Effect Of Chain Stiffness Research Articles

Role of chain stiffness on the conformation of single polyelectrolytes in salt solutions

Conformation of single polyelectrolytes in tetravalent salt solutions is investigated under the framework of a coarse-grained model, using Langevin dynamics simulations. The chain size, studied by the radius of gyration, shows three different variational behaviors with salt concentration, depending on the chain stiffness. According to the size variations, polyelectrolytes of fixed chain length are classified into three categories: (1) flexible chain, for which the variation shows a curve similar to a tilted L, (2) semiflexible chain, whose curve resembles U, and (3) rigid chain, for which the curve is a straight line. The wormlike chain model with persistence length predicted by the Odijk-Skolnick-Fixman theory is found to be able to qualitatively describe the end-to-end distance at low salt concentration not only for semiflexible and rigid chains but also for flexible chain. In a low salt region, a flexible polyelectrolyte extends more significantly than a semiflexible chain, in reference of the size of their uncharged counterparts, and in a high salt region, regardless of chain stiffness, a chain attains a dimension comparable to that of its neutral polymer. The chain stiffness influences both the local and the global chain structures. A flexible chain exhibits a zigzagged local structure in the presence of salt ions, and the condensed structure is a disordered, random globule. A semiflexible chain is locally smooth, and the condensed structure is orderly packed, taking a form such as hairpin or toroid. Moreover, the chain stiffness can also affect the nature of the coil-globule transition. The transition occurred in a discrete manner for semiflexible chain, whereas it occurred in a continuous way for flexible chain. This discrete feature happened not only at low salt concentration when a semiflexible chain collapsed but also at high salt concentration when the collapsed chain is reexpanded. At the end, the effects of chain stiffness and salt concentration on the conformation of single polyelectrolytes are summarized in a schematic state diagram.

First Cumulant of the Dynamic Structure Factor for Flexible Polymers. Excluded-Volume Effects

The first cumulant Ω(k) of the dynamic structure factor as a function of the magnitude k of the scattering vector was determined from dynamic light scattering (DLS) measurements for an atactic polystyrene (a-PS) sample with the weight-average molecular weight Mw=8.04×106 in toluene at 15.0 °C, an atactic poly(methyl methacrylate) (a-PMMA) sample with Mw=1.31×107 in acetone at 25.0 °C, an atactic poly(α-methylstyrene) (a-PαMS) sample with Mw=5.46×106 in toluene at 25.0 °C, and a polyisobutylene (PIB) sample with Mw=6.63×106 in n-heptane at 25.0 °C. It was also determined for the a-PS and a-PαMS samples in cyclohexane in the vicinity of the respective Θ temperatures (34.5 and 30.5 °C). The translational diffusion coefficient D was also determined from DLS measurements for the a-PS, a-PMMA, a-PαMS, and PIB samples in the same solvent conditions mentioned above. From a comparison of the present data with the previous ones for all the samples in the respective Θ temperatures, it is found that the observed dimensionless coefficient C of the k4-term in the expansion of Ω(k) does not almost depend on excluded volume, as predicted by the Tanaka–Stockmayer first-order perturbation theory of C, being in good agreement with the theoretical value for the Gaussian chain with nonpreaveraged hydrodynamic interaction. Further, it is found that the excluded-volume effect is very small on the quantity [Ω(k)/Dk2−1]/<S2>k2 over the whole range of k2 examined, where <S2> is the mean-square radius of gyration. Their first-order perturbation theory of the height of the plateau in the k3-region, however, cannot explain the behavior of the present experimental data. The disagreement between theory and experiment may be regarded as arising from the fact that the theory based on the Gaussian chain model cannot take proper account of effects of chain stiffness and local chain conformation.

