A Molecular Dynamics Study
of the Spreading of Nano-Droplets
CHANGMING
YUAN
Master of Science Thesis
Stockholm, Sweden 2010
A Molecular Dynamics Study
of the Spreading of Nano-Droplets
CHANGMING
YUAN
Master’s Thesis in Numerical Analysis (30 ECTS credits)
at the Scientific Computing International Master Program
Royal Institute of Technology year 2010
Supervisors were Gustav Amberg and Andreas Carlson KTH SCI
Examiner was Michael Hanke
TRITA-CSC-E 2010:144
ISRN-KTH/CSC/E--10/144--SE
ISSN-1653-5715
Royal Institute of Technology
School of Computer Science and Communication
KTH CSC
SE-100 44 Stockholm, Sweden
URL: www.kth.se/csc
A molecular dynamics study of the spreading of
nano-droplets
Abstract
Dynamic wetting of nano-droplet spreading is studied with molecular dynamics simulations.
The modeling frame work has been validated against a well known analytical solution on the
macroscopic level. This indicated a minimum threshold for the number of molecular needed
in the simulations. Spreading of nano-droplets on substrate with different degrees of
wettability is studied, where special emphases are placed on the wall treatment and interactive
cut-off length. It is in particular shown that in order to model such dynamic cases with fidelity
on the macro-scale how many atoms are needed and that the wall treatment has the possibility
to affect the overall flow.
En molekyldynamik studie av spridningen av
nano-droppar
Sammanfattning
Dynamisk vätning genom spridning av nano-droppar är undersökt med simuleringar av
molekylär-dynamik. Ramverket för modelleringen har blivit validerat mot en välkänd
analytisk lösning på makroskopisk nivå. Resultaten indikerade på att ett minimum av antal
molekyler behövs i simuleringen för att väl sätt representera den makroskopiska lösningen.
Spridningen av nano-droppar på ytor med olik vätning är studerat med speciell fokus på
effekten av väggmodelleringen och trunkeringen av den intermolekylära interaktionen. Det
har visat sig att det behövs ett stort antal atomer för att kunna modellera sådana dynamiska
vätningsfall på en makroskopisk nivå, samt att väggmodelleringen kan påverka det
makroskopiska flödet.
Contents
1.Introduction
2.Literature review
2.1 Young-Laplace equation and the radius of the liquid
2.2 The contact angle and contact line
2.2.1 The expression of the contact angle
2.2.2 Thermostat for temperature controlling
2.2.3 The contact line motion
2.3 Energy dissipation
3.Theoretical basis and numerical method
3.1 Molecular Dynamics and Research Methods
3.1.1 Force field
3.1.2 Newton's equation of motion and numerical methods
3.1.3 Ensemble
3.1.4 Boundary conditions
3.1.5 Dimensionless
3.1.6 Thermostat
3.1.7 Ewald summation method
3.1.8 The calculation of temperature
3.2 Wetting
3.2.1 Surface tension and adhesion work
3.2.2 Contact Angle and Wettability
3.3 Energy dissipation
4.Numerical Analysis of Wetting Model
4.1 The Process of Building a Model
4.2 Laplace Law and the Radius of Liquid
4.3 Water model
4.4 Wetting Model with the Fixed Wall
4.5 Wetting Model with the Spring Wall
4.6 Energy Dissipation
5.Conclusion
Alphabetically
Appendix
1
3
5
5
5
6
7
8
9
9
9
10
11
12
12
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13
15
15
15
16
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17
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20
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30
39
41
42
43
List of References
Figure 1.1.1 the contact angle and interfacial energies
Figure 2.1.1 The structure of the literature review
Figure 2.2.1 Wenzel model
Figure 3.1.1 The Lennard-Jones potential
Figure 3.1.2 Ewald summation method
Figure 4.1.1 The simulation procedure of molecular dynamics
Figure 4.1.2 The work procedure of the simulations presented here
Figure 4.2.1 The equilibrium of 4531 molecules
Figure 4.3.1 The structure of TIP4P water model
Figure 4.3.2 The initial water box
Figure 4.3.3 The spherical shape of water
Figure 4.4.1 The states of liquid-solid wetting process in different time steps: 6162ps,
12324ps, 61594ps, 123188ps, 184756ps, 246350ps
Figure 4.4.2 The states of liquid-solid wetting process in different time steps: 6162ps,
12324ps, 61594ps, 123188ps, 184756ps, 246350ps with contact lines
Figure 4.4.3 The size of contact angle at different time steps
Figure 4.4.4 Different contact angle with different coefficients
Figure 4.4.5 Radius with timesteps
Figure 4.4.6 The loglog figure of radius with timesteps
Figure 4.5.1 The states of liquid-solid wetting process in different time steps: 1170ps,
10920ps, 53950ps, 107640ps
Figure 4.5.2 The states of liquid-solid wetting process in different time steps: 1170ps,
10920ps, 53950ps, 107640ps with contact lines
Figure 4.5.3 The contact line are calculated at macroscopic scale and microscopic scale
Figure 4.5.4 The size of contact angle at different time steps
Figure 4.5.5 The contact angles of the first
Figure 4.5.6 The contact angle of each timesteps
Figure 4.5.7 The contact angle with time in linear scaling
Figure 4.5.8 The contact angle with time in log scaling
Figure 4.5.9 The contact angle of liquid with the specific shape
Figure 4.5.10 The errorbar with contact angles
Figure 4.5.11 The errorbar with spreading radius
Figure 4.5.12 The radius and loglog of radius with time
Figure 4.5.13 The contact angles and loglog of contact angles with time
Figure 4.6.1 The temperature at each atoms
Figure 4.6.2 Rescale the temperature of the liquid
Figure 4.6.3 Scale method for calculation of temperature
Figure 6.1.1 VMD main interface
Figure 6.1.2 New Trajectory
Figure 6.1.3 VMD video
Figure 6.1.4 xmovie video
Figure 6.2.1 Create the new file for building water molecules
Figure 6.2.2 The interface of Material Studio
Figure 6.2.3 Scratch the Oxygen atom and add H atoms automatically
Figure 6.2.4 The water molecule
Figure 6.2.5 Construct the amorphous cells
Figure 6.2.6 Amorphous cells construction
Figure 6.2.7 Water box with 1500 molecules
Introduction
Wettability is one of the important properties of solid surfaces which is decided by the
chemical composition and microscopic geometry construction of the surfaces. Generally, the
wettability of the solid surface is measured by the contact angle at an equilibrium by the angle
formed between the solid surface and a tangent extrapolated along the liquid-gas interface. If
the contact angle is greater than 150 degree, the surface is called Super-hydrophobic surface.
And if the contact angle is almost 0 degree, the surface is called Super-hydrophilic surface.
For contact angles between 0 to 150 degree, the surface is considered to be only partially wet.
Wettablility plays an important role in material science and biomedic processes. The
super-hydrophobic material could be used in producing liquid -proof surface and paint, and
the super-hydrophilic material could be used in producing self-cleaning glass.
The earliest research of solid-liquid contact could be traced back to 1805(T. Young,1805).
The first theoretical description of the contact line arises from the consideration of a
thermodynamic equilibrium is defined by Tomas Young. The famous Young‟s equation must
be satisfied in the equilibrium.
