Progress of Theoretical Physics Supplement No.
103, 1991
61
Relaxations, Fluctuations, Phase Transitions
and Chemical Reactions in Liquid Water
Iwao 0HMINE and Masaki SASAI*>
Institute for Molecular Science, Myodaiji, Okazaki 444
(Received April
17, 1991)
Fluctuations and collective motions in liquid water and their effects on chemical reactions
dynamics are analyzed. Liquid water is a 'frustrated' system with multiple random hydrogen
bond network structures, and has anomalous microscopic and macroscopic properties.
Rearrangement dynamics of the hydrogen bond network induces collective motions of water
molecules and energy fluctuations. Vibrational motions of photoexcited molecules strongly
resonate to these water fluctuations, and thus energy dissipation processes in liquid water are
extremely fast. Time scale, spatial scale and energy scale of the collective motions are
analyzed by examining potential energy surfaces involved. A model of hydrogen bond
network based on the functional integral method is presented and spatial and energy scales
of fluctuations are discussed. Instability of hydrogen bond network is studied to understand
the physical origin of these microscopic and macroscopic anomalies of liquid water.
§ 1.
Introduction
Chemical reactions induce large rearrangements of electronic and atomic (geometric) structures. Solvents yield nonlinear response to reaction dynamics. Intrinsic nonlinear nature of solvents thus appears in electronic and dynamical solvent
effects on reactions. Solvent effects are very different for different solvents. The
·most commonly used solvent, liquid water is the very distinctive solvent. It yields
large solvent effects on many chemical reactions such as SN2 reaction/> photochemical
reaction, 2 > electron transfer reaction, 3 > and acts as an extremely efficient energy
absorber in vibrational relaxation processes. 4 > Liquid water is also known to have
anomalous macroscopic properties; 5 >.s> for examples, the decreases of viscosity with
pressure, large heat capacity, density maximum at 4·c. All these distinct microscopic and macroscopic properties of liquid water are due fact that water is a
'frustrated' system thus involving collective motions and fluctuations.
In the first part of the present review, we deal with energy exchange mechanism
between solvent and excited molecules. 4 >. 7 H> Reaction dynamics of excited state
molecules, usually involving faster and larger amplitude motions, induces more
prominent solvent effects than the ground state reactions. 10 > Two extreme examples
of solvents, liquid Argon and liquid water, are compared as the energy absorber.
Interaction among Argon atoms is very week and so a collision between excited
(reacting) molecule and each solvent Arg'on atom can be regarded to be independent
*> Present address: Department of Chemistry, College of General Education, Nagoya University, Nagoya
464-01.
62
I. Ohmine and M. Sasai
from other Argon atoms and, therefore, so called Isolate Binary Collision (IBC)
model 11 >might be valid. Clear threshold behavior in the energy dissipation is found;
the dissipation rate sharply rises from near no dissipation to quick dissipation when
a reacting molecular motion exceeds a certain threshold amplitude. 9>· 12 >
On the other hand, water molecules mutually form strong hydrogen bonds and
they act as the energy absorber as bulk, quite different from simple collision picture.
Energy dissipation of vibrationally excited molecules was found to be extremely fast
in liquid water: 4>the dissipation rate in liquid water is about two orders of magnitude
faster than in liquid Argon. In water, there exist large energy fluctuations caused by
collective molecular motions. The excited molecular motions resonate these fluctuations and quickly transfer their energy to solvent molecules. Very fast vibrational
energy decays in liquid water also have been observed in many other chemical
processes such as the relaxation process in SN2 reactions/>' 13 >solvated electron thermalization14>'15> and solvation shell reorganization in electron transfer reactions. 16>
The second part of the paper deals with the physical origin of large energy
fluctuations induced by collective motions in liquid water.17)-zo> The collective
motions in water are associated with hydrogen bond network rearrangement. In
order to understand this rearrangement dynamics, the potential energy surface
involved is carefully analyzed. A potential energy surface of a many-body system
consists of numerous number of potential energy wells. Characterization of well
depth, distance between well minima (called inherent structures) and barrier heights
connecting the inherent structures is important to understand the liquid dynamics.
Extensive analysis of the liquid and cluster dynamics in terms of their inherent
structures (potential well minima) was first explored by Stillinger and Weber. 21 >
Amar and Berry22 l have applied the same type analysis to Argon cluster dynamics and
'phase transition'. We applied it to investigate the collective motions in liquid water
dynamics. 18 > It was demonstrated that a clear picture of fundamental structure
changes, collective motions, of liquid water can be obtained as the transitions between
inherent structures. 18 H 0> Classification of inherent structure transitions and the
reaction coordinate, and mode analyses are performed. 19 > The mode excitation
dynamics is carried out to investigate the multidimensional nature of the water
potential energy surface. 20l Upon the information of potential energy surfaces, we
need to know how the system undergoes real dynamics. 23 >
As temperature goes down, spatial scale of water collective motions gradually
expands, time scale slows down, and the macroscopic quantities, such as the compressibility, the heat capacity and the expansibility, are diverging near at the temperature
of spinodal instability lines, for an example, about -46' C at the atmospheric
pressure. 24 H 8> It is important to understand the nature of this divergence and relate
it to the hydrogen bond network instability. In the third part of this paper, a model
of hydrogen bond network is presented to explain growing anomalies of liquid water
as approaching instability line. 29 > Correlation functions and correlation lengths are
introduced to describe collective fluctuations. Mean field solution of the model is
derived to predict the divergence of thermodynamic response functions and their
exponents. The fast and large energy fluctuations in liquid water at the room
temperature arise as the mixture of the collective behavior originate from this insta-
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
63
bility at the low temperature and the independent particle behavior at the high
temperature.
Vibrational relaxation processes in liquids are reviewed in§ 2. Fluctuations and
collective motions in liquid water is discussed in § 3. Sections 4 and 5 deal with the
instability of hydrogen bond network in water. Conclusions are given in § 6.
§ 2.
Chemical reactions in liquids; energy dissipation mechanism in liquid water
Recently many investigations have been made to understand chemical reaction
dynamics in solutions. Molecular dynamics simulations have been performed for
several kinds of liquid phase reactions, such as thermal activated barrier crossing
reaction dynamics/>· 30> electron and proton transfer reactions, 31 > electron relaxation
process in water and other liquids. 14>.Is>. 32 >' 33 > Analytical equation predicting rates for
thermally activated barrier crossing reactions was derived by Kramers fifty years
ago. 34 >' 35 > About 40 years ·after this Kramers' classic work, Grote and Hynes, 36>
Hanggi and Mojtabai,37l Pollak38 >and others have developed linear response theory to
include frequency dependent friction,39l and applied to many reactions. The theory
and its applications are well reviewed by Hanggi et al., 40 >Nitzan 41 >and Hynes. 10 >
We here treat dynamics of optically excited molecules in solvents. Photochemical reactions usually involve fast and large amplitude motions, quite different from
the ground state reactions. Kramers' type theory based on an assumption that a time
required for a reaction is much longer than intra- or intermolecular vibrational
periods. But this assumption often does not hold in excited state reactions. Besides,
large amplitude of motions induce significant reorganization of solvent structure, and
thus their relaxation mechanism is to be
E (kcal/mole)
beyond a simple linear response theory.
To examine solvent response to excited
molecular reaction dynamics, trajectory
calculations4> were carried out for two
simple model systems, (1) an optically
excited ethylene in liquid Argon and in
-10
liquid water, and (2) a simple oscillator
in liquid Argon.
Upon absorption of light, a planar
-20
ethylene in the excited state becomes
vibrationally hot and has an excess
L-----L-----~----~----~- t
0.1
0.0
0.2
0.3
0.4
energy relative to the 90" C=C twisted
configuration, and undergoes the C=C
time (psec)
torsional and C=C stretching moFig. 1. Energy relaxation of an optically excited
tions.4>'42>'43> The vibrational excess
ethylene in liquid Argon and in liquid water.
Ethylene energy (in kcal/mol) vs time (in ps).
energies, in the model calculation we
The time zero (t=O) is chosen as the moment
used here, 4> are about 15 kcal/mol for
of the Franck-Condon transitions. The
both the C=C torsion and for the C=C
energies are relative to the average Franckstretching, respectively. These energies
Condon excitation energies. Averaged over
are dissipated into solvent particles.
30-40 trajectories. From Ref. 4).
64
I. Ohmine and M. Sasai
b
0
....
X0
(1)
Ethylene Vibration
Ethylene Rotation
/d
(2)
.......
lt'~
~.,N\r
Rotational Relaxation
Fig. 2. Indirect energy relaxation mechanism in liquid.
