PHYSICAL REVIEW E 66, 011502 ~2002!

Relaxation processes in an epoxy resin studied by time-resolved optical Kerr effect

D. Prevosto,1 P. Bartolini,2,3 R. Torre,2,3,4 M. Ricci,2,3,5 A. Taschin,2,3,4 S. Capaccioli,1 M. Lucchesi,1 P. Rolla11Dipartimento di Fisica, Universita` di Pisa, and INFM, Unita` di Pisa, via F. Buonarroti 2, I-56127, Pisa, Italy

2LENS, Polo Scientifico, Universita` di Firenze, via N. Carrara 1, 50019 Sesto Fiorentino, Italy3INFM, Unita di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy

4Dipartimento di Fisica, Polo Scientifico, Universita` di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy5Dipartimento di Chimica Fisica, Universita` della Basilicata, via N. Sauro 85, Potenza, Italy

~Received 7 February 2002; published 29 July 2002!

The dynamics of the epoxy resin phenyl glycidyl ether, a fragile glass-forming liquid, is investigated in theliquid and supercooled phases by time-resolved optical Kerr effect experiment with an heterodyne detectiontechnique. We tested the mode-coupling theory and found that the predicted dynamic scenario allows toreproduce properly the measured signal, fort.1 ps, in the whole temperature interval investigated. Further-more, the values ofTc andl, obtained from the analysis of three different and independent dynamic regimes(a regime, von Schweidler,b regime!, are in remarkable agreement. Moreover, relaxation times obtained fromoptical Kerr effect and dielectric spectroscopy measurements are compared. The two time scales differ only fora constant factor in the whole temperature interval investigated.

DOI: 10.1103/PhysRevE.66.011502 PACS number~s!: 64.70.Pf, 78.20.Jq, 78.47.1p

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I. INTRODUCTION

Glass-forming systems can be supercooled belowmelting temperature in a metastable liquid phase withcrystallizing. The principal feature of supercooled liquidsthe strong slowing down of the structural relaxation procresulting from a temperature lowering. This phenomenonquite huge near the liquid-glass transition temperatureTg ,where the cooling causes a freezing of the structural dynics in a glassy state. The microscopic mechanisms undeing the slowing down of the structural dynamics are nowdays subject of a heated debate in condensed matter phyIn the last years the development of computational andperimental techniques stimulated new theoretical@1–3# andexperimental works@4–9# in order to reach a deeper insigon this argument, even if a complete knowledge is far toachieved because of the complexity of this phenomenFrom an experimental point of view, the main problem isstudy the evolution of the dynamics measuring over thetire range from times of the order of picoseconds to 100 swhich the glass transition conventionally takes place. Onother hand, none of the theories presented till now accofor all the features characterizing the dynamics of supcooled liquids. In this scenario, more accurate predictionsthe dynamics evolution and effective analytical formulafitting the data are highly desirable. Moreover, it is importato perform experiments over the extremely wide time rantypical of the dynamics of supercooled liquids, as well asprovide more accurate data for a reliable testing of the thretical predictions. At the state of the art, dielectric spectrcopy, employing quasioptical submillimeter spectrometbesides more conventional frequency response analyzethe only technique that can cover the time interval of inter@5#. Alternatively results from different experimental tecniques can be combined to cover the widest dynamrange, provided that a careful analysis takes into accoundifference among the sensed dynamic observables. Howalthough the matching of data provided by different tec

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niques is a difficult job, such an approach can greatly enrthe knowledge of the observed relaxation, helping alsostronger definition of the universal behavior of the supcooled state@10,11#.

In the middle of the 1980s a new theory, called modcoupling theory~MCT! @1#, was proposed attempting to explain the glass transition phenomenon from a purely dynacal point of view. The MCT originates from the solution ofset of nonlinear equations for the evolution of the correlatof density fluctuations r(t)r(0)&. In its simplest form,called ‘‘idealized MCT,’’ the theory predicts a transition athe critical temperatureTc from an ergodic dynamical regim(T.Tc), where the correlator of density fluctuation decato zero, to a nonergodic regime (T,Tc), where the cor-relator remains at a finite value. The theory, even in the idversion, describes in detail the dynamics of supercooleduids in the moderately viscous regime where it predictsexistence of two different regions of relaxation (a and bregime! and, as we will see later, accurate functional formdescribing the evolution of the time correlator. Despite ttheory was formulated for the evolution of density fluctutions taking into account only translational degrees of frdom, in the last years several theoretical@12# and experimen-tal works@4,7,13,14# tested with success the applicability othe theory for density fluctuations involving even the mlecular orientational degrees of freedom, as it happensexperiments based on optical Kerr effect.

The optical Kerr effect~OKE! spectroscopy techniqumeasures the evolution of the derivative of the correlatfunction of dielectric susceptibility related to orientationdegrees of freedom. Based on a pump and probe pulsednique, it allows to measure the relaxation process directlythe time domain, covering a dynamical range from 100 fs4 ns with a good signal to noise ratio. These characterismake OKE spectroscopy particularly important in probithe dynamics of liquids in the hardly supercooled phasesuitable to check directly in the time domain the dynamevolution scenario predicted by the MCT@13#. Moreover, the

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D. PREVOSTOet al. PHYSICAL REVIEW E 66, 011502 ~2002!

possibility of measuring relaxation times as long as hundrof picosecond makes possible the comparison with relaxatimes measured by dielectric spectroscopy.