(1.1.1)
where is the contact angle,
is the solid vapor interfacial energy,
is the liquid vapor
interfacial energy and
is the solid-liquid interfacial energy(T. Young,1805), see
illustrations gives in Figure 1.1.1. This equation is built on the considerable surface which is
uniform, smooth, no deformation and isotropic. But in actuality, the surface is not uniform,
and there will be some diviations from the analytical prediction of Young‟s equation.
Figure 1.1.1 the contact angle and interfacial energies
The task of contact line motion and the energy dissipation from that have attracted the
attention of many researchers. Since 1976, Dussan and other researchers have made a lot of
experiments(Dussan V., E. B., 1979). They tried to find the appropriate boundary condition at
the contact line and they divided the investigations into two groups depending on the contact
line region to derive the position of the interface and its slope.
The first region contained the whole liquid region. The value of the contact angle was used as
a boundary condition at the contact line based on the assumptions that Navier-Stokes
1
equations and the boundary condition at the liquid interface were valid up to the contact line.
In the second region, they pointed out that only one experimentally measurable parameter
which was independent of the geometry of the outer region could completely determine the
motion of the outer region fluids. The parameter could serve as a boundary condition for the
equation in this region.
The energy dissipation could be considered as the energy which was generated from the
contact line overcome the friction between liquid and solid. The energy released into both the
sliding and surface and transferred to heat.
Molecular dynamics has been developed as a convenient tool for simulating the process of
wetting and energy dissipation. With the development of the computer simulation technology,
it is possible to obtain the important information from computer simulation which is
impossible or hard to obtain from experiments. Because of the advantages, like offering
references for scholars, guiding the experiments, decreasing the blindness of the experiments,
lowing cost and expanding using range, computer simulation, especially, the molecules
simulation plays an important role in molecules physics, fluid dynamics, biology and material
science.
In recent years, a lot of software is designed for molecular dynamics simulation, NAMD,
AMBER, CHARMM, TINKER, LAMMPS and so on. In this thesis, the open source
molecular dynamics code LAMMPS has been applied.
With these backgrounds the thesis is structured into the following parts.
 Literature review
 Theoretical basis and numerical methods
 Lennard-Jones model and solutions
 Water model and solutions
 Conclusions and future work
 Appendix
2
Literature review
In the wetting case, there are many topics that could be researched in depth. In the literature
review the emphasis is placed on the following points:
1. Laplace equation and the radius of the liquid in the equilibrium
2. The contact angle in the equilibrium with different parameters
3. Energy dissipation
4. Real physical model and numerical computation
There are many experiments about the liquid-solid contact with different materials,
experimental conditions and parameters in the past years. Many different molecular dynamics
simulation tools and numerical methods have been used, and different contact angle functions
and energy dissipation functions have been obtained. For smooth surfaces, the contact angle is
well defined by a linear function as Young‟s law. If the surface is not smooth, the contact
angle will obtain either a Cassie state or a Wenzel state. For the microscopic level, some work
suggested that the Young‟s law failed(L.D. Landau, E.M. Lifshitz), while some others
opposite of that(G.He, G.Hadjiconstantinou, 2003). Does Young‟s law fit for the microscopic
level? Or does it only fit for the macroscopic level? We will discuss it in the thesis.
3
In Figure 2.1, a description of the structure of the literature review is given.
The structure of the literature review
Laplace equation and the radius of the liquid
The contact angle and contact line
The expression of the contact angle
Thermostat for temperature controlling
The contact line motion
Energy dissipation
The water model
Figure 2.1: The structure of the literature review
4
2.1 Young-Laplace equation and the radius of
the liquid
Thomas Young developed the qualitative theory of surface tension in his historic essay in
1805 (T. Young,1805). Subsequently Pierre-Simon Laplace completed the mathematical
description in the following year (Mécanique céleste, 1806). The use of Young-Laplace
equation is widely and is out of its foundatious applied in the field of two phases flow.
In 2003, G. He and G. Hadjiconstantinou followed the Young-Laplace equation and
Rowlinson-Widom model(J. S. Rowlinson, B. Widom) to estimate the relationship between
the surface tension and coefficients of Lennard-Jones potential[G.He, G.Hadjiconstantinou,
2003]. That is commonly applied in molecular dynamics simulations.
(2.1.1)
Here, is the surface tension, is the depth of potential well and
the inter-particle potential is zero.
is the distance at which
The Young-Laplace equation is very important to describe the capillary pressure sustained
across the interface and it is a macroscopic property of liquid. Thus, if the simulations could
be fit for the Young-Laplace equation, the number of molecules is enough to research and
discuss the properties and motions at macroscopic level. In Chapter 4, we will discuss the
Young-Laplace equation in depth with numerical analysis.
2.2 The contact angle and contact line
2.2.1 The expression of the contact angle
For calculating the contact angle at the equilibrium with different coefficients, we should find
the expression of the contact angle.
Thomas Young stated in his essay: “But it is necessary to premise one observation, which
appears to be new, and which is equally consistent with theory and experiment; that is, for
each combination of a solid and a fluid, there is an appropriate angle of contact between the
surfaces of the fluid, exposed to the air and the solid (T. Young,1805).” This means, the
contact angle could be calculated when the fluid contact with the solid or the vapor.
He asserts that the famous Young‟s equation which became the basis of liquid-solid contact in
researches. But this equation could be only used in an ideal condition, which means the solid
surface must be uniform, smooth, non-deformation and isotropic. (See formula 1.1.1)
In 1936, Wenzel modified Young‟s equation to make it fit for the effects of a rough surface.
He described a homogeneous wetting regime in the following figure.
5
Figure 2.2.1 Wenzel model
The contact angle is described by the following equation in the Wenzel model.
(2.2.1)
where
is the contact angle in the equilibrium state. r is the roughness ratio which is
defined as the ratio of true area of the solid surface to the apparent area.
is the contact
angle in the Young‟s equation.
Cassie and Baxter mixed both the rough and the smooth surface to construct a new model,
which could describe more complicated physics.
The contact angle is explained in the Cassie-Baxter model as,
(2.2.2)
where
is roughness ratio of the wet surface. f is the fraction of the solid surface area wet
by the liquid,
is the contact angle in Young‟s equation.
From the equation (2.2.2), it is easy to see that when f=1, the equation becomes to the Wenzel
model. In addition, if the wet surface is constituted by different materials, each fraction could
be noted as . And the Cassie-Baxter equation could be written as
(2.2.3)
sl
Where
is the solid-liquid interfacial energy and
is the solid-vapor interfacial energy.
Johnson and Dettre summarized the Wenzel model and Cassie-Baxter model and found that
there is a critical value of the roughness ratio in which the contact angle fits for the Wenzel
model. If the roughness ratio is less than the critical value, the contact line will find a Wenzel
state; if it is greater than the critical value , it will follow the model from
Cassie-Baxter(Johnson R. E., Dettre R. H., 1964).
2.2.2 Thermostat for temperature controlling
The motion of the contact line at the macroscopic level will be researched in this thesis
through molecular dynamics method with the nano-droplet. The change of temperature of the
solid wall will be so small and could be neglected at the macroscopic level. Meanwhile the
temperature will affect the result of simulation at the microscopic level significantly. Thus,
the temperature of the solid wall should be fixed.
