Normal Mode Relaxation Times Research Articles

Dissipative particle dynamics simulation of dilute polymer solutions—Inertial effects and hydrodynamic interactions

We examine the accuracy of dissipative particle dynamics (DPD) simulations of polymers in dilute solutions with hydrodynamic interaction (HI), at the theta point, modeled by setting the DPD conservative interaction between beads to zero. We compare the first normal-mode relaxation time extracted from the DPD simulations with theoretical predictions from a normal-mode analysis for theta chains. We characterize the influence of bead inertia within the coil by a ratio Lm/Rg, where Lm is the ballistic distance over which bead inertia is lost, and Rg is the radius of gyration of the polymer coil, while the HI strength per bead h* is determined by the ratio of bead hydrodynamic radius (rH) to the equilibrium spring length. We show how to adjust h* through the spring length and monomer mass, and how to optimize the accuracy of DPD for fixed h* by increasing the friction coefficient (γ ≥ 9) and by incorporating a nonlinear distance dependence into the frictional interaction. Even with this optimization, DPD simulations exhibit deviations of over 20% from the theoretical normal-mode predictions for high HI strength with h* ≥ 0.20, for chains with as many as 100 beads, which is a larger deviation than is found for Stochastic rotation dynamics simulations for similar chains lengths and values of h*.

Dynamic structure of unentangled polymer chains in the vicinity of non-attractive nanoparticles

Using coarse-grained molecular dynamics simulation, we study the motion of unentangled polymer chains dynamically confined by non-attractive nanoparticles (NPs). Both normal mode and dynamic structure factor S(q, t) analysis are adopted to analyze chain's dynamics. Relaxation behaviors of chains are found to be significantly slowed down by NPs. The relaxation times of chain's normal modes are monotonically increasing with the NP volume fraction ϕ. At the same time, chains' dynamics are becoming non-Gaussian. Inspection of S(q, t) reveals that chain's dynamics can be attributed to two 'phases', a bulk polymer phase and a confined polymer phase between NPs. The dynamics of a confined polymer is slower than that of a bulk polymer, while still exhibiting high mobility. The amount of the bulk polymer phase is found to exponentially decay with increasing ϕ. With this figure at hand, we establish a simple relationship between NP and confined/interphase polymer volume fractions. This work seems to provide the first quantitative prediction on the relationship between NP and confined/interphase polymer volume fractions.

Dielectric Chain Dynamics of Side-Chain Liquid Crystalline Polymer.

Both chain dynamics and statistics of side-chain liquid crystalline polymer are experimentally explored. Dielectric measurements over a wide range of frequencies and temperature windows demonstrate that the chain dynamics in the liquid crystalline phase (SmA) is retarded and has higher apparent activation energy compared to that in isotropic melt. The molecular weight dependence of the normal mode relaxation time in the isotropic melt conforms to the Zimm model with excluded volume effect, while it shows Rouse behavior in the liquid crystalline phase. The mean squared end-to-end distance of polymer chain in the liquid crystalline phase decreases compared to that in the isotropic melt. The main chain takes a self-avoiding walk with ⟨Re2⟩ ∼ N2ν, ν = 0.54-0.6 in the isotropic melt, whereas a random walk between smectic layers with ν ≅ 0.5, consistent with the results from chain dynamics.

Thermodynamic scaling of α-relaxation time and viscosity stems from the Johari-Goldstein β-relaxation or the primitive relaxation of the coupling model