The comparison of experimental results obtained byferent techniques has been proved suitable for improvingknowledge on relaxation behavior of supercooled syste@5,10,11# but here it is employed using OKE and dielectrmeasurements on the same compound, the phenyl glycether~PGE!. PGE is an epoxy resin that behaves as a fraglass forming; the presence of the epoxy group and phenrings makes this system particularly suitable for both dieltric and optical study. Moreover, due to the relatively grevalue of fragility, it presents a huge increase in the relaxattime as the temperature decreases resulting in a good syfor testing the glass transition theories.

The present paper is organized in five sections. In Sethe MCT model is presented focusing on those elementinterest for our test. Section III is devoted to the descriptof experimental procedure focusing on the connectiontween the measured signal and the correlation function ofsusceptibilities and also on the description of the experimtal apparatus. In Sec. IV we present the analysis and dission of the results: in the first three parts of the sectionpresent the check of the MCT while in the last part a coparison with dielectric data is proposed. In Sec. V finasome final remarks are added to the discussion.

II. MODE-COUPLING THEORY

Since many reviews and papers about mode-couptheories@15# are present in literature, here we only recsome results relevant for this work. The ‘‘idealized MCTmodel is based on a number of coupled integro-differenequations that define the evolution of the normalized denfluctuations correlation functionsF(q,t). These equationsread

]2F~q,t !

]t21Vq

2F~q,t !1hq

]F~q,t !

]t

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t

dt8mq~ t2t8!]F~q,t8!

]t850, ~1!

whereVq2 is a characteristic frequency of the system,hq is a

friction coefficient due to the contribution of the fast degreof freedom, andmq(t) is the memory function that in thehard-sphere model, also called ‘‘microscopic or full MCTis written as a quadratic function of the density correlatfunctions@16#. The memory function produces the couplinbetween correlators characterized by different wave vecq. The full MCT presents the asymptotic solutions that avalid when the temperature is approachingTc , the criticalvalue. The asymptotic solutions predict some analytical futions that describe the correlators relaxation. These functapply for all the correlators and theq dependence appeaonly in the prefactor parameters and it does affect the mrelaxation features@16#. Since this work is concerning onl

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with a single wave vectorq;0 , see Sec. III A, we do noreport theq dependence in the following MCT predictions

The asymptotic solutions provide a two-step scenariothe relaxation dynamics: a short time region calledb regime,that can be subdivided into the so-called ‘‘critical’’ regiofollowed by the ‘‘von Schweidler’’~vS! one, and a long timeregion calleda regime. The MCTb regime, also quoted ab-fast regime, has different nature and features from thostheb relaxation observed in the dielectric spectra, also cab-slow relaxation@5#. In both the vS anda regions the MCTpredicts that the density correlators should satisfy the ttemperature superposition principle~TTSP!.

HenceF(t), for any temperature, is described by a singfunctional form

Fa~ t !5 f cFa~ t/ta!, ~2!

where f c is the nonergodicity parameter andta is thetemperature-dependent time scale that follows this scalaw:

ta}~T2Tc!2g. ~3!

In this equationTc is the ‘‘critical temperature’’ of the MCTand g is a material-dependent parameter. TheTc and g arekey parameters for the analysis of MCT: in factTc definesthe temperature where the predicted ergodic-nonergodic tsition takes place andg characterizes the relaxation processes present in the glass former.

In the long time part of thea regime the functionalFacan be reproduced by the stretched-exponential functalso called Kohlrausch-William-Watts~KWW!:

Fa~ t !5 f c exp@2~ t/ta!b#, ~4!

whereb is the stretching parameter andta is again the char-acteristic relaxation time in this region. Bothf c and b aretemperature independent, whileta scales according to Eq.~3!.

The b regime, apart from the shorter time microscopdynamics, is described by an universal relaxation functio

Fb~ t !5 f c1husu1/2gl~ t/ts!, ~5!

wheres5(Tc2T)/Tc , h is an amplitude factor, andts is asecond characteristic time scale. Thegl(t/ts) is a functiondependent only on the parameterl and is defined by thescaling equation of motion@15,16#. The parametersf c andhare temperature independent while the characteristic timdescribed by a second scaling law:

ts}~T2Tc!21/2a. ~6!

The parametersa and l are material dependent and, as wwill show in the following, they are unambiguously anuniquely related withg. The scaling equation can be solveanalytically in the asymptotic limit,usu,1. The leading or-der solutions are the ‘‘critical decay’’gl(t/ts)5(t/ts)21/2a

in the very short time regionV21!t,ts , and the vS decaygl(t/ts)52B(t/ts)b in the following time regionts,t!ta . The vS decay can be extended up to longer timet

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RELAXATION PROCESSES IN AN EPOXY RESIN . . . PHYSICAL REVIEW E66, 011502 ~2002!

;ta , thanks to the next-to-leading order correction@16#,hence, forts,t,ta , the density correlator can be describas

Fvs~ t !5 f c2hB~ t/ta!b@12K~ t/ta!b#. ~7!

Theb,B, andK parameters are temperature independent.parametersb,a,g, and l can be related through the Eulefunction G by the equations:g51/2a11/2b and l5@G(11a)#2/G(112a)5@G(11b)#2/G(112b).

An alternative solution of the scaling equation of motiothat applies in the wholeb region forV21!t!ta , can beobtained by a numerical calculation. The calculatedgl(t)and so the relative correlator, see Eq.~5!, are called the fullb-correlator solutions.