(1) Vibrational energy of an excited molecule is converted to the overall rotation of the excited
molecule by the first collision with a solvent particle and then (2) the rotating molecule collides
with many solvent particles, quickly loosing its rotational energy. This type multistep dissipation
process is dominant in liquid Argon.
The energy dissipation in liquid water was found to be extremely fast, as shown in
Fig. 1. 4 > We can see in the figure that relaxation rate is one to two order of
magnitude faster in liquid water (half decay time is 0.05 ps) than in liquid Argon. A
detailed analysis of energy transfer mechanism showed that the ethylene vibrational
energy flows dominantly into the water inter-molecular vibrations, and not much
(only 1% of the total dissipated energy) goes into the intramolecular vibrations. 4 >
That is, there is resonance between the ethylene and inter water molecular vibrations.
Work done by ethylene to a solvent molecule is equal to a product of the ethylene
force acting on the solvent molecule and the solvent displacement. In liquid Argon,
the force acting on Ar atom cannot become large in the most cases, since the Ar-Ar
interaction is weak. So occasionally occurring ethylene-Ar hard collisions induce
only significant energy transfer. Energy transfer here is not direct (see Fig. 2). 4 >
Since right head-on (colinear) collisions are so rare, a hard collision usually, not
colinear one, induces energy transfer from the ethylene vibration to the ethylene
overall rotations. This first collided Ar atom does not get much energy from ethylene. Successive collisions of the 'rotating' ethylene with many surrounding Argon
atoms result in energy dissipation to the solvent. 44>' 45> This multistep energy transfer
mechanism is dominant in liquid Argon, but is different from a simple Isolated Binary
Collision (IBC) model; 11 > an IBC model usually accounts for only head-on collisions.
The energy dissipation rate predicted with this mechanism is thus much faster than
that by the IBC model.
The energy transfer mechanism is more complicated in liquid water. There are
both mechanisms, a multistep (indirect) mechanism just mentioned and a direct
mechanism. 4> Water-water molecular interaction is strong and puts a restoring force
on a water molecule colliding with ethylene, which results in large ethylene-water
interactions (force). Efficient direct energy transfer thus occurs.
It was found that there exists almost no temperature gradient around a hot solute
molecule, meaning that ethylene energy is distributed very rapidly to the entire
solvent.4 > As we will see in § 3 b, coupled see-saw type energy exchange mechanism
exists in liquid water/ 8 >' 19 > due to strong hydrogen bond interactions and to existence
of 'frustrated' interactions. This see-saw type energy exchanges occur relatively
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
C ~ C Bond Length
65
(Al
:;~.
:E:=
45
0
--,·------------,-------
0
~
u
~
t
~
time (psec)
Fig. 3. Vibrational relaxation of each mode. An
excited ethylene in liquid water. The ethylene
C=C bond length and C=C torsional angle vs
time. Time in ps, the bond length in A, and C
=C torsional angle in deg. The C=C bond
length and torsional angle are averaged over
all trajectories (about 30 trajectories). The
standard deviations from these averaged val·
ues are plotted with the dotted lines. From
Ref. 4).
1.0
2.5
2.0
1.5
Initial amplitude (A)
Fig. 4. Threshold behavior of energy relaxation in
liquid. A vibrational excited oscillator in liq·
uid Argon. Total energy loss of the oscillator
in 3ps (E,oss) vs the different initial amplitude
of the oscillator motion (r,nu). An oscillator
has the same initial excited energy for all
different initial amplitude. Clear threshold of
E1oss is seen at r,nu=l.6A.
slowly in the room temperature, occurring only once in each 10 ps for each molecule
in average. 18 > Driven by a hot molecular motion (note that temperature of an
optically excited molecule is a few thousands degree) these see-saw energy exchanges
are enhanced and the excess energy is quickly distributed over the entire system;
liquid water yields nonlinear dissipation mechanism.
Different modes yield different energy dissipation rates. As can be seen in Fig. 3,
a small amplitude motion, C=C stretching, decays much more slowly than a large
amplitude motion, C=C torsion. 4> Water around apolar sites of a solute molecule
usually form a clathrate cage and so that small amplitude motions do not cause large
interaction with water molecules. It is, however, expected that there is no threshold
behavior of energy decay in liquid water (see Fig. 4 for an oscillator in liquid Argon),
since there exists resonance among solute (ethylene) vibrations and solvent (inter
water molecular) vibrations.
Clear threshold behavior in the energy relaxation is, on the other hand, is seen in
liquid Argon (see Fig. 4 for an oscillator in liquid Argon12 >). We can see that the
energy relaxation rate is almost zero for small amplitude motion, even if the oscillator
has large excess energy, but suddenly increases when the amplitude exceeds a certain
threshold value. 4 >' 12>' 46 > For small amplitude motion system does not make any strong
collisions. With high frequency oscillator motion, solvent A,rgon atom cannot penetrate to get close to the oscillator to make hard collisions; the oscillator expels the
solvent particles by making many fast and small collisions. This 'dynamical caging'
by the oscillator motion is loosened when the oscillator frequency decreases to be about
equal to solvent molecular vibrational frequencies. Then, some solvent particles
occasionally can get close to the oscillator during its oscillation period and induce
hard collisions. 4 >' 12 >' 47 > Thus, some energy is transferred even from very small ampli-
66
I. Ohmine and M. Sasai
ro
tude oscillator motion. A schematic
phase diagram for the oscillator amplitude and frequency dependence of the
energy relaxation rate is expected to be
like Fig. 5.
We have found that the rate of the
energy dissipation from an exited molecule to solvent and the heat conduction
of the excess energy released in reacr thres
tions among solvent molecules are very
Fig. 5. A schematic phase diagram of energy loss
fast in liquid water. Similar fast energy
of a vibrationally hot molecule in a simple
dissipation and heat conduction have
liquid. Shaded areas indicate where finite
been also found in many other reactions
energy loss exists. Wo is an oscillator frequency and Ws is an solvent frequency (high
in liquid water. For examples, Rossky
frequency part). r1n1t is the initial amplitude of
and coworker 14 >and others48 >have theooscillator (see caption of Fig. 4).
retically predicted that time required for
a free electron relaxing to a solvated state is about 0.2 ps, in good agreement with
experimental values obtained by Mingus et al. 49 > and Eisenthal et al. 50 > A similar
process takes much longer time in alcohol solution, about 12 ps observed experimentally.51>'52> Energy relaxation from the top of a energy barrier (the transition state)
to product or reactant in an SN2 reaction was calculated by Gertner et al. 1>' 30>and was
found to be also extremely fast, again order of sub-picoseconds. Charge transfer
reactions and other solvation dynamics in aqueous solutions also yield fast energy
relaxation. 16 > In the next section, we deal with collective motions and fluctuations in
liquid water, which are origin of these fast energy dissipation and heat conduction in
water.
§ 3.
Collective motions and fluctuations in liquid water
Upon the melting from ice to liquid, water absorbs latent heat 80 cal/g, about
These frustrations easily move around in liquid water, since water has (deformed) tetrahedral
bonding structure and individual water molecules undergo facile rotations to change
their bonding partners.l7)- 2o> This is quite different from dipole molecular liquids or
other hydrogen bonded liquids where rotations are more constrained. 53 > Until a
classical work by Rahman and Stillinger54 > in 1971, it had been believed that there
exist long-lived cluster structures in liquid water. The anomalies of water thermodynamic quantities were explained by assuming that water molecules form various
sizes of clusters. 55 > Rahman and Stillinger, 54> however, showed in their molecular
dynamics (MD) study that there are not well classified long-lived clusters in water.
Water consists of a random hydrogen bond network including strained and broken
bonds. The hydrogen bond network alters in picosecond time scale with some kind
of 'collective motions'. 56> One of the most difficult problems to treat liquid dynamics
and to deal with its collective motions is how to extract 'fundamental structure
changes' of systems from those buried with thermal noises in real dynamics.
1.4 kcal/mol, and hydrogen bonds are partially broken ('frustrated').
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
67
In order to find out water collective motions, we performed a trajectory calculation of liquid water at the room temperature, and then analyzed its inherent structures.18)-zo) A standard Molecular Dynamics (MD) method is used for a trajectory
calculation; 216 (or 64) water molecules are confined in a cubic box, a periodic
boundary condition and the minimal image technique are used. Truncated TIPS2
potential57 l is used for water intermolecular interactions. Other empirical potentials,
e.g. central force model, 58 l ST2, 54l MCY potential, 59 l were also examined, but the
results presented here 17 )-zo) were found to be independent of the potential parameters
used. A steepest descent method is used to find local minima (inherent structures).
