Reversible atomic processes as basic mechanisms of the glass transition
- Feng Ye*,†,‡,
- Wolfgang Sprengel§,
- Rainer K. Wunderlich¶,
- Hans-Jörg Fecht¶,‖, and
- Hans-Eckhardt Schaefer*
- *Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany;
- †State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, People's Republic of China;
- §Institut für Materialphysik, Technische Universität Graz, Petersgasse 16, A-8010 Graz, Austria;
- ¶Materials Division, Universität Ulm, Albert-Einstein-Allee 47, 89081 Ulm, Germany; and
- ‖Institut für Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany
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Communicated by Manuel Cardona, Max Planck Institute for Solid State Research, Stuttgart, Germany, June 7, 2007 (received for review April 4, 2007)
Abstract
Reversible formation and disappearance of vacant spaces (vacancy-type defects) in bulk Zr57Cu15.4Ni12.6Nb5Al10 glass are directly evidenced by high-resolution, time-differential dilatometry studies. The vacancy kinetics are strongly temperature-dependent, with an effective migration enthalpy of H V M = 3.34 eV. This may explain the strong temperature dependence of glass properties such as viscosity. The results presented here are of general importance for understanding amorphous condensed matter and biomaterials and for the technical development of amorphous steels.
Understanding the formation of glasses by viscous slowdown with decreasing temperature is currently seen as a major intellectual challenge in condensed-matter physics (1–4). Although substantial progress has been made in deciphering the short- and intermediate-range structure of metallic glasses (5–7), the atomic-scale dynamics of their strongly temperature-dependent properties are far less well understood. Thus, specific studies of atomic processes in bulk metallic glasses (BMGs) (8–10) are of particular interest. A pivotal question has been whether the properties of these glasses differ at temperatures above or below the so-called “glass transition temperature,” T g, which is usually deduced from the endothermic overshooting peak in the specific heat, Δc p (11–13). In BMGs (8–10), the viscosity, η (see ref. 9) and atomic diffusivity, D, after long-term annealing at lower temperatures (11, 14) vary continuously at T g, so that no change of mechanism at T g can be detected. Yet, η and D change strongly (9, 11), by 2 to 3 orders of magnitude, in the narrow temperature window of T g ± 20 K. This raises a question regarding the origin of the strong temperature dependence that gives rise to the observation that glasses basically do not flow below T g. Specific studies of atomic processes in BMGs, using time-differential dilatometry as reported here, may help to answer this question.
Atomic processes in crystals in which atoms jump between neighboring lattice sites are mainly controlled by the formation and migration of thermal vacancies. Thermal formation of vacancies in crystals can be specifically identified by studying the three characteristic signatures:
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Upon vacancy formation, the specimen volume expands because of the vacancy formation volume V V F, whereas the volume reversibly shrinks when the vacancy disappears.
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The vacancy formation process is time-dependent because of the diffusion-controlled filling of the specimen volume, after a fast temperature change, by vacancies formed at vacancy sources.
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Vacancies in crystals show the size, Ω, of one missing atom. In some cases, the size may be smaller (Ω/2) or larger (e.g., 2Ω for divacancies).
Time-dependent volume expansion due to thermal vacancy formation has been demonstrated, together with positron lifetime studies
for the local investigation of vacancies, in the case of crystalline intermetallic alloys (15, 16). The kinetics of vacancy formation measured after a fast temperature change from T
i to T
f by the reversible isothermal length changes
can, in the simple case of a regular array of linear vacancy sources or sinks [e.g., dislocations in a crystal (17)], be described by an exponential with the relative equilibrium length change Δl
0/l = 1/3C
V
V
V
F. Here, C
V is the difference of the vacancy equilibrium concentrations at T
i and T
f. The rate constant
with Boltzmann's constant k
B is governed by the vacancy migration enthalpy H
V
M and the preexponential factor
The preexponential factor 1/τ0 is mainly determined by the number density, N
D, of vacancy sources or sinks and the vacancy migration entropy, S
V
M. The mean square jump distance is denoted by x
2, v
0 is the jump attempt frequency, g ≈ 1 is a geometrical factor, and r
0 ≈ 0.3 nm is the source radius within which the vacancy equilibrium concentration can be attained at all times (17).