The thermostat is always used to fix the temperature to a fixed value in molecular dynamics
simulations. In the dynamics simulation it is needed to control the temperature, and therefore
the thermostat is a very important process. The temperature of the solid wall need to be fixed
6
in the wetting model to assess the energy dissipation in wetting. In a recent paper, G. He and
N. G. Hadjiconstantinou used a Langevin thermostat to maintain the temperature to
and get the expression of energy dissipation(G.He, G.Hadjiconstantinou,
2003). In another paper from Micheal Tretyakov(Ruslan L., Davidchack, Richard Handel,
Micheal Tretyakov, 2009), the Langevin thermostat has been used with success to control
the temperature in the simulation of TIP4P model of water and it works well. Thus, the
Langevin thermostat seems to be a good choice to maintain a fixed temperature in the solid in
my simulation.
Anirban found that the Berendsen thermostat is a very effective in preserving temperature in
the SPC/E model of water. The Berendsen thermostat is widely used in molecular simulation
under the constant-temperature, constant-volume ensemble (NVT). There is a function for
fixing the temperature by Berendsen thermostat in LAMMPS, but it will cause a nonzero
velocity of the solid wall under the constant-energy, constant-volume ensemble (NVE) which
will affect the result of simulation and the contact line motion. Therefore, a Berendsen
thermostat is assumed to be less appropriate in my study.
All the thermostat methods mentioned above controlled the temperature by rescaling the
velocity field and keeping the kinetic energy of the system to be the fixed value. The kinetic
energy will be calculated as
(2.2.4)
2.2.3 The contact line motion
J. De Coninck and T.D. Blake gave us a definition of the contact line: the mathematical line at
which solid-liquid, liquid-vapor, and solid-vapor interfaces meet(J. De Coninck, T.D. Blake,
2008). In recent decades, the wetting phenomena have attracted the attention of many
scientists, and great progress has been made both in theoretical and experimental aspects.
Tanner described the shape of oil droplets sliding on horizontal surface in 1978, which used
the Navier-Stokes equations to calculate the contact angle and contact line motion in
2-dimensions (L. H. Tanner, 1979).
Considering the Young‟s equation 1.1.1, some authors argued that the Young‟s equation is a
force balance at the contact line and is known up to the contact line (Johnson R. E., Sadhal
S. S., 1983). Others have pointed out that the parameter describes the energy of the
interface(Dussan V., E. B., 1979).
The contact line motion is a very important physical phenomenon in wetting procedure.
Maruyama integrated the one layer of fcc(1,1,1) lattice and found the potential to be
(2.2.5)
where
is the nearest neighbor distance,
Lennard-Jones potential between solid and argon.
will be minimum when
.
and
are the coefficients of the
To consider about the effect of coefficients of Lennard-Jones potential equation at the
7
microscopic level, the contact angle at the equilibrium will be calculated under each variable
value.
2.3 Energy dissipation
Maruyama mapped the temperature for 2D wetting model and presented the temperature in
the droplet. The figure(see fig 2) looks layer by layer of molecules, the edge of the liquid is
lower than the center and the highest temperature locate near the contact plane(S. Maruyama,
T. Kurashige, S. Matsumoto, Y. Yamaguchi, T. Kimura, 1998).
The energy dissipation has been researched by a lot of scientists in the recent decades. W.
Ren and R. E. give a dissipation energy from the incompressible Navier-Stokes equation for
two liquid contacts with a solid wall.
(2.3.1)
where
is the density of the fluid,
is the viscosity of the fluid, p represent the pressure,
is the unit normal to the interface,
is the friction coefficient,
,
denotes the advancing and receding fluids,
denotes the boundary of
,
denotes the
fluid-fluid interface and
denotes the fluid-solid interface.
Equation 2.3.1 could be calculated with the boundary conditions of fluid-fluid interactions
and fluid-solid interactions.
The constant-energy, constant-volume ensemble is used in the simulation which keep the total
energy of the system to be a constant. We want to find how the energy moves, from solid wall
to liquid or the convert. Energy dissipation could give us an explanation where the energy
goes, thus I rescaled the temperature map and tried to analyze the energy distribution.
8
Theoretical basis and
numerical method
3.1 Molecular Dynamics and Research Methods
Molecular dynamics (MD) is an important simulation method of the microscopic area for
assessing and predicting the structure and properties of materials by simulating the motion
and interaction of atoms and molecules with computers. During the simulation, each atom
will be seen as moving in accordance with Newtonian mechanism under the potential field
which is constituted by all neighbor atoms. Then the structure and properties in the
macroscopic level will be obtained by the methods from statistical physics. In short, MD
method is a simulation by applying the force field and Newton‟s equation of motion.
3.1.1 Force field
The force field is an expression of energy potential surface which is the basis of molecular
dynamics simulation and depends on the molecules potential energy and the distance between
each molecules. For various purposes, force field can be divided into many forms with
different scope and limitations. The reliability of the solutions is dependent on which force
field we choose.
Lennard-Jones potential is the most widely used potential function.
(3.1.1)
Here U(r) is the molecules potential at value r, r is the distance between molecules and
represents the depth of the potential well, and
is the distance at which the inter-particle
potential is zero.
9
F=0, Equilibrium
Attractive
Repulsive
Slope=0, Force=0
Figure 3.1.1 The Lennard-Jones potential
The force field has been developed from monatomic molecules system to polyatomic
molecules system since last decades (O. V. Voinov, 1977). The complexity, accuracy and
scope of application are also extended significantly. Because of the variety of force field, we
need to determine which force field we should use for finding a balance between simulation
time and accuracy depended on the simulation conditions and system characteristics when the
force fields could be used in the physical problem.
3.1.2 Newton's equation of motion and numerical methods
(3.1.2)
where
is the displacement,
is the velocity,
is the acceleration.
System as considered here is only applicable for pure monatomic systems with similar
properties as Argon.
The basis of molecular dynamics simulation is the calculation of the positions and velocities
of molecules stepwise, and finally obtaining the trace of molecules in this way.
There are lots of methods for solving Newton‟s equation of motion, Verlet method is used in
most of simulation program which is called the Leap Frog Method.
1
2
ri t
t
ri t
t
vi t
1
2
t
ai t t
(3.1.3)
t
(3.1.4)
where t is the timestep.
10
1
2
It is easy to find that if vi t
t and ri t are known, we could calculate the accelerate
ai t from the position ri t at time t. Then we could get the velocity at time t
1
2
t from
equations 3.1.1 and 3.1.2. Thus, the velocity at time t could be calculated in following
equation.
vi t
1
2
vi t
1
2
t
1
2
vi t
t
(3.1.5)
This algorithm has been applied in most of simulation program because that it only need to
know the velocity at time t
1
2
t and the position at time t, which reduce the space spending
on computers, and it is a stable method.
3.1.3 Ensemble
Ensemble is the collection of part of the properties of the system which has the same
circumstance. It is a very important concept of statistical mechanism, which is the basis of
almost every statistical property.
Generally, we use the NVT, NVE and NTP ensembles in the simulations where N is the
number of atoms, V is the volume of the system, E is the energy of the system, P is the
pressure of the system, and T is the temperature of the system.
 NVT ensemble
The constant-temperature, constant-volume ensemble (NVT), also known as the canonical
ensemble, is obtained by controlling the thermodynamic temperature.
Assume that we put N atoms in a box whose volume is V and embed it in the thermo bath
whose temperature is constant T. The total energy and system pressure may change around
the average value in this case. Generally, the NVT system could be thought like a system
transport heat with an infinity heat source. The NVT system and the heat source could be seen
as an NVE system. It is an appropriate choice for the models with periodic boundaries and the
energy or the pressure or the volume but not when the temperature needs to be calculated
from the simulation.
