There are very few things that modern science does not understand. One of them is consciousness; the other is water.1 In the case of consciousness, the hard problem is designing good experiments; in the case of water, finding a theory that explains its properties.
Thermodynamic theories assume that any substance may be characterized by its internal energy U(S,V), where S and V are extensive variables, S is a measure of the number of accessible quantum states at a given temperature T, and V is a measure of the spatial extension of the system at a given pressure p. By the first law of thermodynamics dU(S,V) = T·dS – p·dV. Every substance has an equation of state (EoS) p = kBT·[1/V + B2(T)/V2 + …], where kB = 0.0138 zJ·K–1 is a universal constant. B2(T) is negative at low temperature and positive at high temperatures. At B2(T) = 0, attractive and repulsive forces are in balance. For an ideal gas, B2(T) = 0; for a system of hard spheres of diameter σ, B2(T) = 2π·σ3/3; for a van der Waals gas, B2(T) = b – a/kBT, with b and a measuring the repulsive and attractive forces between molecules.
Let MW designate molecular weight. The substance displaying the greatest ratio TB/MW = 80(8) K·Da–1 is water (H2O), followed by hydrogen fluoride (HF), and ammonia (NH3). TB/MW thus corresponds to the density of cohesive energy. The fact that water is an elixir for life is perfectly justified from a thermodynamic point of view. In chemistry, such high densities are usually associated with abnormally high ebullition points.
A phase diagram in the plane represents the domains where a substance may exist as a solid, liquid, or gas. Domain boundaries are marked by coexistence lines. It is there that two phases coexist. Substances undergo first-order transitions at coexistence lines; the system absorbs or releases energy, giving rise to discontinuous changes in volume and density. The sublimation line is the locus of points where a solid coexists with a gas and may extend from zero to the triple-point temperature T3. In the case of water, this triple point is observed at T3 = 0.01°C = 273.16 K and p3 = 611 Pa = 0.006 atm. The melting line is the locus of points from T3 to an infinite temperature. These phases have different symmetries. Full rotational invariance in the liquid phase turns into discrete symmetries in the crystal phase. The condensation line is the locus of points where a gas coexists with a liquid and extends from T3 to a critical point C. Above C, only a single-phase homogenous system exists. In the case of water, this critical point is observed at TC = 373.946°C = 647.096 K and pC = 22.06 MPa = 217.7 atm. This critical point occurs as soon as two phases share the same symmetry group.
The spinodal line separates metastable from unstable states; it corresponds to the locus of points where (∂p/∂V)T = 0. The spinodal line is responsible for the occurrence of metastable states such as superheated, stretched, and supercooled liquids. Metastable states may spontaneously change into a more stable state by nucleation and growth. In nucleation, a critical microscopic embryo will be formed within the metastable region that is doomed to grow to a macroscopic size as time evolves. In the case of unstable homogeneous states, the activation barrier for nucleation disappears; small amplitude fluctuations are able to grow, leading to spinodal decomposition. The thermodynamic response functions of liquid water are measurable at ambient pressure –42°C (231 K) ≤ T ≤ 280°C (553 K).2 Water exists in a solid state at ambient pressure below T = –137°C (136 K), the glassy state becoming unstable relative to hexagonal ice above T = –123°C (150 K). A no-man’s-land exists for metastable states of water between 150 K and 231 K, a domain where spontaneous formation of hexagonal ice prevents measurements on liquid or glassy water.
Thermodynamic response functions, when taken over the observable metastable liquid domain, give rise to a number of mysterious anomalies. Observation reveals a U-shape curve, with a minimum or a maximum upon varying temperature, instead of the expected curve representing a monotonically increasing or decreasing function. Thermodynamic response functions are related to spontaneous fluctuations δX of the state variables (X = U, S, T, V, p, H, …) at equilibrium (dX = 0). Except for a conjugate pair (S, T) and (V, p), fluctuations involving an extensive variable (U, S, V, H), and an intensive variable (V, p), should not be correlated. Thus <(δT)·(δV)> = <(δS)·(δp)> = 0, with <(δS)·(δT)> = kB·T and <(δp)·(δV)> = –kB·T. Fluctuations of entropy and temperature are always correlated; fluctuations of pressure and volume are always anticorrelated, the correlation or anticorrelation increasing with temperature.
