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Relation between Low-Temperature Thermal Conductivity and the Specific Heat of Cesium Borate Glasses G. D’Angelo,*,† C. Crupi,‡ G. Tripodo,† and G. Salvato‡ Dipartimento di Fisica, UniVersita’ di Messina, Salita Sperone 31, I-98166 Messina, Italy, and Istituto per i Processi Chimico-Fisici del C. N. R., Sezione di Messina, Salita Sperone, I-98166 Messina, Italy ReceiVed: July 27, 2009; ReVised Manuscript ReceiVed: December 4, 2009
Low-temperature specific heat and thermal conductivity measurements have been performed on cesium borate glasses as a function of cesium oxide content. We have found experimental evidence of a concurrent growth of specific heat and thermal conductivity with increasing Cs+ content. This finding shows the existence of an uncommon relationship between the peak in Cp/T3 and the plateau in thermal conductivity in glasses and represents the most intriguing result for these alkaline borate glasses. The role of local modes associated with heavy cations on the vibrational dynamics in oxide glasses has been considered. Furthermore, a possible correlation between low-temperature thermal properties and the structure on the nanometer length scale of these glasses is put forward. I. Introduction The nature and the origin of vibrational excitations in the THz frequency range of amorphous materials continues to be a central issue in the field of condensed matter physics.1-7 The main peculiarity is represented by the so-called boson peak (BP), a mysterious broad hump that discloses the existence of an excess of vibrational density of states over the Debye level and is displayed in the inelastic Raman8,9 and neutron scattering spectra10 over the spectral region below 10 meV. The BP is considered to be responsible for a number of universal features observed in thermal properties. In particular, at temperatures above 1 K, an excess specific heat Cp over the Debye prediction is found and is revealed as a peak in a Cp/T3 plot.11,12 Moreover, thermal conductivity k(T) exhibits a more mysterious plateau at around 10 K, which testifies to a stopping of phonon propagation.13 Another relevant observation is that the BP appears in the energy range where acoustic phonons have a wavelength on the order of a few nanometers, calling into question the possibility that the structural heterogeneities, observed in glasses on nanometer lengths,14 are responsible for strong scattering of phonons. Many models competing on the origin of the BP have been proposed,1,3,4,15 but at present, there is no agreement of which microscopic mechanisms inherent in the disorder promote lowenergy vibrations. Recently, interest in this issue has been further stimulated owing to the findings that the boson peak would arise from transverse vibrational modes associated with soft structures in the glass.16,17 A further interesting question concerns the suggestion of a possible connection between the boson peak and a length scale at which the continuum elastic description breaks down.18 This length scale is marked by the existence of heterogeneities in local elastic coefficients, considered as an important ingredient of local disorder.3 New relevant insight * To whom correspondence should be addressed. E-mail: gdangelo@ unime.it. † Universita’ di Messina. ‡ Istituto per i Processi Chimico-Fisici del C. N. R.
