Primary and Secondary Relaxations in Neat and Binary Glass Formers Studied by Means of Dielectric Spectroscopy

Primary and Secondary Relaxations in Neat and Binary Glass Formers Studied by Means of Dielectric Spectroscopy Von der Universit¨at Bayreuth zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung von Robert Kahlau geboren am 3. Juni 1982 in Coburg Erster Gutachter: Prof. Dr. Ernst R¨oßler Zweiter Gutachter: Prof. Dr. Roland B¨ohmer Tag der Einreichung: 10. Januar 2014 Tag des Kolloquiums: 12. M¨arz 2014 Falls Gott die Welt geschaffen hat, war seine Hauptsorge sicher nicht, sie so zu machen, dass wir sie verstehen k¨onnen. Albert Einstein Contents 1 Abstract 5 2 Kurzdarstellung 9 3 Extended Abstract 13 3.1 Introduction.............................. 13 3.2 Properties of Liquids, Supercooled Liquids and Glasses . . . . . . 15 3.3 Molecular Dynamics of a Liquid . . . . . . . . . . . . . . . . . . . 17 3.4 Dynamic Heterogeneities . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Dielectric Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5.1 Dipole Fluctuations and Macroscopic Polarization . . . . . 21 3.5.2 Static Electric Fields . . . . . . . . . . . . . . . . . . . . . 23 3.5.3 Dynamic Response . . . . . . . . . . . . . . . . . . . . . . 25 3.5.4 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . 29 3.6 Contemporary Broadband Dielectric Spectroscopy . . . . . . . . . 32 3.7 Existing Approaches to Describe Generic Relaxation Phenomena, Open Questions, and Major Results . . . . . . . . . . . . . . . . . 35 4 Publications 53 4.1 Generalization of the Cole–Davidson and Kohlrausch Functions to Describe the Primary Response of Glass-Forming Systems . . . . 59 4.2 Evolution of Excess Wing and β-Process in Simple Glass Formers 67 4.3 Quinaldine: Accessing Two Crystalline Polymorphs via the SupercooledLiquid ............................. 79 4.4 Secondary Relaxations in a Series of Organic Phosphate Glasses Revealed by Dielectric Spectroscopy . . . . . . . . . . . . . . . . . 91 4.5 On the Cooperative Nature of the β-Process in Neat and Binary Glasses: A Dielectric and Nuclear Magnetic Resonance SpectroscopyStudy................................ 101 4.6 Dynamics of Asymmetric Binary Glass Formers. I. A Dielectric and Nuclear Magnetic Resonance Spectroscopy Study . . . . . . . . . 115 Bibliography 129 Danksagung 137 1 Abstract The subject of this thesis is the investigation of neat and binary molecular glass formers studied with the help of dielectric spectroscopy (DS). One of two main interests followed by this thesis is to refine the generic picture of the dynamic susceptibility, including the temperature evolution of primary and secondary relaxation phenomena. The other objective is to gain insight into the microscopic origin of the β-process (a secondary relaxation observed for T.Tg) in neat and binary systems, as well as the primary relaxation phenomena in binary systems. The results of all investigations are collected in six partially interrelated publications. Three classes of glass forming systems are analyzed: type-A systems showing no discernible β-process, but in most cases the so-called excess wing (EW) appearing as a power law ε00(ν)∝ν−γin the susceptibility at frequencies beyond the α-peak position (ννmax); type-B systems which have a β-process clearly resolved as a susceptibility peak; and binary systems which are represented by mixtures of the type-A system polystyrene (PS, Mw≈2 kg/mol) and the type-B system tripropyl phosphate (TPP) in this thesis. The analysis of type-A systems as presented in Papers 1 & 2 is based on the application of a specifically introduced three-parameter fit function, which interpolates continuously between Kohlrausch and Cole–Davidson spectral shapes. With the help of this function small temperature-affected spectral changes of the α-peak can be assigned fully to a variation of its low-frequency behavior. In the framework of this ansatz the method of spectral analysis, which has become usual to be applied for type-A systems by our group, can be advanced. The investigation of the type-A glass former quinaldine (2-methyl quinoline) leads to the discovery of a family of substances which have metastable dielectrically active crystalline polymorphs. Dielectric spectra of all observed phases are presented in Paper 3. The phase transitions, occurring in the deeply supercooled state, show temperature-dependent transformation kinetics, which are tracked and quantified by performing long-term DS experiments. As a consequence of the temperature dependence of the transformation kinetics, the metastable phases (including the supercooled liquid) can be kinetically stabilized by lowering temperature. In addition to DS experiments X-ray diffraction (XRD) and differential scanning calorimetry (DSC) measurements are performed, which help to characterize the quinaldine phases and to track their phase transitions. Further, a family of neat, chemically related type-B glass formers, i.e. symmetric phosphoric esters, are investigated in Paper 4 in order to find a possible correlation between their molecular structures and the corresponding molecular dynamics, in 5 1 Abstract particular concerning the β-processes. In two aspects a universal behavior of their β-processes is observed. Firstly, temperature-independent distributions of activation energies g(E), which control the spectral evolution of the β-processes, are found for all systems for temperatures well below the glass transition temperature (T < 0.9Tg). Secondly, the relaxation strength ∆εβ(T) is nearly constant for T < 0.9Tg. When Tgis approached (T > 0.9Tg), a distinct increase of the mean activation energy as well as a strong increase of the relaxation strength is observed. Both mentioned properties are interpreted to be sensitive to the softening of the glassy sample near Tg, pointing, in turn, towards the cooperative nature of the β-processes. Since the behavior of ∆εβ(T) is known for structural as well as for orientational glasses made up by rigid molecules, and since no strong correlation between g(E) and the number of internal degrees of freedom of the phosphoric esters is found, all types of β-processes appear to be mainly determined by intermolecular interactions and are not of merely local nature. Rather they may be considered as a generic feature of the glass transition. The system TPP/PS, a so-called asymmetric binary glass former with a Tg contrast of ∆Tg= 201 K, is investigated in the form of a joint study by DS and 31P/2H NMR, supported by depolarized light scattering (DLS) as well as DSC experiments. The results are collected in Papers 5 & 6. Tripropyl phosphate (TPP), which is also subject of the just summarized study on phosphoric esters, acts now as low-Tgcomponent with a well-resolved β-process. The high-Tgcomponent polystyrene (PS) has a relatively low molecular mass and shows no indication of aβ-relaxation. The β-process found in neat TPP is also observed in the mixtures (Paper 5), and for high TPP concentrations (cT P P ) its properties are nearly concentration-independent. Selective NMR experiments prove the cooperative nature of the β-process: the β-relaxation, which is induced by the TPP molecules, is imposed on the PS monomers in the mixture. NMR experiments show that, when cT P P is reduced continuously, there exist increasing fractions of TPP molecules as well as PS monomers not being involved in the β-process. Due to the high Tgcontrast of the components in TPP/PS, two primary relaxations (α1and α2) and, correspondingly, two calorimetric glass transition temperatures (Tg1and Tg2) are resolved (Paper 6). Selective NMR experiments allow for a general assignment of the α1-relaxation to matrix dynamics and of the α2relaxation to additive dynamics. On the other hand, it is shown that a fraction of TPP molecules is involved in the matrix dynamics, i.e. the molecules relax on the long time scale of the PS monomers. The analysis of the DS spectra shows that this fraction is temperature-dependent and vanishes above a temperature Tc. As a further result, the α2-process is found to change its appearance with varying TPP concentration: at high cT P P its temperature-dependent peak shape resembles the α-peak of a liquid in confinement, while its time constants follow a non-Arrhenian temperature dependence. While, on the one hand, NMR experiments prove that the α2-process is also isotropic at low cT P P , it develops, on the other hand, into a thermally activated process with a temperature-independent distribution of activation energies g(E) when cT P P is decreased. In addition, a further reduction 6 of cT P P leads to a decrease of the mean activation energy of the α2-process. As a consequence, the glass transition temperature related to the α2-relaxation (Tg2) exhibits a maximum at intermediate TPP concentrations. 7 3 Extended Abstract Figure 3.1: (a) Sketch of the dynamic ranges of several experimental methods; figure taken from [14]. (b) Sketch of the dynamic ranges of the dielectric methods applied by P. Lunkenheimer and co-workers in Augsburg; figure taken from [15]. Red marks indicate the dynamic range of the DS setup used for this thesis. the other hand, covering 14 decades or more of a physical property is naturally an experimental challenge, too. Figure 3.1 (a) gives an overview of the accessible correlation time ranges of several experimental methods probing molecular dynamics of liquids [14]. Neutron scattering (NS), optical Kerr effect (OKE) and depolarized light scattering (DLS) consisting of photon correlation spectroscopy (PCS) and the application of a double monochromator in combination with a tandem Fabry–P´erot interferometer (DM/FPI) are taken into account, besides dielectric spectroscopy (DS), solid state NMR methods (2D NMR, stimulated echo) and fast field cycling (FC) relaxometry. Obviously, DS covers the widest dynamical range without any gaps, which is again demonstrated in Fig. 3.1 (b), where the frequency ranges of the dielectric methods used by A. Loidl, P. Lunkenheimer and co-workers in Augsburg are compiled [15]. The dynamical range covered in this thesis is marked red in both parts of Fig. 3.1. Dielectric spectroscopy has the further advantage of a uniquely large resolution, i.e. dielectric losses of 10−4are still measured with high accuracy with up-to-date instruments. Due to this high resolution, even tiny secondary relaxation features or small variations of the main relaxation peak are discovered with the help of dielectric spectroscopy. Often it is difficult to judge if the observed phenomena are qualitatively generic features of the glass transition or rather individual properties of the analyzed substance. Consequently, the establishment of a coherent picture of the temperature evolution of the dynamic susceptibility of supercooled liquids and glasses constitutes a great challenge and is one important issue dealt with in this thesis. Since DS is, compared to NMR methods, basically not selective to different types of molecules within the sample, NMR experiments are indispensable in this thesis for the investigation of the complex dynamics of a binary system. Also for 14 3.2 Properties of Liquids, Supercooled Liquids and Glasses gaining new information on the microscopic origin of secondary (β-) relaxations the combination of DS and NMR is a successful recipe. 3.2 Properties of Liquids, Supercooled Liquids and Glasses A liquid is a form of condensed matter, i.e. an ensemble of particles (e.g. atoms, molecules, monomeric units of polymer chains, particles of colloidal suspensions) with inter-particle distances of the order of the particle size [11]. The structure of a liquid shows short range order, but no long range order. This is demonstrated in Fig. 3.2 (taken from [16]), where the radial distribution function g(r) of liquid argon is plotted, clearly showing broad periodic peaks which lose intensity at higher r. Liquids show a low compressibility and a high density, just like crystals. Figure 3.2: Radial distribution function g(r)of liquid Argon at T= 85 K [16]. Unlike crystals, liquids show a finite viscosity being still higher than the viscosity of gases with much larger inter-particle distances. The reason for the fluid character of liquids lies in the mobility of their particles, which have to rearrange permanently when the liquid flows. In this context the Stokes-Einstein equation D=kT 6πηR (3.1) is usually cited, which connects the diffusion coefficient Dwith the viscosity η, a macroscopic property. Together with hr2i= 6Dt , (3.2) which is valid for diffusional processes in three-dimensional space on long time scales, it follows that the mean squared displacement of a particle of the liquid per unit time increases with decreasing viscosity [11,17]. 15 3 Extended Abstract Cooling a liquid and, at the same time, avoiding any phase transitions like the crystallization below Tm, results in smooth variations of all macroscopic properties. Variables like the thermodynamic potentials, heat capacity or specific volume undergo only small changes, while the viscosity ηincreases over 14 decades [18]. Similar to the increase of η, the structural relaxation time τof the liquid undergoes a tremendous increase until any laboratory time scale is exceeded. Then no rearrangement of the microscopic particles in the sample is possible any more. In the simplest case the sample has become a structural glass with the mechanical properties of a solid (η→ ∞) and the structure of a liquid (cf. g(r) in Fig. 3.2). Figure 3.3 shows the temperature dependence of the entropy S(T) of o-terphenyl. The black line follows the thermodynamically stable behavior of the sample: at the melting point Tmthe first order phase transition between liquid and crystal manifests itself as a step in S(T). If a liquid sample is supercooled below Tmthe entropy evolution of the liquid is continued (blue solid line). Here, although the liquid phase is not the thermodynamically stable state, the sample may be considered as being in thermal equilibrium (i.e. the equilibrium of the liquid phase) as long as any experiment is performed after waiting at least for the structural relaxation time τ(τobs ≥τ). When temperature falls further, τexceeds laboratory time scales (τobs τ). At this point, any measurement will investigate a non-ergodic sample which is structurally arrested or, respectively, a sample still approaching equilibrium (compare so-called aging experiments [19, 20]). The entropy evolution (now representative for any thermodynamic variable) bends over and runs nearly parallel to (or mimics) S(T) of the crystal (red solid line in Fig. 3.3). In this temperature regime the sample is considered to be in the glassy state. The glass temperature Tgis only arbitrarily definable since no phase transition is observable between liquid and glass. Conventionally, the calorimetric Tgis defined as the onset of the “glass-step” in the DSC (differential scanning calorimetry) signal recorded at the heating rate Q= 10 K/min, detecting the temperature where the structural relaxation time is on the order of τ≈100 s (see, e.g., [21]). An often preferred, more precise definition follows the rule τα(Tg) = 102s, i.e. the correlation time of molecular reorientation, which is accessed amongst other methods by dielectric spectroscopy, assumes 100 s at the glass temperature. Further, independent of the applied definition, Tgdepends strongly on the cooling rate [17,22]. In Fig. 3.3 the so-called Kauzmann paradox [23] is sketched, which is not observed in real experiments but which is of theoretical relevance: if the supercooled liquid was further cooled very slowly in order to allow for full relaxation, the entropy evolution of the liquid is assumed to be continued uninterruptedly (dashed line in Fig. 3.3). As a consequence, S(T) of the supercooled liquid would, firstly, intersect the entropy curve of the crystal at the Kauzmann temperature TKand, secondly, finally vanish at finite temperatures. Different theoretical scenarios are possible in order to resolve this paradox, e.g. a phase transition to an ideal glassy state [24, 25]. Yet an answer cannot be found experimantally since τincreases above experimentally accessible time scales already above TK. 16 3.3 Molecular Dynamics of a Liquid Figure 3.3: Entropy S(T)of o-terphenyl, taken from [26]. The picture of the dynamic glass transition given above may be generalized to any supercoolable phase, e.g. plastic crystals and rotor phases [1–9, 27] forming orientational glasses, or colloidal suspensions [28, 29]. Further, a glass transition can be induced by partially removing the solvent of a solution [10]. 3.3 Molecular Dynamics of a Liquid In the following section a closer look on the particle dynamics of supercooled liquids shall be taken by introducing the incoherent intermediate scattering function Fs(q, t), which reflects the (q-dependent) autocorrelation of the position of a tagged particle. In Fig. 3.4 (a) Fs(q, t) of a Lennard-Jones mixture, as obtained by a molecular dynamics simulation [30,31], is shown for different simulation temperatures Tand a constant q=qmax = 7.25, the latter defining the maximum position of the structure factor S(q). Choosing q-values in the vicinity of qmax assures that dynamics on microscopic length scales (comparable to inter-particle distances) on correspondingly short time scales are probed, which further allows for a comparison with other experimental methods, e.g. probing molecular reorientation [32–35]. Correlation times (now and in the following called τ) obtained for such length scales are thus comparable with the structural relaxation time introduced above. Inspecting now Fig. 3.4 (a), a single-step correlation decay at high temperatures is observed, which approximately has the shape of a simple exponential decay. When Tis decreased, the curve shifts to longer times and, simultaneously, develops into a two-step decay. Fig. 3.4 (b) displays the same data as part (a), now scaled to the correlation time τ, here defined as the point of time when Fs(q, t) has reached the value e−1. From this representation one can infer that the long-time parts of the two-step decays collapse to a common master curve. Additionally, they are more stretched than the single-step decays corresponding to highest temperatures. The temperature independence of the curve shapes is 17 3 Extended Abstract Figure 3.4: Incoherent intermediate scattering function of one component of a binary Lennard-Jones liquid (solid lines). Figures adapted from [31]. Simulation temperatures Tindicated. (a) Raw simulation data. (b) Data scaled to correlation time τ. Dashed lines: guides for the eye. a phenomenon usually referred to as time-temperature superposition (TTS) or, in frequency domain, frequency-temperature superposition (FTS). The two-step character of the correlation functions as well as the validity of TTS/FTS and the non-exponentiality (stretching) of the long time behavior are characteristic of “glassy dynamics”, i.e. the molecular dynamics of liquids appearing below the boiling point, yet well above the melting point (Tm< T < Tb) [36]. Figure 3.5 shows the mean squared displacement hr2(t)iof one component of the binary Lennard-Jones liquid just discussed. Here, another typical signature of glassy dynamics is seen: looking at first to high temperature curves in Fig. 3.5 (a), a crossover from hr2(t)i ∝ t2at short times to ∝t1at long times is observed. When Tis decreased, the whole curve shifts to longer times, and the t1-regime becomes more and more separated from the hr2(t)i ∝ t2-regime by a plateau (∝t0) emerging at intermediate times. The t2-regime is called ballistic regime. Here the Figure 3.5: Mean squared displacement of one component of a binary Lennard-Jones liquid. Figures taken from [30]. Simulation temperatures Tindicated. (a) Raw simulation data. (b) Data scaled to long time behavior (see abscissa). 18 3.3 Molecular Dynamics of a Liquid particle simply moves at a constant velocity. On longer time scales the particle inevitably interacts with its neighbors. Thus, the mechanism of translation must transform into a diffusive process at long times, manifesting itself as a t1-law (diffusive regime). When temperature is lowered, the diffusive process is slowed down more strongly than the ballistic one, leading to a separation of both regimes. In between a plateau emerges, because at intermediate times the particle is trapped by its neighbors for a certain period. This is usually referred to as the “cage effect” [11]. At longer times the particle escapes again from its cage via the delayed diffusional process. Figure 3.5 (b) shows the same data as part (a) after scaling the abscissa with the temperature-dependent diffusion coefficient D(T). Similar to the scaling behavior demonstrated for Fs(q, t) in Fig. 3.4 (b) the long time parts of the hr2(t)icollapse. In many cases the microscopic particles of a liquid are no highly symmetric objects like the particles in the simulation just considered; the latter interact via spherically symmetric potentials. Molecular liquids, for example, consist of generally asymmetric molecules. The spatial rearrangement of the molecules, leading finally to the full decay of Fs(q, t), involves also their reorientation. Probing molecular reorientations of supercooled liquids with the help of different methods, thus, constitutes a variety of experimental ways to investigate the glass transition. The experimental method of depolarized light scattering (DLS) probes collective molecular reorientations by measuring the collective reorientational autocorrelation function C(2)(t) of the anisotropic part of the molecular polarizability tensor [36,37]. Figure 3.6 shows the time domain representation of DLS data of m-tricresylphosphate (m-TCP). For all temperatures a monotonic decay of C(2)(t) from 1 to 0 is observed. When Tis decreased, the curve shifts to longer times and, above all, develops into a two-step decay. This emergence of a two-step correlation function upon cooling is comparable to the temperature evolution of the Fs(q, t)- curves in Fig. 3.4 as well as the hr2(t)i-data in Fig. 3.5, and is considered again as the signature of glassy dynamics. The terminal relaxation in Fig. 3.6 will later be attributed to the structural relaxation or α-process of the liquid. On cooling close to Tg, the terminal loss of correlation shifts to long times. The corresponding correlation times increase about twelve orders of magnitude from τ≈10−11 s to τ≈101s, which is again the manifestation of the glass transition described above. From Fig. 3.6 one also infers that TTS is fulfilled over the complete temperature range. This is demonstrated for the T= 290 K ≈Tmcurve, which matches exactly the shape of the T= 207 K ≈Tgcurve (blue dashed line). Lowering temperature below Tgwould result in the terminal decay shifting out of the accessible time range. The sample would fall out of equilibrium, since the experiment takes place on a shorter time scale than the structural relaxation of the sample. Then, the sample may be considered as a glass. As will become clear below, dielectric spectroscopy (DS) experiments reveal a multitude of dynamical processes, which appear on time scales between the short time limit and the terminal step of the correlation function. The analysis of the 19 3 Extended Abstract Figure 3.6: Time domain representation of DM/TFPI as well as photon correlation spectroscopy data of m-tricresyl phosphate (red solid lines), taken from [36]. Dashed blue line: data from T= 290 K shifted on top of the T= 207 K curve. manifestation of these processes as spectral features in the dielectric susceptibility is the actual scope of this work. 3.4 Dynamic Heterogeneities Although the origin of the phenomenon is not clarified, it is a commonly accepted view that supercooled liquids show dynamic heterogeneity [26,38–41]. This means that different subensembles of molecules in the liquid show different mobilities, which causes a distribution of correlation times. Figure 3.7 shows results of a molecular dynamics simulation of a binary Lennard-Jones liquid. Here, each arrow indicates the displacement of a particle after some waiting time on the order of the structural relaxation time. As can be seen clearly, spatially contiguous regions of quite different mobilities are found, demonstrating strong spatial correlations. There are also experimental attempts to explore dynamic heterogeneities. Russel et al. detected the fluctuations of dielectric properties of a polyvenyl-acetate film on nm-scale with the help of a piezo-cantilever of an atomic force microscope [42, 43]. The authors found domains of common dynamical properties on the length scale of about 10 nm, surviving on the time scale of the structural relaxation. Another experimental possibility of monitoring dynamic heterogeneities are, e.g., non-resonant dielectric hole burning experiments [39,44,45]. If a sample shows dynamic heterogeneity, this method leads to a selective heating of dynamically related subensembles of molecules, which leads to a deformation of the distribution of correlation times G(ln τ). Blochowicz et al. [45] found particularly strong effects in binary mixtures. Binary systems analyzed within this thesis by DS and NMR show, likewise, broad distributions of correlation times, which are attributed to strong dynamic heterogeneities. 20 3.5 Dielectric Spectroscopy Figure 3.7: MD simulation results for a binary Lennard-Jones mixture in two dimensions. Arrows of different lengths indicate single particle displacements after a period of the order of the structural relaxation time. Figure taken from [41]. Dynamic heterogeneity is a possible explanation for the decoupling of diffusivity and viscosity close to Tg[38,46], which is subject of many experimental approaches. For example Ediger et al. [47] analyze crystallization kinetics of several organic and inorganic systems in order to find a dependence D∝η−ξwith ξ < 1 (fractional Stokes-Einstein relation, compare Eq. 3.1). Mapes et al. [48] find a violation of Eq. 3.1 by performing vacuum desorption experiments with protonated/deuterated o-terphenyl bilayer films, and Ehlich et al. [49] use forced Rayleigh scattering at holographic gratings made of several glass formers. Important to mention are also the contributions by Stickel et al. [50,51], where the decoupling of the structural relaxation time from the ionic conductivity contribution to the dielectric loss (see below) is interpreted as a decoupling of viscosity from diffusivity. 3.5 Dielectric Spectroscopy 3.5.1 Dipole Fluctuations and Macroscopic Polarization Reorientations of polar particles (i.e. polar molecules of molecular liquids or polar monomeric units in polymer melts) cause fluctuations of the macroscopic polarization ~ Por(t) = 1 VX i ~µi(t) (3.3) of the sample, where Vis the sample volume and ~µiare the permanent dipole moments of the molecules. Note that the time average of ~ Por vanishes when no 21 3 Extended Abstract external field is present. When the liquid under investigation is placed into a capacitor, these fluctuations can be measured. The actual quantity of interest, with a clear atomistic meaning, is the dipole autocorrelation function Cµ(t) = 1 µ2h~µ(0)~µ(t)i,(3.4) with h·i denoting the ensemble average [44,52,53]. Cµ(t) is a single particle autocorrelation function and has to be connected with the (collective) autocorrelation function of the macroscopic polarization CP(t): CP(t) = h~ Por(0)~ Por(t)i h~ Por(0)~ Por(0)i= N P i,j=1 h~µi(0)~µj(t)i N P i,j=1 h~µi(0)~µj(0)i = N P i=1 h~µi(0)~µi(t)i+ N P i,j=1 i6=j h~µi(0)~µj(t)i Nµ2+ N P i,j=1 i6=j h~µi(0)~µj(0)i .