Journal of Non-Crystalline Solids 407 (2015) 246–255

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Journal of Non-Crystalline Solids

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On the uncertain distinction between fast landscape exploration andsecond amorphous phase (ideal glass) interpretations of the ultrastableglass phenomenon

C. Austen AngellDept. of Chemistry & Biochemistry, Arizona State University, Tempe, AZ 85287-1604, USA

http://dx.doi.org/10.1016/j.jnoncrysol.2014.08.0440022-3093/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 2 July 2014Received in revised form 14 August 2014Available online 7 October 2014

Keywords:Ultrastable glasses;Liquid–liquid phase transitions;Liquid–glass phase transitions;Rotator phases with lambda transitions;Fragile–strong transition

This paper is written as a challenge to the ultrastable glass community to distinguish phenomenologically be-tween the fast landscape searching scenario under discussion in the ultrastable glass community, and the alter-native scenario in which the material in the ultrastable state is actually an attempted realization of the lowtemperature ideal glass phase of the system that can be reached in some systems, like ST2 water and amorphoussilicon, and model systems like the attractive Jagla model, by a first order thermodynamic transition. These arespecial in that they exhibit first order phase transitions to the ground state that are accessible above the normalTg of the high temperature phase — and in consequence exhibit vanishingly small excess entropies over crystal,and none of the “ubiquitous” glassy state cryogenic anomalies. In particular we show how, near a liquid–liquidcritical point, the diffusivity can decrease arbitrarily rapidly over several orders of magnitude and that a growthfront transformation from ultraslow phase to normal viscous liquid will be very difficult to distinguish fromnucleation and growth of a new phase of different mobility.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The discovery of glassy phases that have enthalpies far below thevalues characteristic of the glassy phases formed by the normal coolingprocess [1], is certainly one of the most intriguing developments of re-cent glass science. Based on the enthalpy that is regained on warmup,with transformation back to the normal viscous liquid state at thenormal Tg, it has been estimated that the residual entropy of the“ultrastable” (US) glass has been cut to as little as 30% of the residual en-tropy of the “standard” glass, i.e. the glass formed during cooling at thestandard rate of 20 K min−1 (0.33 K s−1).

The original observation, and a schematic summarizing the findings(which will be re-used in modified form subsequently), is presented inFig. 1.

It is interesting that the systems in which the greatest decreases inenthalpy occur seem to be those in which the normal glass is formed,by structural arrest, at the temperature closest to the apparentimpending entropy crisis temperature known as the Kauzmann tem-perature, TK. TK is usually determined by extrapolation of supercoolingliquid entropy to the temperature where S(liquid) = S(crystal). Theseare the glasses that have the most “fragile” precursor liquid phases, touse a term that describes how dramatically the liquid departs fromthe classical Arrhenius equation description. To a good approximationthe variation of liquid relaxation times with temperature is given by

τ ¼ τ0 expðDT0= T−T0ð Þ ð1Þ

where τ0, D, and T0 are constants, the “fragile” liquids being those withsmall values of the parameter D. D is referred to as the “strength” pa-rameter, leading to a fragility parameter F = 1/D. τ0 is a physically rea-sonable inverse vibration frequency, amounting to the time betweensuccessive attempts to cross an energy barrier. To good approximation,T0=TK, though there is somedisagreement between Tanaka [2] and Ri-chet [3] (and the author [4]) about this identity for the case of strong liq-uids where the extrapolations necessary to determine TK are long anduncertain.

From Eq. (1) one quickly derives [5] the alternative and commonlycited definition of fragility, m, which is based on the more difficult-to-measure slope of the Tg-scaled Arrhenius plot at Tg where this temper-ature is calculated as the temperature at which τ = 100 s.

m ¼ dlog10 b τN=d Tg=T� �

ð2Þ

¼ E= ln 10ð ÞRTgh i

¼ log 10ð Þ τg=τ0� �

ð3Þ

where E is the usual Arrhenius activation energy determined from theslope of Eq. (1) at Tg.

Finally there is the relation between m and D (or M and F),

m ¼ mmin þ 590=D ¼ 16þ 590 F ð4Þ

Fig. 1. Original observation of the US glass phenomenon in the case of the molecularglassformer TNB (LHS) showing the major energy absorption starting far above the nor-mal glass transition that is typical of the ultrastable glass phase, and its schematic repre-sentation (RHS) in terms of enthalpy differences between long-annealed “normal”glasses (ΔHann) and the US glass at its fictive temperature. The fictive temperature ofthe US glass is derived from the measured enthalpy uptake during the heating of the USglassy state, and its transformation back to the normal liquid state followed by coolingthrough the standard Tg and down to the starting point, along the ordinary liquid enthalpytrajectory.

