Recent NMR investigations on molecular dynamics of polymer melts in bulk and in confinement

Current Opinion in Colloid & Interface Science 18 (2013) 173–182

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Recent NMR investigations on molecular dynamics of polymer melts in bulk and in confinement E.A. Rössler a, S. Stapf b,⁎, N. Fatkullin c a b c

Universität Bayreuth, Experimentalphysik II, D-95440 Bayreuth, Germany Dept. Technical Physics II/Polymer Physics, TU Ilmenau, PO Box 100 565, D-98684 Ilmenau, Germany Institute of Physics, Kazan Federal University, Kazan, 42008 Tatarstan, Russia

a r t i c l e

i n f o

Article history: Received 17 March 2013 Accepted 18 March 2013 Available online 29 March 2013 Keywords: Polymer melts Polymer dynamics Relaxation Diffusion

a b s t r a c t Polymer dynamics in the melt state cover a wide range in time and frequency, for both molecular weights below and above the entanglement length. Nuclear Magnetic Resonance (NMR) offers a number of techniques that cover a broad section of this frequency range, with frequency dependent (i.e., magnetic field dependent) relaxometry providing the widest window. Combining fast field cycling techniques with frequency–temperature superposition has recently improved the understanding of polymer melt dynamics from the local to global range. At the same time, a detailed theoretical approach that separates intra- and intermolecular contributions to relaxation times has been developed. These methods are shown to improve the description of segmental dynamics in polymers, being related to time-dependent diffusion coefficients, and to distinguish between these two different relaxation contributions for a number of model compounds. The findings represent the foundation for a more thorough understanding of polymers under external restrictions and bear potential to provide a conceptually new access to biopolymer dynamics and interactions. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction In this review, we summarize new experimental results using NMR relaxometry and theoretical descriptions of the spin–spin dipolar coupling effects of polymer melts that have recently been presented in the literature and have shown to converge towards a more detailed description of polymer chain dynamics in the melt state. A particular aspect in these studies covers the relative importance of intra- and intermolecular dipolar interaction to relaxation for the most ubiquitous NMR active nucleus, the hydrogen atom. The relative contributions from local and global dynamics can be accessed by frequency– temperature superposition of experimental data that are compiled in a large temperature range. Typically a Larmor frequency range between 104 and 108 Hz is covered using Fast Field Cycling (FC) relaxometry, where the relative influence of a slow dynamics contributions around and above the entanglement length of the polymer is shown to depend on chain length for linear macromolecules. The different frequency dependence of intra- and intermolecular contributions to the relaxation rate have both been predicted theoretically for the two fundamentally different concepts of isotropic and anisotropic dynamics, and were experimentally proven by recent experiments employing isotopic dilution in perdeuterated polymers of varying concentration. Following a summary of the most important concepts of polymer melt dynamics in the light of NMR investigations (Section 2), the ⁎ Corresponding author. Tel.: +49 3677 693671. E-mail address: [email protected] (S. Stapf). 1359-0294/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cocis.2013.03.005

theoretical background is presented in Section 3. Experimental results for the dipolar correlation function and, derived from it, measures of the mean square displacement and the diffusion coefficient are discussed in Section 4. We conclude with a collection of recent developments and potential applications in bordering fields in Section 5 that discuss the behavior of polymer melts under external restrictions such as cross-linking, geometrical confinement and interfaces. 2. Polymer melt dynamics and relaxometry Field-cycling (FC) NMR relaxometry has become an important source of information on molecular dynamics in condensed matter, in particular, in liquids and polymer melts [1–3⁎⁎]. As will be demonstrated, the method allows to cover all relaxation regimes relevant to polymer dynamics, such as segmental or “local” dynamics, Rouse as well as entanglement dynamics. Regarding the polymer specific dynamics also multi-quantum (MQ) NMR [4,5⁎,6] provides similar information and both FC and MQ NMR have recently been compared in detail [7⁎,8⁎⁎]. Further NMR methods including field-gradient (FG) NMR may also contribute [9–12]. In contrast to rheology [13], for instance, NMR probes the dynamics on a molecular level and the term “molecular rheology” has been coined. Field-cycling can be achieved by electronic or mechanical “switching” of the polarization, relaxation and detection B field. The magnetic field B fixes the Larmor frequency via ω = γB (γ denotes the gyromagnetic ratio) and FC NMR relaxometry allows to measure the frequency dependence (dispersion) of the spin–lattice relaxation time T1 (or T2) by

