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Heterogeneous large amplitude atomic motion in supercooled liquids
Chemical Physics 292 (2003) 325–331 www.elsevier.com/locate/chemphys
Heterogeneous large amplitude atomic motion in supercooled liquids Margarita Russina a,*, Olga Russina b, Ferenc Mezei a,b b
a Los Alamos National Laboratory, MS H805, Los Alamos, NM 87545, USA Hahn-Meitner-Institute Berlin GmbH, Glienicker Str.100, Berlin 14109, Germany
Received 22 April 2003; in final form 21 May 2003
Abstract One of the central questions in glass physics is the dynamic nature of the glass transition. Essential issues are the type of atomic motions involved and their homogeneity or heterogeneity. Previous experimental studies of the dynamic heterogeneity performed by special NMR techniques for times >106 s and by incoherent neutron scattering were restricted to the a-relaxation process. Here we review the results of neutron scattering studies focused on the picosecond time domain showing that fast b-process corresponds to large amplitude cluster like heterogeneous motion. Ó 2003 Published by Elsevier Science B.V.
1. Introduction The concept of heterogeneity in the dynamics around the glass transition was first introduced in 1969 by Addams and Gibbs [1], who suggested the existence of cooperative, cluster-like motion. The idea has been strongly supported by recent theoretical and molecular dynamics developments. The theoretical approach of Colby [2] uses the idea of free volume in a simple mean-field model to point out the existence of this kind of heterogeneous motion. In the glassy state there is no free volume of a sufficient size and the particles are trapped in the ‘‘cage’’ of surrounding particles. On microscopic time scales the atoms vibrate in the cages
*
Corresponding author. Fax: +1-505-665-2676. E-mail address: [email protected] (M. Russina).
around quasi-equilibrium positions. With increasing temperature the free volume grows and opens up the way for rapid cooperative rearrangement of groups of particles in the form of large amplitude motion (i.e., atoms moving to distances comparable to the separation between nearest neighbors). However, only a fraction of atoms in the sample can move at a given time, the rest has to wait for free volume to become available. Hence the heterogeneity in the dynamics: a small fraction of particles is mobile on a short time scale, while the rest follows conventional slow viscous flow. This picture is in agreement with molecular dynamics simulation studies [3] revealing the existence of correlated atomic motion in form of loops or chains (‘‘string’’-like motion) observed by identifying those atoms that moved to large distances compared to the average within a given time. In contrast to the Johari–Goldstein hypothesis [4],
0301-0104/03/$ - see front matter Ó 2003 Published by Elsevier Science B.V. doi:10.1016/S0301-0104(03)00256-8
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this proposed mechanism of heterogeneity in the dynamics is rather a spontaneous one, it does not require the existence of structural heterogeneity or ‘‘islands of mobility’’. Heterogeneous as well as homogeneous (e.g., [5,6]) scenarios have been intensively explored in the last decade on a broad time scale for the different relaxation processes observed around the glass transition. The current picture of relaxation behavior of supercooled liquids is complex. Mode coupling theory [5] introduced slow a- and fast b-processes, whose existence has been confirmed experimentally in almost all known glass forming liquids. The fast b-process takes place on the picosecond time scale at all temperatures where it is observable, while the a-relaxation moves from macroscopic times to picoseconds with increasing temperature. In addition in some systems one finds the slow Johari–Goldstein b-process on a strongly temperature dependent time scale, which is shorter but close to that of the a-relaxation. Previous experimental studies related to dynamic heterogeneity focused on the a- or the slow (Johari–Goldsdtein) b-processes. The question whether homogeneous and heterogeneous scenarios can be distinguished by the momentum transfer dependence of the a-relaxation time has been investigated by incoherent neutron scattering. The authors in [6] conclude that the stretching of the a-relaxation is related to homogenous sublinear diffusion. In NMR studies [7,8] the heterogeneity of a-relaxation is probed by selecting a dynamically distinguishable ensemble and monitoring its return to the equilibrium. The conclusion in these works is that above the glass transition temperature one observes heterogeneities with life times of order of a-relaxation time. It is important to mention that these NMR techniques can only explore macroscopic times scales (ls or longer). The slow b-process was also investigated by NMR [8,9] and dielectric spectroscopy experiments. Although both studies found that all molecules contribute to the process in similar fashion, authors in [8] believe that the non-exponential 2 H spin–lattice relaxation observed at low temperatures (T < 1:1Tg ) and at times t < sa is due to the heterogeneity of the b-processes. However, no ev-
idence of distinguishable ‘‘islands of mobility’’ has been obtained. There is less information available on the fast bprocess, although it may play a most crucial role in the physics of glasses. It is most remarkable, that the fast b-process has been first predicted by the mode coupling theory of the glass transition and discovered later experimentally [10] more than a decade ago. Since then the fast b-process was found to be a universal feature of glassy systems. In contrary to the processes discussed to this point, the fast bprocess cannot be probed by the NMR techniques mentioned above since its time scale varies little with temperature and remains in the picosecond range. The various explanations and theories proposed for the nature of the fast b-process can be roughly divided into two groups. The first group of authors believes that the fast b-process is related to vibrational motion [11,12], while the second group [5] suggests that it is caused by structural relaxation. The momentum transfer Q-dependence of the dynamic structure factor SðQ; xÞ can provide the key to the answer. It has been shown in [13] that the dynamic structure factor of the phonon-like excitations SðQ; xÞ shall be proportional to Q2 SðQÞ, where SðQÞ is the static structure factor. In the case of rigid structural relaxation we expect the approximate law SðQ; xÞ / SðQÞ. The present paper focuses on work aimed at answering the questions about the nature and homogeneity/heterogeneity of the fast b-process by probing glassy systems on the picosecond time scale. In what follows we will review and discuss coherent and incoherent neutron scattering data, both previously published and new ones.
