A tetrahedral entropy for water

Abstract

We introduce the space-dependent correlation function C (Q)(r) and time-dependent autocorrelation function C (Q)(t) of the local tetrahedral order parameter Q identical with Q(r,t). By using computer simulations of 512 waterlike particles interacting through the transferable interaction potential with five points (TIP5 potential), we investigate C (Q)(r) in a broad region of the phase diagram. We find that at low temperatures C (Q)(t) exhibits a two-step time-dependent decay similar to the self-intermediate scattering function and that the corresponding correlation time tau(Q) displays a dynamic cross-over from non-Arrhenius behavior for T > T (W) to Arrhenius behavior for T < T (W), where T (W) denotes the Widom temperature where the correlation length has a maximum as T is decreased along a constant-pressure path. We define a tetrahedral entropy S (Q) associated with the local tetrahedral order of water molecules and find that it produces a major contribution to the specific heat maximum at the Widom line. Finally, we show that tau(Q) can be extracted from S (Q) by using an analog of the Adam-Gibbs relation.

Fig. 1.

Dependence on Q of the probability density function P(Q,T) for nine values of T. At high T, P(Q,T) is bimodal with peaks at high and low values of Q. The magnitude of the high Q value peak grows as it shifts toward the larger values of Q upon decreasing temperature whereas the magnitude of the low Q value peak decreases and, at sufficiently low temperatures, P(Q,T) becomes unimodal. Analogous plots are shown for TIP4P and TIP5P models, respectively, in refs. and . Error bars are shown in green.

Fig. 2.

Average tetrahedral order parameter, its variance and spatial correlations as functions of separation between water molecules for different temperatures. (A and B) The average order parameter Q(r) (A) and its variance, σ_Q ²(r) (B), as a function of the distance r. (C) Temperature dependence of σ_Q ² (r = 0.32nm) −σ_Q ²(∞) shows a maximum at the Widom temperature, suggesting that the local fluctuations in Q at the fifth-neighbor distance increase upon decreasing temperature and have a maximum at the Widom line. (D) Spatial correlation function C _Q(r) of the tetrahedral order parameter Q (Eq. 1) at various temperatures for pressure P = 1 atm. C _Q(r) has positive peaks at the positions of the nearest-neighbor peaks in oxygen–oxygen pair correlation function g _OO(r) at high T. Although the position of the first maximum of C _Q(r) remains fixed for all the temperatures, the position of the second maximum moves slightly to the smaller r as the temperature decreases. A negative minimum at the fifth-neighbor distance r ≈ 0.32 nm for high T implies that the local tetrahedral order parameters of a central molecule and its fifth neighbor are anticorrelated at T > 250 K. Interestingly, the anticorrelation at r = 0.32 nm changes to positive correlation below the T _W ≈ 250 K.

Fig. 3.

Temperature dependence of time autocorrelation function of local tetrahedral order parameter and relaxation times. (A) Autocorrelation function C _Q(t) of tetrahedral order parameter Q at various temperatures. C _Q(t) is exponential at high temperatures but displays a visible two-step decay at low temperatures. (B) Correlation time τ_Q extracted from C _Q(t) (circles). The solid line is the fit using the Adam–Gibbs relation (Eq. 13) between the tetrahedral entropy S _Q(T), and the tetrahedral relaxation time τ_Q. The dotted lines in B show the power-law fit B(T − T _MCT)^−γ with the fitting parameters B = 25.39, T _MCT = 246.18, and γ = 1.17. The behavior of τ_Q deviates from the power-law fit for the temperatures below the Widom-line temperature (indicated by a vertical arrow) T _W where a cross-over to Arrhenius behavior at lower temperature occurs.

Fig. 4.

Temperature dependence of tetrahedral entropy S _Q(T) (A), defined in Eq. 11, (the red solid line is a guide to the eye) and tetrahedral specific heat C _P ^Q = T(∂S _Q/∂T)_P (B), which has a maximum around the same temperature, T ≈ 250 ± 10 K ≈ T _W, where the total specific heat C _P ^Total has a maximum. Moreover, the difference ΔC _P of the two specific heats is a constant within the error bars for all the temperatures, hence suggesting that C _P ^Q is responsible for the Widom-line transformations.

Fig. 5.

Relation of tetrahedral entropy to tetrahedral order parameter and contribution of two-point translational correlations to entropy S ² for temperatures T = 230,240,245,250,260,270,280,290,300,320 K. (A) Test of the relation of Eq. 16 for the relation between the tetrahedral entropy S _Q and the tetrahedral order parameter Q̄, assuming F = −0.40 independent of temperature. (B) S _Q as a function of t* shows that S _Q decreases linearly with t*. The dashed line is a linear fit through the data. (C) Fig. 5 A and B are consistent with the possibility that the average translational order parameter t* varies as log(1 − Q̄). (D) Dependence of S _Q on S _ex. S _Q changes linearly with S ² at low temperatures, but the linearity begins to break down for T > T _W ≈ 250 K.