Secondary relaxation processes in neat and binary glass formers studied by 2H NMR spectroscopy

Secondary relaxation processes in neat and binary glass formers studied by 2H NMR spectroscopy Von der Universität Bayreuth zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung von Björn Micko geboren am 15. August 1980 in Ostfildern-Ruit 1. Gutachter: Prof. Dr. E. Rößler 2. Gutachter: Prof. Dr. M. Vogel Tag der Einreichung: 24.02.2012 Tag des Kolloquiums: 20.06.2012 The cover page shows 2H NMR solid echo spectra of cyanocyclohexane recorded in the plastically crystalline phase for different inter pulse delays ( 146.8K , tp = 10 µs to 300 µs back to front). The β -process imposes prominent fast motion limit line shape effects in this temperature range, which allow for a direct determination of the degree of restriction. Contents 1 Introduction 1 2 The glass transition phenomenon 5 2.1 Supercooled liquids and glasses ..................... 5 2.2 The α-process ............................... 8 2.3 The β-process ............................... 9 2.3.1 Open questions .......................... 11 32H NMR spectroscopy 13 3.1 The nuclear spin and its interactions .................. 13 3.2 Hamiltonians ............................... 14 3.2.1 Density matrix representation .................. 16 3.2.2 The rotating frame ........................ 17 3.2.3 Radio frequency pulses ...................... 17 3.2.4 Spherical tensor operators .................... 18 3.3 Relaxation phenomena .......................... 18 3.4 2H NMR experiments in the solid state ................. 19 3.4.1 Solid echo ............................. 20 3.4.2 Spin-lattice and spin-spin relaxation measurements . . . . . . 22 3.4.3 2D NMR ............................. 23 3.4.4 Stimulated echo .......................... 24 3.4.5 2D NMR in the frequency domain ............... 28 3.5 Development of a low temperature double resonance probe . . . . . . 31 4 Random walk simulations 35 4.1 Introduction ................................ 35 4.2 Isotropic motion ............................. 36 4.2.1 2D NMR in time domain .................... 37 4.2.2 Solid echo line shape ....................... 41 4.2.3 Heterogeneous dynamics ..................... 43 4.3 Anisotropic reorientation ......................... 46 4.3.1 Random jump type motion on a cone .............. 46 4.3.2 Random jump type motion within a cone ........... 49 4.3.3 Multi step motion within a cone ................ 52 4.3.4 Distributions of geometries and correlation times ....... 56 4.4 Summary and conclusions ........................ 57 I Plastic Crystals 59 5 Introduction 61 ii Contents 6 Cyanocyclohexane 65 6.1 Introduction ................................ 65 6.1.1 Chemical and physical properties ................ 66 6.1.2 Dielectric loss spectra ...................... 68 6.2 Structure elucidation ........................... 70 6.3 Experimental results – overview ..................... 72 6.3.1 Experimental details ....................... 72 6.3.2 Spin-lattice relaxation ...................... 73 6.3.3 Solid echo spectra ........................ 75 6.3.4 Solid echo spectra: low temperature regime .......... 78 6.3.5 Solid echo spectra: fast motion limit effects .......... 80 6.3.6 Stimulated echoes T <Tg..................... 83 6.3.7 2D exchange spectra of the α-process .............. 85 6.3.8 Stimulated echoes T >Tg..................... 87 6.3.9 Preliminary conclusions ..................... 91 6.4 Experimental results – refinement .................... 92 6.4.1 Modelling spin-lattice relaxation ................ 92 6.4.2 Modelling solid echo spectra ................... 95 6.4.3 The tp-dependence of solid echo spectra T >Tg......... 105 6.4.4 Dynamics of the α-process – reassessment ........... 107 6.5 Summary and conclusions ........................ 112 II Binary Glass Forming Systems 115 7 Introduction 117 7.1 Digest ................................... 117 7.2 Spectroscopic amendments ........................ 121 8 Toluene in PCB 123 8.1 Introduction ................................ 123 8.2 Dielectric Spectroscopy .......................... 124 8.3 Experimental details ........................... 127 8.4 Thermal analysis ............................. 128 8.5 NMR results – an overview ....................... 130 8.5.1 Spin lattice relaxation ...................... 130 8.5.2 Solid echo spectra – high temperature regime ......... 135 8.5.3 2D exchange spectra ....................... 139 8.5.4 Stimulated echoes – high temperature regime ......... 141 8.5.5 Solid echo line shape – low temperature regime ........ 147 8.5.6 Stimulated echoes – low temperature regime .......... 152 8.5.7 Spin lattice relaxation – revisited ................ 155 8.6 Plausibility check ............................. 160 8.6.1 Modelling the toluene dynamics in mixtures with PCB . . . . 161 8.6.2 Models for two dynamically distinct toluene sub-ensembles . . 165 8.6.3 Discussion ............................. 171 Contents iii 9 Universal behaviour of toluene in binary mixtures 173 9.1 Motivation ................................. 173 9.2 Toluene in picoline ............................ 173 9.2.1 Thermal analysis ......................... 174 9.2.2 Dielectric measurements ..................... 175 9.2.3 NMR results ........................... 177 9.3 Toluene in polystyrene .......................... 181 9.3.1 NMR results ........................... 184 9.4 A model for the concentration dependence ............... 185 9.4.1 Application ............................ 186 10 Discussion and outlook 189 III Summary, Appendix & Bibliography 191 11 Summary 193 12 Summary - German translation 195 A Additional results: cyanocyclohexane 199 A.1 Dielectric spectroscopy .......................... 199 A.2 Spin-lattice relaxation .......................... 200 A.3 Spin-spin relaxation times ........................ 202 A.4 Correlation of time scale and geometry for the β-process . . . . . . . 202 B Additional results: binary glass forming systems 205 B.1 Dielectric spectroscopy .......................... 205 B.2 Methyl group dynamics ......................... 207 B.3 Spin-spin relaxation times ........................ 208 B.4 T1weighted stimulated echo experiments ................ 208 C Details on the random walk simulations 211 C.1 Creation of trajectories .......................... 211 C.2 Calculation of the NMR signal ..................... 212 C.3 Phenomenological model functions ................... 213 C.3.1 Distribution functions for the α-process ............. 213 C.3.2 Distribution functions for the β-process ............. 214 Bibliography 217 List of figures 229 Chapter 1 Introduction At temperatures below the melting point of a liquid, T<Tm , the crystal represents the thermodynamically stable phase. For sufficiently large cooling rates, however, the formation of a crystalline phase can be avoided in practically all systems: the molecules do not achieve spatial order and a supercooled liquid is obtained. At the glass transition temperature TgTm the characteristic time of structural relaxation in the supercooled liquid finally becomes longer than the time scale of a typical experiment – the system falls out of ergodicity. Hence the glassy state distinguishes itself from the crystalline one by a lack of translational symmetry; the structure of the liquid is virtually maintained. The glass transition problem is exceptional in the sense that it features a long standing research history, but remains unsolved to the present day. In recent years the activity in the field is accelerated by the recognition of the fundamental nature of the problem on the one hand, and the importance of supercooled liquids and glasses in everyday life on the other hand. From traditional soda-lime-silica glasses, used in windows and other applications for centuries, to polymer-plasticizer systems, i.e. plastics, which revolutionized consumer goods in the last decades, amorphous substances play a major role in modern life. Yet there exists no commonly accepted theory that describes all features regarded as intrinsic to the phenomenon in the full temperature range from the simple liquid to the glassy state. The outstanding features of the glass transition phenomenon – the viscosity changes by many orders of magnitude in a narrow temperature range – have drawn the attention of many experimentalists and various techniques have been developed to examine the properties of glass forming substances during the last century. Due to the pronounced thermodynamic effects, the changes regarding microscopic dynamics in conjunction with the glass transition have sometimes taken a back seat. The dynamics of the α -process, which causes all correlations within a liquid to disappear and hence governs the structural relaxation, is well described by a diffusive motion in the liquid state, TTm , but becomes more complex in the supercooled liquid and particularly of cooperative nature: glass transition arises from many body effects. In addition faster secondary processes emerge at T>Tg , which display strikingly universal features in vastly different glass forming substances. The main focus of this work is placed on one of these processes: the Johari-Goldstein β -process. Johari and Goldstein discovered that also molecules without internal degrees of freedom show a secondary relaxation that exhibits thermally activated behaviour. Due to this – in contrast to the α -process – low temperature dependence, the β -process dominates the relaxation behaviour in the glassy state. The authors attributed this process to “islands of mobility”, i.e. to a fraction of molecules that maintains enhanced mobility even below Tg . In recent years it was however demonstrated that all molecules 2 Chapter 1. Introduction contribute to the β -process, which consequently represents a restricted local motion. The degree of restriction is rather pronounced at temperatures below Tg , but becomes significantly reduced at higher temperatures before ultimately the time scales of α - and β -process merge in most glass forming substances. Consequently it is of great interest to study the β -process at temperatures above Tg , since the effects are typically rather subtle below the glass transition temperature. For most substances the applicable temperature range is however quite narrow due to a merging with the α-process. In addition to supercooled liquids, a glass transition can also be observed in plastically crystalline phases. Plastic crystals exhibit translational symmetry and orientational disorder: the molecules reorient around their point of gravity on the respective position in the lattice. The orientational degrees of freedom in such a system can be supercooled and subsequently undergo a glass transition that exhibits some or all characteristics of the glass transition observed in structural glass formers, including the arise of secondary relaxations. Consequently plastic crystals are often regarded as “model systems” for the glass transition due to the reduced complexity of molecular dynamics. Apart from neat glass formers also binary systems, i.e. mixtures of two substances with at least one being a “good” glass former, are of industrial as well as theoretical interest. Binary mixtures serve as a model for the widely used polymer-plasticiser systems. Furthermore simulations and mode-coupling theory calculations for mixtures composed of small and large hard-spheres yield a glass transition of the smaller spheres within the rigid matrix of the larger ones. In mixtures of molecular liquids the dynamics of the respective species are however still a matter of debate. The arise of a single Tg for example often served as a criterion for miscibility, recent studies however reported the emergence of two glass transitions even in fully miscible systems, demonstrating a decoupling of time scales between the components. Yet it remains unclear if this decoupling holds for all sub-ensembles or if a fraction of molecules participates in the dynamics of the other component – i.e. a bimodal relaxation pattern arises. Due to the pronounced dynamic heterogeneities inherent to binary mixtures, i.e. the distributions of correlation times become very broad compared to the neat systems, it may often not be feasible to discriminate between the two scenarios in a typical experiment. The present work attempts to provide further insight on the nature of the glass transition with special attention on the Johari-Goldstein β -process. The focus hereby lies on nuclear magnetic resonance (NMR) studies, as the technique provides the ability to elucidate the detailed mechanism of molecular reorientation. The employed 2H NMR technique furthermore has the advantages of selectivity, i.e. the site or molecule of interest can be marked via selective deuteration, and simplicity in data analysis, as the predominant quadrupolar interaction is of effective single-particle type. The selectivity of the method is of significance especially with regard to studies of binary systems, as the components can be monitored individually by 2H NMR. 3 Scope and structure of the present work After the basic features of the glass transition phenomenon and the employed 2H NMR technique are introduced in chapters 2and 3, chapter 4is devoted to random walk simulations of 2H NMR experiments for selected modes of reorientation. Due to the complex nature of dynamics in supercooled liquids and glasses this becomes necessary, as the 2H NMR results can often not be interpreted in straightforward manner and have to be discussed in the framework of random walk simulations to obtain a deeper understanding of molecular motion. This especially holds for the β -process in general and the α -process in binary mixtures, as the inherent pronounced dynamic heterogeneities conflict with the available 2H NMR time window and render a detailed simulation approach a necessity. In part Iof the experimental section we address the dynamics of cyanocyclohexane in the plastically crystalline phase. Cyanocyclohexane exhibits a pronounced β -process that can be studied in a relatively broad temperature regime above Tg (as it does not merge with the α -process) and yields distinct effects in the 2H NMR observables, previously not reported for glass forming systems. This allows us to extend present hypothesis on the microscopic nature of the β -process and gain further insight on the dynamics of the α -process, as the processes are well separated within the 2H NMR time window, which is typically not the case. In part II we selectively address the dynamics of toluene in mixtures with a polychlorinated biphenyl (PCB54), which exhibits a higher glass transition temperature, i.e. toluene serves as a plasticizer. Toluene is a well studied glass former, consequently a 2H NMR assessment of the concentration dependent change in dynamics is facilitated as comprehensive literature data regarding the neat system is at hand. In particular we address the question of possible bimodal relaxation behaviour and attempt to clarify the role of the Johari-Goldstein β -process for the complex dynamics in binary mixtures. If the β -process represents an intrinsic phenomenon of the glass transition and serves as precursor for the structural relaxation – as often speculated – a distinct concentration dependence is anticipated as the structural relaxation of toluene is slowed in the mixtures, even more so if dynamically different toluene sub-ensembles arise. Although refuted in the case of neat glass formers, where the contribution of all molecules to the β -process has been confirmed, a reintroduction of the “islands of mobility” concept may not be argued for binary systems. 10 Chapter 2. The glass transition phenomenon Figure 2.6: Relative relaxation strength 1-S =∆ / ∆ β of the β -process in cyanocyclohexane [ Tschirwitz 2002a ] and toluene [ Benkhof 1999 ]. Whereas the absolute strengths are different, the qualitative temperature dependence with respect to Tg is comparable. 0.01 0.1 1 0.8 1 1.2 1.4 1.6 1-S Tg / T cyanocyclohexane toluene distribution [Kudlik 1997a,Kudlik 1998]: G(log τβ) = 1 2πσ2exp   −log τβ−log τm β2 2σ2  ,(2.6) with a temperature dependent width σ[Kudlik 1998]: ¯σ(T) = σ ln10 ·1 T−1 Tδ.(2.7) The temperature dependence of the peak maximum exhibits thermally activated behaviour and the rule Ea≃ 24 ·Tg holds for many substances [ Kudlik 1999 , Blochowicz 2004 , Ngai 2004a ]. The relaxation strength of the β -process in dielectric spectroscopy is typically quantified via the fraction 1S = ∆ β/ ∆  of the total relaxation strength: 1S is small and virtually constant for temperatures T<Tg , but rapidly grows above Tg until around typically τβ(T)≈ 10 −6s [ Hansen 1997 ] the time scales of α - and β -process merge and only a single peak is observed in the dielectric spectra at higher temperatures. Whereas the temperature dependence of 1S is rather universal, the absolute strength varies among different systems, cf. figure 2.6. Johari and Goldstein introduced the “islands of mobility” concept [ Johari 1976 ] to explain the arise of a β -peak in the dielectric spectrum: it was speculated that certain groups of molecules remain an enhanced mobility with respect to their surroundings even in the glassy state. Williams and Watts on the other hand postulated that all molecules contribute to the β -process, which yields a small correlation loss for all sub-ensembles before any remaining correlation is destroyed at much longer times by the α-process [Williams 1971]. By means of dielectric spectroscopy, solvation [ Wagner 1998 ] and NMR [ Vogel 2001a , Vogel 2001b , Vogel 2002 ] techniques it was demonstrated that the latter assumption holds, also in case of molecules without internal degrees of freedom, i.e. all molecules contribute to the Johari-Goldstein β -process – the “islands of mobility” concept was refuted. On the basis of the latter NMR experiments (an overview is presented in [ Vogel 2005 ]) it was furthermore demonstrated that the β -process is a restricted, i.e. anisotropic, multi-step motion. For toluene a model was proposed wherein the motional process is restricted to the circumference of a cone with half opening angle 2.3. The β-process 11 χ 0 0.1 0.2 0.3 0 10 20 30 G ( χ ) χ / ° (a) 0 10 20 30 40 50 0 60 120 180 240 300 360 t / ms φ / ° (b) Figure 2.7 left: Proposed model of a reorientation on the circumference of a cone for the β -process. (a): Distribution of opening angles from an adaptation of the model to 2H NMR results of toluene [ Vogel 2000a ]. (b): Sketch of the proposed multi step motion for the β-process, φdenotes the orientation on the circumference [Vogel 2000a]. χ , cf. figure 2.7. By means of a distribution G(χ) , which is centred around 4° but incorporates also a small fraction of larger angles (figure 2.7 (a)), and a complex multi-step reorientation by which the circumference is explored (cf. figure 2.7 (b)), all 2H NMR observables for T<Tg in toluene were successfully reproduced. The model has furthermore been applied to different glass formers where comparable results in terms of G(χ)were obtained [Vogel 2005]. 2.3.1 Open questions Whereas the described model has proven successful in reproducing the 2H NMR results obtained in several systems, the proposed scenario certainly is ambiguous and purely phenomenological. A straightforward connection to the energy landscape of the glass, which could serve to explain the universal features of the β -process, is lacking. Numerous questions regarding the nature of the β -process remain and are currently a matter of debate. First of all the description of the β -process as an “intrinsic property of the glassy state” according to Johari and Goldstein apparently conflicts with the afore introduced classification of type-A/B glass formers. Whereas the excess wing in type-A systems was argued by some to represent an inseparable feature of the α -process [ Dixon 1990 , Leheny 1997 ], other authors suggest that the excess wing is nothing but the high frequency flank of a β -process hidden underneath the α -peak [ Hensel-Bielowka 2002 , Ngai 2004b , Schneider 2000 ]. Blochowicz et al. [ Blochowicz 2004 ] demonstrated that the excess wing in 2-picoline is gradually transformed into a β -peak upon a slowing down of the α -process in binary mixtures with tri-styrene, an effect which was also reported for other type-A systems [ Shahin Thayyilab 2008 ]. Furthermore the excess wing in glycerol was demonstrated to proceed via small angular displacements [ Gainaru 2008 ], i.e. the large angular reorientations, which account for the better part of correlation loss due to the α -process, are not observed in the respective frequency range. Yet certain glass formers exhibit both, an excess wing and a resolved β -peak, 12 Chapter 2. The glass transition phenomenon in their dielectric spectra, challenging the interpretation of the first in terms of a submerged representation of the latter. For glass formers with a distinct β -peak in the dielectric spectrum the interdependence of the latter and the α -process is currently also a matter of debate: as mentioned before, the rule Ea≃ 24 ·Tg holds for many substances. 2D 2H NMR experiments demonstrated that a positive correlation between τα and τβ within the respective distributions exists [ Böhmer 2006 ], i.e. molecules with a momentarily fast β -process also exhibit a relatively fast α -process and vice versa. Furthermore the entropy and volume dependencies of the processes (i.e. under pressure and temperature variation) suggest a strong correlation [ Prevosto 2009 ], which is anticipated in the coupling model introduced by K.L. Ngai [Ngai 1979]. As the relaxation strength of the β -process strongly grows above Tg and since all molecules participate already at T<Tg , i.e. the effect can not be rationalised by an increasing number of contributing sub-ensembles, it was argued that the β -process governs relaxation at temperatures above the merging of α - and β -process and the former becomes extinct [ Garwe 1996 ]. The random first-order transition (RFOT) theory [ Lubchenko 2007 ] of glasses also predicts that the secondary process becomes the dominant mode of structural relaxation at high temperatures [ Stevenson 2010 ], yet no agreement has been reached regarding the dominance of either process above the merging temperature. The fast and non-merging β -process in cyanocyclohexane yields the possibility to monitor the β -process at relatively high temperatures by means of 2H NMR in this work and hence quantify the relaxation strength, hence the substance represents a hopeful candidate regarding the elucidation of the high temperature relaxation behaviour. The β -process was assumed to be of non-cooperative nature by Johari and Goldstein [ Johari 1976 ], a perception which was recently challenged by theory [ Stevenson 2010 ]. If the process indeed exhibits a certain degree of cooperativeness, a characteristic concentration dependence is expected to arise in binary mixtures of glass forming substances as the local environment of a molecule is gradually altered. The study of toluene in binary mixtures, presented in part II of this work allows us – amongst other things – to investigate a possible cooperativeness. Chapter 3 2H NMR spectroscopy Contents 3.1 The nuclear spin and its interactions .............. 13 3.2 Hamiltonians ............................ 14 3.2.1 Density matrix representation .................. 16 3.2.2 The rotating frame ......................... 17 3.2.3 Radio frequency pulses ....................... 17 3.2.4 Spherical tensor operators .................... 18 3.3 Relaxation phenomena ...................... 18 3.4 2H NMR experiments in the solid state ............ 19 3.4.1 Solid echo ............................. 20 3.4.2 Spin-lattice and spin-spin relaxation measurements ...... 22 3.4.3 2D NMR ............................. 23 3.4.4 Stimulated echo ........................... 24 3.4.5 2D NMR in the frequency domain ............... 28 3.5 Development of a low temperature double resonance probe 31 In the following we will briefly review the principles and methods of 2H NMR spectroscopy as employed in the experimental part of this work. Unless noted otherwise, the according references for this overview are to be found in standard NMR literature [ Abragam 1973 , Mehring 1983 , Slichter 1990 ], the outline of this chapter in particular follows the presentation given by Schmidt-Rohr and Spiess [Schmidt-Rohr 1994]. 3.1 The nuclear spin and its interactions Nuclei with a spin I 6 =0 exhibit interactions with external magnetic fields and internal magnetic and electric fields. In NMR the external fields consist of a static magnetic field ~ B0 (w.l.o.g. aligned along the z -axis) and an alternating magnetic field ~ B1 with frequencies typically in the MHz regime. The internal fields are governed by the local electronic structure in the sample and allow for a determination of order and dynamics. Nuclei with a non-vanishing spin possess a magnetic dipole moment ~µ =γ~ I, (3.1) 14 Chapter 3. 2H NMR spectroscopy with the characteristic gyromagnetic ratio γ . In zero field the eigenvalues of the spin I are degenerated, if an external field ~ B0 is applied, the levels split up (Zeeman splitting) and the magnetic moment precedes with the frequency ω0=−γB0.(3.2) The characteristic Larmor frequency ω0 hence depends on the type of nuclei and different nuclei can be probed selectively in a resonance experiment. In thermal equilibrium the magnetization is aligned parallel to the ~ B0 field according to Boltzmann distribution. In this work we only consider Fourier transform (FT-) NMR experiments, in which the spin system is modified via short radio frequency pulses. If such a radio frequency pulse is applied to the probe coil under resonance condition ωRF = ω0 , a static field ~ B1 in the frame of the magnetization is created and magnetization is deflected from the z -direction to the x, y -plane for a pulse of suitable strength and length, a so-called 90° -pulse. In the x, y -plane the preceding magnetization yields an alternating magnetic flux which induces a voltage in the probe coil – the free induction decay (FID), which represents the NMR signal. Due to the influence of local fields the resonance frequency of nuclei i differs from ω0 : the distribution of charge carriers, the dipole-dipole interaction of the nuclei etc. yield ω6 = ω0 : the NMR spectrum ω−ω0 is obtained. After the equilibrium magnetization is perturbed by an RF-pulse, the spin system relaxes via a coupling to the so-called lattice, i.e. via statistic fluctuations of the local fields that reflect the dynamics of the sample. In the following we will present a theoretical treatment of the conducted experiments, the extensiveness of which will be kept to a minimum. For a more detailed discussion we refer again to standard NMR literature cited above. 3.2 Hamiltonians Due to the strength of the external magnetic field ~ B0 ( 7.05 T for the experiments conducted in this work), the Zeeman interaction is dominant in NMR. The corresponding Hamiltonian reads1: ˆ HZ=−γ~B0ˆ Iz(3.3) In the following we will use the Hamiltonians in terms of frequency units, i.e. ˆ Hω = ˆ H/~ . As this represents the common notation in NMR literature, we will drop the index ωand equation 3.3 reads: ˆ HZ=−γB0ˆ Iz=ω0ˆ Iz,(3.4) with the Larmor frequency ω0 . In FT-NMR the coupling to the alternating ~ B1 field acts only during the short duration of an RF pulse (typically on the order of 2 to 3µs for a 90°pulse), and is in case of an x-pulse e.g. given via: ˆ HRF,x =−γB1ˆ Ix.