6620
J. Phys. Chem. B 2001, 105, 6620-6627
H2O below 277 K: A Novel Picture†,‡ Daniel Kivelson* Department of Chemistry and Biochemistry, UniVersity of California, Los Angeles, California 90095
Gilles Tarjus Laboratoire de Physique Theorique des Liquides, UniVersite P. et M. Curie, 4 Place Jussieu, Paris 75005, France ReceiVed: January 11, 2001
We present a mesoscopic picture of supercooled water in which we fit the properties of water into a general scheme applicable to all so-called fragile glass-forming liquids. Water is taken as a single liquid phase above its glass transition at about 136 K. It is the only liquid known to have a Kauzmann temperature that lies above (well above) the glass transition temperature; we discuss this phenomenon and its implications. We have also associated the metastable cubic ice Ic with the defect-ordered phase predicted by the theory of frustration-limited domains, thereby developing a general physical and theoretical picture that incorporates both the metastable low-temperature liquid and solid phases at 1 atm.
H2O at 1 Atm below 277 K Pure water below 277 K has many astonishing features that give it the appearance of being unique or at least a member of an eccentric group of tetahedrally bonded liquid materials. A number of studies have focused on the uniqueness of water, but here, we try to fit water below 277 K into a general scheme applicable to all glass-forming liquids, and to attribute the unique properties of water to behavior conforming to a universal scheme. This generality is attained by developing a macroscopic or mesoscopic (in contrast to a molecular) description; consequently, we do not delve into the details of molecular interactions and structure which are so special in water and give it many unique properties. The most evident special feature characterizing water is the minimum in its density at 277 K (at a constant pressure of 1 atm). One can, and we do, interpret this minimum as arising because of a structural crossover from one form of water above 277 K to another form below 277 K. There is no necessary implication that the crossover at 277 K be associated with a phase change. We shall be primarily interested in water below 277 K and not in the crossover at 277 K, although we shall comment on this latter. It is the decrease of density with decreasing temperature, a rather marked decrease,1,2 that may be the most unusual feature of water below 277 K, but rather than trying to develop a molecular structural explanation for this phenomenon, we show that despite it the other evident characteristics of water can be interpreted as quite normal. It is water and a metastable solid phase denoted cubic ice Ic on which we focus and not on the ordinary crystalline hexagonal ice Ih; however, a connection between the decreasing density of water with decreasing temperature and the fact that the ice is † Part of the special issue “Bruce Berne Festschrift”. We dedicate this article to Bruce Berne, a stimulating and engaging colleague who has been a leader in and a major contributor to the study of liquids over the past several decades. ‡ First presented at the ACS Symposium on Water, New Orleans, August 1999.
considerably less dense than water seems evident. We note that below 273 K the stable phase is ice Ih, and that the liquid as well as the Ic ice below this temperature are metastable (supercooled).3 Liquid H2O, i.e., amorphous material with measurable viscosity, can be obtained at temperatures down to 236 K and also in the range between 160 and 136 K,3-7 but in the temperature range between 160 and 236 K there is always solid crystalline phase present. Below 136 K, the amorphous material is a glass, i.e., its viscosity is too large to measure; thus the glass transition, which is a point of dynamic arrest and not a thermodynamic transition, takes place at Tg ≈ 136 K. (Tg is specified in a number of reasonable, arbitrary, and more or less equivalent ways. Conceptually, it is the temperature below which the liquid cannot relax properly within the available experimental times; in conformity with this idea Tg is often specified as the temperature at which the rotational relaxation time becomes 103s or the viscosity becomes 1013 Poise. The calorimetric Tg is taken as the temperature at which the heat capacity of the liquid drops abruptly at a cooling rate of 10 K per min. The calorimetric Tg is simpler to determine, but whereas the relaxation time and viscosity are state functions well specified in linear response, the calorimetric Tg is a measure of a nonlinear dynamic effect replete with hysteresis.) Upon supercooling below the melting point, the liquid always crystallizes to ice Ih below about 236 K. However, it is possible to form H2O glass below 136 K by a number of indirect pathways such as vapor deposition or hyperquenching of liquid droplets, and liquid can be formed by heating the glass to temperatures above Tg;3-5 at about 160 K the liquid crystallizes, forming metastable cubic ice Ic which turns to stable ice Ih upon further heating.8-11 See Figure 1. Liquid H2O in the temperature range 236 to 273 K is usually denoted “supercooled water”. To distinguish supercooled water from that in the 136 to 160 K temperature range, we denote the latter “twice-supercooled water”. (The twice supercooled water is supercooled both below the melting point of the stable crystalline phase and the melting point of the metastable cubic
10.1021/jp010104b CCC: $20.00 © 2001 American Chemical Society Published on Web 06/19/2001
H2O below 277 K
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6621 occur, a phenomenon which could readily explain why some experiments yield solid-like properties. We present a theoretically motivated20 framework that treats supercooled and twice-supercooled water as a single phase and also incorporates metastable cubic ice Ic. Although this approach is somewhat speculative, as are all others that attempt to describe liquid water in the 136 and 273 K range,6,7,13-17,21-24 it is, we believe, the only current model that tries to cover the entire range. Excess Thermodynamic Properties
Figure 1. ∆F(t)/∆F(Tmelt) vs T. Density of ice Ih taken as constant (0.92 g/cm3). Solid line from ref 1, broken line from ref 2 with density of ice extrapolated as constant. Value between 136 and 160 K uncertain; density of liquid taken as that of glass; see ref 18. At sufficiently slow heating rates, Ice Ic forms below 160 K, and at even slower heating or cooling rates, ice Ih presumably always forms below 273 K. Dots represent possible alternative behavior of water in 160 to 236 K regime where it cannot actually be formed.
