4719
J . Phys. Chem. 1994,98, 4719-4725
Thermodynamic Continuity between Glassy and Normal Water C. P. Johari' Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario U S 4L7, Canada
Gerhard Fleissner, Andreas Hallbrucker, and Erwin Mayer Institut f u r Allgemeine, Anorganische und Theoretische Chemie, Leopold- Franzens- Universitiit Innsbruck, A-6020 Innsbruck, Austria Received: January 27, 1994"
The enthalpy, the heat capacity, and Gibbs free energy of glassy water and of metastable water a t 153 K have been examined by measuring the heat evolved on its crystallization to cubic ice. Measurements are made both isothermally and for slow heating, since the total heat evolved on crystallization decreases with the temperature. It is shown that a small fraction of metastable water persists up to 180 K when heated a t 30 K min-l and that the excess enthalpy of water a t 153 K is 1.2 f 0.1 kJ mol-I. Free energy considerations suggest that a thermodynamic continuity is possible only if the residual entropy of glassy water at 0 K is 1 5 . 3 J K-1 mol-', or the excess residual entropy over that of hexagonal ice is 5 1 . 9 J K-* mol-'. Precise estimates have been made by finding a variation of heat capacity with temperature between 153 and 236 K which satisfy the requirements of both the enthalpy and the third law of thermodynamics. These give a value of 3.8-4.5 f 0.1 J K-' mol-' for the residual entropy of glassy water, or 0.4-1.1 f 0.1 J K-l mol-' more than that of hexagonal ice. On this basis, it is concluded that glassy water and normal water can be connected by a continuous thermodynamic path. Recent estimates of the excess residual entropy of water are consistent with this value. Its absolute value can be determined only when its reversible transformation to a state of known entropy becomes observable.
Introduction Recent attempts at inferring whether metastable water below 150 K is thermodynamically connected to normal water at 273 K or not have led to conflictingviews.l-6 It was first pointed out1 that water's heat capacity, C,, at 273 K could not be connected to that at 136 K without invoking an entropy loss greater than required by the third law of thermodynamics. Therefore, water at 136 K may not be thermodynamicallyconnected to that at 273 K by a continuous path.l Careful measurements by differential scanning calorimetry (DSC), which demonstrated the onset temperature of its glass liquid transition (Tg) at 136 K for a heating rate of 30 K min-1,G have shown that the increase in C, AC,, at its TBis much smaller than the earlier value.'s9 The new AC, (= 1.6 J K-I mol-I) led us to conclude that entropy loss on supercooling can no longer be seen as a basis for the earlier conclusion.1 Speedy10 has reconsidered this issue. From an analysis of water's enthalpy at 236 and 150 K, he argued for reversion to the earlier conclusion,l namely, that a thermodynamic continuity between water at 150 K and at 273 K is not possible, and metastable water at 150 K is a new liquid phase, or water 11. This water I1 is thought to be a "strong" liquid," i.e., one which, like Si02, has a relatively small AC, at Tg and has an Arrhenius temperature dependenceof its viscosity and relaxation time. Speedy'sIO analysis is most interesting and the implications for two liquid phases of water are far reaching. Reading his paper, it appears that thermodynamic discontinuity has to be accepted. The experimental value for the excess entropy at 150 K, which would settle this issue, is not known and will be very difficult to obtain because of interference by crystallization. Therefore, in the absence of an experimental value, we attempt in this paper to see if thermodynamiccontinuity,which is the more conventional approach, is still possible. By doing this we will appear to give a one-sided view. Reference 10 can be consulted for collection of arguments in support of thermodynamicdiscontinuity of states.
-
Abstract published in Aduunce ACS Abstracts, March 15, 1994.
In our attempt, we also present new data on the excess enthalpy of metastable water obtained isothermally at 151, 153, and 155 K, and compare those with its Q i , values obtained by slow heating. We then use these data to determine a thermodynamic path, in terms of heat capacity, that satisfies both the enthalpy requirements and the third law of thermodynamics. Finally, we examine the implications of our study for the current views on the state of glassy water.