(3.5) 1A separate notation for vector operators is not used in this work. 3.2. Hamiltonians 15 As the deuterium nuclei considered in this work exhibit spin I = 1, they posses a quadrupole moment Q which interacts with the electric field gradient (EFG) at the site of the nucleus, which results from the charge distribution in the respective chemical bond(s). The EFG tensor V is defined via the second spatial derivatives of the electric potential Φ: Vj αβ =∂2Φ ∂rα∂rβ ;α, β ∈x, y, z (3.6) For a single spin this interaction is described via the corresponding quadrupolar Hamiltonian ˆ HQ: ˆ HQ=eQ 2I(2I−1) ~ˆ IV ˆ I, (3.7) with the elementary charge e . As the Zeeman interaction dominates for the ~ B0 fields commonly used in NMR, the internal interactions can be treated in first order perturbation theory. Therefore only the secular part of ˆ HQ has to be considered, i.e. the part that commutes with ˆ Iz and hence ˆ HZ . This so-called truncated part of the quadrupolar Hamiltonian reads: ˆ HQ=eQ 2I(2I−1) ~Vzz 1 23ˆ Izˆ Iz−ˆ Iˆ I.(3.8) The remainder of treatment is facilitated by a transition from the laboratory frame (LF) to the principal axis system (PAS) of the EFG tensor: VLF zz =1 23 cos2θ−1−ηsin2θcos (2φ)VPAS zz ,(3.9) with the polar angles θ, φ of the external ~ B0 field in the PAS of the tensor. As the tensor has diagonal form in the PAS, it is defined via two principal values: VPAS zz and the asymmetry parameter η: η=VPAS yy −VPAS xx VPAS zz .(3.10) With VPAS zz =e·qequation 3.8 reads: ˆ HQ=eQeq 2I(2I−1) ~ 1 43 cos2θ−1−ηsin2θcos (2φ)3ˆ Izˆ Iz−ˆ Iˆ I.(3.11) For I=1 and ∆ m± 1the corresponding transitions are in 2H NMR (cf. figure 3.1) observed at the frequency: ω(θ, φ) = E0±EQ ~=ω0±ωQ(θ, φ) =ω0±δ 23 cos2θ−1−ηsin2θcos (2φ),(3.12) with the anisotropy parameter δ: δ=3 4 eQeq ~.(3.13) 16 Chapter 3. 2H NMR spectroscopy m=-1 m=0 m=1 ħωL ħωL ħ(ωL+ωQ) ħ(ωLωQ) HZ+HQ HZHZ+HQ+HDD ωL-ωQ ωL+ωQ ωL ωL-ωQ ωL+ωQ B0=0 B0=0 Figure 3.1: Energy level diagram of a spin I =1. Under the influence of a magnetic field and the quadrupolar coupling the otherwise degenerated energy levels split up. The additional (relatively weak) dipolar coupling yields in the ensemble average a broadening of the observed lines. For all deuterated compounds in this work the aliphatic C2 H bonds in first approximation have an axially symmetric charge density, hence the EFG tensor is also axially symmetric in good approximation. In this case the asymmetry parameter η vanishes and the quadrupolar frequency does not dependent on the polar angle φ: ωQ=δ 23 cos2θ−1.(3.14) As for η = 0 the principal value VPAS zz of the EFG tensor is parallel to the direction of the C2 H bond, ωQ observed in the experiment directly yields the angle θ between ~ B0and the respective C-2H bond. 3.2.1 Density matrix representation As the spin systems under consideration are usually not fully polarized, a simple treatment via the Schrödinger equation is often not possible. Instead a convenient description in terms of statistical quantum mechanics via the density matrix representation is employed. The density operator ˆρ describes the state of the spin system (including any incoherent superposition of spin states) and the Hamiltonians represent the internal and external interactions. Analogous to the Schrödinger equation, the operators ˆρ and the total Hamiltonian ˆ H of the spin system are connected via the von Neumann equation: d dt ˆρ(t) = −ihˆ H(t),ˆρ(t)i(3.15) In this formalism the physical observables are obtained from the trace of the density operator: hAi=Tr (ˆρA)(3.16) For a time-independent Hamiltonian ˆ H , a formal solution of the von Neumann equation is given by: ˆρ(t) = e−iˆ Ht ˆρ(0) eiˆ Ht,(3.17) 3.2. Hamiltonians 17 with the propagator ˆ U(t) = e−iˆ Ht.(3.18) In thermal equilibrium the description in terms of the density operator ˆρ is analogous to the classical Boltzmann distribution: ˆρeq =1 Ze−ˆ H/kT Z=Tr e−ˆ H/kT (3.19) As for temperatures above 1K and common ~ B0 field strengths |~γB0/kT|  1holds, an expansion of the exponential operator in equation 3.19 can be truncated after the linear term. This so called high temperature approximation yields: ˆρeq ∝ˆ 1 + ~γB0 kT ˆ Iz.(3.20) Within this framework the evolution of the density matrix under the influence of ˆ H can in principal be calculated, for the consideration of RF-pulses it is however convenient to employ a description in the rotating frame. 3.2.2 The rotating frame For the Hamiltonian ˆ HZ of the Zeeman interaction the time evolution according to equation 3.17 reads: ˆρ(t) = e−iω0tˆ Izˆ Iαeiω0tˆ Iz,(3.21) i.e. any magnetization ˆ Iα is rotated about the z -axis with an angle ω0t . Via the transformation to a rotating frame with ωr=ω0, this precession is eliminated: ˆρr(t) = e−iωrtˆ Izˆρ(t)eiωrtˆ Iz=e−i(ωr−ω0)tˆ Izˆ Iαei(ωr−ω0)tˆ Iz.(3.22) If we now consider the Hamiltonian ˆ H = ˆ HZ + ˆ HQ of the Zeeman and the quadrupolar interaction for example in the rotating frame, the von Neumann equation reads: d dt ˆρ(t)r=−ihˆ HQ,r (t),ˆρ(t)Ri,(3.23) i.e. ˆ HZ does no longer appear. As the experimental NMR frequency is also obtained in terms of the rotating frame, the remainder of this section will be presented in this notation and the index rwill be dropped. 3.2.3 Radio frequency pulses Of the alternating ~ B1 field acting during the duration of radio frequency pulses, only the components perpendicular to the ~ B0field are of interest: BLF 1(t)=2B1cos (ωRF t+φ).(3.24) This produces two rotating fields: one with + ωRF and one with −ωRF . For an RF pulse in resonance, ω0 = ωRF , the field rotating with −ωRF can be neglected in first 18 Chapter 3. 2H NMR spectroscopy approximation. Furthermore the alternating ~ B1 field is transformed to a static one in the rotating frame under this condition (ωr=ω0=ωRF ). In most experiments 90° pulses are employed, i.e. the magnetization is turned by 90° with respect to a certain axis: π 2=γB1tP.(3.25) If we adopt the ’left handed’ convention (as common in NMR literature), the propagator of an +xpulse reads: ˆ Px(t) = eiˆ HP,xt=eiγB1tˆ Ix,(3.26) i.e. +zmagnetization is rotated to the +yaxis. 3.2.4 Spherical tensor operators To circumvent the explicit solution of the von Neumann equation, the symmetry of the system can be exploited in the expansion of the Hamiltonians in terms of spherical tensor operators according to the Wigner-Eckart theorem: ˆ Hλ=cλ 2 X l=0 l X m=−l (−1)mˆ Tλ l,m ˆ Rλ l,−m,(3.27) with the irreducible spin operators ˆ Tλ l,m and the spatial part ˆ Rλ l,−m . The evolution of these operators under the influence of RF-pulses and the quadrupolar coupling is tabulated [ Schmidt-Rohr 1994 ] and can conveniently be employed in the description of the pulse sequences in the remainder of this chapter – without explicitly solving the von Neumann equation. 3.3 Relaxation phenomena The introduced density matrix formalism does not account for relaxation phenomenons, which drive the system to thermal equilibrium. Relaxation is mediated via weak couplings of the spin to the lattice (spin-lattice relaxation T1 ), which provides means of energy transfer and and restores magnetization parallel to ~ B0 , and to other spins (spin-spin relaxation T2 ), which leads to a correlation loss of the transversal magnetization, i.e. no energy transfer is involved. Both components can be described as coupling to a fluctuating local field introduced by e.g. molecular motion in terms of perturbation theory. The spectral density J(ω) connects the correlation function of the molecular motion to the 2H NMR relaxation processes: J(ω) = 1 2 ∞ Z −∞ F(t)eiωtdt. (3.28) For an isotropic motion and an axial symmetric EFG tensor ( η = 0) the spin-lattice relaxation T1is then given via [Abragam 1973]: 1 T1 =2 15δ2[J(ω0)+4J(2ω0)] .(3.29) 3.4. 2H NMR experiments in the solid state 19 The transversal spin-spin relaxation T2reads under these conditions: 1 T2 =1 15δ2[3J(0) + 5J(ω0)+2J(2ω0)] .(3.30) For a simple dynamic process where F2is exponential (i.e. for a spectral density of Debye type) equation 3.29 (and eq. 3.30 accordingly) can be transformed to the so-called BPP equation [Bloembergen 1948]: 1 T1 =2 15δ22τ 1+(ω0τ)2+8τ 1 + (2ω0τ)2.(3.31) In supercooled liquids and glasses the motion is typically characterized by a distribution of correlation times G(log τ) . In the presence of such dynamic heterogeneities equation 3.31 is no longer applicable, instead the average rate h1/T1i has to be expressed in terms of the distribution after Resing et al. [Resing 1965]: 1 T1= ∞ Z −∞ 1 T1(τ)G(log τ)dlog τ, (3.32) where T1(τ) denotes equation 3.31. In this case the corresponding magnetization curve Φ (t) = Mz,∞−Mz(t) Mz,∞ (3.33) is non-exponential and h1/T1iis obtained from the initial decay: 1 T1= lim t→0 ∂Φ (t) ∂t .(3.34) In the experimental part of this work also an alternative approach will be employed to describe non-exponential magnetization curves. Hereby Φ (t) is fitted via a Kohlrausch function: Φ (t) = Mz,∞exp −t T1βK!,(3.35) where the mean spin-lattice relaxation time is then obtained via: hT1i=T1 βK Γ1 βK.(3.36) 3.4 2H NMR experiments in the solid state In the following we present the main pulse sequences used throughout this work. The effects of the radio frequency pulses on the density matrix are thereby given in the formalism of the spherical tensor operators introduced in section 3.2.4. As before, the discussion is restricted to the peculiar features of 2H NMR experiments on supercooled liquids and in the solid, respectively glassy state. 26 Chapter 3. 2H NMR spectroscopy pulse lengths in the experiment, a comparison with equation 3.49 allows to obtain the latter quantity. Furthermore the oscillations in F (te, t0=te;tm→0) can be used to accurately determine the coupling constant δ. For sufficiently long mixing times tm on the other hand, each C2 H bond probes all orientations accessible to the respective motional process. For full isotropic reorientation the conditional probability P(ωQ1, ωQ2;tm) is equal to the a-priori probability P0(ωQ1), and in this limit equation 3.48 reads (figure 3.6 (b)): Fcc te, t0=te;tm→ ∞=1 4   π/2 Z 0 cos (2ω(θ1)te) sin (θ1)dθ1   2 Fss te, t0=te;tm→ ∞=1 4   π/2 Z 0 sin (2ω(θ1)te) sin (θ1)dθ1   2 .(3.50) In the remainder we will use the symbol F∞ for F (te, t0=te;tm→ ∞) . F∞ consequently yields information on the geometry of the molecular motion, i.e. the number of accessible sites n for a C2 H bond. If the process exhibits cubic point symmetry (as does the isotropic motion discussed above) always Fss ∞(te→0) = 0 is obtained [ Fujara 1986 ]. For longest evolution times on the other hand the number of accessible sites is reflected: Fss ∞(te→ ∞) = 1 n,(3.51) i.e. F∞ reflects the fraction of accessible sites that do not yield correlation loss: the initial correlation2. Model free correlation function For sufficiently short evolution times te 1 /δ an expansion of the sine in Fss can be truncated after the linear term: Fss te→0, t0=te;tm=3 4hsin (ωQ1te) sin (ωQ2te)i ∝ hωQ1ωQ2it2 e ∝ hP2(cos θ1)P2(cos θ2)it2 e ∝F2(tm).(3.52) In this limit Fss correlates the frequencies, and hence via the second Legendre polynomial the orientations, during evolution and detection. The obtained correlation function is proportional to the one-particle two-time correlation function F2 , which can be compared to results from other methods in a model free approach. In the experiment, the amplitude of the stimulated echo is typically fitted via a 2If the point group of the model contains the inversion operation, Fss ∞(te→ ∞) = 2 nholds. 3.4. 2H NMR experiments in the solid state 27 two-step decay function: Fi te(tm) = A0"1−Fi ∞,teexp −tm τteβK,te!+Fi ∞,te#exp −tm T1/1QβK,T1/1Q!, (3.53) where i stands for cc or ss. The first Kohlrausch function models the decay from A0 to Fi ∞,te due to correlation loss via molecular motion. The second one describes the signal decay due to longitudinal relaxation during tm : T1 in case of Fcc and T1Q for Fss. The employed Kohlrausch function in many cases is an appropriate description of the correlation functions observed in disordered systems, the mean relaxation time is given via: hτtei=τte βK,te Γ1 βK,te.(3.54) The reorientation angle distribution Whereas Fss yields the correlation function F2 in the limit te→ 0, the stimulated echo technique becomes increasingly sensitive on the elementary step of the motional process for prolonged evolution times. Hence the te -dependece of the stimulated echo allows to determine the geometry of motion, i.e. to discriminate between a random-jump like motion and rotational diffusion for example. A detailed assessment of the te -dependence is most expedient in the framework of random walk simulations, nevertheless we will present a brief introduction to the theoretical treatment of limiting cases. For corresponding random walk simulations and examples see chapter 4. When addressing more complex modes of reorientation equation 3.48 can be separated into two parts: Fcc te, t0=te;tm= π Z 0 Kcc (te, β)R(β;tm)dβ, (3.55) where the integral kernel Kcc represents a geometrical quantity that is independent of any reorientation: Kcc (te, β) = π Z 0 π Z 0 cos (ω(θ1)te) cos (ω(θ2)te)P0(θ1)P(θ1;θ2|β)dθ1dθ2.(3.56) For disordered systems Kcc is determined by the statistical distribution of C2 H bonds with the a-priori probability P0(θ)∝sin θ and the conditional probability P(θ1;θ2|β) to find a pair of angles θ1, θ2 with the intermediate angle β , i.e. Kcc simply reflects the geometry of the powder. The reorientation angle distribution R(β;tm) on the other hand contains information on the dynamical process: R(β;tm) yields the probability that any C2 H bond with initial orientation θ1 at t = 0 has reoriented by the angle β at t = tm . Due to the symmetry in 2H NMR the integration in equation 3.55 can be limited to 0 −π 2 as 28 Chapter 3. 2H NMR spectroscopy 0° 30° 60° 90° -2 -1 0 1 β log(tm/τc) 0° 30° 60° 90° -2 -1 0 1 β log(tm/τc) 0° 30° 60° 90° -2 -1 0 1 β log(tm/τc) Figure 3.7: Distribution of reorientation angles R(β;tm) for the isotropic models of random jump (top left), rotational diffusion (top right) and a γ = 20° jump (bottom). Higher intensities around β = 0° are cut for reasons of clarity. ωQ(θ) = ωQ(π−θ). R (β;tm) is obtained in straightforward manner for simple models of reorientation (cf. figure 3.7) and hence the te -dependence of the stimulated echo is conveniently accessible via equation 3.55. 3.4.5 2D NMR in the frequency domain Instead of analysing only the amplitude at t0 = te + tp in the pulse sequences of the stimulated echo, it is also feasible to Fourier transform the resulting FID and obtain a 2D spectrum. The 2D spectrum yields direct information on the frequencies ωQ1 and ωQ2 before and after the mixing time tm . Consequently the spectral intensity maps if and how the individual molecules have reoriented during tm. As an FID corresponding to te = 0 is required for the 2D Fourier transform, the four pulse sequence of the stimulated echo is typically extended by an echo pulse between 3.4. 2H NMR experiments in the solid state 29 the first and second pulse: ZE: π 2x−tp1−π 2y−tp1+te−π 2x−tm−π 2x−tp2−π 2y−t0 SA: π 2x−tp1−π 2y−tp1+te−π 4y−tm−π 4y−tp2−π 2y−t0. (3.57) To obtain a 2D spectrum, the mixing time tm is held constant and the evolution time te is varied typically with the same stepping as employed in the detection period t0 , hence equidistant points are sampled in either dimension (in NMR this sampling period is often referred to as “dwell time”). A two-dimensional Fourier transform of the signal does – in contrast to the solid echo sequence – not yield a purely absorptive spectrum, hence the so-called States method is employed: after the first Fourier transform the imaginary part in the Zeeman and the real part in the Spin-Alignment sequence is set to zero. After the Fourier transform in the second dimension the obtained spectra are purely absorptive: the Zeeman part is symmetric with respect to ω1 = ω2 = 0 and the Alignment part antisymmetric. The sum of both parts reduces the ambiguity and the 2D 2H NMR spectrum is obtained. To obtain the same amplitude from both sequences, typically the length of the second and third pulse is set to 54.7° : the unwanted spin states created by this deviation are eliminated via appropriate phase cycling. This procedure furthermore brings the merit that both sequences are equal with respect to the times tp and te , as the effective pulse distance depends on the finite pulse lengths. Limiting case spectra The 2D 2H NMR spectrum yields the probability to find the frequency ω2 after tm if it was ω1 at t = 0. Consequently the spectral intensity is again given as product of a conditional and an a-priori probability: P(ω1, ω2;tm) = P(ω2;tm|ω1)P0(ω1).(3.58) If no reorientation takes place during tm the conditional probability P(ω2;tm|ω1) equals the delta function δ(ω2−ω1) and hence the 1D (Pake) spectrum is observed along the main diagonal, cf. figure 3.9 (a). If the molecules on the other hand reorient isotropically during tm , i.e. the orientation after the mixing time is independent of the initial orientation, equation 3.58 reads: P(ω1, ω2;tm) = P0(ω1)P0(ω2),(3.59) and the corresponding spectrum covers the full accessible frequency range, cf. figure 3.9 (b). In-between these limiting cases the evolution of the 2D spectrum under the influence of dynamics can again be expressed in terms of the reorientation angle distribution R(β;tm): S(ω1, ω2;tm) = π/2 Z 0 R(β;tm)S(ω1, ω2|β)dβ, (3.60) 30 Chapter 3. 2H NMR spectroscopy (a) no reorientation (b) full isotropic reorientation Figure 3.8: Simulated 2D 2H NMR spectra for the limiting cases of a static sample and for full isotropic reorientation during tm. where S(ω1, ω2|β) is again a purely geometrical term and R(β;tm) the same as in case of the stimulated echo, i.e. contains the dynamics of the system. Reorientation during evolution and detection As in the case of the solid echo discussed above, also in 2D experiments the effects of molecular motion become more complex and ambiguous if reorientation takes place during evolution and detection time. The resulting 2D line shapes can typically only be addressed by means of random walk simulations in this limit. If a 2D spectrum is however recorded in a system that is characterized by a distribution of correlation times which extends to τ 1 /δ and τ 1 /δ , i.e. for which a two phase spectrum is observed in the solid echo experiment (cf. section 3.4.1), exchange within this distribution G(log τ)during tmcan be monitored. For molecules which are fast during the evolution period (i.e. contribute to a Lorentzian line) and slow during detection (i.e. contribute to a Pake spectrum) or vice versa, characteristic cross peaks arise in the spectrum (figure 3.9 (a)). The spectral intensity is given via (SLrepresents a Lorentzian): Sx(ω1, ω2) = 1 4[(P0(ω1) + P0(−ω1)) SL(ω2) + SL(ω1) (P0(ω2) + P0(−ω2))] . (3.61) A typical spectrum recorded under these conditions yields (depending on the mixing time) a static fraction of molecules contributing to a Pake pattern along the main diagonal, a fraction of molecules which are fast during evolution and detection which yields a central Lorentzian line, exchange or cross-peaks between those fractions according to equation 3.61 and a characteristic reorientation pattern resulting from molecules which are static during te and t0 , but reorient during tm . An example for such a spectral pattern is sketched in figure 3.9 (b). 3.5. Development of a low temperature double resonance probe 31 (a) exchange spectrum (b) “multi phase” spectrum Figure 3.9: Simulated 2D 2H NMR spectra for exchange processes via relatively fast and slow reorienting molecules. 3.5 Development of a low temperature double resonance probe In the course of this work a low-temperature 31 P/ 1 H double resonance probe was designed and built in collaboration with Bruker BioSpin GmbH 3 . Due to the lack of commercially available double resonance probes for the temperature regime T≥4K , a new probe was designed on the basis of the HP.LTSTAT single resonance probe from Bruker BioSpin GmbH and a probe built by I. Roggatz in our lab [ Roggatz 2000 ]. The probe was manufactured in part by Bruker and the University of Bayreuth. The purpose of this probe was to extend the available setup in our laboratory with respect to 31 P NMR studies of glass forming substances and supercooled liquids at low and lowest temperatures, while maintaining the possibility of proton decoupling in the experiments. The previously available (wide bore) double resonance probes are cooled via a stream of cold nitrogen gas, which is only feasible down to approximately 160 K and provides less thermal stability than a typical cryostat setup. The design of the probe was adapted to an available CF 1200 cryostat from Oxford Instruments, which fits inside the shim systems of typical wide bore NMR magnets (i.e. in a 89 mm bore) and offers an inner diameter of 49 mm . The employed NMR magnet provides a field strength of 9.4T , i.e. the resonance frequency of 1 H nuclei equals 400 MHz and 161.9MHz in case of 31P. For the purpose of recording 31 P solid state spectra at low temperatures, the probe has to fulfil certain criteria which are in part contradictory [Roggatz 2000]: • The resonance circuits should generally exhibit a low temperature dependence. Frequency tuning and impedance matching on both channels must be adjustable from outside the cryostat. 3 The exciting and fruitful collaboration with Dr. E. Naumann of Bruker BioSpin is greatly acknowledged. 32 Chapter 3. 2H NMR spectroscopy X H tuning X matching X H lock probe coil tuning H matching H λ 1/2 switch (a) circuit diagram 50 100 150 200 250 300 350 -20 0 20 T / K ν / kHz (b) test measurements Figure 3.10 Left: Circuit diagram of the 31 P/ 1 H double resonance probe developed within this work. The dotted parallel lines mark coaxial cables. Right: Test measurements of the described setup with a sample of 19 % (by weight) trimethyl phosphate in polystyrene. Hahn-echo pulse sequence, 16 scans, with 1 H-decoupling: solid line, without: dashed line. The intersection with the y-axis marks the temperature. • The quality factor of the resonant circuit must be sufficiently low to record broad solid state spectra (typically about 60 kHz) without distortion. • The quality factor must be sufficiently high to assure adequate 90° pulse lengths (below 3µs) and signal-to-noise ratio during recording. • The resonant circuit must withstand pulsed RF power on the order of 1kW and continuous excitation with hundreds of Watts without RF flash-overs in the regions of high field strength. The basic design of the probe is as such that all components of the resonant circuits are located inside of the cryostat. This yields higher reliability and a better signalto-noise ratio, especially at low temperatures. Hence all electric components are incorporated into a frame made of three glass fibre reinforced plastic rods, that were copper plated and soldered to cylindrical brass plates to support the components, cf. figure 3.11. This design provides low heat conductance and a low thermal expansion coefficient, which is crucial for the application at lowest temperatures. The circuit diagram is sketched in figure 3.10 (a): the 1 H part of the resonant circuit consists of two adjustable capacitors, one for frequency tuning and the other for impedance matching to 50 Ω . Both capacitors are adjustable from outside the cryostat by means of axles made from glass fibre reinforced plastics and a vacuumtight duct at the bottom of the probe (cf. figure 3.11). During adjustment the upper 3.5. Development of a low temperature double resonance probe 33 X tuning temp. X matching X tuning CX X matching X H H tuning to probe coil H matching terminal 4x sample Figure 3.11: Sketch of the basic design of the 31 P/ 1 H double resonance probe developed within this work. For details of the wiring see figure 3.10 (a). part of the capacitor body remains fixed, which is essential as both capacitors are coupled inductively via the alignment of this part. The rigid λ/ 2wire connecting the 1 H tuning capacitor to the probe coil provides a λ/ 4switch, which allows for 13 C operation on the X-channel ( ω0 = 100.58 MHz ). The solenoid probe coil made of a copper/silver flat wire is supported by PTFE framework which contains the 5mm NMR tube. The X-channel ( 31 P/ 13 C) consists of an adjustable capacitor for frequency tuning and an inductance for impedance matching, which is adjustable by means of a spindle to bypass the appropriate amount of turns. The connection to the probe coil includes a low-pass filter which efficiently shields this path of the circuit from the proton frequency. The temperature in the area of the NMR sample is monitored by a Cernox ™ resistance temperature sensor from Lake Shore Cryotronics Inc. The sensor is read out via a four-wire measurements, which allows for wiring with low heat conductance. By means of the employed sensor the temperature can be monitored accurately 4 in the region 1.4K to 325 K , due to the four-wire readout however the sensor has to be disconnected during NMR measurements, as it introduces spikes in the NMR signal. Test measurements with the described probe down to approximately 20 K showed stable and convenient operation of the setup. RF flash-overs of the proton channel in helium atmosphere could be efficiently reduced by a slight modification of the PTFE dielectric employed in the tuning and matching capacitors. As the present work focuses on 2H NMR , the probe was however not employed in 4 We however detected a slight temperature offset between the sample and the location of the sensor, which is currently still under evaluation. 34 Chapter 3. 2H NMR spectroscopy the remainder of this thesis – for further results obtained with the described setup see [Bock 2011,Hoff 2010,Pötzschner 2011]. Chapter 4 Random walk simulations Contents 4.1 Introduction ............................ 35 4.2 Isotropic motion .......................... 36 4.2.1 2D NMR in time domain ..................... 37 4.2.2 Solid echo line shape ........................ 41 4.2.3 Heterogeneous dynamics ..................... 43 4.3 Anisotropic reorientation ..................... 46 4.3.1 Random jump type motion on a cone ............. 46 4.3.2 Random jump type motion within a cone ........... 49 4.3.3 Multi step motion within a cone ................ 52 4.3.4 Distributions of geometries and correlation times ....... 56 4.4 Summary and conclusions .................... 57 The raison d’être of the present chapter is to discuss a number of limiting cases for (molecular) motion typically associated with disordered systems from a principal point of view on the one hand, and to analyse their respective imprint on 2H NMR observables on the other. We will therefore expand the discussion of the last chapter in this regard with special attention towards the the solid echo line shape and the stimulated echo decay. The models presented in the following are not to be regarded as realistic or physically plausible for the systems studied in this work – rather they present an arbitrary subset of scenarios, which distinguish themselves by their simplicity (from a mathematical point of view) and serve to raise awareness for the analysis of experimental results presented in the following chapters. The reader familiar with 2H NMR experiments regarding dynamics in supercooled liquids and glasses may skip ahead to the experimental section in chapter 5and eventually return to certain sections of the present chapter when referred to. 4.1 Introduction The angular resolution of solid state 2H NMR is a great vantage of the method, but on the other hand accounts for the challenge of correct data interpretation – with various, often ambiguous and subtle effects. 