2. Experimental results Though the nature of fast b-process has been intensively investigated for quite some time, it remained a subject of controversy. In neutron scattering work this was largely due to experimental difficulties, mainly to the multiple scattering noise. Multiple scattering (MS) means that a neutron can be scattered more than one times while it is inside the sample, and this can lead to spurious noise masking small signals. In order to overcome the
M. Russina et al. / Chemical Physics 292 (2003) 325–331
problem of MS we developed a new method for multiple scattering correction (see details in [14]), which is based on the fact that we can vary the probability of multiple scattering in the same sample by changing the experimental conditions. This method allows us to establish very precisely the single scattering in a broad momentum transfer Q range, especially in the region of low Qs where the single scattering signal is very small. The importance of the low Q domain is that this range corresponds to the length scale of several times the distance between the nearest atomic neighbors. Probing the dynamic structure factor at this length scale gives us the possibility to follow and analyze the collective aspects of the atomic motion. We applied this method to study of the microscopic dynamic of three inorganic glass formingsystems, Ca0:6 K0:4 (NO3 )1:4 (cf. [15]) and most recently Ca0:6 K0:4 (NO3 )1:4 + H2 O and ZnCl2 . Each of these glasses represents an ionic molten salt, however there important differences. Ca0:6 K0:4 (NO3 )1:4 is a typical example of so-called fragile systems. The definition of fragility [22] describes the deviation of the variation of the viscosity as a function of temperature from the Arrhenius behavior: the larger is the deviation, the more fragile is the system. Ca0:6 K0:4 (NO3 )1:4 , shortly called CKN, is a coherent neutron scatterer and provides information about the pair-correlation function, i.e., in particular about the cooperative atomic motion. Ca0:6 K0:4 (NO3 )1:4 + 2–4 at.% H2 O is very similar to the CKN system, with the glass transition temperature slightly (3–5 K) lower that of CKN (Tg ffi 333 K). However, due to the introduction of H atoms this system displays a strong contribution of incoherent neutron scattering and provides information about the self-correlations in the motion of atoms. ZnCl2 exhibits different temperature dependence of viscosity and belongs to the category of the intermediate glass forming systems between fragile and strong [20]. In contrast to the loose ionic structure of CKN, ZnCl2 consists of closely packed Zn–Cl tetrahedra, where Cl anions shield Zn2þ cations and restrict their mobility, leading to a more networked structure [21]. The goal of this particular part of the study is to analyze how the microscopic dynamics changes with more pronounced atomic structure and lesser degree of fragility.
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2.1. Coherent neutron scattering, Ca0:6 K0:4 (NO3 )1:4 Fig. 1 from [15] shows the coherent neutron scattering spectra of the ionic glass forming model system Ca0:6 K0:4 (NO3 )1:4 . All data were normalized to the Bose factor of harmonic temperature dependence of phonon-like excitations as discussed in detail in [15]. The slight shoulder positioned around 4 meV indicates the so-called Boson peak, previously observed in CKN in the light scattering spectra [16]. Below the glass transition temperature the spectra are harmonic and well approximated at low energies by a constant (Debye phonon spectrum). With increasing temperature we observe the onset of anharmonicity at T > Tg , where Tg ffi 333 K is the temperature of the calorimetric glass transition. It is important to note that the temperature dependence of the spectral intensity at the range of small energy transfers x < 2 meV, corresponding to the picosecond time scale, is considerably stronger than the temperature dependence of the intensity of the phonon-like excitations at higher energy transfer values, namely >4 meV. Consequently, the spectra can be presented as a sum of a slightly anharmonic vibrational part and a rapidly varying low energy transfer part (below 4 meV), which can be identified with the fast b-pro-
Fig. 1. Temperature dependence of the coherent neutron scattering spectra in CKN. In the glassy states (T < Tg ¼ 333 K) the spectra are harmonic, while the shape drastically changes with the increase of the temperature at T > Tg . The temperature evolution of the intensity shows clear differences at different energies. In the range of the b-process at x < 2 meV the anharmonic increase of intensity is much stronger than for the phonon-like excitations at higher energies. Figure from [15].
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cess [5] and is actually well described by the predicted power law line shape below 3 meV. We analyzed the dynamic structure factor as a function of Q at different temperatures at constant energy transfers. Fig. 2(a) demonstrates that the phonon-like excitations with the expected Q2 SðQÞ structure factor dominate the dynamics in glassy state (T < Tg ). This is consistent with the Debye like spectrum in Fig. 1. With increasing temperature the fast b-process emerges and we observe pronounced deviations from the Q2 SðQÞ law (Fig. 2(b)). Our next step is to extend the fit by introducing a term proportional to SðQÞ, describing rigid structural relaxation: SðQ; xÞ ¼ BSðQÞ þ AQ2 SðQÞ. In this model the phonon-like excitations contribute to the spectra at low energies rather as a background, which can be evaluated by extrapolating from the temperature dependence at higher energy transfers. This determines the weight factor A of the Q2 SðQÞ term and the weight factor B of the structural relaxation term SðQÞ remains the only free fit parameter. Fig. 3 shows the good agreement between the data and the proposed model and leads to the conclusion that the structural relaxation term dominates the dynamics in this temperature range, in particular in the domain of low Qs. Further crucial information on the picosecond process is provided by the Q-dependence of the effective Debye–Waller factor fq , measured by neutron spin-echo (NSE) [17]. Indeed, in the temperature range studied the a-process is more than
Fig. 3. Dynamic structure factor of CKN and the extended model, that includes a structural relaxation term proportional to SðQÞ. The vibrational term AQ2 SðQÞ was assumed to follow the same temperature dependence as the phonon-like excitations at higher frequencies (Fig. 1). The structural relaxation term dominates the dynamics, in particular at low Qs. The insert shows the static structure SðQÞ used in the fit (cf. [15]).
