Porod Wormlike Chain Research Articles

Autonomous topological time crystals and knotty molecular motors

We show that topology is a very effective tool, to construct classical Hamiltonian time crystals. For this we numerically analyze a general class of time crystalline Hamiltonians that are designed to model the dynamics of molecular closed strings. We demonstrate how the time crystalline qualities of a closed string are greatly enhanced when the string becomes knotted. The Hamiltonians that we investigate include a generalized Kratky–Porod wormlike chain model in combination with long range Coulomb and Lennard–Jones interactions. Such energy functions are commonplace in coarse grained molecular modeling. Thus we expect that physical realizations of Hamiltonian time crystals can be constructed in terms of knotted ring molecules.

Semiflexible Polymers Interacting With Planar Surfaces: Weak versus Strong Adsorption.

Semiflexible polymers bound to planar substrates by a short-range surface potential are studied by Molecular Dynamics simulations to clarify the extent to which these chain molecules can be considered as strictly two-dimensional. Applying a coarse-grained bead-spring model, the chain length N and stiffness as well as the strength of the adsorption potential are varied over a wide range. The excluded-volume (EV) interactions inherent in this model can also be “switched off” to provide a discretized version of the Kratky–Porod wormlike chain model. We study both local order parameters (fraction f of monomers within the range of the potential, bond-orientational order parameter ) and the mean square gyration radius parallel, , and perpendicular, , to the wall. While for strongly adsorbed chains EV has negligible effect on f and , we find that is strongly affected when the chain contour length exceeds the persistence length. Monomer coordinates in perpendicular (⊥) direction are correlated over the scale of the deflection length which is estimated. It is found that , and converge to their asymptotic values with corrections. For both weakly and strongly adsorbed chains, the distribution functions of “loops”, “trains”, and “tails” are analyzed. Some consequences pertaining to the analysis of experiments on adsorbed semiflexible polymers are pointed out.

The effect of side chain length on hydrodynamic and conformational characteristics of polyimide-graft-polymethylmethacrylate copolymers in thermodynamically good solutions

Samples of graft-copolymers with polyimide backbones and poly(methyl methacrylate) side chains synthesized by ATRP of methylmethacrylate on polyimide macroinitiator are investigated by 1H NMR, GPC, static and dynamic light scattering, sedimentation, and viscosimetry in chloroform and ethyl acetate solutions. Hydrodynamic and conformational behavior of graft-copolymers is well-described by the wormlike spherocylinder and Porod wormlike chain models. Samples of graft-copolymers have high equilibrium rigidity, the Kuhn segment length grows from 20 to 95 nm with the lengthening of side chains. The equilibrium rigidity and hydrodynamic diameter of the investigated graft-copolymers depend on the molar mass M of side chains as M 0.7.

Main-Chain Stiffness and Helical Conformation of a Poly(quinoxaline-2,3-diyl) in Solution

Light and small-angle X-ray scattering and viscosity measurements in tetrahydrofuran at 25 °C were made for nine helical poly[5,8-dimethyl-6,7-bis(propoxymethyl)quinoxaline-2,3-diyl] samples ranging in the weight-average molar mass Mw from 8 × 103 g mol–1 to 6 × 105 g mol–1 to determine the particle scattering function, the radius of gyration, and the intrinsic viscosity as a function of Mw. The dimensional and hydrodynamic properties were consistently explained in terms of the Kratky–Porod wormlike chain. The helix pitch per residue h (or the contour length per residue) and the chain stiffness parameter λ–1 (the Kuhn segment length or twice of the persistence length) were estimated to be h = 0.19 nm and λ–1 = 43 ± 3 nm. The former parameter corresponds to an internal rotation angle of about 120° which is substantially the same as the most stable helical structure estimated from the internal rotation potential. The latter one (λ–1) indicates that the poly(quinoxaline-2,3-diyl) has a rigid helical main cha...

