Dynamics of glass-forming liquids. IV. True activated behavior above 2 GHz in thedielectric -relaxation of organic liquidsC. Hansen, F. Stickel, R. Richert, and E. W. Fischer

Citation: The Journal of Chemical Physics 108, 6408 (1998); doi: 10.1063/1.476063 View online: http://dx.doi.org/10.1063/1.476063 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/108/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamics of glass-forming liquids. XI. Fluctuating environments by dielectric spectroscopy J. Chem. Phys. 124, 164510 (2006); 10.1063/1.2191491 Dielectric and shear mechanical alpha and beta relaxations in seven glass-forming liquids J. Chem. Phys. 123, 234511 (2005); 10.1063/1.2136887 Dynamics of glass-forming liquids. VII. Dielectric relaxation of supercooled tris-naphthylbenzene, squalane, anddecahydroisoquinoline J. Chem. Phys. 118, 1828 (2003); 10.1063/1.1531587 Decoupling of the dc conductivity and (-) structural relaxation time in a fragile glass-forming liquid under highpressure J. Chem. Phys. 116, 9882 (2002); 10.1063/1.1473819 Fragility and dynamical properties of glass-forming liquids above their Tg AIP Conf. Proc. 513, 98 (2000); 10.1063/1.1303337

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JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 15 15 APRIL 1998

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Dynamics of glass-forming liquids. IV. True activated behavior above2 GHz in the dielectric a-relaxation of organic liquids

C. Hansen, F. Stickel, R. Richert, and E. W. FischerMax-Planck-Institut fu¨r Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany

~Received 28 October 1997; accepted 13 January 1998!

We have measured the dielectric relaxation of butylbenzene and of the glass-former propylbenzenein the frequency range 1022 Hz to 231010 Hz in order to characterize the variation of relaxationtimes with temperature for these low loss liquids. Additionally, salol has been remeasured above 1GHz with improved resolution. Using the sensitive data representation@2d log10~f maxHz!/d~1/T!] 21/2 vs 1/T we find demarcation temperaturesTA , at which the temperature dependencechanges from a Vogel–Fulcher type law within the limitsTB<T<TA to Arrhenius behavior forT.TA , corresponding to a position of the loss peakf max.2 GHz. The activation energies derivedfrom dielectric relaxation data forT.TA are associated with the energy of vaporization,Eh

}DEvap. A comparison of dielectric relaxation timestD to viscosity data in this wide range oftemperatures suggests the relationtD}h/T rather thantD}h. © 1998 American Institute ofPhysics.@S0021-9606~98!50415-1#

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I. INTRODUCTION

The temperature dependence of a wide range of reation times in liquids and other disordered materials is ofincompatible with simple activated behavior,1–3 which is ex-pressed by

log10~x!5A1B/T, ~1!

where in this Arrhenius~ARR! law x can refer to a relaxationtime t(x}t), to a relaxation frequencyv(x}v21) or to theviscosity h(x}h).4 Here, ‘‘simple activation’’ refers to asite and temperature invariant activation barrier, as requby the ARR law in order to apply to a macroscopic esemble. The observation of data following the ARR law tohigh degree of accuracy is usually limited to temperatuwell above the melting pointTm and therefore seen mostly ihigh temperature viscosity measurements.5–7 Deviationsfrom the form of Eq.~1! are especially noticeable in the caof glass-forming liquids, where their ability to circumvecrystallization gives rise to an increasingly pronouncgrowth of the average relaxation time as the temperaturlowered.1–3 Further cooling to below the glass transitiotemperatureTg leads to the glassy state in which the systis no longer capable of equilibrating within the time windopreset by experimental conditions. The empirical law comonly used to account for the deviations from the ARR bhavior is the Vogel–Fulcher–Tammann8,9 ~VFT! tempera-ture dependence of the form

log10~x!5A1B/~T2T0!, ~2!

whereA, B, andT0 are constants with respect to tempeture. In this expression, the quantity of interest,x or x21, isanticipated to diverge asT approaches the VogeltemperatureT0.0, whereas the caseT050 restores theARR form. Although the VFT law has been put forwainitially to rationalize the data just aboveTg , it has beenrecognized that a further higher temperature range ex