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varying the relaxation field. A wider use of the electronic FC method became possible due to commercial availability of STELAR FC NMR spectrometers [2]. By using such spectrometers one typically can probe nuclear spin relaxation in the range of 10 kHz–20 MHz (for 1H) and relaxation times down to about 1 ms. In special cases including conventional NMR spectrometers the range can be extended up 500 MHz, or employing home-built FC spectrometers with appropriate earth field compensation frequencies even below 100 Hz may be reached [14⁎]. For an introduction to the principles of NMR relaxometry the reader is referred to the pioneering work of Noack [1] and of Kimmich and coworkers [2]. The most common way to employ FC NMR relaxometry is to perform 1H spin–lattice relaxation studies. Other frequently investigated nuclei are 2H and 19F. In the case of 1H and 19F the relaxation processes are driven by magnetic dipole–dipole interactions, while for 2H the relaxation mechanism is quadrupolar [15]. The measured relaxation rates R1 = 1/T1 (or R2) are given as linear combinations of spectral densities, which are Fourier transforms of corresponding correlation functions encoding information on the molecular dynamics. Applying frequency–temperature superposition (FTS) as is usually done in rheological studies of polymers and also found in good approximation in simple glass formers, [3⁎⁎,13,16,17], i.e., assuming that the spectral shape of the susceptibility is virtually not altered with changing temperature, up to ten decades in time/frequency can be covered [3⁎⁎,8⁎⁎,12,18–22]. By this local (“segmental”) and polymer specific relaxation can be probed in particular when molecular mass M is systematically varied. In the case of dipolarly coupled spins, e.g. protons, one has to distinguish intra- and intermolecular relaxation contributions [15]. While the first originates from interactions among the spins within a molecule and their fluctuations reflect reorientational dynamics, the latter stems from translational motion among the polymer segments. In many relaxation studies intermolecular relaxation is ignored so far as it is argued that due to the short range interaction relaxation contributions between different molecules become small. This is, however, in general not the case, and in the present review special attention is drawn to the role of the intermolecular dipolar interaction, in particular, to its frequency dependence which only recently has been systematically studied [23,24⁎⁎,25⁎,26⁎⁎,27⁎⁎]. It allows determination of the long-time diffusion coefficient [28,29⁎–31], and in the case of subdiffusive motion as found in polymer systems it may even provide the mean square displacement as a function of time [23,27⁎⁎]. Moreover, the possibly different frequency dependence of the intra- and intermolecular relaxation contributions allow discriminating different models of polymer dynamics [26⁎⁎]. The review is concerned with recent applications of FC 1H NMR relaxometry in order to study cooperative dynamics of polymers. The dipolar correlation functions and decomposition in its intra (reorientational) and inter (translational) part is described and their interpretation by means of polymer theories. The most successful approach of describing polymer dynamics is given by the tubereptation model of Doi and Edwards [32] which can be considered as a combination of the Rouse model [33] for non-entangled polymer chains (with molecular mass M below the entanglement molecular mass Me) and de Gennes' reptation model [34] for M > Me. The tube-reptation model is successful in reproducing semi-quantitatively the M dependence of transport coefficients like viscosity and translational diffusion coefficient. In order to account for weak deviations between model predictions and experimental or simulation data [35–38⁎] the model has been refined, e.g., by effects of contour length fluctuations and constraint release [31,32,39]. The first ones are due to motions of the chain ends, the latter are caused by the chains forming a non-static tube. However, there exist alternative approaches for describing the dynamics of entangled polymers such as the renormalized Rouse model and the polymer mode–mode coupled model which, as will be shown, lead to clearly different dispersion laws as compared to

the tube-reptation model [2,40–42]. The latter models have, if compared with the tube-reptation model, an analytical character, and can analytically describe continuous transition from the Rouse (i.e. non-entangled) regime of motion to the entangled regime. However, there exists a price for having the benefit of analytical mathematical models: the a priori postulation of an isotropic mechanism of motion for times longer than the segmental relaxation time, i.e., the absence of correlations for polymer segments displacements with their initial conformation. The tube-reptation model in this respect is strongly anisotropic, because it postulates, by means of the concept of diffusion inside the tube, a strong correlation of polymer segments spatial displacements with the initial conformation up to the terminal relaxation time. The tube-reptation model actually is of analytical nature, i.e., it can be defined through dynamical equation of motions, only for times longer than the Rouse relaxation time. It remains an experimental challenge to probe the dynamics on a microscopic scale involving all the relaxation regimes forecast by the tube-reptation model. Important experimental techniques addressing these issues and revealing details of the dynamics on a molecular level are neutron spin echo [43,44] dielectric spectroscopy, [45,46] and NMR, in particular FC NMR relaxometry. 3. Theoretical background Any macromolecule possesses a complex structure, and different interacting spins can also be related by quite different space separations. Even a monomer of macromolecule has a complex structure and contains many degrees of freedom. For example, the repeat unit of the simplest polymer, i.e. polymethylene [−CH2−]n has – neglecting electronic properties – nine degrees of freedom, which is already too much for an analytical description. For this reason, the analytical descriptions of polymer properties, and in particular of its dynamics, focuses on a coarse-grained model in terms of the entire polymer segments, Kuhn segments or Rouse segments [16,45,47–51]. As shown in [52], the Kuhn segments and Rouse segments are essentially the same and we will use all these terms as synonyms. Therefore intramolecular dipole–dipole interactions can be separated into intrasegmental and segment–segment parts. An essential progress has been made in the treatment of these different contributions and their role in different time/frequency regions of proton spin kinetics in polymer melts [23,24⁎⁎,26⁎⁎,52–54⁎]. For several decades in proton NMR of polymer melts it has been postulated that the main contribution to the different proton spin relaxation comes from intramolecular dipole–dipole interactions [5⁎,55]. Moreover, in considering intramolecular dipole–dipole interactions as a rule, with the exception of [56,57], the focus was set on contributions from nearest spins belonging to the same polymer segment. Actually even the treatment of contributions from intrasegmental magnetic dipole–dipole interactions to the spin kinetics was of a too phenomenological character at that stage. The point is connected to the rather complex character of the real atomic structure of the Kuhn segment. On the one hand, the Kuhn segment is the minimal unit for the coarse-grained analytical description of polymer dynamical properties for times t ≥ τs, where τs is the segmental relaxation time. On the other hand, the Kuhn segment consists of many atoms even for flexible polymers and therefore has at least on the order of 10–100 degrees of freedom, which are subject to thermal fluctuations (often called “local” motion although actually part of glassy dynamics) and are not explicitly described by the coarse-grained descriptions in terms of polymer segments. This situation is usually overcome by using a phenomenological concept of “residual part” of magnetic dipole–dipole interaction Hamiltonian [55] or, which is equivalent, the concept of “dynamical order parameter” [4,22]. In [25⁎], using the Mori–Zwanzig projection operator technique, it was shown that it is possible to derive the effective Hamiltonian for spin-segment coupling, which reflects dynamics of the polymer modes having wave length ranging from Kuhn segment up to the

E.A. Rössler et al. / Current Opinion in Colloid & Interface Science 18 (2013) 173–182