A different route for data interpretation is based on coparison of the measured data with the complete solutiobtained from schematic MCT models@17#. Recently Simio-nesco and Krauzman@18# proposed a ‘‘two-correlators’MCT model to fit light-scattering data of some glass formin order to reproduce properly the spectra in the high fquency region. Indeed, they have been able to fit the spefeatures in a very accurate way, including the microscoand boson peaks features where, vice versa, the ‘‘ocorrelator’’ schematic MCT model partially failed. Neverthless some of the approximations used in this model aresafely acceptable. A ‘‘slave-correlator’’ MCT model was sugested by Sjo¨gren @19# for the description of tagged-particlmotion in a glassy environment, and successfully usedFranosch and co-workers@20# and by Go¨tze @21# to fit mea-sured spectra of some fragile glass formers. The main adtage of these approaches is that they do not rely on theplicability of asymptotic formulas and that all the resultscrossover phenomena and preasymptotic correctionslogically included in agreement with the MCT. Furthermothese theoretical models improve the analysis of the speallowing to extend the fitting time and temperature regiobut unfortunately they introduce several substantial numcal and fitting complications. Nevertheless, some authfind that the asymptotic formulas qualitatively give an aequate description of the data@21,22#. In fact, the analysis oflight scattering data based on the MCT asymptotic solutior the two-correlators MCT model give substantially tsame value for the MCT parameters,l andTc .

The MCT was first formulated for correlation function othe density fluctuations, see Eq.~1!, but it should also applyfor any observablex that has superposition with density flutuation, i.e., r(t)x(t8)&Þ0, @15,16#. Further in the last yearsseveral theoretical@12# and experimental works@4,7,13,14#proved with success the applicability of the theory to tmolecular orientational degrees of freedom. For this reasois significant to test the theory with measurements collecby OKE experiment@13#.

Finally we would like to stress that MCT predicts a dtailed relaxation pattern for the glass-former dynamics wthe parametersa,b,g, and l that are strongly related ancharacterized by a defined temperature dependence. Soof MCT predictions on the experimental results is in oopinion meaningful indeed.

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III. EXPERIMENTAL PROCEDURES

A. Time-resolved optical Kerr effect experiment with optical-heterodyne detection

In a transient OKE experiment, an excitation~or pump!polarized laser pulseEex induces an optical anisotropy in thsample. The relaxation towards the equilibrium of theduced birefringence is probed through the polarizatchange of a second polarized laser pulseEpr that propagatesin the liquid nearly in the same direction of the excitinbeam. Increasing the temporal delay between the excitaand the probing pulses allows to measure the relaxationthe induced anisotropy directly in the time domain. In oexperiment, the intensity of the depolarized componentsthe probe beam, which represents the signal fieldEs , is re-vealed through a heterodyne detection scheme. In this cfiguration Es interferes on the detector with an extra fielcalled local fieldEloc . TheEloc is generated byEpr and it isdirectly proportional to it. The detected intensity is the suof three contributions: the intensity ofEloc , the homodynecontribution, and the heterodyne one. The heterodyne t~HD! is the interesting signal in this experiment and it canwritten as@23#

SHD~t!}2ReS ihE2`

1`

dtUEpr~ t2t!U2E2`

1`

dt8R~ t

2t8!UEex~ t8!U2D , ~8!

whereh is a complex constant that depends on the wayElocis generated~see Sec. III B! andR(t) is the response functionof the system.

The response of the system is directly related to the crelation function of the dielectric constant («xy) of the liquid@24,25#

R~q,t2t8!}2u~ t2t8!

KBT

]

]t^«xy~2q,t8!«xy~q,t !&, ~9!

whereKB is the Boltzmann constant,T is the temperatureu(t) the Heaviside step function, andq is the wave vectorcharacterizing the spatial modulation of the anisotropy. Tinduced anisotropy is characterized by a wave vector thaalmost zero (q;0). When the exciting and probing pulseduration is short~in our experiment 50 fs! compared to thecharacteristic response time of the system, the heterodsignal is proportional to the response function of the systSHD(t)}R(t). So the HD-OKE technique measures the mterial response function directly in the time domain andcording to the Eq.~9!

SHD~ t !}21

T

]

]tF««~ t !, ~10!

whereF«« is the correlation function of dielectric constan

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B. Laser system and optical setup

The laser system used for the HD-OKE experimentcomposed of a Kerr lens mode-locked Ti:Sapphire oscillaand a chirped pulse amplification system@26#. The amplifi-cation system is a regenerative Ti:Sapphire cavity pumby a Nd-YLF ~yttrium lithium fluoride! laser, it delivers500 mJ, 50-fs pulses at a repetition rate of 1 kHz. In texperimental setup, see Fig. 1, the laser beam is dividedthe exciting and the probing beam~about 80%-20%! by abeam splitter BS1. The exciting beam arrives directly onsample passing through a polarizerP0; the probing beamgoes through an optical delay line controlled by a computhen it is splitted~BS2! generating the so-called referenbeam and finally it arrives to the sample through a polariP1 that selects the direction of polarization (45° respecthe pump beam!. Both theEex and Epr are directed on thesample by a 600-mm focal length lensL1 that realizes thespatial superposition of the two pulses. A polarizerP2crossed respect toP1 is placed after the sample; a wide bal/4 plate is placed betweenP1 and the sample and adjustefor minimum leakage of theP2 polarizer in presence of thsample and ofL1. Eloc used for the heterodyne detectiongenerated by a small rotation of thel/4 plate~in our experi-ment the rotation is of an angle«;3°) and it is proportionalto the probe field:Eloc5hEpr . In this configuration thehconstant that appears in the heterodyne term, see Eq.~8!, ish5«1 i« @23#. Actually this experimental procedure pro

FIG. 1. Experimental setup used for OKE measurement. Band BS2, beam splitters;P0, P1, andP2, Glann-laser polarizersL1 andL2, quartz lenses;F, optical cleavage; ND, variable neutradensity filter; DA, differential amplifier; ODL, optical delay line.

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duces both a birefringent and a dichroic contribution. Indethe dichroic signal is very small and it can be safely nglected@27#.