two orders of magnitude slower than the fast bprocess, thus the dynamics reflected by the effective Debye–Waller factor determined by NSE at times shorter than the a-relaxation time is dominated by the fast b-process. For small homogeneous mean square displacements hu2 i the Q-dependence of the Debye–Waller factor can be described as eð2 WÞ 1 Q2 hu2 i=3:
ð1Þ
In contrast, the observed behavior (Fig. 4) rather is a superposition of two components. The component following (1) manifests itself at large Q values, while
Fig. 2. Best free fits of the dynamic structure factor of CKN in (a) glassy and (b) liquid states by the approximation for phonon-like excitations. In the liquid state in the range of smaller Qs the model amounts to a fraction of the observed intensity only.
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primary data treatment and correction for multiple scattering we applied the approach described before to analyze the dynamic scattering function and compared it to the Q2 SðQÞ law. This approximate law also holds for incoherent scattering from phonon-like vibrations and with Sinc ðQÞ being a constant it transforms to the simple Q2 law. Our data show that deviations from the Q2 SðQÞ behavior are also observed here, which can be similarly connected to the structural relaxation (Fig. 5). 2.3. System with different degree of fragility, ZnCl2 Fig. 4. Q-dependence of the effective Debye–Waller factor measured by neutron spin-echo (NSE) in CKN. Deviations from the approximation for small homogeneous displacements eð2 WÞ 1 Q2 hu2 i=3 (for example the continuous line with ) indicates the coexistence of different types of ðhu2 iÞ1=2 ¼ 0:36 A asmotion. The dashed line corresponds to ðhu2 iÞ1=2 ¼ 0:2 A suming that 1ß% of the atoms participate in heterogeneous . (In this latter large amplitude motion with ðhu2 iÞ1=2 > 1:5 A case the behavior at smaller Qs depends on geometrical details of structure and motion.)
the low Q-data clearly show deviation from this law, indicating that the displacements cannot be characterized by a single mean square displacement hu2 i. 2.2. Incoherent neutron (NO3 )1:4 + 2–4% H2 O
scattering,
Ca0:6 K0:4 -
To complement the above results with information on self-correlations in the motion of atoms in the temperature and time range of the fast b-process we investigated the mixed system CKN + 2–4% H2 O that displays a strong fraction of incoherent signal, especially at the low Q range. The data have been taken at TOF-spectrometer NEAT in Berlin at three different temperatures (300, 392 and 424 K) below and above Tg 330 K. In order to correct for multiple scattering we made two sets of experiments, with two incoming neu and kI ¼ 8:5 A , retron wavelengths, kI ¼ 6 A spectively. A prominent feature of NEAT is a very flexible instrumental configuration and it allowed us to use the same elastic resolution value at both incoming neutron wavelengths. The spectra were corrected for background and detector efficiency and normalized to the Bose-factor. Above 330 K we observed a marked onset of the b-process. After
ZnCl2 samples were prepared according to the procedure described by Soltwitsch et al. [19] and were kindly provided by C. Dreyfuss. The ZnCl2 has been analyzed by differential calorimetric spectroscopy (DCS) and the glass transition temperature was found to be equal to 387 K and the melting point Tm to be equal to 588 K. ZnCl2 has a strong tendency to crystallize and neutron scattering experiments are not feasible between 410 and 570 K. We have measured neutron scattering spectra at temperatures on both sides of this range on spectrometer NEAT and corrected them for background, detector efficiency as well as the multiple scattering [18]. The spectra (Fig. 6) show harmonic temperature behavior at temperatures even higher then glass transition temperature (cf. 373 and 393 K)
Fig. 5. Dynamic structure factor of the mixed system CKN + 2% H2 O. One observes similar deviations from the Q2 SðQÞ law as in the case of the coherently scattering system CKN.
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Fig. 6. Neutron inelastic spectra in ZnCl2 . The strongly anharmonic increase of spectral intensity below 3 meV indicates the onset of the fast b-process. In order to compare the intensity of the fast b-process in ZnCl2 and in CKN we take the ratio m of quasi-elastic intensity in the x channel 0.40–0.50 meV to the elastic structure factor. The results show that the value of m in CKN is about three times larger than the value of m in ZnCl2 at the equal viscosity.
and exhibit a well-pronounced Boson peak. We applied the approach introduced by A. Sokolov to characterize the strength of the Boson peak in different system [16] and compare the intensity of the Boson peak maximum to the minimum at x ¼ 0:45 meV [18]. The results show that this ratio is 1.8 times larger in ZnCl2 than in CKN. At the higher temperatures we observed the first evidence for a fast b-process in ZnCl2 (Fig. 6) provided by neutron inelastic scattering. Most remarkable is that while vibrational processes, reflected in Boson peak, are more pronounced, the fast b-process in the ZnCl2 is relatively weak. The ratio between the intensity at x ¼ 0:45 meV and the elastic (x ¼ 0) structure factor is about three times larger in CKN than in ZnCl2 at corresponding intermediate range Q values (Q ¼ 0:8 1 for CKN and Q ¼ 1 A 1 for ZnCl2 , respecA tively) and equivalent temperatures defined by approximately equal values of the viscosity.
