Rouse Modes Research Articles

The Hartree approximation in dynamics of polymeric manifolds in the melt

The Martin–Siggia–Rose functional integral technique is applied to the dynamics of a D-dimensional manifold in a melt of similar manifolds. The integration over the collective variables of the melt can be simply implemented in the framework of the dynamical random phase approximation. The resulting effective action functional of the test manifold is treated by making use of the self-consistent Hartree approximation. As an outcome the generalized Rouse equation of the test manifold is derived and its static and dynamic properties are studied. It was found that the static upper critical dimension, duc=2D/(2−D), discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, d̃uc=2duc, distinguishes between the simple Rouse and the renormalized Rouse behavior. We have argued that the Rouse mode correlation function has a stretched exponential form. The subdiffusional exponents for this regime are calculated explicitly. The special case of linear chains, D=1, shows good agreement with Monte-Carlo simulations.

Monte Carlo Simulation of Many-Chain Star Polymer Solutions

Many-chain systems of star polymer molecules in a good solvent have been investigated using a dynamic Monte Carlo algorithm on a simple cubic lattice. Several bead−jump moves are proposed for the central part of stars with three and four arms, in addition to the usual combination of bends, crankshafts, and terminal moves also employed for linear chains. Static properties such as the chain size, or the mean square radius of gyration, are calculated. It is shown that the ratio of the star size to that of a linear chain with the same number of units does not change with concentration. The collective scattering functions are also obtained for different concentrations, and the interchain structure factors obtained from these functions are compared with those corresponding to linear chains. The dynamics generated from the Monte Carlo trajectories yields Rouse modes which exhibit the expected dependencies on the mode number and chain length. Finally, the elastic and rotational relaxation modes of individual chains are shown to obey dependencies on the arm length and the number of arms that are compatible with the predictions of scaling theory.

Relaxation modes of molten polydimethylsiloxane

Although it is known that chain reptation with contour length fluctuations and tube constraint release are the relevant mechanisms of chain relaxation, only applications of the theory to specific polymers could allow one to determine the relative importance of these relaxation modes. With this purpose we study the relation between the shear relaxation modulus of entangled, linear and flexible polydimethylsiloxane (PDMS) samples with narrow molecular weight distribution (MWD) and some of their blends. The linear PDMS are synthesized by anionic polymerization. Binary blends having different concentrations of the linear PDMS with narrow MWD are also prepared. A theory developed by some of the authors has been proposed previously and is based on the double reptation mixing rule and a law for the relaxation time of the chains. Since, calculations in the theory must be carried out with rheometric data of linear viscoelasticity, the time-temperature superposition principle is used at a reference temperature of 20°C to obtain the and master curves. We conclude that the PDMS chains in a matrix with 1.5 relax predominantly by reptation while the tube renewal follows Rouse mode, which is proportional to chain molecular weight raised to the square power. It is also shown that dilution effects due to short chains start to be important in the binary blends of PDMS samples due to the presence of Zimm mode of relaxation, which is suggested by the theory for linear polymers of relatively high polydispersity.

Viscoelasticity of low molecular weight polymers and the transition to the entangled regime

The viscoelasticity of unentangled polystyrene melts has been investigated in terms of terminals parameters: zero-shear viscosity, steady-state compliance and relaxation spectrum. The Rouse model applies well for molecular weights lower than the average molecular weight between entanglements, providing that one takes into account the proper variations of the radius of gyration. Moreover, local motions at the scale of Kuhn segments have to be considered in order to describe correctly the relaxation modes intermediate between the terminal zone and the glassy plateau. On the other hand, reptation models are commonly used for describing the entangled regime. We propose an expression of the shear modulus which accounts not only for the terminal modes (reptation, tube length fluctuations and tube renewal), but also for the relaxation modes responsible for the plateau zone and the transition of the glassy plateau. A crossover region between the unentangled and untangled regimes is located around . When the molecular weight increases, a shift transfer of Rouse modes towards reptation modes occurs. That leads to a continuity of the expression of the shear modulus over the entire range of molecular weights.