Dilute solution properties of semiflexible star and ring polymers

Our recent theoretical and/or Monte Carlo (MC) studies of dilute solution properties of semiflexible stars and rings are briefly summarized. The theoretical results for the intrinsic viscosity [η] of the Kratky–Porod (KP) wormlike three- and four-arm stars are shown, and effects of chain stiffness on [η] of the stars are examined. A comparison of the results for [η] with those for the effective hydrodynamic radius and the second virial coefficient A2 in a good solvent was made for the semiflexible three-arm stars. It was found that [η] is the most suitable object of study to examine the effects of chain stiffness on average chain dimensions of the stars. As for the rings, the MC results for A2 of the ideal KP rings, which is related to the intermolecular topological interactions, are presented and then compared with the data in the literature for ring atactic polystyrene (a-PS) at Θ for large molecular weight M (1 × 104−6 × 105). Even for ring a-PS in such a range of M, the effects of chain stiffness were still remarkable. The effects of the intramolecular topological constraints on the mean-square radius of gyration and the scattering function of the KP rings are also discussed. Theoretical and/or Monte Carlo studies are made of dilute solution properties of semiflexible stars and rings based on the Kratky–Porod wormlike chain model. The behavior of the properties in the range of the crossover from the rigid limit to the random-coil one is clarified. It is shown that effects of chain stiffness affect largely the dilute solution behavior not only of linear polymers but also of stars and rings, and are still remarkable even for typical flexible polymers with large molecular weight (∼105).

Transport Coefficients of Poly(diisopropyl fumarate) in Dilute Solution

Top of pageAbstract The intrinsic viscosity [η] was determined for 13 samples of poly(diisopropyl fumarate), each with the fraction of racemo diads fr = 0.22, in the range of weight-average molecular weight Mw from 4.02 × 104 to 8.59 × 105 in tetrahydrofuran at 30.0 °C. The translational diffusion coefficient D was also determined from dynamic light scattering measurements for 12 of them in the range of Mw from 4.95 × 104 to 8.59 × 105 under the same solvent condition. The data so obtained were analyzed on the basis of the corresponding transport theories for the unperturbed Kratky–Porod (KP) wormlike cylinder model combined with the quasi-two-parameter theory for the intramolecular excluded–volume effect. It is shown that the experimental values are in quantitative agreement with the perturbed KP theory ones by the use of the model parameter values consistent with those determined previously from analyses of the mean-square radius of gyration (S2) and second virial coefficient A2. An examination was also made of behavior of the Flory–Fox factor Φ and reduced hydrodynamic radius ρ−l as functions of Mw. Keywords: Poly(diisopropyl fumarate), Intrinsic Viscosity, Translational Diffusion Coefficient, Kratky–Porod Wormlike Chain Model, Excluded-Volume Effect, Quasi-Two-Parameter Theory

Branched polymeric labels used as drag-tags in free-solution electrophoresis of ssDNA

In the framework of the classical blob theory of end-labeled free-solution electrophoresis of ssDNA, and based on recent experimental data with linear and branched polymeric labels (or drag-tags), the present study puts forward design principles for the optimal type of branching that would give, for a given total number of monomers, the highest effective frictional drag for ssDNA sequencing purposes. The hydrodynamic radii of the linear and branched labels are calculated using standard models like the freely jointed chain model and the Kratky-Porod worm-like chain model. Based on comparisons of the theory with the experimental data, we propose that the design of new branched labels should use either side chains whose length is comparable to the distance between the branching points or two long branches located near the ends of the molecule's backbone.

Determining persistence length by size-exclusion chromatography

Several methods for determining persistence length from size-exclusion chromatography data are evaluated for a stiff polymer, poly(n-hexyl isocyanate) (PHIC), and the more flexible poly[2,7-(9,9 di-n-hexylfluorene)]. The molar mass dependence of the root-mean-square radius obtained from light scattering detection is used to calculate the persistence length of PHIC, based on the Kratky–Porod wormlike chain model. The persistence length estimate is in reasonable agreement with literature values, and the results are relatively insensitive to the sample concentration. Directly solving for persistence length by non-linear regression is more suitable than linear approximations that use ratios of root-mean-square radius and molar mass. The persistence length calculated for the same polymer from viscometry detection, and the Yamakawa–Fujii–Yoshizaki hydrodynamic cylinder model for wormlike chains, is reasonable only when the viscosities measured by viscometry detection, and the molar masses measured by light scattering detection, are extrapolated to zero concentration values. The concentration effects are not significant for the more flexible PHF, and viscometry data are suitable for determining persistence lengths, even by linear approximation methods that use the ratio of intrinsic viscosity and molar mass. Molar masses calculated from a universal calibration curve and the viscometry detector are shown to be erroneous for the freely draining PHIC and should not be used for persistence length determination. Other methods that examine selected regions of viscosity conformation plots are shown to be of limited utility.