A liquid potential energy surface consists of many potential wells, each of which
is represented by its minimum, an inherent structure;21 l the inherent structures are the
local minimum geometries on the total potential energy surface. An inherent
structure represents a fundamental water binding structure. A transition between
inferent structures is reorganization of hydrogen bond network. In this inherent
structure analysis, we first calculate the inherent structures successively visited by the
system in a trajectory, and then determined the reaction coordinates connecting the
successive inherent structures. 18 l The system configurations in the trajectory are
quenched in certain intervals to their local minimum geometries (inherent structures)
by a steepest descent method. The multidimensional nature of water potential
energy is then explored by using a normal mode excitation method, exciting normal
modes at several inherent structures with various kinetic energies. 20l Mode relaxation processes are analyzed to find the anharmonic couplings on the potential surface.
Bifurcations of inherent structure transitions are also examined. 20 l
Inherent structures and their transitions
An inherent structure is a local potential energy minimum of the total system, not
of individual molecules. And water is a 'frustrated' system. Thus, there exist a few
very unstable individual molecules at each liquid water inherent structure. An
energy difference between the most stable and most unstable water molecules reaches
up to 20 kcal/mol. Stability and instability of individual molecules alter in time with
water fundamental binding structure change. 18 l
In order to analyze nature of the inherent structure transitions, we calculated an
inherent structure distance matrix defined,
a.
R(t, t') 2=JQ(t)-Q(t')J 2 ,
(3·1)
if R(t, t') 2 is less than a certain threshold value (here 60 A2), then mark a circle in a
map (see in Fig. 6).19l Here Q(t) and Q(t') are the inherent structures corresponding
to the system configurations in the trajectory at timet and t', respectively. A system
with 64 water molecules is used for the analysis. We can see that the matrix consists
of many islands. The inherent structures belonging to the same island, for an
example, t=0.25-0.57 ps, have similar geometries, and thus can be grouped to an
overall inherent structure. The transitions between overall inherent structures
involve large collective motions. A typical example of collective motion in 216 water
molecule system is shown in Fig. 7. 18 ) We can see that a collective motion consists of
a few tens molecular displacements localized in space. It is found that a minimum
68
I. Ohmine and M. Sasai
r-------------------------------~"
... N
·-····
-.........
! . ·-·..•-:"'!
- ...
•!!!~:!
o....o.........______t._o_ _ _ _ _2___J.o"
0
t (ps)
Fig. 6. The distance matrix [Eq. (3·1)] for inherent structures. The matrix R(t, t') is marked in the
figure with an open circle if the distance between the inherent structures Q(t) and Q(t') is smaller
than a certain threshold value, R( t, t') 2 < 60 A 2 • 200 inherent structures in 10 fs intervals are used
(1 fs=10- 15s). The inherent structures successively visited by the system in a trajectory. From
Ref. 19).
y
X
Fig. 7. Collective motions in liquid water (bottom left half). Water molecular displacements in a
transition from an inherent structure to the next inherent structure in a trajectory. In a system
with 216 water molecules. The heavy solid lines indicate the displacements of individual atoms of
water molecules. The 0-H bond of the individual water molecules after the transition are
indicated by the dotted lines. From Ref. 18).
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
69
system size to have the collective motions (and fluctuations) is about 30 water
molecules to be contained. 18 > A collective motion occurs once in sub-picoseconds,
while small inherent structure transitions occur once in every 13 fs, as the average, for
the system with 64 water molecules; 19 > tens of small transitions occur between a
collective transition. As we will see below, an analysis for reaction coordinates,
connecting between the inherent structures subsequently visited by the system, shows
that an average barrier energy for the inherent structure transitions, including all
small and large transitions, is about 3 kcal/mol. 19 > With this energy, an activation
type theory predicts a frequency of inherent structure transitions in good agreement
with that observed in the trajectory. The barriers for the transitions between overall
inherent structures are, though involve collective motions, found to be also small. 19 >
These facts indicate that the water dynamics can be explained as activated processes.
Searching a path, causing large collective motion but with a small activation energy,
in phase space is a bottleneck for the water fundamental network rearrangement
dynamics. Large parts of water dynamics are explained by such a mechanism, but
there are some time domains where mechanism is different.
There are time domains where inherent structures do not form any groups; for an
example 1.9-2.0 ps in Fig. 6. This happens when the system goes through ridges
between deep potential wells. The well minima, the inherent structures, are away
from the corresponding instant structures along the trajectory, and subsequent inherent structures are mutually apart. The energy barriers for inherent structure transitions in this region are sometimes extremely large, as large as 20 kcal/mole. 19 >
Energy contour lines of the potential wells are curved and intricate and the system
just passes through edges of these well, not trapped into; the system behaves as
ballistic. The water dynamics is thus mixture of activated and ballistic processes.
Instant structure energies and corresponding inherent structure energies along the
trajectory are compared in Fig. 8. 60> An instant structure energy at t is averaged
over t-Llt and t+Llt, where Llt=20 fs. We can see that overall fluctuation profile of
~
20
~
10
~
0
~
~
<a
·p
!
'""'
0
20
~
10
>.
ff
~
<a
.,
-10
0
-10
~
<a
0
E-c
~
(a)
10
Time(in ps)
20
~E-c
-20
0
10
20
Time (in ps)
Fig. 8. Fluctuation of the total potential energy of the system along the trajectory. (a) Instant
structure energies (real energies along the trajectory) averaged over some time intervals (see text);
(b) the corresponding inherent structure energies in 10 fs intervals. Time in ps and energies !n
kcal/mol. Along the 20ps trajectory. In a system with 64 water molecules.
70
I. Ohmine and M. Sasai
the inherent structure energies is mostly similar to that of the instant ,structure
energies, except the latter profile amplitude (Fig. 8(a)) is more suppressed than the
former one (Fig. 8(b)). So the system dynamics is well reflected in the inherent
structure transitions, we analyze here.
Reaction coordinates and energy barrier heights
b.
To obtain detail information of potential energy surface, the reaction coordinates
(RC) connecting inherent structures and their transition states (TS) are to be determined.61> There are few methods to find RC and TS for systems with very many
degrees of freedom like liquids. T\vo types of methods are used to determine RC and
TS. (1) Climbing hill method; among them the most well know methods are a method
proposed by Cerjan and Miller62 > and a gradient extremal method developed by
Ruedenberg et al. 63 > (2) Methods searching RC by starting from approximate paths
or transition states (TS); Elber and Karplus64 > first proposed a practical method to
obtain approximate RC for systems with very many degrees of freedom by minimizing
the average value of the total potential energy along a path. Elber et al. proposed a
new iterative method searching for a steepest descent path. 65 > To locate the exact TS
from approximate TS (RC) obtained by the Elber and Karplus method, we can use the
Mciver and Komornicki method66 > minimizing the norm of the potential energy
gradient. Reaction coordinates are then determined as the steepest descent paths
from the exact TS. 67 >
There are advantages and disadvantages to use each type of methods, as discussed by Case et al. 68 >and Elber et al. 65 >' 69 > Climbing hill methods (1) can easily find RCs
with small energy barriers (TS) and were used in the analysis for cluster dynamics by
Wales et aF0 >.n> Liquids dynamics, however, often passes through relatively high
energy barriers, which are sometimes hard to reach with these methods (1), especially
for systems with many degrees of freedom. The second kind methods (2) start
usually from a least motion (LM) path straightly connecting two minima 19 >' 64 >but, if
the initial path is not carefully chosen and far from seeking RC, sometimes end up
with wrong RC or a path going through extra minimum. A second kind method is yet
quite practical to obtain RCs corresponding to the transitions occurred in the real
~ 20
I
(b)
(a)
CD
Ia
Oi
~
20
r
10
I
10
i
:i
I
0
(d)
(c)
0
c.
~
-10
-10
'"'"
R.C.
R.C.
RC.
R.C.
Fig. 9. Total potential energies of the system along the reaction coordinates (RC) connecting between
the successive inherent structures. Energies are in kcal/mol and relative to a standard energy
(arbitrary chosen as -779.74 kcal/mol). In a system with 64 water molecules. From Ref. 19).
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
71
liquid dynamics.
We here only deal with a global feature of the water potential energy surface.
