Statistics Of Polymer Chains Research Articles

Extended tail states in an imaginary random potential

Non-Hermitean operators may appear during the calculation of a partition function in various models of statistical mechanics. The tail eigen-states, having anomalously small real part of energy $Re(\eps)$, became naturally important in this case. We consider the distribution of such states and the form of eigenfunctions for the particle propagating in an imaginary random potential (the model motivated by the statistics of polymer chains). Unlike it is in the Hermitean quantum mechanics, our tail states are sufficiently extended. Such state appear if the values of random potential turns out to be anomalously close inside the large area. Results of numerical simulations in the case of strong coupling confirm the analytic estimates.

Microscopic theory of protein folding rates. I. Fine structure of the free energy profile and folding routes from a variational approach

A microscopic theory of the free energy barriers and folding routes for minimally frustrated proteins is presented, greatly expanding on the presentation of the variational approach outlined previously [J. J. Portman, S. Takada, and P. G. Wolynes, Phys. Rev. Lett. 81, 5237 (1998)]. We choose the λ-repressor protein as an illustrative example and focus on how the polymer chain statistics influence free energy profiles and partially ordered ensembles of structures. In particular, we investigate the role of chain stiffness on the free energy profile and folding routes. We evaluate the applicability of simpler approximations in which the conformations of the protein molecule along the folding route are restricted to have residues that are either entirely folded or unfolded in contiguous stretches. We find that the folding routes obtained from only one contiguous folded region corresponds to a chain with a much greater persistence length than appropriate for natural protein chains, while the folding route obtained from two contiguous folded regions is able to capture the relatively folded regions calculated within the variational approach. The free energy profiles obtained from the contiguous sequence approximations have larger barriers than the more microscopic variational theory which is understood as a consequence of partial ordering.

Conformational statistics of polymer chains in the interphase of semi-crystalline polymers

The microscopic structure of a semi-crystalline polymer interphase has been investigated by off-lattice Monte Carlo techniques for polyethylene-like flexible chains. In this approach, the conformational space consisting of chain populations of loops, bridges and tails is explored by robust cutting and splicing moves in real space. The simulations capture the most probable equilibrium distributions. The populations of loops and tails follow a truncated exponential distribution and the population of bridges shows a maximum as a function of chain length. For simulations of flexible chains, 40–45% of the chains form adjacent entry folds. The effect of molecular weight has been investigated. The bridge population is found to increase from 5 to 10%, for the interphase thicknesses studied, as the molecular weight of material simulated increases from ∼8000gm/gmol to ∼30 000gm/gmol. A Gaussian model for the interphase has been developed and compared with simulations of non-interacting phantom chains. The distributions match well at long chain lengths and deviate at short lengths.

Lattice-based simulations of chain conformations in semi-crystalline polymers with application to flow-induced crystallization

Polymer fiber processes, such as high-speed spinning of nylon and PET, are highly complex involving a complicated interplay between an evolving internal molecular microstructure, macroscopic transport phenomena, such as fluid mechanics and heat transfer, and also non-equilibrium thermodynamics and kinetics affecting nucleation and subsequent crystal growth. All of the above processes are important in determining the final product's semi-crystalline morphology which is primarily responsible for its mechanical properties. Our approach to attack this problem is a hierarchical one: A macroscopic (continuum) model is developed based on the Hamiltonian formalism of non-equilibrium thermodynamics for flowing systems [Beris and Edwards, Oxford University Press, Oxford, 1994]. This approach allows us to follow the dynamic evolution of several macroscopic (continuum) variables involving both kinematic and structural parameters. To implement this approach successfully, an accurate modeling of the (extended) free energy (Hamiltonian) of the system under consideration and the dissipation therein is necessary. While the later is, at the moment, phenomenological, we are developing a first principles approach for the former based on a microscopic modeling of chain conformations using a lattice model. Lattice models have been used extensively before for the analysis of chain conformations in both purely amorphous and semi-crystalline polymers. We have reinterpreted some of the earlier lattice models by systematically deriving the relevant statistics of polymer chains and by outlining the a priori approximations in those models which are necessary to arrive at closed-form expressions. Although we do make use of such earlier work, we have also extended it through a computer-aided analysis. This analysis has enabled us to generate from first principles free energy surfaces for a system consisting of polymer chains, represented as multiple self- and mutually-avoiding random walks, on a 2-D fully populated semi-crystalline lattice. It is shown that the numerical results for dense semi-crystalline systems can be fitted with low order polynomials that provide closed-form approximations for the configurational entropy in terms of non-equilibrium structural parameters, such as the orientation and stretching of the polymer chains. Finally, chain statistics for bulk amorphous polymers have been validated against their theoretical predictions.

