The Boson peak in structural and orientational glassesof simple alcohols: specific heat at low temperatures

M.A. Ramos *, C. Tal�oon, S. Vieira

Laboratorio de Bajas Temperaturas, Depto. de F�ıısica de la Materia Condensada C-III, Instituto ‘Nicol�aas Cabrera’,Universidad Aut�oonoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Abstract

We review in this work specific-heat experiments, that we have been conducted on different hydrogen-bonded glasses

during recent years. Specifically, we have measured the low-temperature specific-heat Cp for a set of glassy alcohols:normal and fully deuterated ethanol, 1- and 2-propanol, and glycerol. Ethanol exhibits a very interesting polymorphism

presenting three different solid phases at low temperature: a fully ordered (monoclinic) crystal, an orientationally

disordered (cubic) crystal or ‘orientational glass’, and the ordinary structural glass. By measuring and comparing the

low-temperature specific heat of the three phases, in the ‘Boson peak’ range 2–10 K as well as in the tunneling-states

range below 1 K, we are able to provide a quantitative confirmation that ‘glassy behavior’ is not an exclusive property

of amorphous solids. On the other hand, propanol is the simplest monoalcohol with two different stereoisomers (1- and

2-propanol); this allows us to study directly the influence of the spatial rearrangement of atoms on the universal

properties of glasses. We have measured the specific heat of both isomers, finding a noteworthy quantitative difference

between them. Finally, low-temperature specific-heat data of glassy glycerol have also been obtained. Here we propose

a simple method based upon the soft-potential model to analyze low-temperature specific-heat measurements, and we

use this method for a quantitative comparison of all these data from glassy alcohols and as a stringent test of several

universal correlations and scaling laws suggested in the literature. In particular, we find that the interstitialcy model for

the Boson peak [Phys. Rev. Lett. 68 (1992) 974] gives a very good account of the temperature Tmax at which themaximum in Cp=T 3 occurs.� 2002 Elsevier Science B.V. All rights reserved.

PACS: 65.60.þa; 63.50.þx; 61.43.Fs

1. Introduction

After several decades of research on the subject,the universal properties exhibited by glasses at lowtemperatures [1,2] (i.e., their vibrational excita-

tions at low frequencies) continue to be a matter ofinterest and debate [3]. It is well known [4,5] thatglasses or amorphous solids have thermal prop-erties (and also dielectric or acoustic ones) verydifferent from those of crystalline solids. More-over, these properties are very similar among dif-ferent families of glassy materials irrespective ofeither the type of chemical bonding or otherstructural details, hence the name ‘universal’. At

Journal of Non-Crystalline Solids 307–310 (2002) 80–86

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* Corresponding author. Tel.: +34-91 397 5551; fax: +34-91

397 3961.

E-mail address: miguel.ramos@uam.es (M.A. Ramos).

0022-3093/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0022 -3093 (02 )01443 -6

temperatures T < 1 K, the specific-heat Cp dependsapproximately linearly (Cp / T ) and the thermalconductivity j almost quadratically (j / T 2) ontemperature, in contrast to the cubic dependenceobserved in crystals for both properties which iswell understood in terms of Debye’s theory.Quantitatively, the specific heat of non-metallicglasses is much larger and the thermal conductivityorders of magnitude smaller than those of dielec-tric crystals. At T > 1 K, Cp still deviates from theexpected CDebye / T 3 dependence, presenting abroad maximum in Cp=T 3, in the same tempera-ture range where the thermal conductivity exhibitsa universal plateau. It is now clear that this uni-versal feature is closely related to an excess in thevibrational density of states gðmÞ over the crystal-line Debye behavior, leading to a ubiquitousmaximum in gðmÞ=m2 at frequencies m � 1 THzwhich is known as the Boson peak, a dominantfeature in the vibrational spectra of glasses verythoroughly observed and studied [3] by Ramanand inelastic neutron scattering.About 30 years ago, Phillips [6] and Anderson

et al. [7] introduced independently the now well-known tunneling model (TM), which postulatedthe ubiquitous existence of atoms or small groupsof atoms in amorphous solids which can tunnelbetween two configurations of very similar energy.This simple model of tunneling states successfullyexplained the low-temperature properties of amor-phous solids [2], though only for T < 1 K. How-ever, the also rich universal behavior of glassesabove 1 K (the broad maximum in Cp=T 3, thecorresponding Boson peak in vibrational spectra,or the abovementioned plateau in the thermalconductivity) still were unexplained. Among thedifferent approaches proposed since then to under-stand the general behavior of glasses in the wholerange of low-frequency excitations, the phenome-nological soft-potential model (SPM), which canbe regarded as an extension of the TM, is one ofthe best accepted and most often considered. TheSPM [8,9] postulates the co-existence in glassesof acoustic phonons (crystalline-like, extendedlattice vibrations) with quasi-localized low-fre-quency (soft) modes. In the SPM, the potential ofthese soft modes is assumed to have a uniformstabilizing fourth-order term W, an energy scale