By now it is well established that the structural α-relaxation time, τ(α), of non-associated small molecular and polymeric glass-formers obey thermodynamic scaling. In other words, τ(α) is a function Φ of the product variable, ρ(γ)/T, where ρ is the density and T the temperature. The constant γ as well as the function, τ(α) = Φ(ρ(γ)/T), is material dependent. Actually this dependence of τ(α) on ρ(γ)/T originates from the dependence on the same product variable of the Johari-Goldstein β-relaxation time, τ(β), or the primitive relaxation time, τ(0), of the coupling model. To support this assertion, we give evidences from various sources itemized as follows. (1) The invariance of the relation between τ(α) and τ(β) or τ(0) to widely different combinations of pressure and temperature. (2) Experimental dielectric and viscosity data of glass-forming van der Waals liquids and polymer. (3) Molecular dynamics simulations of binary Lennard-Jones (LJ) models, the Lewis-Wahnström model of ortho-terphenyl, 1,4 polybutadiene, a room temperature ionic liquid, 1-ethyl-3-methylimidazolium nitrate, and a molten salt 2Ca(NO(3))(2)·3KNO(3) (CKN). (4) Both diffusivity and structural relaxation time, as well as the breakdown of Stokes-Einstein relation in CKN obey thermodynamic scaling by ρ(γ)/T with the same γ. (5) In polymers, the chain normal mode relaxation time, τ(N), is another function of ρ(γ)/T with the same γ as segmental relaxation time τ(α). (6) While the data of τ(α) from simulations for the full LJ binary mixture obey very well the thermodynamic scaling, it is strongly violated when the LJ interaction potential is truncated beyond typical inter-particle distance, although in both cases the repulsive pair potentials coincide for some distances.

Segmental and Normal Mode Relaxation of Poly(alkylene oxide)s Studied by Dielectric Spectroscopy and Rheology

A series of poly(alkylene oxide)s (PAO’s) with different lengths of the side chains and two different molecular weights were studied by means of differential scanning calorimetry, dielectric spectroscopy (BDS), and rheology. The glass transition temperatures are almost independent of molar mass and the length of the side chains, which is also reflected by the dielectric α-relaxation, which depends only slightly on the different samples. However, the corresponding maxima in the dielectric loss of the samples with longer side chains show a shoulder at high frequencies with a temperature-dependent shape. The shape of the normal mode changes slightly with the molar mass and the length of the side chain. The normal mode relaxation times show a molar-mass-dependent behavior indicating a significant increase of the entanglement molecular weight Me with increasing length of the side group. A Rouse mode analysis incorporating the molar mass distribution was performed but could not reproduce the experimental data of the normal mode process completely. The relaxation times obtained by rheology agree well with the dielectric results, and estimates of Me based on the number of entanglements and on the packing model confirm the values suggested by BDS. A comparison of the estimated chain dimensions with literature data supports the reliability of our calculations.

On the Universality of Chain Dynamics

On the basis of a comparison of the dielectric normal mode relaxation times τ n of unentangled cis-1,4-polyisoprene, poly(propylene glycol), and polyoxybutylene with literature data for the shear viscosity n of polystyrene, atactic polypropylene, and polycarbonate from various sources, Ding and Sokolov [Macromolecules 2006, 39, 3322] concluded that the Tg-normalized temperature dependence of τ n and η is the same for all polymers. Liu et al. [Macromolecules 2006, 39, 8867] reached the same conclusion from an examination of their own dynamic mechanical modulus data for 1,4-polybutadiene, polyisobutylene, polystyrene, polycarbonate, and eight aliphatic copolymers. We verify that the data in these studies ostensibly support a universal Tg-normalized temperature dependence; however, a detailed analysis of viscoelastic data on many different polymers, from creep compliance measurements carried out in the same laboratory, shows unambiguously that the Tg-normalized temperature dependences of τ n and η can vary substantially among polymers. Thus, the behavior of τ n and η is not universal.

Molecular dynamics simulation of temperature and pressure effects on the intermediate length scale dynamics and zero shear rate viscosity of cis-1,4-polybutadiene: Rouse mode analysis and dynamic structure factor spectra