So we use the Elber and Karplus method (2) to obtain approximate RCs, connecting
between the successive inherent structures visited by the system in the trajectory,
without making further improvement for the RC determination. Some examples of
the potential energies along RCs obtained are plotted in Fig. 9. 19 > It is found that the
average barrier height (averaged over 60 TSs so far calculated) is about 3 kcal/mol.
The barrier energies of the transitions between overall inherent structures are found
to be also small; for an example, a TS energy for an overall inherent structure
transition between t=1614 fs and t=1616 fs in Fig. 6, involving large structural
rearrangement, is only 1 kcal/mol. Barrier energy heights along RCs are usually
Gi
0
~
:; -10
u
.¥
>.
2'
Gl
-20
c
w
0
50
100
Time {ps)
Fig. 10. Potential energy of individual water molecule along the trajectory. A water molecule is
arbitrary chosen. Potential energy, U,, in the V-structure (the water binding structure averaged
over fast vibrational modes; see Ref. 18)) is plotted. In the 100 ps trajectory for a system with 216
water molecules. Energy in kcal/mol and time in ps (10- 12s). From Ref. 18).
(b)
(a}
Gi
0
20
j
10
r
~
0
c
i
«i -10
~
Reaction Coordinate
Reaction Coordinate
Fig. 11. Total potential energy of the system (a) and the potential energy of individual water
molecules (b) along the reaction coordinate. In (a) the reaction coordinate energy (RC) and the
least motion path energy (LM) are plotted. In (b) 5 out of total 64 water molecules in the system
are selected. Total energy is relative to a standard energy as zero (see caption of Fig. 9) and the
potential energies of individual molecules are in absolute energy. Energies are in kcal/mol.
From Ref. 19).
72
I. Ohmine and M. Sasai
1/3-1/5 of those along LMs; individual molecular motions are strongly correlated.
Individual molecular potential energies exhibit very large energy fluctuations,
sometimes reach up to 20 kcal/mol, as seen in Fig. 10.m The total system potential
energy yields relatively small energy fluctuation in comparison with these individual
molecular fluctuations, though the total energy fluctuation in liquid water is larger
than those in other liquid systems (note that water has larger heat capacity than other
liquid). This is since there are see-saw type energy exchanges 17>'72 > among individual
molecules (i.e., as a molecule is stabilized, another is unstabilized). 18> The see-saw
type energy exchange, as seen in Fig. 11, causes large individual energy fluctuations
with small total energy change and with small kinetic energy (proportional to the
square of the see-saw motion velocity). We can see, 19 > for an example, in Fig. 11 that
a few individual molecules yield 5-10 kcal/mol potential energy changes, while the
total energy yields small change, 3 kcal/mol along a RC. An individual water
molecule thus becomes unstable, individual molecular energy U, > -12 kcal/mol, i.e.,
its two hydrogen bonds among four are broken, and then usually undergoes rotational
motions to find a new stable bonding structure.m Such destabilization of binding
structure occurs about once in each 10 ps for each water molecule. 18 >
c.
Multidimensional properties of potential energy surfaces
So far we have analyzed, the inherent structures and their transitions along the
trajectory. Liquid water system has many degrees of freedom (e.g., 1293 dimensions
for the system with 216 molecules, and 381-dim for the system with 64 molecules). To
find a reason why the system chooses a certain inherent structure transition in a
trajectory, we need to know multidimensional nature of the water potential energy
surface. 20 > It is to be analyzed (1) how many neighbor inherent structures exist
around an inherent structure (it should be order of exp(aN) where N is the number of
molecules in the system and a is a certain constant)?, (2) what is barrier energy
distribution for their transitions?, and (3) then which are important transitions for the
0
....I'llGl
....0
....0
....>o
·;
c
Gl
Q
0
500
1000
Frequency (cnf1)
Fig. 12. Normal mode distribution. The distribution over 25 inherent structures. Frequency is in
cm- 1 • For the system with 216 water molecules. TIPS2 potential is used. From Ref. 18).
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
73
water dynamics? Only transitions to few neighbors among exp(aN) neighbors can be
energetically accessible at the room temperature? We here make use of the normal
modes at each inherent structure in order to perform such a analysis. The normal
modes of liquid water mainly consist of high frequency, 400-1000 cm-I, rotational
(librational) modes and low frequency, 10-300 cm-I, translational modes. 18J' 19 J,nJ
There are some intermediate frequency, 300-400 em-\ modes localized in space.
Their motions are mixture of mainly translations and some rotations. There are
also few modes fairly localized in bond defects area in the lowest frequency region. 19 J
The mode distribution is shown in Fig. 12. Taking several inherent structures as
initial geometries, we performed many trajectory calculations by exciting individual
modes with different kinetic energies. 20 J' 74 J Different trajectories for different modes,
different kinetic energies, and different inherent structures. Some simultaneous
excitations of two or more modes were also performed. We analyzed how the
excited modes relax through mode mixing and the system undergoes inherent struc·
ture transitions after these mode excitations.20 J Figure 13 shows a typical mode
energy profile when a high frequency mode is excited; here i=lO mode (the lOth
highest frequency mode) is excited. This high frequency mode is small in amplitude,
couples only one or two near frequency modes, and so exhibits the energy recurrence,
yielding almost no decay even after 40 ps. Most of the water normal modes are of
this type 'inactive' modes, unless they are excited extremely large initial energies.
Localized modes in the intermediate frequency region, on the other hand, yield very
fast energy decay, as shown in Fig. 14 for the i=188 mode. The localized modes
couple with many double frequency modes. This can be seen by analyzing a spectrum of displacement correlation function
~
6
8
10
12
14
16
18
Mode
1-----------
0
•
2
I
Tme(ps)
Fig. 13. Mode energy relaxation in the mode ex·
citation trajectory. A high frequency mode
(i=10, the lOth highest mode; 924 cm- 1) at a
certain inherent structure is excited with the
kinetic energy comparable to twice of the room
temperature. The 40 ps trajectory. A num·
ber of 'Mode' axis indicate a group number of
each 20 modes; '1' indicates for the 1-20 modes
counting from the highest frequency, '2' indi·
cates for the 21-40 modes, ···, '19' for the
361-380 modes and 20 for 381 only). Note 381
degrees of freedom for a system with 64 water
molecules in a periodic cubic box. Here the
mode group '1' does not include the excited
mode i = 10. Each curve indicates the total
mode energy summed over all 20 modes includ·
ed in each group. A mode energy includes
both kinetic and potential energy (see Ref. 20)
for the detail). Energy of the excited i=10
mode is plotted as indicated. In the top of the
figure, the mode mixing energy is plotted.
Time in ps and energies are relative to the
excitation energy, E,=lo (t=O), as 1. From
Ref. 20).
74
I. Ohmine and M. Sasai
~ 0.0
Q)
"C
::I
1.0
.~
l
1=188
0.5
0.0
0
500
1000
Frequncy w
Mode
Fig. 14. Mode energy relaxation in the excitation
of an intermediate frequency mode. A localized mode (i=188; the 188th highest mode) is
excited with the kinetic energy comparable to
the room temperature. See caption of Fig. 13.
From Ref. 20).
S(w)= jc:Jr(t) · Llr(t + r)>eiwrdr,
Fig. 15. Fourier spectrum of the displacement time
correlation function [Eq. (3·2)]. For the normal mode excitation trajectory of i=188 at an
inherent structure with the room temperature
order excitation. Frequency w is in em-'.
Correlation function is calculated over 20 ps
and so the frequency resolution is about
1.5 cm- 1• Square amplitude of Fourier
transformation is plotted. Peak heights are
normalized as the highest one is 1. From Ref.
20).
(3·2)
where Llr(t) is the system geometry deviated from the initial inherent structure at
time t and the average< ) is over the time t. The spectrum in Fig. 15 has a peak at
the frequency of the initially excited mode and several peaks near at its double
frequency. Thus, this fast relaxation takes place through 1: 2 Fermi resonance.
There are many modes around the double frequency, but only those modes yielding
strong atomic overlaps with the initially excited mode mainly accept the excited mode
energy. Excitations of the localized modes induce inherent structure transitions
when the initial excitation energies exceed certain threshold values.
The lowest frequency mode excitation easily induces inherent structure transitions. Some inherent structure transitions here are the same as those induced by the
localized mode excitations of intermediate frequency. Indeed, RCs of these inherent
structure transitions consist of vector elements of the lowest frequency modes and the
localized modes. 19l Exciting a certain mode in this lowest frequency region with
different initial energies often results in different inherent structures; bifurcation of
transitions occurs. With small energy excitations, transitions involving very localized molecular displacement in space such as shown in Fig. 16(a) are usually induced.