which is the basic parameter of the model [8–13].In addition, each mode has its own first-order(asymmetry D1) and second-order (restoring forceD2) terms, which can be either positive or negative,hence giving rise to a distribution of double-wellpotentials (tunneling states) and more or lessharmonic single-well ones (soft vibrations). Theparameter W marks the crossover from the tun-neling-states region at the lowest temperatures tothe soft-modes region above it. Indeed, W can beapproximately determined either from the mini-mum Tmin in Cp=T 3 (W ’ 1:8 – 2kBTmin) or from theposition of the maximum Tj;max in a j=T vs T plot:(W ’ 1:6kBTj;max) [13,15]. Similarly to the TM, arandom distribution of potentials is assumed:PðD1;D2Þ ¼ Ps. For a more detailed description ofthe SPM, the reader is referred to the reviews ofRefs. [14,15].In addition to these phenomenological models,

other recent approaches [16–20] have focused onsuggesting general scalings, correlations or uni-versalities in the low-temperature properties ofglasses which could hint at their microscopic origin.In order to gain understanding in this issue, we

have conducted a series of measurements [21–26]of specific heat at low temperatures for a specialfamily of glasses, simple alcohols such as ethanol[21–23], propanol [24] and glycerol [25], whichhave low glass transition temperatures Tg (they areliquid at room temperature) and a molecular, hy-drogen-bonded network. In this work, we proposea systematic method to analyze low-temperaturespecific-heat data, partly based upon the SPM.Then, we employ this method for a critical com-parison of the data obtained by us for the alco-hols and, finally, we make use of the collected setof parameters as a test of several correlations andscaling laws suggested in the literature.

2. Experimental

During last years, we have performed specific-heat measurements on different glassy alcohols, byemploying a calorimetric cell especially designedfor samples which are liquid at room tempera-ture and that allows to prepare in situ differentsolid phases. Firstly, experiments were conducted

M.A. Ramos et al. / Journal of Non-Crystalline Solids 307–310 (2002) 80–86 81

in a 4He cryostat, reaching temperatures down to�1.7 K. Later, we have used a very similar cal-orimetric cell within a 3He cryostat being able tomeasure the specific heat to about 0.5 K.In particular, we have studied [21–23] both

normal hydrogenated and fully deuterated etha-nol, both of which exhibit a rich polymorphism,presenting stable (monoclinic) crystal, orienta-tional glass (OG, an orientationally disorderedcubic crystal, obtained by quenching a rotationallydisordered plastic phase), and structural glass(amorphous) phases [27].We have also investigated [24] the behavior of

the next substance in the series of monoalcohols,propanol, which is the smallest one which has twodifferent stereoisomers, 1- and 2-propanol, henceallowing us to study directly the effect of the spa-tial rearrangement of atoms on the low-tempera-ture properties of glasses.Finally, we have also measured [25] the specific

heat and the thermal conductivity of glassy glyc-erol, probably the most widely studied [28] glass-forming liquid.Further details on the experimental setup em-

ployed for the heat-capacity measurements, as wellas on the experimental procedures followed toobtain and characterize the different solid phasesare given in the corresponding references indicatedabove.

3. Results

The specific heat of several glassy alcohols(deuterated ethanol in either true glass or OGphases, both isomers of propanol, and glycerol) isshown in Figs. 1 and 2, all of them exhibiting theusual ‘glassy’ behavior, with a quasi-linear contri-bution at very low temperature (tunneling states)and a broad maximum in Cp=T 3 (Boson peak).First of all, we remark that both structural glass

and OG (i.e., a crystal with orientational disorder)phases of ethanol show, qualitatively and evenquantitatively, the same glassy features in the low-temperature specific heat [22,23], the Boson peakin inelastic neutron scattering [21] or at the glasstransition [22]. These results provide a quantitativeconfirmation of the fact that glassy behavior is not

an exclusive property of amorphous solids, whichsimply lack translational disorder, but a more gen-eral characteristic of solids where any kind ofdisorder is able to soften the rigid vibrational spec-trum of a crystalline lattice and/or to provide ad-ditional thermodynamic degrees of freedom whichare somehow the basic ingredient of the glass state.It is also noteworthy that 2-propanol has a

much larger specific heat above 1 K than 1-pro-panol (and than any other glassy alcohol). How-ever, the reason for this difference is based on its

Fig. 1. Low-temperature specific heat plotted as Cp=T vs T 2 forseveral glassy alcohols. Solid lines are fits to quadratic poly-

nomials (see text).

