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Origin of dispersion in dipolar relaxations of glasses
Volume 2 16, number 1,2
24 December1993
CHEMICALPHYSICSLETTERS
Origin of dispersion in dipolar relaxations of glasses Ranko Richert Fachbereich PhysikalischeChemie, UniversitdtMarburg, Hans MeerweinStra.we,D-35043 Marburg, Germany
Received 11 October1993
Dispersive relaxationstypically found in glassy systems stem from a distributionof responsetimes, spreadeither heterogeneously or homogeneouslywithin the ensembleof sites. To distinguishbetweenthese two possibilitieswe follow the dynamicsof dipole solvation, a techniquewhich providesvaluable insight on relaxationin both the liquid and the glassyregime.Employing the dynamicmean sphericalapproximation(MSA) theory,with c(w) as an input, we calculatethe solvation time evolution in the homogeneousand heterogeneouslimits. Due to the remarkableagreementbetweenthe homogeneousMSAand the observed solvation in 2-methyltetrahydrofuran at To+ 3 K= 94 K we conclude on the homogeneousnature of dispersion for dipolar relaxations.
1. Introduction Non-exponential relaxation patterns and temperature dependences of the apparent activation energy are typical manifestations of disordered systems
where cooperativity confines the motional freedom of the molecules [ 11. This glassy or supercooled state is unique in combining configurational irregularity with highly frustrated relaxations, which proceed on the time scale of x 100 s at the glass transition temperature TG [2 1. A response $(t) after perturbing such complex media commonly displays a continuous spectrum of relaxation times as represented empirically by dispersive decay laws like the widely applied Kohlrausch-Williams-Watts (KWW) function #kww(t)=exp[-(t/r)OL] [3].Ingoingbeyondthis heuristic approach, one is confronted with the question of whether the dispersion is an intrinsic property in the response of a single site or, rather, originates from the ensemble average over a heterogeneous distribution of exponential responses with site-specific time constants. The related problem of the length scale of cooperatively rearranging regions is of critical importance to theories of relaxations in glassy media [ 4,5]. Discriminating between these pictures of a homogeneous or heterogeneous origin for the dispersion calls for an experimental technique with spatial resolution on microscopic scales.
Due to the absence of spatial information in most relaxation experiments, little information is available on this subject, which is crucial for understanding the dynamics of disordered matter. The two models, the homogeneous (i.e. dynamic or serial) and heterogeneous (i.e. static or parallel) nature of dispersion, are illustrated in fig. 1. Dielectric relaxation spectroscopy provides a powHOMOGENEOUS
HETEROGENEOUS
L
ch
1
AVERAGE
I
IL AVERAGE
=
3 TIME
TIME
Fig. 1. Heterogeneousand homogeneousnatureof ensembleaveragednon-exponentialityregardingan arbitraryrelaxingquantity 0. The spatiallydistributedsites i are representedin termsof their single site relaxationfunction log titversus t. By virtue of the indispensableensembleaverageboth scenarioscan eventually yield identical $(t) patterns.
0009-2614/93/S 06.00 0 1993 Elsevier Science PublishersB.V. All rightsreserved. SSDI0009-2614(93)E1251-B
223
Volume 216, number I,2
CHEMICAL.PHYSICS LETTERS
erful tool for investigating the dynamical behaviour of polar media over a wide frequency range [6,7]. In liquids of low viscosity a Debye-type dielectric function e(w) =e,+ (es-eol)/( 1 fiwrO), the equivalent of an exponential response of the polarization P( t), is usually observed. In the case of a glassforming sample, sufficient cooling will invoke departures from Debye behaviour, quantified for instance by the Cole-Davidson [ 8 ] (CD) dielectric function
with OK/~-Z1 [ 91. A spread of dielectric relaxation times, recognized by /3c 1 in eq. ( 1 ), can be rationalized within either of the two limiting pictures given in fig. 1, i.e. e (0) remains ambiguous in this respect. In the present Letter an alternative route for approaching the dielectric behaviour of a supercooled liquid is presented, which allows us to discriminate between a homogeneous and heterogeneous origin for the dispersion. The necessary spatial information is deduced from solvation dynamic experiments which serve for locally probing the orientational dielectric relaxation in the field of a molecular dipole. Theoretically, this spatial aspect can be cast into the dependence of solvation effects on e (w, R) [ 10,111. Based on dielectric relaxation data and employing the dynamic generalization of the mean spherical approximation (MSA ) theory, the temporal pattern of the Stokes shift a molecular probe is subject to can be predicted for the homogeneous and heterogeneous cases. This first comparison of these two models with experimental findings clearly indicates that the dispersion of dielectric relaxation is homogeneous within the time range of the experiment and for temperatures as low as TG+ 3 K, where the dielectric relaxation time is = 1 s.
