Journal of Molecular Liquids 269 (2018) 38–46
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Molecular dynamics simulations for optical Kerr effect of TIP4P/2005 water in liquid and supercooled states Ping-Han Tang 1, Ten-Ming Wu ⁎ Institute of Physics, National Chiao-Tung University, Hsinchu 300, Taiwan
a r t i c l e
i n f o
Article history: Received 11 January 2018 Received in revised form 15 May 2018 Accepted 31 July 2018 Available online 04 August 2018
a b s t r a c t The optical Kerr effect (OKE) spectroscopy measured with heterodyne detection (HD) is a useful tool to provide information regarding intermolecular vibrations and structural relaxations in liquid water. Recently, the measurements of the OKE spectroscopy have been extended to the supercooled regime of water. Though the measured results can be well described by using a phenomenological model, the time-resolved OKE spectroscopy of liquid and supercooled water still need a comprehensive understanding. In this paper, we investigated the OKE nuclear response functions of this peculiar liquid and their reduced spectral densities by performing molecular dynamics simulations with the TIP4P/2005 water model. The collective polarizability of water was computed via a dipolar induction scheme, which involves the intrinsic polarizability and the first-order hyperpolarizability tensor of water molecule. Our simulation results were qualitatively consistent with the HD-OKE experimental observations for displaying that the polarizability anisotropy relaxation of supercooled water in the highdensity liquid phase was fragile-like by following a stretching exponential decay with an exponent βs insensitive to temperature and the temperature dependence of the relaxation time exhibited a power-law divergence at a singular temperature Ts with a critical exponent γs. Indicated by our quantitative results, Ts was predominately determined by the structural arrest, but βs and γs were not only related to the structural relaxation but also influenced by the collective polarizability of the liquid. For all investigations, the effects due to the first-order hyperpolarizability tensor were examined. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Water, the most abundant liquid on earth, exhibits many anomalous behaviors as observing temperature variations of its density and thermodynamic and physical quantities such as shear viscosity, selfdiffusion coefficient, constant-pressure specific heat, isothermal compressibility, and thermal expansion coefficient, etc. [1–3]. The anomalies of water become more pronounced as in its supercooled states and their temperature dependences are expected to diverge in a power law near −45 °C at ambient pressure [4]. Various scenarios were proposed to explain the anomalies of water [5–10]. Among them, the two-state model is a widely accepted one [11–15]. In this model, water is considered as a mixture of two distinct local structures: the high-density form consisting of strongly distorted H-bonds and the low-density form constituted by tetrahedrally ordered H-bonds [2,16,17], where the two forms of local structure are interconvertible. At high temperature and pressure, water is in the high-density-liquid (HDL) phase for the high-
⁎ Corresponding author. E-mail address:
[email protected] (T.-M. Wu). 1 Present address: Research Center of Applied Science, Academia Sinica.
https://doi.org/10.1016/j.molliq.2018.07.121 0167-7322/© 2018 Elsevier B.V. All rights reserved.
density local structures are predominated, whereas at low temperature and pressure, the low-density local structures become prevalent so water is in the low-density-liquid (LDL) phase. In this two-state scenario, the anomalies of supercooled water were interpreted by the existence of a hypothetical liquid-liquid critical point (LLCP) [6], the termination of the first-order phase transition between the two liquid phases. However, it is difficult to explore the LLCP of bulk water by experiments, because homogeneous nucleation to ice occurs near 232 K at ambient pressure before reaching the LLCP upon cooling [2,3], while the studies of confined water [18] and simulation [19–21] supported the hypothesis of the LLCP. The ultrafast vibrational spectroscopy has been used broadly to extract the relations between structure and dynamics of water [22]. Recently, the optical Kerr effect (OKE) spectroscopy of water has been measured from ambient to supercooled states, and the analysis of the time-resolved OKE signals indicates the coexistence of two distinct local structures in supercooled water [23], which is consistent with the hypothesis of the two-state model for water. The OKE measurement with heterodyne detection (HD) is a third-order pump-probe nonresonant spectroscopy [24,25]. The OKE spectroscopy of a molecular liquid contains an instantaneous electronic component and a subsequent nuclear response arising from the change of its collective polarizability
P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46
anisotropy due to molecular reorientations and the dipolar induced interactions in the liquid. In the frequency domain, the OKE spectroscopy delivers information regarding Raman-active intermolecular and intramolecular modes of the liquid [26]. The HD-OKE technique has been utilized to study water dynamics in nanoscale confinements [27,28]. The time-dependent OKE nuclear response functions of water in liquid and supercooled states can be well fit by using the schematic modecoupling (MC) model [23,29]. Following a stretching exponential decay, the long-time behavior of the nuclear response function of water has a relaxation time exhibiting a power-law temperature dependence, which diverges at a singular temperature Ts with a critical exponent γs [23,30]. This long-time behavior is consistent with the ideal MC theory for the glass transition of fragile molecular liquids [31]. Though the OKE nuclear response function of water can be depicted successfully by using a phenomenological model, it is essential to understand the polarizability anisotropy dynamics that causes the OKE signal of water observed experimentally. The polarizability anisotropy dynamics in molecular liquids is complicated for involving not only the structural relaxation in the liquid but also the dipolar interactions, whose ranges are far beyond the nearest neighbors of molecules [32–34]. The case for water is more subtle due to its rapidly fluctuating H-bond structure [35] and strong induction between molecules in the liquid phase [36,37]. Molecular dynamics (MD) simulation has been a useful tool to scrutinize the underlying mechanisms responsible for the OKE spectroscopy of molecular liquids [38–46], and several simulation studies for liquid water have been reported [47–53]. In reality, the collective polarizability of water involves charge fluxes within and between molecules [37] so that a polarizable water model is required for simulation studies. However, there is no consistent result between the existing polarizable water models so far [54,55]. Also, in consideration of computation time, most MD studies for the OKE spectroscopy of water exploited the