Shear Viscosity and Self-Diffusion Evidence for High Concentrations

Jon M. Sorenson , Greg Hura , Alan K. Soper , Alexander Pertsemlidis , Teresa Head-Gordon. The Journal of Physical Chemistry B 1999 103 (26), 5413-542...
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J. Phys. Chem. 1995,99, 10635-10643

10635

Shear Viscosity and Self-Diffusion Evidence for High Concentrations of Hydrogen-Bonded Clathrate-like Structures in Very Highly Supercooled Liquid Water G. ELWalrafen* and Y. C . Chu Chemistry Department, Howard University, Washington, D.C. 20059 Received: January 17, 1995; In Final Form: April 25, 1 9 9 9

Activation energies from shear viscosity and self-diffusion of highly supercooled liquid water rise very rapidly with decreasing temperature at 1 atm and approach e 7 0 kUmol, or more, at 228 K. These high values may result because the flow of H20 molecules is impeded by passage through hydrogen-bonded pentagonal rings whose opening radius is ~ 0 . 8, 4 smaller than the van der Waals radius of H20. Two principal effects may contribute to the overall energetics: (1) breakage of hydrogen bonds around a given H20 molecule; and then, only if no opportunity exists for free diffusion through hexagonal or larger rings in the immediate vicinity of the freed H20 molecule, (2) impeded, highly endothermic diffusion of that H20 molecule through a hydrogenbonded pentagon, by forced ring expansion, followed by rebinding of the H2O. Large negative activation volumes are also indicated from 273 to 228 K and 1 atm. These negative volumes may arise from activated filling of voids in bulky, clathrate-like, polyhedral structures. A model for the diffusional process is described.

Introduction Several types of experimental'-8 and theoretical9-'' evidence exist for the presence of ~ l a t h r a t e - l i k e ' structures ~.~~ in highly supercooled liquid water. Some of the experimental evidence has recently been reviewed by Sorensen,I4 who presented a strong, and especially lucid, case for sizeable concentrations of clathrate-like structures in highly supercooled water. Of the experimental evidence, the viscosity data of Halfpap and Sorensen' is closely related to the present work, and of the theoretical evidence, the molecular dynamics (MD) simulations of Geiger, Mausbach, and Schnitker" are very compelling. Halfpap and Sorensen measured viscosities of ethanol-water solutions. These solutions exhibit viscosity enhancement. The maximum enhancement occurs for the composition 4.88 mol H20/1 mol ethanol, which is close to the ethanol-water clathrate hydrate composition of 5.75 mol H20/1 mol ethanol.I4 The viscosity enhancement, moreover, increases markedly as the temperature is lowered, e.g., to -25 "C. SorensenI4attributes this viscosity enhancement to clathratelike water cages around the hydrophobic ethyl groups. He also finds that the viscosity data from a 7.33 mol H20/1 mol ethanol solution in the temperature range from =20 to -10 "C can be superimposed on the viscosity data reported for supercooled water between -10 and =-35 "C, if the ethanol data are uniformly displaced downward by 29 "C.I4 This superposition is interpreted to mean that supercooled water at -10 "C has the same clathrate-like content as the ethanol solution at 19 O C . I 4 A further key point is that the curvature of the ethanol-water viscosity versus temperature data is the same as that of the pure water data between -10 and -25 O C , I 4 and probably even to -35 "C, when probable systematic errors in the pure water viscosity data are considered.I5.l6 This equality of curvatures is important in the present work, as seen subsequently, and it does not involve displacement by 29 "C. Geiger, Mausbach, and Schnitker" have conducted extensive MD simulations of water for T = 283 K and a density of 1.0 g/cm3, using the ST-2 potential. Special attention was paid to cavities in water in the MD investigation. @

Abstract published in Advance ACS Abstracts, June 1, 1995.

0022-3654/95/2099-10635$09.00/0

Geiger et al. presented a distribution function of orientations of the OH covalent bond with respect to cavity centers. They emphasized that the shape of this OH orientation distribution function for water is the same as the corresponding distribution functions obtained for the clathrate-like hydration shells of water around various hydrophobic solutes. Note that this theoretical result agrees very well with the conclusion obtained by Sorensen from experimental data.I4 Geiger et al. also obtained cavity-oxygen and cavityhydrogen pair correlation functions for water. These two functions peak within 0.2 A of each other, thus indicating a straddling orientation of the OH bonds around the cavity center. But an additional peak was obtained in the cavity-hydrogen correlation, which refers to some hydrogen atoms (OH groups) pointing radially outward from the cavity center. Both the OH straddling orientation and the radial hydrogen (OH group) orientation would be consistent with the presence of clathrate, e.g., pentagonal dodecahedral or similar structures in liquid water. Geiger et al. further presented a cavity center-cavity center distribution function, which indicated a large eak near 5.5 A. A distance of 5.5 8, is very close to the 6 peak from the same type of distributionfunction obtained by simulating a neon pair distribution in water. They conclude that "the structure of gcc(r) (the cavity center-cavity center distribution function) indicates an arrangement (without dissolved particles) similar to the separation of two adjacent cavities by a shared water layer corresponding to two adjacent cavities in a clathrate crystal". Geiger et al. also report an increasing self-diffusion coefficient with density rise, 0.8 to 1.0 g/cm3, which implies a negative activation volume (283 K, 1 atm), observed experimentally. Further evidence supportive of clathrate-like structures in supercooled liquid water comes from thermodynamics and statistical mechanical fluctuation theory. Equilibrium fluctuation (EF) theory, N , P , T ensemble, indicates that the third moment of the, volume is positive and that it increases with supercooling of the water, while the third moments of the entropy and of the energy simultaneously become increasingly negative. The same conclusions result from nonequilibrium fluctuation (NEF) theory, N , P, Tensemble.I8 These EF and NEF results indicate that highly ordered, low-energy, large-volume structures increase

8:

0 1995 American Chemical Society

10636 J. Phys. Chem., Vol. 99, No. 26, 1995 in concentration as the supercooling of water proceeds to increasingly low temperatures, just what would be expected if increasing concentrations of clathrate structures were to form with increased supercooling of water. Moreover, changes observed between the N , P , T and N , V, T ensembles indicate that pressure rise disfavors (lowers the concentrations of) the bulky structures, as would be demanded by Le ChPtelier's principle. If the pressure of supercooled water is increased, e.g., isochoric conditions, N , V, T ensemble, it is found that the third moment of the entropy becomes weakly positive below 233 K and that the third moment of the energy is also positive, just opposite in sign to the negative third energy moment obtained at 1 atm." Similarly, most of the unusual properties of supercooled water at 1 atm, e.g., the large molal volume, the large Cp, the large negative value of the expansion coefficient, ,9, the large values of the isothermal compressibility, KT, disappear as the pressure is increased to high va1~es.I~Two kilobars of pressure are probably sufficient to break up many of the bulky clathrate structures in highly supercooled liquid water. Apart from the need to understand why the fluidity of very cold water increases with pressure rise,20the desire to explain the enormous activation energies and large, negative activation volumes obtained from the viscosity and self-diffusion data for supercooled water provided the impetus for the present work. These quantities are so large for supercooled water that no viable alternatives to explanations involving a hierarchy of clathraterelated polyhedral structures have yet presented themselves to us. The Arrhenius activation energy from viscosity is very large, roughly 58 kJ/mol at -35 OCG2' Similarly, the apparent activation volumes, VD, from the self-diffusion pressure dependence are negatively large for cold and supercooled water. Weing;irtner22gives VD = -2.8 cm3/mol at 278 K and 1 atm,22 and VD may approach a limit of roughly -4, or more, cm3/mol to 228 K.

