REVISTA MEXICANA DE FÍSICA.aS SUI'I.El\IENTO 1, 11~17

Remarks on the glass transition*

Lcopoldo Garda-Col ín t

f)cpartamcllto de Física, U"il'ersidai/ Autónoma Metro/JO/itll1lll./zllIpa/apaAv. Michnacdn y Purísima, S/N, 09340 México D,F., Mn.ico

Recihido cl26 de febrero de 1998: aceptado cl30 de mayo de 1998

JUNIO 199t.J

Sorne of [he mosl relevant conc:epls associaled \l,'ilh the so called slnw ano f<lstrela.'<:I[ionproccsses occuring duting Ihe formation of glasse~are discussed, Emphasis is placed in those whieh Ihe author thinks are at present the 1110strekvant Olles, In particular, Ihe recenl disc~veriesconcerning possihlc relmionships helwcclI the tcmperature anomalies of the Dchye. \Vallcr faclor, tbe hoson peaks and anharmonicities in Ihelow IClllperalure degrees nf freedom in glasses are brought into their prescnt perspcc[ivc.

Keywo/ll.c Glass lransi[ion; Kauzmann's paradox; relaxation limes: boson peak anharlllonicitics

En este trabajo se disculen algunos de los conceptos mas importantes asociados COIllos llamados procesos dc relajación lentos y rápidos queocurren durante la formación de un vidrio, Especial énfasis se pone en m~uellos que el autor piensa son. hoy en día, los mas relevanles. Enparlicular, los descubrimientos recientes apuntando hacia una posible relación entre ;lIlornalías dd factor de Debye-Wallcr corno función dela temperatura. el pico hosónico y la existencia de posibles anarmonicidades en los grados de I¡henml de los vidrios a hajas temperaturas, seuoican en su perspectiva aClual.

Dl'S('f'ipl0r('.\': Transición vítrea; paradoja de Kauzmann: proceso de relajación: pico oo'\<Ínico; allarmonicidadcs

PAes: 61.42.+h; X2.20 Pe; 05.90.+111

1. Introductinn

It is \Videly recognized that the prohlcm of Ihe glass lransitionis nol only olle ofthc lIlost fascinaling ones in condenscd 111<11-ler physi<.:s hut also, il remains praclically unsolved. lvtanl'nf its chara<.:teristic fcalurcs mal' he lIualitativell' ullderstoodlIsing rather simplislic ll1odcIs. in spile 01' the allnOSI over-whelming <ltnOllnt 01' experimental dala now availahle. Nev-crlheless, the mere qucstion 01' lrying to surnmarize the phc-nomcllological knowledge now cxisting ahoul glasses into amore or less llnified lhennohl'drodynarnical schemc rcrnainsan open (ask.

In Ihe rast few years several exccllent revicw anicles cov-ering mosl 01' Ihese aspecls ranging from the properlics 01'slrong and rragile glass forming Iiquids to poll'meric ones.ha ve hccll written [1-81. The wealth 01' Ihe malerial includedin lhese \vorks is su<.:hthal it \vould he completell' presumplll-ous lo atlempl discussing it or even to summarize il in <1singlerareroflimiled lenglh. The reader interesled in the details 01'ho\\' Ihis f¡eld has evolved in lhe past ten years will incvitahlyhe forccd lo read at leasi sOlTIe 01' Ihcl11. lIere \Ve shall lakclhe more modest approach lO the suhject and attempt to em-phasize in sorne 01' the very important, huI weH estahlishedaspccts nI' glasses and lo hring up other ones which are cithcrnot illclllded in sllch reviews or for reasons Ihat will hecomeclear in lhe texI, deservc a closer examination.

The slructurc 01' the papcr is as follo\Vs: In Sect. 2 wcshal1 review sOllle nI' lhe long slanding prohlems in the glassIransitioll, Sec!. 3 is dcvotcd mainly 10 discuss Ihe very newfcatures brought out hy Raman and neutmn scatlering in-volving the sought relatiollshir hcl\Veen Ihe hoson peak ami

Ihe possihle anharmonicilies in Ihe glass helow rhe transitionlempcralure amI Scc!. 4 is dcvoted to some concluding re-marks.