3. Discussion and conclusions Our analysis of coherent and incoherent quasielastic neutron scattering data (Figs. 2, 3, and 5) leads to the conclusion that the fast b-process is
due to a first, fast step in the structural relaxation, as it was predicted by mode coupling theory. Since this relaxation term dominates the dynamics in the low Q range corresponding to distances, which are much longer than the distances between the nearest atomic neighbors, the physical nature of this relaxation process corresponds to cooperative, correlated motion of several atoms. One can also raise the question whether we have to do with large amplitude motion, where particles move in short time to distances comparable to their own size, or with collective small amplitude wobbling around equilibrium positions, as suggested in [11]. These two models can be distinguished by using incoherent neutron scattering. Large amplitude motion leads to deviations from the Q2 SðQÞ law (Fig. 5), as in case of coherent scattering, at Q values comparable to or larger then 1=r0 (where r0 is the amplitude of the motion), while for small amplitude motion the Q2 behavior holds up to Q values much larger than the inverse of the nearest neighbor distance. On this basis we can estimate that in the first . The second model r0 is in the range of 1–2 A model implies that the number of particles participating in the motion has to be very large [11] in order to cause such high intensities at relatively 1 ), as observed in Fig. 3. While small Qs (Q < 1 A this is difficult to justify in non-network glasses, such as CKN, in the network glass ZnCl2 we observe weaker fast b-process than in CKN. Further piece of information is delivered by the Q-dependence of the Debye–Waller factor, which reveals the coexistence of two types of motion, i.e., dynamic heterogeneity. One component is rather slow motion with small atomic displacements, which can be identified by a drop of type (1) at the high Q end in Fig. 4. Extrapolation to the vertical axis gives the approximate number of particles participating in this type of motion, which in this case ranges from 96% to 85%, depending on the temperature. The rest of atoms (4–15%) participate in a different type of motion characterized by a much larger amplitude hu2 i. The superposition of the two behaviors leads to deviations from (1) in the domain of small Qs. This particular range of Q indicates that we have to do with a cluster-like motion, with characteristic cluster size of 7–10 A at least, in view of the lowest Q we have observed.
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The possibility of such motion in less mobile system ZnCl2 is more limited, leading to the lower intensity of the b-process. To summarize, we conclude that the fast bprocess is a first step in the structural relaxation and it is not related to phonon-like excitations. It originates from the collective, correlated, largeamplitude motion of several atoms, kind of local fast flow. Fast flow appears with the softening of the structure, as the amount of free volume becomes sufficient to allow particles to move rapidly. When one atom leaves its position another will take its place and it comes to fast, cluster-like flow. Only some fraction of particles participates in such motion at a given time, the others stay around slowly flowing equilibrium positions waiting for their ‘‘turn’’ to move. After some period of time exchange of free volume between neighboring clusters takes place and another cluster will be able to move. This dynamic heterogeneity emerges spontaneously, it is not coupled to static structural inhomogeneities, such as the ‘‘islands of mobility’’ first proposed by Johari and Goldstein. By probing the disordered system at the longer time scale of the a-process, essentially all atoms will have participated in the fast b-process, but not at the same time. The measured dynamic structure will be a vast average over a quasi-infinite number of dynamic configurations, and the dynamic heterogeneity at the picosecond time scale will be averaged out by the much longer observation time. On the other hand, deviations from the Q ! 0 asymptotic behavior in both the dynamic structure factor and the Debye–Waller factor are clear signatures of heterogenous dynamics in the neutron scattering observations on the relevant time scale.
Acknowledgements We thank Catherine Dreyfuss for her great help with preparation of the ZnCl2 sample and R.
