Phantom Chains Research Articles

Conformations and Structures of Poly(oxyethylene) Melts from Molecular Dynamics Simulations and Small-Angle Neutron Scattering Experiments

An ensemble of H(CH2OCH2)12H chains has been studied by molecular dynamics simulations as both melt chains and unperturbed phantom chains as a model system to investigate condensed phase effects on chain conformations of poly(oxyethylene) (POE). In addition, conformations of high molecular weight POE chains in the melt have been determined by small-angle neutron scattering (SANS) experiments over a temperature range of 347−459 K. Our simulations show that POE chains in the melt are more extended than the phantom chains which represent the unperturbed chains in ϑ solution. Moreover, the melt chains exhibit a negative temperature coefficient of chain dimensions in contrast to a positive value for the phantom chains. The difference in chain dimensions and the difference in the temperature dependence of chain dimensions between melt and phantom chains are corroborated by the results of our SANS measurements when they are compared with experimental results for POE chains in ϑ solution. We attribute these signi...

Microscopic Nonaffine Deformation of Polydisperse Polymer Networks

The stress-strain behavior of polydisperse networks is investigated using the maximum entropy principle. The network is described as a gas of phantom chains, and the deformation is taken into account by macroscopic constraints. Thus, nonaffine deformation on a microscopic scale is allowed and occurs. The main experimental features are reproduced, without introducing parameters additional to the strand number. In addition, the length distribution of network strands in a polymerization reaction is calculated. An exponential distribution of chain lengths is found.

Modelling the elastic behaviour of real chains in polymer networks

AbstractMonte‐Carlo simulations of the elastic behaviour of PDMS networks, using a realistic (R‐I‐S) network‐chain model, are able to reproduce experimentally observed deviations from Gaussian network behaviour in uniaxial extension. The finite extensibility of the network chains is shown to cause non‐affine deformation of the mean‐square network‐chain end‐to‐end distance, even at moderate sample deformations (λ≈1.5). An increase in the proportion of fully‐extended chains with increasing macroscopic strain gives rise to a steady decrease in the rate of network free‐energy change, causing a reduction in the network modulus at moderate macroscopic strains. There is no need to invoke a transition from affine to phantom chain behaviour as deformation increases.

Molecular modelling of the elastic behaviour of polymer chains in networks. Comparison of polymethylene and poly(dimethylsi1oxane)

It has been shown recently that Monte Carlo (MC) simulations of the elastic behaviour of polymethylene (PM) and poly(dimethylsiloxane)(PDMS) networks, using a realistic rotational-isomeric-state (RIS) network-chain model, are able to reproduce experimentally observed deviations from Gaussian network behaviour in uniaxial extension. An increase in the proportion of fully extended chains with increasing macroscopic strain gives rise to a steady decrease in the rate of network Helmholtz energy change, causing a reduction in the network modulus at moderate macroscopic strains. There is no need to invoke a transition from affine to phantom chain behaviour as deformation increases. In the present paper, the MC simulation method is used to compare the room-temperature behaviour of PM and PDMS network chains. Differences are interpreted in terms of the end-to-end distributions and chain-flexibilities. The stiffer PM chains generally show larger deviations from Gaussian behaviour.

Conformational analysis of double-tethered chain molecules at an impenetrable interface: A Monte Carlo study

Due to tremendous interest in surface modification, numerous theoretical and simulation efforts have sought to elucidate the conformational properties of chain molecules grafted, or tethered, at one end to an impenetrable interface. Few studies have, however, addressed the analogous properties of chains in which both ends are anchored to an interface and behave as ‘‘sticky feet’’ in the presence of a good solvent. In this work, on-lattice Monte Carlo simulations have been performed to examine the characteristics of double-tethered chains in the limit of low interfacial coverage. Phantom and self-avoiding chains possessing up to 200 monomers are considered in 2 D random-flight simulations, as are corresponding non- and single-tethered chains to facilitate property comparison. While single-tethered chains adopt a more extended conformation than their nontethered analogs, segmental density distributions reveal that the existence of a second tethered site exerts little influence on monomers near the chain origin, but promotes overall coil localization near the interface. Scaling relationships for chain dimensions such as the root-mean-square (rms) end-to-end distance and coil radius are derived here in terms of both monomer and tether number.