Brief Overview of the Model and Methods In the present work we present a model for microscopic flow in supercooled water which is an isotropic analog of the Onsager-Runnels (0-R) model for self-diffusion in ice.23In the 0-R model, single H20 molecules are considered to break away from the lattice, to diffuse freely through hexagonal channels, and then to rebind to the lattice.23 A major difference between the 0-R model for ice and our model for supercooled water is that we assume that free diffusion through hexagons gives way to very difficult, highly energetic passage through the pentagonal faces of the bulky, pentagon-based, clathratelike polyhedra, as the pentagon concentration due to such bulky polyhedra dominates over the hexagon concentration upon increased supercooling. The flow impediment, as envisioned in our model, involves pentagonal ring expansion, and thus it requires a large energy in excess of that needed simply to free H20 molecules by hydrogen-bond breakage. Pentagonal ring expansion is necessary because the equilibrium hole size of the pentagonal rings is too small to pass a single water molecule. A large negative activation volume at 1 atm may also be involved because of the densification produced by activated filling of voids in bulky pentagonal dodecahedra or similar pentagon-based polyhedra by H2O molecules after entry through pentagonal rings. This negative activation volume feature of our model explains why the viscosity of very cold water first decreases and then finally increases at x2 kbar; activated void filling is more difficult when the bulky polyhedral structures break up, yielding hexagons, etc.

Walrafen and Chu Interpretations which involve a significant equilibrium population of broken hydrogen bonds to explain the change in the viscosity of very cold water with rise of pressure to 2 kbar disagree with recent low-frequency Raman data,24which give no evidence for significant concentrations of broken hydrogen bonds in non-supercooled water at -18 "C and 2.0 kbar. A description of the procedures which we applied here follows. An activation energy was determined from the shear viscosity and self-diffusion coefficient of supercooled water at the limiting temperature of 228 K, namely, x 8 0 kJ/mol. The hydrogenbond energy, 20 kJ/mol H20, was then subtracted from the activation energy to obtain an excess energy, E(&). E(&) is the amount of energy, beyond the hydrogen-bond energy, needed to overcome the difficult passage through the pentagonal holes. If an H20 molecule is forced through the hole in a hydrogenbonded pentagonal ring, it must, according to van der Waal's radii, stretch the five 0-0 ring distances from 2.84 to 3.34 8, during passage. The 0-0 stretching corresponding to E(dr)/5 was next calculated using a Morse potential, from which the value of 3.38 A (3.26 A self-diffusion coefficient, alone) resulted. This close agreement is consistent with the presence of high concentrations of hydrogen-bonded clathrate-like pentagonal rings in water near the limiting supercooling temperature. The large negative activation volume may conform to the activated filling of voids in bulky pentagon-based polyhedral structures. It should be emphasized that we regard passage of an H20 molecule through pentagonal rings and clathrate void-filling to be sequential activation events. One could argue, nevertheless, that a freed H20 molecule need only break one bond of a pentagonal ring (with some ring bending) to produce an enlarged hole for passage. However, the counterargument is that the totally unbonded H20 molecule would rebind immediately at the breakage point, without moving further, and thus diffusion would not occur. The ring breakage argument merely amounts to the formation of a six-membered ring from a five-membered ring, which is occumng all the time, anyway, as part of the continual formation and breakage of structures in the liquid. The energy involved is already included in the 20 kJ/mol H20, which leaves 40-60 kJ/mol H20 unexplained. Very large densification must also be involved during the activation process to explain the large negative activation volume for self-diffusion in supercooled water. This is just the opposite of the expansion which must occur for water to give positive activation volumes above 47 OC.** Virtually all other liquids have only positive activation volumes for self-diffusion. More mechanistic details are presented subsequently, but because we depend heavily upon the Young-Westerdahl (YW) m e t h ~ d ?as ~ opposed ? ~ ~ to the Arrhenius activation approach, it is necessary to describe the Y-W procedure next.

Details of the Young- Westerdahl (Non-Arrhenius) Thermodynamic Method The Arrhenius activation energy obtained from isochoric shear visocisty data from water between 0 and 200 "C is temperature dependent. However, if the Y-W method is applied to the isochoric viscosity data, a single, temperatureindependent enthalpy of 4.86 kcavmol H20 result^.^' This Y-W enthalpy equals the generally accepted value for hydrogen-bond breakage.28s29Moreover, the constancy of the Y -W enthalpy change indicates that the temperature dependence of the isochoric Arrhenius activation energy is an artifact

J. Phys. Chem., Vol. 99, No. 26, 1995 10637

Hydrogen-Bonded Clathrate-like Structures resulting from the inability to include hydrogen-bond breakage in the Arrhenius approach. It should be emphasized that the Y-W approach is conceptually much different than the Arrhenius method. According to the Arrhenius method, curvature in the logarithm of a transport property versus 1/T corresponds to temperature dependence of the activation energy, an outcome which is far from enlightening. In contrast, the Y-W method removes the curvature in a logical thermodynamic way, by using an additional parameter related to the high-temperature limit of the data, thus yielding a constant enthalpy over a limited temperature range. The Y-W method is, unfortunately, not well-known in spite of its importance; hence it is described here. It was first applied to equilibria between boroxol rings and BO3 triangles in molten boric o ~ i d e . However, ~ ~ , ~ ~ the method is general and may be used for structured liquids under isochoric or isobaric conditions. The Y-W method was designed to treat equilibria between two species, states, populations, etc., in liquids. These two may be designated as B (bonded) and U (unbonded). For water, B and U may refer to hydrogen-bonded and non-hydrogen-bonded molecules,27or B and U may refer, in the specific case of the Raman spectroscopy of water, to OH groups that are either hydrogen bonded or not hydrogen bonded.30 A mass balance is first invoked, namely, [B] -I- [U] = constant = c, where the square brackets refer to concentrations. The method then uses some physical observable, e.g., the isochoric fluidity27 or the integrated Raman intensity, I , of an intramolecular band,30 for the concentration of one of the states; for example, I = constant x [B130 The method is not needed when physical observables are available for both U and B, because the equilibrium constant, [U]/[B], is readily obtained. From the mass balance, the equilibrium constant, K, for the process B = U may be formulated, for example, as K = [U]/(c - [U]), or K' = [B]/(c - [Bl, if more convenient. c is constant and initially unknown, but c may be determined as described below. The enthalpy change for the process B = U may be obtained from the relation (a In Kla T l ) p = -AH/R. In addition, In K = -AHIRT f ASIR. If AH and AS are constant, In K is linear in UT. In many cases, AH and AS are often approximately independent of T for ranges of T which are not large, or altematively, AH and AS are constant if ACp = 0, as seen from (aAH/aT)p = ACP and (aAS/aT)p = ACp/T. ACp values are often small for B = U equilibria for pure liquids. The Y-W approximation involves the assumption that ACp = 0. For a series of values of the physical observable chosen to represent [U], for example, @ = values of c are chosen until a fit of the form ln{[U]/(c - [U])} = a/T b yields a minimum in the least squares standard error of fit. This c value, c(MIN), then represents the appropriate Y-W value of c. c(MIN) may be obtained readily with a digital computer using a program which iterates c, until a minimum in the standard error of fit results. The formal physical meaning of c(M1N) refers to K = -, which requires that B has been completely converted to U. Complete conversion to U does not mean, however, that the U state is in any sense free. Other strong, cohesive interactions are usually present, but these other interactions, e.g., van der Waal's forces, are commonly not included in the Y-W formulation. U does not refer to a gas state. c(MIN) also formally corresponds to a limiting or maximum value of the physical observable, e.g., as T approaches -. This consideration