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where Eq.~2! yields an excellent representation of the teperature dependence,10–13 albeit with parameters which arno longer compatible with the original idea of the free voume picture14,15 or with the theory of Adam and Gibbs,16

both requiring thatT0,Tg .We have previously addressed the temperature de

dence of dynamic quantities in detailed studies of dielecrelaxation, dc-conductivity, and viscosity data.11–13 The pre-condition for unambiguously identifying subtle changesthe temperature dependence is a data precision which alfor obtaining reliable figures regarding the derivativelog10(x) with respect to temperature, where in the casedielectric measurementsx5 f max/Hz represents the peak frequency of the dielectric losse9~v/2p!. More specifically,plots of @2d log10(x)/dT] 21/2 vs temperature and als@2d log10(x)/d~1/T!] 21/2 vs 1/T transform a VFT type de-pendence into a linear graph, and in the latter case an Atype x(T) appears as horizontal line.11 The typical scenarioderived from such an analysis is the existence of a demation temperatureTB , which has been observed in manglass-forming liquids.TB separates the distinct low and higtemperature ranges,Tg<T<TB and TB<T<TA , in whichdifferent VFT laws are appropriate for representing tdata.12 The upper limitTA of the high temperature VFT behavior has not yet been observed in terms of dielectric reation data confined to the frequency rangef <1 GHz. How-ever, the appearance ofTA in terms of a transition toArrhenius behaved dynamics has showed up systematicfor the dielectric relaxation of monohydric alcohols, methnol through octanol.12 On the other hand, we have showna subsequent study that the dominating relaxation peamonohydric alcohols is not related to the structural ora-relaxation, so that these material are exceptional regardtheir temperature dependence of the main dielectric reation peak.13

8 © 1998 American Institute of Physics

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6409J. Chem. Phys., Vol. 108, No. 15, 15 April 1998 Hansen et al.

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In the present paper we address the existence ofTA astransition temperature above which true activated dynamcan be observed by dielectric relaxation spectroscopy, wh‘‘true’’ is meant to indicate that the corresponding activatibarrier is a physically sensible quantity. We have found sbehavior forn-alkyl-benzenes by extending the experimenfrequency range to 18 GHz with a resolution of tand'331023 in the range 650 MHz to 16 GHz. The results acompared with viscosity data, for which the activation eergy forT.TA is a thermodynamically meaningful quantitbecause it is closely linked to the energy of vapization.4,17,18The threshold values of viscosity and dielectrelaxation time at the transition temperatureT5TA turn outto attain common values, i.e.,h(TA)'0.015 Poise andtD(TA)'60 ps.

II. EXPERIMENT

The alkylbenzenes under study,n-butylbenzene (99%1) andn-propylbenzene~98%!, were obtained from Aldrichand used without further purification. Phenyl salicyla~salol!, also from Aldrich, has been purified by distillatioand recrystallization.

Frequency domain dielectric relaxation experimewere performed in the frequency range 1022 Hz to 231010 Hz. For the range up to 1 GHz three different systeare used as reported previously;11,12 a frequency responsanalyzer~Solartron FRA-1260, 1022 Hz to 106 Hz!, an im-pedance analyzer~HP-4192A, 102 Hz to 107 Hz!, and a co-axial line reflectometer~HP-4191A, 106 Hz to 109 Hz!. Forfrequencies from 1022 Hz to 107 Hz a disk sample geometris used, for measurements between 106 Hz and 109 Hz acoaxial arrangement with the sample capacitor mountedpart of the inner conductor is employed. For all setupssample temperature has been controlled by a temperacontroller ~Quatro, Novocontrol! using a heated N2 gasstream. Independently we measure the sample temperwith a Pt-100 sensor at a relative accuracy of better t60.05 K.

For the high frequency measurements on the two loloss alkylbenzenes we used a network analyzer~HP-8510B,53107 Hz to 231010 Hz! employing a 7 mmcoaxial typetransmission line and the ‘‘full 2-port’’ calibration using amm class B.1 calibration kit~open, short, 50V!. A 30 cmairline ~HP-1250-1877! served as sample holder with the iner conductor being supported only by the terminating Pconnectors. The fluid under study was kept between twoflon spacers of 3 mm thickness in order to prevent its conto the connectors. The sample fluid inside the coaxial lwas allowed to exchange volume with an outer reservthrough a 0.5 mm hole in the outer conducter in ordermaintain constant pressure conditions and complete fillinvarious temperatures.

The transmission characteristics of the 30 cm line wmeasured in terms of the four complexS-parameters, whichspecify the complex energy ratios of input to output signaSout,in, regarding the reflection and transmission charactetics of the device under test. These values constituteS-matrix,

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S~v!5S S11 S12

S21 S22D . ~3!

