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![Chapter 1. Introduction 1
Chapter 1
Introduction
1.1 Interaction Energy
Based on the Born-Oppenheimer approximation as the basis for many body the-
ory, it is possible to generate model potentials assuming the absence of external
forces with the total energy of N interacting particles as [3]:
EN = φ1 + φ2 + φ3... + φn (1.1)
where φn is the sum of the n-body interaction energy. The total energy of a
system of N interacting particles may additionally be expressed as the sum of
isolated energies φ1 or
E N = φ1 (1.2)
We can then describe the interaction energy of a system of N interaction parti-
cles as a function of their positions based on the difference of their total energies
EN − E N [25].
Φ = EN − E N = φ2 + φ3 + ... + φn (1.3)](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-9-320.jpg)
![Chapter 1. Introduction 3
1.1.1 Lennard-Jones-Halicioglu-Pound Potential
The interaction potential energy U(rij) of two atoms, i and j, separated by a
distance rij is described by the Lennard-Jones potential as
ULJ(rij) = 4
σ
rij
12
−
σ
rij
6
(1.8)
where is the depth of the potential at the minimum and σ is the closest distance
at which U(rij) is zero. The calculations for these potential energy parameters
were performed through thermodynamic relations on their crystalline properties
as [14]:
5
2
RT − ∆H − E1 − PV = 2N aσ12 gN
V
4
− bσ6 gN
V
2
(1.9)
V
Tα
κ
− V
dE1
dV T
− PV = 4N bσ6 gN
V
2
− 2aσ12 gN
V
4
(1.10)
Where a and b are lattice sums dependent on the geometry of the structure
[22], ∆H is the amount of enthalpy required to isothermally transfer N atoms
from its crystal state to the vapor phase, E1 is the vibrational energy of the crystal
(including zero point energy), and κ is the isothermal compressibility. Parame-
ters were fitted mainly to ∆H the amount of enthalpy required to isothermally
transfer N atoms from its crystalline state to a vapor phase, and V the equilibrium
volume. The relevant potential parameters are given in Table 1.1 [25]:](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-11-320.jpg)
![Chapter 1. Introduction 5
1.1.2 Morse Potential
The interaction potential energy U(rij) of two atoms, i and j, separated by a
distance rij can also be represented by the Morse function
UM(rij) = D[e(−2α(rij−r0))
− 2e(−α(rij−r0))
] (1.11)
where α and D are constants that account for reciprocal distance and the energy
of approach between the two atoms respectively. The dissociation energy D can
be found through this relationship by U(r0) = −D.
The total potential energy of a large crystal with atoms at rest can be done by
choosing one atom in the lattice as an origin and calculating its interaction with
all other atoms nj in the crystal. Accounting for N total atoms within the crystal
is accomplished by multiplying for N
2 [12]. This gives a total energy as
ΦM = −
ND
2
β β ∑
j
njγj
2
− 2 ∑
j
γjnj (1.12)
where β = e(−αr0), and γj = e(−αrj)
with rj is the distance from the origin to the
j-th atom. In a face-centered cubic (fcc) lattice, the separation distance rj in terms
of d0 the distance between nearest lattice points in the crystal is [15, 17]:
rj = d0 j (1.13)
The relation between measurable physical quantities and potential parameters](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-13-320.jpg)
![Chapter 1. Introduction 6
are calculated for the Morse Potential as [19]:
D · β[β ∑
j
γjβj − 2 ∑
j
βj] = 2(U0 − K0) (1.14)
∑
j
rjβj − β ∑
j
rjγjβj = 0 (1.15)
Dα2
β[2β ∑
j
rj
2
βj − ∑
j
rj
2
βj] =
9ca0
3
8κ
(1.16)
with βj = njγj, U0 is the energy of sublimation, a0 is the value of the lattice
constant a by which the lattice is in equilibrium, c = 2 or 4 for face-centered or
body-centered crystals, κ is the compressibility, and finally K0 is the zero point
energy relatable to the Debye characteristic temperature θD by [15, 17]:
K0 =
9
8
NkBθD (1.17)
The potential function has been extrapolated via pseudo-harmonic approx-
imation [23], which allows for a discussion on crystal properties at arbitrary
temperatures to be possible [23, 24]. Following parameters fitted to experimen-
tal data on energy of vaporization, compressibility, and lattice constants taken at
room temperature the relevant potential parameters are shown in Table 1.2 [25].](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-14-320.jpg)
![Chapter 1. Introduction 8
1.1.3 Erkoc Potential
The Erkoc potential separates the pair-interaction function into its repulsive and
attractive terms, expressing the total interaction energy of a system as a linear
combination of two, two-body functions as
Φ = D21φ21 + D22φ22 (1.18)
where φ21 and φ22 are the two-body energies of the form
φ2k = ∑
i<j
U2k
ij , k = 1, 2 (1.19)
and the general form of the two-body atomic interactions in terms of inter-atomic
distances is defined as:
U2k
ij =
Ak
r
−λk
ij
e
−αkr2
ij (1.20)
This gives the repulsive part set of parameters (A1, λ1, α1) in U
(21)
ij and the at-
tractive part parameters (A2, λ2, α2) in U
(22)
ij , which were determined [9] by fitting
the exact pair-potential function to experimentally determined curves [21]. The
Erkoc potential parameters were calculated considering crystalline state proper-
ties, dimer, and small micro clusters [9]. This potential energy function has been
found to satisfy bulk cohesive energy and bulk stability condition exactly, with
the following parameters outlined in Table 1.3 [25].](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-16-320.jpg)
![Chapter 1. Introduction 10
1.2 Non-Crystalline Structure
1.2.1 Pair Distribution
In describing atomic distributions in non-crystalline materials, it typically re-
quires the use of the pair distribution function g(r). This is the probability of
finding another atom at a distance r from an origin atom (at the point of r = 0).
Starting with an atom located at a position r1 in real space, we find the one-body
number density function v1(r) as [26]:
v1
(r) = 0, when r = r1 (1.21)
b
a
v1
(r)dx = 1, for a < r1 < b (1.22)
Which evidently has the same property of the Dirac function δ(r1), so the one-
body number density can be expressed as the following with N atoms located on
a line:
v1
(r) = δ(r − r1) =
N
∑
i=1
δ(r − r1) (1.23)
In the event that we expand our one dimensional number density function for
two dimensional atoms located at (ri, rj), then we can write the corresponding
number density as
v2
(r, r ) =
N
∑
i=1
i=j
N
∑
j=1
δ(r − ri)δ(r − rj) (1.24)](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-18-320.jpg)
![Chapter 1. Introduction 12
It is important to note that for a crystal state, where n(r, r ) is dependent on
the distance and direction within the crystal lattice, that g(r, r ) = g(r) as the
information from g(r) is only one-dimensional. When the crystal is heated to
a point that it is able to melt however, the ordering within the crystal lattice
disappears and g(r, r ) can be represented by g(r).
In extending the number density to three dimensional real space in (ri, rj, rk),
we find [26]
v3
(r, r , r ) = ρ3
(r, r , r ) (1.32)
ρ3
(r, r , r ) =
N
∑
i=1
N
∑
j=1
i=j=k
N
∑
k=1
δ(r − ri)δ(r − rj)δ(r − rk) (1.33)
V
ρ3
(r, r , r )dv = (N − 2)ρ2
(r, r ) (1.34)
where typically the radial distribution function 4πr2ρ0g(r) which represents the
number of atoms in a spherical shell between R and R + dR is used when dis-
cussing non-crystalline systems. Thus by defining the pair distribution function
g(r), we are able to make connections between real measurable thermodynamic
properties to theoretical statistical mechanics averages.](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-20-320.jpg)
![Chapter 1. Introduction 13
1.2.2 Diffraction Theory
When performing diffraction measurements on a sample using x-rays, electrons,
or neutrons, information on the intensity of radiation scattering into a state ψk
(r)
from an incident wave state ψk
(r) is collected (Figure 1.4). As the incident and
scattered beam are created and collected in free space, then the following is as-
sumed:
Figure 1.4: Diffraction of a wave incident on a solid, which will reveal lattice
structure [8].
ψk
(r) =
eik·r
√
V
and ψk
(r) =
eik ·r
√
V
(1.35)
The specimen will behave towards the incident beam as a local potential U(r),
where in the case of elastic scattering |k| = |k |. The scattering amplitude in the](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-21-320.jpg)
![Chapter 1. Introduction 14
Born approximation with respect to the potential is [4]:
k |U(r)|k = ψ∗
k
U(r)ψk
d3
r =
1
V
U(r)ei(k−k )·r
d3
r (1.36)
And if we let q = k − k be our scattering vector, then we find that:
1
V
U(r)e−iq·r
d3
r = U(q) (1.37)
The scattering amplitude thus gives us the Fourier Transform of the potential
U(r). We can take a superposition of potentials for each atom centered at atomic
sites Ri as u(r − Ri) such that:
U(r) = ∑
i
u(r − Ri) (1.38)
Now supposing the one-component system u(r − Ri) is independent of i, with
r − Ri = r such that r = r + Ri then:
U(q) =
1
V ∑
i
u(r − Ri)e−q·r
d3
r =
1
V ∑
i
u(r )e−iq·r
d3
r e−iq·Ri (1.39)
U(q) =
1
V
u(r )e−iq·r
d3
r ∑
i
e−iq·Ri = f (q) ∑
i
e−iq·Ri (1.40)
Where f (q) is called the form factor, that is dependent on the nature of the
interaction-potential U(r), and ∑ e−iq·Ri depends on the structure of the sample.
