Chain Dynamics and Segmental Orientation in Polymer Melts

Article pubs.acs.org/Macromolecules

Chain Dynamics and Segmental Orientation in Polymer Melts Confined to Nanochannels Cornelius Franz,† Frank Lange,† Yury Golitsyn,† Brigitte Hartmann-Azanza,‡ Martin Steinhart,‡ Margarita Krutyeva,§ and Kay Saalwac̈ hter*,† †

Institut für Physik − NMR, Martin-Luther-Universität Halle-Wittenberg, Betty-Heimann-Str. 7, D-06120 Halle, Germany Institut für Chemie neuer Materialien, Universität Osnabrück, Barbarastr. 7, D-49069 Osnabrück, Germany § Jülich Centre for Neutron Science (JCNS) and Institute for Complex Systems (ICS), Forschungszentrum Jülich GmbH, D-52428 Jülich, Germany ‡

ABSTRACT: We study changes in the dynamics of polymer chains confined to cylindrical nanochannels within aluminum oxide membranes. Specifically, a proton time-domain NMR technique is used to assess the effect of transient wall contacts on the time-averaged orientational order of poly(butadiene) segments in melts with different molecular weights (MW). Previous work has evidenced that the weakly interacting polymer, residing in ∼100 μm long, 20 and 60 nm wide channels, shows no significant confinement-related changes in the segmental (α) relaxation time and only weak (less than a factor of 2) changes in the micrometer-scale diffusivity. In the relevant temperature range above 340 K, we here use samples with pores oriented at different angles with respect to the main magnetic field to study the macroscopic anisotropy of segmental rotations and the effect of slower motions in regimes III and IV of the tube model up to the milliseconds time scale. We show that the pore walls exert a significant orientation effect on the chains, measured in terms of a time-averaged order parameter with a related length scale of one to a few nanometers, coexisting for high molecular weight (MW) inhomogeneously with bulk-like behavior in the pore center. Low MW with fewer than about 10 entanglements as well as low MW liquids exhibit a homogeneous response, with an overall residual orientation that represents a diffusively averaged quantity reflecting the pore geometry. We support our findings by a simulation model based upon one-dimensional curvilinear chain diffusion along the primitive path. The study is complemented by deuterium NMR experiments on a labeled poly(dimethylsiloxane) sample, in which strong surface contacts prevent full diffusive averaging.

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samples22,24,26 or from the kinetics of nonequilibrium polymer infiltration into nanochannels.23 More direct assessments on a microscopic level concerning confinement-related changes in tube-confined segmental dynamics27,29,30 or chain conformation44 under equilibrium conditions have so far only provided weak or even absent effects, respectively. Even in contrast, studies on tracer diffusion in thin films13,15,16 or nanocomposites34−36 that are sensitive to large-scale motion (reptation) have consistently revealed a slowdown of the dynamics, which were discussed to depend on the interaction strength in the thin-film cases.13,15,16 Our own previous work has demonstrated that a molecular weight independent slowdown, measured under equilibrium conditions, can be explained by a segmental friction coefficient that is increased due to transient adsorptive interactions with the substrate.42

INTRODUCTION Effects of nanoscopic confinement on the dynamics and rheological properties of polymer melts remain to be under active investigation and debate.1−4 Particularly complex phenomena arise for the segmental dynamics (related to the glass transition temperature, Tg), where it is now established that surface- or interface-related changes and thus Tg gradients can go in both directions, depending on whether free-surface effects5−7 or adsorptive interactions to the substrate2,8−11 dominate. Recent developments stress that aging phenomena2,7,8,12 and the absolute temperature and time scale of the experiments3−6 play important and nontrivial roles. Even in the potential absence of significant Tg effects, further changes in polymer chain fluctuations and relaxations on longer time and length scales have been reported.10,11,13−42 In particular, disentanglement effects (apparent tube dilation) and the related enhanced dynamics, arising from the folding of the chain onto itself close to neutral obstacles,43 are expected to play a relevant role for entangled chains.14,41 Such effects have so far been inferred from the mechanical properties (plateau modulus, creep/flow behavior) of highly entangled thin-film © XXXX American Chemical Society

Received: October 21, 2015 Revised: December 10, 2015

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DOI: 10.1021/acs.macromol.5b02309 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules A specific observation of relevance to the present work is the dielectric response of specific (type A) polymers in either thinfilm38 or cylindrical39,40 confinement. In the entangled case, the so-called normal mode of type A polymers represents a molecular-scale observable of reptation. In confinement, the normal mode was found to be accelerated and reduced in amplitude. This was attributed to the inability to observe a long-time isotropization of the whole chain in oriented samples and to the resulting restriction of the observed remaining relaxation to terminal subchains.38,39 Our results will provide a refined picture of this phenomenon. The focus of our work are segmental orientation fluctuations as probed by nuclear magnetic resonance (NMR). Specifically, we extend upon previous 2H NMR work, which has evidenced that surfaces lead to anisotropic segmental rotations in thin films that are either flat17 or cover the inner wall of cylinders.18 The former case is comparable to ours in that a weakly interacting polymer was studied. For unentangled chains, it was found that the NMR response represents a fast diffusive average across the film,17 while the entangled case could not be studied due to resolution issues. The strong-absorption case (mainly short-chain poly(dimethylsiloxane) on silica or alumina) is characterized by significant dynamic inhomogeneity in thicker films,18 as was also evidenced by different low-resolution 1H relaxometry techniques.19,21,25 Here, we go beyond these studies in that we perform measurements on macroscopically ordered samples with wellentangled chains and use a more quantitative technique. We extend the time range to slow motions up to the milliseconds range by using 1H time-domain multiple-quantum (MQ, or more specifically, double-quantum, DQ) NMR.45,46 This technique was previously shown47−50 to quantitatively reflect the features of and small but significant deviations from the tube model51 of entangled polymer melts. In the relevant temperature range of about 150 K above Tg, we mainly probe the dynamics in the Doi−Edwards regimes III (reptation) and IV (terminal diffusion) in terms of the time averaging (isotropization) of residual orientation correlation arising from the tube constraint.52,53 A first account of our results on poly(butadiene) (PB) confined to powdered self-ordered anodic aluminum oxide (AAO) membranes featuring straight 100 μm long, 20−60 nm wide cylindrical channels28 confirmed the mentioned earlier results on surface-induced anisotropic segmental motions17 for strongly entangled chains and also rather weak surface interactions. Our data suggested an inhomogeneous scenario, with different dynamics in a nanometer-sized surface layer and the cylinder core, the latter behaving bulk-like. Notably, the system under investigation does not exhibit significant changes of the segmental relaxation time (τα), at least not in a wide temperature range well above Tg.28,32,40 Confinement-induced changes in segmental orientation fluctuations are also assumed to be the origin of the so-called “corset effect”, a phenomenon related to confinement-modified T1 relaxation times detected as a function of Larmor frequency by Kimmich and co-workers using field-cycling NMR.20,31 The field-cycling technique was also applied to our samples,32 confirming the absence of significant changes in τα(T ≫ Tg), but showing a somewhat different phenomenology than the corset effect. That data were interpreted to be compatible with a modification of the Rouse spectrum, in a way that regime I (Rouse) and regime II (constrained Rouse) motions appeared delayed in confinement. However, caution has to be exercised