(3.5) Here the sum terms with identical indices (i=j) are autocorrelation contributions, while expressions with i6=jrepresent so-called cross-correlation contributions. If the latter vanish due to some approximation, the expression CP(t) = 1 Nµ2 N X i=1 h~µi(0)~µi(t)i=1 µ2h~µ(0)~µ(t)i=Cµ(t) (3.6) follows. One could argue that this approximation never holds in polar liquids. However, it is usually assumed that the cross-correlation contributions behave similarly to the autocorrelation terms [54] and, thus, do not play a significant role. As a further approximation, the equivalence of ensemble and time average, actually only valid in ergodic systems [55], is extended to temperatures below Tg, i.e. to thermodynamic situations where the liquid has fallen out of equilibrium. Thus, the autocorrelation function of the macroscopic polarization may be generally defined as CP(t) = h~ Por(0)~ Por(t)i= lim T→∞  1 T T Z 0 ~ Por(τ)~ Por(τ+t) dτ .(3.7) 22 3.5 Dielectric Spectroscopy According to the Wiener-Khintchine theorem [56,57], the spectral density of dipole fluctuations can be formulated like [58] SP(ω) = 1 2 ∞ Z −∞ CP(t) exp [iωt] dt= ∞ Z 0 CP(t) cos(ωt) dt . (3.8) The last equality in Eq. 3.8 is due to the fact that CP(t) is an even function. Dielectric spectral densities of glycerol and PVC have been measured above and below Tgby Israeloff et al. [59,60]. As will become clear below, molecular dynamics may be investigated alternatively by measuring the reaction of the sample to an external field. In order to adapt this statement to dielectric spectroscopy, the question of how electric fields interact with polar samples has to be addressed at first. 3.5.2 Static Electric Fields If a static external electric field is present, a non-vanishing time average of the macroscopic polarization ~ Pis found. If the external field is sufficiently low, the linear dependence ~ P=ε0χ : ~ E(3.9) with the generally tensorial susceptibility χ : is a good approximation. The macroscopic polarization consists of the orientational part ~ Por (cf. previous section) and a contribution ~ Pind due to a deformation of the molecular charge density distribution: ~ P=~ Por +~ Pind and (3.10) χ : =χ : or +χ : ind .(3.11) When the dynamic response is discussed (see next section), this has to be kept in mind. The dielectric displacement is then calculated to ~ D=ε0~ E+~ P =ε0 1 :+χ :~ E =ε0ε : ~ E(3.12) 23 3 Extended Abstract and the sapphire disk secure a spacing of about 60 µm between both electrodes, where the liquid sample under investigation is placed. No additional spacer material like, e.g., fiber glass is required for this setup. The diameter of the upper electrode is about 18 mm, which leads to an empty capacity C0≈38 pF. As can be inferred from Fig. 3.9, there is enough space above the upper electrode to collect excess sample material not fitting in the gap between the elecrodes. When binary liquids were measured, the O-ring in the lower electrode was used to prevent evaporation of the volatile component and, thus, to keep the concentration of the mixture constant. The sample capacitor was connected to the “Alpha-A Analyzer” by Novocontrol, which operates in the frequency range ν= 10−3–107Hz. Since measuring below ν= 10−2Hz is very time consuming, the lower bound of the frequency range was limited to ν= 10−2Hz for most measurements. Since measurement artifacts around ν= 106Hz tend to influence the results, data acquired in the range ν= 105–107Hz are partially omitted in the figures. For the frequency-dependent measurement a harmonic voltage ˆ U(ω, t) = U0exp(iωt) (3.36) is applied to the sample. The phase-sensitive detection of the current ˆ I(ω, t) allows for the measurement of the impedance ˆ Z=ˆ U ˆ I.(3.37) For a capacitor I=˙ Q=C˙ Uholds, which can be adapted to the complex properties like ˆ I(ω, t) = ˆ C(ω)·iω ˆ U(ω, t) (3.38) finally leading to ˆε(ω) = 1 iωC0ˆ Z(ω).(3.39) Note that any DC conductivity σof the sample (now thought of as ohmic resistor), which appears solely in the real part of the impedance ( ˆ Z(ω) = R), will contribute exclusively to the imaginary part of the dielectric function ˆε(ω). Introducing the resistivity ρ=RA d=RC0 ε0 =1 σ(3.40) leads to ˆεcond(ω) = 1 iωC0R=σ iωε0 (imaginary) (3.41) and ε00 cond(ω) = σ ε0ω.(3.42) 30 3.5 Dielectric Spectroscopy The conductivity contribution to the dielectric susceptibility given in Eq. 3.42 appears in the form of a power law ε00(ω)∝ω−1. 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 -1 10 0 10 1 ∼ω +1 ∼ω -1 ε ''( ω ) ω / rad s -1 ∼ω -1 ω p =1/ τ 0 5 10 15 ε ∞ ε '( ω ) ε ∞ + ∆ε Debye function with conductivity τ =10 -3 s σ = ε 0 ∆ε =10 ε ∞ =1.5 Figure 3.10: Sum of a Debye function and a typical DC conductivity contribution. Upper panel: real part ε0(ω)according to Eq. 3.45; lower panel: imaginary part ε00(ω) according to Eq. 3.46. The simplest case of a step response function is a single exponential decay ΦP(t) = exp −t τ.(3.43) According to Eq. 3.31, the corresponding dielectric function is given by ˆε(ω) = ∆ε ∞ Z 0 −d dtexp −t τexp(−iωt) dt+ε∞ = ∆ε·1 1 + iωτ +ε∞(3.44) (cf. also Eq. 3.14) and called Debye function. The corresponding real and imaginary parts are ε0(ω) = ∆ε·1 1 + ω2τ2+ε∞and ε00(ω) = ∆ε·ωτ 1 + ω2τ2.(3.45) Adding a conductivity contribution (Eq. 3.42) to the imaginary part is in accord with a common approach, assuming an ohmic resistor connected in parallel to the 31 3 Extended Abstract sample capacitor, and yields ε00(ω) = ∆ε·ωτ 1 + ω2τ2+σ ε0ω.(3.46) Figure 3.10 displays real and imaginary part of a Debye function including a DC conductivity contribution. The corresponding parameters are given in the figure. In the upper panel the real part ε0(ω) according to Eq. 3.45 is shown, decaying monotonically from ε∞+∆εto ε∞. The step (maximum slope) is situated around ω= 1/τ. The lower panel of Fig. 3.10 displays the imaginary part ε00(ω) showing a peak at ωp= 1/τ. Due to the double logarithmic representation the flanks of the peak, power laws ∝ω1and ∝ω−1, appear as straight lines. Some frequency decades below ωpa minimum is observed; left to the minimum the DC conductivity contribution is found. The conductivity contribution also appears as a straight line on the low frequency side of the peak (power law ∝ω−1; according to Eqs. 3.42 and 3.46). When the isotropic molecular reorientation in supercooled liquids is probed, the observed susceptibility peak, the α-process, is broadened asymmetrically on the high-frequency side in comparison with the Debye peak. This broadening is usually interpreted as a signature of the cooperative nature of the α-process. A distribution of correlation times due to dynamic heterogeneities also corresponds to a broadened susceptibility peak. Several phenomenological fit functions quantifying the braodening, such as the Kohlrauschor the Cole–Davidson-function [52], or modeled distributions of correlation times [44, 65] exist; a new function is discussed in Paper 1. Nevertheless, the special case of the exponential correlation decay and the corresponding Debye line shape of ε00(ω) is experimentally observed, too [66, 67]. A relaxation process found in monoalcohols like n-butanol [68], 2ethyl-1-hexanol [69], ethanol [4] or water [68, 70], the mechanism of which is not fully elucidated yet, causes a typical Debye line shape of ε00(ω), and is thus usually referred to as the Debye process. 3.6 Contemporary Broadband Dielectric Spectroscopy Historical dielectric experiments, which were limited to the determination of the temperature-dependent static dielectric constant or the measurement of the dielectric function ˆε(ν=const., T) at single frequencies, have been replaced by broadband techniques with a wide frequency range. Presenting data of isothermal measurements covering eight frequency decades has become the scientific standard. Combining several broadband setups (including time-domain setup, microwave reflectometry and Fourier-transfom infrared spectroscopy), as done for example in the dielectric laboratory at the University of Augsburg run by A. Loidl and P. Lunkenheimer [19], makes it possible to measure the dielectric susceptibility 32 3.6 Contemporary Broadband Dielectric Spectroscopy over the range of 16 frequency decades, which may be considered as the experimental state of the art (cf. also Fig. 3.1 (b)). Important to mention is also the application of high precision capacitance bridges as done, for example, by C. Gainaru et al. in order to obtain highly accurate susceptibility data even at lowest temperatures [53, 71]. The bridge used by Gainaru et al. has a narrower frequency range, but a higher accuracy, or sensitivity to low signals, than the broadband spectrometer applied within the present work. Figure 3.11: (a) Dielectric susceptibility of super-cooled glycerol, measured by the Augsburg group around P. Lunkenheimer. Figure taken from Ref. [72]. Inset: temperature dependence of the susceptibility peak positions. (b) The same data as in (a), rescaled to the susceptibility minimum according to the axes titling [73]. Crosses: data acquired in the temperature range T= 184–273 K. Circles: temperature range T= 289–413 K. (c) Selection of the data shown in (a), now scaled to the maxima [53]. In Figure 3.11 (a) dielectric susceptibility data of glycerol as published by the Lunkenheimer group [72] are shown, covering the frequency range ν≈10−5– 1012 Hz. The signal amplitudes reach from ε00 ≈0.01–20, hence a wide dynamic range is accessed and even tiny spectral features can be resolved. For all isotherms shown in Fig. 3.11 (a) a prominent peak is observed, which shifts from νP= 1010 Hz 33 3 Extended Abstract at T≈400 K to νP= 10−4Hz at T= 184 K. The temperature dependence of the peak position νPis displayed in the inset of Fig. 3.11 and shows a strong curvature, i.e. a super-Arrhenius behavior of the time constant τP= 1/(2πνP) typical for glassy dynamics [18,26, 36, 74] is observed. Note that only in the case of the Debye peak τ=τmax holds exactly. In most cases there is yet no significant difference between the temperature dependences of τmax(T), corresponding to the peak positions, and the mean correlation times τ(T) as defined by Eq. 3.21. Obviously an increase of the correlation time beyond any laboratory time scale is expectable when Tis lowered further (cf. Sec. 3.3). When the DS data are rescaled to the susceptibility minimum between α-relaxation and the so-called boson peak at highest frequencies (Fig. 3.11 (b)), the data of the temperature range T= 289–413 K (circles) have a common envelope [73]. Data of temperatures below T= 289 K (crosses) deviate from the common envelope due to secondary relaxation features emerging between αand boson peak in the form of excess susceptibility. For example, inspecting the ε00(ν) curve for T= 184 K in Fig. 3.11 (a), the high frequency side of the main relaxation peak has the shape of a power law in the range ν= 10−3–10−1Hz. At higher frequencies (ν= 100–102Hz) a distinct positive curvature is observed, marking a crossover to another power law shape in the range (ν= 102–107Hz). The latter feature is usually referred to as excess wing (EW); glass formers showing an EW instead of secondary relaxation peaks are called type-A systems, according to the classification by Kudlik et al. [75]. All peaks shown in Fig. 3.11 (a) have nearly the same spectral shape, as demonstrated in Fig. 3.11 (c), where the data are scaled to the susceptibility maxima [53]. Thus, FTS (frequency domain equivalent of TTS, cf. Sec. 3.3) is obeyed in good approximation by the main relaxation peak. (As will be discussed in Paper 2 a slight sharpening of the peaks with increasing temperature can be quantified anyway.) TTS is also observed for the long-time decay of the correlation functions acquired by broadband DLS shown in Fig. 3.6 in Sec. 3.3. Remarkably, the data collapse shown in the master curve in Fig. 3.11 (c) also involves the EW contribution. Besides the EW, as observed in the glycerol data (Fig. 3.11) and many other type-A glass formers (see Papers 2 & 3), explicit secondary relaxation (or β-) peaks are found in the susceptibility of type-B systems [75] (see Papers 4 & 5). In contrast to the α-relaxations, the time constants of which show non-Arrhenius behavior, the β-processes show thermally activated dynamics, i.e. their time constants follow Arrhenius laws [53,75,76]. Kudlik et al. [75] find mean activation energies of hEi ≈ 24 Tgfor most cases and attempt times in the range τ0= 10−18–10−15 s. The relaxation strengths of the β-processes ∆εβin relation to the respective αrelaxation strengths differ strikingly when several glass formers are compared [44]. Below Tgthe ∆εβ(T) are nearly constant, while they show a strong increase when temperature is increased above Tg[65,77]. In Fig. 3.12 [44] DS data of the type-B systems (a) toluene (TOL) and (b) 3-flouro aniline (or m-fluoro aniline; m-FAN) are collected, showing explicit β-peaks in addition to the α-relaxation peaks, ap34 3.7 Existing Approaches, Open Questions, and Major Results pearing at clearly higher maximum frequencies than the latters. For TOL (Fig. 3.12 (a)) an immediate crossover from αto β-peak is observed. In the case of m-FAN (Fig. 3.12 (b)) indications of an additional EW, appearing as a power law between αand β-peak, are found. Figure 3.12: Dielectric susceptibility of type-B systems; figures taken from [44]. (a) Toluene (Tg= 117 K). (b) 3-flouro aniline (Tg= 172 K). The investigation of secondary relaxations in supercooled liquids has become an important task within the “glass community”, as well it is a significant issue of the present work. In Paper 4 an ensemble of chemically related systems consisting of six type-B glass formers are investigated. Since these systems contain phosphorus, they are very interesting systems for being also analyzed with the help of 31P NMR as done by our group. In the Papers 5 & 6 binary mixtures of one of these systems (tripropyl phosphate; TPP) and polystyrene are investigated with DS as well as 31P and 2H NMR. A major strength of dielectric spectroscopy compared to other methods is the ability to resolve these secondary processes with high accuracy despite of their usually low signal compared to the primary relaxation. Due to the high sensitivity of the method, even subtle variations of the line shapes of primary as well as secondary relaxation features are detectable and quantifyable with the help of DS. Each publication included in this thesis thrives on this fact (Papers 1–6). 3.7 Existing Approaches to Describe Generic Relaxation Phenomena, Open Questions, and Major Results Contemporary broadband DS on supercooled organic glass formers deals with the challenge of understanding the temperature evolution of the dynamic susceptibility, i.e. to phenomenologically comprehend the glass transition. An important step towards this understanding, which has been followed since long, is to establish an extensive data collection of different glass forming systems covering frequencies 35 3 Extended Abstract from 0–1 THz. In this context, it is not less important to learn how to distinguish between system specific and generic phenomena by finding a conclusive way of disentangling the individual relaxation features. Today, several approaches are followed by different working groups, leading to different perceptions of EW, αand β-relaxation. In the following section the most important interpretations are summarized. α-relaxation The main process of molecular rearrangement in a liquid causes the so-called α-relaxation in the case of an external disturbance. This α-relaxation is responsible for the structural relaxation of the sample, which is accompanied by an isotropic and cooperative reorientation of the molecules (or, respectively, monomer units in polymers). This molecular reorientation manifests itself as the α-peak in the dynamic susceptibility. Orientational relaxation in plastic crystals shows a similar phenomenology [2–9]. One of the main experimental controversy concerning the α-relaxation is the question whether frequency-temperature superposition (FTS) holds for the αrelaxation at all temperatures T≥Tg. Of course, any emerging secondary processes appearing close to Tghave to be separated adequately from the αpeak when this question is addressed. Schneider et al. [72] evaluated the data shown in Fig. 3.11 (a) by fitting exclusively the α-peaks and found a systematically temperature-dependent peak width, which is contrary to the results of the scaling analysis mentioned above (Fig. 3.11 (c)) presented ten years later [53] and actually reflecting the interpretation followed by the Bayreuth group since then. There are also earlier publications by this group explicitly showing systematic temperature dependences of the width parameter of the α-peak, yielded by fitting analyses [4,58,65,73,75,78,79]. These results were later put into perspective with the problem of a limited available fitting range (see below). Furthermore, in the context of a scaling analysis used by S. R. Nagel (University of Chicago) and coworkers [80–82], which applies within a certain approximation [83,84], a failure of FTS is reported. On the other hand, Olsen et al. [85] showed, for the system triphenyl phosphite, that FTS holds for the main relaxation peak even down to the glass transition temperature, as long as secondary processes are absent (EW) or at least well-separated (β-relaxation) from the α-peak. Based on this idea, Gainaru et al. introduced the approach of understanding the dielectric susceptibility as being composed additively of α-relaxation, excess wing (EW) and, if necessary, a β-relaxation [53,76] (in this thesis often referred to as the Gainaru approach). Within this ansatz, besides a temperature-invariant line shape of the α-peak a temperature-independent exponent of the EW is assumed. Only ταand the amplitudes of α-peak and EW are temperature-dependent. This approach gives a much simpler picture of the evolution of the dynamic susceptibility than the straight forward fitting procedures presented by Blochowicz et al. [65,79]. Furthermore, Brodin et al. showed that the typical temperature dependence of the line shape parameters of several 36 3.7 Existing Approaches, Open Questions, and Major Results systems presented there may be merely an artifact due to the particular fitting strategy and the limited frequency range of the data [86]. The authors showed with the help of simple scaling representations, that actually FTS holds in good approximation for the α-relaxation peak, and even the comparison of α-peaks of different systems do not show significant differences. This new picture gives new input for, e.g., the advancement of theories like MCT. In addition to that, procedures of any analysis of data with a limited frequency range, e.g. the evaluation of fast field cycling NMR data, is based on FTS. The following open questions concerning the α-relaxation are (re-)addressed in the present work: •To which extent does FTS hold for the α-relaxation close to Tg? (→Papers 1 & 2.) •Can apparent variations of the broadening of the α-peak be explained by a temperature-dependent excess wing amplitude? (→Paper 2.) •How can susceptibility spectra be interpolated adequately on the basis of the refined picture of the α-relaxation? (→Papers 1 & 2.) •How does the main relaxation peak of a plastic crystalline phase formed in the supercooled regime differ from the α-peak of the corresponding liquid? (→Paper 3.) The major results of the present work concerning the α-relaxation shall be summarized in the following. The first part of Paper 2 focuses on the temperature dependence of the dielectric susceptibility of glass formers with no resolved β-peak (type-A systems), i.e. the evolution of α-peak and EW under varying temperature. Plotting the normalized spectra versus ωταleads to a collapse of the data on the low-frequency side of the α-peak (low-frequency scaling). While the line shape of the α-peak was considered to be essentially temperature-independent in the context of Gainaru’s approach [53,76,86], a subtle but systematic variation in the high-frequency region of the master curves becomes obvious here. When the data sets are rescaled as demonstrated for PG in Fig. 3.13 (a), an almost perfect collapse on the high-frequency part of the data is observed, including the high-frequency flank of the α-peak as well as the EW. This high-frequency scaling suggests that temperature-affected changes of the line shape, which cannot be explained solely by the varying EW amplitude, concern merely the overall peak width instead of the high-frequency exponent of the α-peak. The subtle violation of FTS can thus be projected completely to the low-frequency region of the α-relaxation. This result is of high importance for the Gainaru approach itself since for this ansatz a temperature-independent high-frequency exponent of the α-peak is a necessary implication. In order to quantify these findings, a phenomenological three parameter fit function for the normalized α-relaxation peak introduced in Paper 1 is applied. The 37 3 Extended Abstract Figure 3.13: (a) Normalized susceptibility of propylene glycol (PG) plotted versus a reduced frequency (high-frequency scaling; figure taken from Paper 2). (b) Fit function introduced in Paper 1 for several values of the shape parameter α, normalized and plotted versus ωτα. function is defined as step-response function Φg(t) and has two shape parameters αand βbesides the time constant τg. While the Cole-Davidson (CD) and Kohlrausch (K) functions have only one stretching parameter, which affects the broadening of the peak by determining the exponent of its high-frequency power law, the new function has a further shape parameter, which changes the shape of the tip of the relaxation peak without affecting the high-frequency exponent. The specialty of the new function is that the well-established K and CD functions are reproduced in the limiting cases α→βand α→1, respectively. In other words, the function Φg(t) interpolates continuously between K and CD peak shape (Fig. 3.13 (b)). Contrary to the Havriliak-Negami (HN) function, Φg(t) has a welldefined mean relaxation time ταdue to the physically valid low-frequency limit ε00(ν)∝ν1of its susceptibility representation. Finally it is shown in Paper 1 that Φg(t) has a non-negative distribution of correlation times G(ln τ), which is the precondition for a physically valid relaxation function. In Paper 2 dielectric susceptibilities of PC, PG, m-TCP, GLY and 4-TBP are fitted with this function, which is now extended to account also for the EW contribution. The results are displayed in Fig. 3.14. For basically all considered systems the shape parameter γof the EW as well as the high-frequency parameter βof the α-peak are, in the spirit of the Gainaru approach, kept constant for all temperatures. The resulting parameters α(additional shape parameter of the α-peak) and C(relative relaxation strength of the EW) are found to be temperaturedependent: with increasing T α increases from α < 1 to α= 1, reflecting the peak assuming a Cole-Davidson (CD) shape above a certain temperature. At the same time C > 0 is nearly constant over a certain temperature range, before it decreases quickly to 0 when αassumes unity. The system 4-tert-butylpyridine (4-TBP), which is also discussed in Paper 2, cannot be analyzed in the same way. As can be inferred from Fig. 3.13 (inset), the shape parameter βis significantly 38 3.7 Existing Approaches, Open Questions, and Major Results temperature-dependent (inset), further the temperature dependence of its shape parameter αdiffers from the behavior of the other analyzed systems. It turns out that this irregularity is caused by the presence of a weak β-process in the spectra of 4-TBP, which is hidden by the EW (see below). Figure 3.14: Fit parameters αand Cas presented in Paper 2. Quinaldine (2-methyl quinoline) is an exemplary type-A glass former [87]. In Paper 3 the three-parameter fit function introduced in Paper 1 is extended to account for EW and conductivity contribution and is now used to quantify the evolution of the dynamic susceptibility of quinaldine. Again, the high-frequency shape parameter βof the α-peak as well as the power law exponent γdescribing the EW are found to be temperature-independent, while αand Cshow a weak variation with T. Further, the low-frequency scaling of the susceptibility curves reveals a temperature dependence of the line shape, which is limited to the EW region. The α-relaxation peak itself appears to obey FTS almost perfectly. At temperatures well above Tg= 180 K (200 K < T < Tm) two subsequent phase transitions of liquid quinaldine (2mq1) into two dielectrically active phases (2mq2 and 2mq3) are observed. Paper 3 is funded through the project “Schwerpunktprogramm (SPP) 1415: Kristalline Nichtgleichgewichtsphasen - Pr¨aparation, Charakterisierung und in-situ-Untersuchung der Bildungsmechanismen” by Deutsche Forschungsgemeinschaft (DFG). Particular interest of the SPP lies in the mechanism of formation of the metastable crystalline phase. Thus, the analysis of the phase transitions and the phases itself is the actual main concern of Paper 3. Figure 3.15 (a) displays results of a DS long-term measurement tracking the phase transition from the liquid phase 2mq1 to the dielectrically active crystalline phase 2mq2. A quantitative analysis of the time dependence of this phase transition reveals stretched exponential decays (Avrami laws, cf. Fig. 3.15 (b)) with temperature-dependent transformation time constants and exponents, the evaluation of which strongly suggests that the crystallization process is controlled by 39 3 Extended Abstract that the fractions of PS monomers as well as TPP molecules participating in the β-process decrease with decreasing TPP concentration. This is interpreted as the emergence of “islands of rigidity”, as also observed for toluene/aroclor mixtures by our group [96]. Although the concept of “islands of mobility” [89,98] has been rejected for neat glass formers [93,97], these results suggest its re-introduction for binary systems. excess wing (EW) As can be inferred from Fig. 3.11 (Sec. 3.6), the high-frequency flank of the α-relaxation peak bends over to show a lower dispersion at high frequencies. This high frequency feature, which can be described by a power law ε00(ν)∝ν−γ(with γbeing smaller than the high-frequency exponent of the α-peak) is called excess wing (EW), and if there is not a β-relaxation dominating the spectra, i.e. if a type-A glass former [75,78] is concerned, it is usually found in the dynamic susceptibility of supercooled liquids and glasses. There are three possible interpretations: 1. The excess wing is part of the α-process. 2. The excess wing is an individual relaxation process and has to be distinguished from αand β-process. 