Now as our measurements in diffraction experiments measure the intensity
I(q) = U∗(q)U(q) of the diffracted beam, where U∗(q) is the complex conjugate](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-22-320.jpg)
![Chapter 1. Introduction 16
ture such that S(q) = S(q), and g(R) = g(R), then our structure factor S(q)
becomes [4]:
S(q) = 1 + n0
∞
0
(g(R) − 1)d3
R = 1 + n0
∞
0
(g(R) − 1)
sin qR
qR
4πR2
dR (1.46)
We can perform an Inverse Fourier transform to find our pair distribution
function g(R) in terms of the structure of the sample the beam is diffracted off of
from our structure factor S(R) as[4]:
g(R) = 1 +
1
8π3n0
∞
0
(S(q) − 1)
sin qR
qR
4πq2
dq (1.47)
This allows us a means to make real measurements on the underlying atomic
structure of solids, liquids, or gasses with diffraction experiments. In doing so,
the intensity I(q) of the diffracted beam in our detectors allows the calculation
the structure factor S(q). Finally to compare with our calculations on the sta-
tistical mechanics properties for the sample, we can obtain its pair distribution
function g(R) as the probability of finding an atom at rj from an atom ri through
an Inverse Fourier Transform on S(q). The connection between substrate state
with characteristic pair distribution function and structure factor g(r) and S(q)
respectively is demonstrated in Figure 1.5.](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-24-320.jpg)
![Chapter 1. Introduction 17
Figure 1.5: Characteristic diffraction results for pair distribution and structure
factor for solids (c)/(C), liquids (b)/(B), amorphous materials (b’)/(B’), and
gasses (a)/(A) [26].](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-25-320.jpg)
![Chapter 1. Introduction 18
1.2.3 Dense Random Packing of Hard Spheres
Originally thought to be an ideal model for the liquid state, the dense random
packing of hard spheres (DRPHS) was initially carried out experimentally by
compressing solid ball bearings into containers with flexible or irregular walls.
It has been shown that this model more accurately represents the meta stable
amorphous or commonly called glassy state of simple liquids, wherein a liquid
is cooled or compressed until the atoms effectively are immobilized without the
onset growth of a crystal phase [5]. These amorphous alloys are now being pro-
duced by a number of different techniques, such as by rapid cooling (quenching)
of a liquid melt [7, 16], or through vapor deposition [18] and electrodeposition
[6].
Generally simulating the DRPHS begins by taking a seed cluster of 3 spheres
in hard contact with each other, and bringing one additional sphere at a time,
placing each on a surface site of the existing cluster such that it will rest in hard
contact with three spheres already present [2]. After the sphere has been placed
in its new location on the cluster, it becomes part of the substrate that subsequent
spheres are deposited on, and is not allowed to move. The resulting aggregate of
spheres is then based on the type seed cluster used, and the criterion by which
successive surface sites are chosen for deposition. The global criterion in which
new particles are added at the site closest to the center of the original side corre-
sponds roughly to choosing the site with the lowest gravitational or long range
potential. It has been previously demonstrated that this global criterion best fits
experimental results with a sharp first peak and a split second peak [2].](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-26-320.jpg)
![Chapter 1. Introduction 19
The first peak r1 in g(r) plots is reflective of nearest-neighbor atom-atom con-
figurations, and in the dense random packing of hard spheres model it corre-
sponds to spheres in contact with each other. For the splitting of the second
peak characteristic of metallic glasses (Fig 1.5 b’), the first sub-peak at r2 =
√
3r1
originates from a continuum of configurations of two hard-bonded equilateral
triangles sharing common sides (Figure 1.6 b), and should only be visible at high
density when the first peak is very sharp. The second sub-peak r3 = 2r1 geo-
metrically originates from the set of local configurations in which two nearest
neighbors are in contact with an intervening sphere in a linear array (Figure 1.6
a) [27].
Figure 1.6: Geometric origin for splitting of the second peak for amorphous
metallic solids [2].](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-27-320.jpg)
![Chapter 2. Methods 20
Chapter 2
Methods
2.0.1 Importance Sampling
The basic idea behind importance sampling can be demonstrated by supposing
we desire to evaluate numerically a one-dimensional integral I as:
I =
b
a
f (x)dx = (b − a) f (x) (2.1)
where f (x) is an unweighted average of f(x) over the interval [a,b]. In the
instance where our f(x) is negligible for points in the range of [a,b], then it be-
comes preferable to spend more time sampling points in regions where f(x) is
nonzero, and less time elsewhere. This remains just as true in problems where
the dimensionality of our integral increases, requiring greater time to compute.
If we were intent on evaluating the above integral by random sampling, but
with sampling points distributed non-uniformly over an interval [0,1] with some
non-negative probability density w(x), then we can rewrite the integral as:
I =
1
0
w(x)
f (x)
w(x)
dx (2.2)](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-28-320.jpg)
![Chapter 2. Methods 21
Now if we assume that w(x) is the derivative of another (non-negative, non-
decreasing) function u(x), where u(0)=0 and u(1)=1 (implying the normalization
of w(x)) and writing u as the integration variable with x a function of u then [13]:
I =
1
0
f [x(u)]
w[x(u)]
(2.3)
Now we can generate L random values of u uniformly distributed in the interval
[0,1], which gives us an estimate for I as [13]:
I ≈
1
L
L
∑
i=1
f [x(ui)]
w[u(xi)]
(2.4)
The advantages to rewriting our integral in this form comes when estimating the
variance σ2
I in IL, our estimate for I with L random sample points [11]
σ2
I =
1
L2
L
∑
i=1
L
∑
j=1
f [x(ui)]
w[x(ui)]
−
f
w
f [x(ui)]
w[x(ui)]
−
f
w
(2.5)
and as different samples i and j are assumed to be totally independent, the cross
terms vanish such that:
σ2
I =
1
L2
L
∑
i=1
f [x(ui)]
w[x(ui)]
−
f
w
=
1
L
f
w
2
−
f
w
2
(2.6)
This demonstrates that the variance σ2
I in I still goes as 1
L, but we can greatly
reduce the magnitude of this variance if we can choose the non-negative proba-
bility density w(x) such that f(x)/w(x) is a smooth function of x [11].](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-29-320.jpg)
![Chapter 2. Methods 22
2.0.2 Monte Carlo
The Monte Carlo is a means for utilizing computer systems to calculate properties
of substances composed of interacting molecules, equivalent to solving for ex-
pected equilibrium values from difficult statistical mechanics many dimensional
integrals. It’s typically applied to the field of liquid study, but the core method
remains useful in a large variety of complex problems best solved through com-
putational numerical integration [1].
The method is utilized when considering a square system of finite size with N
number of interacting particles. Surface effects are minimized by considering the
complete substance to be periodic, or comprised of many smaller squares with
N particles in the same configuration, by which there will be some distance dAB
that represents the shortest possible distance between particles A and B. Utilizing
a model for the interaction between atoms A and B by which the interaction
energy falls off at large distances, then the minimum distance dAB will be the
most prominent to our considerations (largest contribution to total energy) [20].
To do so we utilize our knowledge of the positions for the N particles in the
square to calculate the potential energy of the system:
E =
1
2
N
∑
i=1
i=j
N
∑
j=1
U(dij) (2.7)
where U is the potential between molecules, dij is the distance between particles
i and j.
In calculating the properties of a system, the canonical ensemble is used, such](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-30-320.jpg)
![Chapter 2. Methods 23
that if F is the equilibrium value of interest, then [11]
F =
Fe
−E
kBT
d3N pd3Nq
e
−E
kBT
d3N pd3Nq
(2.8)
where (d3N pd3Nq) is a volume element in the 6N-dimensional phase space. For
larger systems it would then be impractical to do several hundred dimensional
integrals through numerical methods. The Monte Carlo allows then a method
for many-dimensional integrals simply by integrating over a random sampling
of points instead of over a regular array of points.
Now in the Metropolis method for importance sampling, our canonical en-
semble solved through the Monte Carlo is made computationally easier than
placing particles in random locations. This is accomplished by taking new con-
figurations that are chosen with a probability of e
−E
KBT
and are weighted evenly
[20]. To do so, we can begin with a system of N particles in any configuration,
then attempt to make a trial move for each particle in succession according to
[20]
X → X + α 1 (2.9)
Y → Y + α 2 (2.10)
Z → Z + α 3 (2.11)
where α is the maximum allowable displacement, 1, 2, and 3 are random num-
bers between [−1, 1]. In accordance with our periodicity assumption however, if
the particle trial move would place it outside the square, then we place it back in-](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-31-320.jpg)
![Chapter 2. Methods 24
side by reflection, equivalent to the particle entering the square from the opposite
side.
After each trial move of our N particles, we calculate the change in energy
of our system ∆E caused by the move. If the move brings our system to a state
of lower energy than previously (∆E < 0), then we allow the move and place
the particle in its new position. If however the trial move places our system in
a higher energy state than previously (∆E > 0), then we allow the move with a
probability of e
−∆E
KBT
. We do this by taking a new random number 4 in [0, −1] such
that if 4 < e
−∆E
KBT
we accept the new position, otherwise we return the particle to
its old position. Now whether the particle was allowed to move or not, returning
the particle to its old position is considered as a new configuration and hence
counted towards averaging. Thus we find for Fj as the value for F after the jth
move [20]
F =
1
M
M
∑
j=1
Fj (2.12)
which allows us to utilize the iterative power of computers when evaluating our
desired properties of a system, as opposed to traditional numerical methods.](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-32-320.jpg)
![Chapter 3. Liquids 32
3.2 Discussion
Discussions for the properties of simulated liquid copper, gold, and silver with
their respective calculated pair distributions remain limited to a mostly qualita-
tive nature in comparing the differences between the inter-atomic energy models
described earlier. That is not to say however that although the evolution of a
crystalline metal through melting to a liquid state (where the loss of long range
order removes many useful mathematical tools for analysis and predictions) is
not achieved specifically as change of state in this study, that insights into char-
acteristic liquid structure cannot be appreciated from the computer simulated
clusters. With liquid atomic clusters produced by a canonical ensemble with the
Monte Carlo, wherein the number of particles, temperature, and volume of the
system is held constant, the term of "melting" anywhere in this discussion or
study refers to the achievement characteristic liquid state properties. Regardless,
the use of three different two body inter-atomic potential energy models for run-
ning the Monte Carlo simulations allows an insight into the theoretical behavior
for a cluster of atoms when considering either: parameters calculated based on
crystalline state properties with the Lennard-Jones [14] and Morse potentials [19],
or considerations on crystalline state, dimer, and small micro clusters with the
Erkoc potential [10].