2. The KallZlI1ann paradnx alld related topies

As it has hecn (!Carly stated hy Illalll' aulhors in this field, lheKallzlll<lnn parado.\( [911ies at the very heart orthe glass lran-sition and as 01' loday, it has nol heen resolved. Its originalversion, recently paraphrased hy Angell in his excelcnt re-view 01' the topic ['21 is dirtlclllt lo improve. Bere we shallconlenl ourselvcs with a more pedagogical view hased onIhe traditional (hut not uniqlle~) IIlcthod of forming a glassnamely hy Ihe supercooling (JI' a viscous liquid. In Fig. 1, lhecharacterislic hehavior nI' lile specific heat of a glass formingliquid. in this casc glycerol. is sllown as a funclion of tcm-peralure. Norrnalll' when a liquid reaches its rnelting poinlil cryslalizes ami C,' decrcases its vallle following the curvelahelled crystal. But ir sllpcrcooling is achieved, the matc-rial relllains in lhe liquid rhase anu dcpending on the rate01' cooling a temperalure T~Jis reached where it hccomes aglass. For tempcratures T < 7~ extrapolation of Ihe experi-mental cur\'c would apparcntly exhihit Ihat the specific healnI' the glass is al\Vays I;¡rger Ihan Ihe one 01' the st3hle rhasenamely, the cryslal. Ir Ihis were so a violalion to Ncrnst prin-cipie \\-'ould occur since if rOl' every tcmperature in the rangeO :S T < '''.''' C~I;\.~s > c;;r)'stal thcn since Cp is the sJope01' the elllropy wilh respect to tcmperalllfe, requiring thatcWass --t (J as T --t () in accordance (o Nernst's print:iple,the enlropy 01' the crystal woul£! he negalive at T = O and

LEOPOLDO GARCiA.CoLíN

FIGURE l. Charac.:lcristic behavior of Ihe specitic heat of a glassforming ¡iquid (glyccrol in lhis case) as Ihe tempera!ure Ty isreached and below.

6.05.2 5.61000 IT (K.1 )

4.8

4

1 2-!~

O

.1

-24.4

FIGURE 2, Experimental data for Ihe peak frequcncy glycerol andpropylene-glycol \'ersllJ l/T. The lItted (solid lines) are with aVFf equation [from N.O. Birge.\~1I 163 (1986)1.

T ('K)400

12

0.8

'" '-0.6 ~ .•...• ,DI ....-:::: ,,•• ,, ru ,- ,•• 0.4•..,u

'"10.2

'"

\vhcre lJ is the \'iscosity, '10 == l1(To), To is a lemperature, inprincipie Iying somewhere between Tg and TAo and clearly theone al which lJ => 00, meaning physically the state al whichno configuralional changes in lhe glassy structure are possi-

this is impossihlc. Nc\'cnheless this catastrophc ncvcr oecurs.Al a ccrtain tcmpcraturc Tk < Tg lhe thcrmodynamic vari-ahles dcvialc from lheir equilibrium values forming. via aslO\\' kinctic proccss a long ¡¡ved metastable statc. Thc ex-istence of Tk, rcfcrrcd lo as Kauzmann tcmpcraturc al whichkinclic sluggishncss savcs lhe thermodynamic day is knownas Kauzmann's paradox. \Vhy and how il occurs rcmains asone 01' Ihe hig challcngcs in this field. It is pertillent lo Illcn-(ion hcrc tha! lhe glass translion temperaturc is no! uniquelydCllncd, il depends on the rate of cooling [10]. For practi-cal purposcs <lnd hy convention, sorne authors define il asthe ICll1perature of thc supercooled liquid whcn its viscosityreaehes the value of 1013 poise [1].

