We present molecular dynamics simulations of the optical Kerr effect in liquid and supercooled water and compare with recent time-resolved Kerr spectroscopy measurements [R. Torre, Nature (London) 428, 296 (2004)]. The short time features of the Kerr response, characterized by peaks near 15, 60, and 160 fs, are weakly temperature dependent. The long-time decay is well described by a stretched exponential with a nearly constant stretch parameter and relaxation times that follow a power law approximately (T-T(S))(-gamma), with T(S)=198.3 K and gamma=2.35. Our findings are discussed in the light of the spectroscopy data and previous simulation analyzes of the structural relaxation in supercooled water.9413780

Skaf, Munir S

Sonoda, Milton T

Physical Review Letters

English

Repositorio da Producao Cientifica e Intelectual da Unicamp

PRL 94, 137802 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 APRIL 2005Optical Kerr Effect in Supercooled Water
Munir S. Skaf* and Milton T. Sonoda
Institute of Chemistry, State University of Campinas, Cedex P. 6154, Campinas-SP, 13084-971, Brazil
(Received 7 July 2004; published 7 April 2005)0031-9007=We present molecular dynamics simulations of the optical Kerr effect in liquid and supercooled water
and compare with recent time-resolved Kerr spectroscopy measurements [R. Torre et al., Nature (London)
428, 296 (2004)]. The short time features of the Kerr response, characterized by peaks near 15, 60, and
160 fs, are weakly temperature dependent. The long-time decay is well described by a stretched exponen-
tial with a nearly constant stretch parameter and relaxation times that follow a power law TTS,
with TS  198:3 K and   2:35. Our findings are discussed in the light of the spectroscopy data and
previous simulation analyzes of the structural relaxation in supercooled water.
DOI: 10.1103/PhysRevLett.94.137802 PACS numbers: 61.20.Lc, 61.20.Ja, 82.53.-kThere is a large body of experimental measurements
showing that supercooled water exhibits critical behavior
typical of fragile glass forming liquids [1]. The tempera-
ture dependence of several properties diverges as a power
law T  TSx for T not too close to TS. A variety of
experimental estimates places the singularity temperature
in the range TS  220–230 K, slightly below the homoge-
neous nucleation temperature (235 K), and exponents that
depend on whether the measurement refers to a thermo-
dynamic (x 0:02–0:35) or a transport property (x
1:5–2:5) [2–4]. The nature of this critical behavior, the
aqueous glass transition, and the putative relationship be-
tween structural, thermodynamic, and dynamic super-
cooled anomalies are subjects of intense research activity
currently [5–11].
Molecular dynamics (MD) simulations and experimen-
tal works have been reported in recent years which suc-
cessfully interpret the dynamics of weakly supercooled
water (T * 250 K) within the framework of the mode
coupling theory (MCT) of ordinary glass forming liquids
[12–14]. The underlying physical picture of the MCT is
that the slowing down of the dynamics of supercooled
states arises from density fluctuation modes over length
scales comparable to the distance between nearest neigh-
bors (transient caging effects). In short, the main pre-
dictions of the MCT are: (1) The long-time decay (
relaxation) of dynamical correlations follows a stretched
exponential form,
Ft  A expt=; (1)
with a temperature independent stretch exponent ;
(2) The characteristic relaxation time exhibits a power
law temperature dependence of the type,
T  0