Diffusion at Polymer/Polymer Interfaces Probed by Rheological Tools

Diffusion at the polymer/polymer interface was probed by small-amplitude oscillatory shear measurements carried out on polystyrene (PS)/polystyrene (PS) sandwich-like assembly as a function of the time of welding in the molten state. It was found that the dynamic complex shear modulus G*(t) at a fixed frequency increases with the time of contact in two time regimes. First G*(t) increases proportionally to t1/2 and then a second regime takes place where G*(t) increases proportionally to t1/4. At longer times, G*(t) tends asymptotically toward G* of pure polystyrene. The results were interpreted in terms of reptation theory and the time of transition between the two scaling law regimes was found to be in agreement with the time needed for the transition from the Rouse mode to the reptation mode. Special attention was given to the initial state of the polymer surfaces before contact by performing experiments on (i) freshly prepared surfaces, (ii) presheared samples, (iii) fractured samples, and (iv) corona-t...

Atomistic Molecular Dynamics Simulation of Polydisperse Linear Polyethylene Melts

Well-relaxed atomistic configurations of polydisperse, linear polyethylene (PE) melts, obtained with the end-bridging Monte Carlo algorithm, have been subjected to detailed molecular dynamics simulations in both the canonical (NVE) and microcanonical (NVT) ensembles. Three different systems have been investigated, characterized by mean molecular lengths C24, C78, and C156, and by the same polydispersity index I of about 1.09. Results are presented for the static and (mainly) dynamic properties of these melts at P = 1 atm and T = 450 K. The diffusion coefficient D, determined for various chain lengths, N, is in very good agreement with experimentally measured values. The friction coefficient ζD is extracted from D by invoking the Rouse model; it is seen to increase from a relatively small value characteristic of short alkanes to a chain-length-independent plateau, reached in a region of N = 60−80. The friction coefficient ζτ is also obtained by analyzing the decay of the time autocorrelation function for the normal modes Xp at various chain lengths; the values thus extracted are consistent with those obtained from D for N above 40. Although the decay of the autocorrelation function of the end-to-end vector is very well described by the Rouse model, individual Rouse modes show some deviation from theoretical predictions. Even for chains sufficiently long to be in the asymptotic ζ regime, only the first two normal modes fully conform to Rouse theory in terms of their squared amplitudes and correlation times. Zero-shear viscosities computed from ζD values by means of the Rouse model are in excellent agreement with available experimental data for N = 90.

Monte Carlo simulation of stretched exponential relaxation near the glass transition

The dynamical behavior of a supercooled polymer melt is studied by means of Monte Carlo simulation. The simulation uses a version of the bond-fluctuation (lattice) model in which long bond vectors are favored energetically. The expansion on the length scale of a bond also induces an increase of the chains' size and stiffness during supercooling. In a dense melt the tendency of an individual chain to stretch has to compete with that of all others. This competition between the internal energy of a chain and the density of the melt strongly slows down the structural relaxation at low temperatures. In order to analyze the dynamical behavior of this model different relaxation functions were calculated in a temperature region ranging from the normal liquid-like to the supercooled state of the melt. Among these relaxation functions are the end-to-end vector correlation function and the correlation function of the Rouse modes. Both functions exhibit a time-temperature superposition property at all times. The corre sponding superposition (relaxation) times can be fitted by a Vogel-Fulcher equation, yielding a common Vogel-Fulcher temperature of about T 0 ≈ 0.12–0.13. As in experiments, this temperature is smaller than the critical temperature of mode-coupling theory (T 0 < T c ≈ 0.15). In addition to this temperature dependence another characteristic feature of the correlation functions is that their relaxation is stretched. For the Rouse modes this stretching is unexpected from the Rouse theory. However, as the theory predicts, the modes remain statically uncorrelated for all studied temperatures.

Obtaining molecular-weight distribution information from the viscosity data of linear polymer melts

Based on the model of Bersted and Slee, and later Malkin and Teishev, both differential and integral methods were developed to determine the molecular-weight distribution (MWD) from viscosity data. The more sensitive differential method can detect small inflections in the viscosity data and convert these into MWD information. The integral method is, however, capable of handling moderately incomplete viscosity data. Combining self-consistent differential and integral approaches, we are able to resolve details of a MWD and quantify reasonably broad MWDs from many sets of limited viscosity data. These methods both have very short computation times. A reported overemphasis of the high-molecular-weight end of distribution is due to the excitation of Rouse modes during the rheological measurements at high frequencies, which masks the diffusive contributions of the low-molecular-weight components to the relaxation process.