When examining the Erkoc potential closely to determine connections be-
tween simulated "melting temperatures" (those that achieved characteristic ex-
perimental liquid state properties) and depth of the well, it was observed that
the shallowest to deepest silver, gold, and copper (Figure 1.3) metals directly cor-](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-40-320.jpg)
![Chapter 3. Liquids 34
by copper), that the depth does not portray the overall ability to simulate liq-
uid structures. That is to say, the Erkoc demonstrated a superior ability overall in
terms of matching experimental values for a liquid state, and was closer followed
by the Morse potential than the Lennard-Jones.
It was found that the Erkoc model makes considerations on crystalline state,
dimer, and small micro clusters [10], where as the Morse and Lennard-Jones
models use only crystalline state properties [14, 19]. With the greater success
found in simulating liquid structure thus coming from a models that consider
more than crystalline properties, it is thought that this reflects the need truncate
the many-body expansion after the second (two and three body interactions) or
higher terms. Therefore it is concluded that with the loss of long range order-
ing and thus inability to revert to relatively simplistic mathematical modeling,
the description of a liquid state necessitates considerations on the complicated
interactions between more distant neighbors larger shells of nearest neighbors.
Now in regards to the details of varying fixed temperatures within simula-
tions, the most notable trend found by increasing temperature parameters for all
materials is that of the slowly decreasing and left shifting peak heights (Figures
3.1-3.9 a) . In particular, it was observed during the process of increasing tem-
peratures in 400 ◦C increments across all metals and potentials, that the greatest
barrier to loss of long range order and achievement of liquid properties in agree-
ment with experimental results came from the presence of small sub-peaks after
the first and second peaks.](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-42-320.jpg)
![Chapter 3. Liquids 35
This near transition from a pre-liquid state can be seen with more detail by
turning our attention specifically to copper generated through the Erkoc potential
(Figure 3.3 a). At the lowest run temperature 1200 ◦C, a number of peaks can be
observed at approximately
√
2,
√
3, 2,
√
5, and
√
7 from the largest first peak.
The distances of
√
2,
√
5, and
√
7 were found to be characteristic of face-centered
cubic (fcc) and hexagonal close packed (hcp) [2], and thus play an important
role in defining solid structure. As the temperature moves to a near critical
temperature at 2400 ◦C, it was observed that the peaks located at
√
2, 2, and
√
5 nearly disappear (Figure 3.3 a). Final observations on the liquid properties
of copper generated through the Erkoc potential demonstrated good agreement
with the 1150 ◦C experimental results for a liquid when simulations were run
at 2800 ◦C (Figure 3.3 b) and higher. At this point the smoothly behaving g(r)
results showed only short range ordering of the copper cluster, with the second
and third peaks approximately at
√
3 and
√
7 times the first sharp peak.
Therefore with the disappearance of many characteristic peaks for a face-
centered cubic (fcc) and hexagonal close packed (hcp) lattice, and the overall loss
of long rang ordering, it is concluded that these results reflect the movement of
the atoms within the cluster from its initial fcc cluster to positions that match
the properties of a liquid. Furthermore, it is concluded that the disappearance
of atoms located at
√
2, 2, and
√
5 from the nearest neighbor atom in the g(r)
plots reflects failure of the metal at particular simulation temperatures due to the
inter-atomic potential models to maintain solid properties.](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-43-320.jpg)
![Chapter 4. Amorphous 45
4.2 Discussion
Of primary interest for analyzing simulated amorphous clusters quenched from
their characteristic liquid state, is how the resulting pair distribution g(r) first
peak and split second peaks compare to the theory of Dense Random Packing
of Hard Spheres (DRPHS). In this way we can make comparisons on two very
different computer simulation methods: one with DRPHS by taking a seed of
three hard contact spheres and adding one at a time to create a growing aggregate
of immovable spheres [2], against the current method of calculating a canonical
ensemble through the Monte Carlo with our potential energy models used in this
study.
When looking closely at the results for metallic copper in the Morse and Erkoc
potentials, it was noted that although there was a loss of long range order, the g(r)
plots failed to produce the expected second peak splitting (Figures 3.2 and 3.3).
As this absence of peak splitting was isolated only to the metallic copper, with
gold and silver showing very prominent split sub-peaks, it is currently unknown
why this occurred, and is thought to be independent of the Morse and Erkoc
potentials.
Observations on the remaining g(r) plots generated through re-running the
liquid clusters obtained previously (Figures 3.1 b to 3.9 b) found the maintained
disappearance of atoms at positions
√
2r1 and
√
5r1 that are characteristic of
face-centered cubic and hexagonal close-packed crystal lattices (Figures 4.1-4.9),
suggesting the cluster failed to achieve a crystalline state. More importantly
for classifying the pair distribution plots however was the overall loss of long](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-53-320.jpg)
![Bibliography 49
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St.Catharines ON, 2015.
[5] GS Cargill III. Dense random packing of hard spheres as a structural model
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[7] HS Chen and D Turnbull. Evidence of a glass–liquid transition in a gold–
germanium–silicon alloy. The Journal of Chemical Physics, 48(6):2560–2571,
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[8] Cristales. Diffraction retrieved from http://lacasadeloscristales.trianatech.com/
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[9] ¸S Erkoç. A new class of empirical many-body potential energy functions for
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[10] ¸Sakir Erkoç. An empirical many-body potential energy function constructed
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[11] Daan Frenkel and Berend Smit. Understanding molecular simulation: from
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[12] Louis A Girifalco and Victor G Weizer. Application of the morse potential
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[13] H Gould and Jan Tobochnik. Introduction to computer simulation. applica-
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[14] T Halicioglu and GM Pound. Calculation of potential energy parameters
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New York, 1954.
[16] W Klement, RH Willens, and POL Duwez. Non-crystalline structure in so-
lidified gold–silicon alloys. 1960.](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-58-320.jpg)
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[17] Daniel D Konowalow and JO Hirschfelder. Intermolecular potential func-
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[18] S Mader, H Widmer, FM d’Heurle, and AS Nowick. Metastable alloys of
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[19] C Malinowska-Adamska. Potential energy parameters from crystalline state
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[23] NM Plakida and T Siklós. Theory of anharmonic crystals. iii. three-
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[24] T Siklós. Self-consistent dynamical theory of anharmonic crystals. Acta
Physica Academiae Scientiarum Hungaricae, 34(4):327–336, 1973.](https://image.slidesharecdn.com/38ef955e-5aaf-4458-bdbd-e8de23003ba3-160614163451/85/monte-carlo-simulation-59-320.jpg)
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[25] Dietrich Stauffer. Empirical potential energy functions used in the simulations of
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![[S.L._Kakani]_Material_Science_(New_Age_Pub.,_2006(BookSee.org).pdf](https://cdn.slidesharecdn.com/ss_thumbnails/s-230301071329-4f25d7e9-thumbnail.jpg?width=640&height=640&fit=bounds)
monte-carlo-simulation
- 1. Monte Carlo Simulationof Liquid and Amorphous Metals: A Theoretical Study on Three Two-Body Potential Energy Models by David Wachmann A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE in The Faculty of Mathematics and Sciences Department of Physics BROCK UNIVERSITY June 11, 2016 2016 © David Wachmann
- 2. Abstract ii Abstract Monte Carlosimulations were used comparing the abilities of three various two- body potential energy models in creating metallic clusters that would achieve liquid properties for: copper, gold, and silver at various fixed temperatures. The Erkoc potential demonstrated characteristic short-range ordering closest to real experimental results, followed by the Morse and Lennard-Jones models. This was found attributable to considerations from dimer and small micro clusters, in ad- dition to crystalline properties when calculating the Erkoc potential parameters. A connection between well depth and initial temperatures required to simulate a liquid was found, with shallow wells more quickly matching real experimen- tal results. Simulations of fast quenching liquid clusters to 20 ◦C maintained an absence of long range order, and a greater degree of short range order than in liq- uid experimental results due to a splitting of the second peak in pair distribution plots. Therefore fast quenched simulations were able to achieve characteristic properties of amorphous metals. The split second peaks were found in agree- ment with the Dense Random Packing of Hard Spheres theory within 0.2588% to 3.9562%, thereby demonstrating the ability to generate clusters with characteris- tic amorphous properties through the Monte Carlo.
