Segmental Dynamics of Entangled Poly(ethylene oxide) Melts

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

pubs.acs.org/Macromolecules

Segmental Dynamics of Entangled Poly(ethylene oxide) Melts: Deviations from the Tube-Reptation Model A. Lozovoi,† C. Mattea,† N. Fatkullin,‡ and S. Stapf*,† †

Department of Technical Physics II, Technische Universität Ilmenau, 98684 Ilmenau, Germany Institute of Physics, Kazan Federal University, 420008 Kazan, Tatarstan, Russia

‡

Downloaded via UNIV OF EDINBURGH on December 13, 2018 at 12:38:10 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

S Supporting Information *

ABSTRACT: The dynamics of entangled polymer melts not only is of fundamental theoretical interest but also has wide-reaching consequences for developing a theoretical foundation for investigating biological macromolecules and complex systems relevant to material sciences. Despite several decades of intensive experimental and theoretical research in this field, open questions remain regarding segmental dynamics over the wide range of time scales from local to global motion. This work employs a novel approach based on nuclear magnetic relaxation to scrutinize the character of dipolar interactions in entangled polymer melts, thereby accessing unique information about segmental diffusion and rotation. The main focus is set on the separate consideration of intra- and intermolecular contributions to the proton dipolar interactions, which have been previously shown to possess a different, nontrivial time dependence. A combination of well-established field-cycling T1 relaxometry and recently developed methods based on spin echo is utilized to investigate dipolar couplings in entangled poly(ethylene oxide) melts of various molecular weights. Isolation of the intermolecular contributions to the corresponding experimental quantities provides a means to observe segmental translations taking place during more than 6 orders of magnitude in time. Time dependences of the mean-square displacement obtained in this way revealed apparent exponents of the power laws significantly deviating from predictions of the widely used tube-reptation model of polymer dynamics in the regime of entangled motion. In addition to that, the relative ratio of intermolecular dipolar interactions over the intramolecular counterpart is probed through their corresponding contributions to the transverse relaxation rate. A strong deviation from the tube-reptation model predictions for the evolution of this quantity is observed in the whole range probed experimentally. The obtained data do not reflect the restricted character of segmental motion anticipated in the corresponding time regime. It is emphasized that similar results, both in amplitude and in qualitative behavior, have been previously demonstrated in polybutadiene and polyethylene-alt-propylene, thereby allowing to discuss the universality of the observed deviation.

■

Polymer Melt Dynamics. The behavior of linear chains with molecular weights below a characteristic critical value Mc is well described by the famous Rouse model4,5 which treats a macromolecule as a set of beads in a viscous medium which are connected by springs representing entropically elastic forces between the neighboring chain segments. This formalism is based on the universal coarse-grained description of the macromolecule through the statistical Kuhn segments, ideal character of polymer chain conformations, neglecting longrange hydrodynamic interactions, and memory effects on the time scale of terminal relaxation time6 (see a more detailed discussion in ref 7). Its derivation originates from the fundamental principles of molecular dynamics, being a proven and commonly accepted starting point for any consideration of more elaborate problems, in particular the description of dynamic behavior of a melt of long macromolecules with

INTRODUCTION Polymer melts and concentrated polymer solutions are systems highly relevant from a fundamental molecular dynamics point of view as well as from the prospective of the application to the investigation of biomacromolecular dynamics1,2 and the problems of material flow and extrusion.3 In spite of the conventional use of extremely elaborate polymer-based materials, a seemingly simpler problem of dynamics of a flexible linear macromolecular chain in a viscous polymeric liquid still remains unresolved. A high number of degrees of freedom and multibody interactions between a lot of neighbors at any given moment of time result in the complex dynamics of a polymer melt, which demands the use of sophisticated theoretical models to describe processes developing on the scales from the single-segment (10−9−10−7 s) to the multimacromolecular motion (10−1−101 s). Existing theoretical approaches to this problem are discussed further on in this work, and novel experimental techniques are used to highlight important drawbacks of the current understanding of polymer melt dynamics. © XXXX American Chemical Society

Received: August 28, 2018 Revised: December 1, 2018

A

DOI: 10.1021/acs.macromol.8b01857 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Figure 1. (a) Schematic illustration of the type of motions within the tube-reptation model with corresponding regimes and transition time constants. (b) Time dependences of the segmental mean-square displacement and (c) of the intra- and intermolecular dipolar correlation functions and Ad:intra are shifted in an entangled polymer melt as predicted based on the tube-reptation model in double-logarithmic scale. Lines for Ad:inter 0 0 relative to each other for better visibility and do not represent the actual relative values of these two parameters.