Analytical model for the microscopic nonaffine deformation of polymer networks

A general model for the description of the stress-strain behavior of polymer networks is introduced. We assume that the network consists of a gas of phantom chains. Its deformation is accounted for by macroscopic constraints, which allow nonaffine behavior on the microscopic scale. The advantage of the model lies in the fact that the number of assumptions is smaller than in other models and—at the same time—the main experimental features are reproduced. Comparison to mechanical deformation experiments shows good agreement with the model.

Influence of microscopic non‐affinity and functionality on the deformation of polymeric networks

AbstractA non‐affine model for the description of the stress‐strain behavior of polymer networks is introduced. We assume that the network consists of a gas of phantom chains. The deformation of the chains is taken into account by macroscopic constraints, where the macroscopic deformation is affine and the microscopic one non‐affine. In contrast to existing models, the number of parameters is reduced; nevertheless the main experimental features are reproduced.

Kinetics of chromosome condensation in the presence of topoisomerases: a phantom chain model

We discuss the requirement of type II DNA topoisomerase in the process of mitotic chromosome condensation. Using a known model describing the collapse of homopolymers, we propose that the compaction process necessitates a change in the topological state (i.e., a self-knotting) of the chromosomal chain. We argue that the enzymes are necessary to reach the compact metaphase state in a time interval that is much smaller than the time expected in the uncatalyzed process. The folding process is such that the potential entanglement points are localized at particular regions of the chromosome known as the scaffold-associated regions. The concentration of entanglements in the metaphase chromosome is related to the average size of the radial loops. A phantom chain model for the condensation process, in which each potential entanglement point is dealt with by a topoisomerase II molecule, is proposed.

Monte carlo simulation of polymers in steady potential flows

AbstractThis work presents a study of the configurational properties of bead‐spring chains in steady‐state potential flows. These properties are obtained from off‐lattice Monte Carlo simulations. The results of our simulations compare favourably with theoretical results for phantom chains. We investigate the effects of excluded volume and hydrodynamic interaction on the configurations of the chains, and compare our results to the predictions of scaling arguments. The scaling law observed here for the latter case deviates from theoretical predictions but is in agreement with findings reported from molecular dynamics simulations.

Methods of generating dense relaxed amorphous polymer samples for use in dynamic simulations

A comparison is made between different methods of generating relaxed dense amorphous systems of model linear polymer chains. One of them, referred to as phantom chain growth (PCG), has many advantages when compared with self-avoiding random walk procedures. In PCG chains are grown under non-excluded volume conditions such that successive torsion angles in a chain occur with a probability which is related to immediately adjacent torsion angles only. A series of experiments on a polyethylene-like designed to test the effect of preparational history and heat treatment on the relaxation of the amorphous samples is reported

Chain expansion in the Θ state

The results of a statistical-conformational investigation of real polymer chains in their theta state are reported. The two following issues are particularly pointed out: (i) whether approaches based on the usual two- and three-body interaction model predict expansion or contraction with respect to the ideal phantom chain, for varying chain lengths; (ii) whether or not significant repulsive forces not reducible to the usual two- and three-body interactions exist, which may significantly expand the chain beyond the phantom size

The umbrella shape of star polymers in the theta state

The chain conformation in the theta state is calculated by self-consistent free-energy minimization assuming that the overall long-range two- and three-body interactions mutually compensate. However, the repulsive screened interactions arising from the intrinsic chain thickness do survive, imparting an asymptotically finite expansion to the chain. Because of the medium-range character of these interactions, the proportionality between the unperturbed mean-square radius of gyration 〈S2〉0 and the molecular weight is asymptotically maintained. Compared with the phantom chain, devoid of both intrinsic thickness (i.e. screened interactions) and long-range interactions, the arms experience an increasing expansion upon increasing the number of arms, f. At the same time, the fluctuations of the branch point around the centre of mass do increase as well, since the average angle between the arms decreases, and the macromolecule tends towards an umbrella-like shape. As a consequence, the macromolecular density becomes more uniform than for the phantom chain. This is reflected in the calculated structure factor S(Q) by a sharpening of the peak in the plot of Q2S(Q) versusQ (where Q = 4π sin(θ2)λ), in good agreement with the experimental results by Huber et al. Through cancellation of contrasting effects, the topological index g = 〈S2〉0〈S2〉0,linear is close to the value of the phantom chain for any f, whereas h =RH0RH0,linear, RH being the hydrodynamic radius, and ϱ = 〈S2〉120RH0 are somewhat larger. The negligible influence of the residual three-body interactions in the theta state compared to the screened interactions is finally commented upon considering also recent Monte Carlo simulations.