+

c

LIMITING A

=4 t

J

i

\1 1

-4

-

IATM

-5

"

I

'

' ' ' '

A-

'

I

'

'

-

IO ~ / T Figure 1. Young-Westerdahl plots of isochoric viscosity data (upper) (**' and straight line) and of isobaric viscosity datal6 (lower) (x). Movement from A to B requires high pressures, 2 2 kbar. See ref 31 for equation (lower). Temperatures (in "C) are indicated by parentheses and located by dots; see top border.

is useful, because initial estimates of c(MIN), helpful for computation, may be obtained by estimating the limiting value of the physical observable, roughly, say at some high T. Determination of the single parameter c(MIN) fixes both AH and AS for the equilibrium B = U. AH and AS are not adjusted in the Y-W method. Moreover, after a and b, Le., AH and AS, have been determined, one may use the relation, a/T -I-b = ln(xu/xa), where x refers to mole fraction; see also ref 30 for other relations.

Results Obtained from Thermodynamic Treatment of Fluidity and Self-Diffusion Coefficient Data In Figure 1 we contrast a Young-Westerdahl (Y-W) ln(6) = ln[W(constant - @s)] plot of isochoric fluidity data from water ((*) and upper straight line), previously shown to be linear in (UT) for t from 0 to 200 "C when the constant = c(MIN) = 6.148,27against the nonlinear plot, well-fitted by a cubic in (1/ r ) (lower curved line3'), which results from treating the isobaric fluidity data16,32with the same Y-W ln(6) function. The Y-W in(&)function, including the constant c(MIN) = 6.148, used for the isobaric 1 atm data was identical to that for the isochoric data. However, the resulting ln(6) versus UTplot is not a Y-W plot, because the enthalpy is temperature dependent. Either the isobaric or isochoric fluidity data yield the same line from 0 "C to the upper left comer of the figure, -45 "C, but only the isobaric data curve downward and extend into the region of highly supercooled liquid water from 0 to -35 "C at A. The upper straight line, labeled limiting minimum slope, represents the Y-W equation extended to B. The constant Y-W slope corresponds to a limiting minimum enthalpy for conditions under which pentagonal hydrogen-

Walrafen and Chu

i

AR '

i 4

I

i

ISOBARIC-1 1 ATM

r 1

-5

' 3.1

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'

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I " " 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

I0 3 / ~ Figure 2. Young-Westerdahl plots of 2 kbar self-diffusion coefficient data2' (0)and of isobaric self-diffusioncoefficient data34(0).See ref 31 for equation (lower). Temperatures (in "C) are indicated by parentheses and located by dots; see top border.

bonded rings are absent, as shown experimentally below, because this slope results only from high-temperature, 0-200 "C, data. (7jl = l/@s = viscosity, cP.) The Y-W slope should not represent the subzero, isochoric viscosities (no data available). Such viscosities, obtained from an equation of state,33are larger, and thus the fluidities are smaller, than the calculated Y-W values, because the pressures corresponding to the constant molal volume of 18.02 cm3/mol are thought to be much too small to break significant numbers of pentagonal dodecahedral or similar pentagon-based structures. The Y-W limiting minimum slope, Figure 1, is shown to be equivalent to the high-pressure isobaric slope, 2 kbar, Figure 2. In Figure 2 we contrast a Y-W ln{@s(D)/[6.148 - @s(D)]} = ln(I;) (D = self-diffusion coefficient) versus (UT) plot of highpressure, 2 kbar, low-temperature self-diffusion data (upper)2' against the corresponding low-temperature, 1 atm, self-diffusion plot (lower).34 Both plots of Figure 2 involved a ln(5) formulation identical to that used in Figure 1. Experimentally, the fluidity, @s(D)in (cP)-', is given by 1050 divided by a constant, 1.882 4~ 0.061, for water from 298.2 to 242.5 K,16,34not by 102k~T/2na (Stokes-Einstein). ln(I;) was fitted by least squares, linear in (1/T) (upper) and cubic in (1/ r ) (lower).31 We emphasize that Figures 1 and 2, both of which contain Y-W (upper) plots, are virtual copies of each other. Note that four entirely different 7jl and D data sets were involved. The 2 kbar Y-W enthalpy value, Figure 2, is smaller than all of the 1 atm activation energies at low temperatures. Moreover, the isobaric, 1 atm activation energies increase rapidly as the temperatures decrease. We believe that this increasing divergence between the constant Y-W enthalpy and the 1 atm

activation energies occurs because a large number of bulky clathrate-like structures have been broken by high pressure (Le Chitelier). In regard to clathrate structures it is noted that the selfdiffusion coefficient of water rises to a maximum with increasing pressure from 10 down to z - 2 0 "C, but the pressure required to reach this limit also rises, to 2.5 kbar.2' These effects appear to occur because the clathrate-like concentration increases as the temperature of the supercooled water decreases at 1 atm. The Y -W limiting enthalpy from viscosity, Figure 1,20 344 J/mol H20 (4862.8 cdmol), and the high-pressure self-diffusion Y-W value, Figure 2,21 660 J/mol H20 (5177.3 cdmol), agree within experimental uncertainties of the present methods and data, thus supporting the linear extrapolation to B, Figure 1. In contrast, the 1 atm (non-Y-W) activation energies determined from the cubic least squares equations for 228.15 K are 92 157 and 67 282 J/mol H20, respectively, for viscosity and self-diffusion. Note the very large slopes at A, -35 "C, Figure 1, and at 1000/T = 4.1, -30.7 "C, Figure 2, bottom curve. The least squares, Y-W equation used to fit the 0-200 "C isochoric shear viscosity data is In 5 = 1n[(Ds/(6.148 - QS)] = A T 1 -I- B

(1)

where A = -2446.92 and B = 6.68448. The 2 kbar self-diffusion data yield A = -2605.19 and B = 7.379 18. We emphasize that ln(6) versus (UT) linearity results with the same parameter, 6.148, when scaled values, @s(D) = 105D/1.882, are used.'6,34 Finally, it must be made absolutely clear that the breakdown of the total enthalpy into two parts, namely, the Y-W enthalpy, 20 344-21 660 J/mol, and the remaining enthalpy, roughly 60 000 J/mol, cannot, of itself, have mechanistic significance. Extra-thermodynamic calculations and considerations are required to provide possible mechanistic processes related to the two-part division of the total enthalpy. In fact, it may eventually be possible through molecular dynamics, for example, to find a single mechanism which encompasses both enthalpies of the present formal breakdown; for example, a large-amplitude center-of-mass translation of a fully hydrogen-bonded molecule causes it to break its restraining hydrogen bonds in the on-going act of the forced passage through a pentagonal hole, with concomitant densification by clathrate void-filling. Extrathermodynamic considerations follow in the next section.