The signal, so generated, contains three terms: a conintensity produced byEloc , a homodyneSHO , and a hetero-dyne termSHD . To eliminate the first constant term the sinal is revealed through a differential photodiodes systeone diodeD1 measures the whole signal coming from tsample; the other diodeD2 is balanced with the referencbeam to compensate theEloc intensity. The output of thedifferential is set to zero, in absence of the pump beam,means of a continuously variable neutral density filter placin front of D2 diode. A boxcar gated integrator, synchronizto a mechanical chopper placed on the exciting beam,lects the output of the differential diode amplifier allowinan automatic subtraction of the other backgrounds presenthe signal. The homodyne contribution is eliminated by sutracting two different measurements taken with oppositerection of the rotation of thel/4 plate. In fact, these twomeasurements are characterized by positive terms forSHOand by theSHD signal with opposite signs. A set of calibrateneutral density filters were used in order to increase thenamic range of the apparatus.

C. Data collection

The HD-OKE data are composed by a number of timpoints corresponding to the scan of the optical delay liThe present data have from 1800~high! to 2500 ~low-temperature data! time-points. For each time point the signintensity is the average over 500 laser pulses. The wideperimental time window is collected with three differescans composed of time points spaced in different way: fr0 ps to 4 ps we use a time step of 5 fs, from 2 ps to 18 psuse a time-step of 10 fs, and for the remaining time winda logarithmic time step. The scans are matched to rebuisingle data, using an extreme attention to the signal contency in the overlapping time regions. We used laser puof different energy and temporal length to optimize the sigintensity: pump pulses of 50 fs and 1.2mJ ~transform lim-ited! are employed in the short scan and pump pulses of 1and 12 mJ ~chirped! in the long scan. With longer lasepulses it is possible to transfer a larger quantity of energythe system without increase the instantaneous intensity opulse. The signal is then characterized by a better signoise ratio that allows to reach longer temporal region, ascan see in Fig. 2. Using the heterodyne detectionstretched pulses more than five intensity decades and mthan six temporal decades signals are detected.

PGE sample~Fig. 3!, provided from Aldrich with a puritygreater than 99%, was used without further purifying. Tsample was placed in a 2-mm-thick quartz cell of very gooptical quality. The temperature was controlled by a cryossystem within60.1 K.

The HD-OKE data on PGE sample are collected intemperature interval 265.3 K–352.5 K. In Fig. 3 we repall the experimental data normalized to the instantaneelectronic response peak.

IV. ANALYSIS AND DISCUSSION

OKE measurements show clearly three different temporegions~Fig. 3!: the first one (;02200 fs) is characterized

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RELAXATION PROCESSES IN AN EPOXY RESIN . . . PHYSICAL REVIEW E66, 011502 ~2002!

by an instantaneous response that has to be mainly addreto the electronic dynamics, the second one~from ;200 fs toabout 100 ps depending on temperature! where the responsis weakly dependent on temperature, and the third regwhere the signal exhibits a slow relaxation strongly depdent on temperature. In the first picoseconds, OKE measments show the presence of the oscillatory dynamics duintramolecular vibrational degrees of freedom, correspoing to Raman frequencies:y>245 cm21 and 332 cm21. Aswe have seen in Sec. II, the MCT predictions concerncorrelation function behavior, while OKE measurementsproportional to the system response function, i.e., to the tderivative of the correlation function of the dielectric costant@Eq. 10#. So, in order to check the MCT predictions, thOKE data have to be compared with the derivative of

FIG. 2. Heterodyne detected OKE signals (R in arbitrary unit!of PGE at 322.9 K represented in a semilogarithmic plot. The ctinuous line represents a measurement performed with 50-fs puthe dotted line is the measurement with 1-ps pulses.

FIG. 3. Bilogarithmic plot of OKE signals vs timet of PGE forthe temperatures: 265.3, 273.8, 283.5, 293.1, 303.4, 312.9, 3333.4, 343.0, and 352.5 K.

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MCT functional. In theb region the response function mube reproduced by the time derivative of Eq.~5!, that is,

Rb~ t !52xs

T

]gl~ t/ts!

]~ t/ts!, ~11!

where xs5husu1/2ts21}(T2Tc)

(a11)/2a. In the a regionR(t) should be the time derivative of Eq.~2!:

Ra~ t !52xa

T

]

]~ t/ta!Fa~ t/ta!, ~12!

wherexa} f cta21}(T2Tc)

g. Also in the last part of thearegime the MCT theory predicts an analytical form of tcorrelator, the KWW law@Eq. 4#, so the response functiobecomes

Ra~ t !} f cS b

taD S t

taD b21

exp@2~ t/ta!b#. ~13!

In the vS time region the MCT predicts vS law@Eq. ~7!# thatgives the following response function:

Rvs~ t !}B1tb211B2t2b21, ~14!

where B15hBb/tab and B2522hBKb/ta

2b with referenceto notation of Eq.~7!.

One difficulty of testing the MCT predictions is to estalish the temporal range of validity of thea andb regime. Toovercome this difficulty we used the following procedurFirst we tested the existence of a temporal region in agment with the TTSP: in doing this we have superimposedfinal part of each curve by scaling them by an amplitude aa temporal factor. In this way it was possible to find tvalues of the amplitude factorxa and of the scaling timeta .Within the region of validity of the TTSP we fitted eacexperimental curve toRa(t) @Eq. ~13!#. The temporal win-dow has been selected to optimize the fit quality and the bvalues of the parameters were found through a least sqfitting based on a Levenberg-Marquardt method. To repduce the experimental curve in the faster temporal reglimited at long times by the validity of the KWW law, wetested the validity of the vS law with the next-to-leadinorder correction@Eq. ~14!#. As before, the fit time intervawas chosen to optimize the fit quality. From vS fit it has bepossible to obtain the values of the amplitude parametersB1andB2 and of the exponentb that is connected to exponential parameterl.

