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Diffusion and the Mesoscopic Hydrodynamics of Supercooled Liquids Xiaoyu Xia∗ and Peter G. Wolynes† ∗ Department of Physics, University of Illinois, Urbana, IL, 61801 † Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, CA 92093 The description of molecular motion by macroscopic hydrodynamics has a long and continuing history. The Stokes-Einstein relation between the diffusion coefficient of a solute and the solvent 1 viscosity predicted using macroscopic continuum hydrodynamics, is well satisfied for liquids under 0 ordinary to high temperature conditions, even for solutes as small as the solvent. Diffusion in 0 supercooled liquids near their glass transition temperature has been found to deviate by as much 2 as 3 orders of magnitude from that predicted by the Stokes-Einstein Relation [1]. Based on the n random first order transition theory [2–4], supercooled liquids possess a mosaic structure. The a size- and temperature-dependence of the transport anomalies are quantitatively explained with an J effective medium hydrodynamics model based on the microscopic theory of this mesoscale, mosaic 5 structure. ] n 64.70.Pf n - s According to macroscopic continuum mechanics and formingliquidsatlowtemperaturearemodeledasatwo- i d Einstein’s Brownian motion theory, the diffusion con- componentmedium. Onecomponentisasolid-likeback- . stantofaprobeparticleisdeterminedbytheviscosityof ground in which the diffusion is slow or impossible; the t a theliquidthroughwhichitmovesbythe Stokes-Einstein othercomponentisaliquid-likeexcitationinwhichstress m Relation can be released quickly and the probe can move faster. - By adjusting the parameters like the lifetime, appear- d k T n DtS−E = 4πBηR, (1) ancerate,internalviscosityandthesizeoftheliquid-like o domains, Stillinger and Hodgdon [11,12], and Cicerone, c where DS−E is the translational diffusion coefficient, η Wagner, and Ediger [13] have shown it is possible to get [ t is the viscosity, and R is the radius of the probe. This significant deviations from the Stokes-Einstein law. It 1 has been appreciated [13] that there might not be such relation must hold if the solute is much larger than any v a clear cut distinction between two components; instead relevant length scale in the liquid. Remarkably it is well 3 thereshouldbeadistributionofheterogeneities. Asensi- 5 satisfied even for self diffusion, i.e., where the radius of blequestiontoaskisthenhowtoderivethisdistribution 0 the solvent is the same as the solute. It is actually not a 1 badrelationevenfordiffusioninmoderatelysupercooled from microscopic theory and thereby to explain quanti- 0 liquids (T ≥T ) [5–7]. Viscosity can vary by as much as tativelythedecouplingbetweendiffusionandviscosityin 1 g supercooled liquids. 12 orders of magnitude and diffusion coefficients vary by 0 / nearly as much. Nevertheless there is a noticeable dis- t a crepancywiththe macroscopichydrodynamicprediction m in the vicinity of glass transition temperature T . In su- g - percooled liquids around Tg, the translational diffusion d is enhanced by as much as 3 orders of magnitude out ho n of the 14 orders of magnitude change of viscosity [8,9,1]! o Cicerone and Ediger have found the deviation depends c hi : strongly onthe size of the probe and the temperature at L v which the measurement is taken [1]. For deviations as i X large as a factor of 1,000, small adjustments of hydro- r dynamic boundary conditions arising from the details of a the probe-liquid interface will not suffice. It is reason- able to conclude there is an additional length scale in a supercooled liquid to explain the discrepancy. Such a length scale arises naturally in the random first order transition theory of glasses. In this paper we will sketch FIG. 1. A probe particle (shaded circle) with radius R is the predictions of that theory for the deviations from shown in a mosaic-like supercooled liquids. At the instant the Stokes-Einstein law expected in supercooled liquids. shown, the particle is moving inside oneof thedomains with Ediger and others recently have used the size-dependent size L. The viscosity inside the domain is denoted as ηi and deviations to infer the size of