- 3. Contents iii Contents Abstract .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Lennard-Jones-Halicioglu-Pound Potential . . . . . . . . . . 3 1.1.2 Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Erkoc Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Non-Crystalline Structure . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 Pair Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.2 Diffraction Theory . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.3 Dense Random Packing of Hard Spheres . . . . . . . . . . . 18
- 4. Contents iv 2 Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.0.1 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . 20 2.0.2 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.3 Silver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Amorphous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.1 Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.2 Gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.3 Silver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
- 5. List of Tablesv List of Tables 1.1 Calculated values of Lennard-Jones potential parameters for metal- lic solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Calculated values of Morse potential parameters for metallic solids. 7 1.3 Calculated values of Erkoc potential parameters for metallic solids. 9 4.1 Measured vs calculated distances for amorphous copper in the dense random packing of hard spheres model . . . . . . . . . . . . 38 4.2 Measured vs calculated distances for amorphous gold in the dense random packing of hard spheres model . . . . . . . . . . . . . . . . 41 4.3 Measured vs calculated distances for amorphous silver in the dense random packing of hard spheres model . . . . . . . . . . . . . . . . 44
- 6. List of Figuresvi List of Figures 1.1 Lennard-Jonnes potential . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Morse potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Erkoc potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Diffraction of a wave on a solid . . . . . . . . . . . . . . . . . . . . . 13 1.5 Characteristic pair distribution and structure factor of solids, liq- uids, amorphous materials, and gasses . . . . . . . . . . . . . . . . 17 1.6 Geometric origin for second peak splitting . . . . . . . . . . . . . . 19 3.1 Lennard-Jones pair distribution results for liquid copper . . . . . . 25 3.2 Morse pair distribution results for liquid copper . . . . . . . . . . . 26 3.3 Erkoc pair distribution results for liquid copper . . . . . . . . . . . 26 3.4 Lennard-Jones pair distribution results for liquid gold . . . . . . . 28 3.5 Morse pair distribution results for liquid gold . . . . . . . . . . . . 28 3.6 Erkoc pair distribution results for liquid gold . . . . . . . . . . . . . 29 3.7 Lennard-Jones pair distribution results for liquid silver . . . . . . . 30 3.8 Morse pair distribution results for liquid silver . . . . . . . . . . . . 30 3.9 Erkoc pair distribution results for liquid silver . . . . . . . . . . . . 31
- 7. List of Figuresvii 4.1 Lennard-Jones pair distribution results for amorphous copper . . . 36 4.2 Morse pair distribution results for amorphous copper . . . . . . . . 37 4.3 Erkoc pair distribution results for amorphous copper . . . . . . . . 37 4.4 Lennard-Jones pair distribution results for amorphous gold . . . . 39 4.5 Morse pair distribution results for amorphous gold . . . . . . . . . 39 4.6 Erkoc pair distribution results for amorphous gold . . . . . . . . . 40 4.7 Lennard-Jones pair distribution results for amorphous silver . . . . 42 4.8 Morse pair distribution results for amorphous silver . . . . . . . . 42 4.9 Erkoc pair distribution results for amorphous silver . . . . . . . . . 43
- 8. Acknowledgements viii Acknowledgements First andforemost I would like to thank my supervisor Professor Shyamal Bose. His support in providing the necessary Fortran code to generate results for anal- ysis, guidance throughout the year introducing and explaining higher level the- ories, and overall patience with my "best when written down" memory made completing this study fully and on time possible (albeit in true "work until the end" physics style). Thank you to Professor Edward Sternin for his role in providing all the highly appreciated linux and technical support. Automating the fleet of physics termi- nals to optimize my simulations through bash scripting made life exponentially easier, removing any highly repetitive grunt work so my time could be better spent on research and course work. Also Professor Maureen Reedyk for her tu- torial on working with data sets in xmgrace, which was quickly recognized as the versitile tool I needed and is my new favourite graphing software. Thank you to Professor David Crandles, whose first year mechanics class is the reason I quickly switched into physics and never looked back. Finally, a large thank you to the friends and family that supported me while I locked myself away to discover a few new things about everything. I did it... I’m free! Master has set Sm´eagol free!
- 9. Chapter 1. Introduction1 Chapter 1 Introduction 1.1 Interaction Energy Based on the Born-Oppenheimer approximation as the basis for many body the- ory, it is possible to generate model potentials assuming the absence of external forces with the total energy of N interacting particles as [3]: EN = φ1 + φ2 + φ3... + φn (1.1) where φn is the sum of the n-body interaction energy. The total energy of a system of N interacting particles may additionally be expressed as the sum of isolated energies φ1 or E N = φ1 (1.2) We can then describe the interaction energy of a system of N interaction parti- cles as a function of their positions based on the difference of their total energies EN − E N [25]. Φ = EN − E N = φ2 + φ3 + ... + φn (1.3)
- 10. Chapter 1. Introduction2 Φ = Φ(r1, r2, ..., rN) (1.4) φ2 = ∑ i<j U2(ri, rj), (1.5) φ3 = ∑ i<j<k U3(ri, rj, rk), (1.6) φn = ∑ i<j<...<n Un(ri, rj, .., rn) (1.7) Where U2, U3, Un represent two, three, and n-body interactions respectively to the many body expansion of the total energy of the system Φ. Depending on the system described, Φ can have several definitions. All of which are measur- able quantities and represent different physical situations, such has configuration energy, interaction energy, potential energy, cohesive energy, and more. It is usually believed that this many body expansion of Φ is quickly con- vergent, and that the first two terms (two and three-body contributions) give a reasonable approximation to the interaction potential. This allows for higher mo- ments to be neglected, which would normally drastically restrict the applicable use of this equation for systems greater than a few atoms. Thus the many-body expansion is typically truncated after the first term for initial estimations. This allows simplifications of statistical mechanical formalism’s for thermodynamic properties and computer simulations.
- 11. Chapter 1. Introduction3 1.1.1 Lennard-Jones-Halicioglu-Pound Potential The interaction potential energy U(rij) of two atoms, i and j, separated by a distance rij is described by the Lennard-Jones potential as ULJ(rij) = 4 σ rij 12 − σ rij 6 (1.8) where is the depth of the potential at the minimum and σ is the closest distance at which U(rij) is zero. The calculations for these potential energy parameters were performed through thermodynamic relations on their crystalline properties as [14]: 5 2 RT − ∆H − E1 − PV = 2N aσ12 gN V 4 − bσ6 gN V 2 (1.9) V Tα κ − V dE1 dV T − PV = 4N bσ6 gN V 2 − 2aσ12 gN V 4 (1.10) Where a and b are lattice sums dependent on the geometry of the structure [22], ∆H is the amount of enthalpy required to isothermally transfer N atoms from its crystal state to the vapor phase, E1 is the vibrational energy of the crystal (including zero point energy), and κ is the isothermal compressibility. Parame- ters were fitted mainly to ∆H the amount of enthalpy required to isothermally transfer N atoms from its crystalline state to a vapor phase, and V the equilibrium volume. The relevant potential parameters are given in Table 1.1 [25]:
- 12. Chapter 1. Introduction4 Table 1.1: Calculated values of Lennard-Jones potential parameters for metallic solids. Metals (eV) σ (Å) Cu 0.4093233 2.338 Au 0.4414700 2.637 Ag 0.3447794 2.644 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Distance (Å) -0.75 -0.625 -0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5 Energy(eV) Copper Gold Silver Figure 1.1: Lennard-Jones interaction potential between atoms for copper, gold, and silver.
- 13. Chapter 1. Introduction5 1.1.2 Morse Potential The interaction potential energy U(rij) of two atoms, i and j, separated by a distance rij can also be represented by the Morse function UM(rij) = D[e(−2α(rij−r0)) − 2e(−α(rij−r0)) ] (1.11) where α and D are constants that account for reciprocal distance and the energy of approach between the two atoms respectively. The dissociation energy D can be found through this relationship by U(r0) = −D. The total potential energy of a large crystal with atoms at rest can be done by choosing one atom in the lattice as an origin and calculating its interaction with all other atoms nj in the crystal. Accounting for N total atoms within the crystal is accomplished by multiplying for N 2 [12]. This gives a total energy as ΦM = − ND 2 β β ∑ j njγj 2 − 2 ∑ j γjnj (1.12) where β = e(−αr0), and γj = e(−αrj) with rj is the distance from the origin to the j-th atom. In a face-centered cubic (fcc) lattice, the separation distance rj in terms of d0 the distance between nearest lattice points in the crystal is [15, 17]: rj = d0 j (1.13) The relation between measurable physical quantities and potential parameters
- 14. Chapter 1. Introduction6 are calculated for the Morse Potential as [19]: D · β[β ∑ j γjβj − 2 ∑ j βj] = 2(U0 − K0) (1.14) ∑ j rjβj − β ∑ j rjγjβj = 0 (1.15) Dα2 β[2β ∑ j rj 2 βj − ∑ j rj 2 βj] = 9ca0 3 8κ (1.16) with βj = njγj, U0 is the energy of sublimation, a0 is the value of the lattice constant a by which the lattice is in equilibrium, c = 2 or 4 for face-centered or body-centered crystals, κ is the compressibility, and finally K0 is the zero point energy relatable to the Debye characteristic temperature θD by [15, 17]: K0 = 9 8 NkBθD (1.17) The potential function has been extrapolated via pseudo-harmonic approx- imation [23], which allows for a discussion on crystal properties at arbitrary temperatures to be possible [23, 24]. Following parameters fitted to experimen- tal data on energy of vaporization, compressibility, and lattice constants taken at room temperature the relevant potential parameters are shown in Table 1.2 [25].
- 15. Chapter 1. Introduction7 Table 1.2: Calculated values of Morse potential parameters for metallic solids. Metal D (eV) α (Å −1 ) r0 (Å) Cu 0.3446 1.3921 2.838 Au 0.4826 1.6166 3.004 Ag 0.3294 1.3939 3.096 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Distance (Å) -0.75 -0.625 -0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5 Energy(eV) Copper Gold Silver Figure 1.2: Morse interaction potential between atoms for copper, gold, and silver The above results account for the pairwise interaction of a central atom with a near infinite number of shells of nearest-neighbors, and as such is expected to be more reliable than models built on a single shell of nearest-neighbors.