3. The excess wing is a submerged β-relaxation peak. The first interpretation is found primarily in the context of the scaling procedure applied by Nagel et al. [80–82]. In Ref. [107] it is stated explicitly that the EW exponent γis strictly related to the exponent of the high frequency flank of the α-relaxation peak, and that FTS is not obeyed. The second interpretation is followed in the framework of the analysis by Gainaru and co-workers [53, 76, 86]. Here, the shape parameters of α-peak and EW are found to be constant in the whole temperature range down to Tg, which is in contrast to the findings by Blochowicz et al. [65,79], and which implies that FTS is obeyed by the α-peak. Solely the relative amplitude of the EW in relation to the α-relaxation changes slightly with temperature, causing an apparent variation of the high frequency exponent of the α-peak. The assumption that β-process and EW are separate processes is supported by the fact that the EW is resolved in DLS experiments, where the β-relaxation is invisible [36,108, 109]. Moreover, it is demonstrated in [110] that the EW obscures the g(E)-scaling of the thermally activated β-process described above (Fig. 3.17 (b) and 3.18 (b)), which again points into the direction that EW and β-process have to be distinguished. Further, contrary to the temperature-dependent peak shape of β-processes, the EW exponent γshows no significant temperature dependence. Thus, by means of the Gainaru approach aging experiments performed by Schneider et al. [19,111] can be reinterpreted by assuming the high-frequency susceptibility contribution as being composed additively of EW and β-process (the authors themselves prefer the third 46 3.7 Existing Approaches, Open Questions, and Major Results interpretation; see below). Similar aging experiments on 4-tert-butylpyridine (4TBP) are shown in Paper 2 (see below), where they are also interpreted according to this interpretation. The third interpretation is supported by the results of dielectric aging experiments performed by Schneider et al. [19, 111]. Here, the sample is cooled to temperatures few Kelvin below Tgand, hence, pushed out of thermal equilibrium, which is reached only on time scales of 106s. While waiting the α-relaxation peak (not observed directly below Tg) becomes more separated from the EW. The latter starts to show a distinct curvature and develops continuously into a relaxation peak. The authors conclude that this peak is the “real face” of the EW, which, thus, is nothing else than a β-relaxation, usually submerged below the α-peak. This is supported by Casalini et al. [112], performing pressure-dependent DS experiments on a type-B glass former. With increasing pressure an initially abscent EW emerges and, finally, develops into a further well-resolved peak. Note that up to now the origin of EW as well as β-process are still to be clarified. NMR measurements on type-A glass formers below Tgdo not reveal indications of restricted molecular motion like in the case of type-B systems [36]. However, slightly above Tgthe spectra start to resemble those of type-B glass formers [36, 113], which makes a clear distinction between the two processes again appear difficult. Learning about commonalities, similarities and differences between both processes, and finding the answer to the question if there are two different kinds of secondary processes at all, are issues of current and future research. In the present work the second interpretation in the spirit of the Gainaru approach is favored. Concerning the EW, the following open questions are addressed in the publications: •How can the EW be described within the Gainaru approach refined according to the subtle line shape variations of the α-peak? (→Paper 2.) •Are there further examples of type-A systems which have a β-relaxation covered by the EW? (→Paper 2.) •Which commonalities among type-A systems may indicate that the EW has a microscopic origin different from that of β-processes? (→Paper 2.) •How does the EW found for the liquid phase of quinaldine behave, and is it also observed for the other dielectrically active phases? (→Paper 3.) It was already stated in the context of the α-process that the EW is not observed in the usual way in the crystalline phases of quinaldine, all the same it is pointed out above how dielectric spectra of type-A systems can generally be quantified by accounting for the EW contribution. In the following, the major results presented in the second part of Paper 2 concerning the EW shall be summarized at this point. Aging experiments on supercooled 4-TBP at temperatures closely below Tg monitor the susceptibility approaching thermal equilibrium within a number of 47 3 Extended Abstract Figure 3.19: (a) Equilibrium spectrum of 4-TBP at T= 161 K (circles). Crosses: the same data after subtraction of a power law ∝ν−0.2(dashed line). Inset: DS data of glycerol (full circles; T= 179 K) and propylene carbonate (PC, full diamonds; T= 153 K. Open symbols: data sets after subtraction of a power law ∝ ν−0.2. (b) Temperature dependence of the normalized dielectric susceptibility of several type-A (open symbols) and type-B systems (full symbols) at fixed frequency ν= 1 kHz, plotted versus reduced temperature. (c) Time constants of αand β-processes of several glass formers. days. Since the analysis of type-A systems has shown (high-frequency scaling; EW amplitude C≈const. near Tg, see Fig. 3.14) that the EW contribution is strongly coupled to the α-relaxation peak, the susceptibility variation of 4-TBP during aging (for details see Paper 2) is interpreted, consequently within the ideas of the Gainaru approach, as a vanishing of the spectral contribution of the EW. The latter withdraws together with the α-peak, which shifts slowly to longer time scales. The curvature in ε00(ν) of 4-TBP, which is already present in the nonequilibrium spectra, but gets more pronounced upon aging, is interpreted as the signature of a hidden β-relaxation. Finally, the β-peak is completely revealed by subtracting the EW contribution in the form of a power law ε00(ν)∝ν−γwith γ≈0.2. The resulting β-peak is symmetric, and its time constants show typical Arrhenius behavior with the activation energy hEi= 24 Tgas often found for β48 3.7 Existing Approaches, Open Questions, and Major Results processes (cf. Fig. 3.19 (c)). Figure 3.19 (a) illustrates this procedure of spectral decomposition, displaying the susceptibility of 4-TBP at T= 161 K before and after the subtraction of the EW contribution. Dielectric measurements on several molecular glass formers (type-A and -B) down to lowest temperatures of T≈2 K have been performed in the framework of Paper 2. In Fig. 3.19 (b) a comparison of resulting ε00(T) data at fixed frequency ν= 1 kHz, normalized by the relaxation strength of the α-relaxation near Tg, is displayed. For all systems the signature of the α-peak is found near Tg. At lowest temperatures (0 < T/Tg<0.3) low-temperature anomalies interpretable as thermally activated dynamics in asymmetric double well potentials (ADWP) [71] appear as peaks. In the intermediate temperature range susceptibility contributions of excess wings as well as β-relaxations or, respectively, their low-temperature signatures, are observed. Here, almost identical amplitudes within different type-A systems are found, suggesting that α-relaxation and EW have strongly correlated molecular origins. On the other hand, higher and strongly varying loss amplitudes are found for type-B glass formers, suggesting that the β-relaxation has a microscopic origin different from that of the EW phenomenon. This finding also justifies the Gainaru approach used for the spectral decomposition of 4-TBP: any β-process contribution in Fig. 3.19 (b) appears as additional contribution to a generic, “basic” susceptibility. This result supports strongly the Gainaru approach, i.e. the overall susceptibility may, thus, be interpreted as a superposition of α-process, β-process, and EW. A compilation of time constants describing β-processes of a variety of glass formers, including also 4-TBP and other type-A systems where the β-process was revealed by subtracting the EW contribution, is shown in Fig. 3.19 (c). Their activation energies vary within hEi= 11 Tg–26 Tg. Thus, the hEi= 24 Tgrule introduced by Kudlik et al. [75,78] seems obsolete (cf. also Fig. 3.17 (a) and (b), as well as Paper 4). relaxation processes in binary systems Two-component mixtures of organic glass formers are called binary systems in this thesis. Providing a comprehensive description of the phenomenology of the primary relaxations in binary systems may become quite challenging, as documented by several contributions on polymer-plasticizer systems [114–118] and mixtures of low molecular weight glass formers [119–121]. Only in the case of asymmetric binary systems, i.e. mixtures of components with a high Tgcontrast [35,122], two primary relaxations are well resolved and attributed to the molecular dynamics of each component. Principally, FTS does not hold in binary systems since broad temperature-dependent distributions of correlation times caused by strong dynamic heterogeneity underlie the primary relaxations [38, 45]. Moreover, in the recent work by Blochowicz et al. [35, 122] a temperature-dependent fraction of plasticizer molecules attributed to the matrix dynamics is identified. However, the effect of a low-Tgcomponent on the structural relaxation of the high-Tgcomponent (plasticizer effect) as well 49 3 Extended Abstract as the effect of the high-Tgcomponent on the primary relaxation of the low-Tg component (anti-plasticizer effect) are well known, e.g. from differential scanning calorimetry (DSC) experiments showing two glass transition temperatures for binary mixtures [123–128] (cf. also Fig. 3.20). Besides a complex phenomenology concerning the primary relaxation features of the components “in the mixture”, and the existence of two concentration dependent Tg, binary systems offer another viewpoint on secondary relaxations. If a β-relaxation is present, it may be found for almost all investigated mixing proportions [96,122,129]. In some cases, at high concentrations of the low Tgcomponent (c→1), αand β-peak approach each other and the β-relaxation even submerges below the α-peak, i.e. the β-peak develops continuously into an EW [44,119,122]. The investigation of binary glass forming systems is a quite new, wide field of research. Since many combinations of components are possible and each combination has to be analyzed for several concentrations, much effort is required for the complete charactrization of a binary system, and, hence, comparably little knowledge on the molecular dynamics of binary systems has been accumulated. In the context of the analysis of a highly asymmetric binary system within the present work, the following open questions are addressed: •How are the primary relaxations of each component affected by the presence of the other component in the mixture? (→Paper 6.) •Are there additional dynamics in the mixture, which were not observed in the neat components? (→Paper 6.) •Do the microscopic mechanisms (isotropic vs. spatially hindered) of the primary relaxations or their appearance (thermally activated vs. non-Arrhenian τ(T); FTS) vary with concentration? (→Paper 6.) •How do the Tgdepend on concentration? (→Paper 6.) •How can the geometry of molecular motion be characterized for both components? (→Papers 5 & 6.) The major results concerning the primary relaxation phenomena of the asymmetric binary system TPP/PS (tripropyl phosphate, Tg= 134 K; polystyrene, Mw= 2250 g/mol, Tg= 335 K) shall be summarized here. As already pointed out above, the molecules of the two components possess highly differing permanent dipole moments. As a consequence, DS monitors solely the dynamics of TPP. The results for TPP/PS mixtures of 13 concentrations are presented in detail in Paper 6, which is a joint study by dielectric spectroscopy (DS), 31P and 2H nuclear magnetic resonance spectroscopy (NMR) and, actually, also differential scanning calorimetry (DSC) and depolarized light scattering (DLS) experiments. Due to the high Tgcontrast of the components, two distinct glass steps are observed in the DSC traces of the mixtures, situated at the two concentrationdependent glass transition temperatures Tg1(cT P P ) and Tg2(cT P P ). 2H NMR experiments on the PS monomers and, equivalently, 31P NMR measurements on the 50 3.7 Existing Approaches, Open Questions, and Major Results Figure 3.20: (a) Time constants from DS (full triangles: α1; full circles: α2), NMR (open triangles: α1; open circles: α2), DSC (open diamonds: α1; crossed diamonds: α2) and DLS (left pentagons: α1; right pentagons: α2) for all investigated concentrations (cf. color code). Lines are guides for the eye. For details see Paper 6. (b) Glass transition temperatures Tgas yielded by DSC, NMR and DS for all investigated mixtures. Dashed lines: guides for the eye. TPP molecules assign Tg1to PS (“matrix”) dynamics, and Tg2to TPP dynamics occurring on shorter time scales. DLS experiments demonstrate, that the time scales of the TPP and PS dynamics are separated up to highest concentrations and temperatures. DS measurements probing, as said above, solely dynamics of the polar TPP molecules reveal two primary TPP relaxations on different time scales. The time constants τ2of the faster relaxation (α2-process) correspond with the TPP dynamics observed by NMR and identified as Tg2in the DSC traces for 51 3 Extended Abstract the respective concentrations. The τ1of the slow α1-relaxation agree with the time constants of PS dynamics observed by NMR, as well as the Tg1of the DSC experiments. We conclude that two sub-ensembles of TPP molecules exist, one of which is dynamically “trapped” by (or attributed to) the PS matrix. The other TPP sub-ensemble performs “free” TPP dynamics, yet under the concentration dependent anti-plasticizing effect of the PS matrix. A quantitative analysis of the dielectric relaxation strengths ∆ε1and ∆ε2of the α1and α2-process, respectively, reveals that the fraction of TPP molecules associated with the high-Tgcomponent (i.e. the matrix component) decreases with increasing Tand, finally, vanishes at a temperature Tc. This phenomenon was also observed by Blochowicz et al. [35], who interpreted it as the signature of a so-called type-A glass transition as predicted by MCT. The combination of methods yields time constants of both relaxations in a wide temperature and concentration range and, therefore, gives an unprecedented overview over the quite complex dynamics of the binary system (Fig. 3.20 (a)). The time constants τ2(T) of the α2-process are found to develop from VFT behavior at high TPP concentrations to an Arrhenian temperature dependence at low concentrations, i.e. a transition from fragile to strong behavior under decreasing concentration is observed. This leads, by applying the definition τ2(Tg2) = 100 s, to a maximum in Tg2(cT P P ), which is confirmed by the DSC results, but has, up to our knowledge, not been observed before (Fig. 3.20 (b)). The α2-peak has a typical α-peak shape at high TPP concentrations. Here, the failure of FTS is restricted to the low-frequency side of the peak. When cT P P is lowered, the α2-peak assumes a broad shape increasing its width with decreasing temperature, which reminds of the behavior of a thermally activated β-process. Indeed, the scaling procedure also used in Papers 4 and 5 works, and proves that the α2-process is thermally activated at low additive concentrations, yielding temperature independent distributions of activation energies g(E). However, 31P NMR spectra of the cT P P = 20% mixture identify the α2-relaxation as a process of isotropic, liquid-like molecular reorientation, even in the case of low TPP concentrations. 52 4 Publications List of publications forming this thesis: Paper 1 Generalization of the Cole–Davidson and Kohlrausch Functions to Describe the Primary Response of Glass-Forming Systems. R. Kahlau, D. Kruk, T. Blochowicz, V. N. Novikov, and E. A. R¨ ossler, Journal of Physics: Condensed Matter 22, 365101 (2010). Paper 2 Evolution of Excess Wing and β-Process in Simple Glass Formers C. Gainaru, R. Kahlau, E. A. R¨ ossler, and R. B¨ ohmer, The Journal of Chemical Physics 131, 184510 (2009). Paper 3 Quinaldine: Accessing Two Crystalline Polymorphs via the Supercooled Liquid R. Kahlau, T. Gnutzmann, F. Emmerling, K. Rademann, and E. A. R¨ ossler, The Journal of Chemical Physics 137, 054505 (2012). Paper 4 Secondary Relaxations in a Series of Organic Phosphate Glasses Revealed by Dielectric Spectroscopy R. Kahalu, T. D¨ orfler, and E. A. R¨ ossler, The Journal of Chemical Physics 139, 134504 (2013). Paper 5 On the Cooperative Nature of the β-Process in Neat and Binary Glasses: A Dielectric and Nuclear Magnetic Resonance Spectroscopy Study D. Bock, R. Kahlau, B. Micko, B. P¨ otzschner, G. J. Schneider, and E. A. R¨ ossler, The Journal of Chemical Physics 139, 064508 (2013). Paper 6 Dynamics of Asymmetric Binary Glass Formers. I. A Dielectric and Nuclear Magnetic Resonance Spectroscopy Study R. Kahlau, D. Bock, B. Schmidtke, and E. A. R¨ ossler, The Journal of Chemical Physics 140, 044509 (2014). 53 4 Publications Individual contribution to each publication: Paper 1 I had the idea of applying the introduced modifications to the step-response representation of the Cole–Davidson function. The properties of the resulting fit function were analyzed by me, and I calculated the mean correlation time. I showed that the Kohlrausch function is the second limiting case of the function (besides the Cole–Davidson function). Following the advice given by T. Blochowicz, I applied Bernstein’s theorem to show that the new function is a physically valid relaxation function, and proved its complete monotonicity. The distribution of correlation times was calculated and numerically evaluated by D. Kruk. Paper 2 I reanalyzed previously published DS spectra of molecular glass formers by applying the low-frequency and the high-frequency scaling, as demonstrated in Fig. 2. For this purpose, I normalized the data by the relaxation strength, which is demonstrated in Fig. 3. I finally fitted the data of propylene carbonate (PC), propylene glycol (PG), m-tricresyl phosphate (m-TCP), 4-tertbutylpyridine (4-TBP), and glycerol with the function introduced in Paper 1. The results of this analysis are collected in Fig. 4, corresponding data interpolations are shown in Fig. 3. The second part of Paper 2 is basically the work by C. Gainaru. Data of decahydroisoquinoline (DHIQ) were measured by me, but already shown in my Diploma thesis. Besides some modifications, Fig. 8 as well as related measurements are also part of my Diploma thesis. Paper 3 I executed DS measurements on liquid quinaldine and liquid 3methyl quinoline, which finally led to the discovery of the dielectrically active phases 2mq2 and 2mq3, as well as 3mq2 and 3mq3. I conducted all further experiments, including frequency resolved dielectric spectroscopy on the dielectrically active phases of 2-methyl quinoline and 3-methyl quinoline, time resolved DS experiments monitoring the time dependent phase transformations, time resolved X-ray diffraction (XRD) experiments (together with T. Gnutzmann in the laboratory of Bundesanstalt f¨ur Materialforschung und -pr¨ufung in Berlin) and differential scanning calorimetry (DSC) experiments. All analyses shown in this paper, except the processing of XRD data (T. Gnutzmann), were executed by me. 54 Paper 4 I measured triethyl phosphate (TEP), tripropyl phosphate (TPP) and tributyl phosphate (TBP). Data of TEP and TPP are partially published in [104], data of TPP are further used in Papers 5 & 6. The systems tris(2-butoxyethyl) phosphate (T2BOEP) and tris(2-ethylhexyl) phosphate (T2EHP) were measured, under my guidance, by T. D¨orfler in the framework of his Bachelor thesis. All analyses shown in this paper were done by me. Paper 5 I executed all DS measurements for this publication. Data were also used in Paper 6, data of neat TPP were also shown in Paper 4 and in [104]. I further did all analyses of dielectric data. All NMR experiments and related analyses on TPP, PS, and TPP/PS mixtures were done by D. Bock. Paper 6 I performed all DS measurements for this publication. Data were also used in Paper 5, data of neat TPP were also shown in Paper 4 and in [104]. I further did all analyses of dielectric data. All NMR and DSC experiments as well as related analyses on TPP, PS, and TPP/PS mixtures were done by D. Bock. 55 J. Phys.: Condens. Matter 22 (2010) 365101 RKahlauet al Figure 1. Comparison of Debye, Cole–Davidson (CD) and Kohlrausch (K) susceptibility as a function of reduced frequency; the corresponding stretching parameters are set to 0.6. spectral shape of the αprocess even in simple glass formers, a result already anticipated by Olsen et al [15]aswellas by Blochowicz et al [14]. Moreover, they observed that an interpolation by the CD function works well at high temperatures, whereas usually the Kohlrausch function gives a better description at low temperatures. Thus, looking for a susceptibility function which contains both CD and K functions as special cases is a starting point for an improved description of the measured susceptibilities. In the present contribution we shall introduce a threeparameter step-response function encompassing the CD and K functions, which allows for a straightforward interpolation of the experimental α-relaxation peak (excluding any secondary relaxation processes). As will be demonstrated, the function is sufficiently flexible to provide almost perfect fits for both simple as well as binary glass formers. The most widely applied three-parameter susceptibility is given by the Havriliak–Negami (HN) function [7]. However, the HN function exhibits an unphysical low-frequency (ωτ  1) behavior, i.e. its time constant (or the first moment of the corresponding distribution of relaxation times) diverges. Thus, the function is not adequate to describe the main relaxation in liquids, and there is a need for a physically well-behaved (and simply implemented) three-parameter susceptibility function. As discussed in a preliminary study [16], applying this susceptibility function for simple liquids reveals a crossover from a CD susceptibility at high temperature to a K susceptibility at low temperatures while the high-frequency parameter βmay be kept temperature-independent. 2. Generalization of the Cole–Davidson and Kohlrausch step responses Before introducing a generalization of CD and K susceptibility functions we briefly recall the definitions of both. The CD susceptibility is given by χCD(ω) =1 (1+iωτCD)β0<β⩽1(1) with βdescribing the imaginary part χ CD(ω) of the complex susceptibility. The corresponding pulse response function is givenby[7] ϕCD(t)=−dφCD(t) dt=1 τCD(β) t τCD β−1 exp −t τCD  (2) and the step-response function by φCD(t)=1 (β) ∞ t τCD xβ−1exp(−x)dx=β, t τCD  (β) (3) with (β) =∞ 0 xβ−1exp(−x)dx(4) denoting the Gamma function and (β, y)=∞ y xβ−1exp(−x)dx(5) being the upper incomplete Gamma function [18]. The K function represents a step response: φK(t)=exp −t τKα(6) with the corresponding stretching parameter α. Both the K as well as the CD functions converge to a Debye susceptibility when the corresponding stretching parameter reaches 1. Figure 1shows the CD and K functions, the latter after (numerically calculated) Fourier transformation into the frequency domain, with the same relaxation time ταand stretching parameter α=β. Generally, for a given stretching parameter the Kohlrausch relaxation peak is broader than the Cole–Davidson peak. For the realistic value of α= β=0.6 this difference is expected to be resolvable in experiments (cf below). Most experimental works analyzing, for example, dielectric spectra of glass-forming systems use the CD or K functions, and some deficiencies of the interpolation are accepted when a large temperature range is covered. As mentioned in section 1, indications have been found that actually two lineshape parameters are needed to fully reproduce the primary relaxation peak [14,15]. This is once again demonstrated in figures 2(a) and (b) where the normalized dielectric spectra of propylene carbonate (PC) [5] and propylene glycol (PG) [14] are rescaled to agree at the high-frequency flank of the relaxation peak. Figure 2(a) shows the full spectral range for which the dielectric data have been measured. In figure 2(b) we focus on the data around the relaxation maximum, as the function to be introduced describes the relaxation peak only. Indeed, the data coincide over a substantialfrequency range at high frequencies, indicating that the high-frequency parameter does not change significantly with temperature. We note that at the highest frequencies (figure 2(a)) indications for the so-called excess wing are recognized which may change its amplitude upon cooling [5,6,17]. At the low-frequency flank and around the peak (figure 2(b)) no agreement is found. Here, the peak appears to broaden when lowering temperature. The overall change of the width is also reflected in a decreasing height of the peak. Moreover, as demonstrated by the fits in 2 J. Phys.: Condens. Matter 22 (2010) 365101 RKahlauet al Figure 2. (a) Propylene carbonate (PC, left axis) and propylene glycol (PG, right axis) susceptibility data normalized by the relaxation strength and plotted versus a reduced frequency to allow for a overlapping of the data at high frequencies (PG data: [14]; PC data: [5]). (b) The same plots shown for the frequency range of the main relaxation peak only. In addition Cole–Davidson and Kohlrausch interpolations, respectively, are shown for the indicated temperatures. Figure 3. (a) Interpolation of the main relaxation of propylene glycol (crosses) with a Kohlrausch susceptibility function (red lines, fit weighted, data from [14]); note deviations around peak at high temperature. (b) Main relaxation of glycerol (crosses) interpolated with the Cole–Davidson (CD) function (red lines) without weighting. Blue dotted line: CD fit (weighted fit, data from [1]); systematic deviations occur at low temperatures. figure 2(b) a crossover from a CD spectral shape towards a K shape is suggested, i.e. a CD function works better at high temperature whereas a K function is better at low temperature. Consequently, a fit over a large temperature range by the CD or K functions yields systematic deviations independent of the applied fitting strategy. This is demonstrated in figure 3.Once again we focus only on the data around the relaxation peak. Next, we present a function φg(t)which may be called a generalization of K and CD functions; it includes the CD and K functions as special cases. The function φg(t)can be introduced by modifying φCD(t)in equation (3): φg(t)=∞ t τg xβ−1exp(−xα)dx ∞ 0xβ−1exp(−xα)dx(7) by adding a stretching parameter αto the exponential expression, i.e. changing the exponential term in equation (3) to a Kohlrausch expression (cf equation (6)). An alternative definition of φg(t)which is more suitable for a numerical implementation of the model function is given by φg(t)=1 β α∞ t τgαyβ α−1exp(−y)dy= β α,t τgα β α.(8) This formula has been obtained by substituting y=xαin equation (7). It is obvious that φg(t)=φCD(t)if α=1andβ=βCD (cf equation (3)), and one can easily prove that the condition α=βgives again the K function: φg(t)=1 (1)∞ t τgαexp(−y)dy=exp −t τgα=φK(t) =exp −t τKαfor α=β. (9) By integrating φg(t)(equation (8)) one gets the mean correlation time (cf the appendix): τα=τg 1+β α β α.