heterogeneous domains in ηo for theaveraged outer region. supercooled liquids [10]. In the published models, glass- 1 Recently, we developed a microscopic theory of acti- This suggests regions of size L are separated by domain vation barriers and fragilities in supercooled molecular wallsandcanreconfigurenearlyindependently. Thecon- liquids based on the random first order theory for the figurational entropy, the driving force for reconfigura- liquid-glass transition [4]. Several remarkable quantita- tions,fluctuatesaroundameanvalues =∆c˜(T)T−TK, c p TK tive regularities emerge from theory. The relation be- where ∆c˜(T) is the specific heat jump per unit volume p tween the typical activation barrier in a supercooled liq- at the transition, with a magnitude δs = k ∆c˜/V‡, c B p uid and its configurational entropy is quantitatively ob- whereV‡ = 4πL3isthevolumeofatypicalpdomain. This 3 tained[4]. Furthermorethetheoryyieldsthedistribution results in a corresponding variation in free energy barri- of activation barriers explaining the β parameter of the ersandsizeforeachmosaicelement. Thisthereforegives stretchedexponentialrelaxationfunctions [14]. The the- adistributionofrelaxationtimesanddomainsizes. This ory also yields the size scale of heterogeneous relaxing leads to a quantitative relation between the Kohlrausch- domains [4,14]. We are now ready to investigate the ef- William-Watts exponent β and the liquid fragility [14]. fect of such heterogeneities on molecular hydrodynamics The intricate mosaic structure of the supercooled liq- insupercooledliquids. Tosetthe stagewebrieflyreview uidstatepresentsachallengingprobleminhydrodynam- therandomfirstordertheoryandthemicroscopictheory ics. We can think of each region as having its own vis- of barriers. cosity η determined mostly by the local relaxation time i The Random first order transition framework for the τ . The distribution of τ is determined by the entropy i i liquid-glass transition suggests supercooled liquids relax fluctuations. Of course there will be fluctuations in the inanactivatedmannerbelowacertaintemperature,TA. higher frequency elastic modulus of each region, yi, but –The system has to overcome some free energy barrier these should be small. Thus η ≈< y > τ . If the probe i i to reach another metastable state. The activation bar- is as large or larger than a typical domain it will be sur- rier arises from a competition between the favorable in- rounded several cells of the mosaic with varying η . But i crease in configurational entropy (the large number of the effect of this will be essentially the same as a homo- other minimum energy configurations the local region geneous medium with the average viscosity η =< η >. i could freeze into) and an unfavorable interfacial energy Only probes smaller than L will be affected. The situa- (theenergycostofforminganotherstateintheoriginal). tion should be well described by a two-zone fluctuating Thisinterfacialenergyreflectsthe costofadomainwall- viscosityhydrodynamics: aviscosityη locally,whichhas i like excitation where atoms are strained and are in far a fluctuating value embedded in an infinite region with from local energy minimum configurations. Mathemat- the average viscosity η. A two-zone mean-field hydrody- ically, the free energy change in forming a droplet with namics description where the local viscosity is fixed was radius r is proposed by Zwanzig [15] and in a different context by Goodstein to explain ion mobility in superfluid helium 4 F(r)=− πTscr3+4πσ(r)r2, (2) [16]. This picture was elaborated for glasses by Hodg- 3 don and Stillinger [11]. Much of their calculation can where sc is the configurational entropy density driving now be taken over, but with an additional final averag- the random first order transition. The surface tension is ing being required. If the viscosity has one value η in i stronglyrenormalizedby the multiplicity ofstateswhich the inner zone and a different valueη in the outer zone, o can wet the interface, much like in random field Ising foraincompressiblefluidwithlowReynoldsnumber,the model, σ(r) = σ0(rr0)1/2 [4], where r0 is the interparti- fluidvelocityvi isdeterminedbythelinearNavier-Stokes cledistance. Theshortrangesurfacetensionσ0 isnearly equation, universalsinceitdependsonthedegreeofvibrationallo- ∂p calizationinaperiodicminima. Thisisusuallyexpressed η∆v = , (5) i ∂x as a universal Lindemann-like criterion for vitrification i [4]. The free energybarrierdetermined by Eq.