- 16. Chapter 1. Introduction8 1.1.3 Erkoc Potential The Erkoc potential separates the pair-interaction function into its repulsive and attractive terms, expressing the total interaction energy of a system as a linear combination of two, two-body functions as Φ = D21φ21 + D22φ22 (1.18) where φ21 and φ22 are the two-body energies of the form φ2k = ∑ i<j U2k ij , k = 1, 2 (1.19) and the general form of the two-body atomic interactions in terms of inter-atomic distances is defined as: U2k ij = Ak r −λk ij e −αkr2 ij (1.20) This gives the repulsive part set of parameters (A1, λ1, α1) in U (21) ij and the at- tractive part parameters (A2, λ2, α2) in U (22) ij , which were determined [9] by fitting the exact pair-potential function to experimentally determined curves [21]. The Erkoc potential parameters were calculated considering crystalline state proper- ties, dimer, and small micro clusters [9]. This potential energy function has been found to satisfy bulk cohesive energy and bulk stability condition exactly, with the following parameters outlined in Table 1.3 [25].
- 17. Chapter 1. Introduction9 Table 1.3: Calculated values of Erkoc potential parameters for metallic solids. Parameter Cu Ag Au A1 110.766008 220.262366 345.923364 λ1 2.09045946 1.72376253 1.04289230 α1 0.394142248 0.673011507 0.750775965 A2 -46.1649783 -26.0811795 -38.9245908 λ2 1.49853083 1.81484791 1.05974062 α2 0.2072255507 0.120620395 0.229377368 D21 0.436092895 1.00610152 0.888911352 D22 0.245082238 0.221234242 0.254280292 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Distance (Å) -0.75 -0.625 -0.5 -0.375 -0.25 -0.125 0 0.125 0.25 0.375 0.5 Energy(eV) Copper Gold Silver Figure 1.3: Erkoc interaction potential between atoms for copper, gold, and silver
- 18. Chapter 1. Introduction10 1.2 Non-Crystalline Structure 1.2.1 Pair Distribution In describing atomic distributions in non-crystalline materials, it typically re- quires the use of the pair distribution function g(r). This is the probability of finding another atom at a distance r from an origin atom (at the point of r = 0). Starting with an atom located at a position r1 in real space, we find the one-body number density function v1(r) as [26]: v1 (r) = 0, when r = r1 (1.21) b a v1 (r)dx = 1, for a < r1 < b (1.22) Which evidently has the same property of the Dirac function δ(r1), so the one- body number density can be expressed as the following with N atoms located on a line: v1 (r) = δ(r − r1) = N ∑ i=1 δ(r − r1) (1.23) In the event that we expand our one dimensional number density function for two dimensional atoms located at (ri, rj), then we can write the corresponding number density as v2 (r, r ) = N ∑ i=1 i=j N ∑ j=1 δ(r − ri)δ(r − rj) (1.24)
- 19. Chapter 1. Introduction11 and knowing that N atoms are located in the volume V, then: V v1 (r)dv = N (1.25) V V v2 (r, r )dvdv = N(N − 1) (1.26) Now if we allow for the set of (r1, r2, ..., rN) to have various values in the above arguments, then we must look to statistical mechanics in order to find the probability. In doing so, we consider the averages as: v1 (r) = n1 (r), V n1 (r)dv = N (1.27) v2 (r, r ) = n2 (r, r ), V V n2 (r, r )dvdv = N(N − 1) (1.28) V n2 (r, r )dv = n1 (r)(N − 1) (1.29) By which we can define the pair distribution function g(r, r ) as n2 (r, r ) = n1 (r)n1 (r )g(r, r ) (1.30) and in the case of a homogeneous system such as a liquid, the one-body density n1(r) is independent of r, then n1(r) = n1(r ) = ρ0 where ρ0 = N/V the average number density of the material. Thus we can write n2(r, r ) as a function of r = |r − r | as: n2 (r, r ) = ρ2 0g(r) (1.31)
- 20. Chapter 1. Introduction12 It is important to note that for a crystal state, where n(r, r ) is dependent on the distance and direction within the crystal lattice, that g(r, r ) = g(r) as the information from g(r) is only one-dimensional. When the crystal is heated to a point that it is able to melt however, the ordering within the crystal lattice disappears and g(r, r ) can be represented by g(r). In extending the number density to three dimensional real space in (ri, rj, rk), we find [26] v3 (r, r , r ) = ρ3 (r, r , r ) (1.32) ρ3 (r, r , r ) = N ∑ i=1 N ∑ j=1 i=j=k N ∑ k=1 δ(r − ri)δ(r − rj)δ(r − rk) (1.33) V ρ3 (r, r , r )dv = (N − 2)ρ2 (r, r ) (1.34) where typically the radial distribution function 4πr2ρ0g(r) which represents the number of atoms in a spherical shell between R and R + dR is used when dis- cussing non-crystalline systems. Thus by defining the pair distribution function g(r), we are able to make connections between real measurable thermodynamic properties to theoretical statistical mechanics averages.
- 21. Chapter 1. Introduction13 1.2.2 Diffraction Theory When performing diffraction measurements on a sample using x-rays, electrons, or neutrons, information on the intensity of radiation scattering into a state ψk (r) from an incident wave state ψk (r) is collected (Figure 1.4). As the incident and scattered beam are created and collected in free space, then the following is as- sumed: Figure 1.4: Diffraction of a wave incident on a solid, which will reveal lattice structure [8]. ψk (r) = eik·r √ V and ψk (r) = eik ·r √ V (1.35) The specimen will behave towards the incident beam as a local potential U(r), where in the case of elastic scattering |k| = |k |. The scattering amplitude in the
- 22. Chapter 1. Introduction14 Born approximation with respect to the potential is [4]: k |U(r)|k = ψ∗ k U(r)ψk d3 r = 1 V U(r)ei(k−k )·r d3 r (1.36) And if we let q = k − k be our scattering vector, then we find that: 1 V U(r)e−iq·r d3 r = U(q) (1.37) The scattering amplitude thus gives us the Fourier Transform of the potential U(r). We can take a superposition of potentials for each atom centered at atomic sites Ri as u(r − Ri) such that: U(r) = ∑ i u(r − Ri) (1.38) Now supposing the one-component system u(r − Ri) is independent of i, with r − Ri = r such that r = r + Ri then: U(q) = 1 V ∑ i u(r − Ri)e−q·r d3 r = 1 V ∑ i u(r )e−iq·r d3 r e−iq·Ri (1.39) U(q) = 1 V u(r )e−iq·r d3 r ∑ i e−iq·Ri = f (q) ∑ i e−iq·Ri (1.40) Where f (q) is called the form factor, that is dependent on the nature of the interaction-potential U(r), and ∑ e−iq·Ri depends on the structure of the sample. Now as our measurements in diffraction experiments measure the intensity I(q) = U∗(q)U(q) of the diffracted beam, where U∗(q) is the complex conjugate
- 23. Chapter 1. Introduction15 of our Fourier Transformed potential, then we find: I(q) = |U(q)|2 = |f (q)|2 ∑ i,j e−iq·(Ri−Rj) = N|f (q)|2 S(q) (1.41) S(q) = 1 N ∑ i,j e−iq·(Ri−Rj) = 1 + 1 N ∑ i,j i=j e−iq·(Ri−Rj) (1.42) Where S(q) is our structure factor of the material, by which we can eliminate for the forward scattering term from the delta function singularity at q = 0 such that S(q = 0) = 0 as: S(q) = 1 + 1 N ∑ i,j i=j e−iq·(Ri−Rj) − Nδq,0 (1.43) This definition of the structure factor is valid for a fixed set of Ri, i = 1...N. For a liquid/amorphous solid/dense gas however, we must consider ensemble averages for the structure factor as: S(q) = 1 + 1 N ∑ i,j i=j e−iq·(Ri−Rj) − Nδq,0 (1.44) Now if we let Ri − Rj = R, and take for a given i with Ri chosen as the origin, then: ∑ i ∑ j=i e−iq·(Ri−Rj) ⇒ N e−iq·R n0g(R)d3 R (1.45) Where n0g(R)d3R is the number of particles in a volume element d3R at R, given that there is a particle at the origin. Now if we consider an isotropic struc-
- 24. Chapter 1. Introduction16 ture such that S(q) = S(q), and g(R) = g(R), then our structure factor S(q) becomes [4]: S(q) = 1 + n0 ∞ 0 (g(R) − 1)d3 R = 1 + n0 ∞ 0 (g(R) − 1) sin qR qR 4πR2 dR (1.46) We can perform an Inverse Fourier transform to find our pair distribution function g(R) in terms of the structure of the sample the beam is diffracted off of from our structure factor S(R) as[4]: g(R) = 1 + 1 8π3n0 ∞ 0 (S(q) − 1) sin qR qR 4πq2 dq (1.47) This allows us a means to make real measurements on the underlying atomic structure of solids, liquids, or gasses with diffraction experiments. In doing so, the intensity I(q) of the diffracted beam in our detectors allows the calculation the structure factor S(q). Finally to compare with our calculations on the sta- tistical mechanics properties for the sample, we can obtain its pair distribution function g(R) as the probability of finding an atom at rj from an atom ri through an Inverse Fourier Transform on S(q). The connection between substrate state with characteristic pair distribution function and structure factor g(r) and S(q) respectively is demonstrated in Figure 1.5.
- 25. Chapter 1. Introduction17 Figure 1.5: Characteristic diffraction results for pair distribution and structure factor for solids (c)/(C), liquids (b)/(B), amorphous materials (b’)/(B’), and gasses (a)/(A) [26].