non-polarizable pointcharge water models amended with induced dipoles located at molecular centers of mass, where liquid configurations were generated by the water model and the collective polarizability was evaluated through the intrinsic polarizability of each molecule and the induction between induced dipoles in liquid configurations generated. This approach generally produced the nuclear response function of liquid water in a reasonable agreement with the experimental results, except for a few femtoseconds at initial. In this paper, we performed MD simulations with the TIP4P/2005 force field [56] to study the OKE spectroscopy of liquid and supercooled water at ambient pressure, where the TIP4P/2005 force field is the non-polarizable model that reproduces accurately thermodynamic properties of water, including the temperature of maximum density [57], and shows the termination of the liquid-liquid phase transition at a LLCP [21]. The collective polarizability of water was calculated via a dipolar induction scheme involving the first-order hyperpolarizability of water molecule [36,39], which is essential in liquid water. The rest of this paper is organized as follows: Section II depicts the theoretical background for the OKE nuclear response function of molecular liquids. Section III describes our MD simulations and the collective polarizability models studied in this paper. In Section IV, the simulation results of the nuclear response function and its frequency spectral densities are presented and discussed. The polarizability anisotropy relaxation in simulated liquids was investigated, and the singular temperature Ts and the critical exponent γs for its relaxation time were obtained. The effects due to the first-order hyperpolarizability on the OKE spectroscopy are also examined. Our conclusions are given in Section V. 2. Theoretical background In the OKE spectroscopy, the nuclear response function R(t) of a molecular liquid at temperature T is related to the change rate of
39
polarizability anisotropy of the liquid and, in the classical limit, can be cast into the form [32–34,58]. Rðt Þ ¼ −
θðt Þ dΨðt Þ ; kB T dt
ð1Þ
where kB is the Boltzmann constant and θ(t) is the Heaviside step function. Ψ(t) is the polarizability anisotropy time correlation function (TCF), which measures the anisotropy dynamics of collective polarizability tensor Π of the liquid. The explicit expression of Ψ(t) is given as. Ψðt Þ ¼
15 Πxz ðt ÞΠxz ð0Þ; γ2 N
ð2Þ
where N is the total number of molecules in the liquid, γ is the molecular polarizability anisotropy, and Πxz stands for one of the off-diagonal elements of Π. In this paper, the brackets denote an ensemble average over liquid configurations. The collective polarizability Π of a molecular liquid can be expressed as a sum of the molecular polarizability (MP) ΠM and the induced polarizability (IP) ΠI. Π ¼ ΠM þ ΠI ;
ð3Þ
where ΠM is simply a sum of the intrinsic polarizabilities αM i of individual molecules N
ΠM ¼ Σi¼1 αM i :
ð4Þ
The IP ΠI arises from interactions between induced molecular dipoles. According to the Torri's theory [39], in which the induced dipole of each molecule was assumed to locate at its center of mass (CM), the IP ΠI of a molecular liquid can be given as h i N M ΠI ¼ Σ i; j ¼ 1 αM i T ij α j þ βi T ij μ j ;
ð5Þ
i≠j where μi and αi are the dipole and polarizability of molecule i in the liquid, respectively, βM i is the first-order hyperpolarizability of the molecule, and the dipole interaction tensor Tij is expressed as T ij ¼
3 ^rij ^rij −^I 3 ; r 3ij
ð6Þ
with ^r ij a unit vector along the CMs of molecule i and j separated at a distance rij and ^I 3 a 3 × 3 unit matrix. Comments about the IP ΠI given in Eq. (5) are the following: For a single polar molecule, the intrinsic polarizability αM is the first-order response of its molecular dipole μM to an electric field, and the third-rank hyperpolarizability tensor βM with eleM ments given as βM lmn = ∂αmn/∂El describes the second-order response, where the molecular subscripts are omitted for clarity. The second term in Eq. (5) is resulted from the change in the intrinsic polarizability due to the dipole field created by other molecules in the liquid. The dipole μi and the polarizability αi are enhanced from their gasM phase values μM i and αi , respectively, by an induced part through the induction between molecules in the liquid. They satisfy a set of selfconsistent equations given as [39] μi ¼ μM i þ
N X
αM i T ij μ j ;
ð7Þ
j≠i
αi ¼ αM i þ
N h i X M αM i T ij α j þ βi T ij μ j :
ð8Þ
j≠i
In principle, αj and μj in Eq. (5) should be the solutions of Eqs. (7) and (8). The self-consistent equations can be solved by the
40
P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46
iterative method, and the lowest-order solutions are simply their gasM phase values αM i and μi . For pure water, the collective polarizability Π calculated with the lowest-order solutions did not yield noticeable differences from the results by the iterative solutions [45]. Thus, we simply replaced αj and μj in Eq. (5) with their lowest-order solutions. Therefore, the αTα term in Eq. (5) is usually referred as the dipole-induceddipole (DID) mechanism, and the βTμ term is the so-called extended DID (XDID) mechanism [39]. The XDID mechanism contributes substantially the collective polarizability of anisotropic molecules, like water, in the liquid phase. According to Π given in Eq. (3), the TCF Ψ(t) in Eq. (2) can be separated into three components ΨMM ðt Þ ¼
ΨII ðt Þ ¼
E 15 D M Πxz ðt ÞΠM xz ð0Þ ; 2 γ N
E 15 D I Πxz ðt ÞΠIxz ð0Þ ; 2 γ N
ΨMI ðt Þ ¼
E D Ei 15 hD M Πxz ðt ÞΠIxz ð0Þ þ ΠIxz ðt ÞΠM ; xz ð0Þ γ2 N
ð9Þ
ð10Þ
ð11Þ
where ΨMM(t) and ΨII(t) are the TCFs for the anisotropy dynamics caused by ΠM and ΠI, respectively, and ΨMI(t) describes the cross correlation between the off-diagonal elements of ΠM and ΠI. Correspondingly, ΨMM(t), ΨII(t) and ΨMI(t) generate the molecular (MM), dipoleinduced (II) and cross-term (MI) components of the nuclear response function R(t), which are denoted as RMM(t), RII(t), and RMI(t), respectively [45,52]. In general, RMM(t) and RII(t) are related to the self and collective dynamics of the liquid, respectively, whereas RMI(t) is associated with both of them. In the frequency domain, the OKE spectral density is the imaginary part of the Fourier transform of the nuclear response function [59,60]. So, the OKE spectrum does not contain the electronic contribution but provides information on intermolecular dynamics in a liquid through its nuclear response. Generally, the nuclear response function of a molecular liquid at long times exhibits a slow decay, which is resulted from the diffusive process of molecular reorientations characterized with a decay time τOD, whereas the molecular orientational anisotropy induced originally by a pump laser pulse has a rising time τs [25,59]. The long-time decay of R(t) creates a sharp peak near zero frequency in the OKE spectral density [52,61], which covers up partly the information regarding the low-frequency intermolecular modes in the liquid. Before converting the OKE time signal to the frequency domain, it is usually to remove out the long-time portion in the nuclear response function with the procedure proposed by McMorrow and Lotshaw [59]. Hence, the nuclear response function RðtÞ at short and intermediate timescales is given as −τ t