Fig. 2. (LH panel) The dependence of vapor-deposited glass volume (relative to the mini-mum volume observed), on the temperature of deposition, relative to the standard glasstemperature, Tg. (RH panel) Relation between intersection temperature Ti of the LHpanel relative to the normal Tg (Ti/Tg) and the m fragility parameter, both reproducedfrom Ref. [10] (by permission). The compounds are TL: toluene, EB: ethyl benzene, IPB:isopropyl benzene, PB: propylbenzene, EC:, ethylcyclohexane, and BN: butyronitrile.

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the numerical constants being based on the assumption of a pre-exponent having the physical value 10−14 s.

It is to be noted thatm and F are not always self-consistent becausemis often determined from data taken in a temperature rangemuch closerto Tg than are the data used to obtain the F parameter. A table showingcases of agreement and disagreement is available in Ref [6]. A review ofthe hazards of, and common errors in, the determination of glassformerfragility was recently given in a review of the subject by the author [7] inthe symposium book Fragility of Glassforming Liquids. The F1/2measure offragilitywas developed [8,9] to overcome these hazards, but has been lit-tle adopted. The consequence is that any correlations between them fra-gility parameter and other measured quantities are liable to haveconsiderable scatter, and may also be author-dependent.

2. Correlation of US glass phenomena with fragility

With the latter proviso, we note that two different research groups,one specializing in molecular glasses [10] and the other specializing in

metallic glasses [11], using literature sources of m values, have nowcome to the conclusion that glassformers that are low on the m scaleshow little [11] or no [10] effect, of the temperature of vapor depositionT/Tg, on the value ΔTg, i.e. do not form highly ultrastable glasses. It isfound that the liquids with the largest fragilities yield themost extremechanges in Tg as determined on re-heating. This is illustrated with par-ticular clarity by the dependence of volume on deposition temperature,for liquids of different fragility, shown in Fig. 2 taken froma recent paperfrom the Nakayama–Ishii group. Although this is, so far, only a roughcorrelation, we will assume it is one that will eventually gain credence.Our rationale for doing so is based largely on the fact that it was directlypredicted in an article from 2012 [12]. The prediction was made on thebasis of a theoretical analysis by Matyushov and the author which hasnot yet made great inroads, but whose time might be at hand. It is theonly theory besides the free volume theory of Cohen and Grest [13],that contemplates a first order transition to a glassy ground state i.e. ageneralization of the idea of formation of 'perfect' glasses, by firstorder phase transition from the fragile liquid state [16]. The theory ap-plies to fragile liquids, and underpins the thinking thatwill be presentedin the present communication. Clearly, “time will tell”, is all that can besaid at this moment. It is to be noted that this Fig. 2B plot reaches the

Fig. 3. The strong fragile pattern for the “glassy crystal” or plastic crystal class ofglassformers. Themost fragile case shown is that of Freon 112 first studied by Seki and co-authors and recently revisited by Pardo et al. [24] Note that an evenmore fragile case hasrecently been reported [22]. There are several cases that are ideally strong according to theFig. 3 criterion, only the case of C60 being shown here. (Reproduced from Ref. [21] by per-mission of Amer. Inst. Phys.)

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“zero-effect” condition at m = 40, which implies that there is a con-siderable range of m values that are not susceptible to the mobility ofthe molecules on their surfaces or to any other factor that wouldanomalously reduce their enthalpy.

This is certainly consistent with the finding that glassy water, in theLDA form, is incapable of losing any enthalpy by any such mechanism[14]. In this case it is not at all surprising, since water, like well-annealed amorphous Si, is already in its ground state (in the LDA amor-phous form) [15]. LDA water is known to have an entropy that is indis-tinguishable from that of the Ice Ih crystal and neither LDA water norwell annealed Si show the so-called “ubiquitous” two level systemsthought to be generic to the glassy state [16–18].

In what follows, we will recount some background information onphase transitions in analog “glassy crystal” systems that have provoca-tive characteristics (leading to the belief that they deserve study bythe vapor deposition approach), and will then connect their phenome-nology to that of some “strong” liquids and their possible phase transi-tions. This will then lead on to our principal objectives of the presentpaper, which are as follows:

(1) an analysis of the possible validity of the proposal that the USglass phenomenon is the reflection of a general property of frag-ile systems, namely that, due to cooperative excitation mecha-nisms, the approach to the Kauzmann point passes through avan der Waals like instability, which is relieved by a first orderphase transition to a “superstrong”, or "perfect" glass, state [16].