Interpretation, Nature of the Diffusional Model, and Energetics Two processes may be envisioned by which freed H20 molecules might pass through holes in hydrogen-bonded pentagonal rings, as follows: (1) 0-0 bond compression occurs, in which one 0 refers to the freed H20 molecule undergoing passage through the pentagonal ring, without any ring expansion; and (2) symmetric expansion of the five hydrogen bonds of the ring occurs, without significant 0-0 compression, as the freed H20 molecule forces itself through the ring. Process 1, of course, is highly unlikely, but it is nevertheless useful for exposition. The opening radius formed by a static, regular pentagon of oxygen atoms, in contact, and whose van der Waals radii are taken as 1.42 8,35to give an 0-0 separation of 2.84 8, (the 0-0 X-ray RDF value at 4 0C,36 where E(&) x 0), is 0.9959 8,. (The van der Waals radius of H in a linear 0'-H-02 hydrogen bond overlaps that of atom O2by e 0 . 7 A,35) and the

J. Phys. Chem., Vol. 99, No. 26, 1995 10639

Hydrogen-Bonded Clathrate-like Structures electron distribution in the lone pair region of H20 is nearly spherical.35 Hence, H20 molecules pack very roughly like 0 atoms.) The radius of a circle passing through the five 0 nuclei of the pentagon is 2.416 A, and the uncompressed 0-0 distance is 2.84 8. Thus, dr, the (improbably large) compression necessary for passage of an 0 atom (or water molecule) is dr = r - ro = 2.416 - 2.84 A = -0.424 A, where r and ro are the compressed and equilibrium radii, respectively, process 1. Process 1 can be eliminated with certainty because 2.416 A almost equals the symmetric hydrogen bond, 0-H-0, distance, 2.4 8, which involves pressures of roughly 500 kbar for water,37 and because calculation indicates that E(dr)/5 can only yield a compressive dr of -0.24 A at 228.15 K, x1/2 of the d r needed. In process 2 the 0-0 distance of the five 0-H-O bonds in the pentagonal rings is assumed to be 2.84 8. Without 0-0 compression, the presence of a sixth (free) H20 at the center of the pentagon in van der Waals contact requires all five 0-0 distances of the ring to expand to 3.339 A. The expansive d r is thus +0.499 A, close to the value calculated below. We consider the following impedimentary mechanism to shear flow or self-diffusion: (1) A freed H20 molecule, all hydrogen bonds broken, and immediately adjacent to a pentagonal ring, moves to the center of the ring. (2) The moving H2O molecule then produces a symmetric stretching of the ring as it enters and passes through the ring hole. (3) The H20 molecule next reforms its hydrogen bonds at a new site not far from the ring, with the result that a different H20 molecule is ejected from that site. (4) The microscopic process propagates, in viscous flow, such that flow occurs throughout the liquid in the direction of the external driving force. At this point it should be made absolutely clear that a freed H20 immediately near hexagonal or larger rings will engage in free diffusion through such rings in preference to impeded diffusion of the type described below. Impeded diffusion is thought to occur only when pentagonal rings are the only possibility for diffusion in the immediate vicinity of the freed H20. Free diffusion is likely to occur readily through hexagons, heptagons, etc., at high temperatures, e.g., 4 "C, at which the pentagonal ring concentration is very low. This means that the only energy required in this case is the amount necessary to break the hydrogen bonds around the H20 molecule so that it can move to, and through, the large rings. On the other hand, the concentration of pentagonal rings is probably very high when the water is very highly supercooled and when the pressure is low, e.g., 1 atm. Accordingly, a large, extra energy, beyond that needed just to free the molecule for flow, may come into play, due to the pentagonal ring expansion described above. The sole energy source for the pentagonal ring expansion is E(&), namely, (-S - 2446.92)k~,where S is the slope, not the entropy (see lower curved plots, Figures 1 and 2), as opposed to the total amount of -%e. The value of 2446.924 (from v), or 2506.19ke (from D),eq 1, is the amount needed to break the hydrogen bonds to the H20 molecule, thus freeing it for passage through the ring. Note that the enthalpy of (2#6.92-2605.12)k~, which is %5 kcal/mol H20, is the amount necessary to free an H20 molecule: to break all the hydrogen bonds that hold it, so that it may jump to a new site. This enthalpy is the minimum amount required for self-diffusion or flow, irrespective of whether strong additional impediments, such as passage through pentagonal rings, exist, or if nearly free passage through hexagonal rings occurs, instead. The 5 kcal/mol H20 enthalpy is the same for high pressures, 2 kbar, under which the pentagonal dodecahedral structures, or other pentagon-based polyhedra, are largely broken, or for constant 1 atm pressure, under which the bulky

pentagon-based polyhedral structures may be prevalent in highly supercooled water. However, the fact that the 1 atm slopes of both Figures 1 and 2 correspond to increasingly larger enthalpies as the temperature decreases means that a second, highly energetic process may impact more and more molecules, thus requiring increasingly more enthalpy, in addition to the fixed, minimum amount needed just to free an H20 molecule for possible diffusion. The second, highly energetic process, thought to impede the diffusion of freed H20 molecules, comes into play at subzero temperatures only if the pressure falls from several kilobars to near 0, e.g., 1 atm. Therefore, this additional flow impediment must, according to Le Chkelier's principle, involve largevolume structures, e.g., clathrate-like polyhedral structures. But the only feature of clathrate-like structures that can impede, as opposed to stopping, self-diffusion is the pentagonal ring hole, whose equilibrium diameter is less than the van der Waals diameter for an H20 molecule. The next section thus involves the calculation of the symmetric expansion of the pentagonal rings, namely, dr, for the adjoining 0-0 stretched distance, resulting from the excess energy, E(&), and its comparison with the geometric dr value expected when a central H20 just contacts each of five concentric H20 molecules, using the van der Waals radius for the H20 molecules. This procedure is intended as a test of the present model of self-diffusion.

Anharmonic Potential Approach to Symmetric Expansion of Hydrogen-Bonded Pentagonal Rings We employed the anharmonic Morse potential to calculate d r from the excess energy, E(&). For symmetric expansion of a pentagonal ring, the excess energy, according to the Morse potential, is given by E(&) = DE{ 1 - e~p[P(-dr)])~,where the 5 comes from the fact that five 0-H-O, Le., five hydrogen bonds, are symmetrically stretched. We illustrate the calculation method using the limiting Y-W enthalpy, 20 34.4 J/mol(4862.7 cal/mol), from isochoric fluidity data. Dd2 = hydrogen-bond energyhydrogen bond, 2446.92kd 2. DE(^) = 2446.92kd2 '/zhcAV(r). DE(T) is temperature dependent, because AV( r ) refers to the temperature-dependent Raman 0-0 stretching frequency in c ~ - ' . ~ O p = (K/~DE)"~. Dd2 = 2446.92k~/2= 1.689 x erg/HB (2 HB/molecule H20, fully hydrogen-bonded tetrahedral network). The force constant K was obtained from the T-dependent Raman 0-0 stretching frequency, namely, AV = 284.6191 - 0.381245T(in c ~ - ' ) . ~ OK = 2.07 x lo4 dyn/cm at 228 K, slightly smaller than the K for ordinary ice Ih. The expression which we used for calculating the 0-0 stretched dr, for the case where E(&) is divided by 5, is

+

(-S - 2446.92)(1.38066 x lo-") 1.97188 x

I(

1.97188 x

0.265357 - 3.786677 x

- 3.786677 x

]'/*I>

)'"((0.3812451-

284.619) (2)

10640 J. Phys. Chem., Vol. 99, No. 26, 1995

Walrafen and Chu

TABLE 1: Expansions, 6r > 0, Calculated from an Anharmonic (Morse) Potential Using Eq 2 O self-diffusion 1086r, A T,K Y-W In vb Y-W InP 228.15 238.15 248.15