Flory radius, i.e. the typical polymer coil length scale. The additional part of this Hamiltonian is supplementing it to describe the total magnetic dipole–dipole Hamiltonian of all spins belonging to the same Kuhn segment connected with intrasegmental degrees of freedom and reflects the local motions, i.e. internal rotations about chemical bonds and local thermal vibrations. It was shown that contributions from local motions to the spin–lattice relaxation rate at frequencies ω τs ≤ 1 are negligible compared to contributions coming from the effective spin–segment Hamiltonian, i.e. from polymer modes [24⁎⁎,25⁎]. The importance of the intermolecular dipole–dipole interactions for the proton relaxation in polymer melts was first discussed theoretically based on scaling arguments and was demonstrated experimentally using the technique of isotopic dilution [53]. Afterwards, this approach was essentially developed both in theoretical and experimental aspects [23,24⁎⁎,25⁎,26⁎⁎,53,54⁎]. The proton relaxation rate R1 = 1/T1(ω) can be decomposed by a sum of the intra- and intermolecular rate: [15] R1 ¼ R1

intra

þ R1

inter

ð1Þ

where R1intra is the intramolecular and R1inter is the intermolecular contribution. Both contributions are frequency dependent and are always increasing with decreasing frequency, i.e., at lower magnetic field strength. The rates R1inter and R1intra can be extracted from experiments using isotope dilution technique [23,24⁎⁎,27⁎⁎]. Here, protonated chains are diluted in a melt of fully deuterated chains of similar molecular mass so that the intermolecular contribution to relaxation is considerably reduced (cf. Section 4.1). Regarding the intramolecular part, in the frame of the tubereptation model the dynamics of entangled polymer chains can be attributed to four relaxation regimes (I–IV) depending on the time scale of the motion [2,3⁎⁎,7⁎,8⁎⁎]. At shortest times the “local” or glassy dynamics still has to be added (regime 0) which usually is not included in polymer theories. Fig. 1 shows schematically the segmental mean square displacement br2(t)> and the rank-two reorientational correlation function g (2)(t), the latter probed by R1intra(ω), as predicted for liquid, non-entangled, and entangled polymer [2,3⁎⁎,8⁎⁎,60]. At the

0

I

glassy

free Rouse

t

t

-0.25

t

t

t

IV free diffusion

-1

simple liquid

cage

III

0.5

t

0.25

1

log

log g(2)(t)

t

II

constrained reptation Rouse

0.5

t

-0.5

M < Me

M > Me

2

τs

τe

τR

τd

log t Fig. 1. Schematic time dependence of the logarithm of the segmental reorientational correlation function of rank two g(2)(t) (red line) and the mean square displacement br2(t)> (black line) as a function of logarithm of time as expected from the Doi-Edwards tube-reptation model. For g(2)(t) the behavior of a simple liquid (dotted line), a non-entangled polymer (M b Me, dashed line), and entangled polymers (M > Me, solid line) is distinguished (adapted from ref. [8⁎⁎]). The time constant τs refers to the segmental (local) dynamics, τe to the entanglement time, τR to the Rouse and τd to the disengagement (terminal) relaxation time.

175

shortest times, t ≤ τα ≅ τs (regime 0), the dynamics is governed by the glass transition phenomenon for which the long-time tail of g (2)(t) can be described by a stretched exponential decay. The segmental correlation time τs can be identified with that of the α-process τα [3⁎⁎,8⁎⁎,17,19,20]. The absence of any slower dynamics is the characteristics of a simple liquid. In br2(t)> the ballistic short-time behavior (t ≪ τs) followed by a plateau signaling the cage effect is typical of glassy dynamics [61]. Beyond M > MR (molecular mass of a Rouse unit) chain connectivity starts to govern the segmental dynamics at long times and is described by the Rouse model (regime I) as reflected in the power-law behavior g (2)(t) ∝ t−1 and b r2(t) > ∝ t 0.5. Above the entanglement molecular mass Me and between the entanglement time τe and the Rouse time τR the chain feels the constraints of the tube formed by its neighboring chains (regime II). At longer times the polymer chain moves along the primitive path of the tube (regime III) and finally free diffusion is reached after the terminal or disengagement relaxation time (t ≫ τd, regime IV) leading to an essentially exponential cut-off in g(2)(t) and b r 2(t) > ∝ t 1. Within the tube-reptation model, in regimes II and III the reorientational correlation function independent of rank is given by [2,27⁎⁎,60] g

ð1;2Þ

2

ðt Þ∝br ðt Þ>

−1

:

ð2Þ

We note that an alternative model for describing the dynamics of entangled polymers, namely the renormalized Rouse model, [2] ends up with different relationships between the reorientational correlation functions and the mean square displacement. For example, in contrast to the tube-reptation model which predicts the m-independence (cf. Eq. (2)) the renormalized Rouse model, as any another isotropic model, suggests a relation [2,23,26⁎⁎,53] g

ð2Þ

h i2 ð1Þ 2 −2 ðt Þ∝ g ðt Þ ∝br ðt Þ> :

ð3Þ

Thus, the experimental investigation of the power-law behavior of correlation functions of different ranks together with the mean square displacement provides the chance to validate the different theoretical approaches. The ratio of intermolecular and intramolecular components R1inter/ intra R1 is sensitive to details of polymer dynamics and can thus be used to test concurring models such as the Doi-Edwards tube-reptation model or the renormalized Rouse model, both models aiming at describing polymer dynamics in the melt for high molecular mass, i.e., long chains. For the isotropic polymer dynamical models, i.e. those where the spatial displacements of polymer segments at times exceeding the segmental relaxation time do not correlate with the initial chain conformation (e.g. renormalized Rouse model and polymer mode–mode coupling model) this ratio is increasing with decreasing resonance frequency: [26⁎⁎] D E1=2 r 2 ð1=ωÞ T 1 intra ðωÞ ∝ b T 1 inter ðωÞ

ð4Þ

where 〈r2(t)〉 is the polymer segment mean squared displacement during time t and ω τ1 ≥ 1, τ1 is the terminal relaxation time, b is the Kuhn segment length. The situation is opposite for the strongly anisotropic case, where spatial displacements of polymer segments are correlated with the initial polymer chain conformation (e.g. tube-reptation model) for frequencies corresponding to time scales for which the polymer chain performs reptation inside a stable tube, i.e. for Doi-Edward limits II and III: [26⁎⁎] intra ðωÞ T1 d ∝ 1=2 ; T 1 inter ðωÞ r 2 ð1=ωÞ

where d is the tube diameter.