RðtÞ ¼ RðtÞ−Ae
OD
t 1−e−τs ;
ð12Þ
where A is the magnitude of the contribution due to diffusive molecular reorientations to the long-time decay of R(t). The OKE reduced spectral density (RSD) is defined as the Fourier-sine transform of RðtÞ, which is given as χ } ðωÞ ¼ Im½χðωÞ ¼
Z 0
∞
dt Rðt Þ sinðωt Þ:
ð13Þ
The OKE RSD is related to the Bose-Einstein corrected, lowfrequency Raman spectrum obtained through the depolarized Raman measurement [60,62]. Corresponding to the MM, II, and MI components of RðtÞ, the OKE RSD can be also separated into three components, where the MM and II components reveal the Raman-active modes resulted from ΠM and ΠI, respectively, and the MI component describes
the modes associated with both of them. This is a separation scheme of the OKE RSD according to the polarizability components. 3. MD simulations and the water model In this work, we performed MD simulations with the TIP4P/2005 model by using the package LAMMPS [63]. In our simulations, 864 water molecules were confined in a cubic box with periodic boundary conditions. Each TIP4P/2005 molecule contains four sites [56], two sites of hydrogen with positive charge qH = 0.5564e, one site of oxygen without charge, and a site M with negative charge qM = − 2qH located at a distance 0.1546 Å from the oxygen along the bisector of the HOH bond angle, where the two OH-bonds at a length of 0.9572 Å forms an angle of 104.52°. The total potential energy of the system is a sum of the interactions between every molecular pairs, where the interactions associated with a molecular pair include the Coulombic potentials between every two charges coming from different molecules and the Lennar-Jones (LJ) potential between their oxygen atoms with the parameters of atom size σ = 3.1589 Å and energy depth ϵ/kB = 93.2 K. The time step of our MD simulations was 1.0 fs. The shake algorithm was employed to constrain the OH-bond length and the bond angle of each molecule. The system temperature was controlled through the technique of Nose-Hoover thermostat with the damping parameter of 0.1 ps [64,65]. The LJ potential was truncated at a distance of 9 Å. The Coulombic interactions were handled normally within this cutoff distance but the long-range forces were evaluated by using the PPPM method with an accuracy of 10−4 [66]. The long-range corrections were made for the energy and pressure of the system. Our simulations included the NPT and the NVT simulations. At each temperature, the NPT simulation determined the density of the model liquid at 1 atm, where the system density was an average over liquid configurations generated, and the NVT simulations performed at the average density determined by the NPT simulation generated liquid configurations for further calculations. Both simulations were carried out at seven temperatures from 328 to 230 K. At 230 K, the equilibrium time of each simulation was 9 ns, and the times of the production run were 30.6 and 453 ns for the NPT and NVT simulations, respectively. The simulation details are described in Supplemental Materials [67]. In our computations, the intrinsic polarizability αM and the firstorder hyperpolarizability βM of water molecule were taken from Ref. [41], in which the data were obtained via ab initio quantum chemical calculations. The molecular polarizability anisotropy γ in Eq. (2) is defined as γ2 ¼
2 2 M M M 2 αM þ αM þ αM =2; xx −α yy yy −α zz zz −α xx
ð14Þ
M where αM μμ, with μ = x, y, z, are the principal elements of α . This polarizability model is more anisotropic than the Murphy's model [68], which was unable to reproduce the fast oscillatory structure of the nuclear response function of liquid water observed in the HD-OKE experM iment [69]. The αM i and βi of each molecule in the Lab. frame were obtained, respectively, from αM and βM in the body frame through a similarity transformation between the two coordinate systems [70].
4. Results and discussion 4.1. Thermodynamic states, structure, and dynamics The temperature dependence of the density ρ obtained via our NPT simulations is presented in Fig. 1. Our data display a maximum density at a temperature close to the reported value 278 K of the TIP4P/2005 model [71]. Fig. 1 also shows the Widom line (WL) and the LLCP of the TIP4P/2005 water, taken from Refs. [21,72], respectively, where the WL is the loci of the maxima of thermal response functions as moving away from a critical point into the one-phase region [73]. For the
P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46
1.04
5 328 K 307 K 282 K 261 K 250 K 240 K 230 K
4
1
g(r)
3
ρ (g/cm )
(a)
Density at 1 atm Widom Line LLCP
1.02
3
0.98
2
0.96
1
0.94
41
200
250
0
300
T (K)
1
2
3
4
5
6
7
8
9
10
r (Å) 2
TIP4P/2005 water, the WL has been shown to cross approximately the inflection points of the density-temperature curves from ambient pressure to 1200 bar [21]. Indicated in Fig. 1, the thermodynamic states calculated in this paper are predominated in the HDL phase of the TIP4P/ 2005 water. On the other hand, for systems involving the liquid-liquid phase transition, the WL has been shown to coincide with the crossover between the fragile and strong dynamics, which follows the ideal MC theory and the Arrhenius behavior, respectively [72–74]. This coincidence suggests that the dynamics of water is fragile-like in the HDL phase and strong-like in the LDL phase. Fig. 2 presents the radial distribution functions g(r) and the static structure factors S(q) of oxygen atoms in our NVT-simulated liquids. Indicated by the sharper first peak and the deeper first minimum in g(r) at lower temepratuers, the simulated liquids become more structured as decreasing temperature. The structure change is also manifested by the temperature variation of S(q) in Fig. 2(b). At T = 230 K, which is close to the WL, the S(q) behaves more like that of the LDL [16] by displaying pronounced first and second peaks near 1.86 and 3.03 Å−1, respectively, where the first peak is associated with the H-bond [75]. As increasing temperature, the first-peak and the second-peak positions move closer toward each other so that the two peaks in S(q) mix in an extent. Up to T = 328 K, the mixing of the two peaks results in a very weak maximum around 2.23 Å−1 and a shift of the second maximum to 2.88 Å−1 with a significant decrease in magnitude. Thus, the behavior of S(q) at high temperatures is HDL-like. In dynamics, we calculated the self-intermediate scattering function (SISF) of oxygen atoms with the formula [76]
q1, q2
1.5
3
2 240
280
320
T (K)
1
0.5
0
0
1
2
3
4
5
6
7
8
-1
q (Å ) Fig. 2. Structural variation of TIP4P/2005 water at 1 atm with temperature from 328 to 220 K: (a) the radial distribution function g(r) and (b) the static structure factor S(q) of oxygen atoms. The distributions were calculated with liquid configurations generated via our NVT simulations. In (b), the insert shows the temperature variations of q1 (open) and q2 (filled) of S(q) in the unit of Å−1.