Closcly rel<lled lo lhc bchavior of thc specifk hcat of glassforll1ing liquids as T9 is reached, is the dcpendcncc of vis-eosil)' with tempcralUfc. This has becn extensively sluoieohy Angell [1, 111 for glasses of non-polymcric origin ano forglass forll1ing ano polymeric liquids by many other aUlhors.In lhe formcr class one flnds liquids thal follow a convcn-tional Arrhcnius typc law for such temperature dependencc,rnainly I1lctallic oxides such as B203• Ge02, Si02, cte. lo liq-uids that draslieally depart from sueh hehavior. This lead An-gell lo dislinguish the former which he named slrong ¡¡quidsfrom Ihe latter named fragile Iiquids. Glyeerol, propyleneglycol and othcr alcohols are intermediate I¡quids whereaslactie add, glucose and othcr organic liquids are fragile. Tosummarilc this hehavior, the original Vogel-Fulcher- Tam-mann cmpirical equalion [12,13] first proposed to accounlfor Ihe non-Arrhenius behavior of several glass forming Iiq-uids, has hccn modificd to rcad,

(DTo )'/(T) = '70 exp T _ T

o' ( 1 )

hle and D is a constant which accounls for lhe devialion fromthe Arrhenius type law. lt is known from experimenl thal5 :s D :s 100 where the lower value is for fragile liquidsthe highest for strong liquids. lt is remarkahle Ihat Eq. (1)eonstitutes a rather good lit to all dala availahle for the vis-eosity for non polymerie glass forming liquids. For furtherJClails [he render is urged to consult Ihe current literature onIhe suhjeel [1. 3, 14-1GI. In Fig. 2 we show a lit of Eq. (1)taking DTo = B = COllst. for glycerol and propilene-glyeol.

Jt is rather eonvenient al Ihis stage to saya few wordsahoul To. Presumahly the sluggish kinetie phenomena whiehoccur at T~,and bring lhe liquid into a metastablc state aresomehow related to Ihe laek of mohility of the moleeuleslo reach thcir true equilihrium state, Thus il has been asso-ciated wilh the condition thal the configurational cntropy isle ro, only vibrational degrees of freedom remain in the sys-tcm and perhaps sorne other internal ones as librations. Onerould further associate this condirian with an infinite valuefor the viscosity in which casc To should he laken as Tk. IfIhis is so lhe temperature To = TAo is rcfcrred lo as the isoen-lropie lernperalure and sorne authors do accept lhis equalityas valid [17,18]. Nevertheless il is still a debalihle issue al-though the general resull borne out by expcrimcnts is lhatTe/To ::: 1 [2].

For polymeric I¡quids Ihe story is slightly differenl,sincc Ihe viscosity 3150 depends on (he molecular weightand the VFf cquation does nol prove lo be uscful in fit-ting experimental data. To deal wilh these systems Landel,Williams and Ferry (LWF) introduced in 1955 a differenlapproach [13,19]. Let T(T) he a relaxalion time assoeialedwith the viseosity '7(T) or any other eharaelerislic mode ofIhe glass transilion and 'o a reference rel<lxation time. Thelogarithm of Ihe ratio belween these limes, denominaled thelogarithmie shift faet (LSF), denoted hy ,,(T), is assumed tosatis!"y the equation

_ T(T) Cl (T - T,)- ¡agIO ,,(T) = 10glO --;:;; = C, + (T _ T,) , (2)

Re\'. Mex. rú. ~5SI (1999) 11-17

REMARKS ON THE GLASS TRANSITION 13

j(w) = ,(w)P(w),

\vhere P and j denote force ami effecl, respectivcly and X lhesusceptibilily or inerlia for cach frequency w. h may be thenshown thal [23]

where TO is identified with the experimental relaxation timeand B is a numher whieh mensures lhe dcviation [mm cxpo-nential decay. and the so called stretched exponential equn-tion for q,(I) firsl used hy Kohlrauseh over one hundred and

(6)0(1) = cxp [ - (;y]

3. Dvnamical behavior of glasses

In lhe discussion of Ihe previous seetion it was c1ear that \Ve

wcre dealing with slow diffusional relaxalion proeesses mod-ulated hy lhe viscosity or its corresponding relaxation times.These proccsses are no\\' rcferred to as lhe Q-relaxation pro-ces ses. In the case of nonpolymcrie glass forming liquids thecorresponding experimental data are fitted Ihrough lhe VFfequation [c.f. Eq. (1 )1, whereas for polYl11erie materials thelit is achieved l11ainly via the LWF equation [Eq. (2)].