ð5Þ

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From the frequency dependence of R1inter one can, by applying Fourier transformation, determine the mean squared displacement D E of polymer segments ~r 2 ðt ¼ 1=ωÞ during time interval t = 1/ω. Under the condition of sub-diffusive translation as is the case in polyD E 2 mer melts, i.e., the mean square displacement follows ~r ðt Þ ¼ At α , where α b 2/3 it is found [23,26⁎⁎] " #2=3 D E γ4 ℏ2 ~r 2 ðt ¼ 1=ωÞ ∝ H ρs T 1 inter ðωÞ ; ω

ð6Þ

where γH is the gyromagnetic ratio of the proton, ℏ is Planck's constant, and ρs is the concentration of protons. In addition to the frequency range accessible to FC NMR experiments, time intervals up to about the millisecond range can be reached by proton Free Induction Decay (FID) or proton Hahn Echo (HE), for example. The latter situation has been analyzed recently [54⁎]. In analogy with spin–lattice relaxation it has been shown that the intermolecular part of the FID is likewise determined by the relative mean-squared displacements of polymer segments of different macromolecules. This fact opens up a new way for determining displacements, i.e. translation mobility, from the intermolecular part of FID or HE. For situations when the relative mean squared displaceD E ment is determined by a power-law of the form ~r 2 ðt Þ ¼ At α , where α b 2/3, the latter can be derived from intermolecular part of the FID g~ inter ðt Þ: [54⁎] D E ~r 2 ðt Þ ¼

"rffiffiffiffiffiffi #2=3 2 36πγ 4 ℏ2 ρs t 2   3π 5ð2−3α Þð4−3α Þ ln 1=g~ inter ðt Þ

ð7Þ

under the condition t ≤ 2T2∗ , where T2⁎ is the effective spin–spin relaxation time. Thus, one can attempt to isolate intra- and intermolecular contributions to the FID. 2⁎ The translational dipolar correlation function Cinter(t) = 〈Ym (Ω(t)) 2 Ym (Ω(0))/r 3(t)r 3(0)〉 describes fluctuations of the interspin distance r and the orientation of interspin axis via the solid angle Ω encoded 2 in spherical harmonics Ym . Independent of molecular details its long-time limit follows a power-law, Cint er(t) ∝ t−3/2, which is characteristic of free diffusion [62–64]. As a result, the spectral density Jinter(ω), being the Fourier transform of Cinter(t), depends linearly on pffiffiffiffi the square root of the frequency, ω in the low-frequency limit. As reorientational dynamics are significantly faster than the corresponding translational motion their influence on the dispersion can be neglected in this limit. Then, the low-frequency expansion of the total relaxation rate is given by a universal dispersion law [29–31] intra

R1 ðωÞ ¼ R1

inter

ðωÞ þ R1

ðωÞ ¼ R1 ð0Þ−

B pffiffiffiffi · ω D3=2

ð8Þ

with a numerical factor B depending only the spin density. The intramolecular (reorientational) contribution is included in R1(0) as the rotational contribution is frequency independent in the low-frequency range, i.e. ωτrot b b 1 (τrot denotes the rotational correlation time). Eq. (8) applies for ω b b 1/τt, where τt is the terminal relaxation time. Thus, analyzing the dispersion at lowest frequencies provides the diffusion coefficient. This has been exploited for simple molecular liquids [27⁎⁎,29⁎] as well as polymeric melts as will be shown (cf. Section 4.3). Generally, the Bloembergen, Purcell and Pound (BPP) expression [65] relating relaxation rate R1 and the spectral density J(ω), explicitly R1 ðωÞ∝½ J ðωÞ þ 4J ð2ωÞ

ð9Þ

can be re-written in the susceptibility representation [16,19] ω=T 1 ðωÞ ¼ K ½χ ″ðωÞ þ 2χ ″ð2ωÞ≡ 3Kχ ″DD ðωÞ:

ð10Þ

It allows a direct comparison with results from other techniques and a simple implementation of frequency–temperature superposition (FTS). K is the NMR coupling constant and χ ″ DD ðωÞ is subsequently called “dipolar susceptibility” probed by 1H NMR. Although χ ″ DD ðωÞ is a weighted sum of susceptibilities χ″(ω), both quantities are essentially indistinguishable for a broad relaxation dispersion on logarithmic scales as is the case in polymers. Via χ″(ω) = ω ⋅ J(ω) the susceptibility is associated with the spectral density J(ω) being the Fourier transform of the dipolar correlation function CDD(t) which, in turn, has to be distinguished from g (2)(t) which probes solely rotational motion. The effective frequency range of FC NMR can be extended by applying FTS, an approach well introduced in rheological studies [13,16]. Here, 3K χ ″ DD ðωÞ is plotted as function of the reduced frequency ωτs. It may be remarked that the abovementioned relation using 1H as the most ubiquitous dipolar nucleus also holds for investigating 13C nuclei which have been studied in earlier works, with the added benefit of manipulating intra- and intermolecular couplings by selective labelling. For instance, the analysis of the FID of 13C in polymer melts was reported in [58], and a number of studies were carried out on 13C NOE and T1 measurements at three different fields for polymers below and above the entanglement limit [89,90]. The latter papers also hinted to the potential of applying T1ρ, i.e. T1 in the rotating frame, to allow an alternative access to slow dynamics. T1ρ provides a more limited frequency range and contains a more complicated dependence on molecular dynamics as compared to T1 which makes field-cycling the more flexible technique for measurements of fundamental properties. The much increased sensitivity of modern field-cycling equipment, together with the finding that intermolecular dipole–dipole interactions play an essential, if not dominant role in polymer melts with sufficiently large molecular masses at low frequencies/long times. Because the gyromagnetic ratio of 13C is about four times smaller than that of 1H and due to the natural abundance of 1H of about 1%, the short correlation time limit will be reached at correspondingly larger frequency for 13 C–13C and also 13C–1H interactions. Observing direct 13C– 13C couplings is currently possible at least for isotopically enriched samples, while high-field investigations make natural abundance coupling measurements accessible. This direction of future application NMR to study polymer melt dynamics is very interesting and promising.