With larger energy excitations, nonlocal collective transitions are induced (Fig.
16(b)).
Only about 5% of whole modes, the localized modes and the lowest frequency
modes, can cause inherent structure transitions. Only few neighbor inherent struc-
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
(a)
~/"">..r(\!>
y
'f' _)
k
<tv-
~-( ( /-:
"'~t<l
k ~-4
(J "\
--\-I
-I
/"
X
I
)('
~~ ~~(.J)
y
('
(
~
75
(b)
/t$tt'
)~-((~
~'rt<-,
~~
)~
-~ r.-J
/"
.j/j\
X
Fig. 16. Bifurcation of inherent structure transitions. Molecular motion along the RC of an inherent
structure transition caused by the mode excitations. Exciting the same mode (a) with small
excitation energy; (b) with the larger excitation energy. Different energy excitations of the same
mode result in different inherent structure transitions. The molecular geometries at ten equidistance grid points along RC (see Eq. 1 in Ref. 18) for the definition) are plotted [there are 12
configurations for each molecule; one initial, one final and ten intermediate grid points]. Dotted
line indicates OH bonds of individual water molecules. Thick solid lines indicate those at the
initial inherent structure. Allow indicates major atomic movements along RC. All 64 water
molecules are projected on (x, y) plane.
tures can be reached from each inherent structure by mode excitation trajectory
calculations carried out so far, exciting each mode or some of their combinations with
initial kinetic energies comparable to the room temperature. More intensive study
analyzing high order mode mixing in high energy dynamics, however, is required to
obtain definitive distributions of high energy barriers and available neighbor transitions.
Recently, we have examined the power spectrum of the total potential energy
fluctuation of liquid water. Power spectrum of the inherent structure energy change
along the trajectory yields so called 1/P frequency dependence (! is frequency). 75 J
This spectrum is strong contrast to spectra of simple liquids; for an example, liquid
Argon exhibits a near white spectrum, 1//0 frequency dependence. This fact indicates that there exists extended multiplicity of hydrogen bond network configurations
in liquids water. Details will be presented elsewhere. 60 J
§ 4.
The functional integral model of hydrogen bond network
In preceding sections we saw that fluctuations of water molecules are strongly
nonlinear and collective and far different from simple atomic liquids. It is, then,
interesting to inquire how these molecular fluctuations are related to observed
macroscopic data of liquid water. Water has been known as a peculiar liquid
because its macroscopic properties are so much different from other atomic or
nonpolar molecular liquids. 5 > Density maximum at 4°C at 1 atm, large value of heat
capacity, and decrease of viscosity with increasing pressure are few examples of
76
I. Ohmine and M. Sasai
many unusual properties. These peculiarities, however, become more evident when
water is supercooled below the freezing temperature. 24 J,ZSJ, 76 J' 77 J Especially important
is the fact that heat capacity, Cp, expansivity, a, and compressibility, Kr, show the
diverging behavior,25 J
X =Ax( T/Ts-1)-r + Bx,
(4·1)
where X is Cp, -a or Kr, Ax is a constant, and Bx is a nondiverging background part
of X. Ts is about -46·c at 1 atm, which is slightly below the homogeneous nucleation temperature, TH~ -41·c, at which water freezes homogeneously to be ice.
Using r~ 1/2, Eq. (4 ·1) can fit the experimental data with the suitable choice of Bx. 78 J
Dynamical properties are also unusual around Ts as
X=Ax(T/Ts-1) 8 ,
(4·2)
where X represents the self-diffusion constant, D, inverse viscosity, 1/TJ, or inverse
rotational correlation time, 1/r2, and 8~1.6-1.8.77) The behavior like Eq. (4·2) is
predicted by the mode-mode coupling theory on the glass transition of supercooled
liquids. 79 J Equation (4 ·1), however, clearly shows that anomalies of water are owing
not to the glass transition but to some other thermodynamic instability. Since Ts is
close to TH for the wide region of pressure, the most plausible explanation is that Ts
is the spinodal limit for freezing. From this argument Speedy suggested that
Eq. (4 ·1) should be similar to the one observed near the spinodal instability for boiling
in the superheated region. 78 J
The cooperative fluctuations of hydrogen bond network should be the key to
understand the instability. This can be seen, for example, in the success and the
failure of the random network model of Sceats and Rice. 6l They described liquid
water as a thermally vibrating quasi-static network. This model explained the
properties above the freezing temperature with an excellent agreement with the
experiments, but failed to explain the instability. They argued that this failure is
because the cooperative fluctuations associated with the reorganization of the network are not taken into account in their model. 6 J In order to take account of the
network reorganization effects, the partition function should be calculated by summing over all topologically different network patterns with the proper statistical
weight. This is a highly nontrivial combinatorial problem and difficult to be solved.
The partial summation of network patterns were calculated by Dahl and Andersen80 J
with the cluster expansion method. Their theory also explained liquid water above
the freezing temperature but could not explain the instability. Thus the summation
of network patterns based not on the perturbational approach should be desirable.
Such a nonperturbative model was developed by Stanley and Teixeira81 J by using the
percolation theory. They revealed that four-bonded molecules distribute not
homogeneously but in a correlated way. In the Stanley and Teixeira model, however, thermodynamic quantities are directly related to the nondiverging geometrical
properties, so the divergence in the supercooled water could not be explained.
Thus many theoretical efforts have been devoted to elucidate the instability and
important insights have been gained6J' 27 J,soJ-seJ but the mechanism and origin of the
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
77
instability are still open problems and lots of questions remain to be answered: What
is the order parameter? What are the correct values of exponents, r and a? How
the characteristic time scale and the correlation length of fluctuations grow? In this
and next sections we go into rather detail to discuss these questions. In § 4 we
explain a new statistical mechanical model of the hydrogen bond network which was
introduced in Ref. 29). We develop the technique similar to the one used in the study
of the network polymer. 87l-s9> Entropy and energy are calculated on a unified basis
with the nonperturbative approach. In § 5 we discuss the application of this method
to the problem of the instability. We here concentrate on the static thermodynamic
properties.
First we should clarify the meaning of the words "network" and the "summation
over network patterns". This can be done through the idea of quenching which is
discussed in§ 3. The system of N water molecules is described by a 9N-dimensional
coordinate vector, x=Cx1, xz, ···, XN). By using the steepest descent technique, the
instant (real) structure at time t, x(t), is quenched to be an inherent structure, x(a).
This is a map M(x(t))-+a, where a is an index to distinguish inherent structures.
Different instant structures x(t) and x(t') may be mapped into the same inherent
structure if It- t'l is not so large. Thus the basin B(a) is defined as a set of instant
structures which are mapped into the a-th inherent structure. Then the 9N-dimensional space represented by x is decomposed into the discrete inherent structures as
L!aB(a). 2 1l The partition function for N molecules is also decomposed as
(4·3a)
where AH and tto are the thermal deBroglie wavelength of H and 0, 2NN! is the factor
arising from the permutation, V(x) is the potential energy, Va is the energy of the
inherent structure, and la is the vibrational free energy defined by
e-Pl·= (
dxexp(-,BV(x)+,BVa).
J:rEB(a)
(4·3b)
In instant structures, due to thermal vibrations, the hydrogen bond is rather largely
distorted and there is no clear distinction between the "connected" and "disconnected"
bond. 5> In inherent structures, however, the distinction becomes fairly clear and the
hydrogen bond is defined in a reasonable way with the energetic or geometric criterion.18>'21> Thus the inherent structure can be regarded as the well defined network of
hydrogen bonds and the water molecule in the inherent structure can be regarded as
a "vertex" of the network. In the following discussion, the grand canonical ensemble
is more convenient. Thus, from Eq. (4·3), the grand canonical partition function is a
summation over the network patterns,
8
= L!
( network)
patterns
AH -BNAo -sN exp(- ,8( Va + Ja)) ,
ePW
(4 ·4)
78
I. Ohmine and M. Sasai
where ~<network patterns>=~N~a(2NN!)- 1 . (Notice that the definition of f.l is different
from that in Ref. 29).)
Equation (4 ·4) is an exact representation of liquid water in the classical limit.