In this section \Ve wish lo address ourselves to the dis-cussion 01' mueh more recenlly discovered fast dynamieal re-laxalion processes which oceur in lhe liquid and which. so10 speak. l11ark the lhreshold of the glass transition [29]. Infael, whal has been reeently discovcred is the existence ofn crossovcr lemperature Te, somewhere within the interval1.15Tg < Te < L2Tg in which many properties of glass-fonning liquids have explicit changes in their values. Curi-ously enough. allhough the logarithm of the slow relaxationtime T shows no changc in this range aftemperatures. StickeleL al [30J have pointed out that such a change is elearlyseen if onc examines the quantily [dlogx/dTr1/2, where"' = Um.,/ll" (7<1e',//8-1, ,,/Poise-1) where 1m., isthe 105s peak 01' dicleelrie relaxation data, ade lhe de eOI1-duetivity ancll} lhe viscosity. Far lhe VFf equalion

fifty years ago and later applied to dielectrie relaxation byWilliams and Watls [25, 26J

.. -11"2

(dlogJ.) = (T-To)B-1/2 (7)dT

In Eq. (6). kno\Vn as the Kohlrauseh \Villiam \Vatls (KWW)equation TlI' is the relaxation time and /3 a parameter suchthat O < ¡J ::; 1 hcing larger for strong liquids and smaller forfragilc ones. Table I givcs the values of lhese exponents fordifferent Iiquids (hus exhibiting the non validity of lhe sim-ple exponenti,,1 (Dehye's) decay. Once more, it is relevanlto stress lhe facl ahout the remarkable accuracy with whiehEqs. (5) ,md (6) lit lhe experimental dala. Thejustification ofeither cquatioll through lhe underlying microseopie mee ha-nisms responsihle ror Ihe relaxation times as they appear inthcm remains. as one of Ihe deep mysteries of lhe glass tran-sitioll. h is also worthwhile menlioning lhe faet lhat lhe ex-islcnce of Ihis Iype of relaxation meehanisms in glasses andother kind 01"malerials [27J is an eloquen! testimony of lhefact that macroscopieally speaking. they lie beyond lhe realmal' linear irreversihle thermodynamies [28].

In (he next section v.'e shall lurn our attention to sornemore recent phcnomena ohserved in glass forming liquidswhose naturc remains cqually puzzling as those deseribedherc.

(5)

(3 )'1 = GooT,

where G", is the high frequeney shear modulus and T lhe av-erage relaxation time. In slatistical mechanics T is essential1yrelaled to the stress-stress aUlocorrelation function which canbe cilher measured or cvaluatcd using molecular dynamics. Infact linear response lheory [23J sho\V, that for not very largeperturbations lhe response 01' a syslem subjecl la an exter-nal perturhation and the latter are reJaled through the simpleformula,

x(w) - ,(O) _ -1'" e-;W'~(I) dt. (4),(O) - ,«(X)) o

where \ (O) = ,(w = O). X( (0) = ,(w = (0) and¿(t) = ,!<;,(t)/dt where ,,(t) is the response function re-lated lo the appropriate time correlation funclian of Ihe mi-croscopic propcrties ofthe syslem. Since this funclion has de-featcd analylic evaluation for practically evcry realistic sys-temo il has heen useful in practice to tit the data either directlyfor ,(w) or through the response function 0(1). The two mostuseful and celebrated formulae for this purpose are. lhe Cale-Davison equation [24]

