Water transport in trees - The physiochemical properties of water under negative pressure

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Oct 27, 2013 (3 years and 11 months ago)

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Water transport in trees
-

The physiochemical
properties of water under
negative pressure




Bachelor thesis





Mascha Gehre


Supervised by
Dr.
Ir.
Bernd Ensing







29. 07.

2012













Tabel of contents





1. Introduction


1.1 General
Introduction











3


1.2 Motivation












6


1.2.1 Determination of the equation of state of
water



under negative pressures
.








6




1.2.2 Determination of
the pressure at which cavitation
in








liquid water can be observed
.








7



1.2.3
Creation of

new

reaction paths at higher pressures





7

.


1.2.4 Determination of the transition state and the size of






the critical cavitation nucleus.








8









2. Theory


2.1 Theoretical background










8


2.1.1 Molecular Dynamics simulations



2.1.2 Transition path sampling







10


2.2 Simulation Details









12




2.2.1 Equation of sate of water under negative pressure




1
3



2.2.2 Cavitation Pressure








13



2.2.3 Creation of new reaction paths

at higher pressures









2.2.4 Determination of the transi
tion states and the size




of the
critical cavitation nucleus.






13


3. Results & Discussions

3.1

Equation of state of water at negative pressure





14

3.2

Determination of the spinodal Pressure P
s





15

3.3

Creation of reaction paths at higher pressures and






determination of their transition states






17

3.4

Determination of the transition state
and the size of

the critical cavitation nucleus.







17


4. Conclusion











19


5. Further Research










19


6. References











20






1. Introduction



1.1 General Introduction


The protection of the remaining forests that cover our planet
has beco
me

an important topic during

the
last decades
. This is

because essentially all free energy utilized by biological systems arises from solar
energy that is trapped by the process of photo
synthesis.

This means that

photosynthesis is the source of
essentially all the carbon compounds and all the oxygen that makes aerobic metabolism possible.
1

However, this

is not the only reason why forest
s need to be protected. Trees

also play an essential
role
in the regulation of the global water cycles and in dissipating the incoming solar radiation with a
re
levant cooling effect. Only 1
hectare
of forest can evaporate through the leaves 40.000 liters of water
which makes them extraordinary efficient mach
ineries for adsorbing water from the soil and
transporting it to the leaves.
2

Therefore
,

scientist are eager to understand the processes of water
transport within trees in order to develop novel artifi
cial devices that can draw

up

water

under tension
over
long distances,
equivalent to the process in trees
.
2, 3

However the process of lifting water to the top of very ta
ll trees against gravity is complex and till this
day

not totally understood. The movement of water occurs through a complex network of very n
arrow
“tubes”, the so called xylem conduits. The movement originates from the transpiration process
occurring

in the leaves. When stomata open, the internal leaf mesophyll comes in direct contact
with
the free atmosphere, in which the
water content is gene
rally lower. The water

wetting the cell walls is
forced to evaporate and the surface of the remaining water is drawn into the pores
,

where it forms
concave water menisci.
Because of the surface tension, the
pressure in the water decreases and
becomes negat
ive.

Therefore,
water flows

in trees under a thermodynamically meta
stable state with
respect to its

vapor

phase
.

This resul
t
s in

the nucleation of vapo
r bubbles,
also known as cavitation.
2

However, cavitation
can stop
the
circulation of water in the
vessels of real trees or in synthetic ones that
are used for microfluidic flow t
ransport driven by evaporation.
3

Th
is implies
,

that before

artificial
devices can be developed
, which

draw up water over long distances
,

the process of cavitation has to be
tot
ally understood.

In order to explain the process of cavitation we first have to explain what we mean when we say that
water is in a metastable state.
Any liquid can be prepared in a metastable state with respect to its vapor
in two ways: either by superhe
ating above its boiling temperature T
b
, or by stretching belo
w its
saturated vapor pressure P
sat
.