on the nature of the boson peak has been recently gained by studying the changes of the boson peak following the hardening of the elastic medium by pressure,19 temperature,20 densification,21 or connectivity change.22 These investigations have shown that, in some cases, variations of the BP can be entirely accounted for by the corresponding changes of the Debye level and remark the strict relation between the excess of THz vibrations and the continuum elastic contribution. However, despite the great theoretical and experimental effort, many doubts remain about which type of disorder (structural or elastic, static, or dynamic) causes localization of network vibrations. It should be noted that most of the studies concern the boson peak, while the temperature dependence of the thermal conductivity is not as much studied, although it provides important information on phonon localization. Nevertheless, it is widely believed that a strict correlation exists between the two thermal properties since the plateau in k(T) occurs, for a given material, at roughly the same temperatures where the peak in Cp/T3 is observed.3,23-25 The most accepted interpretations suppose that the plateau in the thermal conductivity of amorphous materials arises from phonon localization due to the scattering of phonons by TLS (two-level systems)13 or to Rayleigh scattering by density fluctuations25 or to scattering by additional nonpropagating vibrational modes.3,6,26 In particular, the last theoretical approaches assume that additional quasilocal modes, which scatter resonantly acoustic phonons, limiting drastically their mean free path, are the same that give rise to the boson peak. In this view, the increasing magnitude of the BP reveals the existence of an increasing number of excess modes which lead to a higher number of scattering events for thermal phonons and hence to a decreasing plateau in thermal conductivity. The experimental confirmation of this expectation is considered as proof of the validity of localization models. In this connection, it is worth noting that studies concerning the dependence on pressure of thermal properties of several polymers have shown that an increase in pressure results in a
10.1021/jp907152y 2010 American Chemical Society Published on Web 02/01/2010
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TABLE 1: Parameters Obtained by Numerical Fits Using Equation 5 (shown in Figure 2) Together with the Mass Density G, Debye Temperature ΘD, Glass Transition Temperature Tg, Longitudinal Sound Velocity WL, Transverse Sound Velocity WT, and Mean Sound Velocity WD sample B 2 O3 (Cs2O)0.04(B2O3)0.96 (Cs2O)0.14(B2O3)0.86 (Cs2O)0.33(B2O3)0.67 a
F (kg/m3) a
1838 2062 2484 3297
VT (m/s) a
1872 1979 1914 1853
VL (m/s) a
3367 3571 3525 3553
VD (m/s) a
2086 2205 2141 2072
Tg (K) 526 545 605 642
ΘD (K) a
267 281 264 244
A (m-1 K-1)
B (10-3 K-2)
D (m-1 K-4)
lmin(Å)
53096 84099 73596 48746
16 1.95 2.42 6.02
2507 1470 1272 960
7.75 12.85 8.24 12.06
Reference 69.
decrease of the excess specific heat and in an increase of thermal conductivity.27 However, the origin of the plateau is still an open issue.6,24,28 Very recently, we studied the dependence of the boson peak on the cation size in a series of alkaline borate glasses by lowtemperature specific heat measurements, Raman, and neutron scattering.17 It has been observed that by adding to boron oxide alkaline cations having a decreasing size, the boson peak decreases in magnitude and shifts to higher energies. Binary alkali borate glasses have been studied for decades, and a clear dependence of different physical properties on the type of alkali modifier has been found. Namely, the viscosity,29 glass transition temperature,30 thermal expansion coefficient,31 sound velocity, and elastic constants32 decrease systematically as the alkali changes from Li to Cs. Moreover, when the composition is changed across a wide range, the elastic behavior of the lightest alkali borate glasses (Li, Na) differs markedly from that of the heaviest alkalies (Cs, Rb, K). In particular, the stiffness of lithium and sodium glasses increases with increasing alkali content, whereas glasses with heavier alkalies show a nonmonotonous trend with composition with a minimum and two local maxima. Moreover, the molar volume of light alkali borate glasses decreases in conjunction with the increase of rigidity. Again, a different behavior showing two minima and a local maximum is found for the composition dependence of the molar volume of heavier alkalies.33 It is believed that these differences have a structural origin lying in dissimilar intermediate range orders.34 In the present study, we investigate the low-temperature thermal properties of binary borate glasses with relation to their elastic properties. The final goal is to gain insight into microscopic mechanisms which give rise to low-energy vibrations in glasses by monitoring the influence of local vibrational modes of heavy metal ions on the vibrational behavior of an oxide glassy matrix. We have performed low -temperature specific heat and thermal conductivity measurements on (Cs2O)x(B2O3)1-x glasses containing different amounts, x, of cesium oxide. Herein, we show that in cesium borate glasses, the addition of an increasing amount of Cs2O enhances the excess of specific heat. This is an atypical result because in alkaline binary glasses, the inclusion of a light alkali leads to a strong reduction of specific heat.35-37 At the same time, thermal conductivity at temperatures above 2 K shows a strong increase with Cs2O, in accordance with the experimental evidence on sodium borate38 and silicate39 glasses. We explain this unusual thermal scenario by supposing that the low-frequency rattling motion of loosely bonded Cs+ ions dominates the low-temperature specific heat. The presence of these local excitations does not appear to influence thermal conductivity, which, on the contrary, is assumed to be ruled by the scattering of phonons by elastic fluctuations on the nanometer length scale that are intrinsic to the glassy structure.