to the stretching exponential decay. According to the MC theory, the behavior of the SISF can be described as " # " # t 2 t βα ϕð t Þ ¼ ð1−Aα Þ exp − þ Aα exp − ; τG τα
ð16Þ
where τG and τα are the relaxation times of the initial Gaussian decay and the α relaxation, respectively, βα is the so-called Kohlrausch exponent of the stretching exponential function, and Aα is the LambMossbauer factor and related to the height of the plateau in the SISF produced by the β relaxation. 1
0.8
ð15Þ
! where r i ðtÞ is the position of O atom in molecule i at time t. The results of the SISF calculated at the first-peak position q1 of S(q) are presented in Fig. 3. At initial, the single-particle dynamics is ballistic and the SISF follows a Gaussian decay. After the initial Gaussian decay, the SISF can be described by the ideal MC theory [31], which predicts two stages of structural relaxation. At high temperatures, the SISF enters directly into the α relaxation and can be depicted by a stretching exponential function. As decreasing into the supercooled regime, a plateau develops in the SISF between the initial Gaussian decay and the α relaxation. This plateau, called the β relaxation, is resulted from the cage effect, which is the trapping of a particle in a cage formed by its nearest neighbors. At long times, the cages relax and the behavior of the SISF changes back
Fs(q1, t)
h i+ * ! ! * iq r i ðt Þ− r i ð0Þ 1 N Σi¼1 e F s ðq; t Þ ¼ ; N
(b)
S(q)
Fig. 1. Density-temperature phase plane of TIP4P/2005 water. The red circles, guided by the dash line, are the densities at 1 atm obtained via our NPT simulations. The green diamonds, referred from Ref. [72], indicate the thermodynamic states corresponding to the maximum of the isochoric specific heat at several pressures. The black square indicates the location of LLCP at T = 193 K and ρ = 1.012 g/cm3 [21]. The green dot line is the WL. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0.6
0.4
0.2
0 -3 10
328 K 307 K 282 K 261 K 250 K 240 K 230 K 10
-2
10
-1
10
0
10
1
10
2
10
3
t (ps) Fig. 3. The SISFs of oxygen atoms in TIP4P/2005 water at indicated temperatures. The solid lines in color are the simulation results calculated at the first-peak position q1 of S(q). The black dash lines are the results fit by using Eq. (16).
P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46
ð17Þ
where Tc is the MC temperature and γc is the critical exponent. Fig. 4 shows the fitting for the τα data by using Eq. (17) with the parameters τα0 = 0.104 ps, Tc = 212 K, and γc = 2.53, where the Tc and γc values are comparable to the TIP4P/2005 supercooled liquid at constant density [72]. Eq. (17) indicates that the relaxation time τα diverges at Tc for the system dynamics is structurally arrested. 4.2. OKE nuclear response function and reduced spectral density The results of nuclear response function R(t) up to 0.5 ps are presented in Fig. 5(a) and (b) for the IP ΠI calculated with and without the XDID mechanism, respectively. The behaviors of the simulated R (t) are generally comparable with the time signals measured by the HD-OKE experiments [23,29]. At all temperatures, the first two peaks in R(t) occur near 15 and 50 fs, which are agree with the experimental values [60,78]. Apparently, the height of the first peak is much enhanced by the XDID mechanism. Regarding the temperature effect, the intensities of the first two peaks increase slightly with decreasing temperature as both DID and XDID mechanisms were involved in ΠI, but are not so sensitive to temperature as the DID mechanism was only involved. Due to their timescales, the first two peaks in R(t) were attributed to librational motions of molecules [79]. Beyond the first two peaks, the behavior of R(t) is complex and sensitive to temperature. At ambient temperature, R(t) generally displays a dip near 0.1 ps and a broad peak near 0.16 ps, and then a slow decay at long times. As lowering temperature, the magnitudes of the dip and the broad peak increase progressively, and weak oscillatory structures appear at later times and their magnitudes are enhanced gradually as getting into the supercooled regime. At temperatures close to the WL, the oscillatory structures become strong enough to be clearly observed and can extend up to 0.5 ps, and after this timescale the decay of R(t), presented in Fig. S1 in Supplemental Materials [67], elongates for extremely long times, accompanying with the much slow structural relaxation in the supercooled regime described in the previous subsection. Table 1 The fit parameters in Eq. (16) for the SISFs of TIP4P/2005 water at 1 atm. T (K)
Aα
τG(ps)
τα(ps)
βα
328 307 282 261 250 240 230
0.7092 0.7288 0.7231 0.7019 0.7064 0.7314 0.7504
0.2077 0.2142 0.2056 0.1971 0.1930 0.1841 0.1795
0.5699 0.8417 1.6555 4.0716 7.7894 16.7731 50.4251
0.9321 0.8592 0.8126 0.8236 0.8250 0.7892 0.7711
1
10
-1
1/τα (ps )
τα (ps)
40 30
10
10
0
-1
-2
20 10
0
10
1
10
2
T - TC
10 0 220
240
260
280
300
320
340
T (K) Fig. 4. Temperature dependence of the relaxation time τα of the SISF. The circles indicate the simulation data. The solid line is the fit by using the inverse power law in Eq. (17). The inset shows the log-log plot for the inverse of τα versus T − Tc.