This can be explained in terms

of the density of the liquid
. Since
b
oth
cases result in an increase in the distance bet
ween the molecules the density of the
liquid decreases
.
When the factor, by which the distances between the molecules are increased

is not too high, the
attractive

forces between the molecules allow the system to remain in its liquid state. The system is
then stated to be metastable with respe
ct to its vapor. However
,

when the intermolecular distances
between the molecules get too
large,

the attractive forces between the molecules are too weak to allow
the system to remain in its liquid state. As a result the system becomes mechanically unstabl
e. The
critical density and pressure at which the system gets mechanically unstable are stated as the spinodal
density
ρ
s

and the spinodal pressure P
s
. Beyond this point the system will eventually return to
equilibrium by nucleation of vapor bubbles.
4


The classical nucleation theory (CNT) is the simplest way to describe the the
rmodynamics of
the
nucleation of a more
stable phase in a metastable phase
.
5

In a liquid that is s
uperheated at constant
pressure,
P
,

to a temperature
,
T
,

above T
b
, or, equi
valently, st
retched at constant temperature,
T
, to a
pressure,
P
,

below P
sat
(T), the minimum work required to create a sphere of vapor of radius R in the
liquid is


where P' is the pressure at which the vapor is at the same chemical potential as
the liquid at P and
σ

is
the liquid
-
vapor surface tension.
The first term in equation

(1) gives the energy gained when forming a
volume of the stable phase, whereas the second term is the energy cost associated with the creation of
an interface. Their comp
etition results in an energy barrier





reached for a critical bubble of radius R
c

=2
σ

/(P'
-
P).
4

A bubble whose radius is larger than R
c

will
grow spontaneously
.
5

Equation (2) can be now rearranged in such a way that we obtain an expression ,
that relates
the height of the

reaction barrier

to the size of the critical bubble
and the liquid
-
vapor
surface tension
as follows:












(1)

(2)

(3)

1.2 Moti
vation


This project is a subproject of the project TENSIWAT. TENSIWAT addresses the problem of water
transport in plants within a multi
-
disciplinary and fundamental approach. This means that theoretical
and experimental physicists, chemists, plant ecologists and
material engineers will collaborate for a full
attack enterprise where all aspects of the water transport in trees are studied and
interrelated. The
ultimate goal

of TENSIWAT is to provide a theoretical an exp
erimental framework
, which
allows one
to

unders
tand how trees are able to easily “handle” te
nsile water over long distances. Therefore, the
physi
ochemical properties of metastable

water and the role
of
conduits characteristics

within trees are
studied
. The obtained information can then be used to devel
op novel artific
ial devices that can draw
up
water
under tension over long distances.
2

The main go
al of this subproject is to gain
more insight and understanding of the physiochemical
properties of
pure
water under tension
.
D
u
ring the three and a half months
of researc
h the following
topics are
treated:


1.

Determination of the equation of state
(EOS)
of

water under negative pressure
.

2.

Determination of the spinodal

pressure (P
s
).

3.

Creation of
new
reaction
paths

at less negative

pressures.

4.

Determ
ination of the transition state
, TS,
of each reaction path,
and determination
of the
size of the critical cavitation
nucleus at the transition state
.



1.2.1 Determination of the equation of state of
water under negative pressures

It is
an experimental fact that each substance is described by an equation of state (EOS), an equation
that interrelates the volume, V, the amount of substance (number of molecules), n, the pressure, P, and
the temperature, T. However, it has been established ex
perimentally that it is sufficient to specify only
th
ree of these variables because

the fourth variable is fixed. The general form of an EOS is








P

= f(T,V,n)
.