ice Ic phase.) Despite the fact that they are both amorphous and liquid, and that no significant structural differences have been detected by means of diffraction,12 it has been suggested that supercooled and twice-supercooled water are distinct liquid phases;13 because the two cannot be interconverted reversibly along a continuous path, it is difficult to resolve the question of whether supercooled and twice-supercooled water are distinct phases, but there is growing consensus that liquid-phase continuity is the most plausible scenario.6,7,14-17 Our principal interest here lies with all the metastable phases of H2O found in the temperature range 136 to 273 K at 1 atm pressure, i.e., the liquid or liquids (supercooled and twice-supercooled) and the cubic ice Ic phase. (Supercooled water has sometimes been called “water I” and twice-supercooled water has been denoted “water II”,13 but this classification tends to prejudge the existence of two distinct liquid phases. The twice-supercooled water has also been called “low-density amorphous liquid” (LDL) and when taken below its glass transition called “low density amorphous ice” (LDA); when the LDA is submitted to high pressure it appears to convert via a first-order transition to “high density amorphous ice” (HDA).18 It has been suggested that not only is the twice-supercooled water distinct from supercooled water19 but that it can be found in several distinct phases;5 we do not pursue this concept.) Although experiments have established the liquid nature of twice-supercooled water in part of the range between 136 and 160 K, other experiments have indicated solid-like behavior.65 It is not possible at present to make a definitive assessment of this problem but we believe the evidence weighs heavily towards the existence of low-temperature liquid. The experiments7 on which our interpretation is based are the most direct studies of water mobility in the low-temperature range; additionally, there appears8-11 to be a liquid-to-cubic ice transition below 160 K. Furthermore, in treating twice-supercooled materials, in particular small samples of the material, unexpected, uncontrollable, and unwanted transitions to the stable crystalline state frequently
We choose to examine excess thermodynamic properties, the value of liquid properties minus those of the stable crystal. We can think of such properties as those inherently characteristic of the liquid, those properties shared by liquid and crystal having been canceled out. In particular, the intramolecular contributions, which are likely to be quantum mechanical in character, should be largely canceled in the excess heat capacity and entropy, ∆C and ∆S, respectively. To extract universal properties, i.e., those properties that exhibit few species-specific characteristics, we are interested in the normalized ratios, ∆C(T)/∆C(Tmelt), ∆S(T)/∆S(Tmelt), and ∆F(T)/∆F(Tmelt), where ∆F is the excess density. (The ratio Cliq(T)/Cxtal(T) has been extensively studied25 but it is species-specific and therefore not optimal for studies of universal properties.) Density. As indicated above, for supercooled water the excess density, ∆F, is positive, and its magnitude decreases markedly with decreasing temperature. For water and for all other substances, ∆F(T)/∆F(Tmelt) decreases as T is lowered below Tmelt. For supercooled water, this ratio drops sharply and is about 0.7 at 236 K;1,2 extrapolation would lead to the unlikely value of zero in the range 200 to 210 K. Because ∆F(T)/∆F(Tmelt) ≈ 0.25 for twice-supercooled water at 136 K,6 we see that indeed this ratio for supercooled water must halt its rapid decline with decreasing T within a few degrees below 236 K. See Figure 1. This is one of a number of indicators that suggests that supercooled water is approaching an important crossover at a temperature between 236 and 190 K. Many studies have focused on the coefficients of isothermal compression and isobaric thermal expansion, but here we take them to be properties derivative of the density. Without probing too specifically into its structure, we note that the low density of ice Ih suggests a diaphanous structure, an open lattice which is in marked contrast with the rather tightly packed structures of most liquids. Furthermore, as water is cooled below 277 K, its density decreases toward that of ice, which suggests that this water is also assuming an increasingly diaphanous structure although this is not true of water at T’s above 277 K. Such diaphanous structures, present in both ice and water below 277 K, require rather specific, very directional intermolecular bonding, but the local structures found in water below 277 K must be distinct from those of ice Ih; were this not the case, the water would not supercool. In fact, water cannot be deeply supercooled without transforming to ice Ih (at 236 K). Heat Capacity. The excess heat capacity ratio, ∆C(T)/ ∆C(Tmelt), increases quite markedly as T is decreased from 273 to 236 K, from 1 to more than 2. It appears26,27 that for all liquids that have been studied experimentally this ratio increases as T decreases although for supercooled water it is extreme,28,29 and for some supercooled liquids all that can be said is that it does not decrease. We look upon this increase or at least constancy of ∆C(T)/∆C(Tmelt) with decreasing T as uniVersal. Of course, all such studies have a low temperature limit below
6622 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Figure 2. Excess heat capacity, ∆Cp(T), vs ln[T] for supercooled water (filled circles, data from refs 28-30) and twice supercooled water (thick line, from refs 5, 13, 15). TK, the T at which the area under the extrapolated curve is equal to ∆Smelt ) 21.8 J K-1mol-1, is illustrated for two extreme extrapolation limits: TK ) 170 K from the high-T data29 only, and TK ) 216 K from a fit with no divergences to all data above 236 K. Tmax must lie between TK and 236 K.