molecular weights M > Mc. In this case, effects of chain overlapping, or “entanglements”, start to play an important role significantly complicating the treatment of the system. Tube-Reptation Model. The existence of entanglements in polymer systems has been identified from rheology data well before the development of the corresponding theoretical framework (e.g., see ref 8 and references therein). A prominent attempt to account for the constraints imposed by the surrounding macromolecules onto the tagged chain was made by Edwards9 when he introduced the concept of tube for the description of elastic properties of polymer networks. Later on, de Gennes in his seminal work of 197110 suggested the mechanism of chain reptation in a system of fixed obstacles, which he treated as diffusion of defects along the chain contour. A combination of the tube concept with the reptation-like motion was extended to the case of polymer melts by Doi and Edwards.12,13 These works established one of the best-known and widely used models for the description of entangled polymer melt dynamicsthe “tube-reptation” (TR) model. A schematic illustration of different regimes of motion and the corresponding time dependences of the segmental mean-square displacement (MSD) predicted by the TR model are shown in Figure 1a,b. The shortest time constant in the model characterizing the local fluctuation on the length scale of a Kuhn segment is the segmental fluctuation time, τs. At times t > τs, when the coarsegrained description is applicable, chain dynamics is well described by the Rouse model mentioned above, and in the case of macromolecules of a molecular weight M < Mc, this type of motion takes place up to the transition to the normal diffusion regime, where the center-of-mass translation becomes dominant. However, for longer chains additional regimes of dynamics associated with the chain entanglements are predicted. The tagged chain begins to sense the restrictions imposed by the surrounding macromolecules of the melt at times t > τe, where τe is the so-called entanglement time: τe = τsNe2. Here, Ne denotes the number of Kuhn segments between entanglements in the chain, typically ranging from 5 to 50 segments for different polymer species.14,15 In this

regime the chain undergoes incoherent reptation, which is a restricted Rouse-like wriggling motion combined with the reptation along the fictitious tube. The conformation of the tube is a random coil16 with a step length and diameter of dt = bNe1/2,13 where b is the Kuhn segment length. Segmental motion in this time range is extremely slow as is indicated by the corresponding MSD power law: ⟨r(t)2⟩ ∝ t1/4. After the N2

longest Rouse mode relaxation time, τR = τs N 2 , where N is the e

number of Kuhn segments of the chain, the motion of different parts of the chain becomes coherent. This leads to the onset of coherent reptation along the primitive path with relatively faster dynamics that is additionally dependent on the length of the chain: ⟨r(t)2⟩ ∝ N−1/2t1/2. This motion is accompanied by a constant renewal of the tube that takes place at the time scale N3

of terminal relaxation, or tube disengagement, time τd = 3τs N . e

It also features the transition from a rubber-like to a liquid-like behavior of the creep compliance J(t).11 Eventually, at times t > τd, chain dynamics becomes dominated by normal diffusion of the center of mass with a diffusion coefficient Dcm(t) dependent on the length of the chain as Dcm(t) ∝ N−2. Experimental Evidence and Discrepancies of the Tube-Reptation Model. Numerous experimental studies of entangled polymer melts have been performed since the introduction of the TR model aimed at checking the corresponding predictions. Interestingly, the molecular weight dependence of the center-of-mass diffusion coefficient Dcm(t) ∝ N−2.3 observed by means of pulsed field gradient nuclear magnetic resonance,18−21 small-angle neutron scattering,22 and infrared microdensitometry23 as well as the viscosity and terminal relaxation time η, τ1 ∝ N3.4 obtained via mechanical relaxation11,17 turned out to be stronger than predicted by the TR model (Dcm(t) ∝ N−2 and η, τ1 ∝ N3). However, additional theoretical consideration of contour length fluctuation24 and constraint release25 effects provided a consistent explanation of this deviation. A comprehensive summary of such measurements in polybutadiene by various techniques as well as an B