This equation tells us that, if we know the values of T, V, and n for a particular substan
ce, then the
pressure has a fixed value. Each substance is described by its own EOS, but we know the explicit form
of the equati
on in only a few special cases.
6

Over time the scientific community proposed different
equations of state of liquid water. The m
ost recent international formulation is the so
-
called IA
PWS
-
95
formulation

(IAPWS)
,

which was pub
lished by the International Association for

properties of Water
and Steam.
7

The formulation at negative pressures is based on experimental
data

obtained

at pos
itive
pressures.
In

Figure
1

a graphical representation of
t
he IAPWS
is

shown. As
can be

seen
,

the IAPWS
predict
s

a spinodal pressure of
-
160
0 bar
. However
,

the validity of the

extrapolation to negative
pressures had been tested only indirectly or w
ith a
weakly metastable liquid.
8


Therefore,
Caupin

and coworkers

recently

determined
,

by acoustic experiments
,

the values on the right
hand side of the vertical in
Figure
1
. T
heir data pr
ove the fidelity of the IAPWS

down to
-
26
0 bar
.

Furthermore
,

by the use of
a fiber optic probe hydrophone (FOPH)
they determined the spinodal
density
ρ
s

from which they calculated
P
s

=
-
28
7
±

1
0.5 bar

at 23.3 °C.
8

The obtained value for P
s

i
s
consistent

with
the majority
of the results
of numerous

other cavitation
experiments.
14


However
,

there
is one exception. In so
-
called inclusion experiments, in which water is trapped in small pockets inside
crystals, spinodal pressur
es down to
-
140
0 bar

were found.
9





Figure 1
: Equation

of state of liquid water
at
23.3 °C from


the IAPWS formulation
extrapolat
ed to their spinodal pres
-


sures.
The range of pressures reached in aco
ustic experi
-


ments is limited to
the right hand side of the vertical line.
8


This

questions whether

the

water samples prepared for

experiments
, except for inclusion

experiments

are totally pure. When we say
, that a water sample is

not totally pure, we mean that
destabilizing
impurities
are present

in the water sample, which trigger the process of cavitation
.

Therefore those
impuritie
s
lower the height of the reaction barrier and lead to a higher spinodal pressure.
10


Caupin and c
oworkers suggest that hydronium ions
,

naturally
occurring

in
neutral
water could
be such
a
destabilizing impurity
.

They also predict that hydronium ions
would be absent
or inactivated in the
inclusion
experiments
. This

wo
uld explain, why the spinodal pressures found for those experiments are
shifted to weigh more negative pressures
.
4


Hydronium ions could have
a destabilizing effect because they occurrence

leads to the
proton charge
transfer from a H
3
O
+

ion to a neighboring H
2
O molecule
. In the first step of the mechanism

of the
proton transfer

the hydrogen
-
bond coordination number of one of the H
2
O molecules in the first
solvation shell is lowered by the b
reaking of a hydrogen bond to the second solvation shell.
11

Therefore
,

the existence of hydronium ions destabilizes
the hydrogen bond n
etwork within the system and could

trigger cavitation.


Due to a lack of data at large negative pressures, the disagreem
ent between experiments and theory
cannot be solved.
4

Therefore
,

we will determine an equation of state for

an

idealized
water
system
under negative pressure by performing
computer
simulations
of the form P=
f(T,V,n)
. The density o
f the
treated water system
var
ies
between 1020 and 300 kg/cm
3
.
That

our system is
idealized

means
that no
stabilizing or destabilizing impurities will be
present. In real trees stabilizing or destabilizing impurities
a
re

for example

present in the form of
ions,

which are
dissolved in the water, that is

transported within

the xylem conduits. Also
the
presence of boundary condition
s

in the form of the
walls of the xylem
conduits could

have

a stabilizing or destabilizing

effect on the metastable water system.

However, i
n t
he
system
,

which is
treated througho
ut this project
,

those

impurities will be
absent.
Besides that
,

the formation of hydronium ions
,

which spontaneously occur
s

in natural
water,

will not
take place during the performed simulations
.

Therefore
,

in accordance with the theory proposed by
Caupin and coworkers we
expect

that

we will find a

spinodal pressure
,

similar to the

spinodal

pressure
predicted by

the IAPWS

formulation
. This means P
s

= ± 1600 bar.