which the liquid inevitably crystallizes or reaches its glass temperature Tg below which it is no longer liquid. If it were possible to keep lowering the temperature and measuring the viscosity while still maintaining the system as liquid, the ratio ∆C(T)/∆C(Tmelt) would necessarily drop and vanish in the limit T f 0 K; such a decrease is required because of the ultimate onset of quantum effects. We look upon this required decrease of ∆C(T)/∆C(Tmelt) at low temperatures as uniVersal. It then follows that if the temperature could be reduced below Tg, the ratio ∆C(T)/∆C(Tmelt) would pass through a maximum, or at least start to decrease below a temperature Tmax. For water, and to date only for water, it is possible to obtain heat capacities below Tmax; as expected, a small value of ∆C(T)/∆C(Tmelt) is observed for twice-supercooled water at temperatures below 160 K. See Figure 2. Note that all these measurements are made at constant pressure. The heat capacity of liquid at constant volume, especially that for water, exhibits much less increase with decreasing T than does the heat capacity at constant volume. Thus, the very marked increase of the constant pressure heat capacity for water with decreasing T is entirely (or almost entirely) a consequence of the properties of the coefficients of isothermal compression and isobaric thermal expansion. However, it is not clear how to study the excess ∆C(T) at constant volume. Entropy. The excess entropy ratio, ∆S(T)/∆S(Tmelt), can readily be calculated from measurements of ∆S(Tmelt) and ∆C(T). For all liquids, the ratio ∆S(T)/∆S(Tmelt) decreases with decreasing T; this is thus a uniVersal property. As a consequence of the increase (or constancy) of ∆C(T) with continuous decrease of temperature down to the lowest temperature at which the liquid can be maintained, the ratio ∆S(T)/∆S(Tmelt) not only decreases but does so rapidly, and if extrapolated to lower temperatures the ratio vanishes at a positive temperature TK, known as the Kauzmann temperature.31 Thus, a positiVe Kauzmann temperature is a uniVersal phenomenon.3 (It has been suggested that for nonfragile liquids such as GeO2, B2O3, and SiO2, the ratio TK/Tmelt lies very close to 0 K,31,32 but our analysis of the heat capacity data for GeO2,25,32 and B2O3,31,33,34 suggest values of about 0.27 and 0.36, respectively. We could not find relevant experimental data for SiO2. For many fragile liquids, the ratio is about 0.6 and for water it is in the range 0.6 to 0,8.) It should be emphasized that the extrapolation to the Kauzmann
Kivelson and Tarjus
Figure 3. log10[viscosity] vs Tg/T for a fragile (OTP, open circles) and a nonfragile (GeO2, open squares) liquid. Data from refs cited in ref 20. Also shown is the log10(1/diffusivity) of water for both the liquid/ supercooled liquid phase (data from refs 37-39) and for the twice supercooled phase (data from ref 7).
temperature does not yield the behavior expected if the system could be truly cooled below Tg. It follows from the discussion above that Tmax > TK. In many cases, the drop in heat capacity with decreasing T below Tmax must be very rapid; this can be deduced from the fact that Tmax lies below the lowest temperature at which liquid properties have been studied, and for many fragile liquids TK does not lie much below this lowest temperature, perhaps no more than 20%; thus, we take as uniVersal (at least for a class of liquids later denoted fragile) that the ratio TK/Tmax is only slightly less than one. It is difficult to determine TK for water but we estimate it to lie between about 170 and 216 K,4 whereas Tmax must lie above it. See Figure 2. Dynamical Properties Fragility. We turn next to the dynamical behavior of liquid H2O. As the temperature is lowered the viscosity (η), the rotational relaxation time, and the inverse of the translational diffusion constant (Dtrans) all increase. At the level of our analysis, these three dynamical quantities depend rather similarly upon temperature. In representing dynamical data, it is customary to make use of Arrhenius plots, such as ln[η] vs T-1, and it is found that for liquids such plots are either linear (Arrhenius) or upward curving (superArrhenius). See Figure 3. A much used dynamical classification is that suggested by Angell35 in which substances represented by curves with appreciable upward curvature are denoted “fragile” liquids, the greater the curvature the greater the fragility. One can alternatively interpret the Arrhenius plots as linear (Arrhenius-like) at high T and upward curving (superArrhenius) below a crossover temperature T*, the measure of fragility being determined by the behavior below T* where the curvature is pronounced.20 It has also been suggested36 that the greater the slope of the Arrhenius plot at T ) Tg, i.e., the greater the value of d ln[η]/d(Tg/T) at T ) Tg, the greater the fragility; this measure also emphasizes the dynamical behavior at T's where the curvature is most pronounced. Note that fragility specified by any of these methods is a dynamical concept and not a structural or thermodynamic concept. See Figure 3.