Experimental Section The method of preparation of glassy water by hyperquenching of micrometer-sized droplets has been already described in our earlier p a p e r ~ . ~ JThese ~ J ~ also include the details on handling the glassy water samples, on their characterization, and on calorimetric measurements with Perkin Elmer's Model DSC-4. Isothermal data were collected by using the software provided for theDSC-4. The samples were first rampheated to theselected temperature at a rate of 10 or 20 K min-l, and then kept at this temperature until completion of thermal events. The sample was thereafter cooled and its isothermal scan was recorded under identical conditions and subtracted from the first run. The weight of the sample was determined thereafter by measuring its heat of melting in a temperature scanning experiment, as in our previous studies.6.7.12,13 Evaluation of the heat evolved by integrating the areas from isothermal or heating at a rate of 1 K min-' experiments is less accurate than from the areas on heating at rates of 5-30 K min-1. Therefore, the heat evolved in the former case is given only to two significant features. The values are then corrected for an impurity of 4 % crystalline ice.
Results
-
Previously reported values for glassy H2O water cubic ice transformation for heating rates of 10 and 30 K min-l were 1.33 f 0.02 kJ mol-' for unannealed samplesI2 and 1.43 f 0.03 kJ mol-' for annealed samples.13 We consider the latter value more reliable and will use it in the following because it was obtained
0022-365419412098-4119%04.50/0 0 1994 American Chemical Society
4720 The Journal of Physical Chemistry, Vol. 98, No. 17, 1994
I-/"
Johari et al.
Wmin:
b
u
E LL
140
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T(K)
Figure 1. (a) DSC thermograms of hyperquenched glassy water heated at 1 K min-1 (curve l), 5 K min-I (curve 2), and 30 K min-I (curve 3). The samples for curves 1 and 2 are from the same batch, and curve 3 is from ref 13. The samples were annealed at 130 K for 60 min. Sample weights were 8.35, 10.2, and 15.5 mg for curves 1-3. The peak height of curve 1 corresponds to 2.2 mJ s-l: the ordinate scales of curves 2 and 3 are shown normalized with respect to sample weight and heating rate and drawn on the same scale by multiplying with 0.164 for 2 and with 0.0180 for 3. The limits of integration are indicated by vertical lines. Vertical scale is arbitrarily shifted to accommodate the various thermograms. The temperature scale is corrected for the thermal lag of the instrument. The glass transition endotherm is not discernible on the scale used in this figure. (b) The fraction of glassy water crystallized at a certain temperature for heating rates of 1, 5, and 30 K min-I. after annealing a t 130 K in order to remove the broad exotherm due to enthalpy relaxation. This allows the low-temperature limit of integration to become well-defined, by eliminating the overlap between the enthalpy relaxation and crystallization exotherms. This limit was necessarily more arbitrary for an unannealed sample (see Figure 1 of ref 12). Results of crystallization of hyperquenched glassy water obtained by slow heating at 1 and 5 K m i d are compared in Figure l a with a previous DSC scan for heating at 30 K min-I. Curves 1 and 2 are for two samples of glassy water from the same batch and heated at a rate of 1 and 5 K min-', and curve 3 is for a sample heated a t 30 K min-I (from ref 13). The samples were annealed at 130 K for 60 min prior to heating and recording of the DSC scans. (We note that the shape of the exotherm changes with increase in the heating rate and a low-temperature shoulder becomes apparent on heating a t 30 K min-I. This means that there are at least two rates at which the heat is released during crystallization.) The figurecontainsalso the limits of integration (vertical lines), and the peak minimum temperatures. The heat evolved is 1.3 kJ mol-' for curve 1, and 1.40 and 1.49 kJ mol-' for curves 2 and 3, after correction for 5% crystalline ice impurity. Heating at 1 K min-I was repeated with another sample from the same batch, and the heat evolved was 1.2 kJ mol-'. Note that the values obtained on heating a t 1 K min-1 are lower than those reported for heating annealed glassy water a t 10 and 30 K min-l,13 but the value obtained for heating a t 5 K min-1 is in between. For the three thermograms, the fraction of glassy water crystallized at a certain temperature is shown in Figure lb. This figure also shows that for the heating rate of 30 K min-I a small
I
o k
-. 4
8
min
12