First Cumulant of the Dynamic Structure Factor for Polymers in Θ Solvents. Effects of Chain Stiffness and Local Chain Conformation

The first cumulant Ω(k) of the dynamic structure factor as a function of the magnitude k of the scattering vector was determined from dynamic light scattering (DLS) measurements for two atactic poly(α-methylstyrene) (a-PαMS) samples with weight-average molecular weights Mw=3.22×106 and 5.46×106 in cyclohexane at 30.5 °C (Θ) and for one polyisobutylene (PIB) sample with Mw=6.63×106 in isoamyl isovalerate at 25.0 °C (Θ). The translational diffusion coefficient was also determined from DLS measurements for the a-PαMS sample with the larger Mw and for the PIB sample in the same solvent conditions. The mean-square radius of gyration was determined from static light scattering measurements for the PIB sample. From a comparison of the present data for a-PαMS and PIB along with the previous ones for atactic polystyrene and poly(n-hexyl isocyanate) with the helical wormlike chain theory, it may be concluded that the universality predicted by the Gaussian chain theory for the plot of η0Ω(k)/kBTk3 against 1/2k cannot be realized for flexible polymers even with very high Mw (\\simeq106–107), and that the height of the plateau in the k3-region depends on chain stiffness and local chain conformation, where η0 is the solvent viscosity, kB the Boltzmann constant, and T the absolute temperature.

Self-Consistent-Field Study of Compressible Semiflexible Melts Adsorbed on a Solid Substrate and Comparison with Atomistic Simulations

The present work addresses the problem of a self-consistent-field (SCF) description of a specified polymer melt/solid substrate interfacial system. Key points of the employed method are the coarse-grained representation and the numerical treatment of the continuous-space SCF theory. Compared to other works on polymer adsorption, the main difference of the current approach is the description of the polymer coil connectivity through the wormlike chain model, which, after incorporating local stiffness, reproduces two characteristic lengths of the studied polymer: the mean-squared end-to-end distance and the contour length. As a test case, polyethylene melts adsorbed on a graphite substrate are considered; recent atomistic simulations of the same systems are used to evaluate the theoretical approach. For comparison and elucidation of some effects of chain stiffness on conformational properties of adsorbed molecules, an alternative (and more common) representation of chain connectivity through the Gaussian model, reproducing the mean-squared end-to-end distance, is also considered. Results refer to local and global chain conformational properties, with an emphasis on the latter. In particular, predictions for the shape of chains are obtained, while the conformations of adsorbed molecules are quantified in terms of tails, loops, and trains. For small chain lengths, both the Gaussian and the wormlike chain models deviate considerably from the simulation data. At intermediate chain lengths, however (such as C400), the predictive power of the wormlike model is very good for several conformational properties. On the contrary, predictions from the Gaussian model, especially for the case of loops, deviate considerably from simulations over a broader range of molecular lengths.

Effect of chain stiffness and entanglements on the elastic behavior of end-linked elastomers

The effect of chain stiffness and entanglements on the elastic behavior and microscopic structure of cross-linked polymer networks was studied using Monte Carlo simulations. We investigated the behavior of entangled and entanglement-free networks at various degrees of chain stiffness and densities. Based on previous results that indicated that trapped entanglements prevent strain-induced order-disorder transitions in semiflexible chain networks, we prepared the entangled networks by end-linking the chains in very dilute conditions so as to minimize the extent of trapped entanglements. We also considered the entanglement-free case by using a "diamond" structure. We found that the presence of even a very small amount of trapped entanglements is enough to prevent a discontinuous strain-induced transition to an ordered phase. In these mildly entangled networks, a nematiclike order is eventually attained at high extensions but the elastic response remains continuous and the cross-links remain uniformly distributed through the simulation box. The entanglement-free diamond networks on the other hand show discontinuities in their stress-strain data. Networks at higher densities exhibit a more stable ordered phase and show an unusual staircaselike stress-strain curve. This is the result of a stepwise extension mechanism in which the chains form ordered domains that exclude the cross-links. Extension is achieved by increasing the number of these ordered domains in the strain direction. Cross-links aggregate in the spaces between these ordered domains and form periodic bands. Each vertical upturn in the stress-strain data corresponds to the existence of an integer number of ordered domains. This stepwise elastic behavior is found to be similar to that exhibited by some tough natural materials.