The MM, II, and MI components of the nuclear response function calculated with both DID and XDID mechanisms involved are displayed in Fig. 6 for high and low temperatures. At each temperature, the MM component only survives for a short period, roughly within 0.1 ps, the II component can extend to longer times, and the MI component is intermediate between the MM and II ones. Our calculations indicate that the three components contribute coherently to the first peak in R(t), which agrees with the results calculated with the same polarizability model for the SPC/E water [45]. The second peak near 50 fs in R(t) arises from a combination of the MM and MI components, and is related to the anisotropy of intrinsic polarizability of water molecule; this peak is not reproducible by a less anisotropic polarizability model [68]. Beyond the first two peaks, the oscillatory structures in R(t) are dominated by the II 0.2
(a) 328K 307K 282K 261K 250K 240K 230K
2
τα ðT Þ ¼ τα0
−γc T −1 ; Tc
10
50
R(t) (ps/amu Å )
60
0.1
0
0 0.1
0.1
0.2
0.3
0.4
0.5
(b) 2
The fit results of the SISF by using Eq. (16) are presented in Fig. 3, and show that the simulated SISFs well follow the ideal MC theory, except for small deviations appearing around 1 ps at 230 K. This confirms the dynamics of the simulated water in the HDL phase to be fragile-like. The fit parameters for the SISFs are given in Table 1. The Aα and τG data associated with the Gaussian decay are generally insensitive to temperature. The decrease of βα from near 1 at ambient conditions to about 0.77 as temperature is close to the WL indicates the change of the α relaxation from the exponential to non-exponential decay. The decrease of βα can be realized through the spatially heterogeneous dynamics scenario [77], which suggests the increase of structural heterogeneity in deeply supercooled liquids and the slower relaxations for regions of local structure more persisted. As decreasing temperature, the relaxation time τα of the α process increases dramatically. The MC theory predicts that the temperature dependence of τα follows an inverse power law given as
R(t) (ps/amu Å )
42
0.05
0
0
0.1
0.2
0.3
0.4
0.5
t (ps) Fig. 5. Nuclear response function of TIP4P/2005 water at 1 atm. The results in (a) and (b) were calculated with both DID and XDID mechanisms and with the DID mechanism only, respectively.
P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46
43
0.0008
(a)
(b) 282K
0.0008
χ"(ω) (arb. unit)
(a) 328K R(t)
0.1
0
(c) 250K
(d) 230K
0.0004 0
0.0004
0
0
200
400
600
0
50
800
100
1000
0.0008
0.1
0 0
0.1
0.2
t (ps)
0.3
0.4
0
0.1
0.2
0.3
0.4
t (ps)
Fig. 6. The MM, II and MI components of the nuclear response functions at (a) 328 K, (b) 282 K, (c) 250 K and (d) 230 K. The results were calculated with the IP model involving both DID and XDID mechanisms. The red dash, blue dot-dash, and green dot lines indicate the MM, II and MI components, respectively, and the black solid line is their sum. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
component arising from the dipole induced interactions, and are generally caused by intermolecular vibrations in water. The amplitudes of the oscillatory structures were found to be larger in the TIP4P/2005 water than in the SPC/E model, since the tetrahedral order in the TIP4P/2005 water is stronger and tends to enhance vibrational motions between molecules [80]. Though not shown here, our observations are generally applied for the polarizability model with the DID mechanism only, except for a much reduction in the MI component. The OKE of water can be further realized through the RSD χ"(ω). To obtain the OKE RSD, the simulated nuclear response functions after 0.2 ps were fit by using a bi-exponential function b1e−t/τ1 + b2e−t/τ2, where τ1 and τ2 are the relaxation times of the fast and slow components, respectively. The fit results and the temperature variations of τ1 and τ2 are presented in Supplemental Materials [67]. The long-time portion of R(t) was removed out according to Eq. (12), in which A and τOD were replaced with b2 and τ2 of the slow component in the biexponential function, respectively, and the rising time τs of the molecular orientational anisotropy was set at 0.35–0.5 ps. The OKE RSD χ"(ω) was obtained through the Fourier-sine transform of RðtÞ in Eq. (13). The RSD χ"(ω) of TIP4P/2005 water is presented in Fig. 7 (a) and (b) for the polarizability model with and without the XDID mechanism, respectively. At 328 K, the RSD calculated with both DID and XDID mechanisms displays three bands and its shape is comparable with the experimental result [81]. The extremely broad band at frequencies larger than 400 cm−1 is associated with librational motions, or hindered rotations, of water molecules [82,83]. The XDID mechanism considerably enhances the amplitudes of the librational band and the temperature influence on this band, similar as what the mechanism does on the first two peaks in the nuclear response function shown in Fig. 5. Though somewhat shifted in position, the bands around 30 and 220 cm−1 in χ"(ω) correspond to the 60 and 175 cm−1 bands observed in Raman spectrum of liquid water [84]. The band around 220 cm−1 in χ"(ω) is associated with stretching vibrational motions parallel to the H-bonds connecting O-atoms, and the influence of the XDID mechanism on the band amplitude is not so effective as that on the librational band. As temperature decreases to 230 K, close to the WL, the stretching band in χ"(ω) has a blue shift to 265 cm−1 and its shape becomes sharper by a large enhancement in magnitude, more like the stretching band in
χ" (arb. unit)
R(t)
(b) 328 K 307 K 282 K 261 K 250 K 240 K 230 K
0.0004
0
0
200
400
600
800
1000
-1
ω (cm ) Fig. 7. OKE RSD of TIP4P/2005 water at indicated temperatures. The results in (a) and (b) were calculated with both DID and XDID mechanisms and with the DID mechanism only, respectively. The OKE RSD was generated from the nuclear response function after removing the long-time portion as described in the text. The inset in (a) shows the lowfrequency portion of the OKE spectral density obtained from the full nuclear response function.