A well-relaxed atomistic configuration of a 32-chain C 128 cis-1,4-polybutadiene ( cis-1,4-PB) system has been subjected to long (on the order of a few microseconds) molecular dynamics (MD) simulations in the NPT ensemble using the united-atom forcefield introduced by Smith et al. [G. Smith, D. Bedrov, W. Paul, A molecular dynamics simulation study of the alpha-relaxation in a 1,4-polybutadiene melt as probed by the coherent dynamic structure factor, J. Chem. Phys. 121 (2004) 4961–4967] on the basis of quantum chemistry calculations. This allowed us to study the temperature and pressure dependences of the Rouse-mode relaxation spectrum of cis-1,4-PB over a wide range of temperature (ranging from T = 430 K down to 165 K) and pressure (from P = 1 atm up to 3.5 kbar) conditions. Results are presented for: (a) the time decay of the autocorrelation function of the normal coordinates (Rouse modes), (b) the single chain intermediate coherent dynamic structure factor, S coh( q, t), and (c) the intermediate incoherent dynamic structure factor, S inc( q, t), for different values of the wavevector q. By mapping our MD simulation results onto the Rouse model, we have been able to extract a prediction for the zero shear rate viscosity of the simulated cis-1,4-PB system as a function of temperature and analyze its fragile character. In agreement with our previous MD simulation studies on the same system [G. Tsolou, V.A. Harmandaris, V.G. Mavrantzas, Atomistic molecular dynamics simulation of the temperature and pressure dependences of local and terminal relaxations in cis-1,4-polybutadiene, J. Chem. Phys. 124 (2006) 084906-1-11] and in contrast to what is experimentally observed [see, e.g., G. Floudas, T. Reisinger, Pressure dependence of the local and global dynamics of polyisoprene, J. Chem. Phys. 111 (1999) 5201–5204; C.M. Roland, R. Casalini, T. Psurek, S. Pawlus, M. Paluch, Segmental- and normal-mode dielectric relaxation of poly(propylene glycol) under pressure, J. Polym. Sci. Part B: Polym. Phys. 41 (2003) 3047–3052], we predict that pressure and temperature influence practically similarly all normal mode relaxation times along the simulated C 128 cis-1,4-PB chain. Furthermore, our MD simulation results predict a transition from a homogeneous to a heterogeneous dynamical behavior in the region of wavevectors near the first (intermolecular) peak in the static structure factor, consistently with recent neutron scattering (NS) measurements [see, e.g., B. Frick, G. Dosseh, A. Cailliaux, C. Alba-Simionesco, Pressure dependence of the segmental relaxation of polybutadiene and polyisobutylene and influence of molecular weight, Chem. Phys. 292 (2003) 311–323; A. Arbe, J. Colmenero, B. Farago, M. Monkenbusch, U. Buchenau, D. Richter, Intermediate length scale dynamics in glass forming polymers: coherent and incoherent quasielastic neutron scattering results on polyisobutylene, Chem. Phys. 292 (2003) 295–309] and previous simulation studies [see, e.g., J. Colmenero, F. Alvarez, A. Arbe, Self-motion and the alpha relaxation in a simulated glass-forming polymer: Crossover from Gaussian to non-Gaussian dynamic behavior, Phys. Rev. E 65 (2002) 041804-1-12].

Dielectric Normal Mode Relaxation of Poly(propylene oxide) and Poly(butylene oxide) in Dilute and Semidilute Solutions

We investigated the dielectric normal mode relaxation in dilute and semidilute solutions of poly(propylene oxide)(PPO) and poly(butylene oxide)(PBO) in benzene which is a good solvent for PPO but a marginal solvent for PBO. The results indicate that the normal mode relaxation times τn in dilute solutions of PPO and PBO are proportional to [η]M in agreement with the Rouse-Zimm theory where [η] and M denotes the intrinsic viscosity and the molecular weight, respectively. With increasing concentration C, τn increases in proportion to C[η]. The dynamic crossover concentration between the dilute and semidilute regimes is about four times higher than the static crossover concentration C*. The relaxation strength Δε divided by C decreases with increasing concentration indicating that the mean square end-to-end distance <r2> decreases on account of the screening of the excluded volume effect. The concentration dependence of <r2> agrees approximately with the scaling theory.

Volume and Temperature Dependences of the Global and Segmental Dynamics in Polymers: Functional Forms and Implications for the Glass Transition