(2) in particular, a comparison of phenomenological expectationsfrom,whatwewill call the “current” interpretation of fast energylandscape exploration due to surface enhancedmobility, and thealternative phase transition-based interpretation, where theaccess to the new phase which exists on a distinct and separateenergy landscape is also facilitated by the high surface mobility.

3. “Glassy crystals” and their relation to phase transitions

It has been known since 1970 [19,20] that all of the phenomenologyof glass-forming liquids, with the likely exception of the US glass phe-nomenon, can be found in the so-called “glassy crystal” phases. Theseare center-of-mass ordered, but orientationally disordered solids, oftensoft and deformable because of relatively high center of mass diffusioncoefficients. However their re-orientation times do not obey any closerelation to the diffusion-based structural relaxation times, being ordersof magnitude faster. The rotational disorder decreases with decreasingtemperature and concomitantly the orientational relaxation time in-creases, until ergodicity is broken at the orientational glass transition.

Fig. 3 displays the strong–fragile pattern as it has been observed forthese phases [21]. It is to be noted that, in contrast with thewell-knownliquid case, the majority of substances studied are in the “strong”glassformer part of the pattern. A single freon and amixture of dinitriles(succinonitrile andmalanonitrile) areweakly in the fragile domainwithm values of 68 and 63 respectively. To these was recently added a veryfragile case studied by Vispa et al. [22]. They identified a Freon 113 withan extraordinary m fragility of 127, comparable to those of propylenecarbonate and ethylbenzene. It would obviously be of interest to deter-mine if this material, but not the others, would yield an ultrastableversion of the normal orientational glass when prepared by vapor depo-sition at some critical fraction of its standard glass temperature. But thatis not themain thrust of this section. Themain thrust is to point outwhatis characteristic of the materials at the strong extreme of the pattern.

Found at the strong end are materials like fullerene 60 and thecarboranes that are globular molecules with little to interfere withtheir rotations. It has been found by calorimetry that, in the exampleswith highermelting points (or nomelting point at all (C60)), the systemheat capacity rapidly diverges at high temperature and passes through alambda transition. The glass transitionwhere the rotations freeze in, liesfar below the lambda point in temperature and is very feeble, in many

cases beyond detection by ordinary scanning calorimetry. It is evidentin these cases that the “orientational glass” behavior is just the low tem-perature dynamics of the well- understood order–disorder transitionwhere the elementary excitations of the system are fairly obvious. Ifthis is the case then there is certainly no possibility of formation ofglasses of ultrastable character since the systems are already essentiallyin their ground states at their Tg.

Let us provide, in Fig. 4, examples of rotator phases with lambdatransitions that are observable before any fusion occurs. One is a molec-ular liquid of the small-branch hydrocarbon class, and the other is a setof globular inorganics, CCl2Br2, etc. Sodium nitrate is another wellknownexample of such transitions, but no reports of ergodicity-breakingat low temperatures have been made to the best of our knowledge.

Itwas this observation that lead the author andHemmati [23] recentlyto examine more carefully the possibility that SiO2, the archetypal strongliquid, might itself be acting under the influence of a nearby order–disorder transition. It is important to note that the peak temperaturesof these transitions are critical points of the same Ising universalityclass as is the gas–liquid critical point. They all have the same lambdashape and to a large extent the same critical exponents.

The strongest liquid of all, according to work published in the pastyear [18], is the case of water itself (as long argued by the author onmarginal evidence [25]). As is well known, there has been intense inter-est, since 1992 [26], in the possibility that, at higher temperaturesduring cooling, water is approaching a critical divergence domain asso-ciated with a second critical point where two different liquid waterphases of different density (due to different and mutually exclusiveO\O\O bond angles), become identical. This second critical point,based on computer simulations using pairwise additive potentialmodels of water [26,27], lies at positive pressures. It follows that the fa-mous anomalies of supercooled water are fallout from the presence ofthis critical point [28–30] and therefore that the strong liquid behaviorseen near its Tg is indeed consistent with being the low temperatureend of an Ising type order–disorder transition (as the liquid–liquid criticalpoint should be). It follows that, as an extreme case of the other strongliquids, it will not have any trace of US glass behavior. Indeed this is

Fig. 4. The phase transitions that accompany the strong rotator phase glassformers, when melting does not pre-empt their observation. Figure 4A, organic rotator phase trans1,4dimethylhexanse:(Huffman,H. M., Todd, S.S and Oliver, G.D. J Amer. Chem. Soc. 71, 584-592 (1949): Figure 4B, inorganic rotator phases CBr2Cl2, open circles, CBrCl3 filled circlesand CCl4 filled triangles (Ohta, T.,Yamamuro, O. & Matsuo, T. J., Phys. Chem. 99, 2403-2407, (1995)). Arrhenius reorientational relaxation times have been identified for the case ofCBr2Cl2 (M. Zuriaga etal. Phys. Rev. Lett. 103, 075701(2009)).