Nuclear Magnetic Relaxation of Poly(n-hexyl isocyanate) in Dilute Solution. Effects of Chain Stiffness

The spin–lattice relaxation time T1, spin–spin relaxation time T2, and nuclear Overhauser enhancement NOE were determined from nuclear magnetic relaxation measurements on five well-fractionated samples of poly(n-hexyl isocyanate) in the range of weight-average molecular weight Mw from 1.65×104 to 5.71×104 in n-hexane at 25 °C. It is found that T1 increases and T2 decreases monotonically with increasing Mw in the range of Mw examined, while NOE is almost independent of Mw. Such behavior may be regarded as arising from effects of chain stiffness. A comparison is then made of the present data for T1, T2, and NOE with the theory developed for the generalized Kratky–Porod (KP) wormlike chain as a special case of the helical wormlike chain, and it is shown that the theory may almost quantitatively explain the data, using the established values of the KP model parameters. A comparison is also made of the present data with the theoretical values calculated from the equations of Woessner for the equivalent, rigid prolate spheroid.

Effect of Chain Stiffness on Polyelectrolyte Condensation

The effect of adding tetravalent counterions to polyelectrolytes of varying stiffness is investigated by a flat histogram Monte Carlo technique that is capable of giving the free energy of the system by direct simulation. The ensemble average of the polyelectrolyte size decreases with the amount of added salt for all chain stiffness. When examining the size distribution functions, flexible chains have narrow end-to-end distance distribution functions for all amounts of added salt. The distribution functions for semiflexible polyelectrolytes are always broader because such chains fluctuate between stretched and collapsed conformations. For stiff chains, the distribution functions show double maxima, which reflect the fact that individual chains prefer to be in elongated or compact toroid conformations, but not inbetween. This coexistence between compact and elongated conformations for stiff polyelectrolytes, when multivalent salt is added, is manifested in force-extension curves which exhibit a plateau regime in which the chain size changes markedly at a constant force.

Some Comments on the Analysis of the Third Virial Coefficient for Polymer Chains

An analysis is made of experimental data for the third virial coefficient A3 of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) in the Θ solvent cyclohexane at 30.5 °C and in three good solvents, toluene, 4-tert-butyltoluene, and n-butyl chloride, at 25.0 °C on the basis of the helical wormlike (HW) chain model. It is found that A3 at Θ, which is denoted by A3,Θ, becomes a constant of 5.0×10−4 cm6 mol/g3 independent of the weight-average molecular weight Mw for Mw ≳ 104 and deviates from it for Mw ≲ 104 because of effects of chain ends. The observed dependence of A3,Θ on Mw may be well explained by the HW theory that takes account of the effects. For the three good-solvent systems, the behavior of the factor g defined by A3/[A2(HW)]2Mw is examined as a function of the cubed gyration-radius expansion factor αS3, where A2(HW) is the part of the second virial coefficient A2 without the effects of chain ends. It is found that the data points for a-PαMS samples with Mw ≳ 105 in all the three good solvents form nearly a single-composite curve, as predicted by the HW theory, and that they follow the Stockmayer–Casassa theoretical curve for αS3 ≳ 2 but deviate upward from it for αS3 ≲ 2 because of effects of three-segment interactions. Effects of chain stiffness are of minor importance for an explanation of this deviation. Some literature data for polystyrene are also examined.

