Model Of Polymer Chain Research Articles

A New Framework of Polymer Solution Science. The Helical Wormlike Chain

The object of this review article is to introduce and digest Yamakawa’s new book, “Helical Wormlike Chains in Polymer Solutions.” A brief description is given of a new model for polymer chains, called the helical wormlike chain, and its applications to dilute solution behavior of polymers, that is, equilibrium, steady-state transport, and dynamical properties. The description follows most of the chapters of the book. It is shown that all theoretical and experimental investigations on the basis of this model provide a new framework of polymer solution science, which takes the place of the Flory–Kirkwood framework.

Monte Carlo and molecular dynamics simulation of the glass transition of polymers

Two coarse-grained models for polymer chains in dense glass-forming polymer melts are studied by computer simulation: the bond fluctuation model on a simple cubic lattice is treated by Monte Carlo methods, and a continuum bead-spring model with a Lennard-Jones potential between the beads is treated by means of molecular dynamics. While the dynamics of the two models differ for short length scales and the associated timescales, the two models behave similarly on mesoscopic spatial and temporal scales. In particular, the mode-coupling theory of the glass transition can be used to interpret the slowing down of the undercooled polymer melt. For the off-lattice model, the approach to the critical point of mode coupling is studied both at constant pressure and constant volume. The lattice model allows a test of the Gibbs-Di Marzio entropy theory of the glass transition to be carried out, and our finding is that although the entropy does decrease significantly, there is no `entropy catastrophe' where the fluid entropy would turn negative. Finally, a forward look at the effects of confinement on the glass transition in thin-film geometry is given.

Conformational statistics of semiflexible polymers: comparison between different models

Recently performed small-angle neutron scattering (SANS) and small-angle X-ray scattering (SAXS) experiments (Macromolecules 30 (1997) 273) had provided experimental data for measurement of the static scattering formfactor which are in very good agreement with the theoretical results for the formfactor obtained with the help of the Dirac model of semiflexible polymer chains. This agreement had prompted us to make additional detailed comparisons between the observables which can be calculated with help of the Dirac and the Kratky–Porod (K-P) models of semiflexible chains. In this work we provide comparison between the closed form exact analytic results for the Dirac formfactor and various approximations of the static formfactor obtained with help of the K-P model. Remarkable agreement between the exact result and various approximations is thoroughly investigated through systematic classical and semiclassical analysis of the K-P model. We found conditions (approximations) under which the K-P model could be obtained from the Dirac model. This enabled us to calculate the observables related to measurements of optical anisotropy and magnetic birefringence and to reanalyze previously obtained exact analytic results for the force–extension relations based on the Dirac model.

Simulations of the Erythrocyte Cytoskeleton at Large Deformation. I. Microscopic Models

Three variations of a polymer chain model for the human erythrocyte cytoskeleton are used in large deformation simulations of microscopic membrane patches. Each model satisfies an experimental observation that the contour length of the spectrin tetramers making up the erythrocyte cytoskeleton is roughly √7 times the end-to-end distance of the tetramer in vivo. Up to modest stress, each brushy cytoskeletal network behaves, consistently, like a low-temperature, planar network of Hookean springs, with a model-dependent effective spring constant, k eff, in the range of 20–40 k B T/ s o 2, where T is the temperature and s o is the force-free spring length. However, several features observed at large deformation distinguish these models from spring networks: 1) Network dimensions do not expand without bound in approaching a critical isotropic tension ( 3 k eff ) that is a characteristic limit of Hookean spring nets. 2) In surface compression, steric interactions among the chain elements prevent a network collapse that is otherwise observed in compression of planar triangulated networks of springs. 3) Under uniaxial surface tension, isotropy of the network disappears only as the network is stretched by more than 50% of its equilibrium dimensions. Also found are definitively non-Hookean regimes in the stress dependence of the elastic moduli. Lastly, determinations of elastic moduli from both fluctuations and stress/strain relations prove to be consistent, implying that consistency should be expected among experimental determinations of these quantities.

Field-induced winding of chiral polymers

We propose a microscopic model of a chiral polymer chain with permanent transverse dipoles interacting with an external electric field. Its behaviour has been investigated by computer simulation in the limit of weak chirality. Large-scale (tertiary) helical winding induced along the field direction has been found above a threshold field Ec, and the helix parameters have been calculated as functions of the field strength. Below Ec there is no coherent helical structure of the chain conformation. We find a characteristic scaling of the threshold and the winding radius a with the chain bending modulus \(\varepsilon ,({E_c}/{k_B}T) \sim {\varepsilon ^{1/4}}\)and \(a \sim ({k_B}T/E)\varepsilon \).

