8794
J. Phys. Chem. B 1997, 101, 8794-8801
Heterogeneity at the Glass Transition: Translational and Rotational Self-Diffusion Inyong Chang and Hans Sillescu* Institut fu¨ r Physikalische Chemie, Johannes Gutenberg-UniVersita¨ t Mainz, D-55099 Mainz, Germany ReceiVed: December 16, 1996; In Final Form: February 19, 1997X
Self-diffusion coefficients, D, have been measured in the glass forming liquids salol, glycerol, phenolphthaleine dimethyl ether (PDE), cresolphthaleine dimethyl ether (CDE), and RRβ-trinaphthylbenzene (TNB) in the supercooled regime. The NMR static magnetic field gradient technique was applied where D >10-14 m2 s-1 can be attained. The results are similar to previous diffusion experiments where an enhancement of translational diffusion was found in comparison with rotational diffusion and shear viscosity. Various models of spatial heterogeneity are related to a phenomenological environmental fluctuation model in view of recent diffusion and relaxation data close to the glass transition.
I. Introduction Various experimental phenomena characterizing the dynamics of supercooled liquids and polymer melts have led to a very extensive literature on cooperative motion and/or spatial heterogeneity at the glass transition.1-5 This cooperativity is said to occur in addition to the well known cage and backflow effects in normal liquids.6 The latter can also change their nature on supercooling and this change has been modeled within the formalism of mode coupling theory.7 However, the heterogeneities discussed in the present paper should be most pronounced close to the caloric glass transition temperature Tg, whereas the dynamical anomaly predicted by mode coupling theory is relevant at temperatures of about 1.2Tg.7 It is not possible to review here the many different theoretical concepts arguing in favor of characteristic length scales of the order of a few nm around Tg.8 Thus we restrict ourselves to the discussion of theoretical work related to recent experiments providing new information to this old problem. Stillinger has arrived at spatial heterogeneity in supercooled liquids by considering properties of local minima of a multidimensional potential function within his “inherent structure” formalism.9 This led to a picture where “the entire material sample would consist of an irregular patchwork of well-bonded amorphous domains separated by domain walls across which the bonding is suboptimal” and to his suggestion that “shear flow would be retarded by the log-jam effect of large wellbonded domains, while self-diffusion might still proceed at a less inhibited rate due to molecular motions within the weakly bonded walls”.9 In order to quantify these predictions one necessarily has to introduce additional assumptions which are arbitrary to some extent and may be disputed (see following Section).10-12 The experimental situation has been stimulated recently by two classes of new experiments. The first relates the rotational correlation function of a selected subensemble of (e.g., “slow”) molecules with the correlation function of the same ensemble after a waiting period twait. Currently, there are three experiments of this type, namely, reduced 4D-NMR,13-15 fluorescence “deep bleach”,16 and dielectric hole burning17 experiments. It is obvious that in the limit of small twait the two correlation functions are identical whereas for long twait the second correlation function might approach that of the full ensemble of all molecules within the sample. New information is obtained X
Abstract published in AdVance ACS Abstracts, September 15, 1997.
S1089-5647(96)04098-9 CCC: $14.00
from the crossover regime where the rotational correlation function changes from the selected to the full ensemble. The second class of experiments is not really new, except that particular NMR and optical methods are now available for investigating “ultraslow” translational and rotational diffusion in supercooled liquids at temperatures down to the glass transition.18-20 Both types of experiments have produced results that were interpreted in terms of spatial or dynamical heterogeneity. In the following section we formulate an environmental fluctuation model (EFM) based on a master equation for translational and rotational motion of molecules which are exchanged between different “environmental” states characterized by a single variable x. Although various aspects of this model have been discussed long ago,21-23 it is particularly well suited for a phenomenological description of the recent experiments mentioned above and for discussing the assumptions underlying related models of heterogeneity.10-12,14,24 In section IV, we present new NMR results on self-diffusion in supercooled liquids, which are then combined with other recent experimental results and discussed in terms of the EFM and other theoretical concepts of heterogeneity at the glass transition. II. Theoretical Considerations Any ab initio type theory accounting for dynamical and spatial heterogeneities has to include higher order time and space correlation functions beyond the usual memory function formalism.6 Since no general derivation of heterogeneities from these higher order correlation functions seems to be possible for real systems at present, one has to introduce some ad hoc assumptions which are hoped to be close to reality. One has repeatedly tried to relate the postulated heterogeneities to thermal equilibrium fluctuations.4,5 However, in all cases characteristic length scales of spatial heterogeneity are derived from additional assumptions, e.g., on the cooperatively rearranging regions of the Adam-Gibbs formalism.1 In the context of nonexponential dielectric relaxation functions, Anderson and Ullman have developed a model of “environmental fluctuations” where each molecule is labeled by a variable x characterizing its environmental state. Any “distribution of relaxation times” determined in a supercooled liquid is then related with a corresponding distribution over the x variable, which is assumed to fluctuate in time by an Ornstein-Uhlenbeck process (see below). Fast environmental fluctuations result in narrowing the apparent correlation time © 1997 American Chemical Society
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distributions to be compared with dielectric relaxation time distributions.22 This model should also be applicable to the analysis of the 4D-NMR, deep bleach, and dielectric hole burning experiments13-17 mentioned above since the selection of a subensemble of (e.g., slow) molecules can be related to a certain range of the variable x and the return to the full equilibrium ensemble is determined by the dynamics of x(t). In the following, we develop a more general environmental fluctuation formalism for analyzing these experiments. It will also account for the comparison of translational and rotational diffusion experiments and it can be used as a basis for the discussion of spatial domain scenarios. Environmental Fluctuation Model. In order to describe the single particle motion in a supercooled liquid we assume that the state of each molecule is given by the center of mass position r(t), the orientation Ω(t), and a variable x(t) associated with its “environmental state”. Furthermore, we assume that the time dependence is given by a stationary Markoff process in the space of these variables and a master equation