Scattering function for wormlike chains with finite thickness

AbstractThe particle scattering function P(k) is approximately evaluated for the Kratky–Porod wormlike chain with a circular cross section to examine the effect of chain diameter d on the scattering curve of k2P(k) versus k, the magnitude of the scattering vector, for stiff chains and also the applicability of the cross‐section plot of ln[kP(k)] versus k2 to them. In the evaluation, series expansions from the rod and coil limits up to the fifth‐order and third‐order deviations, respectively, are combined together. The major results or conclusions derived are as follows. First, the conventional equation, P(k) = P0(k) exp(−k2d2/16), for straight cylinders overestimates k2P(k) at relatively large values of k, whereas its alternative, P(k) = P0(k)[2J1(kd/2)/(kd/2)]2, is a good approximation to exact P(k) unless contour length L is shorter than 10d. Here, P0(k) denotes the scattering function for the chain contour, and J1(x) is the Bessel function of the first order. Second, as d is increased for a fixed value of L (relative to the Kuhn segment length), the k2P(k)–k curve lowers with a pronounced maximum, and the peak position shifts to a lower scattering angle. Third, if the chain is somewhat flexible, the cross‐section plot has an approximately linear region, with a slope fairly close to −d2/16 expected from the aforementioned conventional equation. This plot for rods appreciably bends down, and thus the experimental observation of an approximately linear relation (over a wide k range) implies that, in contrast to the prevailing notion, the polymer examined is not completely rigid but instead is somewhat flexible. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1398–1407, 2004

Rheology, Cryogenic Transmission Electron Spectroscopy, and Small-Angle Neutron Scattering of Highly Viscoelastic Wormlike Micellar Solutions

To understand the strong viscoelastic response showed by aqueous solutions of erucylbis(hydroxyethyl)methylammonium chloride (EHAC) in the presence of potassium chloride (KCl), steady-state rheology, small-angle neutron scattering (SANS), and cryogenic transmission electron microscopy (cryo-TEM) experiments were performed. This cationic surfactant has the ability to self-assemble into giant wormlike micelles. The effect of surfactant concentration, added salt, and temperature were investigated. The surfactant solutions have a gellike behavior at room temperature and become Maxwellian as the temperature is increased. It was found that the low-shear viscosity has a strong dependence on salt concentration and temperature. Small-angle scattering indicated the formation of wormlike micelles. The high-Q range was fitted using the Kratky−Porod wormlike chain model, and a cross-sectional radius of gyration (Rg,xs) of 21 A was obtained. Additionally, cryo-TEM images revealed changes in the structure of the entangl...

Small-Angle Neutron Scattering from Peptide Nematic Fluids and Hydrogels under Shear

Small-angle neutron scattering has been used to study the mesoscopic structure responsible for the nonlinear rheology of aqueous dispersions of self-assembling peptide fibrils (nematic-fluid states) and fibrillar networks (nematic hydrogels). The orientation of the nematic director has been studied as a function of the shear rate (0−500 s-1) in both gel and fluid phases. The powder-averaged scattering data from the fluid nematic phase has been modeled using a Kratky−Porod wormlike chain and the cross-sectional area (a stack of 8−10 tapes) of the self-assembled fibrils measured under flow in the direction of the velocity gradient. A separate observation of the aggregate structure in both the fluid and the gel states was possible in the tangential-velocity direction. An apparent increase in the diameter of the fibrils upon gelation is, in view of the electron micrographs, attributed to fiber formation by the entwining of pairs of fibrils: it is estimated that approximately 10% of the total contour length of the fibrils participates in these fiberlike junctions. The stress-controlled nonlinear rheology has also been studied, and the nematic fluids were found to display Type I shear thinning behavior, typical of liquid-crystalline polymers. Stress relaxation in the form of a loss of orientation only occurred at high concentrations (6 mM) in nematic gels. Furthermore, the gels were observed to display Lozenge shaped scattering at intermediate shear rates, indicative of mobile oriented fibrils in an unoriented network.