Previously, from dielectric relaxation measurements obtained for various temperatures T and pressures P on polypropylene glycol, 1,4-polyisoprene, and poly(oxybutylene) [Roland, C. M.; Casalini, R.; Paluch, M. J. Polym. Sci. Polym. Phys. Ed. 2004, 42, 4313. R. Casalini and C. M. Roland, Macromolecules, 2005, 38, 1779], both the primary α-relaxation time τα and the normal mode relaxation time τn were shown to yield master curves when plotted versus the quantity T-1V-γ . Moreover, the value of γ is the same for the two processes (here V is the specific volume and γ is a material-specific constant). Such a result appears to be consistent with an assumption underlying models for polymer viscoelasticity that the friction coefficient governing motions over large length scales can be identified with the local segmental friction coefficient. However, notwithstanding the superpositioning obtained using the same value of γ, τα and τn differ in their dependences on either the product variable T-1V-γ , V at constant T, or T at constant P. In each case, the difference is more pronounced at longer τα. Such behavior is inconsistent with the Rouse and tube models. However, these differences in the respective dependences of τα and τn can be accounted for quantitatively by the coupling model. The framework of the solution of the problem supports the proposal that the temperature and volume dependences of molecular mobility, which trigger the glass transition, do not originate from the primary α relaxation. Instead, they have their origin in the primitive relaxation of the coupling model.

The optimized Rouse–Zimm theory of excluded volume effects on chain dynamics

Based on the optimized Rouse-Zimm (ORZ) approximation to the Kirkwood diffusion equation, we investigate the effects of excluded volume interactions on the single chain dynamics. By incorporating the nonuniformly expanded moments of interbead distances into the expressions for the diffusion and structure matrices appearing in the ORZ diffusion equation, we obtain the general relaxation spectrum for flexible chains that is valid over the range from theta; solvents to good solvents. The present theory involves four parameters: the Kuhn statistical length b(0), the bead number N, the excluded volume parameter z, and the hydrodynamic interaction parameter h(*). These model parameters are determined from structural data of polymers with the aid of the quasi-two-parameter theory. The set of relaxation times of ORZ normal modes calculated with these bead-and-spring model parameters enables the theoretical prediction of various frictional and dynamical properties of polymers within a unified framework. The present ORZ theory generalizes the Ptitsyn-Eizner-type approaches by incorporating the nonuniform chain expansion effect into the structure matrix as well as the diffusion matrix.

Dielectric study on dynamics and conformation of poly( d, l-lactic acid) in dilute and semi-dilute solutions

Fractionated samples of d, l-poly(lactic acid) (PLA) were prepared and the dielectric normal mode relaxation was studied for dilute and semi-dilute solutions of the PLA in a good solvent benzene. Results indicate that in the dilute regime the normal mode relaxation time is proportional to [ η] M w in agreement with the Rouse–Zimm theory, where [ η] and M w denote the intrinsic viscosity and weight average molecular weight, respectively. The dielectric relaxation strength which is proportional to the mean square end-to-end distance 〈r 2〉 increases with increasing M w with the power of 2 ν, where ν is the excluded volume parameter determined from [ η]. The relaxation time in the semi-dilute regime increases with increasing concentration C due to increases of the entanglement density and the friction coefficient. The relaxation time corrected to the iso-friction state agrees approximately with the dynamic scaling theories. The relaxation strength decreases with increasing concentration indicating that 〈r 2〉 decreases on account of the screening of the excluded volume effect. The concentration dependence of 〈r 2〉 agrees approximately with the scaling theory proposed by Daoud and Jannink.

A molecular-dynamics simulation study of dielectric relaxation in a 1,4-polybutadiene melt

We have carried out atomistic molecular dynamics simulations of a melt of 1,4-poly(butadiene) from temperatures well above the experimentally observed merging of the primary α process and secondary β process down to temperatures approaching the experimentally observed bifurcation temperature. The relaxation strength and maximum loss frequency and its temperature dependence for the combined α-β dielectric relaxation process from simulations were in good agreement with experiment. The maximum loss frequency, melt viscosity, chain normal-mode relaxation times and torsional autocorrelation times were found to exhibit nearly identical non-Arrhenius temperature dependencies well represented by a Vogel–Fulcher fit with parameters in good agreement with experimental values obtained from dielectric and viscosity measurements. The dielectric susceptibility showed increasing intensity at high frequency for the lower temperatures investigated, indicative of a breakdown in time-temperature superposition due to an emerging β process. Comparison of time scales for the chain normal-mode dynamics and dielectric relaxation revealed that the latter is associated with motions on the segmental length scale. The correspondence of time scales and temperature dependence for the dielectric relaxation and the torsional autocorrelation function further confirmed the localized nature of the dielectric relaxation and indicated that the combined α-β dielectric process is fundamentally tied to microscopic conformational dynamics of individual dihedrals. However, the mean conformational transition rates were found to exhibit Arrhenius temperature dependence, leading to a divergence of time scales between the torsional, dielectric, chain and mechanical relaxation processes and the rates of conformational transitions with decreasing temperature. This divergence was associated with the increasingly heterogeneous character of conformational dynamics in the melt with decreasing temperature. Hence, the time scale of the principal (α) relaxation in the melt is fundamentally correlated with the time scale for homogenization of conformational dynamics, and not to the time scale of the conformational transitions themselves.