DOI: 10.1021/acs.macromol.8b01857 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules overview of theoretical developments considering the TR model can be found here.26 It is important to emphasize that the above-mentioned, wellconfirmed predictions of the tube-reptation model, many of which have been confirmed in publications based on experimental results, are connected to the macro characteristics of a bulk polymer melt. They do not provide direct information about the dynamical behavior on the level of the chains’ Kuhn segments. This type of micromotion is described by segments’ displacement in time and their mutual orientation. These properties play an important role in establishing a fundamental physical picture of the tubereptation dynamics. Prominently, it was demonstrated that the segmental mean-square displacement’s time dependence predicted by the TR model (Figure 1b) is reproduced in all the regimes by molecular dynamics Monte Carlo simulation study with the use of a harmonic radial potential.27,28 In spite of that, experimental results obtained in real polymer melts turn out to be more controversial. Possibilities to study this problem are rather limited, since there are only two techniques that are applicable to probe segmental dynamics in entangled polymer systems in the time range of interest (τs < t < τd): neutron spin echo (NSE) and nuclear magnetic resonance (NMR). Both of them provide unique information about the features of segmental motion through the dynamic structure factor obtained by NSE29 and field-gradient NMR diffusometry30 and through the dipolar correlation function encoded in NMR spin relaxation.31 These methods can be considered complementary to each other since they probe different time scales, only overlapping in certain conditions in regimes I and II of the TR model. Corresponding limitations and accessible time and displacement ranges are compared and illustrated in these works.32−34 The meansquare displacement power law ∝t1/2 corresponding to the Rouse regime (I) has been observed by NSE in entangled polyethylene-alt-propylene and polyethylene.29,35 Field-gradient NMR diffusometry has been applied to probe entangled dynamics in the regime of coherent reptation (III), yielding a ∝t1/2 power law in entangled poly(ethylene oxide)36 and polybutadiene melts.37 In this work36 it was also shown that even the regime of incoherent reptation (II) can be accessed for an extremely large molecular weight of poly(ethylene oxide) (5000 kDa) with additional consideration of spin diffusion.38 Interestingly, in that study a clear discrepancy between the experimentally measured MSD time dependence (∝t1/2) and the model prediction (∝t1/4) for this regime was observed.36 Recent advances in the theoretical description of dipolar interactions39−44 in polymer melts allowed to further investigate and to shed more light onto this discrepancy. As is well known, dipolar interaction is the main mechanism of proton NMR relaxation in polymer melts.21,45 In these works39−41 it was theoretically argued and experimentally confirmed that intermolecular dipolar couplings, which are sensitive to the relative segmental translation (Figure 2), can be separated from the intramolecular counterpart by using isotope dilution and can be directly recalculated into the relative segmental mean-square displacement. This gave rise to a comprehensive revisiting of entangled chain dynamics by means of T1 field-cycling relaxometry33,37,40,44 and more recently by T2 relaxation-based methods.34,43,46,47 The extended time range of these measurements as compared to the NSE and FG NMR allowed to confirm the power laws of

Figure 2. Dipolar interactions in the entangled polymer melt consisting of four C50H102 molecules. Conformation is simulated in Chem3D. Intrasegment dipolar coupling is mostly modulated by the rotation of the segment, whereas intermolecular interaction is dominated by the translation of the macromolecules relative to each other.

MSD time dependence in highly entangled polybutadiene, polydimethylsiloxane,37 and polyethylene-alt-propylene33,34 for all the regimes of the TR. In spite of this, a clear discrepancy between the model predictions and the observed MSD behavior in the regimes of entangled dynamics was still found in poly(ethylene oxide),40 corroborating previously mentioned field-gradient NMR results.36 Dipolar Interactions in the Tube-Reptation Model. Generally, dipolar interactions are characterized by the total dipolar correlation function which is defined as21,32 A 0d (t ) =

1 ∑ Ns km

(1 − 3 cos2(θkm(t ))) (1 − 3 cos2(θkm(0))) rkm3(t )

rkm3(0) (1)

where r⃗km is the vector connecting spins k and m, θkm is the angle between the direction of the external magnetic field H⃗ 0 and r⃗km (see Figure 2), Ns is the number of spins in the system, and the summation is taken over all the spin pairs. In these works42,43 theoretical predictions for the intramolecular Ad:intra 0 and intermolecular Ad:inter parts of the total proton dipolar 0 correlation function for isotropic (regime I) and anisotropic (regimes II and III) segmental motion were made. The corresponding time dependences are shown in Figure 1c. An important consequence of these results manifests itself through the obvious qualitative difference in the behavior of the ratio A 0d:inter A 0d:intra

with respect to the character of the segmental motion. In

regime I this ratio increases with time as ∝t0.25, whereas in regimes II and III it decreases as ∝t−0.125 and ∝t−0.25, respectively. In terms of MSD that translates to A 0d:inter A 0d:intra

∝ ⟨r(t )2 ⟩1/2 for regime I and

A 0d:inter A 0d:intra

∝ ⟨r(t )2 ⟩−1/2 for

regimes II and III. This significant contrast provides an opportunity to examine the degree of restriction of the segmental motion in an entangled polymer melt, which is highly relevant with respect to the tube hypothesis lying at the core of the TR model. The time dependence of the intramolecular orientational dipolar correlation function was studied by means of double-quantum NMR spectroscopy48−50 and was shown to be at variance with the TR model predictions for regime II. It should be also taken into account that in these studies the intermolecular part of dipolar coupling was found to be constant in a studied time range contradicting C

DOI: 10.1021/acs.macromol.8b01857 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

dissolving the undeuterated poly(ethylene oxide) sample and its deuterated counterpart in chloroform CHCl3, creating a dilute solution (