We come to the conclusion that, in glasses, the boson peak and the plateau in k(T) do not necessarily stem from the same vibrational excitations. We speculate that the experimental study of the origin of the boson peak in glasses is more complex than is believed. In particular, the question on the localized or propagating nature of the boson peak must be followed with adequate attention in modified and multicomponent glasses, where vibrations of loosely bonded atoms or groups of atoms can have a relevant function in regulating the anomalous behaviors observed in the THz frequency region. II. Experimental Details (Cs2O)x(B2O3)1-x glasses, where 0 e x e 0.33, have been prepared from laboratory reagent 99.99% purity grades of boron oxide and cesium nitrate (Aldrich) following specific procedures in order to minimize the concentration of hydroxyl ions OHin the samples. The mixed powders have been melted in quantities of about 20 g by heating at 1000 °C for about 2 h in a platinum crucible within an electric furnace. After occasional stirring to ensure the homogeneity of the liquid, the melt has been cast into a preheated (500 °C) hollow steel mold, used to realize a parallelepiped sample with a cross-sectional area A of 2.2 × 2.4 mm2 and a height of 4-5 cm for thermal conductivity measurements. A disk with 0.1 mm thickness has been cut for the specific heat measurements. After casting, each glass has been annealed and stabilized at about 20-30 K above its calorimetric Tg in a high-purity nitrogen atmosphere in order to avoid undesired effects arising from different thermal histories. Analysis of the intensity of the MoK X-rays as a function of the diffraction angle has revealed only very broad bands, typical of glasses, and no sign of crystalline peaks. To avoid any possible contamination with moisture, the glasses have been stored in a darkened desiccator. Fourier transform IR analysis of the glasses has revealed no sign of OH- groups within experimental sensitivity; only concentrations of less than 0.1 mol % (corresponding to 1019 cm-3) are expected. The glass transition temperatures Tg of cesium and boron oxide samples have been determined by the thermograms measured by a Perkin-Elmer differential scanning calorimeter (DSC-Pyris). Measurements have been performed with a heating rate of 20 K/min. The specific heat has been measured by means of an automated calorimeter operating by the thermal relaxation method.40 Thermal conductivity measurements have been carried on by using a conductimeter operating with the steady-state heat flow method in the temperature range from 1.5 to 300 K.41 Precaution has been taken to limit the sample contamination by superficial water produced by moisture; samples have been handled in a drybox under a continuous flow of nitrogen. Densities have been measured at room temperature by a Micrometrics Accupyc 1300 gas pycnometer under helium gas, having an accuracy of 0.03%. Room-temperature velocities of
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Figure 1. The experimental low-temperature specific heat plotted as Cp/T3 of pure B2O3 (b), (Cs2O)0.04(B2O3)0.96 (4), (Cs2O)0.14(B2O3)0.86 (O), and (Cs2O)0.33(B2O3)0.67 (0) glasses, together with their corresponding Debye specific heat values plotted as lines. In the inset, the fractional change of ∆Cp/Cmax,B2O3 is reported as a function of Cs2O content.