Raman spectrum of low-density amorphous ice, whose tetrahedral structures are less distorted than in high-density amorphous ice [85]. Thus, the stretching band in χ"(ω) is related to tetrahedral structures in water and its broadness is associated with the distortion of tetrahedrons in the liquid. The band around 30 cm−1 in χ"(ω), which is also significantly enhanced by the XDID mechanism, is predominated by the OO-O bending motions perpendicular to the H-bonds connecting Oatoms [84,86]. This band was also observed in the power spectrum of the velocity autocorrelation function of oxygen atoms in the same water model, and was found to be related to transverse acoustic vibrational modes in crystalline ice [87]. In the full OKE spectral density presented in the inset of Fig. 7(a), this bending band does not show up at ambient temperature but appears only at low temperatures close to the WL, where the diffusive process of molecular orientations is attenuated. At 230 K, the bending band in χ"(ω) has a slight blue shift to 50 cm−1, close to the band frequency observed in Raman spectrum of liquid water, but is reduced in magnitude. Fig. 8 shows the MM, II and MI contributions to the OKE RSD at high and low temperatures. The librational band is contributed mostly from the MM and MI components and partly from the II component, which produces a small peak around 500 cm−1 as in the supercooled regime. As in liquid and supercooled states, the stretching band is strongly dominated by the II component. In reverse to the librational band, the bending band is caused mostly by the II and MI components and partly by the MM component. 4.3. Singular temperature from polarizability anisotropy relaxation To extract the polarizability anisotropy relaxation in water, we returned back to the normalized polarizability anisotropy TCF ΨðtÞ ≡ ΨðtÞ=Ψð0Þ. The simulated results of ΨðtÞ calculated with and without the XDID mechanism are presented in Fig. 9(a) and (b), respectively. The behavior of ΨðtÞ displays a Gaussian decay on a timescale of
44
P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46 0.0008
χ"(ω)
(a) 328K
(b) 282K
0.0004
0
χ"(ω)
(c) 250K
(d) 230K
0.0004
0
0
300
600
900
0
300
-1
600
900
-1
ω (cm )
ω (cm )
Fig. 8. The MM, II and MI components of the OKE RSD at (a) 328 K, (b) 282 K, (c) 250 K and (d) 230 K. The red-dash, blue-dot-dash, and green-dot lines are the MM, II and MI components, respectively, and the black-solid line is their sum. The results were calculated with both DID and XDID mechanisms. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
10 fs but then exhibits complicated oscillatory structures, which are caused by molecular reorientations and intermolecular vibrations as discussed previously and have amplitudes sensitive to the XDID mechanism, up to a timescale depending on temperature. After the oscillatory structures, the behavior of ΨðtÞ is qualitatively similar as the SISF shown in Fig. 3, by directly getting into a slow decay at ambient
1
328 K 307 K 282 K 261 K 250 K 240 K 230 K
Ψ(t) / Ψ(0)
0.8
0.6
temperature or showing a plateau region before the slow decay as in the supercooled regime, where the plateau is realized as the consequence of the cage effect on the system dynamics. Because of the oscillatory structures, the time profile of ΨðtÞ is unable to be described by Eq. (16). But, the long-time behavior of ΨðtÞ can be depicted by a stretching exponential function given as " # t βs ΨðtÞ≅As exp − ; τs
ð18Þ
where βs and τs are the stretching exponent and the relaxation time characterizing the polarizability anisotropy relaxation, respectively, and As is associated with the plateau magnitude of ΨðtÞ at intermediate times for supercooled water. The fit results of ΨðtÞ at times larger than 0.2 ps by using Eq. (18) are plotted with the dash lines in Fig. 9, and the fit data are given in Table 2. The temperature dependence of the stretching exponent βs is shown in Fig. 10(a). Our results indicate that βs generally insensitive to temperature, which is consistent with the experimental observation [31]. However, the value of βs depends on the IP model studied: For TIP4P/2005 water, βs is estimated to be 0.7 for the IP model involving the DID mechanism only but increases to 0.75 as the XDID mechanism is included in the IP model, whereas the two simulation values are larger than the experimental data βs= 0.6 obtained by the HD-OKE measurement for supercooled water. On the other hand, the stretching exponential decay was also observed in the OKE spectroscopy of noble-gas liquids with the value of βs about 0.66 for Ar, Kr and Xe liquids, which are free from rotations and intramolecular vibrations occurring in molecular liquids [88]. However, our simulation results indicate that for water the temperature dependence of βs basically differs from that of βα, the stretching exponent of the α relaxation observed in the SISF, which is also shown in Fig. 10(a). According to our results, the βs of water is related to its liquid structure and is also associated with the collective polarizability of the liquid. Hence, the physical origin for the stretching exponential decay in the OKE nuclear response function of water still needs further investigations. Fig. 10(b) shows the temperature dependence of the relaxation time τs for the two IP models studied in this paper follows an inverse power law, which is given as
0.4
τs ðT Þ ¼ τ s0
−γs T −1 ; Ts
ð19Þ
0.2
(a) 0 -3 10
10
-2
10
-1
10
0
10
1
10
2
10
3
t (ps) 1
328 K 307 K 282 K 261 K 250 K 240 K 230 K
Ψ(t) / Ψ(0)
0.8
0.6
where Ts is the singular temperature and γs is the critical exponent for the polarizability anisotropy relaxation in supercooled water. The fitting of the τs data by using Eq. (19) yields τs0 = 0.259 ps, Ts = 217.6 K, and γs = 1.94 for the IP model involving both DID and XDID mechanisms, and τs0 = 0.188 ps, Ts = 217.1 K, and γs = 2.01 for the model involving the DID mechanism only. These results confirm that the polarizability anisotropy dynamics of the HDL water is fragile-like. The Ts values of the two IP models are almost the same, indicating that Ts is independent of the IP model. Thus, the Ts of TIP4P/ 2005 water was estimated near 217 K, which is lower than the Table 2 The fit values of As, τs, and βs in Eq. (18) for the long-time portion of the normalized polarizability anisotropy TCF ΨðtÞ.