A transmission line of lengthx filled with a dielectric whoseproperties are characterized bye~v! is know to have the fol-lowing S-matrix elements.19

S11~v!5S22~v!5G~12T2!

12G2T2 ,~4!

S12~v!5S21~v!5T~12G2!

12G2T2 ,

with G5~1/Ae21!/~1/Ae11! and T5exp ~2 ivc21xAe!. ThecorrespondingK-matrix is calculated using20

S K11 K12

K21 K22D 5

1

S21S 2det S S11

2S22 1 D . ~5!

The K-matrix of the entire line is the product of the fivindividual segments, i.e.,K tot5KX1•KL1•KL•KL2•KX2, eachKx being defined by its lengthx and thee~v! of the dielec-tric, where we haveeX15eX251 for the air segments aneL15eL252.04 for the Teflon segments. TheS-parametersare regained by

Stot51

K tot 22S K tot 12 det K tot

1 2K tot 21D , with Stot 125Stot 21.

~6!

Because the transmission results turn out to be more accuthan the reflection data, we use only theS12 results for de-termining the unknown dielectric function of the fluieL(v). Finally, eL(v) is calculated from the implicit equation Stot 12(eL(v))5S12, whereS12 is the measured valueusing a Newton–Raphson algorithm.21

Due to the sufficient dielectric strength of salol its higfrequency dielectric properties have been measured withnetwork analyzer~HP-8510B! in the single port mode employing the ‘‘high-temperature dielectric probe’’~HP-85075B! designed to operate in the range240 °C<T<200 °C with a coaxial-open-end geometry. This HP prousing glass as inner dielectric medium is much more reliafor temperature dependent measurements than similar prwith a Teflon filled coaxial line used previously,12 becausethe thermal expansion of teflon did change significantlyrelevant geometry.

III. RESULTS

The measured dielectric datae* (v)5e8(v)2 i e9(v)were subjected to fit procedures according toHavriliak–Negami22 ~HN! function

e* ~v!5e`1es2e`

@11~ ivtHN!a#g , ~7!

which yields an appropriate description of the data invicinity of the peak frequencyf max of the dielectric losse9(v). The dielectric constants in the limits of high and lofrequency are, respectively, denotede` andes andtHN sets acharacteristic time scale. The shape parametersa and g inthe range 0,a, a•g<1 quantify the symmetric and asym

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6410 J. Chem. Phys., Vol. 108, No. 15, 15 April 1998 Hansen et al.

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metric broadening of the loss peak relative to the Debbehavior witha5g51. For delineating the temperature dpendence of the relaxation time scale we chose to focusthe frequencyf max, at which e9(v) shows a maximum.11

This value can be derived from the HN fit parameters via23

f max~a,g,tHN!5~2ptHN!21•sin1/a~ap/~212g!!

•sin21/a~agp/~212g!!. ~8!

Figure 1 displays the high frequency ranges, 650 M< f <16 GHz, of the dielectric loss data in terms ofe9 vslog10( f /Hz) for butylbenzene and propylbenzene as a fution of temperature. Due to the weak polarity of these luids, emax9 ,0.06, the resolution limit around tand'331023

becomes visible in the wings of the loss profiles. Howevthe data quality is sufficient for obtaining reliablef max(T)data in the entire experimental temperature range.

The dielectric activation behavior for butylbenzenlog10( f max/Hz) vs 1/T, is depicted in Fig. 2, together withliterature27,28 viscosity data, log10(k•h21/Poise21), whichhas been shifted by a factor ofk in order to match the di-electric f max data at intermediate temperatures. In this cathe lower frequency limit at log10( f max/Hz)'4.5 stems fromthe strong tendency of butylbenzene to crystallize. In F2–4, the different symbols designate the dielectricf max

~closed circles!, inverse viscosity h21 ~triangles!, andh21/T21 ~squares!. Figure 3 shows the analogous picture fthe glass-forming material propylbenzene, where the posiof the dielectric loss peak covers the entire range frommHz to 20 GHz upon varying the temperature. The datasalol in Fig. 4 is identical to previously published work12 forf max,1 GHz only, whereas the results in thef max>1 GHzrange are improved data for the reasons outlined in theperiment.