(10) Again, the limits of CD and K susceptibility [7,19]are recovered: τα,CD =lim α→1τα=τg (1+β) (β) =τgβ(11) τα,K=lim α→βτα=τg 1+α α (1)=τg1 α+1=τg α1 α. (12) 3 J. Phys.: Condens. Matter 22 (2010) 365101 RKahlauet al (a) (b) Figure 4. (a) Graphs of the susceptibility function χ g(ωτα)(cf equation (8)) for α=0.6,0.7,0.8,0.9,1.0andβ=0.6. Dashed red line: power law proportional to (ωτα)−0.6. (b) Corresponding distribution of correlation times G(ln τ); solid line: α=β=0.5 (Kohlrausch distribution); dashed line: α=0.4,β =0.5; dotted line: α=0.6,β =0.5; dashed–dotted line: Cole–Davidson distribution for β=0.5. Figure 4(a) shows the susceptibility χ g(ω) calculated from equation (8) via Fourier transformation for the parameter β=0.6andseveralαvalues ranging between 0.6 (α=β) and 1.0. A continuous crossover of the spectral shape from CD to K type is illustrated. The parameter βdetermines the high-frequency slope of the peak, in analogy to the CD stretching parameter, and αis responsible for its flattening. The corresponding distribution G(lnτ) is displayed in figure 4(b) (cf the appendix). With respect to the CD distribution with its clear cutoff at long correlation times the new function allows us to vary its long-time flank continuously. Any valid relaxation function φ(t)of a system in thermodynamic equilibrium can be expressed in terms of a non-negative distribution of relaxation times representing φas a sum of single exponential decays [7,20]. This is equivalent to φbeing completely monotonic, which in turn, by virtue of Bernstein’s theorem, is the equivalent of [21–23] (−1)nφ(n)(t)⩾0for0⩽t<∞(13) with φ(n)(t)=dnφ(t) dtnbeing the nth derivative of φ(t).Inthe case of φg(t), it is easily seen that φ(0) g(t)>0for0⩽t<∞(14) (equations (7)and(8)). Using the properties of the regularized incomplete Gamma function [18] the first derivative can be calculated: φ(1) g(t)=d dt1−1 β αt τgα 0 xβ α−1exp(−x)dx =− α τg β αt τgβ−1 exp−t τgα⩽0 (15) for α, β, τg>0and0⩽t<∞. From Bernstein’s theorem it follows that the higher-order derivatives (n⩾2) of φ(n) g(t)will be of opposite signs if and only if the expression t τgβ−1 exp −t τgα(16) Figure 5. Susceptibility data of propylene glycol (crosses) interpolated by the χ g(ω) function (red lines, βfixed to 0.72). Inset: temperature dependence of the parameter α(boxes); straight line: guide for the eye. is completely monotonic in t. Since it is well known [21,22] that the product of two completely monotonic functions is again completely monotonic, and for the factors exp[−(t τg)α] as well as (t τg)β−1equation (13) is obviously fulfilled for τg> 0andα, β ⩽1, the complete monotonicity of φgis proven. Therefore φg(t)is a physically valid relaxation function for α, β, τg>0andα, β ⩽1. 3. Applications In figure 5we show dielectric spectra of propylene glycol (PG) [14] interpolated by the new function φg(t)given as a Fourier transform χ g(ω). A perfect interpolation is provided and the deficiencies of the CD function are removed. Of course, this is of no surprise as a second shape parameter has been introduced. However, the interpolations shown are achieved by keeping the high-frequency parameter β=0.72 temperature-independent and changing only the parameter α. In other words, as suggested by the high-frequency scaling in figure 2the susceptibility keeps its high-frequency flank while it broadens on the low-frequency side. When inspecting the 4 J. Phys.: Condens. Matter 22 (2010) 365101 RKahlauet al Figure 6. α-relaxation peaks of 2-picoline (5%) in tri-styrene (crosses) compared to propylene glycol (PG) at 176 K (blue triangles, shifted arbitrarily). Interpolations by the model function φg(t)(red lines, cf equation (8)): for comparison interpolation by Kohlrausch function (dashed blue line) showing systematic deviations. temperature evolution of the parameter α(cf inset of figure 5), it decreases starting from 0.89 at the highest temperature to 0.72 at the lowest temperature, i.e. a trend towards the K function (α=β) is recognized while cooling. In order to demonstrate the flexibility of the introduced function φg(t)we try to interpolate broad αpeaksastheyare typically observed in binary glass formers [8,9]. Figure 6 shows the dielectric spectra of the binary system 2-picoline (5%) in tri-styrene in comparison to the spectra of the simple liquid PG. Clearly, the relaxation peak in the binary system is much broader than in the simple liquid PG. Again, the interpolation by φg(t)is superior to a fit by, for example, a K function. In contrast to neat glass formers the parameter α stays virtually constant with α=0.57 while βdrops from 0.3 to 0.18 with increasing temperature. 4. Discussion and conclusions The introduced three-parameter step-response function φg(t) provides a highly flexible description which allows for ‘interpolating’ between the CD-type and K-type spectral shapes of the susceptibility in neat as well as binary glass formers. Three-parameter descriptions of the main relaxation have already been introduced, such as the gamma distribution [14,24] or the HN susceptibility[7]. Regarding the HN susceptibility, often applied for interpolating, for example, the spectra of polymers, as already mentioned in section 1,the function does not show the physically correct low-frequency limit, i.e. its time constant is not defined. Thus it is not expected that the HN function provides a correct interpolation of any main relaxation in complex liquids. In the case of the gamma distribution the susceptibility has to be calculated via a Laplace transform of the distribution of correlation times whereas the presently introduced function applies directly for the step-response function. As suggested by the high-frequency scaling of the experimental susceptibility in neat glass formers (cf figure 2(b)), the function enables one to keep the high-frequency parameter βtemperature-independent while varying the parameter αto account for the experimentally documented changes of the linewidth. Thus, it appears that the failure of FTS in neat glass formers might also be reflected by a low-frequency broadening. Generalizing the results, the invariance of the high-frequency flank of the main relaxation peak might be a generic property of neat glass-forming liquids, and could become an essential input for a full lineshape analysis including main and secondary relaxation. Independent of such conclusions the presented function offers a new tool for interpolating relaxation peaks in complex fluids. Appendix A.1. Mean correlation time Starting from the definition of φg(t): φg(t)=1 β α∞ t τgαyβ α−1exp(−y)dy= β α,t τgα β α(A.1) one can calculate the mean correlation time of φg(t)by changing the variable z=(t τg)α: τα=∞ 0 φg(t)dt=1 β α∞ 0 β α,t τgαdt =1 β α∞ 0∞ t τgαyβ α−1exp(−y)dydt = τg α β α∞ 0 z1 α−1β α,zdz.(A.2) This expression can be converted [18]to τα=τg 1+β α β α.(A.3) A.2. Distribution of correlation times A step-response function φ(t)can be explicitly rewritten as φ(t,˜τ) with (˜τ=τCD,τ KKW,τ g) and related to a distribution function G(τ, ˜τ)defined as φ(t,˜τ) =∞ 0 G(τ, ˜τ)exp −t τdτ(A.4) and connected with the often-used form G(lnτ) via G(lnτ, ˜τ) =τG(τ, ˜τ). Substituting s=τ−1one can write equation (A.4)inthe form φ(t,˜τ) =∞ 0 G(s,˜τ) s2exp(−st)ds =∞ 0 k(s,˜τ)exp(−st)ds.(A.5) This implies that the distribution function G(τ, ˜τ) = s2k(s,˜τ) =k(τ−1,˜τ) τ2can be obtained by taking the inverse Laplace transform of the step-response function φ(t,˜τ): L−1 s[φ(t,˜τ)]=k(s,˜τ). Detailed calculations show that it is convenient to express the distribution function in terms of u=τ ˜τ:G(u)=s2k(s,˜τ). The distribution function GCD(u) 5 J. Phys.: Condens. Matter 22 (2010) 365101 RKahlauet al associated with the Cole–Davidson step-response function, φCD(t,τ CD)≡φCD(u), is given as [7] GCD(u)=sin(πβ) π 1 uu 1−uβ (A.6) while the distribution function GK(u)corresponding to the Kohlrausch step-response function φK(t,τ K)≡φK(u)is given by the formula [19] GK(u)=−1 π ∞  k=0 (−1)k k!sin(πkα)(αk+1)uαk−1.(A.7) The φg(t,τ g)≡φg(u)is related to a distribution function Gg(u)which can be obtained in a ‘semi-analytical’ form by taking the following steps: Gg(u)=τgs2kg(s,τ g)=τgs2L−1 s[φ(t,τ g)] =−τgsL−1 sd dtφ(t,τ g) =τgs β αL−1 sd dtt τgα 0 xβ α−1exp(−x)dx =τgsα β αL−1 st τgβ−1 exp−t τgα.(A.8) Since the distribution function anyway depends on u=τ τg,it is convenient at this stage to set τg=1. The inverse Laplace transform can be obtained by applying the rule for a Laplace transform of a product of two functions, which yields L−1 s{tβ−1exp(−tα)}=s 0 g(s−σ)h(σ) dσ(A.9a) where g(s−σ) =(s−σ)−β (1−β) (A.9b) is the inverse Laplace transform of tβ−1, while h(σ) is given by equation (A.4) with σ=u−1. Finally, the distribution function Gg(s)takes the form Gg(s)=sα β α(1−β)−1 π∞  k=0 (−1)k k!sin (παk) ×(αk+1)s 0 (s−σ)−βσ−(αk+1)dσ. (A.10) For the special case of α=0.5 the distribution function is given by a simpler expression: Gg(s)=s 4√π(2β)(1−β) ×s 0 (s−σ)−βσ−3/2exp−1 4σdσ. (A.11) Examples for the corresponding G(lnτ) distribution are plotted in figure 4. References [1] Lunkenheimer P, Schneider U, Brand R and Loidl A 2000 Contemp. Phys. 41 15 [2] Ediger M D 2002 Annu. Rev. Phys. Chem. 51 99 [3] Blochowicz T, Brodin A and R¨ossler E A 2006 Adv. Chem. Phys. A133 127 [4] Johari G P and Goldstein M 1970 J. Chem. Phys. 53 2372 [5] Kudlik A, Benkhof S, Blochowicz T, Tschirwitz C and R¨ossler E 1999 J. Mol. Struct. 479 201 [6] Ngai K L and Paluch M 2004 J. Chem. Phys. 120 857 [7] B¨ottcher C J and Bordewijk P 1978 Theory of Electric Polarization vol 2 (Amsterdam: Elsevier) [8] Blochowicz T and R¨ossler E A 2004 Phys. Rev. Lett. 92 225701 [9] Capaccioli S, Kessairi K, Prevosto D, Lucchesi M and Ngai K L 2006 J. Non-Cryst. Solids 352 4643 [10] B¨ohmer R, Ngai K L, Angell C A and Plazek D J 1993 J. Chem. Phys. 99 4201 [11] Brodin A, Gainaru C, Porokhonskyy V and R¨ossler E A 2007 J. Phys.: Condens. Matter 19 205104 [12] Nielsen A I, Pawlus S, Paluch M and Dyre J C 2008 Phil. Mag. 88 4101 [13] G¨otze W and Sj¨ogren L 1992 Rep. Prog. Phys. 55 241 [14] Blochowicz T, Gainaru C, Medick P, Tschirwitz C and R¨ossler E A 2006 J. Chem. Phys. 124 134503 [15] Olsen N B, Christensen T and Dyre J C 2001 Phys. Rev. Lett. 86 1271 [16] Gainaru C, Kahlau R, R¨ossler E A and B¨ohmer R 2009 J. Chem. Phys. 131 184510 [17] Hofmann A, Kremer F, Fischer E W and Sch¨onhals A 1994 Disorder Effects on Relaxational Processes (Berlin: Springer) chapter 10 [18] Abramowitz M and Stegun I A 1972 Handbook of Mathematical Functions (New York: Dover) [19] Lindsey C P and Patterson G D 1980 J. Chem. Phys. 73 3348 [20] Wagner K W 1913 Ann. Phys. (Leipzig) 40 817 [21] Alzer H and Berg C 2002 Ann. Acad. Sci. Fenn. Math. 27 445 [22] Alzer H and Berg C 2006 Ramanujan J. 11 225 [23] Widder D V 1972 The Laplace Transform (Princeton: Princeton University Press) [24] Nicolai T, Gimel J C and Johnsen R 1996 J. Physique 6695 6 Paper 2 Evolution of Excess Wing and β-Process in Simple Glass Formers C. Gainaru, R. Kahlau, E. A. R¨ ossler, and R. B¨ ohmer, The Journal of Chemical Physics 131, 184510 (2009). c 2009 American Institute of Physics doi:10.1063/1.3258430 67 68 Evolution of excess wing and ␤ -process in simple glass formers Catalin Gainaru,1,2,a兲Robert Kahlau,1Ernst A. Rössler,1and Roland Böhmer2 1Physikalisches Institut, Universität Bayreuth, Bayreuth 95440, Germany 2Fakultät für Physik, Technische Universität Dortmund, Dortmund 44221, Germany 共Received 1 August 2009; accepted 12 October 2009; published online 13 November 2009兲 Dielectric loss spectra of glass forming liquids are analyzed, with emphasis on systems for which a peak due to a secondary relaxation is not immediately obvious. Thus, glass formers are considered for which the high-frequency flank of the ␣ -relaxation peak appears to be dominated by a so-called wing contribution. It is shown that even for such supercooled liquids the shape of the ␣ -peak has to be characterized by two parameters. By performing a series of aging experiments it is demonstrated that the high-frequency flank of the ␣ -relaxation, assumed to follow a power-law behavior, is superimposed by contributions from an excess wing and from a ␤ -relaxation peak. In particular, the excess wing, previously associated with either the ␣ -orthe ␤ -relaxation, is identified as a feature that evolves in its own right. It is argued that excess wing and ␤ -relaxation are always present albeit with relative strengths that may vastly differ from glass former to glass former. ©2009 American Institute of Physics.关doi:10.1063/1.3258430兴 I. INTRODUCTION When cooling or compressing glass forming liquids their dynamic susceptibilities evolve in a complicated fashion. The structural or primary ␣ -relaxation slows down from picoseconds at TⰇTgto about 100 s near the glass transition temperature Tg. This relaxation is markedly nonexponential, i.e., stretched in time, and defines the ultimate long-time decay of structural fluctuations in small-molecule glass formers. At shortest times microscopic dynamics are observed as an oscillation dephasing with typical frequencies in the terahertz range at all temperatures. In the enormous range of more than 14 decades in time or frequency which separate the time scale of the ␣ -relaxation near Tgfrom that of the microscopic dynamics a variety of other relaxation processes emerges. In dielectric loss spectra these features manifest themselves as so-called excess wing 共EW兲and more often as secondary relaxation peaks. The last were first studied systematically by Johari and Goldstein.1,2Primary and secondary relaxations can often, but not always, be distinguished from each other on the basis of their different spectral, aging, pressure, and temperature evolutions. Whether the excess wing is a feature separate from those relaxations is debated,3,4because a generally accepted procedure allowing one to disentangle the wing contributions from those of the primary and/or of the secondary relaxations is currently not available.5–11 Clarifying the role of the secondary peak and/or of the wing contributions is important since any assumption regarding their spectral shapes may affect the result for that of the primary relaxation. For example, taking results from neutron12 or from light scattering measurements13 as well as from other experimental14 or computational work,15,16 frequency temperature superposition 共FTS兲has been reported to be obeyed by the ␣ -process of glass formers at high temperatures for which neither secondary peaks nor wing relaxations significantly interfere. While FTS is assumed to be an essential feature of glassy dynamics by certain theories,17 at temperatures close to Tgit was often found that the spectral shape probed, e.g., by dielectric spectroscopy is temperature dependent.3,18,19 Some studies emphasize that via appropriate spectral analyses FTS can be recovered, at least asymptotically,14,20,21 throughout the supercooled regime. Recently, based on high-pressure experiments as well as on theoretical work it was argued that the spectral shape should remain invariant neither along a line of constant temperature nor under constant-pressure conditions, but merely along an isochronal line in the temperature-volume plane.22–24 However, different phenomenological approaches for disentangling the contribution of the ␣ -peak from the other relaxation processes were proposed,18,19,25–28 without leading to an overall consensus regarding the situation. Disregarding secondary processes for the moment, it may be asked whether a single parameter is sufficient to describe the shape of the dielectric ␣ -relaxation close to its peak frequency at a given temperature. Such a one-parameter description was assumed in studies inquiring into correlations between the super-Arrhenius with the non-Debye character of the ␣ -process.29 In the meantime there are some indications that more than one shape parameter may be necessary to describe the spectral profile of the ␣ -relaxation even in relatively simple low molecular weight glass formers.14,18,27 Such a case obviously complicates the search for straightforward correlations among various parameters and could rationalize reports suggesting shortcomings of simple correlations.21,30 To clarify the nature of ␤ -processes themselves, an unambiguous separation of them from the primary relaxation is sometimes considered desirable, too. In view of the connectivity between the ␣ - and ␤ -process31,32 it is not clear, however, to which extent such a separation can a兲Electronic mail: [email protected]. THE JOURNAL OF CHEMICAL PHYSICS 131, 184510 共2009兲 0021-9606/2009/131共18兲/184510/10/$25.00 © 2009 American Institute of Physics131, 184510-1 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp be achieved not only phenomenologically, but under which circumstances a separate consideration is permissible in principle. The temperature at which secondary relaxation peaks or excess wing first appear upon cooling is sometimes taken to signal a crossover in the dynamics of supercooled liquids.9,33 Yet, its clear-cut manifestation in susceptibility spectra well above Tgremains a matter of debate. In order to disentangle the various processes, dielectric studies monitoring pressure22,34 and aging dependences28,35 turned out to be particularly useful. In aging experiments Schneider et al.35 found subtle spectral changes in the highfrequency wing of the ␣ -peak for glycerol and propylene carbonate 共PC兲. These results were interpreted as arising from an emerging secondary relaxation peak and led those authors to suggest that the excess wing is a relaxation process of the JG type.2,35 The cited aging study was important in pointing out that ␤ -relaxations may exist in glass formers in which their presence is not obvious, at first glance. However, as already suggested on the basis of high-pressure experiments we will argue “that the excess wing and the ␤ relaxation cannot be treated on the same footing.”22 To put these results in a broader perspective and to address the question whether secondary relaxations may be universally present in glass formers, in Fig. 1we compile dielectric loss spectra of several supercooled liquids for scaling temperatures Ts⬇Tg. The data are normalized to their maximum loss, see Table Ifor ␧max ⬙,Ts, and references to the original works. For some substances such as toluene or dimethylphthalate 共DMP兲,a ␤ -relaxation peak is clearly discernible at about 5 decades above the ␣ -peak frequency. These liquids are thus typical examples of type B systems.11 For glass formers like diglycyl ether of bisphenol A 共DGEBA兲or trimethylphosphate 共TMP兲only the low-frequency flank of a secondary peak is recognized in Fig. 1. Here the separation from the ␣ -peak is larger than 8 decades and we will show below that in fact all degrees of separation are observed. For methyl-tetrahydrofuran 共MTHF兲a ␤ -process is barely resolved.36 For other supercooled liquids the ␤ -relaxation amplitude is even smaller and a ␤ -peak seemingly absent. Glass formers in this limit have been called type A systems11 with glycerol, propylene glycol 共PG兲, and propylene carbonate37 usually considered as classical examples.38 We point out that for systems such as m-flouroaniline 共m-FAN兲, TMP, and DGEBA both a well resolved EW and a ␤ -peak are observed, suggesting that the two processes are independent relaxation phenomena. As revealed in Fig. 1, slightly stronger high-frequency contributions are exhibited by 2-picoline 共2-PIC兲and 4 tert-butyl pyridine 共4-TBP兲.Inall type A systems collected in Fig. 1the high-frequency loss asymptotically varies as ␧⬙⬀ ␯ − ␥ with ␥ ⬇0.2. Although 2-PIC and 4-TBP appear relatively similar in this plot, in Sec. IV we will point out that the behaviors of these glass formers are in fact to be distinguished. Here, we mention that the excess wing is equivalent39 to what has been termed the intermediate power-law on the basis of optical Kerr effect studies.21,40,47 There, again a very similar exponent close to ␥ ⬇0.2 has been reported. The main point of the present article is that near Tgthe EW and the ␤ -relaxation are to be considered as universally present but distinguishable features of varying relative strengths. Thus, we do not view the EW as a special 共submerged兲 ␤ -process. Merely, the high-frequency wing in most type A glass former will be interpreted to involve an EW and a very weak ␤ -process. As will be demonstrated this interpretation also allows one to describe the aging experiments presented in Refs. 35 and 51, and in Sec. III, below. Before discussing our aging results, we first present a         m    mA EBA   C B  C  mC ν  εε ν − = FIG. 1. Dielectric loss spectra for various glass formers recorded for comparable peak frequencies and scaled to the same peak height. The spectra exhibit a succession of decreasing high-frequency contributions thus illustrating a continuous transition from type B to type A profiles. This representation is compatible with ␤ -process and excess wing always being simultaneously present, albeit with different relative intensities. The spectral shape of this glass former is thus intermediate between so-called type A and other type B systems. References and information regarding the measurement temperatures and ␧max ⬙are provided in Table I. TABLE I. The scaling parameters ␧max ⬙and Tsused for Fig. 1are summarized. Typically, Tsis slightly higher than Tgas defined by Tg=T共 ␶ ␣ =100 s兲. The line shape parameters ␤ and ␥ from the fits are reported here, if they were kept constant, while the temperature dependence of the free parameters is presented in Fig. 4.The ␥ exponents of m-TCP and TMP are those given in Fig. 9. Glycerol PC 2-PIC PG 4-TBP m-TCP TMP DHIQ Toluene m-FAN MTHF DGEBA DMP ␧max ⬙22.5 30.3 4.5 26.5 4 2 16.6 0.063 0.06 7 6.3 2.7 0.15 Ts共K兲196 163 133 173 168 213 140 183 119 177 94 256 198 Tg共K兲189 158 130 168 166 205 136 179.5 117 172 92 251 195 ␤ 0.66 0.78 0.6 0.74 ¯0.6 ¯¯ ¯0.52 ¯¯¯ ␥ 0.23 0.23 0.2 0.23 0.2 0.19 0.17 ¯¯0.23 ¯¯¯ References 10,11 11 11 27 27 27 68 共This work兲11 28,11 54 69 70 184510-2 Gainaru et al. J. Chem. Phys. 131, 184510 共2009兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp scaling analysis 共Sec. II A兲which implies that a oneparameter description is sometimes insufficient to describe the shape of the ␣ -process. This is even the case in the region where we may disregard the contributions from the excess wing and from the ␤ -process. Then, in Sec. II B we quantify these indications using a phenomenological fitting approach which supports the notion that ␣ -relaxation and excess wing are governed by different behaviors. In Sec. III these indications are further confirmed by investigating the aginginduced change of the spectral shape of 4-TBP, PC, and glycerol, glass formers that were already well studied under equilibrium conditions.27,41 Finally, in Secs. IV and V we discuss and summarize our main findings. II. SHAPE OF THE RELAXATION PEAK T>Tg A. Scaling features Line shape parameters that change with temperature are often used to describe the evolution of the main relaxation over a broad frequency range.2However, if no secondary relaxation peak interferes, temperature independent line shape parameters were extracted from some spectral analyses.14,21 The determination of these parameters depends on the extent to which contributions from the ␣ -peak region and from the EW were included in the analysis. This ambiguity demonstrates the difficulty encountered when using different fitting strategies, a problem that is particularly acute at high temperatures. Here, the spectral range in which secondary relaxation features are observable becomes smaller and smaller so that the power laws originating from the ␣ -peak may become indistinguishable from those of the EW. This situation may adversely affect the outcome of “free fits” in the sense that it is likely that artificial temperature dependences are introduced. To circumvent such problems, let us resort to studying the scaling properties of dielectric susceptibility data from a wide temperature range in a model independent way. In order to detect possible changes of the loss shape we consider normalized susceptibility spectra, ␹ ⬙共 ␯ 兲=␧⬙共 ␯ 兲/⌬␧. Here the relaxation strength ⌬␧ was extracted from the corresponding ␧⬘共 ␯ 兲data by taking their lowand highfrequency limits ␧sand ␧⬁into account. ␧⬁was considered constant in the entire temperature range T⬎Tg. At temperatures close to Tgthe plateau associated with ␧sis no longer fully reached. To obtain the plateau value in those cases the ␧⬘共 ␯ 兲curves were extrapolated to lower frequencies, according to their behavior at higher temperatures. The integral of such normalized spectra yielded ␲ /2⫾0.05 for all the data we will present below. In a next step we plotted them as a function of reduced frequency ␻ ␶ ␣ with ␶ ␣ denoting the time constant of the ␣ -relaxation. ␶ ␣ was determined in a modelfree way as follows: It is an intrinsic property of the normalized spectral density J共 ␻ 兲of simple liquids that 冏d ␹ ⬙共 ␻ 兲 d ␻ 冏 ␻ ␶ ␣ →0 =J共0兲= ␶ ␣ ,共1兲 and hence ␹ ⬙共 ␻ ␶ ␣ 兲兩 ␻ ␶ ␣ →0= ␻ ␶ ␣ .共2兲 Therefore, as a function of temperature, the low-frequency flank of the normalized susceptibility should collapse when plotting ␹ ⬙versus ␻ ␶ ␣ . This low-frequency scaling is demonstrated in Fig. 2共a兲for a variety of systems. Let us focus first on glass formers that were previously categorized as being of type A such as PG, glycerol, 2-PIC, 4-TBP, and PC. For several of these systems the peak systematically becomes narrower and, since the area under the peak is conserved, the peak amplitude becomes higher with increasing temperature. To quantify the width changes we note that, e.g., for PC at half the peak value of ␧⬙the horizontal shift of the loss curves is about 0.2 decades, while for the other systems, like 2-PIC, such effects are less pronounced. All curves in Fig. 2共a兲intersect at a frequency slightly higher than that corresponding to the peak maximum. Furthermore, the high-             ε∆ε ε∆ε () ωτα D A B   C  C           ω/ω     FIG. 2. 共a兲Normalized dielectric susceptibility spectra plotted vs reduced frequency for glass formers previously categorized as type B 共DHIQ and m-FAN兲or type A systems 共all others兲. The data were taken from the references quoted in Table I. After scaling by ⌬␧ the data set for each glass former was shifted vertically for clarity. The red and the black lines refer to the highest and to the lowest temperatures, respectively, for each set of data. While in frame 共a兲a “low-frequency scaling” procedure was implemented, frame 共b兲involves “high-frequency scaling” for the example of PG. 184510-3 Evolution of excess wing and ␤ -process J. Chem. Phys. 131, 184510 共2009兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 15 W. Kob and H. C. Andersen, Transp. Theory Stat. Phys. 24, 1179 共1995兲. 16 S. Kämmerer, W. Kob, and R. Schilling, Phys. Rev. E 58, 2131 共1998兲; 58, 2141 共1998兲. 17 W. Götze and L. Sjögren, Rep. Prog. Phys. 55,241共1992兲. 18 T. Blochowicz, C. Tschirwitz, S. Benkhof, and E. A. Rössler, J. Chem. Phys. 118, 7544 共2003兲. 19 K. L. Ngai, P. Lunkenheimer, C. León, U. Schneider, R. Brand, and A. Loidl, J. Chem. Phys. 115, 1405 共2001兲. 20 A. I. Nielsen, T. Christensen, B. Jakobsen, K. Niss, N. B. Olsen, R. Richert, and J. C. Dyre, J. Chem. Phys. 130, 154508 共2009兲;A.I. Nielsen, S. Pawlus, M. Paluch, and J. C. Dyre, Philos. Mag. 88, 4101 共2008兲. 21 A. 