Statistical mechanics of the extensible and shearable elastic rod and of DNA

We have developed a new statistical mechanical theory for wormlike chains and elastic rods which will account for deformations of bending, twisting, shear, and axial extension/stretching. We have derived a Fokker–Planck equation for Green’s function. We have also obtained an exact expression for the mean square end-to-end distance. Our new theoretical model is the most general statistical mechanical model for wormlike chain polymers available to date. The Kratky–Porod wormlike chain and the Yamakawa–Fujii helical wormlike chain models are found to be special cases of this new model. This new theory may provide deeper understanding of recent experimental data regarding overstretching single DNA molecules.

Kerr Effect and Wide Angle Light-Scattering Studies of a Para-Aromatic Polyamide in Dilute Solution

The geometric and optical properties of a para-aromatic polyamide are characterized using wide angle light scattering and electric birefringence. The polydispersity correction was applied theoretically by assuming the most probable distribution. The aromatic polyamide studied can be satisfactorily modeled as a Kratky−Porod wormlike chain with a presistence length of 220 ± 50 A and an optical unit anisotropy ratio of 2.2 ± 0.3 in tetrahydrofuran. The refractive index of 1.67 ± 0.03 is comparable to that of similar polyamides. The high persistence length of the polymer chain and the high inherent optical anisotropy of the repeat unit are found to be responsible for the exceptionally high birefringence observed in the oriented films. It is also conclusively established that there is no aggregation in the absence of moisture.

Dilute-Solution Behavior of Cellulose Tris(3,5-dimethylphenylcarbamate)

Eleven samples of cellulose tris(3,5-dimethylphenylcarbamate) ranging in weight-average molecular weight Mw from 2×104 to 4×106 have been prepared by extensive fractionation and studied by static light scattering, sedimentation equilibrium, and viscometry with 1-methyl-2-pyrrolidone at 25°C as the solvent. This polymer is found to exhibit pronounced optical anisotropy differing from other cellulose derivatives, and light scattering data are corrected for the anisotropy effect on the basis of the Nagai theory for the Kratky–Porod wormlike chain with cylindrically symmetric polarizabilities. It is shown that the molecular weight dependences of 〈S2〉z (the z-average mean-square radius of gyration), δ (the optical anisotropy factor), and [η] (the intrinsic viscosity) for Mw below 106 are explained consistently by the known theories for the unperturbed wormlike chain. From the comparison between theory and experiment the persistence length and the monomeric length along the chain contour are estimated to be 7.8 and 0.52 nm, respectively. It is also shown that excluded-volume effects on 〈S2〉z and [η] are experimentally visible for Mw above 106 but remain small even at the highest Mw studied.

Dynamic properties of molecular chains with variable stiffness

The dynamics of a free-draining chain of variable stiffness in a dilute solution is investigated. The chain is considered as a differentiable space curve with stretching and bending elasticity. Second moments, like the mean square end-to-end distance, the radius of gyration, and the pair correlation function of the equilibrium distribution exactly agree with those of the well-known Kratky–Porod wormlike chain. The equation of motion of the chain is derived and solved by a normal mode analysis. In the limit of a flexible chain the model exhibits the well-known Rouse dynamics, whereas in the rod limit the eigenfunctions correspond to bending motion only. In addition, the rotational motion in the latter limit is naturally obtained within the model. The relaxation times obtained by the model are compared with experimental transient birefringence and dynamic light scattering data. In addition, electric dichroism measurements are interpreted in terms of the model. All of these experiments are in good agreement with the theoretical predictions.

