Polymer Melts Investigated by Field Cycling NMR Relaxometry: From Simple Liquid to Reptation Dynamics

Polymer Melts Investigated by Field Cycling NMR Relaxometry: From Simple Liquid to Reptation Dynamics Von der Universit¨at Bayreuth zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung von Axel Herrmann geboren am 27.08.1982 in Coburg Tag der Einreichung: 25.09.2012 Tag des Kolloquiums: 29.11.2012 1. Gutachter: Prof. Dr. Ernst R¨oßler 2. Gutachter: Prof. Dr. Franz Fujara Contents 1 Abstract 1 2 Kurzdarstellung 3 3 Extended Abstract 7 3.1 Introduction.............................. 7 3.2 From Simple Liquid to Polymer Melt . . . . . . . . . . . . . . . . 39 3.3 Universal Dynamics for Various High-MPolymers . . . . . . . . . 46 3.4 Protracted Crossover to Reptation Dynamics . . . . . . . . . . . . 49 3.5 Reorientational and Translational Dynamics in Entangled Polymer Melts.................................. 54 3.6 Linear Polymers in Solution . . . . . . . . . . . . . . . . . . . . . 60 4 Publications 65 Bibliography 119 Acknowledgements 131 i 1 Abstract The focus of this thesis is the investigation of linear polymer melts by applying Field Cycling Nuclear Magnetic Resonance (FC NMR) relaxometry. The objective is to understand their microscopic dynamics and its dependence on the molecular mass (M) of the polymer chains. The results are the subject of five interrelated publications; one of them is concerned also with the dynamics of polymers in solution and its modifications with respect to that observed for the bulk melts. With the commercial availability of FC NMR relaxometers, the method gained attraction for studying dynamics of soft condensed matter due to its ability to detect both the structural or α-relaxation (identified with the segmental dynamics) and slower collective dynamics. In the case of polymer melts the latter is described most often by the Rouse model for non-entangled chains and the Doi/Edwards tube-reptation model for entangled polymers. Since 2004 a commercial relaxometer by Stelar has been operated in the R¨ossler group. Its capability to rapidly switch between different magnetic fields allows to measure the spin-lattice relaxation time T1in the proton (1H) frequency range from 10 kHz to 20 MHz. In previous works by the R¨ossler group polybutadienes (PB) and some low molecular liquids have been studied and the pioneering works by Kimmich and co-workers have been extended in order to combine the results of a broad temperature range: Frequency-temperature superposition is applied to construct master curves in the susceptibility representation χ00(ω) = ω/T1. The key benefits are: via the scaling χ00(ωτs), where τsdenotes the time constant of segmental dynamics, an ”isofrictional” representation is achieved; the accessible frequency range is significantly increased; the time constants τs(T) are provided and compared with those obtained by other techniques; the regimes of glassy and polymer dynamics can be easily distinguished; finally, the dipolar correlation function is obtained directly by Fourier transform. In this thesis by employing the above approach, the dipolar correlation function CDD(t) of PB melts is presented and comprises – depending on M– glassy, Rouse and entanglement dynamics. The latter two relaxation regimes can be described by different power-laws ∝t−, which are compared to the predictions of the tubereptation model. A good agreement is found for the Rouse regime (I). For the constrained Rouse regime (II) at long times, a highly protracted crossover to completely established reptation dynamics is discovered. That is, the exponent  depends on Mand reaches = 0.32 only at M= 441000, which is in accord with Double Quantum (DQ) 1H NMR results (= 0.29) by Saalw¨achter and co-workers and very close to = 0.25 predicted for regime II of the tube-reptation model. This is only achieved by additional relaxation experiments in cooperation with the 1 1 Abstract Fujara group at TU Darmstadt, since their home-built FC NMR relaxometer is equipped with an active stray field compensation, which allows to reach extremely low frequencies down to 200 Hz. Consequently, the frequency range is extended by two decades toward lower frequencies with respect to the commercial spectrometer and the obtained correlation function CDD(t) stretches over 10 decades in time and 8 in amplitude for molecular masses up to 220 ·Me. This establishes FC 1H NMR also at long times as competitive with DQ 1H NMR. Furthermore, it is shown for different systems that their separated relaxation spectra of polymer dynamics are very similar, although their overall susceptibility master curves are different. By comparing selectively deuterated PB and polyisoprene, polydimethylsiloxane, and poly(propylene glycol) it is demonstrated that the polymer relaxation strength depends on the orientation of the proton pair vector with respect to the chain contour. Moreover, the characteristic molecular masses of PB are now substantiated from analyses both in the frequencyand the time-domain, i.e., the molecular mass of the Rouse unit MR= 500 g/mol describing the onset of Rouse dynamics, and Me= 2000 g/mol representing the molecular mass between two entanglements. The analyses of the dipolar correlation function CDD(t) render a comprehensive picture of the molecular dynamics of polymers, and the coincidence between the different polymers appears support the applicability of the tube-reptation model. However, CDD(t) always comprises intramolecular and intermolecular contributions and up to now the dominance of the first has been assumed implicitly. Therefore, isotopic blends of high-Mprotonated and deuterated PB are investigated, which allows to decompose the 1H master curves into intramolecular and intermolecular relaxation contributions. They reflect reorientational and translational dynamics, respectively. It is demonstrated that at long times or low frequencies the intermolecular contribution dominates. Consequently, the reorientational correlation function C2(t) obtained from the intramolecular part exhibits a faster decay with the long-time exponent = 0.49 than CDD(t). This is ascertained by the FC 2H NMR relaxation of completely deuterated PB, which detects reorientational dynamics only. The observed exponent is significantly larger than = 0.25 of regime II of the tube reptation model. Concomitantly, the segmental mean square displacement R2(t)is attained from the intermolecular part following an approach by Kimmich and Fatkullin. The predicted power-laws of the tube-reptation model for the Rouse and constrained Rouse regimes are identified for the first time by FC NMR: a transition between the power-laws ∝t0.49 and ∝t0.19 is revealed, respectively. Thus, NMR relaxometry is designated as a method comparable to neutron scattering to study subdiffusion in polymer melts. In conclusion, the power-law predictions of the tube-reptation model are disclosed by the segmental mean square displacement, yet not by the reorientational correlation function, and the relation C2(t)∝R2(t)−1as assumed by the model is not confirmed. Thus, the simple tube-reptation model does not completely describe the microscopic dynamics of polymer melts. 2 2 Kurzdarstellung Den Schwerpunkt dieser Arbeit bildet die Untersuchung linearer Polymerschmelzen mithilfe der Field Cycling Nuclear Magnetic Resonance (FC NMR) Relaxometrie. Das Ziel ist es, die mikroskopische Dynamik der Polymerketten und ihre Abh¨angigkeit vom Molekulargewicht (M) zu verstehen. Die Ergebnisse sind Teil von f¨unf inhaltlich miteinander verkn¨upften Ver¨offentlichungen, von welchen sich eine auch mit der Dynamik von Polymerl¨osungen und den daraus resultierenden Ver¨anderungen bez¨uglich der Dynamik von Schmelzen besch¨aftigt. Seit der kommerziellen Verf¨ugbarkeit von FC-NMR-Relaxometern findet die Methode erh¨ohte Verbreitung zum Studium der Dynamik weicher Materie, da sie in der Lage ist, sowohl die Strukturrelaxation (α-Prozess), die mit der segmentalen Dynamik identifiziert wird, als auch die langsamere kollektive Dynamik zu detektieren. Um Letztere zu beschreiben, werden im Fall von Polymerschmelzen meistens das Rouse Modell f¨ur nicht verschlaufte Ketten und das Reptationsmodell von Doi/Edwards f¨ur verschlaufte Ketten herangezogen. Seit 2004 wird in der Arbeitsgruppe R¨ossler ein kommerzielles Relaxometer der Firma Stelar eingesetzt. Es zeichnet sich dadurch aus, sehr schnell zwischen verschiedenen Magnetfeldern schalten zu k¨onnen, und erm¨oglicht es, die Spin-Gitter-Relaxationszeit T1f¨ur Protonen (1H) im Frequenzbereich von 10 kHz bis 20 MHz zu messen. In fr¨uheren Arbeiten der R¨ossler Gruppe wurden Polybutadien (PB) und einige niedermolekulare Fl¨ussigkeiten untersucht und der urspr¨ungliche Ansatz von Kimmich und Mitarbeiten wurde erweitert, um auch die Ergebnisse eines sehr breiten Temperaturbereichs mit einzuschließen: Das Frequenz-Temperatur-Superpositionsprinzip wird angewandt, um Masterkurven in der Suszeptibilit¨atsdarstellung χ00(ω) = ω/T1zu erstellen, was folgende Vorteile mit sich bringt: Durch die Skalierung χ00(ωτs), wobei τsdie Zeitkonstante der segmentalen Bewegung bezeichnet, wird eine ”isofriktionale” Darstellung erreicht; der zug¨angliche Frequenzbereich wird erheblich erweitert; die erhaltenen Zeitkonstanten τs(T) k¨onnen mit jenen anderer Methoden verglichen werden; die Bereiche der Glasund Polymerdynamik k¨onnen einfach unterschieden werden; schließlich erh¨alt man per Fourier Transformation direkt die dipolare Korrelationsfunktion. In dieser Arbeit werden unter Verwendung des obigen Ansatzes die dipolaren Korrelationsfunktionen CDD(t) von PB pr¨asentiert, die je nach MBeitr¨age der Glas-, Rouseund Entanglement-Dynamik beinhalten. Die letzten beiden Relaxationsbereiche k¨onnen durch verschiedene Potenzgesetze ∝t−beschrieben werden, die mit den Vorhersagen des Reptationsmodells verglichen werden. Im Rouse-Bereich (I) herrscht eine gute ¨ Ubereinstimung; im constrained Rouse-Bereich (II) wird ein stark verz¨ogerter ¨ Ubergang zu vollkommen entwickelter Reptationsdynamik ent3 2 Kurzdarstellung deckt. Dies bedeutet, dass der Exponent von Mabh¨angt und = 0.32 erst f¨ur M= 441000 erreicht wird, was im Einklang mit Doppelquanten (DQ) 1H NMR Ergebnissen (= 0.29) von Saalw¨achter und Mitarbeitern steht und dem vom Reptationsmodell vorhergesagten = 0.25 sehr nahe kommt. Erreicht wird dies durch weitere Relaxationsexperimente in Kooperation mit der Arbeitsgruppe Fujara (TU Darmstadt), deren selbstkonstruiertes FC-NMR-Relaxometer mit einer aktiven Streufeldabschirmung ausgestattet ist, wodurch extrem niedrige Frequenzen bis 200 Hz erreicht werden k¨onnen. Folglich wird der Frequenzbereich um zwei Dekaden zu niedrigeren Frequenzen bez¨uglich des kommerziellen Spektrometers erweitert und die schließlich erhaltene Korrelationsfunktion CDD(t) erstreckt sich ¨uber 10 Dekaden in der Zeit und 8 in der Amplitude f¨ur Molekulargewichte bis zu 220 ·Me. Dies etabliert die FC 1H NMR auch bei langen Zeiten als konkurrenzf¨ahige Methode zur DQ 1H NMR. Außerdem wird anhand verschiedener Systeme demonstriert, dass deren abgetrennter Anteil des Relaxationsspektrums, der die Polymerdynamik darstellt, sehr ¨ahnlich ist, obwohl sich die kompletten Masterkurven voneinander unterscheiden. Der Vergleich von teildeuteriertem PB und Polyisopren, Polydimethylsiloxan und Polypropylenglycol zeigt, dass die Polymerrelaxationsst¨arke von der Orientierung des Vektors des Protonenpaars im Bezug zur Kettenkontur abh¨angt. Weiterhin werden die charakteristischen Molekulargewichte von PB durch Auswertungen in der Frequenzund Zeitdom¨ane best¨atigt, n¨amlich das Molekulargewicht der Rouse-Einheit MR= 500 g/mol, welches das Einsetzen der Rouse-Dynamik beschreibt, sowie Me= 2000 g/mol, welches das Molekulargewicht zwischen zwei Entanglements darstellt. Die Auswertung der dipolaren Korrelationsfunktion CDD(t) ergibt ein umfassendes Bild der molekularen Dynamik in Polymeren und die ¨ Ubereinstimmung zwischen den verschiedenen Polymeren scheint die allgemeine Anwendbarkeit des Reptationsmodells zu bekr¨aftigen. Allerdings enth¨alt CDD(t) stets intramolekulare und intermolekulare Beitr¨age und bislang wurden erstere implizit als dominierend angesehen. Deshalb werden Isotopenmischungen von hochmolekularem protoniertem und deuteriertem PB untersucht, die es erm¨oglichen, die 1H Masterkurven in intraund intermolekulare Relaxationsbeitr¨age zu zerlegen. Diese spiegeln entsprechend die Reorientierungsbzw. die Translationsdynamik wider. Es wird dargelegt, dass zu langen Zeiten bzw. niedrigen Frequenzen der intermolekulare Beitrag dominiert. In Folge besitzt die Reorientierungkorrelationsfunktion C2(t), die aus dem intramolekularem Anteil gewonnen wird, mit ihrem Langzeitexponenten = 0.49 einen schnelleren Abfall als CDD(t). Dies wird anhand der Ergebnisse f¨ur volldeuteriertes PB durch FC 2H NMR belegt, wobei nur die Reorientierungsdynamik detektiert wird. Der gefundene Exponent liegt deutlich h¨oher als = 0.25 von Bereich II des Reptationsmodells. Gleichzeitig wird das segmentale mittlere Verschiebungsquadrat R2(t)einem Ansatz von Kimmich und Fatkullin folgend aus dem intermolekularen Anteil gewonnen. Hierbei werden die vom Reptationsmodell geforderten Potenzgesetze f¨ur den Rouseund constrained Rouse-Bereich erstmalig mithilfe der FC-NMR identifiziert: Ein ¨ Ubergang zwis4 chen den Potenzgesetzen ∝t0.49 bzw. ∝t0.19 ist zu beobachten. Damit ist die NMR-Relaxometrie in der Lage, auf ¨ahnliche Weise wie die Neutronenstreuung das subdiffusive Verhalten in Polymerschmelzen zu studieren. Somit finden sich die vom Reptationsmodell vorhergesagten Potenzgesetze zwar im segmentalen mittleren Verschiebungsquadrat, nicht jedoch in der Reorientierungkorrelationsfunktion, und die Modellannahme C2(t)∝R2(t)−1best¨atigt sich nicht. Folglich ist das einfache Reptationsmodell nicht in der Lage, vollst¨andig die mikroskopische Dynamik von Polymerschmelzen zu erfassen. 5 3 Extended Abstract Figure 3.5: Mean square displacement hr2(t)iof a binary Lennard-Jones (LJ) liquid (dashed line) and a polymer model (solid line) (adapted from [61]). master curves in anticipation of a detailed explanation which follows below. The data which were measured at different temperatures are plotted as a function of the reduced frequency ωταand are well described by a CD susceptibility [56]. The main structural relaxation manifests itself similarly in both methods as αpeak. Together with the agreement of the time constants τα(cf. Figure 3.3) this demonstrates the validity of FTS, which confirms the assumption that the spectral shape of the glassy dynamics does not change with temperature. Some important properties of the cooperative dynamics of dense liquids are also disclosed by the time dependence of the mean square displacement hr2(t)i. Results for a binary Lennard–Jones liquid from MD simulations [62] are presented in Figure 3.5, in which three dynamical regimes can be distinguished. At short times the ballistic regime [26] with a power-law exponent 2 has been observed, where the interactions between particles are negligible and particles ”fly” freely until they hit a neighboring particle. The long-time behavior is linear in time, which indicates normal diffusion. In between these two limits a plateau is present, in which the particle is confined in a cage formed by its neighbors. This ”cage effect” serves as an explanation for the glass transition phenomenon. Phenomenological Manifestation of Polymer Dynamics First of all, an overview is provided how the dynamics of polymers manifests itself in the results of the experimental methods mentioned above, especially DS and rheology. Polymers are macromolecules which consist of several covalently bound units (monomers). There are many different kinds of topologies, compositions, and functionalities such as blends, blockcopolymers, dendrimers, polymers with functionalized side groups, micells, or nanoparticle composites in order to enhance the technically relevant properties of polymers. Their study is an attractive field for colloidal or advanced functional material research. However, the focus of this 12 3.1 Introduction 10-6 10-4 10-2 100102 10-2 10-1 100 M [g/mol] 314000 (T=253K) 157000 (T=232-252K) 110000 (T=232-252K) 47300 (T=232-248K) 21200 (T=230-245K) 14900 (T=236-248K) 13500 (T=236-248K) 9910 (T=230-245K) 4470 (T=224-242K) 3840 (T=224-236K) 1920 (T=218-232K) 1370 (T=218-230K) 1040 (T=218-224K) ε"/ε"αmax ν/ναmax (a) PI Figure 3.6: (a) Molecular weight dependence of the dielectric spectra of the different polyisoprenes investigated; dielectric permittivity 00 rescaled by its α-peak height 00 α,max displayed as a function of frequency νrescaled by the α-peak frequency ν00 α,max (adapted from [63]). (b) Loss modulus G00(ω)for polybutadiene (PB) of different Mas obtained with a mechanical rheometer in the temperature range from 158 K to 353K (adapted from [64]). thesis is to elucidate fundamental physical principles in melts of linear homopolymers, to which the term ”polymers” refers herein. One technique which allows to study polymer dynamics is DS. By applying an electric field to the sample, the induced polarization equals the sums of the molecular dipoles. In case of a type A polymer [65] like polyisoprene (PI) or polypropyleneglycol (PPG) a component of its monomeric dipole moment is parallel to the chain contour. This enables DS to probe in addition to the first rank correlation function g1(t)∝ hP1(t)P1(0)i/[P1(0)]2of the segmental dynamics also a correlation function hRee(t)Ree(0)i/hRee(0)2iof the end-to-end vector [30, 66]. The latter is reflected in the spectra as normal mode relaxation, which represents the M-dependent polymer dynamics. This offers an interesting possibility to study segmental and collective polymer dynamics simultaneously [63, 67]. Figure 3.6a contains spectra of PI with different Mmeasured by DS in a wide frequency range [63]. They are scaled on the peak at high frequencies, which has been identified as the α-peak reflecting the local segmental dynamics. The peak at lower reduced frequencies represents the normal mode relaxation, i.e., polymer dynamics. It is shifted toward lower frequencies while increasing M. In Figure 3.6b the dynamic loss modulus G00(ω) for different high-Mpolybutadienes (PB) obtained by rheological measurements is shown [64]. The shape of the peak at high frequencies is M-independent, while the position of the minor peak depends on M. Again the first is attributed to the glassy dynamics and the latter represents the polymer relaxation. Thus, a qualitatively similar behavior is observed as in DS, except for the fact that the weighting between the two peaks is different. Note that both in DS and rheology the presented spectra are master curves, i.e., FTS has been applied to extend the accessible frequency range. 13 3 Extended Abstract 10-7 10-5 10-3 10-1 101 10-4 10-2 100 DS PPG 18200 (T = 230 - 240 K) PG (T = 208 - 298 K) PPG 18200 (T = 253 - 383 K) ε"/ε"αmax , χ"/χ"max ωτs , ν/ναmax FC NMR normal mode segmental / α-relaxation Figure 3.7: Susceptibility master curves of propylene glycol (PG) [68] obtained by FC NMR, and of poly(propylene glycol) (PPG) [69] with molecular mass M=18200 compiled from FC NMR and dielectric spectoscopy (DS) in the temperature range as indicated. In order to demonstrate the benefits of the representation as susceptibility master curves, results by DS and FC 1H NMR are displayed as master curves in Figure 3.7 in anticipation of a detailed explanation which follows below. The main or structural relaxation (”α-peak” for simple liquids and segmental dynamics for polymers) manifests itself similarly in both methods. However, at lower frequencies specific differences are discernible. The results can be directly compared by scaling them on the amplitude and the time constant of the α-peak. Firstly, by comparing the master curves of poly(propylene glycol) (PPG) [69], a bimodal shape in case of 1H FC NMR and a low-frequency peak (”normal-mode”) in case of DS are seen. Since both features occur at low reduced frequencies, i.e., they represent dynamics slower than the (local) segmental motion, they are spectral characteristics which reflect the polymer-specific dynamics. Secondly, by comparing the NMR relaxation spectra of PPG and its monomer propylene glycol (PG) [68], an excess intensity in the first with respect to the latter is observed. Thus, this additional, low-frequency contributions in the susceptibility represent the relaxation of the polymer chains. Note that NMR relaxometry is capable to reach even lower frequencies than DS, since in the latter case dc conductivity interferes (increase at the lowest frequencies). The applicability of FTS to the whole temperature and frequency range yields a crucial result, on which the presented analysis of NMR relaxation data relies: FTS is not only valid for the α-process alone (cf. Figure 3.4) but also for the polymer dynamics with respect to the α-process. Thus, the temperature dependence of the segmental dynamics drives that of the polymer dynamics. 14 3.1 Introduction Polymer Models The tube-reptation model [70] by Doi and Edwards is most often applied to describe experimental or simulation results of the dynamics of polymer melts. It can be considered as a combination of the Rouse model [71] for non-entangled chains (with molecular mass Mbelow the entanglement molecular mass Me), and Doi’s idea of a tube and de Gennes’s reptation model [72] for M > Me. It assumes a snakelike motion of a chain in a static tube which is set up by the surrounding chains and confines the dynamics of a single chain. However, on long time-scales motion along the contour and an escape from the tube is possible. This explains the transient elasticity and long-term flow. Though the tube model was originally proposed for the problem of rubber elasticity [73, 74] and can be applied to describe the dynamics of a chain in a network [72], it is also successful in explaining the properties of highly entangled polymer melts which do not form a permanent network. In the Rouse model (M < Me) a polymer chain consists of Nsegments (”beads”) which are connected by entropic springs. The beads include a constant number of chemical bonds (or monomers) and their distribution of end-to-end distances is Gaussian. Whereas the spherical beads reflect the frictional properties due to the viscous surrounding, the entropic springs have a temperature-dependent force constant, which accounts for the polymer coil to shrink for higher temperatures. The equation of motion of the overdamped oscillator system (Langevin equation) can be solved analytically and the correlation functions of the pth Rouse normal modes Xp(t) can be calculated [71, 75, 76] Cp(t) = hXp(t)Xp(0)i=b2 8Nsin2(pπ/2N)exp(−t/τp); p= 1,...,N −1 (3.9) where bis the effective Rouse segment length. The relaxation time of the pth mode is given by τp=τ0π2 4 sin2(pπ/2N)(3.10) with τ0=b2ξ/(3π2kBT) (ξ: friction coefficient of a bead). The relaxation time of the slowest mode (p= 1) is denoted as Rouse time τR=τ1=τ0π2 4 sin2(π/2N)(3.11) The time constant τ0is referred to as the segmental time constant in polymers, i.e., τ0≈τs. This is in turn identified with the time constant of main structural relaxation (α-process), i.e., τs≈τα, which will be explicitly demonstrated below in the context of Figure 3.21. Eventually the quantities which are experimentally accessible can be calculated, namely for dielectric spectroscopy the permittivity [63] 00(ω) = 2∆n N(N−1) N−1 X p=1,3,... cot2pπ 2Nωτp 1+(ωτp)2(3.12) 15 3 Extended Abstract 10-6 10-5 10-4 10-3 10-2 10-1 100101102 10-6 10-4 10-2 100 N = 300,200,150,100,70,50,30,20,10,6,5,4,2 N χn" ωτs (a) 10-6 10-5 10-4 10-3 10-2 10-1 100101 10-5 10-4 10-3 10-2 10-1 100 χ"Rouse ωτ0 (b) N = 1000 100 50 20 10 5 4 3 2 Figure 3.8: (a) Dielectric susceptibility (normal mode spectrum) calculated via the discrete Rouse model for different chain lengths N;τsdenotes the segmental time constant (adapted from [63]). (b) Corresponding NMR susceptibility. Dashed line: high-Nlimit (adapted from [77]). and for NMR relaxation the spectral density [77] JRouse(ω) = 4τ0 π2(N−1)2 N−1 X p,q=1 sin2pπ 2N+ sin2qπ 2N 16 π4sin2pπ 2N+ sin2qπ 2N2+ω2τ2 0 (3.13) For the magnitude of the spectral density at ω= 0 it follows JRouse(0) = 1 (N−1)2 N−1 X p,q=1 τpτq τp+τq (3.14) As an illustration how the Rouse dynamics emerges with increasing chain length, Figure 3.8a shows the dielectric normal mode susceptibility χ00 n(ωτs) = 00(ωτs)/∆n (cf. eq 3.12) as calculated for different Nby utilizing the discrete Rouse model [63]. With increasing N, additional low-frequency modes are accumulated, while the slowest one gives the highest intensity. A qualitatively similar picture is rendered by the normal mode spectra of Figure 3.6a. However, in the latter also effects of entanglement have to be considered for M > Me. In order to explore how polymer dynamics are probed by FC NMR relaxometry the same calculation has been performed for the spectral density JRouse (cf. eq. 3.13). In Figures 3.8b the NMR susceptibility χ00 Rouse(ω)∝ω[JRouse(ω) + 4JRouse(2ω)] is presented. Analogously to the DS spectra, the increase of Nprovides a distribution of correlation times and additional intensity at low frequencies. However, the peak position itself is not shifted toward low frequencies, i.e., the fastest Rouse mode yields the highest intensity. This demonstrates the different weightings of the pRouse modes for the observables of DS and NMR and is exemplified by their susceptibility for N= 6 in Figure 3.9 part a and b, respectively. Therein the single modes for p= 1,...,5 calculated from eq. 3.12 and 3.13 and their sum are 16 3.1 Introduction 10-3 10-2 10-1 100101 10-3 10-2 10-1 100 101 sum for N = 6 p = 1 p = 2 p = 3 p = 4 p = 5 χ''n [a.u.] ωτs (a) DS 10-3 10-2 10-1 100101 10-3 10-2 10-1 100 101 sum for N = 6 p = q =1 p = q =2 p = q =3 p = q = 4 p = q = 5 χ''Rouse [a. u.] ωτs (b) NMR Figure 3.9: (a) Dielectric normal mode and (b) NMR susceptibility calculated via the discrete Rouse model for N= 6 (black line), which are composed by the sum of the individual modes with p= 1,...,5and p,q = 1,...,5(colored lines), respectively; τsdenotes the segmental time constant. displayed (for NMR only the modes p=qare plotted for simplicity). Regarding DS, the cotangent in eq. 3.12 provides the p-dependent weighting which gives the highest amplitude for p/N 1, i.e., p= 1 dominates the sum. In case of the NMR susceptibility the modes bear an equal amplitude and the maximum of the sum stems from the mode distribution in time, which is denser for the high modes. Thus, in DS the end-to-end relaxation according to the Rouse model is dominated by the slowest mode p= 1, and for NMR relaxometry the Rouse spectrum is governed by the fastest mode p=N−1. For long chains (N1) the trigonometric functions in eq. 3.9 - 3.13 are usually approximated by the first term of their series expansion (”continuous Rouse model”) [66, 70]. This leads to expressions for the dielectric normal mode susceptibility [63, 66, 78], the NMR susceptibility [77], and the dynamic shear modulus [70, 79, 80] DS: χ00 n(ω) = 2 N2 N X p=1,3,... cot2pπ 2NωτR/p2 1+(ωτR/p2)2(3.15) FC NMR: χ00 Rouse(ω) = N X p,q=1 ωτR/(p2+q2) 1+(ωτR/(p2+q2))2(3.16) rheology: G00 Rouse(ω)∝ N X p=1 ωτR/p2 1+(ωτR/p2)2(3.17) For the latter quantity the Rouse spectrum is similar to that of FC NMR (cf. Figure 3.9b). This can be anticipated also from the rheological results of Figure 3.6b in which the low-frequency wing of the α-peak reflects the contribution of Rouse dynamics. 