The analysis of the first picoseconds OKE measuremeis troubled by presence of intramolecular oscillations tprevents a safe and independent test of the critical decathis time region. In order to extend signal analysis to the ftime region we used the MCT fullb correlator, Eq.~5!.Using a calculus program we produce a numerical cuequivalent to the derivative of @Fb(t)2 f c#51h8usu1/2gl(t8). The program, for any value of the inpuparameters (l andTc), gives numerical curve where the timt8 and amplitudeh8 scale are in arbitrary units and they hato be normalized to the experimental scales. So compa

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D. PREVOSTOet al. PHYSICAL REVIEW E 66, 011502 ~2002!

the full b-correlator curves to the OKE data we found tparametersxs andts for each temperature and hence frotheir scaling law the value ofl andTc . Finally we comparedthe values ofl andTc found from the different analysis (aregime, vS regime, and fullb correlator!.

A. Master plot and time temperature superposition principle

The first objective of the analysis was to check the TTand to define the temporal region where it applies. We hrescaled all the OKE signals, corresponding to different teperatures, in time and amplitude on 265.3 K signal, whichthe lowest measured temperature. As we can see in Fig.the signals rescale onto a master curve. The time scafactors have been subsequently multiplied byta obtainedfrom the KWW fit of the 265.3 K curve in order to get threal values of thea-relaxation times. The scaling proceduallows to get the amplitudexa and the timeta scales of thea regime, see Eq.~12!, and it turns out to be a precise procedure thanks to the signal-noise ratio. Fromxa andta wecan calculate the value of a parameter proportional tof c ; infact from Eq.~12! f c}xataT. Errors affectingta were com-puted propagating the error of the corresponding scatemporal factor with that of the value ofta of the curve at265.3 K, while that off c is computed from the errors of thtwo scaling factors.

The values of the relaxation timeta are presented in Fig5~b! in a linearized plot with the relative fit according to E3. The theoretical curve well reproduces experimental dby yielding Tc5(22963) K and g52.660.2 correspond-ing to l50.7660.04. Also in some other molecular glaformers it was found that the MCT describes the temperadependence ofta in the moderately viscous regime at higtemperature@7,5,13#.

In Fig. 5~a! the calculated values off c are presentedThough they are scattered they do not present a real tand assuming, as expected from the theory, a constant v

FIG. 4. Bilogarithmic plot representing the master plot of OKdata obtained by scaling the time and amplitude of each signalsthe signals are superimposed, in the long time region, takingreference the measurement at 265.3 K.

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it is possible to calculate its mean value:f c5106611 ~onestandard deviation is taken as error!. In a previous analysis oOKE measurements ofm toluidine @13# it was found thatf cbecomes larger as temperature decreases. Deviation fromprediction of the MCT was found in the analysis of dielectdata and polarized Brillouin light scattering on propylecarbonate but the same authors move some doubts atheir evaluation off c @21,28#. However, in recent analysis opolarized Brillouin light-scattering analysis ofm toluidineand toluene a temperature-independent value off c was foundfor T.Tc @22,29#.

B. Slow dynamics and the MCTa regime

The second step of the analysis was testing the KWWthe long time relaxation region. The derivative of the KWlaw @Eq. ~13!# well reproduces the experimental curves in tlong times region as we can see in Fig. 6 where four of thare represented with respective fit in a log-log plot. Itimportant to note that this fitting procedure is independ

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FIG. 5. ~a! Plot of the nonergodicity factorf c ~squares! vs tem-peratureT obtained from the time temperature superposition prciple. ~b! Plot of ta ~squares! vs temperatureT as deduced from thescaling procedure~see text!. The continuous line represents the linear fit according to the scaling law Eq.~3! while the dashed one isits extrapolation down toTc . The errors are not reported becauthey are smaller than the symbols.

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from the verification of the TTSP and then it is a secoindependent test of thea regime of the MCT. Thanks to thegood signal-noise ratio it was possible to extract the valof f c ,ta , andb for each temperature. In Table I we presethe values ofta and f c as obtained from the fit and from thscaling procedure of the master plot and also the valuesbfrom the fit.

As we can see from Table I the parameterb does notpresent any temperature dependence: in fact, thoughspread is wide, all the values, except that at 333.4 K,consistent with each other in the error bar. The mean valuthis set of data resultsb50.91. The temperature indepedence ofb is in agreement with the MCT and it is a necesary condition to thea-relaxation process to obey the TTSAlso in the previous OKE study onm toluidine @13# andlight-scattering studies of toluene and propylene carbonaconstant value ofb was found@28,22#. However, in othercases analysis of dielectric or light-scattering measuremperformed in the frequency domain reveals an increase ob

TABLE I. Values of the parameters related to the MCTa re-gime as from the fit with the derivative of the KWW~last threecolumns! and from the master plot scaling procedure~second andthird columns!.

T (K) f c (a.u.) ta (ps) fc (a.u.) ta (ps) b

265.3 11669 610641 12568 588625 0.8860.04273.8 11767 371634 12065 373611 0.9360.03283.5 10862 21867 11363 21867 0.9260.02293.1 10665 13569 10863 13664 0.9260.01303.4 10167 9267 10363 9663 0.9060.02312.9 10467 6864 11063 6962 0.9360.03322.9 8165 4864 8463 5062 0.9360.02333.4 11569 3963 11767 3862 0.8760.02343.0 111614 3364 11763 3262 0.9060.02352.5 10568 2562 10267 2762 0.9360.03

FIG. 6. Plot of the long time region of the OKE signals with thcorresponding fit with the KWW law. From top to bottom thcurves correspond to 352.5, 312.9, 393.1, and 265.3 K.