4. Experimental studies 4.1. Full dipolar correlation function In a series of papers, the dynamics of linear polymers such as 1,4-polybutadiene (PB) with systematically varied M has been studied by FC NMR relaxometry [17,19–22]. Applying frequency–temperature superposition (FTS) master curves have been constructed which cover the full polymer dynamics including local, Rouse and entanglement contributions. Fig. 2a shows R1(ω) for entangled PB (M ≫ Me ≈ 2000) obtained over a large temperature interval. For each temperature a differently strong dispersion is observed. Arrows in Fig. 2b illustrate FTS which is applied to create a master curve χ ″ DD ðωτ s Þ from the susceptibility χ ″ DD ¼ ω=T 1 (solid red line). Fig. 2a also includes low-frequency relaxation data obtained by compensating earth and stray fields. [8⁎⁎] Almost two orders of magnitude are gained with respect to the commercial FC NMR spectrometer. Fig. 3a shows the master curves of susceptibility data χ ″ DD ðωτs Þ for series of PB with different M. The scaling by the segmental correlation time τs yields “isofrictional” spectra and provides a common peak at ωτs ≈ 1 representing the primary (α-)relaxation (also segmental relaxation denoted) governed by the glass transition. With increasing M a continuously rising excess intensity on the low-frequency side of the peak (ωτs b 1) is discernible which is due to the slower, M-dependent polymer dynamics. For the high-M (M > Me) curves with three relaxation regimes (0, I, II) are distinguished, and they are

E.A. Rössler et al. / Current Opinion in Colloid & Interface Science 18 (2013) 173–182

1010

103

T [K] 223 228 233 238 243 253 273 298 323 363 393

with compensation

109 2

10

ω / T1 [s-2]

1 / T1 [s-1]

177

101

10

102

103

104

107 106

(a) PB 87500, T = 223 - 393 K

0

108

105

106

107

PB 87500 (b)

100 101 102 103 104 105 106 107 108

ν [Hz]

ω, ωaT [s-1]

Fig. 2. (a) Dispersion of the relaxation rate 1/T1 for 1,4-polybutadiene (PB) with M = 87,000 g/mol in the temperature range as indicated measured without (crosses) and (open triangles) earth field compensation. (b) Susceptibility representation ω/T1 of the data in (a), with actual measurements shifted by the temperature-dependent shift factor aT (solid red line). At lowest temperatures the α-peak is discernible and fitted with Cole–Davidson function to provide the segmental time constant τs (dashed red line). From ref. [8⁎⁎].

tentatively attributed to local (0), Rouse (I) and constrained Rouse (II) dynamics of the Doi-Edwards tube-reptation model. Fourier-transforming the susceptibility master curves χ ″ DD ðωτs Þ allows displaying them as the full dipolar correlation function CDD(t/τs) in Fig. 4b. Applying FTS the correlation loss is probed of nine decades in time and eight in amplitude, a result not obtained before. While the low-M system (PB 466, dotted line) exhibits a stretched exponential decay typical of a simple liquid, for higher M the relaxation becomes increasingly retarded. Depending on M characteristic power-laws t−α can be identified (regimes I and II). In the time range up to t/τs b 103 a common envelope with α = 0.85 is found which is not altered at high M. This is close to α = 1 predicted by the Rouse theory [32] expected at short times (cf. Fig. 1, regime I). Above Me entanglement dynamics set in leading to correlation loss decaying even slower at longest times at t > τe (regime II). Here, particularly the M-dependent power-law

100

(a)

χ

DD

'' [scaled]

II

10-2 10-3

I

0

10-2

with compensation

-4

10

M [g/mol] 355 466 777 816 1450 2020 2760 4600 9470 11400

0.32

∝ω

10-5 10-6

1

∝ω

-7

10

10-9

10-7

-0.85

∝t

10-1

10-5

10-3

ωτs

10-1

18000 24300 35300 47000 56500 87500 143000 196000 314000 441000 817000

101

CDD

10-1

t−α(M) is recognized, i.e., the exponent α (of regime II) is reduced with growing M in Fig. 4b. For M ≤ 56,500 the curves at longest times bend down due to the terminal relaxation. Similar results have been reported for other polymers like polyisoprene, polypropylene glycol and polydimethylsiloxane (PDMS) [66]. Moreover, the agreement with the results from double quantum (DQ) 1H NMR is almost perfect [7⁎], see Fig. 3b. The dependence α(M) is plotted in Fig. 4. A continuous decrease of ε from 0.73 to 0.32 can be noted in the M range between 9470 and 441,000. The high-M results obtained by FC 1H NMR agree again well with those from DQ 1H NMR [7⁎]. For highest M the power α determined is slightly larger, yet close to the prediction α = 0.25 of the tube-reptation model for regime II (cf. Fig. 1). Since the theoretically expected value is still not reached and could possibly be attained only for M > 2,000,000, it has been concluded that the crossover to full reptation dynamics is highly protracted [8⁎⁎].

