Longest Rouse Relaxation Time Research Articles

Molecular dynamics study of epoxy/clay nanocomposites: rheology and molecular confinement

To investigate the effect of nanoparticles on molecular confinement, epoxy/clay nanocomposites in 6 % wt were prepared by low shear mixer and ultrasonication methods. According to the XRD results, high energy sonication can significantly increase the intergallery space compared to that of the low shear mixer and, therefore, provide more interfacial area for polymer-particle interaction. Rheological measurements, including dynamic mechanical moduli, continuous relaxation spectrum and zero-shear viscosity, were carried out in the linear region to reveal the solid-like characteristics induced in the pure epoxy. The longest Rouse relaxation time, which was determined from the value of zero-shear viscosity, was employed to study the segmental friction. It is clear that the chain relaxation process was slowed by the polymer-particle interactions, which created very high monomeric friction. We introduce the ratio of monomeric friction coefficients (ζnano/ζepoxy) as the strength of immobilization to correlate the increasing elasticity, longer relaxation time and molecular confinement quantitatively. Because more monomers can be immobilized on a clay surface due to their strong affinity, the friction ratio increased to 359.48 for the sonicated sample, whereas weak polymer-particle affinity resulted in much lower growth, with a friction ratio of 2.11 for the low shear mixing method.

Chain contraction and nonlinear stress damping in primitive chain network simulations

Doi and Edwards (DE) proposed that the relaxation of entangled linear polymers under large deformation occurs in two steps: the fast chain contraction (via the longitudinal Rouse mode of the chain backbone) and the slow orientational relaxation (due to reptation). The DE model assumes these relaxation processes to be independent and decoupled. However, this decoupling is invalid for a generalized convective constraint release (CCR) mechanism that releases the entanglement on every occasion of the contraction of surrounding chains. Indeed, the decoupling does not occur in the sliplink models where the entanglement is represented by the binary interaction (hooking) of chains. Thus, we conducted primitive chain network simulations based on a multichain sliplink model to investigate the chain contraction under step shear. The simulation quantitatively reproduced experimental features of the nonlinear relaxation modulus G(t,γ). Namely, G(t,γ) was cast in the time-strain separable form, G(t,γ)=h(γ)G(t) with h(γ)=damping function and G(t)=linear modulus, but this rigorous separability was valid only at times t comparable to the terminal relaxation time, although a deviation from this form was rather small (within ±10%) at t>τ(R) (longest Rouse relaxation time). A molecular origin of this delicate failure of time-strain separability at t∼τ(R) was examined for the chain contour length, subchain length, and subchain stretch. These quantities were found to relax in three steps, the fast, intermediate, and terminal steps, governed by the local force balance between the subchains, the longitudinal Rouse relaxation, and the reptation, respectively. The contributions of the terminal reptative mode to the chain length relaxation as well as the subchain length/stretch relaxation, not considered in the original DE model, emerged because the sliplinks (entanglement) were removed via the generalized CCR mechanism explained above and the reformation of the sliplinks was slow at around the chain center compared to the more rapidly fluctuating chain end. The number of monomers in the subchain were kept larger at the chain center than at the chain end because of the slow entanglement reformation at the center, thereby reducing the tension of the stretched subchain at the chain center compared to the DE prediction. This reduction of the tension at the chain center prevented completion of the length equilibration of subchains at t∼τ(R) (which contradicts to the DE prediction), and it forces the equilibration to complete through the reptative mode at t≫τ(R). The delicate failure of time-strain separability seen for G(t,γ) at t∼τ(R) reflects this retarded length equilibration.

Entanglement and confinement effects constraining polymer chain dynamics on different length and time scales

Abstract With the time constants usually considered to be characteristic for polymer dynamics, namely τ s (the segment fluctuation time), τ e (the entanglement time), and τ R (the longest Rouse relaxation time), the time scales of particular interest: (i) t τ s ; (ii) τ s t τ e ; and (iii) τ e t τ R will be discussed and compared with experimental data. These ranges correspond to the chain-mode length scales: (i) l b ; (ii) b l d 2 / b ; and (iii) d 2 / b l L , where b is the statistical segment length, d is the dimension of constraints by entanglements and/or confinement, and L is the chain contour length. Based on Langevin-type equations-of-motion coarse-grained predictions for the mean-squared segment displacement and the spin-lattice relaxation dispersion will be outlined for the scenarios “freely-draining”, “entangled”, and “confined”. In the discussion we will juxtapose “local” versus “global” dynamics on the one hand, and “bulk” versus “confined” systems on the other. The experimental technique of particular interest here is field-cycling NMR relaxometry which predominantly probes conformational fluctuations. A comparison with methods sensitive by contrast to translational fluctuations such as field-gradient NMR diffusometry and neutron scattering will be discussed with respect to sensitivity to confinement phenomena, i.e. the so-called corset effect.