17 3 Extended Abstract 10-6 10-5 10-4 10-3 10-2 10-1 100 10-2 10-1 M [g/mol] 9910 4470 3840 1920 1370 1040 ε"/Δεn ν/ναmax PI Figure 3.10: Normalized normal mode spectra for non-entangled PI (M < 9910) obtained from the spectra of Figure 3.6a. Lines: predictions by the Rouse theory (adapted from [63]). τd τd τe τs τR Figure 3.11: Sketch of a polymer chain in a virtual tube with the characteristic time constants of the regimes of the tube-reptation model (see text). A direct comparison between experimental and calculated normal mode spectra is performed in Figure 3.10 for M < Me[63]. The first were attained by subtraction of the α-relaxation from the overall spectra of Figure 3.6a, while the latter were calculated from the discrete Rouse model for a corresponding number of monomers N. Both the experimental and theoretical spectra have been normalized to the same integral π/2, i.e., experiment and theory are compared on absolute scale. Though the spectral features are captured by the Rouse model, systematic deviations are recognized. The experimental normal mode spectra are broader than predicted as it is known from previous studies [67, 81]. Note that concerning the approach for decomposition, cross-relaxation terms have been assumed to be negligible. The essentials of the tube-reptation model are illustrated in Figure 3.11 which displays a single chain within its virtual tube. For very short times the chain 18 3.1 Introduction M < Me t1 t0.5 t0.25 t0.5 t2 t-0.5 t-0.25 log g2(t) log t 0 I II III IV glassy free constrained reptation free Rouse Rouse diffusion cage t-1 M > Me simple liquid log <R2(t)> τs τe τR τd Figure 3.12: Schematic time dependence of the logarithm of the segmental reorientational correlation function g2(t)(red line) and mean square displacement hR2(t)i (black line) as a function of logarithm of time as expected from the Rouse model (non-entangled polymers, M < Me) and Doi–Edwards tube–reptation model (entangled polymers, M > Me) (adapted from [38]). segments are not yet exposed to constraints of the tube. Here the segmental mean square displacement hR2(t)ias predicted by the tube-reptation model, which is displayed for an entangled polymer melt (M > Me) in Figure 3.12, shows a ballistic behavior and then exhibits a plateau. This indicates the cage effect and is typical of glassy dynamics (regime 0). On the time-scales of the segmental correlation time τsthe segments reorient and if the displacements are much smaller than the tube diameter, the chain behaves as a free Rouse chain for times on the order of the entanglement time τe(free Rouse dynamics, regime I). For t≤τRthe influence of the tube becomes effective and restricts the Rouse dynamics perpendicular to the primitive path around which the tube is constructed (constrained Rouse dynamics, regime II). Since Figure 3.11 provides a just snapshot, the primitive path can be obtained by time-averaging these motions and yields the mean positions of the chain segments, i.e., it represents a connection of the chain ends and allows for the topological constraints of the entanglements [53, 82]. For longer times the chain effectively moves along the primitive path, i.e., it performs a one-dimensional reptation-like motion in the tube taking into account that the polymer chains cannot cross each other (reptation dynamics, regime III). For the regimes of polymer dynamics (I - III) characteristic power-laws are predicted for hR2(t)ias indicated in Figure 3.12. Finally, beyond the tube disengagement time τdthe chain leaves 19 3 Extended Abstract the original tube and forms a new one, which leads to the free diffusion behavior hR2(t)i ∝ t1(regime IV). The characteristic times scale with the time constant of segmental motion τsand the number of monomers N:τe∝τsN2 e(Ne: number of monomers between entanglements), τR∝τsN2, and τd∝τsN3[70]. The reorientational correlation function g2(t) is also depicted in Figure 3.12 for a simple liquid, a non-entangled polymer melt (M < Me), and an entangled melt. In regime 0 the decay at shortest times is due to fast relaxation processes and not relevant for polymer dynamics. The non-exponential decay at t≈τscharacterizes glassy dynamics which is a common relaxation feature in simple liquids and polymers. Note that two-step character of the correlation function (cf. also Figure 3.2b) and the plateau of hR2(t)i(cf. also Figure 3.5) appear in the same time range are due to the cage effect. For the high-Mlimit (MMe)g2(t) decays with corresponding power-laws ∝t−in regimes II and III as hR2(t)i ∝ tα. The long-time decay in regime IV is essentially exponential [83]. The power-law exponents of g2(t) are related to those of hR2(t)i. In regime I for the correlation functions of rank l= 2,1g2(t) = [g1(t)]2and the relation g2(t)∝1/hR2(t)ihold; thus, = 2α= 1. In regimes II and III gl(t)∝1/hR2(t)i is independent of the rank land therefore =α. For deriving this relation it is assumed that within the time tthe chain segment is still located in or has returned to same tube section as it was at the time t= 0 (”return-to-origin probability”) provided that the original tube survives [47, 83]. However, deviations from the behavior predicted by the tube-reptation model have been observed, e.g., the molecular mass dependence for M > Meof the steady-state viscosity exhibits an exponent 3.4 instead of 3.0. Thus, refinements of the tube-reptation model were needed which incorporate more sophisticated topological interactions and are subject of ongoing debates [3, 40, 78, 82, 84, 85]. For example, effects of constraint release can be considered, i.e., that surrounding chains reptate by themselves and form a nonstatic tube (or matrix) [86, 87]. Also contour length fluctuations (CLF) can be taken into account, which are due to motions of the chain ends and concomitant shrinking and expanding of the chain inside of the tube [70, 88, 89]. Both mechanisms lead to a reduction of the life time of the tube and thus to a faster decay in terms of a reorientational correlation. Moreover, it is discussed especially in the simulation community whether the concept of a ”continuous” tube is suitable to describe the topological constraints of ”discrete” entanglements or whether a slip-spring model is more appropriate which introduces the confinements by entanglement through virtual springs loosely attached to the chains [90, 91]. Note that Graessley conjectured that that pure reptation behavior might be revealed for very high M[79], which has later been confirmed experimentally [63, 92]. Another model describing the dynamics of entangled polymer melts is the nrenormalized Rouse model, which has been introduced by Schweizer [93–95]. Its mathematical properties are very similar to the original Rouse model [71, 96], however, in order to take entanglement effects into account the relaxation times of the normal modes are modified and depend also on the entanglement molecu20 3.1 Introduction lar mass. The model has been applied by Kimmich, Fatkullin, and co-workers to explain their NMR relaxation results of the spin-lattice relaxation dispersion [47, 96, 97]. Transformed to the correlation function g2(t), two power-laws are predicted: g2(t)∝(t/τs)−0.5for τs≤t(”high-mode number limit”) and g2(t)∝(t/τs)−0.8for τe≤t(”low-mode number limit”, see also [47] and Pub. 1). Since these power-law exponents are different to those of the tube-reptation model (cf. Figure 3.12), the works by Kimmich and co-workers have cast doubt on the applicability of the tube-reptation model to describe the NMR relaxation results of polymer melts, which will be clarified in this thesis. Experimental Evidence for Polymer Dynamics The dynamics of polymers can be studied by various techniques, such as dielectric spectroscopy [30, 66, 78], neutron scattering [3, 98], and several NMR experiments [36, 99–103]. Since this thesis concentrates on field cycling NMR relaxometry [36] and aims at representing the results in a way that they can be compared [38] to those of other methods, the results of the latter and of simulations are briefly reviewed. In Figure 3.13a results for the time dependence of the mean square displacement g1(t/τ) as obtained by MD simulations [104] are displayed for different chain lengths Nwhile Ne≈35. At short times all curves follow a common powerlaw ∝t0.5as predicted for the free Rouse regime (I). Above N= 50 and at t>τe≈1700 deviations from the Rouse behavior are seen and with increasing Na power-law ∝t0.25 clearly emerges, which is expected for the constrained Rouse regime (II) of the tube-reptation model. Note that only the five innermost monomers have been analyzed since at the chain ends fluctuations are expected due to, e.g., CLF [104, 107, 108]. However, Monte Carlo simulations [109] have revealed that the crossover to reptation dynamics is very protracted, i.e., even for M≈14Methe power-law of regime II of the tube-reptation model is not clearly seen. Furthermore, MD simulations [41, 61, 110] have demonstrated that for a simple liquid and a polymer the glassy dynamics (regime 0) are reflected in hR2(t)iin the same way. Yet, in the case of the polymer system, hR2(t)iexhibits between the plateau due to the cage effect and the free diffusion limit a characteristic power-law regime, which is attributed to Rouse dynamics (cf. Figure 3.5 and [61]). Figure 3.13b presents the mean square displacement of a Neutron Spin Echo (NSE) study [98] for a polyethlene melt with M= 190000. NSE probes the segmental mean square displacement averaged over all monomers via the incoherent structure factor assuming a Gaussian distribution of the displacements. The time range of NSE is about 0.01 −300 ns. At T= 509 K a crossover between the power-laws of regimes I and II is observed. Note that due to the high Mof the samples influences of CLF have been treated as negligible. Yet, further NSE results [108] of polyethylene have revealed the relaxation contribution of CLF for intermediate M. In Figure 3.13c the mean square displacement hZ2(t)iof polystyrene with different 21 3 Extended Abstract 250 300 350 400 10-3 10-2 10-1 100 T [K] T1 [s] 20 MHz 1.3 MHz 150 kHz 10 kHz OTP PB 87500 Figure 3.17: Spin-lattice relaxation time T1of o-terphenyl (OTP, black) and polybutadiene with high molecular mass M=87500 (PB 87500, red) as a function of the temperature Tfor some Larmor frequencies ν. Lines: guides to the eye. since the frequency-dependent polymer relaxation processes are detected on the high-temperatures wing with respect to the minimum. Data Representation Since polymer dynamics stretch over many decades in time or frequency [38, 47] (cf. also Figures 3.12 and 3.6), our studies aim at exploiting the dynamic range of the technique as far as possible. Besides extending the frequency range of FC NMR to about 5 decades by employing an active stray field compensation (Section 3.4), frequency-temperature superposition (FTS) is applied in order to combine the relaxation data measured in a very broad temperature range. FTS assumes that the relaxation of different applied temperatures can be shifted through the FC NMR frequency window without alteration of its spectral shape. This holds especially for the susceptibility χ00(ω) as shown in Figure 3.2 and by eq. 3.7. The relaxation rates 1/T1(ν) (cf. eq. 3.18) are transformed to the susceptibility representation χ00(ω) = ω/T1(ω) (3.20) and susceptibility master curves are created by plotting χ00 as a function of the reduced frequency ωτs(τs: time constant of segmental motion). The approach is well known from, e.g., rheology (cf. Figure 3.6b) and reflects a fundamental feature of cooperative dynamics [38]. The susceptibility representation of NMR relaxation data has also been applied by Cohen-Addad and co-workers [144, 145]. The procedure of constructing master curves is illustrated exemplarily for 1,4polybutadiene (PB) with molecular mass M= 87500 g/mol for which the rates and corresponding susceptibilities are displayed in Figure 3.18a and b, respectively. A susceptibility master curve χ00(ωaT) (solid red line) can easily be cre28 3.1 Introduction 102103104105106107 100 101 102 103 1 / T1 [s-1] ν [Hz] PB 87500, T = 223 - 393 K (a) 100101102103104105106107108 106 107 108 109 1010 ω / T1 [s-2] ω, ωaT [s-1] T [K] 223 228 233 238 243 253 273 298 323 363 393 PB 87500 (b) Figure 3.18: (a) Dispersion of the relaxation rate 1/T1for 1,4-polybutadiene (PB) with M=87500 g/mol in the temperature range as indicated measured with (open triangles) and without (circles) compensation. (b) Susceptibility representation ω/T1of the same data as in (a). At lowest temperatures the α-peak is discernible and fitted with a Cole-Davidson function (dashed red line). Arrows illustrate frequency-temperature superposition which is applied to create a master curve (solid red line). The color for temperatures is equivalent in both figures (adapted from Pub. 4). ated by shifting the susceptibilities of different temperatures solely in frequency (FTS, illustrated by arrows) to achieve maximal overlap. At low temperatures the susceptibility exhibits a relaxation maximum, which reflects the main structural relaxation and from which the segmental correlation time τscan be extracted. The α-peak can be interpolated with an appropriate function (for instance ColeDavidson susceptibility, dashed red line) which yields the time constant τsfor one temperature (T= 228 K in Figure 3.18b). Then the temperature-dependent shift factors aT(T) can be transformed to the time constants τs(T)≈τα(T) of segmental reorientation (see below and cf. Figure 3.21). Therefore in the following the master curves are plotted as a function of the reduced frequency ωτs. Note that the construction of the susceptibility master curves presumes the applicability of FTS for both the αand the polymer relaxation [38, 146]. That is, the susceptibility spectra of low and high temperatures, which reflect predominantly glassy and polymer dynamics, respectively, are combined to one susceptibility master curve by utilizing τs(T). Thus, the obtained time constants τsof the segmental dynamics have to be compared to those provided by other methods in order to validate the application of FTS. Moreover, the amplitudes of the α-peak for the susceptibilities of different temperatures are expected to coincide. Yet a slight reduction of the peak amplitude with decreasing temperature can be observed on a linear scale; on logarithmic scales this effect is negligible. This weak violation of FTS might result from a temperature-dependent stretching of the peak as an intrinsic property of the α-relaxation or from the emergence of a faster secondary relaxation, which is an ongoing discussion [32, 33, 147–150]. An extra part of 29 3 Extended Abstract Section 3.2 is dedicated to an investigation of this issue. There are several consequences and benefits of creating master curves in the susceptibility representation: First, the master curves including the relaxation data of the complete temperature interval span over about 9 decades in reduced frequency, which is a significant enhancement with respect to the results acquired for one single temperature. Second, the susceptibility master curves can be compared to spectra obtained by DS (Figures 3.6a and 3.7) or rheology (Figure 3.6b). Third, the applicability of FTS can be confirmed by comparing the obtained time constants with those of other techniques, and consequently the time constant of segmental dynamics is identified with that of the main relaxation, i.e., τs≈τα. This is fundamentally important when comparing different systems, in which the dynamics is depending on chemical structure, molecular mass, or solvent. Moreover, it is demonstrated that FC NMR is well suited to probe the slow polymer dynamics, provided that low-Tgpolymers are investigated (cf. also Section 3.2). At high temperatures the rates (Figure 3.18a) are strongly increased compared to those of a simple liquid as OTP (Figure 3.16b) which reflects the polymer-specific dynamics. This manifests itself in the susceptibility (Figure 3.18b) as an excess intensity between the master curve of the polymer and the CD function representing glassy dynamics. Thus, a separation of the susceptibility into glassy and polymer dynamics is possible (see below). Finally, by Fourier transform of the master curves the dipolar correlation function is accessible over a wide time range which permits a direct comparison with the predictions of polymer models (cf. Section 3.4 and Pub. 3). State of the Art Many different polymer melts such as polyisobutylene (PIB), polydimethylsiloxane (PDMS), polyethyleneoxide (PEO), and 1,4-polybutadiene (PB) have been investigated by Kimmich and co-workers [36, 47] by applying FC NMR. However, only a small temperatures range and a few molecular masses without the low-M systems as a reference have been covered. As the construction of master curves has not been attempted, the frequency range has been quite limited. Neither the influence of the segmental relaxation has been explored or quantitatively estimated experimentally. A new approach has been proposed by the R¨ossler group yielding alternative interpretations and these preceding works are outlined in the following. A first comprehensive study of PB by S. Kariyo has included a broad range of temperatures and molecular masses from the low-Mlimit (M < Me) to well entangled melts (MMe) [56, 77, 151]. Its aim has been to systematically characterize glassy and polymer dynamics and their dependency on M. Note that in the following the molecular masses Mrefer to the weight average molecular mass Mw, i.e., M=Mw/(g/mol). At first, the results of the lowest and highest Mhave been compared [56]. The susceptibility master curves of PB 355, PB 466, OTP, and tristyrene are shown in Figure 3.19a. The spectra are identical for ωτs≤1, i.e., on the low-frequency 30 3.1 Introduction 10-6 10-4 10-2 100102104 10-5 10-4 10-3 10-2 10-1 100 Kohlrausch fit (OTP, βK= 0.61) GG fit (PB 466 , βGG= 0.50) χ'' [a.u.] ωτs PB 355 (T = 203 - 333 K) PB 466 (T = 203 - 363 K) OTP (T = 263 - 393 K) tristyrene (T = 283 - 401 K) ∝ ω1 (a) 10-6 10-5 10-4 10-3 10-2 10-1 100101 10-5 10-4 10-3 10-2 10-1 100IKF IIKF IIIKF χ'' [a.u.] PB 355 PB 466 PB 56500 PB 87000 PB 314000 PB 817000 ωτs high-M limit simple liquid limit pure polymer dynamics T = 223 - 393 K (b) Figure 3.19: (a) Susceptibility master curves resulting from applying temperature-frequency superposition for o-terphenyl (OTP), tristyrene and the two low molecular mass 1,4-polybutadienes PB 355 and PB 466. Lines: corresponding interpolations with the generalized gamma distribution (GG) and Kohlrausch function. (adapted from [56]). (b) Susceptibilty master curves for the low-Mand highMPB as indicated. The difference (red) between the high-Mand low-M spectra is identified with pure polymer dynamics. Straight lines: power-law regimes (IKF, IIKF, IIIKF) as discussed by Kimmich, Fatkullin, and co-workers [36, 47] (adapted from [56]). side of the main relaxation, and can be interpolated by a suitable function for the α-process with a power-law χ00(ωτs<1) ∝ω. Explicitly, the oligomers of PB with M= 355 and 466 exhibit the same relaxation behavior as the glass formers OTP and tristyrene and therefore are regarded as representing the glassy dynamics of a simple liquid (”simple liquid limit”). The differences at ωτs>1 are attributed to a different stretching of the α-peak of OTP and PB. Note that the master curves have been scaled to a common peak amplitude (χ00(ωτs≈1) = 1) via division by the amplitude obtained, e.g., from a CD fit (cf. Figure 3.18b). In Figure 3.19b the master curves of several high-Mpolybutadienes are displayed together with the results for the simple liquid limit. At ωτs<1 an intensity in excess to that of the low-MPB is observed. This is due to polymer dynamics which is slower than the segmental dynamics. In the whole frequency range covered the susceptibilities of PB with M= 56500 - 817000 have the same spectral shape, i.e., the dynamics are independent of Mand thus are considered to be characteristic for a well-entangled melt (”high-Mlimit”). The dispersion data of the high-MPB agree with those reported by Kimmich, Fatkullin, and co-workers who have claimed that the observed power-law regimes (IKF, IIKF, IIIKF) are universal [36, 47] and can be explained by the renormalized Rouse model [152–154]. The power-laws provide a fair approximation of the low-frequency side (ωτs<1) of the master curves for high M. However, in regime IKF a power-law behavior cannot be anticipated unambiguously for PB, since the curve continuously bends while approaching the relaxation maximum. Moreover, in this regime glassy dynamics 31 3 Extended Abstract 10-6 10-5 10-4 10-3 10-2 10-1 100101 10-5 10-4 10-3 10-2 10-1 100PB, T = 223 - 390 K ∝ ω1 χ" [a.u.] M [g/mol] 355 466 777 1450 2020 2760 4600 11400 56500 87000 314000 817000 ωτs M (a) 10-3 10-2 10-1 100101102103104105 10-6 10-5 10-4 10-3 10-2 10-1 100 M [g/mol] 355 1450 11400 CDD t /τs PB, T = 223 - 390 K (b) Figure 3.20: (a) Susceptibility master curves plotted as a function of the reduced frequency ωτsfor PB with different Mas indicated (adapted from [77]). (b) Dipolar correlation function CDD versus reduced time t/τsfor three polybutadienes, obtained from selected master curves of (a). Dashed line: estimate of the plateau value associated with reptation dynamics (adapted from [77]). is expected to influence the spectra. Thus, a separation of the master curves into contributions of glassy and polymer dynamics is necessary and can be achieved as will be described below. Note that the power-laws IKF - IIIKF are not identical to those of the tube-reptation model. Thus, the works by Kimmich and Fatkullin raise doubt about the applicability of the tube-reptation model to describe the NMR relaxation behavior of polymer melts. Furthermore, the Mdependence of the dynamics between the two extreme cases of lowest and highest Mhas also been investigated, i.e., the crossover from glassy through Rouse to reptation dynamics [77]. The susceptibility master curves of PB with several different Mare presented in Figure 3.20a. Only above M= 466 an excess intensity with respect to the simple liquid limit is recognized which indicates the onset of Rouse dynamics. Beginning with M= 777 the excess intensity at ωτs<1 is increasing with M. For PB the molecular mass of the Rouse unit MR≈500 is found. MRis determined by the smallest possible Rouse chain. Note that here a factor 2 may be discussed [77, 151]. For higher Mthe crossover to terminal relaxation, which is characterized by a power-law χ00(ω)∝ω(”Debye-limit”, dashed line) is shifted toward lower frequencies. Thus, with higher Mthe distribution of polymer relaxation times (τterminal(M)≤τ < τs) is getting broader. Above M≈2000 the susceptibility master curves become bimodal which indicates the presence of an additional, slower relaxation process, i.e., entanglement dynamics. Consequently, the entanglement molecular mass of PB is identified with Me≈2000. As mentioned before the master curves of M≥56500 are indistinguishable in the accessible frequency range (high-Mlimit) and M-dependent dynamics is expected to be detected only at ωτs<10−6, i.e., beyond the accessible frequency range. 32 3.1 Introduction 150 200 250 300 350 400 -12 -10 -8 -6 -4 -2 0 2 4 lg(τs/s) T [K] DS FC M [g/mol] 355 466 777 816 1450 2020 2760 4600 9470 56500 87000 314000 817000 PB Figure 3.21: Time constants of polybutadiene (PB) as obtained by dielectric spectroscopy (DS) and FC 1H NMR. Lines: VFT-fit of the joint data (adapted from [155]). The dipolar correlation function CDD(t/τs) can be obtained by Fourier transform of the master curves (eq. 3.4). Results for some Mare depicted in Figure 3.20b. In analogy to χ00(ωτs) the same relaxation features are discovered. For the low-M PB the decay of the correlation function can be described by a stretched exponential, which is typical for the α-process of a simple liquid. At higher M, for the non-entangled PB 1450, the correlation decay is retarded due to Rouse dynamics, and for M= 11400 > Methe bimodal shape indicating the onset of entanglement dynamics can be clearly seen. In other words the α-process does not cause alone a complete loss of correlation; a certain residual correlation, which is referred to as polymer relaxation strength f, or the squared order parameter S2(see also below) survives beyond τsand decays on longer time-scales due to Rouse and reptation dynamics. Thus, 1H FC NMR is able to cover three relaxation regimes: glassy, Rouse, and entanglement dynamics. The long-time decay of PB 11400 can be fitted with a stretched exponential (dashed line in Figure 3.20b) and yields for the relaxation strength of entanglement dynamics fe≈0.0006 which corresponds to Se≈0.025. This emphasizes on the one hand that the amplitudes at which entanglement dynamics are disclosed in the correlation function are very low, and on the other hand the ability of NMR relaxometry to probe the correlation function on logarithmic scales, i.e., down to very low amplitudes. By creating the master curves χ00(ωτs) (Figure 3.20a) the time constants τs(T) have been provided. As shown in Figure 3.21 they complement those measured by DS at low temperatures in good agreement which also verifies that the segmental correlation time is identified with time constant of the α-process, i.e., τs≈τα. The joint data of DS and FC 1H NMR can be interpolated for each Mby a VFT 33 3 Extended Abstract function (eq. 3.8) which is a strong indication that FTS works. The Mdependence of the glass transition temperature has been obtained from the condition Tg=T(τs= 100 s). It has turned out that Tg(M) of PB exhibits a non continuous increase (Figure 15 in [77]) as already reported by Cowie [156] for several polymers. Three linear regimes can be recognized and the crossover molecular masses can be identified with MRand Me[77, 157, 158]. The increase of Tg(M) below Mecan be seen of course as well in Figure 3.21 for the temperatures at lg(τs/s) = 2. For that reason it is important to plot the master curves as a function of the reduced frequency ωτs. They are isofrictional spectra [159, 160] in which the dynamics of polymers with different molecular masses can be directly compared because the Mdependence of the glass transition temperature Tghas been taken into account which is pronounced especially at low M[77, 157]. As mentioned the NMR relaxation of polymer melts comprises glassy (local) and polymer (collective) dynamics. One approach to extract spectra which represent solely the latter (”polymer spectra”) is to decompose the susceptibility master curves [56, 77, 151]. Assuming statistical independence and time-scale separation of glassy and polymer dynamics results in multiplicative contributions, and taking into account the relaxation strength fof polymer dynamics yields [56] χ00(ω) = (1 −f)χ00 glassy(ω) + fχ00 polymer(ω) (3.21) Note that a very similar approximation has been described also by Kimmich and Fatkullin [47, 125], yet it has not been employed for a decomposition. For low M, i.e., for the Rouse regime, the assumptions mentioned above are problematic, however, at least for higher Mthis approach can be utilized for a phenomenological description of the relaxation behavior of different polymer systems (see Section 3.3 and Pub. 2). The polymer relaxation strength fcan be identified with the squared order parameter S2and can be determined by integrating the relative polymer contribution in the susceptibility spectra which is referred to as ”excess intensity” f=S2=Z∞ −∞ χ00 polymer(ω)d ln ω/ Z∞ −∞ χ00(ω)d ln ω(3.22) Thus, the polymer spectra χ00 polymer(ωτs) of all M > MRcan be extracted from the total spectra χ00(ωτs) by subtraction of the glassy spectrum χ00 glassy(ωτs) which is assumed to be given by the master curve of PB 466. Since it has been demonstrated by DS that the spectral shape of the α-peak of PB [77], PDMS [157], and polyisoprene (PI) [63] is M-independent, it is well-justified to subtract from each master curve the spectrum of the low-Mreference. The resulting polymer spectra of PB can be compared to spectra calculated from the discrete Rouse model. While the number of Rouse units Ncan be mapped to the molecular masses Mfor M < Me, at higher Mthe Rouse model does not reproduce the bimodal shape of χ00 polymer(ωτs). This is again a clear indication for a second relaxation process, i.e., entanglement dynamics. Furthermore, the