A model of relaxation in supercooled polymer melts

We present a dynamical mean-field model for molecular motions in a supercooled polymer melt. A macromolecule is represented by a harmonic chain undergoing Brownian motion whose bead mobilities fluctuate between zero and a finite value. These fluctuations mimic the dynamic obstacles formed by the chain segments surrounding a given segment, whose effects become more pronounced as T decreases. The rate of these mobility fluctuations is determined self-consistently by equating it to the asymptotic long-time relaxation rate of the shortest-wavelength Rouse mode. The resulting fluctuating rate vanishes as c, the equilibrium fraction of mobile beads, approaches a threshold value c*. As c→c*, relaxation times become arbitrarily large, permitting the modeling of fluids as T approaches Tg. Calculations of autocorrelation functions of Rouse mode coordinates and of segmental mean-squared displacements are presented and compared to results from recent simulations of melts at low temperatures. The deviations from the Rouse model observed in the simulations are features of this theory.

Creep recovery of acrylate urethane oligomer/acrylate networks. 2. Investigation of different modes of molecular motion

The creep recovery and dynamic mechanical properties of acrylate urethane oligomer/acrylate networks were investigated. The retardation spectra LCR obtained from the creep recovery experiments were significantly different from the corresponding retardation spectra LDMA obtained from the dynamic mechanical measurements. The reduced frequency ω* dependence of LDMA and the relaxation spectra HDMA in the higher ω* region were approximately represented as LDMA ∝ω*−p and HDMA ∝ω*q, although LCR decreased faster than LDMA with an increase in ω*. The exponents q were close to ½ characterizing the Rouse modes in the systems containing an acrylate urethane oligomer of Mw = 5000 but less than ½ in the system containing an acrylate urethane oligomer of Mw = 12,000. For the latter systems, significant thermorheological complexity was observed. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2543–2550, 1997

Dynamics of polymer “isotope” mixtures: Molecular dynamics simulation and Rouse model analysis

We report results of a molecular dynamics simulation of an “isotope” mixture of polymer chains, which are represented by a standard bead–spring model, and whose two species differ only by their monomer masses. Detailed analysis of the Rouse modes shows that for sufficiently short (non-entangled) chains this system can be well described by the Rouse model. Each species is described by its individual monomeric friction coefficient, whose dependence on both mass ratio as well as mixing ratio is studied. The main effect of mixing is an acceleration of the slower chains and a slowdown of the faster ones, while both species remain dynamically different. Some microscopic insight into the mechanism is obtained by studying the short-time behavior of the monomeric velocity autocorrelation function. Studies in the slightly entangled regime (chain length up to N=150, where the typical entanglement chain length is Ne≈35) seem to further corroborate the hypothesis that the “tube diameter” of the reptation model is a quantity which results mainly from the static configurations, i.e., is an equilibrium thermal average. The usefulness of recently suggested analysis methods in this regime is briefly discussed.

Polymer liquid dynamics studied via dielectric normal mode spectroscopy

AbstractDielectric spectroscopy was applied to study dynamics of cis‐polyisoprenes (PI), used as type‐A probe, in blends with polybutadiene (PB) and in block copolymers with polystyrene (PS) of SI‐ and SIS‐type. For dilute high‐molecular weight (M) PI/low‐ M PB blends we identified Rouse mode with M2‐dependent relaxation time τ, while for low‐ M Pi/high‐ M PB blends, we identified pure reptation mode with M3 ‐dependent τ. In between τ ∞ Mα with the exponent α varying from 2 to 3 as MB was increased, as suggested by Graessley with constraint release via tube renewal mechanism. For the blends with the MW ratio MI/MB = 2.5, we found bulk polymer behaviour with τ ∞ M3.5, in which competition between pure reptation and tube renewal appear to be essential and the contribution of contour length fluctuation may be ruled out. For SI‐diblock copolymers between Tg(I) &lt; T &lt; Tg(S) we observed normal modes of I‐block chains tethered on rigid S‐domains. The mode distribution as judged from the dielectric loss ε” curves was dependent on the domain morphology, reflecting restricted motions of crowded I‐tethered chains. For SIS‐triblock copolymers normal modes became appreciable, even below their critical solution temperature, in the range of T &gt; Tg(S), exhibiting broadening due presumably to their micro‐phase‐separated structure. The relaxation mechanisms for such end‐capped I‐chains in SIS‐triblock copolymers could be junction hopping in those with isolated S‐domains but chain rotating in those with S continuous morphologies.