Viscoelastic effects in three-dimensional microphase separation of block copolymers: Dynamic mean-field density functional approach

In the present paper, we extend the dynamic mean-field density functional method which describes microphase separation phenomena in polymer liquids, to account for viscoelastic effects. The effect of simple steady shear on polymer orientation and elongation is taken into account by adapting the polymer configurational distribution function. We propose a simplified model for polymer chains in a simple steady shear flow and show numerically that this model correctly reproduces expected conformational changes. The conformational effect is only of importance for high viscosity liquids and/or high shear rates.

Simulation of the glass transition in polymeric systems: Evidence for an underlying phase transition?

Abstract The bond fluctuation model of polymer chains on sc lattices with an energy that favours long bonds can describe the slowing down of supercooled melts that approach the glass transition in qualitative similarity with various experiments. In this paper we focus on the question of whether there exists a correlation length that increases to large values when the temperature is lowered towards the glass transition. Two types of analysis are presented: firstly density oscillations near hard walls become long range, and the resulting correlation length becomes larger than the gyration radius, secondly oscillations in the pair correlation function in real space also become long range, and a similar correlation length can hence also be deduced in the bulk, without recourse to surface effects. In contrast, a finite-size scaling analysis of the diffusion behaviour of chains in small three-dimensional lattices with periodic boundary conditions does not give evidence for this length, unlike earlier results fo...

Electro-deposition of polymer chains on an adsorbing wall: Density profiles and wall coverage

Growth of polymer density in an electro-deposition model of polymer chains on an impenetrable wall is studied on a two dimensional discrete lattice using a Monte Carlo simulation. Polymer-polymer repulsion and polymer-wall attraction for the adsorbing wall (along with the neutral and repulsive interactions) are considered in an external field. Effects of the field strength (B), temperature (T), and chain length (Lc) on the density profile of the polymer chains and wall coverage are investigated. The spatial density profile shows onset of oscillation near the wall at a characteristic field (Bc) which depends on chain length and temperature. In low field, adsorption-to-desorption transition at the wall appear on increasing the temperature (unlike neutral and repulsive walls). In high field regime, on the other hand, a non-monotonic dependence of coverage on temperature is observed with a maximum at a temperature (Tm) which increases on increasing B. The equilibrium value of the polymer density (pd) shows a power-law decay with the chain length, pd∼Lc−α, at the wall and in the bulk with corresponding values of the exponent αW and αB; these exponents differ substantially and depend on B, T, and Lc. The coverage decays monotonically with the chain length at a constant temperature and field.

Water-in-Oil Emulsions: Role of the Solvent Molecular Size on Droplet Interactions

We prepare monodisperse inverted (water-in-oil) emulsion droplets and observe their phase behavior in various types of nonaqueous continuous phases. By using linear polydimethylsiloxanes or linear alkanes of length n (monomer units), mixed at volume fraction φ with dodecane, we determine the boundary within the φ/n plane which describes the colloidal aggregation. In the limit of large n, the boundary is well predicted by a hard core depletion model of polymer chains. Deviations to this prediction become increasingly pronounced as n is reduced.

Adsorption of a macromolecule in an external field: An exactly solvable model with bicritical behavior

We introduce a model of an adsorbing polymer chain in an external field, which admits exact analytical solution for the partition function for a finite number of the units, $N.$ In the thermodynamic limit, the system has an isotropic and two ordered phases, exhibits continuous and discontinuous phase transitions, and has a bicritical point. We obtain exact expressions for the Landau free energy as a function of an order parameter in the vicinity of the first- and second-order phase transition lines, and compare them to the original theory. The Landau free energy is also calculated as a function of two independent order parameters in the vicinity of the bicritical point. The distribution of complex zeros of the partition function is found for both first- and second-order transitions. In the thermodynamic limit it is described by exact analytical expressions, allowing a comparison to the existing phenomenological results within the framework of the Yang-Lee-Fisher approach. An advantage of the model presented is that the partition function can be calculated analytically not only in the thermodynamic limit, but for finite $N$ as well. The ideas of finite-size scaling analysis are checked against the exact solution, in terms of both the functional form of the free energy and the $N$ dependence of the distribution of Fisher zeros.