0.4
0.2
DID and XDID
(b) 0 -3 10
10
-2
10
-1
10
0
10
1
10
2
10
3
t (ps) Fig. 9. Normalized polarizability anisotropy TCFs of TIP4P/2005 water at 1 atm. The results in (a) and (b) were calculated with both DID and XDID mechanisms and with the DID mechanism only, respectively. The solid lines in color are the simulation results, and the black dash lines are the fits by using Eq. (18) for times larger than 0.2 ps.
DID only
T (K)
As
τs(ps)
βs
As
τs(ps)
βs
328 307 282 261 250 240 230
0.7917 0.7439 0.7132 0.7084 0.7075 0.7311 0.6972
0.7771 1.2927 2.6704 5.9139 10.8482 21.2769 67.7283
0.7251 0.7400 0.7606 0.7491 0.7522 0.7092 0.7617
0.8347 0.7512 0.6960 0.6776 0.6684 0.6873 0.6542
0.5587 0.9629 2.0698 4.6575 8.7344 17.2689 55.4876
0.6561 0.6758 0.6952 0.6857 0.6960 0.6582 0.6992
P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46 1 DID and XDID DID only Expt. data
0.9
βs
0.8
0.7
0.6
(a) 0.5
220
240
260
280
300
320
340
T (K) 80 10
(b)
10
0
τs (ps)
-1
1/τs (ps )
60
1
40
10
10
-1
-2
10
-3
10
0
10
20
1
10
2
T - TS (K) DID and XDID DID only
0 200
220
240
260
280
300
320
340
T (K) Fig. 10. Temperature dependences of (a) the stretching exponent βs and (b) the relaxation time τs of the polarizability anisotropy relaxation in supercooled water at ambient pressure. The black circles are the simulation results for the IP model involving both DID and XDID mechanisms, and the red squares are for the model involving the DID mechanism only. In (a), the blue crosses are the data obtained from the HD-OKE measurements [30], and the green triangles indicate the stretching exponent βα of the α relaxation in the SISF. In (b), the dashed lines are the fits for the simulation data by using Eq. (19), and the inset shows the log-log plot for the inverse of τs versus T − Ts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
experimental value 227 K obtained via the HD-OKE measurement [23]. This is acceptable for the melting temperature and the temperature of the LLCP predicted by the TIP4P/2005 model are lower than the corresponding values estimated from experiments [71,89–91]. According to our results, Ts is close to the singular temperature Tc = 212 K resulted from the structural relaxation, suggesting that the structural arrest is the common origin for the two singular temperatures. On the other hand, the γs values of the two IP models studied are lower than γc = 2.53, which has nothing to do with the collective polarizability. This suggests that the critical exponent γs depends on the IP model of water and the XDID mechanism makes the simulation value of γs more close to the experimental data 1.7 [23]. Like βs, the critical exponent γs is associated with not only the liquid structure of water but also its collective polarizability. The small difference between simulation and experimental values of γs is probably resulted from the constant gas-phase values of the first-order hyperpolarizability βM used in our IP model, whereas a recent study shows that βM of water molecules in liquid phases may vary with their local environments [92]. 5. Conclusions In this paper, we have investigated the OKE of water from liquid to supercooled states at ambient pressure by performing MD simulations with the TIP4P/2005 water model, which reproduces accurately thermodynamic properties of water. The thermodynamic states in our studies are in the HDL phase of water, where the liquid dynamics for
45
structural relaxation is fragile-like. In our calculations, the collective polarizability of water included two parts: the MP contributed from the intrinsic polarizabilities of individual molecules and the IP arising from interactions between induced dipoles, which are located at molecular CMs. We considered two IP models: One involves only the DID mechanism, which has been widely used to study the OKE of molecular liquids, and the other involves both DID and XDID mechanisms, where the XDID mechanism comes from the first-order hyperpolarizability of water molecule [39]. The OKE nuclear response function and its RSD were calculated for the two IP models, and the effects of the XDID mechanism on the OKE spectroscopy of water were examined through comparing the results of the two models. The OKE nuclear response functions calculated with the water model in the liquid and supercooled states behave similarly as those observed experimentally. The first two peaks in the nuclear response functions and the high-frequency band in the OKE RSD are resulted from librational motions of molecules, and their intensities are significantly enhanced by the XDID mechanism. Also, as the XDID mechanism is included in the IP model, the temperature influence on the first two peaks in the nuclear response function and the librational band in the RSD is more effective. After the first two peaks, the oscillatory structures in the nuclear response function are predominately caused by the dipole induced interactions through intermolecular vibrations, which give rise to the stretching and bending bands in the low-frequency region of the OKE RSD. The oscillatory structures in the nuclear response function are sensitive to temperature, but not so much influenced by the XDID mechanism as for the first two peaks. As cooling into the supercooled regime, the oscillations are clearly observed and can extend up to an intermediate timescale, which partly overlaps with the reorientation relaxation of molecules escaping out of from their cages. As approaching to the WL, the structure of supercooled water becomes more tetrahedrally ordered like the LDL as indicated by the static structure factor, whereas the stretching and bending bands in the OKE RSD have a blue shift in frequency and the stretching band intensity is much enhanced, consistent with the observation of the HD-OKE measurements [23]. The polarizability anisotropy relaxation of the HDL water was confirmed to be fragile-like as the prediction of the MC theory, and follows a stretching exponential