- 26. Chapter 1. Introduction18 1.2.3 Dense Random Packing of Hard Spheres Originally thought to be an ideal model for the liquid state, the dense random packing of hard spheres (DRPHS) was initially carried out experimentally by compressing solid ball bearings into containers with flexible or irregular walls. It has been shown that this model more accurately represents the meta stable amorphous or commonly called glassy state of simple liquids, wherein a liquid is cooled or compressed until the atoms effectively are immobilized without the onset growth of a crystal phase [5]. These amorphous alloys are now being pro- duced by a number of different techniques, such as by rapid cooling (quenching) of a liquid melt [7, 16], or through vapor deposition [18] and electrodeposition [6]. Generally simulating the DRPHS begins by taking a seed cluster of 3 spheres in hard contact with each other, and bringing one additional sphere at a time, placing each on a surface site of the existing cluster such that it will rest in hard contact with three spheres already present [2]. After the sphere has been placed in its new location on the cluster, it becomes part of the substrate that subsequent spheres are deposited on, and is not allowed to move. The resulting aggregate of spheres is then based on the type seed cluster used, and the criterion by which successive surface sites are chosen for deposition. The global criterion in which new particles are added at the site closest to the center of the original side corre- sponds roughly to choosing the site with the lowest gravitational or long range potential. It has been previously demonstrated that this global criterion best fits experimental results with a sharp first peak and a split second peak [2].
- 27. Chapter 1. Introduction19 The first peak r1 in g(r) plots is reflective of nearest-neighbor atom-atom con- figurations, and in the dense random packing of hard spheres model it corre- sponds to spheres in contact with each other. For the splitting of the second peak characteristic of metallic glasses (Fig 1.5 b’), the first sub-peak at r2 = √ 3r1 originates from a continuum of configurations of two hard-bonded equilateral triangles sharing common sides (Figure 1.6 b), and should only be visible at high density when the first peak is very sharp. The second sub-peak r3 = 2r1 geo- metrically originates from the set of local configurations in which two nearest neighbors are in contact with an intervening sphere in a linear array (Figure 1.6 a) [27]. Figure 1.6: Geometric origin for splitting of the second peak for amorphous metallic solids [2].
- 28. Chapter 2. Methods20 Chapter 2 Methods 2.0.1 Importance Sampling The basic idea behind importance sampling can be demonstrated by supposing we desire to evaluate numerically a one-dimensional integral I as: I = b a f (x)dx = (b − a) f (x) (2.1) where f (x) is an unweighted average of f(x) over the interval [a,b]. In the instance where our f(x) is negligible for points in the range of [a,b], then it be- comes preferable to spend more time sampling points in regions where f(x) is nonzero, and less time elsewhere. This remains just as true in problems where the dimensionality of our integral increases, requiring greater time to compute. If we were intent on evaluating the above integral by random sampling, but with sampling points distributed non-uniformly over an interval [0,1] with some non-negative probability density w(x), then we can rewrite the integral as: I = 1 0 w(x) f (x) w(x) dx (2.2)
- 29. Chapter 2. Methods21 Now if we assume that w(x) is the derivative of another (non-negative, non- decreasing) function u(x), where u(0)=0 and u(1)=1 (implying the normalization of w(x)) and writing u as the integration variable with x a function of u then [13]: I = 1 0 f [x(u)] w[x(u)] (2.3) Now we can generate L random values of u uniformly distributed in the interval [0,1], which gives us an estimate for I as [13]: I ≈ 1 L L ∑ i=1 f [x(ui)] w[u(xi)] (2.4) The advantages to rewriting our integral in this form comes when estimating the variance σ2 I in IL, our estimate for I with L random sample points [11] σ2 I = 1 L2 L ∑ i=1 L ∑ j=1 f [x(ui)] w[x(ui)] − f w f [x(ui)] w[x(ui)] − f w (2.5) and as different samples i and j are assumed to be totally independent, the cross terms vanish such that: σ2 I = 1 L2 L ∑ i=1 f [x(ui)] w[x(ui)] − f w = 1 L f w 2 − f w 2 (2.6) This demonstrates that the variance σ2 I in I still goes as 1 L, but we can greatly reduce the magnitude of this variance if we can choose the non-negative proba- bility density w(x) such that f(x)/w(x) is a smooth function of x [11].
- 30. Chapter 2. Methods22 2.0.2 Monte Carlo The Monte Carlo is a means for utilizing computer systems to calculate properties of substances composed of interacting molecules, equivalent to solving for ex- pected equilibrium values from difficult statistical mechanics many dimensional integrals. It’s typically applied to the field of liquid study, but the core method remains useful in a large variety of complex problems best solved through com- putational numerical integration [1]. The method is utilized when considering a square system of finite size with N number of interacting particles. Surface effects are minimized by considering the complete substance to be periodic, or comprised of many smaller squares with N particles in the same configuration, by which there will be some distance dAB that represents the shortest possible distance between particles A and B. Utilizing a model for the interaction between atoms A and B by which the interaction energy falls off at large distances, then the minimum distance dAB will be the most prominent to our considerations (largest contribution to total energy) [20]. To do so we utilize our knowledge of the positions for the N particles in the square to calculate the potential energy of the system: E = 1 2 N ∑ i=1 i=j N ∑ j=1 U(dij) (2.7) where U is the potential between molecules, dij is the distance between particles i and j. In calculating the properties of a system, the canonical ensemble is used, such
- 31. Chapter 2. Methods23 that if F is the equilibrium value of interest, then [11] F = Fe −E kBT d3N pd3Nq e −E kBT d3N pd3Nq (2.8) where (d3N pd3Nq) is a volume element in the 6N-dimensional phase space. For larger systems it would then be impractical to do several hundred dimensional integrals through numerical methods. The Monte Carlo allows then a method for many-dimensional integrals simply by integrating over a random sampling of points instead of over a regular array of points. Now in the Metropolis method for importance sampling, our canonical en- semble solved through the Monte Carlo is made computationally easier than placing particles in random locations. This is accomplished by taking new con- figurations that are chosen with a probability of e −E KBT and are weighted evenly [20]. To do so, we can begin with a system of N particles in any configuration, then attempt to make a trial move for each particle in succession according to [20] X → X + α 1 (2.9) Y → Y + α 2 (2.10) Z → Z + α 3 (2.11) where α is the maximum allowable displacement, 1, 2, and 3 are random num- bers between [−1, 1]. In accordance with our periodicity assumption however, if the particle trial move would place it outside the square, then we place it back in-
- 32. Chapter 2. Methods24 side by reflection, equivalent to the particle entering the square from the opposite side. After each trial move of our N particles, we calculate the change in energy of our system ∆E caused by the move. If the move brings our system to a state of lower energy than previously (∆E < 0), then we allow the move and place the particle in its new position. If however the trial move places our system in a higher energy state than previously (∆E > 0), then we allow the move with a probability of e −∆E KBT . We do this by taking a new random number 4 in [0, −1] such that if 4 < e −∆E KBT we accept the new position, otherwise we return the particle to its old position. Now whether the particle was allowed to move or not, returning the particle to its old position is considered as a new configuration and hence counted towards averaging. Thus we find for Fj as the value for F after the jth move [20] F = 1 M M ∑ j=1 Fj (2.12) which allows us to utilize the iterative power of computers when evaluating our desired properties of a system, as opposed to traditional numerical methods.
- 33. Chapter 3. Liquids25 Chapter 3 Liquids 3.1 Results 3.1.1 Copper 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 g(r) T=1150 (Experimental) T=2400 T=2800 T=3200 T=3600 T=4000 T=4400 T=4800 T=5200 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1150 (Experimental) T=5600 T=6000 T=6400 (b) Liquid Figure 3.1: Pair distribution results from Monte Carlo liquid simulations for cop- per using the Lennard-Jones potential from 2400 ◦C to 6400 ◦C in 400 ◦C incre- ments.
- 34. Chapter 3. Liquids26 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4g(r) T=1150 (Experimental) T=1200 T=1600 T=2000 T=2400 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1150 (Experimental) T=2800 T=3200 T=3600 T=4000 T=4400 (b) Liquid Figure 3.2: Pair distribution results from Monte Carlo liquid simulations for cop- per using the Morse potential from 1200 ◦C to 4400 ◦C in 400 ◦C increments. 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 g(r) T=1150 (Experimental) T=1200 T=1600 T=2000 T=2400 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1150 (Experimental) T=2800 T=3200 T=3600 (b) Liquid Figure 3.3: Pair distribution results from Monte Carlo liquid simulations for cop- per using the Erkoc potential from 1200 ◦C to 3600 ◦C in 400 ◦C increments.
- 35. Chapter 3. Liquids27 Copper Monte Carlo simulations found that observable pair distribution peaks and smaller sub peaks decreased in size with increasing temperatures from: 2400 ◦C to 5200 ◦C in Lennard-Jones (Figure 3.1 a), 1200 ◦C to 2400 ◦C in Morse (Figure 3.2 a), and 1200 ◦C to 2400 ◦C in Erkoc (Figure 3.3 a), all in 400 ◦C incre- ments. Due to the presence of short and long range ordering, it is concluded that at the aforementioned temperatures, the simulated clusters failed to reach a liquid state. Characteristic loss of long range ordering and presence of short range or- dering attributable to a liquid state was demonstrated for: 5600 ◦C to 6400 ◦C in Lennard-Jones (Figure 3.1 b), 2400 ◦C to 4400 ◦C in Morse (Figure 3.2 b), and 2400 ◦C to 3600 ◦C in Erkoc (Figure 3.3 b).
- 36. Chapter 3. Liquids28 3.1.2 Gold 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 g(r) T=1150 (Experimental) T=2400 T=2800 T=3200 T=3600 T=4000 T=4400 T=4800 T=5200 T=5600 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 g(r) T=1150 (Experimental) T=6000 T=6400 (b) Liquid Figure 3.4: Pair distribution results from Monte Carlo liquid simulations for gold using the Lennard-Jones potential from 2400 ◦C to 6400 ◦C in 400 ◦C increments. 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 g(r) T=1150 (Experimental) T=1200 T=1600 T=2000 T=2400 T=2800 T=3200 T=3600 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1150 (Experimental) T=4000 T=4400 (b) Liquid Figure 3.5: Pair distribution results from Monte Carlo liquid simulations for gold using the Morse potential from 1200 ◦C to 4400 ◦C in 400 ◦C increments.