258.15 268.15 278.15

0.6746 0.4180 0.2689 0.1651 0.0875 0.0381

0.6684 0.4112 0.2604 0.1512 0.0557 ZO

0.4392 0.3097 0.2198 0.1546 0.1105 0.0921

0.4319 0.3034 0.2113 0.1400 0.0825 0.0384

fs 0.82 0.51 0.29 0.15 0.07

0.04

Slopes from In 7 and In D, fitted by least squares cubic equations in (l/Z+,), were also used to calculate 6 r for comparisons between the Arrhenius and Young-Westerdahl methods. Arrhenius method. a

The stretched 0-0 van der Waals value is 3.34 8,. The stretched 0-0 Morse dr value, calculated from eq 2 using the average slope, -(@, 79 720 J/mol H20, is 0.54 8, at 228.15 K. Hence, the stretched 0-0 Morse distance is 2.84 8, 0.54 8, = 3.38 8,. The van der Waals value and the Morse value thus agree within 1%, but the Morse value is 0.04 8, larger than the van der Waals value. However, if the enthalpy from the self-diffusion slope, 67 282 J/mol H20 at 228.15 K, is used in conjunction with the Y-W self-diffusion enthalpy value, Le., 2446.92, is replaced by 2605.12 in eq 2, we obtain a stretched 0-0 distance of 3.26 A, which is 2% less than the van der Waals value, 3.34 8,. The 1 atm self-diffusion data are considered to be extremely reliable,2'~22 whereas our analysis of the 1 atm viscosity data indicates that the corresponding fluidities become abruptly somewhat too small between -30 and -35 "C. The above 0.08 8, difference is important, and its significance is discussed subsequently. A rapid decline in the d r values is evident from Table 1 for temperatures above 228 K. This decline occurs because the concentration of hydrogen-bonded pentagonal ring holes is very greatly reduced, e.g., near 4 "C, the temperature of maximum density, compared to -45 "C (228 K), the approximate limit of supercooling. Moreover, the decreasing 6 r values above 228 K correspond to situations in which the relatively unhindered passage of H20 molecules through hexagonal rings makes a progressively larger contribution. 6 r = 0 for hexagonal or larger rings. The 0-0 expansion for passage of a water molecule through the pentagonal ring should be essentially constant, as opposed to the highly temperature-dependent 6 r values evident from Table 1. We would not expect the size or shape of a pentagonal ring to vary significantly with temperature. Moreover, the temperature dependence of the dr's of Table 1 should disappear if the temperature-dependent fraction of pentagonal holes corresponding to the changing clathrate concentration were adequately considered. Such constant 6 r values do, in fact, result from the use of the pentagonal hole fraction, fs, defined below, which declines between 228 K and 278 K, Table 1. fs is defined by f 5 f>5 = 1, where f 5 should refer, ideally for our model, to the fraction of regular, pentagonal holes, and 5.f to the fraction of large, regular, nonpentagonal holes, e.g., regular, hexagonal and heptagonal ring holes. We emphasize that this definition is an approximation. For example, squashed, or otherwise severely distorted, hexagons or heptagons, etc., might impede flow just as effectively as a regular pentagon, and hence they would be included infs, unless they stop the flow entirely. Openings which stop the flow entirely are not counted as a hole; for example, a four-membered hole stops the flow entirely. Thus, our definition of fs refers essentially to any opening which impedes flow to the same extent as a

+

+

regular pentagon; that is, the size and shape of the opening is the important mechanistic feature. We believe, neveflheless, that regular pentagons and regular hexagons become the important structures at very low temperatures. Iffs is included explicitly in the Morse approach, the excess energy is reformulated as E(&) = 5v5)&{ 1 - e~p[P(-dr)])~. Then if we assume thatfs = E(dr)/E(dr,228.15K), we find that a constant ring 0-0 expansion results for all temperatures from 277 to 228 K, using, for example, the average activation energy of 79 720 J/mol H20. Of course, the avera e activation energy leads to a stretched 0-0 distance of 3.38 , which is too large by 0.04 8,. Hence, the f5 obtained from this average energy is roughly unity. fs may approach unity, but it is very unlikely that it is ever equal to unity. This conclusion involves the 0.08 8, or 2% difference mentioned above, as shown next. If the self-diffusion enthalpy (2 kbar) and activation energy (1 atm) at 228.15 K are employed, both from the Figure 2 data, we find the calculated (expansive) 6 r to be 0.42 8,. This 6 r yields an 0-0 separation of 3.26 A, which is smaller than the van der Waals value of 3.34 8, by 0.08 8,. This 0.08 8, difference is small, but it is sufficient to give anfs value of 0.82 at 228 K. This f5 value of 0.82 at 228 K is significant and importantly different from 1. X-ray diffraction data have been reported for a solid clathrate38 which indicate a structure involving a bcc pentagonal dodecahedral array. The unit cell contains four hexagons and 24 pentagons. Thefs value is thus 0.86. This X-ray value is in very good agreement with thefs value of 0.82 obtained here from the self-diffusion data and the anharmonic potential calculations. In summary, we can expect that fs is small near 4 "C and that it increases with decreasing temperature, finally reaching a value of ~ 0 . 8 2only , 7% smaller than the clathrate value, at 228.15 K. A series offs values, calculated solely from selfdiffusion data, is presented for various temperatures in Table 1. A rise in f5 with decreasing temperature is also in agreement with conclusions obtained below from volume considerations involving pentagonal dodecahedra.

R

Molal Volume and Activation Volume Evidence for the Model

c,

The molal volume, of a bcc array of pentagonal dodecahedra of H20 molecules is 22.6 cm3/mol when the voids are empty.39 However, the molal volume decreases to 19.3 cm3/ mol if the voids in the bcc array are filled with one H20 per void.39 In addition, ccp and/or hcp arrays of such pentagonal dodecahedra can yield a equal to, or below, 19.3 cm3/mol, even with unfilled voids.40 Accordingly, a network of pentagonal dodecahedra, or related polyhedral structures, might constitute a good model of the structure of very highly supercooled water, as shown next. Speedy4' has reported a value for supercooled water at -45 "C and 1 atm of 19.2 cm3/mol. This value is only 0.01 cm3/mol smaller than the above void-filled bcc clathrate value. The molal volume of supercooled water thus obviously approaches, and then essentially equals, that of a void-filled bcc, or ccp and/or hcp, array of pentagonal dodecahedra, as supercooling proceeds to the limiting value of 228 K. This agreement, of course, must not be construed as a proof that pentagon-based polyhedral structures exist in highly supercooled water. The H20 molecules in the voids of the bcc structure cannot be free (not hydrogen bonded), unlike a guest clathrate molecule, because Raman spectral observations indicate that most or