Figure 1. Schematic representation of the sample setup for orientation-dependent NMR measurements of polymers infiltrated into stacked AAO membranes. The rf coil around the sample tube is not shown.

in interpreting T1 relaxation times. The time needed to measure T1 (i.e., T1 itself) is much longer than the actual correlation time of motion that is probed by it. In inhomogeneous polymer systems (surface vs bulk), molecular exchange on a time scale shorter than T1 itself but longer than the relevant motions affecting it (i.e., Rouse dynamics) leads to averaging effects that are to the best of our knowledge unexplored. This stresses one advantage of the technique used herein, which provides a direct observable of the degree of segmental anisotropy and a realtime measure of slower processes.47−50 In this work, we extend our preliminary studies on a single high-molecular-weight (MW) PB sample confined to AAO28,32 to a large MW range from about 1 to 200 kg/mol, covering the range of about 1−100 entanglements, also including different low-MW liquids. We further study not only powder samples but also stacks of oriented membrane pieces as a function of orientation with respect to the magnetic field B0 (see Figure 1). Relying on our recent finding of the only marginally affected large-scale diffusion coefficient along the channels,42 and on a minimal primitive-path based simulation model, we arrive at a coherent picture of weakly interacting wall confinement effects on entangled chain dynamics. Our findings can be explained by segmental-orientation effects arising from the wall constraint, which are in the low-MW case phenomenologically similar to but larger in magnitude than what we find for low-MW liquids. At high MW, the segmental orientation correlations are more localized and measured as a time average due to motions within the primitive-path segments (reptation tube) close to the confining wall.41,54

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EXPERIMENTS AND THEORY

Samples. AAO membranes with 2055 and 60 nm56 channel diameter and 100 μm thickness (see Figure 1) were melt-infiltrated with low-polydispersity PB of variable weight-average MW (Mw) from 0.8 to 200 kg/mol (Rg = 1.1−17 nm). The PB samples were obtained from PSS (Polymer Standards Service GmbH, Mainz, Germany) and have about 50% cis and 45% trans units. Details on the properties of the higher-MW samples can be taken from refs 48 and 50, and the infiltration process and spectroscopic characteristics of the AAO samples are described in refs 28, 32, and 40. We further investigated one sample of perdeuterated poly(dimethylsiloxane) (PDMS) of 21 kg/mol (Rg = 3.5 nm) infiltrated into 26 nm pores by 2H NMR, which was previously studied by neutron scattering. This sample contained a 6% minority fraction of hydrogenated PDMS of 17 kg/mol, both for contrast matching reasons and in order to observe its chain dynamics. See refs 27, 29, and 37 for details. A part of the samples was crushed and measured as powder, while another part was investigated in the form of stacked oriented membranes (see Figure 1), using Teflon spacers and tape to position B


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Macromolecules the stacks in 5 mm NMR sample tubes. Because of shape irregularity and some corrugation of the membranes, the degree of orientation is generally not perfect, and angle deviations and distributions of the order of ±10° are expected. NMR Spectroscopy. Static MQ NMR experiments were performed on Bruker Avance III spectrometers at 1H Larmor frequencies of 200 (B0 = 4.7 T) and, mostly, 400 MHz (9.4 T), and a 2H Larmor frequency of 61.4 MHz (9.4 T), using Bruker static double-resonance probes. Temperature control was based upon a heated or cooled air or nitrogen flow covering a temperature range from 263 to 383 K with an accuracy of about 0.5 K. The 400 MHz probe was custom-modified with a goniometer for rotation of the sample tube within the solenoid coil by an angle Ω with an angular accuracy of the order of 5°. Additional 1H high-resolution magic-angle spinning (MAS) experiments for sample characterization were performed on a 600 MHz (14.1 T) Bruker Avance III instrument using a Bruker 2.5 mm double-resonance MAS probe. The 90° pulse length was around 2.5 μs on all probes. Sample Characterization. A reliable static low-resolution 1H NMR analysis of the infiltrated polymer requires knowledge and control over the contribution of residual 1H signals related to hydroxy groups or water on the surface of or within the AAO material. 1H fastMAS spectra (Figure 2a) of a filled 20 nm membrane were acquired after a variable Hahn echo filter in order confirm that the broad membrane-associated signal has a rather short T2 of the order of 300 μs. This is due to the rigid nature of these moieties and the correspondingly strong dipolar interactions, as shown previously also by DQ-filtered MAS spectra.32 In contrast, the resolved polymer signals at around 2 and 5.5 ppm hardly decay even for echo delays of several milliseconds. Taking the difference spectrum between the spectra taken at the shortest possible echo delay and a delay of 1.5 ms (Figure 2a, bottom), it is revealed that no PB resonances remain visible, demonstrating that no significant amount of PB is rigidly adsorbed on the substrate. A comparison of integrated signals from static and MAS Hahn echo experiments as a function of echo delay is shown in Figure 2b. The T2 decay integrated over the PB resonances resolved under MAS is seen to be identical to the long-time decay of the overall signal, again proving no significant immobilization of PB. Most importantly, the comparison demonstrates that the ∼70% AAO contribution to the overall signal in the 20 nm membranes decays in the static case (black solid squares) within a total Hahn echo delay of 200 μs. Thus, all MQ NMR experiments reported below were conducted with an initial 200 μs Hahn echo filter. The quality and reproducibility of AAO membrane samples varied samewhat among different batches, with sometimes more background signal distorting the initial DQ buildup (see below) despite the filter. Data from a few samples were thus discarded as outliers. The static data for the 60 nm membrane (red solid circles) reveals a much lower AAO background of only ∼20% in this case, likely due to the lower porosity and different anodization conditions (using oxalic acid instead of sulfuric acid as electrolyte for anodization). Note that the relative intensities in this plot suffer some systematic error due to baseline uncertainties, explaining differences between the static and MAS cases. MQ NMR. The 1H MQ experiment is based on the pulse sequence published by Baum and Pines57 and is described in detail elsewhere.46 Basically, it excites all even quantum orders among the many protons by relying on the residual dipole−dipole coupling Dres, thus probing the intrinsic orientation dependence. Using appropriate phase cycling, the pulse sequence yields an intensity buildup curve dominated by DQ coherences (IDQ) and reference decay curve (Iref) as a function of pulse sequence time τDQ, which is incremented in small steps by adjusting either the interpulse spacings or the number of pulse sequence cycles. Both incrementation schemes were combined to assemble a sufficiently well-resolved time axis covering a range of τDQ between some tens of microseconds (limited by the pulse sequence) and tens of milliseconds. Because of formally identical Hamiltonians, the same experiment can also be applied to excite DQ coherences in single 2H nuclei,46,50,58−60 then relying on the residual quadrupole coupling χres. We note the use of low-field spectrometers, as done in previous

