PHYSICAL REVIEW E JANUARY 2000VOLUME 61, NUMBER 1

Scaling behavior of thea relaxation in fragile glass-forming liquidsunder conditions of high compression

Marian Paluch*Max-Planck-Institut fu¨r Polymerforschung, Postfach 31 48, D-55021 Mainz, Germany

Stella Hensel-Bielo´wka and Jerzy ZiołoInstitute of Physics, Silesian University, ul. Uniersytecka 4, 40-007 Katowice, Poland

~Received 1 June 1999!

The effect of pressure variation on dynamics ofa relaxation process in poly@~phenyl glycidyl ether!-co-formaldehyde# has been investigated both under isothermal (T5293 K) and isobaric (P50.1, 60, 120, 180,and 240 MPa! conditions using broad band dielectric spectroscopy (1022 to 106 Hz). The a relaxation isanalyzed by means of the Havriliak-Negami relaxation function which has two shape parameters~a andg! tocharacterize non-Debye behavior. As a result we found that the shape parameters of the dielectric functioncollected for different pressures fall on master curves constructed by plottinga andag against the logarithmof relaxation time. The scaling of shape parameters for different pressure stems from pressure independence offragility. This provides strong experimental evidence supporting correlation of non-Debye behavior with non-Arrhenius relaxation under high pressure. From an analysis of the shape parameters of relaxation function, interms of the Schonhals and Schlosser model, we drew conclusions that the molecular mobility of PPG iscontrolled in the same way by temperature and pressure. The relaxation times exhibit a clear non-Arrheniusbehavior under isothermal and isobaric condition.

PACS number~s!: 64.70.Pf, 77.22.Gm, 65.50.1m

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INTRODUCTION

The vitrification process both in low molecular weigliquids and in complex systems, like polymers, is usuastudied as a function of temperature at ambient pressAlthough cooling of a metastable liquid is the procedumost often applied to induce the liquid-glass transitiondoes not mean that it is the only route leading to glassy s@1#. Among other ways there is also the possibility to genate glass transition by compressing liquid. The influencepressure on the dynamics of supercooled liquids is in mrespects similar to the effect of temperature@2#. In a previouspaper we showed that the isothermal pressure dependendielectric relaxation times~or viscosity! can be describedanalogously to temperature dependence at atmosphericsure @2,3#. As the glass transition is approached throucompression of the liquid, thea-relaxation timet increasesin the non-Arrhenius fashion and shows a curvature angous to the temperature dependence oft. As a result, it wasfound for fragile glass-forming liquids that isothermal dat(P) does not obey the simple volume activation modelt5t0 exp(PV/RT) @2#.

A non-Arrhenius temperature~or pressure! evolution ofrelaxation times is not the sole characteristic feature ofnamics of supercooled liquids and glasses. It has been knfor a long time that dynamics glass transition also featuthe non-Debye behavior of the relaxation function or equilently nonexponential character of response function@4#. Un-fortunately, only few experimental papers that analyzeinfluence of pressure on the shape dielectric function

*Electronic address: paluch@mpip-mainz.mpg.de

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available @5#. It is still unclear how the pressure variatiowould influence on the nonexponentiality. The explorationthis question is partially possible due to the correlation whhas been established between the shape of relaxation ftion ~nonexponentiality! and the degree to which normalizethe temperature dependence of relaxation time departs fArrhenius behavior~fragility! @6#. Analyzing the pressure effect on fragility one can come to some conclusions regardthe influence of isothermal density changes on nonexpontiality. In particular, if t(Tg /T) dependencies obtained adifferent pressures constitute a master curve~fragility is pres-sure independent! then one would expect that temperatucharacteristics of nonexponentiality for various pressuwill also fall onto a single scaling curve. In these considations we have skipped, however, over a fundamental qtion, whether or not the correlation quoted is still preservunder the condition of high compression. Therefore, it mbe of interest to explore the consequences of pressure vtion on correlation between fragility and nonexponentialit

The aim of this paper is to study the problems discusabove. Among other things we examine if the scaling behior of a relaxation in glass-forming liquid is influenced bhigh pressure. To this effect we carried out the measuremof dielectric relaxation function in poly@~phenyl glycidylether!-co-formaldehyde# over a wide range of pressure, temperature, and frequency.

EXPERIMENT

Poly@~phenyl glycidyl ether!-co-formaldehyde# ~PPG!with an average molecular weight of about 345 was pchased from Aldrich Chemicals. This compound, similarlyother epoxy systems, exhibits a permanent strong dipolement originating from the epoxy ring@7–9#. It is noteworthy

526 ©2000 The American Physical Society

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PRE 61 527SCALING BEHAVIOR OF THEa RELAXATION IN . . .

that the glass transition temperatureTg5264 K at ambientpressure@determined from differential scanning calorimet~DSC!# is relatively high which facilitates the high-pressuexperiments.