249C.A. Angell / Journal of Non-Crystalline Solids 407 (2015) 246–255

Fig. 5.Thephase diagram forWACSiO2 according to Ref. [42]with threeWidom linepoints fromRef. [23], supporting the slopes but suggesting a slightly lower critical pressure. Themiddlepanels show how the isobaric heat capacities change from those characteristic of strong liquid at zero pressure to fragile liquid at high pressure. The right hand panel shows an expandedversion that includes the co-existence line for the ensuing discussion of phase transitions during isobaric cooling above the critical pressure.

250 C.A. Angell / Journal of Non-Crystalline Solids 407 (2015) 246–255

already known from thework of Sepulveda et al. on vapor-deposited thinfilms of water [31].

Let us emphasize here that water, above this critical divergence do-main, is themost fragile liquid known. It is a bad glassformer because offast crystallization to ice Ih, so it does not vitrify unless hyperquenchedat extraordinary rates. But ignoring this problem and making the natu-ral connection between the observed precrystallization p spike and thelow temperature heat capacity above Tg (as shown in Ref. [32]) one rec-ognizes the lambda transition. Thus the notion of a metastable liquidstate version of the lambda transition of the orientational glasses indeedseems viable.

There are further cases. A model system that mimics an idealizedmetallic glassformer [33] is known to have a liquid–liquid critical pointthat lies in the stable liquid domain [34], and it seems likely that asilica-like liquid will also soon be available for model studies of a liq-uid–liquid critical point that is effectively free from interference by crys-tal phases [35]. The liquid–liquid thermodynamic transition, is also afragile-to-strong transition in which a correlation length for order risesas the critical point is approached, and then decreases as temperaturecontinues towards the glass transition. While this all seems a little im-probable at first contact, there is an unambiguous case of a glassformerwhere this phenomenology is known. It occurs in a different sort ofdisordering system — a binary metallic alloy system Co–Fe [36].

The next step in our development is one in which we carry outgedanken physical manipulations on a system that has a critical point

and a line of first order liquid–liquid transitions, originating at the crit-ical point and descending to lower temperatures. Physical relationswill require that the first order transition temperature will be pushedbelow the glass transition temperature, the condition that we wish toconsider in detail in relation to the US glass phenomenon.Wewill illus-trate this with Fig. 5, but some background is needed first.

4. Strong-to-fragile transitions, and liquid–liquid phase transitions

It has been common, with this author at least, to regard “strong” liq-uids as quite boring relative to their fragile cousins. But now with therecognition that at least two of them, low temperaturewater, and liquidsilica, have thermodynamic anomalies and even first order thermody-namic transitions at temperatures well above their glass temperatures,there comes a considerable change of viewpoint. We emphasize againthat critical points lying above the glass transition seem to be the rulefor the cousins of liquid glassformers, the orientational glassformers.Indeed, “Strong liquids are more interesting than we thought” was thesub-title of our last paper on this subject. Let us now consider two fur-ther ways in which strong liquids can be interesting.

The first is that they need not have anomalous densities like waterand silica. Liquid P2O5 is as strong as SiO2 but does not have densitymax-imum or anomalous pressure dependence of viscosity — which meansthat pressure should not convert it to a fragile liquid. The same shouldbe true for the chalcogenide glasses, like Ge–As–Se and Ge–Sb–Se in

251C.A. Angell / Journal of Non-Crystalline Solids 407 (2015) 246–255

which it is known that the strongest liquid compositions are those of thehighest, not the lowest density (see data from Senapati and Varshneya[37]). And at least in the case of As2Se3, there is indeed a high tempera-ture transition known. It is a semiconductor-to-metal transition [38] likethe liquid–liquid transition in silicon, and it has a response functionanomaly in its liquid expansivity [39].