Figure 2. 1H NMR characterization of PB55k in AAO. (a) Hahn-echo filtered 1H MAS (30 kHz) NMR spectra of a 20 nm sample. (b) Hahnecho intensities of 60 nm (circles) and 20 nm (squares and rhombs) samples under static and MAS conditions, referenced against singlepulse excitation. The rhombs show the integrated resolved PB resonances after background subtraction, fitted to a single exponential. The other lines just guide the eye. work,47,48 was not possible due to low overall filling factor. Experiments on bulk PB confirmed our previous low-field results, emphasizing the field independence of the measured phenomena. Neglecting interchain dipole−dipole couplings,47,50 and lumping multiple intrachain couplings into an effective coupling constant in combination with a second-moment approximation,47−49 the two experimental signal functions and two relevant derived functions can be computed in terms of a single-coupling ansatz46,48

IDQ (τDQ ) = ⟨sin ϕ1 sin ϕ2⟩

(1)

Iref (τDQ ) = ⟨cos ϕ1 cos ϕ2⟩

(2)

IΣMQ (τDQ ) = ⟨sin ϕ1 sin ϕ2⟩ + ⟨cos ϕ1 cos ϕ2⟩

(3)

InDQ (τDQ ) = IDQ (τDQ )/IΣMQ (τDQ )

(4)

where the brackets denote a space and time average. The functions depend upon dipolar phases acquired during spin evolution, ϕ1 ≡ ϕ(0, τDQ) and ϕ2 ≡ ϕ(τDQ, 2τDQ), defined as ϕ(ta , tb) = Deff

∫t

a

tb

P2(cos θt ) dt ;

tb − ta = τDQ

(5)

The dynamic information is encoded in the statistically fluctuating orientation of the internuclear vector θt with respect to the external B0 field. As mentioned, Deff is a second-moment type quantity characterizing the average dipolar interaction strength within a monomer unit. On a more coarse-grained level one can presuppose an average over fast local processes which anyways do not influence the signal functions. Then, a reduced value Deff can be used, which, for instance, can be taken to represent a statistical (Kuhn) segment, while θt then describes the local segmental (backbone tangent) orientation.61 C


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Macromolecules The normalized DQ buildup function InDQ(τDQ) is partially relaxation corrected (see eqs 3 and 4). The decay of the full-echo relaxation function IΣMQ(τDQ) used for this purpose is only governed by the time scale of segmental motions, with highest sensitivity to motions in a time range covering a few decades centered around τDQ.49 In the absence of inhomogeneities and motions of the order of τDQ or slower, which is realistic for networks far above Tg (but not for reptating linear polymers), InDQ(τDQ) reaches the value of 0.5 in the long-time limit and can be fitted to46

InDQ (τDQ , Dres) =

⎞ 1⎛ 2 ⎜1 − exp − Dres 2τDQ 2 ⎟ ⎠ 2⎝ 5

{

}

either the accumulated dipolar phase or C(t) according to eq 5 or 8, respectively, by finite-difference interval summation.

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RESULTS AND DISCUSSION Powder Samples. In our previous work,28 we only compared nDQ buildup functions for PB55k in bulk, 20 nm, and 60 nm pores at different temperatures. These data clearly indicated a significant enhancement of time-averaged local chain order as compared to the bulk in regimes III (reptation) and IV (free diffusion) of the tube model,51 while at lower temperatures, where Rouse motions prevail, the confinement did not have a significant effect. The data were tentatively analyzed in terms of a simplistic two-component model, assuming bulk-like behavior in the core region and a “networklike” behavior described by eq 6 for a region close to the pore surface. The latter should not be taken literally: “network-like” merely means that C(t) is assumed to exhibit a long-time plateau, which of course does not arise from cross-links but from a surface-induced anisotropy. The fitting results were compatible with a surface layer thickness of the order of 2.5 nm, which is of the same order as the tube diameter for PB (dT ≈ 4.1 nm). Figure 3a now shows corresponding DQ buildup and ΣMQ decay functions for PB55k and 196k in bulk and 20 nm confinement at the extremes of the studied temperature interval. At 263 K, the NMR response is sensitive to motions in the tube-model regimes I (Rouse) and II (constrained Rouse). While the initial parts of the nDQ curves hardly differ for bulk and confinement, systematic differences in the longtime relaxation behavior of the DQ and ΣMQ signals are observed even at this temperature. This is in line with our fieldcycling relaxometry results,32 from which it was concluded that confinement also affects the Rouse modes (but not τα). Obviously, this modification does not go along with significantly modified orientational order reflected by InDQ. At the highest temperature (383 K), very significant differences, in particular in the DQ buildup curves, once again demonstrate persistent, surface-induced segmental orientation. For a more quantitative analysis, we note that the normalization procedure used to calculate nDQ buildup curves (see eq 4) is not strictly valid in systems with large dynamic heterogeneity, in particular not at long τDQ. This is simply because both constituent functions, IDQ and IΣMQ, are sums of different components, for which a simple division cannot give the sum of the InDQ functions of the components.66 In order to take this into account, we employ a still simplistic simultaneous two-component fit to the individual signal functions,66 assuming for IDQ of the surface region again eq 6, but damped with a stretched exponential ∼exp{−(t/T2)β}, while the associated IΣMQ is just the same stretched exponential. For the bulk-like core, we do not fit any parameters but use interpolated and smoothed, separately measured experimental results for IDQ and IΣMQ from bulk samples.48 This results in four independent parameters: a fraction f, Dres,s, T2, and β of the surface layer. As will become more apparent below, such a crude model can only work for high-MW samples in which reptation motion does not lead to significant exchange between the surface layer and the bulk-like core. We thus fitted only the data for the highest MW samples, but stress that even for these samples the approximation is crude. The fit results in Table 1 demonstrate, again, that 60 and 20 nm confinement data are on average consistent with a surface layer height hs of 2.6 ± 1.2 nm and a