where T, i, the reference temperature al which T, is de-termined and el,el are constanlS. For a long time il wasIhought. hy mere experimental work. lhal Eq. (2) is universalfor polyrncric glass fonning Iiquids ir Ts is chosen in such away that CI = 8.86 and C2 = 10L6(C). Thi, however is nolquile correc!. Since Eq. (2) is known lo he equivalent lo IheVFr Eq. (1). the WLF equalion is not quite so universal asthouchl earlier. Nevertheless. lhe fact that it can be derivedfrom- the Adam.Gihhs model [20J in lherms of the contig-uralional entropy 01' lhe polymcr has Icad lo a \'cry userulformula for dctcrmining lhe relevant pararnclcrs of lhe liq-uid [20-22]. For a more delailed discussion and further in-formation lhe Teader is urgcd to consult the Just three rncn-tioned rcfcrcnccs. A ¡¡¡SI suhjcct thal is wonh slressing inconncction with lhe standard phcnolllcnology about glasscsis lhe so callcd l1on-cxpollcntial dccay ol' inlrinsic responsesto external pcrturhations. Wc rccall thal viscosity is Ihe mca-sure of lhe ¡nenia 01"a Iiquid lo un cxtcrnally imposed shearstress and is related lo ils corresponJing relaxation time by!vlax\\'cll's cquation,

Rev. Mex. Fis. 45 SI (1999) t 1-17

14 LEOPOLlX) GARCíA-CoLíN

T,\BI.E 1. Valucs for (he exponenls ()f lhe Kohlr.lllsch-Williams-Watts anu lhe Colc.Davidsoll formulas for diffcrent systcms. Adaptcd fromRef. 2X.

~1casurclllcnt tcchniquc

C"Spcctroscopy

Diclcctric rcbxation

Ultrasonics

Digital corrclation

Espcctroscopy

Dicleclric rclaxation and

V;SCtlC/aslic rctanJalion

Diclcctrit- loss (rclaxalion)

Material

Glyccrol

Propylcnc-glycol

Glyccrol

Propylcnc-glycol

Glyccrol

Glyccrol

Tri -0-lo1yl- phosphatc

Tri -2-ch lorocthyl-

phosphalc

SorhÍ!ol

¡i(KWW)

O.G:;r 0.03

OGI r 0.01

080 r 0.02

O.7[j

OGO r 005

0.70

0.63

0.6

B(DC)

0.51 :1:0.03

0.44 r 0.04

O:;S r 0.03

O.GGr 0.01

0.42 r 0.0:;

0.10 r 0.0:;

0.4

O.•

0.5

Temperature rangc (K)

195-225

195-230

235-300

195-22523-1--244

197-217

198-208

272

TI"lIIpl'n1fuH' I K

FIGU){ 1: 3 Tcmpcralurc varialioll 01' lhe derivativc 01' lhe rciaxatitllllime r showing c1carly Ihe prcscncc 01' a crossovcr tCllIpcraturc'T:. > T!, ror salol. IFwm Stickcll'l al.. Phys. Rel'. L('t!. 7.\ (1994)2~361

The ()riginal experiments pcrrormed wilh phcnyl-salicylate(salol) sho\\' Ih<ll rOl"T(' > 7~"T clearly the presence 01' re-iaxa(ioll proccsses which no knowll Ihcory can properly ac-count rOL 1I is sOlllewhat illustrative thal the variation 01' T

is of Illore than 1() orders of magnitudc in Ihe r<lnge Tr. >T" 130](See Fig. 3).

1I is at present thoughl that some of (hese fasl dynami-cal processes are not completely uncorrelalcd among lhem-selvcs. Indeed. rcccnl sludy of several 1ll¡¡leri~lIs Ihal rangefmm protcins lo ¡onie, polymeric ami molecular glasses hyneulron and Raman scaltering poinl Ollllhal sllch fasl dyn~lIn-kal processes that occur ror 7~ 2:1~ have a quasielaslic na-Ime and are Ihercforc a relaxalional phenolllcnon [31. :-l31.Spcdfk sludies carried out in polyisohulylenc (PIB). re-gorded as the "leasl fragile" polymer (D - IG) I:¡¡ 1, all1nr-phous ro1ycarhonalcs 132) and holh ch<l!cogenidc glasscs Seand AS1 SCJ ~\Ild the oxide glasses B10J and 5101 (supr<lsi1)have lel! lo ralher stimulaling resulls. \Vc shall try lo give a<,;ullllllary 01"lhe ideas hehind thelll.