Formation and Characterization of Semiflexible Polymer Networks via Monte Carlo Simulations

The effect of chain stiffness and entanglements on deformation properties of end-linked networks was investigated using Monte Carlo simulations. Tetrafunctionally cross-linked monodisperse networks were prepared in the framework of the bond fluctuation model (BFM). The degree of entanglement in these networks was tuned by changing the initial polymer concentration at curing (Φ0). The chain stiffness was controlled by using an adjustable bond angle bending potential. Continuum-space simulations of isotropic swelling and uniaxial deformation were carried out in isobaric and isostress ensembles, respectively. Both equilibrium swelling and stress−strain data indicate that the stiffer chain networks are more entangled, confirming previous simulation results on polymer melts. For such entangled networks, stiffer chains are associated with a higher elastic modulus and smaller equilibrium swelling; the opposite is true, however, for entanglement-free “diamond” networks. The elastic modulus determined from low-strain uniaxial deformations agrees with semitheoretical predictions for moderate chain stiffness, but not for very stiff chains. For the realistically cross-linked (and hence entangled) networks studied here, the theoretically predicted strain-induced discontinuous ordering transition was not observed, although the transition region becomes narrower for the least entangled networks made from the stiffest chains. Entanglement-free diamond networks do exhibit a discontinuous transition from a disordered state to a nematic-like state under uniaxial extension.

Effects of chain stiffness and penetrant size on penetrant diffusion in simple polymers: deduced relations from simulation and PRISM theory

Molecular dynamics simulations in the NVT ensemble were performed for a repulsive system of bead-spring polymer chains with angle constraints. The diffusion coefficients of spherical penetrants were measured for different size penetrants as the angle constraints were varied. The scaling of the diffusion coefficient with penetrant size varies as a function of chain stiffness from liquid-like behavior to polymeric behavior. Free volume distributions were calculated from both simulation and PRISM theory. It is found that free volume distributions and mean void size are constant with chain stiffness although the diffusion coefficient changes by a factor of two. This suggests that while free volume is necessary for diffusion to occur, binary collisions and chain relaxation also play a role in determining penetrant diffusion. The relative contributions of these factors to the diffusion coefficient may change as a function of chain stiffness.

Effects of chain stiffness on the quenching of an excited polymer by small quenchers

We present a theory for analyzing the effects of chain stiffness on the diffusion-influenced quenching kinetics of an excited polymer. We model the polymer as an optimized Rouse–Zimm chain and the quencher molecule as a spherical particle. The excitation is considered to be localized at any one monomer, or to move randomly along the chain backbone. In regard to the dependence on the chain stiffness, we found two distinctive kinetic regimes. When the excitation migration rate is small, the quenching rate decreases as the chain becomes stiffer. On the other hand, when the mobility of excitation is large, the opposite trend is observed. We also investigate the dependence of Stern–Volmer coefficient on the length and stiffness of the chain in the fast excitation migration limit.

A Monte Carlo study of effects of chain stiffness and chain ends on dilute solution behavior of polymers. II. Second virial coefficient

A Monte Carlo (MC) study is made of the second virial coefficient A2 for polymers using two freely rotating chains, each of bond angle 109°, with the Lennard-Jones 6-12 intramolecular and intermolecular potentials between beads in a cutoff version for the number of bonds in the chain ranging from 6 to 1000 in the Θ and good-solvent conditions. It is found that effects of chain ends on A2 are appreciable for small molecular weight M, as was expected, and that the second virial coefficient A2,Θ at the Θ temperature, at which the ratio 〈S2〉/M of the mean-square radius of gyration 〈S2〉 to M becomes a constant independent of M for very large M, remains slightly negative even for such large (but finite) M where the effects of chain ends disappear. Such behavior of A2,Θ, which cannot be explained within the framework of the binary cluster theory, is shown to be understandable if possible effects of three-segment interactions are considered. The present MC data for A2 (along with the previous ones for 〈S2〉) may then be consistently explained by the existent theory based on the helical wormlike chain model only if a minor correction is made to the theoretical A2,Θ in almost the same range where the effects of chain ends are appeciable. The present MC data are also compared with experimental data, and it is shown that the latter may also be similarly explained.