The second point of interest arises when they indeed have theanomalous density behavior that is caused by an anomalous inverse re-lation between volume and enthalpy in the thermal excitations. Becauseof this inversion,whenwe compress the liquid silica and examine its be-havior between Tg and the fluid state, we find that the liquid becomesmore fluid and more fragile. We will use the silica case for the rest ofthis section. The increase in fragility is shown by the change in form ofheat capacity temperature dependence in the middle section of theFig. 5. Elsewhere [35] we estimate that a fragility m of about 60 can bereached at high compression in the WAC model of silica.

In the study of this model by the author and Hemmati [23], a liquid–liquid critical point like that of water, was estimated to occur at about5 GPa and 3500 K [40,41]. From this point a line of first order liquid–liquid transitionswould slope away to lower temperatures. The calcula-tions lent support to a phase diagramprojected from an earlier study bySaika-Voivod et al. [42] and this is all that is needed to develop the plau-sibility argument that we wish to make here. We therefore bring for-ward a modified version of Fig. 5 of Ref. [23] for discussion.

Fig. 5 shows the phase diagram for theWAC silicamodel deduced bySaika Voivod et al. [42] using a combination of simulations and theorybased on the Rosenfeld–Tarazona theorem [40]. Superimposed on thephase diagram are three points of maximum compressibility fromthree different isotherms of pressure vs volume, obtained in Ref. [23]These points should lie on the extension of the coexistence line, as aline of maximum correlation length. This is now known as aWidomline. The displacement stands as a minor correction to the Ref. [42]phase diagram which is confirmed by the more extensive assessmentsof response functions for thismodel currently reported [35]. The behavioris like that observed in water, with the advantage that crystallizationdoes not pre-empt the study of the phase equilibria.

The important point to be gleaned from Fig. 5 is that, at pressuresabove the critical pressure in a case where the coexistence line has anegative slope (as in Fig. 5 and in water), the cooling of the liquid willresult in the occurrence of a liquid–liquid transition to a low density

Fig. 6.The distinction between anomalous and normal systems (a) The scenario based on silica-lbelow the second critical point, to fall below themelting point. (b) The same scenario, seen in aslope. In this case the isobaric cooling trajectory is depicted as a low pressure horizontal line w

phase (see the horizontal line at 10 GPa in the RHS panel). The negativeslope of the coexistence line, and the flattening out of the diffusion coef-ficient, taken jointly,mean that, at some point, the LL transition temper-ature will disappear below the Tg line.

Now the negative slope of the coexistence line is a consequence ofthe open (low density) nature of the structure of the strong liquids con-sidered so far. It does not have to be like this, as already mentioned, cit-ing the chalcogenide glass case. The plastic crystal high temperaturetransitions also have positive slopes for the transition temperatures, solet us suppose that there is some symmetry in nature and that thestrong liquid phenomenology, which basically results from the cooper-ative nature of the configurational excitations, can exist in the oppositesense, even if not uncommonly observed. We depict this symmetry inFig. 6.

The equivalent of Fig. 1 for this case is shown in Fig. 7(A). The differ-ence is fairly minor. At a certain distance below the normal glasstemperature, judged to lie around 0.9 Tg in a fragile liquid case like tol-uene, the excitation profile develops a van-der Waals-like instability(due to excitation–excitation attractions, (see below)) which is re-lieved, as in the classical gas–liquid case, by a first order transition tothe low enthalpy state. This scenario is depicted in the experimentalcase of toluene by the relative density vs deposition substrate tempera-ture in Fig. 2, and it is predicted theoretically (in Refs. [43,44] byMatyushov and the author) for toluene and a variety of other very frag-ile liquids. Since it is happening under non-equilibrium conditions, itcannot be discussed in terms of entropy changes. However, if the exper-iment could be carried out on long enough time scales for equilibrium tobe maintained throughout, then the alternative to the common resolu-tion of the Kauzmann paradox would be seen as in Fig. 7(B). In the caseof toluene, the substrate temperature that produces the optimum low-ering of enthalpy below that of the standard glass, is 0.9 Tg.

While toluene is an extremely fragile liquid, it is not the most fragileknown. Both decalin and decahydroisoquinoline are more fragile. Cis-decalin and its mixtures with the trans isomer have been studied, andare found to be comparable in behavior to toluene but extreme inminor respects such as the diminution of the normal glass heat capacity[45]. Then there is the case of the highly fragile liquid triphenylphosphitediscovered by Kivelson and coworkers [46,47] that has been studied bymanyworkers as an example of a liquidwith an LL transition that occursabove Tg. It has been controversial because of the presence in the low

ike systems fromFig. 5,where increase of pressure caused the phase coexistence that existsn inverted form, asmight be expected for the casewhen the coexistence line has a positiveith the liquid–liquid coexistence line again falling below the Tg curve.