(8)

Bul ror gla<>symalcrials Ihis doesn'l occur. Al frequcncicsl'orrcsponding lo cncrgy Jitfcrences hetween 0.1 and 5 rncV.<In inclaslic pcak occurs in S(i}.""") called Ihe "hoson peak".This mighl indicate a pronollnced cxcess intensity over IheDchyc level ("...'2. Thc aprearance of this pcak is not well un-derslood and is currently the suhject of hoth experimenlal anJlheoretical rescarch. However. it also happcns Ihat lhe peak issomehow rciJted lo Ihe lemperalure dependen ce of the D\VF.This qucstion was dosely examined hy Frick ami Richlcr 131 J

3Nf, _"11" n(w) + I ( )--f' - (r----g w2;\[ w '

where ,,-"" is Ihe Dehye- \Valler faclor (D\VF), 11' =(ljG)J,::.!{r:!) whcrc (,.1) is the Illcan-squarc displacelllcnl,I/(W) = U,flwjkT_1 )-1 is the Hose factor . .i.\l lhe mass oflhe scaltcrers and N its IlUmheL Also Ifl = 2rr,\-I, .\ hc-ing the w<lvclcngth. Thereforc. lhe two imrorlant p~lrame-tcrs in Eq. (H) are Ihe dcnsity 01"vihrational statcs Y(w) <lndIhe D\VF. ,,-'IV Al low frequencies, n(w) - kT/w and.rJ[)(w) ~ w1, Ihe Dchyc dcnsily of vihrational states, so lha!according In Eq. (8), Sine('], ""',)does no! dcpcnd 011 lhc frc-qllcncy.

lhe firs( importanl and common reature of these sludiesis Ihal (he fas! rclJx<ltion mcch<lnisms appear. nol only al Ihecrossover lempcraturc 1~huI also Jeep in Ihe gl(lssy rhase.long hefore !lo\\' processes becollle observable, in healingfrom hclow, on an experimental seale. lhe features of (heseproccsses are quile different dcpcnding if we are clase 10J~ from ahoye or fmm hclow. Thus lhe hroad quasiclasticpeak ohscrvcd in Ihe spcclra mus! be an intrinsic fC<lturc 01'Ihe glass ilself [31, 321. Lel us Ihen recalllhallhe incoherenlscaltcring fUllctiollS ror <lvihrating syslcrn Sillc:(ij,W) can hewrillen as, [~II32(130u260 280

• I SALOL I~,//<i

6,,.! ," ~o-'<1 3 /2

220

Rl'I'. Me.\". Fú. 45 SI (1999) 11-17

REMARKS ON TIIE GLASS TRANSITION 15

o-1-2

BPA- pe. (d,)

o JOOK• 200Ko nOK• 53K

-JO

200

300250200150

T(K)

100

o

0.05

-0.05O

0.55 ---....-r "'lar~tMlIIlo

t0.45 -- 00

oo

'"00

~ 0035 t';,: o

B~A o""~ 0.25 B8'••• o~oo~~. o

V~o

0.15

Energy trans1er (meV)

wherc (,'0 is a conslant used as n fitting paramenter.ln lhe

ralher dilTcrcnl nalure [.33). Thc intensity of light in Ramanspcclra aS:l funcliol1 of the frequency is given by

()Go/(,,)

q /1 =:; ----. [,,(,,)+ 1]

(9)1(1') = !J(u)C(u)_[n_('_,)_+_II,u

r-r(;URE 5. Bose amI Dchyc.Wallcr faelor scaled ncutron spcclrafm BilA-pe al four (cmpcraturcs bclow Tg. The sharpness of (heho.\o/l pe;l!.; i'i enhanccd as \Ve go lo lo\Ver temperalllrcs. (FromReL .12).