∂ P(r,Ω,x,t) ) ∂t
∫dr′∫dΩ′∫dx′ Π(r,Ω,x|r′,Ω′,x′)P(r′,Ω′,x′t)
P(r,Ω,x,0) ) δ(r - r0)δ(Ω - Ω0)δ(x - x0)
(1)
P(r,Ω,x,t) ≡ P(r,Ω,x,t|r0,Ω0,x0) is the probability density of finding a molecule at time t in the state r, Ω, x under the condition that it was in the state r0, Ω0, x0 at time t ) 0. The rate of transitions r′, Ω′, x′ f r, Ω, x is given by Π(r, Ω, x|r′, Ω′, x′). (The ranges of all integrals in eq 1 and the following equations are omitted for simplicity.) Although the master equation, eq 1, will be shown to have remarkable flexibility and allows for a phenomenological description of all recent experiments on dynamics in supercooled liquids, it is important to realize what it cannot describe and this will be discussed first. An equation for single particle motion cannot account for possible cooperative motion of adjacent molecules. Though it is plausible to assume that a molecule in a “slow” environmental state xslow (with diffusion coefficients Dtrans(xslow) and Drot(xslow) much smaller than the average values) is surrounded by densely packed molecules which are also “slow” and move cooperatively in a “slow domain”, this is not required by eq 1 but must be added through further assumptions. It should also be noted that the transition rates Π(...|...) in eq 1 are time independent and thus cannot account for scenarios of “anomalous diffusion in fractal space” as discussed in the literature in relation with nonexponential relaxation functions.25,26 In particular the coupling model of K. Ngai starts from scenarios where certain assumptions on coupling between molecules lead to cooperativity and spatial heterogeneity.27,28 However, at the stage of the “coupling scheme” used for fitting experimental data the corresponding stretched exponentials are derived from a rate equation having a time dependent rate and the adjustable parameters are already averaged over the heterogeneities. Although the Ngai coupling scheme leads to nonexponential relaxation functions, it cannot account for the 4D-NMR, deep bleach, and hole burning experiments13-17 mentioned above. The transition rates Π(...|...) can, in principle, account for processes where a translational displacement r′ f r depends on the molecular orientation and simultaneous transitions Ω′ f Ω occur. However, the Ω-dependence of translational displacement is usually ignored since it is averaged out during the long diffusion times in current experiments. The rdependence of molecular reorientation Ω′ f Ω would be essential in models where spatial heterogeneity is treated
explicitly. However, since the environmental state variable x has no explicit r-dependence, that of Ω′ f Ω should also be ignored in the EFM and eq 1 should be replaced by two master equations, one for Ω′ f Ω and one for r′ f r transitions. Translational Molecular Motion. The transition rates Π(r,x|r′,x′) can describe situations where a change of the environmental state, x′ f x (e.g., by crossing an interface between spatial domains) induces a simultaneous translational jump r′ f r. If this coupling is irrelevant one can assume that r′ f r occurs at constant x′ ) x. If we further assume Fickian diffusion the master equation for translational displacement becomes
∂ P(r,x,t) ) Dtrans(x)∇2r P(r,x,t) + ∂t
∫dx′ Π(x|x′)P(r,x′,t) (2)
Since present experimental techniques for investigating translational diffusion in supercooled liquids average over all heterogeneities in a spatial range up to about 100 nm, eq 2 is reduced to Fick’s law, ∂P(r,t)/∂t ) 〈Dtrans〉∇r2P(r,t), with a diffusion coefficient given by the average
〈Dtrans〉 )
∫dx D′trans(x)P′trans(x)
(3)
In principle, D′trans(x) and P′trans(x) depend both upon Dtrans(x) and Π(x|x′), and they can be obtained from the solution of the master equation, eq 2. The problem simplifies in the limit of slow exchange, Π(x|x′) , ∆-2|D(x) - D(x′)| where x is assumed to be constant over the small translational displacement ∆. In this case, P′trans(x) ) Ptrans(x) is the stationary distribution obtained from eq 2 for ∂P/∂t ) 0 and we obtain D′trans(x) ) Dtrans(x). Thus eq 3 describes an average29 over the equilibrium distribution Ptrans(x). (For a particular domain structure, this average has recently been discussed by Tarjus and Kivelson.30) It should be noted that a “translational correlation time”, τtrans(x), can be defined by τtrans(x)-1 ) Dtrans(x)/∆2. The average, 〈Dtrans〉 ) ∆2 〈τtrans-1〉, is then taken over inverse correlation times whereas the usual average rotational correlation time, 〈τrot〉, is a linear average. For fast exchange, P′trans(x) is an “exchange narrowed” distribution and D′trans(x) is an apparent diffusion coefficient which depends upon Π(x|x′) and Dtrans(x). For only two states, x1 and x2, explicit expressions for these apparent quantities have been given.21 Rotational Molecular Motion. The case of molecular reorientation in a fluctuating environment is most interesting since it is relevant for many recent experiments. 13-18 Therefore, we discuss in some detail possible assumptions for simplifying the transition rates Π(Ω,x|Ω′,x′). In the original Anderson-Ullman approach an Ornstein-Uhlenbeck process was assumed for changes of the environmental states which results in22,23
∂ P(Ω,x,t) ) Drot(x)∇2ΩP(Ω,x,t) + ∂t ∂ ∂2 γ x + Dγ 2 P(Ω,x,t) (4) ∂x ∂x
(
)
The first term on the right-hand side describes rotational diffusion with an x-dependent rotational diffusion coefficient which was arbitrarily assumed22 as Drot(x) ) D0 exp(xR) with R g 1. The second term describes diffusion in x-space with a diffusion coefficient Dγ and an attractive force centered at x ) 0. The solution of eq 4 yields nonexponential rotational correlation functions of the Legendre polynomials PL which can be interpreted as a superposition of exponentials and a corresponding “distribution of correlation times”. The rotational
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correlation time τL defined by the integral over the correlation function 〈PL(0)PL(t)〉 is given by the average
L(L + 1)τL ) 〈D-1rot〉 )
∫dx D′rot(x)-1P′rot(x)
(5)