Fig. 2. Low-temperature specific heat of the same glasses in

Fig. 1, scaled to the cubic Debye contribution CDebye, as afunction of temperature normalized to hD. The arrow indicatesthe position of the maximum at T ¼ hD=35 predicted within theinterstitialcy model.

82 M.A. Ramos et al. / Journal of Non-Crystalline Solids 307–310 (2002) 80–86

significantly larger Debye contribution (see Fig. 1and Table 1). This difference between both isomersof propanol also occurs in their crystalline states[24], which have been recently found to be differentindeed, a monoclinic crystalline structure for 1-propanol and a triclinic one for 2-propanol [29].Therefore, the influence of the position of the hy-droxyl (OH–) on the elastic constants of the hy-drogen-bonded network seems to be very relevant.In contrast, the ‘excess’ specific heat attributableto tunneling states and quasi-localized vibrationsin glasses is much less affected by these changes inthe atomic arrangement.In addition, we have concurrently measured the

specific heat and the thermal conductivity of glassyglycerol [25], and have used those combined dataas a more reliable test of the SPM, which has beenshown to successfully explain the specific heat andthe thermal conductivity in a wide temperaturerange. In this work, we will only use the specific-heat data for a general comparison with otherglassy alcohols.In Fig. 1, the specific heat of five glasses (four

truly amorphous and one disordered crystal) isdisplayed below �2.5 K in a typical Cp=T vs T 2

plot. In the simplest version of the TM [2,6,7], therandom distribution of tunneling states can be

regarded as a constant density of two-level systems(TLS), which neglecting some logarithmic correc-tions gives a linear contribution to the specific-heatCp ¼ CTLST . Taking also into account the Debyecontribution due to extended long-wavelength vi-brations of the amorphous lattice, CDebye ¼ CDT 3,it is clear that Cp=T vs T 2 should plot linearly withan intercept CTLS at T ¼ 0. Although this simplemethod has been traditionally used to determineCTLS and CD, it poses some problems. As Fig. 2reminds, there is an additional source of specificheat arising from the low-frequency vibrationsresponsible for the Boson peak and the maximumin Cp=T 3. This contribution is not completelynegligible below 1 K, that brings as a consequencethat many linear fits Cp ¼ C1T þ C3T 3 found in theliterature unavoidably provide cubic coefficientsclearly exceeding the true Debye one, obtainedfrom elastic measurements in those cases wherethey are available, i.e., C3 > CD. Indeed, C1 andespecially C3 depend on the chosen range for thelinear fit. In order to solve these inconveniences,we propose a systematic method to analyze low-temperature specific-heat measurements. The basicpoint is to realize that the difficulties originate fromthe lower-energy side of the additional vibrationsresponsible for the Boson peak. These have been

Table 1

Measured data and fit parameters obtained for several studied glassy alcohols (see text)

H-ethanol D-ethanol 1-Propanol 2-Propanol Glycerol

Glassa OG Glass OG glass glass glass

Pmol (g/mol) 46.1 46.1 52.1 52.1 60.1 60.1 92.1

Tg (K) 95 95 95 95 98 115 185

Tmin (K) 2.3 2.6 2.1 2.3 1.8 1.6 2.0

Tmax (K) 6.1 6.8 6.0 6.4 6.7 5.0 8.7

ðCp=T 3Þmax (mJ/molK4) 2.4 2.2 2.8 2.6 2.7 3.6 1.4

CTLS (mJ/molK2) 1.2 1.27 1.05 1.13 0.424 0.516 0.157

CD (mJ/molK4) 1.55 1.45 1.80 1.72 1.77 2.54 0.855

Csm (mJ/molK6) 0.0432 0.0288 0.0572 0.0419 0.0367 0.0845 0.0139

W (K) 4.1 4.7 3.8 4.1 3.3 2.9 4.3b

Ps (mol�1) 4:0 1019 5:2 1019 3:6 1019 3:8 1019 1:1 1019 1:4 1019 1:6 1019

ðCexc=CDÞmax 0.55 0.52 0.56 0.51 0.53 0.42 0.64

hD (K) 224 229 213 217 236 209 317

hD=Tmax 37 34 35 34 35 42 36

aData for the glass phase of H-ethanol were taken in a 4He-cryostat, only down to 1.7 K.bThe value of W for glycerol has been determined from thermal conductivity data [25].