(2)canbe with two values of the viscosity as shown in Fig.(1). p is expressed as the pressure, and the equation of continuity, 3πσ2r ▽·~v =0, (6) ∆F‡ = 0 0, (3) Tsc must be satisfied. For a probe particle with radius R centered in such a spherical domain with size L, solving When the Lindemann value is substituted into the mi- Eq.(5) and (6) with the proper boundary conditions at croscopic expression for σ this results in an excellent 0 r = R and r = L, Hodgdon and Stillinger as well as description of the activation barriers in a wide range of Goodstein find, liquids. Thesizeofthedroplet(ordomain)isdetermined by the condition F(L)=0 giving f =4πη RuC , (7) o 5 3σ r0.5 wheref isthetranslationaldragforceontheprobe. This L=( 0 0 )2/3. (4) Ts isobtainedbyintegratingthepressureoverthesurfaceof c 2 thesphere,uisthefluidvelocityfarfromthefluctuating to be expected for the approximated two-zone model we domain, and C is a constant determined by the ratios applied. It could be connected by a very modest change 5 of inner/outer viscosity and L/R. Using slip boundary (∼10%)inthe localdomainsizeanddoubtlesslyreflects conditions, at the surface of the probe molecule, the crudeness of the simple geometry we have chosen. C =l[−3+3l5+ζ(3+2l5)]/d , 5 s ds =−2+2l5+ζ(4−3l+l5+3l6) 3 +ζ2(l−1)3(1+l)(2+l+2l2), (8) ) E 2.5 S- where l =L/R and ζ =ηo/ηi. (cid:13)/D 2 From Eq.(7), we have the translational diffusion coef- t D 1.5 ficient in a given domain as ( g o 1 D˜ = kBT , (9) l t 4πη RC 0.5 o 5 Inheterogeneoussupercooledliquidmosaic,the effective 1 2 3 4 5 6 7 translational diffusion coefficient for the whole system r R / 0 mustbeaveragedoverthedistributionoflocalrelaxation FIG. 2. The deviation from Stokes-Einstein Relation times, highly depends on the size of the probe particle as shown on thegraph. Theradiusoftheprobeparticleisexpressedinthe Dt =Z D˜tP(sc)dsc. (10) unitof molecular diameters in theliquid. Foraprobewith a sizeofseveralmoleculardistance,thedeviationdropsrapidly, The deviationfromthe homogeneoushydrodynamicsre- i.e., the Stokes-Einstein Relation is retained. The points are fromexperiments[1]andthesolidlineisthepredictionmade sults, defined as the ratio between “real” diffusion coef- byrandom first order theory for glass transition. ficient and the one predicted by the macroscopic Stokes- Einstein Relation, Eq.(1), is Thetemperaturedependenceofthedeviationfromthe D 1 macroscopicStokes-EinsteinlawisshowninFig.(3). The t = P(s )ds . (11) DtS−E Z C5(ζ,l) c c measurementwasconducted forterraceneino-terphenyl [1]. Again, there is a very good agreement between the With Eq.(3)and (4), we may express ζ and l in terms of theory and experiment over a large temperature range. s , c η 3 ζ = o =eβ∆F(1−∆F/∆F) =eβ∆F(1−sc/sc); (12) η i 2.5 ) E S- L LL¯ L¯ s D 2 l = R = L¯R = R(scc)2/3. (13) D/t1.5 ( where ∆F and L¯ are the typical values for free energy g 1 o barrier and domain size respectively, which can be ob- l 0.5 tained by substituting s into Eq.(3) and (4). Using a c Gaussiandistributionofs withthe givens andδs ,we c c c 0 (cid:13) readily calculate the effective diffusion coefficient for a 1 1.1 1.2 1.3 1.4 finite size probe immersed in the heterogeneous mosaic. T/ Tg The experimental data are available only for a very FIG.3. Thetemperaturedependenceofthedeviationfrom limited number of materials due to the difficulty of such Stokes-EinsteinRelationforterracenediffusinginsupercooled measurementsnearT . (AtT thetypicalrelaxationtime o-terphenylisshown. Thetheoreticalresults(solidline)agrees g g is 1,000-10,000seconds.) Using the results from our pre- excellently with experimental data (points) [1]. vious papers for the distribution of relaxationtimes, the size dependence of the deviation factor for a probe mov- The deviation factors depend on the fragility, D, of ing in o-terphenyl, the most-studied glass-forming ma- the liquids if they are all measured at the glass tran- terial, at Tg can be calculated. The result, which does sition temperature Tg. The degree of the heterogene- not contain any adjustable parameters, is in excellent ity, as measured by δsc/sc, decreases as the liquid gets agreementfor small probes as shown in Fig.