Theoretical Equilibrium Moduli and Swelling Extents for Elastomers Crosslinked in Solution

Abstract Equilibrium elongation moduli and swelling properties have been calculated for amorphous polymer networks prepared in a diluted reference state of polymer volume fraction υ2C. The three molecular elasticity theories employed are based on (i) an idealized affine deformation model (in which the constraints on junction fluctuations are infinite), (ii) an idealized phantom chain model (in which the constraints are absent), and (iii) the more realistic Flory-Erman theory (in which the constraints vary with crosslinking conditions, degree of swelling, and extent of elongation). A discussion of the dependence of the constraint parameter κ on υ2C is given, and is applied to the calculation of stress-strain isotherms in elongation specifically for cis-polyisoprene cured with dicumyl peroxide. The experimental dependence of κ on υ2C investigated by Erman and Mark was found to be similar to the dependence of the plateau modulus on dilution, and this is consistent with the constrained contour model proposed by Erman and Monnerie.

Polydisperse polymer networks : elasticity, orientational properties, and small angle neutron scattering

The properties of polymer networks may be expected to depend upon the distribution, f(S), of chain lengths between crosslinks. We develop a formalism for treating phantom networks which allows the explicit inclusion of an arbitrary f(S) distribution. In bimodal networks we investigate chain orientation in the strained network by calculating and for the long and the short chains separately. The responses of the two species differ markedly from each other, although it is found that the overall network birefringence is dependent only on the mean chain length. Similarly, although the typical degree of extension of the chains upon straining the network differs widely between the species, the network modulus is formally independent of f( S). We investigate scattering from a long labelled chain crosslinked in a network at many randomly-positioned points, and explicitly account for the Poisson distribution of lengths between crosslinks. The scattering function is calculated exactly if the simplification of affine deformation of the junction points is made, and if the junctions are allowed to fluctuate freely, then an approximate solution is possible. The former model (Polydisperse Junction Affine model) agrees much better with the experimental data than the existing theories, both as regards the shape of the Kratky plot and the iso-intensity contours. There remain, however, features of the data which cannot be reproduced by any of the models considered. It is shown by the calculation of limit curves that these differences can only be accounted for by a treatment which goes beyond the idea of phantom chains.

Orientation‐strain behaviour of crosslinked rubbers

AbstractProcedures for the evaluation of quantitative orientation parameters using wide‐angle x‐ray scattering data are outlined. Data are presented for crosslinked natural rubber. a copolymer of polyethylene and polyvinylacetate and for polymethylmethacrylate. These quantitative orientation data are compared to the predictions of models of rubber elasticity. Critical comparisons are made with affine and phantom chain models and the variable network constraint model recently introduced by Erman and Flory. The effects of entanglements and segmental interactions are also considered. An approach is introduced which allows the evaluation of the level of intersegmental interaction contribution to the orientation‐strain behaviour.

Topological theory of entanglement (two body topological interaction approximation). I. Friction coefficients of the center of polymer chains

The topological theory of entanglement presented in the previous papers [J. Phys. Soc. Jpn. 37, 1413, 1423, 1429 (1974)] is generalized to a system composed of N three dimensional (D3) linear polymer chains and a system composed of a two dimensional (D2) linear polymer chain and N fixed barriers. The fundamental assumption of the present theory is that entanglement states of the systems are described by a set of (slightly modified) Gauss linking coefficients, ea, each of which represents topological states of two D3 polymer chains or of a D2 polymer chain and a point. These ea’s are called ’’(two body) entanglement coordinates.’’ The Verdier–Stockmayer model is used for representing the Brownian motion of the polymer chains, and diffusion equations are derived for simultaneous distribution functions in regard to the entanglement coordinates {e} and the Rouse coordinates {q}. It is found that the previous theory leads two contradictions: (i) When it is applied to the so-called phantom chains, nonzero interactions, which must be essentially zero, appear among the D3 chains or between the D2 chain and the barriers. (ii) The friction coefficient of the center of the polymer chains derived before diverges in the present systems, when N is infinity. These contradictions are removed by considering the non-Fokker–Planck terms explicitly in the diffusion equations, and nondivergent expressions of the friction coefficient are derived.