1.2.2 Determination of the pressure at whi
ch cavitation in liquid water can be observed.

After determining the equation of state in the form:












P

= f(T,V,n)








we are interested in determining the highest pressure
,

at which cavitation can be observed

in the
performed simulations
.
This pressure is

equal to

the spinodal pressure, P
s
.
In order to

do so we will
perform simulations of the type:








V = f(T,P
,n)



(4)

(5)


In accordance with the IAPWS formulation, we
expect
that
the

spinodal pressure will

lie
somewhere
in
the area
of the minimum of the equation of state
of water
under negative pressure.



1.2.3 Creation of

new

reaction paths at less negative

pressures

After

we determined the highest pressure, at which cavitation can

be observed, we obtain a reaction

path
, which shows how a

metastable water

phase

returns to equilibrium by the formation of water
bubbles.
From the obtained reaction path lots of information
s about the process of cavitation at the
determined spinodal pres
sure

can be obtained.
However, we expect that the determined spinodal

pressure

will be substantially

more negative

than pressures found

in natural

systems
, like trees.
2

Therefore
,

we are interested in determining
reaction path
s

at
higher
pressures
,

which are more likely to
be found in nature. In order to do so we will use the method of Transition Path Sampling (TPS). The
method will be explained in full detail in section 2.1.2.



1.2.4 Determination of the transition state and the size of the criti
cal cavitation nucleus.

As was stated in equation (3), the classical nucleation theory states, that the radius of the critical
nucleus R
c

is related to the height of the energy barrier, E
b
,

in the following way:






According to equation 3,

The CNT predicts that

the critical nucleus increases as the energy barrier of
the system increases.

By determining
the size of the critical nucleus

at the transition state

of each reaction path

we will
determine, if the size of the critical nucleus does i
ndeed increase as the height of the reaction barrier,
E
b
, does increase.
As can be seen in Figure 2, t
he transition state is

the
moment when our sy
stem
crosses the energy barrier and

is

therefore

equal to
the height of the reaction barrier.
16








Figure

2
:

Change in the

free energy along the




transition

of the


initial state A


into the final





state B. The

height

of the reaction

barrier is








defined as the transition state.



Once we located the precise position of the transition
state

for each reaction path, we
are

therefore

able
to obtain information of the
precise config
uration of the system

at the transition state.
This means that
we can determine the size of the critical nucleus of our system, which is nothing less than the size of
the
cavitation nucleus
at the transition state.

An increase in the size of the critical nucleus

as the pressure becomes less negative f
or our system,
would also support the theory proposed
by Caupin and coworkers
. The presence of
desta
bilizing
impurities

in the water system

would in fact

decrease the height of the reaction barrier

for cavitation
and would therefore result in a less negati
ve spinodal pressure, P
s
.




2. Theory



2.1
Theoretical background


2.1.1 Molecular Dynamics Simulations

Gromacs

The method of choice for this study is the Gromacs Molecular Dynamics Simulation method. Gromacs
uses classical mechanics to describe the
motion of atoms.
12

This means that Newton’s laws of motion













F = ma






a = dv/ dt






v = dr/ dt,



are used, where the vector F is the force on a particle, a its acceleration, v the velocity and r the

position. M is the mass of a particle and

t is time.
13

For every time step of the simulation
,

Newton’s
equations of motion for a system of N interacting atoms








and the potential function V (r 1 , r 2 , . . . , r N ), which is a negative derivative of the forces
,





are solved
simultaneously
. During the simulation
,

we made sure

that the temperature and pressur
e
remain at the required values. Moreover,
the coordinates, velocities and forces
which are
calculated
after every time step are written to an output file at reg
ular intervals.
The coordinates as a function of
time represent a trajectory of the system.