(2). The de- stronger [14]. For very strong glass-forming liquids (like viation from the experimental data for large probes is SiO2 with D = 150), one would expect the macroscopic 3 Stokes-Einstein Relation to hold well. The theory pre- proximation we use will hold well. However, for probes dicts large deviations only for very fragile liquids (with with very large size comparable to the typical domain D approximately smaller than 20) as shown in Fig.(4). size, such an approximation may pose a problem as the The plot is for self-diffusion (or probe with similar size diffusion of rubrene in o-terphenyl indicates. When the withthe liquidmolecule). Unfortunately presently,most probe particle occupies multiple domains, an additional of the experiments have been reported only for a very average over these domains should be taken to get more narrowrangeofsubstanceswithD ∼10. Wepresentthe accurate results. Nevertheless, the model developed in fragility dependence thus as a prediction to encourage this paper should serve as a goodstarting point even for investigations over a wider range of liquids. larger probe diffusion. Insummary,themosaicpictureofthesupercooledliq- OTP uids based on the random first order transition theory 3 explainstheunusualdecouplingoftranslationaldiffusion 2.5 from the macroscopic hydrodynamics of glass-forming ) E S- liquids. The picture has as an essential element the D 2 new length scale of dynamically correlatedregions. This (cid:13)/ Dt1.5 lengthisdeterminedtoagoodapproximationbyamicro- (g SiO2 scopic theory of the glassy state itself. The mosaic pic- o 1 ture also has interesting consequences for the frequency l dependenceandnonlinearityoftheresponseofatranslat- 0.5 ing probe. Similarly the mesoscale correlationwill affect 0 rotational hydrodynamics and relative motion of probes 0 20 40 60 80 100 120 140 inthe liquidasisimportantfordiffusioncontrolledreac- D tions. We leave these problems for the future. FIG.4. Thefragility ofthesupercooled liquidshasimpor- This work is supported by NSF grant CHE-9530680. tanteffectontheobservedStokes-EinsteinRelationdeviation. We are happy to dedicate this work to Bruce Berne a For fragile liquids like o-terphenyl, such deviation is as large pioneer of not only computer simulation but also of the as 3 ordersof magnitude, while in strongliquids likeSiO2,it hydrodynamic view of molecular motion in the modern should be only minor. The plot is for self-diffusion at glass transition temperature Tg. era. Thereareacoupleofimportantpointstobemadeclear before we conclude our remarks. One is that we use the “average” or the measured viscosity as the outer viscos- ityη inthecalculation. Amorethoroughtheoryshould o treat simultaneously the fluctuations inside and outside [1] M. T. Cicerone and M. D. Ediger, J. Chem. Phys. 104, the domain. Nevertheless the approximation to sepa- 7210 (1996). rate these two terms turns out to be a sound one. The [2] T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. A 35, physicsunderlyingthesimplificationistheasymmetryin 3072 (1987). the contribution to the diffusion from the “fast” regions [3] T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes, and “slow” regions. From the structure of Eq.(10), it is Phys. Rev.A 40, 1045 (1989). clear fast regions with relatively smaller local relaxation [4] X. Xia and P. G. Wolynes, Proc. Natl. Acad. Sci. U. S. A. 97, 2990 (2000). times (or viscosities) have much larger contributions to [5] E. Rossler, Phys.Rev.Lett. 65, 1595 (1990). the averaged effective diffusion coefficient. A little alge- [6] J.Jonas,D.Hasha,andS.G.Huang,J.Phys.C84,109 bra shows the largest contribution is from regions with (1980). s ∼ 0.67s . For these regions the local relaxation time c c [7] R. Robinson and R. Stokes, Electrolyte Solutions (But- is more than 5 orders of magnitude faster than the mea- terworths, London, 1965), p. 12. sured relaxation time (about 1,000-10,000seconds). 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[16] D.Goodstein, Phys. Rev.B 15, 5210 (1977). 5