24 December 1993
shift will proceed gradually in time as does polarization in dielectric spectroscopy. Since absolute energies are redundant for focusing on the response function, we follow common practice and normalize the scale of mean emission energies to obtain the socalled Stokes shift correlation function (SCF) C(f)=
v(t) - da) Y(0) - v(a)
*
Experimental access to the dynamical aspect of Stokes shifts [ 12,13 1, simulation work [ 14,lS 1, ab initio type theories [ 111, and calculations [ 16- 18 ] which relate the energetic equilibration of a probe molecule to solvent dielectric properties has greatly improved the understanding of solvation dynamics [ 191. In context with the more recent fmdings on solvation in supercooled liquids the MSA theory has turned out to be extremely successful in predicting the absolute emission energies [20,21] and the temporal relaxation patterns [ 221. For such systems of highly cooperative dynamics, simulations and non-c (a)based theories are not applicable. The dynamic version of the MSA (DMSA) predicts the SCF C(t) for the spatially local solvation process of a dipole in terms of the macroscopic e(o) in an analytical fashion, but without calculating e (0, R) explicitly [ 171. In its original formulation, the dipolar DMSA assumes site-independent dielectric relaxation [ 17 1, thereby referring only to the homogeneous limit, which certainly is the appropriate picture regarding low viscosity liquids. If the dispersive case of eq. ( 1), /3< 1, is believed to stem from a distribution of site specific r it is convenient to express e(o) in terms of the probability density of In t:
where the choice 2. Model considerations The emission energy of a chromophore subject to a change in dipole moment upon excitation and embedded in a polar solvent is shifted with respect to the corresponding gas-phase energy, due to electrostatic solute-solvent coupling. If orientational dipole relaxation of the solvent is involved this Stokes 224
GB(In 7)
=
0,
7370,
(4)
reproduces eq. ( 1) [ 71. For the Debye case expressed in terms of eq. (3 ) one would use G, (In T) =6(ln r-ln
to)
.
(5)
CHEMICAL PHYSICS LETTERS
Volume 216, number 1,2
In the homogeneous limit the DMSA calculates C( t ) , denoted C,,,(t) in the following, based on e(w) as in eq. (3), so that Chom(t) depends only on e_, es and GB(ln 7): Chom(t)=Chomtt,~ol,es,
GB(ln r)
--oo
~Ch~~[t,~~,~~,G~(ln7,)ldln7.
(7)
Note that &, ( t) in eq. ( 7 ) refers to the Debye case of eq. ( 6 ) by inserting G, = GB_i. Model calculations along the lines of eqs. (6) and (7 ) for a CD liquid of various polarities and for two dispersion parameters B are shown in fig. 2. Comparing C,,(t) and C,, ( t ) indicates that the discrepancies increase with increasing AE and with decreasing 8. Note that the MSA predicts a non-exponential C(t) for a polar Debye system [ 17,23 ], so that in general Chet(t ) is more dispersive than the Laplace transform of GB(In 7). For example, a CD dielectric characterized
-2""."."' 0
0.5
by e== 3, e,= 19,
70= 1, and jI= 1 (resembling the polarity of MTHF) would result in a Ch,,*( t) decay similar to the KWW function with (rKww=0.85 and 7K-=o.41.
(6)
Wn7)l.