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with T @5,29,30# with a tendency to almost saturate at a ctain value at high temperature. Finally it is noteworthynote that recently some photocorrelation measurementsPGE were published corresponding to a temperature rabelow 211 K, i.e., belowTc @31#. Even at this low temperature the long time region of the relaxation process canfitted to a stretched exponential law with a constantlower value of the parameterb. The difference in thebvalue obtained from OKE and photocorrelation measuments can be ascribed to the different experimental teniques used or to the two different temperature intervalvestigated, which could be separated by a temperature recharacterized by a decreasing ofb. Some further studyingwould be necessary to choose between these two possties.

The values off c are presented in Fig. 7~a!: it is peculiar tonote that nearly the same spread is present in the set ofues of f c obtained both from the master plot@Fig. 5~a!# andthe fit. This evidence seems to reveal a slight dependencthe parameter from the temperature. However the accuof the values does not allow to confirm this hypothesis.fact, while the values at lower temperatures seems to prea linear increase we must note that the increasing overwhole temperature range is only of 6.8% that is of the saorder of the error of the value at higher temperatures. Weconclude that according to the experimental uncertaintyf cpresents no temperature dependence in this region asdicted from the theory. The mean value obtained from theis f c5110612 and it is consistent with that obtained frothe master plot.

Finally the characteristic timesta obtained from the fitgive a second test of the scaling law provided for by tMCT. As we can see in the Table I these values of the relation time are very similar to those obtained from the masplot scaling procedure as expected from the theory. Onlyvalues at 265.3 K disagree in a more pronounced way thothey are consistent in the error bar. However, we must nthat the measurement at this temperature is on the limitthe time window of the experiment and this fact results inless good evaluation of the timeta as evident from the greaerror assigned to it. Values ofta(T) from the KWW functionwere fitted to Eq. 3 that led toTc5(22765) K and g52.760.2 corresponding tol50.7860.04 @Fig. 7~b!#. Allthese values of the parameters are completely consistentthose found by the master plot. This is an important proofconsistency because though the master plot gives less prvalues ofta , it is a fitting independent and simple methodobtainingta .

C. Fast dynamics and the MCTb regime

In the intermediate time region we fitted the signal to tvS law with the next-to-leading-order correction@Eq. ~14!#.In Fig. 8 four measurements with related fitting curves arepresented; we can note that the vS law reproduces aporal region that increases with decreasing temperatFrom a least square fitting procedure we extracted the vaof the parameters. The parameterb presents a set of valuescattered around the mean valueb50.5960.03 that corre-

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sponds tol50.7360.02. This value is in good agreemewith that obtained from thea-regime analysis. Moreoverfrom the amplitude parametersB1 andB2 we can extract thevalues of the second-order correction parameterK, in factK52B2 /B1ta

b @Eq. ~14!#. In this calculation we have usethe values ofta obtained from the fit with the KWW law andb from the previous analysis. This procedure gives valueK affected by a great error but at the moment this is the oexperimental method to obtain theK parameter. We find, inthe experimental uncertainty, no temperature dependenctheK parameter@Fig. 9# as predicted from MCT theory. Thmean value from this analysis isK50.2760.03 that is inagreement with the value of the same parameter as froprevious work on them-toluidine @13#.

Finally a deeper analysis of the fast time region has bperformed using the MCT fullb correlator. We used a calculus program to produce numerical curves equivalent totime derivative of theb correlator. For each couple ofl andTc parameters the program produces a family of numercurves each of them defined by the experimental tempera

FIG. 7. ~a! Plot of the nonergodicity factorf c ~squares! vs tem-peratureT. The values off c are obtained from the fitting proceduremployed with the derivative of the KWW law.~b! Linearized plotof ta ~squares! vs temperatureT with the correspondent fit according to the scaling law, Eq.~3!. The dashed line represents the etrapolation of the fit down toTc . The errors are not reported because they are smaller than the symbols.

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T. These curves should reproduce the experimental data afrom an amplitude and a temporal factor independent oftemperature. With the OKE data this was only partly posible: indeed we found a single temporal factor, valid fortemperatures, but the amplitude factor has to be variedeach temperature. Finally we have chosen theb-correlator curves corresponding to an unique value oland Tc , that are able to reproduce the largest numberexperimental data in the widest temporal range. Fromanalysis we found that according to our judgementl50.73and Tc5230 K give the best fit. However, curves relativto l50.76 andTc5230 K and l50.69 andTc5233 Kalso fit the experimental data but in a shorter temporalgion. Since this analysis is not based on a least square fiprocedure it is not easy to choose the best values forparameters, nevertheless we estimated a 5.4% error foexponential parameter (l50.7360.04) and a 1.3% error fothe critical temperature (Tc523063K). As we can see in

FIG. 8. Plot of the intermediate region of the OKE signal for ttemperature 352.5, 312.9, 293.1, and 265.3 K with the correspdent fit according the vS law.

FIG. 9. Values of the amplitude parameterK of the next-to-leading-order correction of the vS law@see Eq.~7!#.