The dielectric data for salol below 1 GHz and thef max(T)fit results for propylene-carbonate are taken from our preous publications.11,12 The viscosity data for salol have beecompiled from different sources.24–26 The viscosity data forthe n-alkylbenzenes~C1 to C7!, n-alkanes ~C1 to C24!,

FIG. 1. Dielectric loss resultse9(v) shown for frequencies between 65MHz and 16 GHz. Upper frame, data for butylbenzene in the range 29<T<358 K in steps of 5 K in the order of increasing peak frequencyLower frame, data for propylbenzene in the range 278 K<T<353 K insteps of 5 K in the order of increasing peak frequency.

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n-alkenes~C2 to C16!, and n-alkylacetates~C1 to C6! aretaken from the literature,27,28 as well as the values for thheat of vaporizationDHvap and for the boiling temperatureTb .29

IV. DISCUSSION

Before addressing the results in detail we wish to coment on the present philosophy of analyzing transport dat

KFIG. 2. Temperature dependence of the dielectric peak frequency~d, x5 f max/Hz! and of the inverse viscosity~n, x5k•h21/Poise21, log k57.57! for butylbenzene. The solid line is a VFT fit to theTB<T<TA data,the dashed line is an ARR fit to theT.TA dielectric data~see Table I forparameters!. The dotted line is an ARR fit to theT.TA viscosity data,log10(k•h21/Poise21)511.812655 K/T. The inset shows the Arrhenius regime on enlarged scales and including the data forx}T•h21 ~h! in addi-tion to x}h21 ~n!. The viscosity data is taken from the literature~Refs. 27,28!.

FIG. 3. Temperature dependence of the dielectric peak frequency~d, x5 f max/Hz! and of the inverse viscosity~n, x5k•h21/Poise21, log k57.70! for propylbenzene. The solid line is a VFT fit to theTB<T<TA

data, the dashed line is an ARR fit to theT.TA dielectric data~see Table Ifor parameters!. The dotted line is an ARR fit to theT.TA viscosity data,log10(k•h21/Poise21)511.922620 K/T. The inset shows the Arrhenius regime on enlarged scales and including the data forx}T•h21 ~h! in addi-tion to x}h21 ~n!. The viscosity data is taken from the literature~Refs. 27,28!.

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6411J. Chem. Phys., Vol. 108, No. 15, 15 April 1998 Hansen et al.

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general. Model functions of temperature dependent transdata like those of Arrhenius or Vogel–Fulcher–Tammaare usually associated with a particular physical basis, likwell defined activation barrier or the free volume pictuThese models imply certain restrictions on the parametfor instance phonon frequency like prefactors or a VotemperatureT0 confined to the range 0,T0,Tg . Especiallythe fit function VFT is presently employed as a data redtion tool on a purely phenomenological basis, i.e., to this ewe refrain from any physical interpretation of the resultiparameters. Accordingly, we do not expect unphysicasharp transitions from one fit regime to any other. Althoua parametrization of temperature dependencies as condubelow is appropriate for assessing and classifying transproperties, the eventual goal of a physical understandinterms of a single and continuous model function with phycally relevant parameters should be kept in mind. In partilar, Eq.~2! has been applied in the intermediate temperatrange,10–13 TB<T<TA , where ‘‘VFT’’ is a convenient butsomewhat inappropriate designation. On the other handwill argue that the ARR regime at temperaturesT.TA isassociated with true activated behavior.

A. Dielectric relaxation

A demarcation temperatureTA intending to separateVFT type f max(T) at T,TA from an ARR typef max(T) at T.TA has already been put forward previously30 on the basisof analyzing dielectricf max(T) data in a log10( f max/Hz) vs1/T graph. Because such an assessment off max(T) curves isnot very sensitive to deviations between data and fit,former ‘‘TA’ ’ is not equivalent to the present findings,corresponds more closely to the current value ofTB . Forinstance, the ‘‘TA’ ’ 5283 K reported30 for salol is signifi-cantly below the current findingTA5348 K. Numerous other

FIG. 4. Temperature dependence of the dielectric peak frequency~d, x5 f max/Hz! and of the inverse viscosity~n, x5k•h21/Poise21, log k57.75! for salol. The solid line is a VFT fit to theTB<T<TA data, thedotted line is an ARR fit to theT.TA dielectric data~see Table I forparameters!. The inset shows the Arrhenius regime on enlarged scalesincluding the data forx}T•h21 ~h! in addition tox}h21 ~n!. The viscos-ity data is taken from the literature~Refs. 24–26!.