Within the heterogeneous picture, each site is believed to be surrounded by a Debye-type solvent region, but in this case with GB(ln 7) mapping the probability of fmding a region with time constant 7 within the ensemble. In this limit the prediction regarding C(t) thus reads CL(t) = 7
24 December 1993
1
time/s
Fig. 2.Modelcalculationscomparing homogeneous (-_) and heterogeneous (- - -) limits of the MSA for CD dielectrics with h-O.8 (upperframe) and&,=O.4 (lowerframe)where&=3 in all cases. The three pairs of curves in each frame refer to &=c,-c,valuesof 1, and 100fromtop to bottom.
3. Experiment and results As an experimental model system for discriminating the homogeneous and heterogeneous viewpoint, we chose the aprotic low-molecular-weight glass former 2-methyltetrahydrofuran (MTHF, T,= 9 1 K) doped with the chromophore quinoxaline (QX). The dynamics of the solvation process are monitored by time-resolved (1 ms-1 s) phosphorescence spectroscopy regarding the So+Tl (O0) emission associated with a lifetime of rph= 0.25 s. The quantity u(t) is evaluated in terms of the mean emission energy of the well-separated O-O band and then normalized according to eq. (2) to yield C(t). At temperatures below TO the orientational contribution to the dielectric behaviour is frozen and a time invariant emission energy is observed which corresponds to the Y(t=O) emission at higher temperatures. Recalling from fig. 2 that the short time domain is crucial for resolving the differences between Chti( t) and Cbom(t), previous C(t) data [ 22,241 for QX/MTHF has been extended in time scale ( 1 ms to 1 s) and energy resolution for the present purpose. Additionally, the critical slope dC( t ) /dt at t z 0 could be confumed by resealing C(t) traces obtained at lower temperatures according to the time-temperature superposition principle, whose premise of a temperature-invariant dispersion is satisfied in the present experiment [ 221. Since the energetic relaxation process can be suppressed at T-z TG the normalization value leading to C( t =O) = 1 can easily be confinned by data observed at a slightly lower temperature, thereby assuring that no fast relaxation component is disregarded. The experiment was conducted as described previously [ 22 1, measuring the optical C(t) and the dielectric e(w) (20 Hz-l MHz) data simultaneously for a single sample where temperature mismatch and stability are B 30 mK. The optical result in terms of C(t) is plotted in fig. 3. At the temperature of interest, TG+ 3 K= 94 K, where C(t) is resolved best, the dielectric properties of MTHF are well characterized by a CD function 225
Volume 216, number 1,2
CHEMICAL PHYSICS LETTERS
I
0.4
0.8
time/s Fig. 3. Solvation dynamic C(t) data (dots) for QX/MTHF at 3 KabovethekineticglasstransitionT((r)=lOOs)=To=91K. Theoretical curves compare homogeneous (-) and hetcrogeneous (- - - ) cases of the MSA calculations based on the ( 0 ) experimental dielectric data c( w ) as acquired simultaneously with C(t).
with c,=3, ~~0.35 s pzO.49 [22], outlining all which enter MSA calculation. resulting predictions C,,,(t) and for MTHF 94 K included in 3. Clearly, decay pattern reproduced by homogeneous version the MSA a remarkfashion. We conjecture that in solvation and, concomitantly, dielectric is of nature. 4. Discussion A brief comparison to a recent result on the nature of loss of orientational correlation appears in order. In a fascinating “C exchange 4D NMR experiment, Schmidt-Rohr and Spiess were able to analyze the origin of dispersion for PVAc at 20 K above the caloric TG= 300K [ 251.By following the orientation and correlation time of specific molecules as a function of time the authors come to the conclusion that within the time window of the experiment ( 6 1 s) the nature of dispersion gradually shifts from heterogeneity to a more homogeneous character. The straightforward explanation is that motional freedom tends to average out the heterogeneity on the time scale of observation. Therefore, it must be expected that heterogeneity prevails on this time scale for temperatures below T,+20 K. For a proper comparison of the two results the term homogeneity 226
24 December 1993
has to be quoted with respect to the spatial scale set by the individual experiment, which is one molecule in the NMR case. According to simulation work [ 15,19 ] on Brownian dipole lattices, the approximate scale of coupling in solvation extends over 73 lattice points, i.e. approximately four solvent shells are involved. Within such a cluster effective electrostatic coupling is likely to result in highly cooperative motion, thereby ruling out that adjacent molecules relax on time scales which differ by orders of magnitude. Most probably, it is the lack of such longrange coupling in the NMR experiment on PVAc which allows for heterogeneity on molecular scales. In the case of homogeneity of dielectric relaxation down to the length scale of intermolecular distances the dispersion would be intrinsic in the response of each molecule. Homogeneity in this sense reduces dispersion to a hierarchy of time scales, i.e. serial rather than parallel processes prevail [ 261. Interestingly, it is this serial type of process which can be made responsible for obtaining decay patterns of the KWW form [27]. Although the function h-(t) is impractical for fitting dielectric data obtained in the frequency domain, its cvspace analogue Y-’ [ - d&ww (t)/dr] has been shown to yield excellent results in representing dispersive dielectric data [28,29 1. It is appealing to rationalize the common KWW-type relaxation behaviour in terms of the homogeneous and thus serial nature of the underlying processes as presently stated for dielectric relaxation in molecular glass formers.