Monte carlo calculations of second virial coefficients of chain molecules

AbstractMonte Carlo calculations have been performed for different types of chain molecules whose units interact through Lennard‐Jones potentials. From the averaged Mayer function, we have evaluated the intermolecular two‐body cluster integral, obtaining results for second virial coefficients. We have investigated the following points: a) the site modelization of alkanes by comparison of our results with gas phase data of different linear and branched alkanes and their mixtures. b) the prediction of interpenetration factors for flexible linear and star polymer chains in a good solvent (or excluded volume conditions). c) the determination of the theta point for a model of flexible polymer chains and the comparison of data for finite chains with theoretical predictions.

Semiflexible polymers in straining flows

We introduce a model for semiflexible polymer chains based on the integral of an appropriate Gaussian process. The stiffness is characterized physically by adding a bending energy. The degree of stiffness in the polymer chain is quantified by means of a parameter and as this parameter tends to infinity, the limiting case reduces to the Brownian model of completely flexible chains studied in earlier work. The calculation of the partition function for the configuration statistical mechanics (i.e., the distribution of shapes) of such polymers in elongational flow or quadratic potentials is equivalent to the probabilistic problem of finding the law of a quadratic functional of the associated Gaussian process. An exact formula for the partition function is presented; however, in practice, this formula is too complicated for most computations. We therefore develop an asymptotic expansion for the partition function in terms of the stiffness parameter and obtain the first-order term which gives the first-order deviation from the completely flexible case. In addition to the partition function, the method presented here can also deal with other quadratic functionals such as the “stochastic area” associated with two polymer chains.

Inertia tensor and size of a polymer chain

Guided by an approach used in the study of membranes we construct the partition function for a discrete Gaussian chain model of polymers. The Hamiltonian is , where describes the chain connectivity, is the inertia tensor, which gives the shape and size of the polymer chain, and tr denotes the trace. Here we consider only the size, , which is obtained from the partition function by differentiating with respect to the conjugate variable . The partition function is expressed in terms of the hypergeometric function , an orthogonal polynomial, where n is the number of monomers and is the ratio of and the coefficient in . The limit corresponds to the random coil, while the limit describes a compact object. We also study the excluded volume problem by discretizing Edwards' continuous self-avoidance term. We obtain dimensionally regularized expressions for the radius of gyration and the end-to-end distance. The short distance cut-off dependence of the continuous model is reproduced.

Tricritical behavior of an off-lattice flexible polymer model: Monte Carlo calculation of second virial coefficients

Simulation data of second virial coefficients have been obtained for an off-lattice model of linear polymer chains by means of Monte Carlo method with different values of the variable that describes intramolecular attractions between units. The data have allowed us to study the theta, or tricritical, region where binary interactions cancel out. These Monte Carlo results are quantitatively compared with numerical predictions from the tricritical renormalization group theory. With this purpose, we have employed the same values for the three-body and the cutoff theoretical parameters that were previously used to reproduce the mean size data.

PH-Sensitive hydrogels based on non-ionic acrylic copolymers

Hydrogels based on ethoxytriethyleneglycol monomethacrylate/methyl methacrylate copolymers were prepared by free radical polymerization at 70 °C in bulk with 2,2′-azobisisobutyronitrile as initiator. The swelling behaviour of the hydrogels was studied by immersion of the films in water and in buffered solutions at various pHs. The hydration process was followed gravimetrically measuring the water uptake of the films with the time of treatment. The kinetics fit adequately a Fickian law for reduced sorption coefficients, M t / M ∞, lower than 0.6. The values of the diffusion coefficient were lower at neutral pH than at acidic or alkaline pHs, and they seemed to be independent of the copolymer composition. The results obtained were consistent with the stress relaxation model of polymer chains predicted for a relatively fast hydration process. The swelling process is reversible and the polymeric films are stable after repeating cycles of hydration and desorption promoted by the change of pH in appropriate buffered solutions.

Modification of Continuum Chain Model of Surface-Interacting Polymers To Describe the Crossover between Weak and Strong Adsorption

Continuum models of flexible polymers give physically incorrect results in situations where polymers are “localized” to restricted regions of space. Limitations of continuum chain theories for confined polymers are illustrated for surface-interacting polymers where both lattice and continuum calculations can be performed exactly for ideal chains. This comparison reveals the details of the breakdown of the continuum chain model in the regime of strong polymer adsorption. As a remedy to this difficulty, we formally introduce a corrected continuum chain theory by taking the spatial dimension d to depend on the surface interaction in such a way that the lattice and continuum model calculations are required to be consistent for weak and strong polymer adsorption.