H2O below 277 K
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6623 quite good. See Figure 4. If the high T data, say above 250 K, are fitted to an Arrhenius expression
Dtrans ) D∞exp[-E∞/T]
(2)
where D∞ and the empirical activation energy E∞ are constants, one can detect a crossover to superArrhenius behavior below a temperature T* in the 195 to 230 K range. See Figure 4. If below T* the diffusion data are fitted to the expression
Dtrans ) D∞exp[-E∞/T]exp[-B(T*/T)(1 - T/T*)-8/3] (3)
Figure 4. log10[1/D] vs 1/T for water with three different FLD theory fits. (eq 3): (T* ) 200 K, B ) 458, E∞ ) 2378 K), (T* ) 205 K, B ) 387, E∞ ) 2229 K), (T* ) 215 K, B ) 268, E∞ ) 2229 K). Also shown is a power-law fit to the data above 236 K with a divergence at Ts ) 223.4 K. (from ref 41); more recent data,42 not included here, yield similar conclusions.
Fragility and Continuity of Phases for Water. In those cases where the viscosity can be determined in a large temperature range from above melting down to Tg it is usually quite evident whether one is studying a fragile or nonfragile liquid.40 Determination of the fragility of water is, however, somewhat uncertain because it cannot be supercooled below 236 K, a temperature at which the viscosity is 12 orders of magnitude below the value required at Tg. However, because liquid H2O has also been studied as twice-supercooled water in part of the temperature range 136 to 160 K,7 one can place both the supercooled and twice-supercooled water data on a single Arrhenius plot; if this is done with the best available data for water, the result suggests considerable fragility as well as phase continuity between supercooled and twice-supercooled water. See Figure 4. Fits of the combined supercooled and twice-supercooled water diffusion constants, Dtrans(T), to the Vogel-Fulcher-Tamman formula (VFT)
Dtrans ) Doexp[-DTo/(T - To)]
(1)
where Do, D, and To are constants, have yielded {D ) 7.6, To ) 118 K}7 and {D ) 7.3, To ) 119 K};17 the small value of D signals a very fragile liquid.4 (It has been proposed that twicesupercooled water is very nonfragile because the heat capacity of the liquid minus that of the glass at Tg is extremely small.201 Although, in some cases, this quantity seems to track the dynamically defined fragility,25 it does a poor job in other cases. A nonfragile character of water has been inferred from a study of the glass transition width measured by scanning calorimetry, but this is a very indirect, model-dependent analysis.) The VFT fit though less than perfect when carried out over the entire temperature range for which data are available is nevertheless
the expression given by the theory of frustration-limited domains (FLD),20,40 then for choices of T* ranging from 200 to 215 K, the value of B for water ranges from 460 to 270 which is suggestive of a liquid ranging from fragile to very fragile. See Figure 4. The fits are quite good, and furthermore, this formula seems to fit the relaxation data of most (or all) liquids well.40 Note that D∞ and E∞ are determined from high T data (well above T*) and T* and B from the low T data (below T*). There is an alternative view, not endorsed by us, which envisages an algebraic divergence in η at a temperature just below 236 K, an interpretation that calls for a Tg and a divergence point Ts for supercooled water in the inaccessible 160 to 236 K range; this Tg and Ts are distinct from the Tg and To of twicesupercooled water at 136 and 118 K, respectively. See Figure 4. (This analysis is supported by the observation of systematic positive deviations (above 236 K) from the 2-parameter Arrhenius expression fitted in the range above 250 K and from the 3-parameter VFT expression. Better fits were obtained above 236 K with the 3-parameter algebraic expression,41-43 D ) Ds(T-Ts)γ, with Ts ) 223-228 K, but, of course, this power-law expression (which is suggestive of a phase change from supercooled to twice-supercooled water) is not applicable to the twice-supercooled water data. See Figure 4. It should be noted that critical slowing near Ts does not imply that the twice-supercooled water be a solid.) Although it is usually assumed that Dtrans is proportional to T/η, as specified by the Stokes-Einstein relation, it has recently been shown that this relation breaks down at temperatures not far above Tg.44 This effect would enhance the apparent fragilesuperArrhenius character of twice supercooled liquid water. However, within the uncertainties of the present discussion it is sufficient merely to note this discrepancy. Characteristic Dynamic Temperatures There is an assortment of characteristic temperatures that have been used in studying supercooled liquids, including water. These temperatures, about which data are often scaled to highlight their universal properties, play fundamental roles in the various theories or models. Glass Temperature, Tg. Although the most used characteristic temperature is Tg, it represents an arbitrary experimental point and as such is not suited for model building. For water, the difficulties associated with Tg are especially noticeable; if supercooled water has its distinct Tg (which must lie well above that of twice-supercooled water), then the approach to this putative high Tg is totally different than that of other liquids. The Tg for twice-supercooled water is about 136 K. Ideal Glass Temperature, To. Extrapolation of dynamical data for fragile liquids to temperatures below the lowest one attainable with the system still retaining liquidlike properties can be envisaged as leading to a divergence at an “ideal glass” temperature To that lies above 0 K, but below Tg. The value of To depends on the functional form (such as the VFT expression)