2D NMR methods were often employed to study simple motional processes with fixed geometries like π -flips, methyl group rotations etc., whereas isotropic dynamics were typically modelled via simple modes of reorientation like rotational diffusion [ Pschorn 1991 ], as the evolution of R(β;tm) within these models can be readily calculated and an adaptation to the experimental 42 Chapter 4. Random walk simulations -7 -6 -5 -4 -3 -2 -100 0 100 log( τc / s ) ν / kHz rand. jump τj ≈ 0.3µs 10ms -100 0 100 ν / kHz γ = 35° 0.15µs 5ms -100 0 100 ν / kHz 10° 15ns 0.5ms -100 0 100 ν / kHz 3° 1.3ns 40µs -100 0 100 ν / kHz 2° + 40° 0.5ns 18µs Figure 4.6: Random walk simulations of the solid echo line shape according to different models of motion in the intermediate motional regime, τ≈ 1 /δ . The position of the baseline with respect to the y-axis denotes the respective correlation time τc , in all cases the inter pulse delay was set to 20 µs . Rightmost spectra comprise from a model where 5% of 40° -jumps are added to a 2°γ -jump motion. The dotted spectrum represents the limiting Pake pattern and is included in all models for reasons of comparison. nature of those changes is different amongst the employed models: for all γ -jump scenarios the intensity in the centre of the spectra decreases, whereas in the random jump type model a slight increase is observed. Line shape parameters As the calculated spectra for short τc significantly differ from a rigid limit Pake spectrum, we will introduce phenomenological line shape parameters to quantify the change in spectral intensity in the style of S. Lusceac [ Lusceac 2005a ]. The latter change in central intensity is monitored via the quantity R(tp) that measures the relative intensity around zero frequency, cf. figure 4.7 (b). The angular sensitivity of the solid echo spectrum is largest around ω = 0, i.e. a small angular displacement imposes the most prominent effect here. Corresponding R(tp) values to the spectra presented in figure 4.6 are plotted in figure 4.8 (a): motions with γ≤10° exhibit a prominent minimum in R(tp) around τc≈ 7 · 10 −5 s, whereas models proceeding via large angular jumps yield constant R(tp) in this region. The behaviour of R(tp) in the model comprised of a mixture of 2° and 40° jumps is dominated by the small angular displacements. 4.2. Isotropic motion 43 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 RSE ( τc ) log( τc / s ) (a) rj 10° 3° 2° and 40° h1 h2 h3 R=2h3(h1+h2)-1 C80 (b) Figure 4.7 (a): Reduction factor RSE of the solid echo spectra presented in figure 4.6 ( tp = 20 µs ). (b): Definition of the line shape parameters R(tp) and Cx : R(tp) measures the relative intensity at zero frequency. Due to slight distortions sometimes arising in experimental spectra, the average height of the singularities is used. Cx measures the apparent spectral with at height x, e.g. at 80 % for C80. 0 0.2 0.4 0.6 -4 -3 -2 R( τc, 20µs ) log( τc / s ) (a) rj 35° 10° 3° 2+40° 110 120 130 140 -4 -3 -2 Cx / kHz log( τc / s ) (b) C50 C80 C100 Figure 4.8 (a): Relative intensity at zero frequency R(tp) of the solid echo spectra presented in figure 4.6. (b): Apparent spectral width Cx of the simulated solid echo spectra presented figure 4.6. The line shape effects are however not limited to the centre of the spectra. Therefore we introduce the apparent spectral width Cx , which measures the width of the spectra at relative height x , i.e. at 80 % in case of C80 (cf. figure 4.7 (b)). Cx of the simulated spectra is presented in figure 4.8 (b): the reduction in Cx of the γ = 10° , 3° and the 2° + 40° models is rather universal. The spectra calculated for the random and the γ = 35° jump model on the other hand maintain their width C80 down to shorter correlation times τc and exhibit an increase of C50 for τc< 1 · 10 −3 s. A broadening of the spectra in the lower half is hence a fingerprint of a motion dominated by large angular jumps. 4.2.3 Heterogeneous dynamics So far we have only considered motional processes for a single correlation time τc , i.e. the effects of a possible distribution G(log τ) have been neglected. With regard to the study of binary glass formers in part II of this work, we will briefly review 44 Chapter 4. Random walk simulations -8 -7 -6 -5 -4 -3 -2 -100 0 100 log( τm / s ) ν / kHz σ=1 log-Gaussian ideal heterogeneous -100 0 100 ν / kHz σ=1.5 -100 0 100 ν / kHz σ=2 -100 0 100 ν / kHz σ=3 -100 0 100 ν / kHz σ=4 Figure 4.9: Random walk simulations of the solid echo line shape according to a random jump type motion with an underlying (ideal heterogeneous) log-Gaussian distribution of correlation times. The position of the baseline with respect to the y-axis denotes τm (the median of the distribution) for each spectrum. Simulations for various widths σ are displayed, in all cases the inter pulse delay tpwas set to 20 µs. the effects of a broad distribution G(log τ) , as typically present in these systems, on the solid echo line shape. Therefore we assume a random jump type motion with an underlying ideal heterogeneous log-Gaussian distribution of correlation times, i.e. no exchange within the distribution is considered. With regard to the reduction factor presented for the case of a single correlation time in figure 4.7 (a), it becomes obvious that a superposition of a liquid line and a solid state spectrum is expected if the distribution spans more than a decade in τc . This behaviour is demonstrated in figure 4.9: for a log-Gaussian distribution with σ = 1 such two phase spectra are observed in a very narrow range for τm . In case of σ = 4 pronounced two phase spectra are observed for about 4.5 decades in τm. We determined the relative weight W of the liquid line in the individual spectra via a fit with a superposition of a Lorentzian and a rigid-limit Pake pattern, the resulting W ( τ )is plotted in figure 4.10 (b). As this weighting factor is often exploited in the analysis of experimental data, we will compare the results of our simulation with a theoretical prediction in terms of G(log τ) . Conventionally the liquid line is attributed to molecules with a correlation time faster than the inverse of the coupling 4.2. Isotropic motion 45 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 -1 RSE ( τ ) log( τ / s ) (a) σ = 0 1 1.5 2 3 4 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 W ( τ ) log( τ / s ) (b) σ = 1.5 2 3 4 Figure 4.10 (a): Reduction factor RSE of the solid echo spectra presented in figure 4.9 ( tp = 20 µs ). (b): Weighting factor of the Lorentzian line from a simple “two phase” fit to the spectra in figure 4.9. The lines represent the weighting calculated via equation 4.7 (dotted line) and 4.8 (full line). constant τc<1/δ: W(τm) = log 1/δ Z −∞ G(log τ)dlog τ, (4.7) which is represented by the dotted line in figure 4.10 (b). With regard to the simulation results the predicted weighting factor from equation 4.7 is shifted towards longer times τm and significantly over-estimates the regime in which two phase spectra are observed. The deviations arise because the reduction factor RSE is neglected in eq. 4.7: if we account for the fraction of molecules not observed in the experiment due to RSE ≈ 0, significantly better agreement can be achieved. In figure 4.10 (a) it is seen that molecules with a correlation time on the order of τ =0.513.5µs in first approximation do not contribute to the solid echo intensity (if relatively faster or slower fractions with RSE ≈ 1are present). Consequently equation 4.7 is modified to account for this fraction: W(τm) = 1 N log 0.5µs Z −∞ G(log τ)dlog τ(4.8) with the normalization factor N= ∞ Z log 13.5µs G(log τ)dlog τ+ log 0.5µs Z −∞ G(log τ)dlog τ. (4.9) The results are given by the solid lines in figure 4.10 (b), which are in agreement with W( τ ) obtained from the simulated spectra. We will exploit this result in the analysis of the two phase spectra in mixtures of toluene and PCB54 in part II of this work. 46 Chapter 4. Random walk simulations Figure 4.11: Sketch of a random jump type motion on the circumference of a cone with half opening angle χ . The (initial) direction of the C2 H bond is marked by the dashed line. After each jump a random position on the circumference of the base circle is occupied. � � 1 2 3 4 4.3 Anisotropic reorientation The remainder of this chapter is devoted to studies of anisotropic dynamics and their imprint on 2H NMR experiments, which will serve as foundation for a more advanced discussion of the β-process in the experimental parts of this work. Although 2H NMR studies of the β -process, for example in ethanol, were reported to be sensitive on the placement of the “active” bond (i.e. via selective deuteration) [ Schneider 2001 ], the majority of glass forming systems exhibit a remarkably universal manifestation of the β -process in 2H NMR – despite vast differences in molecular size and shape [ Micko 2007 ]. Therefore it does not appear justified to derive jump geometries from molecular symmetry arguments, hence we begin the discussion with the most general forms of restriction, i.e. for the rotational motion probed by 2H NMR the confinement of the C2 H vector to a certain area on the unit sphere, in the most simplest case the circumference of a cone. The impact of the latter cone models on oneand two-dimensional 2H NMR experiments has been intensively discussed by M. Vogel [ Vogel 2000b , Vogel 2000a , Vogel 2005 ] and S. Lusceac [ Lusceac 2005a ] in the framework of random walk simulations, which have been successfully applied to the experimental results of numerous systems. Nevertheless it is necessary to refine and broaden the analysis with respect to the fast and non-merging β -process found in cyanocyclohexane and the binary mixtures of toluene presented in this work. For the sake of clarity we reproduced some of the previously reported results 3 , hence the following sections cater to provide a comprehensive overview. 4.3.1 Random jump type motion on a cone To demonstrate the basic effects common to all the models based on a conical restriction, we begin this section with the simplest thereof: a random jump type motion on the circumference of a cone with a fixed jump time τj . Subsequently we will focus on more complicated (in terms of geometry and computational effort) but more general and hence plausible models for the β -process in supercooled liquids and glasses. The model proposed in this section is sketched in figure 4.11: after each individual jump a random position on the circumference of a cone with half opening angle χ is occupied, hence (the χ dependent fraction of) correlation is completely lost after each step (the assumptions made in all calculations are the same as in the 3 The simulation routine developed within this work was shown to provide consistent results with previously published data. 4.3. Anisotropic reorientation 47 -100 0 100 ν / kHz τ=14ms ( τc=τj ) -100 0 100 ν / kHz τ=0.8ms -100 0 100 ν / kHz τ=0.4ms -100 0 100 ν / kHz τ=0.2ms -100 0 100 ν / kHz τ=2.6µs -100 0 100 ν / kHz τ=0.6µs tp=10µs 20µs 50µs 200µs -100 0 100 ν / kHz τ=0.3µs -100 0 100 ν / kHz τ=0.07µs -100 0 100 ν / kHz τ=0.01µs Figure 4.12: Simulated solid echo spectra for a random jump motion on the circumference of a cone with half opening angle χ = 7° . For each correlation (i.e. jump-) time the spectra corresponding to tp =10, 20, 50 and 200 µs (top to bottom) are plotted. The calculated spectra have been convoluted with a Gaussian and damped with respect to pulse excitation effects to resemble the experimental spectra in part I and II, this routine was kept constant throughout this work. 3·106trajectories were simulated in the present case. 48 Chapter 4. Random walk simulations 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 -1 R( tp ) / R0 log( τ / s ) (a) 0.98 0.99 1 1.01 -8 -7 -6 -5 -4 -3 -2 -1 C80( tp ) / C0 log( τ / s ) (b) tp = 10µs 20µs 50µs 200µs Figure 4.13 (a): Normalized intensity around zero frequency R(tp) / R0 of the spectra displayed in figure 4.12 ( χ = 7° , including additional data) for tp =10200 µs (key see figure (b)). The dashed line marks τ = 1 /δ . (b): Corresponding normalized apparent spectral width C80 . The dashed line marks the fast motion limit plateau predicted by equation 4.10. previous sections, i.e. Ivanov model, Markov process and δ = 129.6kHz4 ). In virtue of this property the correlation functions are exponential and τj = τc holds, which significantly reduces simulation effort with respect to the more complex scenarios propagating via small angular displacements. As the resulting correlation functions F1/2 are hence trivial apart from F∞ (which may not be the case in the experimental stimulated echo due to time window effects), the considerations will be limited to the solid echo line shape in the present case, which is displayed in figure 4.12 for selected τj and a cone of half opening angle χ = 7° at different inter pulse delays tp . The line shape changes predicted in chapter 3are rather articulate: for jump correlation times on the order of 10 −6 s the intensity in the central part of the spectrum decreases, an effect which can be intensified by prolonging the inter pulse delay tp . The spectra for τj = 0.01 µs and different inter pulse delays tp are again indistinguishable and resemble the slow motion limit case observed for τj≥14 ms – albeit with a slightly reduced apparent coupling constant ¯ δ . For all models in which the reorientation is restricted to the circumference of a cone (i.e. independent of the elementary jump) the reduced apparent coupling is given via: ¯ δ=δ1 23 cos2(χ)−1.(4.10) As the changes in apparent width are hardly observed in this representation, the line shape parameters R(tp) and C80 are displayed in figure 4.13 for the sake of a more quantitative discussion. The apparent spectral width Cx is used as the coupling constant ¯ δ is often not directly accessible – as being the case experimentally, where distributions in τ and χ and hence distorted line shapes are found, which do not resemble a Pake pattern. The spectral intensity around zero frequency R(tp) vs. log (τ) displays a (symmetrical) minimum around τj =1 /δ ≈8µs which is about 3.5 ( tp = 10 µs ) to 5.5 ( tp = 200 µs ) decades wide. The position of the minimum itself exhibits little tp -dependence, with the minimum condition τj =1 /δ being best fulfilled for tp = 20 µs . As with further reduction of the jump time the motion enters the 4Data treatment is sketched in the caption to figure 4.12. 4.3. Anisotropic reorientation 49 �1 2 3 4 �j (a) motion within a cone 0 0.5 1 0 30 60 90 δ / δ χ / ° motion on a cone motion within a cone (b) reduced apparent coupling Figure 4.14 (a): Sketch of a random jump type motion within a cone of half opening angle χ . The (initial) direction of the C2 H bond is marked by the dashed line. After each jump a random position within (or on) the base circle is occupied. (b): Normalized reduced apparent spectral widths according to equations 4.10 and 4.11. fast motion limit regime and consequently the R(tp) effects vanish, the apparent spectral width drops about 2% , approaching the theoretical value given via equation 4.10 at τj≈ 0 . 1 /δ for all simulated inter pulse delays tp . Whereas C80 decreases monotonically for short tp ( 10 µs and 20 µs ), a local maximum is traversed for tp≥50 µs . This effect is also visible in the spectra displayed in figure 4.12, as the slope of the inner singularities becomes steeper for the spectra with lowest intensity around ν=0Hz. 4.3.2 Random jump type motion within a cone Albeit the results already mimic the experimental low temperature findings in systems with a β -process quite well, the “motion on a cone” model however appears rather artificial: a fast motion of on the circumference of a cone would demand for a certain molecular (or in case of plastic crystals probably lattice) symmetry which accounts for such dynamics. Naively one would rather expect to find some sort of diffusive motion, limited by constrains imposed by the local energy landscape in the heterogeneous glass. A behaviour which is better mimicked by the model of a motion within a cone, which will also be considered as of random jump type in a first step. If a C2 H bond freely moves on the unit sphere within a cone of half opening angle χ, a similar relation for ¯ δas in case of equation 4.10 is found [Batchelder 1983]: ¯ δ=δ1 2cos (χ) [1 + cos (χ)] .(4.11) The resulting apparent coupling ¯ δ/δ is plotted in figure 4.14 (b): as, on average, the reorientation angle of a single C2 H bond is smaller than in the case of a motion on the circumference of a cone with the same opening angle χ , the effects are more subtle in the present case – the qualitative behaviour is however the same in both models for χ < 54.7°. 50 Chapter 4. Random walk simulations 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 -1 R( tp ) / R0 log( τ / s ) (a) 0.99 1 1.01 -8 -7 -6 -5 -4 -3 -2 -1 C80( tp ) / C0 log( τ / s ) (b) tp = 10µs 20µs 50µs 200µs Figure 4.15 (a): Normalized intensity around zero frequency of simulated spectra for a motion within a cone ( χ = 7° , tp =10200 µs , key see figure (b)). The dashed line marks τ = 1 /δ . (b): Corresponding normalized apparent spectral width C80 . The dashed line marks the fast motion limit plateau given via equation 4.11. This trend is reflected in all extracted quantities (cf. figure 4.15 for R(tp) , where the same opening angle χ = 7° as in figure 4.13 was chosen): the minimum in the tp = 20 µs trace appears again at τj≈ 1 /δ , the minimum value is however significantly higher within this approach (approx. 0.55 opposed to about 0.3 for a motion “on the cone”). The reduction in C80 is in good agreement with the theoretical prediction of eq. 4.11 and about half the magnitude as in the previous case. Again a maximum in C80 is traversed for longest tp values, this particularity being however more pronounced in the present scenario than for the “motion on a cone” model: at tp = 200 µs the apparent spectral width around τj =10 −4 s has grown roughly 1% and therefore exceeds the effect found in the previous model, albeit the overall changes in C80 (i.e. from the static to the fast motion limit case) are smaller in the present case. Even though the centre of the 2H NMR solid state spectrum is most sensitive to the small angular displacements considered here, it was already seen in figure 4.12 that the changes are not limited to this frequency range: the outer ridges of the spectrum are also clearly affected for intermediate jump times – which explains the peculiarity arising in C80 in this region. To monitor the overall loss in intensity, the reduction factor RSE is displayed in figure 4.17: the curves resemble the dependence of R(tp) on τj : the position and width of the minimum is almost identical to the one arising in R(tp) and the difference between the employed models is also qualitatively similar. This however follows naturally as both quantities are closely related and in the present case probe mainly the same spectral changes. Apart from the subtle differences between the two models discussed so far, which furthermore only become relevant when extracting quantitative information from the experimental solid echo line shape, there arises a clear qualitative difference for vanishing restriction, i.e. growing opening angle χ between the models. Although the models were chosen to mimic a highly restricted motion (like the β -process below Tg ), it is worth discussing what predictions can be made for the case of vanishing restriction, as it is known (e.g. from dielectric spectroscopy) that the amplitude of the β -process universally grows above the glass transition temperature, albeit the observation in the line shape is in most systems hampered by the onset 4.3. Anisotropic reorientation 51 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 -1 R( tp ) / R0 log( τ / s ) (a) RO tp = 20µs 7° 14° 28° 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 -1 R( tp ) / R0 log( τ / s ) (b) RW 0.6 0.7 0.8 0.9 1 -8 -7 -6 -5 -4 -3 -2 -1 C80( tp ) / C0 log( τ / s ) (c) RO 0.8 0.9 1 -8 -7 -6 -5 -4 -3 -2 -1 C80( tp ) / C0 log( τ / s ) (d) RW Figure 4.16 (a): R(tp) values simulated in the “random jump type motion on a cone” model for different opening angles χ , tp = 20 µs in all cases. (b): R(tp) values for a simulation of the “random jump type motion within a cone”, otherwise the same parameters as in (a) were used. (c): Corresponding apparent spectral widths C80 for the data displayed in (a), same symbols denote same cone angles. (d): Corresponding C80 values for figure (b). 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 -1 RSE( tp = 20µs ) log( τ / s ) (a) RO 7° 14° 28° 0 0.2 0.4 0.6 0.8 1 -8 -7 -6 -5 -4 -3 -2 -1 RSE( tp = 20µs ) log( τ / s ) (b) RW Figure 4.17 (a): Reduction factor for the “random jump type motion on a cone” model for different opening angles χ and tp = 20 µs . (b): Reduction factor for the “random jump type motion within a cone” model for different χ, otherwise same parameters as in (a). 58 Chapter 4. Random walk simulations minimum for tp = 20 µs provides an independent measure of the correlation time and hence the ability to extract τβ from 2H NMR solid echo measurements. This holds independently of the elementary step considered and is also valid in the presence of a distribution of correlation times G(log τβ) . For a motion proceeding via small angles γ the line shape changes for longer values of tp become visible at relatively longer times τc . The fast motion limit regime is however unaltered - hence a shift of the R(tp) minimum with tp represents an indicator for small angular displacements. The fast motion limit plateau value in C80 directly reflects the degree of restriction, i.e. the cone opening angle in the present models. For previously studied glass forming systems the observation of this plateau value was in all cases obstructed by first line shape changes due to the α -process at T>Tg . The fast and non-merging β -process of cyanocyclohexane studied in part Iis however a hopeful candidate to quantify the relaxation strength of the β -process above Tg directly by means of 2H NMR. Part I Plastic Crystals Chapter 5 Introduction Apart from various works regarding the glass transition phenomenon in molecular glass formers, polymers and ionic liquids that have been published in recent years, the literature also contains numerous studies focusing on supercooled plastic crystals. These systems are of special interest, as the complexity of dynamics and therefore the effort needed in complex modelling to interpret typical bulk experiments is reduced, but yet they display some or all features regarded as intrinsic to the glass transition phenomenon. Most plastically crystalline systems consist of molecules with almost globular shape [ Sherwood 1979 ] and form a cubic lattice, the crystallization to which is characterized by a relatively low entropy of fusion [ Timmermans 1961 ]. The molecules are hence centred on fixed lattice points, their orientation however maintains random to at least some degree. Consequently the systems exhibit translational order but orientational disorder, the degrees of freedom are reduced to three. Interestingly the orientational degrees of freedom can be supercooled in many of these substances and subsequently undergo a calorimetric glass transition which resembles the features known from structural glass formers: a non-exponential main ( α -) relaxation following a nonArrhenius temperature dependence [ Brand 2002 , Tschirwitz 2002a ] and for some systems in addition a thermally activated secondary relaxation: the Johari-Goldstein β -process [ Singh 1982 , Ramos 1996 , Bonjour 1981 ]. Concerning the dynamics of the α -process it is noteworthy that the crystalline packing in virtually all systems does not allow for a free rotation of a single molecule around its center of gravity: the tumbling motion is hindered via the constrains imposed by neighbouring molecules on the lattice [ Sherwood 1979 ]. Hence any isotropic motion in plastically crystalline systems – if present – has to be of cooperative nature. crystalplastic crystalliquid Figure 5.1: Sketch of the phases accessible to a plastic crystalline system below the boiling point, adapted from N. Petzold [Petzold 2008]. 