Results and Discussion
In the present dilatometry studies of bulk Zr57Cu15.4Ni12.6Nb5Al10 glass specimens (Fig. 1), signatures 1 and 2 of thermal vacancy formation are verified by examining the time-dependent expansion and shrinkage (see Fig. 2 Inset). The size of the vacancies is estimated to ≈Ω/2, as discussed below. From the temperature dependence of the rate constants 1/τ (T f) measured after heating or cooling (see Fig. 2), a vacancy migration enthalpy H V M = 3.34 eV and 1/τ0 = 5.25 × 1021 s−1 for bulk Zr57Cu15.4Ni12.6Nb5Al10 glass are derived according to Eqs. 1 and 2.
Sketch of the time-differential dilatometry setup. A BMG rod with a diameter of 4 mm and a length of 40 mm was spark-cut into a particular shape, as shown. Two parallel mirror surfaces separated by a distance of 20 mm were carefully polished. The change in specimen length between the mirror surfaces could be measured by a two-beam Michelson laser interferometer (SP120D; SIOS Messtechnik, Illmenau, Germany) to an accuracy of ≈20 nm. The specimen temperature, measured by three chromel–alumel thermocouples welded on the specimen, was stable within 0.1 K, with a gradient <20 K and a temperature setting time of 200 s after a change of temperature.
Temperature variation of the time constant of the length change of Zr57Cu15.4Ni12.6Nb5Al10 BMG after fast temperature changes, as derived from time-differential dilatometry (Inset). For measuring the isothermal time-dependent length change due to vacancy formation (heating) or disappearance (cooling), the temperature is changed rapidly from an initial temperature, T i, to a final temperature, T f. The changes in isothermal time-dependent length (see Inset with reversible isotherms after heating and cooling; note the different scales) are described by an exponential, according to Eq. 1. The temperature variation of the rate constant τ−1 yields a high vacancy migration enthalpy, H V M = 3.34 eV, and a high preexponential factor, 1/τ0 = 5.25 × 1021 s−1.
The formation of vacancy-sized species in BMGs similar to lattice vacancy formation can be readily envisaged by considering the strong short- and intermediate-range order in amorphous metals (5). Information on the size and chemical surroundings of the vacancies may be obtained from local probe experiments by positron annihilation (see below).
A high concentration, C V ≈ 6 × 10−3, of thermal vacancies in the BMG, which indicates a low vacancy formation enthalpy H V F, is deduced from the expansion amplitude Δl/l 0 ≥ 10−3 at 653 K (Fig. 2 Inset), which is well below the glass transition temperature T g = 672 K if a vacancy formation volume of one-half of a mean atomic volume (18, 19) is assumed. The appearance of different types of vacancies is likely in the present five-component BMG but is not taken into account in the kinetics of Eq. 1. This effect may, however, play a role in the full understanding of the temperature dependences of viscosity (9) or Fe diffusivity (11) in BMGs.
The dilatometry effects are fully reversible on the present time scale, as shown by repeated heating and cooling and by the time constants after heating or cooling (e.g., to 642 K). Indications of a reversible behavior of BMGs have been reported previously (11, 19).
The most striking result of the present investigation is the high value of the vacancy migration enthalpy H V M. High H V M values were found in crystalline (15) and quasicrystalline (20) intermetallic compounds and were confirmed by ab initio theory (21); the high values were due to high barriers for opening windows for atomic jumps.