We here introduce the idealized model of the network in order to evaluate Eq. (4·4)
in an approximate way. We assume that each vertex (water molecule) has four
"functional sites", two of them are on hydrogen nuclei (hydrogen sites) and two others
are on the oxygen nucleus (oxygen sites). We refer to the functional sites which are
not involved in the hydrogen bond as "nonbonding sites". We assume that the
network pattern is specified by the number of hydrogen bonds, Nb, the number of
nonbonding-hydrogen sites, NH, and the number of nonbonding-oxygen sites, No. We
also assume that two oxygens at positions x andy interact with the potential v(x, y)
regardless of whether two molecules are hydrogen bonded or not. We thus approximate Va by the sum of the contribution from hydrogen bonds, Nbvb, and the contribution from the oxygen-oxygen interactions, Wa=~v(x, y),
(4·5)
where Vb is the energy gain due to the hydrogen bond formation and Wa includes the
oxygen-oxygen short range repulsion, the van der Waals interaction, and the long
range Coulomb repulsion. In the same spirit, we approximate ]a as
(4 ·6)
Equation (4·6) is equivalent to the assumption that vibration modes are determined
only by the local coordination number of the molecule. In this simplified model of
Eqs. (4·5) and (4·6), the grand canonical partition function is
(4·7)
where
A= AH - 6Ao- 3exp(/3(f.l- jN)) ,
B=e-P.m,
C=e-P.m,
K=exp(-!3(vb+jb)).
Analyses on MD calculations show that most of the molecules are 4 or less coordinated.26> Thus we assume that each hydrogen site can be bonded with at most one
oxygen site of the other molecule and that each oxygen site can be bonded with at
most one hydrogen site of the other molecule. Then there are 0, 1, 2, 3 or 4 coordinated molecules but there is no 5 or more coordinated one. This assumption leads to the
relation
(4·8)
so that only three of N, Nb, NH and No are independent. If we take Nb, NH and No as
independent variables, then Eq. (4·7) is
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
H
v
79
H
where ro=A- 112K-I, n =A - 114 BK- 1 and
r2=A-li4CK- 1. Equation (4·9) is the
starting point of our network model.
Summation over the network patterns in Eq. (4·9) can be carried out by using the
functional integral method. 89 ) We first consider the easier case that Wa=O. We
define the scalar field variable ¢(x) and its complex conjugate ¢(x )* at the spatial
position x. In Ref. 29) the n-component vector field was used but the scalar (n=1)
field is enough if the different clusters in the disconnected graph are not distinguished.
Corresponding to the water molecule at position x, the vertex, X(x), is introduced,
Fig. 17. Representation of the water molecule
with field variables, </J and¢*.
(4 ·10)
This correspondence is shown in Fig. 17; ¢(x) represents the hydrogen site and ¢(x)*
represents the oxygen site. The factor 1/4 is a symmetry factor coming from the
exchange between X1 and X2 and between X3 and X4. w(x; X1X2X3X4) represents the
structure of the water molecule and has nonzero value only when the following
geometric condition is satisfied,
w(x; X1X2X3X4)=1 when all the following conditions are satisfied:
(1) lx~-xl=lx2-x1 and lx3-xl=lx4-xl,
(2) four vectors, x1-x, x2-x, x3-x and x4-x are pointing
toward the tetrahedral vertices ,
(3) lx~-xl+lx3-xl~the average length of hydrogen bond
~2.sA
=0 otherwise.
(4 ·11)
Thus the integral in Eq. (4·10) represents the summation over all possible
configurations of the water molecule located at x. We then define the functional
integral average,
Jn¢D¢*(···)exp(- jdxro¢(x)¢(x)*)
(4·12)
jD¢D¢*exp(- jdxro¢(x)¢(x)*)
where D¢D¢*= IlxdRe¢(x)dim¢(x). We have
<¢(x)¢(y)*>o=(1/ro)o(x-y),
<¢(x)¢(y))o=<¢(x)*¢(y)*>o=O.
(4 ·13)
In Fig. 18 we show an example of the small·patch of the network. Using Eq. (4 ·13),
the statistical weight of this network can be calculated by operating the average
80
I. Ohmine and M. Sasai
Eq. (4 ·12) to the product of ¢'sand 1/J*'s.
Thus the summation of network patterns
is calculated by operating Eq. (4 ·12) to
the polynomial of ¢ and ¢*. Equation
(4·9) with Wa=O is expressed as
E( W=O)=<I1(1 + X(x))(1 + r1¢(x)*)
X
X
(1 + r2¢(x))(1 + r1r2/ro)- 1>o.
(4 ·14)
Using Eq. (4 ·13), Eq. (4 ·14) is transformed to
E(W=O)=R fD¢D¢*e-i,
F=
Fig. 18. A small patch of the hydrogen bond network with N=4, NH=4, Na=4, and N.=4 (a)
and its representation with field variables (b).
The statistical weight of this network is
<r1 tjJ*(xJ) r1 t/J*(xz)X(xa) r1 t/J*(xa)rzt/J(x.)X(x.)
X r1 tjJ*(xs) rzt/J(xs)X(xc) rzt/J(X7) rzt/J(xs)X(xd)>o
= ( rdro) 4 ( rz/ro) 4 (1 /ro) 4 •
Jdx{ro¢(x)¢(x)*- r1¢(x)*
- r2¢(x)-log(1 + X(x))},
(4 ·15)
where R is a Gaussian integral and readily calculable.
Next we consider the case Wa=FO. The oxygen-oxygen interaction v(x, y) has to
be taken into account. Since the water molecule is represented by X(x ), this can be
done with the cumulant expansion, 90 >
exp{ \Jdxlog(1+X(x)))cJ
=exp{jdxlog(1 + X(x))+
+
~ jdx jdyv2(x, y)log(1 + X(x))log(1 + X(y))
~ jdx jdy jdzv3(x, y, z)log(1 + X(x))log(1 + X(y))log(1 + X(z))+--·},
(4·16a)
where vn(Xl, ···, xn) is the n-th cumulant interaction due to the n-body interaction,
v2(x, y)=exp(- /3v(x, y))-1,
V3(x, y, z)=exp(- /3v(x, y)- /3v(y, z)- /3v(z, x))-3exp(- /3v(x, y))+2.
(4·16b)
Using Eq. (4·16), we have
E=R fn¢D¢*e-F,
81
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
F= J dx{ro¢(x)¢(x)*- r}lj1(x)*- rzt/l(x)}- \Jdxlog(1 + X(x))
)cv.
(4 ·17)
The partition function of the network is now expressed by the functional integral of
field variables with the effective action F. Equation (4 ·17) is a formally correct
expression of our network model of Eq. (4·9).
From Eq; (4 ·17), many kinds of physical quantities are calculated. For example,
PV=ksTlogE,
S=ks(1-/3 a~)logE,
alog8
li.T><lVH
-Yl
arl
(
( )
)dx(pH X>,
(No>=rz
alogE
arz
(4 ·18a)
where <· · · > is the average with F
jD¢D¢*(···)e-F
<···>
(4 ·18b)
jD¢D¢*e-F
and PH(x), Po(x) and Pm(x), are the local density of nonbonding-hydrogen sites,
nonbonding-oxygen sites, and water molecules, respectively,
PH( X)= Y1 t/J(x)*- Y1 rz/ro
,
Po(x)=rz¢(x)-rlrz/ro,
(4·18c)
Various correlation functions are also interesting; <Pm(x)pm(Y)> gives the distribution
function of water molecules, (pH(x)po(y)> corresponds to the probability to find the
nonbonding-hydrogen site at x and the nonbonding-oxygen site at y. Since PH(x)
- Po(x) corresponds to the local polarization, <(pH(x)- po(x))(pH(Y )- Po(Y ))>gives the
information on the static polarizability of liquid water. In Eq. (4·17) r1 and rz can be
regarded as "sources" or "external fields" to act on ¢*and ¢. These sources control
the ability to form nonbonding sites by disconnecting hydrogen bonds. Therefore the
effect of solute ions should be included in the value of n and rz. When r1 and rz
depend on the spatial position, r1(x) and rz(x), the hydration structure around solute
ions can be studied through response functions, (pH(x)pH(y)>, (po(x)po(y)> and
(pH(x)po(y)>. The field theoretic description of the hydrophobic interaction is also
an interesting problem to be pursued. Thus, in this functional integral model, many
important problems are translated into the language of field variables ¢ and ¢*.
Lastly we comment on the similarity between the present model and the model of
the network polymer. When N/V is small, the log term in Eq. (4·17) can be truncated
at the order of O(X2). The resultant action F has the structure similar to the one
used in the study of the sol-gel transition. 89l Though the field theoretic models made
82
I. Ohmine and M. Sasai
a great success in polymer physics, it has been known that the major drawback of the
polymer field theories is their inability to control the chain length distribution; the
chain length and the network size depend on the monomer concentration and temperature.87> We should stress that in liquid problems, on the contrary, this dependence of
the network size on temperature and other physical quantities is not a drawback but
the desired property of the theory.