Figure 2. (a) Heat evolved during isothermal crystallization of a 17.5mg sample of glassy water held at 153 K plotted against time, and (b) its fraction crystallized at a certain time. The broken line in (a) encloses the integrated area. fraction of metastable water persists up to =180 K. (In principle, metastable water can be heated up to 273 K provided the heating rate is high enough.) In view of the fact that the enthalpy difference between a supercooled liquid and its crystalline phase decreases on cooling,14 it was necessary to examine whether the heat released during isothermal crystallization is less than the weighted average of 1.43 kJ mol-' obtained for a heating rate of 30 K min-I, and of 1.3 or 1.2 kJ mol-' obtained for heating a t 1 K min-l. Figure 2a shows the typical heat released when hyperquenched glassy water was kept at 153 K. (Here also evidence for two rates of heat release is seen as a change in slope a t ~ 0 . min.) 5 Figure 2b shows the fraction of water crystallized as a function of time. For Figure 2a a value of 1.2 kJ mol-' was obtained when using the broken line as limit of integration. Four other isothermal experiments at 155, 153, and 151 K gave values between 1.2 f 0.1 kJ mol-'. While this is close to the values of Qirrcvobtained from the two temperature scans for heating a t 1 K min-I, it is lower than Qimev of 1.43 kJ mol-l obtained for heating at 10 and 30 K min-I and reported earlier. I 3
Discussion Crystallization of metastable water a t a rate too rapid to allow measurements of its C, prevents us from resolving the continuity issue decisively. Therefore, inferences on how water's C, would change with temperature are drawn. This is done within the bounds of the laws of reversible thermodynamics by using the excess thermodynamic functions: excess heat capacity, C,,ex,= Cp,waterICpjce;ezcessenthalpy, Hcxc = Hwater-Hijirx; excess entropy, Sex,= - Sice; and excess free energy, G,, = Hex,- TS,,,. Here H = H - H0o and = S - S0o, Hoe, and S O 0 being the enthalpy and entropy of the thermodynamically stable state at 0 K. (Note that R a n d 3, and not H a n d S, are measured from calorimetry.) At the freezing point, Hex,= AHf, Sex,= A&, and G , = 0, where AHf and A& are the latent heat and entropy of freezing a t 273 K, respectively. Both Hex,and Sex,are functions of temperature; they decrease on supercooling, or increase on heating the metastable liquid towards the melting/freezing point. Since they represent the sum of excess configurational and
s
Thermodynamics of Glassy and Normal Water
The Journal of Physical Chemistry, Vol. 98, No. 17, 1994 4721
vibrational contributions: cp,cxc
-- Cfonf
+
p,cxc
1400
pibr 1200
T
T
I
'
I
-
I
'
l
-
. ~JWJ.~ ...
T
30
600
(Here Hex,, Sex,, and G,,, are the same as AH,AS,and AG in ref 10. We hold that the notation A be used exclusively for reversible phase transformations and reversible thermodynamic paths. Since water can be heated and cooled through its Tg,the path is reversible for the same thermal history and ACp needs to be used for the change of heat capacity at Tg, as is common in text books. He, and Sex, become equal to AH and AS at 273 K. At all other fixed temperatures, crystallization of water to ice is irreversible, and therefore Hex, and Sex, are used.)
-
In the temperature scanning mode of DSC, the heat of an irreversible transformation, e.g. water cubic ice is equal to the integral
where q is the heating rate and TOand TIare the transformation's onset and end temperatures. It is important to note that this heat is released over a range TOto T1, which increases with increasing q and is -30 K wide for metastable water heated a t a rate of 30 K min-I . He,, which can only be measured isothermally, increases with increasing temperature. Strictly speaking, Qm, is therefore the sum of the increasing Hex,weighted by a decreasing fraction of the untransformed sample as the temperature is increased during the scanning. This means that Qmw determined by temperature scanning is higher than Hex,a t a temperature closer to the onset temperature TO,and lower than Hex,a t TI.This also means that an increase in the heating rate would increase the measured Qirrwtoward a limiting value, AHf (i.e., on heating to 273 K, when q m, Qi,, AHf,and superheating of crystals does not occur; when q 0, Qrrw H,,,, as in the isothermal scans), and a decrease in heating rate would lower the measured Qrrwtoward Hex, at To. We use He,, = 1.2 kJ mol-' at 153 f 2 K to analyze the issue of thermodynamic continuity of water but do so in terms of excess Gibbs free energy instead of entropy or enthalpy, although the answers from the three considerations should be identical when a correct value of Sex,of water at 0 K, SO,,,, is used. Similar procedures have been used by Whalley et al.I5 and by SpeedylO for drawing inferences on the residual entropy of amorphous solid water.