v

v

Hydrogen-Bonded Clathrate-like Structures virtually all of the water molecules in very highly supercooled water are fully hydrogen b ~ n d e d ,but ~ ~interpenetration .~~ may help to resolve this difficulty. Moreover, the pentagonal dodecahedra of the ccp and/or hcp structures must share edges and/or faces to yield the correct Interpenetration and distortion may both help to resolve such difficulties with the bcc, ccp, and hcp structures. With regard to interpenetration, an especially interesting, bulky structure formed by 17 H20 molecules and containing six pentagons, with two pentagons interpenetrated, is shown in Figure 3-1 of ref 13. Here, several side-shared pentagons are involved. This 17 molecule structure may be one of a hierarchy of pentagon-containing, distorted, interpenetrated, bulky, polyhedral structures which may be important in highly supercooled water. Such distorted, interpenetrated, bulky polyhedral structures may not be so objectionable to some workers, as perfectly regular pentagonal dodecahedra which they consider much too ordered to exist even in strongly supercooled liquid water. We employ the void-filled bcc structure extensively here. It is emphasized, nevertheless, that we regard it only as a useful construct, some of whose properties mimic those of supercooled water. For example, the agreement between the molal volume reported by Speedy and the molal volume of the void-filled bcc array at 228 K suggests that progressively larger supercooling produces relatively greater numbers of pentagonal rings, thus decreasing the self-diffusion coefficient, according to the details of our model. Similar \. olume effects might be expected from the pentagonal dodecahedral structures or the bulky, interpenetrated polyhedral structure, described above. The preceding static structural considerations, of course, are incapable of characterizing the disorder expected for liquid water, even when it is very highly supercooled. Temporal considerations are essential, which means that it is important to realize that all pentagon-based polyhedral structures must be continually reforming, existing for some finite residence time and then breaking down, just as all other structures, including the hydrogen bonds themselves, are similarly being transformed. Under such temporal conditions, the pentagon-pentagon centerto-center distance for tilted, side-sharing pentagons would be extant during the residence time of, say, a pentagonal dodecahedron; or not, during the ensuing down time. Such behavior is precisely that involved in a correlation length, because the correlation length is the distance over which the fluctuations are correlated. The correlation length of highly supercooled water is discussed in a subsequent section. In regard to temporal structures, Pauling concluded that clathrate structures of the kind which he invoked for liquid water would be very labile.39 However, he also emphasized that such clathrate structures are energetically more stable than ice structure^,^^ which would explain one of the reasons for invoking them so prominently in supercooled water. A more compelling argument for the presence of bulky clathrate-like structures in very highly supercooled liquid water involves the large negative activation volumes which result from the isothermal pressure dependence of the self-diffusion coefficient. Such negative activation volumes are virtually unique to water. It has been known for some time that the apparent activation volumes for self-diffusion associated with a rise in the pressure of supercooled water are strongly negative below 1-2 kbar.22 This means that the self-diffusion coefficient of supercooled water increases with rising pressure at very low, but constant, temperatures and finally reaches a plateau or maximum, beyond which normal behavior (a decrease) ensues with continued pressure rise.

v.

J. Phys. Chem., Vol. 99, No. 26, 1995 10641 The lowest temperature at which the pressure dependence of the self-diffusion coefficient has been studied is x-20 0C.21A temperature of -20 "C is the lowest temperature, at a pressure of 2.0 kbar, at which bulk water can be held in an equilibrium, non-supercooled, state. However, lower temperatures may be achieved by the use of small droplets or emulsions under high pressure, e.g., -90 "C at 2 kbar,21 and thus such conditions were assumed in previous extrapolations and in the extrapolations to be described below. The molal volume, at 228 K and 1 atm is given by Speed?' as 19.2 cm3/mol. The molal volume for water at 228 K and 2.0 kbar may be calculated from the HGK (Haar, Gallagher, Kell) equation of state;33 it is 16.5 cm3/mol. The change in the molal volume for the pressurization of water from 1 atm to 2.0 kbar at 228 K may thus be about -2.7 cm3/mol. This negative volume is one measure of the voids in the water structure at 228 K and 1 atm. Limiting values of the activation volumes for self-diffusion are now presented, where VD, the apparent activation volume, is given as follows:

v,

VD = -RT(a In DlaP), = -RT(aD/aP),/D

(3)

The slopes, (l/D)(aD/aP)T, needed for calculation of VD are only available, and then with very limited accuracy, to -20.1 0C.21 Nevertheless, a plot of a number of VD values calculated from eq 3, using slopes from ref 21 and VD values from ref 22, suggests than an extreme upper order-of-magnitude limit for VD at 228 K and 1 atm is -10 cm3/mol. A similar estimate may be calculated, as shown next. Integration of eq 3 requires a knowledge of VD as a function of pressure at constant temperature. This function is unavailable, but examination of ref 21 reveals that VD is negative at -20 "C and 1 atm and that it rises nonlinearly to 0 near 2.0 kbar. A rough order-of-magnitude estimate of the change in VD may thus be obtained from the following equation: 1n(D2/Dl) = (AVD)(Pi - P2)/RT

(4)

The (AVD) term in eq 4 is an integrated average change in the apparent activation volume, VD, over the pressure range from PIto P2. (The subscripts 1 and 2 refer here to 1 atm and to 2.0 kbar, respectively). The value of (AVD) which results solely from the least square equations involving the self-diffusion data, Figure 2, is x-12.6 cm3/mol at 228.15 K, Le., -10 cm3/mol. It is easy to show, however, that the above -10 cm3/mol limits are much too large. A volume change of -9.5 cm3/mol corresponds to the compression from 19.2 cm3/mol to the ccp volume, using van der Waal's radii for the water molecules. Such a ccp density for the activated state would correspond to pressures well in excess of 500 kbar. Hence, we can reject it. Better activation volume limits might come from the highpressure ices, e.g., ice VI, because they involve interpenetrated lattices. The molal volume of ice VI1 is 11.5 ~ m ~ / m o land ,~' that for ice VI is 13.4 ~ m ~ / m o lThe . ~ ice ~ - VI ~ molal volume yields the more conservative VD limit of roughly -6 cm3/mol. We therefore can infer, at least; that VD should be negatively about 2 times larger at 228 K and 1 atm than at 278 K and 1 atm, where it is -2.8 ~ m ~ / m o l .Any * ~ viable model of the activation process must be capable of yielding negative VD values of such magnitudes. The bcc pentagonal dodecahedral array has six ( X ) voids, two (Y) voids, and 46 water molecules not in voids.38 The minimum included sphere volume in the ( X ) void is 58.3 A3; the minimum included sphere volume in the (Y) void is 78.8 A3. The volume of a water molecule is 12 A3. The ( X ) voids might be able to

10642 J. Phys. Chem., Vol. 99, No. 26, 1995

hold two water molecules, and the (Y) voids, which are oblate, might be able to hold three. (Both minimum included sphere volumes neglect some volume available for activation processes.) The addition of a second water molecule into the eight voids of a singly filled pentagonal dodecahedral array produces a change in the molal volume (19.3 cm3/mol with one molecule per void) of -2.5 cm3/mol, and the addition of a third water molecule into the two (Y) voids produces an additional change of -0.5 cm3/mol, for a total change of -3.0 cm3/mol. The addition of three water molecules in all voids gives a total change of -4.4 cm3/mol, which is probably the negative limit. Of course, no clathrates are known which can contain two host molecule^,'^ but this conclusion only refers to equilibrium structures. The above densification from -2.5 to -4.4 cm3/ mol would refer to the activated state, which is short-lived. We presume that all but one water molecule per void would be forced out of the pentagonal dodecahedron upon its retum to equilibrium. Similar densifications should apply to all bulky, pentagon-based, polyhedral structures. Finally, a vital clue which relates void-filling to the negative activation volumes for water comes from the fact that VD approches 0 at 47 0C,22which is about the temperature at which the isothermal compressibility is minimal, namely, 46 “C at 1 atm.45 VD is obviously negative for all temperatures below 47 “C at 1 atm, because activated-state densification can occur by filling any voids of the proper size. In contrast, VD becomes positive, like other “normal” liquids, when a water molecule must push through its neighbors to diffuse, thus momentarily increasing the molal volume during the activation process. Rising thermal amplitudes with the ensuing collisions should also cause VD to become even larger (positive) at elevated temperatures, e.g., 11.1 cm3/mol at 473 K.22

Correlation Length and MD Simulations for Supercooled Water A correlation length of -3.8 8, was recently reported from low-angle X-ray scattering for water supercooled to temperatures of -34 0C.46 We suggest that this correlation length may be understood from the pentagon-pentagon radial correlation function, obtained in both MCY and ST-2 molecular dynamics (MD) simulation^:^ as follows. The MD simulations yield an intense, broad, major peak which is centered between 2.5 and 4.0 A. This peak was found to correspond to the pentagon-pentagon center-to-center distance for tilted pentagons sharing a single side.47 The tilt angle is the same as that for side-sharing pentagons in a pentagonal dodecahedron. Moreover, the correspondence in distance between the MD peak and the observed correlation length of -3.8 8, is very close. This indicates that the observed correlation length may refer to the bulky, pentagonal dodecahedral or clathrate-like structures invoked here. One of the authors of the correlation length study,46Ludwig, however, has recently developed a picture somewhat different from that originally published?8 This involves large, noninteracting, nearly spherical clusters of H20 molecules randomly arranged in a matrix of other hydrogen-bonded HzO molecules. The radius of gyration of these clusters is e 5 A, close to that of polyhedral voids. Ludwig’s new picture strengthens the present clathrate picture, and thus it strengthens the present model of impeded diffusion.