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Lechner, A. Desmedt and J. Pieper for their help in the neutron scattering experiments. References [1] G. Addams, J.H. Gibbs, J. Chem. Phys. 43 (1969) 139. [2] R. Colby, Phys. Rev. E 61 (2000) 1783. [3] C. Donati, J.F. Douglas, W. Kob, S.J. Plimpton, P.H. Poole, S.C. Glotzer, Phys. Rev. Lett. 80 (1998) 2338. [4] G.P. Johari, M. Glodstein, J. Chem. Phys. 53 (1970) 2372. [5] W. G€ otze, Z. Phys. B 60 (1985) 195; W. G€ otze, W. Sj€ ogren, Rep. Progr. Phys. 55 (1992) 242. [6] A. Arbe, J. Colmenero, M. Monkenbusch, D. Richter, Phys. Rev. Lett. 81 (1998) 590. [7] H. Sillescu, J. Phys. Condens. Matt. J. Non-Crystal. Solids 307 (2002) 16, and references therein. [8] R. Bohmer, G. Hinze, T. Jorg, F. Oi, H. Sillescu, J. Phys. Condens. Matt. 12 (2000) A383. [9] A. Doss, M. Paluch, H. Sillescu, G. Hinze, J. Chem. Phys. 117 (2002) 6582. [10] F. Mezei, W. Knaak, B. Farago, Phys. Rev. Lett. 58 (1987) 571. [11] W. Schmidt, M. Ohl, U. Buchenau, Phys. Rev. Lett. 85 (2000) 5669; U. Buchenau, A. Wischnewski, M. Monkenbusch, W. Schmidt, Philos. Mag. B 79 (1999) 2021. [12] S. Kojima, V. Novikov, Phys. Rev. B 54 (1996) 222; A. Sokolov et al., Philos. Mag. B 77 (1998) 349. [13] J.M. Carpenter, C.A. Pelizzari, Phys. Rev. B 12 (1975) 2391. [14] M. Russina, F. Mezei, AIP Proc. 479 (1998) 47–54. [15] M. Russina, F. Mezei, R. Lechner, S. Longeville, B. Urban, Phys. Rev. Lett. 84 (2000) 3630. [16] A.P. Sokolov et al., Phys. Rev. Lett. 71 (1993) 2062. [17] F. Mezei, M. Russina, J. Condens. Matt. 11 (1999) A341. [18] O. Russina, M. Russina, F. Mezei, Appl. Phys. A 74 (2002) 1192. [19] M. Soltwisch, J. Sukmanowski, Q. Quitmann, J. Chem. Phys. 86 (1987) 3207. [20] J.D. Mackenzie, W.K. Murphy, J. Chem. Phys. 33 (1960) 366; A.J. Easteal, C.A. Angell, J. Chem. Phys. 56 (1972) 4231. [21] J.A.E. Desa, A.C. Wright, J. Wong, R.N. Sinclair, J. Non Cryst. Solids 51 (1982) 57. [22] C.A. Angell, J. Phys. Chem. Sol. 49 (1988) 863.
Heterogeneous large amplitude atomic motion in supercooled liquids Margarita Russina a,*, Olga Russina b, Ferenc Mezei a,b b
a Los Alamos National Laboratory, MS H805, Los Alamos, NM 87545, USA Hahn-Meitner-Institute Berlin GmbH, Glienicker Str.100, Berlin 14109, Germany
Received 22 April 2003; in final form 21 May 2003
Abstract One of the central questions in glass physics is the dynamic nature of the glass transition. Essential issues are the type of atomic motions involved and their homogeneity or heterogeneity. Previous experimental studies of the dynamic heterogeneity performed by special NMR techniques for times >106 s and by incoherent neutron scattering were restricted to the a-relaxation process. Here we review the results of neutron scattering studies focused on the picosecond time domain showing that fast b-process corresponds to large amplitude cluster like heterogeneous motion. Ó 2003 Published by Elsevier Science B.V.
1. Introduction The concept of heterogeneity in the dynamics around the glass transition was first introduced in 1969 by Addams and Gibbs [1], who suggested the existence of cooperative, cluster-like motion. The idea has been strongly supported by recent theoretical and molecular dynamics developments. The theoretical approach of Colby [2] uses the idea of free volume in a simple mean-field model to point out the existence of this kind of heterogeneous motion. In the glassy state there is no free volume of a sufficient size and the particles are trapped in the ‘‘cage’’ of surrounding particles. On microscopic time scales the atoms vibrate in the cages
*
Corresponding author. Fax: +1-505-665-2676. E-mail address: [email protected] (M. Russina).
around quasi-equilibrium positions. With increasing temperature the free volume grows and opens up the way for rapid cooperative rearrangement of groups of particles in the form of large amplitude motion (i.e., atoms moving to distances comparable to the separation between nearest neighbors). However, only a fraction of atoms in the sample can move at a given time, the rest has to wait for free volume to become available. Hence the heterogeneity in the dynamics: a small fraction of particles is mobile on a short time scale, while the rest follows conventional slow viscous flow. This picture is in agreement with molecular dynamics simulation studies [3] revealing the existence of correlated atomic motion in form of loops or chains (‘‘string’’-like motion) observed by identifying those atoms that moved to large distances compared to the average within a given time. In contrast to the Johari–Goldstein hypothesis [4],
0301-0104/03/$ - see front matter Ó 2003 Published by Elsevier Science B.V. doi:10.1016/S0301-0104(03)00256-8
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M. Russina et al. / Chemical Physics 292 (2003) 325–331
this proposed mechanism of heterogeneity in the dynamics is rather a spontaneous one, it does not require the existence of structural heterogeneity or ‘‘islands of mobility’’. Heterogeneous as well as homogeneous (e.g., [5,6]) scenarios have been intensively explored in the last decade on a broad time scale for the different relaxation processes observed around the glass transition. The current picture of relaxation behavior of supercooled liquids is complex. Mode coupling theory [5] introduced slow a- and fast b-processes, whose existence has been confirmed experimentally in almost all known glass forming liquids. The fast b-process takes place on the picosecond time scale at all temperatures where it is observable, while the a-relaxation moves from macroscopic times to picoseconds with increasing temperature. In addition in some systems one finds the slow Johari–Goldstein b-process on a strongly temperature dependent time scale, which is shorter but close to that of the a-relaxation. Previous experimental studies related to dynamic heterogeneity focused on the a- or the slow (Johari–Goldsdtein) b-processes. The question whether homogeneous and heterogeneous scenarios can be distinguished by the momentum transfer dependence of the a-relaxation time has been investigated by incoherent neutron scattering. The authors in [6] conclude that the stretching of the a-relaxation is related to homogenous sublinear diffusion. In NMR studies [7,8] the heterogeneity of a-relaxation is probed by selecting a dynamically distinguishable ensemble and monitoring its return to the equilibrium. The conclusion in these works is that above the glass transition temperature one observes heterogeneities with life times of order of a-relaxation time. It is important to mention that these NMR techniques can only explore macroscopic times scales (ls or longer). The slow b-process was also investigated by NMR [8,9] and dielectric spectroscopy experiments. Although both studies found that all molecules contribute to the process in similar fashion, authors in [8] believe that the non-exponential 2 H spin–lattice relaxation observed at low temperatures (T < 1:1Tg ) and at times t < sa is due to the heterogeneity of the b-processes. However, no ev-