T
TS
 1


; (2)
divergent at some temperature related to the liquid’s struc-
tural arrest.05=94(13)=137802(4)$23.00 13780Through extensive MD simulations on the extended
simple point charge (SPC/E) potential, Sciortino, Stanley,
and co-workers investigated the time-dependent structural
correlations of supercooled water over a wide range of
temperatures [15], the diffusivity along different isobars
[16], and the spatially heterogeneous dynamics [17] in
connection to the Adam-Gibbs theory of supercooled dy-
namics [3]. The structural relaxation in supercooled states
shows two time separated processes [15], the slowest of
which is identified with the  relaxation. In this regime, the
relaxation is well described by Eq. (1) in which the wave
vector dependence of the parameters A, , and  are nicely
correlated with the oscillations in the static structure factor,
in agreement with the MCT predictions. The first experi-
mental confirmation of these MD predictions were ob-
tained by Bellissent-Funel and co-workers through time-
domain neutron spin-echo spectroscopy, although only two
supercooled temperatures were studied [18]. Other experi-
mental measurements based on time-resolved ultrafast
laser techniques, such as time-domain terahertz (THz)
and Raman induced optical Kerr effect (OKE) spectros-
copies, have been reported in recent years for various
temperatures in the range 270–370 K. The analysis and
interpretation of the observed dynamics differ: The THz
dynamics [19] has been characterized as an intermolecular
structural relaxation with a singular point estimated at
228 K, whereas the OKE relaxation was interpreted as
single particle diffusive reorientational motions [20].
In contrast, Torre et al. [21] have very recently reported
new state-of-the-art OKE measurements, which fully sup-
port the scenario predicted by the MCT. The observed
OKE relaxation rates are notably consistent with the
MCT structural relaxation behavior: Stretched exponential
decay with   0:6, independently of temperature, and a
power law dependence for the relaxation time with TS 
221	 5 K and   2:2	 0:3.
In this Letter, we report MD simulations for the OKE
response of liquid and weakly supercooled water and
compare the results directly to the most recent experimen-
tal Kerr study [21]. We find that the short time features of2-1  2005 The American Physical Society
2
3
4
5
Total
Induced
Cross
Intrinsic
0 0.1 0.2 0.3 0.4
0
1
2
3
4
5
10
0 
x 
k B
 R
nu
c (
t)
 / 
K
-1
ps
-1
323 K
270 K
250 K
215 K
(a)
(b)
PRL 94, 137802 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 APRIL 2005the Kerr response have a limited temperature dependence,
whereas the behavior of the long-time relaxation is well
described by Eqs. (1) and (2), with a nearly constant stretch
parameter  0:6, TS  198:3 K, and   2:35, in very
good agreement with experiments. Our simulations were
performed in the NVE ensemble on systems containing
N  500 SPC=E [22] water molecules at a fixed density of
1 g=cm3 and several (average) temperatures in the range
215–323 K. We use a cutoff at half the box length for
Lennard-Jones forces and Ewald sums for Coulomb
interactions.
The dynamics of interest here is the relaxation of the
polarizability anisotropy expressed by the time correlation
function (tcf) of the off-diagonal elements of the collective
polarizability tensor [23]:
t  hxztxz0i=N
2=15; (3)
with 2  1=21  22  1  32  2  32
being the square of the ideal polarizability anisotropy
and j the principal polarizability components. The col-
lective polarizability tensor,   M I, is comprised
by the sum of gas phase molecular polarizabilities and
induced polarizability contributions computed according
to [24]:
I 
XN
i
X
ji
Mi  Tij  j  
M
i  Tij  j: (4)
The enhanced polarizability tensor and dipole vectors are
obtained by iteratively solving the equations:
 i  
M
i 
X
ji
Mi  Tij  j; (5)
 i  
M
i 
X
ji
Mi  Tij  j; (6)
where Mi , 
M
i , and 
M
i are the gas phase dipole, polar-
izability, and hyperpolarizability of molecule i, respec-
tively. Tij  rirjjri  rjj1 is the dipole tensor between
molecules i and j. The molecular M and M tensors
needed in this scheme were computed from ab initio quan-