5. Conclusions Simultaneous measurements of dielectric e(u) and optical Stokes shift C(t) data for a model glass former have allowed us to discriminate between the heterogeneous and homogeneous nature of the dispersion regarding dipolar relaxation on a length scale of = 20 A. Employing the DMSA theory for obtaining the spatially local solvation dynamics C( t ) gives rise to distinct predictions for the cases C,,(t) and C,,(t), the homogeneous case being in excellent agreement with the experimental data. Therefore, the dispersion of dielectric relaxation in polar supercooled molecular liquids is of homogeneous origin, i.e. the non-exponentiality arises from serial rather
Volume 2 16, number 1,2
CHEMICAL PHYSICS LETTERS
than parallel processes. This effect can be viewed as a consequence of long-range cooperativity, presumably promoted by effective dipolar coupling. Referring to previous results on the polymer PVAc the length scales of cooperatively rearranging regions appear not to be universal quantities in glass-forming materials.
Acknowledgement I am grateful to I. Rips for pointing out the DMSA calculation in the static limit. This work has been financially supported by the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie, and the Minerva-Gesellschaft.
References [l] J. J&kle,Rept.Progr. Phys. 49 (1986) 171. [ 21 J. Wong and C.A. Angell, Glass structure by spectroscopy (Dekker, New York, 1976). [ 31 G. Williams and D.C. Watts, Trans. Faraday Sot. 66 (1970) [4] gAdam
and J.H. Gibbs, J. Chem. Phys. 43 (1965) 139.
[ 51 K.L. Ngai, R.W. Rendell and D.J. Plazek, J. Chem. Phys. 94 (1991) 3018. [ 61 A. Schbnhals, F. Kremer and E. Schlosser, Phys. Rev. Letters 67 (1991) 999. [ 71 C.J.F. Bottcher and P. Bordewijk, Theory of electric polarization, Vol. 2 (Elsevier, Amsterdam, 1978).