idence of distinguishable ‘‘islands of mobility’’ has been obtained. There is less information available on the fast bprocess, although it may play a most crucial role in the physics of glasses. It is most remarkable, that the fast b-process has been first predicted by the mode coupling theory of the glass transition and discovered later experimentally [10] more than a decade ago. Since then the fast b-process was found to be a universal feature of glassy systems. In contrary to the processes discussed to this point, the fast bprocess cannot be probed by the NMR techniques mentioned above since its time scale varies little with temperature and remains in the picosecond range. The various explanations and theories proposed for the nature of the fast b-process can be roughly divided into two groups. The first group of authors believes that the fast b-process is related to vibrational motion [11,12], while the second group [5] suggests that it is caused by structural relaxation. The momentum transfer Q-dependence of the dynamic structure factor SðQ; xÞ can provide the key to the answer. It has been shown in [13] that the dynamic structure factor of the phonon-like excitations SðQ; xÞ shall be proportional to Q2 SðQÞ, where SðQÞ is the static structure factor. In the case of rigid structural relaxation we expect the approximate law SðQ; xÞ / SðQÞ. The present paper focuses on work aimed at answering the questions about the nature and homogeneity/heterogeneity of the fast b-process by probing glassy systems on the picosecond time scale. In what follows we will review and discuss coherent and incoherent neutron scattering data, both previously published and new ones.
2. Experimental results Though the nature of fast b-process has been intensively investigated for quite some time, it remained a subject of controversy. In neutron scattering work this was largely due to experimental difficulties, mainly to the multiple scattering noise. Multiple scattering (MS) means that a neutron can be scattered more than one times while it is inside the sample, and this can lead to spurious noise masking small signals. In order to overcome the
M. Russina et al. / Chemical Physics 292 (2003) 325–331
problem of MS we developed a new method for multiple scattering correction (see details in [14]), which is based on the fact that we can vary the probability of multiple scattering in the same sample by changing the experimental conditions. This method allows us to establish very precisely the single scattering in a broad momentum transfer Q range, especially in the region of low Qs where the single scattering signal is very small. The importance of the low Q domain is that this range corresponds to the length scale of several times the distance between the nearest atomic neighbors. Probing the dynamic structure factor at this length scale gives us the possibility to follow and analyze the collective aspects of the atomic motion. We applied this method to study of the microscopic dynamic of three inorganic glass formingsystems, Ca0:6 K0:4 (NO3 )1:4 (cf. [15]) and most recently Ca0:6 K0:4 (NO3 )1:4 + H2 O and ZnCl2 . Each of these glasses represents an ionic molten salt, however there important differences. Ca0:6 K0:4 (NO3 )1:4 is a typical example of so-called fragile systems. The definition of fragility [22] describes the deviation of the variation of the viscosity as a function of temperature from the Arrhenius behavior: the larger is the deviation, the more fragile is the system. Ca0:6 K0:4 (NO3 )1:4 , shortly called CKN, is a coherent neutron scatterer and provides information about the pair-correlation function, i.e., in particular about the cooperative atomic motion. Ca0:6 K0:4 (NO3 )1:4 + 2–4 at.% H2 O is very similar to the CKN system, with the glass transition temperature slightly (3–5 K) lower that of CKN (Tg ffi 333 K). However, due to the introduction of H atoms this system displays a strong contribution of incoherent neutron scattering and provides information about the self-correlations in the motion of atoms. ZnCl2 exhibits different temperature dependence of viscosity and belongs to the category of the intermediate glass forming systems between fragile and strong [20]. In contrast to the loose ionic structure of CKN, ZnCl2 consists of closely packed Zn–Cl tetrahedra, where Cl anions shield Zn2þ cations and restrict their mobility, leading to a more networked structure [21]. The goal of this particular part of the study is to analyze how the microscopic dynamics changes with more pronounced atomic structure and lesser degree of fragility.
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2.1. Coherent neutron scattering, Ca0:6 K0:4 (NO3 )1:4 Fig. 1 from [15] shows the coherent neutron scattering spectra of the ionic glass forming model system Ca0:6 K0:4 (NO3 )1:4 . All data were normalized to the Bose factor of harmonic temperature dependence of phonon-like excitations as discussed in detail in [15]. The slight shoulder positioned around 4 meV indicates the so-called Boson peak, previously observed in CKN in the light scattering spectra [16]. Below the glass transition temperature the spectra are harmonic and well approximated at low energies by a constant (Debye phonon spectrum). With increasing temperature we observe the onset of anharmonicity at T > Tg , where Tg ffi 333 K is the temperature of the calorimetric glass transition. It is important to note that the temperature dependence of the spectral intensity at the range of small energy transfers x < 2 meV, corresponding to the picosecond time scale, is considerably stronger than the temperature dependence of the intensity of the phonon-like excitations at higher energy transfer values, namely >4 meV. Consequently, the spectra can be presented as a sum of a slightly anharmonic vibrational part and a rapidly varying low energy transfer part (below 4 meV), which can be identified with the fast b-pro-
Fig. 1. Temperature dependence of the coherent neutron scattering spectra in CKN. In the glassy states (T < Tg ¼ 333 K) the spectra are harmonic, while the shape drastically changes with the increase of the temperature at T > Tg . The temperature evolution of the intensity shows clear differences at different energies. In the range of the b-process at x < 2 meV the anharmonic increase of intensity is much stronger than for the phonon-like excitations at higher energies. Figure from [15].
