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 Introduction3
1.2 Motivation6
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 sampling10
2.2 Simulation Details12
2.2.1 Equation of sate of water under negative pressure13
2.2.2 Cavitation Pressure13
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 & Discussions
3.1Equation of state of water at negative pressure14
3.2Determination of the spinodal Pressure Ps15
3.3Creation of reaction paths at higher pressures and
determination of their transition states17
3.4Determination of the transition state and the size of
the critical cavitation nucleus.17
4. Conclusion19
5. Further Research19
6. References20
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 itsvapor phase.This results in the nucleation of vapor bubbles, also known as cavitation.2However, 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.3This 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. 5In 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).4A bubble whose radius is larger than Rc will grow spontaneously.5Equation (2) can be now rearranged in such a way that we obtain an expression , that relates the height of the reaction barrierto the size of the critical bubble and the liquid-vapor surface tension as follows:
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 tounderstand 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:
- Determination of the equation of state (EOS) of water under negative pressure.
- Determination of the spinodal pressure (Ps).
- Creation of new reaction paths at less negative pressures.
- 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.6Over 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.7The formulation at negative pressures is based on experimental data obtained at positive pressures. InFigure 1a graphical representation of the IAPWS is shown. As can be seen, the IAPWS predictsa 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 recentlydetermined, 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.8The obtained value for Ps is consistentwith 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
Thisquestions whether the water samples prepared for experiments, except for inclusion experimentsare 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 occurringin 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.11Therefore, 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 anidealized 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 aspinodal 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:
V = f(T,P,n)
In accordance with the IAPWS formulation, we expect that the spinodal pressure willlie 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 phasereturns 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 pressurewill be substantially more negative than pressures found in natural systems, like trees.2Therefore, 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 isthe moment when our system crosses the energy barrier and istherefore 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 = 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 theposition. M is the mass of a particle and t is time.13For 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 remain in their ground state.12
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 thermodynamically 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.6The 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.15In Figure 3an example is given of a trajectory that connects an initial state Ato 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.