longitudinal and shear ultrasound waves, measured using conventional pulse-echo ultrasonic techniques at a frequency of 10 MHz, have been used to determine the Debye temperatures (see Table 1). III. Experimental Results In Figure 1, the specific heat divided by T3 of (Cs2O)x(B2O3)1-x glasses with x ) 0.04, 0.14, and 0.33 is plotted. This representation allows us to emphasize the deviation of Cp from the Debye phonon contribution indicated in the same figure by the respective horizontal line. Experimental data of pure B2O3 glass42 are also shown. A clear peak, the boson peak, is observed in all glasses, revealing the existence of excess vibrational states above the expected Debye levels. The addition of an increasing quantity of cesium oxide to pure B2O3 glass shifts the position of the BP from 5.3 to ∼6.5 K. More importantly, its magnitude shows a nonmonotonic dependence on Cs2O concentration; at first, for x ) 0.04, the excess specific heat decreases with respect to that of B2O3 glass, whereas for a higher content of the alkaline oxide, it increases, reaching larger and larger values than that of pure boron oxide. The thermal conductivity of (Cs2O)x(B2O3)1-x glasses, with x ) 0.04, 0.14, and 0.33, is shown in Figure 2 and compared with that of pure boron oxide.43 At the beginning, a fast increase marks the temperature dependence of k(T), more clearly observable in B2O3 than in cesium borate glasses, after which k(T) appears to be only weakly temperature-dependent (plateau region). For even higher temperatures, the thermal conductivity starts to rise again. The plateau, readily discernible in each glass, appears to be less emphasized in cesium borate glasses than that in vitreous B2O3. Furthermore, the k(T) values in the whole investigated temperature range clearly increase with cesium content. IV. Discussion The observed increase of Cp with increasing Cs2O amount (see Figure 1) is an unusual finding. At the same time, the invariance of the peak position with the cesium content is also atypical. Both results contrast with the findings on sodium borate
Figure 2. Thermal conductivity of B2O3 (b),43 (Cs2O)0.04(B2O3)0.96 (4), (Cs2O)0.14(B2O3)0.86 (O), and (Cs2O)0.33(B2O3)0.67 glasses (0). The solid lines are obtained by numerical fits of eq 5 to the data.
glasses35 where the Cp/T3 peak shifts toward higher temperatures, visibly decreasing its intensity with increasing Na2O content. It is worth noting that a boson peak depending markedly on composition has been revealed also in lithium borate glasses by Raman spectroscopy.44 On the other hand, the same authors report a quite different behavior in cesium44 and barium45 borate glasses, which exhibit a boson peak having a peak frequency almost independent of the composition. A similar result has been found in Raman spectra of silver borate glasses.9 In these systems, an unusual nonmonotonous behavior of the excess specific heat with the Ag2O content has been also disclosed.46 All of this evidence leads us to believe that the vibrational properties of borate glasses modified with heavy metallic ions are deeply different from those containing light alkalies. A look at the structure of modified borate glasses can shed light on the composition dependence of vibrational dynamics in these systems. The formation of an alkaline borate glass (M2O)x(B2O3)1-x (where M indicates the alkaline ion and x the molar fraction) implies strong modifications of the structure of the pure vitreous B2O3. Such structural changes depend on the
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nature as well as on the concentration of the metallic ion introduced in the borate matrix. It has been found that the addition of alkaline oxide to B2O3 mainly assists the formation of charged BØ4 tetrahedral groups (Ø ) bridging oxygen atom) by cross-links between the BØ3 planar units building up the borate network. The concentration of BØ4 groups increases up to about x ) 0.20,47-49 which corresponds to the creation of two BØ4 groups for each oxygen introduced by the metallic oxide. For x > 0.20, the BØ4 production rate decreases, and nonbridging oxygens (NBO) appear, whose number should be less significant at lower concentration.48 All of these transformations promote the formation of new borate complexes hosting alkali ions and based on four-fold-coordinated boron atoms and a number of polyhedra involving three-fold-coordinated boron atoms. The number and the type of borate complexes change with the alkali content and are quite similar to those found in the corresponding crystals.50 In particular, it has been proven that the short-range structure of alkali borate glasses is almost independent of the alkali ion