which corresponds to eq 3 for translational diffusion. Again, we obtain an “exchange narrowed” apparent distribution for fast exchange x ′ f x where D′rot(x) and P′rot(x) are functions of Drot(x), γ, and Dγ. Whereas eq 4 can account for nonexponential 2-time correlation functions if compared, say, with dielectric relaxation22 (see below) it fails for the 4-time correlation functions determined in 4D-NMR unless one introduces a particular xdependence of Dγ and γ.15 However, extensions of eq 4 based on transition rates of the form Π(Ω,x|Ω′,x′) ) Πx(Ω|Ω′) + ΠΩ(x|x′) have shown that rather different scenarios of environmental fluctuations can fit the known 4D-NMR data provided that the distribution of exchange rates ΠΩ(x|x′) exceeds a certain minimum width.31 Enhanced Translational Diffusion. An important difference between the averages of eqs 3 and 5 is that the rotational average is over the inverse Drot-1 which is dominated by the “slow” portion of the distribution P′rot(x) whereas the translational average over Dtrans is dominated by the “fast” portion of P′trans(x). If P′rot(x) ∼ P′trans(x) the different averaging can explain the “enhancement” of translational over rotational self-diffusion seen in supercooled liquids on approaching Tg (see section IV).10,16,18,20,24,30,32 The essential aspects of this enhancement can be discussed in a simplified picture where the environment fluctuates between only two states, xs and xf, for “slow” and “fast” motions, respectively. The corresponding master equations are then simply pairs of coupled equations having the same form for translational and rotational diffusion:
∂Pf/∂t ) D(f)∇2Pf - ΠsfPf + ΠfsPs ∂Ps/∂t ) D(s)∇2Ps - ΠfsPs + ΠsfPf
(6)
D(f) and D(s) are the (translational or rotational) diffusion coefficients in the “fast” and “slow” state, respectively, and the exchange rates are subject to detailed balance, Πsf φf ) Πfs φs, where φf and φs ) 1 - φf are the fractions of molecules in the “slow” and “fast” states which can be identified with the corresponding volume fractions for simplicity. The mean residence times, τf ) Πsf-1, and τs ) Πfs-1 in the “fast” and “slow” states, are then related by
τf ) τsφf/(1 - φf)
(7)
In order to demonstrate the translational enhancement let us assume that the Stokes-Einstein-Debye (SED) relation
Dtrans ) kBT/6πηRH ) 4/3RH2Drot
(8)
is valid where η is the shear viscosity of the liquid and RH the hydrodynamic radius of the diffusing molecule. In the following, we assume for simplicity that τf ) τs (or φf ) φs ) 1/2). We assume further that close to Tg the condition
over a diffusion time much longer than the lifetimes τf ) τs is given by
〈Dtrans〉 ) 1/2Dtrans(f) + 1/2Dtrans(s) ≈ 1/2Dtrans(f)
and dominated by the “fast” state. The ratio of these appropriately averaged translational and rotational diffusion coefficients becomes
〈Dtrans〉/〈Drot-1〉-1 ) 4/3RH2 (Drot(f)/Drot(s))
(9)
holds and that the SED relation is valid, separately, in the states xf and xs. Thus we obtain
〈Drot-1〉 ) 1/2[Drot(f)]-1 + 1/2[Drot(s)]-1 ≈ 1/2[Drot(s)]-1
(10)
demonstrating that the rotational average is dominated by the “slow” state. On the other hand, the translational average taken
(12)
Since we can assume that Dtrans/Drot ) 4/3RH2 (eq 8) is valid far above Tg where no heterogeneity is observable, the factor (Drot(f)/ Drot(s)) g1 in eq 12 is a measure of the “enhanced” translational diffusion on approaching the glass transition. If the condition, eq 9, of very slow environmental fluctuations is no longer valid and is replaced by
[Drot(s)]-1 . τs . [Drot(f)]-1
(13)
it is obvious that the rotational correlation function cannot decay as long as a molecule is in the “slow” state xs until it undergoes a transition to xf after a time of order τs, but then its decay to zero occurs on the very fast time scale of [Drot(f)]-1. In other words, the decay of the rotational correlation function is dominated by the lifetime τs of the “slow” state xs and the rotational correlation times (for all L) are given by32
τL ≈ τs
(14)
This was also assumed in a model of Stillinger and Hodgdon10 where, in addition, the lifetime in the “fast” state is very short,
τf , τs
(15)
which results in a correspondingly small fraction of “fast” molecules, φf (∼10-5 in ref 10). It is important to realize that at this stage nothing is said about spatial heterogeneity. All experiments discussed so far13-20,24,32 can be described phenomenologically by fluctuations of an environmental state variable x(t) which, per se, is independent of position or spatial arrangement of the molecules. All scenarios of spatial heterogeneity must be added by way of additional assumptions which are discussed further in section IV. Viscoelastic Liquids. In Newtonian liquids, the shear viscosity, η, is independent of the shear rate if the applied stress refers to a volume containing many molecules (hydrodynamic limit). On approaching the glass transition, a supercooled liquid is gradually transformed into an elastic solid and η is replaced by a frequency dependent complex viscosity η*(ω) ) η′(ω) i η′′(ω) ) -iG*(ω)/ω. Here G*(ω) ) G′(ω) + iG′′(ω) is the complex shear modulus as obtained in mechanical relaxation experiments determining the shear stress σ(ω) in response to a shear deformation γ(ω) ) γ0 sin ωt. According to these conventions33 one can identify η with the zero shear viscosity
η ) lim η′(ω) ) lim G′′(ω)/ω ) ωf0
τf ) τs . [Drot(s)]-1 . [Drot(f)]-1
(11)
ωf0
∫0∞G(t) dt ) G0τG
(16)
The relaxation function G(t) is usually formulated as a superposition of exponentials defining a “distribution of relaxation times” H ˆ (τrel) and the average relaxation time becomes34
τG ) 〈τrel〉 )
∫0∞τrelHˆ (τrel) dτrel
(17)
where τG and G0 are defined by eqs 16 and 17. It is well known33 that in experiments where the shear deformation γ(ω)
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J. Phys. Chem. B, Vol. 101, No. 43, 1997 8797
is determined in response to an applied shear stress σ(ω) the complex compliance J*(ω) ) 1/G*(ω) corresponds to a “distribution of retardation times” Lˆ (τretard), which is shifted to larger times with respect to H ˆ (τrel) and defines an average retardation time
τJ ) 〈τret〉 )
∫0∞τretLˆ (τret) dτret ) J0η
(18)
In simple Debye model systems, we have 〈τrel〉 ) τG ) 〈τret〉 ) τJ and G0 ) 1/J0, however, in polymers, 〈τret〉 can exceed 〈τrel〉 by several decades at temperatures close to Tg.33 It is important to realize that the complex dielectric function34 *(ω) as determined in response to a periodic electric field corresponds to a compliance and the usual “distribution of dielectric relaxation times” gˆ(τdiel) should more accurately be termed a retardation time distribution4,35 with a mean retardation time
〈τdiel〉 )
∫0∞gˆ(τdiel) dτdiel
Figure 1. Self-diffusion coefficients D in glass forming liquids: salol ([), glycerol (0), PDE (b), CDE (4). The full lines were obtained from the Stokes-Einstein relation, eq 8, using the hydrodynamic radii, Rtrans ) RH, from Table 1, and the shear viscosities, η, from the literature quoted in section III.