Semidilute and Concentrated Polymer Solutions near Attractive Walls: Dynamic Monte Carlo Simulation of Density and Pressure Profiles of a Coarse-Grained Model

Using a bead−spring model of flexible polymer chains, we study polymer adsorption from solutions onto attractive planar walls, varying both the strength of the adsorption potential e and the concentration of the solution over a wide range. Treating the case of good solvents, the profiles of density and pressure are computed and it is shown that thermal equilibrium between the adsorbed layer and the bulk solution is obtained. The case of a wall with purely repulsive potential under otherwise identical conditions is treated for comparison. It is shown that for the strongly adsorbing wall there is a pronounced layering, while a layered structure at the repulsive wall occurs only for high concentrations, and this layering is also much weaker. These features carry over to the profile of the total pressure as well. From the difference in the pressure components in the parallel and perpendicular directions we compute also the interfacial energy between the polymer film and the repulsive wall as a function of the...

Dewetting of thin polymer films adsorbed on solid substrates: A Monte Carlo simulation of the early stages

Using an off-lattice bead–spring model of flexible polymer chains containing N=32 beads under bad solvent conditions, thin films of polymer melts are simulated. The films are confined between two parallel plates, the upper plate being purely repulsive while the lower plate exerts a short range attraction on the polymer layer so that a dense thin film is adsorbed on this plate for large enough attraction strength ε. Then “quenching experiments’’ are simulated by suddenly reducing |ε| at time t=0 and monitoring the time evolution of the polymer film. While for large enough final values of |ε| only the density in the film decreases somewhat, but the film stays laterally homogeneous, for |ε| less than a critical value |εc| it is found that the film breaks up into droplets. The early stages of the time evolution of this process in the framework of a dynamic Monte Carlo simulation are studied both by recording the time dependence of the adsorbed amount, the average thickness of the layer, the distribution function of meansquare displacements, and with the help of snapshot pictures of the system configurations. Also equilibrium properties of the films are investigated including both collective properties such as density profiles and radial distribution functions, and single-chain properties such as parallel and perpendicular parts of meansquare gyration radii, in dependence on the adhesive strength |ε| of the substrate.

Spectrum of self-avoiding walk exponents

A short range interaction is incorporated into the self-avoiding walk (SAW) model of polymer chains by partitioning SAW's into equivalence classes of chain configurations having m nearest-neighbor contacts, and performing an energetically weighted averaging over these restricted SAW configurations. Surprisingly, there have been limited studies of the geometrical properties of 'contact-constrained' SAW configurations, which contrasts with the well studied unrestricted SAW"s. Accordingly, we generate Monte Carlo data for the total number of SAW configurations ${\mathrm{C}}_{\mathrm{n},\mathrm{m}}$ having a fixed number of contacts m for chains of length n on square and cubic lattices. Applications of the standard ratio method to the ${\mathrm{C}}_{\mathrm{n},\mathrm{m}}$ data shows a unique connectivity constant \ensuremath{\mu} (NAW), corresponding to neighbor-avoiding walks (m=0), and a 'spectrum' of \ensuremath{\gamma} exponents which depend on the contact number m. The asymptotic scaling of the number of contact-constrained SAW's is found to be similar to the number of lattice animals and random plaquette surfaces having a fixed cyclomatic index c and genus g, respectively. The existence of this common structure is promising for the development of an analytic theory of interacting polymers and surfaces.

Electron Paramagnetic Resonance and Proton Nuclear Magnetic Resonance Relaxations in Polyanilines

Magnetic resonance measurements, including nuclear magnetic resonance T1 and T1 and electron paramagnetic resonance T1 and T2e relaxation times are presented for polyanilines prepared according to a modification of the conventional synthesis and doping methods, showing a conductivity higher than that of standard HCl-polyaniline polymers. The results, obtained as a function of the doping rate, are interpreted in terms of one-dimensional diffusive motions of spin and charge carriers. High anisotropy in the spin diffusion rate is found. In the framework of the model of single metallic polymer chains, this leads to the conclusion that in our polyanilines the mechanism of conduction is more markedly one-dimensional. © 1997 John Wiley & Sons, Ltd.