After initial changes, the system will usua
lly reach an
equilibrium state.
12



The Born
-
Oppenheimer approximation

During the Gromacs Molecular Dynamics simulations
a conservative force field
, which

is a function of
the positions of atoms
,

is used. This means that the electro
nic motions are not considered, implying
that
the electrons are supposed to adjust their dynamics instantly when the atomic positions change, and

remain in their ground state.
12




(6)

(7)

Periodic Boundary conditions

Since the system size is small
,

there are lots of unwanted boundaries with its environment (vaccum).
This condition is avoided by the use of perio
dic boundary conditions to evade

real phase b
oundaries.
Since real liquids are not composed of period units, such as crystals, one must be aware that something

unnatural remains.
12




2.1.2 Transition Path Sampling

The theory

Almost all reactions consist of the rapid transition
of
long
-
lived stable
states. By “stable” also therm
o-
dynamically metastable states are designated.
Such transition events are rare

because the stable states
are separated from each other by high potential energy barriers.
13

An example of such an energy barrier
separating the tw
o stable states
,

A and B
,

is given in
Figure

2.

But

while being rare, these transitions
proceed swiftly when they occur.
13

Transition Path Sampling (TPS
) is a technique that allows
one
to
compute the rate of

such

a barrier
-
crossing process without a priori knowledge of the reaction coordinate or the transition state.
6

The basic
idea of transition path sampling is to focus only on those parts of the trajectory that connect both the
initial and final states, and hen
ce

those that

are cr
ossing the free energy barrier.
15


The method

Since

a trajectory crosses the free energy barr
ier an infinite number of times

an ensemble of crossing
paths is formed, the so
-
called transition path ensemble (TPE).
In order to obtain the p
ath ensemble
a
sampl
ing scheme is used, in which an existing pathway connecting the initial and final state is altered,
so that new pathways are created. The creation of new reaction path
s

is followed

by
accepting or
rejecting new trial pathways according
to
the following

acceptance rule:


First the in
i
tial

and final state
s

of the
reaction of interest are

defined. Then a trajectory is created that
conne
cts the initial to final state.
15

In
Figure

3

an example

is given

of a
trajectory that connects an

initial
state

A

to
a
final state

B
.
The initial state A

in Figure

3 represents
a metastable water phase and
the final state B represents the simultaneous existence of a stable water phase and a vapor phase. The
formation of a stable water and vapor phase
from a metastable water phase is indicated by an increase
in the
box size
,

w
hich is represented on the y
-
axis

in
Figure

3
.





Figure

3
:

Trajectory that connects the initial


s
tate

A

to the final s
tate B.



After the init
ial and
final states are defined
,

certain
time slice
s

on the c
urrent trajectory that connect
the
initial to
the final state are randomly

s
e
l
ected
,

as shown

in
Figure

4
.
Then
the momenta
of the time
slices are changed slightly and a new trajectory of the same lengt
h is created by integrating the
equations of motion both forward and backward in time. The new trial trajectory is accepted if it
connects t
he initial
with the final state. Otherwise it is rejected and the old path is kept. The shooting
move is then repeat
ed with a different shooting time slice. In fact, for complex systems only shots
initiated from the barrier region rather than the stable state regions have a chance of creating acceptable
pathways.
15

Figure 5 shows

that in our example o
nly the trial traje
ctory created from frame 82 may be
accepted. Those created from Frame 60 and Frame 128
has

to be rejected because they do not connect
the initial with the final state.




Figure

4
:

Random

selection
of time

slices


Figure

5
:

Integration

of

the equations of

mo
-


along
the trajectory displayed in
F
igure
3
.


tions
forward

and

backward

in

time for the








three time
slices selected in
F
igure
4
.



2.2 Simulation Details


For the simulation
s

performed in the scope of this project,

a cubic box with periodic boundary
conditions
was con
structed. T
he box was

then

filled with

360
water molec
ules

and all of the
simulations were performed at room temperature (298 K).