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cess [5] and is actually well described by the predicted power law line shape below 3 meV. We analyzed the dynamic structure factor as a function of Q at different temperatures at constant energy transfers. Fig. 2(a) demonstrates that the phonon-like excitations with the expected Q2 SðQÞ structure factor dominate the dynamics in glassy state (T < Tg ). This is consistent with the Debye like spectrum in Fig. 1. With increasing temperature the fast b-process emerges and we observe pronounced deviations from the Q2 SðQÞ law (Fig. 2(b)). Our next step is to extend the fit by introducing a term proportional to SðQÞ, describing rigid structural relaxation: SðQ; xÞ ¼ BSðQÞ þ AQ2 SðQÞ. In this model the phonon-like excitations contribute to the spectra at low energies rather as a background, which can be evaluated by extrapolating from the temperature dependence at higher energy transfers. This determines the weight factor A of the Q2 SðQÞ term and the weight factor B of the structural relaxation term SðQÞ remains the only free fit parameter. Fig. 3 shows the good agreement between the data and the proposed model and leads to the conclusion that the structural relaxation term dominates the dynamics in this temperature range, in particular in the domain of low Qs. Further crucial information on the picosecond process is provided by the Q-dependence of the effective Debye–Waller factor fq , measured by neutron spin-echo (NSE) [17]. Indeed, in the temperature range studied the a-process is more than
Fig. 3. Dynamic structure factor of CKN and the extended model, that includes a structural relaxation term proportional to SðQÞ. The vibrational term AQ2 SðQÞ was assumed to follow the same temperature dependence as the phonon-like excitations at higher frequencies (Fig. 1). The structural relaxation term dominates the dynamics, in particular at low Qs. The insert shows the static structure SðQÞ used in the fit (cf. [15]).
two orders of magnitude slower than the fast bprocess, thus the dynamics reflected by the effective Debye–Waller factor determined by NSE at times shorter than the a-relaxation time is dominated by the fast b-process. For small homogeneous mean square displacements hu2 i the Q-dependence of the Debye–Waller factor can be described as eð2 WÞ 1 Q2 hu2 i=3:
ð1Þ
In contrast, the observed behavior (Fig. 4) rather is a superposition of two components. The component following (1) manifests itself at large Q values, while
Fig. 2. Best free fits of the dynamic structure factor of CKN in (a) glassy and (b) liquid states by the approximation for phonon-like excitations. In the liquid state in the range of smaller Qs the model amounts to a fraction of the observed intensity only.
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primary data treatment and correction for multiple scattering we applied the approach described before to analyze the dynamic scattering function and compared it to the Q2 SðQÞ law. This approximate law also holds for incoherent scattering from phonon-like vibrations and with Sinc ðQÞ being a constant it transforms to the simple Q2 law. Our data show that deviations from the Q2 SðQÞ behavior are also observed here, which can be similarly connected to the structural relaxation (Fig. 5). 2.3. System with different degree of fragility, ZnCl2 Fig. 4. Q-dependence of the effective Debye–Waller factor measured by neutron spin-echo (NSE) in CKN. Deviations from the approximation for small homogeneous displacements eð2 WÞ 1 Q2 hu2 i=3 (for example the continuous line with ) indicates the coexistence of different types of ðhu2 iÞ1=2 ¼ 0:36 A asmotion. The dashed line corresponds to ðhu2 iÞ1=2 ¼ 0:2 A suming that 1ß% of the atoms participate in heterogeneous . (In this latter large amplitude motion with ðhu2 iÞ1=2 > 1:5 A case the behavior at smaller Qs depends on geometrical details of structure and motion.)
the low Q-data clearly show deviation from this law, indicating that the displacements cannot be characterized by a single mean square displacement hu2 i. 2.2. Incoherent neutron (NO3 )1:4 + 2–4% H2 O
scattering,
Ca0:6 K0:4 -
To complement the above results with information on self-correlations in the motion of atoms in the temperature and time range of the fast b-process we investigated the mixed system CKN + 2–4% H2 O that displays a strong fraction of incoherent signal, especially at the low Q range. The data have been taken at TOF-spectrometer NEAT in Berlin at three different temperatures (300, 392 and 424 K) below and above Tg 330 K. In order to correct for multiple scattering we made two sets of experiments, with two incoming neu and kI ¼ 8:5 A , retron wavelengths, kI ¼ 6 A spectively. A prominent feature of NEAT is a very flexible instrumental configuration and it allowed us to use the same elastic resolution value at both incoming neutron wavelengths. The spectra were corrected for background and detector efficiency and normalized to the Bose-factor. Above 330 K we observed a marked onset of the b-process. After
ZnCl2 samples were prepared according to the procedure described by Soltwitsch et al. [19] and were kindly provided by C. Dreyfuss. The ZnCl2 has been analyzed by differential calorimetric spectroscopy (DCS) and the glass transition temperature was found to be equal to 387 K and the melting point Tm to be equal to 588 K. ZnCl2 has a strong tendency to crystallize and neutron scattering experiments are not feasible between 410 and 570 K. We have measured neutron scattering spectra at temperatures on both sides of this range on spectrometer NEAT and corrected them for background, detector efficiency as well as the multiple scattering [18]. The spectra (Fig. 6) show harmonic temperature behavior at temperatures even higher then glass transition temperature (cf. 373 and 393 K)
Fig. 5. Dynamic structure factor of the mixed system CKN + 2% H2 O. One observes similar deviations from the Q2 SðQÞ law as in the case of the coherently scattering system CKN.