onset of entanglement dynamics limits the number of participating Rouse units which can be supposed as the formation 34 3.1 Introduction 102103104105106 0.00 0.05 0.10 0.15 entanglement Rouse f M [g/mol] Mc ≈ 4000 MR ≈ 500 simple liquid (a) 103104 50 100 150 200 Jpolymer(0) M [g/mol] Me experiment Rouse theory (b) Figure 3.22: (a) Relaxation strength fof polymer dynamics as a function of molecular mass. MRand Mc= 2Medenote the Rouse unit and entanglement molecular masses, respectively. Dynamic regimes are indicated. (adapted from [77]). (b) Spectral density of the separated polymer contribution at lowest experimentally accessed frequency as a function of Mand comparison with Rouse theory. Squares: experimental data. Pluses and crosses correspond to MR= 500 and 250, respectively, Meindicates mass of onset of entanglement (adapted from [77]). of a transient network. Note that the discrete Rouse model has to be applied, since a certain approximation cannot be used for small N, i.e., a limited number of Rouse modes [77]. Moreover, the polymer spectra need to be normalized in amplitude by their integral [161]. By inspecting the dependence of the polymer relaxation strength fon M(Figure 3.22a) again three regimes show up [77]. The transitions between the regimes mark the characteristic molecular masses MRand Mc.Mcdenotes the critical molecular mass as it is observed, e.g., for the M-dependence of the viscosity [53, 80, 86, 162], and is usually twice the entanglement molecular mass Me [114, 163]. In the simple liquid regime f= 0 by definition, in the Rouse regime f(M) is continuously increased, and in the entanglement regime it levels off. The maximum value of f≈0.11 is essentially determined by Rouse dynamics, although a weak increase feis still observed above Mc, which corresponds to that estimated from the long-time decay of the correlation function (cf. Figure 3.20b). frepresents the squared order parameter S2[116, 164–166] and may be taken as a measure of spatial restriction of glassy dynamics. It is worthwhile to note that the quantity probed by NMR [56] is not related to structural order in the sense of long-range or static orientational order as known from, e.g., nematic liquid crystals [167]. Thus, the restriction of local segmental dynamics is a result of some segments within a single chain being pinned down between two entanglements. However, the opposite is not necessarily true, i.e., structural order may not be deduced from restricted local motion. Moreover, an important conclusion can be drawn from fregarding the analysis of 35 3 Extended Abstract 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100101 103 104 105 106 107 108 109 M [g/mol] 162 311 860 1600 2490 5940 11000 21600 41400 128000 232000 PB 466 ν/T1 [s-2] ωτs ∝ ω1 PDMS, T = 163 - 363 K Figure 3.23: Susceptibility master curves of PDMS with Mand in a temperature range as indicated. PB 466 includes as a reference for a simple liquid. Vertical lines: regime of the M-independent power-law (adapted from [155]). the dispersion data. The susceptibility χ00(ω) = ω/T1(ω) does only reflect polymer dynamics, if (1 −f)χ00 glassy(ω)fχ00 polymer(ω) holds (cf. eq. 3.21). In consequence especially if f1, e.g., for short chains exhibiting solely Rouse dynamics or for systems with a generally weak relaxation strength, the assumption of time-scale separation is not sufficient to ensure that polymer dynamics actually dominate even at ωτs1. This is illustrated in Figure 3.19b in which also the polymer spectrum of the high-Mlimit is shown. Therefore, also when evaluating dispersion data in the T1or 1/T1representation, the fast (here: α) relaxation processes have to be taken into account [168], which is an important insight with respect to previous reports [36, 47]. An additional way to determine Meis to directly compare the NMR relaxation results with the predictions of the Rouse model and to determine the Mat which they begin to deviate from each other. In Figure 3.22b the spectral densities of the polymer contribution at the lowest accessible frequencies ωτs→0 are displayed together with those calculated from the discrete Rouse model (cf. eq. 3.14) assuming MR= 500 or 250. While the lower Mappear to follow the Rouse behavior Jpolymer(0) ∝log M, above a certain Mthe experimental spectral density increases stronger, which is due to the onset of entanglement dynamics. This transition defines Me≈2000 and is in good agreement with literature [169]. As a second polymer PDMS has been investigated in a wide molecular mass (M= 162 −232000) and temperature range (T= 163 −363 K) [170]. The susceptibility master curves are displayed in Figure 3.23. Concerning the polymer dynamics at ωτs<1 the relaxation behavior is in general similar to that of PB. Nevertheless a clear difference between the master curves of PDMS and PB can be 36 3.1 Introduction recognized: Whereas in the case of PB the excess intensity due to polymer dynamics sets in at frequencies just below the main relaxation peak (cf. Figure 3.20a), for PDMS the polymer dynamics are shifted to lower frequencies and in between (marked by vertical lines) a power-law is discernible, which is a common feature for all M. This result has been explained as an anomalous α-process [170]. Yet, the effect has not been completely clarified, since possible influences of intermolecular relaxation were believed to be negligible. In summary, the following principles set the approach of the R¨ossler group apart from that of Kimmich and co-workers: The measured temperature range should be as broad as possible in order to detect both the fast segmental and the slower polymer-specific dynamics. Low-Msystems are investigated as a reference for the glassy spectrum which represents the segmental dynamics also for higher M. FTS is applied to create master curves in the susceptibility representation, which significantly enlarges the accessible frequency range. The master curves are Fourier transformed to the correlation function, which allows for a comparison with the results from other experimental techniques and simulations, and predictions by polymer models. As a consequence, the interpretations go far beyond what has been reported in the literature for NMR relaxation results, e.g., the emergence of polymer dynamics with Mcan be clearly traced and the characteristic molecular masses MRand Mecan be determined. Recently [171] the term ”molecular rheology” was coined with respect to FC and DQ 1H NMR and their joint capability of exploring the time range from glassy dynamics to terminal relaxation in entangled polymers like ”conventional” rheology. The mentioned principles are continued and enhanced in this thesis; especially lower frequencies (or longer times) will be accessed by technical means in order to experimentally ascertain as an ultimate goal the reorientational correlation function g2(t) and the mean square displacement as they are schematically rendered in Figure 3.12 over several decades. Objectives of this Thesis Based on the above findings more profound questions were raised which establish the principal part of this thesis. Moreover, from the studies of polymer melts the topics ”polymer dynamics in confinement” and ”intramolecular and intermolecular relaxation in low-molecular systems” have evolved and are continued by M. Hofmann [69] and R. Meier [172–174], respectively. The following items motivate the next sections. The sections themselves provide compact introductions and cross-relations to the corresponding publications Pub. 1 - 5 (Section 4). They also give a perspective of recent developments or further studies which are currently underway. •Section 3.2 and Pub. 1 What can be learned from a quantitative evaluation of the correlation functions of different Mand a comparison to results from simulations? When does a molecule become a polymer and can the molecular mass MRof the Rouse unit be reliably determined? 37 3 Extended Abstract included. Considering first the results of PS, a non-systematic trend with Mis observed. The master curve of M= 370 already exhibits an excess intensity, whereas that of M= 690 clearly traces the Debye behavior. The susceptibility of PS 1380 shows a low-frequency intensity, which is larger than that of PS 370. Note that the molecular masses provided by the supplier have been confirmed by oligo-GPC (gel permeation chromatography) and MALDI-TOF (matrix-assisted laser desorption/ionization time of flight mass spectrometry) measurements. In the case of PB the effect is similar. The lowest M= 355 is not identical with the Debye curve, the slightly larger M= 466 exhibits a little lower intensity, which is yet not identical to the Debye behavior, and at higher Ma stronger low-frequency contribution is noticeable. Thus, for PB the previously determined molecular mass of the Rouse unit MR≈500 is essentially confirmed. In the case of PS, MRis found to be higher than that of PB, say MR≈1000. This is slightly larger than the mass of the Kuhn segment (about 8-9 monomers [175]), yet smaller than the mass of the random step Mr≈5000 reported by Sokolov and co-workers [175]. Given the currently available data, an explanation of the non-systematic behavior at very low Mis rather speculative without further investigations. However, it is known from studies of low-Mglass formers that an intermolecular relaxation contribution which reflects translational diffusion is always present at low frequencies (ωτs<1) [68, 173, 174]. This might also be the reason for the anomaly observed for PDMS (cf. Figure 3.23). Since the intermolecular relaxation and the onset of polymer dynamics occur in the same frequency range, they are probed simultaneously by 1H FC NMR and cannot be readily isolated. Regarding this issue more details and the extracted intermolecular relaxation contributions for PB and PDMS are presented in Section 3.5. Since at this stage this phenomenon cannot be resolved, we refrain from applying the second and third scaling, and create the master curves as described in Section 3.1, i.e., allowing minor FTS violations which are negligible on logarithmic scales (cf. inset of Figure 3.25a) and even smaller in case of PB. In summary, presently the answer to the question ”When does a molecule become a polymer?” by evaluating results of FC NMR relaxometry remains the one given above in this section. It may be refined in a future study, which explores the development of the intermolecular relaxation in oligomers with different M by isotopic dilution experiments. Thereby intermolecular and intramolecular relaxation contributions can be separated. Consequently, this would facilitate the identification of the polymer-specific relaxation as reflected in the intramolecular contribution. With increasing M, it would be expected for the intramolecular contribution that beginning at MRthe excess intensity with respect to a Debye susceptibility is discernible, while the intermolecular contribution is successively shifted toward lower reduced frequencies, as diffusion is slowed down. Furthermore, the study of PS, whose Tgchanges by more than 250 K from its monomer to high M[157], emphasizes that only low-Tgpolymers can be investigated by FC NMR given the currently available temperature rage of the Stelar relaxometer; of course also the temperature stability of the polymer has to be 44 3.2 From Simple Liquid to Polymer Melt considered. For say Tg>340 K the frequency regime of the polymer relaxation can not be reached anymore at highest temperatures (T≈400 K). For example, the relaxation maximum of PS 1380 (Tg= 314 K [157]) is observed in the range T= 368 −388 K (cf. Figure 3.25a). For the next higher M, PS 3250 with Tg= 347 K [157], the α-peak can be expected at temperatures T≈400 −420 K. This underlines the necessity to access higher temperatures and also lower frequencies, which requires several hardware modifications. Note that the results of PS 1380 at T= 408 K (cf. inset of Figure 3.25a) have been obtained for frequencies down to 1 kHz with the FC NMR relaxometer in Darmstadt (see Section 3.4 for further details). 45 3 Extended Abstract 3.3 Universal Dynamics for Various High-M Polymers 30 For Table of Contents use only Universal polymer dynamics revealed by field cycling 1H NMR A. Herrmann §, S. Kariyo ☼, A. Abou Elfadl §, R. Meier §, J. Gmeiner §, V.N. Novikov+, E.A. Rössler § * § Experimentalphysik II, Universität Bayreuth, 95440 Bayreuth, Germany ☼ Faculty of Science and Technology, Yala Islamic University, 135/8, M.3, A. Yarang, Pattani 94169, Thailand + IA&E, Russian Academy of Sciences, Novosibirsk, 630090, Russia Universal polymer spectra for PB, PI and PDMS revealed by field cycling NMR 10-6 10-5 10-4 10-3 10-2 10-1 10-4 10-3 10-2 10-1 100 M >> Me Rouse regime terminal relaxation entanglement regime ω0.5 ω1 polymer susceptibilty ωτs M = Me PB, PI, PDMS Figure 3.27: Table of Content Graphic (adapted from Pub. 2). The emergence of polymer dynamics has been thoroughly investigated for PB in a broad range of temperatures and molecular masses by an quantitative analysis of both susceptibility master curves and the dipolar correlation functions (Pub. 1). While the glassy dynamics are intrinsic to the relaxation behavior for all M, for the high-Mlimit two power-laws have been observed, which have been attributed to Rouse and entanglement dynamics. However, for polymer melts the predictions for the power-law exponents of the tube-reptation model have not been confirmed by the results of the FC 1H NMR experiments. The exponent of regime II has been discovered by Kimmich and co-workers [36, 183, 184] merely for a linear polymer confined in a solid matrix, i.e., for chain dynamics in a static tube. Yet, the authors have considered the low-frequency power-law observed for the 1H relaxation of linear polymer melts as universal [47], although some reports show a different behavior [185]. Thus, the goal of Pub. 2 is, first, to compare the 1H relaxation of different polymers in order to review whether the low-frequency behavior as it is reflected in the T1dispersion or the susceptibility is in fact universal. For that purpose high-MPI was measured and the resulting susceptibility master curve is analyzed together with the existing ones of PB and PDMS. Second, under the assumption that the entire relaxation can be separated into contributions of glassy and polymer dynamics, the polymer spectra can be extracted and the relaxation strength fcan be determined (Section 3.1). Moreover, it has been discussed that the order parameter in entangled systems depends on the specific proton-proton dipolar couplings [102, 116, 165], i.e., for a high-Mpolymer melt fdepends on the chemical structure of the monomer unit (eq. 5 of Pub. 2). Therefore partially deuterated PB and PI (the latter from [185]) were investigated and are compared with the corresponding completely protonated polymers. The central results of Pub. 2 are summarized in the following: 1. For PB, PDMS, and PI with M > Methe dispersion of T1(Figure 1 in Pub. 2) and consequently the susceptibility master curves (Figure 2 in 46 3.3 Universal Dynamics for Various High-MPolymers Pub. 2) do exhibit a non-universal power-law behavior at low frequencies in contrast to the claim in literature [47]. The master curves of PB have a larger excess intensity than those of PI. In the case of PDMS polymer dynamics sets in at lower frequencies than in PB and PI, and an M-independent power-law ∝ω0.5is observed between polymer dynamics and the α-peak (Figure 4a in Pub. 2). A similar power-law is also seen in other low-M systems at ωτs<1 (Figure 4b in Pub. 2). 2. Different selectively deuterated PB and PI show other power-laws in T1(ν) than the corresponding completely protonated polymer chains. Concomitantly, the shape of the susceptibility master curves at ωτs<1, i.e., in the range where polymer dynamics are located, is not uniform for all polymers studied. 3. A common shape for high-MPB, PDMS, PI, and the partially deuterated samples is only disclosed by taking into account the contribution of glassy dynamics along eq. 3.21. That is, at lowest frequencies their normalized polymer spectra show a common power-law ∝ω0.5(Figure 3 in Pub. 2). Furthermore, at intermediate frequencies (10−3< ωτs<1) the high-M spectra have the same shape as those with M≈Me. This demonstrates that in the Rouse regime (I) the dynamics of non-entangled polymers with M≈ Meand entangled polymers are essentially identical. Thus, the apparent discrepancy in the overall susceptibility spectra is resolved by isolating the polymer spectra confirming the decomposition approach. 4. The relaxation strength fof polymer dynamics depends on the orientation of the internuclear vectors of the protons with respect to the chain contour (Table 2 in Pub. 2). For PB which has the highest values of f, the differently deuterated samples can be clearly distinguished: For the one with the protons at the carbon atoms of the double bond, fis higher than for the fully protonated one, whereas it is vice versa for the one with the protons at the methylene groups. In the first case the 1H–1H direction is parallel to the chain contour given by the double bond. This causes just a small reduction of the order parameter of the chain. In the case of the methylene group protons, the angle between their axis and the chain contour is larger, thus the reduction is much stronger. For the completely protonated PB a value in between is found. Ad 1.: The anomalous relaxation behavior of PDMS with the M-independent power-law (10−2< ωτs<1) has been outlined in Section 3.1 already. In Figure 5 of Pub. 2 it is shown explicitly that it is also found for the 1H relaxation of the low-Mglass formers glycerol and propylene glycol, and it has been attributed to an anomalous α-process. Yet, the relaxation feature does not show up in the dielectric spectra of PDMS and glycerol [186], which indicates an intermolecular origin (cf. Section 3.5). This triggered a thorough investigation of different low-M 47 3 Extended Abstract systems [68, 172, 173, 187] with the aim to separate intramolecular and intermolecular contributions. Since the latter can be related to translational motion a quite simple way of determining the diffusion coefficient by FC NMR relaxometry has been reported [174]. Ad 3.: The power-law ∝ω0.5observed in polymer spectra at lowest frequencies corresponds to the power-law ∝t−0.5found in the dipolar correlation function of high-MPB (Figure 2 of Pub. 1). The fact that a common power-law is revealed for all high-Msystems including the selectively deuterated ones, emphasizes the importance of taking into account also the different relaxation strength of polymer dynamics. Therefore it is eminently important to study a broad temperature or frequency range, especially to access also sufficiently low temperatures including glassy dynamics. Since the glassy dynamics is a generic feature of the relaxation behavior of polymers, it is necessarily required to have a reference (τs) for the time scale to account for a change of τs, and to estimate fin order to draw conclusions from comparing different polymers. Note that an alternative approach of determining fis presented in Figure 5 in Pub. 2. Furthermore, it is demonstrated in Figure 6 in Pub. 2 that the subtraction of glassy dynamics from the total susceptibility yields essentially the same result for the polymer spectrum as a multiplicative decomposition of the dipolar correlation function. Regarding the above mentioned results of polymer dynamics in a static confinement by Kimmich and co-workers [36, 183, 184] two comments are worthwhile. Firstly, the conclusion that the power-law of regime II of the tube-reptation model has been revealed has been drawn at first [184] from deuterated PEO in a protonated matrix, i.e., by applying FC 2H NMR in order to detect solely the NMR signal of the confined polymer. Thereby also intermolecular relaxation contributions are excluded. Later [183] the power-law has also been confirmed by 1H relaxometry which comprises intramolecular and intermolecular contributions. Although usually a different dispersion can be expected for 1H and 2H relaxation, here it appears that the intermolecular relaxation either has the same spectral shape as the intramolecular contribution or does not play a role. Secondly, from these results the corset effect has been discovered [183], i.e., that confinement effects have been observed even for confinement diameters much larger than the size of a single chain. The discussion is still ongoing [188, 189] and involves also new results by NS [190, 191] and DQ NMR [192]. However, recent FC 1H NMR experiments for PB in nanoscopic matrices of anodic aluminum oxide by our group [69] have not shown a clear indication for the corset effect. Instead it appears that the relationship between τsand τRis changed in confinement. Very recently we have reported [155] on a similar M-dependence of the polymer dynamics for PB, PDMS, PI, and polypropylene glycol (PPG) melts. Especially the protracted transition to full reptation dynamics has also been observed. 48 3.4 Protracted Crossover to Reptation Dynamics 3.4 Protracted Crossover to Reptation Dynamics 1 Corrections of galley proofs of manuscript ma202489y Protracted crossover to reptation dynamics: a field cycling 1H NMR study including extremely low frequencies A. Herrmann, B. Kresse, J. Gmeiner, A.F. Privalov, D. Kruk, F. Fujara, E.A. Rössler ll. 766 ref 40 is now published, replace “DOI: 10.1016/j.ssnmr.2011.10.002.” by “40, 134137.” Please replace the TOC graphic, since the legend on the right was missing. 10-1 100101102103104105106107108 10-6 10-4 10-2 100M / Me ≈ 1, 5, 18, 220 polybutadiene M << Me correlation function reduced time 220 (DQ) 0.32 earth field compensation close to tube-reptation model in accord with DQ NMR Figure 3.28: Table of Content Graphic (adapted from Pub. 3). The previous studies [56, 77, 151, 161] and Pub. 1 and 2 have clearly shown that the most prominent features of polymer dynamics for high-Mmelts are revealed by FC NMR at high temperatures and low frequencies. While the Rouse regime (I) has been well covered in the master curves of PB, at lower reduced frequencies the constrained Rouse regime (II) reflecting influences of entanglements has been found to be merely in a state of development (see Figure 1 in Pub. 1). Especially the power-law exponent of regime II characterizing the long-time slope of the correlation function has not reached = 0.25 as expected for M > Meby the tubereptation model. Therefore the two limitations, highest temperature (T≈410 K) and lowest frequency (ν≈10 kHz) which actually stem from the specifications of the Stelar spectrometer, should be reconsidered in order to further augment the accessible dynamic range of the method. An additional increase of the temperature above T= 408 K (the highest Tat which PB was measured in Pub. 1) is principally possible. For this purpose a home-built probehead together with a more powerful heating system could be employed which have been developed recently [134]. First test runs have indicated that higher temperatures (T≈500 K) than with the commercial equipment can be achieved while the signal-to-noise ratio of the new solenoid coil design still has to be improved. However, since the temperature dependence of the time constants τs(T) is very weak at such high temperatures (see Figure 3.21), the dynamics is expected to be shifted just minimally with respect to the already measured temperature, i.e., after applying FTS the master curves would be extended only insignificantly. Moreover, in the case of PB sample degradation is an issue [193], i.e., it inhibits experiments above say T≈410 K. Thus, an enhancement toward lower frequencies (ν < 10 kHz) is more promising. As the sample inside of the probehead is still exposed to the magnetic field of the earth and stray fields of the superconducting magnets in the lab which are on the scale of a few kilohertz, a feasible solution to perform relaxation experiments at defined lower fields is shielding the sample from these influences. The Stelar spectrometer offers the possibility to compensate magnetic fields along the axis 49 3 Extended Abstract Abbildung 3: Schema einer „Nutation“ Grund von leicht adiabatischen Schaltvorgängen oder bei hohen Frequenzen fvon der Totzeit verursacht werden kann. Mz(t) = a·cos(2π·f·t+ϕ)·exp(−t/T∗ 1) + b(26) Über die Messung von „Nutationen“ kann das Evolutionsfeld für kleine Frequenzen wie folgt kalibriert werden: Ist der Offset bei einer „Nutation“ viel kleiner als die Amplitude bzw. ist die Oszillation symmetrisch um null, so ist das transversale Feld Bx y viel größer als das Feld in z-Richtung. Auf diese Weise kann der Offset des Evolutionsfeldes auf null eingestellt werden und zusätzlich das Magnetfeld in transversaler Richtung Bx y absolut bestimmt werden. Stellt man nun verschiedene Evolutionsfelder ein und misst bei jedem die Frequenz fder „Nutation“, so kann über eine einfache trigonometrische Überlegung das Evolutionsfeld präzise bestimmt werden: B2 ev= (2πf/γ)2−B2 x y (27) Eine weitere Möglichkeit besteht in der In-situ Bestimmung des Evolutionsfeldes während einer T1Messung. Dazu wird wie oben beschrieben ein bekanntes Feld in transversaler Richtung angelegt. Dieses sollte kleiner als das Evolutionsfeld sein. Wenn das T1der Probe lang genug ist sowie der Zeitbereich und die Punktdichte bei der Aufnahme richtig gewählt werden, so sollte man „Nutationen“ auf dem Plateau der Magnetisierungskurve vor dem T1-Abfall sehen. 10 Figure 3.29: Scheme of a nutation caused by a non-compensated component of the magentic field in the x,y-plane (adapted from [196]). of the main coil by adjusting the offset current of the magnet with the help of a Teslameter which is inserted at the sample position before the actual relaxation experiments are started. Components of magnetic stray fields in the plane perpendicular to the axis of the main coil may be canceled out by additional coils placed around the main coil. However, it has turned out that after installing the saddle coils provided by Stelar it is still not possible to acquire reasonable magnetization curves at lower frequencies nor to successfully perform nutation experiments (see below and [194, 195]). The observation of the latter would demonstrate that the low fields are precisely controlled and stable in time. The origin of a nutation is illustrated in Figure 3.29. If there is a component Bxy of the magnetic field in the x,y-plane which is not negligible compared to Bz, then the effective field Bres is not parallel to zanymore. As a result the magnetization Mprecesses on a cone around Bres. The acquired signal (projection on the x,y-plane) shows an oscillating exponential decay with a frequency of the Larmor frequency corresponding to Bres and the time constant T1, respectively. This allows measuring both quantities simultaneously (cf. also [133, 196]). The fact that the nutation cannot be observed with the commercial relaxometer is due to the power supply of the magnet, since the current exhibits random spikes (some of them might be attributed to the rectified three-phase current). As a result the relaxation fields below ν= 10 kHz are neither stable nor reproducible in time which leads to an unpredictable evolution of the magnetization. Thus, the drawbacks of the Stelar spectrometer are, firstly, that by construction it operates solely the single main coil for generating high (≈1 T), low (≈1 mT), and (allegedly) ultralow (≈10 µT) magnetic fields and, secondly, the bipolar design of the power supply. The first requires digital-to-analog-converts which operate linearly over say five decades and the second a sophisticated compensation for leakage currents of the MOSFETs. Unfortunately up to now a solution by Stelar is lacking and relaxation experiments below ν= 10 kHz are not possible with the current design of the commercial spectrometer. 