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examples exist where an Arrhenius behavior is claimedobeyed only to a rough approximation.

In trying to find a suitable data reduction as regardscomplex manner with which characteristic relaxation timvary with temperature, we have observed in previous studthat the common application of the empirical VFT law tobroad range of temperatures in which the relaxation ticovers'102 s to '1029 s is inappropriate.11–13 Instead, byanalyzing the derivatives d log10( f max/Hz)/dT andd2 log10( f max/Hz)/dT2 we have found that the dynamics ithe frequency range'1022 Hz to '109 Hz subdivides intotwo distinct VFT regimes related to the temperature ranTg<T<TB andT.TB ,11,12 as has been noted earlier.10 Therelaxation frequency atTB is typically near 106 Hz and thetemperatureTb at which a Johari–Goldstein31,32 type b-process merges into thea-relaxation coincides withTB .13

For those glass-formers investigated up to frequenciesGHz no indication of a transition to ARR behavior could bdetected.12 However, propylene-carbonate displays such aTA

at f max'3 GHz,12 while the main dielectric peaks on-propanol and ethanol indicate aTA at peak frequenciesbetween 1 MHz and 10 MHz already.12 A subsequent scrutiny of n-propanol has clarified that the dielectric signatuof the structural relaxation is the much less intense highquency shoulder, whereas the microscopic origin ofdominant dielectric peak of monohydric alcohols remainsbe clarified.13 Therefore, we address the problem whethechange atTA towards simple activated dynamical behavioat sufficiently high temperatures can be found as a univefeature of dielectric relaxation in organic liquids of low molecular weight.

The activation plots for butylbenzene, propylbenzeand salol, Figs. 2–4, all indicate the usual high temperatVFT behavior in the rangeT.TB as indicated by solid lineswhere againf max(TB)'106– 107 Hz. As emphasized by theinsets of these figures, a significant deviation from this Vfit becomes noticeable in the 109– 1010 Hz range. For allthree materials thef max(T) and h21(T) data beyondlog10(x)'9.3(23109 Hz) display linearity in the activationplots. In this range the data deviates from the concomitARR fits ~dashed and dotted lines! by 60.01 decades onlyThe demarcation temperatures,TA and TB , delineating thetemperatures at which the activation behaviour chanqualitatively, are seen more clearly in Figs. 5–6 for the alklbenzenes. In these graphs,@2d log10(x)/d(1/T)#21/2 vs1/T, a VFT typex(T) appears linear and ARR behavior cabe identified as horizontal lines. From the intersections oflinear fits in this representation we derive the values forTA

andTB referring to the dielectric relaxation data. The resuing temperatures and frequencies,TB , log10( f max(TB)/Hz),TA , log10( f max(TA)/Hz), and the fit parametersA, B, T0 forboth regimes are compiled in Table I. The conclusion to tend is that materials differing significantly in their chemicconstitution, polarity, glass-transition temperature, aboiling-point all display the common scenario of a low temperature VFT behavior (Tg<T<TB), a higher temperatureVFT behavior (TB<T<TA), and an ARR regime (T.TA), with the only exeption that theTg<T<TB regime isnot accessible for butylbenzene due to crystallizati

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6412 J. Chem. Phys., Vol. 108, No. 15, 15 April 1998 Hansen et al.

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The results of the above analysis also indicate sevinteresting quantitative features seen in Table I. Forpresent materials, the peak frequencies atT5TB are found toattain values in the typical range, log10( f max(TB)/Hz)57.060.3, in accord with other experimental evidence indicatlog10( f max(TB)/Hz)'662.11–13The novel observation is thaalso TA is associated with a universal relaxation frequenlog10( f max(TA)/Hz)59.460.1 according to Table I, whereacorrelations betweenTA and the boiling temperatureTb arenot obvious. The simple activated behaviors observedfrequencies in excess off max'2.5 GHz extrapolate to similaattempt frequencies,AARR511.860.5, which correspond totypical phonon frequencieskT/h'1012 Hz. On the otherhand, the trend ofAARR as a function of alkyl chain lengthderived from viscosity data~see below! does not support theinterpretation in terms of attempt frequencies. In any cathe values off max(TA) and AARR appear not to be materiaspecific, although the activation parameterBARR varies from

FIG. 5. Temperature dependence of the dielectric peak frequency and oviscosity ~Refs. 27, 28! for butylbenzene, plotted as@2d log10(x)/d~1/T!#21/2 ~in units of K21/2! vs 1/T. The data, fits, and symbols correspondthose in Fig. 2. The indicated characteristic temperatures areTA(h)5300 K, TA(D)5268 K, andTB5172 K. The inset shows the Arrheniuregime on enlarged scales and including the data forx}T•h21 ~h! inaddition tox}h21 ~n!.