 NVE ensemble
The constant-energy, constant-volume ensemble (NVE), also known as the microcanonical
ensemble, is obtained by solving the standard Newton equation without any temperature and
pressure control.
Most of simulations could use this ensemble, while the NVE models are isolated systems
which have constant energy. The energy of the system is conserved, so that transfer of the
energy inside the system can be more clearly distinguished.
 NTP ensemble
The constant-temperature, constant-pressure ensemble (NPT) allows control over both the
temperature and pressure.
The NTP ensemble is always chosen in the cases when the correct pressure, volume and
11
density are very important in the simulations.
In my simulations, the pressure should not be fixed for proving the Young-Laplace equation
could be used in nano-droplets and the temperature should not be fixed for considering about
the energy dissipation. Thus, NVE ensemble is used in my simulations.
3.1.4 Boundary conditions
The number of molecules is fixed in the simulation box in the molecular dynamics simulation,
which means if some molecules run out of the box, the same number of molecules will enter
into the box to assure that the density of the system must not be changed. Usually, periodic
boundary, when an object passes through one face of the unit cell, it appears on the opposite
face with the same velocity is applied in the simulation. Without periodic boundary
conditions, volume, pressure, and density are not defined constant-pressure dynamics cannot
be carried out. But the constant-volume is very important in the molecular dynamics
simulations.
3.1.5 Dimensionless
Physical models describe the inner characters of actual problems. Dimension is the additional
measure method of variables which depend on the choice of the basic physical variables. The
discipline described in the physical models should be independent of the dimensions of
physical quantities. Thus, the dimensionless physical quantities are important to describe the
physical laws objectively. Dimensionless also could simplify the numerical computing
equations to be a specific mathematics equation which could be solved easily.
3.1.6 Thermostat
 Langevin thermostat
The Langevin thermostat uses a Langevin equation of motion instead of the Newton‟s
equation of motion(Adelman, S.A., J.D. Doll, 1976).
ma
v f v
f
(3.1.6)
where
is the frictional constant, f is an additional frictional force to conserve the
temperature by adjusting the velocity and let the kinetic energy match the set temperature.
In LAMMPS, we use „fix Langevin‟ function to keep the temperature as a constant in the
solid. For conserving the temperature, LAMMPS uses
(3.1.7)
c
f
r
where c is a conservative force depending on usual inter-particle interactions; f is a
12
viscous damping term computed via
m
damp
f
v and
r
is the force calculated from
recent temperature T. Comparing with equation (3.1.6) and (3.1.7), we could see that
v, r f v and c f .
f
 Berendsen thermostat
To maintain the temperature of the system, the simulation system could be seen as coupled
with an external heat source. The velocities are rescaled at each step.
dT t
1
T0 T t
dt
where is the coupling parameter which determines the strength of the coupling(Victor
Ruhle, 2007).
The function „fix temp/berendsen‟ is used in LAMMPS, and it gets the target temperature
with Ttarget
Tend Tstart t and uses a mask for rescaling the velocity map. The scaling
factor is
1
t
T0
T t
1 , where
t
2
determines how tight the system is coupled with
the heat source.
 Velocity scaling
This is a simple way to alter the temperature of the system. As we know,
1
2
T
2
i 1
mi v2i
Ns kB
where T is the temperature, mi is the mass, vi is the velocity, Ns is the dimension and kB
is the Boltzmann constant.
Add a coefficient
T0 T t , we could rescale the velocity map.
T
1
2
2
i 1
mi v2i
Ns kB
1
2
2
i 1
mi v2i
Ns kB
But there is a problem that this approach does not allow fluctuations in temperatures under the
canonical ensemble.
3.1.7 Ewald summation method
Ewald summation is a method for computing the interaction energies, between all atoms,
which rewrites the interaction potential into two parts,
r
s r
l r
where s r is the short range interaction, and l r is the long range interaction.
The short range interaction is summed in real space, and the long range interaction summed in
Fourier space.
Let‟s consider a cube with its side length L containing N atoms,
13
L
2
L
Fourier vector is k
kx ky kz and the interparticle vectors are ri j n
rj
ri
nL.
Thus, the Ewald sum is:
ri
where
z
2
z
e
t2
4
L3
N
qj
j 1
ikrij
e
2
k e
k2
4 2
k
ri j n
n
dt.( Ewald P, 1921)
We could use function kspace_style ewald to use ewald summation method. It gives order
O N2 time consuming.
Particle-Particle-Particle Mesh is a Fourier-based Ewald summation method which is used to
calculate the potentials.
The long range potential is evaluated in Fourier space using
long
where ^ indicates the Fourier transform,
k
k
k
k
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
Figure 3.1.2 Ewald summation method
The particle mesh is described in the following steps shown in Figure 3.1.2,
14

Mesh grid for simulation space. Several mesh methods could be chosen depending on the
accuracy and time consuming desire.

Use Fourier Transform over the grid to get the function

Apply the inverse Fourier Transform to obtain
long
k , and calculate
long
k .
k .
 Differentiating the potentials numerically to obtain the grid-defined forces.
 Intepolate forces back to particles and rebuild the coordinates.
The function kspace_style pppm could be used to apply pppm method to calculate the sum. It
gives O NlogN in computational time consuming.
3.1.8 The calculation of temperature
In Lammps, the temperature is calculated by the formula ke
3
NkB T
2
in 3-Dimensional
model, where N is the number of atoms, kB is the Boltzmann constant, ke is the kinetic
energy and T is the temperature.
We could get ke
2
N mi vi
i 1 2
from the dump file and then calculate the temperature by
following the famous uniform distribution formula.
3.2 Wetting
Wetting is the ability for a liquid to wet a solid and can be described by the equilibrium given
by the force balance between adhesive and cohesive forces.
3.2.1 Surface tension and adhesion work
Actually liquid-solid wetting is the physical process that the vapor is replaced by the liquid
above the solid.
The adhesion work is the energy which is needed to divide the two interfaces.
Wa r1 r2 r12
Which r1 and r2 are the surface tension surface 1 and surface 2, r12 is the interfacial
tension between surface 1 and surface2.
15
3.2.2 Contact Angle and Wettability
The wettability can be expressed in terms of the contact angle , which is the intersection
angle between the tangent line of the liquid and the solid surface. When the liquid reaches an
equilibrium, it should satisfy the following function
S
SL
L cos
Where S is the interfacial tension between the solid surface and vapor, SL is the
interfacial tension between the solid surface and liquid, L is the interfacial tension between
liquid and vapor and is the contact angle.
3.3 Energy dissipation
Energy dissipation is a physical process that the dissipative system lose energy over time,
typically due to the action of friction or turbulence. The kinetic energy is converted to heat
and the temperature of the system will increase.
In this thesis, I calculate the kinetic energy of every area and try to find the process of the
energy dissipation in the nano-droplet.
16
Numerical Analysis of Wetting
Model
Molecular dynamics is a form of computer simulation following the known laws of physics to
determine how the atoms and molecules moves under the interactions.
After the introduction of basic theories, we should begin to see what happened in the
simulations. Very briefly, let us summarize the basic steps of the simulations.