24 December 1993
[8]D.W.DavidsonandR.H.Cole, J.Chem.Phys. 19 (1951) 1484. [ 91 J.G. Berbetian and R.H. Cole, J. Chem. Phys. 84 (1986) 6921. [ lo] T. Fonseca, B.M. Ladanyi and J.T. Hynes, J. Phys. Chem. 96 (1992) 4085. [ 111 L.E. Fried and S. Mukamel, J. Chem. Phys. 93 ( 1990) 932. [ 121 W.R. Ware, P. Chow and SK. Lee, Chem. Phys. Letters 2 (1968) 356. [ 131 M. Maroncelli, J. MacInnis and G.R. Fleming, Science 243 (1989) 1674. [ 141 M. Maroncelli, J. Chem. Phys. 94 ( 1991) 2084. [ 15 ] A. Papazyan and M. Maroncelli, J. Chem. Phys. 95 ( 1991) 9219. [ 161 P.G. Wolynes, J. Chem. Phys. 86 (1987) 5133. [ 171 I. Rips, J. Klafter and J. Jortner, J. Chem. Phys. 89 ( 1988) 4288. [ 18 ] I. Rips, J. Klafter and J. Jortner, J. Chem. Phys. 88 ( 1988) 3246. [ 191 H.-X. Zhou, B. Bag&i, A. Papazyan and M. Maroncelli, J. Chem. Phys. 97 (1992) 9311. [ 201 R.F. Loring, J. Phys. Chem. 94 ( 1990) 5 13. [21] R. Richertand A. Wagener, J. Phys. Chem. 97 (1993) 3146. [22] R. Richert, Chem. Phys. Letters 199 (1992) 355. [ 231 C.F. Chapman, R.S. Fee and M. Maroncelli, J. Phys. Chem. 94 (1990) 4929. [24] R. Richert and A. Wagener, J. Phys. Chem. 95 ( 1991) 10115. [25] K. Schmidt-Rohr and H.W. Spiess, Phys. Rev. Letters 66 (1991) 3020. [ 261 J. Klafter and M.F. Shlesinger, Proc. Natl. Acad. Sci. US 83 (1986) 848. [27] R.G. Palmer, D.L. Stein, E. Abrabamsand P.W. Anderson, Phys. Rev. Letters 53 (1984) 958. [28] G. Williams, D.C. Watts, S.B. Dev and A.M. North, Trans. FamdaySoc. 67 (1971) 1323. [29] C.P. Lindsay and G.D. Patterson, J. Chem. Phys. 73 (1980) 3348.
227
24 December1993
CHEMICALPHYSICSLETTERS
Origin of dispersion in dipolar relaxations of glasses Ranko Richert Fachbereich PhysikalischeChemie, UniversitdtMarburg, Hans MeerweinStra.we,D-35043 Marburg, Germany
Received 11 October1993
Dispersive relaxationstypically found in glassy systems stem from a distributionof responsetimes, spreadeither heterogeneously or homogeneouslywithin the ensembleof sites. To distinguishbetweenthese two possibilitieswe follow the dynamicsof dipole solvation, a techniquewhich providesvaluable insight on relaxationin both the liquid and the glassyregime.Employing the dynamicmean sphericalapproximation(MSA) theory,with c(w) as an input, we calculatethe solvation time evolution in the homogeneousand heterogeneouslimits. Due to the remarkableagreementbetweenthe homogeneousMSAand the observed solvation in 2-methyltetrahydrofuran at To+ 3 K= 94 K we conclude on the homogeneousnature of dispersion for dipolar relaxations.
1. Introduction Non-exponential relaxation patterns and temperature dependences of the apparent activation energy are typical manifestations of disordered systems
where cooperativity confines the motional freedom of the molecules [ 11. This glassy or supercooled state is unique in combining configurational irregularity with highly frustrated relaxations, which proceed on the time scale of x 100 s at the glass transition temperature TG [2 1. A response $(t) after perturbing such complex media commonly displays a continuous spectrum of relaxation times as represented empirically by dispersive decay laws like the widely applied Kohlrausch-Williams-Watts (KWW) function #kww(t)=exp[-(t/r)OL] [3].Ingoingbeyondthis heuristic approach, one is confronted with the question of whether the dispersion is an intrinsic property in the response of a single site or, rather, originates from the ensemble average over a heterogeneous distribution of exponential responses with site-specific time constants. The related problem of the length scale of cooperatively rearranging regions is of critical importance to theories of relaxations in glassy media [ 4,5]. Discriminating between these pictures of a homogeneous or heterogeneous origin for the dispersion calls for an experimental technique with spatial resolution on microscopic scales.
Due to the absence of spatial information in most relaxation experiments, little information is available on this subject, which is crucial for understanding the dynamics of disordered matter. The two models, the homogeneous (i.e. dynamic or serial) and heterogeneous (i.e. static or parallel) nature of dispersion, are illustrated in fig. 1. Dielectric relaxation spectroscopy provides a powHOMOGENEOUS
HETEROGENEOUS
L
ch
1
AVERAGE
I
IL AVERAGE
=
3 TIME
TIME
Fig. 1. Heterogeneousand homogeneousnatureof ensembleaveragednon-exponentialityregardingan arbitraryrelaxingquantity 0. The spatiallydistributedsites i are representedin termsof their single site relaxationfunction log titversus t. By virtue of the indispensableensembleaverageboth scenarioscan eventually yield identical $(t) patterns.