Fig. 7. (A) The enthalpy temperature relation, during extended equilibrium cooling below the normal glass temperature, for a fragile liquid, showing theway (marked 2), alternative to thatin Fig. 1 (marked 1), that the Ultrastable Glass enthalpy state can be accessed. This is a first order phase transition which is a possibility in the thermodynamic behavior of cooperativelyexcited molecular systems. The center panel shows how the excess heat capacity of liquid over crystal continues to rise during the equilibrium cooling. The divergence typical of a criticalpoint or of spinodal limit to the high temperature liquid stability, is cut off by first order liquid-to-liquid phase transition. (B) The panel completes the diagrammatic argument for theequilibrium (ultra slow cooling) case, by showing how the usual resolution of the Kauzmann paradox by second order transition to the ground state, is replaced by first order transitionto the low enthalpy, low entropy phase. Some unusual properties of this phase are anticipated below.

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temperature liquid of small amounts of crystallinematerial that seem al-ways to be present (a characteristic also of the Y2O3–Al2O3 case that wasthe first LL transition to be reported [48]). The presence of some crystal-line traces is not a surprising observation when it is appreciated that theLL transition in the metastable domain is essentially an Ostwald “stage”,on the way to the crystalline state. It differs from the familiar (and orig-inal) Ostwald stages only by being amorphous rather than also crystal-line. The important point is that it shares, with its crystalline analogs,the feature of being closer topologically to the stable state than the pre-cursor phase — and therefore lower in interface tension, and so morelikely to nucleate the stable phase in close time sequence.

The transition in TPP has been given particular attention by Tanakaand co-workers, who have been guided by the same author's twoorder-parameter model [49] developed initially for the anomalous liq-uids of tetrahedral character. The prolific output of the Tanaka laborato-ry has been the subject of major recent major review [50] of liquid–liquid phenomenology but does not deal with the problem of sub-Tgcases, which may be more common, for reasons discussed furtherbelow. The reports by Kurita and Tanaka [51] on TPP deserve special at-tention because of themanner inwhich theywere able to study the pro-cess under spinodal transformation conditions. These studies greatly

strengthened the case for a true liquid–liquid phenomenon, not onlyby avoiding any generation of crystals, but also by demonstrating theconformity of the transformation to the rules of spinodal processes[51]. The paucity of cases that are found in the metastable liquid statewas also considered by these authors [52]. The evidence of the presentcontribution is that transitions of this type occur mostly below Tg.

Wemust ask ourselves what will be the subsequent phenomenologyto be observed during an isobaric cooling of a liquid at, say, 10 GPa in theleft hand system of Fig. 6, or an isobaric cooling at ambient pressure inthe second (RHS) system of Fig. 6. The second panel of Fig. 7(A) showshow the excess heat capacity of liquid over crystal will continue to riseduring the equilibrium cooling but the divergence at a critical point ofhigh temperature liquid spinodal (instability with heat capacity diver-gence), is cut off by the first order liquid–liquid transition. The righthand panel completes the diagrammatic argument for the time-independent case by showing how the usual resolution of the Kauzmannparadox by second order transition to the ground state, is replaced by afirst order transition to the low enthalpy, low entropy glass phase.

We now need to ask how this scenario might be manifested innormal experimental observations. We can answer the question intwo stages, the first by direct observation of kinetic characteristics of

Fig. 8. Variations in diffusivity near critical points in two different types of systemwith liquid–liquid critical points (ST2 water, LHS panel adapted from Ref. [57], and S–Wsilicon adaptedfrom Ref. [53] (RHS panel). The red arrows represent changes of three orders of magnitude in each case.

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the phase transitions and comparison with what is known on theultrastable glasses, and the second by consideration of the thermody-namics of liquid–liquid transitions predicted by simple models.

5. What is to be expected in the vicinity of a sub-Tg liquid–liquidtransition or a near critical point?

5.1. Observations of mobility plunges near a critical point. The cases ofwater in the ST2 model, and silicon in the Stillinger–Weber model

Wenote that, in the literature on US glasses, there are frequent refer-ences to the striking three orders of magnitude lengthening of the

Fig. 9. Illustration of theway the formation of a glassy phase can convert to afluid phase byformation of a layer of greatly reduced viscosity, in the case of humidification of a glassyaqueous solution.With permission of Thomas Koop.

relaxation time of the US state, relative to that of the normal alpha relax-ation time at the standard Tg. Sowe now introduce the observations thathave been made on mobility variations in the vicinity of a liquid–liquidphase transition for different cases. The first is that of ST2 water sincethis is now a firmly established case of a liquid–liquid critical point[30]. The second is that of liquid Si at, and above, its recently establishedcritical point [53]. A third is the newly published case of SiO2 in theWACmodel near its pseudo-critical point [54]. A fourth possibility is the caseof the attractive Jagla model where the critical point is actually in thethermodynamically stable liquid state [55] (as mentioned earlier), butthe available data have not yet been collected in the needed form.