50 3.4 Protracted Crossover to Reptation Dynamics The situation is more favorable in Franz Fujara’s group at TU Darmstadt (Institut f¨ur Festk¨orperphysik). Their home-built field cycling NMR spectrometer (FC-I) [132] has been continuously upgraded during the last 10 years and in the meantime separately controls six different coils. There are three pairs of coils for compensating static fields in x-, y-, and z-direction and two B0-coils for the frequency ranges 30 MHz - 250 kHz and below 250 kHz. For attaining frequencies below 4 kHz there is yet another coil which is equipped with an active compensation system, i.e., the relaxation field is stabilized by simultaneously measuring the magnetic field at the sample position with a fluxgate sensor and returning this value to the PID controller. Further details are described by B. Kresse et al. in ref. [133]. Therein a reproducible nutation at very remarkable 12 Hz has been reported. Consequently, relaxation experiments at significantly lower fields than accessible with the commercial spectrometer can be reliably performed with the FC-I spectrometer and the NMR relaxation of polymers is of course an attractive area for application, which is the topic of a joint DFG project. The goal is now to investigate the polymer dynamics on even slower time-scales than reported in Pub. 1 and 2, and to compare the obtained correlation functions to model predictions. For this purpose six PB samples with M > Mewere measured both in Darmstadt and Bayreuth, and the existing master curves of three other PB samples (previously measured in Bayreuth) were supplemented by measurements in Darmstadt. The key results of Pub. 3 are: 1. The master curves now cover 10 decades in frequency (Figure 5a in Pub. 3) and are obtained by having succeeded in extending the frequency window for two decades toward lower frequencies (Figure 2 in Pub. 3). 2. The dipolar correlation functions, which now include results up to M≈ 220Me, clearly reveal the protracted transition to reptation dynamics (Figure 6a in Pub. 3). This is also reflected in the very slow decrease of the M-dependent power-law exponent of regime II from = 0.73 down to 0.32 (Figure 6b in Pub. 3), which however is not identical with = 0.25 predicted for regime II of the tube-reptation model. 3. The FC 1H NMR results agree well with those of DQ 1H NMR by Vaca Ch´avez and Saalw¨achter [117, 171, 197] (Figure 7 in Pub. 3) which demonstrates that FC 1H NMR has turned from a complementary method to DQ 1H NMR into a competitive one. Ad 2.: The discussed influence of constraint release and contour length fluctuations may be assessed in a forthcoming study in which ”isotopic tri-block-copolymers” are investigated both by DQ and FC 1H NMR; that is, the polymer chains have a defined number of protonated monomers in the center while the chain ends are deuterated. In this way the dynamics probed by 1H NMR originates from the center monomers which are exposed to entanglement effects to a higher degree than the chain ends as it has been observed by simulations [104, 107, 181] and neutron 51 3 Extended Abstract 10-1 100101102103104105106107108 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 DQ NMR (T = 243 K - 363 K) M = 23600 - 2000000 FC NMR (T = 223 K - 408 K) M = 466 - 18000, without comp. M = 9470 - 441000, with comp. CDD (t/τs) t/τs PB ∝ t-0.32 Figure 3.30: Dipolar correlation functions of PB as a function of the reduced time t/τs as obtained by DQ 1H and FC 1H NMR in the temperature and Mrange as specified. The DQ data are from [171]. Dotted line represents glassy dynamics. Dashed line: power-law of regime II observed for M= 441000. scattering experiments [108]. It can be expected that thereby the exponent of regime II is further diminished with respect to that for completely protonated chains and comes closer to = 0.25 as predicted by the tube-reptation model. In addition, blends in which the protonated and deuterated chains have different Mcan be investigated, e.g., to study effects of constraint release. Presumably  will decrease with respect to the isotopic blend with equal M, i.e., come closer to the prediction of the tube-reptation model, if the matrix has a significantly larger Mthan the probe chain. However, first of all the influence of the intermolecular contribution, which is located in the same frequency range as polymer dynamics, to the total 1H relaxation has to be investigated. This is the topic of the next Section and Pub. 4. Ad 3.: In Figure 3.30 the dipolar correlation functions CDD(t/τs) obtained by FC and DQ 1H NMR for PB are compared on absolute scale. The FC data without stray field compensation correspond to Figure 2 in Pub. 1, cover about 8 decades in time, and contain molecular masses up to M= 18000 ≈9Me. They are very well complemented by the DQ data, which however begin just at t≈τeτs, to longer times and higher M. By employing the stray field compensation and including higher M, FC 1H NMR is now able to cover also the range which was previously reserved by DQ 1H NMR. This illustrates the strength of low-field FC 1H NMR relaxometry: The dipolar correlation function can be traced over about 10 decades in time and 8 in amplitude, comprises also the regime of glassy dynam52 3.4 Protracted Crossover to Reptation Dynamics 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100101 10-6 10-5 10-4 10-3 10-2 10-1 100 ∝ ω1 ∝ ω0.75 ∝ ω0.50 M [g/mol] 162 - 21600 41400 128000 232000 χ''DD [a.u.] ωτs PDMS T = 163 K - 408 K ∝ ω0.50 with compensation Figure 3.31: Susceptibility master curves of PDMS as a function of the reduced frequency ωτsin the temperature and Mrange as specified. The data below ωτs≈10−7 were acquired with the FC-I spectrometer in Darmstadt. Lines: observed power-laws. ics, and, depending on M, spans all regimes of the tube-reptation model. Note that the protracted transition to reptation behavior has been disclosed also in very recent MD simulations [181] which were inspired by the joint results of FC and DQ 1H NMR (Pub. 1, [117]). Recently we have reported [155] a comparison of the relaxation behavior of different polymers (PB, PDMS, PI, and PPG). It remains the task of future work to study also the very low frequencies in all these systems. In perspective even lower frequencies may be accessible [133, 198]. First results for PDMS are shown in Figure 3.31. Three samples of PDMS (M= 41400, 128000, and 232000) were measured with the FC-I spectrometer in Darmstadt and the corresponding master curves include about 1.5 decades below ωτs≈10−7which is the low-frequency limit in Bayreuth. While for M= 41400 the crossover to terminal relaxation (∝ω1) is observed, the susceptibilities of the two samples with higher Mbear the same spectral shape, i.e., they constitute the high-Menvelope for PDMS in the accessible frequency range. Interestingly, they reveal a low-frequency power-law which can be only vaguely anticipated from the Bayreuth data alone (cf. Figure 4a in Pub. 2) and its power-law exponent = 0.50 at low reduced frequencies is different from that of PB (= 0.32) and that predicted by the tube-reptation model for regime II (= 0.25). This result will be discussed in the context of the findings of the next section. 53 3 Extended Abstract 3.6 Linear Polymers in Solution 5 Dynamics of Linear Polybutadienes in Solution Studied by Field Cycling 1H NMR Axel Herrmann, Ernst A. Rössler For TOC use only. 100101102103104105106107 10-6 10-5 10-4 10-3 10-2 10-1 100 30% 50% 76% dipolar correlation function reduced time PB 18200-h6 in toluene-d8 simple liquid 100% ∝ t−ε f power-law exponent depends on molecular mass and polymer mass fraction relaxation strength of polymer dynamics reduced upon dilution Figure 3.36: Table of Content Graphic (adapted from Pub. 5). Up to now neat polymers have been studied by NMR relaxometry in this thesis and the results have been compared to the predictions of the tube-reptation model for high-Mpolymer melts. How polymer dynamics is modified upon dilution with respect to that of the melt has been subject to many studies by rheology and is now considered as textbook knowledge [80, 159, 169]. The relaxation behavior of polymers in solution has been analyzed also in reports by dielectric spectroscopy [66]. First experiments by FC NMR were performed by Kimmich and co-workers and the T1dispersions of PDMS diluted in CCl4at different concentrations have been compared [180]. The authors have claimed to observe a transition from entanglement dynamics for high Mto Rouse dynamics upon dilution [36]. However, their study included results only for one temperature and the analysis did not take the concentration dependence of the time constant τsof segmental dynamics into account, which also drives the polymer dynamics. Preliminary own measurements [170] of PB in solution have already indicated that the polymer relaxation strength fis reduced upon dilution. That is, the low-frequency part of the master curves reflecting the relaxation contribution due to polymer dynamics is diminished. However, the PB samples studied therein had a wide distribution of molecular masses and the solvent (CCl4) turned out to be inappropriate. The aim is now to investigate PB with well-defined molecular masses systematically for different M > Me, various concentrations, and in a broad temperature range, and to utilize the same approach for analyses which has been successfully used for bulk polymer melts. Deuterated toluene was chosen as solvent and samples with 10 different concentrations for PB 9470, three for PB 18200, and one for PB 47000 were prepared and measured with the Stelar relaxometer. Note that the given concentrations crefer to the mass fraction of the polymer. The data of the pure PB melts are from Pub. 3. The obtained relaxation data are transformed to the susceptibility representation, FTS is applied, and master curves χ00 DD(ωτs) are constructed. The polymer relaxation features are analyzed both in the frequency domain and in the time domain by accessing the dipolar correlation function CDD(t/τs) via Fourier transform. Note that the master curves of the 60 3.6 Linear Polymers in Solution pure PB 18200 and PB 18000 are identical in the frequency range accessible by the Stelar relaxometer – as expected due to the tiny difference in M. Since the latter sample has also been measured toward lower frequencies in Darmstadt (cf. Pub. 3) both are used equivalently. The central results are summarized in the following: 1. The susceptibility master curves χ00 DD(ωτs) (Figure 1 in Pub. 5) reflect the regimes of glassy, free Rouse, and entanglement dynamics as determined from the undiluted PB. The relaxation features of polymer dynamics, i.e., the excess intensity at ωτs<1, are continuously diminished upon dilution. That is, the relaxation strength f(c) of polymer dynamics is reduced with decreasing cin a similar way for all M(Figure 3a of Pub. 5). For high cthe decay is similar to that of the plateau modulus G0 N(c) = G0 N(1)c2.3, which is a comparable quantity obtained by rheology. Moreover, it appears that in the applied concentration range the intermolecular contribution is not reduced, since amplitude of the relaxation maximum which is associated with the dipolar coupling constant (cf. eq. 3.19) is not systematically changed. 2. By construction of the master curves the time constants τs(T) of segmental motion are provided (Figure 2 in Pub. 5) which show an acceleration of the dynamics upon dilution (plasticizer effect) due to the concentration dependence of the monomeric friction coefficient. This results in a decreased glass transition temperature Tg(c) for lower polymer mass fractions. 3. Evaluating the susceptibility χ00 DD(ωτs) of PB 9470 in solution at lowest frequencies yields the concentration dependence of the terminal relaxation time τd(c) (Figure 3b in Pub. 5). It is decreased upon dilution and essentially follows the behavior τd(c)∝c2.3expected from rheology [159]. 4. The dipolar correlation functions CDD(t/τs) (Figure 4a in Pub. 5) obtained by Fourier transform of the master curves exhibit at long times two powerlaw regimes ∝t−, which are attributed to the regimes I and II of the tube-reptation model. In order to consider only the contribution of polymer dynamics, i.e., to remove the influence of the segmental dynamics, the separated correlation function of the polymer dynamics Cpolymer(t/τs) is attained (Figure 4b in Pub. 5) and the c-dependent long-time exponent (regime II) is extracted (Figure 5 of Pub. 5). It is increased from its bulk value upon dilution up to values close to = 1 as expected for the Rouse regime and thus demonstrates the diminished contribution of entanglement dynamics. 5. From the increase of it is deduced that the entanglement molecular mass Me(c) increases for lower polymer mass fractions (inset of Figure 5 of Pub. 5) as expected from rheology [169] via Me(c) = Me(1)c−1.3. If Me(c) for low 61 3 Extended Abstract 104105106107 100 101 102 103 104105106107 323 293 253 233 223 213 204 193 R1,DD [s-1] ν [Hz] 183 c = 50% (a) 50% 30% 76% 100% PB 18200-h6 in toluene-d8 T = 298 K (b) 10-6 10-5 10-4 10-3 10-2 10-1 100 104 105 106 107 108 109 PB 466 PB 18000 T = 298 K 100% 76% 50% 30% T = 178 K - 393 K PB 18200-h6 in toluene-d8 ωτs χ''DD [s-2] (c) Figure 3.37: 1H relaxation rate R1,DD(ν) = 1/T1,DD(ν)of PB 18200-h6diluted in toluened8(a) at c= 50 % polymer mass fraction and in the temperature range (in K) as indicated, and (b) for T= 298 K and different concentrations. (c) Susceptibility master curves of PB 18200-h6including all concentrations and temperatures as indicated. The susceptibilties obtained for T= 298 K are highlighted in color. concentrations is not small compared to Manymore, the feature of entanglement dynamics, i.e., the long time shoulder, is suppressed in the polymer correlation function Cpolymer(t/τs). Ad 1.: The measured proton relaxation rates R1,DD(ν) reflect features which depend on temperature, polymer mass fraction, and molecular mass. In Figure 3.37a the proton relaxation rates R1,DD(ν) of PB 18200-h6diluted in toluene-d8at c= 50 % for various applied temperatures are displayed. For comparison the rates at different concentrations and fixed temperature are displayed in Figure 3.37b. They reflect differently strong dispersion regimes depending on temperature and c-dependent changes of the spectral shape, respectively. Note that Kimmich and co-workers have concluded from the results [36, 180] at a single temperature that with increasing dilution the interval between segmental motion and entanglement dynamics is enlarged yielding space for Rouse dynamics. This is apparently supported by Figure 3.37b. However, the concentration dependence of the monomeric friction coefficient (or equivalently τs, cf. Figure 2 of Pub. 5) has to be taken into account. Therefore a representation with a reference time is needed which is achieved by the susceptibility master curves and the scaling on the segmental correlation time τs. In Figure 3.37c it is explicitly shown that the relaxation processes at fixed temperature (T= 298 K, in color) and different concentrations are differently separated from the α-process. Thus, from this representation it can be concluded that the observed changes of R1,DD(ν) (Figures 3.37 a and b) are due to both a shift toward lower reduced frequencies and a diminished polymer relaxation strength. Ad 2.: Extending the obtained time constants τs(T) toward lower temperatures 62 3.6 Linear Polymers in Solution 103104105106107108 104 105 106 107 108 Weber et al., T = 293 K 50% PDMS 423000, 50% CCl4 213 K 233 K 298 K 323 K ∝ ω0.78 χ''DD [s-2] ντshift ∝ ω0.75 49% PDMS 112000 51% CCl4 Tref = 233 K Figure 3.38: Susceptibility master curves as a function of the reduced frequency ντshift for PDMS with M= 112000 in CCl4at polymer mass fraction c= 49 % in the temperature range as indicated. Triangles: literature data [180] for comparison. Dashed line: power-law behavior of the undiluted high-MPDMS melt. Line: power-law behavior observed for PDMS 112000 at c= 49 %. with complimentary measurements by dielectric spectroscopy (DS) could, firstly, provide a verification of τs(T) by a different technique; secondly, it might also reveal how τs(T) of the lowest concentration approaches that of the pure diluent. In addition the investigation of polyisoprene (PI) in solution by broadband DS appears very attractive, as both the time constants ταand τnreflecting the segmental dynamics and the terminal relaxation time, respectively, can be extracted from the spectra (cf. Section 3.1). However, a combined study by DS and FC NMR may be hampered by the low polymer relaxation strength fof PI ([155] and Pub. 2). Ad 4.: It appears that is reduced at higher M/Meonly for much lower polymer mass fractions. Thus, in future studies also very high Mshould be investigated, i.e., for which the predicted = 0.25 for the bulk melt is almost reached. A similar trend as observed for PB can be anticipated from the preliminary results of PDMS with M= 112000 and c= 49 % in CCl4which are shown in Figure 3.38. Measurements below T= 213 K were impeded by crystallization, i.e., it was not possible to reach the the relaxation maximum which would be expected at higher frequencies and lower temperatures. With the available relaxation data a master curve is created by applying FTS for the temperature range T= 213 K - 323 K and the reference temperature Tref = 233 K. For the reduced frequencies 104< ντshift <5·106a power-law behavior ∝ω0.78 is observed, which has a slightly higher exponent than the power-law ∝ω0.75 found for the undiluted high-MPDMS (Figure 7 in Pub. 4 and cf. Figure 3.23). A direct comparision of the master curves of undiluted and diluted PDMS in the isofrictional representa63 3 Extended Abstract tion χ00 DD(ωτs) is unfeasible, since for the latter the the time constant τscould not be extracted. However, it appears that also for PDMS in solution the relaxation strength is reduced upon dilution, as the low-frequency part (ωτs<1) of the master curve with diluted polymer decreases steeper than that of the pure melt. This is corroborated by the agreement with the results by Weber and Kimmich [180] for high-MPDMS with similar cat T= 293 K. The fact that the difference between the exponents is so small on the one hand may be explained in analogy with the finding for PB, that is significantly reduced at higher M/Meonly for much lower polymer mass fractions (cf. Figure 5 of Pub. 5). On the other hand it has been shown in Pub. 4 by an isotopic dilution experiment that the power-law ∝ω0.75 in the case of PDMS is due to the dominating contribution of the intermolecular relaxation, which is just reduced in amplitude yet not changed in terms of the power-law exponent in the relevant frequency range. In conclusion, by utilizing the susceptibility representation and applying FTS, the results obtained on a microscopic level by NMR relaxometry for linear polymers in solution corroborate those of rheology which are considered as textbook knowledge. Thus, FC NMR is again proven as a competitive technique and may be referred to as ”molecular rheology”. 64 4 Publications List of included publications as referred to in this thesis Pub. 1 Dipolar and Bond Vector Correlation Function of Linear Polymers Revealed by Field Cycling 1H NMR: Crossover from Rouse to Entanglement Regime. Herrmann, A.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2009,42, 2063-2068. Pub. 2 Universal Polymer Dynamics Revealed by Field Cycling 1H NMR. Herrmann, A.; Kariyo, S.; Abou Elfadl, A.; Meier, R.; Gmeiner, J.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2009,42, 5236–5243. Pub. 3 Protracted Crossover to Reptation Dynamics: A Field Cycling 1H NMR Study Including Extremely Low Frequencies. Herrmann, A.; Kresse, B.; Gmeiner, J.; Privalov, A. F.; Kruk, D.; Fujara, F.; R¨ossler, E. A. Macromolecules 2012,45, 1408-1416. Pub. 4 Mean square displacement and reorientational correlation function in entangled polymer melts revealed by field cycling 1H and 2H NMR relaxometry. Herrmann, A.; Kresse, B.; Wohlfahrt, M.; Bauer, I.; Privalov, A. F.; Kruk, D.; Fujara, F.; R¨ossler, E. A. Macromolecules 2012,45, 6516–6526. Pub. 5 Dynamics of Linear Polybutadienes in Solution Studied by Field Cycling 1H NMR Herrmann, A.; R¨ossler, E. A. ACS Macro Letters 2012,1, 1339–1342. 65 4 Publications Individual contributions to joint publications Pub. 1 I conducted all experiments for the samples listed in Table 1 therein. The other samples were measured by S. Kariyo and previously published in [77]. I performed all analyses during my Ph.D. studies. Pub. 2 I conducted all experiments for the samples listed in Table 1 therein as part of my preceding diploma thesis except for: Propylene glycol was measured by R. Meier, PB 466-h6and PB 2020-h6were measured by S. Kariyo and previously published in [151], PB 35300-h6was already contained in Pub. 1, PI 111000-h5and PI 127000-h3were measured by S. Kariyo and previously published in [185], PI 157000-h8was measured by A. Abou Elfadl. I performed all analyses during my Ph.D. studies. Pub. 3 I conducted all experiments with the Stelar spectrometer in Bayreuth for the samples listed in Table 1 therein. The low-frequency measurements were done by B. Kresse and me in Darmstadt. All other samples were already contained in Pub. 1. I performed all analyses during my Ph.D. studies. Pub. 4 I conducted all experiments with the Stelar spectrometer in Bayreuth for the PB samples. The low-frequency measurements were done by B. Kresse in Darmstadt. PB 24300-h6and PB 196000-h6were already contained in Pub. 3. The PDMS blends were measured by M. Wohlfahrt [134]. I performed all analyses during my Ph.D. studies. Pub. 5 I conducted all experiments. The completely protonated PB samples were already contained in Pub. 3. I performed all analyses during my Ph.D. studies. 66 Other publications •From Simple Liquid to Polymer Melt. Glassy and Polymer Dynamics Studied by Fast Field Cycling NMR Relaxometry: Low and High Molecular Weight Limit. Kariyo, S.; Brodin, A.; Gainaru, C.; Herrmann, A.; Schick, H.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2008,41, 5313–5321. •From Simple Liquid to Polymer Melt. Glassy and Polymer Dynamics Studied by Fast Field Cycling NMR Relaxometry: Rouse Regime. Kariyo, S.; Brodin; A.; Gainaru, C.; Herrmann, A.; Hintermeyer, J.; Schick, H.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2008,41, 5322–5332. •Molecular Weight Dependence of Glassy Dynamics in Linear Polymers Revisited. Hintermeyer, J.; Herrmann, A.; Kahlau, R.; Goiceanu, C.; R¨ossler, E. A. Macromolecules 2008,41, 9335–9344. •Molecular Weight Dependence of Fragility in Polymers. Abou Elfadl, A.; Herrmann, A.; Hintermeyer, J.; Petzold, N.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2009,42, 6816–6817. •From Rouse to Fully Established Entanglement Dynamics: A Study of Polyisoprene by Dielectric Spectroscopy. Abou Elfadl, A.; Kahlau, R.; Herrmann, A.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2010,43, 3340–3351. •Polymer Dynamics of Polybutadiene in Nanoscopic Confinement As Revealed by Field Cycling 1H NMR. Hofmann, M.; Herrmann, A.; Ok, S.; Franz, C.; Kruk, D.; Saalw¨achter, K.; Steinhart, M.; R¨ossler, E. A. Macromolecules 2011,44, 4017–4021. •Glassy, Rouse, and Entanglement Dynamics As Revealed by Field Cycling 1H NMR Relaxometry Hofmann, M.; Herrmann, A.; Abou Elfadl, A.; Kruk, D.; Wohlfahrt, M.; R¨ossler, E. A. Macromolecules 2012,45, 2390–2401. •Field-cycling NMR relaxometry of viscous liquids and polymers. Kruk, D.; Herrmann, A.; R¨ossler, E. A. Progess in Nuclear Magnetic Resonance Spectroscopy 2012,63, 33-64. 67 4 Publications •Long-Time Diffusion in Polymer Melts Revealed by 1H NMR Relaxometry. Meier, R.; Herrmann, A.; Kresse, B.; Privalov, A. F.; Kruk, D.; Fujara, F.; R¨ossler, E. A. ACS Macro Letters 2012,submitted. 68 Publication 1 Dipolar and Bond Vector Correlation Function of Linear Polymers Revealed by Field Cycling 1H NMR: Crossover from Rouse to Entanglement Regime. Herrmann, A.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2009,42, 2063-2068. Copyright 2009 by The American Chemical Society DOI: 10.1021/ma802818j 69 (2) Angell, C. A.; Ngai, K. L.; Mckenna, G. B.; McMillan, P. F.; Martin, S. W. J. Appl. Phys. 2000,88, 3113–3157. (3) Lunkenheimer, P.; Schneider, U.; Brand, R.; Loidl, A. Contemp. Phys. 2000,41, 15–36. (4) Blochowicz, T.; Brodin, A.; Ro¨ssler, E. A. AdV. Chem. Phys. 2006, 133 Part A, 127–256. (5) Binder, K.; Baschnagel, J.; Paul, W. Prog. Polym. Sci. 2003,28, 115– 172. (6) Paul, W.; Smith, G. D. Rep. Prog. Phys. 2004,67, 1117–1185. (7) Go¨tze, W.; Sjo¨gren, L. Rep. Prog. Phys. 1992,55, 241–376. (8) Kariyo, S.; Brodin, A.; Gainaru, C.; Herrmann, A.; Schick, H.; Novikov, V. N.; Ro¨ssler, E. A. Macromolecules 2008,41, 5313–5321. (9) Kariyo, S.; Brodin, A.; Gainaru, C.; Herrmann, A.; Hintermeyer, J.; Schick, H.; Novikov, V. N.; Ro¨ssler, E. A. Macromolecules 2008, 41, 5322–5332. (10) Kariyo, S.; Gainaru, C.; Schick, H.; Brodin, A.; Ro¨ssler, E. A. Phys. ReV. Lett. 2006,97, 207803-1-207803-4. Erratum: Kariyo, S.; Herrmann, A.; Gainaru, C.; Schick, H.; Brodin, A.; Ro¨ssler, E. A. Phys. ReV. Lett. 2008,100, 109901-1. (11) Hintermeyer, J.; A. Herrmann, A.; Kahlau, R.; Goiceanu, C.; Ro¨ssler, E. A. Macromolecues 2008,41, 9335–9344. (12) Kreer, T.; Baschnagel, J.; Mu¨ller, M.; Binder, K. Macromolecules 2001,34, 1105. (13) Sommer, J.-U.; Saalwa¨chter, K. Eur. Phys. J. E 2005,18, 167–182. (14) Kimmich, R.; Fatkullin, N. AdV. Polym. Sci. 2004,170, 1–113. (15) Bloembergen, N.; Purcell, E. M.; Pound, R. V. Phys. ReV.1948,73, 679–715. (16) Graf, R.; Heuer, A.; Spiess, H. W. Phys. ReV. Lett. 1998,80, 5738– 5741. (17) Fatkullin, N.; Kimmich, R.; Weber, H. W. Phys. ReV.E1993,47, 4600–4603. (18) de Gennes, P.-G. J. Chem. Phys. 1971,55, 572–579. (19) Kimmich, R.; Fatkullin, N. J. Chem. Phys. 1994,101, 822–832. (20) Schweizer, K. S. J. Chem. Phys. 1989,91, 5802–5821. (21) Schweizer, K. S. J. Chem. Phys. 1989,91, 5822–5839. (22) Ball, R. C.; Callaghan, P. T.; Samulski, E. T. J. Chem. Phys. 1997, 106, 7352. (23) Faller, R.; Mu¨ller-Plathe, F.; Heuer, A. Macromolecules 2000,33, 6602–6610. (24) Smith, G. D.; Borodin, O.; Bedrow, D.; Paul, W.; Qiu, X.; Ediger, M. Macromolecules 2001,34, 5192–5199. (25) Smith, G. D.; Borodin, O. J. Chem. Phys. 2002,117, 10350. (26) Faller, R.; Mu¨ller-Plathe, F. Polymer 2002,43, 621–628. (27) Kimmich, R.; Fatkullin, N.; Seiter, R.-O.; Gille, K. J. Phys. Chem. 1998,108, 2173–2177. (28) Dollase, T.; Graf, R.; Heuer, A.; Spiess, H. W. Macromolecules 2001, 34, 298–309. (29) Saalwa¨chter, K. Progr. NMR 2007,51, 1–35. (30) Saalwa¨chter, K. Private communication. (31) Read, D. J.; Jagannathan, K.; Likhtman, A. E. Macromolecules 2008, 41, 6843–6853. (32) Fatkullin, N.; Kimmich, R.; Kroutieva, M. J. Exp. Theor. Phys. 2000, 91, 150–166. MA802818J 2068 Herrmann et al. Macromolecules, Vol. 42, No. 6, 2009 Publication 2 Universal Polymer Dynamics Revealed by Field Cycling 1H NMR. Herrmann, A.; Kariyo, S.; Abou Elfadl, A.; Meier, R.; Gmeiner, J.; Novikov, V. N.; R¨ossler, E. A. Macromolecules 2009,42, 5236–5243. Copyright 2009 by The American Chemical Society DOI: 10.1021/ma900625x 77 78 pubs.acs.org/Macromolecules Published on Web 06/25/2009 r2009 American Chemical Society 5236 Macromolecules 2009,42, 5236–5243 DOI: 10.1021/ma900625x Universal Polymer Dynamics Revealed by Field Cycling 1 H NMR A. Herrmann, † S. Kariyo, ‡ A. Abou Elfadl, † R. Meier, † J. Gmeiner, † V. N. Novikov, § and E. A. R€ ossler* ,† † Experimentalphysik II, Universit€ at Bayreuth, 95440 Bayreuth, Germany, ‡ Faculty of Science and Technology, Yala Islamic University, 135/8, M.3, A. Yarang, Pattani 94169, Thailand, and § IA&E, Russian Academy of Sciences, Novosibirsk 630090, Russia Received March 23, 2009; Revised Manuscript Received May 6, 2009 ABSTRACT: We apply fast field cycling 1 H NMR to study segmental reorientation dynamics in melts of linear polybutadiene, polyisoprene, and polydimethylsiloxane in the high molecular weight limit. Measuring fully protonated as well as partially deuterated polymers, we show that in contrast to previous reports the relaxation behavior at low frequencies, for which polymer-specific contributions show up, is not universal but depends on the particular internuclear vectors of the 1 H spin pairs in the monomer unit. Only after extracting the polymer specific contributions from the overall susceptibility spectra by accounting for the glassy contribution, the “polymer spectra” reveal universal behavior which can be described by two power law regimes: one attributed to free Rouse dynamics and one, at lower frequencies, to entanglement effects. Yet the predictions of the tube-reptation model are not observed. I. Introduction The dynamics of melts of linearpolymerscomprisesboth polymer specific dynamics and glassy dynamics. Whereas polymer dynamics originate from collective dynamics of Rouse and reptation type, 1,2 glassy dynamics 3-5 are attributed to “segmental” or “local” relaxation of the polymer chain. Here, one has to keep in mind that the glass transition phenomenon itself includes cooperative motion, and the segmental correlation time τ s may be identified with that of the R-process, τ R , characterizing glassy dynamics. The latter drives the polymer dynamics via determining the monomeric friction coefficient and is responsible for the non-Arrhenius temperature dependence of the relaxation times in polymer melts. From an experimental point of view it is of interest whether one can separate polymer dynamics from glassy dynamics in order to test predictions by respective theories. For example, it may be possible that polymer dynamics modify glassy dynamics. 