0009-2614/93/S 06.00 0 1993 Elsevier Science PublishersB.V. All rightsreserved. SSDI0009-2614(93)E1251-B
223
Volume 216, number I,2
CHEMICAL.PHYSICS LETTERS
erful tool for investigating the dynamical behaviour of polar media over a wide frequency range [6,7]. In liquids of low viscosity a Debye-type dielectric function e(w) =e,+ (es-eol)/( 1 fiwrO), the equivalent of an exponential response of the polarization P( t), is usually observed. In the case of a glassforming sample, sufficient cooling will invoke departures from Debye behaviour, quantified for instance by the Cole-Davidson [ 8 ] (CD) dielectric function
with OK/~-Z1 [ 91. A spread of dielectric relaxation times, recognized by /3c 1 in eq. ( 1 ), can be rationalized within either of the two limiting pictures given in fig. 1, i.e. e (0) remains ambiguous in this respect. In the present Letter an alternative route for approaching the dielectric behaviour of a supercooled liquid is presented, which allows us to discriminate between a homogeneous and heterogeneous origin for the dispersion. The necessary spatial information is deduced from solvation dynamic experiments which serve for locally probing the orientational dielectric relaxation in the field of a molecular dipole. Theoretically, this spatial aspect can be cast into the dependence of solvation effects on e (w, R) [ 10,111. Based on dielectric relaxation data and employing the dynamic generalization of the mean spherical approximation (MSA ) theory, the temporal pattern of the Stokes shift a molecular probe is subject to can be predicted for the homogeneous and heterogeneous cases. This first comparison of these two models with experimental findings clearly indicates that the dispersion of dielectric relaxation is homogeneous within the time range of the experiment and for temperatures as low as TG+ 3 K, where the dielectric relaxation time is = 1 s.
24 December 1993
shift will proceed gradually in time as does polarization in dielectric spectroscopy. Since absolute energies are redundant for focusing on the response function, we follow common practice and normalize the scale of mean emission energies to obtain the socalled Stokes shift correlation function (SCF) C(f)=
v(t) - da) Y(0) - v(a)
*
Experimental access to the dynamical aspect of Stokes shifts [ 12,13 1, simulation work [ 14,lS 1, ab initio type theories [ 111, and calculations [ 16- 18 ] which relate the energetic equilibration of a probe molecule to solvent dielectric properties has greatly improved the understanding of solvation dynamics [ 191. In context with the more recent fmdings on solvation in supercooled liquids the MSA theory has turned out to be extremely successful in predicting the absolute emission energies [20,21] and the temporal relaxation patterns [ 221. For such systems of highly cooperative dynamics, simulations and non-c (a)based theories are not applicable. The dynamic version of the MSA (DMSA) predicts the SCF C(t) for the spatially local solvation process of a dipole in terms of the macroscopic e(o) in an analytical fashion, but without calculating e (0, R) explicitly [ 171. In its original formulation, the dipolar DMSA assumes site-independent dielectric relaxation [ 17 1, thereby referring only to the homogeneous limit, which certainly is the appropriate picture regarding low viscosity liquids. If the dispersive case of eq. ( 1), /3< 1, is believed to stem from a distribution of site specific r it is convenient to express e(o) in terms of the probability density of In t:
where the choice 2. Model considerations The emission energy of a chromophore subject to a change in dipole moment upon excitation and embedded in a polar solvent is shifted with respect to the corresponding gas-phase energy, due to electrostatic solute-solvent coupling. If orientational dipole relaxation of the solvent is involved this Stokes 224
GB(In 7)
=
0,
7370,
(4)
reproduces eq. ( 1) [ 71. For the Debye case expressed in terms of eq. (3 ) one would use G, (In T) =6(ln r-ln
to)
.
