Thermodynamic mechanism of the density anomaly of liquid water

1. Introduction

Most liquids become monotonically denser when cooled from room temperature, but liquid water reaches its maximum density at approximately 4°C, below which it expands to become less dense as it is cooled further. To explain this density anomaly, several ideas have been proposed.

Röntgen [1] considered water to be a mixture of molecular complexes. Some of these complexes are less dense and ice-like, whereas others are denser, proper-water complexes. A change in the proportions of these complexes with temperature could explain the non-monotonic dependence of some water properties on these parameters. However, there is no explanations of what induces such a change in the proportions.

Bernal and Fowler [2] put forward the idea that a three-dimensional network exists in liquid water. They reasoned that at temperatures close to the melting point, this network is a mixture of two structures: one denser and quartz-like and the other less dense and tridymite-like (or ice-like). This was a concretization of Röntgen's idea of water as a mixture of ice-like and water-like particles.

The next version of the network model was formulated by Pople [3], who proposed that hydrogen bond bending is the reason for the greater density of water in comparison with ice. Bernal [4] returned to the problem of liquid water. He applied the notion of a random tetrahedral network to water. The concept of the random tetrahedral network was developed quantitatively by Sceats and Rice [5–7].

Two other groups of models were developed and discussed in the 1950 and 1960s, the first being the model put forward by Samoilov [8], who proposed that water retains an ice-like network, and cavities in this network are partially filled by water molecules. The network itself is perturbed by thermal motion. According to this model, ice melting must be accompanied by molecules falling into the cavities; this is why the density increases. Unlike Samoilov, [9] supposed that the network is not ice-like, but gas-hydrate-like. These models can be described as clathrate because their main feature is the filling of cavities.

The second group of models is a direct development of Röntgen's two-state idea. These models suggest that water is a microheterogeneous system consisting of two microphases: one consisting of fragments of a hydrogen-bonded tetrahedral framework and another in which the molecules are not involved in hydrogen bonds. The most popular among these models is that proposed by Némety and Scheraga [10]. They considered that hydrogen-bonded water clusters are floating in a “sea” of non-bonded molecules. Both microphases of the classical two-state models came to be regarded as containing hydrogen bonds; the possibility of hydrogen-bond formation between the cavity and framework molecules began to be admitted by proponents of clathrate models. Thus, a picture of water as a continuous three-dimensional network predominated on the eve of the computer simulation era. The impossibility of Némety and Scheraga type models was ultimately demonstrated when models based on percolation theory were applied to water [11]. The three dimensional network was accepted by practically all scientists involved in the study of water [12].

In addition, Monte Carlo and molecular dynamics calculations have been performed for liquid water by using the BNS, MCY, and ST2 potential functions [13], and by using TIP3P [14], TIP4P [15], TIP5P [16], SPC [17] and SPCE [18]. There is an enormous literature on the properties of water. Stanley et al. [19] have presented an overview of recent research that applies ideas of statistical mechanics to try to better understand the statics and in particular the dynamic puzzles regarding liquid water. They have discussed recent molecular dynamics simulations using TIP5P, which is closer to real water than previously-proposed classical pairwise additive potentials, and have shown some interesting results. Guillot [20] has appraised in his review paper what have been accomplished during the last three decades and what remains to be improved: one has a taste of incompletion if one considers that not one water model is available in the literature that is able to accurately reproduce all the properties of water. Despite great efforts to improve this situation, there has been little significant progress. Molinero and Moore [21] developed a coarse-grained model of water called “mW” that essentially models water as an atom with tetrahedrality intermediate between carbon and silicon. The mW model mimics the hydrogen-bonded structure of water through the introduction of a non-bonded angular dependent term that encourages tetrahedral configuration. This model reproduces the experimentally obtained density-temperature curve at 1 bar better than any other water model presented so far.

Although realistic models can provide a description of the density maximum and reproduce a number of the other anomalies of water, they make the direct causes obscure and difficult to elucidate because the models should include a number of properties of water, and some of them would not be the immediate causes. Therefore, it is impossible to capture the essential physics via the study of realistic models that include the miscellaneous properties of water, even if they can reproduce all of the anomalies of water. To capture the physics underlying the density anomaly, one should use simplified models that include only the properties crucial to explaining the density anomaly of water. Such a traditional method is frequently used in many sciences. It is important to clear up the mysteries one by one, accumulate knowledge, and develop ideas. It is impossible to illuminate the direct causes of all the anomalies of water simultaneously.

To clarify the physics underlying the density anomaly, a number of simplified (core-softened) potentials have been presented by many different authors. Those can be divided into two groups; one composed of a hard-core (HC) plus a purely repulsive tail, and another composed of an HC plus a soft repulsion and an attraction. Models from both groups have presented us with many fruitful results regarding the anomalous behavior of liquids. Although the shapes of the potential tails of both groups are very different, they exhibit water-like anomalous behaviors (see e.g., [22–32] and references quoted therein). It has recently been shown that even weakly softened potentials are able to yield water-like anomalies, i.e., in this case anomalies arise without ever invoking (as usually done) the interplay between two repulsive length scales [33, 34]. However, the simplified potential models presented so far have not been shown to reproduce quantitatively the experimental behaviors of liquid water sufficiently well. We think that this prevents one from obtaining a conclusive explanation of the density anomaly because the thermodynamic properties of liquids depend strongly on the interparticle interactions. Furthermore, numerical results obtained by using a limited number of particles via a Monte Carlo method or molecular dynamics simulation occasionally show a mixture of less dense and denser regions of particles, and it is claimed that a change in the proportions of these regions causes the density anomaly. However, there are other claims that such a mixture is never observed. At any rate, it is not known what thermodynamic mechanism causes a change in the proportions and results in negative thermal expansion. Therefore, a sophisticated method that reproduces the experimental data with high accuracy without using such computer simulations can bring great progress to the solution of this problem.

A self-consistent Ornstein-Zernike approximation (SCOZA) is frequently used as one of the sophisticated methods mentioned above. The SCOZA is known to describe the overall thermodynamics very well and provide a remarkably accurate critical point and coexistence curve. This scheme is entirely self-contained, that is, no supplementary thermodynamic or other input is necessary. Up to now, the SCOZA has been applied to the Yukawa, Sogami-Ise, square-well, triangle-well, and screened-power series potentials, and many fruitful results have been obtained (see [35–40] and references quoted therein). Because any smooth potential tail can be approximated with sufficient accuracy by Yukawa terms, the SCOZA with Yukawa terms is applicable to a variety of liquids.

In our previous paper [41], we determined many functional representations of interparticle interactions between water molecules, all of which reproduce the experimentally measured density-temperature relation at 1 bar with an accuracy better than that obtained by previous models. Numerous similar descriptions of pair interactions will be discovered in the coming years, and these will help us to understand why solid water has polymorphic structures and why liquid water has a large number of anomalies. We mentioned earlier that nearly all of the ideas put forward so far do not explain the immediate causes of the density anomaly of liquid water.

Our present aim is to find the force that appears attractive above 4°C to condense water, but seems repulsive to expand water below 4°C, with reducing temperature. Such a force (hereafter for simplicity referred to as the “anomaly force”) is the immediate cause of the density anomaly of liquid water [41]. Our study is blind to freezing and, more generally, to the solid phases of the model system (see e.g., [42]), but it gives significant insight into the thermodynamic properties of water in these phases. We apply the SCOZA to our core-softened models with an HC repulsion and a tail that is composed of a soft-repulsion and an attraction. We present a new phase diagram (or thermodynamic mechanism) that elucidates the anomalous behavior of liquid water: unusual negative thermal expansion is observed when the value of the potential tail at the hard-core contact is in some proper range that depends on the shape of the attraction; otherwise negative thermal expansion is never observed. It is also found that the behavior of excess internal energy plays an essential role in the density anomaly and that the behavior is mainly determined by the value of the soft-repulsive potential, especially near the hard-core contact, depending on the attraction. Our results show that most of the ideas put forward up to now, such as the second critical point hypothesis, simple two-state model, liquid-liquid phase transition model, clathrate model, network model, hydrogen bonding, orientation-dependent potential, and so on, do not explain the direct causes of the density anomaly of liquid water, as we discussed in our previous article [41].

In the next section, we consider nine cases of pair interactions to illuminate what characteristics of the interactions cause the density anomaly. In Section 3, we clarify the thermodynamic mechanisms that induce the anomaly, and finally, we present conclusions and discussion in Section 4.

2. Models

We consider a fluid of spherical particles with a pair potential given by an HC repulsion and some tail ϕ(r), which is expressed as

$\begin{array}{cc}\varphi (r)=\{\begin{array}{l}\infty r<1\hfill \\ -{\displaystyle \sum _{n=2}^N{a}_n}\frac{\exp [-z_n(r-1)]}{r}r\ge 1\hfill \end{array}& (1)\end{array}$

where N is an arbitrary integer and z_n and a_n are arbitrary constants.

To clarify the physics underlying the density anomaly, we consider nine cases of tails ϕ_i(r) (i = 1–9). The model parameters z_n and a_n are listed in Tables 1, 2. Figure 1 shows these potentials except for ϕ₆(r). The tail ϕ₆(r) is the same as ϕ₅(r) in Yasutomi [41]. The diameter of the hard-core σ_u is used as the unit of length and the depth of the potential ε_u as the unit of energy.

TABLE 1

Table 1. Parameters for ϕ₁(r), ϕ₂(r), …, and ϕ₆(r).

TABLE 2

Table 2. Parameters for ϕ₇(r), ϕ₈(r) and ϕ₉(r).

FIGURE 1

Figure 1. Interaction potentials. Thick full lines: ϕ₁(r), ϕ₂(r), …, and ϕ₅(r) from the top to the bottom at r = 1 of the figure, respectively. Thin full line: ϕ₇(r). Dotted line: ϕ₈(r). Dot-dash line: ϕ₉(r). Full and open circles show ϕ₁(1⁺), ϕ₉(1⁺), ϕ₈(1⁺), ϕ₂(1⁺), ϕ₃(1⁺) = ϕ₇(1⁺), ϕ₄(1⁺), and ϕ₅(1⁺) from the top to the bottom of the figure. Here, 1⁺ = lim_ε→0(1 + |ε|).

The thermodynamic properties of the models are obtained by using the SCOZA. We express the physical quantities by the same symbols used in Yasutomi [41], and the numerical computations are performed by using the same method described in Pini et al. [35, 41]. Tables 3, 4 show the density grid Δρ, the temperature grid Δβ, the density ρ₀ at which we made use of the so-called high-temperature approximation [43], and β_f; numerical computations are performed in the range of 0 < β < β_f.

TABLE 3

Table 3. Parameters used in numerical computations for Models 1–6.

TABLE 4

Table 4. Parameters used in numerical computations for Models 7–9.

We first present the numerical results for the HC-ϕ₂(r) liquid (hereafter referred to as “Model ϕ_i” (i = 2) for simplicity) as a representative of the models that show the density anomaly. The pressure isotherms at kT_c = 1.235, kT₁ = 1.135, kT₂ = 0.925, kT₃ = 0.735, and kT₄ = 0.543 of Model ϕ₂ are shown by thin full lines in the P−ρ plane in Figure 2. The thick full line shows the binodal line, which has a critical point (ρ_c, P_c) = (0.2464, 0.0974) exhibited by the full circle and a maximum density ρ = 0.5633. The dot-dash line shows the spinodal curve, which has the largest density ρ = 0.3615 and a minimum pressure of P = −0.5196. It is found that for similar to normal substances, the pressure isotherms shift to lower pressure as the temperature is reduced in the range of T ≥ T₄. The density ρ_i(P = const, T_i) in the liquid phase can be obtained from the intersection of the pressure isotherm at T_i with a line of constant P: for example, the dotted line of P = 0.063. Figure 2 shows that the density increases with reducing temperature at a specified P in the liquid phase, as in normal substances. We note here that the intersection point of each pressure isotherm in the liquid phase with the binodal line indicates the boiling point.

FIGURE 2

Figure 2. Pressure isotherms of Model ϕ₂ in the P−ρ plane: Thin full lines: pressure isotherms at kT_c = 1.235, kT₁ = 1.135, kT₂ = 0.925, kT₃ = 0.735, and kT₄ = 0.543 from the top to the bottom of the figure, respectively. Thick full line: binodal curve, which has a critical point (full circle) of (ρ_c, P_c) = (0.2464, 0.0974) and a maximum density ρ = 0.5633. Dot-dash line: spinodal curve, which has the largest density ρ = 0.3615, and a minimum pressure P = −0.5196. Dotted line: P = 0.063.

Figure 3 is the same as Figure 2 but at kT₅ = 0.390, kT₆ = 0.260, kT₇ = 0.198, kT₈ = 0.177, and kT₉ = 0.160. The binodal line (thick full line) in the liquid phase has a maximum density ρ = 0.5633 and the lowest density ρ = 0.3615, which ends at the spinodal line. The spinodal line (dot-dash line: ρ = 0.3615) in the liquid phase has a minimum pressure of P = −0.5196. All isotherms and the binodal overlap with the spinodal on the scale of the figure for gas branches. Contrary to Figure 2, the pressure isotherms for liquid branches shift toward higher pressure with reducing temperature. As a result, the liquid expands to become less dense as it is cooled further in the range of kT < kT₄ at a specified pressure. The behavior of the isothermal pressure-density curves shown in Figures 2, 3 is just what one can infer from the shape of the experimentally measured isobaric density-temperature curve of liquid water, which is exhibited in Figure 2 in Yasutomi [41] by open circles.

FIGURE 3

Figure 3. The same as Figure 2 but for kT₅ = 0.390, kT₆ = 0.260, kT₇ = 0.198, kT₈ = 0.177, and kT₉ = 0.160 from the bottom to the top of the figure, respectively. The binodal line (thick full line) in the liquid phase has a maximum density ρ = 0.5633 and ends at the spinodal line. The spinodal line (dot-dash line : ρ = 0.3615) in the liquid phase has a minimum pressure P = −0.5196. All isotherms and the binodal curve overlap with the spinodal curve on the scale of the figure for gas branches.

The density-temperature curves for liquid branches at the critical pressure P_c = 0.0974 (dotted line) and P = 0.0341 (dashed line) in Figure 4 both show density anomalies. The intersection point of the dashed line with the binodal curve marked by the full circle indicates the boiling point at P = 0.0341.

FIGURE 4

Figure 4. Density-temperature curves for liquid phase at specified pressures for Model ϕ₂. Dotted and dashed lines: ρ−T curves at P_c = 0.0974 and P = 0.0341, respectively. Full line: binodal curve. Dot-dash line: spinodal curve. Full circle: the boiling point at P = 0.0341.

We next study what characteristics of the interparticle interaction cause the density anomalies or, equivalently, the pressure anomalies. Figure 5 shows the density as a function of temperature for Models ϕ_i (i = 2–5) at P_c = 0.0974, 0.1216, 0.1173, and 0.1025, respectively, by full lines. The isobaric ρ−T curves depend so weakly on the pressure that they are indistinguishable or very close to each other on the scale of the figure in the range of 0 < P < P_c.

FIGURE 5

Figure 5. Density-temperature curves at P_c. Full lines are for Models ϕ_i (i = 2–5), and dashed lines are for ϕ_i (i = 9-7) from the bottom to the top of the figure, respectively. P_c = 0.0974, 0.1216, 0.1173, 0.1025 for Models ϕ_i (i = 2–5), and P_c = 0.384, 0.441, 0.571 for ϕ_i (i = 9–7), respectively. Full circles show experimental densities of liquid water at P = 1 bar.

All models except for Models ϕ₁ and ϕ₅ exhibit density anomalies, which are qualitatively induced by the same thermodynamic mechanism as that already shown for Model ϕ₂. Model ϕ₁ has only a gas phase and no liquid phase. In Model ϕ₅, the density increases monotonically with reducing temperature and never exhibits negative thermal expansion.

Next, we examine the thermodynamic properties of Models ϕ_i (i = 7–9), whose attractive potentials are shifted to values lower than those of Models ϕ_i (i = 1–5) to study what the effects of such attractions are on the thermodynamic properties of the models as shown in Figure 1 by the thin full, dotted, and dot-dash lines, respectively. In Figure 5, the density as a function of temperature is shown for Models ϕ₇, ϕ₈, and ϕ₉ by dashed lines at P_c = 0.571, 0.441, and 0.384, respectively. The ρ−T curves depend so weakly on the pressure that they are indistinguishable on the scale of the figure in the range of 0 < P < P_c. Models ϕ₈ and ϕ₉ show density anomalies, which are caused by the same qualitative mechanism as that in Model ϕ₂. To the contrary, Model ϕ₇ exhibits no density anomaly. As seen from Figure 1, both ϕ₇ and ϕ₃ have nearly the same soft repulsion, but the attractive part of ϕ₇ is shifted to a value lower than that of Model ϕ₃, which exhibits the density anomaly. It is clear that the difference in thermodynamic behavior between Models ϕ₃ and ϕ₇ is ascribed to the difference between the attractive parts of ϕ₇ and ϕ₃.

Thus, we can conclude that negative thermal expansion is exhibited when the value of ϕ(1⁺) of the potential tail at the hard-core contact is in some proper range that depends on the shape of the attraction; otherwise the negative thermal expansion is never observed. Here, 1⁺ = lim_ε→0(1 + |ε|).

3. Thermodynamic Mechanism Underlying the Density Anomaly

The derivative of pressure P with respect to temperature T at a fixed density ρ is written as

$\begin{array}{cc}\alpha \equiv \frac{\partial P}{\partial (kT)}=\frac{1}{kT}\left(u-\rho \frac{\partial u}{\partial \rho }+P\right)& (2)\end{array}$

where k is the Boltzmann constant and u is the excess internal energy per unit volume defined by

$\begin{array}{cc}u=2\pi {\rho }^2{\displaystyle {\int }_1^{\infty }d}r{r}^2\varphi (r)g(r)& (3)\end{array}$

where g(r) is the distribution function. When the coefficient α is positive, the pressure increases with temperature as in usual substances as shown in Figure 2. To the contrary, when α < 0, the pressure increases with reducing temperature, and the pressure anomaly, or equivalently the density anomaly, occurs as shown in Figures 3, 4. Equation (2) shows that competition among u, ρ(∂u/∂ρ), and P determines the sign of α. The value of the pressure P is always positive and contributes to positive α. In contrast, u is able to become positive or negative depending on the value of ϕ(1⁺), as seen from Equation (3); at high density and low temperature, the distribution function g(r) has a positive large value at the hard-core contact, which makes u positive or negative depending, on the sign of ϕ(1⁺). The behavior of u determines ρ(∂u/∂ρ). In this way, the value of ϕ(1⁺) and the behavior of u play important roles in determining the sign of α at low temperature and high density.

The dotted lines in Figure 6 show the energy u as a function of density ρ at T(°C) = +20, +4, and −20 from the bottom to the top of the figure, respectively, for Model ϕ₆, which is the same as Model ϕ₅ in Yasutomi [41]. The function u shifts to larger values with reducing temperature, and it is accompanied by an increase in the term ρ(∂u/∂ρ). As a result, the function α shifts from positive to negative around ρ = 0.3 with reducing temperature as exhibited in Figure 7 by dotted lines at T(°C) = +20, +4, and −20 from the top to the bottom of the figure, respectively. These induce the pressure anomaly and result in the unusual negative thermal expansion similar to the case of Model ϕ₂ as shown in Figures 2, 3.

FIGURE 6

Figure 6. The excess internal energy u per unit volume as a function of density ρ. Dotted lines: u−ρ curves at T(°C) = +20, +4, and −20 from the bottom to the top of the figure, respectively, for Model ϕ₆. Thin full, dashed, and thick full lines are for kT_c = 0.896, kT = 0.372, and kT = 0.0324, respectively, for Model ϕ₅.

FIGURE 7

Figure 7. The differential coefficients α = ∂P/∂(kT) as a function of density. Dotted lines are at T(°C) = +20, +4, and −20 from the top to the bottom of the figure, respectively, for Model ϕ₆. Thin full, dashed, and thick full lines are α/100 for kT_c = 0.896, kT = 0.372, and kT = 0.0324, respectively, for Model ϕ₅.

For comparison, the energy u and the coefficient α are also shown in Figures 6, 7, respectively, for Model ϕ₅ by thin full, dashed, and thick full lines for kT_c = 0.896, kT = 0.372, and kT = 0.0324, respectively. This model has no soft repulsion and has the smallest negative value of ϕ₅(1⁺). It causes the energy u and ρ(∂u/∂ρ) to have small negative values at the largest density at low temperature to keep α positive at any temperature. This is the reason why the pressure anomaly, or equivalently negative thermal expansion, is never induced in Model ϕ₅. Figure 6 shows the great difference in the behaviors of u between Models ϕ₅ and ϕ₆ and confirms that the behavior of the energy u plays an essential role in the density anomaly in liquids.

To study the behavior of the integrand f(r, ρ, T) = r²ϕ(r)g(r) in Equation (3), let us define Δf(r, ρ, T) as follows:

$\begin{array}{cc}\Delta f(r,\rho ,T)=f(r,\rho ,T)-f(r,\rho ,+20°\text{C}).& (4)\end{array}$

Figure 8 exhibits the difference Δf(r, ρ, T) as a function of distance r. Full and dashed lines show Δf(r, ρ₀, +4°C) and Δf(r, ρ₀, −20°C), respectively, where ρ₀ = 0.7535. The figure shows Δf(r, ρ₀, T) has a much larger value near the hard-core contact than in other regions, which can be shown to be true at any density. Therefore, the value and behavior of energy u (which induce the density anomaly of liquid water) are mainly determined near the hard-core contact where the pair potential has large positive values of the soft-repulsive potential.

FIGURE 8

Figure 8. The difference Δf(r, ρ, T) as a function of distance r. Full and dashed lines show Δf(r, ρ₀, +4°C) and Δf(r, ρ₀, −20°C), respectively, where ρ₀ = 0.7535.

4. Conclusions and Discussion

In our previous paper [41], we determined many functional representations of interparticle interactions between water molecules, all of which reproduce the experimentally measured density-temperature relation at 1 bar with an accuracy better than obtained by previous models. Numerous similar descriptions of pair interactions will be discovered in the coming years, which will help us to understand why solid water has polymorphic structures and why liquid water has a large number of anomalies. We also discussed the fact that nearly every idea put forward so far does not explain the immediate cause of the density anomaly of liquid water.

In the present paper, we illuminated the physics underlying the density anomaly of liquid water by using the thermodynamically self-consistent Ornstein-Zernike approximation (SCOZA). The SCOZA is known to describe the overall thermodynamics very well and provide a remarkably accurate critical point and coexistence curve. This scheme is entirely self-contained, that is, no supplementary thermodynamic or other input is necessary. We considered a fluid of spherical particles with a pair potential given by a hard-core repulsion and a tail. The tail is composed of a soft-repulsion and an attraction. Our scheme is one of the most optimum methods for clarifying the thermodynamic mechanism that induces the density anomaly of liquid water because it contains only the crucial physics that causes the density anomaly.

We showed that negative thermal expansion is exhibited when the value of the potential tail at the hard-core contact is in some proper range that depends on the shape of the attraction; otherwise negative expansion is never observed. Furthermore, we showed that the behavior of the excess internal energy plays an essential role in the density anomaly and is mainly determined by the values of the soft-repulsive potential, especially near the hard-core contact, and is very different from those of usual liquids that never show the density anomaly.

It is impossible to capture the crucial physics via the study of realistic models because these include many of the miscellaneous properties of water, and not all of them are relevant to explaining the immediate causes. Numerical results obtained by using a limited number of particles via a Monte Carlo method or molecular dynamics simulation occasionally show a mixture of less dense and denser regions of particles, and one result claims that a change in the proportions of these regions causes the density anomaly. Another, however, claims that such a mixture is never observed. At any rate, it is not known what thermodynamic mechanism causes a change in the proportions and results in negative thermal expansion.

Thermodynamic anomalies are found qualitatively even with purely repulsive tails. However, it still has not been shown that the tails can reproduce quantitatively the experimentally measured density anomaly of liquid water with sufficient accuracy.

As discussed in our previous work [41], the present results show that nearly all of the ideas that have been proposed, such as the second critical point hypothesis [44], a simple two-state model [1], the clathrate model [9], the network model [2], tetrahedral structure, and the orientation-dependent potential [21, 45–47] do not explain the direct cause of the density anomaly in liquid water [32]. It has long been pointed out that these ideas tell us nothing about what causes the negative thermal expansion at temperatures below 4°C. Let us argue that the orientation-dependent potential is not the immediate cause of the density anomaly in liquid water. The anisotropy of the potential is considered to come from hydrogen bonding or the electronic dipole-dipole interaction between two water molecules. These are attractive and are able to reduce the distance between molecules in thermodynamic equilibrium. Each molecule has a thermal motion around the equilibrium orientation that minimizes the potential. The thermal motion around the equilibrium orientation becomes small with reduced temperature. Therefore, it is impossible to consider that the dipole-dipole interaction or hydrogen bonding turns into a repulsive force below a certain temperature that causes a negative thermal expansion of liquid water. Instead, we believe that the ideas proposed up to now would be formed as a result of negative thermal expansion (see discussion below).

Recently, Russo and Tanaka [48] introduced a novel structural order parameter, which quantifies the degree of translational order of the second shell in water. They show that this parameter is extremely helpful and accurate in describing water properties, including the density anomaly. They used a two-state model to describe the behavior of liquid water over a wide region of the phase diagram. In one of the two states, denoted by S, local structures have low energies, high specific volumes, and low degeneracy. In contrast, structures in the other state, labeled ρ, are thermally excited and characterized by a high degree of disorder and degeneracy, low specific volumes, and high energies. However, the authors do not explain what causes the decomposition of water into the two states. We can explain the cause on the basis of the thermodynamic mechanisms described in Section 3 as follows. Cooling of water generates the higher density ρ state structures, makes their excess internal energies higher owing to the high value of the soft-repulsive potential near the hard core-contact due to condensation, and makes the coefficient α defined by Equation (2) negative, resulting in negative thermal expansion that generates the S state structures. In this way, we can understand that the decomposition of water into the two states is not the cause of the anomaly but just an accompanying effect.

Furthermore, no idea mentioned in the Introduction tells us anything about what induces negative thermal expansion. However, we can explain the cause in the same way as mentioned above. The cooling of water may generate denser proper-water complexes [1], a denser quartz-like structure [2], hydrogen bond bending [3], or the filling of cavities [8, 9]. It makes the excess internal energies higher because of the high value of the soft-repulsive potential near the hard core-contact due to condensation, and makes the coefficient α negative, resulting in negative thermal expansion that may generate less dense and ice-like complexes [1], less dense tridymite-like ones [2], stretching of the hydrogen bond bending [3], or preventing of the filling of cavities and causing negative expansion [8, 9].

Our results now enable us to consider that the thermodynamic mechanism we present induces negative thermal expansion, which is accompanied by rearranging of the orientations of molecules and results in polymorphic structures of solid water that depend on the shape of the orientation-dependent potential, or are accompanied by second critical point, two-state structures, or clathrate structures. In this way, although our study is blind to freezing and, more generally, to the solid phases of the system, it presents significant insights into the thermodynamic properties of water in these phases. We believe that there is a high possibility that our main remarks are applicable to density anomalies in other liquids.

We note here that for some of the isobars shown in Figure 5, the density of the maximum is very high (that of Model ϕ₄ nearly coincides with close-packing), and likely to lie inside the region where fluid-solid coexistence or a solid phase is expected. In this case, the occurrence of the solid may preempt the density anomaly. However, this does not have any major influence on the present conclusions.

Finally, we note that all of the density anomalies shown for models with different tails here and in our previous work [41] are caused by the same thermodynamic mechanism we describe.

Conflict of Interest Statement

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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