(b)

-ε(M)

∝t

10-3

PB

10-4

FC NMR (T = 223 K - 408 K) M = 466 - 441000

10-5 10-6

-0.25

∝t

DQ NMR (T = 243 K - 363 K) M = 23600, 35000, 55300, 2000000

10-7

0

I

II

-8

10

10-1

101

103

105

107

t/τs

Fig. 3. (a) Susceptibility master curves as a function of the reduced angular frequency ωτs for polybutadiene of different M. The frequency range in which data has been acquired while employing the compensation for stray fields is marked by arrows. Vertical dotted lines: relaxation regimes 0, I, and II, i.e., glassy dynamics, Rouse and entanglement dynamics, respectively. (b) Dipolar correlation function CDD(t/τs) obtained from the data in (a) via Fourier transforming compared to those from double quantum (DQ) 1H NMR.[7⁎] Dotted curve: low-M system representing glassy dynamics. Dashed lines: observed power-laws of regime I and II. Gray area illustrates variation of power-law exponent α of regime II for 24,300 ≤ M ≤ 441,000. Solid line: predicted power-law of regime II by tube-reptation model.

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0,8

PB 1

FC H NMR 1 DQ H NMR

α

0,6

≈ 1000⋅Me

0,4

0,2 104

105 M [g/mol]

106

to R1 must not be ignored suggesting that α might also be sensitive to influences of the intermolecular relaxation [23,24⁎⁎]. In order to separate intra and intermolecular relaxation contributions mixtures of protonated and deuterated PB and PDMS have been studied, i.e., the isotope dilution technique is applied [27⁎⁎]. In Fig. 5 the correlation functions CDD(t/τs), Cintra(t/τs), g (2)(t/τs) (from FC 2H NMR) and Cinter(t/τs) are plotted for two values of M. At short times where glassy (or “local”) dynamics dominate (regime 0) all correlation functions coincide. In regime I only weak differences among the correlation functions are observed which become, however, significant in regime II. Whereas the power-law exponent α of Cinter(t/τs) is always lower than the corresponding one of CDD(t/τs) that of Cintra(t/τs) is always higher. Moreover, for both M the reorientational correlation function Cintra(t/τs) obtained from the intramolecular contribution agrees well with g (2)(t/τs) from FC 2H NMR, the latter by its very nature probing solely reorientational dynamics. Due to the strong intermolecular contribution at low frequencies, the intramolecular part significantly changes with respect to the total relaxation, and it is

Fig. 4. Power-law exponent α as a function of molecular mass M obtained in the time domain; for comparison DQ 1H NMR results [7⁎] are included. Even at M ≈ 2,000,000 the prediction of the tube-reptation model α = 0.25 (dashed line) is not fully approached yet. From ref. [8⁎⁎].

10

10

2

10

3

10-2

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107

108

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t

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t

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-3

106

101

α = 0.53 α = 0.70 α = 0.45

total intra inter

105

24300 196000

100

10-1

104

102

[nm2]

When comparing the observed power-law exponents in Fig. 3(b) to the predicted ones the dipolar correlation function CDD(t) probed by FC 1 H NMR does not necessarily have to be identical to the rank-two reorientational correlation function g(2)(t) since intermolecular contributions to CDD(t) may also contribute. Indeed, as discussed (cf. theoretical background), it has been shown that the intermolecular contribution

t /τs 1

100

101

102

103

104

105

106

t (Tref) [ns]

10-4

t /τs

10-5

101

102

103

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106

107

108

2

H-NMR 22800-d6

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-7

10

-1

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α= 0.32 α = 0.49 α = 0.28

total intra inter

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α

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10-5

t 101

101

102

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PDMS, Tref = 408 K

PB 196000

10 103

104

105

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107

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t /τs Fig. 5. Different correlations functions for PB with M = 24,300 (a) and 196,000 (b): CDD(t/τs) (total: comprising intra- and intermolecular contributions), Cintra(t/τs), CQ(t/τs) (from FC 2H NMR), and Cinter(t/τs). For regime II their power-law exponents α are indicated. From ref. [27⁎⁎].

(b)

-1

10-2 100

0.25

100

H-NMR 191000-d6

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t

2

10-4

21600 (inter) 41400 (total) 232000 (total)

103

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10-6

10-1

100

101

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104

105

t (Tref) [ns] Fig. 6. (a) Segmental mean squared displacement bR2(t/τs)> for polybutadiene (PB) with M = 24,300 and 196,000 calculated from χ ″ inter ðωτ s Þ according to Eq. (4). Upper axis: reduced time, lower axis: time t at the reference temperature Tref = 393 K. (b) Corresponding representation of bR2(t/τs)> for polydimethyl siloxane (PDMS) with M = 21,600. Lower axis: time t at Tref = 408 K. From ref. [27⁎⁎].

E.A. Rössler et al. / Current Opinion in Colloid & Interface Science 18 (2013) 173–182

found that the exponent of C2(t/τs) in regime II is rather high and does actually not agree with the prediction α = 0.25 of the tube-reptation model. Explicitly, while the long-time exponent in the total dipolar correlation CDD(t) is α = 0.32 ± 0.02 it becomes α = 0.49 ± 0.05 for Cintra(t/ τs) ≅ g(2)(t/τs). While the observed trend may tentatively be interpreted as support for the existence of an isotropic model of segmental motion as was discussed in Section 3, it is too early to derive such a conclusion based on the available data, and it would be a realistic approach to assume an intermediate scenario in between the existing isotropic and anisotropic models. In particular, the assumption of a universal behavior for all types of linear polymers needs will be subject to detailed investigation in the near future.