The complex dielectric permittivity«* 5«82 i«9 wasmeasured in the frequency range from 1022 Hz up to 106 Hz.In this frequency range two systems were used: a frequeresponse analyzer~Solartron SI-1260! and an impedanceanalyzer~HP-4192!. The experimental setup used for higpressure dielectric studies has been described in detailwhere @10#. The pressure was measured by a Nova Swtensometric pressure meter, with resolution60.1 MPa andaccuracy60.4 MPa. The temperature was varied in interv270–330 K and was controlled within60.1 K by means of aliquid flow provided by a thermostatic bath.

RESULTS

Measurements of complex dielectric permittivity habeen carried out for five different isobars~0.1, 60, 120, 180,and 240 MPa! and one isotherm~293 K!. Typical dielectricspectra measured in PPG for various temperatures anconstant pressure are displayed in Fig. 1. The presented stra exhibit two different relaxation modes. The most promnent relaxation process, occurring as a maximum of lcurve, is associated with thea process whereas an upgointrend in«9(v) at the lowest frequencies is identified as d

FIG. 1. Real and imaginary parts,«8 and «9, of the dielectricpermittivity of PPG versus frequency at constant pressure andferent temperatures. The experimental data have been fitted bsuperposition of a conductivity contribution and a relaxation fution according to Havriliak-Negami.

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conductivity relaxation which occurs due to the presencecharged carriers in liquid. In order to extract informatioabout the shape of the relaxation function of thea-relaxationprocess, the experimental data«8(v) and«9(v) have beenanalyzed by an empirical relaxation function introducedHavriliak and Negami~HN! @11#. Since loss spectra alscover dc conductivity, the appropriate term,s0 /v«0 , whichaccounts for such behavior has been added to the HN foTherefore, the fit function representing the frequency depdence of the complex dielectric permittivity takes the following form:

«HN* ~v!5 is0

v«01«`1~«s2«`!@11~ ivtHN!a#2g, ~1!

wheres0 and «0 denote dc electrical conductivity and pemittivity of free space,«` represents the asymptotic valuethe permittivity at high frequency,«s is the value of theopposite limit,tHN is the characteristic relaxation time, ana andg are parameters ranging between 0 and 1.

Equation ~1! yields satisfactory fits~solid line! to datagiven in Fig. 1. At very high frequencies, deviations froassumed behavior were only observed. These deviatcould be attributed to the second relaxation process,maximum of which is situated beyond our measurementquency range~the peak of the secondary process wasseen even in the lowest temperatures studied here!. Since ithas been assumed that its influence on the result ofanalysis can be neglected, therefore, this additional reation process was excluded from the fitting procedure. Tfits by means of Eq.~1! were not made simultaneously fothe real and imaginary parts of complex permittivity. However, the similar values of parameters in Eq.~1! have beenobtained in both cases. Finally, we used parameters resufrom fits of the imaginary parts of the dielectric permittivitfor further analysis. An independent analysis of dispersand absorption spectra should lead to the same result duthe fact that«8(v) and «9(v) are related by KramersKroning relations@12#.

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FIG. 2. Parametersa andag determined from isobaric and isothermal data versus logtHN . The solid straight lines are guides fothe eyes. The symbols are the same as in Fig. 3.

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528 PRE 61PALUCH, HENSEL-BIELOWKA, AND ZIOŁO

DISCUSSION

The main results of our study are presented in Figs. 2The parametersa andag, determined from isobaric and isothermal measurements, are shown in Fig. 2 as a functiodielectric relaxation time. It is evident that values of paraeters that characterize the shape of the relaxation functio~aand ag!, collapse onto two masterplots, respectively. Tonly region of small discrepancies in the scaling of shaparametera is at the longest relaxation times, where trelaxation peak is at very low frequencies. These appadiscrepancies result from limited frequency range~the low-frequency part of the dielectric response is truncated!. Itleads to an increase in size of the error bars of thea param-eter for logt.21. On the contrary, error bars are no largthan the size of symbols for logt,21. Figure 2 also showsthat value a slightly decreases with the rise of logtHNwhereas the second parameterag seems to attain a constavalue of 0.4. A slight drop in the value ofag for shorterrelaxation times is probably caused by the second relaxaprocess, which gives a contribution to high-frequency tailsthe primary process in the high range of temperatures.well established that the majority of low molecular weigglass-forming liquids follow the behavior witha'1 whereasthe behavior witha,1 features mainly polymers. In thtested systema,0.7 is observed in whole range of logt.