Models and equilibrium properties of stiff molecular chains

The partition functions of discrete as well as continuous stiff molecular chains are calculated using the maximum entropy principle. The chain is described by mass points, and their connectivity is taken into account by harmonic constraints (flexible segments) in addition to the bending restrictions. For comparison and as a test of the formalism the freely rotating chain as well as the Kratky–Porod wormlike chain (rigid segments) are reexamined treating the bending restrictions as constraints. It is shown that the second moments for the chain of flexible segments agree exactly with those known from the freely rotating chain for the discrete as well as the continuous chain and for all stiffnesses. Moreover, the Green’s function for the continuous chain is determined, which allows to obtain any desired two point distribution function. The influence of various bending restrictions on equilibrium properties is discussed. Furthermore, a comparison to other existing models, especially the Harris and Hearst model, is given and the validity of the various models is examined.

Chain stiffness of poly‐methoxypropyl[1.1.1]propellane

AbstractThe chain stiffness of a methoxypropyl‐substituted poly[1.1.1]propellane in solution is investigated by static and dynamic light scattering. The Kuhn statistical segment length is determined as lk=150Å utilizing the Kratky–Porod wormlike chain model. It is, however, not yet clear whether the derived Kuhn length characterizes the chain stiffness of a perfect polypropellane chain or rather originates from “kinks” in the polymer backbone due to the presence of possible side reactions during polymerization.

Moments for DNA topoisomers: the helical wormlike chain.

AbstractThe mean‐square radius 〈S2〉N of gyration of the DNA topoisomer with the linking number N is evaluated as a function of N and chain length L on the basis of a (circular) twisted wormlike chain, i.e., a special case of the helical wormlike chain. Evaluation of 〈S2〉N and also of the moment 〈Wr2〉 of the writhe Wr is carried out over a wide range of L, following the Monte Carlo procedure of Frank‐Kamenetskii et al. It is found that the present Monte Carlo values of 〈Wr2〉 for large L are appreciably larger than the known Monte Carlo values for freely jointed chains. Thus, the empirical interpolation formula for 〈Wr2〉 previously constructed on the basis of the theoretical values for small L along with the latter Monte Carlo values for large L is revised with the present Monte Carlo values. By the use of the revised formula, a reanalysis of the experimental data for the distribution of topoisomers is made, and it is found that the present estimates of the torsional constant and the stiffness parameter are equal to and somewhat larger than the previous ones, respectively. It is shown that 〈S2〉N decreases with increasing |ΔN|, where ΔN = N − N, with N the number of helix turns in the linear DNA chain in its undeformed state. The mean‐square radii of gyration 〈S2〉Wr and 〈S2〉 of the original circular Kratky–Porod wormlike chain with and without Wr fixed are also evaluated.

Dynamics of helical worm-like chains. IX. Dynamic intrinsic viscosity

The dynamic intrinsic viscosity of flexible chain polymers in dilute solution is studied on the basis of the discrete helical worm-like chain. The correlation function formalism of the complex intrinsic viscosity [η] is given, taking account of the effect of the finite hydrodynamic volumes of subbodies of the chain. A new basis set, which is a hybrid of the one- and two-body excitation basis functions, is introduced. Then the eigenvalue problem for the representation of the diffusion operator may be reduced to N six-dimensional eigenvalue problems with N the number of subbodies. Among the six branches of the eigenvalue spectrum, one global and two local branches make contribution to the dynamic intrinsic viscosity. The theory predicts the existence of the high-frequency plateau which is distinguished from the infinitely high-frequency viscosity. The plateau arises from the interaction between the global and local chain motions (caused by the helical nature of the local chain contour), the constraints, and the finite hydrodynamic volumes of the subbodies. The interaction above vanishes for the Kratky–Porod worm-like chain possessing no helical nature. A comparison of theory with experiment is made with respect to the plateau value, and it is found that its dependence on the chemical structure of the chain may well be explained.

Stiffness and excluded-volume effects in polymer chains

In order to examine the correlation between stiffness and excluded-volume effects in polymer chains, polymethylene-like rotational isomeric state chains with excluded volume are simulated and their mean-square radii of gyration are evaluated by Monte Carlo methods. The present results and also experimental data for ordinary flexible chains may well be explained by the Kratky–Porod worm-like chain, or more generally the helical worm-like chain, with the excluded-volume effects incorporated in the Yamakawa–Stockmayer scheme. For typical stiff chains with excluded volume, however, there is clear disagreement between their theory and experiment, and the problem remains unsolved.