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541 K to 1100 K, equivalent to activation barriers betwe1.1 and 2.2 kcal/mol. This observation indicates thatBARR

reflects a thermodynamically relevant quantity, the true brier height for orientational dynamics, instead of some appent activation energy. This issue will be addressed onbasis of viscosity data in more detail below.

B. Viscosity

The simple activated behavior forT.TA observed forthe dielectricf max(T) is paralleled by a concomitant changin the temperature dependence ofh21(T) from VFT to ARRbehavior, albeit at a slightly higher temperature and leadto an activation energy which is lower than that off max(T).Comparing the dielectric (D) with the viscosity ~h! dataaboveTA , we find for the ratio of the activation energieBD/Bh5ED/Eh51.2(Eact5 ln(10)•R•BARR) for butylben-zene, propylbenzene, and salol.

FIG. 6. Temperature dependence of the dielectric peak frequency and oviscosity ~Refs. 27, 28! for propylbenzene, plotted as@2d log10(x)/d~1/T!#21/2 ~in units of K21/2! vs 1/T. The data, fits, and symbols correspondthose in Fig. 3. The indicated characteristic temperatures areTA(h)5264 K, TA(D)5240 K, andTB5170 K. The inset shows the Arrheniuregime on enlarged scales and including the data forx}T•h21 ~h! inaddition tox}h21 ~n!.

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TABLE I. Characteristic temperatures and activation parameters for the distinct Vogel–Fulcher and Arrregimes inferred from dielectric relaxation data. The fit parameters refer to log10( f max/Hz)5A2B/(T2T0)with T050 in the ARR case. The values for salol belowTA and those for propylene-carbonate are taken frour earlier work~Refs. 11, 12!. Tb denotes the boiling temperature.

Material

VFT: TB<T<TA ARR: T>TA

Tb (K)TB (K)log10( f maxHz)

at T5TB A, B (K), T0 (K) TA (K)log10( f max/Hz)

at T5TA A, B (K)

salol 265 6.80 10.43, 149, 224 348 9.52 12.42, 1100 46butyl-benzene

172 6.69 10.64, 182, 126 268 9.36 11.40, 541 456

propyl-benzene

170 7.22 10.44, 130, 130 240 9.31 11.52, 512 432

propylene-carbonate

200 7.30 10.70, 158, 153 290 9.55 11.75, 643 513

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6413J. Chem. Phys., Vol. 108, No. 15, 15 April 1998 Hansen et al.

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Various models have been proposed for relating chateristic relaxation times~e.g. a dielectrictD}1/f max! to theviscosityh. One of these is given by the Maxwell model,

tS5h/G` , ~9!

which relates the relaxation timetS of shear stress to thviscosityh and the infinite frequency shear modulusG` .33

However,tD}h is recovered only if one assumes propotionality between dielectric and shear stress relaxation timtD}tS , and a temperature invariantG`ÞG`(T), where es-pecially the latter condition is unrealistic.34

According to the Kubo equation,35 the shear viscosityhcan be written in terms of a correlation function regardithe off-diagonal elementss0

xy of the Fourier componentsskxy

of the stress tensor fork→0, i.e.,

h51

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`

^s0xy~0!•s0

xy~ t !&dt}^tS&/T. ~10!

Again anticipatingtD}tS , Eq. ~10! translates intotD}h•T.

An alternative approach concerns the hydrodynamlimit and focuses on the rotational diffusion constantDr de-fined via the torques acting on a rotating sphere.Dr reflectsthe equilibrium fluctuations which should dominate thesponse function of dipolar relaxation in a dielectric expement. For spherical particles of radiusr one obtains theStokes–Einstein–Debye relation,

Dr5h21kT

8pr 3 5~2tD!21, ~11!