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Rev. Lett. 97, 135701 共2006兲. 32 K. Kessairi, S. Capaccioli, D. Prevosto, M. Lucchesi, S. Sharifi, and P. A. Rolla, J. Phys. Chem. B 112, 4470 共2008兲. 33 V. N. Novikov and A. P. Sokolov, Phys. Rev. E 67, 031507 共2003兲. 34 R. Casalini and C. M. Roland, Phys. Rev. Lett. 91, 015702 共2003兲. 35 U. Schneider, R. Brand, P. Lunkenheimer, and A. Loidl, Phys. Rev. Lett. 84, 5560 共2000兲. 36 F. Qi, T. El Goresy, R. Böhmer, A. Döß, G. Diezemann, G. Hinze, H. Sillescu, T. Blochowicz, C. Gainaru, E. Rössler, and H. Zimmermann, J. Chem. Phys. 118, 7431 共2003兲. 37 This does not necessarily imply that local 共internal兲motions are absent. From NMR and calorimetry a CH3rotation was identified for PC, see F. Qi, R. Böhmer, and H. Sillescu, Phys. Chem. Chem. Phys. 3, 4022 共2001兲. 38 A pattern similar to that shown in Fig. 1is also observed for a series of polyalcohols, see A. Döß, M. Paluch, H. Sillescu, and G. Hinze, Phys. Rev. Lett. 88, 095701 共2002兲; A. Döß, M. Paluch, H. Sillescu, and G. Hinze, J. Chem. Phys. 117, 6582 共2002兲. 39 A. Brodin and E. A. Rössler, J. Chem. Phys. 125, 114502 共2006兲;126, 244508 共2007兲. 40 M. Ricci, P. Bartolini, and R. Torre, Philos. Mag. B 82, 541 共2002兲. 41 K. Kessairi, S. Capaccioli, D. Prevosto, M. Lucchesi, and P. A. Rolla, J. Chem. Phys. 127, 174502 共2007兲. 42 R. Kahlau et al. 共unpublished兲. 43 Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun 共National Bureau of Standards, Washington, D.C., 1964兲, Chap. 26. 44 C. J. F. Böttcher and P. Bordewijk, Theory of Electric Polarization: Dielectrics in Time-Dependent Fields 共Elsevier, Amsterdam, 1978兲, Vol. 2. 45 Note that this connection of ␣ -process and EW does not imply that these processes are identical. Recent studies show that while the ␣ -process is isotropic, the excess wing originates from a spatially highly restricted motion, see M. Vogel, C. Tschirwitz, S. Schneider, C. Koplin, P. Medick, and E. Rössler, J. Non-Cryst. Solids 307–310, 326 共2002兲; C. Gainaru, O. Lips, A. Troshagina, R. Kahlau, A. Brodin, F. Fujara, and E. A. Rössler, J. Chem. Phys. 128, 174505 共2008兲. 46 The Fourier transformation was carried out numerically using a program written by Dr. A. Brodin whom we thank for making this program available to us. 47 The emergence of the EW at high temperatures is best recognized in optical Kerr effect as well as in light scattering experiments, see Ref. 21 and H. Cang, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 118,2800 共2003兲. 48 Note that for polymeric glass formers two parameters are in general required for a description of the main relaxation, albeit there for a different reason, see A. Schönhals, F. Kremer, and E. Schlosser, Phys. Rev. Lett. 67, 999 共1991兲. 49 The conditions 共i兲through 共iii兲can be fulfilled with ␤ =0.63 and ␥ =0.21 for glycerol and with ␤ =0.78 and ␥ =0.23 for PC. Here the ␤ -parameters are those resulting from previous scaling analyses of glycerol 共Ref. 68兲and of PC 共Ref. 27兲, respectively. 50 A recent study on glycerol at ultrahigh pressure also reports more “normal” ␤ time constants below the ␣ - ␤ merging region, see A. A. Pronin, M. V. Kondrin, A. G. Lyapin, V. V. Brazhkin, A. A. Volkov, P. Lunkenheimer, and A. Loidl 共unpublished兲. 51 For xylitol extended aging led to a resolved ␤ -peak, see R. Wehn, P. Lunkenheimer, and A. Loidl, J. Non-Cryst. Solids 353, 3862 共2007兲. 52 It is clear that at the lowest temperatures the high-frequency part of the ␣ -process is irrelevant for determining ␶ ␤ . 53 G. P. Johari, Ann. N. Y. Acad. Sci. 279,117共1976兲. 54 For MTHF the E/Tgratio was reported to be 1650/91=18.1, see Ref. 36. Since in that article a different method of analysis was employed the data are not included in Fig. 7. 55 T. Blochowicz and E. A. Rössler, Phys. Rev. Lett. 92, 225701 共2004兲. 56 R. Brand, P. Lunkenheimer, U. Schneider, and A. Loidl, Phys. Rev. B 62, 8878 共2000兲共and references cited therein兲. 57 See, e.g., T. El Goresy and R. Böhmer, J. Phys.: Condens. Matter 19, 205134 共2007兲and references cited therein. 58 K. L. Ngai, Phys. Rev. E 57, 7346 共1998兲. 59 D. Pisignano, S. Capaccioli, R. Casalini, M. Lucchesi, P. A. Rolla, A. Justl, and E. A. Rössler, J. Phys.: Condens. Matter 13, 4405 共2001兲. 60 The same value for the exponent ␥ was used by U. Buchenau, J. Chem. Phys. 131, 074501 共2009兲; see also R. Casalini and C. M. Roland, Phys. Rev. B 69, 094202 共2004兲. 61 The behavior of TPP was rationalized in Ref. 21. 62 A. Rivera-Calzada, 共private communication兲, see also Fig. 3in A. RiveraCalzada, K. Kaminski, C. Léon, and M. Paluch, J. Phys.: Condens. Matter 20, 244107 共2008兲. 63 C. Gainaru, A. Rivera, S. Putselyk, G. Eska, and E. A. Rössler, Phys. Rev. B 72, 174203 共2005兲. 64 Apart from the data of Refs. 63 and 65 we included data on 3,3,4,4benzophenonetetracarboxylic dianhydride 共2PC兲from Ref. 2and data on DHIQ from this work. 65 C. Gainaru, Dissertation, Universität Bayreuth, 2007. 66 The peaks observed for T⬍0.3Tgin Fig. 8are due to the relaxation in asymmetric double well potentials, see the discussion in Ref. 63. 67 A. Kudlik, C. Tschirwitz, T. Blochowicz, S. Benkhof, and E. Rössler, J. Non-Cryst. Solids 235-237, 406 共1998兲. 68 S. Adichtchev, T. Blochowicz, C. Gainaru, V. N. Novikov, E. A. Rössler, and C. Tschirwitz, J. Phys.: Condens. Matter 15, S835 共2003兲. 69 C. Gainaru et al. 共unpublished兲. 70 A. Brodin, R. Bergman, J. Mattsson, and E. A. Rössler, Eur. Phys. J. B 36, 349 共2003兲. 184510-10 Gainaru et al. J. Chem. Phys. 131, 184510 共2009兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Paper 3 Quinaldine: Accessing Two Crystalline Polymorphs via the Supercooled Liquid R. Kahlau, T. Gnutzmann, F. Emmerling, K. Rademann, and E. A. R¨ ossler, The Journal of Chemical Physics 137, 054505 (2012). c 2012 American Institute of Physics doi:10.1063/1.4738583 79 80 THE JOURNAL OF CHEMICAL PHYSICS 137, 054505 (2012) Quinaldine: Accessing two crystalline polymorphs via the supercooled liquid Robert Kahlau,1Tanja Gnutzmann,2,3 Franziska Emmerling,2Klaus Rademann,3 and Ernst A. Rössler1 1Physikalisches Institut, Universität Bayreuth, Bayreuth 95440, Germany 2BAM Federal Institute for Materials Research and Testing, Berlin 12489, Germany 3Department of Chemistry, Humboldt-Universität, Berlin 12489, Germany (Received 6 March 2012; accepted 6 July 2012; published online 3 August 2012) Quinaldine (2-methyl quinoline) is a liquid at room temperature, which can be supercooled to reach finally the glassy state. By heating the glass above the glass transition temperature Tg=180 K the sample performs two subsequent transitions into, likewise, dielectrically active phases. Thus, the reorientational relaxations of these phases as well as the kinetics of the phase transitions can be tracked in a highly resolved way by dielectric spectroscopy. X-ray diffraction analysis clearly shows two structurally different crystalline phases in addition to the supercooled liquid. Calorimetric measurements support the notion of first order phase transitions, occurring irreversibly in the supercooled regime, and suggest that the intermediate crystalline phase is metastable, too. Analyzing the quite distinct dielectric relaxation strengths, we discuss the possible nature of the two crystalline phases. Additionally, a very similar behavior to quinaldine is observed for 3-methyl quinoline, indicating a broad field of polymorphism among the quinoline derivatives. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4738583] I. INTRODUCTION Organic compounds are commonly known for showing the phenomenon of polymorphism,1–8i.e., they may exist in structurally different crystalline phases. For example, in the case of the compound ROY, at least seven polymorphs have been identified.2Under the given conditions, only a single phase can be thermodynamically stable whereas the others are all metastable. Finding the routes to obtain and capture such metastable phases has thus become an important issue in materials science. Many liquids can be supercooled leading to a strong increase of their transport coefficients like viscosity or diffusion coefficient. At temperatures around the glass transition temperature Tg, the crystallization process of the deeply supercooled liquid is typically diffusion controlled and therefore avoided on laboratory time scales.1Then the liquid is in a metastable state. Considering the chemical potential μl(T) of the highly supercooled liquid as a function of temperature, usually a large gap opens with respect to the potential μc(T) of the thermodynamically stable crystalline phase (cf. Fig. 1). A metastable crystalline phase with its potential μm(T), if present, will appear within this gap. In other words, by increasing the temperature of the highly viscous liquid and thus again facilitating diffusion the usually occurring phase transition does not necessarily have to end up with the thermodynamic equilibrium phase but may yield metastable phases. Thus, strongly supercooling a liquid opens the possibility for searching unknown metastable phases of molecular systems. A prominent example is liquid ethanol,9,10 which transforms into a metastable crystal if it is kept close to Tg=97 K, i.e., well below the melting point Tm=159 K. Regarding its orientational degrees of freedom, this crystal exhibits “glassy” dynamics, i.e., highly cooperative molecular rotation above the corresponding Tg. Upon heating, the glassy crystal phase of ethanol transforms into the orientationally ordered and thermodynamically stable phase, which melts at Tm. Studying the regime of a supercooled liquid or even the glass (T<Tg) with respect to possible phase transitions is also an important issue in understanding how to stabilize glasses against crystallization. For some applications, for example, pharmaceuticals, the glassy state is advantageous over the crystalline state, and it is important to avoid crystallization or at least to minimize the rate of crystal growth. In a recent application, Ediger and co-workers11 have investigated the diffusion coefficient Dof the supercooled liquid indomethacin (Tg=315 K). The authors have found that the temperature dependence of the diffusion is significantly weaker than that of viscosity, and that it is indeed D(T), which controls the transformation into the three identified12,13 crystalline polymorphs of indomethacin. In the present contribution, the polymorphism of the compound quinaldine (2-methyl quinoline, 2mq, see Fig. 3)is examined. Quinaldine is a liquid at room temperature, which can easily be supercooled below its melting temperature Tm≈264–270 K as demonstrated by Capaccioli et al.14 Like in other glass-forming liquids, the structural relaxation becomes very slow upon cooling, i.e., the reorientational correlation time of a molecule rises from τ= 10−12 s at the melting point to τ=100 s at the glass temperature Tg. Since the quinaldine molecule is polar, this behavior is well verifiable by means of dielectric spectroscopy,15 and we have determined the glass transition temperature to be Tg≈180 K. We demonstrate that 0021-9606/2012/137(5)/054505/10/$30.00 © 2012 American Institute of Physics137, 054505-1 Downloaded 06 Aug 2012 to 132.180.21.94. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054505-2 Kahlau et al. J. Chem. Phys. 137, 054505 (2012) μ m (T) μ c (T) T m μ (T) T T g liquid stable solid phase metastable solid phase μ l (T) FIG. 1. Searching for metastable phases: chemical potential of stable and metastable solid phases together with that of the (supercooled) liquid phase; melting (Tm) and glass transition temperatures (Tg) are indicated. two phase transitions can be monitored with the help of dielectric spectroscopy, x-ray diffractometry (XRD) and differential scanning calorimetry (DSC) upon heating the highly supercooled liquid. Here, we benefit from the fact that all the three phases of quinaldine are dielectrically active and thus allow to probe the dynamics in each phase as well as both phase transitions. Moreover, as quinaldine is a rigid molecule of low symmetry, any dynamics has to involve the total molecule. This is likely to result in some kind of cooperative dynamics in the crystalline state, the nature of which we are interested in. Finally, we show that similar polymorphs are identified for 3-methyl quinoline. II. EXPERIMENTAL A. Chemicals 2-methyl quinoline (“quinaldine,” CAS 91-63-4, 97+%) was purchased from Alfa Aesar, 3-methyl quinoline (CAS 612-58-8, 99%) was delivered by Sigma-Aldrich. Both substances were used as received without any further treatment. B. Dielectric spectroscopy Dielectric measurements were carried out with the Alpha-A analyzer by Novocontrol, which allows for frequency resolved measurements of the dielectric susceptibility in the range of ν=10−2–106Hz. During a frequency scan, the sample temperature was kept constant within ±0.2 K with the help of a Quatro-H temperature controller by Novocontrol. In order to interpolate the main relaxation peak in the dielectric spectra of the (supercooled) liquid phase (2mq1), we used a step response function, which generalizes the Kohlrausch and the Cole-Davidson spectral shape and has turned out to be a versatile response function for supercooled liquids.16 Explicitly, φg(t)=1 β α∞ t τgαyβ α−1exp(−y)dy = β α,t τgα β α(1) was used for the structural relaxation or α-peak. Here (...) denotes the gamma function and (...,...)theupper incomplete gamma function as defined by Eq. (1), respectively. This (normalized) step response function is connected with the susceptibility spectra in the frequency domain by the one-sided Fourier transform F(...) ε (ω)=εχ (ω)=εωRe {F[φ(t)]}(2) with ε being the dielectric relaxation strength. The integral of φg(t)(Eq.(1)) yields the mean relaxation time τα=τ0 β+1 α β α.(3) The interpolation of the so-called excess wing (EW), appearing on the high-frequency flank of the α-relaxation peak as a kind of power-law, was obtained by including the step response of the Cole-Davidson function (CD function) φCD (t)=γ, t τCD  (γ)(4) together with the Williams-Watts approach17 φ(t)=[CφCD (t)+(1−C)]φg(t)(5) with Cbeing a measure for the amplitude of the EW. The time constants of both contributions are set equal (τg=τCD) so that only the high-frequency power-law with the exponent γof the Cole-Davidson contribution remains visible.18 The second observed quinaldine phase (2mq2) exhibits a relaxation peak, which has again the spectral shape of a structural relaxation peak rather than the shape of a secondary process. Since its low-frequency flank, however, has an exponent slightly smaller than 1 we used the normalized HavriliakNegami susceptibility function as fit function, ˆχHN (ω)=1 [1 +(iωτ)α]β.(6) The third phase (2mq3) shows a broad relaxation peak, which is best interpolated by the distribution of correlation times usually applied for secondary relaxations in glasses.19 Explicitly, Gβ(ln τ)=Nβ(a,b)1 bτ τβa+τ τβ−ab (7) with the normalization factor Nβ(a,b)=a(1+b) πbb 1+bsin πb 1+b.(8) The parameter acauses a symmetric broadening of the distribution while bonly affects its short time flank. Therefore bcan also be called “asymmetry parameter.” After Laplace transforming Gβ(ln τ) into frequency domain, the resulting susceptibility peak is broadened symmetrically by a, which is also the exponent of the power-law approached asymptotically by the peak’s low frequency flank. The exponent of the power-law asymptote of the high-frequency flank is given by the product ab. Downloaded 06 Aug 2012 to 132.180.21.94. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054505-3 Kahlau et al. J. Chem. Phys. 137, 054505 (2012) For all phases, a power-law accounts additively for the dc conductivity contribution χ DC (ω)=Aω−1.2···−0.8.(9) C. Differential scanning calorimetry For the DSC measurements, we used a Netzsch DSC 200. Its temperature scale was calibrated within the range of interest with the help of the first order phase transitions of six reference samples (cyclohexane, chloroform, mercury, carbon tetrachloride, water, benzene). The DSC-signal intensity was not calibrated; yet, any conclusions can be drawn in a qualitative way. Throughout the DSC studies within this work, temperature has either been kept constant (isothermal measurement) or heating/cooling rates of 20 K/min (ramp experiments) were applied. Any temperature scan shown below was compiled of a cooling stage from room temperature to 120 K, an isothermal stage lasting 10 min and a heating run back to room temperature. If additional stages were applied in order to induce phase transitions, it will be explained explicitly. D. X-ray structure analysis The x-ray diffraction (XRD) experiments were carried out using Cu-Kα1 radiation and a curved position sensitive detector (INEL CPS120) and a furnace (MRI) as sample environment. The sample was kept on a temperate stage in a thermally isolating vacuum. Using this setup, it was possible to gain diffractograms on short time scales in order to obtain several snapshots of the phase transition processes in the sample. In order to gain higher signal qualities for characterizing the neat crystalline phases, the data accumulation time could be increased and adapted to the experimental situation. III. RESULTS A. Dielectric spectra 1. Supercooled liquid (2mq1) Typical dielectric susceptibility spectra of the supercooled quinaldine’s liquid phase (2mq1, T>Tg)areshown in Fig. 2. In addition to a main (α-) relaxation peak, an EW is observed on the high-frequency flank of the relaxation peak. Since no secondary β-relaxation peak can be resolved, the system can be categorized as type-A glass former.15 Both spectral features shift to lower frequencies with decreasing temperature without any significant spectral changes. In order to fit the full spectra, including the EW, the relaxation function is made up of a convolution of the generalized Kohlrausch/Cole-Davidson function (α-process, Eq. (1)) with a CD function with width parameter γ(Eq. (4)) applying the Williams-Watts ansatz (Eq. (5)); the CD function accounts for the EW, which follows a power-law behavior (εEW(ν) ∝ν−γ). The decomposition of the spectra is indicated in Fig. 2. The high-frequency shape parameter of the α-peak is found to be temperature independent with β=0.71, and also 10-2 10-1 100101102103104105106107 10-3 10-2 10-1 100 101 ~ ν−β 185K 192K ε''(ν) ν / Hz β=0.71 γ =0.30 liquid phase (2mq1) 186K 190K 195K 200K 210K 215K ~ ν−γ FIG. 2. Dielectric spectra of supercooled liquid quinaldine (2mq1, open symbols: this work, connected dots: data taken from Ref. 14). Continuous lines: fits including relaxation peak and EW (Eqs. (1)–(5)). Dashed lines: decomposed fit of the spectrum at 185 K. The shape parameters βand γhave been kept fixed for all temperatures (numbers); the shape parameter αand the EW amplitude Cvary weakly (inset in Fig. 3). the exponent of the EW γ=0.30 can be kept constant at all measured temperatures. The parameters α(low-frequency shape parameter) and C(amplitude of the EW) vary weakly with temperature (cf. inset Fig. 3). In other words, frequencytemperature superposition (FTS) holds in good approximation for the full spectra. This is verified once again with the help of the corresponding master curve shown in Fig. 3where the normalized susceptibility is plotted versus ωτα.Nosignificant changes in the spectral shape of both main relaxation and EW are observed. The spectral shape is in almost perfect agreement with that reported by Capaccioli et al.14 as demonstrated in Fig. 2. A small discrepancy in amplitude (factor 1.1) and frequency position (factor 1.4) exists, which is barely recognized on logarithmic scale and probably due to a tiny mismatch in temperature calibration. The extracted relaxation times τα(cf. Eq. (3)) are shown in Fig. 4(a) (squares). Their non-Arrhenius temperature 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 -5 10 -4 10 -3 10 -2 10 -1 180 190 200 210 0.0 0.2 0.4 0.6 0.8 1.0 ε''/Δε ωτ α 2mq1 2mq2 liquid phase (2mq1) scaled by ωτ α T / K α C FIG. 3. Susceptibility spectra (182 K– 215 K) of supercooled liquid quinaldine normalized by the relaxation strength ε and plotted versus ωτα(continuous lines). For comparison: main relaxation of the second phase 2mq2 (at 215 K, dashed line). Inset: temperature dependence of the fit parameters α and C(cf. Eqs. (1),(4),and(5)). Downloaded 06 Aug 2012 to 132.180.21.94. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054505-4 Kahlau et al. J. Chem. Phys. 137, 054505 (2012) 4.44.85.25.6 10-6 10-3 100 103 2mq1 2mq2 2mq3 τ / s 1000K / T Tg1=(180±0.5)K Tg3=(180±2)K Tg2=(193±1)K (a) quinaldine time constants 180 190 200 210 220 101 102 103 104 2mq3 2mq2 TΔε T / K 2mq1 quinaldine relaxation strengths (b) FIG. 4. (a) Time constants τof the three dielectrically active phases of quinaldine; τof 2mq1 (liquid, squares) are interpolated by a VFT law (Eq. (10)); τ of 2mq2 (triangles) and 2mq3 (crosses) are fitted by Arrhenius laws. The corresponding values of Tg(defined by τ(Tg)=100 s) are indicated. Larger open symbols: time constants as obtained by DSC. (b) Testing the Curie law: product Tε versus temperature for each phase. dependence being typical of supercooled liquids, indicating cooperative molecular dynamics, can be interpolated by a Vogel-Fulcher-Tammann (VFT) law, explicitly τ=τ0exp B T−T0.(10) We define the glass transition temperature as Tg=T(τα =100 s), which is determined as Tg1=180 K (cf. Fig. 4(a)). In order to compare the effective dipole moments μeff participating in the relaxation processes of the different quinaldine phases, we analyze the relaxation strengths (see also Eq. (2)) via the Curie law ε =Nμ2 eff 3kBT(11) with Nbeing the number density of dipoles. In Fig. 4(b),the product Tε for phase 2mq1 is plotted versus temperature and compared to that of the other phases (cf. below). In agreement with Eq. (11), a constant value is obtained all over the analyzed temperature range. 2. Second phase (2mq2) When the supercooled liquid of quinaldine is kept at temperatures above, say, 200 K, the dielectric spectra become time dependent, cf. Fig. 5. Here the sample was kept at T=210 K. The time interval between two shown data sweeps was about 35 min. With advancing time, the amplitude of the α-relaxation of the liquid decreases monotonically, while the dc-conductivity contribution at low frequencies is lowered similarly. At intermediate frequencies a second, weaker relaxation peak is revealed. Its amplitude increases monotonically with time (arrows in Fig. 5(a)). Finally, the dielectric response becomes again temperature independent, and the transformation was completed after 6 h. Since the amplitude of the relaxation peak in the dielectric susceptibility is proportional to the density of rotationally mobile dipoles (cf. Eq. (11)), the strong decrease of the signal in the frequency range of the main relaxation of the liquid indicates the disappearance of the liquid phase. We conclude that the sample is performing a phase transition to another phase (2mq2) with different rotational degrees of freedom, which results in a new relaxation peak at lower frequencies. 100101102103104105106 10-2 10-1 100 101 ε''(ν) ν / Hz 2mq2 relaxation dc conductivity 2mq1 relaxation 2mq1 to 2mq2 T=210K t=0...6h (a) 4.50 4.75 5.00 5.25 5.50 5.75 10-20 10-16 10-12 10-8 10-4 100 104 108 103104105 0 600 1200 1800 τtr τα time / s 1000K / T (b) TΔε(2mq1) t / s 200K 210K 215K FIG. 5. (a) Monitoring dielectric spectra (symbols) at T=210 K over 6 h demonstrating the phase transition from the liquid (2mq1) to a second phase (2mq2); arrows: progress of time. Lines: fits made up of the sum of the contributions of 2mq1 relaxation (Eqs. (1),(4) and (5)), of 2mq2 relaxation (Eq. (6))and of conductivity contribution (Eq. (9)). (b) Transition times τtr (open diamonds) compared to the structural relaxation times ταof 2mq1 (full squares). Inset: decrease of the property Tε(t) of the liquid phase 2mq1 during transition to phase 2mq2 versus time at different temperatures (squares). Lines: fits according to the Avrami law (Eq. (12)). Downloaded 06 Aug 2012 to 132.180.21.94. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054505-5 Kahlau et al. J. Chem. Phys. 137, 054505 (2012) In Fig. 5(a), the overall susceptibility curves at different times are compared and interpolated by the sum of three spectral contributions: the 2mq1 relaxation described by Eqs. (1), (4), and (5), the 2mq2 relaxation (Havriliak-Negami, Eq. (6)), and the conductivity contribution given by Eq. (9). During fitting, all time constants and spectral shape parameters were kept constant. Only the relaxation strength parameter ε of both phases and the conductivity parameters were adjusted. For the sake of clarity, only every sixth susceptibility scan is shown in Fig. 5(a). In order to access the phase transition kinetics, the time dependence of the relaxation strength of the liquid’s spectral contribution shall be quantified. In the inset of Fig. 5(b) the obtained decay curves Tε(t) are interpolated with the Avrami law Tε(t)=Tε(t=0)exp −t τtr n,(12) which is often used to describe crystallization kinetics.20 Any induction period is neglected since the preparation time of each experiment is short compared to the transformation process, which itself set in already during the first frequency scans. Here Tε(t) is assumed to be proportional to the fraction of untransformed liquid. The resulting exponents nshow some variation: n(200 K) =2.2, n(210 K) =2.6 and n(215 K) =2.9. Dantuluri et al.21 have found values of the same range for the crystallization kinetics of supercooled celecoxib. In Fig. 5(b), the obtained transformation times τtr (open diamonds) are compared to the structural relaxation times τα (full squares) of the liquid phase 2mq1 (cf. Fig. 4(a)). Obviously, τtr shows a temperature dependence different from that of the structural relaxation, following the relation τtr ∝ τα0.44. According to Ref. 22, the Avrami transition time τtr is connected with the linear crystallization velocity uvia u∝τ−n n−1 tr ∝τ0.44 α−n n−1=τ−0.72 α,(13) assuming a time and cluster-size independent rate of thermal nucleation at constant temperature. The last equality was obtained after inserting the average exponent of the three Avrami interpolations n=2.57 (see text above). Since a decoupling of the diffusion coefficient Dfrom viscosity η,explicitly D∝η−ξ∝τ−ξ αwith ξ≤1,(14) is well established,23 diffusion controlled crystal growth is expected to lead to u∝D∝τα−ξ. This has been confirmed by Sun et al.2for the crystallization of different (supercooled) liquids (ξ=0.7–0.8). As we find a similar exponent, we assume that the phase transition from 2mq1 to 2mq2 is controlled by diffusion, too. The dielectric spectra of the phase 2mq2 at different temperatures can be monitored as demonstrated in Fig. 6. In order to account for the low-frequency exponent of the relaxation peak, which is smaller than one, interpolations have been done by applying the Havriliak-Negami function (Eq. (6)). The dc conductivity contribution has been included in the fitting procedure by adding a power law according to Eq. (9). At low temperatures, the susceptibility deviates from the Havriliak-Negami shape due to an emerging highfrequency contribution reminding of an EW. Since this feature is not resolved well in the whole temperature range, 10-1 101103105 10-3 10-2 10-1 205 210 215 220 0.3 0.6 0.9 ε''(ν) ν / Hz 206K 209K 212K 215K 218K 2mq2 α αα αβ ββ β β ββ β T / K α αα α FIG. 6. Dielectric susceptibility spectra of the second phase of quinaldine 2mq2 at indicated temperatures; interpolation (solid lines) by HavriliakNegami function (Eq. (6)) including a dc-conductivity contribution (Eq. (9)). Inset: temperature dependence of the corresponding shape parameters α(triangles; exponent of the low frequency peak flank) and β(dots). The product αβ (squares) represents the high-frequency flank’s exponent. it has been excluded from the fits. At higher temperatures, the conductivity contribution deviates from a simple powerlaw behavior. Here the fitting range was limited to frequencies one decade below the minimum position. As shown in Fig. 6, a small increase of the low-frequency exponent β of the relaxation peak is observed. This is reflected once again by the increase of α(low frequency exponent) with a constant product αβ (high-frequency exponent, cf. inset of Fig. 6). The resulting time constants τHN are included in Fig. 4(a). In contrast to the supercooled liquid, τHN of 2mq2 seems to show an Arrhenius temperature behavior yielding an activation energy Ea/kB=24 100 K. Considering the exponential prefactor τ0=7.10 ×10−53 s of the Arrhenius fit, which is not a physically reasonable value for a single particle attempt time, we expect the τ(T) curve to flatten at higher temperatures. Since the permittivity curves are presented on logarithmic scale, it shall be pointed out that the second phase’s relaxation strength is strikingly smaller than the one of the liquid. In Fig. 4(b) the product Tε is plotted in order to be compared with the findings for the liquid (2mq1). We find again a temperature independent value with Tε(2mq2) ≈0.06 Tε(2mq1). 