tum chemical calculations [25] and are shown in Table I.
The hyperpolarizability contributions are very small inTABLE I. Molecular polarizability and first hyperpolarizabil-
ity tensors obtained from restricted Hartree-Fock electronic
structure calculations at the MP2=6 311Gd; p level.
The z and y directions are along the main symmetry axis and
normal to the molecular plane, respectively. y33 is smaller than
0:05 A=e and the remainders are zero.
M11 
M
12 
M
22 
M
33
 A3 1.04    1.00 1.17
Mx11 
M
x13 
M
x22 
M
y23 
M
z33
 A5=e 0.80 -0.19 0.06 0.05 0.86
13780liquid water, but some degree of nonlinear electronic re-
sponse is incorporated through the ‘‘T’’ term. The
optically heterodyne-detected OKE signal obtained from
the time-resolved spectroscopy provides the intermolecu-
lar dynamics through the nuclear response function, given
by the time derivative of the polarizability anisotropy tcf
[26,27]:
Rnuct  
t
kBT
@t
@t
; (7)
where t is the Heaviside step function.
We start by discussing the short time behavior of the
OKE relaxation, shown in Fig. 1(a) for several tempera-
tures. The response functions show a fast rise to a maxi-
mum at 15 fs, followed by a satellite peak near 60 fs and
a third, broader, local maximum at 160 fs. The remaining
decay is comprised by slowly relaxing components. The
short time features are in agreement with previous simu-
lation results for near ambient conditions [28,29] and
experimental OKE signals [30,31], which report peaks
near 20, 60, and 200 fs. The first peak is mainly due to
inertial and librational motions of the Hydrogen atoms,
while the peaks at 60 and 200 fs have been ascribed to a0 0.1 0.2 0.3 0.4
t / ps
-1
0
1
FIG. 1. (a) Simulated Kerr nuclear responses for different
temperatures. The effect of T on the short time features is
restricted to a sharpening of the peaks without affecting their
positions. (b) Total and separate contributions to the Kerr re-
sponse at ambient conditions. For comparison, the intrinsic and
induced contributions at 250 K are also shown (dotted lines).
2-2
PRL 94, 137802 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 APRIL 2005superposition of librational and hindered translational
modes within the cage of neighbors [31]. This picture is
supported by the behavior of the separate contributions to
Rnuc depicted in Fig. 1(b). The intrinsic part is mainly
responsible for the rapid rise of the response, whereas the
collision induced contribution accounts for most of the
librational and hindered translation peaks. In frequency
domain, such translational motions appear at 60 and
180 cm1 [31,32]. The temperature dependence of the
short time oscillations in Rnuct [Fig. 1(a)] is limited to an
enhancement in the definition of the translational peaks,
which are sharper at lower temperatures. Essentially no
changes in the peak positions are observed, indicating that
the underlying ‘‘intracage’’ motions are not affected by T.
This trend is very consistent with the experimental Kerr
signals obtained by Vöhringer and co-workers [20] and the
earlier light scattering spectra by Sokolov et al. [33].
The slow relaxation of the response is best seen from the
polarizability anisotropy tcf itself. The normalized tcfs
were computed for times up to 12 ps at several different
temperatures and are shown in Fig. 2 (symbols). The t
tcfs are very well described by stretched exponentials
(lines) and our best fit parameters are listed for each
temperature, accordingly. The stretch parameter   0:6,
independent of T within the range of temperatures consid-
ered, is in very good agreement with the OKE experiments
[21]. Other studies, theoretical and experimental, of the
structural relaxation of supercooled water based on the
framework of MCT report very similar values of 
[15,18]. The time scale of the simulated OKE relaxation
corresponds to the early part of the structural  relaxation
and is roughly 2 orders of magnitude smaller than the time
scales involved in previous studies of the intermediate
scattering functions and diffusion constants. This is an0 2 4 6 8
t / ps
0
0.5
1
Ψ
(t
) 
/ Ψ
(0
)
215       81.55       0.62
220       40.34       0.56
225       28.05       0.64