Start
Initialize
positions and
velocities
Calculate forces
for all molecules
using potential
Till termination condition
Apply thermostat
and volume charges
Update position and
velocities
Analyze the data
Figure 4.1.1 The simulation procedure of molecular dynamics
17
In this thesis, Lennard-Jones potential is chosen for computing the atom interaction s of
considerable model. It is a mathematically model that describes the interaction between a pair
of atoms.
U r
4
r 12
r 6
U(r) is the potential at value r, r is the distance between molecules, is the energy well depth
and
is the interaction length scale.
Because of computational limitations, due to the summation of influence at all atoms, the
potential is usually written as,
r
if x
c
c
L
L
0
if x
c
where c is the cut-off radius.
In cases, simulated here
1 05, is fixed in each models.
The LJ type units are used in the simulation, thus the physical quantities are dimensionless.
 mass = m
 distance = , where x
x

time = , where
t
t
m
2
 energy = , where
 velocity =
, where v
v
 temperature = LJ temperature, where T
T b
3
 pressure = LJ pressure, where P
P
In the LJ simulation, the timestep size dt 0 005 , m is the particle mass in the LJ model.
4.1 The Process of Building a Model
Following figure 4.1.1, we need to initialize the position, velocities and parameters. In
LAMMPS, we should initialize the parameters for the models in the following way.
#Choose the LJ model, so we use
units lj
#Specify atomic style in considerable model
atom_style atomic
#Determines fcc as the lattics, the faced centered cubic which have atoms on both the vertex
and center of the faces of the cubic.
lattice fcc
#Set boundary conditions
#Set the x, z direction‟s boundary as a periodic boundary and the y direction‟s boundary as a
mirror boundary.
boundary p s p
#Initialize the position of atoms
#Create boxes for the model to bound atoms in the boxes.
create_box
#Create region for each items in the model.
18
region
#Create atoms in the region and we could get the initial atom position.
create_atoms
#Initialize the velocities of atoms
#Generate an ensemble averaging of velocities using a random number generator for the
liquid.
velocity lo create 0.4 395467
#For each model, we set the different coefficients for liquid and solid. Then we will see what
happened and try to understand why it will like that.
#After initialize the position and velocities of atoms, calculate forces for all atoms using
potential.
#Set the parameters of Lennard-Jones potential
#Set pair style and cut radius for the model
pair_style lj/cut 2.5
#Set coefficient of Lennard-Jones potential between liquid and liquid, liquid and solid.
pair_coeff
#Set the timesteps
timestep 0.01
#Calculate interactions under NVE essemble using pppm summation method.
#Apply thermostat in the simulation.
#Apply Langevin thermostat in the simulation
fix langevin
Thus, figure 4.1.1 will changed to following figure
Till termination condition
Calculate forces
for all molecules
using potential
Lennard-Jones Potential
pppm summation
Apply thermostat
and volume charges
Langevin thermostat
Update position and
velocities
Verlet Algorithm
Figure 4.1.2 The work procedure of the simulations presented here
Following the steps mentioned above, we could complete a wetting simulation and then
analyze the data to explore the physical phenomenon in wetting.
19
4.2 Laplace Law and the Radius of Liquid
First of all, a liquid cubic as a fcc lattice of atoms was built with the side length 5, 10, 17, 23,
30 which have 108, 500, 1099, 2457 and 4631 molecules.
I build the programme with the LJ coefficients being set to
1 05 and
4 5, which
21
means
0 45nm,
1 05 10
, cutoff radius is 2.5 , timestep size is about 0.13ps
and change the size of liquid cubic to find when the Young-Laplace equation will be fit for
the simulation.
At the equilibrium, the liquid cubic becomes a liquid sphere. See Figure 4.2.1,
Figure 4.2.1 The equilibrium of 4531 molecules
Figure 4.2.1 shows that the liquid cubic gathered into a spherical shape at few nano seconds
where a few atoms surrounds the bulk liquid. Because there are only thousands of atoms, it is
not perfectly spherical. But in further research, we could find the number of atoms is enough
to simulate wetting process and find the physical law in the simulation.
From the equation
, we could get the surface tension
=0.05704
Table 4.2.1 Radius and pressure of the equilibrium droplet
108
500
1099
2457
4531
100000
100000
100000
200000
200000
0.05704
0.05704
0.05704
0.05704
0.05704
Not a sphere
Much atoms
out of the
sphere
Not stable
0.2041
0.2566
∆p
0.5600
0.444
error
0.39%
0.034%
Steps
R
From the Table 4.2.1, we could see that when the number of molecules is large enough, the
pressure and radius will fit for the Laplace‟s law.
20
pout
pin
2
(4.2.1)
Where pout is the outside pressure, pin is the inside pressure,
is the radius of droplet.
The error in the table is calculated from
P
2
P
is the surface tension and R
(4.2.2)
R
internal pressure p
Figure 4.2.2 Laplace’s law
As we know, the Laplace‟s law is appropriate in the macroscopic level which means, if the
model fits to the Laplace‟s law, it would show the same physical properties with the case in
the macroscopic level. From table 4.1.1, we can see when the number of atoms of the liquid is
more than 2457, it could be used in simulations to try to find the macroscopic properties of
the physical process. When the number of atoms is so small, like 108 atoms, the atoms will go
around everywhere and not form a sphere. It is better when the number of atoms increases to
500. The atoms gathered to be a sphere-like shape, but it is not stable, a lot of atoms go out
and then go back. If there are about 1000 atoms, the edge of the atoms looked like a sphere,
but the pressure does not stabilize.
21
4.3 Water model
Then, set the parameters by following the table 4.3.1
Table 4.3.1 The parameters of TIP4P water model
3.15365
0.6480
0.9572
0.15
0.5200
-1.0400
104.52
52.26
k mol
l1
l2
q1 e
q2 e
Figure 4.3.1 The structure of TIP4P water model
Set the coefficients of the water model based on Table 4.3.1. We could find the water cube
becomes to a water sphere. Because of the so high cost of the water simulation. It was not
possible due to computational limitations to study wetting with such models. Therefore, the
study has been restricted to only LJ potentials.
The model is built as a water box at first.
22
Figure 4.3.2 The initial water box
It becomes to be a spherical shape, but it is not perfect because we only have about 2000
molecules in the simulation.
Figure 4.3.3 The spherical shape of water
The green atoms are the oxygen atoms and the blue ones are the hydrogen atoms. We could
see that the water molecules of the water box gathered to be a spherical shape which means
Laplace law could be applied in the microscopic scale of water molecules.
From equation
, we could get the surface tension
23
Calculate the radius from MATLAB.
Scale the liquid sphere to 20 layers at y direction and find all the atoms at each layer, then
find the atoms which density bigger than 20 to form to be a circle. We could calculate the
radius of the circle in using the maximum value and minimum value at x direction.
The pressure is calculated from the LAMMPS
which means Young-Laplace equation (4.2.1) works in the water simulation.
4.4 Wetting Model with the Fixed Wall
Use fix 2 lo setforce 0.0 0.0 0.0 command to fix the wall which set all the force to the wall
going to zero at each 1000 time steps. The timestep size is about 0.13ps. The coefficients
1 05, L 1 00, S 6 9, L 1 5, cut-off radius=2.5.
S
Figure 4.4.1 The states of liquid-solid wetting process in different time steps: 6162ps,
12324ps, 61594ps, 123188ps, 184756ps, 246350ps
The green squares in the figure 4.4.1 are the fixed wall atoms and the blue circles are the
liquid atoms. We could see the water spread quickly in the first 61ns and then slow down. The
24
last figure is the equilibrium state and we could get the spread radius and the contact angle
from it. Find the contact line in MATLAB by the following steps.