- 37. Chapter 3. Liquids29 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5g(r) T=1150 (Experimental) T=1200 T=1600 T=2000 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1150 (Experimental) T=2400 T=2800 T=3200 T=3600 (b) Liquid Figure 3.6: Pair distribution results from Monte Carlo liquid simulations for gold using the Erkoc potential from 1200 ◦C to 3600 ◦C in 400 ◦C increments. Gold Monte Carlo simulations demonstrated observable pair distribution peaks and smaller sub peaks decreased in size with increasing temperatures from: 2400 ◦C to 5600 ◦C in Lennard-Jones (Figure 3.4 a), 1200 ◦C to 3600 ◦C in Morse (Figure 3.5 a), and 1200 ◦C to 2000 ◦C in Erkoc (Figure 3.6 a), all in 400 ◦C incre- ments. Due to the presence of short and long range ordering, it is concluded that at the aforementioned temperatures, the simulated clusters failed to reach a liquid state. Characteristic loss of long range ordering and presence of short range or- dering attributable to a liquid state was demonstrated for: 5600 ◦C to 6400 ◦C in Lennard-Jones (Figure 3.4 b), 2400 ◦C to 4400 ◦C in Morse (Figure 3.5 b), and 2400 ◦C to 3600 ◦C in Erkoc (Figure 3.6 b).
- 38. Chapter 3. Liquids30 3.1.3 Silver 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 g(r) T=1000 (Experimental) T=2400 T=2800 T=3200 T=3600 T=4000 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1000 (Experimental) T=4400 T=4800 T=5200 T=5600 T=6000 T=6400 (b) Liquid Figure 3.7: Pair distribution results from Monte Carlo liquid simulations for silver using the Lennard-Jones potential from 2400 ◦C to 6400 ◦C in 400 ◦C increments. 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 g(r) T=1000 (Experimental) T=1200 T=1600 T=2000 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1000 (Experimental) T=4400 T=4800 T=5200 T=5600 T=6000 T=6400 (b) Liquid Figure 3.8: Pair distribution results from Monte Carlo liquid simulations for silver using the Morse potential from 1200 ◦C to 4400 ◦C in 400 ◦C increments.
- 39. Chapter 3. Liquids31 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5g(r) T=1000 (Experimental) T=1200 T=1600 (a) Pre-liquid 0 2 4 6 8 10 12 14 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 g(r) T=1000 (Experimental) T=2000 T=2400 T=2800 T=3200 T=3600 (b) Liquid Figure 3.9: Pair distribution results from Monte Carlo liquid simulations for silver using the Erkoc potential from 1200 ◦C to 3600 ◦C in 400 ◦C increments. Silver Monte Carlo simulations demonstrated observable pair distribution peaks and smaller sub peaks decreased in size with increasing temperatures from: 2400 ◦C to 4000 ◦C in Lennard-Jones (Figure 3.7 a), 1200 ◦C to 3600 ◦C in Morse (Figure 3.8 a), and 1200 ◦C to 2000 ◦C in Erkoc (Figure 3.9 a), all in 400 ◦C incre- ments. Due to the presence of short and long range ordering, it is concluded that at the aforementioned temperatures, the simulated clusters failed to reach a liquid state. Characteristic loss of long range ordering and presence of short range or- dering attributable to a liquid state was demonstrated for: 5600 ◦C to 6400 ◦C in Lennard-Jones (Figure 3.7 b), 2400 ◦C to 4400 ◦C in Morse (Figure 3.8 b), and 2400 ◦C to 3600 ◦C in Erkoc (Figure 3.9 b).
- 40. Chapter 3. Liquids32 3.2 Discussion Discussions for the properties of simulated liquid copper, gold, and silver with their respective calculated pair distributions remain limited to a mostly qualita- tive nature in comparing the differences between the inter-atomic energy models described earlier. That is not to say however that although the evolution of a crystalline metal through melting to a liquid state (where the loss of long range order removes many useful mathematical tools for analysis and predictions) is not achieved specifically as change of state in this study, that insights into char- acteristic liquid structure cannot be appreciated from the computer simulated clusters. With liquid atomic clusters produced by a canonical ensemble with the Monte Carlo, wherein the number of particles, temperature, and volume of the system is held constant, the term of "melting" anywhere in this discussion or study refers to the achievement characteristic liquid state properties. Regardless, the use of three different two body inter-atomic potential energy models for run- ning the Monte Carlo simulations allows an insight into the theoretical behavior for a cluster of atoms when considering either: parameters calculated based on crystalline state properties with the Lennard-Jones [14] and Morse potentials [19], or considerations on crystalline state, dimer, and small micro clusters with the Erkoc potential [10]. When examining the Erkoc potential closely to determine connections be- tween simulated "melting temperatures" (those that achieved characteristic ex- perimental liquid state properties) and depth of the well, it was observed that the shallowest to deepest silver, gold, and copper (Figure 1.3) metals directly cor-
- 41. Chapter 3. Liquids33 related to a lowest to highest initial melting temperatures at 2000 ◦C, 2400 ◦C, and 2800 ◦C respectively (Figure 3.3 b, 3.6 b, and 3.9 b). This ordering of "melting" for the three simulated metals with respect to the depth of the well was shared by the Morse and Lennard-Jones potentials (Figure 1.4, 1.5), going from lowest to highest initial liquid state temperature as silver (2400 ◦C vs 4400 ◦C), copper (2800 ◦C vs 5600 ◦C), and gold (4000 ◦C vs 6000 ◦C) (Figures: 3.1/2 b, 3.4/5 b). It is therefore concluded that the depth of the potential well has a direct relation to simulating liquid structures at lower temperatures, and as such it is expected that a deeper well would contribute to hindering an atomic cluster from making the necessary movements to that would demonstrate short range ordering found in real liquid experiments. Observations on the ability of the three different potentials overall to induce a lower (and closer to the real experimental at 1150 ◦C for copper/gold, and 1000 ◦C for silver) liquid temperature demonstrated that for all three metals, the Erkoc was the most successful. Often the Erkoc was followed closely by the Morse potential, and far removed from both in terms of realistic temperature simulations was the Lennard-Jones. In examining the energy vs distance plot for gold (Figure 1.4), it was observed that the Morse potential actually goes deeper then the Lennard-Jones, which would contradict the afore mentioned ordering of melting for the metals in terms of well depth. It is thus thought that although we have found within an energy model (say within the Erkoc, and looking at the ordering of metals silver, gold, and copper) that depth of the well is a good in- dicator of the initial melting temperature (silver melts before gold, then followed
- 42. Chapter 3. Liquids34 by copper), that the depth does not portray the overall ability to simulate liq- uid structures. That is to say, the Erkoc demonstrated a superior ability overall in terms of matching experimental values for a liquid state, and was closer followed by the Morse potential than the Lennard-Jones. It was found that the Erkoc model makes considerations on crystalline state, dimer, and small micro clusters [10], where as the Morse and Lennard-Jones models use only crystalline state properties [14, 19]. With the greater success found in simulating liquid structure thus coming from a models that consider more than crystalline properties, it is thought that this reflects the need truncate the many-body expansion after the second (two and three body interactions) or higher terms. Therefore it is concluded that with the loss of long range order- ing and thus inability to revert to relatively simplistic mathematical modeling, the description of a liquid state necessitates considerations on the complicated interactions between more distant neighbors larger shells of nearest neighbors. Now in regards to the details of varying fixed temperatures within simula- tions, the most notable trend found by increasing temperature parameters for all materials is that of the slowly decreasing and left shifting peak heights (Figures 3.1-3.9 a) . In particular, it was observed during the process of increasing tem- peratures in 400 ◦C increments across all metals and potentials, that the greatest barrier to loss of long range order and achievement of liquid properties in agree- ment with experimental results came from the presence of small sub-peaks after the first and second peaks.
- 43. Chapter 3. Liquids35 This near transition from a pre-liquid state can be seen with more detail by turning our attention specifically to copper generated through the Erkoc potential (Figure 3.3 a). At the lowest run temperature 1200 ◦C, a number of peaks can be observed at approximately √ 2, √ 3, 2, √ 5, and √ 7 from the largest first peak. The distances of √ 2, √ 5, and √ 7 were found to be characteristic of face-centered cubic (fcc) and hexagonal close packed (hcp) [2], and thus play an important role in defining solid structure. As the temperature moves to a near critical temperature at 2400 ◦C, it was observed that the peaks located at √ 2, 2, and √ 5 nearly disappear (Figure 3.3 a). Final observations on the liquid properties of copper generated through the Erkoc potential demonstrated good agreement with the 1150 ◦C experimental results for a liquid when simulations were run at 2800 ◦C (Figure 3.3 b) and higher. At this point the smoothly behaving g(r) results showed only short range ordering of the copper cluster, with the second and third peaks approximately at √ 3 and √ 7 times the first sharp peak. Therefore with the disappearance of many characteristic peaks for a face- centered cubic (fcc) and hexagonal close packed (hcp) lattice, and the overall loss of long rang ordering, it is concluded that these results reflect the movement of the atoms within the cluster from its initial fcc cluster to positions that match the properties of a liquid. Furthermore, it is concluded that the disappearance of atoms located at √ 2, 2, and √ 5 from the nearest neighbor atom in the g(r) plots reflects failure of the metal at particular simulation temperatures due to the inter-atomic potential models to maintain solid properties.