We face the first crucial anomaly when looking at the fluctuation correlations involving two extensive variables <(δV)·(δS)> = V·kBT·αP or two intensive variables V·<(δp)·(δT)> =·kBT2·γV. Here αP = (1/V)·(∂V/∂T)p is the isobaric thermal expansion coefficient measuring the degree to which a substance changes its volume ΔV after temperature variations ΔT at constant pressure p. The other response function γV = V·(∂p/∂U)V is the Grüneisen thermo-acoustic coefficient measuring changes in pressure Δp after internal energy variations ΔU at constant volume V. These two coefficients are not independent since γV·cp = αP·c2, where cp is the specific heat capacity at constant pressure and c is the speed of sound. For a normal liquid, one expects that both αP and γV take only positive values. This is not the case with liquid water where αP = γV = 0 around T ≈ 4°C. Lowering the temperature below 4°C leads to an increase in volume V or to a decrease in density ρ = m/V. This is the reason that ice floats.
Strange. A decrease in temperature suggests a decrease in the average speed of molecules in water; and this, in turn, suggests a decrease in volume as molecules are attracted to one another. With water, the opposite is observed. A decrease in speed induces repulsion between water molecules and not attraction.
A second anomaly concerns the isobaric heat capacity Cp = T·(∂S/∂T)p = <(δS)2>/kB coordinating a change in entropy ΔS and a change in temperature ΔT at constant pressure p. Writing H = U + p·V yields Cp = (∂H/∂T)p = <(δH)2>/kB·T2, where Cp measures the mean square fluctuations in entropy S or in enthalpy H. In a normal substance, when temperature is lowered, one expects lower fluctuations in entropy or enthalpy, and thus a decrease in Cp. This is not what is observed in water below 37°C = 310 K: the lower the temperature, the higher Cp. This means that square fluctuations of entropy are minimized around T ≈ 37°C in water, a crucial temperature for biology.
And then there is the matter of isothermal compressibility κT = (1/V)·(∂V/∂p)T, which measures the change in volume ΔV as the pressure Δp is varied at constant temperature T. The requisite response function is a measure of the mean square fluctuations in volume <(δV)2> = V·kBT·κT and should decrease with temperature for a normal substance. Liquid water deviates from this behavior below 46°C (319 K), showing a large increase in κT after decreasing temperatures. This means that mean square fluctuations in volume are minimized around T ≈ 46°C in water, another crucial temperature for biology.
Yet another response function is the isochoric heat capacity, where CV·(δT)2 = kB·T2 and (δU)2 = kB·T2·CV. CV = T·(∂S/∂T)V = (∂U/∂T)V measures the degree to which a substance changes its entropy ΔS, or its internal energy ΔU, in response to a change in temperature ΔT at constant volume V. For a normal liquid, CV should increase as temperature is lowered. In water, this is observed above –13°C (260 K), but below this critical temperature, where mean square fluctuations in temperature or internal energy are minimized, CV decreases instead of increasing. Since κT·(Cp – CV) = T·V·αP2, this anomaly may be viewed as the consequence of the already anomalous behavior of Cp, αP, and κT.
The same consideration apply to mean square fluctuations in pressure, where V·<(δp)2>·κS = kB·T, with κS = (1/V)·(∂V/∂p)S measures a change in its volume ΔV as a function of a change in pressure Δp at constant entropy S. The speed of sound c is governed by ρ·c2·κS = 1. For a normal liquid, κS is expected to decrease as temperatures decrease. After all, fluctuations in pressure should become greater as V decreases. Such behavior is indeed observed in water above 64°C (337 K), but below this critical temperature, where mean square fluctuations of pressure are maximized, κS increases instead of decreasing. Given κS·Cp = κT·CV, this anomaly may be viewed as the consequence of the anomalous behavior of CV, CP, and κT.
What makes water unique for biology is this wide palette of minima or maxima between –42°C and 100°C. It remains a theoretical challenge to explain why large mean square fluctuations in entropy, volume, and temperature, and small mean square fluctuations in pressure are observed as the average speed of water molecules decreases.
Chemists are not without clues. A first: all thermodynamic response functions seem to diverge toward a singular temperature of about 228 K at ambient pressure. A second: all water anomalies are linked to the existence of hydrogen bonds (H-bonds) between water molecules. The idea of performing molecular dynamics simulations in order to predict phase diagrams is, if not inevitable, then, at least, natural.3 Under simulation, H-bonds are characterized by a van der Waals component ε describing the universal attraction between molecules, a directional component J describing the decrease in potential energy specific to hydrogen bonding (two-body interactions), and a component Jc describing the cooperative character of H-bonds (three-body interactions).
At Jc = 0, the ensuing scenario is called singularity free (SF). Water form a homogeneous liquid at all temperatures and pressure. Response functions do not diverge.4
If Jc ≤ J/2, the second-critical-point (SCP) scenario emerges. Under the SCP scenario, water should exist in two different liquid forms at low temperatures and high pressure. A coexistence line ends at a second critical point C' (liquid–liquid critical point, or LLCP) at positive pressure.5 Here, a Widom line is thought to emanate from C' into the one-phase region. The liquid–vapor spinodal remains monotonic. Response functions anomalies arise in a neighborhood of LLCP reaching a peak near the Widom line.