- -- -
- TSCXC
(3)
Both He,, and Sex,increase with temperature according to eq 1 . By assuming different values for Sex,from 0 to 3 J K-'mol-' at 153 K and for Hex,at 153 K = 1.25 kJ mol-' (which includes the 0.050 kJ mol-' evolved during phase transition of cubic to hexagonal icel6), we calculate Gexcat153 K and show thesevalues in the plot of G,,, against the temperature in Figure 3. We also include in this figure calculated values of G,,, a t 158 K for various values of Scxc, and for He,, at 158 K = 1.38 and 1.48 kJ mol-' (from refs 12 and 13). G,,, from 236 to 273 K are the calculated values from C, data of water emulsions.l7J* The standard deviation in G,,, was recalculated from the data in Table I of ref 18, for errors of &OS% in He,, and f O . 1 in S,. (For example, for Hex, = 4290 f 20 and Sex,= 15.2 f 0.1 at 236 K, G, = 703
1
'
1
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supercooled water
I I
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.
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-
LE
Gcxc = H e x ,
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1 . 0.....,_. - 0 . 5 0 002.0 ... .... 2& 1000 -- O0 00 23 .00 2.00- - - - _ _ _ _ x 03.0 --- .... .... 8M 800 - 3 . 0 8 . .
p,cxc
T
-
I I
400
I
f
0
-
'
"
"
'
I
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'
1
4722 The Journal of Physical Chemistry, Vol. 98, No. 17, 1994 I
t
Johari et al. by Raman spectroscopy,26 is 194.1 cm-I, with (dvT/dT) = -0.375 cm-' K-I for measurement from 313 to 253 K. Extrapolation of the peak maximum to 153 K gives a value of 240 cm-1. This is larger than the value for hexagonal ice a t the same temperature which is 225.9 ~ m - ' .This ~ ~ difference increases with decreasing temperature because of different temperature dependencies, and at, e.g., 60 K the extrapolated and measured value for UT is 275 and 230.1 cm-I. Now, if C, was determined mainly by these lattice modes, would even be slightly negative. We adopt in the following a C$Lc(153 K) value of zero which makes the vibrational integral in eq 1 negligible, or zero, in magnitude and the configurational integral or Gfequal toScxca t 153 K. From = 0.11 J K-' mol-'. This was calculated our C, data Sex,= by integrating over the increase in heat capacity on a logarithmic temperature scale from 136 to 153 K, for a AC, = 1.6 J K-1 mol-' and a width of 12 K for the glass t r a n ~ i t i o n . This ~ , ~ means that water loses a further configurational entropy of 0.11 J K-1 mol-' on cooling from 153 to 0 K and, if sub-T, relaxations made an insignificant contribution to C,, whatever valueofS,,iscalculated from constructing C, paths (between 153 and 273 K) in a In T plane, the residual entropy, SO,,,, will be 0.11 J K-l mol-' less than that of Sex,a t 153 K. Curves 1-3 of Figure 4a are drawn such that He,, at 153 K is 1.25 kJ mol-', that is, that the loss of enthalpy from 273.16 to 153 K is 4.76 kJ mol-' (6.008 - 1.25, 6.008 kJ mol-' being the Hex,at 273.16 K). The curves in Figure 4b lead to the following values of Sex,at 153 K: 1.21 J K-I mol-' for the path of C, in curve 1,0.65 J K-1 mol-' for that in curve 2 and 0.47 J K-1 mol-' for that in curve 3 (these were determined using ASf = 22.00 J K-I mol-').'09*5 Other paths gave either negative values or values less than 1.21 J K-' mol-' within the combined uncertainty of ~ 0 . J2 K-' mol-', which comes from the uncertainty in Hex,at 153 K. It is intriguing-and remarkable-that the maximum valueofS,,at 153 Kisobtainedwhen C, at 228 K is used;18,28i.e., C,isallowed toshow a A-typeanomaly without causing the viscosity of water or its structural relaxation time toapproachinfinityat=228K. Any path for C,that bypasses this A-type anomaly leads to a lower S,,,at 153 Kin our analysis. The temperature of the A-type anomaly is given by Angel126 as 228 f 3 K, and by Speedy18 as 227 f 2 K. We have further tested the influence of small temperature shifts of the A-type anomaly within the error estimates. For 230 and 225 K and the same maximum value of C, used in Figure 4, S,,( 153 K) is 1.26 and 1.23 J K-I mol-l which is close to the value of 1.21 J K-1 mol-' obtained for path 1 and 228 K. The above-given observations and the precise variation of Hex, and S,,, with temperature are shown in Figure 5 . The He,, and Sex,a t the limiting temperatures are as described before. From 273.16 to 236 K, the variation (shown by dots) is the same in the three cases. For path 1, the rate of decrease