The low- and high-frequency tails of the primary relaation peak follow the simple power-law relations:

«9~v!;vm, v!vp ~2a!

«9~v!;v2n, v@vp ~2b!

where m5a, n5ag. Therefore, the values ofa and agdetermine the asymptotic behavior of permittivity at frequecies lower and higher than the maximum dielectric loss fquency. Such behavior agrees with the theoretical modwhich have been developed to explain the shape of theelectric relaxation function at the glass transition of polym@13,14#. The model proposed by Scho¨nhals and Schlosse@14#, which is based on the idea that the molecular mobiat the glass transition is controlled by intramolecular a

FIG. 3. ~a! Isobaric and~b! isothermal data of relaxation timesSolid lines indicate the quality of the fit according to Eqs.~3a! and~3b!, respectively. The dashed line represents the Arrhenius la

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intermolecular interactions, provides interpretation for shaparameters in polymeric systems. According to this modthe parametern describes the local chain dynamics of amophous polymers whereas the other parameterm is related tothe intermolecular correlation of segments of differechains. Applying this scheme to our data we can find tmolecular mobility at the glass transition is controlled in tsame way by both thermodynamic variables:P and T. It isalso worth noting that simple scaling couples of isobaric aisothermal dielectric spectra have also been observed inother epoxy system, namely EPON 828@11#. In view of theabove results one can expect an adequate scaling behalso for relaxation times.

The temperature and pressure dependence of the reation time is shown in Fig. 3. It has been observed that bisothermal and isobaric relaxation times systematically deate from the Arrhenius behavior on approaching the gltransition. They can be very well reproduced by the tempeture Vogel-Fulcher-Tammann~VFT! law @Eq. ~3a!# @15# andits pressure counterpart~3b! @16#

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FIG. 4. Data of Fig. 3 after normalizing temperatureTg at whichdielectric relaxation time equals 10 s. The inset shows thatsteepness index~fragility! of PPG is pressure independent withexperimental error.

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PRE 61 529SCALING BEHAVIOR OF THEa RELAXATION IN . . .

where t0 is a characteristic time constant of the ordermagnitude 10213, t0P is the relaxation time at atmospherpressure,T0 and P0 are the temperature and pressure ofideal glass transition,DT and DP are dimensionless parameters.

The displacement of isobart(T) in Fig. 3 with increasingpressure along the temperature axis reflects the fact thaglass transition temperatureTg of PPG is strongly dependenon pressure. In such cases,Tg normalized Arrhenius plotsbecome the common method for comparison of temperadependencies of primary processes at different press@17–20#. Such an approach is frequently applied for the coparison of segmental relaxation of polymers with a differemolecular weight@17,21#. To find out whether relaxationtimes superimpose we reanalyzed the isobaric data from3~a! in terms of the reduced temperatureT/Tg . From Fig. 4it can be seen that the temperature dependence of relaxtimes for different pressures collapses onto a single scacurve. Thus, aT/Tg normalized scheme provides evidenfor the pressure independence of fragility of PPG.

To extract the value of the fragility parameterm(P) fromthe dielectric data shown in Fig. 3 the standard procedurebeen applied. The starting point is the definition of fragilfor relaxation data introduced by Angell and co-workers@6#

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whereTg is determined and identified as the temperaturewhich t510 s. We have chosen a 10 s time scale in definTg due to the fact~i! the dielectric data are available onover a limited range of frequencies and~ii ! the glass transi-tion temperature,Tg5263.3 K, determined from dielectridata at ambient pressure agrees with the data fDSC (Tg5264 K!. The values of the steepness index~fra-gility ! calculated for each isobar are presented~as a functionof pressure! in the inset in Fig. 4. These data confirm thprevious result that fragility is pressure independent witexperimental error. Due to the large value of the steepnindex and the strong non-Debye character of relaxation fution PPG can be classified as a fragile system.

The question that arises immediately is whether or notresults obtained for PPG are in quantitative agreement wcorrelationm(b) from Ref. @6#. In order to bring to effectsuch comparison, at first, we have to determine the stretcexponent –b. Therefore, the frequency-dependent dielecspectra have been transformed by means of inverse Fotransform to time domain and subsequently fitted byKohlrausch-Williams-Watts ~KWW! relaxation function:Q(t);expb2(t/t)bc. From KWW fitsb50.39 is found whichcorrespond to fragility parameterm5116. This result is inaccord with the correlation cited above and agrees perfewell with narrower correlation for polymer systems~see Fig.4 in Ref.@6#!. It should be stressed that the value of fragil

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parameterm5116 has been calculated applying the sadefinitionTg as in@6#, i.e., the glass pointTg was defined astemperature at which the relaxation time equals 100 s.