Graph-like state of matter: 12. Static and dynamic entanglements

It is shown that the contribution of chain entanglements to the equilibrium shear modulus, recently deduced theoretically and measured experimentally, cancel almost quantitatively the downward adjustment in the front-factor g of the equation of state used in such theories. This downward revision of g is contested theoretically on algebraic grounds, and the classical value of unity is recovered as a result. This classical value is supported by reinterpreting literature data on silicone rubbers by Valles and Macosko, and by new measurements (which confirm less accurate ones obtained earlier by a more difficult method) on a well-tried polyester system near its gel point. There are indications that the contributions of entanglements to the equilibrium shear modulus are all too easily overestimated by theory, and the technologist is advised to prefer the remarkably simple formulae of the graph-like-state theory of phantom chains. Dynamic entanglements can readily explain qualitatively the anelastic effects here reported, viz. frequency shifts in the plot of the real part of the dynamic shear modulus against conversion in the range α α c ⩽ 1.01 (where α c is the critical conversion at the gel point). Here too it seems as if graph-like-state theory will ultimately account for the observations, but at least certain eigenvalue problems will have to be solved first. An intriguing observation, that at the gel point of the polyester system the spectrum of relaxation frequencies approaches a block distribution, so that every finite value is equally likely, may have general validity.

Bulk viscosity of polymer solutions

The theory of the bulk viscosity of polymer solutions is considered for cases of low and moderate concentrations of phantom chains under ϑ conditions. It is shown that the familiar preaveraging approximation for the hydrodynamic interactions imposes the constraint that the polymer contribution to the bulk viscosity is minus one-third of its contribution to the shear viscosity, thereby providing an experimental means for testing this widely used approximation. Particular application is made to the theory of sound attenuation in polymer solutions as one potential means for measuring the bulk viscosity. At higher frequencies the longitudinal (sound) modes of the solvent introduce an additional exponentially decaying contribution to the hydrodynamic interactions which leads to a renormalization of the relaxation times of the polymer.

Excluded volume effect on polymer dynamics in concentrated solutions

We consider the dynamics of polymers in concentrated solutions under the combined effects of excluded volume interactions and hydrodynamical interactions (including all inter- and intrachain effects). Exact dynamical equations are derived which describe the motion of a single chain with screened hydrodynamics in the presence of the instantaneous excluded volume field of all other chains, an exact equation representing the precursor to all reptationlike models. Analogous equations are also obtained for the dynamics of a pair of correlated chains, e.g., ones involved in a temporary entanglement in a concentrated system. For the uncorrelated case of phantom chains, Edwards’s results for the excluded volume effect on equilibrium properties are incorporated into the averaged dynamics to determine the excluded volume effect on the polymer viscosity and the effective bead friction coefficient in concentrated solutions. Experimental viscoelastic data for concentrated solutions and melts are shown to provide a basis for the use of phantom chain models when the polymer chain lengths do not become too large. The effective bead friction coefficient is used in an accompanying paper, along with the dynamical equations derived here, to study quasielastic neutron scattering cross sections from concentrated polymer solutions.

The theory of rubber elasticity

This paper attempts to improve several weaknesses in the classical theories of rubber elasticity. It develops a formulation of the statistical thermodynamics of amorphous materials analogous to the Gibbs formalism for conventional statistical mechanics. This then permits the replacement of ‘phantom chains’, i.e. long polymer molecules with the fictitious property that they experience no forces except at cross link points and are transparent to one another, by realistic molecules which do experience forces and which can become entangled. The crosslinked points are no longer assumed to deform affinely with the gross behaviour of the solid. Under the simplest conditions forms like the classical are recovered but with a different coefficient, and the term representing the degrees of freedom lost by crosslinking, over which the classical theories are in dispute, is found to lie between the previous values in a formula which can reproduce the classical results by making different assumptions. The entanglements give rise to more complicated forms than the classical sum of squares of strain ratios, which under certain circumstances can reproduce the Mooney-Rivlin term which when added empirically to the free energy usually improves the fit with experiment. The general expression is complicated, but is nevertheless an explicit function of the density of crosslinks, the density of the rubber and the interchain forces.