in both Hcxcand Sex, reaches a maximum value (at the point of inflection of a sigmoidalshaped curve) at a higher temperature than for paths 2 and 3, as is expected from the drawings in Figure 4. In our analysis of C, seen in Figures 4 and 5 we have intended to provide answers to two issues: first, the path that the C, of water would follow during its hyperquenching and then on reheating from its glassy state. This would establish the thermodynamic continuity within the bounds of thermodynamics. Second, an estimate of the residual entropy of glassy water from the knowledge of its Sex,at 153 K. The plots in Figures 4 and 5 seem to provide a satisfactory answer to the first issue. On hyperquenching C, of water may follow either of the three paths (or any other within the bounds of path 1 and 3) from 273.16 to 153 K. The shape of the plots in Figure 5 reveals an important aspect of supercooled water's thermodynamics both before and after its hyperquenched glassy state has been obtained. We discuss the effects of two different cooling rates: one, where the rate was
q!::,
0'
150
200
I
300
250
Temperature [K]
200
5,O
$1
5,2
5,3
5,4
5,5
5,6
5,7
In [T/K] Figure 4. Variation of heat capacity of supercooled water between 153 and 236 K which satisfies the thermodynamic requirements of both the
enthalpy and the third law of thermodynamics. Measured values are shown by solid notations and the required values by open notations. Heat capacity of water at 153 K exceeds that of ice by 2 J K-' mol-I. In the upper drawing (a), the area confined by curves 1-3 for water and ice is 4.76 kJ mol-' (6.008-1.25). In the lower drawing (b), the area confined is 20.8 J K-'mol-' (22.00-1.2) for curve 1, 21.3 (22.0-0.7) for curve 2 and 21.5 ( 2 2 . 0 4 5 ) J K-I mol-' for curve 3. question by pointing out that the endothermic step in C, which was reported to be 35 J K-1 mol-' near 135 Kin ref 9, is nonexistent, and (ii) the values at e.g. 133.6 K (sample 1 in ref 9) are surprisingly higher than those in refs 6 and 7. C, for the lowdensity amorphous solid made on compression from Handa and Klug2' also should not be used here because for the same rate of heating its T, characteristically differs from that of glassy water.22 Wealso note that Handa and Klug's C,for low-densityamorphous water exceeds that of hexagonal ice by 0.6 J K-I mol-I at 90 K and 0.9 J K-1 mol-' at 120 K,21 and Sugisaki et ale's C, for vapordeposited amorphous solid and hexagonal ice are the same within experimental errors from 60 to 110 K (read for sample 3 from Tables 2 and 3 in ref 9). It seems that, within the measurement errors of C, given in refs 6, 7 , 9 , 21, and 22 and the differences in its magnitude noted by different workers, it is not certain whether glassy water's C,below Tgis more than, equal to, or even slightly less than that of hexagonal ice. The latter can happen, as Kauzmannz3has pointed out, because of a "tighter binding of the molecule in the highly strained liquid structure, with consequent higher frequencies of vibration and a lower density of vibrational levels". The density of glassy water is slightly higher than that of hexagonal ice at the same temperature,24 and therefore, in the glass the more strained bonding between the water molecules can increase the vibrational frequencies and its C,can be slightly less than that of hexagonal ice at the temperature where the density is higher. Some information can however be deduced by comparing the frequency of (translational) lattice vibration modes, VT, in the two solids. At 273 K the peakmaximum of VT ofwater, determined
%a'
The Journal of Physical Chemistry, Vol. 98, No. 17, 1994 4723
Thermodynamics of Glassy and Normal Water
'oool
1200
-
5000
supercooled water
180
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Temperature [K]
=; I
800
-
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4001
180
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220
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240
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~
A
A
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supercooled water
l -\*
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2ool 0
160
180
200
220
240
260
280
300
Temperature [K] Figure 6. Excess Gibbs free energy of water plotted for the three paths against temperature. Solid circles are the measured and further analyzed values given by Speedy.IOJ8 Notations and numbering are the same as in Figure 4.