6 In many cases one chooses experimental conditions which suggest that separation of time scales holds, and one assumes that solely polymer dynamics or solely glassy dynamics are probed as the contribution from the other relaxation is believed to be ignorable. Such an approach has also been taken in the case of fast field cycling (FFC) NMR experiments, a technique well suited to investigate the low-frequency motion in polymers. 2 FFC NMR monitors the dispersion of the spin-lattice relaxation time T 1 (ω). To a fair approximation, the dispersion of 1 HT 1 reflects the spectrum of reorientational dynamics of a polymer segment in terms of the dipolar or second Legendre polynomial correlation function F 2 (t) of the spin pairs within a monomer unit. By converting the dispersion data into the time domain, the bond vector correlation function becomes accessible. 7 Measuring at frequencies ωτ s ,1, experimental results on several polymers have been reviewed by Kimmich and Fatkullin 2 which appear to indicate universal dispersion behavior of the spin-lattice relaxation in linear polymers. The T 1 dispersion manifests itself in characteristic power-law regimes which have been interpreted within Rouse theory (nonentangled polymers) and renormalized Rouse theory (entangled polymers). 2 Thus, the experiments seem not to disclose the power laws expected for the Doi-Edwards tube-reptation model; the latter have only been observed by FFC NMR for polymers confined to tubelike pores formed by a solid matrix. 8 However, there are reports in the literature that do not confirm such universal relaxation behavior, e.g., for the case of polyisoprene. 2,9 Here, the question is whether these deviations from the presumably universal relaxation behavior at ωτ s ,1 are significant and may cast doubt on the previous approach 2 taken to interpret the FFC NMR data. The present publication attempts to answer this question. In a series of papers, 7,10-12 we recently have reinvestigated the 1 HT 1 dispersion behavior of polybutadiene (PB) covering a broad range of molecular weight (M) including the low Mlimit; i.e., we have studied the crossover from simple liquid, to Rouse, and to entanglement dynamics. Experiments have been performed with a commercial spectrometer STELAR FFC 2000. 13 Applying frequency-temperature superposition (FTS), we have obtained master curves covering 6 decades in frequency at ωτ s ,1, and the T 1 dispersion data have been interpreted in a new fashion. The results relevant for the present context can be summarized as follows. (i) In the frequency regime attributed to Rouse dynamics the influence of the spectral contribution of the glassy dynamics cannot be ignored. (ii) The entanglement regime (M>M e ) manifests itself at lower frequencies or longer times than Rouse dynamics, and here the contributions from glassy dynamics can be ignored. This is the case for PB, which exhibits a comparatively strong relaxation strength of the polymer dynamics, but it may not be necessarily the case for other polymers. (iii) The bond vector correlation function shows striking similarity with those obtained from simulations; 14 however, up to Z=M/M e =9no indication of the power laws of the tube-reptation model can be identified, and also the renormalized Rouse theory does not fully apply. One possibility to explain these findings is to assume that the crossover to full entanglement dynamics occurs only at higher M; i.e., the crossover is “very protracted”. 7,14 Investigating now different polymers by FFC 1 HNMR,the present contribution aims at demonstrating that in contrast to previous reports 2,13 the T 1 dispersion at low frequencies in polymer melts is not universal but rather depends on the *Corresponding author. Downloaded by UNIV BAYREUTH on July 21, 2009 Published on June 25, 2009 on http://pubs.acs.org | doi: 10.1021/ma900625x Article Macromolecules, Vol. 42, No. 14, 2009 5237 orientation of the internuclear vectors with respect to the contour of the chain in the particular monomer, a fact also discussed by Spiess and co-workers in the context of their high-resolution double-quantum NMR experiments. 15-17 Only by taking into account the spectral contribution from the glassy dynamics, indeed universal polymer relaxation behavior is revealed for all the polymers investigated. These results are further substantiated by examining partially deuterated polymer samples. In particular, we will discuss results for polybutadiene (PB), polyisoprene (PI), and polydimethylsiloxane (PDMS) in the high Mlimit for which entanglement effects are well established and the T 1 dispersion has become Mindependent (M.M e ). We will show once again that the universal relaxation behavior revealed after isolating the polymer contribution is not described by current polymer theories, at least in the frequency range accessible by FFC NMR at the present time. II. Theoretical Background The spin-lattice relaxation time T 1 describes the evolution of the nuclear magnetization toward its equilibrium value. Its frequency dependence (or dispersion) can be studied by applying the FFC technique where the external magnetic field is switched between a variable relaxation field Band a constant detection field. The angular frequency is defined by the Larmor frequency ω=γB,whereγdenotes the gyromagnetic ratio of the nucleus. In the case of 1 H nuclei, the spin-lattice relaxation is determined by fluctuations of the dipolar interaction of the proton spins. Usually it is argued that T 1 is dominated by intramolecular contributions. 2,13 Accordingly, 1 H FFC NMR relaxation data are expected to reflect mainly reorientation dynamics. However, the role of intermolecular or intersegmental contributions in the case of polymers must not be ignored. 2 As introduced before, we rewrite the Bloembergen-PurcellPound (BPP) expression 18 for the relaxation rate 1/T 1 in the susceptibility form 7,10-12 ω=T 1 ¼C½χ 00 ðωÞþ2χ 00 ð2ωÞ  3C~ χ 00 ðωÞð1Þ where Cis the NMR coupling constant and χ 00 (ω)= ωJ(ω)the susceptibility representation of the fluctuation spectrum, with the spectral density J(ω) being given in first approximation by the Fourier transform of the second rank orientational correlation function F 2 (t) of a polymer segment, more precisely of the internuclear vectors of the spin pairs in the monomer. 2,11 The factor 3 appears in order to keep the integral over the susceptibility” χ~ 00 (ω)normalizedtoπ/2 as usual. As we cover a large temperature range, glassy dynamics as well as polymer dynamics are probed, and one is able to construct master curves χ~ 00 (ωτ s ) for the NMR susceptibility extending over many decades in time, assuming frequency-temperature superposition (FTS). For details about obtaining the master curves, the reader may consult our previous publications. 7,10-12 The master curves χ~ 00 (ωτ s ) combine the results from a broad temperature range (say ΔT=180 K) and present “isofrictional” spectra which allow comparing the results for different M. For simple liquids and oligomers with low molecular weight M<M R (M R denoting the molecular weight of the Rouse unit 12 ), no spectral contribution at ωτ s <1 in excess to the Debye behavior χ~ 00 (ω)µω 1 is observable, and their susceptibility master curves solely represent glassy dynamics (“glassy spectrum”). For samples with higher M, i.e., M>M R , additional intensity on the low-frequency side of the R-peak reflects polymer specific dynamics that involve time scales longer than τ s . As a guideline for a phenomenological decomposition of the spectral contributions from polymer and glassy dynamics, one assumes that both are statistically independent, so that their contributions to F 2 (t) are multiplicative: 2,7,10-12 F 2 ðtÞ¼F glass ðtÞF polymer ðtÞð2Þ Depending on molecular weight M, the polymer part F polymer (t) may contain contributions from Rouse as well as entanglement dynamics. Introducing the relative magnitude fof polymer dynamics, we write F 2 ðtÞ¼½ð1-fÞφ glass ðtÞþfF polymer ðtÞð3Þ where φ glass (t) denotes the normalized correlation function describing the glassy dynamics alone. Assuming time scale separation (cf. also ref 19), the contributions to the total susceptibility are approximately additive ~ χ 00 ðωτ s Þ=ð1-fÞ~ χ 00 glass ðωτ s Þþf~ χ 00 polymer ðωτ s Þð4Þ Under such conditions the “polymer spectrum” χ~ 00 polymer (ωτ s ) containing only the spectral contributions attributed to polymer dynamics can be extracted from the master curve χ~ 00 (ωτ s )ofeach sample by subtracting the glassy spectrum χ~ 00 glass (ωτ s ). Thereupon the relaxation strength f(M)ofpolymerdynamicsis obtained, which is the relative correlation loss due to polymer dynamics on time scales t.τ s . As found for PB, its dependence on Mreflects three dynamic regimes, namely simple liquid, Rouse, and entanglement regime. Specifically, f(M) strongly increases in the Rouse regime (M R <M<M e ) but saturates beyond the entanglement molecular weight M e12 or at M c =2M e . In other words, the glassy dynamics are more and more impeded by the emerging polymer dynamics. The relaxation strength f is connected to what has been called the local order parameter S=√f. 11,12,15,16,20 Though the applicability of eq 4 may be questioned as time scale separation may not always apply in a strict sense the results appear physically reasonable and allow a comparison of the polymer contribution for different polymers what is needed when searching for presumably universal polymer relaxation. In the Appendix we compare the results from applying eq 4 with those from eq 3. In the latter case a full deconvolution is applied, and the differences can essentially be ignored in the case of the PB data. As discussed, for example, by Spiess and co-workers 15,16 considering the entanglement regime, the particular value of the relaxation strength for the order parameter Sdepends on the direction of the internuclear vector between a spin pair with respect to the contour of the chain. Given the order parameter of the chain S chain , the order parameter S ij (or f ij ) of a particular spin pair ij oriented with an angle ϑ ij toward the contour direction is found 16 by tensor calculus S ij ¼ffiffiffiffi f ij p¼1=2ð3cos 2 ϑ ij -1ÞS chain ð5Þ Thus, the measured order parameter S ij is smaller than that of the chain. Moreover, the magnitude of spectral contribution attributed to polymer dynamics depends on the structure of the particular monomer. In other words, no universal dispersion for the overall spin-lattice relaxation time is expected. Only if the susceptibility spectra are decomposed along eq 4 the so-extracted “polymer spectra” will be expected to show universal features. As will be demonstrated, these considerations do not only hold for the entanglement regime but also for the Rouse regime. Here we add that, in contrast to double-quantum (DQ) NMR, FFC NMR usually provides only average values for fas the method cannot discriminate the contributions from different spin pairs in the monomer. However, by measuring partially deuterated polymers, the situation becomes more favorable, and this is Downloaded by UNIV BAYREUTH on July 21, 2009 Published on June 25, 2009 on http://pubs.acs.org | doi: 10.1021/ma900625x 5238 Macromolecules, Vol. 42, No. 14, 2009 Herrmann et al. exploited in the present contribution. Still, the particular f ij may depend on the configuration of the monomer unit. For example, in the case of PB samples an average over cis and trans isomers is measured. Finally, we mention that although assumed to be negligible, intermolecular coupling may still play a role. Thus, a quantitative analysis of the relaxation strength fmeasured by FFC 1 H NMR appears to be not straightforward. III. Experimental Section We investigated samples of differently deuterated as well as fully protonated 1,4-polybutadiene (PB) and 1,4-polyisoprene (PI) with M>M e .TheT 1 dispersion data of partially deuterated 1,4-polyisoprene (PI) from ref 9 were transformed to the susceptibility representation. Moreover, we measured samples of polydimethylsiloxane (PDMS) with different M(see Table 1). Note that the sample name of the polymers reflect the weight-average M w and the number of proton nuclei in the monomer unit. The fully protonated samples (PB-h 6 ,PI-h 8 ,PDMS-h 6 )werepurchased from Polymer Standards Service PSS, Mainz, Germany, while the partially deuterated samples (PB-h 2 ,PB-h 4 ,PI-h 3 , PI-h 5 ) were kindly provided by D. Richter and L. Willner, Institut f€ ur Festk€ orperforschung, Forschungszentrum J€ ulich, Germany. The concentration of cis, trans, and vinyl units is 51%, 42%, and 7%, respectively, for the deuterated PB samples. Propylene glycol (PG, purity g99.5%) was purchased from Sigma-Aldrich. The spin-lattice relaxation time T 1 (ω) was determined with a STELAR FFC 2000 relaxometer which allows measurements in the temperature range of 160-400 K and 1 HLarmorfrequency range ν=10 kHz-20 MHz. For temperature control at the sample position we used a thermocouple in a test tube and inserted it into the probe. The accuracy of temperature measurements was better than (1 K, and temperature stability was better than (0.3 K. T 1 was obtained by a monoexponential fit of the magnetization curve. IV. Results 1 HT 1 Dispersion. Figure 1 displays the dispersion of the spin-lattice relaxation time T 1 (ν) for the investigated polymers PB, PI, and PDMS measured at a single temperature. The respective temperature is selected to allow the best comparison among the different polymers. As the glass transition temperature T g increases in the order of PDMS, PB, and PI, 6 similar temperatures with respect to T g are chosen. In the low-Msystem (PB466-h 6 ) the dispersion profile for low frequencies (ν<500 kHz) is virtually constant, and it has been shown that the relaxation behavior (i.e., its master curve) is indistinguishable from that of a simple liquid such as o-terphenyl. 11 Thus, still no polymer specific dynamics are found for PB at M= 466. In contrast, the dispersion data of the polymers in the high Mlimit (M.M e ) can be approximated at low frequencies by a power law ν a with a>0 (as indicated in Figure 1). For the fully protonated PB (PB35300-h 6 ) an exponent a=0.23 is observed. However, this power law is not found for the fully protonated PI (PI157000-h 8 ); instead, here a=0.17. In the case of PDMS128000-h 6 a quite large exponent a=0.29 is found. Moreover, for the partially deuterated PB samples (PB18000-h 4 and PB20000-h 2 ) power laws with a=0.21 and a=0.27, respectively, are obtained as shown in Figure 1. Clearly, the three differently protonated PB samples show significant variation in the exponent a. In the case of PI, partial protonation only weakly changes the exponent of the corresponding power law. Here, we emphasize that interpolation by power laws is only an approximation for specifying the differences in the relaxation behavior of the investigated polymers. Because of a different 1 H NMR coupling constant Cthe level of the T 1 (ν) curves of the samples varies. Since the apparent exponent adiffers by a factor of about 2 among the differently protonated polymers PB, PI, and PDMS, it can be concluded that no universal dispersion is observed. Here, we note that Kimmich and co-workers 2,13 introduced three presumably universal power law regimes (I, II, III, starting from high frequencies) for the T 1 dispersion behavior of linear polymers, and the frequency range considered in Figure 1 is that which has been called regime II with an universal exponent a= 0.25 (0.05. 2 Clearly, the variation of afound in the present contribution exceeds this margin. Concerning the power laws for PI the deviations have already been recognized in refs 9 and 13. It is the goal of our contribution to understand the variation of the exponent ain the light of interplaying spectral contributions from glassy and polymer dynamics. Susceptibility Master Curves. Measuring the T 1 dispersion in the temperature range of 223-393 K enables us to construct susceptibility master curves χ~00(ωτ s ) for the samples of PB and PI assuming FTS. The corresponding master spectra are plotted in Figure 2 as a function of the reduced frequency ωτ s and are scaled by the corresponding amplitude of R-relaxation peak to account for the individual NMR coupling constant of the different polymer samples; i.e., they agree in the frequency range ωτ s g1. The susceptibility representation of the relaxation data allows to clearly distinguish between glassy and polymer specific dynamics. Whereas around the relaxation peak with ωτ s =1 contributions of glassy dynamics dominate, the excess intensity on the low-frequency side of the peak (ωτ s ,1) compared to the spectrum of the simple liquid limit (PB466-h 6 or o-terphenyl) represents the spectral contribution for which polymer dynamics more and more dominate. The corresponding Table 1. Details on the Samples sample M w [g/mol] M w /M n reference propylene glycol (PG) 76 this work PB466-h 6 466 1.06 10 PB2020-h 6 2020 1.07 10 PB18000-h 4 18000 1.02 this work PB20000-h 2 20000 1.02 this work PB35300-h 6 35300 1.02 7 PI111000-h 5 111000 1.03 9 PI127000-h 3 127000 1.03 9 PI157000-h 8 157000 1.01 this work PDMS860-h 6 860 1.41 this work PDMS5940-h 6 5940 1.15 this work PDMS128000-h 6 128000 1.13 this work Figure 1. Dispersion of spin-lattice relaxation time T 1 (ν) measured for polybutadiene (PB) at T=253 K, for polyisoprene (PI) at T=296 K, 9 and T=298 K and for polydimethylsiloxane (PDMS) at T=233 K. Power laws ν a with abetween 0.13 and 0.29 (straight lines) at low frequencies can be observed for the polymers with M>M e .TheT 1 (ν) curve of PB466-h 6 does not exhibit dispersion toward low frequencies (simple liquid limit). Downloaded by UNIV BAYREUTH on July 21, 2009 Published on June 25, 2009 on http://pubs.acs.org | doi: 10.1021/ma900625x Article Macromolecules, Vol. 42, No. 14, 2009 5239 amplitude is different in the entire range ωτ s <1 for each of the samples. In particular, the master curve of PB20000-h 2 with protons at the double bond exhibits a higher intensity than that for the fully protonated PB35300-h 6 , whereas the situation is vice versa for the spectrum of PB18000-h 4 in which only the protons of the methylene groups are present. Obviously the spectra depend on the particular proton spin pair probed in the 1 H relaxation experiment. In the case of PI, however, this effect is quite small; i.e., the excess intensity does not change strongly when partially deuterated PI is considered. We note that the different spectral intensity is not related to Msince saturation is well established at M. M e , 9 i.e., in the high-Mlimit which is considered within the experimentally accessible frequency range. The power law behavior discussed for the T 1 (ν) data in Figure 1 transforms into a susceptibility behavior χ~00(ω)µ ω 1-a , and the power law for regime II (discussed in Figure 1) with the corresponding exponent for PB-h 6 is included in Figure 2 (green dashed line). From this it becomes obvious that at lowest frequencies the susceptibility bends over to a weaker frequency dependence, and a second power law regime may be identified: Here, apparent exponents a= 0.43-0.48 and a= 0.28-0.35 are found for PB and PI, respectively; the values vary less than those found at higher frequencies for regime II (cf. Figure 1). Nevertheless, they again do not appear to be universal in contrast to the value a=0.45 (0.05 reported by Kimmich and Fatkullin for this frequency range called regime III. 2 For convenience, we indicate in Figure 2 the corresponding regimes according to the Kimmich and Fatkullin classification. At high frequencies close to the susceptibility maximum, it is obvious that in addition to polymer dynamics glassy dynamics contribute significantly to the relaxation. This regime has been called regime I by Kimmich and co-workers, and a power law exponent a=0.5 (0.05 has been claimed to be found. 2,13 However, such an exponent cannot be clearly recognized, and as said the corresponding relaxation contributions cannot be attributed to polymer dynamics alone. Polymer Spectra. Separating glassy and polymer spectral contributions along eq 4 by subtracting the “glassy spectrum” of PB466-h 6 from each of the total susceptibility curves χ~00(ω) of Figure 2 yields spectra reflecting only polymer dynamics (“polymer spectra”, cf. Figure 3). This allows extracting the relaxation strength fof polymer dynamics by calculating the relative integrated intensities of the individual polymer spectra (see Table 2). For PB the highest value f=0.16 is obtained for the sample PB20000-h 2 . As discussed before, for PB18000-h 4 fis smaller ( f=0.06); the fully protonated sample PB35300-h 6 is somewhat in between ( f=0.11). For the differently deuterated PI samples we find similarly small values ( f=0.03). In Figure 3 we show the polymer spectra χ~00 polymer (ωτ s ) which have been obtained by normalizing the polymer spectrum to provide an integral which equals π/2 (cf. ref 12). In this representation it is clearly seen that the high-M polymer spectra for the differently deuterated and the fully protonated PB and PI are virtually identical. In particular, power laws with the same exponents are revealed, and therefore the spectra reflect the same dynamic process, which was not evident from inspecting the total relaxation spectra. Only by accounting for the contribution of glassy dynamics universal polymer spectra are revealed. Such spectra may now be compared to predictions of polymer theories which usually exclude fast segmental dynamics. We note that at ωτ s g 10 -1 a maximum appears in Figure 3 which signals the cutoff of the polymer spectra at high reduced frequencies. Special Case PDMS. The susceptibility master curve of PDMS is not included in Figure 2 as particularities show up which we want to discuss separately. In Figure 4a the master curve for PDMS128000-h 6 (full circles) is displayed together with that for M=5940 =M e =8100 21 (open circles) and for the lowest Mmeasured (PDMS860-h 6 ; crosses). For comparison, again the low Mdata for PB (PB466-h 6 ; pluses) are included. When comparing the data also with those in Figure 2 for PB or PI significant differences show up. For example, the data for PDMS860-h 6 show a different behavior to that of the low Mlimit for PB (PB466-h 6 ); specifically, the relaxation peak for PDMS860-h 6 has a shoulder at low frequencies (ωτ s e1), i.e., significant excess intensity with Figure 2. Master curves of the susceptibility χ 00 =ω/T 1 as a function of the reduced frequency ωτ s for polybutadiene (PB) and polyisoprene (PI) in temperature range as indicated. Curves of PB with M=466 (PB466-h 6 )ando-terphenyl (OTP) are included as a reference for a “glassy spectrum”. Green dashed line: χ~ 00 (ω)µω 1-a with a=0.23 as obtained for PB35300-h 6 in regime II. Numbers and vertical dashed lines mark the different relaxation regimes according to refs 2 and 13. Figure 3. Normalized polymer spectra χ~ 00 polymer (ωτ s ) obtained by subtracting the contribution of glassy dynamics (PB466-h 6 for PB and PI; PDMS860-h 6 for PDMS) from the total susceptibility spectra of Figures 2 and 4 coincide for the high-Mlimit. Dashed lines: power laws at low frequencies ω 1 representing terminal relaxation and ω 0.5 for high-Mpolymers. Note that the polymer spectra of PDMS are shifted by a single factor in frequency (see text). Table 2. Relaxation Strength fof Polymer Dynamics in Fully and Partially Protonated Samples of Polybutadiene (PB) and Polyisoprene (PI) as Well as Polydimethylsiloxane (PDMS) with M>M e , As Obtained by Subtractive Decomposition and by Comparing Amplitudes in Figure 5 sample f(from spectra decomposition) f(from Figure 5) PB20000-h 2 0.16 (0.02 (0.16) PB18000-h 4 0.06 (0.006 0.064 (0.003 PB35300-h 6 0.11 (0.01 0.109 (0.003 PI127000-h 3 0.03 (0.01 0.031 (0.002 PI111000-h 5 0.02 (0.01 0.025 (0.001 PI157000-h 8 0.03 (0.01 0.036 (0.001 PDMS128000-h 6 0.02 (0.002 Downloaded by UNIV BAYREUTH on July 21, 2009 Published on June 25, 2009 on http://pubs.acs.org | doi: 10.1021/ma900625x 5240 Macromolecules, Vol. 42, No. 14, 2009 Herrmann et al. respect to the spectrum of PB466-h 6 . Presumably, the susceptibility data of PDMS860-h 6 show very weak contributions from polymer relaxation, if any. We have not been able to measure PDMS samples with lower Mto fully check this assumption as crystallization always interferes. Inspecting all the three curves for PDMS (high and low Mlimit as well as M e ), this shoulder appears in all cases leading to a virtually common relaxation behavior at 10 -2 <ωτ s e1, i.e., in a frequency range for which Mdepending spectral contributions from polymer dynamics are already found in PB as well as in PI. This common relaxation regime is found for all investigated PDMS samples independent of M;M-dependent contributions only appear at ωτ s <10 -2 . Thus, the shoulder cannot originate from some polymer specific relaxation. We will attribute it to a particular relaxation feature already observed in the low-Mlimit, for which only glassy dynamics control the relaxation (cf. below). It is worthwhile to note that a similar anomaly as for PDMS860-h 6 is also observed in the low molecular weight glass formers glycerol 22 and propylene glycol (PG). In order to show this effect more clearly, we display the corresponding master curves together with those for PDMS860-h 6 , PB466-h 6 , and o-terphenyl (OTP) in Figure 4b. Indeed, the curves for glycerol, PG, and PDMS860-h 6 are very similar, showing in all the cases a low-frequency shoulder with respect to the data of PB466-h 6 or OTP; the latter may be taken as reference systems for FFC spectra of simple liquids. 