Molecular conformation of a polyaramid in nematic solution from small angle neutron scattering and comparison with theory

Zero-average contrast small angle neutron scattering measurements have been performed on an 11%(w/w) nematic aramid polymer (PPTA) solution in concentrated sulphuric acid. The radii of gyration parallel and perpendicular to the director were determined to be 250 and 70 Å, respectively, at room temperature. In addition, the temperature dependence of the radii of gyration was investigated. The values for the radii of gyration are compared to a worm-like path model for the polymer chain statistics where the segment orientation is described using a Maier–Saupe-type orientational distribution function. From this analysis, it is found that the persistence length of the aramid molecules is the same in the nematic phase as in a dilute isotropic phase, about 290 Å. We conclude that there is no effect of the nematic environment on the polymer chain statistics in this main chain polymer solution. As a consequence, the nematic phase of aramid solutions is expected to contain polymer chains with a substantial number of hairpin defects.

The conformational statistics of polymer chains as obtained from their energy landscape: a conceptual view of the rotational isomeric state approach

In the rotational isomeric state (RIS) approach the conformational statistics of a polymer chain can be conveniently described in terms of the statistical matrix. In this work it is shown from a conceptual point of view to which degree the projection onto a few states, inherently present in any RIS approach, may lead to systematic deviations for estimation of the a priori probabilities. It is shown that these deviations scale with some power of the inverse temperature. Furthermore an algorithm is presented, deriving the statistical matrix from computer simulations. This algorithm is applied to polyethylene. Comparison with the conformational statistics as obtained from Monte Carlo simulations allows the quality of the statistical matrix elements to be estimated. It turns out that the systematic deviation does not hamper practical applications. Questions like the relevance of the anharmonicity of the energy landscape or consideration of more than nearest-neighbor correlations in the RIS approach are discussed.

The conformational statistics of polymer chains as obtained from their energy landscape: a conceptual view of the rotational isomeric state approach

The conformational statistics of polymer chains as obtained from their energy landscape: a conceptual view of the rotational isomeric state approach

Polymer chain statistics and conformational analysis of DNA molecules with bends or sections of different flexibility

The worm-like chain model has often been employed to describe the average conformation of long, intrinsically straight polymer molecules, including DNA. The present study extends the applicability of the worm-like chain model to polymers containing bends or sections of different flexibility. Several cases have been explicitly considered: (i) polymers with a single bend; (ii) polymers with multiple coplanar bends; (iii) polymers with two non-coplanar bends; and (iv) polymers comprised of sections with different persistence lengths. Expressions describing the average conformation of such polymers in terms of the mean-square end-to-end distance have been derived for each case. For cases (i) and (iv), expressions for the projection of the end-to-end vector onto the initial orientation of the chain are presented. The expressions derived here have been used to investigate DNA molecules with sequence-induced bending (A-tracts). Mean-square end-to-end distance values determined from a large number of A-tract containing DNA molecules visualized by scanning force microscopy resulted in an average bend angle of 13.5° per A-tract. A similar study was performed to characterize the flexibility of double-strandedDNA molecules containing a single-stranded region. Analysis of their mean-square end-to-end distance yielded a persistence length of 1.3 nm for single-stranded DNA.

A general simulation method for computing conformational properties of single polymer chains

Rotational isomeric state (RIS) theory is the standard method for computing the conformational statistics of polymer chains. It applies to chains under theta conditions, either in the melt or in theta solution. RIS statistical weights can also be used in a Monte Carlo scheme for generating independent chain conformations with the correct statistics. One practical drawback of RIS methods is that statistical weights must be derived before any chain properties can be calculated. This can be tedious for all but relatively simple chains. Here, an efficient method is presented for computing the properties of theta chains without the intermediate step of deriving statistical weights. The method—‘RIS’ Metropolis Monte Carlo (RMMC) simulation-allows computation of the same type of properties as does RIS theory. It shares certain approximations with RIS theory but is not, strictly speaking, a rotational isomeric state method, as it allows bond torsion angles to vary continuously.