wheretD is the dipolar relaxation time according to the Dbye model.35 It should be noted that Eq.~11! has been de-rived for the situation of a large sphere within a fluid fwhich continuum approximations apply on the length scof the sphere’s radiusr . Therefore, the application of a hydrodynamic model to molecular motion in pure liquidsdoubtful36 and in such a caser will not necessarily reflect atrue molecular property. In Fig. 7 we demonstrate this prlem for the Stokes–Einstein relation,D5kT/6prh, by con-fronting the Stokes radiusr S–E5kT/6pDh based on self-diffusion data with the molecular radiusr r . We estimate theradius on the basis of a sphere which encloses the molecmass at the given density,r r5(3MW/4pr)1/3, althoughother measures like the average van der Waals radius odius of gyration are equally appropriate. Empirically, we fifor a series of alcanesr S–E'1 Å10.2•r r , instead ofr S–E

5r r . The causes forr S–E varying with the molecular sizemuch less than expected are probably the deviations frospherical geometry and the badly defined boundary cotions around a real molecule relative to the hydrodynamlimit. Only for the longer alcanes, beyond;C100H202, onecould anticipate a molecular weight invariant value forDh

by assuming Rouse dynamics.37

Within the framework of rate process theorWeymann38 and later Macedo and Litovitz39 have arguedthat the temperature dependent rate (}t21) associated withviscous flow takes the formT/h instead of 1/h employedearlier.4,17 Therefore, it is of interest to investigate thetD –hrelation on the basis of decisive experimental data.

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For practical purposes, a relation liketD}h/T is oftenwell approximated bytD}h becauseh(T) varies muchmore pronounced with temperature than doesT21 or T. FortemperaturesT,TA the casestD}h/T andtD}h cannot bediscriminated on the basis of the present data. At high teperatures, where the apparent activation energies becsmall, the factorT21 can no longer be disregarded relativethe changes ofh(T) such that the experiment can discrimnate betweentD}h/T andtD}h. For a confrontation withthe dielectricf max(T) data, Figs. 2–6 thus include the resufor x5T•h21 as open squares, which follow thef max(T) datamore closely relative to thex5h21 values. This demon-strates thatf max(T)}T•h21(T), or equivalentlytD}h/T, isthe appropriate rationale for relating dielectric relaxation aviscosity data in the entire experimental temperature raT.TB .

From the dielectric data we have inferred that a commrelaxation frequency is observed atT5TA , log10~f(TA!/Hz!59.460.1. Moreover, the shift factorsk invoked tomatch theh21/Poise21 values to thef max/Hz scales alsoattain similar values, log10(k)57.660.1. Consequently, thetypical frequencyf max(TA)'2.5 GHz translates into a typicaviscosityh(TA)'0.015 Poise at the threshold temperaturethe Arrhenius range, log10(h(TA)/Poise)521.860.2. Wehave conducted a derivative type analysis of viscosity dfor the homologous series ofn-alkylbenzenes ranging fromtoluene ton-hexadecylbenzene~C1 to C16! in order to assessthe variation of the activation parameters as a functionalkyl chain length. A graphic compilation of the key parameters regarding the temperature dependent viscosity inrange T.TB is shown in Fig. 8. The values ofAARR

52 log10(h/Poise) andBARR from Arrhenius fits in therangeT.TA , andBVFT and T0 from VFT fits in the rangeTA<T<TB , are all systematic functions of the alkyl chalength, and the viscosity based transition temperatureTA(h)increases gradually with increasing molecular weight with

FIG. 7. Stokes radiir S–E5kT/6pDh based on viscosity~Refs. 27, 28! andself-diffusion~Ref. 41! literature data for a series of the tabulated alcanesa function of the molecular radiusr r estimated from density data,r r

5(3MW/4pr)1/3. The numbers in the table refer to molecular weightsMW .The data followr S–E'1 Å10.2•r r , instead ofr S–E5r r .

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6414 J. Chem. Phys., Vol. 108, No. 15, 15 April 1998 Hansen et al.

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this series. Figure 8 also demonstrates the minor dependof h(TA) on alkyl chain length with practically identical statistics as found for the four materials of Tablelog10(h(TA)/Poise)521.7560.18 for the C1 to C16 alkyl-benzenes. This observation again emphasizes that the VARR transition atT5TA appears to be dictated by the abslute values of viscosity or structural relaxation time scarather than occurring at a common position on the relaTg ...Tm ...Tb temperature scale.