(6)

Its short-time behavior is parabolic InDQ (τDQ ) ≈

1 Dres 2τDQ 2 5

(7)

Even in the presence of slower motions, fits to the initial rise yield the time-averaged residual dipole−dipole coupling constant Dres, which is a measure of the residual segmental anisotropy at time τDQ. Formally, it can be shown49,62 that the signal functions depend only upon Deff (see eq 5), and the orientation autocorrelation function (OACF) of the segmental orientation C(t ) = 5⟨P2(cos θt + τ)P2(cos θτ)⟩τ ,ensemble ∝ InDQ (t = τDQ )/τDQ 2

(8)

The second line represents a short-time approximation49 valid for DQ times up to about 0.5 ms at the lowest experimental temperatures. It is thus seen that InDQ(τDQ ≤ 0. 5 ms) ∼ Dres2 is a direct measure of the residual segmental orientation as described by a dynamic order parameter of the backbone Sb = C(t = τDQ) 1/2 ∼ Dres, which represents an average over all times t < τDQ. In order to probe the complete C(t) over many decades in time, time−temperature superposition (TTS) was applied in our previous work on homo-PB melts.47−50 A similar approach was used to analyze field-cycling T1 relaxation data on a pore-confined sample32 to obtain and interpret apparent shift factors that were at higher temperatures found to be different from bulk melt values. Because of the restrictions due to the short-time approximation, TTS as applied to our data requires a priori knowledge of the shift factors, which have not been measured precisely enough. Also, despite the T2 filter, residual background signal still distorts the initial rise of InDQ to some degree, resulting in data that are not good enough for TTS. Finally, as we are probably dealing with an inhomogeneous scenario, with interface- and bulk-related C(t) being potentially different, the effect of timeaveraging effects on the overall apparent C(t) are as yet unexplored. Thus, we refrained from applying TTS in the present work. Computer Simulations. A simulation program calculating the MQ NMR response functions (eqs 1−4) of a chain undergoing pure reptation (ignoring Rouse motions) was implemented in standard C language and run on a Windows desktop computer. For details see ref 63. The program is based upon a finite-difference integration over trajectories generated by a Monte Carlo algorithm; its features are described in the Simulation Model section and in the Appendix. Technically, one-dimensional random-walk dynamics64,65 of a chain moving along a primitive path was simulated by equal-probability forward or backward jumps of all chain segments, using a Poisson distribution of waiting times in each “raster” position i of a single tagged segment which obeys

tw, i = − τw ln(1 − zi)

(9)

where zi is a random number evenly distributed between 0 and 1. The absolute time scale of the simulation is set by the parameter τw. MQ NMR data as well as correlation functions C(t) were generated as averages over 104 to 5×104 individual trajectories, also varying randomly the position of the tagged segment within the chain. The calculation is based upon the assignment of a specific dipolar tensor orientation to each position on the primitive path and then calculating D


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Dres,s/2π of the order of 360 ± 93 Hz. The significant scatter is likely related to fitting instabilities related to the fact that the data are not quantitatively represented by the fits, partially because there are still distortions of the initial DQ buildup related to variable sample quality. The confinement-induced coupling is to be compared with a maximum value of Dres/2π ≈ 150 Hz for bulk PB196k at 343 K48 and even significantly lower values found for lower MW and higher temperatures. The wall surface thus shows a more significant effect on PB segmental orientation than the polymer chain tube constraint. In order to illustrate the trends for a large MW range, Figure 3b shows nDQ buildup curves for the highest studied temperature of 383 K in bulk vs confinement. For short τDQ, the initial-rise approximation is reasonably valid for InDQ, such that we focus in Figure 3c on single-point intensities for τDQ = 0.5 ms. It must be kept in mind that this choice represents a compromise between using an as-small-as-possible value, needed to ensure the validity of eqs 7 and 8, but avoiding initial-rise distortions arising from residual membrane-related background signal. The bulk data tend to zero for MW below 30K, due to the isotropization in regime IV. In confinement, residual order also decreases first upon reducing MW but reaches a minimum and rises again for the shortest MW. Notably, the minima for 60 and 20 nm confinement are found at different MW. At lower temperature (343 K), the minimum for the shown 20 nm case shows a trend toward lower MW. For a first interpretation, we note that the mean-square displacement at 0.5 ms, as estimated from the diffusion coefficients measured previously,42 is of the order of the respective pore diameters for the MW at which the minimum is found. This suggests that the minimum in the apparent Dres is due to an intermediate exchange-type phenomenon. For lower MW, the chains obviously sample the whole pore in transverse direction within 0.5 ms, whereby the nDQ intensity reflects a fast diffusive average. On the other extreme, in the high-MW limit the reptation motion is sufficiently slow, such that MQ NMR is able to reveal the dynamic heterogeneity of the samples. The observed temperature variation, with overall slower diffusion at lower temperature and the resulting shift of the minimum to lower MW, is in keeping with this interpretation. Oriented Samples. The above interpretation will now be reinforced by data taken on oriented sample stacks at variable orientation Ω of the membrane normal, which coincides with the pore long axis (see Figure 1). The nDQ buildup curves in Figure 4 should first provide a demonstration of the relevant differences between low- and high-MW samples. Figure 4a shows that the apparent residual couplings of PB in the canonical orientations (0° vs 90°) differ by a factor of about 2 for low MW, while they are hardly different for high MW. Also, the magic-angle orientation (Ωm = 54. 7°) is significant in that rather low signal is found for the low MW. In order to rationalize the low-MW behavior, we have also studied different molecular liquids in 60 nm pores (see Figure 4b). Just as in the case of PB (see Figure 2), we did not find any indication of immobile, long-time adsorbed species. Also, qualitatively similar results were measured at elevated temperature. This indicates that we are also in the fast-exchange limit. Octane represents a short-chain system which interacts, in the same way as PB, with the pore wall mainly by van der Waals interactions. The data in Figure 4b show an indication of twostep buildup, which we interpret as contributions from the core and the terminal part of the n-octane chain. Yet, the orientation

Figure 3. Results from powder-averaged samples: (a) 1H MQ NMR signal functions of PB55k and 196k in bulk and confined to 20 nm pores at two different temperatures. For 263 K, the lines are corresponding nDQ curves, and for 383 K they represent fits as explained in the text. (b) Normalized DQ buildup curves measured at 383 K for various MW in bulk and in confinement. (c) Short-time single-point nDQ intensities at τDQ = 0.5 ms as a function of MW. The arrows indicate the MW with the lowest values. The error bars represent variations in cases where more than one (typically 2−3) sample batches were prepared. The sometimes significant error, and deviations from the general trend, can be explained by residual background signal and/or variations in pore size.