§ 5.
Instabilities of hydrogen bond network
In order to apply Eq. (4·17) to the anomalies of cold water, we here introduce the
lattice-gas approximation. The network is assumed to be embedded in the lattice
and the field variables rf;(x) and rf;(x)* are defined only on the discrete lattice sites.
There are two merits to use the lattice-gas method. First, w in Eq. (4·11) has a
nonzero value only when the proper combination of lattice sites are chosen. Equation (4 ·10) becomes the sum of finite number of configurations, which is much easier
to treat than the original integral form. Second, the strong short range repulsion
between oxygen atoms is already taken into account by the restriction that molecules
can occupy only the discrete lattice sites. Then v(x, y) includes the weak residual
interactions and thus Vn in Eq. (4·16) can be expanded by the small quantity (3v. By
taking only the first order of (3v, we have v2(x,y)-:::::-/3v(i,j), and higher order
cumulants can be neglected, Vn(x, y)-:::::0 for n:;:::3.
The body-centered-cubic (bee) lattice is most commonly used to study water. 91 >
When all the bee lattice sites are occupied by water molecules, the high density ice,
ice VII or ice VIII can be formed. When the half of the sites are occupied, the low
density cubic ice, ice Ic can be formed. Ice Ic has a diamond structure and its
physical properties are very similar to the usual hexagonal ice, ice Ih. Therefore ice
Ic is used as an approximation of ice Ih. Density of liquid water is in between ice VII
and ice Ic, so that the bee lattice can represent as many phases as gas, liquid,
high-density ice, and low-density ice. In Figs. 19 and 20 we show the bee lattice used
in our study, where the lattice constant is a, and the number of the bee lattice sites is
2No. Here we introduce additional sites (ij) and (ji) on the line between the i-th site
m
Fig. 19. The bee lattice used in the lattice-gas
model of liquid water. i is the nearest neighbor of j, m is the 2nd neighbor of j, and k is the
3rd neighbor of j.
(a}
(b)
Fig. 20. Four sublattices in the bee lattice, the
lattice 1 (0), the lattice 2 (.0.), the lattice 3 (e),
and the lattice 4 (.&) (a), and the diamond
structure formed when the lattice 1 and the
lattice 3 are occupied and the lattice 2 and the
lattice 4 are unoccupied (b).
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
83
and its nearest neighbor j-th site_ The position of the site (ij) is chosen to be close
to the i-th site_
We define¢ and¢* on sites (ij). Equation (4·17) is now
(5·1)
E=R Jn¢D¢*e-F,
+ X(i))
F= "2.{ro¢(ij)¢(ij)*- r1¢(ij)*- r2¢(ij)}- "2.log(1
i
ij
j)log(1 + X(i))log(1 + X(j)),
+(P/2)"2,"2,v(i,
i j
(5·2)
where D¢D¢*=[L;dRe¢(ij)dim¢(ij), and Eqs. (4·10) and (4·11) are
X(i)=
~ i.tt,m w(i; jklm)¢(ji)¢(ki)¢(il)*¢(im)*
(5·3)
and
w(i; jklm)=1 when all the following conditions are satisfied;
(1) j, k, l and m are nearest neighbors of i
(2) j is the 3rd neighbor of k
(3) l is the 2nd or 3rd neighbor of m
(4) l or m does not belong to the same lattice plane which contains j and k
(5·4)
=0 otherwise.
Here X and w are defined so as to reproduce the geometrical situation of the
hydrogen bond network. Equation (5·3) is the sum over 72 configurations. This is
not the unique way to define X and w but other alternative definitions are possible.
As approaching Ts, however, the characteristic length of fluctuations becomes much
larger than a, so that differences in the detailed definitions of X and w are irrelevant
to the large scale behavior near Ts.
Using the mean-field (steepest descent) approximation, we show that the liquid
phase in the lattice-gas model of Eqs. (5·1)~(5·4) becomes unstable both in the
supercooled region and in the stretched (P< 0) region. Steepest descent values
<¢(ij)>mr and <¢(ij)*>mr are obtained by taking variations,
aF
a[Rerf;(ij)]
aF
a[Imrf;(ij)]
0.
(5·5)
When r1=r2=0, Eq. (5·5) has a trivial solution, Re<¢(ij)>mr=Im<¢(ij)>mr=O. In
liquid water, however, r1 =FO and r2=FO and in pure water without solute ions we should
put n=r2. Then symmetry breaking sources, r1 and r2, make Re<rf;(ij)>mr=FO. We
put Im<¢(ij)>mr=O. In the liquid phase there is no long range structural order and
therefore Re<rf;(ij)>mr should not depend on the position, Re<rf;(ij)>mr=M/4, where the
factor 4 is introduced in order to make the coefficient so in Eq. (5 · 6) to be of order of
one. Thus M is determined from aFo/aM =0, where Fo is
84
I. Ohmine and M. Sasai
~::wM·
-0.4-2 0 2
Fo /No= roM 2 - 8 rrM- 2log(1 + soM 4 )
+!3vm[log(1+soM 4 )]2,
(5·6)
where
so=72/4 4 and Vm=(1/2No)"2.iiv(i, j).
In Fig. 21 we show Fo/No as a function of M with various values of ro. In
general Fo/No has three minima at Ma
>0, Mb>O and Mc<O. From Eq. (4·18)
-2
0
2
we see that M < 0 means <NH> < 0 and
<No><O, so that the minimum at Me has
no physical meaning. Ma is propor1.0
(c)
tional to the "external field", Ma ~ rdro
o.o_2
o
2
M
but Mb is not directly related to rdro, Mb
~(1 + f3vm)- 114 • Thus the minimum at
Fig. 21. The mean-field free energy Fo(M)/No.
Mb can be regarded as the "symmetry
Parameters are kBT=0.5 kcal/mol, r1=r2=Z
xlo-•;r., v(i,j)=-O.lkcal/mol for nearest
breaking" state. With the parametrizaneighbors, v(i, j)=0.2 kcal/mol for 2nd neightion which is adequate for low temperabors, and v(i,j)=O for others and ro=0.36 (a),
ture, we can see Ma<{Mb ~ 1, so that
ro=0.52 (b) and ro=0.74 (c).
from Eq. (4 ·18) <N>, <NH> and <No> are
much smaller at Ma than at Mb. On the analogy of the theory of the network
polymer, the high density phase at Mb is the collapsed phase with the network
percolated through the system. Thus the dilute phase at Ma corresponds to the gas
phase and the dense phase at Mb corresponds to the liquid phase.
In Fig. 21 the energy of the liquid phase is shown to increase as ro increases.
This is because, as shown in Eq. (4·9), the increase of ro is the decrease of the bond
forming tendency. At the certain point (Fig. 21(b)) the first order transition from
liquid to gas takes place, and by further increasing ro, liquid becomes unstable against
the small fluctuation toward gas (Fig. 21(c)). Since F(Ma)~O in the low temperature
region, this spinodal transition takes place with the condition F(Mb)>O. The rough
relation PV ~ ks Tln{exp- Fo} tells us that the spinodal transition toward gas takes
place in the negative pressure (stretched) region.
The partition function is evaluated by taking account of the Gaussian fluctuations
around the mean field,
(a)
::LL(b)
2.0u
F= Fo+
~ ~x o¢(i;;~(kl) 1~!6"~o. ¢(ij)¢(kl)
+ ~x
+
o¢(ij~i;"(kl) 1~!6"~0. ¢(ij)¢"(kl)
~ ~x o¢"(s;~"(kl) 1~!6"~0. ¢"(ij)¢"(kl)'
(5·7)
where ¢(ij)=Re¢(ij)-Mb/4 and ¢"(ij)=Im¢(ij). ¢(ij) represents the fluctuation in
the concentration of nonbonding sites and ¢"(ij) represents the fluctuation in the
direction of the hydrogen bond or the position of the hydrogen nucleus along the
85
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
hydrogen bond. Positions of hydrogen nuclei are random in both liquid and solid
phases, so that the ¢"(ij)-fluctuation does not become singular at the liquid-ice transition or the liquid-gas transition. Thus ¢"(ij) should be safely integrated out,
(5·8)
In Ref. 29) Fo and Fij,kt were obtained in the approximated way by renormalizing the
parameters ro, r1 and rz.