Dynamics of strongly entangled polymer systems: activated reptation

The effect of excluded-volume interactions on the reptation dynamics of long polymer chains is considered theoretically. It is shown that interactions give rise to an exponential increase of the reptation time, , if polymer chains are long enough: , where is the number of monomers per entanglement. We propose a novel dynamical mechanism of activated reptation implying that neighboring chains exchange conformations of their terminal fragments. It is shown that the exchange mechanism is compatible with the equilibrium polymer chain statistics and that it provides a bridge between the previous theories.

Conformational statistics of polymer chain terminally attached to wall (III) —NRW model loop chain

Conformational statistics of polymer chain terminally attached to wall (III) —NRW model loop chain

Conformational statistics of polymer chain terminally attached to wall (II)

The SAW tail chains were studied. The permitted conformational number and the mean square end-to-end distance as a function of the chain length N for such a model tail chain were obtained by computer simulations, including the exact enumeration and Monte Carlo method. These two basic quantities obeyed the relations deduced from the scaling law. The critical exponents and the lattice indexes were given by fitting the data of the computer experiments. It has been shown that there is a certain extension in the size of the SAW tail chains as well as the NRW tail chains in the direction normal to the wall. The normal component of the mean square end-to-end distance is almost twice as large as the parallel component of the short chain SAW. However, as N →∞, the effect of the wall on the chain conformation becomes a little weak because of the self-avoiding behavior for the model. That is quite different from the case of the NRW tail chain.

Conformational statistics of polymer chain terminally attached to wall (III)——NRW model loop chain

When the two end groups of a linear polymer chain are absorbed on a solid surface, the polymer chain forms the “loop” conformation. Investigation has been made on the conformational statistics of a model loop chain by the normal random walk (NRW) on a lattice confined in the half-infinite space. Based on the conformational distribution function of the NRW model tail chain, it is easy to deduce an analytical formula expressing the conformational number of the model loop chain. It was found that the ratio of the conformational number of the model loop chain to that of the free chain varies with the power functionN -2/3 when the chain lengthN→οο. The same result -was obtained by means of the recursion equation. The ratio of the mean square end-to-end distanceh 2 for the model loop chain to its mean square bond lengthl 2 is 2N/3. Compared with the free chain with the same lengthN, the mean square end-to-end distance of the model loop chain contracts to a certain extent. The basic relationships deduced were supported by the exact enumeration and Monte Carlo simulations.

Hydrophobic Polymer Melt, Tethered to the Water SurfaceMonolayers of Polyisoprenes with Sulfonate Head Groups

Hydrophobic polymers with low glass transition temperature (polyisoprenes) and a single head group (sulfonate) have been synthesized and characterized as insoluble monolayers on a water surface. The films are considerably thicker (10−50 nm) than conventional Langmuir monolayers of low molecular weight substances or polymers with surface-active repeating units. The thickness is inversely proportional to the area per head group. Films can be transferred via the Langmuir−Blodgett technique with transfer ratios close to 1. We use these monolayers as model systems to investigate the influence of the polymer chain statistics on the properties of solvent-free tethered polymers. We assume that each polymer chain is bound to the water phase with the head group and has the conformation of a distorted three-dimensional coil. The distortion increases with decreasing area per head group (A/n) and influences the isotherm of the polymer monolayers. We develop a theoretical description of this effect and conclude that, at a given area per head group, the surface pressure increases linearly with the chain length. In the case of Gaussian statistics, the slope of this line is proportional to the third power of the area per head group. In the case of non-Gaussian statistics, this slope can be expressed as a polynomial of A/n. The experimental data confirm these theoretical predictions. Deviations from Gaussian statistics can be neglected, and the description as a nonuniformly stretched brush (Semenov theory) comes closer to the experiment than the model of a uniformly stretched brush. Chains shorter than 300 repeat units systematically deviate from the theory.