10,11 Interestingly, this low-frequency anomaly is only observed in the NMR data but not in the dielectric data of PDMS860-h 6 , 6 which is also displayed in Figure 4b (a similar behavior is observed for glycerol 22 ). Converting our dielectric results for PDMS128000-h 66 into a master curve (line), it is seen that this curve coincides very well with the dispersion data of PB466-h 6 or o-terphenyl (also included in Figure 4b). We emphasize that PDMS is a type B polymer; 23 thus, dielectric spectroscopy only probes glassy dynamics. Up to now this relaxation feature is not understood. We conclude that in PDMS contributions from polymer dynamics are only found at reduced frequencies lower than say ωτ s <10 -2 . In order to obtain the polymer spectrum in the case of PDMS, we have subtracted the master curve of PDMS860-h 6 from the susceptibility data of the higher M. For comparing the data with the polymer spectra of PB and PI in Figure 3, however, the polymer spectra for both PDMS128000-h 6 and PDMS5940-h 6 have been shifted in frequency by a factor of 17. This is understood by the fact that as discussed polymer dynamics set in for PDMS only at significantly lower frequencies as compared to those in PB or PI. Again, a good agreement is found. Essentially, all the polymer spectra agree after correcting for the contribution of glassy dynamics. We note that for the high-Mlimit of PDMS an apparent power law behavior, χ~00(ω)µω 1-a , with a= 0.5 can be observed over a quite extended frequency range 10 -3 <ωτ s < 510 -2 (dashed line in Figure 4a). Thus, in the case of PDMS indeed a clear-cut power law regime is recognized as expected for regime I of the Kimmich-Fatkullin classification scheme. However, it does not reflect solely polymer dynamics; instead, it has its origin in interplay of the contributions from (anomalous) glassy and polymer dynamics. In the case of the data for the low-molecular-weight systems displayed in Figure 4b a similar power law with an exponent a=0.60 (0.03 is observed (dashed line in Figure 4b). We emphasize that the experimental data for PDMS (as well as for PB) agree with those compiled by Kimmich and co-workers. 24,25 Comparison of Rouse and Entanglement Contribution. In order to understand the relaxation behavior of linear polymers in the limit of high M, i.e., with fully established entanglement dynamics, it is of interest to compare the polymer spectra (after subtracting the glassy spectrum from the overall relaxation spectra) with the corresponding spectrum from a sample with M=M e for which solely Rouse dynamics are expected and entanglement dynamics are not established yet. Therefore, we include in Figure 3 the polymer spectrum of PB2020-h 610 and PDMS5940-h 6 . Clearly, a quite different behavior is observed at ωτ s e10 -3 between the master curves of high Mand M=M e whereas for ωτ s > 10 -3 they coincide. We attribute the much stronger amplitude below ωτ s <10 -3 of the high-Msamples to contributions specific to entanglement effects; i.e., as expected, entanglement results in a significant retardation of the correlation loss which leads to a kind of bimodal relaxation. We find a power law behavior, χ~00 polymer (ω)µω 1-a , with an exponent a=0.50 (0.05 as already concluded from discussing Figure 2 for which the contribution from glassy dynamics have not yet been accounted for. In other words, at lowest frequencies the glassy contribution in χ00(ω) can be ignored indeed. 7 Regarding the frequency range ωτ s >10 -3 the susceptibility curves for the sample with M=M e and for the high Mlimit are virtually the same (cf. Figure 3). This means that the polymer dynamics of entangled polymers at such comparatively high frequencies are essentially the same as those of nonentangled polymers with M=M e . As previously shown for PB, 12 the spectra of the latter can be semiquantitatively reproduced by the discrete Rouse model assuming Figure 4. Master curves of the susceptibility χ 00 =ω/Tas a function of the reduced frequency ωτ s for (a) different polydimethylsiloxanes (PDMS) as well as a master curve from dielectric studies (orange line). 6 (b) Master curves for low-Mpolydimethylsiloxane (PDMS860-h 6 ), glycerol, 22 and propylene glycol for temperature ranges as indicated: each spectra exhibits an anomaly for ωτ s e1 compared to the master curve of polybutadiene (PB466-h 6 )oro-terphenyl as reference systems of simple liquids. Orange line: master curve for PDMS860-h 6 from dielectric spectroscopy (DS). Downloaded by UNIV BAYREUTH on July 21, 2009 Published on June 25, 2009 on http://pubs.acs.org | doi: 10.1021/ma900625x Article Macromolecules, Vol. 42, No. 14, 2009 5241 a few Rouse modes being activated (note M e /M R =4-5 for PB and 9-10 for PDMS). Within the tube-reptation model we can identify the crossover at the ωτ s =10 -3 reflecting the ratio τ s /τ e . At the entanglement time τ e =10 3 τ s the polymer chain “feels” first entanglement effects. We note that for PB2020-h 6 and PDMS5940-h 6 the crossover to the Debye limit χ00(ω)µω 1 is observed at lowest frequencies, indicating that contributions up to the slowest (Rouse) relaxation mode are detected. 7,12 We take the fact that the shapes of the polymer spectra of PB and PDMS for M=M e agree so well even at intermediate frequencies (ωτ s <10 -1 ) as an additional argument that indeed the additive separation of contributions from polymer and glassy dynamics is appropriate. V. Discussion and Conclusions Extending our previous FFC 1 H NMR measurements and analysis on PB 7,9-12 to different polymers including partially deuterated systems and concentrating on the entanglement regime (M.M e ), it turns out that the low-frequency behavior (ωτ s <1)oftheT 1 dispersion does not show universal power law characteristics, in contrast to what was claimed before. 2,13 If at all power law regimes can be identified, their apparent exponents differ (cf. Figure 1). Comparing the dispersion behavior in form of the susceptibility (cf. Figure 2) to that of the low-M systems (M<M R ) with no polymer specific contribution but only glassy contribution, the extent of excess intensity at ωτ s <1 with respect to the simple liquid limit χ 00 (ω)µω 1 (ωτ s <1) is very different for the investigated polymers. It may even depend on the extent of protonation of a given monomer, i.e., on the particular spin pairs probed by 1 H NMR. In other words, the relaxation strength fin terms of eq 3 is different in the polymers. We recall that within the present definition fis a measure of the spatial restriction of glassy dynamics for a given spin pair (or groups of spin pairs) imposed by Rouse as well as entanglement dynamics. Referring in particular to the case of PDMS with its M-independent anomalous glassy dynamics in order to understand the polymer specific contribution in the T 1 dispersion data of polymers, our results emphasize the importance of including a study of the low-Mlimit for each polymer. However, this may not always be possible as crystallization may occur. As discussed in section II, in the high-Mlimit (M.M e )the transient entanglement network points are believed to introduce topological constraints impeding the reorientation involved in the glassy dynamics at short times (teτ s ), and for the dynamic order parameter Sis expected to depend on the direction of the internuclear vector between a spin pair with respect to the contour of the chain (cf. angle ϑ ij in eq 5). For example, in their DQ NMR study on PB melts, Spiess and co-workers determined quite different values f ij for the different spectrally resolvable spin pairs of the monomeric segment. 15,16 Although their values of f ij are significantly higher than those estimated from our FFC experiments (for a discussion of this discrepancy cf. ref 11), the same trend is observed in the present study (cf. Table 2): f CHdCH determining the dispersion behavior of PB-h 2 is significantly larger than f CH2 , the latter essentially fixing the dispersion of PB-h 4 . Qualitatively, this difference may be understood by recalling that in PB-h 2 with its protons attached to the carbon atoms at the double bond (in cis and trans configuration) the angle ϑ ij with respect to the contour direction, the latter presumably fixed by the direction of the double bond, is rather small and thus leading to a small reduction of the order parameter with respect to that of the chain itself, S chain .In contrast, a much stronger reduction is expected for the methylene group in PB-h 4 . In the case of PI with its absence of a spin pair along the double bond, different protonation only slightly changes f(cf. Table 2). Refraining to explain the behavior of fin detail, we rather conclude the following. (i) It appears that the considerations regarding the relative values of fhold for both the entanglement regime and the Rouse regime (M R <M<M e ), and they can qualitatively explain the nonuniversal dispersion behavior of T 1 (ω) in the different polymers and in selectively deuterated samples of a given polymer. (ii) Subtracting the corresponding contribution of glassy dynamics from the total dispersion via eq 4 yields universal polymer spectra (cf. Figure 3). Searching in T 1 (ω) for generic polymer effects without accounting for the glass process is misleading as the latter is always a large contribution with 1 -f.f. This holds especially for small Min the Rouse regime in which fdecreases with decreasing M. 12 Here two comments are worthwhile. First, subtracting the “glassy spectrum” from the total susceptibility spectrum of polymers assumes that the first does not change with M,afact well-known from dielectric spectroscopy on type B polymers for which only the glassy transition is probed. 23 It has also been specifically tested for PB. 6 Second, although the spectral contribution of the glassy dynamics and of Rouse dynamics are not fully separated, a simple additive separation yields universal polymer spectra. Thus, we think this separation procedure is essentially a good approximation although some distortion of the polymer spectra may be expected at ωτ s e1. In the Appendix we demonstrate that indeed the results will remain essentially unchanged if one performs a decomposition of the correlation function in the time domain instead of applying the subtraction approach of the susceptibility in the frequency domain. As an alternative approach to the additive decomposition, one can estimate the contribution from glassy and polymer dynamics, that is f, by a simple scaling procedure. In Figure 2 the susceptibility master curves are normalized to agree in the frequency range of the glass transition. Hence, the amplitude of the susceptibility at lowest frequencies for which influence of contributions from glassy dynamics can be ignored may be taken as measure of f. Vice versa, we may plot the data in a way that they agree at lowest frequencies (cf. Figure 5). Assuming universal polymer spectra, this is nothing else than plotting χ~ 00 (ω)/fC except for an unknown factor. Then, the different heights of the relaxation maximum reflect the different strength 1 -fof the contribution from glassy dynamics assuming similar shape of χ~ 00 glassy (ω). Explicitly, the maximum height in Figure 5 is a measure for (1 -f)/f. In order to get absolute values for f,we choose that of PB20000-h 2 obtained by the subtraction method as reference. In Table 2 we compare the results of this estimate to that of the additive separation. Very similar values are obtained, again demonstrating that the additive decomposition of the spectra yields reliable values of f. The spectra in Figure 5 agree at lowest frequencies as expected for a universal entanglement contribution (note that PDMS is excluded from this analysis due to its low-frequency anomaly of the R-process). First, deviations Figure 5. Susceptibility spectra for the different polymers plotted to agree at lowest frequencies. Downloaded by UNIV BAYREUTH on July 21, 2009 Published on June 25, 2009 on http://pubs.acs.org | doi: 10.1021/ma900625x 5242 Macromolecules, Vol. 42, No. 14, 2009 Herrmann et al. from the common spectrum appear at ωτ s >510 -5 for PI which exhibits the strongest contribution χ~ 00 glassy (ω). Thus, this plot directly shows the non-negligible influence from spectral contributions associated with glassy dynamics. The latter is smallest in PB-h 2 , and the corresponding spectrum reflects polymer specific contributions over the largest frequency range. This again reveals an advantage of the susceptibility representation, and it becomes obvious that extracting power law exponents in the frequency range of Rouse dynamics ωτ s >10 -4 (cf. below) without taking into account the contribution from glassy dynamics yields erroneous results. The universal polymer spectra of entangled polymers (after separating glassy and polymer specific contributions) show two power law regimes in addition to a cutoff regime at highest frequencies (cf. Figure 3). This is in contrast to previous statements by Kimmich and co-workers, who have reported three power law regimes (I-III) 2,13 although deviations were discussed, too. 9,13 Their interpretation, however, suffers from the fact that glassy dynamics have not been taken explicitly into account. From the present analysis as well as from our previous works, 7,11,12 we conclude that only two power law regimes exist for entangled polymers (at least in the currently accessible frequency range), and the crossover between them may be associated with the entanglement time τ e . Whereas the first power law regime at high frequencies is attributed to free Rouse dynamics since it is identical with the relaxation behavior in a nonentangled polymer with M=M e , the second one at lowest frequencies reflects contributions specific for entanglement dynamics. It is identical with regime III of the Kimmich-Fatkullin classification scheme. Since here the influence of spectral contributions from glassy dynamics can be neglected in the total dispersion spectra, the result agrees in both interpretations, i.e., a power law χ~ 00 polymer (ω)µω 1-a with a=0.5 is recognized at lowest frequencies. Kimmich and Fatkullin attributed this dispersion regime to intersegmental relaxation. 2,27 This has become possible by comparing 1 Hand 2 HFFCNMRresultsforPB. 25 2 H NMR does not probe intersegmental relaxation, and they have not observed regime III in this case. As the fraction of intermolecular or intersegmental relaxation is expected to depend on structural details of the monomer, it is surprising that taking this interpretation for granted this regime is found to be universal. More experimental data are needed to settle this point. In Figure 3, at higher frequencies, the free Rouse regime is characterized by an apparent power law χ~ 00 polymer (ω)µω 1-a with a=0.30 (0.05, which is similar but not identical to that observed in the total dispersion for PB-h 67 and PDMS-h 6 (cf. Figures 2 and 4a), and it corresponds to regime II of the Kimmich-Fatkullin classification scheme. Both polymers show a quite large excess intensity compared to their low-Msystem, and consequently the difference between the total relaxation spectra and the “polymer spectra” is small. However, this is not the case for PI and PB-h 4 with their small relaxation strength (cf. Table 2). Here, the spectral contributions of the glassy dynamics cannot be ignored. Regarding the investigated polymers, regime I of the Kimmich-Fatkullin classification observed at highest frequencies in the total spectra χ 00 (ω) (cf. Figure 2 or 4) clearly is dominated by spectral contribution from glassy dynamics and thus cannot be attributed to polymer specific relaxation alone. Finally, we note that as discussed thoroughly in ref 7, the polymer specific relaxation regimes can be explained neither by the tube reptation nor by the once renormalized Rouse model. On the one hand, assuming that the spectra at lowest frequencies (regime III) are indeed dominated by intersegmental relaxation, then the twice renormalized Rouse model predicts an exponent a=0.5 as experimentally observed. 27 On the other hand, very similar results are reported by Monte Carlo (MC) simulations, 14 and as mentioned already before, it has been concluded that the crossover from nonentangled to entangled dynamics with tube-reptation dynamics is “very protracted”; i.e., it may be observed at M.M e (for a comparison of FFC NMR and MC simulation results cf. ref 7). This is also supported by very recent results from DQ NMR. 28 The intriguing agreement of results from FFC NMR and simulations, again, may cast doubts on the interpretation that regime III can be attributed to intersegmental correlation effects as MC simulations do not include such correlations. In any case, FFC NMR provides a unique opportunity for elucidating dynamics in polymer systems and even lower frequencies may be accessible in the near future. The program of the Kimmich group to unravel universal polymer relaxation in chemically quite different polymers by FFC NMR appears to be indeed possible provided that the spectral contribution of the glassy dynamics is well accounted for. Acknowledgment. The authors are grateful to D. Richter and L. Willner (Institut f€ ur Festk€ orperforschung, Forschungszentrum J€ ulich, Germany) for kindly providing the deuterated polyisoprene and polybutadiene samples. Support from Deutsche Forschungsgemeinschaft (SFB 481) is highly appreciated. We thank S. Stapf (Technische Universit€ at Ilmenau) for valuable comments on the manuscript. Appendix Subtraction versus Deconvolution. Figure 6 displays the polymer spectrum χ~00 polymer (ωτ s ) of PB18000-h 6 (data from ref 7) as obtained on the one hand (black solid line) by subtracting the glassy spectrum of PB466-h 6 from the total susceptibility master curve χ~00(ωτ s ) and on the other hand (red dots) by decomposing the orientational correlation function F 2 (t) along eq 3. In the latter case, decomposing F 2 (t) by dividing out φ glass (t) consisting of a Kohlrausch function with a stretching parameter β K =0.4 and a relaxation strength f=0.16 (cf. Table 2) yields F polymer (t). Fourier transform back to the frequency domain again gives χ~00 polymer (ωτ s ). As it can be concluded from Figure 6, the two approaches lead to a very similar result, in particular the shape of spectrum and consequently the apparent power laws coincide for both methods provided that M.M e and ωτ s <10 -2 . Therefore, the additive decomposition leading to the polymer spectra reflecting a universal behavior in Figure 3 is well justified in the present context. References and Notes (1) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford Sci. Publication: New York, 1986. (2) Kimmich, R.; Fatkullin, N. Adv. Polym. Sci. 2004,170, 1–113. Figure 6. Polymer spectrum χ~ 00 polymer (ωτ s ) of polybutadiene with M=18 000 (PB18000-h 6 ) obtained by subtracting the glassy spectrum from the total susceptibility spectrum 7 (black line) and by decomposition of the correlation function along eq 3 (red dots). Downloaded by UNIV BAYREUTH on July 21, 2009 Published on June 25, 2009 on http://pubs.acs.org | doi: 10.1021/ma900625x extreme narrowing range the relation R1∝ωDD2τ=(ωDDτ)2τ−1 holds, and then ωDDτ≪1 implies that R1≪τ−1, i.e., T1≫τ. As far as the second condition is concerned, in the low-field range ωDD can approach or even exceed the Zeeman splitting frequency ω, and then the dipolar interaction which is responsible for the relaxation cannot be treated as a small perturbation of the Zeeman interactions. Then the classical relaxation formulae break down, and the SLE approach has to be applied as well. 52,53 It can also happen that residual dipolar interactions (a part of the dipolar coupling which does not vanish due to motional averaging) become comparable with the Zeeman interaction, thereby considerably altering the proton energy level structure. In such a case one can still apply the perturbation relaxation theory (if other conditions are not violated), but the resulting relaxation expressions are different from the “classical”ones 52-54,57 (eq 1) which is not taken into account in the theoretical predictions for the characteristic polymer relaxation slopes. This is a typical situation in polymer melts where a residual coupling exists due to the slow anisotropic dynamics of entangled chains. We will return to these issues in the discussion part, however, a satisfactory consideration will remain the task of future theoretical work. 3. EXPERIMENTAL SECTION Samples of linear 1,4-polybutadiene (PB) with a narrow molecular mass distribution have been investigated over a large range of molecular masses (M). The polymers have been purchased from Polymer Standards Service, Mainz, Germany, thoroughly degassed in 5 and 10 mm NMR tubes and sealed under vacuum. Table 1 gives an overview over molecular masses and polydispersities Mw/Mnin addition to those Malready studied in ref 30. 1H NMR spin−lattice relaxation experiments have been performed with two different electronic field cycling (FC) relaxometers: a commercial one (Stelar Spinmaster FFC2000) at Bayreuth University and a home-built one at Darmstadt University. In Bayreuth, experiments were performed at temperatures from 223 up to 408 K; in Darmstadt, temperatures from 355 up to 393 K were measured. Let us, at this point, mention the most important characteristics and the specific differences between the two FC relaxometers: The Bayreuth Stelar relaxometer covers a 1H frequency range from ν=ω/ 2π= 10 kHz up to 20 MHz. Times of about or less than 3 ms are achieved for switching from a high polarization field to any desired evolution field, thereby avoiding undershoots or strong field oszillations. Lower 1H frequencies can only be reached at the homebuilt Darmstadt relaxometer 58 which, since about 10 years, has been continuously upgraded. For the present project, frequencies down to 200 Hz and up to 30 MHz have been reached. The low frequencies can be attained by utilizing a three-dimensional resistive coil arrangement for compensating for the earth field and other magnetic stray fields. For stabilizing fields corresponding to 1H frequencies below 1 kHz an active field drift and fluctuation compensation tool is activated. Field switching is performed in a controlled way reaching switching times of about 3 ms if the active compensation tool is not used (above 1 kHz) and 6 ms if the compensation tool is used (below 1 kHz). For technical details the reader is referred to ref 40, where it is demonstrated that in favorable cases 1H frequencies down to 12 Hz can be reached. Spin−lattice relaxation timesespecially for high-Mpolymers at very low fields, yet at a temperature far above Tgmay eventually become even shorter than the switching time of the employed relaxometer. As long as the magnetization decay is monoexponential, this does not cause a conflict with the finite switching time except for a corresponding signal reduction. Exemplifying magnetization decay curves are shown in Figure 4 for a couple of 1H frequencies. For each relaxation field the magnetization decays exponentially characterized by the spin−lattice relaxation time T1. Note that the abscissa assignment of the variable time starts from a point safely after the field switching has been done. A comparison of the relaxation rate 1/T1(ν) and susceptibility χDD ′′ =ω/T1measured with both the Stelar and the home-built spectrometer is shown in Figures 2a and 2b, respectively, for PB 87500 at T= 363 and 393 K. The curves coincide well for the two temperatures, indicating also that the temperature control agrees for both instruments. Furthermore, it demonstrates the benefit of a compensation for stray fields: almost 2 decades can be gained at low frequencies, and consequently the relaxation rate dispersion covers more than 5 decades. Figure 3. Schematic illustration of the validity range of the perturbation approach to relaxation. When at a frequency νxthe relaxation rate approaches or exceeds the value of 2πνx,the perturbation treatment breaks down (gray area). Table 1. Details on the Samples a sample Mw[g/mol] Mw/Mn PB 24300 24300 1.01 PB 47000 47000 1.04 PB 87500 87500 1.05 PB 143000 143000 1.02 PB 196000 196000 1.02 PB 441000 441000 1.07 a The sample name reflects the molecular mass Mw. Figure 4. Magnetization decay curves of PB 87500 at T= 393 K in various relaxation fields as indicated. Solid lines: corresponding monoexponential fit. Macromolecules Article dx.doi.org/10.1021/ma202489y |Macromolecules 2012, 45, 1408−14161411 As outlined in section 2, Fourier transform is applied to obtain the dipolar correlation function. Since many decades in frequency are involved, instead of fast Fourier transform, we utilize an algorithm based on the Filon algorithm. 59 When at low frequencies the terminal relaxation, i.e., the crossover to χDD ′′∝ω1, is reached, the susceptibility master curve can be transformed as it is. For the master curves with M ≥35 300, for which the terminal relaxation is beyond the experimentally accessible frequency window, it is essential to extend the susceptibility with the corresponding power-law ω ε(M)over some decades. Otherwise, truncation errors of the algorithm would cause a misleading decay of the correlation function at longest times, which can be avoided by the described extension and subsequently discarding the previously added points. This, in contrast, is not necessary for the evaluation of the DQ 1H NMR experiments measuring dipolar couplings directly in the time domain, 19,33,34 however, being able to probe the correlation function only for t>τe. We will demonstrate that low-field FC NMR adds 2 decades to our prior results 30 in CDD(t/τs) and also allows investigating higher M, in total providing a mutual verification of the results of both NMR techniques. 4. RESULTS According to the procedure outlined in section 2, the T1 dispersion data have been converted to the susceptibility representation χDD ′′=ω/T1, and by merging the data of the different temperatures, a master curve of the reduced frequency ωτshas been created for each M(Figure 5a). The scaling by the segmental correlation time τsas said yields “isofrictional” spectra and provides a common peak at ωτs≈1 representing the primary α-relaxation governed by the glass transition. At ωτs≥1 the spectral shape of the peak is independent of M. With increasing Ma continuously rising excess intensity on the low-frequency side of the peak (ωτs< 1) is discernible which is due to the slower, M-dependent polymer dynamics. For the high-M(M>M e≈2000) master curves three different relaxation regimes (0, I, II) can be distinguished and are phenomenologically in accordance with those from Kimmich’s classification. 24 Below we will attribute them to glassy (0), Rouse (I), and constrained Rouse (II) dynamics of the Doi− Edwards tube-reptation model which is however different from Kimmich’s interpretation. 24,25,30,31 At lowest frequencies in regime II, especially in the range in which the compensation tool has been employed (arrows in Figure 5a), M-dependent power-laws can be identified. The susceptibility master curves (Figure 5a) reflect segmental (“local”) as well as collective polymer dynamics and will be described in the following by going from high to low reduced frequencies. While a simple liquid (PB with M= 466) only exhibits glassy dynamics in form of the α-peak with a power-law ω1at ωτs<1,atM= 777 the contribution of polymer dynamics begins to increase with Min regime 0 and saturates around M= 4600. Yet, since for frequencies ωτs> 3×10−3and around the relaxation maximum glassy dynamics still contributes, regime 0 is designated as glassy dynamics (cf. Figure 1). Note that the frequency position of the lines separating the regimes in Figure 5a differs slightly from our previous works. 