An important achievement in the understanding of rprocesses is Eyring’s observation on the relation betweenactivation energyEh involved in the viscous flow and thenergy of vaporizationDEvap.4,17 For a large number of hydrocarbons it has been demonstrated thatEh/DEvap attainsvalues between 1/3 and 1/4, whereDEvap can be gained fromexperimental data viaDEvap5DHvap2p•DV'DHvap2R•Tb . The understanding is that the work required for tformation of free volume sufficient to allow viscous flow isparticular fraction of the energy of vaporization. Figureplots this fraction for several series of hydrocarbons, whEh5R•BARR• ln(10) has been inferred from the activatioparameters forT>TA by reanalyzing the viscosity data othe basis of@d log10(h/Poise)/d(1/T)#21/2 vs 1/T plots. TheEh/DEvap data in Fig. 9 confirms the previous findings, idicating thatEh and alsoED are thermodynamically meaningful quantities, instead of apparent activation energonly. An analogous assessment regarding the activationergy of h/T, i.e., Eh/T/DEvap, would lead to very similarresults, becauseEh/T5Eh1R•T'1.15•Eh, where the esti-mate of115% stems from the observationBARR'2.5•TA inTable I. The additional information derived from the preseinvestigation concerns the temperature range ofEh/DEvap'0.25 validity. Such a simple understandingviscous flow (Eh) or equivalently of orientation relaxatio(ED,Eh/T) must break down for temperaturesT,TA , i.e.,where the viscosity attaines values in excess ofh(TA)'0.015 Poise or where the dielectric or structural relaxattime exceedstD(TA)'60 ps, usingtD

2152p f max.

FIG. 8. Parameters derived from fits to literature~Refs. 27, 28! viscositydata as a function of alkyl chain length ranging from toluene to hexadebenzene~C1 to C16!. The upper frame showsAARR52 log10(h/Poise) andBARR from Arrhenius fits in the rangeT.TA , andBVFT andT0 from VFTfits in the rangeTA<T<TB . The lower frame plots the transition temperture TA(h) from VFT to Arrhenius behaviour, and the viscosityh (TA) interms of2 log10(h(TA)/Poise)5AARR2BARR /TA .

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V. SUMMARY AND CONCLUSIONS

The dielectric relaxation of butylbenzene, propylbezene, and salol has been measured with emphasis onfrequency range 109 Hz to 231010 Hz in order to seek fortruely activated orientational dynamics known to occurviscosity data well above the melting temperatureTm . For acritical and sensitive data analysis we have focusedthe data representation@2d log10(x)/d(1/T)#21/2 vs 1/Tregarding the characteristic dielectric relaxation timex5 f max/Hz) and the fluidity or inverse viscosity (x5h21/Poise21). For all diverging materials under study, wfind a well pronounced demarcation temperatureTA , atwhich the temperature dependence changes from a VFTlaw obeyed within the limitsTB<T<TA to an Arrheniusbehavior forT.TA . At the onset of the Arrhenius rangeT5TA , both structural relaxation time and viscosity are nmaterial specific and attain values ofh(TA)'0.015 Poiseand tD(TA)'60 ps or f max(TA)'2.53109 Hz. Emphasizedby the results obtained aboveTA , a detailed confrontation odielectric and viscosity data indicates a quantitative couplof these quantities forT.TB according totD}h/T, ratherthan followingtD}h. Therefore, not only the viscosity bualso the orientational relaxation times can be rationalizeda true ~rather than apparent! activation energy which is aparticular fraction of the energy of vaporization,Eh/DEvap

'0.25, and which is associated with reasonable attemptquencies of orderkT/h. Below this clear cut temperatureTA

a further mechanism sets in which gives rise to a VFT tytemperature dependence within the limitsTB<T<TA . Therole of TB when assessing dynamics in terms of the AdamGibbs theory16 is the subject of a further paper.40

l-

FIG. 9. Ratio of activation energiesEh5R•BARR• ln(10), derived fromArrhenius fits to literature~Refs. 27, 28! viscosity data forT.TA(h), to theenergy of vaporization~Ref. 29!, DEvap5DHvap2p•DV'DHvap2R•Tb ,plotted as a function of the boiling temperatureTb . The different symbolsrefer ton-alkylbenzenes~l, C1 to C7!, n-alkanes~h, C1 to C24!, n-alkenes~s, C2 to C16!, andn-alkylacetates~n, C1 to C6!.

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6415J. Chem. Phys., Vol. 108, No. 15, 15 April 1998 Hansen et al.

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ACKNOWLEDGMENTS

Financial support by the Deutsche Forschungsgemschaft~Sonderforschungsbereich 262! is gratefully acknowl-edged.

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