decay with an exponent βs insensitive to temperature, which is consistent with the results of the HD-OKE measurement. The values of βs calculated for the two IP models studied in this paper are higher than the experimental data, and the value somewhat increases as the XDID mechanism is involved in the IP model. Thus, the βs of water is certainly related to its structural relaxation and also associated with the collective polarizability of the liquid. The singular temperature Ts for the divergence of the polarizability anisotropy relaxation time τs was found to be close to the singular temperature Tc caused by structural relaxation. Thus, the structural arrest is suggested to be the common origin for the two singular temperatures of the HDL water. Our results indicate that the critical exponent γs for the power-law divergence of τs is related to the collective polarizability of water, in addition to the structural relaxation. As the XDID mechanism is included in the IP model, the value of γs obtained by simulations is close to the experimental data, where their difference is probably resulted from the constant molecular hyperpolarizability tensor in our IP model. In regard to the significance of the XDID mechanism to the OKE of water, two comments are given in the following: First, the smallness of the polarizability anisotropy of water molecule, with a magnitude roughly in one tenth of its isotropic component, results in a weak contribution to the collective polarizability anisotropy of water in liquid phases. It is, therefore, necessary to consider the XDID mechanism, which is the next order beyond the DID mechanism in the expansion of the dipole induced interactions. Secondly, the OKE of a molecular liquid is resulted from the modulation on the collective polarizability anisotropy caused by liquid dynamics, rather than the collective polarizability itself [39]. Thus, it is reasonable to consider the
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P.-H. Tang, T.-M. Wu / Journal of Molecular Liquids 269 (2018) 38–46
contribution from the XDID mechanism, which arises from water dynamics, to the OKE of water. For further investigations, an improvement more close to the reality in the calculation with the non-polarizable water model for the OKE of supercooled water may need to consider the local-structure effect on the hyperpolarizability tensors of individual molecules. Secondly, the physical origin for the stretching exponential decay in the polarizability anisotropy relaxation of supercooled water is still an open but complicated question. Finally, it would be interesting to examine the OKE spectroscopy of supercooled water as crossing over the WL into the LDL phase, which has been shown to be a weakly fragile state through MD simulation studies with the TIP4P/2005 water model for the specific heat under constant pressure condition [93]. Acknowledgment This work is supported by Ministry of Science and Technology, Taiwan R.O.C. under Grant No. MOST 106-2112-M-009-017. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.molliq.2018.07.121. References [1] P.G. Debenedetti, J. Phys. Condens. Matter 15 (2003) R1669. [2] P.G. Debenedetti, H.E. Stanley, Phys. Today 56 (2003) 40. [3] P. Gallo, K. Amann-Winkel, C.A. Angell, M.A. Anisimov, F. Caupin, C. Chakravarty, E. Lascaris, T. Loerting, A.Z. Panagiotopoulos, J. Russo, J.A. Sellberg, H.E. Stanley, H. Tanaka, C. Vega, L.M. Xu, L.G.M. Pettersson, Chem. Rev. 116 (2016) 7463. [4] R.J. Speedy, C.A. Angell, C. A. J. Chem. Phys. 65 (1976) 851. [5] F. Franks, Water: A Matrix of Life, Royal Soc. of Chemistry, Cambridge, 2000. [6] P.H. Poole, F. Sciortino, U. Essmann, H.E. Stanley, Nature 360 (1992) 324. [7] S. Sastry, P.G. Debenedetti, F. Sciortino, H.E. Stanley, Phys. Rev. E 53 (1996) 6144. [8] R.P. Rebelo, P.G. Debenedetti, S. Sastry, J. Chem. Phys. 109 (1998) 626. [9] G. Pallares, M.E.M. Azouzi, M.A. Gonzalez, J.L. Aragones, J.L.F. Abascal, C. Valeriani, F. Caupin, Proc. Natl. Acad. Sci. U. S. A. 111 (2014) 7936. [10] C.A. Angell, Science 319 (2008) 582. [11] W.K. Röntgen, Ann. Phys. 281 (1892) 91. [12] C.M. Davis Jr., T.A. Litovitz, J. Chem. Phys. 42 (1965) 2563. [13] C.A. Angell, J. Phys. Chem. 75 (1971) 3698. [14] V. Holten, M.A. Anisimov, Sci. Rep. 2 (2012) 713. [15] J.W. Biddle, R.S. Singh, E.M. Sparano, F. Ricci, M.A. Gonzáiez, C. Valeriani, J.L.F. Abascal, P.G. Debenedetti, M.A. Anisimov, F. Caupin, J. Chem. Phys. 146 (2017) 034502. [16] A.K. Soper, M.A. Ricci, Phys. Rev. Lett. 84 (2000) 2881. [17] A. Nilsson, L.G.M. Pettersson, Chem. Phys. 389 (2011) 1. [18] L. Liu, S.H. Chen, A. Faraone, C.W. Yen, C.Y. Mou, Phys. Rev. Lett. 95 (2005) 117802. [19] D.A. Fuentevilla, M.A. Anisimov, Phys. Rev. Lett. 97 (2006) 195702. [20] Y. Liu, A.Z. Panagiotopoulos, P.G. Debenedetti, J. Chem. Phys. 131 (2009) 104508. [21] J.L.F. Abascal, C. Vega, J. Chem. Phys. 133 (2010) 234502. [22] F. Perakis, L. De Marco, A. Shalit, F. Tang, Z.R. Kann, T.D. Kühne, R. Torre, M. Bonn, Y. Nagata, Chem. Rev. 116 (2016) 7590. [23] A. Taschin, P. Bartolini, R. Eramo, R. Righini, R. Torre, Nat. Commun. 4 (2013) 2401. [24] J.T. Fourkas, in: M.D. Fayer (Ed.), Ultrafast Infrared and Raman Spectroscopy, Marcel Dekker, New York, 2001. [25] Q. Zhong, J.T. Fourkas, J. Phys. Chem. B 112 (2008) 15529. [26] N.T. Hunt, A.A. Jaye, S.R. Meech, Phys. Chem. Chem. Phys. 9 (2007) 2167. [27] A. Taschin, P. Bartolini, A. Marcelli, R. Righini, R. Torre, Faraday Discuss. 167 (2013) 293. [28] A. Taschin, P. Bartolini, A. Marcelli, R. Righini, R. Torre, J. Phys. Condens. Matter 27 (2015) 194107. [29] A. Taschin, P. Bartolini, R. Eramo, R. Righini, R. Torre, J. Chem. Phys. 141 (2014) 084507. [30] R. Torre, P. Bartolini, R. Righini, Nature 428 (2004) 296. [31] W. Gӧtze, Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory, Oxford University Press, Oxford, 2009. [32] L.C. Geiger, B.M. Ladanyi, J. Chem. Phys. 87 (1987) 191.
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