The size of th
e box and the pressure were

chose
n in such a way, so that
liquid
water was
in a
metastable state with respect to its vapor. The time

scale of the performed simulations varies
between 100 ps and 100 ns.








Figure

6
:

R
epresentation

of a periodic







box containing 360

water mo
le
cules




2.2.1
Equation of sate of water under negative pressure

The equation of state of water will be determined by the performance of simulations in which
the
volume of the box (V),

the temperature (T) of the system and the number of molecules (n) will be held
constan
t. In that way we will obtain the pressure that corresponds to a certain density of water.

The
simulation time
for those
simulations will be 100 ns.


2.2.2 Cavitation Pressure

In order to determine the pressure at which pure water
separates

spontaneously
in
a water phase and a
vapor phase,
we will
perform

simulations in which the pressure (P), the temperature (T) of the system
and the number of molecules (n) will be held constant. We will perform simulation
s

between
-
10
0 bar
and
-
500
0 bar
.

The simulation t
ime for those simulations will be 100 ns.




2.2.3 Creation of new reaction paths

at less negative pressures

New reaction paths will be created by the me
thod of TPS
, which is described in section 2.1.2. In that
way we will try to create reaction paths

within a
pressure
range

of
-
2148 bar to
-
18
00 bar.
The
simulation time for those simulations will be 100 ps.


2.2.4 Determination of the transition states and the size of
the critical cavitation nucleus


The transition

state of the obtained reaction

paths at different pressures will be determined by
determi
ning the probability that

cavitation will occur along the path. The probability of cavitation will
be determined by selecting different frames along each reaction path.
For each frame the equations

of
motion will be solved in 10 different simulations for which the initial velocities will be randomly
chosen. In that way 10 different reaction paths will be obtained that will eventually end up in the initial
state or the final state. Depending on the r
elative positions of the selected frames to the transit
ion state,
the ratio of reaction paths that will end up in the
initial state A or B will vary
.
For those frames which
are exactly at the transition state

or nearby,

the amount of reaction paths that wi
ll e
nd up in the initial
state A or

the final state B will be equal.
As can be seen in Figure
7, i
n our case this means that we will
obtain 5 reaction paths that will end up in A and 5 that will end up in B.












Figure
7
:

I
ntegration

of

the equ
ation

of






motion
s

for

Frame 82 is performed 10 times.






For

every single

integration

random

initial






velocities are chosen

but

the initial position






of the atoms are the same.




3. Results

& Discussion


3.1 Equation of state of

water at negative pressure


The EOS

of water

(solid line)
, wh
ich we determined by performing
computer simulations

of the type
P
= f(T,V,n)

is given
in
Figure
8.
By the use of a Visual Molecular Dynamics Software,
VMD
,

we were
also able

to determine

the pressure region in which the spinodal pressure can be found. We
able to
observed cavitation at a density of 880 kg/m
3
,
but no more at a density of 860 kg/m
3
.
This means

that
the spinodal pressure lies in the range between
-
2044.34 and
-
2220.44 bar.







Fig
ure
8
:
Equation of state of liquid pure water at 298 K deter
-




mined by computer simulations (solid line).Equation of state of li
-




quid water at 23 °C from the IAPWS formulation
extrapolat
ed to




the spinodal pres
sure (dashed line).
The range of pressures reached




in aco
ustic experiments is limited to
the right hand side of
the ver
-




tical
line
.



As shown

in
F
igure
8
,

the EOS

of water

(solid line)
, which we determined by computer simulations,
has

the same course as the IAPWS

(dashed line)

dow
n to a density of 940 kg/
m
3
. Beyond this

density
the

valu
es are shifted to more
negative p
re
ssures and have a minimum at a density of 880 kg/m
3
.

