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 13 2.2.2 Cavitation Pressure 13 2.2.3 Creation of new reaction paths at higher pressures 2.2.4 Determination of the transition states and the size of the critical cavitation nucleus. 13

3. Results & Discussions3.1 Equation of state of water at negative pressure 143.2 Determination of the spinodal Pressure Ps 153.3 Creation of reaction paths at higher pressures and

determination of their transition states 173.4 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 become 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 photosynthesis. 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 forests 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

relevant cooling effect. Only 1 hectare of forest can evaporate through the leaves 40.000 liters of water

which makes them extraordinary efficient machineries 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 artificial 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 tall trees against gravity is complex and till this

day not totally understood. The movement of water occurs through a complex network of very narrow

“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 generally 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 negative. Therefore, water flows in trees under a thermodynamically metastable state with

respect to its vapor phase. This results in the nucleation of vapor 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 transport driven by evaporation.3 This implies, that before artificial

devices can be developed, which draw up water over long distances, the process of cavitation has to be

totally 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 superheating above its boiling temperature Tb, or by stretching below its

saturated vapor pressure Psat. This can be explained in terms of the density of the liquid. Since both

cases result in an increase in the distance between 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 respect 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 unstable. The

critical density and pressure at which the system gets mechanically unstable are stated as the spinodal

density ρs and the spinodal pressure Ps. 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 thermodynamics of the

nucleation of a more stable phase in a metastable phase. 5 In a liquid that is superheated at constant

pressure, P, to a temperature, T, above Tb, or, equivalently, stretched at constant temperature, T, to a

pressure, P, below Psat(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 competition results in an energy barrier

reached for a critical bubble of radius Rc =2σ /(P'-P).4 A bubble whose radius is larger than Rc 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 Motivation

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 experimental framework, which allows one

to understand how trees are able to easily “handle” tensile water over long distances. Therefore, the

physiochemical properties of metastable water and the role of conduits characteristics within trees are

studied. The obtained information can then be used to develop novel artificial devices that can draw up

water under tension over long distances.2

The main goal of this subproject is to gain more insight and understanding of the physiochemical

properties of pure water under tension. During the three and a half months of research the following

topics are treated:

1. Determination of the equation of state (EOS) of water under negative pressure.

2. Determination of the spinodal pressure (Ps).

3. Creation of new reaction paths at less negative pressures.

4. Determination 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 experimentally that it is sufficient to specify only

three 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 substance, then the

pressure has a fixed value. Each substance is described by its own EOS, but we know the explicit form

of the equation in only a few special cases.6 Over time the scientific community proposed different

equations of state of liquid water. The most recent international formulation is the so-called IAPWS-95

formulation (IAPWS), which was published by the International Association for properties of Water

and Steam.7 The formulation at negative pressures is based on experimental data obtained at positive

pressures. In Figure 1 a graphical representation of the IAPWS is shown. As can be seen, the IAPWS

predicts a spinodal pressure of -1600 bar. However, the validity of the extrapolation to negative

pressures had been tested only indirectly or with 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. Their data prove the fidelity of the IAPWS down to -260 bar.

Furthermore, by the use of a fiber optic probe hydrophone (FOPH) they determined the spinodal

density ρs from which they calculated Ps = -287 ± 10.5 bar at 23.3 °C.8 The obtained value for Ps is

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 pressures down to -1400 bar were found.9

Figure 1: Equation of state of liquid water at 23.3 °C from the IAPWS formulation extrapolated to their spinodal pres- sures. The range of pressures reached in acoustic 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

impurities lower the height of the reaction barrier and lead to a higher spinodal pressure.10

Caupin and coworkers 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 would 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 H3O+ ion to a neighboring H2O molecule. In the first step of the mechanism of the

proton transfer the hydrogen-bond coordination number of one of the H2O molecules in the first

solvation shell is lowered by the breaking of a hydrogen bond to the second solvation shell.11 Therefore,

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

trigger cavitation.