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Fig. 10~a! the fit with the full b correlator is fairly good inthe temperature range from 265.3 K to 303.4 K while taccordance is less reasonable on the upper temperatureticularly for T.322.9 K@Fig. 10~b!#. On the other hand theMCT predicts the fullb correlator as a law effective intemperature region nearTc while these last three curves corespond to measurements at temperature more than 10higher than the critical temperature@ u(Tc2T)/Tcu.0.45#.So it does not surprise if the predictions of the theory areverified for the high-temperature range. Furthermore,note that the fullb correlator superimpose to the vS law onin the measurements corresponding to lower temperat(T,293.1 K). The values found forl and Tc are in verygood agreement with those obtained from the analysis oflong time (a region! and intermediate time~vS region! re-gions. As we already explained the scaling procedure offull b correlator needed one temporal rescaling factor obut different amplitude factors. However, calculating the n

FIG. 10. Bilogarithmic plot of the short time region of OKsignals with the fit according to the derivative of the fub-correlator function~continuous line! for temperature from 265.3to 303.4 K@panel~a!# and for temperature from 312.9 to 352.5@panel~b!#. In panel~a! we can note a good superposition betweexperimental and theoretical curves, while for the curves in pa~b! the agreement is less reasonable.

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amplitudesxs of theb correlator for each measurement witemperatureT,322.9 K, we can still represent them inlinearized plot,xs

2a/(a11) vs T @Fig. 11#, by using the valuea50.31 corresponding tol50.73. The best value ofTc thatallows to reproduce by a linear fit isTc5233 K @continuousline in Fig. 11#, which is greater than those found in thprevious analysis. Indeed, as we showed, the fullb cor-relator is not able to reproduce properly the experimencurves forT.303.4 K and so we tried to fit only the valueof xs

2a/(a11) up to this temperature. The new linear fit giveTc5230 K @dashed line in Fig. 11#, which is in reasonableagreement with that found from the analysis of thea region.The agreement of the various values ofl found from theanalysis of the different temporal regions proves the reliaity of the connection between the parameters characterithe long time regiong and the vS regionb and their depen-dence from the unique parameterl.

D. Comparison between results obtained by OKE and bydielectric spectroscopy

Comparison between experimental results obtaineddifferent spectroscopic techniques can improve the undstanding of glass-former dynamics. Several studies alreexist on this subject but, unlike its importance, the genescenario is still unclear. The difficulty of this kind of analysoriginates both from considering data that often coverdiffent spectral ranges and from the fact that the dynamics ofsystem is investigated by different probes. The present stto the best of our knowledge, is the first attempt of perforing such a comparison between OKE and dielectric measments. There are two main difficulties in the present coparison: the data are collected in different doma~frequency and time!; both techniques measure collective r

el

FIG. 11. Values of the amplitude parameterxs vs temperatureTas obtained from the fit with the fullb correlator~see text!. Thevalues are represented in a linearized plot. Only the values foT,322.9 K are shown. Two fits are presented~straight lines!: thefirst one is over the whole temperature range (Tc5233 K) whilethe second one is performed on data forT,303.4 K (Tc

5230 K). The dashed and the dotted lines represent the extraption of the curve fit down toTc .

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orientation times but these are related to different correlafunctions characterized by different orders of the Legenpolynomials. Nevertheless a comparison between the chateristic time scales, obtained by these two techniques, cindicate the consistency of the relaxation processes.

Dielectric measurements on PGE were carried out usthe frequency domain reflectometry technique. The measments were carried out on the frequency interval betw100 kHz and 30 GHz at temperatures between 353.1 K245.6 K. We employed two different experimental appatuses: the network analyzer HP8753ES withC0>3.5 pFwas employed in the frequency ranges 53104 Hz–1.83109 Hz and the network analyzer HP8722D withC0

>0.15 pF in the range 53107 Hz–431010 Hz. More de-tails about experimental apparatuses are elsewhere rep@32#. In the investigated temperature range, the spectra odielectric constant« clearly show the presence of the strutural relaxationa which shifts towards lower frequencies atemperature decreases, according to the behavior of thetical measurements. Indeed, our dielectric spectral raneven covering more than 6 decades, does not allow to rethe high frequency region that corresponds to theb regionpredicted by the MCT. The experimental curves were fitto the phenomenological Cole Davidson function@33# «(v)2«`5D«/@11( ivta

cd)bcd#, where«` is the high frequencylimit of «(v),bcd is a shape parameter,D« is the relaxationstrength, andta

cd is the characteristic time of the relaxatioThe complete analysis of the dielectric parameters is beythe aim of the present paper and will be presented in a focoming paper.

In order to connect the dielectric and OKE resultstransformed the dielectric relaxation timeta

cd , measured inthe frequency domain, into the corresponding decay timetaof the KWW function using an appropriate numerical retion @34#. By comparing the transformed dielectric relaxatitimes with OKE ones, we found the former to be appreciashorter than the latter@Fig. 12~a!#. Nevertheless the temperature dependence of the two parameters is almost the swhich would confirm that the two techniques detect the sastructural process. In fact, we get a good superposition oftwo time scales rescaling the OKE relaxation times bytemperature independent factor,J50.58. This superpositionapplies in the whole temperature range where data from bthe techniques are available@Fig. 12~b!#.

The Debye rotational diffusion model makes precise pdictions about the dynamical properties of a symmetric-molecule diffusing through a Brownian rotational motio@35#. In particular it predicts that the rotational correlatiofunction relaxes exponentially with a relaxation time definby the order of the Legendre polynomial. Accordingstrong approximations, the dielectric response can be cnected with the first-order Legendre polynomial andOKE to the second-order Legendre polynomial. Followithe Debye model we should findtdiel /toke53, but this pre-diction is not supported by our experiments since we fitdiel /toke50.58. A different model for the rotational dynamics of a symmetric-top molecule has been made by Iva@36#, describing the process as a continuous jump proc

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According to this theoretical model the ratio of the two timscales should betdiel /toke51. So again our experiment doenot support the model. However these disagreements dosurprise, since the approximations used in these theoremodels are indeed too strong.

