Dense Polymer Melts Research Articles

Molecular-dynamics simulations of the thermal glass transition in polymer melts: α-relaxation behavior

We present Molecular Dynamics simulations of the thermal glass transition in a dense model polymer liquid. We performed a comparative study of both constant volume and constant pressure cooling of the polymer melt. Great emphasis was laid on a careful equilibration of the dense polymer melt at all studied temperatures. Our model introduces competing length scales in the interaction to prevent any cristallisation tendency. In this first manuscript we we analyse the structural properties as a function of temperature and the long alpha-relaxation behaviour as observed in the dynamic structure factor and the self-diffusion of the polymer chains. The relaxation can be consistently analysed in terms of the mode coupling theory (MCT) of the glass transition. The mode coupling critical temperature and the exponent gamma, defining the power law divergence of the alpha-relaxation timescale both depend on the thermodynamic ensemble employed in the simulation.

Small-Angle Excess Scattering: Glassy Freezing or Local Orientational Ordering?

We present Monte Carlo simulations of a dense polymer melt which shows glass-transition-like slowing-down upon cooling, as well as a build up of nematic order. At small wave vectors q this model system shows excess scattering similar to that recently reported for light-scattering experiments on some polymeric and molecular glass-forming liquids. For our model system we can provide clear evidence that this excess scattering is due to the onset of short-range nematic order and not directly related to the glass transition.

The diagonal bond method: A new lattice polymer model for simulation study of block copolymers

A new lattice model for Monte Carlo simulations of dense polymer melts, developed in the spirit of Verdier–Stockmayer algorithm on square and simple cubic lattices, is presented. By introducing diagonals of squares and cubes as bonds, the lattice model acquires a large number of configurations and wiggling local moves. While it maintains the excluded volume interactions of monomers, it allows bond crossings and phantom moves, which result in a high mobility of polymers. For an application, we carry out simulations of symmetric A–B block copolymer melts and observe a first-order transition. We also show the stretching of the chains, namely, the non-Gaussian character, as a function of temperature. A quicker evolution towards thermal equilibrium enables us to form an ordered tricontinuous double-diamond (OTDD) phase for linear A–B–C triblock copolymers and a new cylindrical phase for star A–B–C triblock copolymers.

Dynamics of Glassy Polymer Melts in Confined Geometry: A Monte Carlo Simulation

Dynamic properties of a dense polymer melt confined between two hard walls are investigated over a wide range of temperatures by dynamic Monte Carlo simulation. The temperature interval ranges from the ordinary liquid to the strongly supercooled melt. The influence of temperature, density and confinement on the polymer dynamics is studied by various mean-square displacements, structural relaxation functions and quantities derived from them (relaxation times, apparent diffusion coefficients, monomer relaxation rates), yielding the following results: The motion of the monomers and polymers close to the walls is enhanced in parallel, but reduced in perpendicular direction. This dynamic anisotropy strongly increases during supercooling and extends into the bulklike inner region of the film over a length scale which is larger than the bulk radius of gyration at low temperatures. However, the absolute freezing of the melt occurs in each layer at the same temperature for both the parallel and the perpendicular direction.

On the influence of hard walls on structural properties in polymer glass simulation

Structural properties of a dense polymer melt confined between two hard walls are investigated over a wide range of temperatures by dynamic Monte Carlo simulation using the bond-fluctuation lattice model. The temperatures studied vary between those of the ordinary liquid and those of the strongly supercooled melt. At high temperatures the configurational properties of the polymers close to the walls exhibit the following features : The density profiles of the monomers and of the end monomers are enhanced at the walls and decay toward the bulk value on the length scale of the bond. Both the bonds and the chains tend to align preferentially parallel to the wall. The radius of gyration is the largest length scale characterizing the extent of the interfacial region. When the melt is progressively supercooled, all features of the profiles and the tendency of parallel orientation with respect to the walls become more pronounced. For the strongly supercooled melt the perturbation introduced by the presence of the walls penetrates deeply into the bulk, decaying on a length scale which is larger than the radius of gyration.

Glass transition relaxations in 2D polymer melts — a Monte Carlo simulation study

The glass transition in a lattice model (bond-fluctuation model) of dense two-dimensional polymer melt has been studied using the Monte Carlo technique. Attention has been paid to the diffusive motion of the polymers in the system and its dependence on the size of the lattice. At any temperature, the melt is equilibrated so that the results correspond to those in the limit of an infinitely slow cooling rate. A striking size dependence is observed in the dynamical properties of the model, suggesting the presence of a characteristic length of strongly cooperative regions in the system at low temperature.