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Fig. 6. Neutron inelastic spectra in ZnCl2 . The strongly anharmonic increase of spectral intensity below 3 meV indicates the onset of the fast b-process. In order to compare the intensity of the fast b-process in ZnCl2 and in CKN we take the ratio m of quasi-elastic intensity in the x channel 0.40–0.50 meV to the elastic structure factor. The results show that the value of m in CKN is about three times larger than the value of m in ZnCl2 at the equal viscosity.
and exhibit a well-pronounced Boson peak. We applied the approach introduced by A. Sokolov to characterize the strength of the Boson peak in different system [16] and compare the intensity of the Boson peak maximum to the minimum at x ¼ 0:45 meV [18]. The results show that this ratio is 1.8 times larger in ZnCl2 than in CKN. At the higher temperatures we observed the first evidence for a fast b-process in ZnCl2 (Fig. 6) provided by neutron inelastic scattering. Most remarkable is that while vibrational processes, reflected in Boson peak, are more pronounced, the fast b-process in the ZnCl2 is relatively weak. The ratio between the intensity at x ¼ 0:45 meV and the elastic (x ¼ 0) structure factor is about three times larger in CKN than in ZnCl2 at corresponding intermediate range Q values (Q ¼ 0:8 1 for CKN and Q ¼ 1 A 1 for ZnCl2 , respecA tively) and equivalent temperatures defined by approximately equal values of the viscosity.
3. Discussion and conclusions Our analysis of coherent and incoherent quasielastic neutron scattering data (Figs. 2, 3, and 5) leads to the conclusion that the fast b-process is
due to a first, fast step in the structural relaxation, as it was predicted by mode coupling theory. Since this relaxation term dominates the dynamics in the low Q range corresponding to distances, which are much longer than the distances between the nearest atomic neighbors, the physical nature of this relaxation process corresponds to cooperative, correlated motion of several atoms. One can also raise the question whether we have to do with large amplitude motion, where particles move in short time to distances comparable to their own size, or with collective small amplitude wobbling around equilibrium positions, as suggested in [11]. These two models can be distinguished by using incoherent neutron scattering. Large amplitude motion leads to deviations from the Q2 SðQÞ law (Fig. 5), as in case of coherent scattering, at Q values comparable to or larger then 1=r0 (where r0 is the amplitude of the motion), while for small amplitude motion the Q2 behavior holds up to Q values much larger than the inverse of the nearest neighbor distance. On this basis we can estimate that in the first . The second model r0 is in the range of 1–2 A model implies that the number of particles participating in the motion has to be very large [11] in order to cause such high intensities at relatively 1 ), as observed in Fig. 3. While small Qs (Q < 1 A this is difficult to justify in non-network glasses, such as CKN, in the network glass ZnCl2 we observe weaker fast b-process than in CKN. Further piece of information is delivered by the Q-dependence of the Debye–Waller factor, which reveals the coexistence of two types of motion, i.e., dynamic heterogeneity. One component is rather slow motion with small atomic displacements, which can be identified by a drop of type (1) at the high Q end in Fig. 4. Extrapolation to the vertical axis gives the approximate number of particles participating in this type of motion, which in this case ranges from 96% to 85%, depending on the temperature. The rest of atoms (4–15%) participate in a different type of motion characterized by a much larger amplitude hu2 i. The superposition of the two behaviors leads to deviations from (1) in the domain of small Qs. This particular range of Q indicates that we have to do with a cluster-like motion, with characteristic cluster size of 7–10 A at least, in view of the lowest Q we have observed.
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The possibility of such motion in less mobile system ZnCl2 is more limited, leading to the lower intensity of the b-process. To summarize, we conclude that the fast bprocess is a first step in the structural relaxation and it is not related to phonon-like excitations. It originates from the collective, correlated, largeamplitude motion of several atoms, kind of local fast flow. Fast flow appears with the softening of the structure, as the amount of free volume becomes sufficient to allow particles to move rapidly. When one atom leaves its position another will take its place and it comes to fast, cluster-like flow. Only some fraction of particles participates in such motion at a given time, the others stay around slowly flowing equilibrium positions waiting for their ‘‘turn’’ to move. After some period of time exchange of free volume between neighboring clusters takes place and another cluster will be able to move. This dynamic heterogeneity emerges spontaneously, it is not coupled to static structural inhomogeneities, such as the ‘‘islands of mobility’’ first proposed by Johari and Goldstein. By probing the disordered system at the longer time scale of the a-process, essentially all atoms will have participated in the fast b-process, but not at the same time. The measured dynamic structure will be a vast average over a quasi-infinite number of dynamic configurations, and the dynamic heterogeneity at the picosecond time scale will be averaged out by the much longer observation time. On the other hand, deviations from the Q ! 0 asymptotic behavior in both the dynamic structure factor and the Debye–Waller factor are clear signatures of heterogenous dynamics in the neutron scattering observations on the relevant time scale.
Acknowledgements We thank Catherine Dreyfuss for her great help with preparation of the ZnCl2 sample and R.
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