The variation of diffusivity with pressure, as the pressure is changedthrough the critical pressure in the first two of these cases, is shown inFig. 8. Similar changes can be seen in the diffusivity as temperature ischanged across this same region [56]. It can be seen that diffusivitydecrease across the critical domain is some 2–4 orders of magnitude.Changes of this order would be predicted from the Adam–Gibbs equa-tion [58],

τ ¼ b x N2=6D ¼ τ0 exp −C=TScð Þ

where τ0 is a constant of order of a vibration time, x is a jump distanceof order of 0.1 nm, D is the diffusion coefficient, C is a constant and Scis the configurational entropy. Sc changes discontinuously at an equilib-rium isobaric first order phase transition, by ~10% of the entropy of fu-sion [59]. The relaxation time gap should be larger the closer to thenormal Tg that the transition occurs, i.e. the larger is the configurationalentropy change ΔSc, relative to the entropy remaining at the transitiontemperature.

While ST2water is the best characterized case (because its LL transi-tion temperature is in a range where its behavior is accessible on com-putational time scales, and also because it has recently been thesubject of strong controversy [60] which has lead to exceptional effortsat clarification [27,30,61]), it is notable that there are water models thatare less structured (less anomalous) such as SPC-E water, in which the

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critical point (if it exists at all) is predicted to lie below the estimatedglass temperature. Then if the same mobility changes observed in ST2in Fig. 9 were to apply, there would be a 2–3 orders of magnitude differ-ence in mobility between the low temperature and high temperaturephases.

There have been a variety of cooperative two-state models [62–68],starting with Stroessler and Kittel in 1965 [62], that have critical pointsand phase transitions for suitable choices of the excitation interactionparameter. It is worth considering what they predict.

Angell and Moynihan pointed out that the position in temperatureof the LL transition is a systematic function of the ratio of the basicexcitation enthalpy and excitation entropy parameters. This is theratio ΔH/ΔS of the simple non-cooperative version of the model towhich the cooperative effects are added. For instance, a simple bond-lattice model with ideal mixing of the bonded and broken bond latticeelements [69,70], leads to a Schottky (rounded) heat capacity, but thesame model with regular solution mixing of the two states (excitedstates attract) leads to a critical point and liquid–liquid coexistence forsufficient attraction between excitations. It is the same thermodynam-ics (quadratic dependence of energy on fraction of excitations) as in-voked even earlier in the famous Bragg–Williams approximation ofearly critical point theory. In the cooperative bond lattice model it be-comes the quadratic dependence on fraction of excited lattice sites.(The broken bond is the equivalent of a lattice defect in defect latticemodels of crystal thermodynamics).

Thus the larger the entropy generated per excitation, the lower intemperature the second critical point should lie. Since the same ratio de-termines the fragility of the liquid, it is not unexpected that the LL tran-sition should be found below the glass temperature except in liquidsthat have very “strong” behavior near the glass transition. What is lessclear is the status of the intermediate liquids. There is a lack of system-atic knowledge of how the Adam–Gibbs kinetic (energy barrier) termvaries across the different classes of liquid, since this determineswhere, in temperature, liquids of the same excess entropy will becometrapped in the glassy state.

5.2. Expectations if a first order liquid–liquid transition can be provoked

The most significant expectation for a system that has successful-ly accessed its ground state by a first order phase transition, canagain be judged by reference to the behavior of the two systems ofFig. 8, namely water and silicon. In each case it is known that cryo-genic anomalies (two level systems), that were at one time thoughtto be ubiquitous to the glassy state, are missing [16,71] (to an extentthat depends on the heat treatment in the case of silicon), and thatthe entropy in excess of the crystal entropy is close to zero [72,73],at least in the case of water. A comparable diminution in the intensityof the boson peak can also be expected. There may be a rich field ofnon-existent cryogenic anomalies waiting to be investigated for theultrastable glasses. Indeed a reviewer of this paper has kindly drawnour attention to two additional examples of such behavior that have ap-peared in print since our article was submitted. In the first, Liu et al. [74]have confirmed that the absence of two-level systems is not dependenton the presence of hydrogen, but is also true in properly annealed purea-Si. The second, by Perez-Castaneda et al. [75] is evenmore relevant asit concerns measurements made on one of the most intensely studiedexamples of ultrastable glasses, indomethacin. It is already known[71] that hyperquenched glasses (that retain high fictive temperatures— i.e have many quasi–lattice defects) have more intense boson peaksthan normal or annealed glasses, and that the boson peak is evenweak-er still for the ultrastable glasses investigated so far. Furthermore, in thecase of low Tg molecular liquids, the excess heat capacity is increasingexponentially as Tg is approached [76] as if approaching a phase transi-tion instability.