4.2. Mean square displacement obtained from the intermolecular relaxation Having extracted the intermolecular susceptibility χ ″ inter ðωτ s Þ, the mean square displacement br 2(t/τs)> can be calculated applying Eq. (6) [23,27⁎⁎]. The result is shown in Fig. 6. Clearly two regimes are recognized. The first one, at short times, yields a power-law t 0.49 ± 0.03 for both M in accordance with the Rouse model prediction of t 0.5. At long times a power-law t 0.19 ± 0.03 is observed for the high-M PB which is close to t 0.25 expected for the constrained Rouse dynamics (regime II), while PB 24,300 shows a tendency to crossover to a similar behavior but free diffusion interferes at longest time. Comparable results are found for PDMS [27⁎⁎] and agree well with FC NMR results reported before [23]. Thus, the mean square displacement can equally well be obtained from FC NMR as in the case of neutron scattering [43]. As demonstrated above, the reorientational correlation function of PB yields g(2)(t/τs) ∝ t−0.49 ± 0.05 for regime II. Together with the result for the mean square displacement, explicitly b r 2(t/τs) > ∝ t0.19 ± 0.03 (following essentially the tube-reptation model), it can be concluded that in regime II the fundamental assumption of the tube-reptation model, namely g (1,2)(t) ∝ b r 2(t) >−1 is not fulfilled. Simulations by Wang et al. [38⁎] have shown a power-law relationship of C2(t) ∝ [C1(t)]m for m = 2 and 1 in the regimes I, and II, respectively, i.e., they essentially have confirmed the tube-reptation model. Yet, the present results seem to suggest C2(t) ∝ b R 2(t) >−2. Regarding the Rouse regime (I) the prediction C2(t) ∝ C1(t) 2 ∝ t −1 (at τs b t b τe) is in good agreement with our experiments. Simulations have shown that it is important to average only over the innermost monomers [35] in order to reveal the power-laws of the tube-reptation model; this remains a future experimental task to be checked experimentally.

(a)

179

4.3. Determining the diffusion coefficient The limiting low-frequency behavior of the total relaxation rate R1 (ω) provides model independent the long-time diffusion coefficient D (cf. Theoretical Background). First applied to simple liquids [29⁎] this approach allows to determine D(T,M) also for polymers [30]. Following pffiffiffi Eq. (8), Fig. 7(a) shows the relaxation rates R1 versus ν for PB of M = 2020 which is close to the entanglement molecular mass Me ≅ 1800. The solid lines indicate the linear part of the relaxation dispersion observed at low frequencies which reflects the translational contribution to R1. In Fig. 7(b) the diffusion coefficient of PB is displayed as a function of M. The FC NMR data well complement those from field gradient NMR [67]. In particular, the crossover from Rouse to entanglement is well recognized by the change of the power-law exponents. 5. Possible applications: polymers under external influences Understanding and quantifying dynamics of polymers in the melt state not only is of fundamental interest by itself, but also has farreaching consequences for the description of macromolecules under external boundary conditions. Vice versa, analyzing those conditions can give useful insight into the relative importance of dynamic parameters in the undisturbed melt, and/or the possibility of universality in dynamics among different types of polymers. Geometrical confinement in narrow pores appears to be the most obvious change to the mobility of chain molecules. An ideal scenario would see a pure geometrical effect with neither attractive nor repulsive interaction of polymer chains with the pore wall, a situation that is unlikely to be encountered but is often reasonably well approximated. Polymer-wall interactions, if existent, will usually lead to a change of conformation, and therefore a shift of the relative weight of different intrasegmental degrees of freedom to the total segmental mobility, i.e. to the relaxation mechanisms; a general slowing down of mobility is likely. While theoretical work and numerical simulations for confined polymers, albeit rarely in the melt state, are abundant – see [59⁎] which provide an excellent review of the field –, experimental approaches remain scarce. A number of experimental results were discussed recently in [68⁎] for pore-filling polymers, whereas dynamical changes correlated with topological changes for wetting and non-wetting polymers of nanometer films in large pores had been discussed earlier [69⁎]; for the same materials PDMS and polybutadiene [70⁎], describes an increase of anisotropy in motion which appears to extend to much larger distances for PDMS than in PB in qualitative agreement with [68⁎]. Authors of [68⁎], see also a more recent work [71], named a dramatic slowing of polymer dynamics in confined systems the corset effect. In most general words, the basis of the corset effect is as follows. The dynamic properties

(b)

pffiffiffi Fig. 7. (a) 1H spin–lattice relaxation rates, R1, of polybutadiene (PB) of M = 2020 plotted as a function of the square root of the Larmor frequency ν. (b) Molecular mass dependence of the diffusion coefficient D as obtained by FC 1H NMR relaxometry and by field gradient (FG) NMR;67 solid lines: power-laws with exponents as indicated; dashed line marks crossover from Rouse to entanglement dynamics at the molecular mass Mc~Me.