Pair Correlation Functions Experimentally determined pair correlation functions for highly supercooled water or low-density amorphous (LDA) ice might, at least in principle, provide a test of clathrate-based

Walrafen and Chu

(C-B) versus continuous random network (CRN) models. Both models involve a random, hydrogen-bonded,tetrahedral network having a wide range of bond angles and distances, but the C-B model may be viewed as having additional intermediate-range order involving bulky, pentagonal dodecahedra or related clathrate-like polyhedra, albeit distorted and temporal in the liquid case. Any unique features of these polyhedra might offer a small possibility for distinguishing between the C-B and CRN models. D ~ r has e ~discussed ~ the pair correlation functions for LDA ice which were obtained by Chowdhury et al.50-52from neutron scattering. Bellissent-Funel et a1.53-55 have obtained pair correlation functions from highly supercooled water emulsions at 1 atm as well as at high pressures using neutron scattering. Dore apparently tends to favor a CRN interpretation of the neutron scattering pair correlation functions for LDA ice. However, no workers have yet attempted a serious and exhaustive examination of the LDA pair correlations with the specific aim in mind of either confirming or eliminating the possibility of pentagonal rings and clathrate structures, i.e., the C-B model (but see ref 51). In the following discussion we describe the features of the C-B model which agree with pair correlation functions. The first to fifth nearest neighbor 0-0 distances, designated lNN, 2NN, ..., up to 5NN, may be calculated from published X-ray data for a solid clathrate.38 There are three lNN, six 2NN, six 3NN, three 4NN, and one 5NN distances in a pentagonal dodecahedron. These distances are 1NN = 2.84 A, 2NN = 4.57 A, 3NN = 6.35 8, 4NN = 7.27 A, and 5NN = 7.92 8,. The 5NN distance corresponds to opposite comers, both of which lie on the same 3-fold axis; that is, 7.92 A is the diameter of the concentric sphere. In addition, the bcc unit cell edge distance, which is the nearest, center-to-center, pentagonal dodecahedral distance, is 12.03 The two higher nextnearest center-to-center distances are 17 and -21 %,. The X-ray g(r) from LDA ice displays peaks at 2.8,4.5, -6.8, and ~ 8 . A, 4 plus considerable, broad, filling-in centered near 5.8 A.56 The filled-in region near 5.8 8, could easily include the 4.57 and 6.35 8, pentagonal dodecahedral features, and the broad 8.4 8, X-ray peak may readily encompass the 7.27 and 7.92 A features. The neutron dL(r) shows four broad peaks, at distances beyond the nearest neighbor OD and DD peaks, near -4.5, -8, -11-12,androughly 15-168,. Thewidthofthe-88,peak in dL(r) is large enough to accommodate all of the 6.35, 7.27, and 7.92 A, 3NN-5NN, pentagonal dodecahedral distances described above. The 11-12 and 15-16 %,peaksin &(r) can easily accommodate the 12.03 and 17 8, bcc pentagonal dodecahedral features also described above. Note that the structural correlation length (SCL) for LDA ice is -17 A?9-52 The neutron dL(r) function is reasonably well-fitted by the CRN model, as demonstrated by D ~ r e However, .~~ the agreements between the pentagonal dodecahedral distances and the features of the various pair correlation functions mean that the random C-B model is also a reasonable candidate for the structure of LDA ice and highly supercooled liquid water. Finally, it should be emphasized that the pentagonal dodecahedron is an archetype for bulky, but more highly disordered, interpenetrated, and side-sharing ring structures whose nearest and higher neighbor distances may also provide good fits to the pair correlation functions. However, the common feature of all such structures is that they contain hydrogen-bonded pentagonal rings, and it is the holes of these rings which constitute the key impediment to diffusion, according to the present model.

J. Phys. Chem., Vol. 99, No. 26, 1995 10643

Hydrogen-Bonded Clathrate-like Structures

Summary A new model has been proposed to explain the high activation energies and large negative activation volumes of very highly supercooled liquid water. The first energetic step in this model involves the total freeing of an H20 molecule from its restraining hydrogen bonds to allow for flow. This freed H20 is then presented with two choices. Namely, it can undergo free diffusion if hexagonal and larger rings are present in its immediate vicinity, or, if not, it must force itself through smaller holes, such as pentagonal ring holes, if it is to flow. This latter situation, which is the second energetic step, requires a very high, additional activation energy. But, whether the molecule more often encounters pentagonal rings than the larger, hexagonal rings depends upon two joint conditions: (1) a very low temperature plus ( 2 ) a very low pressure. Highly supercooled water may be viewed as a flexible sponge, with only cwo hole sizes, big and little, distributed randomly, through which “hard spheres”, namely, the freed H20 molecules, being only of the big size, either pass readily or must be strongly forced, thus stretching the little holes, before passage can occur. The concentration of the little holes increases, at the expense of the big holes, with decreasing temperature and decreasing pressure, which means that correspondingly more energy is needed for passage. However, the energy needed to produce the “hard spheres”, Le., to free the H20 molecules by breaking the hydrogen bonds, is essentially independent of temperature and pressure. Hence, diffusion of H20 molecules always requires a minimum of roughly 5 kcaVmol H20, but it may require much more energy, in addition, as the temperature and pressure of supercooled water both decrease, to 228 K, and from very high to 1 atm pressure. The sponge acts like a temperatureand pressure-dependent resistance, thus requiring variable amounts of energy for flow, but a minimum constant energy is always required to free the water molecules. The negative activation volumes arise when the cavities in the sponge are filled during the activation step, thus producing a momentary densification. Acknowledgment. We are grateful to Dr. D. C. Douglass, AT&T Bell Laboratories, to Professor C. M. Sorenson, Kansas State University, and to Professor E. Grunwald, Brandeis University, for reading an early version of the manuscript and for making insightful comments. We also thank Dr.D. A. McKeown for making calculations related to pentagonal dodecahedral distances in the bcc unit cell of a clathrate hydrate. This work was supported by contracts from the Office of Naval Research. References and Notes (1) Halfpap, B. L.; Sorensen, C. M. J . Chem. Phys. 1982, 77, 466. (2) Wada, G.; Umeda, S . Bull. Chem. SOC.Jpn. 1962, 35, 646. (3) McDonald, D. D.; Maclean, A.; Hyne, J. B. J. Solution Chem. 1W9, 8, 97. (4) Franks. F.: Reid, D. S . Water, A Comurehensive Treatise; Plenum: New‘ York, 1973; Chapter 5. (5) Scheiber, E.; Gutmann, G. Monatsh. Chem. 1993, 124, 277. (6) Anisimov, M. A,; Zaugol’nikova, N. S . ; Ovodova, G. I.; Ovodova, T. M.; Seifer, A. L. JETP Lett. 1975, 21, 220. (7) Sorensen, C. M. J. Chem. Phys. 1983, 79, 1455. (8) Euliss, G. W.; Sorensen, C. M. J . Chem. Phys. 1984, 80,4767. (9) Stillinger, F. H. Science 1980, 209, 451. (10) Speedy, R. J.; Madura, J. D.; Jorgensen, W. L. J. Phys. Chem. 1987, 91, 909. (11) Geiger, A.; Mausbach, P.; Schnitker, J. Water and Aqueous Solutions; Hilger: Bristol, 1986; Chapter 2. (12) -, Davidson. D. W. Water. A ComDrehensive Treatise: Plenum: New York, 1973; Chapter 3. (13) Sloan, E. D.. Jr. Clathrate Hydrates of Natural Gases; Dekker: New York, 1990. (14) Sorensen, C. M. Proceedings of 12th International Conference on the Properties of Water and Steam, Orlando, FL, September 1994. \