(19)
Since dielectric relaxation times as determined from the decay of the electric field in response to a dielectric displacement are shorter than the corresponding retardation times, they have recently been investigated in a study of ultraslow dielectric relaxation at T < Tg.35 Of course, the molecular processes occurring in supercooled liquids cannot depend on whether a relaxation or retardation technique is applied. However, their difference becomes important in comparisons of mechanical or dielectrical relaxation with experiments probing single particle motions, e.g., the translational self- and tracer-diffusion experiments as well as the NMR and fluorescence bleaching experiments probing rotational correlation functions of well defined molecular vectors.13-16,18-20,24,32 These single particle properties can readily be related with an environmental fluctuation variable x(t) and compared with EFM simulations. Since x(t) is assigned to the single molecules, one has to assume that it is the same for all molecules in a certain domain volume V(x) in order to treat mechanical or dielectric relaxation with specified retardation or relaxation times of the corresponding distributions. After this annotation has been made, the usual formalism33 can be reproduced. However, any influence of correlated molecular motions which may be responsible for the difference between, say, dielectric relaxation times and rotational correlation times determined by NMR (see below) is then not accounted for since eq 1 is restricted to single particle motion. III. Experimental Section Samples. Details of the preparation of the glass formers phenolphthaleine dimethyl ether (PDE) and cresolphthaleine dimethyl ether (CDE) are given in ref 19. The viscosity data of salol, o-terphenyl (OTP), and m-tricresylphosphate (TCP) were the same as quoted in ref 19. Additional literature data were used for glycerol36-39 and trinaphthylbenzene (TNB).40 The sample of TNB used for the self-diffusion measurements was taken from the same stock as in our previous NMR studies.41,42 We agree with Ediger et al.43 that this substance is 1,3-bis(1-naphthyl)-5-(2-naphthyl)benzene, RRβ-TNB, and not RRR-TNB as was erroneously assumed previously. We also encountered the problem of TNB degrading43 after prolonged exposure at high temperatures since the sample showed a tendency of crystallization after some hours during the diffusion runs and repeatedly had to be molten and, again, supercooled to the same diffusion temperature. However, we are confident that this did not result in substantial errors since the self-diffusion coefficient, D, determined at the highest temperature could be
Figure 2. Comparison of self-diffusion and shear viscosity. Symbols: (a) OTP,18 (b) salol, (c) PDE, (d) CDE, (e) m-TCP71 (f) glycerol. The horizontal lines are drawn through the experimental points in the high temperature region.
reproduced after completing all diffusion measurements (see Figure 5). The samples of all glass formers were vacuum sealed in an NMR tube of 4 mm inner diameter. For glycerol, the D values measured in three different samples prepared separately yielded identical results within experimental error and since our values are the lowest in comparison with literature values, we assume that our glycerol samples were free from H2O. NMR Experiments. The self-diffusion coefficients, D, were determined from 1H-NMR stimulated echoes in a static magnetic field gradient.44,45 Using a (π/2)-τ-(π/2)-t-(π/2)-τ-echo three pulse sequence (with evolution time τ and diffusion time t . τ) the echo height is proportional to exp(-q2Dt) with the generalized “scattering vector” q ) τ γ g (gyromagnetic ratio γ, magnetic field gradient g). All measurements were performed on a home built NMR spectrometer with a Magnex cryomagnet system by using a specially designed anti-Helmholtz coil system with available magnetic field gradients up to 180 T/m.45 IV. Results and Discussion Translational Diffusion. In Figures 1 and 2, the selfdiffusion coefficients D of glass forming liquids are compared with the corresponding shear viscosities η. (The self-diffusion of TNB will be discussed separately, see below.) The expression
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TABLE 1: van der Waals Radii and Apparent Hydrodynamic Radii of Glass Forming Liquids glass former
Tg/K
RvdW/nm
Rtrans/nm
Rdiel/nm
RNMR rot /nm
OTP salol PDE CDE m-TCP glycerol TNB
243 218 294 310 205 193 342
0.38 0.35 0.41 0.43 0.42 0.27 0.46
0.23 0.21 0.34 (0.41) 0.24 0.16 0.32
(0.10) 0.22 0.34 0.35 0.25 0.08
0.23
0.23 0.096 0.31
Figure 4. Comparison of self-diffusion and dielectric relaxation. (See legend of Figure 2 for letters a-f.) Only the data of Stickel48,60 (O) are compared with the D values of Figures 1 and 2.
Figure 3. Comparison of dielectric relaxation and shear viscosity. (See legend of Figure 2 for letters a-f.) The dielectric relaxation times τdiel were taken from the literature: Stickel et al.48,60 (O), Dixon et al.69 (2), Hofmann70 (×). τdiel ) 1/(2πfmax) was determined from the maximum of the dielectric function ′′(ω).