Dynamics of a Polymer Test Chain in a Glass Forming Matrix: The Hartree Approximation

In this paper the Martin-Siggia-Rose formalism is used to derive a generalized Rouse equation for a test chain in a matrix which can undergo the glass transition. It is shown that the surrounding matrix renormalizes the static properties of the test chain. Furthermore the freezing of the different Rouse modes is investigated. This yields freezing temperatures which depend from the Rouse mode index.

Dynamics of Polymer Melts above the Glass Transition: Monte Carlo Studies of the Bond Fluctuation Model

The bond fluctuation model on the simple cubic lattice with a bond-length dependent potential energy favoring long bonds exhibits a glassy freezing in as the temperature is lowered, many properties being qualitatively similar to experiment. The present paper studies the dynamical properties of the model (as they result from the “random hopping” algorithm), using configurations of undercooled polymer melts that have been carefully equilibrated by the “slithering snake” algorithm. In this way quantitatively reliable data can be obtained for distinctly lower temperatures than in the previous work on the dynamics of this model that used the random hopping algorithm for equilibration as well. If various mean-square displacements, the self-diffusion constant, autocorrelation functions of the end-to-end vector, intermediate incoherent scattering functions, and single chain dynamic structure factors are analyzed, it is found that the slowing down of relaxation times occurs according to the Vogel−Fulcher law. Within the uncertainties of the analysis the Vogel−Fulcher temperature of all these times is the same, 0.12 ≤ T0 ≤ 0.13. Particularly illuminating is an analysis of the Rouse modes, which supports this conclusion and furthermore yields compelling evidence for the time−temperature superposition principle. It is shown that no static cross-correlations develop between different Rouse modes in the undercooled melt.

Parameter-Free Theory for Stress Relaxation in Star Polymer Melts

We present a theory of stress relaxation in star polymer melts with no adjustable parameters beyond those measurable in linear melts. We incorporate the effect of higher Rouse modes on star arm retraction and the Colby−Rubinstein scaling of entanglement length within “dynamic dilution”. Our results for G‘‘(ω) compare well with experimental data, with excellent agreement in shape and within a factor of 2 in time and modulus scales.

Viscoelastic properties of amorphous polymers. 5. A coupling model analysis of the thermorheological complexity of polyisobutylene in the glass-rubber softening dispersion

Isothermal data of high molecular weight polyisobutylene obtained by mechanical measurements with a spectral range over eight decades and additional photon correlation measurements have found that there are three distinct viscoelastic mechanisms in the glass-rubber transition zone. Theoretical considerations have helped to identify these three mechanisms to originate separately from local segmental (α) modes, sub-Rouse (sR) modes, and Rouse (R) modes. The temperature dependences of the shift factors of these mechanisms, aT,α, aT,sR and aT,R, determined over a common temperature range are found to be all different. The differences in temperature dependences are explained quantitatively by the coupling model. The local segmental motion contributes to compliances ranging from the glassy compliance, Jg, up to 10−8.5 Pa−1. The sub-Rouse modes contribute in the compliance range, 10−8.5 ≤ J(t) ≤ 10−7 Pa−1. The Rouse modes account for the compliances in the range of 10−7 Pa−1 ≤ J(t) ≤ Jplateau, where Jplateau is the plateau compliance. The magnitudes of the bounds given here are only rough estimates. Shift factors, aT, obtained by time-temperature superpositioning of viscoelastic data taken in the softening transition over a limited experimental window are shown to be a combination of the three individual shifts factors, aT,α, aT,sR, and aT,R. Consequently, care must be exercised in interpreting or using the WLF equation that fits the shift factors of the entire softening dispersion, because the latter do not describe the temperature dependence of any one of the three viscoelastic mechanisms. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys, 35: 599–614, 1997