Statistics of stiff polymer chains near an adsorbing surface

AbstractThe exact solution of the problem of adsorption of a long ideal polymer chain with variable degree of stiffness on a plane surface is presented. It is shown that the adsorption of stiff polymer chains is a second‐order phase transition; in the adsorbed state “train” (i.e. adsorbed) sections are relatively longer and loop sections relatively shorter than for flexible chains. This effect is very pronounced: already for moderately stiff chains the number of Kuhn segment lengths in one “train” section at the temperature T = Tcr/2 (Tcr is the critical temperature for adsorption transition) can reach several thousands, and deviation from the surface occurs only in the form of small “hairpins”. The maximum length of the chain, which at the given conditions would flatten completely on the surface, is estimated.

Lorentz lattice gases, abnormal diffusion, and polymer statistics

Diffusive behavior in various Lorentz lattice gases, especially wind-tree-like models, is discussed. Comparisons between lattice and continuum models as well as deterministic and probabilistic models are made. In one deterministic model, where the scatterers behave like double-sided mirrors, a new kind of abnormal diffusion is found, viz., the mean square displacement is proportional to the time, but the probability density distribution function is non-Gaussian. The connections of this mirror model with the percolation problem and the statistics of polymer chains on a lattice are also discussed.

Glass transition temperature and molecular parameters of polymer

A new relationship, which correlates the glass transition temperature ( T g) with other molecular parameters, is developed by using Flory's lattice statistics of polymer chain and taking the dynamic segment as the basic statistical unit. The dependences of T g on the chain stiffness factor ( σ 2), dynamic stiffness factor ( β = − d ln σ 2 dT ) and molecular weight of polymer are discussed in detail based on the theory. The theory is compared with experimental data for many linear polymers and good agreement is obtained. It is shown that T g is essentially governed by the chain stiffness factor at T g. Moreover, a simple correlation between the parameter K g of the Fox-Flory equation ( T g = T ∞ g − K g M n ) and other molecular parameters is deduced. The agreement between theoretical predictions and experimental measurements of K g has been found to be satisfactory for many polymers.

Statistics of polymer chains near a notch and consequences for twin polymer single crystals

It has been argued that flaws exist in the standard nucleation theory of polymer crystal growth because this theory seems to predict that the notches observed in twin crystals of polyethylene should fill out so rapidly as not to be observable. However, I argue that the observed growth in the notch is reconcilable with standard nucleation theory. When proper consideration is taken for the ability of polymer chains to diffuse into the notch, the observed nucleation rates are not surprising. I consider four separate cases. In the first case, I assume that polymers, although attracted by the growth front, are not attracted strongly enough to be adsorbed prior to crystallization. In the second case, I assume that polymers adsorb readily on the crystal face, and then crystallize very near the point of first contact. In the third case, I assume that chains adsorb on the growth front prior to crystallization and that they are able to diffuse about on the growth front through large distances prior to crystallization. In the fourth case, polymers adsorb rather weakly, and reversibly, attaching and reattaching a number of times prior to crystallization. I am able to rule out the second case, while all the others predict a lower than expected nucleation rate and also predict the observed molecular weight and temperature trends.

Finite extensibility and density saturation effects in the polymer brush

We study the equilibrium statistics of flexible neutral polymer chains attached by one end onto a flat interface (the polymer « brush »). We account for the finite extensibility of the polymers, adopting a form for the free energy of a stretched chain that diverges as its fully-extended length is approached. Adopting also a Flory-Huggins equation of state for the local chemical potential, we obtain monomer density profiles and chain-end density distributions under various solvent conditions. We use an analytic self-consistent field approximation for long, highly stretched chains, as developed by Milner, Witten and Cates (Macromolecules 21 (1988) 2610), who found a parabolic density profile for brushes at « moderate densities » in a good solvent. Our work generalizes their approach to allow very high coverages and/or poor solvents to be considered ; in each case the density profile is much flatter than a parabola.

Small-angle neutron scattering from amorphous polymers

The application of small-angle neutron scattering (SANS) to the study of polymers has revolutionized the approach to investigating polymer chain statistics, thermodynamics and dynamics. Some of the significant developments in this field during the past 15 years are described, including the demonstration of Gaussian coil behavior in polymer melts, derivation and application of the scattering theories for homogeneous polymer mixtures, and the recently discovered polymer isotope effect. Specific examples have been selected from a vast literature for the purpose of demonstrating the unique advantages and potential complications associated with the combined use of deuterium labeling and SANS.