3. Third phase (2mq3) Similar to the above described transformation of the supercooled liquid, the dielectric spectra of 2mq2 become time dependent at high temperatures, say, T=210 K– 230 K. The phase transformation evolves on a slower time scale compared to that from phase 2mq1 to 2mq2, therefore, slightly higher temperatures are considered. The transformation leading to the third phase (2mq3) at T=218 K is displayed in Fig. 7. The time interval between two shown spectra is now about 158 min. The complete transformation took 130 h. In contrast to the transformation shown in Fig. 5, the appearing relaxation peak of phase 2mq3 is situated at slightly higher frequencies, compared to the diminishing peak of phase 2mq2. Downloaded 06 Aug 2012 to 132.180.21.94. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054505-6 Kahlau et al. J. Chem. Phys. 137, 054505 (2012) 10 2 10 3 10 4 10 5 10 6 10 -2 10 -1 ε ''( ν ) ν / Hz 2mq2 to 2mq3 T=218K t=0...129h FIG. 7. Time evolution of the dielectric spectra documenting the phase transition from phase 2mq2 to 2mq3 at T=218 K; (arrow) progress of time. The spectra of phase 2mq3 at different temperatures are shown in Fig. 8. Here the best interpolations of the quite broad peaks can be obtained with the help of a distribution of correlation times Gβ(ln τ)(Eq.(7); constant b=0.36), typically applied to interpolate a secondary relaxation process in molecular glasses.19 The resulting time constants τβshow approximate Arrhenius behavior with a mean activation energy of Ea/kB=19 100 K (cf. Fig. 4(a)). For a thermally activated process governed by a distribution of activation energies, the width of the relaxation peak is expected to diminish until the peak assumes Debye shape at infinite temperature, since here the temperature dependent Gβ(ln τ) becomes a δ-distribution. The time constant then approaches the attempt time τ0of the Arrhenius law. As shown in the inset of Fig. 8, the reciprocal width parameter 1/adoes not vanish at infinite but rather at a finite temperature. This may be explained15 by writing the Eyring expression for each correlation time of the distribution τ=τ00 exp −Sa kBexp Ea kBT(15) 10 -2 10 0 10 2 10 4 10 6 10 -3 10 -2 0 1 2 -10 -8 -6 -4 -2 4.00 4.25 4.50 4.75 5.00 ε ''( ν ) ν / Hz τ00 =10-9s 1000K / T log( τ β/s) 1/a 216K 196K 212K 206K 2mq3 T δ =235K 1/a b=0.36 τβ FIG. 8. Relaxation peaks of the third quinaldine phase 2mq3 at temperatures as indicated (2K-increment, open symbols). Dashed lines: Interpolations with a distribution of correlation times Gβ(ln τ)(Eqs.(7) and (8)) and a dc conductivity contribution (Eq. (9)). Inset: Temperature dependence of the time constant τβ(squares) and the width parameter 1/a (triangles). The asymmetry parameter could be kept fixed to b=0.36. with Eabeing the activation enthalpy and Sathe activation entropy. Introducing now the so-called Meyer-Neldel rule,24 which assumes a linear relationship between entropy and enthalpy, explicitly Sa=Ea Tδ ,(16) leads to the expression τ=τ00 exp Ea kB1 T−1 Tδ,(17) which explains the limit τβ(T)=τ00 to be reached at T=Tδ. The width of the relaxation peak, and thus the property 1/a, follows the temperature dependence19 1 a∝1 T−1 Tδ (18) and consequently vanishes when Tδis reached. We find for 2mq3 Tδ=235 K, being actually quite a low value, and τβ(T =Tδ)=τ00 =10−9s, which is at least somewhat closer to a physically reasonable value for a single molecule attempt time (cf. inset of Fig. 8). B. Differential scanning calorimetry The transitions observed by dielectric spectroscopy were investigated further with DSC. DSC heating scans applying a heating rate Q=20 K/min of the supercooled quinaldine (2mq1) in the temperature range from 120 K to room temperature revealed an endothermal step with an onset at 183 K (“glass step,” see Fig. 9, topmost line). Additionally, an exothermal peak was observed at 235 K followed by an endothermal peak at 255 K. Finally, at 266 K the endothermal melting peak is found in accordance with the literature (Tm≈264–270 K). Tentatively, we attribute the two peaks observed in the DSC scan below the regular melting point to phase transitions to the phases 2mq2 and 2mq3, respectively. The relaxation times probed by DSC can be compared with those from dielectric spectroscopy after calculating the 120 140 160 180 200 220 240 260 280 300 185 195 205 185 195 205 2mq3 2mq2 T g =(183±0.5)K DSC signal (arb. units) T / K quinaldine 2mq1 Tg=(195±2)K Tg=(191±2)K FIG. 9. DSC scans of the quinaldine phases 2mq1 (topmost line), 2mq2 (central line), and 2mq3 (lowermost line)—see text. Straight dotted lines: Guides for the eye, indicating the onsets of the endothermal steps. Downloaded 06 Aug 2012 to 132.180.21.94. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 054505-7 Kahlau et al. J. Chem. Phys. 137, 054505 (2012) 10 15 20 25 30 35 40 45 50 Intensity (a. u.) 2θ / ° 2mq1 200 K, t = 17 h 205 K, t = 2 h 205 K, t = 1 h 200 K, t = 0 2mq2 (a) 10 20 30 40 50 2mq2, 200K Intensity (a. u.) 2θ / ° (b) 2mq3, 200K FIG. 10. (a) XRD data of quinaldine monitored during the phase transition from 2mq1 to 2mq2 (evolution times and recording temperatures as indicated). (b) Comparison of the long-time diffractograms (15 h accumulation time) of both crystalline quinaldine phases 2mq2 and 2mq3 after presumably complete transformation. calorimetric time constant via25,26 τcal =kBT2 g HeffQ.(19) Heff denotes the activation enthalpy taken from the dielectric data at the calorimetric Tg. Thus, Heff =kB ∂ln τ ∂1 T 1 Tg =kBB 1−T0 Tg2(20) holds in the case of a VFT temperature dependence (Eq. (10)), and Heff =Ea=const. for the case of an Arrhenius temperature dependence. The extracted DSC time constants for all three phases (see below) are plotted in Fig. 4(a) as open symbols and show fair accordance with the dielectric time constants. In order to further clarify the nature of the phase transition, in the next step the heating scan starting below Tg is interrupted at, e.g., T=218 K. Now the phase transition from 2mq1 to 2mq2 starts and will be completed in less than 30 min. Afterwards, a cooling to 120 K, 10 min of waiting, and reheating to room temperature results in the central trace in Fig. 9. As expected, the exothermal peak is missing now, only a very weak step in the same temperature region remains. The endothermal peaks at higher Tare found at the same positions as in the former measurement. This proves that the central curve in Fig. 9results from probing the 2mq2 phase of quinaldine. At T=195 K, a weak step is found in the DSC signal (cf. inset) which we interpret, similar to the glass transition of the liquid, as an indication of a “freezing” of motional degrees of freedom within phase 2mq2. Calculating the corresponding calorimetric time constant according to Eq. (19) leads to the result included in Fig. 4(a). The experiment can be repeated with 2mq2 by introducing an additional isothermal stage at, for instance, T=255 K. After subsequent cooling, waiting, and heating again to room temperature, the lowermost line in Fig. 9is obtained. The missing of the first endothermal peak is proof of the second phase transition from QN2 to QN3 to have already proceeded during the isothermal stage. We now find a weak DSC-step at T=191 K (cf. inset), which is situated about 4 K below the step in 2mq2 and in accordance with the dielectric findings (cf. Fig. 4(a)). C. X-ray diffraction XRD experiments provide a straightforward way to identify the observed quinaldine phases. The experiment is started with cooling the liquid below Tg=180 K. Afterwards, the sample temperature was increased in 5 K steps. After each increment, XRD data were accumulated isothermally for ca. 15–20 minutes. In Fig. 10(a), the amorphous “halo” of the supercooled liquid (2mq1) is shown (“t=0”). After increasing the temperature up to 205 K, the sample was kept for 1 h while several diffractograms were recorded. The last one is shown in Fig. 10(a) (“t=1 h”). As can clearly be seen, Bragg reflexes from some crystalline phase, most probably 2mq2, have emerged on top of the diffractogram of the liquid. After 2 h, the amorphous contribution has decreased further (“t=2 h”). Since this last signal did not seem to evolve anymore even after further temperature increments up to 215 K, the sample was cooled again to 200 K in order to conserve the present phase of the sample. Here another diffractogram was accumulated for 15 h (Fig. 10(a),“t=17 h”). There is no recognizable change but an improved S/N ratio due to longer accumulation time. Subsequently, the same strategy as described above was repeated with the present crystalline sample at higher temperatures. At 220 K– 225 K, the signal becomes time dependent again: new peaks appear and old peaks start to shrink or even vanish, respectively. This is interpreted as the phase transition from 2mq2 to 2mq3. After keeping the sample at 225 K for ca. 6 h and at 230 K for another hour, no more change in the signal could be observed. The sample was then cooled again to 200 K, where another diffractogram was accumulated for 15 h. Figure 10(b) shows the comparison of the measurements with long accumulation times of both, significantly different, crystalline phases of quinaldine. Due to the isolation vacuum in the chamber where the sample is situated during the measurement, it was not posDownloaded 06 Aug 2012 to 132.180.21.94. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 134504-2 Kahlau, Dörfler, and Rössler J. Chem. Phys. 139, 134504 (2013) secondary process. Organic phosphate glass formers are also well suited to be investigated by 31PNMR. 15,19 Due to their high molecular dipole moment these systems show a strong dielectric response, too. The spectral analysis will focus on scaling the data in an appropriate way to allow for extracting the distribution of activation energies g(E)aswellastherelaxation strength underlying the dynamics of the β-processes in a rather model independent way, i.e., without applying any particular fitting procedure. We will demonstrate that a strong increase of the relaxation strengths of all β-processes above Tgis observed, which again signals the “non-local” character of secondary processes in glasses. In addition, comparing the different glasses containing ester groups with increasing number of internal degrees of freedom, we do not see systematic changes in the spectral evolution, i.e., in the distribution g(E); in some cases even identical distributions g(E) are found. II. EXPERIMENTAL DETAILS Triethyl phosphate (TEP),19 tripropyl phosphate (TPP),15,19 tributyl phosphate (TBP), tris(2-butoxyethyl) phosphate (T2BOEP), and tris(2-ethylhexyl) phosphate (T2EHP) were purchased from Sigma-Aldrich and used without further treatment (cf. Table I). Data of trimethyl phosphate (TMP) is taken from Ref. 20. For the dielectric measurements performed within this work an Alpha-A analyzer by Novocontrol was used in combination with a sample cell design described in Ref. 21. Absolute temperature was assumed to be accurate within ±1 K and was kept constant within ±0.2 K by using a Quatro-H temperature controller by Novocontrol in combination with an Oxford cryostat, cooled with liquid Nitrogen. Dielectric measurements on TMP20 were performed with a SI1260 spectral analyzer by Schlumberger with a similar temperature controlling and sample cell setup. III. RESULTS A. Measurements The dielectric susceptibility data of TBP, TPP, T2EHP, and TEP (cf. Table I) are compiled in Fig. 1. In all four cases the α-relaxation peak is observable as the most prominent relaxation feature above Tg, while below Tgaβ-relaxation is recognized. The maximum value ε max of the α-peak is about 2 for T2EHP, while in the other cases values around 10 are found. When temperature is increased, the α-relaxation shifts to higher frequencies without essentially changing its spectral shape. At the same time a loss contribution caused by ionic conductivity (for most datasets in Fig. 1omitted for the sake of clarity) moves into the frequency window and shows the same temperature dependence as the α-peak. At least as long as it is observable its onset is situated 1-2 frequency decades below the frequency of the α-peak in the cases TBP, TPP, and TEP. T2EHP shows an exceptionally low conductivity contribution at frequencies three decades below the α-relaxation peak frequency. At high temperatures, when the α-relaxation reaches peak frequencies in the kHz regime, TBP, TPP, as well as TEP start to crystallize, and no more permittivity data of the supercooled liquid can be acquired. Again T2EHP is exceptional TABLE I. Molar masses and structural formulae of the investigated glass formers. System M(10−3kg/mol) Tg(K) Structure TMP 140 137 P O CH 3 OCH 3 O CH 3 O TEP 182 137 P O CH 2 CH 3 OCH 2 CH 3 O CH 2 CH 3 O TPP 224 134 P O CH 2 CH 2 CH 3 OCH 2 CH 2 CH 3 O CH 2 CH 2 CH 3 O TBP 266 140 P O CH 2 CH 2 CH 2 CH 3 OCH 2 CH 2 CH 2 CH 3 O CH 2 CH 2 CH 2 CH 3 O T2EHP 435 159 P O O OO T2BOEP 398 167 P O O O O O O O since no crystallization tendency is observed up to highest peak frequencies. Clearly, all systems in Fig. 1show a secondary (β-) relaxation peak on the high-frequency side of the α-peak, which survives below Tgdown to lowest measured temperatures (about 80 K). Thus, all systems of Fig. 1may be called type-B systems. Slightly above Tg, when the α-peak is situated close to the edge of the accessible frequency range, this β-peak is situated about 3-4 frequency decades above the α-peak frequency of TBP, TPP, and T2EHP; in the case of TEP both peaks are spectrally more distant. Figure 2shows the dielectric data of TMP and T2BOEP (cf. Table I). The temperature evolution of the α-relaxation of these systems is again similar to the others discussed above. The maximum value ε max is about 10 for TMP and 5 for T2BOEP. TMP starts to crystallize when the α-peak is situated in the kHz regime, while T2BOEP shows no crystallization tendency. In contrast to the systems shown in Fig. 1 TMP and T2BOEP show a more complex secondary relaxation pattern. In the susceptibility data of TMP an EW is clearly observable in addition to a β-relaxation, i.e., on the high-frequency side of the α-peak (about 2 frequency decades right to the peak position) ε(ν) follows a power law with an exponent of, as typical for the excess wing phenomenon, −γ∼ =−0.2 (cf. Fig. 2(a)). At even higher frequencies this EW is followed by a β-relaxation peak similar to the one of TEP (cf. Fig. 1(d)). T2BOEP even shows two secondary relaxation peaks, which are well resolved at temperatures below Tg(cf. Fig. 2(d)), so TMP as well as T2BOEP, and hence all measured systems, may be called type-B systems. It shall be mentioned that all presented systems may have an excess wing contribution which is obscured by the explicitly observable β-peaks. Since this cannot be clarified without a model dependent quantitative analysis, we focus on the relaxation peaks of αand β-relaxation. We assume further that the dielectric spectra below Tgare dominated by the contribution of the β-relaxation. Downloaded 08 Oct 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 134504-3 Kahlau, Dörfler, and Rössler J. Chem. Phys. 139, 134504 (2013) 10 -2 10 0 10 2 10 4 10 6 10 -2 10 0 138K 132K 110K 120K 100K 90K 80K 142K144K 148K152K156K ε''(ν) ν / Hz TBP 162K (a) 10 -2 10 0 10 2 10 4 10 6 10 -3 10 -1 10 1 114K 130K 122K 104K 94K 82K 154K 150K 144K 140K ε''(ν) ν / Hz 136K TPP (b) 10 -2 10 0 10 2 10 4 10 6 10 -2 10 0 220K 210K 200K 190K180K 170K 165K 135K 105K ε ''( ν ) ν / Hz T2EHP 95K (c) 10 -2 10 0 10 2 10 4 10 6 10 -2 10 0 74K 92K 116K 128K 134K 138K 140K 144K 148K ε''(ν) ν / Hz TEP 152K (d) FIG. 1. Dielectric loss data of (a) TBP (Tg=140 K), (b) TPP (Tg=134 K), (c) T2EHP (Tg=159 K), and (d) TEP (Tg=137 K) at indicated temperatures. See Table Ifor chemical structures. Time constants of all αand β-relaxation peaks from Figs. 1and 2as given by τmax =1/(2πνmax) are compiled in Fig. 3. Due to similar glass temperatures the α-relaxation times ταof TMP, TEP, TPP, and TBP almost coincide. A similar statement can be made for ταof T2EHP and T2BOEP, yet their Tgdiffers considerably in comparison with the other systems. The time constants τβfollow Arrhenius temperature dependences at temperatures well below Tg(straight lines in Fig. 3, cf. Sec. III B). The corresponding activation energies (given in K) vary from E≈2100 – 4200 K, while the attempt times are settled at ν0=10−13 –10 15 s−1(cf. Table II). Note that a representation on Tg/T scale does not lead to a collapse of the data and does not yield further information. When Tgis approached the temperature dependence of τβ 10 -2 10 0 10 2 10 4 10 6 10 -2 10 -1 10 0 10 1 138K 135K 131K 120K 101K 141K 144K 147K ε''(ν) ν / Hz 150K TMP ~ ν-0.17 (a) 10 -2 10 0 10 2 10 4 10 6 10 -2 10 -1 131K 66K 71K 81K 91K 101K 120K ε''(ν) ν / Hz 135K TMP (b) 10 -2 10 0 10 2 10 4 10 6 10 -1 10 0 10 1 ε ''( ν ) ν / Hz 230K 220K 210K 200K 190K 180K175K 170K T2BOEP (c) 10 -2 10 0 10 2 10 4 10 6 0.02 0.04 0.06 ε ''( ν ) ν / Hz 160K 150K 140K 130K 120K 110K 100K T2BOEP (d) FIG. 2. Dielectric loss data of TMP ((a) and (b), Tg=137 K) and T2BOEP ((c) and (d), Tg=167 K) at indicated temperatures (cf. Table Ifor chemical structures). Dashed lines in (a) represent power laws ∼ν−γwith γ=0.17 attributed to the excess wing. Downloaded 08 Oct 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 134504-4 Kahlau, Dörfler, and Rössler J. Chem. Phys. 139, 134504 (2013) 4812 10 -7 10 -5 10 -3 10 -1 10 1 TMP TEP TPP TBP T2EHP T2BOEP τ max / s 1000K / T FIG. 3. Time constants of αand β-process determined from the relaxation peak positions of all investigated systems (cf. symbol map). Small symbols: α-relaxations. Here, curved lines are guides for the eye. Large symbols: βrelaxations. Here, straight lines are calculated from the distributions of activation energies g(E) introduced in Sec. III B. Note the change of the τβ(T) around Tg. becomes weaker (actually a kink is observed near Tg, best recognized for TBP), which is caused by an apparent increase of the mean activation energy near Tg(see Sec. III B,cf.also Refs. 22 and 23). Interestingly, the time constants τβ(T)of TMP and TEP as well as TPP and TBP coincide, which may suggest that the mean activation energy as well as the attempt time τ0are determined rather by intermolecular interactions than by intramolecular interactions dependent on the different (flexible24,25) ester groups. As we will demonstrate in Sec. III B, the spectra of the β-process can be scaled according to the assumption that it is governed by a broad distribution of activation energies, i.e., the β-relaxation is governed by a multitude of thermally activated processes well below Tg. B. Scaling analysis of the β-process spectra In this section a scaling procedure for thermally activated processes will be applied to the β-relaxations found in the investigated systems.26–28 If a temperature independent distribution of activation energies g(E) forms the basis of a relaxation process, i.e., for a given “site” in the glass with activation energy E(given in K) a simple Arrhenius law TABLE II. Scaling parameters (maximum activation energy Emand attempt frequency ν0) and fit parameters (Em,c,andbaccording to Eqs. (5) and (6)) of the investigated β-relaxations. Scaling gβ(E)fits System Tg(K) Em(K) ν0(s−1)Em(K) cb TMP 137 2113 1014 2074 0.00411 0.5067 TEP 137 2143 2 ×1014 2127 0.0043 0.45646 TPP 134 3350 1015 3333 0.00417 0.5318 TBP 140 3251 7 ×1014 3250 0.00356 0.5908 T2EHP 159 3458 1013 3472 0.00277 0.49246 T2BOEP 1 167 2541 4 ×1013 2526 0.00176 1.32432 T2BOEP 2 167 4284 3 ×1014 4267 0.0015 1.05335 τ=τ0exp E Tholds, the corresponding distribution of correlation times can be calculated via G(ln τ)d(ln τ)=g(E)d(E),(1) G(ln τ)=T·g(E).(2) The corresponding susceptibility measured by dielectric spectroscopy is then calculated to ε(ν)=ε ∞  −∞ G(ln τ)2πντ 1+(2πντ)2d(ln τ)∼ =Tεπ 2·g(E). (3) Here, the approximation in the last part of the equation can be made in the case of a broad distribution G(ln τ) (leading to a broad g(E)), which is usually observed for β-processes in molecular glass formers.5After substituting the variable E with the Arrhenius expression τ=τ0exp( E T), the distribution of activation energies can be expressed like28 g(E)=g(Tln(ν0/ν)) =2 π ε(ν) Tε.(4) In other words, plotting the right hand side expression of Eq. (4) versus Tln(ν0/ν) reveals a master curve which yields the distribution of activation energies g(E). Here, the attempt frequency ν0is assumed to be constant for all E. Since the dependence of g(E)onν0is logarithmic, any dependence of ν0 on Eas, e.g., assumed for a kind of Meyer-Neldel relation,5,29 can be neglected in the scaling procedure. In the following this scaling is applied to the dielectric loss spectra of all βpeaks of the investigated systems in order to clarify if indeed a temperature independent distribution of activation energies is present. Figure 4displays the susceptibility of TBP after the scaling along Eq. (4). Instead of dividing the susceptibility ε(ν) by Tand εβ, the latter of which would have had to be determined in advance independently, the data was normalized only by Tas a first step. Afterwards, the ε(ν)/T curves were plotted vs. Tln(ν0/ν), and remaining systematic deviations in 2000 3000 4000 5000 0.0 0.2 0.4 0.6 0.8 1.0 E=3480K 72-134K 136-148K 1000*g(E) ~ ε ''/(T Δε ) E = T ln( ν 0 / ν ) [K] TBP ν0=7*1014s-1 E=3251K FIG. 4. Susceptibility data of TBP (72-134 K; Tg=140 K) scaled according to Eq. (4) (black lines); susceptibility data for T =136-148 K (dashed lines). Arrows indicate maximum positions, the corresponding values are given. Attempt frequency ν0indicated. Downloaded 08 Oct 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 134504-5 Kahlau, Dörfler, and Rössler J. Chem. Phys. 139, 134504 (2013) 2000 3000 4000 5000 ν 0 =4*10 13 s -1 ν 0 =3*10 14 s -1 ν 0 =10 15 s -1 ν 0 =7*10 14 s -1 ν 0 =10 13 s -1 ν 0 =2*10 14 s -1 g(E) [ a. u. ] E=T ln( ν0 / ν ) [K] TMP TEP T2BOEP T2EHP TBP TPP T2BOEP ν 0 =10 14 s -1 FIG. 5. Distribution of activation energies g(E) for the indicated systems. Red lines: interpolations serving as guides for the eye. Note that T2BOEP exhibits two secondary relaxations. height were corrected by dividing the curves by their maximum values. The latter are proportional to εβ(T) and will be discussed below. Actually, this allows to determine the relative temperature evolution of εβ(T) not assuming any particular spectral shape of the β-relaxation. As a third step, the master curve was normalized by its area. As can be inferred from Fig. 4, a very good collapse of the data for T=72–134 K is observed for an attempt frequency ν0=7 ×1014 s−1. The scaling works very well for the spectra below Tgfor all investigated systems, even in the cases of T2BOEP, for which two β-processes are identified. All master curves are compared in Fig. 5. In order to cross check the validity of the scaling procedure, for all β-relaxations the mean logarithmic τβ(T) were calculated along the Arrhenius law using τ0 =1/(2πν0) and the most probable Evalues taken from g(E) displayed in Fig. 5. They are included in Fig. 3, and good coincidence is found for all datasets well below Tg, i.e., all straight lines interpolate the measured points. Above 134 K, 6 K below Tg, the scaling (Fig. 4)startsto fail and the relaxation peaks shift to higher Ewith increasing temperature, which may correspond to a distinct increase of the mean activation energy near Tg. Alternatively, a decrease of ν0near Tgmight also be a possible reason for the bend in τβ(T)inFig.3(cf. Sec. IV). The change of g(E)isalso observed for TPP, T2EHP, and T2BOEP, when temperature gets close to Tg. Figure 5compiles the distributions g(E) of all investigated β-processes as obtained by applying the discussed scaling procedure. Here, only the datasets well below Tgare 0 1000 2000 3000 4000 5000 6000 7000 0.0 0.2 0.4 0.6 0.8 1.0 TMP TEP TPP TBP T2EHP T2BOEP (2x) g β (E) 1000*g(E) normalized E=T ln( ν 0/ ν ) [K] FIG. 6. gβ(E) function (dashed lines) fitted to the experimentally obtained g(E) distributions. Solid lines taken from Fig. 5; cf. symbol map). Note: the glass T2BOEP exhibits two β-processes. shown. In all cases, clearly asymmetric distributions g(E)are revealed. Most probable energies Em=2100–4200 K are found, while ν0varies from 1013 s−1to 1015 s−1. Again, striking coincidence in peak position and shape of g(E) between TMP and TEP, as well as between TBP and TPP, respectively, is observed (cf. also Figs. 3and 6). We note that a further plot of all g(E) versus a reduced energy scale E/Tg does not yield additional information. Instead, the coincidence between TMP/TEP and TPP/TBP is corrupted. The distribution g(E) of T2EHP shows some excess contribution at E≈2000 K, potentially indicating another β-relaxation (cf. Figs. 5and 6). In the following the distributions of activation energies shall be analyzed in a more quantitative way. Since the distributions of activation energies are found to be asymmetric, the model function for g(E) introduced in Ref. 30 (there also applied to interpolate secondary relaxation spectra) gβ(E)=Nβ(c, b)1 bexp[c(E−Em)] +exp[−cb(E−Em)] (5) with the normalization factor Nβ(c, b)=c(1 +b) πbb 1+bsin πb 1+b(6) is adequate and shall be fitted to the normalized g(E)ofFig.5. Note that a susceptibility function such as Havriliak-Negami is not suitable for describing a relaxation process determined by thermally