Molecular dynamics study of entangled hard-chain fluids

By applying efficient computational algorithms to the simplest off-lattice polymer model–the freely-jointed tangent hard-sphere chain–we have been able to perform molecular dynamics simulations long enough to probe chain dynamics in the entangled regime. Chain lengths range from 8 to 192 segments while volume fractions range from 0.3 to 0.45. Analysis of the mean-square displacement (MSD), Rouse modes, scattering functions, and end-to-end vector correlations provides information about chain motion. Chain dynamics are compared with predictions of the Rouse model for short chains and the tube model of Doi and Edwards for long chains. The mean-square displacement for the inner segments of the longest chains are consistent with predictions of the tube model, reproducing the three scaling regimes that are postulated to occur. In addition, anomalous diffusive behavior in the atomic MSD of the inner segments is observed at long times as the inner segments cross over into the free diffusion limit. Rouse-mode autocorrelation functions decay non-exponentially and do not exhibit scaling consistent with the tube model. Definitive plateau-like behaviors are observed in the density–density correlations, normal coordinate decay, and end-to-end vector relaxation of the 192-mer fluids at the highest density.

Synthesis and Dynamic Rheological Behavior of Polybutadiene Star Polymers

Polybutadiene star polymers, of nominal functionality 3, 4, 8 and 12, all with arm lengths of approximately 30 000 g mol-1, have been synthesized using chlorosilane coupling agents. Dynamic mechanical studies have been carried out using parallel-plate rheometry at various temperatures, and the data frequency−temperature shifted to produce master curves. Fits to the Ball−McLeish constraint release theory indicate an effective entanglement molecular weight somewhat higher than the literature value for a linear polymer. This has been interpreted by considering a dilution of constraints effect caused by the Rouse diffusion of the terminal section of the star arm, leading to a predicted modified entanglement molecular weight similar to that found experimentally. Evidence for the Rouse-like behavior of the terminal section has been obtained from deuterium NMR spectra of a selectively deuterated sample. The presence of small amounts of residual linear chains is shown to be responsible for a further dilution of entanglements. An extension of the theoretical fits to include the high-frequency Rouse mode spectrum shows a divergence between theory and experiment which is indicative of the onset of the glass transition.

Probe rotation near and below the glass transition temperature: Relationship to viscoelasticity and physical aging

AbstractReorientation times ťc for two probes in several amorphous polymers near the glass transition temperature Tg are reported. Tg for these polymers ranges from 205 to 459 K. Probe reorientation was measured in the time window from 10−2 to 104 s with a recently developed photobleaching method. ťc for a given probe at the Tgs of the different polymers varies more than three decades. Viscoelastic relaxation times characteristic of the Rouse modes of the matrix polymers are closely related to probe rotation times and thus also not constant at Tg. The characteristic length scale of motions responsible for the glass transition varies significantly for the three polymers studied. Preliminary physical aging results indicate that probe reorientation in polystyrene ages slightly faster than the volume.

Theory for long time polymer and protein dynamics: Basis functions and time correlation functions

We develop methods for alleviating the major impediment in the extension to larger and more complex systems of our matrix method theory for describing the long time dynamics of flexible polymers and proteins in solution. This impediment is associated with the enormous growth in size of the required basis set with the addition of higher order mode coupling basis functions, which are needed to describe the influence on the dynamics of the ‘‘internal friction,’’ or equivalently of the memory function matrices. We use the first order eigenfunctions (the generalized Rouse modes) to construct an approximate mode coupling basis. Specific applications are made to united atom models of alkanes with a white noise structureless solvent, where the theory is compared with Brownian dynamics simulations to provide a no-parameter stringent test of the theory. Good convergence is found to the full second order treatment with the new basis set whose size scales more nearly with the size of the system rather than the cube of the system with the previous full basis. These technical improvements enable us to test the need for third order contributions to the dynamics of the longer alkanes and to compute the orientational time correlation functions probed by fluorescence depolarization and NMR experiments. Additional symmetry considerations provide further reductions in the required basis set sizes.