Due to a lack of data at large negative pressures, the disagreement 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 of the

treated water system varies between 1020 and 300 kg/cm3. That our system is idealized means that no

stabilizing or destabilizing impurities will be present. In real trees stabilizing or destabilizing impurities

are 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 conditions in the form of the walls of the xylem

conduits could have a stabilizing or destabilizing effect on the metastable water system.

However, in the system, which is treated throughout this project, those impurities will be absent.

Besides that, the formation of hydronium ions, which spontaneously occurs 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 Ps = ± 1600 bar.

1.2.2 Determination of the pressure at which 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, Ps. In order to do so we will

perform simulations of the type:

(4)

V = f(T,P,n)

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 informations about the process of cavitation at the

determined spinodal pressure 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 paths 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 critical cavitation nucleus.

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

nucleus Rc is related to the height of the energy barrier, Eb, 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 indeed increase as the height of the reaction barrier,

Eb, does increase. As can be seen in Figure 2, the transition state is the moment when our system

(5)

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 configuration 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 for our system,

would also support the theory proposed by Caupin and coworkers. The presence of destabilizing

impurities in the water system would in fact decrease the height of the reaction barrier for cavitation

and would therefore result in a less negative spinodal pressure, Ps.

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 = maa = dv/ dtv = 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 pressure

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 regular intervals. The coordinates as a function of

time represent a trajectory of the system. After initial changes, the system will usually 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 electronic motions are not considered, implying that

the electrons are supposed to adjust their dynamics instantly when the atomic positions change, and re-

main 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 periodic boundary conditions to evade real phase boundaries.

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 thermo-

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 two 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 hence those that are crossing the free energy barrier.15

The method

Since a trajectory crosses the free energy barrier an infinite number of times an ensemble of crossing

paths is formed, the so-called transition path ensemble (TPE). In order to obtain the path ensemble a

sampling 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 paths is followed by accepting or

rejecting new trial pathways according to the following acceptance rule:

First the initial and final states of the reaction of interest are defined. Then a trajectory is created that

connects 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, which is represented on the y-axis in Figure 3.

Figure 3: Trajectory that connects the initial state A to the final state B.

After the initial and final states are defined, certain time slices on the current trajectory that connect the

initial to the final state are randomly selected, as shown in Figure 4. Then the momenta of the time

slices are changed slightly and a new trajectory of the same length is created by integrating the

equations of motion both forward and backward in time. The new trial trajectory is accepted if it

connects the initial with the final state. Otherwise it is rejected and the old path is kept. The shooting

move is then repeated 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 only the trial trajectory 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 Figure 3. tions forward and backward in time for the

three time slices selected in Figure 4.

2.2 Simulation Details

For the simulations performed in the scope of this project, a cubic box with periodic boundary

conditions was constructed. The box was then filled with 360 water molecules and all of the

simulations were performed at room temperature (298 K). The size of the box and the pressure were

chosen 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: Representation of a periodic box containing 360 water molecules

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

constant. 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 simulations between -100 bar

and -5000 bar. The simulation time 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 method 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 -1800 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

determining 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 relative positions of the selected frames to the transition 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 will end up in the initial

state A or the final state B will be equal. As can be seen in Figure 7, in 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: Integration of the equation of motions 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), which 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/m3, but no more at a density of 860 kg/m3. This means that

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

Figure 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 extrapolated to the spinodal pressure (dashed line).The range of pressures reached in acoustic experiments is limited to the right hand side of the ver- tical line.

As shown in Figure 8, the EOS of water (solid line), which we determined by computer simulations,

has the same course as the IAPWS (dashed line) down to a density of 940 kg/m3. Beyond this density

the values are shifted to more negative pressures and have a minimum at a density of 880 kg/m3.

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 proposed by the IAPWS. This supports the theory

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 for the IAPWS formulation 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 totally 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 Ps

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 of a Visual Molecular Dynamics Software, VMD, and by analyzing the 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 observe 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 phase. In stage b) the formation of a vapor bubble in the left, low

corner of the box is observed. 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 water 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 the left, low corner. In snapshot c) a stable water phase and a stable vapor phase have formed.