Firstly, choose the liquid atoms whose type is equal to 1.
Secondly, separate the atoms to 10 layers and find the edge of each layer by controlling the
distance between the atom and the nearest one of it.
Thirdly, fix a line with several edge points, 3 or 4.
Finally, check if the contact line is fit for the circle as form
.
From the equation of the contact line, we could calculate the contact angle
25
Figure 4.4.2 The states of liquid-solid wetting process in different time steps: 6162ps,
12324ps, 61594ps, 123188ps, 184756ps, 246350ps with contact lines
The red lines in figure 4.4.2 are the contact lines which are calculated in Matlab. From the
Figure 4.4.2, we could find that the contact angle will decrease until the liquid goes to the
equilibrium. But in the fourth figure, the contact line is smaller because of the shape of the
liquid. I use the circle fit method to calculate the contact angle which means that the edge of
the liquid will be found from each timestep and then use least square analysis to fit them on a
circle(Guowang Xu, Mingchao Liao, 2002). The contact line in time could be found in this
way.
Figure 4.4.3 The size of contact angle at different time steps
In Figure 4.4.3 the contact angle relaxation from the simulation presented in Figure 4.4.2 is
shown. It is clear that from which we could see the contact angle decrease rapidly with time.
Obviously, the droplet spreads on the solid surface, and it will reach the equilibrium in the
end. The step size of time is 0.13ps, thus it takes about 260ns to get the equilibrium
properties.
Changing the coefficient of Lennard-Jones potential, we will find that the contact angle at the
equilibrium is different.
26
Figure 4.4.4 Different contact angle with different coefficients
We set the periodic boundary for both x and y directions, and set the mirror boundary
for z directions. The nearest neighbor distance are 2.5 and 4.5 in our model. The wall
is fixed 3 layers. Figure 4.4.4 shows that the relationship with the cosine of the contact angle
with s l . The blue line with circles gives the relationship under the cutoff radius is equal to
2.5 and the black line with squares shows the relationship under the cutoff radius is equal
4.5 . Obviously, the slope of the two lines are not the same and the contact angle is different
at the equilibrium when only the cutoff radius has been changed. Thus, the cutoff radius affect
the wetting process and we can find that the bigger the cutoff radius, the larger the contact
angle at the equilibrium state.
Changing parameters to make the contact angle of equilibrium to be different angles, for there
rather small simulation
27
Figure 4.4.5 Radius with timesteps
There are five curves given in Figure 4.4.5. The light blue curve with circles represents the
liquid which reaches 30 degree at the equilibrium state. The curve with yellow stars, the curve
with green squares, the red curve and the curve with black triangles represents the radius at
each timestep from 45 degree to 120 degree at the equilibrium state as shown in the legend
above. The radius of the liquid increases fast at the first 13ns and than increases much slower.
As we only consider how small system of nano-size behaves, the cut-off length influences the
results. By increasing the number of molecules, the effect of increased cut-off length is
expected to disappear.
28
Figure 4.4.6 The loglog figure of radius with timesteps
By plotting the same quantity in loglog scale the slope is clearly illustrated, from figure 4.4.6
we could see that the slopes(about 30 degree) are very similar in different simulations with
the fixed wall. The legend shows the color of curves which represent the slopes with time
under different contact angles at the equilibrium state. We can see that there are some
oscillations in the figure. It is because the radius is not easily calculated in the 3D model, and
the error is inevitable.
The wall is fixed in the above model, thus there is no heat exchange between the liquid
droplet and the solid wall which means some energy will disappear during the simulation and
it cannot show us the physical laws at the macroscopic scale very well.
29
4.5 Wetting Model with the Spring Wall
Use the fix 2 lo spring/self 250.0 command to set the wall as 3 layers harmonic wall which
the spring constant k 250 0 N m. With the same research process, we get the liquid-solid
contact states at different time steps.
Figure 4.5.1 The states of liquid-solid wetting process in different time steps: 1170ps,
10920ps, 53950ps, 107640ps
The green squares in the figure 4.5.1 are the fixed wall atoms and the blue circles are the
liquid atoms. We could see the water spread rapidly in the first 10ns and then slow down. The
last figure is the equilibrium state and we could get the spread radius and contact angle from it.
The solid wall atoms are bounded with their locations by the spring. The energy will transfer
to the elastic potential energy and then transfer to heat. This simulation is under the NVE
ensemble which means the total energy is a constant and the energy transfers to different
types. We could calculate the heat map of the liquid and find the energy dissipation.
Find the contact line in MATLAB,
30
Figure 4.5.2 The states of liquid-solid wetting process in different time steps: 1170ps,
10920ps, 53950ps, 107640ps with contact lines
The red line in Figure 4.5.2 is the contact angle which is calculated with three atoms at the
edge. Derive the y axis of model to 6 parts, choose the atoms at the edge of the lowest 3
layers and use cftool of Matlab to fit them to a linear function. The contact angle will
decrease until the liquid goes to the equilibrium.
It is hard to determine the contact line because that the number of atoms is very small.
Considering Figure 4.5.3, we could see, different contact line could be calculated when we
choose different atoms at the edge.
a)The contact line at macroscopic scale
a)The contact lines at the macroscopic scale
31
b)The contact lines at the microscopic scale
Figure 4.5.3 The contact line are calculated at the macroscopic scale and microscopic scale
Figure a) in the Figure 4.5.3 shows that it is easy to determine the contact line at macroscopic
scale and the contact line is unique. But in Figure b), we could find that the contact line
cannot be calculated unique, it depends on which atoms we choose and the contact angle
changes frequently.
Figure 4.5.4 The size of contact angle at different time steps
The contact angle at the equilibrium are different with different coefficients. The simulation
32
runs 130ns. From figure 4.5.4, we could find the contact angle decrease very quickly in the
first 5 ns, the details shows in the following figure. Because the shape of the liquid is not a
sphere in the spread process, I use three points at the edge of the liquid droplet to calculate the
contact line of the liquid. With this method, the curve is not smooth, the contact angle may be
not accurate, but the overall trend fits for the physical law.
Figure 4.5.5 The contact angles of the first
From Figure 4.5.5, we can see, the contact angle decrease very fast in the first 5000ps. Red
circles in the figure are contact angles with time.
The contact angles in the whole wetting process illustrated in Figure 4.5.6.
Figure 4.5.6 The contact angle of each timesteps
Which means the contact angle decreases by time, but at some specific time point, it will
oscillate around the mean value because there are only thousands of atoms and the error will
be generated in the calculation process or simulation process.
I did the same simulation with same parameters several times, and the contact angle is
oscillated around the mean values.
33
Figure 4.5.7 The contact angle with time in linear scaling
Figures 4.5.7 gives us four simulations with the same coefficients. The contact angle decrease
quickly in the first 5ns and then much slower. At the end of the simulation, we can see that
the contact angles are almost same. The contact angle decrease so fast at the beginning of the
simulation, so I set the watch points intensively at first. I set a watch point each 433 ps and
then after 8.67 ns I set a watch point each 8.67ns and then I set a watch point each 86.67ns.
Rescale the x axis, and we could see the result clearly.