- 44. Chapter 4. Amorphous36 Chapter 4 Amorphous 4.1 Results 4.1.1 Copper 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1150 (Experimental) Quenched T=5600 Quenched T=6000 Quenched T=6400 Figure 4.1: Pair distribution results for Monte Carlo amorphous simulations of copper using the Lennard-Jones potential quenched to 20 ◦C, with contrasting crystalline face-centered cubic (Fcc) and experimental liquid results.
- 45. Chapter 4. Amorphous37 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1150 (Experimental) T=2800 T=3200 T=3600 T=4000 T=4400 Figure 4.2: Pair distribution results for Monte Carlo amorphous simulations of copper using the Morse potential quenched to 20 ◦C, with contrasting crystalline face-centered cubic (Fcc) and experimental liquid results. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1150 (Experimental) T=2800 T=3200 T=3600 Figure 4.3: Pair distribution results for Monte Carlo amorphous simulations of copper using the Erkoc potential quenched to 20 ◦C, with contrasting crystalline face-centered cubic (Fcc) and experimental liquid results.
- 46. Chapter 4. Amorphous38 Measurements for copper on the first split peak as r2 in the DRPHS model demonstrated a 1.1884 to 1.6757%, 1.1884 to 3.9562%, and 2.5724 to 3.9562% de- viations for ∆r2 = √ 3r1 − r2 in the Lennard-Jones, Morse, and Erkoc simulations respectively (Table 4.1). Additional observations on the second split peak as r3 demonstrated 1.1148 to 1.3604%, 0.6263 to 3.2597%, and un-measurable peak de- viations for ∆r3 = 2r1 − r3 in the Lennard-Jones, Morse, and Erkoc simulations respectively (Table 4.1). Table 4.1: Measured vs calculated distances for amorphous copper in the dense random packing of hard spheres model Trial r1 (Å) r2 (Å) √ 3r1 (Å) r3 (Å) 2r1 (Å) ∆r2 (Å) ∆r3 (Å) LT5600 2.55637 4.48039 4.42776 5.05637 5.11274 -0.05262 0.05637 LT6000 2.55637 4.48039 4.42776 5.04412 5.11274 -0.05262 0.06862 LT6400 2.54412 4.48039 4.40654 5.04412 5.08824 -0.07384 0.04412 MT2800 2.55637 4.48039 4.42776 5.05637 5.11274 -0.05262 0.05637 MT3200 2.55637 4.60294 4.42776 4.94608 5.11274 -0.17517 0.16666 MT3600 2.54412 4.59069 4.40654 5.05637 5.08824 -0.18414 0.03187 ET2800 2.55637 4.54167 4.42776 - - -0.11390 - ET3200 2.55637 4.60294 4.42776 - - -0.17517 - ET3600 2.55637 4.59069 4.42776 - - -0.16292 - *Notation for Trials as ET goes: (L)ennard-Jones, (M)orse, (E)rkoc in Energy, with temperatures denoting original liquid state temperature quenched to 20 ◦C. **±0.01226 (Å) for all measurements on r1, r2, and r3
- 47. Chapter 4. Amorphous39 4.1.2 Gold 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1150 (Experimental) Quenched T=6000 Quenched T=6400 Figure 4.4: Pair distribution results for Monte Carlo amorphous simulations of gold using the Lennard-Jones potential quenched to 20 ◦C, with contrasting crys- talline face-centered cubic (Fcc) and experimental liquid results. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1150 (Experimental) T=4000 T=4400 Figure 4.5: Pair distribution results for Monte Carlo amorphous simulations of gold using the Morse potential quenched to 20 ◦C, with contrasting crystalline face-centered cubic (Fcc) and experimental liquid results.
- 48. Chapter 4. Amorphous40 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1150 (Experimental) T=2400 T=2800 T=3200 T=3600 Figure 4.6: Pair distribution results for Monte Carlo amorphous simulations of gold using the Erkoc potential quenched to 20 ◦C, with contrasting crystalline face-centered cubic (Fcc) and experimental liquid results. Measurements for gold on the first split peak as r2 in the DRPHS model demonstrated a 0.4385 to 0.6824%, 0.1615 to 0.6824%, and 0.4385 to 0.6824% de- viations for ∆r2 = √ 3r1 − r2 in the Lennard-Jones, Morse, and Erkoc simulations respectively (Table 4.2). Additional observations on the second split peak as r3 demonstrated 1.1835%, 1.1835 to 2.0119%, and 1.1835 to 1.4144% deviations for ∆r3 = 2r1 − r3 in the Lennard-Jones, Morse, and Erkoc simulations respectively (Table 4.2).
- 49. Chapter 4. Amorphous41 Table 4.2: Measured vs calculated distances for amorphous gold in the dense random packing of hard spheres model Trial r1 (Å) r2 (Å) √ 3r1 (Å) r3 (Å) 2r1 (Å) ∆r2 (Å) ∆r3 (Å) LT6000 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 LT6400 2.89951 5.04412 5.02209 5.73039 5.79902 -0.02202 0.06863 MT4000 2.92402 5.05637 5.06455 5.73039 5.84804 0.00818 0.11765 MT4400 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 ET2400 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 ET2800 2.89951 5.04412 5.02209 5.73039 5.79902 -0.02202 0.06863 ET3200 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 ET3600 2.89951 5.05637 5.02209 5.71814 5.79902 -0.03427 0.08088 *Notation for Trials as ET goes: (L)ennard-Jones, (M)orse, (E)rkoc in Energy, with temperatures denoting original liquid state temperature quenched to 20 ◦C. **±0.01226 (Å) for all measurements on r1, r2, and r3
- 50. Chapter 4. Amorphous42 4.1.3 Silver 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1000 (Experimental) T=4400 T=4800 T=5200 T=5600 T=6000 T=6400 Figure 4.7: Pair distribution results for Monte Carlo amorphous simulations of silver using the Lennard-Jones potential quenched to 20 ◦C, with contrasting crys- talline face-centered cubic (Fcc) and experimental liquid results. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1000 (Experimental) T=2400 T=2800 T=3200 T=3600 T=4000 T=4400 Figure 4.8: Pair distribution results for Monte Carlo amorphous simulations of silver using the Morse potential quenched to 20 ◦C, with contrasting crystalline face-centered cubic (Fcc) and experimental liquid results.
- 51. Chapter 4. Amorphous43 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Distance (Å) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 g(r) Fcc T=1000 (Experimental) T=2000 T=2400 T=2800 T=3200 T=3600 Figure 4.9: Pair distribution results for Monte Carlo amorphous simulations of silver using the Erkoc potential quenched to 20 ◦C, with contrasting crystalline face-centered cubic (Fcc) and experimental liquid results. Measurements for silver on the first split peak as r2 in the DRPHS model demon- strated a 0.2588 to 0.6824%, 0.6824 to 3.3154%, and 0.4385 to 1.8451% deviations for ∆r2 = √ 3r1 − r2 in the Lennard-Jones, Morse, and Erkoc simulations respec- tively (Table 4.3). Additional observations on the second split peak as r3 demon- strated 0.9721 to 1.1835%, 0.7639 to 1.1835%, and 1.1835 to 3.0853% deviations for ∆r3 = 2r1 − r3 in the Lennard-Jones, Morse, and Erkoc simulations respectively (Table 4.3).
- 52. Chapter 4. Amorphous44 Table 4.3: Measured vs calculated distances for amorphous silver in the dense random packing of hard spheres model Trial r1 (Å) r2 (Å) √ 3r1 (Å) r3 (Å) 2r1 (Å) ∆r2 (Å) ∆r3 (Å) LT4400 2.91176 5.05637 5.04331 5.74265 5.82352 -0.01305 0.08087 LT4800 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 LT5200 2.89951 5.04412 5.02209 5.73039 5.79902 -0.02202 0.06863 LT5600 2.89951 5.05637 5.02209 5.74265 5.79902 -0.03427 0.05637 LT6000 2.89951 5.04412 5.02209 5.73039 5.79902 -0.02202 0.06863 LT6400 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 MT2400 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 MT2800 2.89951 5.16667 5.02209 5.73039 5.79902 -0.14457 0.06863 MT3200 2.89951 5.16667 5.02209 5.73039 5.79902 -0.14457 0.06863 MT3600 2.88725 5.16667 5.00086 5.73039 5.77450 -0.16580 0.04411 ET2000 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 ET2400 2.89951 5.04412 5.02209 5.6201 5.79902 -0.02202 0.17892 ET2800 2.88725 5.09314 5.00086 5.63235 5.77450 -0.09227 0.14215 ET3200 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 ET3600 2.89951 5.05637 5.02209 5.73039 5.79902 -0.03427 0.06863 *Notation for Trials as ET goes: (L)ennard-Jones, (M)orse, (E)rkoc in Energy, with temperatures denoting original liquid state temperature quenched to 20 ◦C. **±0.01226 (Å) for all measurements on r1, r2, and r3
- 53. Chapter 4. Amorphous45 4.2 Discussion Of primary interest for analyzing simulated amorphous clusters quenched from their characteristic liquid state, is how the resulting pair distribution g(r) first peak and split second peaks compare to the theory of Dense Random Packing of Hard Spheres (DRPHS). In this way we can make comparisons on two very different computer simulation methods: one with DRPHS by taking a seed of three hard contact spheres and adding one at a time to create a growing aggregate of immovable spheres [2], against the current method of calculating a canonical ensemble through the Monte Carlo with our potential energy models used in this study. When looking closely at the results for metallic copper in the Morse and Erkoc potentials, it was noted that although there was a loss of long range order, the g(r) plots failed to produce the expected second peak splitting (Figures 3.2 and 3.3). As this absence of peak splitting was isolated only to the metallic copper, with gold and silver showing very prominent split sub-peaks, it is currently unknown why this occurred, and is thought to be independent of the Morse and Erkoc potentials. