# Combined Description of the Equation of State and Diffusion Coefficient of Liquid Water Using a Two-State Sanchez–Lacombe Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{90}Sm

_{10}metal glass); recently, it has been successfully described using a two-state framework [20,42]. Even though in a real material a potential energy landscape is a rugged hypersurface with multiple minima, one can simplify the analysis by only considering the two lowest energy minima and neglecting the contributions of the higher ones. For water in particular, Shi, Russo, and Tanaka [7,17,20] developed a simple combined two-state model for both density and transport properties, capturing many important details; however, many questions remained.

_{α}) depends not on the temperature (T) and pressure (P) separately, but on some combination thereof, i.e., τ

_{α}= f(Tv

_{sp}

^{γ}), where v

_{sp}is the specific volume (inverse of the density) at the current T and P, f(x) is some (material-dependent) function, and γ is a material-dependent constant whose value can be related to the effective interparticle interaction potentials [56]. Within SL-TS2, the relaxation time is assumed to be a function of the “state variable” Z = T/T

_{X}(P), while the specific volume has the form v

_{sp}= v

_{0}(P)g(Z). It is further assumed (in the limit of low pressures) that

_{LLCP}~200–220 K, P

_{LLCP}~180–200 MPa) [58]. We therefore concentrate on the low-pressure region, P < 100 MPa or 1000 bar, but consider a sufficiently broad temperature region—between 160 K and 360 K—to capture both the density anomalies and the fragile-to-strong transition. In Figure 1, this region is shown as a blue rectangle.

## 2. Results

#### 2.1. Master Curves and Scaling

_{X}(P) decreases with pressure according to Equation (16) (see below, in the Section 4). This contrasts with many other amorphous materials, where the glass transition temperature (T

_{g}) increases with P; there, the transition temperature (T

_{X}) and the glass transition temperature (T

_{g}) track one another quite closely. For water, this is not the case, and the exact location of its T

_{g}is still debated, although it should be in the order of 160 K or even less. [59] Thus, the density data for amorphous water at T > 160 K can be considered “equilibrium” (if we disregard the effects of crystallization) and modeled using an equilibrium theory. This analysis is discussed below.

#### 2.2. Density vs. Temperature and Pressure—Model and Experiment

_{HH}= α

_{LH}= 1. The best fit parameters are summarized in Table 2. The error in the parameter determination was about 5%.

#### 2.3. Diffusion Coefficient vs. Temperature and Pressure—Model and Experiment

_{0}= 1.3 Å, so that τ

_{α}(T = 273 K) = 17 ps, consistent with the dielectric relaxation measurements of Bertolini et al. [60] and Buchner et al. [61] The model was then fitted to the experimental data using the generalized reduced gradient (GRG) nonlinear optimizer in Excel, and the best values for the TS2 parameters log(τ

_{∞}), E

_{L}, and E

_{H}were obtained. The error bars for log(τ

_{∞}) were estimated to be ±0.2, and the errors for E

_{L}and E

_{H}were about 15%. The best TS2 fit for the atmospheric pressure diffusion coefficient is given in Figure 5a (dashed line). The agreement between the model and the experiment results was good at temperatures between T = 200 K and T = 360 K, but it worsened significantly at lower temperatures. To specifically address the low-temperature behavior, one can stipulate that there is an additional dependence of E

_{L}on ψ, as follows:

_{L}on ψ is depicted in Figure 5b. The model parameters are summarized in Table 3. Note that the “modified TS2” is consistent with the dynamic scaling, given that the relaxation time depends only on ψ, albeit this dependence is no longer linear. The functional form of Equation (2) is empirical; one could also use other forms, e.g., a two-state sigmoidal function. This would suggest a separate low-temperature transition caused by a possible rearrangement of hydrogen bonds within the L-clusters. At this time, there are not enough data to explore this topic further.

## 3. Discussion

_{X}) is a decreasing function of pressure, while in most conventional glass-formers the glass transition temperature (T

_{g}) and the melting temperature (T

_{m}) are increasing functions of pressure. This anomaly is connected to the fact that the L-state has a lower density and the H-state has a higher density; thus, increasing the pressure makes the H-state more favorable (at constant temperature) and causes the L-H transition to shift towards lower temperatures. The scaling exponent g is estimated to be ~(−1.7), while for most other materials obeying the “τTV” scaling, g is usually positive. There are several theories relating g to the effective interaction between the molecules or clusters; however, those theories apply for positive values of g. A consistent molecular-theory-based understanding of the “τTV” scaling for water is still a task for future research.

_{g}/T

_{X}(where T

_{g}is the glass transition temperature, and T

_{X}is the point where the free energies of the high- and low-temperature states are equal), it would be very close to 1 for polymers and organics, but only ~0.5–0.6 for water.

## 4. Materials and Methods

#### 4.1. The Model Density and Specific Volume

_{L}is the volume fraction of the L-clusters, φ

_{H}is the volume fraction of the H-clusters, and φ

_{V}is the volume fraction of the voids or empty spaces (recall that we utilized the Sanchez–Lacombe [63,64,65,66,67] lattice framework); V is the total volume, Y = Vφ

_{V}is the total number of voids, v

_{0}is the volume of one lattice site, z is the coordination number, and ε

_{ij}(i, j = L, H) represents the van der Waals energies between the neighboring sites. The “internal” entropy of an L-cluster is labeled S

_{H,L}, and that of an H-cluster is labeled S

_{H,H}; the first index “H” refers to the origin of this entropy term as arising from the combinatorics of hydrogen bonds within the cluster (see the Section 3 for more details). Finally, T is the temperature, P is the pressure, and k

_{B}is the Boltzmann constant.

_{L}lattice units, while that of the H-cluster is assumed to be equal to r

_{H}lattice units. Notably, for water, the higher-temperature state is denser than the lower-temperature state; thus, r

_{L}> r

_{H}(unlike most other materials). Finally, $\Delta {S}_{H}={S}_{H,H}-{S}_{H,L}$ is the entropy difference between the high- and low-temperature states.

_{H}< r

_{L}; this means that the parameter $\Delta \tilde{V}$ is negative, and the increase in pressure promotes the high-temperature (not low-temperature) state. Second, because of the above, the translational entropy of the high-temperature (H) state is lower than that of the low-temperature (L) state. This is offset by the hydrogen-bonding combinatorial term $\Delta {S}_{H}$ > 0. The number of possible arrangements of various hydrogen bonds within the H-cluster is assumed to be significantly greater than that in the L-cluster.

_{g}~130–160 K) [2,38,40,68], the state of the matter is equilibrium for all temperatures and pressures considered here. The specific volume (i.e., the inverse of the density) is given by ${v}_{sp}={v}_{sp0}\left(P\right)\frac{\u2329r\u232a}{{r}_{L}\nu}$, where ${v}_{sp0}\left(P\right)$ is the (pressure-dependent) specific volume at zero temperature. Thus, we can determine the density or specific volume unambiguously for any T and P after solving Equations (7) and (8) for y and ν.

#### 4.2. The Model’s Relaxation Time and Diffusion

_{H}is the activation energy for the H-state, and E

_{L}is the activation energy for the L-state. Similar to the case of conventional (non-hydrogen-bonding) amorphous materials, E

_{L}> E

_{H}.

_{0}is some characteristic length that is assumed to be temperature-independent but pressure-dependent. Strictly speaking, there is a linear temperature dependence of D on T in addition to the Arrhenius or super-Arrhenius dependence built into the relaxation time, but here we neglect this weaker dependence. Thus, log(D) and −log(τ

_{α}) differ only by an additive constant, to be determined if both the diffusion coefficient and relaxation time are known at any given temperature.

#### 4.3. The Dynamic “τTV” Scaling

_{1}and T

_{2}such that the right-hand sides are the same, but the left-hand sides are drastically different. The question is whether the corresponding states concept applies to water at all.

^{*}, E

_{H}, E

_{L}) depend on the pressure, as follows:

^{*}, E

_{H}, E

_{L}), and α

_{Τ}=1/P

_{0,T}[48]. Note that Equation (16) implies that the energy-type parameters decrease upon increasing pressure; this is counterintuitive, but it seems to be the only way to describe the pressure dependence of the density-temperature curves.

_{r}=1/P

_{0,v}is the isothermal compressibility. Finally, we further hypothesize that

_{D}. Here, and below, we set a

_{0}(0) = 0.14 nm = 1.4 Å.

_{X}is defined as the temperature for which ψ = 0.5 (equal numbers of H- and L-states). The pressure dependence of T

_{X}is given by Equation (16), such that T

_{X}decreases as P is increased. The state variables y and ν then depend on the combined variable T/T

_{X}(P)—not on T and P independently.

#### 4.4. Experimental Data

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The qualitative phase diagram for amorphous water. Here, LDL is low-density liquid water, HDL is high-density liquid water, TP is the triple point, and CP is the critical point; for other abbreviations, see Ref. [57]. The blue rectangle highlights the region considered in this modeling study. Note that on the pressure scale, 1 bar ≈ 0.1 MPa (adapted from Ref. [57] with permission from AIP Publishing).

**Figure 2.**Experimental data for liquid water as a function of temperature at different pressures (see text for more details): (

**a**) diffusion coefficient; (

**b**) density.

**Figure 3.**The data from Figure 2 rescaled according to Equations (1)–(3) (see text for more details): (

**a**) density; (

**b**) diffusion coefficient.

**Figure 4.**Density vs. temperature for multiple pressures: 0.1 MPa (blue), 40 MPa (orange), 80 MPa (grey), and 120 MPa (gold). Circles represent experimental data; lines are model predictions. See text for more details.

**Figure 5.**(

**a**) The dependence of the relaxation time on temperature; circles are experimental data (converted from the diffusion coefficient data using Equation (8)), the dashed line is the TS2 fit, and the solid line is the “modified TS2” fit. (

**b**) The dependence of the L-state activation energy on the H-state molar fraction, 1−ψ.

**Figure 6.**Diffusion coefficient vs. temperature for different pressures. Symbols are experimental data (corresponding to multiple pressures, as shown in the legend), the dotted line is the TS2 fit, and the solid line is the mTS2 fit.

**Figure 7.**The schematic representation of the L and H clusters within the SL-TS2 model. See text for more details.

Parameter | Units | Value |
---|---|---|

a_{r} | (MPa)^{−1} | 5.48 × 10^{−4} |

a_{T} | (MPa)^{−1} | 9.35 × 10^{−4} |

a_{D} | (MPa)^{−1} | 7.78 × 10^{−3} |

Parameter | Units | Value |
---|---|---|

T* | K | 734.1 |

r_{L} | 56.5 | |

r_{H} | 50.6 | |

α_{HH} | 1.0 | |

α_{HΛ} | 1.0 | |

ΔS_{H} | 18.5 | |

V_{sp,0} | cm^{3}/g | 1.073 |

Parameter | Units | Value |
---|---|---|

_{τ∞} | s | 4.0 × 10^{−13} |

E_{H} | kJ/mol | 6.7 |

TS2 | ||

E_{L} | kJ/mol | 20.1 |

mTS2 | ||

E_{L,min} | kJ/mol | 19.5 |

E_{L,max} | kJ/mol | 28.1 |

k | 25.0 |

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**MDPI and ACS Style**

Ginzburg, V.V.; Fazio, E.; Corsaro, C.
Combined Description of the Equation of State and Diffusion Coefficient of Liquid Water Using a Two-State Sanchez–Lacombe Approach. *Molecules* **2023**, *28*, 2560.
https://doi.org/10.3390/molecules28062560

**AMA Style**

Ginzburg VV, Fazio E, Corsaro C.
Combined Description of the Equation of State and Diffusion Coefficient of Liquid Water Using a Two-State Sanchez–Lacombe Approach. *Molecules*. 2023; 28(6):2560.
https://doi.org/10.3390/molecules28062560

**Chicago/Turabian Style**

Ginzburg, Valeriy V., Enza Fazio, and Carmelo Corsaro.
2023. "Combined Description of the Equation of State and Diffusion Coefficient of Liquid Water Using a Two-State Sanchez–Lacombe Approach" *Molecules* 28, no. 6: 2560.
https://doi.org/10.3390/molecules28062560