Figure 4.5.8 The contact angle with time in log scaling
Figure 4.5.8 shows the details of contact angle at each time, the tendency is very clear. The
contact angle decreases rapidly and then start to increase, after the highest point it will
34
decrease monotonically until the equilibrium state. The four simulations shows a similar
tendency with the same coefficients.
The droplet will looked like,
Figure 4.5.9 The contact angle of liquid with the specific shape
The contact angle is very small in this shape. Thus, the curve of contact angle is not
monotonically decreasing.
Calculate the error bar for each examination point, we could get the average value and errors
in Figure 4.5.10.
Figure 4.5.10 The errorbar with contact angles
The error is quite noticeable at the second point because that the liquid has not contacted with
the solid at that time in some simulations. The error at the eleventh point is also significant
because the liquid becomes a semi-sphere at that time in some simulations, but not in others.
The spreading radius increased with time and the curve is a fit for a power function.
35
Figure 4.5.11 The errorbar with spreading radius
The number of atoms of the simulation is small, so that the radius is not very accuracy. But
from the Figure 4.5.11, we could see that the tendency of the radius is clear and it fits for a
power function.
Run three simulations with different coefficients.
a) sigma=4.5, contact angle=43.0652,
; sigma=2.9, contact angle=57.2408,
2
; sigma=6.9, contact angle=39.5706,
4
.
36
b) The loglog figure of contact angles with different sigma
Figure 4.5.12 The radius and loglog of radius with time
Figure 4.5.12 shows that the change of radius with time. The first figure has three curves,
green curve with circles represents the change under
6 9, the red curve with stars
represents the change under
4 5, the blue curve with squares represents the change under
2 9. We can see the curves are not smooth, the errors exists because it is the 3D model
and the points used to calculate the radius are not on a plane. The second figure shows the
slope of radius. We could find that the slopes are not same, the bigger the sigma, the sharper
the slope at the beginning part.
37
Figure 4.5.13 The contact angles and loglog of contact angles with time
The errors are rather pronounced because that using different ways to calculate contact angle
will get different values. I get the average of some ways (choose different points of the liquid
to calculate the contact line) and obtain this curve looks smooth. From figure 4.5.13, we could
38
see the bigger the sigma, the sharper the slope at the beginning.
Then, I ran a series of simulations with different parameters and obtained the contact angle
and radius at the equilibrium.
Table 4.5.1 The contact angles and the radius with different parameters
neighbor
Radius
Natoms
con
Sim 1
7694
1.05
6.9
39.6
2.5
7.24
Sim 2
7694
1.05
4.5
43.1
2.5
4.32
Sim 3
7694
1.05
2.9
57.2
2.5
3.11
Sim 4
7694
1.05
6.9
35.4
4.5
7.82
Sim 5
7694
1.05
4.5
38.2
4.5
5.62
From this table, we could find that the
inversely affects the contact angle of the
equilibrium. When the
increases, the contact angle of the equilibrium decreases. And with
the same coefficients of Lennard-Jones potential model, the bigger the cutoff radius, the
smaller the contact angle.
4.6 Energy Dissipation
Read the kinetic energy of each atoms and then plot it in different colors depends on the
values of kinetic energy. We will get the following graph.
Figure 4.6.1 The temperature at each atoms
This is a 3D figure and the kinetic energy looks so random, so I rescale the temperature map
by building the cylinders.
The kinetic energy is calculated with the formula 4.6.1.
(4.6.1)
From the equation 4.6.2
ke
3
NkB T
2
(4.6.2)
39
Thus, the temperature is equal to
(4.6.3)
kB
Figure 4.6.2 Rescale the temperature of the liquid
I calculate the temperature of each atoms in LAMMPS, then rescale them in the following
method.
Figure 4.6.3 Scale method for calculation of temperature
Distribute the whole system to cylinders with same height but different locations and radius.
Calculate the sum of kinetic energies in each cylinder and obtain the temperature, then
average to each molecule.
From figure 4.6.2, we could see that the temperature of the top of the liquid and the inner part
of the liquid is higher than the other part. But it looks somewhat random, because there are
only a few molecules in the model and we cannot get the macroscopic physical values well. If
the number of molecular is increased to 100 times, we may find the trend of temperature
during the wetting process.
40
Conclusion
In this thesis dynamic wetting of droplet spreading is studied with molecular dynamics
simulation. Our simulations have been performed with the open-source molecular dynamics
code LAMMPS. First the MD framework was assessed in order to show where the simulation
results obtained a well known macroscopic two-phase flow solution, the Laplace law. It was
found that a lower threshold around 3000 atoms was needed to obtain the macroscopic
surface tension. The result for the surface tension coefficient was found in agreement with He
and Hadjiconstantinou. In the dynamic wetting simulation, liquid and solid were defined with
a Lennard-Jones potential and the solid wall was represented in two types of models: fixed
atoms and harmonic atoms attached by springs. With different coefficients and cutoff, the
contact angle at equilibrium would be changed. The contact angles were observed to increase
followed by the decrease of
or the increase of cutoff length.
The kinetic energy would transform to heat and the temperature of liquid solid interfacial
would increase during the wetting process. In the harmonic wall model, some of the energy
would be converted to elastic potential energy. The total energy in the whole system would be
conserved.
LAMMPS is a effective software for molecular dynamics simulations, and it fits for a lot of
different models and can be reprogrammed by users who have basic programming skills.
With the video tools, the physical process could be observed carefully and conveniently.
41
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43
Appendix
Software Application
LAMMPS and Material Studio
Recently, some software can run dynamics simulations very well. Comparing with other
software, LAMMPS is fit for beginners because the source file and the manual are easy to
understand. There are a lot of commands for running simulations under different physical
conditions. Group command could help determine a group of atoms which will be added
forces or velocities. Fix command can set the basic properties for some groups of atoms.
LAMMPS can be used in various potential energy models and run the simulation faster than
most of other simulation software. In my case, I need to change some source files to fit for my
models and due to insufficient time. Thus, LAMMPS is the best choice in my situation.
1.
The installation and application of the LAMMPS
1.1.
The installation of the LAMMPS
Before installing the LAMMPS, we need to install mpich and fftw. Generally, we will install
mpich and fftw in the same folder.
Then, install the LAMMPS as the manual, we could start to explore LAMMPS.
1.2.
The computational methods of LAMMPS
The LAMMPS sets the initial environment for molecules and gives the molecules random
velocity, then it solves the Newton‟s equations for simulating the situations of molecules at
the next time step. With different parameters and the initial environment, LAMMPS can be
used in various potential energy models and get the equilibrium state or the intermediate state.
In our model, we use it to compute the pressure, energy and position of each molecules to get
the contact angle and understand the motion of the contact line.
The in.* files are input files and the dump.* files are output files in LAMMPS. The physical
model will be built in the in file with the initial position, velocity, atom style, mass and other
parameters. The variables which need to compute and output are determined in the in file too.
From the dump file, any variables have been computed in the programme can be obtained in
each timestep.
1.3.
Commands of LAMMPS and how to build a physical model
The commands of LAMMPS can be divided to four parts: initialization, atom definition,
settings and run.
 Initialization
 units: Determines the type of atoms and set the appropriate value unit for each
physical quantity. For style lj, all quantities are unitless. For the considerable model,
the style lj will be chosen. But for the water model, the style real will be chosen.
 dimension: Set the simulation to 2D or 3D.