The instability of the present lattice-gas model is demonstrated with the method
similar to the theory of the antiferromagnetic transition: In the supercooled region,
fluctuations which have the same wavelength as the lattice constant of ice would be
most significant. In the lattice-gas model these are the fluctuations with the wave
vector k=(± rr/a, ± rr/a, ± rr/a). To extract out these fluctuations we decompose the
bee lattice into four sub-lattices, the lattice 6, 6=1, 2, 3 and 4. The four sub-lattices
are designated in Fig. 20 with four different symbols. Corresponding to these sublattices, ¢(ij) is decomposed into four components, ¢"(ij).
4
¢(ij)= L: ¢"(ij)'
(5·9)
11=1
where ¢"(ij) has nonzero value only when the site i belongs to the lattice 6. When
fluctuations become anomalously large, correlation length of fluctuations should be so
much larger than a. We are then allowed to take the cqntinuum limit, a~ 0 and
¢"(ij)~ ¢"(x). Thus Fmr is
(5·10)
See Ref. 29) for the explicit values of
unstable when
!11v
det(JI!v)=O.
and
ff11v.
The lattice-gas model becomes
(5·11)
The solution of Eq. (5·11) determines a line on the temperature-pressure (T-P) plane.
We call this line the instability line, which is a boundary of the stability region for the
liquid phase. There are two instability lines which satisfy Eq. (5 ·11), T1(P) and
Tz(P). T1(P) lies in the stretched region and the instability which takes place at
T1(P) is the liquid-gas spinodal transition discussed in the paragraph below Eq. (5·6).
Tz(P) lies in the supercooled region and should be identified with Ts in Eq. (4 ·1). By
neglecting v(i, j) farther than the 3rd neighbors, we have kBTz~6vz, where Vz is v(i, j)
between the next neighbor sites. Due to the repulsive Coulomb interaction, v2>0 and
thus vz is the driving force of the instability at T2.
We define the uniform field, B1(x), and the staggered field B2(x),
B1(x)=¢1(x)+¢2(x)+¢3(x)+¢ix),
Bz(x)=(¢I(x)-¢2(x))/2.
(5·12)
B1(x) represents the fluctuation in the concentration of non-bonding sites and B2(x)
86
I. Ohmine and M. Sasai
represents the fluctuation of the extent
of the ice-likeness. In the long-wavelength limit, correlation functions are
P(MPa)
100
1 Iexp( -lx-yl/c;a)
<Ba(x)8a(y))~ Ix-y
0
for
100
0"=1, 2.
(5·13)
In some parts of the hydrogen bond
network, the concentration of nonFig. 22. The instability lines T1(P) and T2(P)
bonding sites is larger than the average.
obtained from Eq. (5·11). The liquid phase is
Such parts are weak or fragile parts of
unstable in the shaded region. Dotted lines are
the
network and .;1 is the characteristic
the experimental stability limits of liquid
radius of the fragile part. In other parts
water (Refs. 78) and 92)).
of the network, the crystalline order is
more developed than the average. 6 is the characteristic radius of the part in which
ice-likeness is more developed than other parts. From Eqs. (5 · 6) and (5 ·11) it is
shown that c;a diverges as approaching Ta,
-100
0
100
T (OC)
200
(5·14)
where LiT= T- Ta, and v=1/4 for 0"=1 and v=1/2 for 0"=2. Using Eqs. (4·18) and
(5·10)~(5·14), Cp, -a and KT, are shown to diverge as
Cp,
-a and KT-::::::(LJT)- 7
(5·15)
,
where r=2v for 0"=1 and r=(4-d)v for 0"=2. Here dis the spatial dimension and
for d=3 we have r=1/2 for both 0"=1 and 2. It would be interesting if experiments
for d<3 can be designed. r=1/2 for d=3 is in agreement with the experimental
data. These mean-field values of exponents do not depend on the detailed definition
of the lattice-gas model.
In Fig. 22 we show on the T-P plane the numerical estimate of the location of Ta
obtained from Eq. (5·11). Thus our network model provides a zeroth order answer
to questions raised in the beginning of § 4. In Fig. 22, however, the quantitative
agreement with the experiment is not satisfactory, but this is not surprising considering the rough approximations used in this section. There are many points to be
improved in the present treatment. Especially, in the region where two instability
lines cross each other, two modes of fluctuations compete and thus the mode-mode
coupling effect, which is the effect beyond the naive mean-field approximation, should
be considered.
§ 6.
Conclusions
Energy relaxation, collective motions and fluctuations in liquid water were
analyzed. Mechanism of vibrational energy relaxation processes in liquids was
examined. It is found 4 l that multiple step energy conversion of solute vibrationoverall rotation-solvent translation is a main path of energy dissipation of excited
Relaxations, Fluctuations, Phase Transitions and Chemical Reactions
87
molecules in simple solvent, like liquid Argon. Extremely fast energy dissipation
occurs in liquid water. This is due to resonance energy transfer from solute vi·
brational motions to solvent water fluctuations. Extremely fast energy dissipation
and thermalization have been also found in many other reactions in liquid water.
We then examined physical origin of energy fluctuations in liquid water by
performing the inherent structure analysis of the potential energy surface involved in
liquid water dynamics.m-zoJ It was shown that there exist collective motions of few
tens water molecules localized in space occurred in sub-picosecond time scale. These
collective motions are associated with the hydrogen bond network rearrangement
dynamics. See-saw type energy exchanges among individual water molecules exist
in the collective motions, inducing large local energy fluctuations. It was shown that
the water dynamics is mixture of activated process and ballistic process, depending on
the modes excited and on domains of potential energy surface the system locates in.
The inherent structures are grouped to overall inherent structures; water undergoes
tens of small inherent structure transitions between a large (collective) overall inher·
ent structure transitions. We found that only few types of transitions to the neigh·
bors inherent structures are accessible with the kinetic energy available at the room
temperature. Since only limited number of trajectory calculations were performed,
however, the distribution of high energy barriers, important in liquid dynamics, is not
yet fully understood. Multiplicity of the hydrogen bond structures of liquid water
was found to yield so called 1// frequency dependence of the power spectra of energy
fluctuations. 60 J
We have investigated water dynamical behavior by analyzing the potential
energy surfaces involved. Liquid water usually has kinetic energy higher than most
energy barriers involved and thus sees only global feature of potential energy surface;
small energy barriers are not important. Methods to analyze such global feature of
potential energy surface and dynamics in high energy, influenced mainly by high
energy barriers, are to be developed. Intensive mode excitations, including simultaneous excitations of various modes, are to be performed to find high order mode
coupling mechanism and a high energy barrier distribution.
Instability of hydrogen bond network of liquid water was investigated by deriving
a mean field solution of the partition function for a water model. 29 J Anomalous
behavior of the water static quantities near instability lines was analyzed and exponents of divergence were predicted. Two fluctuating field variables, the uniform field
and the staggered field, were introduced and the' existence of 'fragile' regions (where
concentration of broken bonds is large) and 'strong' regions (where the short range
crystalline order is relatively well developed) was shown. Here we should stress that
the network static fluctuations are continuous in sjJace, so that there is neither a
definite "cluster" nor a clear boundary between the fragile and the strong regions.
Beyond the static properties of these regions obtained from the partition function
calculation, we need extend our analysis to their dynamical properties. In 'fragile'
regions frequent bond rearrangement and molecular movements are expected. In
'strong' regions, on the other hand, molecules may tend to move collectively like rigid
body rotations. There may exist a certain scaling relation between the characteristic
relaxation time and the correlation lengths, representative radii of the 'fragile' and
88
I. Ohmine and M. Sasai
'strong' regions. In order to understand such dynamical nature of instability, we need
to extend our static statistical theory, presented here, to a new analytical statistical
theory of dynamics. Combining this new analytical statistical theory and molecular
dynamics calculation will lead to deep understanding of the network dynamics in the
molecular scale; both microscopic and macroscopic anomalies of liquid water will be
revealed in a synergetic way.
Acknowledgements
I. 0. thanks Dr. H. Tanaka for longtime cooperation on a water dynamics study
and to Dr. 0. Kitao for the Argon liquid work, reviewed here. Authors thank to
Professor P. G. Wolynes, and Professor R. Ramaswamy for the stimulating discus·
sions and cooperative works. They also thank to Dr. N. Shida, Mr. S. Saito and
T. Komatsuzaki for their important help and discussions in the course of preparing
this manuscript. The present work is supported partially by the Grant-in-Aid for
Scientific Research on Priority Area of 'Molecular Approaches to N onequilibrium
Processes in Solutions' and other funds from the Ministry of Education, Science and
Culture, Japan. Calculations were carried out by the supercomputers at Institute for
Molecular Science and Kyoto University Computer Centers.
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