~

(15) Detailed examination (this work) of the viscosity data for supercooled liquid water reported by Osipov et all6 indicates that the viscosity data are systematically and increasingly too high for temperatures below about -30 “C. (16) Osipov, Yu. A,; Zheleznyi, B. V.; Bondarenko, N. F. Zh.Fiz. Khim. 1977, 51, 1264. 1 atm viscosity data, water, 0 to -35 “C. (17) Walrafen, G. E.; Chu, Y . C. J . Phys. Chem. 1992, 96, 3840. (18) Walrafen, G. E.; Chu, Y. C, Physica A 1994, 206, 93. (19) Walrafen, G. E. Encyclopedia of Earth System Science; Academic Press: Orlando, FL, 1992. (20) Bett, K. E.; Cappi, J. B. Nature 1965, 207, 620. (21) Angel], C. A. Water, A Comprehensive Treatise; Plenum: New York, 1982; Chapter 1. (22) WeingWner, H. Z. Phys. Chem. (Frankfurt) 1982, 132, 129. (23) Onsager, L.; Runnels, L. K. Proc. Nat. Acad. Sci. 1963, 50, 208. (24) Walrafen, G. E.; Chu, Y. C.; Piermarini, G. J. JRDC Forum on Multi-Disciplinary Researches, Kyoto, Japan, March 13- 18, 1994. (25) Young, T. F.; Westerdahl, R. P. ARL 135, Office of Aerospace Research, US.Air Force; 1961. (26) Walrafen, G. E.; Hokmabadi, M. S . ; Krishnan, P. S . ; Guha, S . J. Chem. Phys. 1983, 79, 3609. (27) Walrafen, G. E.; Chu, Y. C. J. Phys. Chem. 1991, 95, 8909. (28) Walrafen, G. E.; Blatz, L. A. J. Chem. Phys. 1972, 56, 4216. (29) Worley, J. D.; Klotz, I. M. J . Chem. Phys. 1966, 45, 2868. (30) Walrafen, G.E.; Fisher, M. R.; Hokmabadi, M. S . ; Yang, W.-H. J . Chem. Phys. 1986, 85, 6970. (31) A factor of 1.067 is needed, according to ref 21, to convert D data from ref 34 to D data from ref 21. 1 atm ln(6) values were fitted by cubics in (llr), A BIT C / p iD / F ; (Figure 1, viscosity) A = 202.573, B = -166 984, C = 4.60849 x lo7, and D = -4.30453 x lo9, and (Figure 2, self-diffusion) A = 129.524, E = -104 664, C = 2.84834 x lo7,and D = -2.65672 x lo9. The low-temperature viscosity and self-diffusion data, refs 16, 21, and 34, are accurate only to a few percent. The six figures used in eqs 1 and 2, and elsewhere, are not significant figures; they are used because the calculations involved differences. Distances computed using eqs 1 and 2 were always rounded off to three significant figures. (32) Kaye, G. W. C.; Laby, T. H. Tables of Physical and Chemical Constants; Longman: London, 1986; 1 atm, from 5 to 40 “C. (33) Haar, L.; Gallagher, J. S . ; Kell, G. S. NBSNRC Steam Tables; Hemisphere: New York, 1984. Isochoric viscosity estimates for supercooled water to 238 K were obtained from a NIST, H20SPIN.EXE program supplied by J. S . Gallagher. (34) Gillen, K. T.; Douglass, D. C.; Hoch, M. J. R. J. Chem. Phys. 1972, 57, 5177. (35) GiguBre, P. A. J. Chem. Phys. 1987, 87, 4835. (36) Narten, A. H.; Levy, H. A. Water, A Comprehensive Treatise; Plenum: New York, 1972; Chapter 8. (37) Walrafen, G. E.; Abebe, M.; Mauer, F. A.; Block, S . ; Piermarini, G. J.; Munro, R. G. J . Chem. Phys. 1982, 77, 2166. (38) McMullan, R. K.; Jeffrey, G. A. J . Chem. Phys. 1965, 42, 2725. (39) Pauling, L. Hydrogen Bonding; Pergamon: London, 1959; Chapter 1. (40) Lawson, A. W.; Hughes, A. J. High Pressure Physics and Chemistry; Academic: New York, 1963; Chapter 4. (41) Speedy, R. J. J. Phys. Chem. 1987, 91, 3354. (42) Chu, Y. C.; Walrafen, G. E. Hydrogen Bond Networks; Kluwer Academic: Dordrecht, 1994; Chapter 17. (43) Walrafen, G. E. J. Solution Chem. 1973, 2, 159. (44) Abebe, M.; Walrafen, G. E. J. Chem. Phys. 1979, 71, 4167. (45) Walker, W. A. Thermodynamic Properties of Water to One Kilobar. NOLTR66-217; Explosion Dynamics Division, Explosions Research Department, U.S. Naval Ordinance Laboratory: White Oak, MD, Jan 30, 1967. (46) Xie, Y.; Ludwig, K. F., Jr.; Morales, G.; Hare, D. E.; Sorensen, C. S . Phys. Rev. Lett. 1994, 77, 2166. (47) Speedy, R. J. J. Phys. Chem. 1984,88,3364. Speedy, R. J.; Mezei, M. J . Phys. Chem. 1985, 89, 171. (48) Ludwig, K. F. Private discussions. (49) Dore, J. C. Water and Aqueous Solutions, Colston Papers, No. 37; Adam Hilger: Bristol, 1986. (50) Chowdhury, M. R.; Dore, J. C.; Chieux, P. J . Non-Cryst. Solids 1981, 43, 267. (51) Chowdhury, M. R.; Dore, J. C.; Montague, D. G. J. Phys. Chem. 1983, 87, 4037. (52) Chowdhury, M. R.; Dore, J. C.; Wenzel, J. T. J . Non-Cryst. Solids 1982, 53, 247. (53) Bellissent-Funel, M.-C.; Teixeira, J.; Bosio, L.; Dore, J. C. J . Phys. Condens. Matter 1989, I , 7123. (54) Bellissent-Funel, M.-C. Hydrogen-Bonded Liquids; Kluwer Academic: Dordrecht, 1988; Chapter 7. (55) Bellissent-Funel, M.-C.; Bosio, L. J. Chem. Phys., accepted. (56) Narten, A. H.; Venkatesh, C. A.; Rice, S . A. J. Chem. Phys. 1976, 64, 1106.

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