Therefore, our results suggest

that
for
an idealized water system, which contains no boundary
conditions and in which no destabilizing or stabilizing
impurities are present,
the spinodal pressure

is

shifted to

even

more negative pressure

than

those

pro
posed by the IAPWS. This
support
s
the
t
heory

that in all experiments
,

except in the inclusion experiments
,

destabilizing impurities are

present
,

which
shift

the spinodal pressure to less negative

pressures.
10

That the values we measured are shifted to even
more negative pressures than those found fo
r the IAPWS formula
t
ion can

be explained by the fact

that
we are using a periodic box in our simulations
. Therefore
,

our system has no boundary conditions at all,
whereas in real experiments boundary conditions can only be minimized but not total
ly avoided
.
Another explanation
could be
the absence of the formation of hydronium ions in our system. Those are
formed spontaneously in real systems but in our system the formation can only take place when a
single proton is added to the system.




3.2

Determination of the spinodal

Pressure
P
s

In order to determine the highest pressure at which spontaneous cavitation takes place we performed
simulation between
-
100 and
-
5000 bar. We tried to observe the process of cavitation by two methods.
By the use o
f

a

Vis
ual Molecular Dynamics Software, VMD,
and by analyzing t
he potential and total
energy,
the pressure, the box size, the density and the temperature in the course of the performed
simulations.


VMD

The highest pressure at which we were able to observ
e cavitation by the use of VMD is
-
2148 bar. In
Figure
9

the three different stages of the simulation

at
-
2148 bar are shown
. In stage a)
one observes

a
homogeneous

metastable

water ph
ase. In stage b)

the
formation of

a
vapor bubble in the left
, low

corner of the box

is observe
d. In stage c)

a
vapor and water phase have
formed which exist
simultaneously.




Figure

9
:

Snapshots of

the simulation of a periodic

box containing 360 wat
er molecules at a pressure

of
-
2148 bar. Snapshot a)
shows a homogeneous
metastable
water phase. In snapshot b) a vapor bubble
can be

observed in t
he left
,


low

corner. In

snapshot c)
a

stable

water

phase and a


stable


vapor phase

have formed
.



Gromacs Energies

Since the forma
tion of a vapor bubble is
difficult

to see

with the naked eye,
VMD
it is not the best
identification method.
Therefore
,

we
also interpreted the
potential energy, pressure, temperatu
r
e,
boxsize and density in the course of the performed simulations. In
Figure

10

and
Figure
11
the

obtained
values
are
shown as
function
s

of time f
or the simulations at
-
2147
and
-
2148 bar respectively.
The
simulation at
-
2148 bar is the simulation
,

at which we were still able to observe cavitation. In the
sim
ulation at
-
2147 bar and

in all of the simul
ations at

less negative
pressures we were not able to
observe cavitation

by the use of VMD
.




Figure
10
:

The potential and total energy,

pres
-



Figure
11
:

The potential

and total

energy,


sure, temperature , box size and density in the


pres
sure, temperatu
r
e
, box size and density


course of the simulation at
-
2148 bar.



in the course of the simulation at
-
2147 bar.




By the comparison of
Figure
10

with
Figure
11

we

find no noticeable differences between 0 and 250
ps.
However
,

beyon
d 250 ps the obtained values for the two simulations

are quite different.
As
shown

in
Figure
11
,

the density of the system decreases at about 250 ps
,

which is in accordance with an
increase in the

distances between the water molecules. Then at 300 ps the boxsize, the pressure, the
temperature and the total and potential energy increases, which is in accordance with the formation of a

vapor bubble. At about 320 ps
however
,

we
noticed a small decreas
e in the temperature and the total
and potential energy, which is in accordance with the formation of a s
table vapor and water phase.
While d
uring
the formation of the stable water phase, the dist
ances between the molecules are

decreasing again
, which resu
lts in

slight decreases

in the values
.



3.3

Creation of reaction paths at higher pressures

and determination of their transition states

By the use of TPS we were able to create
reaction path
s at
-
2000 and
-
1800 bar.