A high vacancy migration enthalpy has important consequences for the glass transition at T g deduced from the endothermal overshooting in differential scanning calorimetry measurements (see, for example, ref. 11), which has been discussed in terms of the kinetic retardation of the formation of free volumes (12). When we now consider the thermal formation of vacancies detected here, the c p overshooting peak upon heating arises from a retarded filling of the specimen volume with thermal vacancies, due to low mobility (high H V M) and high concentration (low H V F).
Therefore, the glass transition phenomenon is basically attributable to the high vacancy migration enthalpy H V M giving rise to a steep temperature change of the vacancy mobility by orders of magnitude within a narrow temperature range (see Fig. 2). An even higher H V M value is obtained when the kinetics are analyzed in terms of a higher reaction order, namely 2 instead of the reaction order 1 used in Eq. 1. The high H V M value is the reason for a steep but continuous temperature variation of, for example, the Fe diffusivity (11) and viscosity (9) in BMGs, which resembles a phase transition. A high preexponential factor, 1/τ0, has already been reported from earlier annealing studies of amorphous metals (13).
In the present Zr57Cu15.4Ni12.6Nb5Al10 BMG, structural vacancies with a size of ≈Ω/2 and an atomic concentration of >10−4 have been detected, as deduced from the positron lifetimes in BMGs lying between the values of lattice vacancies and defect-free lattices (Fig. 3). Thermal formation of vacancies cannot be detected by positrons, as concluded from the time-independent positron lifetime and Doppler-broadening W parameter after a temperature jump from 640 to 593 K (Fig. 4). However, multivacancy clusters as a source of the length change can be excluded because this would give rise to longer positron lifetimes. If the vacancies were smaller than Ω/2, then C v would increase but the vacancy nature would be retained because of specimen expansion and the time-dependent equilibration.
Positron lifetimes (27) measured at ambient temperature for BMGs, pure metals, the quasicrystal i-Al70.2Pd21.3Mn8.5, and intermetallic compounds vs. the mean valence electron density, ρel. For pure metals and intermetallic compounds both, the positron lifetimes in the free delocalized and vacancy-trapped states are depicted. For the BMGs, the positron lifetimes 175 ps of Zr57Cu15.4Ni12.6Nb5Al10 (vit106) (this article), 176 ps of Zr41.25Ti13.75Cu12.5Ni10Be22.5 (vit1) (22), and 185 ps of Zr65Cu17.5Ni10Al7.5 (23) are located between the free lifetimes of delocalized positrons and the positron lifetimes in vacancies, indicating that the size of the structural vacancies in the BMG is slightly less than the size of a lattice vacancy.
Positron lifetime and coincident measurement of the Doppler-broadening W parameter of the positron–electron annihilation radiation of bulk amorphous Zr57Cu15.4Ni12.6Nb5Al10 at 593 K (after cooling from 640 K). During the measuring times, no significant changes of the positron lifetime or the W parameter are detected within experimental uncertainty, whereas the length change occurs with a time constant of τ = 3.7 × 106 s at 593 K (see Fig. 2). All data points have the same error bar of ±1.5 ps.
Finally, we would like to briefly reflect on how thermal vacancy formation and mode-coupling theory (24–26) are correlated. In the ideal mode-coupling theory, the increase in viscosity η and the slowing of the configurational atomic motions in a glass upon lowering of the temperature vs. the glass transition temperature T g are described by the time-dependent correlation function, with η diverging at the critical temperature T c > T g. However, in real systems, some residual motion of atoms is maintained below T c. This may simply be due to the thermal formation and migration of vacancies, as shown in this study. In this sense, thermal vacancy formation upon heating may be the initial step in a viscosity decrease, as a prerequisite for the flowing of glasses, and a fundamental precursor for configurational atomic motion. Without the availability of thermal vacancies, glasses stop flowing at low temperatures.
Materials and Methods
Sample Preparation.