Dη/T shown in Figure 2 should be constant if the StokesEinstein relation, eq 8, were valid. The increase observed on lowering the temperature below ∼1.2Tg is similar to that observed previously in OTP18 and for the diffusion of tracers having about the same size as the solvent molecules.19 Since the comparably high viscosity of glycerol has precluded selfdiffusion experiments at T < 1.3Tg, we could not check whether a similar enhancement occurs in this liquid. Although the hydrodynamic limit assumed in the SED relation, eq 8, does not apply to self-diffusion, it is interesting to discuss the numbers obtained, which are given in Table 1 (Rtrans ) RH). Here, only the high temperature regime was evaluated by drawing the horizontal lines in Figure 2. The van der Waals radii RvdW also listed in Table 1 were obtained from atomic radii as proposed by Edward.46 The fact that Rtrans < RvdW in most liquids has been amply discussed in the literature.47 The effect is largest for liquids with internally flexible molecules (m-TCP, TNB) whereas for rigid molecules both radii are of comparable magnitude, in particular, if slip boundary conditions are used resulting in RH values larger by a factor of 1.5.47 If the enhancement of D on approaching the glass transition is also expressed through apparent RH values, these may be reduced by perhaps two decades or more at Tg as was inferred from tracer diffusion experiments.24,32 Below, we shall discuss this enhancement in relation to different models of heterogeneity. Rotational Diffusion. If dielectric relaxation times, τdiel, are identified with rotational correlation times, τdiel ) τrot(L)1) ) 1/ D , one can check to what extent the expression η/Tτ 2 rot diel follows the Debye relation, eq 8. In Figure 3, we have plotted the corresponding expression versus T/Tg using literature values of τdiel. The corresponding apparent hydrodynamic radii
obtained from the horizontal lines through the high temperature values are listed in Table 1 (Rdiel ) RH). Surprisingly, the values of Rdiel as estimated in OTP48 and glycerol are reduced below the Rtrans value by as much as a factor of ∼2. In glycerol, this is paralleled by a similar reduction of Rrot determined from 2H NMR and this has been discussed before.42,49 However, the enhancement of dielectric relaxation even at T ∼ 1.2Tg in OTP is more difficult to understand. Since the rotational correlation time τrot(L)2) obtained from 2H NMR in deuterated OTP18,49 scales with η in this T regime and follows the Debye equation, eq 8, we find that τdiel is reduced by about one decade in comparison with τrot. Interestingly, an enhancement of the same order of magnitude has recently been discovered in supercooled toluene (at T < 1.15Tg), which is also a very fragile van der Waals glass former.50 It should be noted that the difference may be even larger if the different order of the Legendre polynomials (L ) 2 in NMR, L ) 1 in diel.Rel.) is taken into account. Furthermore, τdiel corresponds to a mean retardation time, which is generally larger than the relaxation time that should be compared with τrot (see discussion following eq 18). The increase of η/Tτdiel observed in Figure 3 on approaching Tg parallels that of Dη/T for self-diffusion which is demonstrated in Figure 4 where we have plotted the product Dτdiel, which is constant for all liquids (except for OTP where the enhancement of η/Tτdiel occurs at lower temperature, see Figure 3). At present, we do not understand this coincidence. The explanation of enhanced translational diffusion discussed above in terms of the environmental fluctuation model is certainly not applicable to dielectric relaxation. Furthermore, the product Dτrot is not constant but increases as in Figure 2 if τrot is determined from single particle correlation functions as was demonstrated for self-diffusion in OTP and for a number of different tracers in various glass forming liquids.20,24 Thus, we have to look for other explanations in order to understand why Dτdiel remains constant although Dτrot increases on approaching Tg. In this context it is interesting to note the increase of η/TτLS observed in OTP for the mean correlation time τLS determined by photon correlation spectroscopy.51,52 Between ∼1.2Tg and Tg, about the same increase by a factor of ∼3 is found for η/TτLS as for η/Tτdiel (see Figure 3a). In the other liquids listed in Figure 3 the increase of η/Tτdiel is generally larger. This is paralleled by a similar increase of η/TτLS in salol.53,54 τrel ≈ τLS has been found in other glass forming
Heterogeneity at the Glass Transition
Figure 5. Self-diffusion coefficients of TNB (9) compared with tracer diffusion coefficients43 of rubrene (O) and tetracene (0). Literature data40 of T/η (full line) are shifted (right scale) in order to fit the selfdiffusion coefficients of TNB at high temperatures. (See text for horizontal and vertical arrows.) Inset: Plot of Dη/T for TNB corresponding to that of Figure 2.
liquids at temperatures close to Tg.52,54-56 Fischer and coworkers51 have related the increase of η/TτLS in OTP with cooperative molecular motion by postulating that the correlation volume Va of cooperative motion agrees with the hydrodynamic volume associated with the τrel values of the relaxation time distribution H ˆ (τrel) defined in eq 17 (or τret of eq 18). By relating dielectric tensor fluctuations with shear angle fluctuations and shear compliance4 they arrive at an expression (eq 9 in ref 51) relating the T dependence of Va with the increase of η/TτLS. Their arguments can be carried over to the case of dielectric relaxation in order to describe the increase of η/Tτdiel. It should be emphasized that these arguments do not apply to the increase of Dη/T shown in Figure 2. Thus, we are left with the result that the correlation times τdiel and τLS, which are sensitive to cooperative molecular rotational motion, differ considerably from the single particle rotational correlation times τrot determined in 2H NMR, if a liquid is supercooled toward Tg. Although, one can associate the different temperature dependence with that of a volume Va of cooperative motion51 a theoretical explanation is still missing. aab-Trinaphthylbenzene (TNB). The self-diffusion coefficients D of TNB are shown in Figure 5 along with the tracer diffusion coefficients of rubrene and tetracene determined by Ediger and co-workers43 and the shear viscosity40 plotted as log(T/η) and shifted vertically in order to fit the self-diffusion data of TNB in the high temperature region. The translational enhancement found at T < 1.2Tg is comparable to that in the other glass forming liquids investigated as can be seen by comparing the inset of Figure 5 with Figure 2. Since no dielectric relaxation times are available for TNB, we have omitted these data from Figures 2-4. However, the relaxation times τLS determined from photon correlation spectroscopy57 are smaller by about one decade than the rotational correlation times determined by 13C 2D-NMR 41 (at T J Tg), which parallels the difference between τdiel and τrot in OTP and toluene (see above). At T J 450 K, rotational correlation times τrot could be determined from 1H-NMR41 and 2H-NMR42 spin-lattice relaxation times.58 Both data sets were interpreted by using the Debye relation (eq 8) for determining apparent hydrodynamic radii Rrotnmr ) (kBT/8πηDrot)1/3 with Drot ) 1/6τrot(L)2) and τrot(L)2) being the average over a Cole-Davidson distribution33 with width parameters βCD of 0.42 and 0.40 for 1H and 2H NMR, respectively, and equal values RrotNMR ) 0.31 nm. In 2H NMR, a nonexponential magnetization decay was observed over the whole temperature range, T J 450 K, that could be described by a biexponential fit yielding two different T1 values. By