62 Chapter 5. Introduction -100 0 100 ν / kHz 29.3 K 74.5 K 94.2 K 112.5 K 159.6 K (a) (b) Figure 5.2 (a): Solid echo spectra of cyanoadamantane ( tp = 20 µs ): at lowest temperatures the rotation of the molecule around its C 3 axis is slow with respect to the 2H NMR time scale, hence a broad solid state spectrum is observed. At higher temperatures a superposition of the latter line shape and a spectrum with reduced apparent coupling constant ¯ δ due to a fraction of fast rotating molecules is obtained [ Roggatz 2000 ]. (b): 2D exchange spectrum of cyanoadamantane recorded in the supercooled plastic crystalline phase: T = 245 K , tm = 3ms . The 90° -ridges reflecting a jump process according to the cubic symmetry of the lattice are well seen [Lusceac 2005a]. Although promising for numerous reasons, NMR studies – especially employing two-dimensional techniques – are still quite rare [ Böhmer 2001 , Winterlich 2003 ]. As a prominent example we present selected results from the plastic crystalline phase of cyanoadamantane, that were previously obtained in our group [ Roggatz 2000 , Lusceac 2005a ]. The cyanoadamantane molecule is of almost globular shape and exhibits a C 3 symmetry axis around which the molecule rotates even at relatively low temperatures, as the motion requires negligible free volume. This behaviour is demonstrated via the solid echo spectra presented in figure 5.2 (a): at lowest temperatures the rotation of the molecule around its C 3 axis is slow with respect to the 2H NMR time scale, hence a broad solid state spectrum is observed. At higher temperatures a superposition of the latter line shape and a spectrum with reduced apparent coupling constant ¯ δ due to a fraction of fast rotating molecules is obtained. The fact that those characteristic “two phase” spectra appear over a quite large temperature range indicates a broad distribution of correlation times for the rotation of cyanoadamantane, reflecting the heterogeneous structure of the glass. With respect to the remainder of dynamics observed, the system represents an archetype concerning (lattice-) symmetry induced dynamics: the cubic symmetry of the lattice is directly reflected in the elementary jump of the α -process: two dimensional 2H NMR experiments in time and frequency domain clearly exhibit the characteristics of a 90° jump. This is well seen in figure 5.2 (b), as the 2D exchange spectrum (which exhibits the described “two phase” spectrum along the main diagonal at the given temperature) shows characteristic ridges typical for a γ = 90° jump motion. This finding is supported by the stimulated echo measurements, 63 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 F∞ τe / µs (a) 235 K 240 K simulation 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 τ( te ) / τ( 3µs) τe / µs (b) 235 K 240 K Figure 5.3 (a): Residual amplitude of the Fss te stimulated echo decay in cyanoadamantane at two temperatures [ Lusceac 2005a ]. (b): Corresponding normalized correlation times τc [Lusceac 2005a]. as the extracted correlations times τ ( te ) become slightly longer with te (cf. figure 5.3 (b)), as expected for large angular jumps ( γ > 54.7° ). Furthermore the residual amplitude F∞ saturates around a value of 1 3 (figure 5.3 (a)), reflecting the cubic symmetry of the motion (cf. chapter 3). Whereas similar dynamics reflecting the structure and symmetry of the lattice were observed in other plastically crystalline systems, this description does not hold up to the melting point. For various systems it was observed – by means of different techniques – that at relatively high temperatures TgT<Tm a crossover exists between the symmetry induced motion found in the deeply supercooled plastic crystal and isotropic rotational diffusion as observed in the liquid T>Tm . This behaviour was reported for example by Bée et al. from a incoherent quasielastic neutron scattering study on chloroadamantane [ Bee 1983 ], which crystallizes in the same space group as cyanodamantane (Fm3m) but exhibits considerably faster dynamics due to the smaller substituent. These results are supported by Affouard et al., who found a dynamical transition via proton spin-lattice relaxation and DSC measurements on the same system [ Affouard 2000 ]. The transition temperature was reported in a region where the correlation times of the α -process are on the order of 10 −12 − 10 −9 s, i.e. inaccessible to multidimensional NMR methods as employed for cyanodamantane at relatively lower temperatures. Due to the fast dynamics this region is however suitable for molecular dynamics (MD) simulations (as opposed to the deeply supercooled and glassy regime) and here also a dynamic crossover was observed: MD simulations of the plastic crystals norbornene [ Affouard 2003 ] and chloroadamantane [ Affouard 2000 ] demonstrated that at low temperatures the system exhibit a restricted “cage rattling” motion within the constrains imposed by the neighbouring molecules (attributed to the β -process of mode-coupling theory by the authors) accompanied by a scarce large angular tumbling motion from one lattice direction to another (cf. figure 5.4 (b)). In consequence two-step correlation functions are observed in the low temperature regime; the fraction τ1/τ2 – readily accessible in the simulations – is close to one. At higher temperatures the dynamics change in the sense that the symmetry constrains of the lattice diminish and the observed motion is comparable to the model of 64 Chapter 5. Introduction −20 −10 0 10 20 −30 −20 −10 0 10 20 30 y / Å x / Å T = 280K (a) −20 −10 0 10 20 −30 −20 −10 0 10 20 30 y / Å x / Å T = 125K (b) Figure 5.4: Molecular dynamics simulation of plastically crystalline norbornene: snapshots of the molecular orientations projected on the x,y-plane at different instants for two temperatures. Figure from [Affouard 2003]. isotropic rotational diffusion with τ1/τ2≈3(cf. figure 5.4 (a)). Cyanoadamantane, for which the prominent 2H NMR results were presented, is a type-A glass former as it displays no discernible β -process in dielectric spectroscopy. Apart from the previously outlined advantages of plastically crystalline systems with regard to studies of glassy dynamics, it appears therefore promising to compare the results of cyanoadamantane to a plastically crystalline system with a pronounced β -process. Even more so as the relaxation strength of the latter process is known to grow above Tg and presumably can be studied in a rather broad temperature range here due to the typically strong nature of the α -process (i.e. low fragility m ) in plastic crystals – yet 2H NMR studies of the β -process in plastic crystals are sparse. Chapter 6 Cyanocyclohexane - dynamics in a plastic crystal Contents 6.1 Introduction ............................ 65 6.1.1 Chemical and physical properties ................ 66 6.1.2 Dielectric loss spectra ...................... 68 6.2 Structure elucidation ....................... 70 6.3 Experimental results – overview ................ 72 6.3.1 Experimental details ....................... 72 6.3.2 Spin-lattice relaxation ...................... 73 6.3.3 Solid echo spectra ........................ 75 6.3.4 Solid echo spectra: low temperature regime .......... 78 6.3.5 Solid echo spectra: fast motion limit effects .......... 80 6.3.6 Stimulated echoes T<Tg..................... 83 6.3.7 2D exchange spectra of the α-process .............. 85 6.3.8 Stimulated echoes T>Tg...................... 87 6.3.9 Preliminary conclusions ...................... 91 6.4 Experimental results – refinement ............... 92 6.4.1 Modelling spin-lattice relaxation ................ 92 6.4.2 Modelling solid echo spectra ................... 95 6.4.3 The tp-dependence of solid echo spectra T>Tg........ 105 6.4.4 Dynamics of the α-process – reassessment ............107 6.5 Summary and conclusions .................... 112 6.1 Introduction Following the general introduction regarding the dynamics in plastically crystalline phases in the last chapter, the remainder of this part will focus on the substance cyanocyclohexane, which exhibits a plastic crystalline phase that is in several aspects promising with regard to studies of the α - and β -process. Before discussing the 2H NMR results obtained within this work, we will summarize prior DSC and dielectric studies to provide an overview regarding the phase diagram and the dynamical processes (yet known) in cyanocyclohexane to emphasize the promising character of 2H NMR measurements in the system. The results from dielectric spectroscopy will furthermore turn out be crucial for a more elaborate analysis of the 2H NMR data in a joint approach of the techniques. 66 Chapter 6. Cyanocyclohexane 6.1.1 Chemical and physical properties The molecular structure of cyanocyclohexane (also known as cyclohexanecarbonitrile, cyclohexanecarboxylic acid nitrile or cyclohexyl cyanide, C 6 H 11 CN) is sketched in figure 6.1. Cyanocyclohexane is not a rigid molecule, hence different conformations with respect to the orientation of the carbonitrile group relative to the cyclohexane ring can be observed: of particular interest are the equatorial (e-) and axial (a-) chair conformations (cf. figure 6.2) which have almost the same potential energy [ Schneider 1978 ] and can inter convert crossing an energy barrier of ∆E/kB≈4500 K [ Corfield 1964 ], hence the abundance of the two conformers is almost equal in the liquid at room temperature [ Raber 1977 ] 1 . The boat conformation exhibits a higher potential energy and therefore its abundance can be neglected at room temperature and below. Calorimetric measurements [ Gonthier-Vassal 1986 , Tschirwitz 2002a ] show two disliquid phase T =285K m phase I - plastic crystal T =217K t supercooled plastic crystal T =134K g glassy crystal phase II OOC Figure 6.1 Left: Molecular structure of cyanocyclohexane. White represents hydrogen (in case of the 2H NMR sample deuterons), light grey edges are carbon atoms and marked in blue / dark grey is the triple-bonded nitrogen atom of the cyano group. Right: Phase-diagram of cyanocyclohexane. tinct signals: an exothermic one at the melting temperature of Tm = 285 K with an entropy of fusion of ∆Sm= ∆Hm/Tm= (13 ±0.8) J/mol K and a prominent step like signal at the glass transition temperature Tg = (134 ±1) K. According to the low entropy of fusion Gonthier et al. speculated in 1986 that cyanocyclohexane forms a plastic crystalline phase (phase I) below 285 K according to Timmermans’ law as ∆ Sm< 21 J/mol K [ Timmermans 1961 ] – X-ray scattering studies to confirm the supposed translational order in the respective phase were however lacking to this point. At atmospheric pressure the plastic crystalline phase is stable above Tt = (217 ±3) K [ Reuter 1993 ] and consists of a mixture of the axial and equatorial conformers. Below Tt the supercooled phase I persist alongside the orientationally ordered crystal (phase II), but is still sufficiently stable on experimental time scales. It has been reported that phase II can be obtained via sufficiently long annealing times at temperatures 1At 303 K the abundance of the e-conformer is reported to be (54.5±3.5) % in a CCl4solution. 6.1. Introduction 67 slightly below Tt or by applying elevated pressure [ Woldbaek 1982 ] 2 . In this work we were not able to observe a transition to phase II of cyanocyclohexane, as also reported by Tschirwitz et al. in case of dielectric measurements [ Tschirwitz 2002a ]. The exceptional stability of phase I at temperatures below Tt , i.e. in the metastable regime, can be explained via the structural properties of the ordered crystalline phase: it is known that cyanocyclohexane crystallizes only in the axial conformer [ Woldbaek 1982 ], hence requiring a substantial number of molecules to overcome the energy barrier of 4500 K for crystallites to form. Regarding thermal energy this would presume for long annealing times at temperatures below Tt , the transition yet being even more unfavourable as the equatorial conformer exhibits slightly lower potential energy. Owing to these favourable circumstances, the supercooled plastically crystalline phase CN C N 0 0.5 1 120 160 200 240 280 DSC / a.u. T / K 0.075 0.1 160 180 200 T / K Figure 6.2 Left: The two chair conformations of cyanocyclohexane. Right: DSC curves of cyanocyclohexane [ Tschirwitz 2002b ]. The melting peak at Tm = 285 K and the prominent glass transition step at Tg = 134 K are observed, the inset shows the weak step-like signal around 170 K, attributed to the conformational change of cyanocyclohexane. can easily be obtained and studied on long time scales. Upon further cooling the phase undergoes a calorimetric glass transition of the rotational degrees of freedom accompanied by a step in the DSC measurements at Tg = (134 ±1) K. Apart from afore mentioned features, the calorimetric measurements (cf. figure 6.2) evince two weaker, step-like features: one at 55 K and one at 170 K , the latter was attributed by Pinvidic et al. [ Pinvidic 1988 ] to a dynamic process with an activation energy of ca. ∆E/kB=4300 K that freezes and was identified by the authors with the free chair-chair transformation of cyanocyclohexane. These findings are supported by Raman scattering experiments from Woldbaek et al. [ Woldbaek 1982 ]: in a sample quench-cooled to 90 K the equilibrium of aand e-conformers of the room temperature liquid was maintained. Upon gradually heating no effect was observed until around 170 K the abundance of the e-conformer was slightly enhanced due to its lower potential energy and the fact that the chair-chair conversion becomes observable on experimental time scales. Previous light scattering experiments in our group [ Surovtsev 2003 ] gave hint to the existence of a second plastic crystalline phase or an incomplete transition to phase I as the α -relaxation times τα exhibited a dependency on the cooling rate 2Woldbaek et al. reported a slow transition to phase II at 10 −15 ·108Pa. 74 Chapter 6. Cyanocyclohexane fit to the magnetization curves yields βK< 1 in this regime, cf. figure 6.7. In this temperature regime the structural relaxation is no longer effective on the time scale of T1 , instead a faster process ( β -process) dominates spin-lattice relaxation. Regarding the temperature dependence of hT1i in cyanocyclohexane one immediately detects the arise of two distinct minima around 240 K and 170 K6 . A minimum in T1 occurs if the corresponding dynamic process approaches correlation times on the order of τ(Tmin)≈ 1/ ωL and hence becomes most effective with regard to spin-lattice relaxation. In most structural glass formers α - and β -process have merged before either of them has approached the nanosecond regime, resulting in a single minimum in hT1i somewhat above Tg , depending on the fragility of the system. The kink (universally) appearing at Tg in deuteron spin lattice relaxation measurements has to be attributed to the growing relaxation strength of the β -process however. As α - and β -process do not merge in cyanocyclohexane due to a relatively fast β -process and - as found in most plastic crystalline systems - a rather strong α -process (fragility index m = 55 . 1[ Tschirwitz 2002a ]), the time constants are still well separated at temperatures where the minima are expected to arise. In a first approach it is therefore reasonable to connect the hT1i minimum at lower temperatures to the β -process in cyanocyclohexane. Applying the condition τ =1/ ωL for both minima yields time constants in good agreement with τα respectively τβ from dielectric spectroscopy, cf. figure 6.7 (b). Regarding the absolute values of hT1i one notices that the spin-lattice relaxation times at the respective minima for α - and β -process are very similar: naively one would expect a substantial lower rate for a restricted motion like the β -process with its relatively small relaxation strength (c.f. figure 6.4). We will however show in section 6.4.1, that this finding is in agreement with results from other techniques. To our knowledge this is the first time that a hT1i minimum in 2H NMR is reported for the β-process. To illustrate this peculiarity, figure 6.8 (a) displays the spin-lattice relaxation times of cyanocyclohexane in comparison with data of toluene and ethanol on a reduced temperature scale Tg/T . The spin-lattice relaxation times of toluene represent an archetype of a supercooled liquid: a kink at Tg followed by a single minimum at higher temperatures. In dielectric spectroscopy α - and β -process of toluene can no longer be separated around τα/β ≈ 10 −7 s: due to the fragile nature of the α -process and the rather large activation energy of the β-process the time scales merge and hence one finds a single minimum in T1 . Ethanol on the other hand exhibit a relatively fast β -process with a low activation energy of Ea / kB≈ 15 ·Tg – yet the temperature dependence of T1 is comparable to the one in toluene: only one distinct minimum arises at T>Tg which corresponds to the α -process 7 . Considering τβ of ethanol in figure 6.8 (c) it becomes obvious that the condition τβ≈ 1/ ωL is already approached at temperatures below Tg : as previously mentioned, one may speculate that the small relaxation strength of the β -process in this region hampers the observation of clear effects in T1 , further discussions thereof will be postponed to section 6.4.1 where a more quantitative analysis is conducted. From this comparison it becomes evident that a reduced temperature scale with 6For a discussion of the small apparent step in this temperature region see appendix A.2 7 The minimum itself is not shown in figure 6.8 since data in the region of interest is not available in literature. 6.3. Experimental results – overview 75 10-2 10-1 100 101 102 0.5 0.75 1 1.25 〈 T1 〉 / s Tg / T (a) cyanocyclohexane ethanol-d2 toluene-d5 -10 -5 0 5 log( 〈 τβ 〉 / s ) (b) τβ=1/ωL 10-9 10-6 10-3 100 103 0.5 1 1.5 2 2.5 τ / s Tg / T (c) Figure 6.8 (a): Average spin-lattice relaxation times hT1i of type-B glass forming systems plotted versus reduced temperature Tg/T (cyanocyclohexane: Tg = 134 K , toluene-d5 : Tg = 117 K [ Vogel 2000a , Rössler 1984 ], ethanol-d 2 : Tg = 97 K [ Schneider 2001 ]). (b): Same data plotted versus an isodynamic temperature scale with respect to the β -process, the lines serve as guide for the eye. (c): Corresponing time constants extracted from dielectric spectroscopy [Tschirwitz 2002a,Kudlik 1997b,Benkhof 1999]. respect to the α -process (i.e. Tg/T ) may not be indicated when comparing the evolution of 2H NMR measurements in different glass forming substances with regard to the β -process: large differences in fragility and relative activation enthalpies hHβi can promote hasty conclusions. Therefore a different reduced temperature scale will be aspired in the remainder of this work, when temperature dependent measurements of different glass formers are to be compared. The iso-dynamic scale with regard to the thermally activated nature of the β -process is a more expedient choice for those comparisons, as seen in figure 6.8 (b) where the spin-lattice relaxation times of the discussed systems are plotted with respect to log( τβ ) obtained from dielectric spectroscopy [ Tschirwitz 2002a , Kudlik 1997b , Benkhof 1999 ]. In this representation is becomes obvious that the low temperature minimum in hT1i of cyanocyclohexane forms around τβ≈ 1/ ωL whereas the β -process is too fast / too slow with regard to the α -process in ethanol / toluene. Apart from this trivial observation it is an interesting finding that the low temperature spin-lattice relaxation times of the three systems collapse on a single envelope when plotted in this representation – again pointing to the remarkable universality of the β -process (in 2H NMR ), as it implies that width and relaxation strength of the secondary relaxation are comparable in all three systems with respect to τβ. 6.3.3 Solid echo spectra Following the overview on glassy dynamics in cyanocyclohexane illustrated by T1 , which already demonstrated the advantageous situation in cyanocyclohexane for 2H NMR measurements, we will now focus on the discussion of the solid echo line shape, which is consequently expected to exhibit prominent features also. Figure 6.9 (a) presents an overview of the solid echo line shape ( tp = 20 µs ) from the 76 Chapter 6. Cyanocyclohexane lowest measured temperature ( 34 K ) to the line shape collapse around 172 K . In this representation it becomes evident that the temperature dependence of the solid echo spectrum in cyanocyclohexane can be forked in three different temperature regimes in an obvious manner: T<TgTg<T.165 K and above. •T<Tg = 134 K : at lowest temperatures the line shape resembles the (rigid limit) Pake spectrum and changes with temperature are very subtle over an experimentally observable range of 100 K . The dominant line shape changes in this regime are limited to the centre of the spectrum and can be emphasized by prolonging the inter pulse delay tp of the solid echo pulse sequence, as demonstrated in figure 6.9 (d). •Tg<T.165 K : at temperatures above Tg the line shape in cyanocyclohexane differs considerably from a rigid limit Pake spectrum. This peculiar line shape prevails over a large temperature interval of about 30 K and exhibits little but continuous temperature dependence from Tg to the line shape collapse. Comparable line shape effects have not yet been observed in supercooled liquids and potentially offer new insights on the mechanism of relaxation in this regime. •T≈ 165175 K : in this temperature regime the broad solid state spectrum collapses to a liquid line due to the fast isotropic tumbling motion of the α-process. Figure 6.9 (c) exemplarily illustrates the line shape in the second – most interesting – temperature regime: not only is the line shape for short inter pulse delays highly distorted, the spectra also exhibit considerable tp dependence at the given temperature. The line shape hence exhibits the fingerprint of a dynamic process and according to the results of chapter 4can be attributed to a motion faster than about 0.5ms . For a preliminary allocation the fraction W(T) = −3.3 Z −∞ G(log τ)dlog τ(6.3) extracted from fits to the dielectric spectra (cf. appendix Cfor details) is plotted in figure (b) for α - and β -process respectively. Within this purely heterogeneous (i.e. exchange processes are neglected), phenomenological approach W ( T )serves only as an conservative estimate, but illustrates that the α -process is only expected to exhibit first, subtle effects on the line shape at temperatures above ca. 155 K , whereas the β-process is virtually in the fast motion limit above Tg. Before going into detail on the line shape effects in the region Tg<T.165 K we will focus on the high and low temperature regimes, as the dominant effects observed here are well known from a variety of other glass forming systems and can be analysed in a straightforward manner. In the high temperature regime of the line shape collapse the correlation time of the α -process approaches 10 −6 s, i.e. is on the order of the inverse coupling constant 1 /δ . The fast isotropic dynamics attributed to the structural relaxation averages over all orientations of the electric field gradient tensor relative to the external magnetic 6.3. Experimental results – overview 77 135 140 145 150 155 160 165 170 175 -100 0 100 T / K ν / kHz 34 K (a) Tg β α 0 0.5 1 W ( T ) (b) τc < 0.5 ms β α -100 0 100 ν / kHz 156.86 K (c) tp=10µs 20µs 50µs 100µs -100 0 100 ν / kHz (d) 76.20 K 20µs 50µs 100µs 150µs 200µs 250µs 300µs 400µs Figure 6.9 (a): Evolution of the line shape above Tg = 134 K . The intersection with the ordinate marks the temperature at which each spectrum was recorded. For comparison a spectrum at lowest temperatures ( 34 K ) is plotted. (b): W ( T )from equation 6.3, i.e. fraction of τα and τβ faster than 0.5ms . The temperature scale of figure (a) is maintained. (c): tp dependence of the line shape in the high temperature regime. (d): tp dependence at lowest temperatures. field and the solid state spectrum collapses to a central line at around 172 K8 . The line shape of cyanocyclohexane in the vicinity of the collapse however exhibits a peculiarity: at temperatures where the solid state spectrum is still well established, it coexists with a central feature at intermediate temperatures ( T =168171 K , cf. figure 6.9 (a)). For systems with a broad distribution of correlation times this is usually explained in the context of a “two phase” model: the broad dynamic heterogeneities lead to a scenario where the number of molecules exhibiting correlation times on the order of the inverse coupling constant are (due to the reduction factor being virtually zero) negligible in contrast to the relatively fast portion of the sample producing a liquid line and the slow fraction giving rise to a solid state spectrum, cf. figure 4.9. If no fast exchange mechanism is present, a simple superposition of the latter line shapes is observed. For cyanocyclohexane however this is not expected since the 8A VFT-fit to dielectric data [Tschirwitz 2002a] yields τDS α(172 K)≈4µs≈1/δ. 78 Chapter 6. Cyanocyclohexane distribution G(log τ) for the α -process is rather narrow and comparable to numerous other structural glass forming systems ( βCD =0.65, cf. section 6.1.2), which do not exhibit such behaviour at the line shape collapse. We will address this apparent “two phase” behaviour in the scope of discussion. 6.3.4 Solid echo spectra: low temperature regime In the regime T<Tg any dynamics that is influencing the solid echo line shape has to be attributed to secondary relaxations, as the α -process is virtually extinct in the glassy state. Typically the line shape changes at temperatures below Tg and short inter pulse delays tp are very subtle and concentrate on the features of the spectrum around zero frequency, where the angular dependency is largest (cf. figure 6.10 (b) and chapter 3). Since the distribution G(log τβ) extends already at these temperatures substantially towards times faster than the inverse of the quadrupolar coupling constant δ , one may immediately declare that the corresponding dynamics must be of highly restricted nature, as the alterations to the limiting Pake spectrum are very subtle (cf. section 4.3). The systematic change of spectral intensity around zero frequency seen in figure 6.9 (d) is anticipated for a restricted multi-step motion as demonstrated in chapter 4. This tender effect can be amplified by prolonging the time tp between the echo pulses and hence the time during which a phase shift (occurring after reorientation) is accumulated. As displayed in figure 6.10 the line shape changes with tp in cyanocyclohexane resemble the behaviour reported for type-B structural glass formers, i.e. the intensity in the middle of the spectrum decreases for longer inter-pulse delays in a characteristic manner. The given spectra of different glass forming substances are recorded at similar reduced temperatures Tg/T and indeed the tp -dependence in cyanocyclohexane, toluene and ethanol – which all exhibit a pronounced β -peak in dielectric spectroscopy – is very similar. Whereas for glycerol, which shows no distinct β -process in dielectric spectroscopy, consequently no systematic changes in the line shape with varying tp are found. As in case of the random walk simulations presented in chapter 4, we will continue the discussion of the latter effect in terms of the R(tp) parameter, which allows for a more direct comparison of the evolution with temperature as it maps the spectral changes around zero frequency to a single quantity (cf. section 4.2.2 for details). As seen in figure 6.11 (a) the R(tp) values for tp≥50 µs traverse a pronounced and broad minimum around 115 K , whereas the values at Tg again roughly resemble the ones found at lowest temperatures. This mimics the behaviour found in the random walk simulations of chapter 4, where in case of restricted dynamics a minimum in R(tp) was observed for τc =1 /δ independent of the details of the dynamical process. And indeed this holds for the minimum found in R ( tp = 50 µs )of cyanocyclohexane, as the corresponding time constant is in agreement with τβ from dielectric spectroscopy, cf. figure 6.19. Within the random walk simulations the criterion τc =1 /δ was found to be fulfilled best in case of R ( tp = 20 µs )– here however no minimum is observed in cyanocyclohexane: the trace traverses a maximum above Tg where the central intensity in the spectra is almost raised by a factor of two with respect to 34 K . As this effect is already pronounced well below Tg , the observation of a minimum is hampered. A raise in R(tp) is generally found for a process proceeding via large 6.3. Experimental results – overview 79 -100 0 100 ν / kHz cyanocyclohexane (a) tp = 20µs 100µs 200µs -100 0 100 ν / kHz toluene -100 0 100 ν / kHz ethanol -100 0 100 ν / kHz glycerol -100 0 100 ν / kHz T = 48 K (b) tp = 20µs 100µs 200µs -100 0 100 ν / kHz T = 84 K -100 0 100 ν / kHz T = 112 K -100 0 100 ν / kHz T = Tg = 134 K Figure 6.10 (a): Solid echo spectra of glass forming substances recorded at similar reduced temperatures Tg / T≈ 1.2. For each substance the spectra corresponding to solid echo inter pulse delays of tp = 20 µs , 100 µs and 200 µs are shown. Cyanocyclohexane: T = 112 K , Tg = 134 K ; toluene-d 5 : T = 97 K , Tg = 117 K [ Vogel 2000a ]; ethanol-d 2 (glassy phase): T = 73 K , Tg = 97 K [ Schneider 2001 ]; glycerol: T = 156 K , Tg = 189 K [ Vogel 2000a ]. (b): tp -dependence of the solid echo line shape in cyanocyclohexane for different temperatures T≤Tg. angle jumps, at the discussed temperatures below Tg the α -process is however not observable at experimental time scales and the β -process is suspected to proceed via small angles. As G(log τβ ) extends to the fast motion limit regime already at Tg another explanation is however possible: as demonstrated by the random walk simulations in figure 4.18, R(tp) also exhibits a distinct raise for a fast, restricted motion upon vanishing restriction, i.e. large opening angles in case of the cone models discussed in chapter 4. We will address this feature in the scope of section 6.4.2. As the line shape changes below Tg are most dominantly reflected in the intensity around zero frequency, the R(tp) values provide a measure closely related to the spin-spin relaxation time T2 in the solid state: in figure 6.11 (b) the quantities are plotted again vs. log τβ for reasons of comparison with other glass forming systems (for additional T2 data see appendix A.3). It becomes imminently evident that R ( tp = 200 µs )and hT2i exhibit the same evolution with temperature, i.e. traverse a minimum around τβ =1 /δ before strongly decreasing in the vicinity of Tg . As 80 Chapter 6. Cyanocyclohexane 0 0.2 0.4 0.6 0.8 30 60 90 120 150 R ( tp ) T / K Tg (a) tp=20µs 200µs 400µs 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 0 200 400 600 R ( tp ) 〈 T2 〉 / µs log( 〈 τβ 〉 / s ) τβ=1/δ (b) cnc tol eth Figure 6.11 (a): Relative spectral intensity at zero frequency in cyanocyclohexane for different inter pulse delays tp ( tp =20, 50, 100, 150, 200, 250, 300 and 400 µs top to bottom). (b): R ( tp = 200 µs )(open symbols) and hT2i (corresponding full symbols) of cyanocyclohexane, toluene-d5[Vogel 2000a] and ethanol-d2[Schneider 2001] plotted versus logτβ. discussed before the experimental observation of this minima in cyanocyclohexane is possible due to the fast and non-merging nature of the β -process – as opposed to the vast majority of glass forming systems. With regard to the deuteron spin-spin relaxation times and R ( tp = 200 µs )values of toluene-d5 plotted in figure (b), the same arguments as in the discussion of T1 hold (cf. figure 6.8 (b)): the line shape collapse due to the α -process appears at relatively lower temperatures (with regard to τβ ), hampering the observation of a minimum in either quantity. The behaviour at lower temperatures is however very similar to the one found in cyanocyclohexane, i.e. the onset of the minimum is discernible. In ethanol-d 2 also no stark effects in T1 were observed, which we tentatively attributed to the small relaxation strength below Tg , were the condition τβ =1/ ωL is approached in this system. This argument does not hold for the present analysis as T2 and R(tp) are more sensitive on the highly restricted dynamics found in this region and consequently the progression of both quantities closely mimics the one found in cyanocyclohexane, i.e. a distinct minimum is observed at τβ=1/δ. Before conducting a more quantitative analysis of the effects in section 6.4, we can already conclude at this point that the β -process in phase I of cyanocyclohexane displays all effects found in structural glass formers of type-B and hence the plastic crystalline phase appears to impose no peculiarities on the process. 