The bulk Zr57Cu15.4Ni12.6Nb5Al10 glass samples were prepared by co-melting of the components and casting the melt into a Cu mold. The glass transition temperature T g = 672 K and the crystallization temperature T x = 773 K were determined by differential scanning calorimetry with a heating rate of 0.33 K/s. X-ray diffractograms of the Zr57Cu15.4Ni12.6Nb5Al10 BMG sample were taken with a Siemens (Erlangen, Germany) D500 diffractometer before and after the dilatometry experiments. The data show that the amorphous structure of the BMG specimen is unchanged, within the uncertainty limits, during the dilatometry studies (Fig. 5).
X-ray diffractogram taken with a Siemens D500 diffractometer of the Zr57Cu15.4Ni12.6Nb5Al10 BMG sample before and after the dilatometry experiments. The data show that the amorphous structure of the BMG specimen is unchanged within the uncertainty limits during the dilatometry studies. However, if we ascribe the experimental uncertainty ±Δ (2θ) of the intensity I (2θ) to a volume fraction f ≤ [Δ(2θ)/S 0 cos θ] × [0.9λ/d c]k p of crystallites with a size of d c = 10 nm, then we estimate f = 1.4%. Here, S 0 is a scale factor, 2θ is the peak position, λ is the x-ray wavelength, and k p is a constant dependent on the shape of the profile functions. The experimentally observed length change of ≈10−3 at 653 K (see Fig. 2 Inset), which is even larger at higher temperatures, can hardly be ascribed to this small crystalline fraction because, in this case, the crystallite would exhibit an unreasonably high volume change equivalent to an atomic vacancy concentration of ≈0.5.
Dilatometry.
It has been shown previously (15) for the case of intermetallic compounds that thermal vacancy formation in solids can be studied by isothermal time-differential dilatometry after a fast temperature change because the vacancy formation process gives rise to specimen expansion. This expansion can be separated from the conventional anharmonic thermal expansion that occurs instantaneously with the temperature change because it is delayed due to the diffusion time necessary for the vacancies to fill the sample volume after their formation at sources such as dislocations in crystals. From time-differential dilatometry data, the concentration and the equilibration kinetics of vacancies can be deduced (15). The expansion is measured interferometrically by a dilatometer employing two laser beams (see Fig. 1).
Positron Annihilation.
Positrons injected into a solid are annihilated with the electrons of the solid in characteristic states. The positron lifetime, which depends on the valence electron density, exceeds in lattice vacancies that in a defect-free lattice and is even longer in vacancy agglomerates. Therefore, the positron lifetime can be used to determine the size of a vacancy (27). By measuring the Doppler broadening of the two positron–electron annihilation quanta in coincidence, the core electron momenta of the atoms adjacent to the annihilation site can be measured. This technique provides a tool for an atomic-scale chemical analysis, e.g., in the vicinity of vacancies (28). In the present study, the radioactive isotope 58Co, diffused into the BMG sample, was used as a positron emitter.
Acknowledgments
We thank F. Sommer (Max Planck Institute for Metals Research, Stuttgart, Germany), E. Ma (Johns Hopkins University, Baltimore, MD), K. Lu (Institute of Metals Research, Shenyang, China), and Y. Shirai (Osaka University, Osaka, Japan) for fruitful discussions. This work was supported by The Max Planck Society, Munich, Germany; H. Dosch of The Max Planck Institute for Metals Research, Stuttgart, Germany; National Natural Science Foundation of China Grant 50501002; New Star Program for Science and Technology of Beijing City Grant 2005B19; and European Union Ductile BMG Composites Grant MRTN-CT-2003-504692.
Footnotes
- ‡To whom correspondence should be addressed. E-mail: yefeng{at}skl.ustb.edu.cn
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Author contributions: F.Y. and H.-E.S. designed research; F.Y., W.S., and R.K.W. performed research; F.Y., W.S., and H.-E.S. analyzed data; and F.Y., H.-J.F., and H.-E.S. wrote the paper.
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The authors declare no conflict of interest.
- Abbreviation:
- BMG,
- bulk metallic glass.
- © 2007 by The National Academy of Sciences of the USA