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



boundary: Determines the type of boundaries of the simulation. Type p is the
periodic boundary, whereas type s is the non-periodic and shrink-wrapped
boundary.
 atom-style: Define the types of atoms which are used in the simulation. For type
atomic, only positions need to be given. But for type full, bond, angle and other
parameters are all needed.
Atom definition
 read_data: Read the structure and position of the molecules.
 read_restart: Read the information of molecules in one timestep of last simulation
and run it continually with the same conditions or different conditions.
 lattice: Determines the lattice of atom.
 region: Set a geometrical area for adding the parameters or computing them.
 create_box: Build boxes in the specific area. Generally, the number of boxes is
equal to the number of types of the atoms.
 create_atom: Build atoms in the specific area based on lattice.
Settings
 pair_coeff: Determine the pairwise force field coefficients for one or more pairs of
atoms types based on which type the atoms are.
 bond_coeff: Set the atom bond coefficients for atoms.
 angle_coeff: Set the angle coefficients for atoms.
 neighbor: Set parameters which affect the building of the pairwise neighbor list.
Style bin creates the linear list for all neighbors which will affect one atom.
 group: Specify some atoms as belonging to a group for setting the physical
quantities for them without affecting other atoms.
timestep: Set the timestep of simulation.
 fix: This command can be used in many cases. It can set velocity, pressure,
temperature, viscosity and other physical quantities for a group of atoms. It also can
set the running style or atoms structure. It is the most widely-used function in
LAMMPS.
 thermo: Computer and output the thermodynamic information on every N timesteps.
 compute: Compute specific physical quantities of each atom or the whole system.
 dump: Set the parameters of output file.
Run
 run: Set the number of steps of the simulation.
An example will explain the functions better than thousands of sentences.
%Choose Lennard-Jones potential model
units
lj
%Choose the type of atom to be atomic which means there‟s no bond between the atoms
atom_style atomic
%Apply the face-centered cubic to be the lattice
lattice
fcc 1
%Create a box whose range of x, y, z are all -20 to 40 to be the edge of the simulation
environment
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region
box block -20 40 -20 40 -20 40
create_box 2 box
%Create regions, one is a sphere whose center is x(10,8,5) and radius is 5
%The other one is a cuboids whose range of x is -5 to 25, range of y is 0 to 1, range of z is -5
to 25
region
liquid sphere 10 8 5 5.0
region
solid1 block -5 25 0 1 -5 25
%Create different type of atoms to both sphere and cuboids
create_atoms
1 region liquid
create_atoms
2 region solid1
%Set the mass of the liquid to be 1.0 and mass of solid to be 10
mass
1 1.0
mass
2 10
%Set the cutoff value to be 2.5
pair_style
lj/cut 2.5
%Set the coefficient of LJ model between liquid atoms to be 1.05 and 6.9, the coefficient of
LJ model between liquid atoms and solid wall atoms to be 1.05 and 4.5, the coefficient of LJ
model between solid wall atoms to be 0 and 0.
pair_coeff
1 1 1.05 1.05 6.9
pair_coeff
1 2 1.05 1.05 4.5
pair_coeff
22000
%Build a group for liquid atoms
%Build a group for solid wall atoms
group
hi region liquid
group
lo region solid1
%Give the liquid atoms initial velocity between 0.4 to 395467
velocity hi create 0.4 395467
%Create the list of atoms with linear scale and build a new list every 20 steps
neighbor 0.3 bin
neigh_modify every 20 delay 0 check no
%Fixed the solid wall with harmonic spring whose spring constant is 250
fix
2 lo spring/self 250.0
%Apply NVE ensemble for the simulation
fix
1 all nve
%Record the result each 1000 timesteps to file named dump.springself6.9
dump
id all atom 1000 dump.springself6.9
restart
1000000 restart.*.springself6.9
%Set the timestep to be 0.01
timestep 0.01
%Print the result on the screen each 1000 timesteps
thermo
1000
%Run the simulation for 3000000 timesteps
run
3000000
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1.4.
Tools for play the output video
 VMD
VMD is a molecules visualization program for displaying and analyzing. It can display
various output files as a video. With different colors and styles, it is easy and clear to
observe the whole physical process.
Before installing VMD, we need to install two packages.
sudo apt-get install csh
sudo apt-get install libstdc++5
Then, tar xvzf vmd.gz.tar
cd vmd-1.8.7
./configure LINUX
cd src
sudo make install
vmd
When you start to use the VMD, you will see the following interface.
Figure 6.1.1 VMD main interface
From VMD Main, we can click File, New to start a new video as the following figure.
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Figure 6.1.2 New Trajectory
Choose the Filename by clicking the button Browse, and then choose the appropriate File
type as LAMMPS Trajectory. Click the button Load, we will see the video.
Figure 6.1.3 VMD video

We can change the shape and color of the atoms. We also can rotate or zoom the video to
observe some physical process more clearly.
xmovie
Before installing xmovie tool, we need to install some software package in Synaptic
Package Manager.
libxaw7, libxaw7-dev, libxt6 and libxt6-dev
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Then, access to /tools/xmovie, run the command “make”
./xmovie dump.3Dwetting
Figure 6.1.4 xmovie video
xmovie is a x-based 2D viewer which can be used to visualize the LAMMPS video. It is
easy and clear to observe the physical process by using different colors to mark different
types of atoms.
 MATLAB
As commonly known, MATLAB is a very good mathematics program. There are some .m
files in LAMMPS tools for reading data in MATLAB. We need to change some lines of the
program to fit for our requirements based on the format of dump files.
2. The installation and application of material studio
Material Studio is a material computational program for researching molecular dynamics
which is produced by Accelrys. It can help us with various problems of chemical and material
industries. We can use it to build 3D model for any molecules by inputting the structure,
angle, bond and other information. We also can simulate the physical or chemical process in
this program. Material Studio can be used under any Linux or Windows environments.
2.1 The installation of Material Studio
 Install as a parallel edition: ./install –type cluster
 Install the software by following the prompt
 Choose the directory for MS
 Choose the models which you need
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



Specify the directory of the license
Determine the temp directory
Copy the license file to /home/msi/ms40/License_Pack/licenses
Run gateway to access /home/msi/ms40/gateway and run the following commands:
 ./msgateway_control_18888 start
 Change the number of nodes in /home/msi/ms40/share/data/machines_LINUX
 Run nohup ~/location/Discover/bin/RunDiscover.sh –np 2 7-1-1.inp &
2.2 The application of Material Studio
Through the Material Studio, I constructed the water molecule.
Firstly, create a new 3D Atomistic.xsd
Figure 6.2.1 Create the new file for building water molecules
Figure 6.2.2 The interface of Material Studio
Secondly, build the water molecule
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Figure 6.2.3 Scratch the Oxygen atom and add H atoms automatically
From this, we get the water molecule
Figure 6.2.4 The water molecule
Then, build the water box by constructing the Amorphous Cells
Figure 6.2.5 Construct the amorphous cells
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Figure 6.2.6 Amorphous cells construction
Configure the parameters of water box and construct it.
Figure 6.2.7 Water box with 1500 molecules
Use command ./msi2lmp watercube.xsd –class I –frc cvff > data.watercube to covert the
material studio file to LAMMPS data file. watercube.lammps05 is the data file we can read in
the in.watercube file.
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TRITA-CSC-E 2010:144
ISRN-KTH/CSC/E--10/144--SE
ISSN-1653-5715
www.kth.se