3.4 Determination of the transition state
and the size of the critical cavitation nucleus.


By the method
described in section 2.2.4 we

were able to determine
the transition state of the reaction
paths at
-
2148 ,
-
2000 and
-
1800 bar. Once we allocated the
position of the transition state we could
obtain information of the precise configuration
of the system
at the trans
ition state. The values of the
v
olum
es

of the box at the transition

state

which are stated the critical volumes, V
c
,

are listed in table 1.


Pressure (bar)

critical box size length (nm)

Critical volume of the box (nm
3
)

-
2148

2,3532

13.03096

-
2000

2,3587

13.122547

-
1
800

2,3633

13.199472

Tabel 1
: Critical volume,
Vc
,

of the simulation box containing 360 water molecules at 298 K for the
rea
ction paths
at
-
2148,
-
2000 and

-
1800 bar
.



Figure 12 shows that

the size of
the box
at the transition state
and
,

therefore
,

also the volume of the box
at the transition state
,

increases as the pressures become
s

less negative. Since the volume of the box at
the transition state is proportional to the size of the critical nucleus at the transition state we also know
that the height of the reaction barrier does increase in accordance with equation (3).
Since the he
ight of
the reaction barrier increases as the pressure increase, the event of cavitation is less likely to take place
as the CNT predicts.
4,5

Therefore, these results

also

support

the theory that
destabilizing
impurities are
present in those experiments
,

f
or which less negative spinodal pressures are found.
4,9,10





l



Figure
12
: Relation between the critical box size at the transi
-



tion state for reaction paths at different pressures.





2.34800
2.35000
2.35200
2.35400
2.35600
2.35800
2.36000
2.36200
2.36400
2.36600
-2148
-2000
-1800
box

size (nm)

pressure

(bar)

4. Conclusion


We were able to determine an equation of
state that strongly supports the IAPWS formulation. The shift
of the spinodal pres
sure to more negative pressures, which was also confirmed by simulations of the
type
V = f(T,P,n)
can be explained in the absence of boundary conditions and in the absence of

the
formation of hydronium ions in our system.

The obtained reaction path at the spinodal pressure, Ps, was used for the creation of new reactions paths
at less negative pressures by using the method of transition path sampling. In that way we were able t
o
obtain reaction paths at
-
2000 and
-
1800 bar. For each path we were able to allocate the position of the
transition state. Therefore we were able to determine, that the size of the critical nucleus and, hence, the
height of the reaction barrier incre
ases

as the pressure increases, meaning that the event of cavitation is
less likely to take place as the pressure increases. Our results support therefore strongly the theory of
Caupin and coworkers, that destabilizing
impurities are present in the

experiments
, for which less
negative pressures were found. However,
since we were
only
able to make
a rough estimation of the
size of
the critical nucleus, more accurate results are needed in order to determine that the deviations in
the size of the critical nucleus
are not caused by an error in the measuring method.



5. Further research


During this project simulations of pure water under negative pressure were performed.
The spinodal
pressure
,

which was obtained during the performed simulations lies at around
-
21
4
8 bar, which is a
n

even less negative

pressure
,

than

the one

predicted by the IA
PWS formulation. Fu
r
thermore we were
able to determine that the height of the reaction barrier increases
, as the pressure increases. This means
that
the event of cavitation is

less likely to take place as

the

pressure becomes less negative.

However,
since the determined differences in the critical volumes, V
c
, are rather small, it will be necessary to
develop a method, that measures the precise size of the critical nucleus. In
that way one can make sure,
that deviations in the size of the critical nucleus at different pressures are not caused by an error

in the
measuring method.

Besides that,

it would be interesting to determine which effect impurities would have on our system.

Our results support

namely only

the theory of Caupin and coworkers that destabilizing impurities d
e-
crease the height of the reaction barrier for cavitation, but

they do not identify the impurities
.
Therefore
there is need for computer simulations, in whic
h hydronium ions are added to the system and
need for
simulations, in which boundary conditions of all different kinds are added.

6.
References



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