Furthermore, our findings about the comparison betwdielectric and optical characteristic time scales qualitativagree with previous light-scattering and dielectric expements on polystyrene and diglycidyl ether of bisphenoA~DGEBA! @11,37#. Comezet al. @11# proposed to single outhe cause of such behavior in the different microscopic prthat allows the determination of the dielectric and of toptical measurements, namely, for DGEBA~and also forPGE!, the epoxy ring and the aromatic ring, respectively.particular, they suggested that the different sizes of suchlecular subunits cancorrespond to a difference of the eftive hydrodynamic volumeVh , which enters in the DebyeStokes-Einstein~DSE! equation,

t5Vhh/kT,

whereh is the shear viscosity of the system. Consequenone obtains different relaxation times for the response fu

FIG. 12. ~a! Plot of OKE ~up triangle! and dielectric~opencircle! ta vs temperatureT. We note that the former are larger thathe latter.~b! Plot of rescaled values of thetoke ~up triangle! andvalues of the dielectric time~open circle!. We can note that the twotime scales superimpose by scaling for a constant factor~0.58!.

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tion of the two observables@11#. In particular, for light-scattering and dielectric relaxation time of DGEBA thfound an almost constant factorJ50.13 between the twotime scales in a restricted temperature range extending aTc.1.2 Tg . The authors speculated that approachingTg thehydrodynamic volume characteristic of the dipolar oriention remains constant, which was supported by the validof the DSE equation for DGEBA, in a temperature intervof more than 100 K. On the contrary, the optical effectivolume could decrease with decreasing temperature, dmining the approach of the dielectric and optical characistic time scales observed for DGEBA for temperatureT,1.2Tg . This fact cannot be verified on PGE, due to the lebroad temperature interval where the OKE time scale hbeen measured.

V. SUMMARY AND FINAL REMARKS

In this paper we have presented measurements of threfringence relaxation carried out by OKE experiment antest of the idealized MCT. We have also compared the cacteristic times of thea-relaxation process present in OKmeasurements with those of the structural process of dietric spectra. The use of picosecond pulses and of the hedyne detection system in the OKE experiment allowed msuring the relaxation processes over a large temporal wintill 4 ns with a good signal-noise ratio. So it has been psible to test the MCT in a wide temperature interval coverthe liquid and supercooled region.

As we have seen in Sec. IV OKE signal presents thtemporal region. We showed that the functional forms pdicted by the idealized MCT~full b correlator, vS next-to-leading-order corrected and KWW laws! reproduce the sig-nal in the whole time region, from about 1 ps, almost forthe temperatures investigated. Only the microscopic dynics due to electronic response of the system and to the etation of internal degrees of freedom cannot be reprodusince the idealized MCT does not include such degreefreedom. As an example of the ability of the theory to repduce experimental data, in Fig. 13 we show the OKE sigat 273.8 K and the corresponding fitting curves. Fromdata analysis we obtained, for each temperature, the vaof the parameters characterizing the relaxation processeparticular, from the TTSP and from the KWW fits we otained agreeing values ofta and f c , whose temperature dependence is well predicted by the MCT. Moreover tstretching parameterb does not depend on temperature asis provided by the theory and as it is required for the validof the TTSP. The vS fit gives us values ofb and of theamplitude parameterK independent from temperature as itstated from MCT. It is noteworthy that until now this is thsecond experimental measurement ofK parameter and thetwo available K values, corresponding to different glaformers, are substantially identical. These preliminary ressuggest a very slight dependence ofK from the molecularproperties of the glass former. Finally we also noted tb-correlator solution applies in a shorter temperature rathan the TTSP. We performed the data analysis using dif

01150

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ent checks of the MCT: the TTSP and the KWW law in tlong times relaxation region, the fullb correlator in the fasttimes region. So we extracted in three independent wayscharacteristic parameters of the MCT theory:l and Tc .Their values are reported in Table II: they are in very goagreement. This is indeed a severe check of consistencMCT that has been completely passed in the present wo

Here a comparison of structural relaxation measuredOKE and dielectric spectroscopy is presented. In a wide teperature region, aboveTc , we found the structural relaxatiotimesta of optical processes larger than the dielectric onNevertheless the temperature dependence of these two srelaxation times is substantially identical, in fact it is posible to superimpose them by a temperature-independenttor. These results suggest the presence of a unique slowdown effects that shows up on different time scaldepending on the observable probed. So concerning thedynamics, the present work is confirming a universal aspof the structural relaxation properties. Undoubtedly, the coparison between results obtained by different experimetechniques is a very interesting topic that only recenstarted to be systematically studied. However, this is a vcomplex issue that needs many further theoretical andperimental works in order to give a complete understandi

TABLE II. Values of the characteristic parameters of the theoas from the master plot scaling procedure, the fit with the derivaof the KWW, and the fit with the derivative of the fullb correlator.

TTSP KWW Full b correlator

Tc (K) 22963 22765 23063

l 0.7660.04 0.7860.04 0.7360.04

FIG. 13. Bilogarithmic plot of the OKE signal at 273.8 K vtime t and of the corresponding fit according to MCT laws: tderivative of the fullb correlator~dashed line!, the derivative of vSlaw ~dotted line!, and the derivative of the KWW law~dashed-dotted line!.

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ACKNOWLEDGMENTS

We thank W. Go¨tze, Th. Voigtmann, and F. Sciortino fovery helpful suggestions and discussions, and for providthe computer program used to generate the fullb correlator.

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This work was supported by the Commission of the Eupean Communities through Contract No HPRI-CT19900111, by MURST cofin 2000, and by INFM through ProjeNo. TREB-Sez.C-PAISS1999.

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