J. Phys. Chem. B, Vol. 101, No. 43, 1997 8799 assuming that the three naphthyl groups of TNB reorient rapidly around their bonds to the central benzene ring one could determine the rotational correlation time τrot(total) and τrot(int) of the whole molecule tumbling and the internal rotation, respectively, the latter being smaller by about one decade.42 By applying the Debye relation to τrot(total), a larger hydrodynamic radius Rrot(total) ) 0.38 nm was determined. The apparent hydrodynamic radius of TNB, Rtrans ) 0.32 nm, obtained from the Stokes-Einstein relation (eq 8), Rtrans ) kBT/ 6πηD, agrees with the value of Rrot ) 0.31 obtained from 1H and 2H NMR without the correction for internal rotation but is smaller than the value Rrot(total) ) 0.38 nm. A similar behavior was found by Ediger et al. for the tracer diffusion of rubrene in TNB 43 where the values of Rtrans ) 0.18 nm and Rrot ) 0.32 nm can be determined by applying the SED relation (eq 8) to their data. It should be expected that the fluorescence bleaching technique yields Rrot(total) of the tracer, indicating that for both, TNB and rubrene, Rrot(total) is larger than Rtrans and both values are surprisingly small in comparison with the van der Waals radius RvdW ) 0.46 nm for TNB. On the other hand, a radius of Rrot(total) ) 0.49 nm is obtained for rubrene in OTP from ref 59 of Ediger et al. which is very close to RvdW. Apparently, the fast internal rotation of the bulky naphthyl groups provides a mechanism that speeds up translational and rotational motion in TNB in comparison with OTP.24,59 Heterogeneity. The horizontal arrows in Figure 5 indicate that the tracer diffusion coefficients of tetracene can approximately be mapped upon those of rubrene (and the T/η curve) by application of a T shift by ∆T ≈ 12 K. In a previous study of the size dependence of tracer diffusion19 we have found that these ∆T shifts (suggested by Stickel48,60) are generally possible (at T < 1.3Tg) if the size of the tracers is smaller than that of the solvent molecules. This could be related to spatial heterogeneity via the time temperature superposition principle33 as follows: Translational diffusion is dominated by molecular motion in fluidized (“fast”) domains having a lower Tg than the less mobile (“slow”) domains which determine the mean shear relaxation times 〈τrel〉 ) τG ) G0η (see eq 17). If we relate η with a diffusion coefficient Dη in the “slow” domains by Dη ) kBT/6πRvdW we obtain D(T) ) Dη(T + ∆T) where ∆T is of the order of the Tg difference between the “fast” and “slow” domains.61 This picture is confirmed by the behavior of the mean rotational correlation times (Figure 5 of ref 43) dominated by motions in the “slow” domains. The values for tetracene and rubrene cannot be superimposed by a horizontal ∆T shift but rather by a Vertical shift of log (T/η) representing the difference of their molecular sizes. On the other hand, the different vertical arrows in Figure 5 of the present paper (from Figure 7 in ref 43) demonstrate that the enhancement of translational diffusion increases much faster for tetracene than for rubrene on lowering the temperature. The observation that the ∆T shift vanishes for tracers having sizes larger than the solvent molecules19,24,32,43 indicates that any spatial heterogeneity described by “fast” and “slow” domains results in relatively small domain sizes. Otherwise, the translational diffusion of the larger tracers would also be dominated by the “fast” domains. A further indication that the domain sizes cannot be large is given by the observation43,59,62 that the width of the rotational correlation time distribution (related with ∆T by the time temperature superposition principle4) decreases with increasing tracer size and a monoexponential decay is already found for tracers not much larger than about twice the solvent molecules. Apparently the translational and rotational diffusion of large tracers is averaged over the “fast” and “slow” domains in a way where the difference
8800 J. Phys. Chem. B, Vol. 101, No. 43, 1997 between 〈τ〉 and 〈τ-1〉-1 vanishes and this already occurs for relatively small tracers. So far, we have discussed an essentially static domain picture where the domains have a lifetime that is longer than the slowest rotational correlation times of the τrot distribution. This picture is supported at T ∼ Tg by the “deep bleach” experiments of Ediger et al.16 and was the basis for an estimate of a mean domain size J2.5 nm from the relation of the width of the τrot distribution with the apparent hydrodynamic volume of the tracers.59 Since the τrot distribution width decreases on increasing the temperature one has to assume a corresponding reduction of the domain size which qualitatively corresponds to the T dependence of the cooperative volume in the scenario of Donth et al.4,51 However, the reduced 4D-NMR14,15 and dielectric hole burning17 experiments performed at T J Tg + 10 K have led to the conclusion that any “slow” domains must have a finite lifetime τs which is of the same order of magnitude as the “slow” rotational correlation time τrot(slow) (corresponding to the “slow” state xs in the EFM, see discussion below eq 13). This implies an alternative possibility for explaining the reduction of translational enhancement as well as the narrowing of the width of the τrot distribution with increasing temperature:59 The mean domain lifetime is longer than 〈τrot〉 at Tg but decreases more rapidly with increasing T and becomes much shorter than 〈τrot〉 at T . Tg. Thus, the lifetime of the domains (of constant size) becomes of the order of the “fast” rotational correlation times τrot(fast) (corresponding to the fast exchange limit in the EFM) at T . Tg and the apparent τrot distribution narrows to a δ-function. This behavior is also in harmony with the “deep bleach” experiments at Tg.16 However, we should emphasize that this tentative explanation must be substantiated by further experiments. In particular, it is necessary to study to what extent the different experiments probing a selected subensemble really measure the same quantities. For example, the process of “filling” a dielectric hole might differ from the exchange of single particle rotation rates in reduced 4D-NMR. It should also be noted that the qualitative explanations given above for the narrowing of correlation time distributions with increasing T provide no quantitative description of experimental results (say, on the width of relaxation time distributions33,47) which apparently depend upon details of molecular shapes and intermolecular interactions not considered in the qualitative discussion above. There is still a wide range of possibilities as to how spatial domains fluctuate between “slow” and “fast” states. Stillinger and Hodgdon10 have proposed a “flickering” fluidized domain mechanism where around Tg one has essentially an amorphous solid but a small fraction (φf ∼ 10-5) of the total volume consists of highly mobile domains flickering at a fast rate and there may be a spatial correlation of successive flickerings within a certain region.12 On the other hand, one can also assume that the fractions of “slow” and “fast” domains are of the same order and the domain lifetime is given by the time needed for crossing the fast domains by translational diffusion.11 This implies that the interface between the “slow” and “fast” domains diffuses on the same time scale. Still further possibilities for domain fluctuations can be inferred from phenomenological rate exchange models within the EFM.31 We should also note that there are further possibilities of assuming spatial heterogeneity, e.g., based on the Adam-Gibbs model,1,3,63,64 or models of “dynamically correlated” domains,65 or the thermodynamic model of Kivelson and co-workers66 postulating the existence of a “narrowly avoided” critical point far above Tg which explains the complex behavior of supercooled liquids via “structural frustration”. The latter model has