(5)
CHEMICAL PHYSICS LETTERS
Volume 216, number 1,2
In the homogeneous limit the DMSA calculates C( t ) , denoted C,,,(t) in the following, based on e(w) as in eq. (3), so that Chom(t) depends only on e_, es and GB(ln 7): Chom(t)=Chomtt,~ol,es,
GB(ln r)
--oo
~Ch~~[t,~~,~~,G~(ln7,)ldln7.
(7)
Note that &, ( t) in eq. ( 7 ) refers to the Debye case of eq. ( 6 ) by inserting G, = GB_i. Model calculations along the lines of eqs. (6) and (7 ) for a CD liquid of various polarities and for two dispersion parameters B are shown in fig. 2. Comparing C,,(t) and C,, ( t ) indicates that the discrepancies increase with increasing AE and with decreasing 8. Note that the MSA predicts a non-exponential C(t) for a polar Debye system [ 17,23 ], so that in general Chet(t ) is more dispersive than the Laplace transform of GB(In 7). For example, a CD dielectric characterized
-2""."."' 0
0.5
by e== 3, e,= 19,
70= 1, and jI= 1 (resembling the polarity of MTHF) would result in a Ch,,*( t) decay similar to the KWW function with (rKww=0.85 and 7K-=o.41.
(6)
Wn7)l.
Within the heterogeneous picture, each site is believed to be surrounded by a Debye-type solvent region, but in this case with GB(ln 7) mapping the probability of fmding a region with time constant 7 within the ensemble. In this limit the prediction regarding C(t) thus reads CL(t) = 7
24 December 1993
1
time/s
Fig. 2.Modelcalculationscomparing homogeneous (-_) and heterogeneous (- - -) limits of the MSA for CD dielectrics with h-O.8 (upperframe) and&,=O.4 (lowerframe)where&=3 in all cases. The three pairs of curves in each frame refer to &=c,-c,valuesof 1, and 100fromtop to bottom.
3. Experiment and results As an experimental model system for discriminating the homogeneous and heterogeneous viewpoint, we chose the aprotic low-molecular-weight glass former 2-methyltetrahydrofuran (MTHF, T,= 9 1 K) doped with the chromophore quinoxaline (QX). The dynamics of the solvation process are monitored by time-resolved (1 ms-1 s) phosphorescence spectroscopy regarding the So+Tl (O0) emission associated with a lifetime of rph= 0.25 s. The quantity u(t) is evaluated in terms of the mean emission energy of the well-separated O-O band and then normalized according to eq. (2) to yield C(t). At temperatures below TO the orientational contribution to the dielectric behaviour is frozen and a time invariant emission energy is observed which corresponds to the Y(t=O) emission at higher temperatures. Recalling from fig. 2 that the short time domain is crucial for resolving the differences between Chti( t) and Cbom(t), previous C(t) data [ 22,241 for QX/MTHF has been extended in time scale ( 1 ms to 1 s) and energy resolution for the present purpose. Additionally, the critical slope dC( t ) /dt at t z 0 could be confumed by resealing C(t) traces obtained at lower temperatures according to the time-temperature superposition principle, whose premise of a temperature-invariant dispersion is satisfied in the present experiment [ 221. Since the energetic relaxation process can be suppressed at T-z TG the normalization value leading to C( t =O) = 1 can easily be confinned by data observed at a slightly lower temperature, thereby assuring that no fast relaxation component is disregarded. The experiment was conducted as described previously [ 22 1, measuring the optical C(t) and the dielectric e(w) (20 Hz-l MHz) data simultaneously for a single sample where temperature mismatch and stability are B 30 mK. The optical result in terms of C(t) is plotted in fig. 3. At the temperature of interest, TG+ 3 K= 94 K, where C(t) is resolved best, the dielectric properties of MTHF are well characterized by a CD function 225
Volume 216, number 1,2
CHEMICAL PHYSICS LETTERS
I
0.4
0.8
time/s Fig. 3. Solvation dynamic C(t) data (dots) for QX/MTHF at 3 KabovethekineticglasstransitionT((r)=lOOs)=To=91K. Theoretical curves compare homogeneous (-) and hetcrogeneous (- - - ) cases of the MSA calculations based on the ( 0 ) experimental dielectric data c( w ) as acquired simultaneously with C(t).