6.3.5 Solid echo spectra: fast motion limit effects So far the effects of the β -process in the slow motion limit at temperatures below Tg and the line-shape collapse due to the α -process around Tg + 40 K were discussed. The fact that the line shape changes below Tg (at least for short tp ) are hardly observable is linked to the small relaxation strength of the β -process (1S ), i.e. high spatial restriction of the motional process under consideration – this is expected to change above Tg however as 1S grows significantly in this regime (cf. figure 6.3. Experimental results – overview 81 -150 -100 -50 0 50 100 150 ν / kHz C80 C50 C100 (a) -80 -60 -40 -20 ν / kHz 40K 137K 146K 163K (b) Figure 6.12 (a): Symmetrized experimental solid echo spectrum at 34.5K ( tp = 20 µs , open symbols) and numerical fit (solid line, δ = 129.6kHz , the numerical spectrum has been damped with respect to experimental pulse excitation effects). The arrows define the width parameters Cx for monitoring the evolution of the line-shape with temperature. (b): Detail around the inner singularity of solid echo spectra for tp=20 µs at different temperatures. 6.4 (a)). Figure 6.12 (b) displays solid echo spectra for tp = 20 µs at four different temperatures: compared to the spectrum at lowest temperatures ( 40 K ), which resembles the limiting Pake spectrum, the overall width of the spectrum at 137 K > Tg is slightly reduced, but yet more dominant is the deviation around the singularities: the spectrum tapers in the upper half and the singularities are smeared out, which also holds for the outer singularities, which have transformed to a more or less structureless decay of intensity towards higher frequencies. The effects are even more pronounced in the spectra recorded at higher temperatures. The intensity deviations that arise around zero frequency (i.e. changes in R(tp) ) are presumably due to slower dynamics ( τα , τβ& 1 /δ ) and will be neglected at first. While the tapering of the spectrum could also result from a non-vanishing asymmetry parameter η that changes with temperature, the remainder of spectral changes can not be explained within this scope. Furthermore we have no reason to expect η > 0for the aliphatic C2 H bonds in cyanocyclohexane. Yet we know that at Tg already more than half of the sub ensembles can be attributed a correlation time τβ in the fast motion limit regime, τβ 1 /δ , i.e. prominent line shape effects are anticipated. Figure 6.12 (a) displays the (symmetrized 9 ) spectrum at lowest temperatures ( 34.5K ) where the line shape is well described by a Pake spectrum, as demonstrated by the fit symbolized by the solid line. Also marked in this figure are the measures Cx , defined as the apparent spectral widths at x =50, 80 and 100 % of maximum intensity which will serve to quantify the spectral evolution with temperature as the spectra above Tg no longer can be described by a Pake pattern. In figure 6.13 the quantities Cx are given for all recorded solid echo spectra. The right part shows an enlarged plot at low temperatures: in this representation it becomes immediately obvious that the spectrum for short inter pulse delays tp 9 Due to minor problems with the resonant circuit of the the probe appearing at lowest temperatures, some of the spectra for T<70 K exhibit intensity deviations with respect to the singularities. 82 Chapter 6. Cyanocyclohexane 120 122 124 126 128 130 132 40 80 120 160 C / kHz T / K Tg (b) 0 20 40 60 80 100 120 140 40 80 120 160 C / kHz T / K Tg (a) C50 C80 C100 Figure 6.13 (a): Apparent spectral width of solid echo spectra for tp = 20 µs at different relative intensities. (b): Enlarged plot of the low temperature regime T<Tg. changes with temperature (aside from the R(tp) value) also below Tg . Whereas the apparent spectral width C80 exhibits almost no temperature dependence at lowest temperatures, it starts to lessen around T = 60 K and decreases approximately linear until Tg . At the glass transition temperature the temperature dependence changes within a few Kelvin and contracts more pronounced towards higher temperatures until the solid state spectrum collapses. Apart from the overall narrowing of the spectra 10 , also systematic deviations in the widths taken at different relative intensities are observed. At low temperatures the three arbitrary Cx values display the same temperature dependence within the error margins, which means that the spectra retain their shape with respect to the singularities (Pake pattern). Starting at 7080 K however, the values gradually diverge, leading to a reduction in the flank slope of the spectra. The effect is rather subtle for temperatures T<Tg but is clearly observed around Tg and grows continuously towards higher temperatures until at around 170 K the solid state spectrum collapses to a liquid line. This behaviour clearly is the fingerprint of molecular dynamics – and, as will be shown, related to the β-process in cyanocyclohexane. The discussed changes in the line shape in the temperature region Tg<T<Tg+40 K can not be analysed further retaining a straightforward, model-free approach. Since the effect sets in well below Tg and becomes more pronounced at the glass transition temperature (where the relaxation strength of the β -process grows) it however is tempting to associate it with fast motion limit effects of the β -process growing in amplitude. Before a more in-depth analysis of the peculiarities in the high temperature solid echo line shape of cyanocyclohexane will be presented in section 6.4.2, the experimental overview will be completed with a discussion of results from two-dimensional 2H NMR methods. 10 Which could also be related to a temperature dependence of the quadrupolar coupling constant δ for T<Tg. 6.3. Experimental results – overview 83 0 0.2 0.4 0.6 0.8 1 10-5 10-4 10-3 10-2 10-1 Fte cos tm / s toluene 97 K (a) te = 3 µs 30 µs 80 µs 0 0.2 0.4 0.6 0.8 1 10-5 10-4 10-3 10-2 10-1 Fte cos tm / s cyanocyclohexane 101 K (b) Figure 6.14: Stimulated echo decays of the cosine-cosine correlation in cyanocyclohexane (this work, (b) ) and toluene ([ Vogel 2000a ], (a) ) for different evolution times te at temperatures T<Tg. Same symbols denote same evolution times te. 6.3.6 Stimulated echoes T <Tg Apart from measurements of T1 , T2 and the solid echo line shape, the β -process was also previously studied by 2D 2H NMR methods below Tg . With the discussion of the latter we will consummate our overview on 2H NMR measurements related to the β-process, as the method directly maps the mechanism of the motion. Due to the broad distribution of correlation times G(log τβ) , the restricted nature of the underlying dynamics and the limited time window of the experiment, it is not possible to measure the full correlation loss due to the β -process below Tg . Nevertheless the stimulated echo decay in toluene-d5 and polybutadiene at low temperatures was intensively studied by M. Vogel [ Vogel 2000a ] and successfully explained via a cone model for the β -process. Figure 6.14 displays the decay curves of the cosine-cosine correlation function Fcc te for three different evolution times te11 at one temperature: for an evolution time of 3µs toluene exhibits a very weak, almost not discernible, more or less structureless decay until about tm = 100 ms when another process decays the stimulated echo amplitude finally to zero. M. Vogel identified this slower process as spin diffusion, as it displays almost no temperature dependence and the extracted time constants are on the order of reported spin diffusion rates for deuterated substances. The initial decay for longer evolution times ( 30 µs and 80 µs ) grows in amplitude, yet retains its overall shape – the fingerprint of a motion which agitates via small angular displacements. As the decay extends over several orders of magnitude in time and is almost not resolvable for short evolution times, it has to be attributed to a highly restricted process with a broad distribution of correlation times – the β-process. The stimulated echo decays of cyanocyclohexane at comparable reduced temperature Tg/T are plotted in figure 6.14 (b). Here the qualitative picture is the same: a broad, 11Fcc was recorded for reasons of the straightforward correction for spin-lattice relaxation decay during the mixing time tm , as opposed to the T1Q decay in the sine-sine correlation Fss . All curves displayed in this section have already been corrected for T1effects during tm. 90 Chapter 6. Cyanocyclohexane Figure 6.19: Average correlation times of the α -process extracted from Fss te for short evolution times te in comparison with dielectric time constants [ Tschirwitz 2002a ]. The 2H NMR results appear slightly shifted with respect to ταfrom dielectric spectroscopy. -12 -9 -6 -3 0 3 4 6 8 10 12 14 log τ / s 1000 K / T α β (a) DS Fsin (3µs) T2 min. evolution times te→ 0and te→ ∞ , hence any lower symmetry can be clearly ruled out (n→ ∞). The stretching parameter βK obtained from the fit via equation 6.6 is displayed in figure 6.18, the mean value for all observed evolution times increases from about 0.55 at lowest temperatures to approx. 0.7 at 157.8K . The fact that the stretching parameter exhibits little dependence on te at all temperatures could be taken as a hint for the predominance of large angular displacements, but it is known that βK ( te ) holds also a strong dependence on the width of the distribution G(log τα) [ Geil 1998 ], not allowing for any straightforward conclusions to be made. The τ ( te ) values however have proven [ Geil 1998 ] to be much less sensitive on any underlying distributions G(log τ) and therefore provide a more non-ambiguous “fingerprint” of the microscopic dynamics. As seen in figure 6.20 (b), the correlation time hτi(te→0) changes by almost two orders of magnitude within the investigated temperature range of 10 K due to the non-Arrhenius behaviour of τα and is in good agreement with results from dielectric spectroscopy, cf. figure 6.19. More noteworthy though is the drastic change in hτi(te) for longer evolution times: whereas hτi ( te = 2µs ) and hτi ( te = 80 µs ) differ by more than an order of magnitude at 147.3K , the difference at higher temperatures becomes much more subtle. This trend is better seen in the right plot of figure 6.20 where hτi ( te, T )is normalized with respect to τα ( T ) = τ ( te→ 0 , T ) for comparison 14 . In a first approach the experimental values are presented in comparison with simple random walk simulations for several discrete γ -jump motions, cf. chapter 4. This representation readily allows for some conclusions regarding the microscopic dynamics in the investigated temperature regime: at 147.3K the plateau approached for long evolutions times is approximately described by a 10° jump, whereas at 157.8K a mean jump angle of roughly 35° (and therefore dynamics close to the random-jump type) is needed to model the experimental data. Hence either the mean jump angle increases dramatically within 10 K or the number of molecules undergoing large angular reorientations detectable by the experiment changes as G(log τα) traverses the 2H NMR time window. This behaviour was also found for structural glass formers (e.g. ortho-terphenyl [ Jörg 2000 ] or m-tricresylphosphate 14τα from dielectric spectroscopy was chosen since τ ( te→ 0) is not accessible in a strictly model free approach from the experimental data. 6.3. Experimental results – overview 91 -4 -3 -2 0 20 40 60 80 log( 〈 τ ( te ) 〉 / s ) τe / µs (a) 157.8 K 155.2 K 152.5 K 151.3 K 147.3 K 0.1 1 0 20 40 60 80 〈 τ ( te ) 〉 / 〈 τc 〉 τe / µs 35˚ 20˚ 15˚ 10˚ (b) Figure 6.20 (a): Experimental hτi values from a fit to Fss te of cyanocyclohexane at different temperatures and evolution times te in absolute representation. (b): Same results normalized to the corresponding correlation time hτci obtained from dielectric spectroscopy [Tschirwitz 2002a]. [Baldus 2005]). Although the behaviour towards long evolution times is described quite well by a fixed γ -jump type motion, it is evident that the model fails at shorter evolution times te : at all temperatures the experimental values are substantially larger than in the according simulations. This discrepancy is a result of keen simplifications in the model: only a fixed jump angle γ was chosen – this obviously contradicts the 2D exchange spectra presented in section 6.3.7 as a single γ = 35° jump e.g. would result in characteristic ellipses arising in the exchange intensity. If a distribution of jump geometries is introduced which substantially extends towards smaller angles – as previously employed for structural glass formers – the experimentally found behaviour is reproduced in more detail. A more in-depth analysis in this manner, that furthermore accounts for the limited time window of the experiment, is conducted in section 6.4.4. 6.3.9 Preliminary conclusions Before we will refine the discussion of the presented results in the next section by applying random walk simulations and via a more quantitative approach incorporating G(log τ) obtained by dielectric spectroscopy, we briefly summarize the obtained findings at this point. Regarding the α -process we support the conclusions drawn from dielectric spectroscopy by Tschirtwitz et al. [ Tschirwitz 2002a ] in the sense that also in 2H NMR the α -process appears even in detail identical to the processes found in supercooled liquids and structural glasses: no signs of the translational symmetry of the plastic crystalline lattice were observed. The two dimensional experiments in time and frequency domain yield results closely related to those found in toluene e.g., hence the elementary jump processes are comparable. The same arguments hold for mea- 92 Chapter 6. Cyanocyclohexane surements regarding the β -process: its manifestation in the 2H NMR observables of cyanocyclohexane exhibits all features known from structural glass formers – with the only difference being the favourable separation in time scales of τα / τβ , which gives raise to the appearance of new features like the (second) minimum found in T1.. The scope of next section will be a more detailed analysis of those features. 6.4 Experimental results – refinement The presented peculiarities in the 2H NMR observables, arising from the well separated nature of α - and β -process in cyanocyclohexane, are hopeful candidates with regard to further insight into the microscopic manifestation of the dynamics. This however brings the necessity of more elaborate data analysis. The following part will address the most prominent findings in detail: first the spin-lattice relaxation and in particular the arise of two distinct minima in T1 will be discussed in a non-model free approach, thereafter the peculiar solid echo line shape above Tg and the apparent temperature dependence of the microscopic mechanism for the α-process observed in the stimulated echo will be further analysed. 6.4.1 Modelling spin-lattice relaxation In section 6.3.2 we attributed the minima in T1 to α - and β -process via their respective temperature in comparison with correlation times from dielectric spectroscopy. To further exploit the prominent features in T1 with regard to the fast and non-merging β -process we will divert from this general, model-free approach by employing the spectral density as obtained from dielectric spectroscopy. This approach benefits from the fact that the spectral density – although only at a fixed frequency, ωL – is probed by NMR with great accuracy. The ansatz is as follows: as long as the time scales of the two processes are well separated, the imaginary part of the normalized complex susceptibility χ00 (ω) can be described within a Williams-Watts [ Williams 1971 ] approach. We employed the sum of a Cole-Davidson function for the α -process and a log-Gaussian distribution for the β -process, as used by Tschirwitz et al. [ Tschirwitz 2002a ] in the analysis of the dielectric spectra: χ00 (ω) = S=(1 (1 + iωτCD α)βCD )+ (1 −S)√π 2Wln 10 exp "−(log ω/2π−log 1/2πτβ)2 W2#,(6.7) the susceptibility χ00 (ω) can subsequently be expressed in terms of spectral density via the fluctuation dissipation theorem [Böttcher 1978a], χ00 (ω) = ωJ(1) (ω).(6.8) Assuming for the moment that the individual contributions of α - and β -process to the spectral density of the system are in first approximation independent of the rank 6.4. Experimental results – refinement 93 0.01 0.1 1 100 150 200 250 300 〈 1 / T1 〉-1 / s T / K αβ S const. S(T) best fit Figure 6.21: Experimental and calculated spin-lattice relaxation times. In contrast to hT1i in figure 6.7 here the inverse of the average rate, h1/T1i , is plotted. The dotted line was calculated using parameters from dielectric spectroscopy and modelling 1S . The solid line results from the same calculations with a slight shift of τα/β towards shorter times, cf. figure 6.22 (b). l , hence J(1) α(ω, T)≈J(2) α(ω, T) and J(1) β(ω, T)≈J(2) β(ω, T) respectively hold 15 , we are now able to calculate the spectral density employing the time constants τα , τβ and shape parameters W , βCD from dielectric spectroscopy 16 . The presumably l -dependent relaxation strength 1 −S will serve as fit parameter and hence provides insight in the microscopic geometry of the process under consideration. Via equation 3.29 it is now possible to calculate T1 for a number of selected models concerning 1-S. Since the magnetization curves (and therefore hT1i ) are affected by spin diffusion at low temperatures, the results of eq. 3.29 can only be compared to the the average spin-lattice relaxation rates h1/T1i (for details see chapter 3). h1/T1i is obtained from the initial slope of the magnetization curves 17 and is displayed in figure 6.21, as opposed to figure 6.7 where hT1i was given. Furthermore we have to remind that T1 of a single spin depends on the orientation of the electric field gradient tensor relative to the external magnetic field and hence in the case of a restricted motion in solids (e.g. a two-site jump process or a rotation of the molecule like in the case of benzene) the general treatment for l =2 does not strictly hold, instead h1/T1i has to be calculated from the powder average within the specific motional model considered. Following the general approach of Torchia and Szabo [ Torchia 1982 ] it can be shown however that in the case of free diffusion on or within a cone of opening angle χ , 15 Regarding the α -process this approximation has proven to be valid in the case of glycerol [Blochowicz 1999a]. 16 The parameters τα , τβ and W were obtained directly from equations 2.1,2.3 and 2.7 with the parameters given in section 6.1.2, βCD was interpolated from dielectric results as shown in figure 6.4. The coupling constant obtained at low temperatures δ0 = 129.6kHz and the Larmor frequency ωL=46.07 MHz were used in all calculations 17 It is noteworthy to mention that by fitting the experimental curves in this manner the discontinuity in hT1i around 170 K , caused by a peculiar behaviour of the stretching parameter βK is eliminated. 94 Chapter 6. Cyanocyclohexane the general treatment for l =2 remains valid [ Lipari 1982 , Woessner 1962 ]. As these models are generally applied to the β -process throughout this work and are also proven to be applicable for cyanocyclohexane in the discussion of the solid echo line shape in the next section, we will – for the sake of simplicity – restrict ourselves to these scenarios for the present discussion of spin-lattice relaxation. First h1/T1i was calculated for a temperature independent relaxation strength of the β -process (1S ), chosen in a way to mimic the experimental rates observed below Tg . The result is given as dashed line in figure 6.21: whereas the qualitative progression below Tg and the position of the high temperature minimum for the α -process are roughly captured within this approach, the absolute values above Tg differ by almost an order of magnitude and the low temperature behaviour exhibits a somewhat lower slope than found experimentally. The present calculation clearly demonstrates that a distinct minimum for the β -process in h1/T1i−1 can only be observed if the relaxation strength in the concerned temperature region is significantly enhanced with respect to the values typically found in the glassy regime below Tg – i.e. the results are in accordance with the experimental findings in ethanol, where no second minimum is observed. Consequently 1S was modelled according to the behaviour observed in dielectric spectroscopy: subtle, linear temperature dependence below Tg where 1S exhibits a sharp bend which was modelled by a phenomenological function (comprised of a linear, an exponential and a quadratic term). The dotted line in figure 6.21 is calculated in this manner. Due to the enhanced relaxation strength of the β -process above Tg , modelled by the temperature dependence of 1S in this approach, a second minimum in h1/T1i−1 arises. The model provides excellent agreement with experimental data in the temperature region below about 160 K and exhibits the two desired minima for Tg<T<Tm with the same rates as found in the experiment, fails however to accurately reproduce the position of those minima. Regarding the subtle mismatch observed between correlation times of the α -process determined via Fss te and dielectric spectroscopy, cf. figure 6.19, the parameters in equations 2.1 and 2.3 were slightly altered to obtain best agreement with respect to the experimental T1 values – the resulting model is represented by the solid line in figure 6.21, which reproduces h1/T1i−1 within the experimental error in the whole accessible temperature range T<Tm . The hereby employed correlation times are plotted in figure 6.22: τα is in good agreement with correlation times obtained in the stimulated echo experiment, i.e. exhibits the same amount of deviation from the dielectric results, whereas τβ only significantly deviates from the dielectric data at high temperatures. As the introduced changes in τα/β are of reasonable small magnitude, they will be accepted at this point. Hence the sole free fitting parameter within this approach is 1S , plotted in figure 6.22 (a): the qualitative behaviour is very similar to the results obtained in dielectric spectroscopy. Quantitatively the obtained relaxation strengths differ however by more than a factor of three, i.e. 1S(2) = 3 ·1−S(1) , as expected for a highly restricted process progressing via small angular displacements, is not fulfilled (as demonstrated via the dotted line in figure 6.22 (a)). This discrepancy will be discussed at the end of this chapter, when further results from the line-shape analysis are at hand. 6.4. Experimental results – refinement 95 0 0.2 0.4 0.6 0.8 1 60 90 120 150 180 210 240 270 1-S(l) T / K (a) Tg DS (l=1) T1 model (l=2) T1 model / 3 10-12 10-9 10-6 10-3 100 4 6 8 10 12 14 τ / s 1000K / T (b) α β DS Fss (te=3µs) dielectric fit T1 best fit Figure 6.22 (a): Relaxation strength 1 −S of the β -process used for calculating h1/T1i in figure 6.21 (solid line) in comparison with dielectric results. (b): Time constants of α - and β-process used in the calculation of h1/T1i. From our present efforts we can conclude that the spectral density J(1) α/β (ω, T) from dielectric spectroscopy is (with minor alterations) sufficient to fully describe the experimentally observed spin-lattice relaxation times at all temperatures, therefore accordance between techniques probing l =1 and l =2 was found. Furthermore the assumptions made by employing the Williams-Watts ansatz in the spectral density appear valid even at high temperatures. The evolution of the relaxation strength with temperature obtained from our model is compatible with the one found in dielectric spectroscopy, i.e. the – at first – peculiar fact that the h1/T1i minima of α - and β -process exhibit comparable heights is deduced in a straightforward manner. Furthermore no plateau is observed in 1S even at higher temperatures. To model the experimental h1/T1i data a steady growth of the relaxation strength of the β -process is needed: at the temperature of the T1 minimum for the α -process 1S(2) is already about 0.7. In summary we have shown that the strong effects observed in the 2H NMR spinlattice relaxation times of cyanocyclohexane can be explained by the growing relaxation strength of the β -process above Tg , which is qualitatively in accordance with other techniques. 