M.A. Ramos et al. / Journal of Non-Crystalline Solids 307–310 (2002) 80–86 83

well accounted for by the SPM [14,15] as harmonicsoft modes giving rise to a specific heat below themaximum Cp ¼ CsmT 5 ¼ ð2p6=21ÞPskBðkBT=W Þ5.Therefore, a quadratic polynomial fit in Cp=T vsT 2 seems the most appropriate solution. The ques-tion however remains how to decide the tem-perature range to fit the data, with physicalmeaning. It is clear that the distribution of softmodes and correspondingly the specific heat can-not grow Cp / T 5 unlimitedly. Gil et al. [11] pro-posed a Gaussian distribution in the asymmetry ofthe soft potentials, which without any further fit-ting parameter allowed them to account for thespecific heat, thermal conductivity and vibrationaldensity of states gðmÞ=m2 in the whole relevantrange, including the Boson peak. At least forpractical reasons, let us assume that distributionfunction. It is easy to find that the simple CsmT 5

approximation starts deviating �5% from theCpðT Þ curve accounting for experimental data ap-proximately at T > 0:75W ’ ð3=2ÞTmin. Therefore,we suggest to fit specific-heat data in a Cp=T vs T 2

representation by using a quadratic polynomialCp ¼ CTLST þ CDT 3 þ CsmT 5 in the temperaturerange 0 < T < ð3=2ÞTmin. The results from thesefits for the studied glassy alcohols are shown inTable 1. As a proof of consistency, we want tomention that the obtained CD for glycerol agreesbetter than 1% (hence within the experimentalerror) with the Debye coefficient estimated [25]from elastic measurements. Unfortunately, this isthe only glass from those studied here for whichelastic data are available. Nevertheless, we believethat the method proposed is a reasonable alter-native to determine the Debye coefficient of a glassfrom low-temperature specific-heat measurements,especially when elastic data are lacking. The so-obtained values of CD and hD have been used inFig. 2 to scale Cp data for the reasons discussedbelow.For sake of completeness, we also show in

Table 1 the SPM parameters W (determined fromW ’ 1:8kBTmin [11,15], with the exception of glyc-erol which has a flat minimum and has been betterdetermined from available thermal-conductivitydata [25]) and Ps (determined from the given ex-pression for Csm). It is to be stressed that allstudied glasses present comparable values of Ps

(basically, the distribution density of quasi-local-ized excitations, either tunneling states or softmodes), not existing any significant difference be-tween amorphous glasses and OG.

4. Discussion

In this section, we will make use of the compiledset of data for the studied glassy alcohols taken asa model system to critically discuss several corre-lations or scaling laws which have been suggestedin the literature to be universal for glasses oramorphous solids.First, we will address very briefly the scaling

proposed by Liu and L€oohneysen [16]. They sug-gested a general correlation between the mecha-nisms leading to the Cp=T 3 maximum in crystallineand amorphous solids. They plotted the height Pcof the maximum in Cp=T 3 (i.e., ðCp=T 3Þmax in ournotation) vs its position Tmax for a wide set ofmaterials, mainly amorphous polymers and met-als, as well as typical network glasses, togetherwith their corresponding crystalline solids. Theyfound an approximate general correlation Pc /T�1:6max , somehow indicating a close relation be-tween the Cp=T 3 maxima in glasses and crystals.However, we have included in such a graph (seeFig. 2 of Ref. [26]) specific-heat data for molecularglasses and crystals (both our hydrogen-bondedmaterials and van der Waals glasses from Lindq-vist et al. [30]), finding that these molecular solidsdeviate systematically about a factor 5 from thegeneral scaling proposed by those authors. More-over, for the different phases of H- and D-ethanolwe have plotted ðCp=T 3Þ=Pc vs T =Tmax, as also sug-gested by Liu and von L€oohneysen [16]. We found(see Fig. 11 of Ref. [22]) that data for the disor-dered solids (either glass or OG) superimpose inthe whole temperature region but, in contrast, thecurves for the ordered crystals showed a far nar-rower shape. This fact points out again to thedifferent nature of the low-energy vibrational spec-tra of glasses and crystalline solids.Other authors have suggested that the low-

energy excitations of glasses may be correlated withthe fragility of the glass-forming liquid, a para-meter proposed by Angell [31,32] to characterize