with c,=3, ~~0.35 s pzO.49 [22], outlining all which enter MSA calculation. resulting predictions C,,,(t) and for MTHF 94 K included in 3. Clearly, decay pattern reproduced by homogeneous version the MSA a remarkfashion. We conjecture that in solvation and, concomitantly, dielectric is of nature. 4. Discussion A brief comparison to a recent result on the nature of loss of orientational correlation appears in order. In a fascinating “C exchange 4D NMR experiment, Schmidt-Rohr and Spiess were able to analyze the origin of dispersion for PVAc at 20 K above the caloric TG= 300K [ 251.By following the orientation and correlation time of specific molecules as a function of time the authors come to the conclusion that within the time window of the experiment ( 6 1 s) the nature of dispersion gradually shifts from heterogeneity to a more homogeneous character. The straightforward explanation is that motional freedom tends to average out the heterogeneity on the time scale of observation. Therefore, it must be expected that heterogeneity prevails on this time scale for temperatures below T,+20 K. For a proper comparison of the two results the term homogeneity 226
24 December 1993
has to be quoted with respect to the spatial scale set by the individual experiment, which is one molecule in the NMR case. According to simulation work [ 15,19 ] on Brownian dipole lattices, the approximate scale of coupling in solvation extends over 73 lattice points, i.e. approximately four solvent shells are involved. Within such a cluster effective electrostatic coupling is likely to result in highly cooperative motion, thereby ruling out that adjacent molecules relax on time scales which differ by orders of magnitude. Most probably, it is the lack of such longrange coupling in the NMR experiment on PVAc which allows for heterogeneity on molecular scales. In the case of homogeneity of dielectric relaxation down to the length scale of intermolecular distances the dispersion would be intrinsic in the response of each molecule. Homogeneity in this sense reduces dispersion to a hierarchy of time scales, i.e. serial rather than parallel processes prevail [ 261. Interestingly, it is this serial type of process which can be made responsible for obtaining decay patterns of the KWW form [27]. Although the function h-(t) is impractical for fitting dielectric data obtained in the frequency domain, its cvspace analogue Y-’ [ - d&ww (t)/dr] has been shown to yield excellent results in representing dispersive dielectric data [28,29 1. It is appealing to rationalize the common KWW-type relaxation behaviour in terms of the homogeneous and thus serial nature of the underlying processes as presently stated for dielectric relaxation in molecular glass formers.
5. Conclusions Simultaneous measurements of dielectric e(u) and optical Stokes shift C(t) data for a model glass former have allowed us to discriminate between the heterogeneous and homogeneous nature of the dispersion regarding dipolar relaxation on a length scale of = 20 A. Employing the DMSA theory for obtaining the spatially local solvation dynamics C( t ) gives rise to distinct predictions for the cases C,,(t) and C,,(t), the homogeneous case being in excellent agreement with the experimental data. Therefore, the dispersion of dielectric relaxation in polar supercooled molecular liquids is of homogeneous origin, i.e. the non-exponentiality arises from serial rather
Volume 2 16, number 1,2
CHEMICAL PHYSICS LETTERS
than parallel processes. This effect can be viewed as a consequence of long-range cooperativity, presumably promoted by effective dipolar coupling. Referring to previous results on the polymer PVAc the length scales of cooperatively rearranging regions appear not to be universal quantities in glass-forming materials.
Acknowledgement I am grateful to I. Rips for pointing out the DMSA calculation in the static limit. This work has been financially supported by the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie, and the Minerva-Gesellschaft.
References [l] J. J&kle,Rept.Progr. Phys. 49 (1986) 171. [ 21 J. Wong and C.A. Angell, Glass structure by spectroscopy (Dekker, New York, 1976). [ 31 G. Williams and D.C. Watts, Trans. Faraday Sot. 66 (1970) [4] gAdam
and J.H. Gibbs, J. Chem. Phys. 43 (1965) 139.
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