Gromacs Energies

Since the formation 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, temperature,

boxsize and density in the course of the performed simulations. In Figure 10 and Figure 11 the obtained

values are shown as functions of time for 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

simulation at -2147 bar and in all of the simulations 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 pressure, temperature, 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, beyond 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 decrease in the temperature and the total

and potential energy, which is in accordance with the formation of a stable vapor and water phase.

While during the formation of the stable water phase, the distances between the molecules are

decreasing again, which results 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 paths 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 transition state. The values of the

volumes of the box at the transition state which are stated the critical volumes, Vc, are listed in table 1.

Pressure (bar) critical box size length (nm) Critical volume of the box (nm3)-2148 2,3532 13.03096-2000 2,3587 13.122547-1800 2,3633 13.199472

Tabel 1: Critical volume, Vc, of the simulation box containing 360 water molecules at 298 K for the reaction 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 becomes 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 height 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, for which less negative spinodal pressures are found. 4,9,10

-2148 -2000 -18002.348002.350002.352002.354002.356002.358002.360002.362002.364002.36600

b o x s i z e ( n m )

pressure (bar) l

Figure 12: Relation between the critical box size at the transi-tion state for reaction paths at different pressures.

4. Conclusion

We were able to determine an equation of state that strongly supports the IAPWS formulation. The shift

of the spinodal pressure 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 to

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 increases 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 -2148 bar, which is an

even less negative pressure, than the one predicted by the IAPWS formulation. Furthermore 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, Vc, are rather small, it will be necessary to de-

velop 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 de-

crease the height of the reaction barrier for cavitation, but they do not identify the impurities. Therefore

there is need for computer simulations, in which hydronium ions are added to the system and need for

simulations, in which boundary conditions of all different kinds are added.

6. References

1. Atkins, P; De Paula, J Physical Chemistry, 8th ed.,Oxford Press, Oxford, 2006

2. Maritan, A TENSIWAT, A fundamental approach to tensile water transport in natural and artifical trees,

Eropean Research Council

3. Vincent, O; Marmottant, P; Quinto-Su, PA ; Ohl, CD Birth and Growth of Cavitation Bubbles within Water

under Tension Confined in a Simple Synthetic Tree American Physical Society 2012, 108, 184502,

4. Caupin, F; Hebert, E; Cavitation in water: a review C. R. Physique 2006, 7, 100-1017

5. Arvengas, A; Herbert, E; Cersoy, S; Davitt, K; Caupin F Cavitation in Heavy Water and Other Liquids J.

Phys. Chem. 2011, 115, 14240–14245

6. Atkins, P; De Paula, J Physical Chemistry, 8th ed.,Oxford Press, Oxford, 2006

7. Baltasar, EH; Taravillo, M; Baonza, VG; Sanz, PD; Guignon, B Speed of Sound in Liquid Water from

(253.15 to 348.15)K and Pressures from (0.1 to 700) Mpa J.Chem. Eng. Data, 2011, 56, 4800-4807

8. Davitt, K; Rolley, E; Caupin, F; Arvengas, A; Balibar, S Equation of State of water under negative pressure,

American Institute of Physics 2010, 133, 174507

9. Zheng, Q; Durben, DJ; Wolf, GH; Angell CA Liquids at large negative pressures: water at the homogenous

nucleation limit Science, 1991, 254, 829-832

10. Davitt, K; Arvengas, A; Caupin, F Water at the cavitation limit: Density of the metastable liquid and size of

the critical bubble EPL 2010, 90, 16002

11. Woutersen, S; Bakker, HJ, Utrafast Vibrational and Structural Dynamics of the Proton in Liquid Water, 2006, American Physical Society, 96, 138305

12. Gromacs Manual

13. Leenders, E Proton transfer in the photocycle of the photoactive yellow protein, 1st ed., 2008

14. Dellago, C; Bolhuis PG Transition Path Sampling and Other Advanced Simulation Techniques for Rare

Events Springer-Verlag , Berlin Heidelberg, 2008

15. Juraszek, J; Vreede, J; Bolhuis, PG Transition path sampling of protein conformational changes Chemical

Physics 2012, 396,30-44

16. Atkins, PW en Jones, LL Chemical Principles, 3rd ed., W.H. Freeman, New York, 2005