25,31 In the frequency range 3 ×10−5<ωτs< 3×10−3the susceptibility of M= 4600 ≈2Merepresents an envelope with a power-law ωαeven for the high-M(M≫Me) systems, indicating that the dynamics remains unchanged if M is increased beyond Me. In this regime the dispersion data can be well described by applying the Rouse model; 29 therefore, it is attributed to free Rouse dynamics (regime I). Approaching lower frequencies (ωτs<3×10−5, regime II), a crossover to a second power-law ωε(M)with ε<αis observed. While increasing Mthe master curves progressively approach the enveloping shape constituted by the highest M. Whereas for M ≤11 400 a crossover to the power-law ω1(Debye limit) characterizing terminal relaxation is accessible in the frequency range of the Stelar spectrometer (ωτs≥5×10−7), for larger M the slowest relaxation mode is more and more shifted to lower frequencies. Together with the fact that the power-laws of the tube-reptation model (regimes II and III) are expected to be very protracted, i.e., to be disclosed only for very high M(M> 100Me= 200 000), 15,34,37 this rendered the need for low-field relaxation experiments. Within the frequency range marked by the arrows in Figure 5a (ωτs<5×10−7) the relaxation data have been obtained with the home-built spectrometer equipped with the active compensation system. Two limiting power-laws can be distinguished in the susceptibility of PB with different M: For M≤35 300 the power-law ω1indicates that the terminal relaxation has been detected (in a 1/T1plot such as Figure 2a this would manifest itself in a horizontal line, i.e., ω0), and for M= 441 000 a power-law ω0.32 is revealed extending over more than 1 decade at lowest frequencies. In between the power-law exponent continuously decreases for increasing M, which will be evaluated and discussed below (see also Figure 6b). Figure 5. (a) Susceptibility master curves as a function of the reduced frequency ωτsfor all investigated Mof PB in the temperature range as indicated. The frequency range in which T1has 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) Loss modulus G″(ω) for PB (with M as indicated) as obtained with a mechanical rheometer and applying FTS (figure adapted with permission from ref 60). Macromolecules Article dx.doi.org/10.1021/ma202489y |Macromolecules 2012, 45, 1408−14161412 However, this already shows that the dynamics in regime II is dependent on Meven if M≫Me. For comparison, Figure 5b contains master curves of the dynamic loss modulus G″(ω) for different high-MPB from rheological measurements. 60 In that study a mechanical spectrometer with cone−plate and torsional configuration was used in the temperature range from 158 to 353 K, and FTS has been applied as well, thereby increasing the accessible frequency range from 3 (as in FC NMR using the Stelar relaxometer) to 17 decades. The master curves of different M exhibit a common high-frequency peak and an M-dependent minor peak at lower frequencies. The first is attributed to the glassy dynamics, and the latter represents the slowest relaxation mode. Since a low-frequency peak as in the rheological data is not observed in the FC 1H NMR master curves, obviously the relaxation mode spectrum is weighted differently for the two techniques. However, the susceptibility representation offers the possibility to directly compare the results from FC 1H NMR (Figure 5a) and rheology (Figure 5b), i.e., microscopic and macroscopic dynamics as reflected in the T1(ω) and G″(ω) measurements, respectively. Recently, 34 the term “molecular rheology”was coined with respect to FC and DQ 1H NMR and their capability of exploring the time range from glassy dynamics to terminal relaxation in entangled polymers like “conventional”rheology. Fourier-transforming the susceptibility master curves χDD ′′(ωτs) of Figure 5a allows displaying them as the full dipolar correlation function CDD(t/τs) in Figure 6a (cf. Experimental Section). While the low-Msystem (PB 466, dotted line) exhibits a stretched exponential (Kohlrausch) decay typical of a simple liquid, for higher Mthe relaxation becomes increasingly retarded as already discussed in ref 30. Depending on Mcharacteristic power-laws t‑αcan be identified with their exponents (above Me) having the same absolute value than in the frequency domain (Figure 5a). In the time range up to t/τs<10 3a common envelope with α= 0.85 is found which is not altered at high Msimilarly to the susceptibility representation. This is close to α= 1 predicted by the Rouse theory (cf. Figure 1), and therefore it can be regarded as the high-Menvelope for the free Rouse dynamics (regime I). First, the free Rouse dynamics leads to a distribution of correlation times in the range τs<τ<τe, and then above Meentanglement dynamics provides relaxation on increasingly slower time scales t>τe(regime II), leading to a bimodal shape. Here, particularly the M-dependent power-law t‑ε(M)is recognized. Analogous to Figure 5a, εis reduced with growing Mas indicated by the gray sector (for 24 300 ≤M≤ 441 000) in Figure 6a. This dependence ε(M) is plotted in Figure 6b. From the correlation functions (Figure 6a) the minimum slope εhas been determined for each Mby a linear fit in the doublelogarithmic representation over at least 1 decade in time. A continuous decrease of εfrom 0.73 to 0.32 can be noted in the Mrange between 9470 and 441 000. Note that our previous data 30 were restricted to M= 18 000 or ε> 0.5, and now the M range in the time domain is extended by a factor of about 25. The new high-Mresults obtained by low-field FC 1H NMR agree well with those from DQ 1H NMR, 34 which are included in Figure 6b for comparison. This demonstrates that low-field FC 1H NMR is capable to cover the full Mdependence of ε. For the highest Mthe power εdetermined by FC and DQ 1H NMR is slightly larger, yet close to the prediction ε= 0.25 of the tube-reptation model for regime II (cf. Figure 1). Since this theoretically expected value is still not yet reached with both NMR techniques and could possibly be attained only for M> 2 000 000, it can be concluded that the crossover to full reptation dynamics is very protracted. Figure 7 shows a direct comparison of the correlation function for several high-MPB measured by FC and DQ 1H NMR plotted in a way to further test the relaxation behavior in regime II. It displays the long-time end of the FC data of Figure 6a focusing on the time window of the DQ data which have been inserted from ref 34. Samples with comparable Mare displayed in the same color. The amplitude is divided by the predicted power-law t−0.25 for regime II of the tube-reptation model illustrating how the theoretically expected behavior (horizontal line) is approached with increasing M, yet not completely attained. Whereas in the correlation function of PB with M= 9470 ≈5Meno indication for the power-law of regime II can be seen; above M= 24 300 the curves come closer to the model prediction and trace the envelope shape of the highest M= 441 000 continuously longer with higher M before bending away at long times. Moreover, a good agreement among the FC and DQ 1H NMR data can be observed. In order to achieve the best match among them, the DQ data have been scaled by a factor of 0.5 in amplitude. Vaca Figure 6. (a) Dipolar correlation function CDD(t/τs) of PB for various Mobtained by Fourier transforming the susceptibility master curves of Figure 5a. Dotted curve: low-Msystem representing glassy dynamics. Dashed lines: observed power-laws of regime I and II. Gray area illustrates variation of power-law exponent εof regime II for 24 300 ≤M≤441 000. Solid line: predicted power-law of regime II by tube-reptation model. (b) Power-law exponent εas a function of Mobtained in the time domain. For comparison, DQ 1H NMR results 34 are included. Even at M≈2 000 000 the prediction of the tube-reptation model ε= 0.25 (dashed line) is not fully approached yet. Macromolecules Article dx.doi.org/10.1021/ma202489y |Macromolecules 2012, 45, 1408−14161413 Cha vez and Saalwachter 61 applied a factor of 0.7, ascribing it to shortcomings of their model determining the absolute value of the correlation function. We note that the investigated polymers were obtained from the same supplier and in the case of PB with M= 196 000 and 441 000 the identical polymer was examined by both NMR methods. 5. DISCUSSION In Figure 7, a second power-law regime (II) can be clearly identified extending over 3 decades in time for M≥143 000. While increasing Mthe protracted transition to almost completely established reptation dynamics is characterized by the very slow approach of the exponent εtoward ε= 0.25 predicted by the tube-reptation model (Figure 6b). The latter has been reported for the mean-square displacement by, e.g., neutron spin echo (NSE) 17 and simulations. 4,37,38 The observed exponent ε(441 000) = 0.32 corresponds in the 1/T1representation to −0.68, which is larger than the value −0.42 found by Kimmich et al. 62 for PB with M= 65 500 at T = 313 K. This difference is due to the larger Minvestigated at both higher temperatures and lower frequencies in the present study. Concerning the identification of regime III of the tubereptation model, the situation is less convincing. For PB with M ≤56 500 the correlation function exhibits just a smooth transition to terminal relaxation, and the predicted power-law t−0.5 (cf. dotted line in Figure 7) may merely be assumed over 1 decade. Therefore, according to our current results, it cannot be decided whether the dynamic range of regime III is quite narrow, resulting in a smooth transition to regime IV (as observed in the DQ 1H NMR study 34 ), whether there is an interplay between the dynamics of different regimes, or whether the power-law would be more pronounced for higher Mand at yet longer times beyond our experimentally accessible range. The first is corroborated by a theoretical work 41 yielding an analytical expression for regimes III and IV together. However, the direct comparison of the correlation functions from both NMR methods demonstrates that FC 1H NMR is indeed capable of reaching the time scales which formerly have been restricted to DQ 1H NMR. We emphasize the agreement among the results of DQ and FC 1H NMR, since the theoretical concepts involved in these methods are quite different, i.e., coherent spin-evolution measuring dipolar correlations in the time domain and longitudinal relaxation probing a spectral density in the frequency domain, respectively. Regarding the limitations of the perturbation-based relaxation theories (cf. Theoretical Background), we are aware that our experimental results collected at lowest frequencies need to be treated cautiously. For the high-Mpolymers one sees from Figure 2a that the relaxation rate still grows up at low frequencies even though it has already reached (3 ms)−1at ν= 200 Hz. Consequently with τ=(2πν)−1= 0.8 ms the condition T1≫τis not strictly fulfilled anymore. Even more, since a relaxation dispersion is still observed the condition will be violated further as frequency decreases. In order to provide exact theoretical predictions for the characteristic polymer relaxation, a treatment based on a full (nonperturbative) solution the stochastic Liouville equation should be adopted. To our knowledge, such a general description has not been formulated yet. Therefore, we base our analysis on the secondorder perturbation theory, even though we are aware of its limitation. We note that the Larmor frequencies are still above the typical residual dipolar coupling of about 100 Hz observed for high-Mpolymer melts which was the second condition to be fulfilled (see above). 34,63 Together with the consistency between FC and DQ 1H NMR data (Figures 6b and 7) this provides some confidence that our results are still unaffected by low-frequency effects exceeding second-order perturbation theory. When comparing the observed power-law exponents to the predicted ones, we underline that the dipolar correlation function CDD(t) probed by FC 1H NMR does not need to be identical to the rank-two reorientational correlation function g(2)(t) of a polymer segment. A common assumption in the analysis of FC 1H NMR experiments is that the intramolecular contribution to relaxation dominates, and therefore the influence of intermolecular relaxation can be neglected. 26,27,30,34 As mentioned, results by Kehr et al. 43,44 indicate that this is not necessarily the case at low frequencies, suggesting that εmight also be sensitive to influences of the intermolecular relaxation. However, DQ 1H NMR results for isotopic dilution of PB 196000 with deuterated PB of the same Mand thereby suppressing the intermolecular interaction have shown 33 that the value of εremains the same as in the pure PB though the absolute values of CDD(t) are different. Very recent own measurements point into the same direction. The εvalues in Figure 6b are notably higher than 0.25 predicted by the tube-reptation model. This may be caused by the early onset of terminal relaxation for M≤56 500 or by additional relaxation mechanisms. Even at highest M= 441 000 ≫Methe theoretical exponent is not reached. In other words, although M≫Mea static tube appears to be not formed yet. All approaches to refine the Doi−Edwards model such as constraint release (CR) or contour length fluctuations (CLF) cause additional correlation loss, i.e., a steeper decay which results in εbeing higher than expected from the original tubereptation model, and are candidates for interpreting our findings. We note that the (twice) renormalized Rouse theory has been used by Kimmich and Fatkullin to describe the experimentally observed power-laws. 24,62 In the light of the recent DQ 1H NMR results together with our data measuring a large frequency and temperature range and exploiting FTS, the applicability of the renormalized Rouse model as proposed in ref 24 can be ruled out, especially in the low-frequency range. 25,44 Figure 7. Correlation function for PB with Mas indicated measured by FC and DQ 1H NMR 34 (dots and lines, respectively) divided by the power-law t−0.25. Dashed horizontal line and dotted line: expectation for regimes II and III of the tube-reptation model, respectively. Macromolecules Article dx.doi.org/10.1021/ma202489y |Macromolecules 2012, 45, 1408−14161414 The protracted transition to full reptation dynamics has also been observed in Monte Carlo simulations by Kreer et al. 37 and Paul. 38 Actually, Kreer et al. seem to be the first ones to discuss this phenomenon; however, such a delayed crossover has not been recognized by more recent simulation work. 64 A possible explanation may be due to the fact that the NMR signal is averaged over all segments in the chain. Whereas the chain ends participate in forming a new tube, the inner monomers are exposed to the topological confinement of the tube to a higher degree. Therefore, their reorientation is more hindered, which is closer to the model picture of a static tube. Kremer et al. 65,66 already pointed out by means of MD simulations that the outer monomers fluctuate stronger, and in order to observe actual entanglement effects, one should inspect the innermost monomers. Zamponi et al. 67 have concluded from a NSE study of polyethylene (PE) with M= 25 000 that the dynamic structure factor of a completely protonated chain differs from that of a partially deuterated chain with protons only at the inner part due to CLF. Effects of CR were believed to be negligible. Yet, complementing the NSE experiments, Vaca Cha vez and Saalwachter 33 directly proved that εis influenced indeed by CR effects, since diluting PB with M= 55 000 in a deuterated PB matrix with both lower and higher Mchanges ε. Because diluting PB with M= 196 000 in the same deuterated Mdoes not modify ε, an influence of intermolecular relaxation on εfor the highest Mcan be excluded. However, a very recent study 68 combining viscoelastic and dielectric relaxation spectra claims to have proven that the dielectric spectra are determined by reptation and CLF motion only and not by CR contributions. Thus, for a final interpretation additional evidence addressing the relevance of CR, CLF, and NMR specific intermolecular relaxation effects is needed and part of ongoing work. Whereas the above discussion implies a continuous transition to full reptation, there are several indications for a discontinuous change of the dynamics at another characteristic molecular mass Mr≫Me. For PB Colby et al. 50 disclosed at Mr ≈200Mea departure from the viscosity power-law η(M)∼M3.4 to M3.0, for PE Vega et al. 35 found Mr≈440Me, and for polyisoprene (PI) Abdel-Goad et al. 36 discovered Mr≈44Me. Therefore, the relation Mr/Meis not constant for different polymers. 69 In a recent study 15 by dielectric spectroscopy on PI we reported a saturation behavior at Mr≈20Meboth from the shape and time constant of the dielectric normal mode. 6. CONCLUSIONS We have investigated the dynamics of PB melts in a series of different Mby FC 1H NMR focusing on the low-frequency behavior for M>Me. The frequency window provided by the commercially available Stelar FC spectrometer has been extended significantly for 2 decades toward lower frequencies (Figure 2) with the aid of a home-built spectrometer 40 in order to explore polymer dynamics on even slower time scales than in our previous study. 30 The susceptibility representation of the relaxation dispersion offers a comparison with rheological data (Figures 5a and 5b). As a result of applying FTS, the susceptibility master curves and the corresponding dipolar correlation functions (Figure 5a and 6a, respectively) embrace over 10 decades in frequency or time and can be analyzed in a range which before was restricted to DQ 1H NMR experiments. 33 Consequently, for all Mstudied FC 1H NMR is now able to cover the relaxation regimes 0, I, and II (glassy, free Rouse, and constrained Rouse dynamics, respectively; cf. Figure 1) and for M≤56 500 also regimes III and IV (reptation dynamics and free diffusion, respectively) of the tube-reptation model. The particular goal of this contribution has been to study the transition from Rouse to fully established reptation dynamics at M≫Me. From the slow decrease of the power-law exponent ε(M) in regime II from 0.73 to 0.32 with increasing Mtoward the predicted ε= 0.25, it can be concluded that only for very high Mthe characteristic dynamics of the tube-reptation model is disclosed. The Mdependence of εis very similar to the values reported by DQ 1H NMR 34 for high M(Figure 6b), and a direct comparison of the correlation functions obtained by FC and DQ 1H NMR shows a good agreement (Figure 7). Hence, FC NMR has turned from a complementary method to DQ NMR into a competitive one. Moreover, we anticipate that a further enhancement of the stray field compensation tool will contribute 1 more decade at low frequencies. This could further illuminate the reptation dynamics of regime III and provide the basis for a joint description of dielectric, viscoelastic, and NMR relaxometric data. ■AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. ■ACKNOWLEDGMENTS 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. The authors thank F. Vaca Cha vez and K. Saalwachter for valuable discussions and providing their data for Figure 7. ■REFERENCES (1) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford Science Publications: Oxford, 1986. (2) Rouse, P. E. J. Chem. Phys. 1953,21, 1272−1280. (3) de Gennes, P. G. J. Chem. Phys. 1971,55, 572−579. (4) Kremer, K.; Grest, G. S. J. Chem. Phys. 1990,92, 5057−5086. (5) Lodge, T. P.; Rotstein, N. A.; Prager, S. Adv. Chem. Phys. 1990, 79,1−132. (6) Watanabe, H. Prog. Polym. Sci. 1999,24, 1253−1403. 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(69) Fetters, L. J.; Lohse, D. J.; Milner, S. T.; Graessley, W. W. Macromolecules 1999,32, 6847−6851. Macromolecules Article dx.doi.org/10.1021/ma202489y |Macromolecules 2012, 45, 1408−14161416 98 Publication 4 Mean Square Displacement and Reorientational Correlation Function in Entangled Polymer Melts Revealed by Field Cycling 1H and 2H NMR Relaxometry. Herrmann, A.; Kresse, B.; Wohlfahrt, M.; Bauer, I.; Privalov, A. F.; Kruk, D.; Fatkullin, N.; Fujara, F.; R¨ossler, E. A. Macromolecules 2012,45, 6516–6526. Copyright 2012 by The American Chemical Society DOI: 10.1021/ma301099h 99 100 Mean Square Displacement and Reorientational Correlation Function in Entangled Polymer Melts Revealed by Field Cycling 1H and 2H NMR Relaxometry A. Herrmann, † B. Kresse, ‡ M. Wohlfahrt, † I. Bauer, † A. F. Privalov, ‡ D. Kruk, § N. Fatkullin, ∥ F. Fujara, ‡ and E. A. Rossler †, * † Experimentalphysik II, Universitat Bayreuth, 95440 Bayreuth, Germany ‡ Institut fur Festkorperphysik, TU Darmstadt, Hochschulstrasse 6, 64289 Darmstadt, Germany § Faculty of Mathematics & Computer Science, University of Warmia & Mazury Olsztyn, Sloneczna 54, 10710 Olsztyn, Poland ∥ Institute of Physics, Kazan Federal University, Kazan 420008, Tatarstan, Russia ABSTRACT: Mixtures of protonated and deuterated polybutadiene and polydimethylsiloxane are studied by means of field-cycling (FC) 1H NMR relaxometry in order to analyze the intraand intermolecular contributions to spin−lattice relaxation. They reflect reorientational and translational dynamics, respectively. Master curves in the susceptibility representation χ″(ωτs) are constructed by employing frequency−temperature superposition with τsdenoting the segmental correlation time. The intermolecular contribution is dominating at low frequencies and allows extracting the segmental mean square displacement ⟨R2(t)⟩, which reveals two power-law regimes. The one at short times agrees with t0.5 predicted for the free Rouse regime and at long times a lower exponent is observed in fair agreement with t0.25 expected for the constrained Rouse regime of the tube-reptation model. Concomitantly the reorientational rank-two correlation function C2(t/τs) is obtained from the intramolecular part. Again two power-law regimes t−εare identified for polybutadiene. The first agrees with t−1of free Rouse dynamics whereas at long times ε= 0.49 is obtained. The latter is corroborated by the 2H relaxation of deuterated polybutadiene, yet, it does not agree with ε= 0.25 predicted for constrained Rouse dynamics. Thus, the relation C2(t)∝⟨R2(t)⟩−1as assumed by the tube-reptation model is not confirmed. 1. INTRODUCTION The commonly accepted model for the dynamics of entangled polymers is referred as the tube-reptation model, 1 which is a combination of the Rouse model 2 for nonentangled chains (with molecular mass Mbelow the entanglement mass Me) and de Gennes’reptation idea 3 for M>Me. The model predicts four different power-law regimes (I−IV) for the time dependence of the mean square displacement ⟨R2(t)⟩of a polymersegment. Regarding the short-time dynamics MD simulations 4−6 as well as neutron scattering experiments 7 have essentially confirmed the model by identifying a crossover from ⟨R2(t)⟩∝ t0.5 to ⟨R2(t)⟩∝t0.25 forecast for the transition from free Rouse (regime I) to constrained Rouse dynamics (or incoherent reptation dynamics, regime II). On long time scales the final crossover from ⟨R2(t)⟩∝t0.5 typical of reptation (regime III) to ⟨R2(t)⟩∝tof the terminal regime of free diffusion (regime IV) has been observed in polymer melts 8,9 as well as polymer solutions 10 by field gradient NMR. Another regime (regime 0) which is usually not included in polymer theories reflects ‘glassy dynamics’at very short times, i.e., local motions connected with fluctuations of intrasegmental degrees of freedom on the time scale of the segmental correlation time τs. Regarding the time correlation function characterizing the segmental reorientations a corresponding set of power-laws is provided by the tube-reptation model. In regime II and III the reorientational correlation function Cl(t) is given by Cl(t)∝ ⟨R2(t)⟩−1independently of its rank l, whereas in regime I Cl(t) is l-dependent with C2(t)=[C1(t)]2∝t−1and C1(t)∝ ⟨R2(t)⟩−1. The argument used for deriving the power-laws for regime II and III follows the idea that the correlation function Cl(t)reflects the probability that at time tthe chain segment has not left the original tube segment and thus is given by the return-to-origin probability which is provided by ⟨R2(t)⟩−1.We note that alternative models for describing the dynamics of entangled polymers, namely the (n-)renormalized Rouse model and the more complicated polymer-mode-coupling model, 11−15 end up with different relationships between the reorientational correlation functions and the mean square displacement (cf. Discussion). 16 Thus, the experimental investigation of the power-law behavior of correlation functions of different ranks Received: May 30, 2012 Revised: July 17, 2012 Published: August 2, 2012 Article pubs.acs.org/Macromolecules © 2012 American Chemical Society 6516 dx.doi.org/10.1021/ma301099h |Macromolecules 2012, 45, 6516−6526 5. DISCUSSION We have separated the intraand intermolecular relaxation contribution to the total 1H relaxation for PB and PDMS. This has been corroborated in the case of PB by comparing the spectral shape of the intramolecular 1H relaxation contribution with that of 2H. No significant difference is observed. It has been shown clearly that the intermolecular relaxation contribution is dominating at low frequencies. As already emphasized 19−22 it is not possible to ignore this contribution. However, benefiting from the strong intermolecular contribution the segmental mean square displacement ⟨R2(t/τs)⟩has been calculated from χ″ inter(ωτs). For PB as well as PDMS two power-law regimes tαare found. The one at short times (regime I) agrees with the prediction t0.5 of the Rouse model while the one at long times exhibits a significantly lower exponent in fair agreement with α= 0.25 forecast by the tube-reptation model for constrained Rouse dynamics (regime II). We find α=0.19±0.03 for PB and α=0.34±0.03 for PDMS. Because of the strong intermolecular contribution, the intramolecular part of PB significantly changes with respect to the total relaxation. It is found that the exponent of C2(t/τs) in regime II is rather high and does not agree with the prediction ε= 0.25 of the tube-reptation model. Explicitly, while the long-time exponent in the total dipolar correlation CDD(t)isε= 0.32 ±0.02 it becomes ε= 0.49 ±0.05 for C2(t/ τs). We emphasize that the exponent agrees with the one extracted from the 2H relaxation results which demonstrates the reliability of the separation procedure via the isotope dilution technique. A similar trend is observed in very recent DQ 1H NMR results by Saalwachter and co-workers. 44 Next we compare the results for C2(t/τs) and ⟨R2(t/τs)⟩. Since the shape of the susceptibilities for high-MPB is considered to be Mindependent and representatively described by M= 196000 (cf. Figure 2b), we will restrict further discussion to those results. As said, the reorientational correlation function yields C2(t/τs)∝t−0.49±0.05 for regime II. Together with the result for the segmental mean square displacement, explicitly ⟨R2(t/τs)⟩∝t0.19±0.03 (following essentially the tubereptation model), it can be concluded that in regime II the fundamental assumption of the tube-reptation model Cl(t)∝ ⟨R2(t)⟩−1for t>τeis not fulfilled (τeis the entanglement time). This is confirmed by recent MD simulations by Meyer et al. 45 which have shown that in regime II the slopes of the reorientational correlation functions of different rank lare neither identical: C1(t)∝t−0.25 and C2(t)∝t−0.34 have been observed in a regime where ⟨R2(t)⟩∝t0.25 is well identified. In other words, the presumed relation only holds for l=1, explicitly C1(t)∝⟨R2(t)⟩−1. However, recent simulations by Wang et al. 29 have shown a power-law relationship of C2(t)∝ [C1(t)]mfor m= 2 and 1 in the regimes I, and II, respectively, i.e., they essentially have confirmed the tube-reptation model. In contrast to the results of the tube-reptation model the experimentally observed relation between Cl(t) and ⟨R2(t)⟩ suggests an l−dependence as forecast by the (n)-renormalized Rouse model and the polymer-mode-coupling model, 11−16 and a relation C2(t)∝[C1(t)]2∝⟨R2(t)⟩−2is predicted. The present results seem to suggest C2(t)∝⟨R2(t)⟩−2which is, however, at variance with the result of the MD simulations of Meyer et al. 45 Regarding the Rouse regime (I) the prediction C2(t)∝C1(t)2∝t−1(at τs<t<τe) is in good agreement with our experiments. Simulations have shown that it is important to average only over the innermost monomers 4 in order to reveal the power-laws of the tube-reptation model. This remains a future experimental task in order to thoroughly examine also effects constraint release or contour length fluctuations. 