6624 J. Phys. Chem. B, Vol. 105, No. 28, 2001 used for the extrapolation. The concept of an ideal glass transition has led to consideration of nonconventional critical theories45,46 built about this low-temperature “critical point”. The reason the theory must be “non-conventional” is because the behavior of the system in the To limit is not that predicted by simple critical theories. The VFT value of To for water is 118 K. Arrhenius-SuperArrhenius Crossover, T*. For fragile liquids, it is often possible to identify a crossover from Arrhenius to superArrhenius behavior, although it is more difficult to do so for nonfragile liquids that are Arrhenius-like over the entire temperature range or for liquids which have been studied only over a small temperature range. This crossover temperature (T*), which when identifiable lies well above both Tg and To, has been used as the characteristic temperature in models and theories of supercooled liquids, the special properties of supercooled liquids developing below T*.20,47 For water, the crossover behavior is quite distinct but because no data are available at temperatures near the presumed T*, an exact value is difficult to obtain; empirically, it is found that in most cases, including water, Tg/T* ranges from about 0.70 to 0.60. Special Water Critical Point, Ts. It has been implied in some studies21,41-43 that supercooled water has a dynamical critical point (Ts) at about 228 K, just slightly below the lowest temperature (236 K) at which water is irrepressibly crystallized. This dynamically inspired scenario is supported by the fact that the thermodynamic quantities (heat capacity at constant pressure, compressibility, and coefficient of thermal expansion) all exhibit what could be interpreted as algebraic divergences as T is lowered toward 236 K, all with divergent temperatures more or less equal to Ts.48 If there is a critical point at Ts, then the supercooled and twice-supercooled water are indeed different phases. This extreme picture can be modified by assuming that there is a critical point near Ts but at a higher pressure of say several kbar;6,24 in this case one might expect protocritical behavior of both the dynamical and the thermodynamic properties at 1 atm, but with no divergences and no phase changes. If there is truly such a nearby critical point, then the protocritical behavior should be treated as a distortion of the universal properties of water at 1 atm, distortions that have only to be subtracted in order to recapture the universal fragile liquid properties. Specifically, this implies that supercooled and twicesupercooled water are the same liquid phase. Additionally, it means that the slight deviations from universal relaxation-fits to the VFT or FLD functions occur only in a small temperature range about Ts. MCT Critical Point, Tc. Although it has played only a small role in the analysis of supercooled water,49 the mode coupling theory (MCT) of glasses, which incorporates a dynamical critical point Tc, has been used for studying supercooled liquids in general.50 For orientation purposes, we point out that usually T* > Tc > Tg > To. Cubic Ice Ic We have already indicated that upon heating above 160 K, liquid H2O transforms to cubic ice Ic which consists of very small crystallites.8-11,51 The presence of solid above 160 K is signaled by a great decrease in the diffusion constant,7 and the identity of the solid as cubic ice is established by X-ray diffraction peaks;11 The smallness of the crystallites is inferred from the width of the diffraction peaks11 and by transmission electron microscopy.9 The density of the cubic phase appears similar to that of the Ih phase as determined by the fact that most of the diffraction peaks are similar and overlapping for
Kivelson and Tarjus the two phases.11 Along with the cubic ice the X-ray data suggest the presence of amorphous material which may be liquid9 or may be such small crystallites of the cubic phase for which the crystalline peaks have been broadened beyond recognition. As the system is heated above 160 K the cubic ice, and presumably the amorphous material as well, convert to stable ice Ih.8,51 Conversion to ice Ih, even at quite rapid heating rates, is completed by 240 K, possibly at lower temperature because the samples are not true bulk samples and are therefore subject to surface and confinement effects and to possible differences between sample and measured temperatures. If liquid does in fact exist throughout the 160 to 236 K range, this would be further evidence for continuity of the liquid phases; however, if the apparently amorphous material really consists of very small cubic ice crystallites, then one can envisage the cubic ice as composed of polydisperse small crystallites, only the larger ones giving rise to sharp diffraction peaks. It appears that at lower temperatures, near or below 160 K, the cubic and hexagonal ice forms can coexist for appreciable periods of time, the rate at which the irreversible transformation from cubic to hexagonal ice increasing as T is increased above 160 K.8,10,51,52 It is possible that this transformation takes place via partial premelting of cubic ice, but no calorimetric signature of premelting would be observed if the subsequent crystallization process were very rapid, and indeed none is observed.11,51 Interpretation of Cubic Ice as Defect Ordered Crystal: FLD Theory We suggest that the cubic ice phase is representative of a universal tendency of fragile liquids to form metastable solid phases, i.e., that its existence is not merely a special specific characteristic of H2O with its very special hydrogen-bonding and local tetrahedral bonding. We base this