When J/2 ≤ Jc ≤ 0.3×ε + 0.36×J, the LLCP is located at negative pressure giving rise to the critical-point-free (CPF) scenario.6 Here the line of density maxima keeps its negative slope while a second spinodal line associated to the liquid–liquid coexistence line meets the liquid–vapor spinodal at negative pressure. Under such conditions, response functions are expected to diverge close to this liquid–liquid spinodal with no minima or maxima.
And, finally, if Jc ≥ 0.3×ε + 0.36×J, the liquid–liquid coexistence line reaches the liquid–vapor spinodal; reentrant behavior follows, leading to a continuous line that bounds the metastable superheated, stretched, and super-cooled states of water.7 In this stability-limit conjecture (SLC), the reentrant liquid–vapor spinodal line and the line of density maximum meet at some negative pressure: response functions are expected to diverge.
Experimental data suggest that ε ≈ 9 zJ, J ≈ 20 zJ, and Jc ≈ 2 zJ.8 Jc ≈ 2 zJ < J/2 ≈ 10 zJ, a result that favors the SCP scenario and yields the prediction that TC' ≈ 235 K and pC' ≈ 200 MPa.9 Because values are in the no-man’s-land of metastable states, experimental confirmation was long delayed.
The experiment reported by Kyung Hwan Kim et al. was designed to rebut the SF, SLC, and CPF scenarios and experimentally prove the existence of an LLCP in supercooled water.10 In order to avoid nucleation of hexagonal ice, it was mandatory to use very pure and very small samples of water. Kim et al. used femtosecond X-ray laser pulses with photon energy of 5.5 keV to probe micrometer-sized water droplets that were cooled down to 227 K in vacuum. It was thus possible to measure the static structure factor S(q) of supercooled liquid water in the no-man’s-land (160 K–232 K) supposed to shelter the LLCP.
As the static structure factor S(q) is the Fourier transform in momentum, Kim et al. also decided to use a large area detector having the capability to cover an extended q-range. This allowed recording simultaneously the wide-angle X-ray scattering spectrum (WAXS) around 1.85 Å–1 and the small-angle X-ray scattering (SAXS) regions down to 0.12 Å–1. The WAXS part of the spectrum made it possible to probe the local order around water molecules related to the growth of tetrahedral structures; the SAXS part of the spectrum is sensitive to density fluctuations. It was thus possible to detect the occurrence of Bragg peaks corresponding to small hexagonal ice crystals.
Results in the low-q region showed an increase in S(q) upon cooling and a passing trough reaching a maximum at 229±1 K for H2O and 234±1 K for D2O. In the WAXS part of the spectrum, Kim et al. observed a continuous shift to lower q-values with decreasing temperature, which could be associated with growth of tetrahedral structures. The temperatures at which the liquid underwent the most rapid change were again 229±1 K for H2O, and 235±1 K for D2O. The tricky job was to extract from these data the isothermal compressibility κT as well as the correlation length ξ, corresponding to the damping factor in the asymptotic decay of the pair-correlation function of the system.
For the isothermal compressibility κT, an extrapolation at q = 0 was needed since S(0) = kBT·(ρ/m)·κT, where kB is Boltzmann’s constant, T the absolute temperature, ρ the liquid density, and m the molecular mass. To retrieve the correlation length ξ, the anomalous excess scattering at low-q was fitted to a Lorentzian shape leading to a fitted S(q) that could be extrapolated at q = 0 after a suitable extrapolation of the density below 240 K. A maximum of correlation length was thus observed at 229±1 K for H2O and 233±1 K for D2O. Since the maximum of ξ is around 4 Å for both isotopes, it follows an average fluctuation length scale d ≈ 4,5·ξ ≈ 2 nm. As the Widom line emanating from a critical point may be defined as the locus of correlation length maxima in the pressure–temperature (p, T) plane, it then follows that the existence of such a Widom line is experimentally proved in supercooled water, validating in a definitive way the SCP scenario with its LLCP.
For the isothermal compressibility computed from S(0), Kim et al. observed maxima at 229±1 K for H2O and 233±1 K for D2O instead of a divergence towards infinity, thus validating SF and SCP scenarios. They suggested that the SF scenario was associated with a null cooperative component of the H-bonding scheme, a conjecture not consistent with a rapid increase of tetrahedral order observed in the WAXS part of the spectrum. Another reason for discarding the SF scenario: the observation of a discontinuous transition between liquid states above the glass transition, an experimental fact not consistent with the continuous transitions predicted at all temperatures and pressures under the SF scenario.11 Using molecular dynamics simulations, Kim et al. showed that the maxima increase with pressure up to 170 MPa, where isothermal compressibility reaches an extremely high value, one that is compatible with the existence of a LLCP at T ≈ 229 K and p ≈ 80 MPa for H2O and T ≈ 233 K and p ≈ 95 MPa for D2O.