Hybrid Monte Carlo simulations of dense polymer systems

We investigate the efficiency of two recently introduced hybrid Monte Carlo algorithms for the off-lattice simulation of dense polymer melts. Their performance is assessed both with respect to the molecular dynamics method and new single-chain Monte Carlo techniques proposed recently. Our findings show that the long-time behavior of both hybrid algorithms offer the possibility of calibration with respect to real time.

Simulation of viscoelasticity in polymer melts: effect of torsional potential

Methods have been developed for the computer simulation of stress relaxation in dense polymer melts using non-equilibrium molecular dynamics. In previous applications of these techniques we have used chain models that incorporate various features of real polymers. In the present work we extend the formulation to include a torsional potential that hinders free rotation about backbone bonds of the chain. The principal effect of the torsional potential in the present context is to shift the temperature level of the relevant phenomena upward to more reasonable levels. The principal conclusions drawn from the present simulations, as from the previous simulations in which the torsional potential is absent, are as follows. (i) Except at very early times, the non-bonded interactions make the primary contribution to the deviatoric stress during the constant strain-rate loading process simulated. (ii) The results for the stress relaxation modulus for simulations at various temperature levels may be reduced to a single master curve by suitable time shift. (iii) The shift factors required to produce a single master curve are found to obey the WLF equation, with reasonable values for the parameters in that equation.

Structural aspects of a three-dimensional lattice model for the glass transition of polymer melts: a Monte Carlo simulation

The glass transition of a dense three-dimensional polymer melt is simulated by the bond-fluctuation model on a simple cubic lattice. Combining this model with a two-level Hamiltonian which favours long bonds, a competition between the energetic and topological constraints in the system is created which drastically reduces the tendency of the melt to crystallize and makes the measured quantities depend upon the cooling rate. Our study concentrates on those quantities which are sensitive to structural changes on all length scales in the melt and we address the following questions: How does the precise choice of the cooling rate influence the structure of melt? Does the melt preserve its high temperature fluid-like state when it vitrifies? To what extent does the underlying lattice of the bond-fluctuation model emerge in the calculated quantities? In order to answer these questions we determined the collective static structure factor, the pair-correlation function and the static structure factor of the polymer chains. All of these quantities emphasize that the melt retains its liquid-like, short-range structure when it is cooled through the glass transition. The onset of this freezing can most clearly be distinguished by the isothermal compressibility which we obtained by both the extrapolation of the collective static structure factor to small q-values and by the “block-analysis” method. The results from the isothermal compressibility suggest that the density fluctuations start to freeze in a temperature range around T ≈ 0.25 (the temperature is measured in units of an energy parameter ϵ of the Hamiltonian), which is compatible with our previous analysis of the local properties in this model.

Analysis of the incoherent intermediate scattering function in the framework of the idealized mode-coupling theory: A Monte Carlo study for polymer melts.

In this Monte Carlo simulation, we calculate the incoherent intermediate scattering function ${\mathrm{\ensuremath{\varphi}}}_{\mathit{q}}^{\mathit{s}}$(t) for a three-dimensional dense polymer melt after having made long relaxation runs in order to eliminate the history of the cooling procedure sufficiently. This function shows the signature of a two-step process in the temperature interval T\ensuremath{\in}[0.16,0.21] (the temperature is measured in units of an energy parameter introduced in the Hamiltonian of the model) whose time evolution was quantitatively analyzed in the framework of the idealized mode-coupling theory (MCT) within the \ensuremath{\beta}-relaxation regime. As a result of this analysis the temperature interval splits into high- and low-temperature parts. In the high-temperature part (T\ensuremath{\ge}0.19), the idealized theory accounts very well for the decay of ${\mathrm{\ensuremath{\varphi}}}_{\mathit{q}}^{\mathit{s}}$(t) over about three decades in time, whereas ${\mathrm{\ensuremath{\varphi}}}_{\mathit{q}}^{\mathit{s}}$(t) relaxes much faster than the idealized MCT anticipates in the low-temperature region (T0.19). Since this discrepancy between the idealized MCT and the simulation data can qualitatively be rationalized by taking hopping processes into account, we try to estimate the critical temperature ${\mathit{T}}_{\mathit{c}}$ from the fits with the idealized MCT, yielding ${\mathit{T}}_{\mathit{c}}$\ensuremath{\approxeq}0.150.