proposal both upon the theoretical predictions of the theory of frustration-limited domains,20 a theory designed to describe fragile supercooled liquids, and upon the parallels we perceive between cubic ice and the glacial phase of the fragile glass-former triphenyl phosphite (TPP).53 The theory envisages a supercooled liquid as composed of equilibrated aggregates known as frustrationlimited domains (FLDs) that grow modestly as T is decreased below a crossover temperature T*; above T*, in the absence of FLD’s, the relaxation is more or less Arrhenius-like (although the theory has little to say about this), and below T*, due to the growing size of the polydisperse FLDs the relaxation becomes superArrhenius.54 Although the structural nature of the FLDs remains an open question, there is considerable evidence that the relaxation properties are correlated over spatial domains55,56 and that these domains are intimately connected with the superArrhenius behavior of the fragile liquids.54 The theory (from which the regular molecular crystal is specifically excluded) predicts that at a temperature Tdo that lies below T* there is a first-order transition to a defect-order crystal. The FLD theory envisages the formation of domains as a consequence of an aborted transition to a phase built upon the locally preferred structure in the high temperature liquid. The reason the transition is aborted is that space cannot be tiled with this structure; this inability to tile space, termed “frustration”, gives rise to increasing superextensive strain which limits the extension of growing nuclei with the extended locally preferred structure. This is the origin of the frustration-limited domains. It is important to note that formation of these domains are phasedriVen and that, consequently, they can equally well be found in weakly interacting systems such as the fragile hydrocarbon o-terphenyl as in hydrogen-bonded systems such as water. The
H2O below 277 K degree of frustration limits the size of the domains, the weaker the frustration the larger the domains, and the larger the domains the greater the activation energy for relaxation and the slower the relaxation. (The domains are equilibrated but have finite lifetimes.) Although the degree of frustration determines the size to which the domains grow, the crossover temperature T* is the point at which the aborted phase sets in. The increasing size of the domains with decreasing temperature gives rise to superArrhenius behavior, and the polydispersity of the domains gives rise to nonexponential relaxation.54 It is the ordering within domains rather than the break up into domains that principally affects the thermodynamic properties such as the entropy but it is the properties of the domains that determine the dynamical properties (the empirical activation energy); thus, the frustration directly affects the dynamics but has little effect upon the thermodynamics. The frustration, which still has not been well described in molecular terms, is a structural quantity but in the theory it is inVersely proportional to the fragility (described by the coefficient B introduced above), the fragility being a dynamical quantity. Of special interest to us here is the theoretically predicted property that the temperature Tdo of the transition to the defectordered phase be less than T*. In some ways, one might think of the defect-ordered phase as freezing of the superArrhenius liquid, i.e., of the liquid below T*, a liquid composed of frustration-limited domains rather than of individual molecules. Thus, one could envisage the melting point of the defect ordered phase as the freezing point of the superArrhenius liquid below T*; seen in this light, the defect ordered phase is not simply a transient specialized metastable phase but a necessary consequence of domain-formation in supercooled liquids, the domainformation being a consequence of the structural frustration. One prediction of the theory is that the ratio Tdo/T* increases with decreasing frustration, i.e., with increasing fragility.57 Although the theory predicts the existence of a defect-ordered phase, it is not certain, even if the theory is correct, that such a phase will always be detected; if Tdo lies below Tg or if the system crystallizes above Tdo, the defect-ordered phase would be hard to uncover. Furthermore, the transition rate may be very slow. Cubic Ice and Glacial TPP as Defect Ordered Phases of Fragile Liquids The cubic ice form of H2O and the glacial phase of triphenyl phosphite (TPP)53,58 are both metastable phases formed from the twice-supercooled liquid upon heating. They both seem to be composed of very small polydisperse crystallites, the smaller ones giving the appearance of amorphous material (as seen in X-ray diffractograms). Both water and TPP are fragile liquids. We tentatively propose that both cubic ice and glacial TPP are defect ordered phases associated with supercooled liquids (below T*), i.e., superArrhenius systems composed of domains (frustration-limited domains). Both cubic ice and glacial TPP undergo similar transformations upon heating and cooling. Although glacial TPP gives strong evidence (calorimetric59 and nmr53) of premelting over a temperature range below Tod, no such direct evidence is available for cubic ice, but premelting followed by rapid crystallization is compatible with the calorimetric observations. It is especially for fragile liquids such as H2O and TPP that one would expect Tdo to be sufficiently close to T* so that Tdo would lie above Tg thereby allowing the defect ordered phase to be detected. An uncertainty in this analysis for cubic ice is that its apparent melting point, i.e., the highest temperature at which it is detected before being fully transformed to hexagonal