6.4.2 Modelling solid echo spectra In the last section we have demonstrated that the simple model of a motion on or within a cone is versatile enough to reproduce the strong effects observed in T1 of cyanocyclohexane due to the fast β -process in the system. As previously pointed out, the peculiar solid echo line shape of cyanocyclohexane associated with the β -process in the temperature region Tg<T<165 K (cf. figure 6.9) is also very promising with regards to further insights on the mechanism of reorientation. Consequently the afore mentioned models will now be tested for compliance with the line shape effects observed in this regime. Prior to this we will however briefly return to lower temperatures ( T<Tg ) to validate 96 Chapter 6. Cyanocyclohexane Figure 6.23: Random walk simulations for the β -process employing G(log τβ) from dielectric spectroscopy. A random jump motion within a cone of χ = 11° is assumed. The experimental behaviour at low temperatures is significantly different from the results within this simple model. 0.1 0.2 0.3 0.4 0.5 0.6 30 60 90 120 150 R ( tp ) T / K RJ tp=20µs 200µs exp. logGauss a number of assumptions for the present approach and furthermore estimate the effects of the α -process on the solid echo line shape above Tg to define the applicable temperature range. Figure 6.23 displays the line shape parameter R(tp) obtained from random walk simulations in comparison with the experimental results discussed in section 6.3.4. For the calculations – again in a phenomenological, purely heterogeneous approach – the distribution of correlation times G(log τβ) from dielectric spectroscopy 18 was used to obtain R(tp) within the simplest scenario: a random jump motion within a cone of fixed (temperature independent) opening angle χ = 11° (this roughly corresponds to the angle of 7° found in toluene within the “motion on a cone” model [ Vogel 2000a ]). The simulations exhibit a minimum in R( tp = 20 µs ) around τβ =1 /δ not observed in the experiment due to the raise of R(tp) around Tg . More dominant however is the discrepancy at lowest temperatures: R ( tp = 20 µs )and R ( tp = 200 µs )gradually diverge only around 80 K , whereas the experimental spectra exhibit line shape effects down to lowest temperatures, T = 34 K . As demonstrated in chapter 4, the R(tp) effects at longest inter pulse delays are extended towards longer times τc if the motion proceeds via small angles (albeit maintaining the “fast” side of the minimum, which is already approximately captured by the approach). Hence, as in the case of toluene [ Vogel 2000a ], a mechanism is observed that agitates via small angular displacements – this is however rather unfavourable for random walk simulations at temperatures above Tg , as due to the separation of time scales τjτc very short jump times τj have to be considered, resulting in large computational effort 19 . As the better part of G(log τβ) is in the fast motion limit regime above Tg anyway, we will therefore divert from a random walk analysis and model the fast motion limit line shape in a simplified approach, as the elementary step of the motion does not affect the results in this limit. As mentioned before, the influence of the α -process at temperatures above Tg on the solid echo line shape will briefly be considered, before we proceed to develop and apply a suitable model for the β -process in this regime. Therefore G(log τα) 18 Both distributions used in [ Tschirwitz 2002a ] were employed: a log-Gaussian and Gβ . For the present case however no significant differences in the results are observed. 19 At 160 K for example τβ is 2ns , which according to equation 4.5 corresponds to a jump time of τj≈4ps for a 2° jump. One C2 H bond would hence reorient, on average, 10 8 times during the pulse sequence for a solid echo spectrum of tp = 200 µs . Typically 10 5− 10 6 single trajectories have to be calculated to obtain a spectrum. 6.4. Experimental results – refinement 97 0 0.2 0.4 0.6 0.8 100 120 140 160 R ( tp ) T / K (a) γ = 2° α-process Tg 120 125 130 135 100 120 140 160 C80 / kHz T / K (b) experimental tp=20µs 200µs Figure 6.24 (a): Line shape parameter R(tp) from random walk simulations (lines) of the α -process employing G(log τα) from dielectric spectroscopy. An isotropic motion proceeding via γ=2°jumps is assumed. (b): Corresponding apparent spectral width C80. from dielectric spectroscopy was employed in random walk simulations of the solid echo line shape for an isotropic motion proceeding via small angles ( γ = 2° , which roughly corresponds to the smallest jump angles extracted from the stimulated echo experiments in section 6.3.8). The calculations do not aim to provide a realistic model, rather a conservative estimate is obtained as for more complex models, i.e. with fractions of larger angles γ or exchange within G(log τα) , the line shape changes will be shifted towards higher temperatures. The results are displayed in figure 6.24 and allow to draw two important conclusions: any influence of the α -process on the apparent spectral width ( C80 , figure (b)) below about 155 K can be neglected. Even at 160 K the observed changes are rather subtle in case of tp = 20 µs and completely fail to reproduce the experimental results. The R(tp) values are however strongly influenced within the model at temperatures slightly above Tg already, which explains the experimentally found reduction in this regime (as the β -process is already too fast to significantly influence the slow motion limit line shape here). Hence the large fractions of small reorientation angles observed in the stimulated echo experiments for the α -process are in agreement with the R(tp) changes of the solid echo line shape in the same temperature regime. The fast motion limit effects (i.e. changes in Cx for tp = 20 µs ) observed in the solid echo line shape at T≤165 K on the other hand can be adequately modelled in a scenario which only accounts for the β-process. Following the introductory remarks we will now focus on the region Tg<T<165 K and describe the line shape changes observed here within a model for the β -process. In consideration of our previous results, it is obvious that any applicable model has to incorporate an adjustable degree of restriction, i.e. the ability to account for the strong temperature dependence of 1S above Tg . Hence again the models discussed in chapter 4, i.e. a motion on or within a cone of (temperature dependent) semi angle χ – which both haven proven to work well in case of the T1 analysis in the last section – are chosen for the present approach. In the anticipated model only the fast motion limit effects will be considered (since in the temperature range under discussion the better part of the distribution G(log τβ) exhibits correlation times in the limit τβ 1 /δ ), this however renders the situation quite favourable 98 Chapter 6. Cyanocyclohexane -100 0 100 ν / kHz 146.5 K (a) 0 0.02 0.04 0° 20° 40° 60° 80° G (χ) χ (b) Figure 6.25 (a): Solid echo spectrum at 146.5K (open symbols), simulation (solid line) according to equation 6.9 and comprising sub-spectra of different ¯ δ ( χ )(grey lines). A step size of 5° was chosen in χ for the displayed sub-spectra, the simulated spectrum consists of sub-spectra with a step-size of 1° however. (b): Corresponding distribution of cone opening angles G(χ)in the “motion on a cone” model. as the details of the dynamic process, i.e. the elementary step of the motion, has no influence on the line shape any more. The proposed models differ therefore only by the distribution of angles θ that are accessible after infinite jumps, hence it is impossible to differentiate between uniaxial rotational diffusion and a random jump motion on a cone for example. Consequently only two cases have to be considered, which both yield a Pake spectrum with reduced apparent coupling ¯ δ in this limit: the “motion on a cone” model, i.e. the fast motion around a C n axis with n>2, where ¯ δ is given by equation 4.10 and the more general case of a bond freely moving on the unit sphere within a cone (where ¯ δis obtained from equation 4.11). Regarding the peculiar line shape of cyanocyclohexane in the discussed temperature region (cf. figure 6.12 (b) or 6.25 e.g.) it is obvious that none of the models is sufficient to describe the experimentally observed spectra via a fixed opening angle χ , as the spectra of cyanocyclohexane – apart from the overall narrowing – exhibit a significant broadening of the singularities which can only be reproduced via a distribution of cone angles G ( χ ). Since glassy dynamics is of heterogeneous nature, it seems straightforward not only to take into account a distribution of correlation times G(log τβ) , but also to assume such a distribution of restricting geometries 20 , as found in the analysis of toluene below Tg [ Vogel 2000a ]. For the thermally activated β -process the log-Gaussian distribution G(log τβ) is assumed to be connected to a Gaussian distribution of energy barriers g ( E ). Indeed a Gaussian distribution of opening angles χ proves to be versatile enough to reproduce the spectral intensity in cyanocyclohexane above Tg : the calculated spectrum represented by the solid line in figure 6.25 is obtained via a weighted sum of Pake spectra with different apparent 20 Both assumptions are only justified as long as τβτα , i.e. the isotropic motion of the α -process does not lead to exchange between the different sub ensembles of G(log τβ) and G ( χ )within the experimental time window. 6.4. Experimental results – refinement 99 -100 0 100 ν / kHz 111.7 K -100 0 100 ν / kHz 165.7 K -100 0 100 ν / kHz 137.0 K -100 0 100 ν / kHz 143.4 K -100 0 100 ν / kHz 152.7 K 0° 45° 90° 0° 45° 90° 0° 45° 90° 0° 45° 90° 0° 45° 90° Figure 6.26: Solid echo spectra for short evolution times tp = 20 µs (open symbols) at different temperatures with according simulations (solid line) within the “motion on a cone” model. All spectra have been symmetrized for reasons of clarity. Above each individual spectrum the corresponding distribution G(χ)is plotted. coupling constants ¯ δ(χ): S0(ω;T) = W(T) 90◦ X χ=0◦ G(χ, T)Sω;¯ δ(χ)+ [1 −W(T)] S(ω;δ0).(6.9) with a Gaussian distribution of opening angles G(χ) and the weighting factor (determined from the log-Gaussian distribution of correlation times G(log τβ ) observed in dielectric spectroscopy [Tschirwitz 2002a]): W(T) = ∞ Z log 1/δ G(log τβ, T)dlog τβ= 1 −1 2erfc −log δ−log 1 τβ(T) 2W(T)!.(6.10) which maps the fraction τβ> 1 /δ to a rigid limit Pake spectrum S(ω;δ0)21 . Hence only molecules in the fast motion limit are considered as any other approach would imply so far unjustified correlations between time scale and geometry of the β -process. The “motion on a cone” model was used for the example given in figure 6.25, yet a single Gaussian distribution of opening angles χ has proven to be flexible enough to reproduce the main features of the spectra within both models – by means of slightly different underlying distributions G(χ) however. Hence it is not possible to exclude either geometry from further considerations at this point. 21S(ω;δ0)is represented by the (rigid limit) solid echo spectrum recorded at T=34 K. 106 Chapter 6. Cyanocyclohexane -60 -40 ν / kHz 142.70 K (a) tp=10µs 100µs 200µs 400µs -60 -40 ν / kHz 146.80 K -60 -40 ν / kHz 156.86 K -60 -40 ν / kHz 161.15 K -60 -40 ν / kHz 163.38 K Figure 6.32 (a): Solid echo spectra for different echo delays tp at temperatures T > Tg . A blow-up of the inner lower singularity is shown to demonstrate the tp -dependence of the apparent coupling constant (solid line: tp = 10 µs , dashed line: 100 µs ). (b): Corresponding C 100 - values. 60 80 100 120 140 145 150 155 160 165 C100 / kHz T / K (b) tp = 10 µs 100 µs Prolonging the inter-pulse delay tp recovers partially the spectrum observed at lower temperatures: the apparent width Cx grows with tp and the outer flank (gradually) regains the rigid limit slope. In the same figure the C100 values at tp =10 and 100 µs are plotted: whereas the value taken at 10 µs decreases steadily towards higher temperatures (due to the dynamics of the β -process discussed in the last section), the values at 100 µs are almost constant. As the results in chapter 4have pointed out, such an effect can be attributed to a process with correlation times in the ms to µs regime, hence the secondary relaxation in cyanocyclohexane is already substantially too fast at the discussed temperatures. If the α -relaxation however proceeds via small angular jumps, as assumed in the simulations displayed in figure 6.24, the described effects are also observed in this case (represented by the “overshoot” in C80 observed for tp = 200 µs ) – and in contrast to the β -process the correlation times of the α -process account for the afore mentioned criterion in this regime, i.e. are on the order of 10−4−10−7s. If the observed behaviour is related to the dynamics of the α -process – which exhibited no other peculiarities in the present study – one could speculate that the effect is inherent to the structural relaxation, not observed in other glass forming systems due to the lack of fast motion limit line shape effects of the β -process above Tg : if the width of the solid echo spectrum for tp = 20 µs is not substantially reduced above Tg, the effect becomes very subtle and is easily overlooked. Indeed we found a similar line shape in meta-fluoroaniline, a structural glass former 6.4. Experimental results – refinement 107 -150 -100 -50 0 50 100 150 ν / kHz 156.86 K (a) 10µs 20µs 50µs 100µs -150 -100 -50 0 50 100 150 ν / kHz mFan - 185.45 K (b) 20µs 100µs 200µs -80 -60 ν / kHz (c) Figure 6.33 (a): Solid echo spectrum of cyanocyclohexane at 143 K for different interpulse delays tp . (b): Solid echo spectra of meta-fluoroaniline (m-FAN) at 182 K > Tg for different tp values. (c): Blow-up of the lower singularity to illustrate the analogy in tp -dependence. previously studied by S. Lusceac [ Lusceac 2005a ]. Meta-fluoroaniline also exhibits a fast β -process with G(log τβ) partially extending to the fast motion limit before merging with the α -process. In contrast to cyanocyclohexane the α -process is more fragile, the relaxation strength of the β -process (as measured by dielectric spectroscopy) substantially smaller and furthermore the observation of the discussed line shape changes is somewhat obstructed by the η6 = 0 spectrum of fluoroaniline. Nevertheless analogous tp -dependence of the line shape at the singularities can be found at temperatures above Tg (cf. figure 6.33), which was previously overlooked. In other systems like toluene e.g., such an effect is not observed due to the lack of pronounced line shape effects of the β -process (i.e. the spectrum still resembles a Pake pattern at high temperatures where the α -process becomes effective on the time scale of 1 /δ ). We will reassess this point in part II of this work, where binary mixtures of toluene are presented. 6.4.4 Dynamics of the α-process – reassessment After we have successfully modelled the β -process in cyanocyclohexane throughout the whole accessible temperature range in the previous sections, one last dynamic imprint on the 2H NMR observables remained speculative and therefore will be subject to a refined discussion: the te -dependence of the stimulated echo measurements and hence the microscopic dynamics of the α -process. In section 6.3.8 it was shown that the normalized hτ(te)i values are approximately described by a combination of small and large angular reorientations, with a strongly temperature dependent weighting of the latter fractions. As such a drastic change in the mechanism of reorientation within a narrow temperature range of about 10 K appears rather peculiar, we will discuss other possible reasons for the observed behaviour. In section 6.3.6 we have demonstrated (in case of the β -process) that the time window of the stimulated echo experiment is a key value in understanding the results of the technique, consequently we will test its influence on our measurements regarding the 108 Chapter 6. Cyanocyclohexane 0.1 1 0 20 40 60 80 〈 τ ( te ) 〉 / 〈 τc 〉 τe / µs (b) 0.1 1 10 100 〈 τ ( te ) 〉 / 〈 τc 〉 τe / µs α = 0.78 α = 0.23 (a) 157.8 K 155.2 K 152.5 K 151.3 K 147.3 K Figure 6.34 (a): Double logarithmic representation of hτ(te)i/hτci : for long evolution times a power law hτ(te)i ∝ t−α e emerges, whereby the exponent α exhibits a strong temperature dependence. (b): Normalized stimulated echo decay time constants resulting within the model of a distribution of small angular jumps and a fraction of 40° jumps (lines, fraction of large angles: 0.7, 0.3, 0.19, 0.11, 0.036 from top to bottom) in comparison with experimental results. α-process also. Figure 6.34 (a) displays the te -dependent correlation times of cyanocyclohexane at different temperatures in a log (hτ(te)i/hτci) vs. log te representation. On this scale it becomes evident that at all temperatures a power law hτ(te)i ∝ t−α e emerges for te≥10 µs , whereby the exponent changes from α = 0 . 78 at 147.3K to α = 0 . 23 at 157.8K . Hence the motion resembles at neither temperature the limiting cases of rotational diffusion ( α = 2) or random jump type reorientation ( α = 0). Consequently the scenario of reorientation via fractions of large and small angular jumps introduced in previous sections and employed in the literature [ Hinze 1998 , Jörg 2000 , Baldus 2005 ] appears applicable. To quantify the change in mechanism for the present assessment, we present a refined fit of the normalized hτ(te)i values within this model in figure 6.34 (b). Hereby a (temperature independent) distribution of small angular jumps (centred around γ = 2° ) was used to model the initial decay in hτ(te)i , whereas the fraction of large angular reorientations ( γ = 40° ) again defines the plateau observed at longest evolution times te . In addition the employed coupling constant has been altered to account for the reduced apparent spectral width observed due to the β -process in this temperature regime (best agreement was found for ¯ δ = 127 kHz ). This approach is versatile enough to model the experimental data for all evolution times and throughout the studied temperature range by adjusting the fraction of large angles from 0.036 at 147.3K to 0.7 at 157.8K. Figure 6.35 (a) displays the distributions of correlation times in cyanocyclohexane as obtained by dielectric spectroscopy at the latter temperatures: for the lowest temperature studied, 147.3K , the better part of the distribution G(log τα) is within the 2D NMR time window marked by the grey area. At the highest temperature, 157.8K , the distribution has almost left the window, with only the terminal relaxation time remaining accessible to the technique. Furthermore it becomes evident in this 6.4. Experimental results – refinement 109 0 0.2 0.4 0.6 0.8 1 146 148 150 152 154 156 158 A(T) T / K (b) G(lnτα) G(lnτβ) fraction of 2° jumps 10-2 10-1 100 10-12 10-10 10-8 10-6 10-4 10-2 100 G(lnτ) τ / s β α (a) 147.3 K 157.8 K Figure 6.35 (a): Distributions of correlation times G(log τ) for the experimentally accessible temperature window in 2D 2H NMR . GG and Gβ distributions were used for α - and β -process respectively, distribution parameters from Tschirwitz et al. [ Tschirwitz 2002a ]. (b): Integral of G(log τα) over the 2D 2H NMR time window A ( T )in comparison with the small angular jump fractions obtained in the afore described model from experimental data. representation that the β -process of cyanocyclohexane is not expected to contribute to any 2D NMR observables in the temperature regime under discussion. To quantify the fraction of G(log τα) within the time window of 2D NMR we will follow the approach presented in section 6.3.6: A(T) = Zlog T1 log 100µs dlog τG (log τα, T).(6.15) As in case of the β -process before we assume a certain degree of heterogeneity in the α -process here, i.e. exchange within the distribution G(log τα) is neglected: if the respective dynamics of a certain sub-ensemble is too slow, no decay other than T1 or T1Q is observed. If τα is too fast, a constant is obtained for all mixing times (one in case of the cosine-cosine correlation and zero in case of sine-sine). The quantity A ( T )for cyanocyclohexane is plotted in figure 6.35 (b): the distribution G(log τα) traverses the 2D NMR time window in the temperature region from 145 K to 160 K , i.e. A ( T )drops from one to zero in this regime. The temperature dependence of A remarkably resembles the one found for the small angle jump fraction from the fit in figure 6.34 (b), represented by the points in the same figure. Consequently the experimentally observed hτ(te)i dependence can be explained via A ( T )without the need for a change in the mechanism of motion: the stimulated echo amplitude is characterized by a temperature independent fraction of large angular reorientations and a fraction of small angular jumps that is weighted via A ( T ). Whether this assumption holds and possible reasons for the different impact of A ( T )on the respective fractions will be discussed in the remainder of this section. Comparison with literature data for structural glass formers First we will however repeat the present analysis for literature data of structural glass formers to ascertain that the proportionality of A ( T )and the fraction of small angular 110 Chapter 6. Cyanocyclohexane 0 0.2 0.4 0.6 0.8 1 140 160 180 200 220 240 260 A(T) T / K (a) A(T) otp mtcp cnc 0 0.2 0.4 0.6 0.8 1 0.85 0.9 0.95 1 A(T) Tg / T (b) Figure 6.36 (a): Integral of G(log τα) over the 2D NMR time window ( A ( T ), lines) for three glass forming substances in comparison with the small angle fraction extracted from corresponding stimulated echo experiments (symbols): ortho-terphenyl ( 2H NMR , [ Jörg 2000 ]), m-tricresylphosphate ( 31 P NMR, [ Baldus 2005 ]) and cyanocyclohexane (this work). (b): Same data represented on the reduced temperature scale Tg/T. jumps does not represent a peculiarity of cyanocyclohexane. We have chosen the systems m-tricresylphosphate (m-TCP) and ortho-terphenyl, as for both compounds temperature dependent measurements of Fss te are available. Ortho-terphenyl was studied by means of 2H NMR by Jörg et al. [ Jörg 2000 ], corresponding dielectric data for the calculation of A ( T )was taken from Gainaru et al. [ Gainaru 2007a ]; m-TCP 31 P NMR data originates from Baldus et al., corresponding dielectric results from Gainaru et al. [ Gainaru 2007b ]. In both cases the te -dependence of the stimulated echo was fitted in a similar model as employed for cyanocyclohexane in this work, hence the extracted small angular jump fractions are directly comparable. Figure 6.36 (a) displays the reported fractions alongside with A ( T )from equation 6.15, calculated from the corresponding G(log τα) extracted from dielectric results. For both system the agreement of A ( T )with the fraction of small angular jumps observed in the stimulated echo is equally well as in the case of cyanocyclohexane. Consequently the proportionality of the latter quantities does neither represent a peculiarity of cyanocyclohexane nor of the employed 2H NMR technique, as the results from 31 P NMR are virtually the same. Figure 6.36 (b) present the same data on the Tg/T scale: the change in the te -dependence of the stimulated echo is apparently not linked to Tg , it solely reflects the time scale of the α -process and is hence shifted according to the fragility min this representation. Consequently the present results render the assumption of a universal change in the mechanism of the α -process in the regime τα≈ 10 −4. . . 10 −1 s rather unlikely and are clearly in favour of an explanation via the time window of the employed technique. Discussion To conclude the present analysis, we will briefly review its implications on the microscopic mechanism of the α -process. When discussing a motion connected with 6.4. Experimental results – refinement 111 0 0.1 0.2 0.3 0.4 0.5 0.6 -6 -5 -4 -3 -2 -1 0 1 2 Fss( tm=0 ) log( τc ) (a) te = 3µs tp = 15µs γ = 2° 10° 35° 0 0.1 0.2 0.3 0.4 0.5 0.6 -6 -5 -4 -3 -2 -1 0 1 2 Fss( tm=0 ) log( τc ) (b) te = 100µs tp = 15µs Figure 6.37 (a): Initial amplitude of Fss te in the four-pulse sequence with te = 3µs and tp = 15 µs for random walk simulations of different discrete γ -jump motions. (b): Same representation for an evolution time of te=100 µs. the α -process that agitates via small angular displacements, at first the excess wing comes to mind. This feature, observed as a power law on the high frequency flank of the α -process in dielectric spectroscopy, was shown to exhibit such a motional process [ Gainaru 2008 ] and is also found in cyanocyclohexane (cf. figure A.1 (b)). Yet the excess wing was neglected in the calculation of A ( T ), but still the agreement with the observed small angle fraction is remarkable. Furthermore it remains questionable that this subtle feature may account for the large amount of small angular reorientations observed at low temperatures, where A(T)≈1. On the other hand the observed behaviour arises naturally if the distribution G(log τα) is heterogeneous with regard to the geometry of the process. Therefore we assume a scenario in which the α -process of molecules that are found in a rather “flat” area of the energy landscape, is fast and agitates via small angular displacements, whereas a small fraction of molecules are “trapped” and only reorient on longest times via large angular jumps when the surrounding structure has relaxed. Then the observed temperature dependence of the stimulated echo arises directly due to the suppression of the fast fraction, which exits the time window first. The random walk simulation discussed in chapter 4offer however an even simpler explanation: due to the separation of time scales τj/τc for a motion progressing via small angles, the contribution of such a fraction is inherently reduced at longer times τc even if the correlation times are identical to the fraction progressing via large angle reorientations. Figure 6.37 (a) presents the initial amplitude ( tm→ 0) of Fss te from random walk simulations for different discrete γ -jump motions: for an evolution time of te = 3µs the contribution from molecules undergoing 2° -jumps is reduced for τc< 10 −2 s, whereas in case of 35° -jumps the reduction sets in only around 10 −3 s. For te = 100 µs the effect becomes even more pronounced, as the reduction factor is shifted more than two decades in τc between the two cases (figure 6.37 (b)). With regard to these findings the te -dependence in the stimulated echo for the α -process can be explained in the previously introduced model without any change in the mechanism of reorientation with temperature or further undue assumptions. A detailed study of the experimentally observed te -dependence by means of random walk simulations is however beyond the scope of this work: too many free parameters 112 Chapter 6. Cyanocyclohexane arise in the model and the calculations are time consuming as fast dynamics during te and tp have to be considered in the approach, opposed to the usually employed simulations in the slow motion limit. It remains unclear if the underlying distribution of jump geometries is indeed bimodal or continuous but strongly asymmetric. Furthermore possible spatial and temporal correlations between the small and large angle jumps remain an open question, as does the exchange rate between the two fractions. Nevertheless we were able to rationalise the peculiar temperature dependence of the motional process within this work. A finding which is of great significance with regard to the universal nature of glassy dynamics: previously differences in the microscopic dynamics of miscellaneous glass-forming systems were identified via measurements at a single temperature (cf. [ Böhmer 2001 ] and references therein), which – as we have demonstrated – can lead to undue conclusions. 