activated dynamics.30,31 The results are displayed in Fig. 6and allow a direct comparison of the g(E)of the systems investigated. All in all, good fits with few minor deviations are obtained. The strongest deviations are found for TMP, TEP, and, due to the excess amplitude at E=2000 K, for T2EHP. Table II summarizes all parameters characterizing the β-process. Obviously the parameter c, which reflects the inverse overall width of the distributions, is lowest for both relaxation processes of T2BOEP and the process of T2EHP. The parameter bis a measure for the asymmetry of the distribution. Values around b =0.5 are found for TMP, TEP, TPP, TBP, and T2EHP, reflecting the high energy slope being steeper than the low energy slope. The relaxation of T2BOEP Downloaded 08 Oct 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 134504-6 Kahlau, Dörfler, and Rössler J. Chem. Phys. 139, 134504 (2013) 1.0 1.5 2.0 2.5 1 2 3 Δεβ(T) / Δεβ(0.9 T g ) T g / T TMP (extrap.) TEP TPP TBP T2BOEP 1 T2BOEP 2 T2EHP 1 FIG. 7. Relaxation strengths εβ(T) of the investigated β-relaxations (cf. Table I) normalized with εβ(0.9 Tg) and plotted versus the reduced temperature scale Tg/T. As the TMP values are nearly constant, εβ(T)was extrapolated linearly to T=0.9 Tg. with Em=2500 K has a b>1, which reflects a converse tilt of the peak. The other T2BOEP peak is rather symmetric and thus has a b=1. As explained above, the temperature dependence of a quantity which is proportional to the relaxation strengths εβ(T) is obtained by producing the master curves of Fig. 5.InFig.7the εβ(T)ofallβ-relaxations are compiled, normalized to the respective value εβ(T=0.9Tg), and plotted versus a reduced temperature scale Tg/T.BelowTgthe values are either constant (TMP and TEP) or decrease slightly with decreasing temperature. The only exception is the relaxation process of T2BOEP with Em=4300 K (“T2BOEP 2”, cf. Table II), the εβ(T) of which increases slightly with decreasing T. Strikingly, all εβ(T) data accessible at higher temperatures increase strongly when Tgis approached. This increase sets in already 13–16 K below the conventionally defined glass transition temperature, which is given by Tg=T(τα=100 s). As discussed in the Introduction, we take the strong increase of εβabove Tgas an indication for all βprocesses to be coupled to the α-process, in the sense that g(E) changes above Tgwhen the glass “softens.” According to the interpretation suggested by the 2H NMR studies10,11,13,18 the increase of εβreflects the increasing angular displacement of the molecules involved in the β-process. IV. SUMMARY AND DISCUSSION Dielectric loss spectra of six phosphate glass formers with different ester groups have been analyzed. A model independent scaling procedure reveals a temperature independent distribution of activation energies g(E) for all investigated βrelaxations well below Tg. We note that the scaling procedure yielding g(E) has already been applied successfully to several molecular glass formers.15,26–28 In particular, a version of the scaling has been used to identify the low temperature dielectric and mechanical losses assumed to originate from defects leading to the tunneling systems at very low temperatures (socalled Gilroy-Phillips approach).32 The distributions g(E) of different β-relaxations vary in maximum position, width, and asymmetry and are categorized quantitatively with an adequate model distribution gβ(E). Yet, for two pairs of systems identical distributions are found, although different ester groups are involved. Already somewhat below the glass temperature (T≈0.9 Tg)theg(E) appears to change. Here, an apparent increase of the mean activation energy (at constant attempt frequency ν0) can be inferred from the systematic shift of the scaled data peaks. This is formally in accord with a reduced temperature dependence of the time constants τβ, a phenomenon well observed in the data and which has also been reported in, e.g., Refs. 22 and 23. The effect can alternatively be interpreted as a reduction of the attempt frequency of the Arrhenius law (at constant activation energy E), which may be explained by a softening of the potential landscape caused by the thermally induced increase of molecular mobility near Tg. One may think of broadened potential wells, leading to lower oscillation frequencies in a harmonic approximation. This apparent change of g(E) is accompanied by a drastic increase of the relaxation strengths εβ(T), which has also been observed for many structural glasses as well as glassy crystals.17 Thus, the appearance of the change of ν0 or mean Eand the strong increase of εβ(T) may be considered as a generic behavior of β-processes observed in molecular glasses. The idea of the surrounding glassy matrix governing the relaxing center has already been presented by Johari and Goldstein.14 Here, the authors state that intermolecular interactions make up the potential barrier landscape, no matter if the relaxing groups of molecules have internal degrees of freedom or move rigidly. The phosphate systems investigated in this work are known to have different conformational states, depending on the length of the side chains,24,25 which implies the presence of internal motional degrees of freedom. One could expect each (chemical) prolongation of the alkyl side chain, leading to an increase of internal degrees of freedom, to cause a change of the activation energy distribution of the β-relaxation, if the latter is significantly determined by intramolecular potentials. Instead, the g(E) is sometimes affected (Emax(TPP) >Emax(TEP)) and sometimes not (Emax(TMP) =Emax(TEP); Emax(TPP) =Emax(TBP)). Hence, the interpretations given above concerning the temperature independent g(E) and the influence of the softening glassy matrix on the relaxation process appear to be unaffected by the character of the intramolecular degrees of freedom being present. Clearly, since the relaxation strength changes above Tg,theβ-process is not governed by a simple thermally activated process as it is paradigmatically found in crystalline rotor phases (e.g., solid benzene). Here, the motional mechanism is caused by the (high) symmetry of the molecule, thus the quality of the motion does not change above and below Tg.33 We think that in the dense matrix of the glass as well as the super-cooled liquid a significant, yet hindered motion is only possible if some extent of cooperativity is present as in the case of the α-process. This is signaled by the generic change of εβ(T) above Tg. 1P. Lunkenheimer, U. Schneider, R. Brand, and A. Loidl, Contemp. Phys. 41, 15 (2000). 2F. Kremer and A. Schönhals, Broadband Dielectric Spectroscopy (Springer, Berlin, 2003). Downloaded 08 Oct 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 134504-7 Kahlau, Dörfler, and Rössler J. Chem. Phys. 139, 134504 (2013) 3T. Blochowicz, C. Gainaru, P. Medick, C. Tschirwitz, and E. A. Rössler, J. Chem. Phys. 124, 134503 (2006). 4K. L. Ngai, Relaxation and Diffusion in Complex Systems (Springer, New York, 2011). 5A. Kudlik, S. Benkhof, T. Blochowicz, C. Tschirwitz, and E. Rössler, J. Mol. Struct. 479, 201 (1999). 6C. Gainaru, R. Kahlau, E. A. Rössler, and R. Böhmer, J. Chem. Phys. 131, 184510 (2009). 7M. Vogel and E. Rössler, J. Phys. Chem. B 104, 4285 (2000). 8M. Vogel and E. Rössler, J. Chem. Phys. 114, 5802 (2001). 9M. Vogel and E. Rössler, J. Chem. Phys. 115, 10883 (2001). 10B. Micko, S. A. Lusceac, H. Zimmermann, and E. A. Rössler, J. Chem. Phys. 138, 074503 (2013). 11B. Micko, D. Kruk, and E. A. Rössler, J. Chem. Phys. 138, 074504 (2013). 12H. Wagner and R. Richert, J. Non-Cryst. Solids 242, 19 (1998). 13B. Micko, C. Tschirwitz, and E. A. Rössler, J. Chem. Phys. 138, 154501 (2013). 14G. Johari and M. Goldstein, J. Chem. Phys 53, 2372 (1970). 15D. Bock, R. Kahlau, B. Micko, B. Pötzschner, G. J. Schneider, and E. A. Rössler, J. Chem. Phys. 139, 064508 (2013). 16M. Vogel, C. Tschirwitz, G. Schneider, C. Koplin, P. Medick, and E. Rössler, J. Non-Cryst. Solids 307–310, 326 (2002). 17C. Tschirwitz, S. Benkhof, T. Blochowicz, and E. Rössler, J. Chem. Phys. 117, 6281 (2002). 18M. Vogel, P. Medick, and E. A. Rössler, Annu. Rep. NMR Spectrosc. 56, 231 (2005). 19S. Adishchev, D. Bock, C. Gainaru, R. Kahlau, B. Micko, N. Petzold, B. Pötzschner, and E. A. Rössler, Z. Phys. Chem. 226, 1149 (2012). 20S. Adichtchev, T. Blochowicz, C. Gainaru, V. N. Novikov, E. A. Rössler, and C. Tschirwitz, J. Phys.: Condens. Matter 15, S835 (2003). 21H. Wagner and R. Richert, J. Phys. Chem. B 103, 4071 (1999). 22N. B. Olsen, J. Non-Cryst. Solids 235-237, 399 (1998). 23N. B. Olsen, T. Christensen, and J. C. Dyre, Phys.Rev.E62, 4435 (2000). 24I. N. Smirnova, A. Cuisset, F. Hindle, G. Mouret, R. Bocquet, O. Pirali, and P. Roy, J. Phys. Chem. B 114, 16936 (2010). 25A. Cuisset, G. Mouret, O. Pirali, P. Roy, F. Cazier, H. Nouali, and J. Demaison, J. Phys. Chem. B 112, 12516 (2008). 26L. Wu and S. Nagel, Phys. Rev. B 46, 11198 (1992). 27J. Wiedersich, S. V. Adichtchev, and E. Rössler, Phys. Rev. Lett. 84, 2718 (2000). 28C. Gainaru, R. Böhmer, R. Kahlau, and E. Rössler, Phys.Rev.B82, 104205 (2010). 29W. Meyer and H. Neldel, Z. Tech. Phys. 12, 588 (1937). 30T. Blochowicz, C. Tschirwitz, S. Benkhof, and E. A. Rössler, J. Chem. Phys. 118, 7544 (2003). 31C. F. J. Böttcher and P. Bordewijk, Theory of Electric Polarization (Elsevier, Amsterdam, 1996), Vol. II. 32K. S. Gilroy and W. A. Phillips, Philos. Mag. B 43, 735 (1981). 33E. Rössler, M. Taupitz, and H. M. Vieth, J. Phys. Chem. 94, 6879 (1990). Downloaded 08 Oct 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 100 Paper 5 On the Cooperative Nature of the β-Process in Neat and Binary Glasses: A Dielectric and Nuclear Magnetic Resonance Spectroscopy Study D. Bock, R. Kahlau, B. Micko, B. P¨ otzschner, G. J. Schneider, and E. A. R¨ ossler, The Journal of Chemical Physics 139, 064508 (2013). c 2013 AIP Publishing LLC doi:10.1063/1.4816374 101 102 THE JOURNAL OF CHEMICAL PHYSICS 139, 064508 (2013) On the cooperative nature of the β-process in neat and binary glasses: A dielectric and nuclear magnetic resonance spectroscopy study D. Bock,1R. Kahlau,1B. Micko,1B. Pötzschner,1G. J. Schneider,2and E. A. Rössler1 1Experimentalphysik II, Universität Bayreuth, 95440 Bayreuth, Germany 2Jülich Centre for Neutron Science JCNS, Outstation at FRM2, Forschungszentrum Jülich GmbH, 85747 Garching, Germany (Received 28 May 2013; accepted 9 July 2013; published online 14 August 2013) By means of dielectric as well as 2H and 31P nuclear magnetic resonance spectroscopy (NMR) the component dynamics of the binary glass tripropyl phosphate (TPP)/polystyrene (PS/PS-d3)is selectively investigated for concentrations distributed over the full range. We study the secondary (β-) relaxation below Tg, which is found in all investigated samples containing TPP, but not in neat polystyrene. The dielectric spectrum of the β-process is described by an asymmetric distribution of activation energies, essentially not changing in the entire concentration regime; its most probable value is E/k∼ =24 Tg. Persistence of the β-process is confirmed by 31P NMR Hahn-echo and spinlattice relaxation experiments on TPP, which identify the nature of the β-process as being highly spatially hindered as found for other (neat) glasses studied previously, or re-investigated within this work. The corresponding 2H NMR experiments on PS-d3confirm the absence of a β-process in neat PS-d3, but reveal a clear signature of a β-process in the mixture, i.e., polystyrene monomers perform essentially the same type of secondary relaxation as the TPP molecules. Yet, there are indications that some fractions of PS-d3as well as TPP molecules become immobilized in the mixture in contrast to the case of neat glasses. We conclude that in a binary glass the β-process introduced by one component induces a highly similar motion in the second component, and this may be taken as an indication of its cooperative nature. © 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4816374] I. INTRODUCTION Since the works of Johari and Goldstein (JG)1a second (or β-) relaxation peak is a well documented relaxation feature observed in many liquids at frequencies higher than those of the primary α–relaxation upon (super-) cooling.2–7In some cases, for type-A glass formers8(in contrast to type-B systems with a well resolved secondary process), such a β-peak is missing and only an “excess wing” appears on the highfrequency flank of the α-peak. Even two resolved secondary relaxation peaks may be found.9,10 Photon correlation spectroscopy (PCS) studies of type-B glass formers have identified only an excess wing, which is masked by a strong β-peak in the dielectric spectrum.10–12 Why PCS does not probe the β-process while it is clearly detected in dielectric and mechanical relaxation as well as in nuclear magnetic resonance (NMR) experiments remains a challenge to be understood. Thus, the situation is quite puzzling since there exists no final conclusion concerning the nature of these processes and their relevance with regard to the glass transition phenomenon. As a β-process is also observed for molecules without internal degrees of freedom, it is assumed to be generic to the glassy state. In contrast to the α-process the β-process is often loosely called a “local” process. This classification may suggest that the extent of cooperativity of the dynamics typical of the α-process is not found for secondary relaxations. Actually, however, all correlation lengths discussed for the αprocess are, if at all present, on the order of a few nanometers which still is rather local. A prominent feature which might point to a local character of the dynamics is the fact that below Tgthe most probable relaxation time τβof the β-process follows an Arrhenius temperature dependence, the peak width increases with reciprocal temperature as expected for a temperature independent distribution of activation energies, and the relaxation strength changes only weakly.8,13 Yet, the mean activation energies are quite high, usually in the range E/k ∼ =11–26 Tg.5,7Its generic nature is also signaled by the fact that the relaxation strength strongly increases above Tg, i.e., the β-process probes the “softening” of the glass above Tg. Important to note is that a β-process is also observed in super-cooled plastic crystals (glassy crystals) with activation energies almost not altered with respect to that of the corresponding structural glass.13,14 Systematic 2H NMR solid-echo studies on structural glass formers such as toluene, decalin, and polybutadiene15–18 as well as on glassy crystals like ethanol19 or cyano cyclohexane20,21 have been carried out in the past. Due to its high sensitivity on small-angle reorientations, the solidecho technique yields a clear picture of the nature of the βprocess in terms of its single-particle dynamics.22,23 Spatially highly restricted reorientation of essentially all molecules prevails in the glassy state of neat systems. As suggested by random walk simulations,22,24 the β-process is a multi-step process (like the α-process), so that the overall loss of correlation is not achieved before a number of elementary steps with a jump time τJτβare performed. The wobblingon-a-cone model allows to reproduce the salient features of the NMR spectra as well as to quantify the extent of spatial 0021-9606/2013/139(6)/064508/12/$30.00 © 2013 AIP Publishing LLC139, 064508-1 Downloaded 14 Aug 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 064508-8 Bock et al. J. Chem. Phys. 139, 064508 (2013) 1000 2000 3000 4000 5000 6000 10 -5 10 -3 2546 ν 0 =5×10 13 Hz 10% E / k = T ln(ν 0 /ν) 10 -5 10 -3 2582ν 0 =5×10 13 Hz 20% 10 -5 10 -3 3030 ν 0 =4×10 14 Hz 30% 10 -5 10 -3 3090 ν 0 =4×10 14 Hz 45% 10 -5 10 -3 3198 ν 0 =6×10 14 Hz 60% g(E) ~ ε '' / (T Δε β ) 10 -5 10 -3 3319 80% 10 -5 10 -3 3404 ν 0 =9×10 14 Hz ν 0 =2×10 15 Hz 90% 10 -5 10 -3 3367 ν 0 =10 15 Hz 95% 10 -5 10 -3 100% ν 0 =10 15 Hz E max /k=3367K α (TPP) FIG. 9. Distribution of activation energies g(E) obtained from scaling the susceptibility (cf. text) of the sub-Tgmeasurements of all investigated mixtures (concentrations, attempt rates, and most probable activation energies indicated). Dashed lines: Guides to the eye. Note: At high E/k deviations are observed due to the α-process of TPP. governed by asymmetric but temperature independent distributions of activation energies g(E). As a consequence, all time constants τβfollow Arrhenius laws. All susceptibility data can be collapsed in order to reveal g(E), making a detailed discussion of shape parameters, as obtainable by data fitting, needless in the context of this work. D. Characterizing the β-process in the TPP/PS-d3 mixtures by NMR experiments As documented by the dielectric spectra, the β-process in the TPP/PS mixtures is well separated from the α-process. This is an important prerequisite for probing it by NMR, in particular, by applying a Hahn-echo (31P) or a solid-echo (2H) two-pulse sequence. The concentration selection of TPP/PSd3mixtures investigated with these techniques can be inferred from Table I. Figure 10(a) shows the Hahn-echo spectra of TPP of the c=20% mixture. A series of spectra, again normalized to their maximum values, is collected at three selected temperatures with a pulse delay tp=40 μs, 80 μs, and 200 μs. While at high as well as low temperature no spectral changes are recognized, a weak but well discernible decrease of intensity around ν=−δCSA/4 is recognized at the intermediate temperature T=121.0 K. Again, the characteristic spectral changes associated with a β-process are well identified. In the case of the c=50% sample a similar evolution of the 31P spectra is observed (cf. Fig. 10(b)), here, even longer tpvalues have been applied at T=120.3 K (solid lines) and the intensity around ν=−δCSA/4 almost vanishes at longest time tpas in the case of neat TPP (cf. Figs. 4(b) and 5(b)). As before for the neat components the spectral effect can be quantified by measuring the intensity close to the center of the spectrum with respect to the singularity, i.e., the quan- -15 0 15 -15 0 15 -15 0 15 121.0K 101.5K 20% TPP / PS-d 3 31P (a) ν / kHz 166.0K -15 0 15 -15 0 15-15 0 15 148.5K ν / kHz 120.3K 50% TPP / PS-d3 31P (b) 96.2K FIG. 10. 31P Hahn-echo spectra of TPP/PS-d3glass at indicated temperatures, (a) c=20%, each set with tp=40 μs, 80 μs, and 200 μs (b) c=50%, 20 μs and 200 μs each, at T=120.3 K very long interpulse delays were applied additionally (solid lines, tp=400 μs, 800 μs, and 1600 μs) and the intensity at ν=−δCSA/4 almost vanishes. For shortest tpa fit by a CSA powder spectrum is included (solid line). tity R(tp) (cf. Eq. (2)). In Fig. 11(a) the Rvalues for long tp (200 μs) (solid symbols) measured for three TPP concentrations (c=10%, 20%, 50%, and for comparison 100%) are plotted versus temperature. Except for c=100% (as discussed before) a distinct minimum, essentially not shifting, is displayed. This signals immediately that the time constant of the process does not significantly change in the mixtures, a result already known from our DS study (cf. Fig. 2). In contrast to the c=100% sample the minimum in R(tp)iswell resolved due to the larger separation of αand β-process in the mixtures. Also the depth of the minimum is very similar, at lowest concentration c=10% it is somewhat less deep. The decays of R(tp) for the different TPP concentrations at very similar temperatures are displayed in Fig. 11(c) and compared to the data of neat TPP. While for the c=50% sample the decay of R(tp) is similar to that of c=100%, the situation for c=20% is different. The R(tp) appears not to relax completely down to zero, indicating that at low TPP concentrations only a fraction of TPP molecules participates in the β-process, i.e., some molecules have become immobilized. Yet, in all cases the decay is described by a Kohlrausch function (cf. Eq. (3), below) with β=0.95, and the apparent Downloaded 14 Aug 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 064508-9 Bock et al. J. Chem. Phys. 139, 064508 (2013) 0.2 0.3 0.4 100 150 200 250 300 0.0 0.2 0.4 2 H (a) R 10% 20% 50% 100% t p =200 μ s 31 P T / K R 0% 10% 20% 50% 80% 90% TPP / PS-d 3 (b) 0 500 1000 1500 2000 0.00 0.25 0.50 0.75 1.00 0.0 0.5 1.0 250 500 750 R / R(t p =0) t p / μ s 20%, 121.0K 50%, 120.3K 100%, 123.0K 31 P NMR (c) τ [ μ s] c FIG. 11. Temperature dependence of line-shape parameter Rat tp=200 μs for (a) TPP and (b) PS-d3; mass fractions as indicated. Lines serve as guides for the eye. (c) Dependence of R(tp), normalized to short tpvalues,on the inter-pulse delay tpas revealed by 31P Hahn-echo; mass fractions, and temperatures as indicated; inset: time constant as a function of concentration, dashed lines: a possible interpretation. time constant (Fig. 11(b), inset) becomes somewhat shorter as also observed in the dielectric spectra (cf. Fig. 2). Neat PS is a type-A glass former not showing any spectrally resolved β-process in the DS spectra. This is also confirmed by 2H spin-lattice relaxation measurements, which display only a weak temperature dependence in the glassy state as discussed before (cf. Fig. 3). Here, the question arises whether polystyrene in the mixture exhibits a β-process. Due to the selectivity of 2H NMR probing solely the dynamics of the PS-d3molecules 2H solid-echo spectra can give a clearcut answer. Figure 12 shows a series of solid-echo spectra of TPP/PS-d3with c=50% taken at different tpvalues for three temperatures. A pronounced spectral change is observed at intermediate temperature T=121.5 K. Similar results are observed for the c=20% sample, in particular, the largest spectral change is again observed at 123.0 K. Moreover, the effect is found at similar temperatures as in the case of TPP studied by 31P NMR. It seems that PS in the mixture shows some secondary relaxation, too, which passes through the solid-echo time window well below Tgat similar temperatures as in the case of TPP. The corresponding R(T) values (at long interpulse delay tp=200 μs) are included in Fig. 11(b). While no distinct temperature dependence is observed in neat PS-d3 (c=0%), a minimum emerges when TPP molecules are added. Up to c=50% the minimum depth increases monotonously. Remarkably, the minima do not shift with concentration and they are positioned roughly at the same temperature as the minima resulting from the 31P Hahn-echo experiment on TPP (Fig. 11(a)). Actually, the minimum of R(T)is slightly shifted to higher temperatures in the case of the 2H spectra of PS-d3, which is expected due to the higher coupling constant δQ. Taking δQ=122.4 kHz from an analysis of the low-temperature 2H solid-state spectra of PS-d3, the extracted time constant τβagrees well with those determined by DS as well as by 31P Hahn-echo experiment (cf. Fig. 2). These findings strongly suggest that in the mixtures polystyrene monomers participate in the highly hindered reorientation of the β-process introduced by the TPP molecules. In other words, the β-process is not solely an intramolecular process. It appears that the TPP molecules cause the monomer units of polystyrene to wobble in a rather similar way and on the same time scale as the TPP molecules do. In order to further investigate the tpdependence of the 2H solid-echo spectra, Fig. 13(a) shows the solid-echo spectra at very similar temperatures for the different investigated - 200 0 200 -150 0 150 -150 0 150-150 0 150 87.6 K 50% TPP / PS-d3 ν / kHz 121.5 K 156.2 K 102.2 K (a) 2 H NMR -150 0 150 -150 0 150 -150 0 150 -150 0 150 ν / kHz 80.0K 98.8K123.0K 173.0K 20% TPP / PS-d 3 (b) 2 H NMR FIG. 12. 2H NMR spectra of TPP/PS-d3, normalized to maxima, at indicated temperatures. (a) c =50%; each set with tp=20 μs, 200 μs (solid lines), 40 μs and 80 μs (dashed lines), (b) c=20%, tp=20 μs and 200 μs. For lowest temperatures fits with Pake spectral shape are included (solid red line, δQ=122.4 kHz). Downloaded 14 Aug 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 064508-10 Bock et al. J. Chem. Phys. 139, 064508 (2013) -150 0 150 -150 0 150 -150 0 150 -150 0 150 -150 0 150 121.5K 50% TPP / PS-d 3 ν / kHz 121.6K 0% (a) 123.0K 20% 2 H NMR 121.5K 10% 123.0K 90% 0 200 400 0.00 0.25 0.50 0.75 1.00 R / R(t p =0) t p / μ s 0%, 121.6K 10%, 121.5K 20%, 123.0K 50%, 121.5K 80%, 122.0K 90%, 123K TPP / PS-d 3 2 H NMR (b) FIG. 13. (a) 2H NMR spectra of TPP/PS-d3for various concentrations of TPP at indicated temperatures; each set with tp=20 μs, 40 μs, 80 μs, and 200 μs. (b) Rvalues as a function of inter-pulse delay, normalized to short tpvalue. Lines are fits according to Eq. (3). concentrations. For neat PS-d3one recognizes only weak spectral changes, which is actually due to some other relaxation mechanism, while the spectral changes induced by large tpvalues significantly grow with increasing TPP concentration. Explicitly, at tp=200 μs the spectral intensity at zero frequency is the lower, the higher cis. It seems as if the fraction of PS molecules participating in the β-process grows with the fraction of TPP molecules present in the mixture. Figure 13(b) displays the corresponding evolution of R(tp). For all concentrations c >0R(tp) decays on similar tptime scale (τ=(170 ±20) ms and β=1.28), while for c=0 (neat PS-d3) a qualitatively different, slower decay is observed. The latter finding is in accordance with the fact that actually neat PS does not exhibit a β-process and another relaxation process may be active. For c=90% and c=80% R(tp) decays down to very low values at long delay times, while for c≤50% R(tp) appears to level off at different plateaus at longest tp. In particular, a systematic trend of the final plateau to increase with decreasing concentration is recognized. To access quantitatively the decay R(tp) and in particular the plateau at longest times tpwe describe the normalized decay by the following expression25 Rn(tp,c)=fβ(c)·exp −tp τβ+(1 −fβ(c)),(3) where fβ(c) is interpreted as a fraction of molecules which contributes to the β-process, and τand βare parameters describing the effective time evolution of the echo spectra. We note that the latter time constant is not identical with τβas determined from the DS spectra. In the case of TPP, a free fit by Eq. (3) to R(tp)(cf.Fig.11) provides similar relaxation times τ(and similar β=0.95 ±0.05) with a small trend to become somewhat shorter at low c=20%. Since a correlation between τand τβis expectable, this is in accordance with the DS result where a weak trend to shorter τβis revealed for c<60% (cf. Fig. 3). The long-time value 1 – fβis not any longer zero but increases with decreasing TPP concentration. The corresponding value fβ(c) reflecting the fraction of TPP molecules participating in the β-process is found in Fig. 14. The higher is the TPP concentration, the higher is the fraction of TPP molecules participating in the β-process. We note that in the case of ethanol, cyano cyclohexane, and toluene (Fig. 5(a)) we find a plateau value 1 – fβ=0.0 ±0.04 while in the case of neat TPP we find fβ=0.93, i.e., 1 – fβ=0.07 ±0.04 putting our above given statement on a quantitative basis: in neat glasses essentially all molecules take part in the β-process while this is no longer the case in a binary system. In the case of PS-d3a corresponding analysis of R(tp) along Eq. (3) (solid lines in Fig. 13(b)) provides essentially the same time constants, but, nevertheless, varying fractions of PS molecules participating in the β-process. The result is included in Fig. 14. With increasing TPP concentration also the fraction of PS molecules participating in the β-process grows quickly. It even appears that the fraction fβof PS is higher than that of TPP. This excess fraction becomes most conspicuous at c=0, where a plateu value of fβ(c=0) =0.25 is found. This is somewhat unreasonable, since, due to the absence of TPP molecules, no contribution to the β-relaxation is expected at all. One can argue that, due to some further 0.0 0.2 0.4 0.6 0.81.0 0.00 0.25 0.50 0.75 1.00 ( ) 31 P NMR 2 H NMR, norm. to t p → 0 2 H NMR, norm. to c=0, t p →∞ TPP / PS-d 3 f β c ( ) FIG. 14. Fraction of molecules of TPP (crosses) and PS-d3(full squares: normalized to behavior of neat PS-d3; open squares: normalized to short tp value) participating in the β-process as estimated by the Hahn-echo and solidecho experiments. Dashed straight lines: A possible interpretation. Downloaded 14 Aug 2013 to 132.180.10.