type and its concentration. On the contrary, the structure over the nanometer length scale changes strongly by changing the alkali ion, revealing a spatial distribution of borate groups strongly dependent on the metallic cation.51,52 Alkaline ions do not participate in the network formation, acting mainly in a charge-balancing capacity. They can occupy cavities in the boron-oxygen random network and are placed in the environment of tetrahedra or NBOs.51,53 The progressive transformation of BØ3 units in linked BØ4 tetrahedra, following the addition of a metallic oxide, results in an increase of the glassy network coherence and in a stiffening of the structure. This is proven in lithium and sodium borate glasses by the marked increase of the elastic moduli.54 On the other hand, the appearance of NBOs causes a decrease of the glassy network coherence and a softening of the structure limiting the rate of increase of the elastic moduli. In particular, in cesium borate glasses, it is believed that the mechanism of formation of NBOs is effective also at low metallic oxide content (x > 0.10), competing against the formation of BØ4 tetrahedra and giving rise to the series of two maxima spaced out by a minimum in the composition dependence of elastic moduli.55 It is worth emphasizing that systems having a more stiffened structure are also characterized by higher values of sound velocities and that this state is descriptive of higher values of the dominant frequency of acoustic phonons. Hence, if the excess vibrations contributing to the boson peak in borate glasses were acoustic-like or, equivalently, were strongly mixed or coupled with acoustic phonons, a shift of the peak in Cp/T3 toward higher temperatures would be expected as a natural consequence of higher sound velocity. At the same time, higher sound velocities result in higher values of the Debye temperature and hence in lower Debye specific heat. For these reasons, the observation of a decreased excess specific heat shifted to higher temperatures could reflect the mere changes of the elastic medium (sound velocity and density) following the addition of a different amount of metallic oxide. In agreement with these statements, the opposite trends revealed in the specific heat of sodium and cesium borate glasses would be explained respectively as due to the increasing and decreasing rigidity upon the addition of metallic oxide. In order to infer the role of elastic properties in the changes of the lowtemperature specific heat, we rescaled the specific heat to the elastic Debye contribution, CD, evaluated by the following equation
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Figure 3. C/CD as a function of T/ΘD of (a) B2O3 (+), (Na2O)0.06(B2O3)0.94 (O), (Na2O)0.16(B2O3)0.84 (4), and (Na2O)0.25(B2O3)0.75 (]) glasses and (b) B2O3 (+), (Cs2O)0.04(B2O3)0.96 (b), (Cs2O)0.14(B2O3)0.86 (2), and (Cs2O)0.33(B2O3)0.67 glasses (9).
CD )
( )
4 nkB 2π2 kB ) 234 3 3 3 5 p FV FΘD D
(1)
where ΘD is the Debye temperature, kB is the Boltzmann constant, n is the numerical density of atoms, F is the mass density, and VD is the average Debye sound velocity defined by
3 1 1 ) 3 + 3 3 VD VL VT
(2)
VL and VT being the longitudinal and transverse sound velocity, respectively. The rescaled specific heats of sodium and cesium borate glasses, reported as a function of T/ ΘD, are compared with that of B2O3 in Figure 3a and b, respectively. It can be observed that the rescaling removes almost completely the differences between the specific heats in sodium glasses, proving that the corresponding decrease of the Debye contribution compensates for the observed decrease of the low-energy vibrational states. On the contrary, the change of the excess specific heat in cesium glasses, with the questionable exception of (Cs2O)0.04(B2O3)0.96 glass, cannot be accounted for by the variation of the elastic medium, showing that in these systems, the addition of a high amount of cesium oxide brings about an increasing number of low energy vibrations. Hence, a different mechanism has to be invoked to explain the increased Cp in glasses having a high Cs+ content. Concerning the possible microscopic origin of the excess specific heat, we discard the possibility that both the NBO’s formation and the presence of water or OH- hydroxyl ions can