Table 1. Component Decomposition of 1H MQ NMR Signal Functions for Confined PB sample

T (K)

f (%)

hsa (nm)

Dres,s/2π (Hz)

196k @ 60 nm

343 383 343 383 343 383 343 383 343 383 383

13 34 10 13 7.5 16 45 57 38 44 55

2.0 5.6 1.5 2.0 1.1 2.5 2.6 3.4 2.1 2.5 3.3

390 255 407 230 273 326 442 399 510 449 270

88k @ 60 nm 55k @ 60 nm 196k @ 20 nm 88k @ 20 nm 55k @ 20 nm a

The surface layer height hs is calculated from the fraction f based upon a cylindrical-shell geometry.

E


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between different samples. This can explain the variability of the results for the magic-angle orientation, which is rather vulnerable to angle uncertainties (in contrast, P2(cos Ω) is flat for the canonical orientations). In order to interpret the similarity of the nDQ data for differently oriented large-MW samples (see Figure 4), we highlight the results of ref 18, where deuterated PDMS absorbed as thin film in oriented but less well-defined disordered AAO pores (Whatman Anopore) was studied by 2 H NMR. For the case of submonolayer coverage, Ω-dependent spectra, in particular those at 90°, were found to be consistent with a radial (i.e., planar 2D powder) distribution. Such a situation arises if the surface-confined segmental dynamics is uniaxial with respect to the pore surface normal.10,11,17,18 On this basis, comparing theoretical buildup curves based upon eq 1 for Ω = 0° vs 90° for a planar powder, a factor of 0.85 between the apparent Dres values taken from the initial rise is expected. This is compatible with the PB196k data of Figure 4a, for which we need to keep in mind that half of the polymer resides in the isotropic bulk-like core region. Turning again to the general trend for a large MW range, Figure 5a shows nDQ buildup curves for all 10 investigated polymers in the two canonical orientations of 20 nm pores. We point out the fact that the long-time intensities of the 24k sample are the lowest among all samples. We remind that for MW of 24k diffusive averaging between surface and core region occurs on the τDQ time scale. Short-time intensities (τDQ = 0.5 ms) are plotted in Figure 5b. Note that the oriented 24k@20 nm sample exhibited an unexpectedly strong initial DQ build-up probably related to residual background signal (not readily visible in Figure 5a). These points were thus discarded for Figure 5b. Overall, the data reproduce the trend and in particular the minimum observed for the powder samples (Figure 3c). Notably, and in agreement with our assessment above, we observe a transition from nearly equal values for the two orientations of large-MW samples to reduced values in the 90° orientation at low MW.

Figure 4. Results from oriented membrane stacks: Normalized 1H DQ buildup curves measured at variable orientation for (a) PB11k and PB196k samples in 20 nm pores at 383 K fitted to eq 6 for τDQ < 0.5 ms and (b) n-octane and methylene chloride in 60 nm pores at room temperature. A simultaneous fit to the latter data set (solid lines) based upon eq 1 gave Dres/2π = 8.2 Hz and a standard deviation of 13° from the average orientation, fixing the average angles for the canonical orientations at 10° and 80°.

effect on the buildup curves, with a factor of about 2 in the apparent residual coupling taken from the initial rise, is rather similar to PB11k, but with the residual coupling reduced by roughly a factor of 10 (note the x scale). This demonstrates that polymer chains experience a comparably enhanced wallinduced orientation effect, possibly due to topological restrictions arising from the chain structure. Methylene chloride (CH2Cl2), in turn, features an isolated 1 H spin pair, little shape anisotropy, and an electric dipole moment. Its interaction mode with the surface is thus expected to differ, but the overall residual-orientation phenomenon; i.e., the angle dependence is the same. Details on the differences between these and other low-MW liquids will be reported elsewhere. Here, we take advantage of the spin-pair response, which means that the nDQ buildup can be described exactly by eqs 1 and 5. This immediately explains the observed sin2 oscillations and the factor of 2 between 0° vs 90° orientations, which suggests uniaxial symmetry following P2(cos Ω). The damping at longer times and the nonzero intensity observed for Ω ≈ 54.7° can be explained by an orientation distribution in our imperfect membrane stacks. We fitted the CH2Cl2 data for the three angles simultaneously with a numerical average of eq 1 over a normal distribution of angles with standard deviation ΔΩ centered around Ω, also adjusting the latter in order to account for an overall tilt (angle inaccuracy) of the stack. The fits are not perfect, which is not surprising considering that the actual orientation distribution function is not known. Since only stacks of a few membrane pieces with irregular shapes are studied, the orientation parameters must be expected to differ

Figure 5. (a) Normalized 1H DQ buildup curves measured for PB in 20 nm pores at 383 K and 0° and 90° nominal orientation for various MW. (b) Corresponding short-time single-point nDQ intensities at τDQ = 0.5 ms as a function of MW. F


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Figure 7. Normalized 2H DQ buildup and ΣMQ decay data measured for PDMS21k in 26 nm pores (same sample as in ref 37) at 383 K at 0° and 90° nominal orientation. A simultaneous fit to the nDQ data (solid lines), based upon a linear combination of eqs 6 and 1 including an orientation distribution of the latter, gave a nonoriented networklike fraction of 24% with χres/2π = 148 Hz (dashed line) and a diffusively averaged oriented fraction with χres/2π = 26 Hz and a standard deviation of 16° from the average orientation, fixing the angles for the canonical orientations at 10° and 80°. The orientation distribution was neglected for the dotted lines.

