the details of the specific polymer monomer structure and topology, as well as the structure, surface energy, and stiffness of the substrate. 49−53 Recent molecular dynamics studies have shown that the glass-formation can be significantly altered in ring 54 and star polymers, 55 and there are ongoing measurements by many groups showing that polymer topology (rings, 56,57 stars, 58,59 bottlebrush polymers 60) greatly affects the T g of thin films, even changing the sign of T g deviation relative to their bulk values with confinement. Motivated by these considerations and observations, we consider the role of monomer structure in altering the interfacial mobility gradients in glass-forming ultrathin polymer films. We examine the degree to which a monomer structure alters dynamics near interfaces (free interface or substrate). Understanding how monomer structure affects the structure and dynamics in thin films should enhance the tailoring of their use in particular applications. Furthermore, to efficiently design polymer materials for applications, a fundamental understanding of the structure−property relationships is required. Article pubs.acs.org/Macromolecules
... Ideas from topology have been applied to the physical properties of real-world knots as well as forensic knot analysis [42][43][44][45][46], but there is a difference between practical knots and theoretical knots analysed by mathematicians. Real knots have ends and can be tied and untied. ...
... Knottability is a practical measure that depends on rope or cord deformability. (In contrast, what is known as the ideal configuration of a theoretical knot presupposes that the idealised knot cannot tighten any further owing to knot energy [32,[42][43][44][45][46].) Knottability is a standardised test (European Standard EN 892:1996) required for high-stretch climbing ropes. ...
Homicides and suicides involving ligatures, and accidents resulting from failed safety systems, sometimes necessitate the careful analysis of knots. Aside from performing accurate knot identifications, establishing knot chirality, and determining whether those knots are commonplace or require specialised training, the experienced analyst evaluates various knot characteristics and is guided by underlying principles and evolving standards, which draw on multiple disciplines. These include the Gordian Maxim as well as the efficiency, ownership and parsimony principles. Subsequent to distinguishing between accidental and deliberate knots, an assessment of knot structure, function and effectiveness sometimes necessitates an understanding of Ashley’s principle and Asher’s law. Crossing number, writhe, strength, security and concatenation can be valuable analysis characteristics. Knot stability may be evaluated using established counting number algorithms. Methods of determining sinuosity and knottability are useful as well. Knots occur with structural variation and they can change accidentally or through deliberate action. Capsizement, flipping, flyping, migration, release, reptation, trambling, transference and enhancement are knot change processes. Along with tier behaviour and material characteristics, these details may require assessment during evidence analysis.
... Investigation of folding kinetics of proteins with a supercoiling motif or a link, sheds also some light on the influence of non-trivial topology on protein stability. First, computer simulation studies of knotted polymers (and also star polymers) have shown that fluctuations of variables such as hydrodynamic radius, radius of gyration and intrinsic viscosity, decrease with the topological complexity, and at the same time, the rigidity of such polymers increases [96,97]. In the case of knotted polymers, reduction in fluctuations of R g for increasing knot's complexity (and hence crossing number), is a consequence of a global increase in topological constraints. ...
Around 6% of protein structures deposited in the PDB are entangled, forming knots, slipknots, lassos, links, and θ-curves. In each of these cases, the protein backbone weaves through itself in a complex way, and at some point passes through a closed loop, formed by other regions of the protein structure. Such a passing can be interpreted as crossing a topological barrier. How proteins overcome such barriers, and therefore different degrees of frustration, challenged scientists and has shed new light on the field of protein folding. In this review, we summarize the current knowledge about the free energy landscape of proteins with non-trivial topology. We describe identified mechanisms which lead proteins to self-tying. We discuss the influence of excluded volume, such as crowding and chaperones, on tying, based on available data. We briefly discuss the diversity of topological complexity of proteins and their evolution. We also list available tools to investigate non-trivial topology. Finally, we formulate intriguing and challenging questions at the boundary of biophysics, bioinformatics, biology, and mathematics, which arise from the discovery of entangled proteins.
... The topological complexity measure mc can be expected to influence both the average molecular shape and rigidity, while leaving other molecular parameters unchanged, providing an opportunity for studying purely topological effects on glassformation. Knotted polymers are also interesting scientifically since rings synthesized in poor solvents can be expected to have knots 6 that influence the properties of bulk materials made from this class of polymers. ...
... Because of this compaction of rings in the melt, the average crossing number, ⟨mc⟩, does not provide as good of an indicator of knot complexity as this property does in good solvent polymer solutions. 6 Star polymers are another class of topologically constrained polymers that exhibit many parallel properties to ring polymers, both in solution and in the melt, where increasing the number of star arms f plays a role similar to mc in knotted ring polymers. 4,7,8 In particular, both types of polymers exhibit topological rigidification and become more particle-like with increasing topological complexity at a fixed chain length, as quantified by mc and f, factors that are known to increase Tg and influence the fragility of glass-formation. ...