Second Virial Coefficient of Oligo- and Poly(α-methylstyrene)s. Effects of Chain Stiffness, Chain Ends, and Three-Segment Interactions

The second virial coefficient A2 was determined from light scattering measurements for atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with the fraction of racemic diads fr = 0.72 over a wide range of weight-average molecular weight Mw from 6.48 × 102 to 5.46 × 106 in cyclohexane at 30.5 °C (ϑ) and in toluene at 25.0 °C, and for Mw ≥ 2 × 105 in 4-tert-butyltoluene at 25.0 °C, in n-butyl chloride at 25.0 °C, and in cyclohexane at 35.0, 45.0, and 55.0 °C. It is shown that the observed characteristic features of A2 at ϑ, which is denoted by A2,ϑ, as a function of Mw in cyclohexane at ϑ, i.e., a sharp increase with decreasing Mw for small Mw and a rather long-tailed negative region for large Mw, may be well explained by the theory that takes account of the effects of chain ends and also of three-segment interactions. It is also shown that the observed dependence of A2 on Mw in toluene may be quantitatively explained by the theory that takes account of the effects of chain stiffness, local chain conformatio...

A Monte Carlo study of effects of chain stiffness and chain ends on dilute solution behavior of polymers. I. Gyration-radius expansion factor

A Monte Carlo (MC) study is made of the mean-square radius of gyration 〈S2〉 and the gyration-radius expansion factor αS for the freely rotating chain of bond angle 109° and with the Lennard-Jones (LJ) 6-12 intramolecular potentials between beads in a cutoff version for the number n of bonds in the chain ranging from 10 to 1500 at the reduced temperature ranging from 3.6 to 8.0, which is defined as the absolute temperature multiplied by the Boltzmann constant and divided by the depth of the well of the LJ potential. It is shown that the ratio 〈S2〉/n approaches asymptotically a constant independent of n for very large n at the value 3.72±0.05 of the reduced temperature, which value is equal to the reduced Θ temperature Θ* of the MC model system, and that possible effects of chain ends on 〈S2〉 and therefore on αS are negligibly small. Taking the values of 〈S2〉 at Θ* as the unperturbed ones, αS2 is evaluated from those at various reduced temperatures higher than Θ*. It is then found that the behavior of αS2 may be well explained in the quasi-two-parameter scheme or is in good agreement with that of real experimental data. Further, the binary cluster integral for a bead in the chain is found to be much smaller in magnitude than that for a single isolated bead at reduced temperatures higher than Θ*, the result being consistent with a previous finding.

Effects of Chain Stiffness on the Dynamics of Loop Formation in Polypeptides. Appendix: Testing a 1-Dimensional Diffusion Model for Peptide Dynamics

Quenching of the triplet state of tryptophan by cysteine provides an important new tool for measuring the rate at which a specific intramolecular contact is formed in disordered polypeptides. By measuring the viscosity dependence of the quenching rate, both the reaction-limited and the diffusion-limited quenching rates can be determined. The diffusion-limited rate corresponds to the rate of forming a short-range contact. The reaction-limited rate, which depends solely on the equilibrium end-to-end distribution, becomes essentially length-independent for short chains, providing clear-cut evidence that the chain is stiff. The length dependence of the reaction-limited rate can be accurately calculated using the distance dependence of the quenching rate determined at room temperature in a rigid glass, together with the end-to-end distance distribution for a wormlike chain having a persistence length of 0.6−0.7 nm. In addition, the length dependence of the diffusion-limited rate can also be reproduced by treat...

Partitioning of Semiflexible Macromolecules into a Slit in Theta Solvent

Investigation of semiflexible coil-like chains in the process of partitioning with a slit in a solvent of variable thermodynamic quality has shown two key results. For semiflexible chains in a good solvent, the effect of chain stiffness played a role only at low concentrations. However, the situation is different with interacting chains where the effect of stiffness is also observed at higher concentrations. For theta chains, the partitioning into a slit in dilute solutions is lower for semiflexible than for flexible chains, while for higher concentrations this order is reversed. The packing ability of semiflexible chains at higher concentrations is enhanced in the theta system. Interestingly, stretching of chains on penetration into the slit is observed at higher concentrations. Decomposition of free energy change on partitioning into entropy and energy contributions gives more information on the details of partitioning of these systems, especially the differentiation between dilute and moderate concentration regimes. A positive change of partitioning entropy in the theta solvent in the semidilute regime for both flexible and semiflexible chains is noteworthy. This is related to breaking favorable interactions between chains on penetration into the slit.