wherc y(v) is lhc density 01"vihr<ttional states, C(v) a co-eflicicllt cOllpling vibralions lo ¡¡ght n(v) + 1 lhe BoseraelOr ami [1 Ihe freqllcllcy. Equatioll (9) thus allows thcca1culalioll 01' Y(I/) if C(I/) is knowl1. The queslion lo hehcre seHled is welhcr 01' nol w¡lh Ihe information availablcb011l fmm neulron ami Raman scatlering OilCcan decide ifY(I/) ••.•• 11"2 as slipulaled by Dchyc's moJel. lt aprears how-evcr fmlll Ihe former sel of data as wcll as from low lelll-peralurc heal capacily data Ihat in the low frequency regimc(11 "-' 20 CIIl- J) 51(f.I) in :unorrhous material s does nol fol-lowa Dchye hehavior. This leaves room lo determine its fre-quency dependance if C{lI) is determineJ using Rarnan scal-lering. The results ohlaincd by Sokolov el al. [33] are shownin Fig. (5) in their papee. By pJotling I(u)/[n(u) + Jiu' ilis clear fmm their results Ihat at low frcquencies is linear orsOI1lCwhalslmnger in 1/.This implies Ihal if g(v) ,....,v"2, C(lJ)musl be al leasI nI' Ihe salIle arder in the frequency. Lack 01'reliable dala frolIl neulron scattcring in this frequency rangehinders a more calhegorical conclusion. Thc mosl cfficienl\vay 01'dClcrmining the coefficient C(/I) fram Raman dala ishy direcl cOlllparison with lhe heat capacily Jata. AssumingIhat C(u) ~ "one may rewrile Eq. (9) as

FI(IURE ot. Anomalous bchavior of(hc MSD fm polyisoblllylcnc .IS

a fUllction of tcmpcraturc showing c1carly thc discontinuity 01'lheslopcs al tcmpcralUrcs hclow Ty (From Rcf. JI).

in PIB hy neulron scattering. Taking Ihe oala for Ihe q-dc-penoence 01'the total elaslic intensily they were ahle lo makcplots of (r"2 (T» l'S. lemperature T in different lempcralureranges ano found a hchavior which is qualilalively shown inFig. 4. The sJope in Ihe range 30 < T < 1::>0K (1~~ 200 K)is approxim:llely 8.35 x 10-1 K-1 A"2 whereas for <1< T < 50il is llluch srnaller, 01"the oruer or 9.5 x 10-5 K-IA"2. AlTy = T. (,."2) ,....,0.2 A2 a value which is much Iarger lhan IheoOlained fmm Ihe: linear explrapolalion al low lemperalures.Thc uplurn 01' Ihe (,."2(T) curves al lelllperatures ,....,50 Klower than T'J indicalcs lhe onsel offasl dynalllic al proccsscswhich appear lo he related 10 the appcarance 01' lhe bosollpeak.

In 1~u.:I.hy cxamining Ihe: dynamic slruclure factorS((J. w). scaling all spe:clra lo Ihe reference 1e:l1lperalure7~,and laking Ihe DWF frolll Ihe low lemperature hchavior (sce[oigo5) one observes lhal Ihe inelastic scattering inlensily COl"-responding lo dirrcrent Icmperalurcs oughl rall on a maslercurve if its origin were due to harmonic vihralions. This is Ihecase rnr all cllcrgy Iransfers 6.E at T < T!JhuI fnr T > Ty iloecurs only at 6.E 2: 3 mcV. Thc curves nol only 110 longercollapsc onlo a master curve huI have fcaturcs cleady indiocaling Ihat Ihe scatlering dcvialcs from Ihe Icmpcralure be.:.havior expcclcd fmm harmonic vibrations. PlIlting l!lis rcslIltlogclhcr wilh the Oile Illenlioned in Ihe prcviolls paragraphseellls 10 indicale thal a possiblc conneclioll hctwccn fasl dy-llamical processes and Ihe vihrational molles relalcd lo Ihetmson peak are: nol far fClched [.1,8,3:2) (see Fig. 5).