Left out of this discussion are glassformers such as those of low fra-gility, illustrated in Fig. 2, that are neither very strong nor very fragile,

and give no sign of either ultrastable glass formation or sub-Tg phasetransition status. Leaving them aside to concentrate on the more inter-esting cases, let us ask what would be the difference between systemswith a single landscape under exploration, and those which reachtheir lower energy states by first order transition, as considered in thepresent paper. The key will lie in the manner in which the originalhigh temperature state is recovered. Does it occur by classical nucleationand growth, as would be expected in a phase transition, or does it occurby simple growth front kinetics as asserted in the majority of papers inthis area or, is there simply noway to tell thedifference. A good exampleof the simpler, growth front, case is provided by the hydration of aque-ous glasses subject to high relative humidity that hydrates the surfacecausing a high mobility layer which then migrates into the glass bygrowth front with no suggestion of phase transition character. This is il-lustrated in Fig. 9 by a schematic from the work of Koop and coauthors[77]. On the other hand, careful work on the case of toluene, bySepulveda et al. [31] lead them to state that the “above results showstrong evidence that the transformation from the highly stable glass tothe supercooled liquid in toluene is heterogeneous and proceeds througha nucleation and growth (NG) mechanism”. Of course a nucleation thatoccurs at the surface also quickly becomes a single growth front, asseen in the movies of triphenylphosphite phase change by Kurita andTanaka that are available on their website.

The most recent development in the ultrastable glass area, is thedemonstration that even m-methyl toluate, which is one of the lessfragile molecular glassformers, can be obtained in the ultrastable state,though the deposition temperature that gives maximum stabilizationis a little lower than for others. This case is interpreted as indicatingthat exhibition of the phenomenon is not fragility limited, as theoreticalconsiderations had tended to suggest. On the other hand, the existenceof the (“superstrong” [16]) glassformers mentioned earlier, that donot transform to ultrastable glasses because they are already inthat state as they vitrify, suggests there may be a cutoff in fragilitybeyond which the route to the ultrastability is too weakly drivenfor the ultrastable state to be realized, or perhaps to be distinguishedfrom the ground state. This group, of course, contains the molecularliquid case of water.

6. Summary and concluding remarks

In this article, we have developed a case for the likelihood of firstorder transitions between states of liquids of different order parametervalues. The order parameter quantifies the concentration of relevantstructural motifs. The postulate is that in liquids of a range of types,there is a range, in the excitation of these motifs in which continuousevolution is prohibited by unstable states, which is resolved by the in-tercession of a discontinuity. The system can jump across a domain ofinstability thatmay be associatedwith unfavorable angular relations be-tween sets of interacting particles as in the case of water, or with unfa-vorable defect concentrations in the case of less structured liquids. Weargue that, in general, the low temperature states are likely to be inac-cessible, because they lie below Tg, butmay be revealed if favorable con-ditions such as surface mobility during vapor deposition at favorablesubstrate temperatures, are provided. Since the stable state is a highlyviscous liquid, the return to the stable state must be inhibited in thesame way as is the melting of a crystal like albite that is very viscousat its Tm, (causing the crystal melting point to be falsely high by hun-dreds of K). An even larger overshoot was noted for the case of quartzwhere the superheating can be as much as 450 K [78].

If there is indeed a phase transition made accessible by the highsurface mobility, there should be other ways of letting it manifest it-self, such as short wavelength mechanical excitation, controlledtemperature mechanical milling, etc., and we advocate a broadeningsearch for such alternative routes as a means of moving towards aresolution of some of the deeper enigmas of the “glass problem”.

255C.A. Angell / Journal of Non-Crystalline Solids 407 (2015) 246–255

Acknowledgments

We acknowledge the support of the NSF Chemistry Division undergrant no. CHE 12-13265. This MS is based on a talk invited for theSymposium on Ultrastable Glasses at the 7th IDMRS conference,Barcelona 2013.

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