... glass-formation in linear polymer melts at constant volume, variable pressure, variable cohesive interaction, polymer nanocomposites, thin films, nanofibers, etc. 9-14 This molecular model, described in detail in the supplementary material, is an extension of the wellestablished "bead-spring" polymer chain model, 15 and we introduce knot topological constraints, i.e., varying mc, based on the methods described in Ref. 6. Here, each knotted ring is represented by L = 32 connected beads having a knot complexity mc in the range 0 ≤ mc ≤ 7, and we show initial configurations of these rings in Fig. 1(a). ...
We perform molecular dynamics simulations on a coarse-grained polymer melt to study the dynamics of glass-formation in ring polymer melts of variable knot complexity. After generating melts of non-concatenated polymeric rings having a range of minimum crossing number values, mc, we compute the coherent intermediate scattering function, the segmental α-relaxation time, fragility, and the glass transition temperature as a function of mc. Variation of knot complexity is found to have a pronounced effect on the dynamics of polymer melts since both molecular rigidity and packing are altered, primary physical factors governing glass-formation in polymeric materials.
... 1,2 Here, we comparatively investigate knotted ring and star polymers having the same molecular mass and chemistry by using molecular dynamics (MD) simulations on a wellestablished coarse-grained polymer model 10 utilized before to study linear polymer chains both in solution and in the melt state. We focus our attention on computing basic solution properties such as hydrodynamic radius, R h , radius of gyration, R g , and intrinsic viscosity, η , of the knotted ring polymers having knot complexity m c in the range, 0 ≤ m c ≤ 9, where m c is the minimum number of crossing points on the polymer, 7 and star polymers having a number of arms f in the range, 2 ≤ f ≤ 20, attached to a star core particle. We note that m c does not fully a) jack.douglas@nist.gov ...
... As a final point, we mention that the approach of g-ratios to their long chain limit tends to be much slower when the polymers are semi-flexible. This phenomenon is illustrated in our previous study of the g-ratios of knotted semi-flexible polymers, such as double-stranded DNA, 7 and we further illustrate this effect for unknotted rings in the supplementary material. ...
... 16,[24][25][26][27] In this study, we consider 10 8 different chain configurations for each topologically constrained polymer structure. 7,8,24 We report the average property as well as its standard deviation as a measure of property fluctuation. The uncertainties in our calculations are obtained by computing the uncertainties related to the path-integral calculations, which in all cases presented in this manuscript are smaller than the point size shown in the figures. ...
We computationally investigate the good solvent solution properties of knotted ring and star polymers by combining molecular dynamics (MD) simulation and path-integral calculations. We consider knotted rings having a minimal crossing number mc in the range, 0 ≤ mc ≤ 9, and star polymers having a range of f star arms, 2 ≤ f ≤ 20, attached to a common core monomer particle. After generating configurational ensembles of these polymers by MD, we use the path-integration program ZENO to calculate basic configurational properties, i.e., radius of gyration, hydrodynamic radius, intrinsic viscosity, as well as fluctuations in these properties. Our simulations indicate that the configurational properties of knotted rings and star polymers in solution show a similar decrease with increasing mc and f. Moreover, fluctuations in these properties also decrease with increasing topological complexity. Our findings should be helpful in polymer characterization and more generally for understanding the role of polymer topology in polymer material properties.
... These are then expected to increase excluded volume repulsion between different subchains, compared to polymer solutions. [68,69] For ionic gels, this effect has been hypothesised by Winkler and co-workers [70] to be the main source of electrostatic excluded volume repulsion. Data by Horkay et al [66] on polyethylen glycol (PEG) networks show that the mixing free energy of polymer solutions and gels at the same volume fraction is similar, and the difference can be attributed to minor structural heterogeneities in the gel. ...
Thermoresponsive polymeric gels are a class of smart materials with the ability to absorb or release large amounts of solvent when changes in their environment occur. Gel swelling scaling theories (de Gennes c* theorem and Obukhov's model) predict that the swelling of a gel correlates with that of a linear chain with equal degree of polymerisation equal to that of a network strand. We examine data on linear and cross linked poly(N-isopropylacrylamide) (PNIPAM) in aqueous solutions of different molar masses at the theta temperature and under good solvent conditions. While the power law exponent for the swelling equilibrium as a function of degree of cross-linking is reasonably close to the scaling predictions, the swelling observed in cross linked networks is two to three orders of magnitude larger than that expected from single chain swelling, in strong contradiction with scaling models.
For micelles, “shape” is prominent in rheological computations of fluid flow, but this “shape” is often expressed too informally to be useful for rigorous analyses. We formalize topological “shape equivalence” of micelles, both globally and locally, to enable visualization of computational fluid dynamics. Although topological methods in visualization provide significant insights into fluid flows, this opportunity has been limited by the known difficulties in creating representative geometry. We present an agile geometric algorithm to represent the micellar shape for input into fluid flow visualizations. We show that worm-like and cylindrical micelles have formally equivalent shapes, but that visualization accentuates unexplored differences. This global–local paradigm is extensible beyond micelles.