Figure 6. Model explaining the two MW domains with qualitatively different angle dependence (Figure 5b). For low MW, fast diffusive averaging leads to an overall residual dipolar tensor orientation parallel to the pore. For high MW, a bulk-like core region with an isotropic distribution of tube segments and the corresponding local order tensors coexists with an interphase layer, within which the localized segmental motions lead to an effective tensor orientation perpendicular to the wall.

consequence of the lack of isotropic large-scale motions on a time scale up to milliseconds. Note that ultimately, due to the similarity of the respective responses, the data do not allow for a distinction between an isotropic distribution of residual tensor orientations and a planar powder.18 In any way, the comparably larger residual coupling of ∼150 Hz is indicative of end-fixed chains, as unentangled PDMS21k does not show any measurable nDQ intensity.48 On the other hand, the freely diffusing fraction with a residual coupling of only 26 Hz probably resides mainly in the pore center. The existence of a well-defined residual coupling along with the clear-cut angle dependence suggests that the residual orientation of these segments must arise from the pore constraint, but its low value suggests that the orientation effect does not arise from direct interactions with the pore wall, but from transient interactions with the more strongly adsorbed surface layer. These results are in tune with and nicely complement the NSE findings of ref 37 measured at about the same temperature. That data, characterizing dynamics on the rather fast time scale of less than 150 ns, were analyzed to consist of a 25% bulk-like contribution, and a surface-related interphase within which the dynamics appears tube-confined (despite the absence of entanglements) and long-wavelength Rouse modes are suppressed to variable degrees. The latter was interpreted to comprise partially adsorbed chains forming loops and other chains in dynamic interaction with these loops. Our data now show that about two-thirds of the interphase chains as identified by NSE (50% of all chains) must be able to undergo large-scale diffusion, while the actually adsorbed subfraction is identified by NMR to comprise only 25% (corresponding to a ∼2 nm layer). These chains are most probably transiently adsorbed on a time scale of up to a few milliseconds. For NMR, the 50% fraction of freely diffusing but on a shorter time scale dynamically interacting chains of course undergoes fast exchange with the 25% bulk-like fraction. Thereby the apparent orientation correlations due to interaction with adsorbed chains arises. It is noted that we were not able to detect a fraction of more strongly adsorbed PDMS behaving as a nearly rigid solid, which

The minimum, indicating the onset of increasingly fast surfacecore exchange upon decreasing MW, goes along with the onset of angular discrimination. On the basis of these observations, we thus suggest a model for the observed phenomena, as sketched in Figure 6. We distinguish the fast-exchange regime at low MW and the slowexchange (quasi-static) regime at high MW. In the former case, the residual, time-averaged dipolar tensor orientation is uniaxial and parallel to the pore long axis, while in the latter case a core region with an isotropic distribution of tube segments and the associated residual dipolar tensors coexists with a surface region (of ∼2.5 nm thickness) within which the residual dipolar tensor is oriented normal to the surface. The latter is of course also a consequence of time averaging due to faster localized Rouse dynamics. This model in phenomenological agreement with experimental results on thin-film samples17,18 and recent simulation studies of PB films confined by graphite walls.10,11 Stronger Surface Interactions: PDMS. PDMS, with its locally polar backbone, is well-known to adsorb rather strongly to hydrophilic oxide surfaces such as silica or alumina.19,37,67 This could prevent the diffusive exchange and is here demonstrated on the example of a PDMS sample previously studied by neutron spin echo (NSE) spectroscopy.37 PDMS21k is effectively unentangled; thus, fast diffusive exchange is a priori expected. The 2H DQ buildup data plotted in Figure 7, which are also expected to follow eq 1 exactly, indeed show the expected features, again with features of an orientation distribution. Note that in the caption we report the residual quadrupolar rather than dipolar coupling (χres). However, a stable two-component fit to the data (see the caption of Figure 7) reveals that the data are only compatible with a PDMS fraction of 25% that does not change its response upon changing the sample orientation, while the remaning 75% is undergoing fast exchange and thus exhibits a strong angle dependence in analogy to the low-MW limit in PB (see Figure 4). The former fraction thus appears to be isotropic, and could be fitted by eq 6, suggesting “network-like” behavior, as a G


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tube segment,48 whose long axis is parallel to the time-averaged chain contour. The simulation parameters are calibrated so as to reproduce the known correlation functions C(t) of bulk PB for the relevant MW range in regimes III and IV of the tube model.48 In fact, a crossover effect also provides a partial match in regime II. Simulation runs using the so-obtained parameters provide a rather good representation of NMR signal functions of bulk melts. Details on the calibration procedure and a comparison of experimental and simulated NMR data are provided in the Appendix. It must be stressed that a relevant limitation of the simulation is the restriction to a single coupling. Normally, abundantly protonated segments feature multiple dipolar couplings, rendering the spin dynamics homogeneous. This leads to a washing out of oscillations and to DQ buildup curves that reach a maximum intensity of 0.5 and show little to no oscillations, in particular also for oriented samples. This is readily apparent from the data in Figure 4, where PB11k and octane (multiple couplings) should be compared with CH2Cl2 (spin pair). Powder averaging alleviates a part of the problem. This is why the simulated data are not expected to provide a good description of long-time data, in particular not for oriented samples. To capture the effect of pore confinement, the PP is restricted to a cylindrical region simply by excluding randomly generated orientations of PP segments that would cross the cylinder wall (see Figure 8). Following our model idea, we assign a specific dipolar tensor orientation (parallel to the wall normal) and a variable, possibly increased residual coupling to those points on the PP that lie within a surface layer region of defined distance hs from the cylinder wall. This is in tune with experimental results on thin-film samples17,18 and recent simulation studies of PB films confined by graphite walls.10,11 All other simulation parameters are kept at the values calibrated for the respective bulk polymer. We did not implement the fact that the chains are slowed down in the surface layer due to increased friction.42 The reason for this is that within the confines of the given model (in which the whole chain follows the random walk of the tagged point) it is not possible to consistently implement a locally slower segmental motion, not speaking of the fact that the neglected Rouse dynamics within the surface tube segment is complex, as noted above. As the overall effect of enhanced surface friction on long-time reptation was found to be moderate,42 we expect an only moderate systematic error. A comparison of experimental and simulated DQ and ΣMQ signal functions for two different molecular weights in bulk vs 60 and 20 nm confinement (powder samples) is shown in Figure 9a. We show results from two different simulation models which differ in the two free parameters, i.e., the surface layer height hs and the interphase-related residual coupling Dres,s. The shown cases roughly cover the range of value pairs for which reasonable qualitative agreement with the data was found. Generally, the ΣMQ relaxation is overestimated by the simulation, which we attribute to the simplification related to neglecting the complex Rouse dynamics within the surface region (in reality, we suppose that only the first molecular surface layer has a rather high Dres,s in dynamic exchange with weakly coupled segments). The general overestimation of the DQ buildup is due to the mentioned limitation to pair couplings, as shown below. The single-point intensities (τDQ = 0.5 ms) in Figure 9b show a trend that compares very favorably with the experimental data