Within the framework of this work it is instructive todemonstrate the effect of pressure and temperature onintensity of thea-relaxation process. The information abothe relaxation strength is comprised inD«5«s2«` . As canbe seen from the plot given in Fig. 5, temperature and psure variables when scaled on the same relaxation timenot give identical results. The change ofD« with the relax-ation time is stronger for isobaric than isothermal data. Frdata given in Fig. 5 we can infer that pressure suppressesstrength of the primary relaxation process. Comparchanges in relaxation strength brought about byT andP oneshould take into account that pressure affects only the dsity whereas the temperature alters the volume as well asthermal energy.

The effect of pressure on the glass transition temperain PPG is shown in Fig. 6. The change inTg with pressurecan be attributed to an increase of molecular packing induby squeezing. The results for glass transition temperashow thatTg is strongly dependent on compression. It can

FIG. 5. D« versus logtHN for both isothermal and isobaric dataThe symbols are the same as in Fig. 3.

FIG. 6. Pressure dependence of Tg~main plot! and T0 ~inset!.The solid lines represent a second-order polynomial fit to theperimental point. The open symbol denotes data from isothermeasurements.

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530 PRE 61PALUCH, HENSEL-BIELOWKA, AND ZIOŁO

also noted that changes inTg brought about by changingpressure have a nonlinear character. Such behavior is oexperimentally observed in many glass-forming liqui@18,22–24#. By fitting the second-order polynomial toTg(P)dependence we could estimate the value of the(dTg /dP)P50.1 MPa50.154 K/MPa60.004 for the tested material. A similar value of (dTg /dP)P50.1 MPa50.156 K/MPa60.001 has been reported by us for another epoxy sysnamely poly~bisphenolA-co-epichlorohydrin!, glycidyl end-capped@20#.

According to arguments presented in a recent paperRichert and Angell@25# the ideal glass transition temperatuT0 , which for the theoretical approach is a more fundamtal quantity thanTg , appears to be close to the KauzmtemperatureTK . However, the results of analysis ofT0(P)dependence become far less precise than ofTg(P) data ow-ing to a large extrapolation, which is required to determT0 . The character of pressure dependence ofT0 seems to bevery similar toTg(P) dependence~see the inset in Fig. 6!.The value of (dT0 /dP)P50.1 MPa50.1 K/MPa and thus isnearly the same as forTg(P) dependence.

CONCLUSION

Introducing pressure as an additional thermodynavariable to study a liquid-glass transition valuable informtion about the effect of molecular packing on dynamics oarelaxation in poly@phenyl~glycidyl ether!-co-formaldehyde#has been obtained. It has been shown that the shape paeters of the dielectric function collected for different presures fall on master curves constructed by plottinga andagagainst logt. The scaling of shape parameters for differepressure stems from the pressure independence of fragThis finding provides strong experimental evidence thatcorrelation between fragility and the shape of the relaxatfunction at atmospheric pressure may be also preserved

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der high pressures. Furthermore, the shape parameters orelaxation function have been analyzed in terms ofSchonhals and Schlosser model. From this analysis we dthe conclusion that the molecular mobility of PPG is cotrolled in the same way by temperature and pressure.

A non-Arrhenius behavior of relaxation times is a common feature of both isothermal and isobaric data. This reshows the isomorphism of the dielectric relaxation timeshavior when approaching the glass transition along the psure and temperature path. Similar to our previous resuother glass formers we also find nonlinear character presdependence of glass transition temperature.

In order to provide a comprehensive description of tbehavior of dynamics of the primary process under hpressures further studies are required. First of all the stushould focus on the materials that exhibit a strong pressdependence of the steepness index~fragility!. In such sys-tems, the change of fragility with pressure should beflected by a lack of the scaling pattern for shape parametIn agreement with the correlation between fragility and noexponentiality, one can expect that an increase of fragiwith pressure should result in an increase of noexponentiality. On the other hand, the drop of fragilishould correspond to a decrease of non-exponentialitygood candidate for such studies seems to be glycerol. Hever, due to the very small value ofdTg /dP, a much higher-pressure range~up to a few GPa! has to be used in such aexperiment to obtain meaningful results.

ACKNOWLEDGMENTS

We would like to thank Professor T. Pakula and DToemsak Srikhirin for helpful discussion. The research wpartially supported by the State Committee for Scientific Rsearch~KBN, Poland! under Project No. 2 P03B 043 15 anProject No. 2 P03B 030 12.

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