Chang and Sillescu gained support by the discovery of a “glacial” amorphous phase which may be related with “slow” dynamics close to Tg.67 However, in all scenarios of spatial heterogeneity the dynamics of the domain fluctuations must be supplemented by additional assumptions. Finally, we should note that there may be spatial heterogeneities that cannot be treated by the EFM. For example, consider a system with “fast” regions which are fixed in space and isolated from each other by very “slow regions”. In this case, the lifetime of a molecule in a “slow” region depends upon the distance to the nearest “fast” region. In other words, the transition rates from “slow” to “fast” states depend explicitly upon position in space. Although this is covered by eq 1 in principle, it has been excluded in eq 2 where the transition rate Π(x|x′) and the diffusion coefficient Dtrans (x) are assumed independent of the position r. A simple model with spatially dependent diffusion in a fluctuating environment was treated via an effective medium assumption by Zwanzig.68 V. Conclusions The self-diffusion coefficients determined in a number of glass forming liquids support our previous finding of enhanced translation on lowering the temperature below T/Tg ∼ 1.2.18 This enhancement can be described by a phenomenological environmental fluctuation model where the translational diffusion coefficient given by the average D ) 1/2∆2〈τtrans-1〉 is dominated by the “fast” portion of the environmental states, whereas the mean rotational correlation time 〈τrot〉 is dominated by the “slow” portion (see eqs 3 and 5). This explanation does not apply to our observation that η/Tτdiel is also enhanced close to Tg and Dτdiel remains constant at temperatures where Dτrot increases on lowering T (Figure 4). For a number of glass forming liquids one observes that dielectric relaxation times τdiel and correlation times τLS determined from photon correlation spectroscopy are of comparable magnitude and both are smaller than τrot as determined from NMR or fluorescence bleaching experiments. This behavior is still unexplained and deserves further study.56 Finally, we have discussed our results in comparison with studies on the size dependence of translational and rotational diffusion of dye tracers in glass forming liquids.19,43,59 From the observation that the translational enhancement vanishes and the rotational correlation function becomes exponential if the tracer size is larger than the solvent molecule size (in TNB, the size of the mobile methyl group), we conclude that any spatial heterogeneity must be characterized by relatively small domain sizes. These must be further reduced on raising the temperature in order to explain the decrease of translational enhancement in a scenario where the lifetime of the domains is large. If the lifetime decreases rapidly with increasing T it may become much shorter than 〈τrot〉 at ∼1.2Tg and the system then appears homogeneous due to the fast time fluctuations of any domains which may even have a constant size. Since there is no unambiguous experimental information on the domain size (except for its upper limit as discussed above) we favor the phenomenological description by an environmental state model which is sufficiently flexible in order to account for all present experimental results and allows us to distinguish between well defined scenarios of dynamical heterogeneity. Acknowledgment. We are grateful to M. D. Ediger for sending us a preprint of ref 43 and the diffusion coefficients of tetracene and rubrene in RRβ-TNB prior to publication. We appreciate helpful discussions with R. Bo¨hmer, G. Diezemann, and G. Hinze. Support by the Deutsche Forschungsgemeinschaft (SFB 262) is gratefully acknowledged.
Heterogeneity at the Glass Transition References and Notes (1) Adam, G.; Gibbs, J. H. J. Chem. Phys. 1965, 43, 139. (2) Ja¨ckle, J. In Disorder Effects on Relaxational Processes; Richert, R., Blumen, A., Eds.; Springer: Berlin, 1994; p 233; J. Non-Cryst. Solids 1994, 104, 72 and references therein. (3) Fredrickson, G. H. Annu. ReVs. Phys. Chem. 1988, 39, 149. (4) Donth, E.-J. Relaxation and Thermodynamics in Polymers, Glass Transition; Akademie Verlag: Berlin, 1992. (5) Mohanty, U. AdV. Chem. Phys. 1995, 89, 89. (6) Hansen, J. P.; McDonald, I. R. Theory of simple Liquids, 2nd ed.; Academic Press: London, 1986. (7) Go¨tze, W.; Sjo¨gren, L. Rep. Prog. Phys. 1992, 55, 241. (8) Sillescu, H., in preparation. (9) Stillinger, F. H. J. Chem. Phys. 1988, 89, 6461. (10) Stillinger, F. H.; Hodgdon, J. A. Phys. ReV. E. 1994, 50, 2064. (11) Sillescu, H. Phys. ReV. E 1996, 53, 2992. (12) Stillinger, F. H.; Hodgdon, J. H. Phys. ReV. E 1996, 53, 2995. (13) Schmidt-Rohr, K.; Spiess, H. W. Phys. ReV. Lett. 1991, 66, 3020. (14) Heuer, A.; Wilhelm, M.; Zimmermann, H; Spiess, H. W. Phys. ReV. Lett. 1995, 75, 2851. (15) Bo¨hmer, R.; Hinze, G.; Diezemann, G.; Geil, B.; Sillescu, H. Europhys. Lett. 1996, 36, 55. (16) Cicerone, M. T.; Ediger, M. D. J. Chem. Phys. 1995, 103, 5684. (17) Schiener, B.; Bo¨hmer, R.; Loidl, A.; Chamberlin, R. V. Science 1996, 274, 752. (18) Fujara, F.; Geil, B.; Sillescu, H.; Fleischer, G. Z. Phys. B 1992, 88, 195. (19) Heuberger, G.; Sillescu, H. J. Phys.Chem. 1996, 100, 15255. (20) Cicerone, M. T.; Blackburn, F. R.; Ediger, M. D. Macromolecules 1995, 28, 8224. (21) Beckert, D.; Pfeifer, H. Annu. Phys. (Leipzig) 1965, 16, 262. (22) Anderson, J. E.; Ullmann, R. J. Chem. Phys. 1967, 47, 2178. (23) Sillescu, H. J. Chem. Phys. 1971, 54, 2110. (24) Cicerone, M. T.; Ediger, M. D. J. Chem. Phys. 1996, 104, 7210. (25) Orbach, R. Science 1986, 231, 814. (26) Sillescu, H. J. Chem. Phys. 1996, 104, 4877. (27) Ngai, K. L.; Rendell, R. W. J. Mol. Liquids 1993, 56, 199. (28) Ngai, K. L.; Rendell, R. W.; Plazek, D. J. J. Chem. Phys. 1991, 94, 3018. (29) The average 〈Dtrans〉 can also be interpreted (Hinze, G. Ph.D. Dissertation, Univ. Mainz 1993) as an average over the mean residence times, τres(xi) [which may be estimated as τres(xi)-1 = ∑xjΠ(xj|xi)] of molecules in the states xi during their spatial displacement: 〈Dtrans〉 ) 〈r(t)2〉/ 6t ) ∑xiDtrans(xi)τres(xi)/t, where τres(xi)/t is the fraction of time spent in the state xi. (30) Tarjus, G.; Kivelson, D. J. Chem. Phys. 1995, 103, 3071. (31) Diezemann, G.; Bo¨hmer, R.; Hinze, G.; Sillescu, H. ACS Symp. Ser. 1997, 676, in press. (32) Chang, I.; Fujara, F.; Geil, B.; Heuberger, G.; Mangel, T.; Sillescu, H. J. Non-Cryst. Solids 1994, 172-174, 248. (33) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York 1980. (34) Due to the conventions on p 60 of ref 32 one defines G*(ω) ) iω ∫∞0 G(t) exp(-iωt) dt and G(t) ) Ge + ∫∞0 H(ln τ)exp(-t/τ)d ln t where H(ln τ) ) τH ˆ (τ) and Ge ) 0 in viscoelastic liquids. In dielectric relaxation, *(ω) ) ∞ + (0 - ∞)∫∞0 (1 + iωτdiel)-1gˆ(τdiel) dτdiel where 0 and ∞ denote the low and high frequency limits, respectively, and ∞ corresponds to Je ) 1/Ge in a viscoelastic solid. (35) Richert, R.; Wagner, H. J. Phys. Chem. 1995, 99, 10948. Polymer, in press. (36) Piccirelli, R.; Litovitz, T. J. Acoust. Soc. Am. 1957, 29, 1009. (37) Vand, V. Research 1947, 1, 47. (38) Tammann, G.; Hesse, W. Z. Anorg. Allg. Chem. 1926, 156, 245. (39) Segur, J. B.; Oberstar, A. E. Ind. Eng. Chem. 1951, 43, 2117. (40) Plazek, J. D.; Magill, J. H. J. Chem. Phys. 1966, 45, 3038; 1968, 49, 3678.