Kim et al. seem to close a debate opened some 400 years ago by Galileo Galilei.12 The molecular description of high-density (ρ ≈ 1,18 g·cm–3) liquid water (HDW) and low-density (ρ ≈ 0,91 g·cm–3) liquid water (LDW) can be derived from spectroscopic data.13 Absorption of X-rays occurs at 535 eV and 537–538 eV for HDW and at 540–541 eV for LDW; X-ray emission from the 1b1 level occurs at 527 eV for HDW and at 526 eV for LDW. O–H bonds vibrations have been observed at 3,400 cm–1 for HDW and at 3,200 cm–1 for LDW. Collective modes deduced from optical Kerr responses were located at 180 cm–1 for HDW and at 225 cm–1 for LDW. And, finally, X-ray scattering shows water-water correlations at 3–4 Å for HDW, but correlations at 4.5 Å and 11 Å for LDW.
It follows that LDW has a more rigid tetrahedral structure based on linear H-bonds than HDW. Recent experiments have shown that a high-pressure phase ice VIII transforms to a low-density liquid under rapid and complete release of pressure at 140–165 K.14 X-ray scattering data suggests that the structure of this liquid was very similar to that of low-density amorphous ice, with no variation of the spectrum with temperature consistent with a fully developed tetrahedral network. By contrast, deeply supercooled water showed a spectrum that displayed strong temperature dependences. These changed continuously due to structural fluctuations between HDW and LDW, and caused water anomalies around the Widom line.
Above T = 320 K, water exhibits normal liquid behavior dominated by HDW, whereas at temperatures far below the Widom line, water may exhibit normal liquid behavior.
Does this mean that the saga of water is over? Obviously not. In order to analyze experimental data, approximations are mandatory. To prove the existence of a maximum of isothermal compressibility at T = 229 K, it is necessary to know the density of supercooled water in no-man’s-land.15 It is there that no measurements are possible. It is for this reason that experimental results involve an extrapolation from data recorded above T = 240 K.16 From a purely methodological point of view, the evidence reported by Kim et al. is rather weak. Extrapolation from one point towards another and unknown point are not as safe as interpolations between two known points.
There is also the problem of measuring accurately the temperature of a micrometer-sized water droplet. Kim et al. adjusted the distance between the dispenser nozzle and the interaction region with x-ray pulses, which allowed for varying the temperature by changing the cooling-time duration. The temperature was not experimentally measured. It was computed using a ballistic evaporation model, which has been calibrated by molecular dynamic simulations. Having used guessed-at temperatures and densities for their analysis, Kim et al. might have been more cautious in their conclusions. The controversy between advocates of the CPF and SCP scenarios, and advocates of SCL and SF scenarios, is far from finished.
Just after the publication by Kim et al., measurements of the sound velocity in liquid water stretched to negative pressure indicated the existence of a line of maxima of isothermal compressibility along isobars.17 These experiments demonstrated that a bundle of extrema lines exist, as well, and that all converge toward the line of maximum density as pressure decreases. This proves the existence of a turning point in the line of maximum density, ruling out definitively the SLC and CPF scenarios.
And more. Aqueous solutions of hydrazinium trifluoroacetate permit supercooling down to the point of vitrification without destruction of liquid-state anomalies.18 In such ideal solutions, the second critical point is displaced toward lower pressures and temperatures in such a way that the ambient-temperature cooling leads the system to cross reversibly a liquid–liquid coexistence line without ice crystallization. These water-rich solutions have a local hydrogen bond structure surrounding a water molecule quite similar to that found in neat water at elevated pressure. If such experiments prove that the concept of an LLCP is perfectly sound, it remains open whether such an LLCP exists in pure water.
What pushes water molecules to form two liquids of different densities having quite different local structures and dynamics? This is not yet understood. Water anomalies reflect the properties of hydrogen bonding, and it is just the question why hydrogen bonds exist in the first place that remains unclear. This problem has already been treated in this journal, and the key point seems to be the capability of water molecules to undergo electronic self-excitations using vacuum virtual energy.19 Quantum field theory allowing direct coupling between matter and photons predicts that two different shapes could be expected for water molecules. When the water molecule is not excited by the vacuum’s virtual energy, it adopts an almost spherical shape that is perfectly suited for strong van der Waals interactions typical of HDW-type liquids. On the other hand, self-excitation toward Rydberg states localized onto the oxygen atom using the vacuum’s virtual energy leads to the tetrahedral structure typical of hydrogen bonding. It is this second tetrahedral shape that displays a strong cooperative behavior associated with quantum coherence typical of the other, LDW type of liquid.