,*-'
160
"
k?~.~.' Q',
I
140
20
140
-
. E LL
160
1000
'a a 8,f
x
m
140
-
260
280
300
Temperature [K] Figure5. Excessenthalpyandentropyofwater for the threepathsplotted against the temperature. Solid circles are the measured and further analyzed values given by Speedy.10J8Notations and numbering are the same as in Figure 4. such that the configurational freezing of the water's structure may have occurred at 200 K, and a lower one where it may have occurred at 180 K. In these cases, an extra amount of configurational enthalpy and entropy, equal to the difference between the excess values a t the configurational freezing temperature and those a t 153 K, are frozen-in. For the first case at 200 K, the extra frozen-in configurational enthalpy and entropy it would have at 78 K (i.e., the temperature to which it has been hyperquenched) will be 0.29 kJ mol-' and 1.6 J K-' mol-' if its C, followed path 1, 0.72 kJ mol-' and 3.9 J K-I mol-' if its C, followed path 2, and 0.89 kJ mol-' and 4.8 J K-1 mol-1 if its C, followed path 3, as is clear from Figure 5. When it is then heated from 78 K, the structural relaxation from 78 to 136 K would cause this frozen-in enthalpy and entropy to decrease. The value to which it would decrease depends upon the heating rate in a DSC scan. For high heating rate this value is low and for low heating rate it is high. For a given heating rate, the enthalpy released will be lower if hyperquenching occurred along path 1 than if it occurred along path 2 or 3. The minimum enthalpy released will be a0.29 kJ mol-' for path 1, assuming that the decrease in the enthalpy from 153 to 136 K is negligible. Enthalpy release on heating from 100K a t a rate of 10 K min-' was observed in DSC scans in the form of a broad exotherm extending from 125 K up to the temperature region where crystallization occurs (see Figure 1 in ref 12). The heat evolved was =0.16 kJ mol-'. This value is expected to be a lower-bound value for experimental reasons because the DSC instrument requires stabilization of the system at 100 K before scanning can bestarted. It is to be expected that during this period part of the enthalpy has been released already. The lower-bound value of =O. 16 kJ mol-' is about half of 0.29 kJ mol-' that would be observed if configurational freezing occurred a t 200 K and path 1 in Figures 4 and 5 was followed. For paths 2 and 3, the frozen-in configurational enthalpy at 200
K is even higher than for path 1. As pointed out above, for experimental reasons the heat released on heating from 78 K is not known, but it can be measured eventually. Therefore, a t present we do not know if the heat released on heating is less or equal to the frozen-in configurational enthalpy at 200 K for paths 1-3, and it is not possible to decide if the configurational freezingin of water had occurred at 200 K. For 180 K, however, the extra frozen-in configurational enthalpy is 0.09, 0.15, and 0.26 kJ mol-' for paths 1-3. The values for paths 1 and 2 are lower than mO.16 kJ mol-l reported as lower-bound value for heat released on heating, but the value for path 3 is higher. Therefore, we conclude that the water structure can be frozen-in during hyperquenching above 180 K if it follows path 1 or 2. The G,,, plots in Figure 6 precisely show the changes that occur on hyperquenching water. In this construction, each plot corresponds to a unique variation of C, with temperature and thus removes the ambiguity associated with the type of construction given in Figure 3. It also shows that a first-order phase transformation is not likely to occur during the hyperquenching as G,,, can vary with temperature continuously along a course leading to S,,,(153 K) of up to 1.2 J K-I mol-). The second issue is the residual entropy of glassy water. The answer from our analysis is that it is between 3.8 and 4.5 J K-1 mol-I (after deducting 0.1 J K-1 mol-' which glassy water would lose on cooling from 153 to 0 K), or So, is 0.4-1.1 J K-1 mol-', with an uncertainty of e k 0 . 1 J K-' mol-I. These values are much lower than those calculated from a construction of random tetrahedral network models where, on the basis of certain assumptions, it was necessary to simplify mathematical procedure~.~,~ According to the calculations for a random tetrahedral network structure for water without broken bonds, SO,,, is 6.05 J K-1 m ~ l - I . * , ~But this value has been put in doubt in a recent discussion29 that also points out that the measured value for vitreous silica, on which the validity of the calculations of a topologically similar tetrahedral arrangement is based, is considerably uncertain.29 For example, in Bell and Dean's30 calculation of the residual entropy for a random arrangement of tetrahedra in a model of vitreous silica, positional disorder alone contributes between 3.3 and 5.8 J K-' mol-'. Inclusion of Si0-Si ring statistics reduces this value. On the other hand, the random network model for the structure of water which gives a value of 6.05 J K-' mol-' 'is constructed from a central configuration which has the connectivity of a continuously branching loop-free tree with a local topology of a distorted ice I str~cture".~bIt is conceded that a more realistic value would be obtained if the effects of five-, six-, or higher-membered rings were included.4b