Monte Carlo simulation of the glass transition in three-dimensional dense polymer melts

We determine the incoherent intermediate scattering function φ s q ( t) for a three-dimensional dense polymer melt. This function shows the signature of a two-step process which was quantitatively compared to the idealized mode coupling theory (MCT) within the β-relaxation regime. A major result of this analysis is that the studied temperature interval splits in a high temperature part, where the idealized theory describes φ s q ( t) over about three decades in time, and a low temperature part, where it strongly overestimates the freezing tendency of the melt. Since one can qualitatively attribute this discrepancy between the idealized MCT and the simulation data to hopping processes, the critical temperature T c was estimated from the fits with the idealized MCT, yielding T c ≈ 0.150 (the temperatures is measured in units of the energy parameter of the model).

Dynamics near the glass transition in two-dimensional polymer melts: a Monte Carlo simulation study

The glass transition in a lattice model (bond-fluctuation model) of a dense two-dimensional polymer melt has been studied using the Monte Carlo technique. Temperature is introduced in this model by associating energies with the bonds connecting the monomers in the chains. This creates competition between the energetic and geometric constraints within the system at low temperatures and the system becomes frozen. The frozen monomers and the persistence of such states are investigated in terms of an appropriately defined autocorrelation function. Attention has been paid to the various relaxation processes at different length scales. In particular, the global relaxation by the diffusion of the polymers, the Rouse relaxation at the scale of the chain length and the relaxation of the above-mentioned autocorrelation function involving the frozen monomers have been studied. The system is equilibrated at various temperatures down to T=0.22 K and the results thus correspond to those in the limit of an infinitely slow cooling rate. No phase transition is observed in the temperature range studied. On all length scales the relaxation times exhibit a crossover from an Arrhenius behaviour at high temperatures to a behaviour closely fitted to the form tau approximately exp(A/T2) at low temperatures, where A is constant independent of the relaxation processes. The increase of the relaxation times does not seem to be fast enough to explain a finite-temperature transition observed on quenching the system.

Monte Carlo studies of static properties of interacting lattice polymers with the cooperative‐motion algorithm

AbstractIn this paper we review some applications of the cooperative‐motion simulation algorithm to dense polymer melts. The basic idea behind the algorithm and its implementation on a computer are discussed. Furthermore we show how to include intra‐and intermolecular interactions in the simulation. Exemplary results are presented concerning on one hand the static properties of athermal and semiflexible lattice polymers and on the other hand the phase behaviour of polymer mixtures and diblock copolymers.

The influence of the cooling rate on the glass transition and the glassy state in three-dimensional dense polymer melts: a Monte Carlo study

In this Monte Carlo simulation the authors study the glass transition of a dense three-dimensional polymer melt using a lattice model (a bond-fluctuation model on a simple cubic lattice) that was highly optimized for a vector supercomputer. Long bonds are energetically favoured to create a competition between bond energy and packing constraints, which prevents the melt from crystallizing when it freezes. The onset of this freezing can be monitored by the temperature variation of various static quantities that probe both the length scale of a bond vector, such as the mean bond length and mean energy per bond, and that of the whole chain, such as the radius of gyration. As the melt vitrifies, these quantities gradually become independent of temperature in a narrow range around T approximately=0.2 (the temperature is measured in units of an energy parameter, epsilon , introduced in the model Hamiltonian) and their value at low temperatures is strongly influenced by the cooling rate. It is thus possible to infer from these curves the cooling-rate dependence of the freezing temperature Tg. Despite the difference in the detailed dependence of Tg on the cooling rate, the extrapolated value TK coincides with the Vogel-Fulcher temperature T0, obtained from the temperature variation of the diffusion coefficient, within the error bars.