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6625 ice is somewhat high, perhaps as high or even slightly higher than the empirical T*. In addressing this question, we note that the values of T* and Tdo are uncertain, lying between about 195 and 230 K and between 160 and 230 K, respectively. Characteristic Thermodynamic Temperatures Melting Temperature, Tmelt. The melting point enters our discussion only as defining an upper bound to the state of “supercooled”. Actually, we take this upper bound for water to be 277 K, below which the liquid becomes increasingly more diaphanous as T is lowered. Cubic Ice Melting Temperature, Tdo. Although somewhat difficult to determine because it represents a metastable-tometastable first-order transition, the melting of cubic ice (Tdo) can be considered a thermodynamic transition. We have proposed that cubic ice can be treated as a defect ordered phase with Tdo well above 160 K and below 240 K. Kauzmann Temperature, TK. The Kauzmann temperature TK is obtained by extrapolation of the excess entropy; therefore TK is a thermodynamic property. For fragile liquids, with the notable exception of water, TK lies close to the extrapolated divergence in the relaxation time at To. The proximity of To to TK has been taken as suggestive of a direct connection between thermodynamics and dynamics; this has led to the development of critical theories, as mentioned above, in which To and TK both represent the critical point, and it has served as motivation for the Adam-Gibbs theory,60 mentioned below. For supercooled and twice-supercooled water taken as a single phase, the situation is unique; TK ≈ 170-216 K lies significantly above To ≈ 118 K. One can summarize these results by stating that water is the only substance for which Tg is known to lie below TK. This would suggest that either water is exceptional17 or that there exists no simple connection between the thermodynamics and the dynamics; we believe the latter to be true. That TK is not merely a rogue thermodynamic property, the creature of extrapolations, and so not significant in comparing water with other liquids, we confirm by noting that the thermodynamic ratio TK/Tmelt ≈ 0.7 ( 0.1 for water is not much different than that of many other fragile liquids. Temperature of Heat Capacity Maximum, Tmax. We have defined Tmax as the temperature below which the thermodynamic excess heat capacity ∆C(T) must start to decline. Although no data on liquids has been obtained near Tmax, it represents a necessary crossover from classical to quantum behavior. Tmax must lie above TK, and except for water, below Tg; because for fragile liquids TK/Tg ≈ 0.8, one expects TK/Tmax ≈ 0.9, which means that a uniVersal property of fragile liquids is that their heat capacities must drop Very rapidly as T is decreased below Tmax. For water, with a reasonable interpolation of the data in the region between 160 and 236 K (see Figure 2), one also finds that the thermodynamic ratio TK/Tmax ≈ 0.9, but as explained above, the mixed thermodynamic-dynamic ratio Tg/Tmax ≈ 0.6 for water and is of the order of 1.1 for other fragile liquids. This again suggests that there is no direct connection between the dynamics and thermodynamics or else that water is exceptional. (The ∆C(T) discussed here should not be confused with the difference between the heat capacities of the liquid and glass at Tg.) Arrhenius-SuperArrhenius Crossover, T*. We first introduced T* as an indicator of a change in dynamics, but later through the FLD theory, we introduced it as a thermodynamicstructural quantity that ultimately controls the dynamics. It is a feature of the FLD theory that although T* plays a fundamental thermodynamic-structural role in determining when phase-
6626 J. Phys. Chem. B, Vol. 105, No. 28, 2001 driven ordering sets in, it has little direct effect on the entropy and heat capacity. On the other hand, because it is only after this phase-driven ordering sets in that the frustration-limited domains form, the signature of T* is prominent in the dynamics. Thus, despite the fact that T* is basically a thermodynamic parameter, it is evaluated via the dynamics, and we expect ratios such as Tg/T* and To/T* to be only weakly material-dependent for all fragile liquids, but do not expect this to be true for Tmelt/ T*, Tmax/T*, and TK/T*. For water the first two ratios conform with the values for other fragile liquids, whereas the last three do not. Relation to Melting Point For many glass-formers the empirically determined dynamical T* lies in the close vicinity of, usually a bit above, Tmelt. Although normal crystallization is explicitly excluded from the FLD theory,20 a physical explanation within the context of the theory can be formulated; it is that the same forces that tend to organize the liquid into the stable crystal are effective in organizing the liquid into various other competing structures. In the context of our analysis, of fragile supercooled liquids water seems quite ordinary in most ways, but its T* is far less than its Tmelt. We have already seen that this gives rise to anomalous Tg/Tmelt and To/Tmelt ratios for water. Although this is not a phenomenon to be addressed by our present analysis, it does call for physical rationalizing at the level used to explain the similarity of T* and Tmelt for other liquids. The special characteristic of water on which we focus in discussing the ratios above is the crossover from what might be called a normal liquid above 277 K to an increasingly “diaphanous liquid” below 277 K. The crystallization to hexagonal ice at 273 K could then be considered the crystallization of normal liquid. The