Local and global dynamics of polylactides.: A dielectric spectroscopy study

Polylactides (PLAs) are a family of degradable plastics having a component of the dipole moment both perpendicular and parallel to the polymer backbone (i.e. is a type-A polymer). We have studied the sub-glass, segmental and global chain dynamics in a series of fully amorphous samples having an L: D ratio of 80:20, with molecular weights in the range: 3.8×10 3< M n<4.7×10 4. Finite size effects have been observed for both the segmental and the sub-glass process. The segmental mode was found to have a steeper temperature and pressure dependence compared to the longest normal mode in accord with findings from other type-A polymers. The molecular weight dependence of the longest normal mode relaxation times ( τ∼ M n 3.1) reflects on the intermediate molecular weight regime ( M e∼6×10 3) from Rouse to entangled chains. These findings are compared with the viscoelastic results on the same system. Agreement in the c 2 value of the WLF equation is found but the scaling of the normalized longest relaxation time derived from the rheological measurements shows a stronger than expected dependence on the molecular weight.

Dielectric normal mode relaxation of undiluted poly(propylene glycol)s

Dielectric measurements were carried out for linear and star-shaped poly(propylene glycol)s (PPGs) of different molecular weights (M). A small loss peak due to the normal mode relaxation was observed on the low-frequency side of the main loss peak. Two peaks were resolved and the normal mode relaxation time τn, the distribution of relaxation times and relaxation strength were examined. The effect of OH groups attached at the chain ends on the normal mode relaxation was also investigated by using PPG samples whose end groups were substituted by acetyl groups. The results indicate that τn is proportional to M2 in accordance with the Rouse theory. The mode distribution of star-shaped PPGs is slightly broader than that of linear PPGs. The effect of hydrogen bonding between the OH groups on the normal mode relaxation time is negligibly small.

Dielectric study of dynamics of subchains and distribution of normal mode relaxation times in dilute and semidilute solutions of miscible block copolymers

We report on the dielectric normal mode relaxation of the isoprene block chains in dilute and semidilute solutions of butadiene–isoprene (BI) diblock copolymers and butadiene–isoprene–butadiene (BIB) triblock copolymers. The dielectric loss ε″ curves in dilute solutions with a concentration close to the overlapping concentration C* agreed approximately with those predicted by the bead–spring model. Above a concentration about three times higher than C*, the ε″ curves broadened with increasing concentration indicating the strong concentration dependence of the distribution of the normal modes. Two maxima of the ε″ curves were observed for BIB solutions. In semidilute solutions of BIB, the ε″ peak located in the high frequency side shifted little with concentration but the ε″ peak in the low frequency side shifted towards low frequency with increasing concentration. This suggests that the lower order normal modes are affected by entanglement effects but the higher order modes are not, for semidilute solutions. We have attempted to explain the distribution of the normal modes in terms of the blob model.

Concentration and Size Dependence of Dielectric Strength and Dielectric Relaxation of Polymers in Solutions of a ϑ Solvent via Molecular Dynamics Simulations

We consider the dielectric strength and dielectric relaxation of dilute and semidilute solutions of polymers in a ϑ solvent. A molecular dynamics technique is employed and the simulation results are contrasted with dielectric spectroscopy measurements, providing an insight into the molecular factors that affect the experimental behavior. The relation between the dielectric strength and the end-to-end distance of the chains, used in experiments to determine polymer dimensions, is established via equilibrium and nonequilibrium simulations. The dielectric loss spectra are computed from the end-to-end vector autocorrelation function of the simulated systems. At infinite dilution the simulation spectra conform well with the Rouse model prediction. The spectra broaden with increasing concentration, and a normal-mode analysis suggests that the overlapping of the chains leads to a broader distribution of relaxation times. Finally, normal mode relaxation times are found to be exponential in density.