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of polymer melts in porous systems considerably differ from the corresponding properties of polymer melts without any spatial constraints at high frequencies (small times t ∝ 1/ω), when the polymer segments' spatial mean squared displacements 〈r 2(t)〉 are much smaller than the squared characteristic tube diameter. Note that the specified differences were observed even for tubes whose radii were several times larger than the characteristic lengths of polymer chains. In other words, a mutual noncrossability of polymer chains and a low compressibility of the polymer melt lead to the specific collective effect: polymer segments span the medium via rapidly propagating acoustic waves and respond to the presence of impermeable tube walls by changing the character of thermal fluctuations relative to those of the common melt much earlier than the direct physical contact with the tube wall ensues. At the initial steps of theoretical realization of this idea [72⁎,73] the tube-reptation model was used as a possible molecular mechanism for describing the frequency dispersion of the spin–lattice relaxation of polymer melts confined in nanoporous systems. Recent simulations [71] confirmed that polymer chains noncrossability combined with low compressibility produce the collective effect in slowing polymer dynamics. Some, but not all, predictions of the dynamics according to tube reptation model were observed in mentioned simulations. Future investigations will provide additional insight into the details of the corset effect. Another important technique for investigating polymer dynamics, neutron scattering, did not find evidence for this phenomenon in earlier publications [74,75], but in a most recent work [76⁎] the polymer dynamics in confinement was attributed to “two phases, one fully equal to the bulk polymer and another that is partly anchored at the surface. By strong topological interaction, this phase confines further chains with no direct contact to the surface.” Actually a strong topological interaction of chains with no direct contact to the surface is essentially an observation that describes the abovementioned behavior of the corset effect. Recently, applying FTS a FC 1H NMR study on PB in self-ordered aluminum oxide membranes with pore channel diameters of 60 nm and 20 nm has been carried out [77]. Although the confinement size is larger than the Flory radius, confinement effects have been found. The susceptibility spectrum of the collective polymer dynamics remains essentially the same in the accessible frequency range, however, the time scale of polymer dynamics slows down whereas that of segment dynamics remains unchanged. This observation may provide an alternative explanation of the corset effect, but investigations are currently under way that aim at the identification of a common description of confined polymer dynamics for a range of model linear polymers with different measurement techniques. A related situation to confinement by the inner wall of pores is the presence of dispersed solid phases, an important field of application being polymers reinforced by clay or nanoparticles with the aim of affecting their mechanical properties. In ref. [78], the residual mobility of PEG chains attached between clay nanoparticles was investigated from the solution state and a strongly restricted mobility was found, not unlike the one at solid PEG; strong adsorption of polymers must therefore be considered to support a transition towards a solid-like order. More generally speaking, long chains tend to form a layer of severe motional restriction, with chain segments farther away from the interface more and more approaching free bulk dynamics, as was found for PEG/silica nanocomposites [79,80]. A more extreme case of this scenario is the permanent, i.e. chemical bonding of chains by grafting to a solid surface; in this case, the motional degrees of freedom are restricted but one would expect a transition towards bulk-like dynamics in the limit of infinite chain lengths. In [81] the deviation of the long-time dynamics – in the range of FC experiments – of grafted PDMS was found to be remarkably small compared to bulk chains. Polymers cross-linked to form rubber or, in the presence of a solvent, a gel network maintain many properties of the uncrosslinked chains; for instance, a rubber is liquid-like when NMR relaxation times are

considered and only a small shift of the frequency-dependent relaxation dispersion is found when compared to the corresponding melt. Rubber– filler interactions may thus be regarded as a mere special case of polymer–interface interactions so that this technologically important field, for which a large body of empiric NMR relaxation data has been generated for decades, may be available for a more thorough re-interpretation taken the segmental mobility of chain molecules into account. Last but not least, while the concept of polymer solutions [82] is beyond the scope of this review, the crossover between melt dynamics and concentrated solution dynamics is encountered in the vast field of biomacromolecules, such as hydrated proteins and, in fact, any kind of live tissue. Early observations of the relaxivity of the residual signal of lyophilized proteins with only a hydration layer present were reported in [83,84], and experimental evidence of backbone fluctuations that are related to the fractal structure of the interproton coupling distribution in natural dry proteins were discussed in [85]. The dynamics of water molecules, and their effect on the proton relaxation properties, remain an issue of ongoing research, and methods similar to approaches discussed earlier in this review, i.e. the use of isotopic dilution by deuteration, have been employed for separating the relevant interactions [91]. Together with – to this point predominantly qualitative – descriptions of water relaxation dispersion in tissue [86–88] this may open up a new access to the in vivo assessment of local biophysical processes. In the near future the described findings, combining fundamental descriptions gained for polymer melts with those results discussing restricted mobility and polymer-solvent interactions, are expected to become available for another field of remarkable current activity, i.e. the understanding of the molecular dynamics of proteins in their natural, functional state over a broad frequency range.

6. Conclusions In recent years, NMR relaxometry has been successfully applied to increase the experimentally accessible range of Larmor frequencies for which longitudinal relaxation times can be obtained. The theoretical framework that has been developed at the same time was able to explain the importance of intra- and intermolecular interactions in their contribution to relaxation rates. The experimental access to these two different factors is obtained most directly by isotopic dilution of polymers, or alternatively by determining relaxation dispersion information of quadrupolar nuclei such as 2H attached to the polymer chains — both approaches facing preparatory and experimental challenges, but having been demonstrated to be entirely feasible with current technology. Significant progress has been achieved to date in generating reliable dipolar correlation functions over an increasing range of times, and first approaches have been carried out to combine various NMR methods on the one hand, and results from alternative techniques such as dielectric relaxation and neutron scattering on the other hand, for a common description of molecular dynamics. With first successes being presented for model materials – i.e. unbranched, well-characterized artificial polymers –, future developments will naturally depend on a more extended integration of complementary methods, and need to be applied to further classes of polymers, eventually biopolymers, in order to test which concepts of polymer dynamics are universal, and to which degree chemical structure possibly affects even long-time correlations of orientation and motion. While numerical molecular dynamics simulations can be expected, for at least another decade or two, to remain insufficient to cover the broad timescale which is required to fully describe the properties of long chain molecules, the current state of the art of fundamental theoretical description, experimental design and coarse-grained numerical simulations bears the best promise to generate significant progress in understanding this fundamental problem which remains an outstanding challenge in the field of soft matter physics.

E.A. Rössler et al. / Current Opinion in Colloid & Interface Science 18 (2013) 173–182 Acknowledgement The financial support of Deutsche Forschungsgemeinschaft (DFG) through priority program SPP 1369 “Polymer-Solid Contacts: Interfaces and Interphases” (RO 907/16), and grant FU 308/14 is acknowledged by ER. NF appreciates support by RFBR grant no. 10-03-00739-a. The authors are grateful to R. Kimmich for valuable discussions.

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