T* for this diaphanous liquid can be quite distinct from that expected for the normal high-T liquid; that for the latter is no longer relevant once the liquid crosses over to the diaphanous species. Consequently, the T* for the diaphanous liquid is expected at a temperature far below the Tmelt which relates to the crystallization of the normal liquid. Relaxation and Entropy The view of Adam and Gibbs60 that the empirical activation energy, E(T), associated with relaxation processes is inversely proportional to the configurational entropy has had wide support, but it seems to be starkly contradicted by the data on water. The configurational entropy has often been taken as a measure of the number of accessible local potential energy minima (inherent structures),61 and though not directly measurable, is thought to be the dominant component of the excess entropy, ∆S. Despite numerous complications,62,63 the fact that for most supercooled liquids the product E(T)∆S(T) is reasonably independent of temperature has lent support to the Adam-Gibbs hypothesis. But the obvious lack of constancy of the product E(T)∆S(T) for water is a challenge to the hypothesis.17 Why does the physically appealing Adam-Gibbs hypothesis fail? In the regime above Tmax, the picture of the number of inherent structures determining the entropy and the hopping between inherent structures determining relaxation is plausible, but in the quantum mechanical regime below Tmax this need not be so because entropy is no longer determined by the number of inherent structures but by the number of occupied eigenstates which are delocalized over the inherent structures. Summary Discussion From the available data on H2O at 1 atm and at temperatures below 277 K, we have extracted and summarized in a consistent
Kivelson and Tarjus manner what we consider to be universal or nonspecies specific properties, properties which when compared with the corresponding ones for other systems emphasize the sameness rather than the uniqueness of water. By its very direction, such an approach cannot be molecular, i.e., molecular specific, and it is therefore not adapted to the task of describing the truly unique properties of H2O, those arising from the strong tetrahedral hydrogen bonding. In seeking water-specific information, researchers have combined theory and computation in order to develop and apply quasi-molecular intermolecular potentials, and others have combined experimental and computational techniques to study water clusters. As motivation for our approach, we propose that the properties of H2O below 277 K are more universal (and hence more collective) than generally assumed. There have been studies (several mentioned above) at the mesoscopic level of H2O below 277 K; although the overall picture that we draw overlaps in some ways with those previously presented, it is in some respects quite novel. Furthermore, our analysis is the first and only one that attempts to integrate the metastable cubic ice phase into a unified and coherent model of fragile liquids. The underlying concepts are those associated with the theory of frustration limited domains which has had considerable success in its application to supercooled liquids but above all provides a consistent physical picture for a range of low-temperature phenomena, both thermodynamic and dynamic.53,63 There are speculative aspects of the analysis and theory, but we believe it to be more solidly based than other existing approaches that have ventured beyond straightforward reporting of the somewhat uncertain experimental results. In summary, we take liquid H2O above the glass transition at 136 K as a single fragile-liquid phase. Like with all fragile liquids, one can distinguish an Arrhenius-superArrhenius crossover at a temperature T*, a Kauzmann point at a temperature TK, a glass transition at a temperature Tg, an extrapolated ideal glass transition at a temperature To, and a temperature Tmax at which the excess heat capacity must start to decrease rapidly. Like some fragile liquids, H2O forms a metastable solid phase composed of rather small crystallites which are possibly polydisperse in size and presumed to melt at a temperature Tdo below T*. The dynamical ratio Tg/T* and the thermodynamic ratios TK/Tmax and TK/Tmelt do not vary much from substance to substance, but the dynamical/thermodynamic ratios such as TK/ Tg and T*/Tmelt are very different for water; in particular, whereas for all other liquids, TK < Tg, for water TK > Tg, and while for all other liquids T* ≈ Tmelt, for water Tmelt . T*. It is these errant dynamic/thermodynamic ratios that we interpret in terms of the uniquely interesting species-specific behavior of water, in particular in terms of the crossover from ordinary to “diaphanous water” at 277 K. We accept that the rather greater than normal increase of ∆C(T) of supercooled water with decreasing T and the small deviation from expected “universal” behavior of the diffusion constant just above 236 K as possibly due to a nearby critical point at higher pressures, but we see this near critical point as having minor influence on the behavior of water at 1 atm. The aspects of the FLD theory that are directly relevant to this study are the prediction of a defect-ordered phase that we associate with cubic ice (and with the glacial phase of triphenyl phosphite), and the association of the dynamical property of fragility with the structural property of inverse frustration. The crossover temperature T* enters as a structural crossover point, but its presence in thermodynamic quantities is minor64 while its presence in dynamical quantities is marked;54 this in some
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