for the case of submonolayer coverage on alumina was shown to be characterized by a residual quadrupolar coupling of about 10 kHz.18 Such a comparably broad spectral line (as compared to the narrow signal providing the data in Figure 7) would have been observable given our signal-to-noise. In fact, the data for higher surface coverage18 suggest that such a component is in dynamic exchange with or is swollen and thus further mobilized by excess PDMS. As a spectrally broad component is, in contrast, observable for the case of PDMS on silica,67 it is concluded that PDMS adsorbs more weakly and transiently to the pore walls of the AAO membranes used here. Simulation Model. Considering the model introduced above (Figure 6), it is clear that the two-component fitting approach used to analyze the high-MW PB data (Figure 3a and Table 1) is necessarily crude. Using eq 6, thus implying longtime stable orientation correlations within the surface layer, must be unrealistic, as some degree of averaging due to reptative back-and-forth motions into the bulk-like core region is expected at the investigated temperatures. Thus, the surfacerelated Dres,s and the layer height hs are subject to systematic error. We were thus interested whether data as those shown in Figures 3 and 5 could be generated on the basis of a coursegrained model as suggested in Figure 6. The principle of our simulation approach is sketched in Figure 8. In order to capture the NMR signature of a reptating chain, the program generates a primitive path (PP) with the appropriate number of segments Npp, subdivided into ain fact arbitrarynumber of steps, which we varied between 2 and 50. This step number per PP segment, together with the waiting time given by eq 9, determines the absolute time scale of the simulation. A randomly chosen tagged position on this PP is then subjected to a 1D random walk, which moves the whole chain back and forth. Upon reptation, new PP segments with random orientation are created at the termini in forward direction, while those vacated by the chain end are deleted, thus removing the tube memory. For the NMR calculation on the basis of eqs 1−5, each point on the PP is assigned a dipolar tensor orientation that is taken to be parallel to the actual PP segment. This is consistent with the notion that Rouse motions preaverage a segmental dipolar tensor to ultimately reflect the orientation of the respective

Figure 8. Sketch of our simulation model based upon a 1D random walk on a primitive path, generated as a 3D random walk within a pore, taking the effective dipolar tensor orientation to be either parallel to the PP segments for bulk-like parts, or perpendicular to the pore wall within the interphase. H


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Figure 10. Simulation results for oriented samples, assuming 10° and 80° for the average canonical orientations and 15° standard deviation. (a) Short-time single-point nDQ intensities at τDQ = 0.5 ms as a function of MW, to be compared with Figure 5b. (b) Simulated normalized DQ buildup curves for PB196k in 20 nm pores (lines) for different orientations as compared to experimental data (symbols).

for many samples, shorter-time data was still somewhat distorted by the rigid 1H background of the AAO.

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SUMMARY AND CONCLUDING DISCUSSION We have studied a wide range of unentangled and entangled poly(butadiene) confined to weakly interacting hydrophilic cylindrical channels of 60 and 20 nm diameter in alumina.28,32,40 Our NMR method is sensitive to the degree of time-averaged orientational order of chain segments and thus to the dynamic regimes of the tube model. Confined samples exhibit a notable increase of average segmental orientation, and the data can be fitted on the basis of two-component model, in which a bulk-like cylinder core coexists with a surface region of a few nanometers thickness, within which the segmental orientation is considerably enhanced due to the surface constraint. Because of the roughly consistent length scale, this surface region is likely associated with the tube segment located at the surface.54 Orientation-dependent experiments on macroscopically oriented samples, and a comparison with confined representative low-molecular liquid samples, have revealed that the NMR response of short chains represents a diffusive average with correspondingly uniaxial overall symmetry of the residual segmental orientation. In contrast, for long chains the confined surface region and the bulk-like core coexist on the NMR experimental time scale of the order of milliseconds. The surface region is characterized by a residual orientation tensor that is parallel to the surface normal, again as a result of averaging over fast, local Rouse motions that are not resolved by the technique. Our previous results on the only weakly affected large-scale diffusion42 were interpreted in terms of transient direct wall contacts in a single molecular layer, which lead to an enhanced apparent friction. Such direct wall contacts are necessarily characterized by a “flat-on” orientation of the respective chain segment, for which an order tensor orientation perpendicular to the surface normal can be assumed. Yet, Rouse motions (tube model regime I) lead to a localized fast-exchange situation,

Figure 9. (a) Experimental powder-averaged 1H MQ raw data for PB55k and 196k at 383 K in bulk and confinement (symbols) as compared to simulation results (lines) assuming two different interphase models (A: hs = dT,PB = 4.1 nm, Dres,s/2π = 480 Hz; B: hs = 1.5 nm, Dres,s/2π = 1.6 kHz). (b) Simulated powder-averaged short-time single-point nDQ intensities at τDQ = 0.5 ms for 383 and 343 K as a function of MW, to be compared with Figure 3c.

in Figure 3c; in particular, the shallow minima for 60 and 20 nm confinement are found at about the same MWs. Also, lowering the temperature shifts the minima to lower MW. Only the depth of the minima is underestimated, which we attribute again to the lack of realistic account of the local chain dynamics within the surface layer. Both limiting model cases, spanning a hs of 1.5−4 nm and a Dres,s/2π of 1.6−0.48 kHz, compare favorably to the fitting results of the simple two-component model (see Table 1). Finally, simulations of oriented-sample data are shown in Figure 10. The single-point DQ intensities in Figure 10a can be compared to the corresponding experimental data in Figure 5b. We have tried to account for the on-average imperfect ordering of the real samples with an orientation angle distribution as estimated from the experimental data in Figures 4b and 7. The agreement is less good, in particular for the 0° orientation. Again, the main reason for the now more serious discrepancies is the still largely oscillatory behavior, due to the restricted range of dipolar couplings in the 0° orientation (even for the 2D powder case, all interphase tensors are then in the 90° orientation), which results in only little interference. This is corroborated by a comparison of experimental and simulated data for the high-MW PB196k (slow-exchange limit) in Figure 10b. Deviations are already apparent at the short τDQ of 0.5 ms. A comparison of data for shorter times was not conclusive as I


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Figure 11. Mapping of simulation results onto experimental 1H MQ NMR data for entangled PB melts at different temperatures.47,48 Orientation autocorrelation function C2(t/τe > 1) (upper row) and MQ NMR signal functions (lower row) for three different molecular weights. The simulated C2(t/τe > 1) are scaled by a factor of 2.5 to obtain the best match with the (unscaled) experimental data.