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The Journal of Physical Chemistry, Vol. 98, No. 17, 1994
There are also considerable uncertainties in the experimental estimates of the residual entropy of vitreous silica, a comparison against which is used for justifying the calculations for a random tetrahedral network model. The best experimental estimate for vitreous silica is 2 . 8 4 5 J K-1 m0l-I.3l.3~ But even this is debatable because such estimates require values of C, at a temperature above the melting point of a crystalline phase (quartz), which are not easily measured. Therefore, both theoretical and experimental estimates of the residual entropy of vitreous silica are approximate and do not as such provide a support for the structure of either vitreous silica or water. Obviously, more accurate values are needed for testing any structural models. More recently inferred rough estimates for the SO,,, of vapordeposited amorphous water are =3 J K-1 mol-' (ref 10) and for low-density amorphous ice made on compression =1 J K-l mol-' (ref 15). Thesevalues are at least consistent with our notion that the magnitudes of both the residual entropy and AC, at T,are a reflection of the number of entropically different configurations available to a liquid a t Tg,Z9and that residual entropy and AC, seem to change in the same manner; Le., when AC, is large, the residual entropy is also large as for H 2 S 0 ~ 3 H 2 0 and , ~ ~when AC, is small, the residual entropy is also small, as for selenium.34 When there is a higher degeneracy of the entropically different configurations, both AC, a t T, and SO,,, are low. This also suggests that water's SO,,, would be low since its AC, at T, is low. Even with So,,, = 1.1 J K-1 mol-I, which seems to be the upper bound value, the number of configurations is ~ 0 . 2 per 1 molecule more in glassy water than in ice (calculated from the equation SO,,, = R(ln Ww,,,,) - R(ln Wi,),with Wi, = 3/2, ref 25). It is also possible that the entropy from orientational disorder in water at 0 K is less than R ln(3/2) (this value for hexagonal ice is deduced for a structure in which the six orientations of each water molecule are equally probable, but this is unlikely to be the case for glassy water where there is a much greater distribution of 0-H--0 distances and therefore of H-bond energies), and the substantial degeneracy of the topologically different arrangements does not greatly add to its total S0o. For S O , , , of 1.1 J K-I mol-', Sooof glassy water is 4.5 J K-1 mol-': it is important to note that this value can be apportioned between topological and orientational disorder in several ways. In this view, SO,, of 1.1 J K-I mol-' should be seen as a minimum contribution from topological disorder. The contribution from topological disorder can be greater than 1.1 J K-1 mol-l if the contribution from orientational disorder was less than R ln(3/2), thevalue for ice, or the protons were not entirely randomly distributed. We now discuss the recent inference that glassy water below T, and viscous water above T, up to ~ 1 5 K0 is a new phase different from normal water,IO and be seen as a "strong liquid"." Diffraction studies have shown that the structure of glassy water differs from that of normal water at, e.g., 298 K.3s,66 There is little doubt that the structure of metastable water above its T,, e.g., up to 153 K, and of water at 298 K are different, because metastable water has gone through a set of changes in its H-bonded structure on supercooling before its vitrification and recovery at 77 K. Therefore, the inference for a new phase of water (water 11) must necessarily be based on the observation that its G,, at 153 KcannotbeconnectedtotheG,,at 236K.10 Theconstruction of the relevant G, curves has relied on assumed values of Sex, at 153 K, as shown in Figure 3 here. The value of the integral and the reliance on the assumed residual entropy values (although calculated from random tetrahedral network models, as discussed earlier) is at the root of the suggestion for the existence of a new phase of liquid water.10 We note that the lack of continuity originally pointed out' did not rely on any residual entropy value. The paths envisaged and shown in Figures 4-6 are invalid if S,,,(153 K) is more than 2 J K-1 mol-'. It seems to us that we are faced with two alternative, but unequally plausible, conclusions. First, if we assume that the