6.5 Summary and conclusions In the present part of this work we have studied the molecular motion in the plastic crystalline phase (PC) of cyanocyclohexane by means of different 2H NMR methods. Cyanocyclohexane was affirmed to form a plastic crystal below Tm = 285 K with a (possibly) face-centred cubic structure and a lattice constant of a = 9.1Å . This translational order is however not reflected in the molecular motion in the supercooled and glassy PC phase of the system: 2D 2H NMR experiments for the α -process have demonstrated that the motion progresses via the same mechanism as reported for structural glass formers, i.e. a model comprised of small and large angular jumps is applicable. This finding is in contrast to other plastic crystalline system like cyanoadamantane for example, where a 90° jump process is observed which directly reflects the symmetry of the cubic lattice [ Lusceac 2004 ] – although also in the case of cyanocyclohexane the motion of a molecule around its lattice position has to be of cooperative nature, as it is constrained by the neighbouring molecules. By means of random walk simulations and a joint approach with dielectric spectroscopy, we were able to demonstrate that the fractions of small and large angular jumps in the model for the α -process not necessarily exhibit a temperature dependence slightly above Tg , although such a behaviour is clearly reflected in the te -dependence of the stimulated echo. The observed change can rather be rationalized by time window effects of the employed 2D 2H NMR techniques, which yield a suppression of the small angular motion in the experiment at relatively lower temperatures. This explanation was also shown to hold in the case of previously studied structural glass formers. Due to the fast and non-merging characteristics of the β -process in cyanocyclohexane pronounced fast motion limit effects were observed in the 2H NMR observables at high temperatures. Below Tg however the mechanism of the β -process in the present system was shown to be virtually identical to the one observed in structural glass formers like toluene for example, i.e. also the secondary relaxation does not exhibit any peculiarities reflecting the translational order of the PC phase. Above Tg a pronounced minimum in T1 was observed for the β -process – for the first time in a glass 6.5. Summary and conclusions 113 0 0.5 1 -12 -10 -8 -6 -4 -2 0 2 F2 log t / s Tg 150K 170K 190K 270K Figure 6.38: Sketch of single particle correlation functions F2 in cyanocyclohexane. A purely heterogeneous scenario was assumed, wherefore the decays due to α - (long time decay) and β -process (decay at short times) are exponential with τ = τα/β at the respective temperatures. The relaxation strength 1S of the β -process is obtained from the T1 analysis. The dashed line marks F2 at 150 K for the largest opening angles χ observed in the analysis of the fast motion limit line shape. forming system. T1 was successfully modelled in the whole accessible temperature range by means of G(log τ) from dielectric spectroscopy and the assignment of the second minimum to the β-process was verified. Furthermore the solid echo line shape showed strong and characteristic deviations from a Pake pattern in a large temperature regime above Tg . The spectra were successfully modelled via a motion restricted to the circumference of a cone with a Gaussian distribution of opening angles G ( χ )within an approach that directly arises from the behaviour observed below Tg by a growing relaxation strength, i.e. via changes in the maximum and width of the distribution G ( χ ). Consequently the distribution of restricting geometries for the β -process, G ( χ ), was for the first time directly accessible to 2H NMR via the fast motion limit line shape, which allowed for a confirmation of previously developed models for structural glass formers below Tg , where G ( χ )can not be quantified in straightforward manner. Whereas the temperature dependence of the dielectric relaxation strength was shown to be in agreement with our 2H NMR results, the magnitude of 1 −S(1) strongly varies among glass formers. As 2H NMR detects a rather universal β -process in the studied systems, our results indicate that 1 −S(1) = ∆ β/ ∆  may not represent a valid quantity to determine the relaxation strength of the β-process. Apart from an extension and refinement of the microscopic model for the dynamics of the β -process, the present study demonstrates the significance of the latter for glassy dynamics. No plateau was observed in the extracted 1S : the relaxation strength was shown to grow until Tm in the analysis of T1 . Figure 6.38 presents a simple sketch of the correlation function F2 in a purely heterogeneous scenario employing 1S as obtained from the T1 analysis: at Tg the β -process accounts in accordance with common perception only for a subtle correlation loss, at higher temperatures however 1S grows and the correlation function is rendered distinctively bimodal. The dashed line demonstrates that already at 150 K the β -process yields a decay of F2 to about 0.3 for a small fraction of molecules. Around the melting point the β -process finally accounts for the better part of correlation loss in this representation. 114 Chapter 6. Cyanocyclohexane This finding is in accordance with the random first-order transition theory of glasses, which predicts that the β -process becomes the dominant mode of structural relaxation at high temperatures [ Stevenson 2010 ], and again raises questions with regard to the dynamics in structural glass formers at temperatures above the merging of α - and β-process. Part II Binary Glass Forming Systems 122 Chapter 7. Introduction of the neat component A . Furthermore we have neglected a potential contribution of species Bto the β-process of A, nevertheless equation 7.4 will be employed (and extended to account for the latter scenario when appropriate) when comparing prior dielectric results to our 2H NMR studies in the following chapters. Chapter 8 Toluene in PCB Contents 8.1 Introduction ............................ 123 8.2 Dielectric Spectroscopy ...................... 124 8.3 Experimental details ....................... 127 8.4 Thermal analysis .......................... 128 8.5 NMR results – an overview ................... 130 8.5.1 Spin lattice relaxation ...................... 130 8.5.2 Solid echo spectra – high temperature regime ......... 135 8.5.3 2D exchange spectra ....................... 139 8.5.4 Stimulated echoes – high temperature regime ..........141 8.5.5 Solid echo line shape – low temperature regime .........147 8.5.6 Stimulated echoes – low temperature regime .......... 152 8.5.7 Spin lattice relaxation – revisited ................ 155 8.6 Plausibility check ......................... 160 8.6.1 Modelling the toluene dynamics in mixtures with PCB . . . . . 161 8.6.2 Models for two dynamically distinct toluene sub-ensembles . 165 8.6.3 Discussion ..............................171 8.1 Introduction The present chapter focuses on mixtures composed of two small molecular glass formers: toluene and a polychlorinated biphenyl (PCB). The compounds are inD D D H H H H H C C C C C C C H H H D D D D D C C C C C C C (Cl)n(Cl)n Figure 8.1 left: Sketch of a toluene-d3 molecule deuterated at the methyl group. Center: ring deuterated toluene-d5 . Right: general sketch of a polychlorinated biphenyl. Here n denotes the number of chlorine atoms per phenyl ring, for the present samples composed of PCB54 2n= 4 . . . 5holds. 124 Chapter 8. Toluene in PCB toluene-d3toluene-d5PCB54 full name toluene-α, α, α-d3toluene-2,3,4,5-d5aroclor 1254 Mn[g/mole] 95.16 97.17 327 ρ[g/cm3] 0.895 0.912 1.51 Tg[K] 117 [Levy 1983] 117 [Levy 1983] 241 chemical purity [%] 99.7 99.2 tech. % atom D 99.9 99.6 - Table 8.1: Material characteristics of the components. For PCB54 the molecular weight denotes an average value, as the number of chlorine atoms per molecule varies. teresting candidates for an in-depth 2H NMR study for numerous reasons: the components are miscible in the whole concentration range, both consist of rigid molecules and are good glass formers that provide a relatively large contrast in Tg (∆ Tg = 124 K ). Furthermore there exist contemporary dielectric measurements from Cangialosi et al. [ Cangialosi 2008 ], which however lack to provide conclusive answers to the questions raised in the introduction to this part, as the results regarding toluene are somewhat ambiguous due to the low dipole moment of the system. The high mobility component in the mixtures, the glass former toluene, was intensively studied by different techniques in our group before – besides dielectric measurements [ Kudlik 1997b , Kudlik 1999 , Blochowicz 2003 ] there exists a detailed NMR study on the β -process in toluene by M. Vogel [ Vogel 2001a ], which provides us with the aptitude to monitor even subtle changes in the secondary relaxation of toluene upon mixing with PCB – if present. The low mobility component is Monsanto’s Aroclor 1254 or PCB54: a polychlorinated biphenyl with a chlorine weight fraction of 54 % , which corresponds to four to five statistically distributed chlorine atoms per biphenyl ring. This somewhat random structure of the PCB molecules has the benefit that the tendency towards crystallization of the component is reduced in comparison to isomers with well defined chlorine bonding sites. PCB54 is toxic and carcinogenic but was widely used in capacitors and as plasticizer in paints and plastics 1 before being banned under the Stockholm convention in 2001. This chapter is organized as follows: after a brief discussion of the dielectric results from Cangialosi et al., we will present our own DSC and NMR results – a refined discussion and interpretation of the data will be part of the next chapters, where the results are compared to other binary glass forming systems containing toluene as mobile component, in the attempt to draw conclusions of more general validity. 8.2 Dielectric Spectroscopy In the following section we briefly review the results from dielectric spectroscopy on mixtures of toluene and PCB54 as on the neat components themselves. Dielectric spectra and time constants of toluene are presented and discussed in the introduction and part Iof this thesis, as previously all dielectric results on neat toluene are adapted 1 In the present mixture PCB54 is the low mobility component, whereas it was mainly used as high mobility additive to polymers with an even higher Tg. 8.2. Dielectric Spectroscopy 125 10-3 10-2 10-1 100103106 ε’’ ν / Hz 133 K 143 K 153 K 163 K xtol = 0.87 α α’/ β (a) -12 -10 -8 -6 -4 -2 0 2 3456789 log ( τ / s ) 1000K / T xtol=0.00 PCB54 0.60 0.78 0.87 0.93 toluene 1.00 (b) α α’/ β Figure 8.2 (a): Loss part of the dielectric permittivity for a concentration of xtol = 0.87 toluene in PCB54 [ Cangialosi 2008 ] at different temperatures. For all temperatures two distinct loss peaks are discernible: a low frequency ( α -) peak and a high frequency peak with lower intensity (denoted α ’/ β ). (b): Corresponding time constants of both processes for different toluene concentrations [ Cangialosi 2008 ] including neat PCB54 [ Casalini 2002 ] and neat toluene data [Kudlik 1999] (DS,) [Rössler 1984] (NMR,). from Kudlik et al. [ Kudlik 1997b , Kudlik 1999 ]. Neat polychlorinated biphenyls were studied by Casalini et al. [ Casalini 2002 ]: the systems are so-called type-A glass formers [ Kudlik 1999 ], i.e. the dielectric loss exhibits an α -peak with an excess-wing but lacks a distinct β -process. The time constants of the α -process follow a VFTbehaviour (cf. figure 8.2 (b)) and are highly dependent on the degree of chlorination: Tg drops about 48 K with the chlorine content from PCB62 to PCB42 (i.e. from 62 to 42 weight percent). The high frequency wing in contrast appears almost unchanged, which may be explained via an interpretation of the excess wing as a submerged β-process [Blochowicz 2004]. Dielectric measurements on mixtures of toluene and PCB54 were performed by Cangialosi et al. [ Cangialosi 2008 ], the discussion of which makes up the remainder of this section. At this point we have to remind the reader that dielectric spectroscopy on binary mixtures – in contrast to NMR – always measures a bulk quantity, i.e. the response of both components in the mixture. As the dipole moment of PCB54 is however significantly larger than the one of toluene, the main response is expected to arise from PCB even at relatively high toluene concentrations. Figure 8.2 (a) displays the loss part of the dielectric permittivity for a mole fraction of xtol =0.87 toluene in PCB54. At all displayed temperatures two discernible peaks are observed in the spectra of the mixture: a low frequency peak with a strong temperature dependence (denoted “ α ”) and a high frequency peak of lower intensity (denoted “ α ’/ β ”), exhibiting a weaker temperature dependence. The time constants corresponding to the α -peak follow a VFT-law and are highly dependent on concentration, cf. figure 8.2 (b). The relaxation strength of the process approximately scales with the PCB54 content, as seen in figure 8.4 (b). Consequently the peak can be identified with the α -process of PCB54 in the binary mixtures, which becomes faster upon higher concentrations of toluene due to the plasticizer effect imposed by the latter molecules. The width 126 Chapter 8. Toluene in PCB Figure 8.3: Loss part of the dielectric permittivity of toluene / PCB54 [ Cangialosi 2008 ] mixtures at different concentrations (including neat toluene [ Kudlik 1997b ] and PCB62 [ Casalini 2002 ]). The data is scaled in frequency with respect to the dominant loss peak and normalized in amplitude. The temperatures were chosen to correspond to similar relaxation times (in all cases the applied frequency shift is less than 1.5 orders of magnitude). Full lines are fits via a HN-function. 10-1 100 10-2 100102104 ε’’ / ε’’max ν / νmax xtol = 1.00, 119K 0.93, 138K 0.87, 148K 0.78, 158K 0.60, 178K 0.00, 282K of the α -process exhibits a distinct concentration dependence, cf. figure 8.3. For concentrations xtol≥ 0.87 the high frequency flank is similar to the behaviour found in neat toluene, whereas a slight broadening is observed on the low frequency side of the peak. For lower concentrations the peak becomes very broad and almost symmetric, which contradicts the limit xtol→ 0, as neat PCB54 exhibits a rather narrow loss peak. The fast process is only resolved at lowest temperatures, where it follows an Arrhenius temperature dependence for T<140 K and coincides for all concentrations with the β -process of neat toluene (cf. figure 8.2 (b)). For the two samples with highest toluene content also time constants at temperatures above the merging of α - and β -process in neat toluene are reported: in this regime they follow the α -process of neat toluene, albeit slightly shifted towards higher temperatures in case of xtol =0.87. The universal (i.e. concentration independent) nature of this process is demonstrated in figure 8.4 (a): at 118 K in first approximation neither the maximum position, nor the width of the fast process depends on the concentration of PCB54 – instead the response of neat toluene is observed, i.e. the β -process of toluene persists in the mixtures. Its relaxation strength scales with the toluene content for concentrations above xtol =0.80 (figure 8.4 (b)), at lower concentrations the experimentally observed relaxation strength is significantly higher than predicted by a linear extrapolation of ∆  according to equation 7.2. In the same concentration range ∆  of the α -peak is somewhat smaller than predicted from equation 7.2. Whereas the dielectric measurements serve to provide a convenient overview of the PCB54 related dynamics and were furthermore able to resolve the β -process of (neat) toluene in the mixtures, the technique fails to differentiate between the scenarios depicted in the introduction to this part (cf. figure 7.2): it remains unclear if the toluene related dynamics observed at relative high temperatures in the mixtures with xtol =0.93 and 0.87 toluene content accounts for all toluene molecules or if a fraction reorients on the time scale of PCB54 in the mixtures (not observed due to the low dipole moment). Furthermore, as no time constants for the α -process of toluene are reported below the merging temperatures of α - and β -process in the mixtures, the attribution of the high temperature, high frequency loss peak to the structural relaxation of toluene by the authors remains speculative – it might still 8.3. Experimental details 127 10-3 10-2 10-1 100103106 ε’’ ν / Hz (a) xtol = 1.00 0.93 0.87 0.78 0.60 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 ∆ε xtol (b) 10 × ∆εβ ∆εα Figure 8.4 (a): Loss part of the dielectric permittivity of toluene / PCB54 mixtures for different concentrations at T = 118 K [ Cangialosi 2008 ] including data of neat toluene [ Kudlik 1999 ]. (b): Relaxation strength of the high ( # ) and low ( ) frequency loss peak in the toluene / PCB mixtures [ Cangialosi 2008 , Cangialosi 2009 ]. The lines represent a linear extrapolation of the relaxation strengths of PCB54 and toluene via equation 7.2 respectively. be the β-process with the α-peak submerged under the loss peak of PCB54. The 2H NMR measurements carried out in this work, which selectively probe the dynamics of the deuterated toluene molecules, will help to elucidate the behaviour of toluene in the present mixtures as well as provide further insight on the mechanism of reorientation. 8.3 Experimental details Having summarized the dielectric results of Cangialosi et al., we will now focus on our own studies of the aforementioned mixtures by means of differential scanning calorimetry (DSC) and 2H NMR . After a brief parenthesis on experimental details and sample preparation, the section will begin with a discussion of the DSC results, as they provide rapid means of comparison with regard to the sample preparation process of Cangialosi et al. and the one used in this work. The deuterated toluene samples were obtained from CDN isotopes and used without further purification (for purities and other material properties see table 8.1). The first set of PCB54 samples (dissolved in the mixtures with xtol =0.87, 0.74 and 0.36 toluene-d3 ) are courtesy of Daniele Cangialosi and originate from Ultra Scientific, the remainder of PCB54 was obtained from Supelco Analytical with the same purity as the aforementioned. For sample preparation we pursued the following scheme: PCB was supplied in ampoules of 50 mg , the contents of which were dissolved with the appropriate amount of toluene-dx2 to obtain the desired concentration. After agitation and annealing for several hours at 50 ◦C in the closed ampoule to assure thorough mixing, the 2 Whenever in order we will differentiate between the various labelled toluene samples by denoting the degree of deuteration, the term toluene-dx will be employed when discussing NMR arguments of general validity, i.e. independent of the labelling site. 128 Chapter 8. Toluene in PCB -1.4 -1.2 -1 -0.8 120 150 180 210 240 270 DSC (shifted) T / K 30 K / min 20 K /min 10 K / min xtol = 0.59 (b) 0.1 0.2 0.3 0.4 0.5 120 140 160 180 200 DSC (shifted, a.u.) T / K 0.590.770.86 20 K / min xtol = 1 (a) 137 K 147 K 173 K 117 K Tg Figure 8.5 (a): Temperature dependent heat flow from DSC scans of toluene / PCB54 mixtures for different concentrations and a heating rate of 20 K/min . The dashed lines mark Tg defined as the onset of the step for xtol =0.59. (b): DSC scans of x=0.59 toluene in PCB54 for different heating rates. A second step in the heat flow (marked by the arrow) can be anticipated for higher heating rates. samples were transferred to a 5mm NMR glass tube which was subsequently sealed under light vacuum. Afterwards the ampoules were purged with acetone and dried to acquire the accurate initial weight of PCB for a refined declaration of the obtained concentration. All concentrations given in the remainder of this part will denote the toluene mole fraction. DSC measurements were conducted on a Netzsch DSC 200 cooled with liquid nitrogen and using helium as purging gas. All heat flow patterns displayed in this work were obtained upon heating after annealing the sample for 2min at 8090 K . 2H NMR measurements on the binary mixtures were conducted with the same experimental setup as employed in chapter 6, i.e. at a magnetic field strength of 7.05 T corresponding to a deuterium Larmor frequency of 46.07 MHz . Again a Bruker Avance DSX 300 console and a home built low temperature probe placed in a Cryovac static cryostat, which was controlled via an Oxford ITC 503 temperature controller were used 3 . For details cf. section 6.3.1. 8.4 Thermal analysis Glass formers are conveniently characterized by DSC measurements, as a step in the temperature dependent heat flow, obtained while heating the sample, provides a convenient and reliable way of determining the glass transition temperature Tg . Following the discussion in chapter 7, DSC studies on the toluene / PCB54 mixtures are of interest for mainly two reasons: as mentioned above, the 2H NMR measurements are only sensitive on the deuterium labelled toluene molecules - as the dielectric studies provided mainly data on the PCB related dynamics, we have no direct means 3Temperature stability was again better than 0.2K. 8.4. Thermal analysis 129 -10 -8 -6 -4 -2 0 2 4 3 4 5 6 7 8 log (τ / s) 1000K / T 0.59 0.77 0.860 1 xtol = (a) DS DSC 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 120 160 200 240 280 320 1000 K / Tg Tg / K toluene conc. (b) 1000K/Tg vs. xtol 1000K/Tg vs. wt. %/100 Tg vs. xtol Tg vs. wt. %/100 Tg C vs. xtol Figure 8.6 (a): Time constants extracted from the glass transition steps in figure 8.5 (a) ( τ(Tg) := 100 s , full symbols) in comparison with dielectric results [ Cangialosi 2008 ] (corresponding open symbols) for different concentrations of toluene in PCB54. For xtol =0.59 the Tg from the second step in figure 8.5 (b) for a heating rate of 30 K / min is also given. (b): Concentration dependence of Tg in the mixtures of toluene with PCB54 on various scales, see text. of comparing the results in a straightforward manner. The DSC scans allow us to check the sample preparation process by comparing the Tg s of mixtures prepared by Cangialosi et al. and us, as we studied the same concentrations to employ the dielectric results in our analysis. Furthermore the measurements could give first hints on the toluene related dynamics via the possible arise of a second step in the heat flow, due to a process submerged under the dominant loss peak of PCB in the dielectric spectra. Figure 8.5 (a) displays the DSC curves of toluene / PCB54 mixtures at different concentrations for a heating rate of 20 K/min . The anti-plasticizer effect, i.e. the slowing down of dynamics due to higher concentrations of PCB54 is clearly observed. Furthermore the glass transition steps in the mixtures appear to broaden with decreasing toluene concentration4. For heating rates above 20 K/min a small, second step at T = 157 K becomes discernible in the sample with xtol =0.59 toluene content, cf. figure 8.5 (b) - a finding previously reported for other binary glass forming systems [ Taniguchi 2004 , Blochowicz 2011 ]. Due to problems with the baseline of the apparatus at low temperatures however, the measurements are not unambiguous – although the second step appeared in several different cooling runs. This feature was not observed in the samples with higher toluene content, possibly due the fact that the second step is submerged under the large high temperature step, which appears at lower temperatures as the PCB54 content is reduced. Assuming τ ( Tg ) = 100 s5 , the Tg ( x )values (defined as the temperate of the onset of 4 The reproducibility of the base line in the apparatus is however not sufficient for a quantitative discussion of the effect. 5 This convention holds for a heating rate of 10 K/min . The measurements were done however with 20 K/min to improve the signal to noise ratio. For a comparison with dielectric results we will 130 Chapter 8. Toluene in PCB the step) indicated in figure 8.5 (a) are plotted in figure 8.6 (a) alongside with the correlation times obtained by dielectric spectroscopy for the same concentrations ( xtol =0.59, 0.77 and 0.86 [ Cangialosi 2008 ]) and literature data of the neat components [ Casalini 2002 , Kudlik 1999 , Rössler 1984 ]. The Tg values resulting from the large, upper step in the DSC curves are in good agreement with the VFT fits to the dielectric data, which validates agreement in the sample preparation processes (the point resulting from the small step in the xtol =0.59 sample will be discussed later in the framework of the NMR measurements). The concentration dependence of Tg is plotted in figure 8.6 (b) in terms of weight and mole fraction: it becomes obvious that the Fox equation [Fox 1956] 1 Tg =wA TA g +wB TB g (8.1) is not fulfilled, i.e. no linear dependence of 1/ Tg on the weight fraction is observed ( # ). In first approximation we found however a linear relation between Tg and the toluene mole fraction (). 8.5 NMR results – an overview We investigated mixtures of PCB54 with two different spin labelled toluene molecules: toluene-d3 and toluene-d5 . Since toluene is apart from the methyl group rotation a rigid molecule, the samples are expected to reflect similar dynamics except for the latter rotation, which is fast with respect to the NMR time scale at all investigated temperatures. The first set of samples was composed of toluene-d3 , which due to fast rotation of the methyl group exhibits considerable shorter spin-lattice relaxation times T1 at lowest temperatures (cf. figures 8.7) and therefore provides the benefit of convenient measurement times in this temperature range. This feature is however also a drawback of toluene-d3 : T1 and the tp -dependence of the solid echo spectra are governed by the methyl-group rotation at low temperatures and hence provide no direct insight on the dynamics of the molecule as a whole. Due to the short spin lattice relaxation time at low temperatures it is furthermore inapplicable to study the β -process of toluene-d3 by means of 2D NMR, as the available time window is too small. To overcome these obstacles, samples with toluene-d5 were prepared to conduct concerted measurements at low and lowest temperatures. The following overview presents all data from the employed 2H NMR experiments, including preliminary analysis and a brief discussion of the results within each technique. This approach will facilitate a refined discussion in section 8.6, where all results are combined in the attempt to provide a coherent picture of the dynamics in mixtures of toluene and PCB54. 8.5.1 Spin lattice relaxation We will begin the review of 2H NMR results with a discussion of spin-lattice relaxation measurements. The spin lattice relaxation times hT1i of toluene-dx in PCB54 for accept the resulting error for the time being. 