J. Phys. Chem. B, Vol. 101, No. 43, 1997 8801 (41) Zemke, K.; Schmidt-Rohr, K; Magill, J. H.; Sillescu, H.; Spiess, H. W. Mol. Phys. 1993, 80, 1317. Zemke, K. Diplomarbeit, Univ. Mainz, 1990. (42) Schnauss, W. Ph.D. Dissertation, Univ. Mainz, 1991. (43) Blackburn, F. R.; Wang, C. Y.; Ediger, M. D J. Phys. Chem. 1996, 100, 18249. (44) Fleischer, G.; Fujara, F. NMR Basic Principles and Progress; Springer-Verlag: Berlin, 1994; Vol. 30. (45) Chang, I.; Fujara, F.; Geil, B.; Hinze, G.; Sillescu, H.; To¨lle, A. J. Non-Cryst. Solids 1994, 172-174, 674. (46) Edward, J. T. J. Chem. Educ. 1970, 47, 261. (47) Tyrrell, H. J. V.; Harris, K. R. Diffusion in Liquids; Butterworths: London, 1984. (48) Stickel, F. Ph.D. Dissertation, Univ. Mainz, 1995. (49) Schnauss, W.; Fujara, F.; Sillescu, H. J. Chem. Phys. 1992, 97, 1378. (50) Do¨ss, A.; Hinze, G.; Schiener, B.; Hemberger, J.; Bo¨hmer, R. J. Chem. Phys. 1997, 107, 1740. (51) Fischer, E. W.; Donth, E.; Steffen, W. Phys. ReV. Lett. 1992, 68, 2344. (52) Steffen, W.; Patkowski, A.; Meier, G.; Fischer, E. W. J. Chem. Phys. 1992, 96, 4171. (53) Sidebottom, D. L.; Sorensen, C. M. Phys. ReV. B 1988, 40, 461. (54) Ruths, T. H. Ph.D. Dissertation; Univ. Mainz, 1996. (55) Hagenah, J. U. Ph.D. Dissertation; Univ. Mainz, 1988. (56) Simultaneous dielectric relaxation and dynamic light scattering experiments in supercooled liquids are currently being carried out by E. W. Fischer, R. Richert, and their co-workers at the Max Planck-Institut fu¨r Polymerforschung in Mainz. (57) Zhu, X. R.; Wang, C. H. J. Chem. Phys. 1986, 84, 6086. (58) The 2H-relaxation data of ref 41 could only be interpreted in a consistent way if the temperature read from the instrument was reduced by 13.5 K. Since the origin of this T shift could not be unambiguously clarified the results were not published. However, the conclusions drawn in the present paper are not affected by this T shift. (59) Cicerone, M. T.; Blackburn, F. R.; Ediger, M. D. J. Chem. Phys. 1995, 102, 471. (60) Stickel, F.; Fischer, E. W.; Richert, R. J. Chem. Phys. 1995, 102, 6251. (61) We refer to ref 19 for further discussion of ∆T shifts but wish to point out here the close correspondence with applications of time temperature superposition in rheology. In these experiments, pieces of a relaxation function, say G(t), are determined at different temperatures above Tg within the time window of the instrument and afterward shifted onto a master curve which corresponds to a single “reference” temperature.33 Since this is only possible if all components of the relaxation time distribution have the same T dependence, one can also shift these components with respect to one another, which is also done in constant frequency experiments in order to determine the average relaxation time. (62) Davies, M.; Hains, P. J.; Williams, G. J. Chem. Soc. Faraday II 1973, 69, 1785. (63) Moynihan, C. T.; Schroeder, J. J. Non-Cryst. Solids 1993, 160, 52. (64) Mountain, R. D. J. Chem. Phys. 1995, 102, 5408. (65) Chamberlin, R. V. Phys. ReV. B 1993, 48, 15638 and references therein. (66) Kivelson, D.; Kivelson, S. A.; Zhao, X.; Nussinov, Z.; Tarjus, G. Physica A 1995, 219, 27. (67) Cohen, I.; Ha, A.; Zhao, X.; Lee, M.; Fischer, Th.; Strouse, M. J.; Kivelson, D. J. Phys. Chem. 1996, 100, 8518 and references therein. (68) Zwanzig, R. Chem. Phys. Lett. 1989, 164, 639. (69) Dixon, P. K.; Wu, L.; Nagel, S. R.; Williams, B. D.; Carini, J. P. Phys. ReV. Lett. 1990, 65, 1108. (70) Hofmann, A. Ph.D. Dissertation, Univ. Mainz, 1993. (71) Fujara, F., unpublished results.