To a,:\sess Ihe possihilily Ihat Ihe anomalOllS hehavior nflhe D\VF as well as Ihe appearance 01' lhe huson peak IllCIl-

tionell ahove are indect.l due lo anharmonicilies in Ihe vi-bralional modes, careful experimenls have been underlakcnIllcasuring ami analyzing Raman SreClra of variolls glasses of

Re". Me.\. Fú. '¡S SI (1999) 11-17

in Fig. 6. It appears lhen lhal far all glasses lhal were in.\'cstigalcd by lhis Illcthod the linear frequency dependence01' C{II) is a very good approximation for calculating C'J'Morco\'cr, the rcsults shown in the figures indicate that thisnnsalz is a major impnn'crncnl with rcspcct lo those obtaincdwith Debyc's moJel. A still open question is how to deter-mine the correet absolule \.alue of g(v) sinee the value ofGo = C(u)1.I-1 is not really kno\l,'n exaetly. This qucstionis n currenl topie of research anJ we refer the reader lo Ihelilcralure ror more l!elails [33J .

SO if anharmonicilies are an impoTlant mode at temper-atures helow the glass transitions temperature Tg• so thal inlhe low frequency speelra g(") is a non-Dehye funelion lheymay playa majar role in cxplaining the three [aets that wehave Jealt wilh in (his section nnmcly, the anomalous behav-ior of Ihe D\VF. the hoson peak and the thermal propcTlies 01'glasscs.

A last comment is rather pertinent at this stage, As wehave sccn. the faet thal relaxational proeesses in supereoolcdliquids as well as in Ihe glassy state are either of the a or/3 lype, lhe corresponding lime sea les are ralher weH de-tlnel!. Thc qllcstion tlwt ohviously arises is whether or no!ncw length sea les are important as \Ve approach Tg and he-lo\\'. So faro experiments and simulations which have beenperformed on hoth structural and dynamical properties havenot providcd any cvidence of such new length seales. Theundcrlying reason ror this could be that as has been arguedhy many authors, the laboratory glass transition is a kineticphenolllcnon and not a Ihcrmodynamic one, The reader is re-fcrred to the ¡ilerature for fUTlherinformation [35].

4. Conclllding Remarks

LEOPOLDO GARCiA.COLÍN

•

10

8

12

11

,.

8 10 12 14 115 18 10°

T(K)(b)

••

•2

• • 1 10 12 14 l. 18 200

T(K)(a)

20

I1

l.

14

12

10

8

••

•."'+.. ... .•. +++.

++++............

O O 2

20

18

~•::t••-..,'"~- ••U •

16

20

I1

,.~ 14•::t 11••- 'o..,~ 1.>r-'

f-- ••U •2

00 2

FIGURE 6. lIca! capacity calculated from Raman spectrn (Iines)aod from expcrimcntnl points for AS2SC3 (a) aod for 8203 (h).(From Rer. 33).

Icmpcraturc rnnge 2 K < T < 20 K where the maio contri~hutioo 10 CJl is givcn by the vibrational contribution, Cp wascalculaled iromlhe well known formula far C,. ('" Cp far allglasscs) namely,

("D ('w)' ehv/'TC,. = 3.Yk Jo g(") kT (ehv/kT _ 1)' d",

"here "o = kfc)n/h is lhe Debye frequeney and lhe resl ofIhe syrnnols huve Ihcir usual rneaning. Thc calculated heatcapacities reproduce well the experimental data as shown

Looking at the enormous wea1th of experimental dalaprescntly availahle ror all kinds of glassforming Iiquids. 01'which unly a very minor fraction has heen eovered in thispaper. there are a numher ol' qllestions, many of which haveheen raiscd along the text, that appear lo be rather impor-tan!. For instancc, ean slow relaxalional process be fitted intoa unified thcrmohydrodynarnical model based on the Adam-Gibos equalion and a reduced number (one ar lWO) of ex.perimental rneasurcments? \Vhat is the true nature of lhefasl dynamical processes oecuring in lhe neighoorhood of lhecrossover lemperature Te? Is mode coupling theory a real-istie alternative in lile understanding of the glass transition?Thcse and other topics are still behind glass formation as amajar problem in condensed matter physics,

This papel' i.••h.:lsed on a L;)jkgiven allhe New Horizons in Ma-tcrials SCierK'(', Juriquilla, Qro .. México, Jan. 1998.

t Also a! El Colegio Nacional.

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Rc\'. Me.':. Fú. 45 SI (1999) 11-17

REMARKS ON THE GLASS TRANSITION 17

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Re!'. Mex. Fe,. 45 SI (\999) 11-17