A molecular dynamics study of macromolecules in good solvents: Comparison with dielectric spectroscopy experiments

Molecular dynamics simulations are employed to study the static and dynamic properties of macromolecules in dilute and semidilute solutions of a good solvent. The results are compared with dielectric spectroscopy experiments. Crossover concentrations, ρ*, that demarcate the dilute and semidilute regimes are identified. The shift from self-avoiding-walk to random-walk behavior is also studied. An investigation is conducted of the normal-mode dielectric relaxation of type-A polar polymers. In dilute systems, a power law molecular weight dependence of the normal mode relaxation times τ∝N2.2 is observed, in accordance with DeGennes’s scaling analysis for self-avoiding-walk chains. This result does not agree with the experimental dependence of the normal mode relaxation time on the size of the polymers. The differences between the simulations and the experimental dynamic results in the dilute regime can be ascribed to the leading assumption of the model, the neglect of hydrodynamic interactions. For higher concentrations ρ&amp;gt;8ρ*, hydrodynamic and excluded volume effects are effectively screened, and the simulation results conform well with experiments. The dielectric spectrum broadens with increasing density, and for low densities the broadening can be explained in terms of chain overlapping. For densities higher than the entanglement density, entanglement effects broaden the spectrum considerably. A mapping of the model parameters to real polymer properties is also proposed.

Transient entanglement behaviour in a poly(propylene glycole) melt: A field gradient NMR self-diffusion study

We report measurements of the temperature dependence of the self-diffusion of a poly(propylene glycole) in the melt. Two kinds of magnetic field gradient NMR are used: pulsed field gradient NMR with large field gradients and static field gradient NMR in a specially designed cryomagnet. The emphasis is put on large field gradients. The (true) longtime self-diffusion coefficients are compared with those calculated from normal-mode relaxation times in dielectric spectroscopy using the Rouse model. They agree very well. For temperature below about 270 K, a time-dependent self-diffusion coefficient appears which is an indication of anomalous diffusion. At 253 K we observe complete transient entanglement behaviour of the PPG melt. These results are discussed in comparison with dielectric and other experimental data.

Crossover to entangled dynamics in polymer solutions and melts

A statistical dynamical theory of the crossover from unentangled Rouse dynamics to entangled behavior is constructed for chain polymer solutions and melts. Both time and spatial crossovers in long chain fluids, and the degree of polymerization crossover for short polymers, are treated. The analysis is based on a microscopic theory of the perturbative dynamical corrections to Rouse theory arising from chain connectivity and intermolecular excluded volume forces. The dependence of crossover properties such as the plateau shear modulus and entanglement time and length scale on solution density, solvent quality, and chain statistical segment length are derived by combining the dynamical theory with equilibrium liquid state integral equation methods. Scaling relations are obtained which appear to be in general accord with most experiments on both solutions and melts. The physical origin of the predicted scaling behaviors is the fractional power law temporal decay of the entanglement friction memory function on intermediate time scales, and power law reduced density dependence of the equilibrium force correlations. The theory is also applied to compute the dependence of the chain normal mode relaxation times on polymer density and chain length. Favorable qualitative comparisons with recent neutron spin echo experiments are made.

Concentration fluctuation effects on chain orientation dynamics of polymer blends in the critical region

Dielectric relaxation spectroscopy has been employed to investigate the normal mode relaxation of polyisoprene I chains in binary polystyrene/polyisoprene (S/I) blends over a broad temperature range in the one and two phase region. The normal mode relaxation time in the homogeneous phase near the transition is faster than expected on the basis of the average bulk composition and the distribution of relaxation times becomes bimodal in the transition region. A concentration correlation model is proposed to account for the new experimental findings in the framework of Rouse chain dynamics and mean-field static behavior. The computed normal mode relaxation times compare well with the experiment for an effective composition averaged over a microscopic scale O(nm).