84 M.A. Ramos et al. / Journal of Non-Crystalline Solids 307–310 (2002) 80–86

how fragile or strong is a supercooled liquid toresist the structural degradation induced by tem-perature, and which is usually measured by thedeviation of the shear viscosity from an Arrheniuslaw. Zhu [19] has suggested a general correlationbetween the density of tunneling states and thefragility, presenting data for a variety of glasses.More fragile glasses would have a larger numberof minima on the potential-energy hypersurface,hence explaining a higher density of tunnelingstates and a higher value of CTLS. On the otherhand, Sokolov et al. [18] have correlated the fra-gility of the system with the excess of vibrationalexcitations normalized to the Debye level. Specif-ically, they showed that the ratio ðCexc=CDÞmax ¼ðCp=CD � 1Þmax decreases with increasing fragility.More recently, Zhu and Chen [20] have arguedagainst the universality of such a correlation.Unfortunately, accurate fragility values using

the same criteria are not available for the fourglass-forming liquids studied in this work. Never-theless, it can be seen from viscosity data [31,32]that all alcohols possess an intermediate degree offragility. Liquid ethanol seems to be slightly morefragile than glycerol, and 1-propanol is less fragile.There are no published data for 2-propanol to ourknowledge. In this framework, the large differencein CTLS between ethanol and glycerol (almost oneorder of magnitude, see Table 1) for two systemsof very similar fragility claims against the corre-lation proposed by Zhu [19]. Moreover, 1-propa-nol has a contribution to Cp from tunneling statesalmost a factor of 3 higher than that of glycerol,being less fragile. The correlation with ðCexc=CDÞmax is more difficult to ascertain. The obtainedvalues are all around ðCexc=CDÞmax � 0:5 as ex-pected for intermediate fragilities, but the smallerdeviations do not show up the expected trend.Neither suggest normalized Cp=CD curves shownin Fig. 2 any kind of universal behavior. Fur-thermore, the very similar values of CTLS andðCexc=CDÞmax for either true glasses (quenching asupercooled liquid) or OG (quenching a plasticcrystal) of ethanol cast doubts about the relevanceof the fragility of the glass-forming liquid to un-derstand the low-temperature properties of glasses.Finally, we would like to consider the inter-

stitialcy model proposed by Granato [33,34]. Ac-

cording to this model, liquids can be considered ascrystals containing a few interstitials in thermalequilibrium, which become frozen in the glassystate. Self-interstitial resonance modes would bethe physical realization of the soft modes andtunneling states of the SPM, giving rise to theBoson peak. For the simplest approximation,taken a single frequency for the resonance modes,the maximum in Cp=T 3 is predicted [34] to appearat a temperature Tmax � hD=35. Several authorshave tested this relation for different collectionsof experimental data, finding empirically Tmax hD=40 [35] or Tmax hD=38 [20]. It can be seen inTable 1 that this correlation works very well for allglassy alcohols studied by us. On the contrary, themaxima of the corresponding crystalline states[22,24,25] take place at higher reduced tempera-tures (Tmax=hD � 1=25 for these alcohols), as itseems always be the case (see Fig. 3.3 in Ref.[2]). However, we have tested this correlation be-tween Tmax and hD=35 from published data [36] foranother model system: vitreous B2O3 submitted todifferent thermal treatments producing significantchanges in mass density, elastic constants, etc. [37].Variations of up to 17% in Tmax and hD wereachieved with this method. We find that, althoughðCexc=CDÞmax ranges uncorrelatedly from 1.18 to1.70 for the seven different B2O3 glasses, the ratiobetween Tmax and hD remains constant for all ofthem: hD=Tmax ¼ 49� 2. Let us notice that forother oxide glass as SiO2, hD=Tmax ¼ 46 [20]. So, itmay well be that the exact Tmax=hD ratio couldsomewhat depend on the kind of glass, what is notin conflict with the interstitialcy model [34], but fora given ‘system’, fixing some secondary parame-ters, the Boson peak and Debye temperaturesseems to be clearly correlated.

5. Conclusions

In summary, we have reviewed and compara-tively discussed specific-heat experiments, that wehave conducted on different hydrogen-bondedglasses during last years: normal and fully deu-terated ethanol, 1- and 2-propanol, and glycerol.We have proposed a systematic method partlybased upon the SPM to analyze these low-tem-

M.A. Ramos et al. / Journal of Non-Crystalline Solids 307–310 (2002) 80–86 85

perature specific-heat measurements, and to testseveral universal correlations and scaling lawssuggested in the literature. In particular, we havefound that the correlation between the tempera-ture Tmax at which the maximum in Cp=T 3 occursand the Debye temperature hD proposed by theinterstitialcy model for the Boson peak is very wellfulfilled.

Acknowledgements

This work was supported by MCyT (Spain)within project BFM2000-0035-C02-01.

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