230       17.12       0.62
240       10.71       0.64
245        7.19        0.62
250        5.36        0.65
260        4.67        0.60
270        2.64        0.61
298        1.00        0.59
323        0.65        0.68
T / K τ / ps β
FIG. 2. Collective polarizability anisotropy relaxation for
SPC/E water at   1:0 g=cm3 and different temperatures (sym-
bols). The lines are expt= fits to the data (t > 0:2 ps) and
the best-fitting parameters are tabulated.
13780important feature, not only because of the shorter MD
runs that are required here, but mainly because it has
enabled experimental observation [21] of the structural
relaxation through a technique that works best for times
shorter than a few tens of picoseconds due to the nearly
isotropic polarizability of the water molecule, which
renders OKE signals of very low intensity.
The relaxation time  is strongly temperature dependent
and spans a range substantially larger than the correspond-
ing experimental range, which goes from 0.22 ps at 314 K
to 2.20 ps at 254 K [21]. Nevertheless, the temperature
dependence of  is very well reproduced by the power law
of Eq. (2), with 0  0:242 ps, TS  198:3 K, and  
2:35, as shown in Fig. 3. These results are in good agree-
ment with the time-resolved Kerr experiments (Texp:S 
221	 5 K and exp:  2:2	 0:3). The experimental char-
acterisitc time is exp:0  0:03 ps. The TS obtained from the
simulations is 22 K lower than the experimental esti-
mate. This shift is comparable to the maximum density
temperature difference (  35 K) between experiments
and MD simulations using the SPC/E model under similar
conditions (P  0 MPa). Our results for TS and  agree
remarkably well with the MD results (199 K,   2:73)
reported by Sciortino et al. [15] from the temperature
dependence of the incoherent scattering function and
mean square displacement. Our findings therefore provide
unambiguous support to the interpretation of the Kerr
experiments in terms of an underlying dynamics that is
closely tied to the liquid’s structural relaxation.
In summary, we have reported for the first time an MD
simulation analysis of the optical Kerr effect relaxation for
weakly supercooled water and compared directly to very
recent time-resolved Kerr spectroscopy measurements.
The short time dynamics is quite insensitive to cooling,
whereas the long-time relaxation follows the behavior
predicted by the MCT in the temperature range considered,200 220 240 260 280 300 320 340
T / K
0
20
40
60
80
100
τ 
/ p
s
200 220 240 260 280 300 320
T / K
0
0.2
0.4
0.6
0.8
(τ
 / 
τ 0
)−
1/
γ
τ(T) = τ
0
 (T/T
S
 - 1)
-γ
T
S
 = 198.3 K      γ = 2.35
τ
0
 = 0.24 ps
FIG. 3. Temperature dependence of the Kerr relaxation time
(symbols). The lines correspond to the power law fit to T with
TS  198:3 K and   2:35.
2-3
PRL 94, 137802 (2005) P H Y S I C A L R E V I E W L E T T E R S
week ending
8 APRIL 2005in a remarkable agreement with the experimental observa-
tion. The characteristic parameters , TS, and  we obtain
agree well with the parameters obtained from other de-
scriptors of the structural relaxation in supercooled water,
thus demonstrating that MCT is a useful framework to
analyze the temperature dependence of the post-librational
relaxation as probed by time-domain ultrafast spectros-
copies. The close relationship between the liquid’s re-
sponses obtained from the optically heterodyne-detected
OKE signal and the solvation responses extracted from the
well-known time-resolved fluorescence spectroscopy tech-
nique, opens the interesting possibility of investigating the
structural relaxation of supercooled water and other liquids
by means of fluorescence up-conversion experiments using
soluble dyes as spectroscopic probes.
This work is supported by the Brazilian agencies
FAPESP (01/09374-3 and 03/09361-4) and CNPq. We
thank Professor Renato Torre for sending us Ref. [21].*Corresponding author.
E-mail: skaf@iqm.unicamp.br
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Optical Kerr Effect In Supercooled Water.

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Optical Kerr Effect In Supercooled Water.

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