Stress relaxation in a polymer melt of freely-rotating chains

It has been generally assumed that the nonbonded interactions in a dense polymer melt make only an isotropic contribution to the stress. Recent computer simulations of stress relaxation in systems of idealized chain models have shown that this is not the case. In fact, the nonbonded interactions are primarily responsible for the anisotropic stress during a stress relaxation process. To check that this result is not due to the idealized character of the chain models, simulations are performed for a melt employing a more realistic model, namely that representing a freely-rotating chain. The parameters employed correspond to the freely-rotating chains (FRC) model melt studied in extensive simulations by Takeuchi and Roe. We find that the primary role of the nonbonded interactions remains unchanged. Simulations are performed at several temperature levels at constant pressure. It is found that the temperature–time equivalence principle applies to the simulated shear relaxation modulus. The required shift factor aT satisfies the WLF (Williams–Landel–Ferry) equation with fairly reasonable parameter values.

Simulation of Dense Polymer Systems in Two and Three Dimensions

Dense polymer systems are modeled by self- and mutually avoiding walks on lattices. Both simple models where the step length is one lattice spacing and more complicated models where the step length is distinctly longer and may fluctuate (“bond fluctuation model”) are discussed, and it is shown that the computer simulation of such models gives useful insight to understand the thermodynamic phase behavior and the relaxational dynamics of dense polymer solutions and polymer melts. The huge demands in computing power needed for a successful simulation of such systems can be covered by parallel computers such as the multitransputer facility of the University of Mainz.

Integral Equation Theory of Homopolymer Melts

Abstract A general theory is developed for the equilibrium structure of dense polymer melts. This theory is based on an integral equation approach developed by Chandler and coworkers for molecular liquids. We are able to construct a tractable formalism for the polymer problem by employing the fact that a polymer molecule in a melt is ideal. This leads to a set of integral equations for the intermolecular radial distribution functions. In the case of a long linear chain this set approximately reduces to a single integral equation which we have solved numerically for the case of Gaussian and freely-jointed chain intramolecular statistics with repeat units interacting via hard core repulsions. From this solution we obtained the radial distribution function, structure factor, and compressibility as functions of liquid density and degree of polymerization. Unlike the random phase approximately (RPA) approach of deGennes, the present theory allows for density fluctuations. These density fluctuations, which decay on a length scale comparable to a few monomer units, are crucial for the calculation of the structure factor and thermodynamic properties such as the equation-of-state of the polymer fluid. Generalizations of the present theory to include chain stiffness effects and attractive interactions are possible.

Static and dynamic properties of two-dimensional polymer melts

Dense linear two-dimensional polymer chains are simulated by the recently developed bond fluctuation method. The chain length and densities considered cover the crossover from the expanded single chain limit to a dense polymer melt. The chains completely segregate and follow a power law of the chain extension ∼N 2 ν, N being the number of bonds of the chains. Relaxation functions and mean square displacements display typical Rouse-like bahavior, with the distinction, that the prefactors for the various relaxation processes differ

Melted flux liquids in high-Tc superconductors

A theory of the entangled appears because of the weak interplanar couplings and high critical temperatures in these materials. Flux line wandering melts the conventional Abrikosov flux lattice over large portions of the phase diagram, and leads to a novel entangled vortex state. We suggest that a heavily entangled flux liquid could exhibit glassy behavior on experimental time scales, in analogy with viscoelastic behavior in dense polymer melts.

Intrinsic atomic-level stresses in polymer melts and networks

By means of the virial stress formula, the macroscopic stress tensor in a polymer system may be expressed as a sum of atomic-level stress tensors, with each of the latter associated with a single atom and with all tensors, macroscopic and atomic, referred to the same fixed laboratory reference frame. In order to gain insight into the interplay between the covalent and noncovalent interactions in such systems, we refer the atomic-level stress tensor associated with a given atom to a moving local frame which maintains a fixed orientation with respect to the covalent structure attached to that atom; we term this the intrinsic atomic-level stress. We compute, by the method of molecular dynamics, the intrinsic stresses for model polymer melts and networks based on systems of freely rotating chains. The noncovalent interactions are through a truncated Lennard-Jones potential which is taken as either purely repulsive or has an attractive portion. We observe that (i) the intrinsic atomic-level stresses are highly anisotropic, even in an equilibrium melt in which the macroscopic stress is isotropic; (ii) the intrinsic atomic-level stress is the same in a polymer melt and in the corresponding polymer network model, if the latter is only moderately deformed; (iii) the intrinsic stresses in dense polymer melts are very different in the presence of noncovalent interactions from those in ideal systems; (iv) the addition of an attractive portion to the noncovalent potential has large effects on the deviatoric part of the intrinsic stress, thus casting doubt on the utility of the van der Waals’ picture for such systems.