8.5. NMR results – an overview 131 10-2 10-1 100 101 3 4 5 6 7 8 9 10 11 12 〈 T1 〉 / s 1000 K / T (c) d5 Tg xtol = 1.00 0.86 0.77 0.59 0.27 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 1000 K / T βK (d) 10-1 100 3 4 5 6 7 8 9 10 11 12 〈 T1 〉 / s 1000 K / T (a) d3 xtol = 1.00 0.87 0.74 0.41 0.36 0.6 0.8 1 4 6 8 10 12 14 1000 K / T βK (b) Figure 8.7 (a): Average spin lattice relaxation times hT1i for different concentrations of toluene-d3 in PCB54 at ωL = 46.07 Mhz (neat toluene-d3 ( × ) [ Rössler 1984 ] recorded at ωL = 55 MHz ). The lines serve as guide for the eye. (b): Corresponding stretching parameter βK . Again, the lines are guide for the eye. (c): Average spin lattice relaxation times hT1i for different concentrations of toluene-d5 in PCB54 at ωL = 46.07 MHz (neat toluene-d5 ( × ), above Tg = 117 K : [ Rössler 1984 ], ωL = 55 MHz , below Tg : [ Vogel 2000a ], ωL = 46.07 MHz ). The lines serve as guide for the eye, the vertical bars mark the respective Tg of the mixture. (d): Corresponding stretching parameter βK. Again, the lines are guide for the eye. different concentrations are displayed in figure 8.7. As T1 becomes non-exponential at temperatures slightly below the minimum, the average spin lattice relaxation time hT1i = T1/βK· Γ β−1 K from a Kohlrausch fit to the experimental magnetization curves is plotted. The hT1i minimum, which according to the BPP condition τ = 1 /ωL marks an iso-kinetic point, is shifted more than 100 K towards higher temperatures as the toluene concentration is lowered from xtol =1 to 0.27. The corresponding time constants are displayed in figure 8.8. In contrast to the DSC measurements, where a mixed response from both components is probed, the selective 2H NMR T1 measurements show a distinct decoupling of the time scale of toluene motion from the PCB54 dynamics observed in dielectric spectroscopy: for each concentration the average correlation time of toluene molecules at the temperature of the respective T1 minimum is about two orders of magnitude faster than the corresponding τα 138 Chapter 8. Toluene in PCB -8 -6 -4 -2 0 2 3 4 5 6 7 8 log (τ / s) 1000K / T xtol = 0 1 (a) d3 0.36 0.41 0.74 0.87 DSC 1D W(T) T1 min DS -8 -6 -4 -2 0 2 3 4 5 6 7 8 log (τ / s) 1000K / T xtol = 0 1 (b) d5 d3 0.59 0.77 0.86 0.27 Figure 8.12 (a): Time constants extracted from the line shape weighting factor W (the errorbar marks beginning and end of the two-phase region: W ( T )=0.01–0.99) for different concentrations of toluene-d5 in PCB54 in comparison with corresponding VFT-interpolations from dielectric spectroscopy (cf. section B.1). For comparison previous results and literature data is repeated from figure 8.8. (b): Same representation for mixtures of toluene-d5 in PCB54. of PCB serves as an upper bound for the correlation time of toluene in the mixtures. The samples with xtol =0.87 toluene-d3 and xtol =0.86 toluene-d5 exhibit a slightly different behaviour: here the time constant for the high-temperature end of the two phase region does not coincide with τα from DS. Instead a consistent decoupling of dynamics also for the slowest toluene sub ensembles is observed in both mixtures. As it may seem plausible that the motion of the PCB molecules in the mixtures provides a terminal relaxation time for the toluene molecules, this finding is nevertheless non-trivial. It demonstrates clearly that a number of toluene molecules find themselves in the vicinity of PCB molecules with a residence time defined by the motion of the highTg component. Whereas the majority of toluene molecules reorient on a significantly faster time scale, i.e. the first coordination shell of toluene on PCB appears not to be subject to fast exchange any more in the microsecond regime. Fast motion limit effects In the last paragraph the solid echo line shape observed in mixtures of toluene with PCB was analysed with regard to the “two phase” characteristic. In this approach any (subtle) change in the solid fraction of the spectra, i.e. the Pake pattern, was neglected. Figure 8.13 displays the apparent spectral widths Cx (cf. section 4.2.2) for mixtures of toluene-d3 , which was determined from the solid fraction of the spectra, i.e. by neglecting the central line in the two phase spectra at high temperatures. For lowest temperatures (below about 120 K ) the line shape parametrized by the apparent spectral widths displays a universal, concentration independent behaviour. The change in Cx with temperature is more pronounced than in the case of toluene-d5 (cf. 8.5. NMR results – an overview 139 28 30 32 34 36 38 40 42 44 40 80 120 160 200 240 Cx / kHz T / K (a) C50 C80 C100 xtol = 1.00 0.87 0.74 0.41 0.36 0 2 4 (C50-C80) / kHz (b) 0 2 4 0.8 1.2 1.6 2.0 2.4 (C80-C100) / kHz Tg C / T Figure 8.13 (a): Apparent spectral width Cx of solid echo spectra for toluene-d3 in PCB54 (full symbols bottom: C100 , open symbols: C80 , full symbols top: C50 ). (b): Slope of the outer flank of the spectra quantified via C80 −C100 (bottom) and C50 −C80 (top) on a reduced temperature scale. figure 6.31 for example), since in toluene-d3 the fast methyl group rotation imposes a strong effect on the width of the spectra at lowest temperatures, whereas the line shape in toluene-d5is dominated by the β-process in this regime. Around 120 K the values for the highest concentration ( xtol =0.87) start to deviate from this common envelope as Tg of the mixture is approached and the spectral width is reduced more effectively by a stronger β -process. At higher temperatures the α-process leads to a more effective narrowing of the spectra before the solid part of the two phase spectra finally vanishes. This behaviour is found in all concentrations, albeit shifted towards higher temperatures for lower toluene concentrations. Consequently the temperature of the “kink” observed in Cx yields a measure analogous to Tg : TC g . The finding that TC g≈Tg holds for all concentrations ( 4 symbols in figure 8.6 (b)), again demonstrates that a fraction of toluene molecules reorient on the time scale of PCB in the mixtures. The shape of the spectra determined by relative differences in Cx taken a different intensities neglects the influence of the methyl group rotation, as the latter only affects the overall width. Hence when plotted on a reduced temperature scale TC g / T (cf. figure 8.13 (b)), the spectral evolution with temperature appears concentration independent. Consequently the mechanism of motion appears not to be concentration dependent for the slowest toluene molecules in the mixtures. 8.5.3 2D exchange spectra In the next step we address the time scale and mechanism of toluene dynamics in the millisecond regime for the present mixtures, i.e. by means of 2D 2H NMR . Hereby 2D exchange spectra will serve to provide a first overview on the mechanism of the reorientation. In figure 8.14 the spectra of xtol =0.74 toluene-d3 in PCB54 recorded at two tem- 140 Chapter 8. Toluene in PCB (a) T = 157.5K, tm=200 µs(b) T = 157.5K, tm=100 ms (c) T = 162.5K, tm=1ms (d) T = 162.5K, tm=10 ms Figure 8.14: 2D exchange spectra of xtol =0.74 toluene-d3 in PCB at two temperatures for various mixing times tm. peratures and for various mixing times tm are displayed. At 157.5K (figures (a) and (b)) the corresponding solid echo line shape still resembles a Pake pattern, as observed along the main diagonal of the 2D spectra. The spectrum recorded for a short mixing time of tm = 200 µs exhibits no discernible exchange intensity, apart from a weak broadening of the diagonal, hence the α ’-process has imposed no significant correlation loss on the latter time scale. At tm = 100 ms on the other hand the exchange intensity covers the full plane of the spectrum: the observed process is isotropic (as all possible orientations of a C2 H bond with respect to ~ B0 are captured) and the spectra resemble those known from neat structural glass formers (cf. the results of cyanocyclohexane in section 6.3.7 for example). With regard to limitations due to T1 it is not feasible to obtain spectra for longer mixing times at the present temperature, i.e. we can not determine if and on which time scale full exchange of all molecules is observed. At 162.5K the solid echo line shape in the xtol =0.74 mixture exhibits already a two phase characteristic, i.e. a liquid line is observed in combination with a solid state spectrum (cf. figure 8.10), as seen again along the main diagonal in figures (c) 8.5. NMR results – an overview 141 and (d). The observed exchange pattern in the spectra allow to draw the following conclusions regarding the dynamics of toluene in the mixtures: • Also at 162.5K the slowly reorienting toluene molecules undergo isotropic motion, as the arising exchange pattern covers the full 2D frequency range. The relatively faster motion, contributing to the central liquid line and intermediate spectral patterns, can however not be tracked by this type of experiment. • The sub-ensembles with relatively fast (central line) and slow dynamics (Pake pattern) show exchange on the millisecond time scale, as distinct cross peaks between the central line and the singularities of the Pake pattern arise for both mixing times. • Exchange between the sub ensembles takes place on the time scale of the slow motion, as the relative growth of cross peak and exchange intensity between the studied mixing times is comparable. In conclusion the 2D exchange spectra have shown that essentially all toluene molecules reorient isotropically in the mixtures with PCB54, and that exchange between relatively fast and slow sub ensembles takes place on the same time scale. From the present analysis it is however not possible to differentiate between a scenario in which those sub ensembles are both part of a broad, continuous distribution G(log τ)on the one hand, or reflect a bimodal scenario on the other hand. 8.5.4 Stimulated echoes – high temperature regime For each of the prepared samples (except for the xtol =0.27 toluene-d5 / PCB mixture) stimulated echo decays were measured for an evolution time of te = 3µs to obtain correlation times for the tumbling motion of the toluene molecules in the millisecond to second regime. Phase cycling was chosen as such that the sine-sine correlation was measured, as it allows for an extraction of τ2 in the limit te→ 0(cf. section 3.4.4 for details). As typically found in mixtures of glass forming substances, a broadening of the correlation functions is also observed in the present case. This is well seen in figure 8.15 (a), where the Fss te decays for three different toluene-d3 / PCB54 mixtures are plotted: in all cases the correlation function spans the whole accessible time window, i.e. no distinct initial or final state plateau value is observed. For all three combinations of concentration and temperature T1 is approx. 40 ms and nearly exponential. T1Q , the relaxation time of the spin alignment state prepared in the present pulse sequence, which can not be determined individually, is typically shorter. As Fss te has not decayed to F∞ in any of the samples on this time scale, a reliable determination of the latter quantity is not feasible. Since the 2D exchange spectra presented in the last section however exhibited clear signs of isotropic motion in the same temperature regime and an assessment of the residual amplitude via a free fit in the stimulated echo decays of toluene-d3 and toluene-d510 in the present 10 It is noteworthy to mention here that toluene-d5 – in contrast to the low temperature regime – provides no advantage in terms of the available time window for the present experiments: T1 and T1Q are typically an order of magnitude shorter in the temperature regime where 100 µs < τ < T1Q holds (i.e. in which Fss tecan be measured), as compared to T1in mixtures with toluene-d3. 142 Chapter 8. Toluene in PCB 0 0.2 0.4 0.6 0.8 1 -5 -4 -3 -2 -1 0 int. / a.u. log ( tm / s ) (a) d3 T1 xtol = 0.36 - 215 K 0.74 - 160 K 0.87 - 136 K 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 int. / a.u. log ( tm / s ) T1Q corrected (b) d3 T1 Figure 8.15 (a): Stimulated echo decay curves for three toluene-d3 / PCB mixtures (sine-sine correlation, te = 3µs ) at different temperatures but comparable correlation times. The solid line represents approximately T1 in all cases, the dashed lines are fits via equation 8.3. (b): Data of figure (a) corrected for T1Q decay during tm . The dashed lines represent (single) stretched exponential fits. mixtures purveyed values well below 0.2, and hence on the order of what is expected for an isotropic motion (cf. figure 3.6), F∞ was fixed to the theoretical value for an effective evolution time of te = 4.5µs (cf. section 6.3.8) to reduce the number of free parameters. As furthermore no distinct bi-modal behaviour was observed in any of the experiments, the stimulated echo decays were fitted via a function comprised of two stretched exponential decays: Fss te(tm) = A0h(1 −Fss ∞)·e(−τ/tm)βτ+Fss ∞i·e(−T1Q/tm)βT1Q,(8.3) including four parameters: A0, τ, βτ and T1Q , as βT1Q was fixed to the stretching observed in T1 at the same temperature. Figure 8.15 (b) displays the resulting correlation functions, i.e. the stimulated echo decays in figure (a) have been fitted via equation 8.3 and subsequently corrected for T1Q decay during tm : in this representation it becomes evident that the correlation loss spans a significantly larger time window than accessible to the experiment. Nevertheless consistent correlation times could be extracted for all investigated concentrations, cf. figure 8.16: interestingly hτi of xtol≤ 0.77 coincides with τα (of the PCB54 component) from dielectric spectroscopy. This experimental finding is in contrast to our previous 2H NMR results, as the correlation times determined from the T1 minimum and the two phase spectra in the region of the line shape collapse exhibited a distinct decoupling from τα of PCB54. In the analysis of the solid echo line shape we found that τα of PCB54 provides an upper bound for τα0 of toluene, whereas the present experiment yields hταi ≈ hτα0i . The finding that this relation does not hold for the mixtures with xtol =0.86-0.87 is however in accordance with our previous results: the apparent decoupling between hταi and hτα0i observed here is consistent with the observations made in the line shape analysis, where the temperature for the first 8.5. NMR results – an overview 143 -8 -6 -4 -2 0 2 3 4 5 6 7 8 log (τ / s) 1000K / T xtol = 0 1 (a) d3 0.36 0.41 0.74 0.87 DSC F2 1D W(T) T1 min DS -8 -6 -4 -2 0 2 3 4 5 6 7 8 log (τ / s) 1000K / T xtol = 0 1 (b) d5 d3 0.59 0.77 0.86 0.27 Figure 8.16 (a): Average correlation times from stimulated echo decays ( τ in the millisecond to second regime, sine-sine correlation, te = 3µs ) for mixtures of toluene-d3 in PCB. Additionally plotted are the corresponding correlation times from W ( T )and the T1 minimum for each concentration. The dotted lines represent an interpolation of τα , cf. figure B.1. (b): Same representation for toluene-d5 in PCB in comparison with dielectric results. The results of xtol=0.87 toluene-d3from figure (a) are repeated as partially filled squares. appearance of a solid state spectrum did also not coincide with τα ( T ) = 1 /δ from dielectric spectroscopy for concentrations of xtol =0.86-0.87. τα0 furthermore exhibits lower temperature dependence, i.e. appears less fragile, in the latter concentrations compared to mixtures with lower toluene content. As mentioned at the beginning of this section, the correlation functions of toluene in the mixtures are more stretched with respect to the data of neat toluene: figure 8.17 displays the stretching parameters βK from the Fss measurements vs. the corresponding correlation time hτi . Although the data exhibit rather large spread, a distinct concentration dependence can be observed: the samples with highest toluene content ( xtol =0.86 and 0.87) exhibit slightly broader correlation functions than observed in the neat system ( βK≈ 0.45 opposed to 0.5 in neat toluene). For 0.2 0.3 0.4 0.5 0.6 -4 -3 -2 -1 0 1 2 βK log ( 〈 τ 〉 / s ) (a) d3 xtol = 0.87 0.74 0.41 0.36 0.2 0.3 0.4 0.5 0.6 -4 -3 -2 -1 0 1 2 βK log ( 〈 τ 〉 / s ) (b) d5 xtol = 1.00 0.86 0.77 0.59 Figure 8.17 (a): Stretching parameters βK from a fit via equation 8.3 to Fss for mixtures of toluene-d3 in PCB54, plotted versus the corresponding correlation time hτi . (b): Same representation for the toluene-d5 containing mixtures. Open symbols from Fcc measurements, data of neat toluene from Hinze et al. [Hinze 1998]. 144 Chapter 8. Toluene in PCB -8 -6 -4 -2 0 2 4 5 6 7 8 log (τ / s) 1000K / T xtol = 0.77 -8 -6 -4 -2 0 2 4 5 6 7 8 log (τ / s) 1000K / T xtol = 0.59 DSC DS T1 min W(T) F2 ss F2 cc Figure 8.18: Repetition of figure 8.16 (b) with additional time constants extracted from Fcc te ("+", cosine-cosine correlation, te = 3µs ) for two concentrations of toluene-d5 in PCB54. The key holds for both figures. intermediate concentrations the broadening effect becomes more pronounced, with stretching parameters βK of about 0.25 to 0.35 for xtol =0.74 and 0.77 – at lower concentrations however this trend reverses and the mixtures with xtol≤ 0.59 yield again βK values of about 0.4 and above. This effect can also be seen in the correlation functions displayed in figure 8.15, as the Fss te decay in the mixture with xtol =0.74 appears broader than in the case of xtol=0.87 and 0.36. This experimental finding is rather surprising as it contradicts the results from dielectric spectroscopy and the analysis of the two phase spectra: in DS the width of the α -peak increases 11 towards lower toluene concentrations, cf. figures 8.4 and B.2 12 , the same holds for the temperature range in which two phase spectra are observed, cf. figure 8.11 (a). The mixture of xtol =0.59 toluene-d5 in PCB54 for example yields stretching parameters comparable to the ones found in neat toluene-d5 , cf. figure 8.17 (b), the mixture however exhibits two phase spectra over a broad temperature range, whereas the neat system (presumably) does not. One plausible explanation of this finding lies in the time window of the stimulated echo technique: as demonstrated in figure 8.15 we are not able to ascertain that the full correlation loss of all toluene molecules is tracked by the experiment. To slightly extend the time window of the experiment towards longer times tm , also cosine-cosine correlation functions Fcc were measured in the toluene-d5 containing samples. Apart from T1 being (slightly) longer than T1Q , the former can be obtained via a different experiment – whereas T1Q is only accessible from Fss . Therefore a straightforward correction of the damping via spin-lattice relaxation during tm is obtained and hence the correlation function can be monitored towards longer times and the stretching parameter βK is determined with better accuracy. One drawback of the cosine-cosine correlation in the stimulated echo experiment is however, that it 11 Again: the α ’-process of toluene was not unambiguously resolved by means of dielectric spectroscopy due to the low dipole moment of toluene. 12 Which however poses a contradiction to the rather narrow dielectric loss peak observed in neat PCB54. 8.5. NMR results – an overview 145 does not yield F2 in the limit te→ 0 13 . We measured the cosine-cosine part of the stimulated echo for an evolution time of te=3µs14 and fitted the decay via Fcc te(tm) = A0h(1 −Fcc ∞)·e(−τ/tm)βτ+Fcc ∞i·e(−T1/tm)βT1.(8.4) As again the theoretical value of F∞ was employed and T1 , βT1 at the given temperatures are known, equation 8.4 contains only three free parameters: τ, β and A0 . The resulting correlation times are displayed in figure 8.18: at high temperatures we found hτi and βK (open symbols in figure 8.17 (b)) from Fcc to be in agreement with our previous results from Fss , i.e. also Fcc te yields hταi≈hτα0i at high temperatures. At lower temperatures (where in the case of Fss a stable correction for T1Q is no longer possible) the time constants from the cosine-cosine decays deviate from the VFT behaviour of τα and follow an almost Arrhenius temperature dependence, whereas the stretching βK remains comparable to the high temperature values. This clearly demonstrates that also at low temperatures a fraction of toluene molecules exhibits dynamics on a time scale considerably shorter than the PCB motion observed in dielectric spectroscopy. Albeit reported for other binary glass forming systems [ Blochowicz 2011 ], the observed temperature dependence of hτi not necessarily reflects a fragile-to-strong transition of the toluene dynamics in this regime: also in case of Fcc we can not ascertain that the full correlation loss is captured by the experiment, the temperature dependence of hτi may as well be explained by time window effects. Interestingly however, an extrapolation of hτi from Fcc in the mixture with xtol =0.59 is in agreement with the second Tg obtained from the low temperature step in the heat capacity of the same concentration. Geometry of the toluene dynamics upon mixing In section 8.5.3 we demonstrated by means of 2D exchange spectra that the dynamics of toluene in the mixtures with PCB54 remains isotropic. To obtain further information on the detailed mechanism of reorientation, we will employ the evolution time dependence of the stimulated echo experiment (cf. sections 3.4.4 and 4.2.1 for details) for the sample with xtol =0.77 toluene-d5 in PCB54. Again the cosine-cosine correlation was chosen, as Fcc te data for neat toluene is available in literature [ Hinze 1998 ]. Due to the broadening of the correlation functions in the present mixtures, it is again not feasible to extract the residual amplitude F∞ of the stimulated echo due to interference with T1 , as demonstrated in figure 8.19 (a). Hence again the theoretical value of Fcc ∞(te;δ) (cf. figure 3.6) for each effective evolution time 15 was employed to reduce the number of free parameters and obtain a stable fit. Consequently all stimulated echo curves were fitted via equation 8.4 13F∞ of the cosine-cosine correlation approaches one in this limit and hence no correlation loss can be observed, cf. section 3.4.4 for details. 14 Which for the used pulse lengths corresponds to an effective evolution time of 4.5µs , i.e. the theoretical residual amplitude of Fcc for an isotropic motion is about 0.05 and hence correlation loss can be measured. 15 For the applied sequence and pulse lengths the effective evolution time was determined via Fss ∞(te;δ) of cyanocyclohexane in section 6.3.8. We found that for the present set-up the effective evolution time is 1.5µs longer than teapplied in the pulse sequence. 146 Chapter 8. Toluene in PCB 0 0.2 0.4 0.6 0.8 1 -5 -4 -3 -2 -1 F2 cc ( te ; tm ) log ( tm / s ) te = 75 µs5 µs T1 (a) 0.1 1 0 20 40 60 80 100 〈τ (te)〉 / 〈τ (5µs)〉 te / µs (b) xtol = 1 (122 K) xtol = 0.77 (153 K) 0.3 0.6 βK (c) Figure 8.19 (a): Stimulated echo decay curves (Zeeman) for different evolution times te of the xtol =0.77 toluene-d5 in PCB54 sample recorded at 153.3K (the dashed line symbolises the magnetization curve at that temperature) . (b): Normalized te dependent correlation times from stretched exponential fits to the curves presented in (a) ( hτci ≈ 240 ms ) in comparison with data of neat toluene [ Hinze 1998 ] recorded at a different temperature (hτci ≈ 15 ms). (c): Corresponding stretching parameters βK. with the parameters τ , βK and A0 . The resulting hτi and βK values obtained in this manner are displayed in figure 8.19 (b) and (c) respectively. hτ(te)i of neat toluene shows the typical behaviour of a supercooled liquid, which is similar to the one discussed in the case of cyanocyclohexane, i.e. can be modelled by a weighted superposition of smalland large-angular reorientation, as suggested by Hinze et al. The obtained βK values for the xtol =0.77 toluene-d5 / PCB mixture are again considerably lower at all evolution times with respect to the neat system and in agreement with our previous results. The individual correlation times hτ(te)i = τ/β · Γ β−1 are normalized with respect to hτ(5µs)iin figure 8.19 (b): in this representation it becomes obvious that both samples exhibit almost identical evolution time dependence in the stimulated echo decay. Hence the microscopic mechanism of toluene reorientation appears unaltered in mixtures with PCB54: any model applicable for the α -process of neat toluene also holds for the observed dynamics in the present mixtures16. Conclusions In summary the presented 2D 2H NMR experiments elucidate a number of points regarding the dynamics of toluene in the mixtures with PCB54, but also raise further questions: in the millisecond regime a substantial fraction of toluene molecules reorients on the time scale of the α -process, i.e. a sub ensemble of toluene is no longer decoupled from the PCB54 motion at this temperature. This does however not hold for all toluene molecules, as the Fcc measurements detected dynamics substantially 16 In part Iof this work we have demonstrated that the te -dependence of the stimulated echo is very sensitive with regard to the time window of the experiment, as hτi and T1 of neat toluene and the xtol =0.77 mixture are however comparable at the selected temperatures, the results nevertheless clearly rule out any drastic change in the motional process. 8.5. NMR results – an overview 147 -100 0 100 ν / kHz xtol = 0.86 tp = 20µs 50µs 100µs 200µs -100 0 100 ν / kHz xtol = 1.00 Figure 8.20: Solid echo spectra of xtol =0.86 toluene-d5 in PCB54 and neat toluene-d5 [ Vogel 2000a ] at T = 107 K for different inter pulse delays tp . The tp - dependent line shape changes imposed by the β-process are similar in both samples. faster than τα at lower temperatures. The mechanism via which the tumbling motion of toluene agitates (in the ms regime) appears unchanged in the xtol =0.77 mixture with respect to the neat system. The correlation functions in the mixtures are slightly broadened with respect to neat toluene, an effect which appears to retract again at low toluene concentration. Albeit we were unable to observe any bi-modal behaviour in Fss , this does however not exclude a scenario with two distinct toluene relaxations in the mixtures: due to the narrow time window of the technique potentially not all toluene molecules are observed in a single experiment. Before any conclusions can be drawn, we will analyse the dynamics of the β -process at low temperatures. Due to the lack of a fast exchange mechanism, this analysis is promising with regard to the observation of a distinct bimodal behaviour of toluene in the present mixtures. 8.5.5 Solid echo line shape – low temperature regime So far we have intensively discussed the high temperature 2H NMR results in the mixtures of toluene with PCB54, i.e. at T>Tg , where a pronounced decoupling of the toluene dynamics from the dielectric results of PCB54 was observed for a fraction of molecules in all experiments. The two phase spectra in section 8.5.2 and the stimulated echo measurements in section 8.5.4 on the other hand demonstrated that some toluene molecules reorient on the time scale of τα , i.e. of PCB54, whereas the 2D exchange spectra in section 8.5.3 showed that exchange between the different sub ensembles exists. Although the results point towards a bi-modal scenario with the arise two toluene fractions, it is possibly due to this effective exchange mechanism at high temperatures in combination with the limited time window of 2H NMR that no distinct bi-modality was observed so far. In the remainder of this chapter we will now focus on the low temperature dynamics in the system, i.e. the concentration dependence of the β -process. In the temperature regime T≤Tg no effective exchange takes place on the experimental time scale and as the α - and β -process in glass formers are suspected to be coupled [ Böhmer 2006 ], a detailed study of the β -process could serve to answer some of the open questions with regard to the microscopic