Length and Time Scales of Entanglement and Confinement Effects Constraining Polymer Chain Dynamics

AbstractWith characteristic time constants for polymer dynamics, namely τs (the segment fluctuation time), τe (the entanglement time), and τR (the longest Rouse relaxation time), the time scales of particular interest (i) t<τs, (ii) τs<t<τe, and (iii) τe<t<τR will be discussed and compared with experimental data. These ranges correspond to the chain-mode length scales (i) l<b, (ii) b<l<d2/b, and (iii) d2/b<l<L, where b is the statistical segment length, d is the dimension of constraints by entanglements and/or confinement, and L is the chain contour length. Based on Langevin-type equations-of-motion coarse-grained predictions for the mean-squared segment displacement and the spin-lattice relaxation dispersion will be outlined for the scenarios “freely-draining”, “entangled”, and “confined”. In the discussion we will juxtapose “local” versus “global” dynamics on the one hand, and “bulk” versus “confined” systems on the other.

Unified application of the coupling model to segmental, Rouse, and terminal dynamics of entangled polymers

The temperature dependence and relaxation function breadth of segmental dynamics ( α-relaxation) for 1,2-polybutadiene and 1,4-polybutadiene are used to predict their respective temperature-dependent terminal relaxation times by unified application of the Ngai coupling model. Literature results for the terminal flow of near-monodisperse linear polybutadienes having widely varying molecular weights are successfully represented using the coupling model by variation of only a single parameter, C, which is the proportionality constant between the longest Rouse relaxation time and the primitive relaxation time which underlies the cooperative segmental process. The value of C varies with molecular weight ( M) according to C ∝ M b where b is found to range from 1.8 to 2.1, in close agreement with the expected exponent of 2. Contrary to experimental data and coupling model predictions, reptation theory predicts identical influences of temperature on Rouse and terminal relaxation processes; we suggest that invoking a temperature dependence for contour length fluctuations, and hence number of effective entanglements per chain, may resolve this deficiency of the tube model.

A critical evaluation of step strain flows of entangled linear polymer liquids

A critical evaluation of step strain flow experiments on entangled, linear polymer liquids is performed. Roughly one-half of the published shear stress relaxation modulus data for these systems are consistent with the predictions of the well-known tube model. A model of step strain flow experiments is developed to determine whether the remaining published data, which are qualitatively different from tube model predictions, are simply artifacts caused by slip, an imperfect strain step strain history, or transducer compliance. Modeling results suggest that these factors are capable of producing the types of behavior observed in experiments deemed anomalous. New criteria based on the retraction time τR, or longest Rouse relaxation time, for avoiding anomalies caused by imperfect strain step strain history and transducer compliance are proposed. These simple criteria are, in a majority of cases, found to be capable of predicting the type of observed stress relaxation behavior for 60 published step strain experiments on entangled, linear polymer liquids.

Viscoelasticity of an Entangled Polymer Solution with Special Attention on a Characteristic Time for Nonlinear Behavior

Nonlinear viscoelasticity was studied for a polystyrene solution in tricresyl phosphate; molecular weight = 5480 kg mol-1; concentration = 49 kg m-3. The longest Rouse relaxation time, τR, was estimated by fitting the Rouse theory to dynamic modulus at high frequencies. The Doi−Edwards tube model theory presumes that 2τR is the characteristic time for equilibration of chain contour length; (2τR)-1 is the rate for an extended chain to shrink back to equilibrium. The τR value was much lower than that evaluated with more widely accepted methods. However, the shear stress (σ) and the first normal stress difference (N1) in the start-up of shear flow with low rate of shear (<(2τR)-1/5) were consistent with the assumption that the contour length is always at equilibrium value. At high rate of shear (>8(2τR)-1), the maxima of σ and N1 were located at t = 2τR and 4τR, respectively, in accord with the interpretation that 2τR is the characteristic time for chain shrink. The strain-dependent relaxation modulus, G(t, γ), was also studied. At high magnitudes of shear, γ = 4 or 5, the ratio G(t, γ)/G(t, 0) decreased rapidly around t = 2τR and leveled off at t = 20(2τR): the chain shrink process plays the main role in damping but a slower process may also be involved. At γ = 1 or less, G(t, γ)/G(t, 0) decreased only at times much longer than 2τR. The damping of relaxation modulus involves some secondary process with a characteristic time longer than that of the chain contraction process.

Evaluation methods of the longest Rouse relaxation time of an entangled polymer in a semidilute solution

Evaluation methods of the longest Rouse relaxation time of an entangled polymer in a semidilute solution

Differential constitutive equation for entangled polymers with partial strand extension

Beginning with a formal statement of the conservation of probability, we derive a new differential constitutive equation for entangled polymers under flow. The constitutive equation is termed the Partial Strand Extension (PSE) equation because it accounts for partial extension of polymer strands in flow. Partial extensibility is included in the equation by considering the effect of a step strain with amplitude E on the primitive chain contour length. Specifically, by a simple scaling argument we show that the mean primitive chain contour length after retraction is L=L 0 E 1/2, not the equilibrium length L 0 as previously thought. The equilibrium contour length is infact recovered only after a characteristic stretch relaxation time λ s that is bounded by the reptation time and longest Rouse relaxation time for the primitive chain. The PSE model predictions of polymer rheology in various shear and extensional flows are found to be in good to excellent agreement with experimental results from several groups.