specific (and long-time averaged) observation of peculiar T1 NMR relaxation time dispersions for different kinds of confined melts is consistent with largely enhanced orientation correlations as studied in detail herein. Yet, the detailed picture remains to be not fully conclusive.27,31,32

resulting in a symmetry that is uniaxial with respect to the surface normal.17,18 Within the finite volume of the respective surface-related tube segment,54 the time-averaged orientation is found to be governed by the surface rather than the local tube orientation, as concluded from the rather large surface-related degree of orientation and its orientation distribution derived from the data. The only 10-fold enhanced friction coefficient within the first transiently adsorbed molecular layer concluded from our diffusion data42 still ensures sufficiently fast dynamics. The Rouse dynamics within the surface region is thus likely complex, and it remains to be seen whether a transient-pinning picture is consistent with recent neutron scattering results on other polymers confined to alumina.29,37 Our findings for weakly interacting chains are corroborated by simulation data, implementing our course-grained model picture into a random-walk reptation simulation of NMR signal functions. For the case of a more strongly interacting but unentangled polymer, poly(dimethylsiloxane), we found a clear two-component response associated with a larger fraction of freely diffusing and a smaller fraction of transiently pinned chains. These results are in nice agreement with and complementary to a neutron scattering study of the same sample, in which faster Rouse dynamics (tube model regimes I and II) was probed.37 As our approach only provided a time-averaged picture on the length scale of the tube diameter, a remaining challenge for modeling the observed confinement effect is certainly the account for the likely complex Rouse dynamics within the surface layer as influenced by direct segment-wall contacts. This is ultimately possible only with sufficiently fine-grained computer simulations, the result of which could then be fed into the appropriate NMR and neutron-scattering theory. Such an endeavor would be particularly valuable with regards to elucidating the origin of the debated “corset effect”.20 This

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APPENDIX. MAPPING OF REPTATION SIMULATIONS Reptation is simulated as a 1D random walk on the primitive path (PP). The number of PP segments Npp (which is identical to the number of entangled segments Z) and their length are taken from the samples’ MW using the known Me ≈ 1.9 kg/mol and from their end-to-end distance based upon the known unperturbed coil dimensions in the melt (4.1 nm), respectively. 48,68 The program generates a PP as an unperturbed (not self-avoiding) random walk of fixed-length PP segments by randomly choosing the orientation of each new PP segment. A reptation trajectory is then generated by equalprobability jumps of a tagged point on the PP (varying the number of steps per PP segment), pushing the whole chain backward and forward. The simulation time scale arises directly from the randomized waiting time in each point calculated by eq 9. In order to match multiple trajectories with random time points, each time axis was projected (interpolated) onto a common discrete time axis. The relevant simulation parameters are adjusted so as to reproduce the experimental C(t/τe) published in ref 48 (see Figure 11, top row). This concerns its magnitude as well as the time scale. As to the time scale, τw in eq 9 was determined to provide optimal overall match of the disentanglement times τd in the relevant MW range. The latter is determined by the point on C(t/τe) at which it follows a power law ∼t−1.5,48 this being equivalent to a tangent slope of −1.5 in a log−log plot (see J


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experimental and simulated MQ NMR signal functions is remarkable. The slight overestimation of the initial DQ buildup at the lower temperature is due to the simplification of considering only a spin-pair response.

Figure 11, top row). Based upon the largest-MW sample (PB196k) with Npp = 101.55 as a reference, τw was computed as τw(Npp) = 0.26 μs × (Npp/101.55)1.2

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(10)

This specific relation is valid for the case of 10 steps per PP segment and 383 K. At 383 K the entanglement time τe equals 0.04 μs.48 This time is used to normalize the time axis of the orientation autocorrelation function C(t/τe), eq 8, and marks the time above which our simulation provides valid results. For other temperatures we used the temperature dependence τe(T) as given by the rheological shift factor48,50 to rescale τw. The Npp scaling in eq 10 is unusual and requires an explanation. The adjustment of the simulation time scale is based upon the argument that the time scale of chain dynamics is governed by the segmental relaxation time τs, which is in turn proportional to monomeric friction coefficient.51 In the simple reptation model,52,69 the famous scaling prediction for the disentanglement time τd ∼ Npp3 arises from the argument that the curvilinear diffusion time of the whole chain along the PP is proportional to the added friction of all monomers; i.e., it is proportional to the length of the PP (∼Npp). The tagged point on the PP must follow this time scale, which would require an exponent of 1 in eq 10. Experimentally, of course, a relation τd ∼ M3.2−3.4 is commonly found and explained by shortcomings of the simple reptation picture (requiring additional mechanisms such as constraint release or contour length fluctuations). In order to accommodate this, the exponent of 1.2 was determined to provide the best overall match of the simulated τd and the NMR values published in ref 48. Since τw takes low values for shorter chains, the simulation becomes more time-consuming because a correspondingly longer trajectory must be simulated. In order to speed up the simulation, we reduced the step number per PP segment down to the value of two. Since we simulate a 1D random walk and the step number measures distance (rms displacement) along the PP, τw is correspondingly multiplied by the square of the reduction factor. The data in the top row of Figure 11 present an overlay of C(t/τe) simulated for different step numbers and rescaled τw, showing good overlap and thus a coverage of many decades in time. The top row of Figure 11 also shows that the best match of the C(t/τe) is obtained by another empirical scaling of its absolute value by a factor of 2.5. Even though the simulation exhibits the expected −0.5 scaling exponent of C(t/τe) for regime III of the tube model51 as initially predicted by de Gennes,52 it also shows a range of lower slope at shorter times, which is an onset/crossover phenomenon. Conveniently, given the additional y-scaling, this leads to an enhanced range of agreement also for the flatter part at short times corresponding to regime II (constrained Rouse), the physics of which is not part of the model. For the simulation of the NMR signal functions (Figure 11, bottom row), the last parameter to be fixed is the effective residual coupling constant in eq 5. With C(t/τe) covering approximately regimes II−IV, the coupling constant must include the pre-averaging due to free Rouse motions in regime I. Empirical adjustment for best match between simulated and experimental MQ signal functions lead to a value of Deff/2π ≈ 400 Hz corresponding to C(t/τe = 1) . This is rather close the experimentally determined value of 360 Hz48 and shows that the crude simulation model captures the NMR response of a reptating polymer rather well. The agreement between

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (K.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank S. Ok for his help in the initial stages of the project, L. Willner, J. Maiz, and C. Mijangos for the preparation of the PDMS-based AAO sample, and M. Sattig and M. Vogel for their advice concerning the random-walk simulation. Funding was provided by the DFG (SA982/4, STE1127/9, and KR 3929/1) in the framework of the priority programme SPP1369 “Polymer−Solid Contacts”.

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REFERENCES

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Chain Dynamics and Segmental Orientation in Polymer Melts

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