integrated term and the residual entropy estimated from model calculations are, despite the weaknesses of such calculations as outlined above, reli.able, then metastable water at 153 K is a new phase and needs be referred to as water 11. Since it is formed without a thermodynamic discontinuity by heating glassy water from 100K, hyperquenching of normal water must have produced this new phase of water. But we also fear that in doing so we overlook the fact that random tetrahedral network calculation makes no provision for a structural change a t the C, anomaly of supercooled water, and as it now stands, is not related toa structure formed after this transition has occurred. The second conclusion is that glassy water is thermodynamically continuous with normal water and that its residual entropy, estimated from the measured excess enthalpy and a plausible C, variation with temperature, is 14.5f 0.1 J K-1 mol-'. Its structure is known to differ from that of normal water. But, it may not be referred to as a new liquid phase for the same reason that water at 277 K (wheredensity maximum resulting from a different H-bond structure results) is not seen as a separate phase from water at 298 K. The issue of the thermodynamic continuity of states between metastable water at 1 1 5 3 K and normal water at, e.g., 298 K can be resolved unambiguously only when the C, and the residual entropy of glassy water are known. Thecurrently availablevalues are based upon models with inherent limitations of completely tetrahedral bonding and dendritic topology. But, even when an exact and reliable value from such calculations becomes available, it would not provide a unique path between the Cp's of water at 150 and 236 K. We believe that experimental determination of C, of glassy water and its residual entropy is necessary. It can be done when extremely fast (microseconds) methods for measuring C, become possible so that C, can be measured before metastable water crystallizes. Our current knowledge of the properties of glassy water does not rigorously support arguments for the existence of a new phase of liquid water at 153 K. Acknowledgment. We are grateful for financial support by the Forschungsforderungsfondsof Austria (project P9 175-PHY). References and Notes (1) Johari, G. P. Philos. Mug. 1977, 35, 1077. (2) Sceats, M. S.; Rice, S. A. J . Chem. Phys. 1980, 72, 3260. (3) Rice, S. A.; Bergren, M. S.;Swingle, L. Chem. Phys. Lett. 1978,59,
14. (4) (a) Sceats, M. S.; Rice, S. A. In Water, a Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1982; Vol. 7, Chapter 2; (b) Sceats, M. S.; Rice, S. A. J . Phys. Chem. 1981,85, 1108. (5) McFarlane, D. R.; Angell, C. A. J . Phys. Chem. 1984,88, 759. (6) Hallbrucker, A.; Mayer, E.; Johari, G. P. Philos. Mag. 1989, 60B, 179. (7) Johari, G. P.; Hallbrucker, A.; Mayer, E. Nature 1987, 330, 552. (8) Johari, G. P.; Astl, G.; Mayer, E. J. Chem. Phys. 1989, 92, 809. (9) Sugisaki, M.; Suga, H.; Seki, S. Bull. Chem. Sot. Jpn. 1968, 41, 259 1. (10) Speedy, R. J. J. Phys. Chem. 1992, 96, 2322. (11) Angell, C. A. J . Phys. Chem. 1993, 97,6339. (12) Hallbrucker, A.; Mayer, E. J . Phys. Chem. 1987, 91, 503. (13) Johari, G. P.; Hallbrucker, A.; Mayer, E. J. Chem. Phys. 1990, 92, 6742. (14) This puts in doubt the use of a fixed value of "H(or HcXshere) in ref 10. (15) Whalley, E.; Klug, D. D.; Handa, Y. P. Nature 1989, 342, 782. (16) Handa, Y. P.; Klug, D. D.; Whalley, E. J . Chem. Phys. 1986, 84, 7009. (17) Angell, C. A.; Sichina, W. J.; Oguni, M. J . Phys. Chem. 1982, 86, 998. (18) Sueedv. R. J. J . Phvs. Chem. 1987. 91. 3354. (19) Mille; J. C.; Miller: J. N. Statisticsfor Analytical Chemistry; Ellis H o r w d : England, 1989; p 46. (20) Giauque, W. F.; Stout, J. W. J. Am. Chem. Sot. 1936, 58, 1144. (21) Handa, Y. P.; Klug, D. D. J . Phys. Chem. 1988, 92, 3323. (22) Hallbrucker, A.; Mayer, E.; Johari, G . P. J . Phys. Chem. 1989,93, 7751. (23) Kauzmann, W. Chem. Reu. 1948, 43, 244. (24) Hofer, K.; Astl, G.; Mayer, E.; Johari, G. P. J. Phys. Chem. 1991, 95, 10777.
Thermodynamics of Glassy and Normal Water (25) Reviewed by: Eisenberg, D.; Kauzmann, W. In The Structure and Properties of Water: Clarendon Press: Oxford, U.K., 1969; p 102. (26) Krishnamurthy, S.;Band, R.; Wiafe-Akenten, J. J. Chem. Phys. 1983, 79, 5863. (27) Johari, G.P.; Chew, H. A. M.; Sivakumar, T. C. J. Chem. Phys. 1984,80, 5163. (28) Angell, C. A. In Water, a Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1982; Vol. 7, Chapter 1. (29) Johari, G.P. J. Chem. Phys. 1993, 98, 7324.
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