Thermal Conductivity and Specific Heat of Cesium Borate be responsible for the observed behaviors. As a matter of fact, alkali borate glasses are hygroscopic and can contain a substantial amount of water or OH- hydroxyl ions. Structural water even in small amounts (no more than 1 mol %) has a profound effect on the particular properties of borate glasses.56 It has been also observed that the specific heat of boron oxide shows significant variations with changing of the water content below 12 K, even if without any well-defined trend. However, the water amount estimated in the investigated samples is too small ( 0.14, where a well-defined plateau is observed. Concerning the causes for the Rayleigh scattering and consequently the plateau in k(T), it has to be considered that in oxide glasses, the main source of disorder comes from a wide distribution of bond-bending angles between network-forming atoms. The corresponding force constants are associated with the softest modes in the structure.66,70 In vitreous boron oxide, BO3, units are largely corner-linked by sharing common O atoms, forming rings.48,51 Very recently, we have performed neutron diffraction measurements41 and molecular dynamics simulation on cesium glasses,52 showing that the presence of cesium oxide is associated with the formation of larger voids, which replace the pre-
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existing voids in the boron skeleton in a ratio of two to one. Moreover, it has been found that the addition of alkali ions with decreasing ionic radius favors the formation of voids having a mean diameter progressively smaller.52 From diffraction experiments, four-membered rings have been identified in boron oxide. They are substituted with seven-membered rings in cesium borate glasses. Obviously, the shape of these structural voids is expected to be highly irregular and asymmetric as a consequence of the structural arrest taking place during the glass formation. Larger rings have been shown to be less strained than small ones.71 We speculate that network-forming atoms overlooking voids are subjected to wider local bond fluctuations and that the force constant for B-O-B bond bending in wider voids is lower than that in smaller ones, as a consequence of the formation of wider B-O-B angles. Force constant fluctuations are an expedient of system to achieve a greater energetic stability. Thinking about a possible correlation between thermal transport dynamics and the structure for these systems, we propose that fluctuations in sound velocity are associated with the force constant fluctuations of B-O-B bond-bending angles between network-forming atoms overlooking voids. We also suppose that ξ, the length scale over which fluctuations in this bond-bending angle are correlated, corresponds to the mean distance between oxygen atoms lying on the outline of these voids (namely the O-second O and O-third O distances). Therefore, the decrease of the strength of D in the Rayleigh scattering (see Table 1) with growing amount of Cs2O mainly arises from the constant force fluctuations associated with B-O-B bond bending increasingly reduced by the gradual replacement of smaller voids with larger voids housing alkaline ions. The reduction of bending fluctuations in larger voids could explain the increase of the plateau in k(T) after the addition of Cs2O to B2O3. Judging from the above arguments, we conclude that the hindrance of thermal propagation in the plateau region arises from Rayleigh scattering of phonons by frozen-in structural disorder in low atomic density regions, specifically, molecular rings occurring as a general characteristic in all oxide glasses. This result stresses the need for a widening of this relationship between structural features in the intermediate length scale and transport dynamics in glasses. A quantitative evaluation of the participation of local modes (such as rattling modes) to vibrational dynamics would be necessary in order to get the contribution to the thermal properties due to vibrational modes inherent to the glassy state so as to test the validity of different theoretical approaches. V. Conclusions In summary, low-temperature specific heat measurements, performed on cesium borate glasses by varying the content of alkaline oxide, have shown a boson peak marked by distinct signs of localization. The observed behaviors are believed to result from the presence of highly localized vibrational modes attributed to the isolated rattling of the Cs+ ions filling the structural voids in the boron oxide network. The increased number of these atoms corresponds to an enhancement of the specific heat. Otherwise, the presence of these vibrations does not appear to limit thermal conductivity, which increases with the Cs+ content as well. We find that Rayleigh scattering by frozen structural disorder, namely, disorder induced by fluctuations of bending force constants related to the network-forming atoms overlooking structural voids, represents a physical mech-
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