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions 064508-11 Bock et al. J. Chem. Phys. 139, 064508 (2013) relaxation process in neat PS, R(tp) decays to some plateau which has to be taken into account also in the mixture. Thus renormalizing 1 – fβby 1 – fβ(neat PS), the fraction fβdecreases and becomes similar to that of TPP. Thus the fractions of TPP and PS molecules participating in the β-process coincide for each concentration. As discussed above, the dynamics of neat PS-d3and TPP have also been characterized by the spin-lattice relaxation monitored as a function of temperature (cf. Fig. 3(a)). No indication of a β-process shows up for PS-d3while below Tgin TPP the 31P relaxation is clearly controlled by the βprocess. The findings for the TPP/PS mixture with c=50% are shown in Fig. 3(b). With respect to the neat components, the results are now quite different. Below Tgthe temperature dependences T1(T) of TPP and PS-d3run parallel, i.e., the same strong temperature dependence is observed in both methods. Above Tga minimum is found for TPP which reflects the isotropic reorientation of the TPP molecules in the mixture. A similar minimum occurring yet at higher temperatures is found for PS-d3. This difference directly reflects the decoupling of the primary (isotropic) dynamics of the components: in the mixture PS reorients much slower than TPP, a fact well known from results on asymmetric binary glass formers.42,43 There is a further feature the detailed discussion of which is postponed to a forthcoming publication: At temperatures for which the T1minimum of TPP occurs, particularities are also observed in the 2H relaxation of PS-d3. This shows that the fast isotropic dynamics of TPP affects the polystyrene monomers leading to a highly hindered (i.e., nonisotropic) dynamics, however, occurring at similar time scale as the TPP molecules. These findings demonstrate that a plasticizer molecule does not only change the Tgbut also induces additional dynamics on the polymer. As in the case of neat TPP the temperature dependence of the spin-lattice relaxation can be understood on a quantitative level by taking the dielectric results into account. These provide the dynamic susceptibility determined by a temperature independent distribution of activation energies, which actually does not change significantly in the mixture. Analogously to the case of 31P dielectric susceptibility data can be compared to T1(T)of2H NMR after extrapolating it to higher frequencies. As can be seen in Fig. 3(b) (dashed line) the slope of T1(T) of TPP as well as PS-d3is reproduced. IV. DISCUSSION AND CONCLUSION We have studied the secondary (β-) relaxation process in the binary glass mixture TPP/PS by dielectric spectroscopy as well as TPP/PS-d3by 31P and 2H NMR. While neat TPP exhibits a β-process (type-B glass former) and neat polystyrene shows none (type-A), in the mixture also PS-d3molecules clearly participate in the β-relaxation. Here, for both TPP and PS-d3, NMR spectra reveal a spatially highly restricted motion as identified by NMR in other type-B systems, like toluene16,22 or ethanol.17,23 Up to our knowledge a Hahnecho sequence (here for 31P) has been applied for the first time to monitor the subtle spectral changes deep in the glass characteristic of the β-process. The dielectric spectra reflect a distribution of activation energies which is temperature independent; yet, its asymmetric shape is rather unusual for β-processes and does not change with concentration, a phenomenon observed also in other binary glass formers.25,43,44 Even at a TPP concentration of c=10% the mean activation energy is still similar to that of neat TPP. The β-process introduced by the type-B component survives in the mixture and induces the type-A molecule to participate in the relaxation process. Yet, in contrast to neat systems not all molecules participate in the β-process; islands of rigidity or immobility appear. The higher the concentration of the typeB component the higher is the fraction of both components which participate. We emphasize, as we do not find any indication for the mixed glasses to decompose, the immobilized molecules are not part of crystalline regions. Instead the mixtures become glasses with inhomogeneously distributed dynamics. In a recent 2H NMR study of another binary system (toluene/aroclor) an indication of a threshold concentration has been found below which immobile type-B molecules appear.25 Although not sufficient NMR data at various concentrations has been collected in the present study a similar behavior can be anticipated also for TPP/PS-d3. Only below, say, c=60% some relevant fraction of TPP or PS-d3appear to become immobilized. This corresponds with the (slight) change of the mean activation energy below 60% (cf. Figs. 7 and 9). All together the presented experimental findings point into the direction that also the β-process exhibits some cooperative nature. When mixed with type-B molecules, type-A molecules do react to the highly hindered motion introduced by the β-process, actually a behavior expected in (dense) condensed matter. A similar phenomenon is observed for the decoupled isotropic reorientation of the TPP molecules in the vitrified matrix of polystyrene. Whether the extent of spatial hindrance is the same for the two molecules is yet to be investigated. In any case both components show dynamics on the same time scale. ACKNOWLEDGMENTS The authors acknowledge financial support by Deutsche Forschungsgemeinschaft (DFG) under Grant No. RO 907/10. 1G. Johari and M. Goldstein, J. Chem. Phys 53, 2372 (1970). 2G. Williams and D. C. Watts, Trans. Faraday Soc. 67, 1971 (1971). 3L. Wu, Phys.Rev.B43, 9906 (1991). 4A. Arbe, D. Richter, J. Colmenero, and B. Farago, Phys.Rev.E54, 3853 (1996). 5A. Kudlik, C. Tschirwitz, S. Benkhof, T. Blochowicz, and E. Rössler, Europhys. Lett. 40, 649 (1997). 6K. L. Ngai and M. Paluch, J. Chem. 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Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions Paper 6 Dynamics of Asymmetric Binary Glass Formers. I. A Dielectric and Nuclear Magnetic Resonance Spectroscopy Study R. Kahlau, D. Bock, B. Schmidtke, and E. A. R¨ ossler, The Journal of Chemical Physics 140, 044509 (2014). c 2014 AIP Publishing LLC doi:10.1063/1.4861428 115 116 THE JOURNAL OF CHEMICAL PHYSICS 140, 044509 (2014) Dynamics of asymmetric binary glass formers. I. A dielectric and nuclear magnetic resonance spectroscopy study R. Kahlau, D. Bock, B. Schmidtke, and E. A. Rösslera) Experimentalphysik II, Universität Bayreuth, 95440 Bayreuth, Germany (Received 30 October 2013; accepted 20 December 2013; published online 29 January 2014) Dielectric spectroscopy as well as 2H and 31P nuclear magnetic resonance spectroscopy (NMR) are applied to probe the component dynamics of the binary glass former tripropyl phosphate (TPP)/polystyrene (PS/PS-d3) in the full concentration (cTPP) range. In addition, depolarized light scattering and differential scanning calorimetry experiments are performed. Two glass transition temperatures are found: Tg1(cTPP) reflects PS dynamics and shows a monotonic plasticizer effect, while the lower Tg2(cTPP) exhibits a maximum and is attributed to (faster) TPP dynamics, occurring in a slowly moving or immobilized PS matrix. Dielectric spectroscopy probing solely TPP identifies two different time scales, which are attributed to two sub-ensembles. One of them, again, shows fast TPP dynamics (α2-process), the other (α1-process) displays time constants identical with those of the slow PS matrix. Upon heating the α1-fraction of TPP decreases until above some temperature Tc only a single α2-population exists. Inversely, below Tca fraction of the TPP molecules is trapped by thePSmatrix.AtlowcTPP the α2-relaxation does not follow frequency-temperature superposition (FTS), instead it is governed by a temperature independent distribution of activation energies leading to correlation times which follow Arrhenius laws, i.e., the α2-relaxation resembles a secondary process. Yet, 31P NMR demonstrates that it involves isotropic reorientations of TPP molecules within a slowly moving or rigid matrix of PS. At high cTPP the super-Arrhenius temperature dependence of τ2(T), as well as FTS are recovered, known as typical of the glass transition in neat systems. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4861428] I. INTRODUCTION The evolution of the dynamic susceptibility in neat glass formers is well documented starting at temperatures close to the boiling point and reaching the glass transition temperature Tg,where the liquid becomes an amorphous solid.1–7In particular, light scattering, dielectric, and NMR spectroscopy have provided a wealth of information. In contrast, binary glass formers are less studied, and no broadly accepted picture of the rather complex dynamics has established so far. Of special interest are so-called asymmetric glass formers, which are characterized by a large difference of the Tgvalues of the components. They are most conveniently prepared by blending a polymer with a low-molecular mass additive,8–13 yet also purely low-molecular weight mixtures have been studied.14–18 It is well established that such systems exhibit two glass transition temperatures albeit they are fully miscible.19–25 In other words, such binary liquids display pronounced dynamic heterogeneities,5,26,27 which are in particular well documented by NMR.28–30 For example, intrinsic confinement effects are expected when the mobile (low-Tg) component still relaxes in a matrix of an arrested (high-Tg) component. Indeed, the NMR phenomenology is similar to that of neat glass formers embedded in porous systems.31–34 Binary systems consisting of soft or hard spheres have also been investigated by simulations35–37 as well as by mode a)Author to whom correspondence should be addressed. Electronic mail: [email protected]. coupling theory (MCT).38–41 In contrast to neat systems, for which a type-B glass transition scenario is expected by MCT, which is triggered by cage formation and characterized by a discontinuous change of the non-ergodicity parameter ffrom zero to f>0 at a critical temperature Tc, the mobile molecules in binary liquids are expected to exhibit a type-A transition, and fshould increase continuously from zero upon cooling below Tc. Here, the mobile (small) particles undergo a localization transition, while the less mobile (large) ones are still arrested due to the cage effect. In a recent paper by Blochowicz et al.13 experimental hints have been given that indeed such type-A transitions might be observed in mixed molecular liquids. We note that such dynamic heterogeneities have also been explained either by concentration fluctuations42,43 or so-called self-concentration effects.44 It is the aim of the present contribution to dwell on this issue by applying several experimental methods to study selectively the dynamics of each component on a (short-chain) polymer-additve system. The binary glass tripropyl phosphate (TPP)/(deuterated) polystyrene (PS/PS-d3,Mw≈2×103g/mol) is characterized by means of dielectric spectroscopy (DS), 2H and 31P NMR as well as by dynamic depolarized light scattering (DLS) and differential scanning calorimetry (DSC). Thirteen concentrations equally spread over the full concentration range are investigated. The system is characterized by a large Tgcontrast of the pure components (Tg∼ =200 K).45,46 Due to the choice of this system the application of 31P and 2H NMR allows for probing selectively the dynamics of TPP 0021-9606/2014/140(4)/044509/12/$30.00 © 2014 AIP Publishing LLC140, 044509-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.180.10.132 On: Wed, 29 Jan 2014 16:11:23 044509-2 Kahlau et al. J. Chem. Phys. 140, 044509 (2014) TABLE I. TPP mass concentrations cTPP of mixtures studied by the different methods, including the assumed errors (see text). NMR data of 80% will be discussed in Paper II.47 cTPP [%] nom. 0 10 20 30 36 45 50 60 70 80 90 95 100 DS 0 8 ±318±329±2 ... 45±1 ... 60±1 ... 80±190±195±1 100 DSC 0 10 ±120±1 ... 36±145±1 ... 60±170±183±190±1 ... 100 DLS ... ... ... ... ... ... ... ... ... 80±190±1 ... 100 2HNMR 0 10±120±1 ... ... ... 50±1 ... ... 80±1 (Paper II)47 90 ±1 ... ... 31PNMR ... 10±120±1 ... ... ... 50±1 ... ... 80±1 (Paper II)47 90 ±1 . . . 100 and PS-d3, while dielectric spectroscopy provides essentially information on the dynamics of the mobile component TPP, since its molecular dipole moment is significantly higher than that of PS. Regarding the pronounced β-process present in the mixed system, it has been studied thoroughly by our group.45 Its time constant shows the typical Arrhenius behavior with an activation energy E/k∼ =24 Tg, which virtually does not vary due to mixing, and which is associated with a spatially highly hindered dynamics as found in other glasses. Actually, throughout this work, only a spatially highly restricted process shall be understood as a β-process. Although introduced by TPP, in the mixture both components participate in the βprocess. This has been taken as an indication of its cooperative nature. In the present contribution we focus on the dynamics of both components above Tg, more precisely above Tg2of the mobile component. By performing DSC experiments we can identify two glass transition temperatures. We will demonstrate that the high-Tgcomponent PS shows liquid dynamics similar to that of neat systems while TPP displays rather complex heterogeneous dynamics. For example, the temperature dependence of the correlation time changes from superArrhenius at high cTPP to Arrhenius behavior at low concentrations. Thus, the additive process may be confused with a β-process, and it is up to NMR to proof whether the process is still liquid-like (isotropic) or β-process-like. This contribution consists of two parts, the present one essentially deals with the results collected by dielectric spectroscopy, complemented by time constants provided by NMR and DLS as well as DSC. In Paper II47 continuative NMR experiments are reported and analyzed in accordance with the dielectric results. II. EXPERIMENTAL DETAILS AND DATA ANALYSIS A. Systems A polystyrene sample with the molecular mass Mw=2250 g/mol (PS), and another polystyrene sample, partially deuterated at the backbone, with very similar mass Mw=2440 g/mol (PS-d3) were purchased from Polymer Standards Service (Mainz, Germany) and used without further treatment. For the DS experiments PS was used for the preparation of the mixtures, while PS-d3was used for the NMR measurements. Tripropyl phosphate (TPP, 99%) was bought from Sigma Aldrich and used as received, too. We do not find any indication that phase separation or crystallization occurs in the mixtures. Among other tests, light scattering experiments show a homogeneous sample. NMR and DLS samples were prepared in the measurement tubes and cells, while the DSC and DS samples had to be prepared in separate test tubes (exact concentrations valid for different methods are listed in Table I). Generally a concentration error of ±1% is assumed for the sample preparation. After the preparation all sample vessels were left at elevated temperatures for one or two days in order to guarantee a maximum possible spatial homogeneity. In the case of DSC and DS the sample had to be transferred afterwards from the preparation vessel to the measurement cell. In the case of DS the samples of low concentrations had to be heated carefully during the transfer, because they are highly viscous at room temperature. Since some fraction of the TPP content may have evaporated during this process, the assumed actual error for the cTPP =10%–30% samples is somewhat higher (Table I). In contrast, the DSC samples could be transferred to the measurement cells without heating. For the sake of clarity the nominal (nom.) concentrations are discussed throughout the paper. B. Dielectric spectroscopy Dielectric measurements were carried out with the Alpha-A Analyzer by Novocontrol while temperature was kept constant within ±0.2 K by using the Quatro-H temperature controller by Novocontrol. The absolute accuracy is assumed to be better than ±1 K. The sample cell has the design described by Wagner and Richert and assures a constant sample volume.48 In order to extract time constants from the dielectric susceptibility data, unless described differently in the text a Kohlrausch stretched exponential was used to fit the α1-relaxation peaks (on the timescale of PS dynamics), and the mean relaxation time τ=(1/βK)τ0/βKis discussed. For the α2-peaks (reflecting faster TPP dynamics) a HavriliakNegami (HN) function had to be used due to stretching parameters significantly less than unity on the low-frequency side of the peak. In these cases the maximum correlation times given by τmax =1/(2πνmax) are discussed. C. Depolarized light scattering Depolarized dynamic light scattering measurements were performed with a vertically polarized Coherent Verdi-V2 laser at a wavelength of 532 nm and 200 mW optical power in combination with a tandem Fabry-Pérot interferometer (TFPI; JRS Scientific, triple-pass-tandem Etalon) working parallel with a double monochromator (DM; Jobin Yvon, U1000). The TFPI was operated at horizontal polarization in almost This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.180.10.132 On: Wed, 29 Jan 2014 16:11:23 044509-3 Kahlau et al. J. Chem. Phys. 140, 044509 (2014) 150 200 250 300 350 400 0.05 0.10 0.15 0.20 f = dq/dt [W/g] T / K 0% 10% 20% 36% 45% 60% 70% 80% 90% 100% Q = 10K/min (a) 140 160 180 200 220 240 260 0.00 0.01 36% 45% df/dT [W/(gK)] (shifted) T / K 70% Q = 30K/min (b) FIG. 1. (a) DSC traces (heat flow per sample mass f=dq/dt) for mixtures and neat systems (cTPP =0%–100%) at the heating rate Q=10 K/min. (b) Temperature derivative of DSC traces df/dT at heating rate Q=30 K/min for indicated concentrations. backscattering geometry, whereas the DM was operated at orthogonal geometry (for details, see Refs. 49 and 50). The TFPI measurements were done with three different free spectral ranges, and the DM measurements with two combinations of slits and frequency intervals. The spectral parts are then adjusted in amplitude to match together and form a smooth spectrum. D. DSC DSC experiments were performed with a Q1000 analyzer by TA Instruments. By using a liquid nitrogen cooling system the temperature range T=120–400 K was covered. Experiments presented were run at heating rates Q =10–40 K/min. Figure 1(a) displays DSC traces of neat PS and TPP as well as mixtures with cTPP =10%–90% TPP in PS (nominal concentrations, see Table I), recorded with a heating rate of Q=10 K/min. For both of the neat samples a distinct glass step is found. For the mixtures a comparably broad glass transition temperature range is observed, consisting of two individual, more or less separated steps. This is best seen in Fig. 1(a) for intermediate concentrations. In order to achieve a better resolution of both contributions the temperature derivative of the DSC traces13 are considered (Fig. 1(b)). The best results were obtained here by choosing a heating rate of Q=30 K/min for all samples. Glass temperatures as yielded by DSC experiments were defined as the peak temperatures of the temperature derivatives just described (cf. Fig. 10). In order to calculate calorimetric time constants the formula, τcal ≈RT 2 g Heff ·1 Q,(1) was used.17,51 It turns out that for all systems a constant prefactor RT 2 g Heff ≈1.7 could be used in good approximation, leadingtoanerrorinτDSC smaller than a factor of two. The extracted time constants are included in Fig. 9. E. NMR Regarding the NMR analysis we refer to Paper II.47 In the present paper we only report some of the NMR results concerning correlation times. III. RESULTS A. Neat components – dielectric spectra The susceptibility spectra of PS (Tg=335 K) are shown in Fig. 2and fit well into the collection of dielectric data on polystyrene samples of different molecular weights given in Ref. 52. Above Tga pronounced peak is visible, which is identified as structural or α-relaxation shifting to high frequencies with increasing temperature. The low amplitude reflects the rather non-polar nature of the PS monomer. Close to Tg,the high-frequency side of the α-peak is made up of a crossover from one power-law behavior to another one, the latter often being called excess wing.1,7,53 When the sample is cooled below Tgthe α-peak moves out of the frequency window and the signal, now consisting only of the excess wing contribution, drops close to the resolution limit of the spectrometer. No indications for a secondary (β) relaxation peak are observed for PS (cf. Ref. 45). The dielectric spectra of neat TPP (Tg=135 K) are also displayed in Fig. 2. As in the case of PS, above Tgan αrelaxation peak can be identified, the amplitude of which exceeds the one of PS by a factor of 1000, i.e.,the TPP molecule carries a high dipole moment. At frequencies several decades above the maximum position of the α-peak a secondary relaxation is well resolved (type-B glass former). This secondary (β-) peak survives at temperatures below Tgwhen the α-peak has already left the frequency window. When temperature is increased above, say, T=150 K, both peaks approach each 10 -3 10 -1 10 1 10 3 10 5 10 7 10 -4 10 -3 10 -2 10 -1 10 0 10 1 142K 138K 345K 148K 290K 375K 365K 355K 156K 330K 335K ε ''(ν) ν/ Hz 395K PS 152K 144K 140K 136K TPP FIG. 2. Susceptibility spectra of neat TPP (top, circles; temperatures indicated). Dielectric spectra of neat PS; T=290 K and T=330–355 K in 5 K steps, T=365–395 K in 10 K steps (bottom, triangles; some temperatures indicated). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.180.10.132 On: Wed, 29 Jan 2014 16:11:23 044509-10 Kahlau et al. J. Chem. Phys. 140, 044509 (2014) 10-2 100102104106 0.00 0.02 0.04 270K 169K 330K 310K ε ''( ν ) ν / Hz 298K 20% TPP / PS FIG. 13. Data from Fig. 4(b); triangles: α1-relaxation for T=290–330 K, circles: α2-relaxation for T=169–280 K. Dashed lines: α2-contributions calculated from the g(E) scaling result shown in Fig. 12(a) along Eq. (2). Solid lines: fits (cf. text). frequencies. In this temperature range, each spectrum was fitted with the sum of a Kohlrausch function with βK(T) =0.28–0.38 and the corresponding (fixed) α2-contribution calculated from the temperature independent g(E). The results (solid red lines in Fig. 13) agree with the data. Resulting time constants τ1,as well as τ2,calc =(1/2πν0)exp(Emax/T) calculated with Emax =6000 K and ν0=2.3 ×1015 s−1 (Fig. 12(a)), are included in Fig. 5(crosses). Good correspondence with the experimental τ2,as well as, respectively, the τ1yielded by the direct fitting analysis of the α1-peak (Secs. II B and III B), is observed. The resulting relaxation strengths ε1(T) are included in Fig. 6(open triangles). APPENDIX B: REVEALING THE α1-PROCESS AT INTERMEDIATE CONCENTRATIONS Figure 14 shows dielectric data for the (a) 30% and (b) 45% TPP/PS mixtures. In contrast to the 10% and 20% data (Fig. 4) only one relaxation peak is explicitly observable. Note that between 30% and 100% TPP only a single peak is observed in the susceptibility (besides the β-process; cf. Appendix C), and with higher concentrations this peak increases further in amplitude and develops continuously into the α-process of neat TPP. In the case of the 30% and 45% mixtures still traces of the α1-relaxations can be identified. 10 -2 10 0 10 2 10 4 10 6 10 -3 10 -2 10 -1 10 0 10 1 β K =0.23 308K 298K 288K 278K 268K 258K 248K ε''(ν) ν / Hz ∼ν -1 238K 30% TPP / PS FIG. 15. Susceptibility of the cTPP =30% sample (squares). Red lines: conductivity contribution. Blue lines: susceptibility after subtracting conductivity contribution. Dashed lines: Kohlrausch functions with βK=0.23. By plotting the data on linear scale, one can infer from Fig. 14(a) that a very broad minimum is located between conductivity and α2-peak. When temperature is increased beyond about 272 K up to 346 K the initially broad minimum becomes narrower and decreases its amplitude. A similar, yet weaker effect is observed for cTPP =45% between T=230 K and 250 K (Fig. 14(b)). Since in the case of the 10% and 20% mixtures a decrease of ε1(T) with increasing Tis observed, we speculate that the decreasing minimum reflects a hidden α1-relaxation peak. In order to reveal the α1-relaxation peak, the conductivity contribution has to be subtracted. This allows for estimating relaxation time and strength. In Fig. 15 this is demonstrated exemplarily for the susceptibility data of the 30% sample; for the 45% data a similar procedure was applied. The resulting τ1(T) correspond well with the NMR findings (Fig. 5). APPENDIX C: FURTHER SUSCEPTIBILITY DATA Fig. 16 shows further spectra for the mixtures with the TPP concentrations cTPP =60%, 80%, 95%. At these high concentrations no reasonable estimation of the dielectric time constant τ1of the α1-relaxation is possible any more. FIG. 14. Dynamic susceptibility data of TPP/PS mixtures. (a) 30% TPP in PS, T=190–260 K in 10 K steps and T=262–346 K in 2 K steps. (b) 45% TPP in PS, T=200–230 K in 10 K steps and T=232–250 K in 2 K steps. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.180.10.132 On: Wed, 29 Jan 2014 16:11:23 044509-11 Kahlau et al. J. Chem. Phys. 140, 044509 (2014) 10 -3 10 -1 10 1 10 3 10 5 10 7 10 -1 10 0 β 60% TPP / PS 250K 210K200K 190K 180K170K ε ''( ν ) ν / Hz 160K α 2 (a) 10 -3 10 -1 10 1 10 3 10 5 10 7 10 -1 10 0 β 80% TPP / PS ε ''( ν ) ν / Hz 146K 152K 158K 164K 170K 176K 182K 188K 194K 200K α 2 (b) 10 -3 10 -1 10 1 10 3 10 5 10 7 10 -1 10 0 10 1 β 95% TPP / PS 154K 162K 220K 210K 198K 170K 166K 158K 150K 146K 142K ε ''( ν ) ν / Hz 138K α 2 (c) FIG. 16. Susceptibility spectra of mixtures with a TPP concentration of (a) cTPP =60%, (b) cTPP =80%, and (c) cTPP =95%. APPENDIX D: LINE SHAPE PARAMETERS OF THE α2-PROCESS AT HIGH CONCENTRATIONS As is demonstrated in Fig. 11(b), at high TPP concentrations the main temperature effect regarding the shape of the spectra is a broadening on the low-frequency flank of the α2130 140 150 160 170 180 190 200 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 c TPP = 80% c TPP = 90% c TPP = 95% a(T) T / K α2 -relaxation peak low freq. parameter c TPP = 100% FIG. 17. 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