Primary cyclization in the polymerization of bis-GMA and TEGDMA: a modeling approach to understanding the cure of dental resins

An optimal dental restorative polymeric material would have a homogeneous cross-linking density giving it consistent mechanical strength throughout the material. When multifunctional monomers are polymerized, a pendant double bond can react intramolecularly with the radical on its propagating chain to form a loop, which results in a primary cyclization reaction. Primary cyclization does not contribute to overall network structure, causes microgel formation, and leads to heterogeneity in the polymer. Knowledge of how cure conditions control the degree of primary cyclization and cross-linking in the polymer is important in developing better dental materials. To gain more understanding about the evolving polymer network, the photopolymerization of a typical dental resin (75/25 wt% bis-GMA/TEGDMA) is modeled using a first principals approach. The overall polymerization rate behavior of 75/25 wt% bis-GMA/TEGDMA is predicted using experimentally obtained propagation and termination kinetic rate constants. The effect of chain stiffness and light intensity on the polymerization kinetics is also explored. Furthermore, the model predicts the extent of cross-linking and primary cyclization in the growing polymer network. At 45% conversion, the fraction of bis-GMA and TEGDMA pendant double bonds created that have cycled is 11 and 33%, respectively. The model shows that using a stiff monomer, like bis-GMA, in dental resins diminishes the extent of cyclization and increases the cross-linking density of the polymer. Therefore, better mechanical properties are obtained than if more flexible monomers were used.

Monte Carlo Simulations of the Interface between Polymer Melts and Solids. Effects of Chain Stiffness

This paper presents the results of a comparative Monte Carlo study of the influence of chain stiffness on the properties of dense systems of chain molecules near solid surfaces. All relevant properties of the interface (molecular packing, conformational distribution, dynamies) are modified by the inclusion of an intrinsic stiffness. Among the others, the density and degree of order of the first layer of units in contact with the surfaces and the thickness of the interfacial region increase with increasing chain stiffness, while the center of mass mobility near the surfaces and in the bulk shows a maximum for model chains with stiffness comparable to polymethylene.

Monte Carlo Simulations of the Interface between Polymer Melts and Solids. Effects of Chain Stiffness

This paper presents the results of a comparative Monte Carlo study of the influence of chain stiffness on the properties of dense systems of chain molecules near solid surfaces. All relevant properties of the interface (molecular packing, conformational distribution, dynamics) are modified by the inclusion of an intrinsic stiffness. Among the others, the density and degree of order of the first layer of units in contact with the surfaces and the thickness of the interfacial region increase with increasing chain stiffness, while the center of mass mobility near the surfaces and in the bulk shows a maximum for model chains with stiffness comparable to polymethylene.

Mechanical Properties and Swelling Behavior of End-Linked Poly(diethylsiloxane) Networks

Linear α,ω-vinyl-terminated poly(diethylsiloxane) (PDES) chains were synthesized and fractionated to obtain melts of low polydispersity. The fractionated samples were characterized using gel permeation chromatography and dilute solution viscometry. PDES elastomers were also prepared from the melts using hydrosilylation end-linking. The effects of chain stiffness on solution viscometric properties and on network swelling in toluene were compared to previously obtained results with poly(dimethylsiloxane) (PDMS). The mechanical behavior of the PDES elastomers in uniaxial extension was studied with a Perkin-Elmer dynamic mechanical analyzer. The dry networks exhibit a stress-induced amorphous-to-mesophase transition, which results in a discontinuous stress−strain relationship. The effects of molecular-level structure on network mechanical properties were deduced by studying PDES elastomers with well-defined molecular weight between chemical cross-links.