27,29 As shown in Figure 5 where the NMR results are compared to those from neutron scattering a very similar behavior is observed for ⟨R2(t/τs)⟩which demonstrates the potential of FC 1H NMR. In other words, NMR relaxometry has become the second method to probe subdiffusive translational motion in polymer melts. The entanglement time τeand the tube diameter a0=(⟨R2(t=τe)⟩)1/2 can be estimated by following the mean square displacement until at t=τethe slope departs from the initial power-law regime. 1 From the intersection (arrows in Figures 5a and 5b) of the power-laws we find for PB τe/τs≈105and a0≈3 nm, and for PDMS τe/τs≈106and a0≈ 8 nm. Introducing a reference temperature (Tref = 393 and 408 K) and using the corresponding τsyields τe(Tref)≈0.8 μs and 2 μs for PB and PDMS, respectively. From rheological experiments 42 at T= 413 K, the tube diameters a0= 4.4 nm and 7.9 nm have been determined for PB and PDMS, respectively. A neutronspinechostudy 46 found for PB a0=4.4nmandτe=5ns at T= 509 K, and also a recent study 47 analyzing atomistic MD simulations of PB with M≈11000 has reported very similar values a0≈3nm. We emphasize that our conclusions rely on the lowfrequency relaxation results (cf. Appendix A) and the possibility to extract intramolecular and intermolecular relaxation contributions from the overall relaxation by isotopic dilution. The demonstrated coincidence between the shapes of χ″ intra(ωτs) and χ″ Q(ωτs) determined by 1H and 2H relaxation, respectively, and the fact that latest results 44 by 1H DQ NMR indicate a similar trend for εmake the results even more trustworthy. We assume cross-relaxation terms to be negligible, since an exponential magnetization decay is observed and the intermolecular proton−deuteron coupling is below 5% of the intermolecular proton−proton coupling. 19 The evaluation of ⟨R2(t/τs)⟩presumes a Gaussian probability distribution to link the intermolecular relaxation rate with the relative mean-squared displacement. Gaussian statistics is a Figure 8. Zoom into Figure 2b with the 1H master curves of the PB 196000 with xH= 100% and 16% and the corresponding intramolecular contribution. The 2H susceptibilities at T= 355 and 393 K of PB 191000-d6are displayed in different colors. Literature data for PB 191000-d6(from ref 22) and PB 196000 with xH= 15% (from ref 19), both at at T= 355 K, are included for comparison. Macromolecules Article dx.doi.org/10.1021/ma301099h |Macromolecules 2012, 45, 6516−65266523 reasonable assumption for describing relative displacements of segments from different molecules in first approximation 16 at least for t>τsand has also been used in the analysis of the NS data. 7 Yet, non-Gaussian statistics may apply for describing the ordinary segmental, i.e., not relative, displacements as assumed by the tube-reptation model. 1,7,16,21,29,48 Note that the application of Gaussian statistics is only important for a numerical coefficient in eq 4. For any reasonable model of polymer dynamics at t>τseq 2 holds, because very general arguments without using Gaussian statistics yield Cinter(t)∝ W(r=0,t), 16,19 i.e., the probability density of a pair of spins of different molecules to recover in the time ttheir initial spatial separation. It follows from scaling arguments that this quantity decays ∝⟨r2(t)⟩−3/2,as⟨r2(t)⟩1/2 is the only important characteristic length for the relative displacements during the time t. Concerning the particularities of PDMS they are attributed to a strong intermolecular contribution dominating already at comparatively high frequencies. In our previous publication, 36 in which we presented results without the decomposition into intraand intermolecular contributions, we concluded that this M-independent effect originates from a particularity of the intramolecular relaxation contribution. Given the results from the decomposition into intraand intermolecular parts we have to revise this interpretation. Although at the moment we do not understand the reason for the extremely strong intermolecular contribution in PDMS, the finding once again emphasizes that the intermolecular relaxation contribution must be taken into account when analyzing FC 1H NMR results. Finally, a note regarding the application of the isotope dilution technique is necessary. When testing for given temperatures the linearity of the relaxation rate R1,DD(xH) (or equivalently the susceptibility) no satisfactory results are observed (cf. also ref 33), which we attribute to an isotope effect. The segmental time constant τsof the protonated polymer is slightly changed by dilution with deuterated polymer (Figure 9 in Appendix B). Although the effect is rather small it renders the construction of susceptibility master curves χ″ DD(ωτs) obligatory to allow the extrapolation to xH→0 and to reveal the linearity in xH(Figure 10 in Appendix B). Figure 9. Correlation times τs(T) for the PB blends with M= 24300 and 196000 (a) and the PDMS blends with M= 21600 (b) of different molar fractions xH. Lines: guide for the eye. Figure 10. Dependence of the 1H susceptibility on the molar fraction xHat some reduced frequencies ωτsfor the PB 24300 blends (a) and the PDMS 21600 blends (b). Lines: linear extrapolation to determine the intramolecular contribution at xH=0. Figure 11. 1H susceptibility χ″ DD(ω)=ω/T1(ω)ofthePDMS 21600 blends with molar fractions xH= 100% and 47% (left axis) and xH= 75% (right axis) for the temperatures indicated. Lines: fits of the α-peak with a Cole−Davidson function. Dashed lines illustrate the shift of the peak position at a constant temperature due to different xH. Macromolecules Article dx.doi.org/10.1021/ma301099h |Macromolecules 2012, 45, 6516−65266524 6. CONCLUSIONS By applying the isotope dilution technique we have separated the relaxation contributions from reorientational and translational dynamics in entangled melts of PB (Figure 2 and 3) and PDMS (Figure 7). For PB, we have provided the corresponding time correlation functions (Figure 4). Because of an isotope effect (Figures 9 and 11) the isotope dilution technique can only be applied in combination with employing susceptibility master curves. The shape of the reorientational correlation function as obtained from the isotope dilution series is reproduced by the 2H relaxation results. Its long-time powerlaw exponent is higher than that of the dipolar correlation function and that forecast by the tube-reptation model due to the dominating influence of intermolecular relaxation. The time-dependence of the segmental mean square displacement (Figure 5) determined from the intermolecular contribution yields power-laws, which essentially accord with the predictions for regime I and II establishing FC 1H NMR as a competitive technique to probe also translational dynamics of polymers. From the fact that for PB in regime II the power-law exponents ε= 0.49 ±0.05 and α= 0.19 ±0.03 are found for the reorientational correlation function and the mean square displacement, respectively, it can be concluded that also the relation C2(t)∝ ⟨R2(t)⟩−1as assumed for the tube-reptation model does not hold. ■APPENDIX A. 2H Relaxation at Low Frequencies As discussed in context of Figure 2b, the 2H master curve of PB 191000-d6exhibits increased scatter at low frequencies and therefore we compare it to literature data. An enlargement of the results of Figure 2b is displayed in Figure 8. By inspecting the susceptibilities (red and blue) which constitute the 2H master curve of PB 191000-d6it can be seen that the data of the two temperatures do not exactly coincide at ωτs<10 −7. Moreover, the 2H data by Kehr et al. 22 (orange) deviate at ωτs<5×10−7from both our 2H data and the intramolecular contribution obtained from 1H relaxation. At such high Mand low frequencies the following situation has to be considered: The 2H spin−lattice relaxation time is on the order of the switching time of the relaxometer (which is not the case for PB 22800-d6, cf. Figure 1) and below ν= 1 kHz the 2H magnetization curves feature a discernible non-exponential decay. In addition it has been already pointed out by Kehr et al. 22 that in this frequency range the correlation times approach the relaxation times, i.e., the Redfield condition might be violated. This contradicts the fundamental assumption of the relaxation theory: one can express the relaxation rates in terms of spectral densities only if the motion leading to the relaxation (or the inverse resonance frequency) is much faster than the relaxation time itself. 18 In the low frequency range, this requirement limits the validity of the relaxation theory and is equivalent to the Redfield condition ωDDτc≪1 where ωDD is the amplitude of the dipole−dipole interaction in angular frequency units and τcis the correlation time. Furthermore, at low frequencies the quadrupolar coupling becomes comparable with the Zeeman interaction and that changes the quantization axis of the proton magnetization. 49 The standard relaxation theory is only valid if the Zeeman interaction dominates (cf. also ref 26). As a consequence the last decade of the 2H relaxation at low frequencies needs to be treated cautiously. Both the literature and our 2H results bend away from the expected power-law behavior, yet for our data this occurs about one decade lower in frequency than for the data by Kehr et al. Therefore, we have confidence in our data at least down to ωτs≈10−7. However, future investigations are necessary to elucidate these issues when such low frequencies are approached by state-of-the-art equipment. B. Isotope Effect The correlation times τs(T) of the PB and PDMS blends for the different molar fractions xHare shown in Figure 9, parts a and b, respectively, as obtained by the construction of the susceptibility master curves. For a given temperature τs(T)is quite similar, however, a small trend depending on xHcan be seen, which is in the case of PB more distinctive for M= 24300. At high temperatures τs(T) is increased by a factor of 3 while reducing xH. This isotope effect is observed for both PB and PDMS; i.e., it appears that the dynamics of the polymer melt is somewhat altered by the addition of deuterated polymer. As the molecular masses of the protonated and deuterated samples are similar, i.e., the glass transition temperature Tg(M) has reached its saturation value for high M, and it has been ensured that the solvent is completely removed (cf. Experimental Section and Figure 1a) we can neglect an influence from this side. It is well known that mixtures of protonated and deuterated polymers may phase separate. Although we do not see any indication for such phase separation, e.g., a non-exponential spin−lattice relaxation, still there may be small changes of the intrinsic friction coefficient caused by, e.g., changes of the coil dimensions. It may be this isotope effect which renders the necessary linearity of R1,DD=R1,DD(xH) at a given frequency to be not fully satisfactory; however, in the master curves χ″ DD(ωτs) the linearity with xHis restored (Figure 10). The shift of the time constant can be directly recognized in Figure 11 in which the “raw”data of different xHare displayed as susceptibility χ″ DD(ω)=ω/T1(ω) for some applied temperatures at which the relaxation maximum is observed. As a guide for the eye the dashed lines mark the frequency shifts of the main relaxation peaks at constant temperatures between different xH, which of course translates into the time constants τsdescribing the peak position. ■AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Notes The authors declare no competing financial interest. ■ACKNOWLEDGMENTS We would like to thank Deutsche Forschungsgemeinschaft (DFG) for funding through grants FU 308/14 and RO 907/16 (SPP 1369) and Russian Foundation for Basic Research (RFBR) for support through Fund 10-03-00739-a. We are very grateful to R. Kimmich, Universitat Ulm, for valuable discussions and cooperation, and N. Popp, Inorganic Chemistry III, Universitat Bayreuth, for help with sample preparation. ■REFERENCES (1) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford Sci. Publications: Oxford, 1986. (2) Rouse, P. E. J. Chem. Phys. 1953,21, 1272−1280. (3) de Gennes, P. G. J. Chem. Phys. 1971,55, 572−579. (4) Kremer, K.; Grest, G. S.; Carmesin, I. Phys. Rev. Lett. 1988,61, 566−569. (5) Paul, W.; Smith, G. D. Rep. Prog. Phys. 2004,67, 1117−1185. Macromolecules Article dx.doi.org/10.1021/ma301099h |Macromolecules 2012, 45, 6516−65266525 (6) Barrat, J.-L.; Baschnagel, J.; Lyulin, A. 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ACS Macro Letters 2012,1, 1339–1342. Copyright 2012 by The American Chemical Society DOI: 10.1021/mz3004924 113 114 Dynamics of Linear Polybutadienes in Solution Studied by Field Cycling 1H NMR Axel Herrmann and Ernst A. Rossler* Experimentalphysik II, Universita t Bayreuth, 95440 Bayreuth, Germany ABSTRACT: Field cycling 1H NMR relaxometry is utilized to investigate dynamics in solutions of monodisperse polybutadienes of different molecular mass (M) and deuterated toluene. Broad temperature and polymer mass fraction ranges (c=5−100%) are studied. By applying frequency−temperature superposition, susceptibility master curves χ″ DD(ωτs) are constructed. They cover the segmental relaxation and polymer chain dynamics, and provide the concentration dependence of the segmental time constant τs(T). The relaxation strength of polymerdynamicsisreducedsimilarlyforallMwith decreasing c; for the lowest c, almost no polymer dynamics shows up, that is, the dipolar correlation function obtained via Fourier transform decays almost completely due to segmental dynamics. The dipolar correlation function is decomposed into contributions of segmental and polymer dynamics. Its long-time power-law exponent associated with entanglement dynamics is increased from its bulk value with reduced c. This is interpreted as a continuous increase of the effective entanglement molecular mass. The dynamics of polymer melts is most often described by the tube-reptation model, 1 which yields predictions for the subdiffusive behavior of a chain segment. For polymers in concentrated and semidilute solution, modifications with respect to the bulk behavior have been reported, for example, by rheology 2−4 or analyses of the dielectric normal mode spectrum. 5 Expected are, for example, a concentration dependence of the monomeric friction coefficient ζ0and of frictioninsensitive properties, for example, the plateau modulus GN0,a shift of the terminal relaxation time τdtoward shorter times and an increase in the entanglement spacing Me. Pioneering works of field cycling (FC) NMR relaxometry by Kimmich and coworkers have qualitatively disclosed a transition from entanglement dynamics for a high Mmelt to Rouse-like behavior for high dilution. 6,7 FC NMR relaxometry has been established as a powerful technique to probe the microscopic dynamics of polymers. 7,8 By electronically controlling the magnetic relaxation field B, the dispersion of the spin−lattice relaxation time T1is measured over 3−5 decades in frequency, which is given by the Larmor frequency ω=γB(γ: gyromagnetic ratio). A very helpful concept in analyzing the NMR relaxation dispersion is the construction of master curves in the susceptibility representation: 8 first, the measured relaxation rate R1(ω)=1/T1(ω)is transformed to the susceptibility representation χ″ DD(ω)=ω/ T1(ω), which at low temperatures allows to extract the time constant τsof segmental motion. Second, by applying frequency−temperature superposition (FTS), the susceptibility is shifted solely in frequency to finally provide a master curve as a function of the reduced frequency ωτs. Thereby, the frequency window of the technique is significantly extended. FTS is well-known from, for example, rheology and reflects a fundamental feature of cooperative dynamics in condensed matter. 8 The master curves χ″ DD(ωτs)(“NMR susceptibility”in the following) exhibit a peak at ωτs≈1, which is identified with the segmental or local dynamics governed by the α-process of the glass transition. For polymers at ωτs< 1, an excess intensity with respect to the spectrum of a simple liquid is observed, which is due to the slower, M-dependent polymer dynamics. Its integral is a measure for the polymer relaxation strength f, that is, the residual correlation that relaxes at t>τsdue to Rouse and entanglement dynamics. 8 It is dominated by the Rouse contribution and is an analog to GN0obtained by rheology; the latter, however, is determined solely by the terminal relaxation caused by entanglement dynamics. Our previous studies 8−11 have focused on understanding the NMR relaxation of simple liquids, nonentangled and entangled polymer melts, that is, the emergence of polymer dynamics with M. The dipolar correlation functions CDD(t/τs) have been extracted by Fourier transform of the NMR susceptibility, cover 10 decades in time, and exhibit different power-law regimes, which have been compared to the predictions of the tubereptation model. The aim of this Letter is to investigate how the addition of a low-Mdiluent to a linear polymer melt (polybutadiene with M >Me) is probed by 1H FC NMR. As a reference for the behavior of bulk melts the previous results 10 are used. A comprehensive picture is rendered by employing the above approach for the first time to polymers in solution, because the susceptibility master curves χ″ DD(ωτs)reflect the spectral Received: October 9, 2012 Accepted: October 26, 2012 Letter pubs.acs.org/macroletters © XXXX American Chemical Society 1339 dx.doi.org/10.1021/mz3004924 |ACS Macro Lett. 2012, 1, 1339−1342 changes upon dilution in a similar way as those of rheology. A large temperature range is studied and FTS is applied in order to include both the (fast) segmental and the (slow) chain dynamics in the master curves. The monodisperse 1,4-polybutadienes (PB, PDI ≤1.04) were purchased from PSS, Mainz, Germany, and M=Mwin g/ mol denotes their mass average molecular mass. Solutions of PB-h6and deuterated toluene (Sigma-Aldrich, degree of deuteration above 99.96%, used without further purification) were prepared by degassing both components, dissolving different mass ratios, and flame sealing the glass tube with the frozen compound under vacuum. The given concentration c is the mass fraction of the polymer, which is very similar to the volume fraction, because the mass densities are rather equal. The volume fraction associated with the critical concentration at the overlap limit can be estimated by c*<Nm −4/5, where Nm is the number of monomers per chain. 13 For the PB samples, this yields values c*< 0.02 (for M= 9470), thus, the regimes of concentrated and semidilute solutions are covered. The samples were measured with a commercial Stelar FC spectrometer in the frequency range ν=ω/(2π) = 10 kHz to 20 MHz and T1 was determined by an exponential fit of the magnetization curve. Note that by applying 1H FC NMR only the dynamics of the protonated PB is probed. In Figure 1a−c, the master curves χ″ DD(ωτs)inthe susceptibility representation are compiled by applying FTS for PB with M= 9470, 18200, and 47000, respectively, and different polymer mass fractions c. A temperature range of T= 168−393 K has been covered depending on cin order to create the master curves. The relaxation regimes of glassy (0), free Rouse (I), and constrained Rouse (II) dynamics are indicated as obtained for the undiluted melts (black curves). 10 Regime 0, in fact, contains an interplay of segmental (glassy) and Rouse dynamics. Note that the master curves are an isofrictional representation, as changes in the time scale due to a concentration-dependent friction coefficient are scaled out by plotting the data as a function of the reduced frequency ωτs. This allows of monitoring the concentration dependence of the spectral shape of χ″ DD(ωτs): The segmental dynamics is represented by the α-peak at ωτs≈1. At ωτs< 1 the excess intensity reflecting the polymer relaxation contribution with respect to the simple liquid (PB 466) is diminished for all M with decreasing concentration c. For the lowest concentration of PB 9470, the susceptibility almost approaches that of the simple liquid, that is, the polymer character as probed by 1H relaxation gets lost as the dipolar correlations are relaxed almost completely by the segmental dynamics. Upon addition of a deuterated diluent it is expected on the one hand that the polymer dynamics is modified; on the other hand for 1H NMR the dipolar coupling of the protons which comprises intramolecular and intermolecular contributions is reduced due to the latter. Isotope dilution experiments by 1H FC NMR, that is, blending protonated and deuterated polymers with similar M, allow attaining both relaxation contributions. Thereby, it has been demonstrated 11,12 that, while the intermolecular part dominates at low reduced frequencies (ωτs≪1), also in the frequency range of the αpeak (ωτs≈1) the amplitude is reduced. However, this is surprisingly not observed in the present case (cf. Figure 1). The segmental time constants τs(T) are displayed in Figure 2 for the dilution series of PB 9470 with different cas yielded from the construction of the susceptibility master curves. The curves can be interpolated with a Vogel−Fulcher−Tammann (VFT) function 8 and are shifted toward lower temperatures with decreasing c. This indicates the well-known plasticizer effect, that is, the segmental dynamics of the polymer gets faster by addition of the solvent. The glass transition temperature is usually defined as Tg=T(τs= 100 s). Because FC NMR detects Figure 1. Susceptibility master curves χ″ DD(ωτs) of PB-h6with M= 9470 (a), 18200 (b), and 47000 (c) diluted in toluene-d8for the polymer mass fractions and in the temperature range as indicated. PB 466 and PB 441000: reference for the simple liquid and undiluted high-Mlimit, respectively. 10 Dashed lines: regimes of glassy (0), free Rouse (I), and constrained Rouse (II) dynamics. The undiluted PB samples have been measured also toward lower frequencies, 10 where the susceptibility of PB 18000 perfectly extends that of PB 18200. Figure 2. Time constants τs(T) of segmental motion for PB 9470 with different concentrations. Lines: VFT interpolations. Inset: glass transition temperature Tg*as a function of concentration; line: fit with Fox eq (1/Tg*=c/Tg,polymer +(1−c)/Tg,solvent). ACS Macro Letters Letter dx.doi.org/10.1021/mz3004924 |ACS Macro Lett. 2012, 1, 1339−13421340 dynamics on much faster time scales, we define Tg*=T(lg(τs/s =−8)). The corresponding dependence Tg*(c) is shown in the inset of Figure 2 for PB with M= 9470 and 18200. A continuous decrease of about 70 K with reduced cis observed that follows the Fox equation. 2 In Figure 3a the relaxation strength fof polymer dynamics is plotted as a function of c. It is given by the difference between the integrated susceptibilities of the polymer and the simple liquid (e.g., PB 466). With decreasing concentration, fis reduced similarly for all M. This demonstrates that for lower c less correlation survives beyond glassy dynamics, that is, on time scales t>τs. Obviously due to the enhanced segmental mobility at lower c, the α-process is more efficient at the expense of the relaxation contribution of polymer dynamics. As noted, in FC NMR fis dominated by Rouse dynamics (regime I, cf. Figure 1). The plateau modulus GN0of rheology is expected 3,4 to be decreased upon dilution along GN0(c)=GN0(c =1)c2.3, which is included in Figure 3a (dashed line). A similar trend is observed at high cfor fand GN0. However, the latter is determined solely by the integral of the terminal relaxation spectrum associated with entanglement dynamics. For PB 9470 the NMR susceptibility χ″ DD(ωτs) exhibits a behavior close to ∝ωat lowest reduced frequencies (Figure 1a), that is, the slowest or terminal relaxation process is detected. This is equivalent to an essentially constant dipolar spectral density JDD(ω), as χ″ DD(ω)=ωJDD(ω)holds. 8 Therefore, at lowest frequencies, the mean correlation time ⟨τ⟩of polymer dynamics is provided, that is, ⟨τ⟩=JDD(0) ∝ χ″ DD(ωτs=2×10−6). The mean correlation time comprises a weighted sum of the individual correlation times of the different Rouse and entanglement modes and, for M>Me, its slowest contribution is given by the terminal relaxation time τd.In Figure 3b the ratio ⟨τ⟩(c)/⟨τ⟩(c= 1) is depicted. It decreases with lower c, as expected 4 by rheology for the terminal relaxation time τd(c)∝c2.3 (dashed line). A crossover (solid lines) to a weaker c-dependence can be anticipated below c= 40%, which we interpret as a transition from Rouse and entanglement dynamics to solely Rouse-like relaxation (see also below). The dipolar correlation function CDD(t/τs) obtained by Fourier transform of the susceptibility master curves (Figure 1) is presented in Figure 4a for PB 18200 with different c. As a reference for the bulk melt behavior, which is discussed first, CDD(t/τs) of undiluted PB with different M(black symbols) including PB 466 10 as the low-Msystem and PB 441000 10 for M≫Me= 2000 is presented. Analogously to the susceptibility representation, CDD(t/τs)reflects segmental, free Rouse, and entanglement dynamics. 10 The latter two can be described by power-laws ∝t−ε, which exhibit the exponents ε= 0.85 and ε= 0.32 (M≫Me) for the regimes I and II, respectively. In regime II, εdepends on M, and only for very high Mthe prediction ε= 0.25 of the tube-reptation model is almost approached, which indicates a highly protracted transition to reptation dynamics. 10 For the decay of CDD(t/τs) of PB 18200 with different c,a similar effect is observed: the exponents εdepend on c. Its origin cannot be unambiguously clarified due to the interplay of segmental and polymer relaxation contributions which is difficult to interpret. To eliminate the influence of the segmental dynamics, Figure 4b shows the correlation functions Cpolymer(t/τs), which contain only the polymer relaxation contribution of PB with different M and comparable c. Statistical independence between segmental and polymer dynamics, that is, a multiplicative approach CDD(t) =Csegmental(t)·Cpolymer(t), is assumed and the polymer relaxation strength f(cf. Figure 3a) is introduced. Finally, Cpolymer(t/τs)is obtained by dividing each CDD(t/τs) by the contribution Csegmental(t/τs) of the segmental dynamics, as given by PB 466, explicity: Cpolymer(t/τs)=CDD(t/τs)/[(1 −f)ϕsegmental(t/τs)+f], where ϕsegmental(t/τs) denotes the normalized segmental correlation function. While εremains essentially unchanged in the Rouse regime (I), effects on dilution are observed for the entanglement dynamics at long times; in regime II, εis increasing with lower cfor M= 9470 and 18200. The values ε(c) obtained by a power-law fit are displayed in Figure 5 together with ε(c= 1) of the undiluted PB melts with different Figure 3. (a) Relaxation strength fof polymer dynamics as a function of polymer mass fraction cfor PB with different M. Dashed line: expectation for GN0(c). (b) Ratio ⟨τ⟩(c)/⟨τ⟩(c= 1) of PB 9470 obtained from the susceptibility at lowest frequencies. Dashed line: expectation for τd(c) from rheology. Figure 4. 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Journal of Magnetic Resonance Series A 1995,117, 53–61. 130 Acknowledgements First I would like to thank my supervisor Ernst R¨ossler for his constructive suggestions, kindness, and inspirations. By providing the freedom to organize one’s work for oneself and through his open-door policy, he created a very pleasant atmosphere. I am indebted to Franz Fujara for the opportunity to work in his lab in Darmstadt, and for his valuable advises and enthusiasm throughout the project. The most significant interpretations were suggested by the low-frequency Darmstadt relaxation data. I am also very grateful to Alexei Privalov and Benjamin Kresse for introducing me to the Darmstadt FC I relaxometer and the Damaris software, and the repetitive adjustment of the compensation system. Their outstanding technical knowledge made the ”50 Her(t)z project” become a reality. Many thanks to Irene Bauer and Herrn J¨urgen Gmeiner for their help with preparing the – in most cases highly viscous – samples. 131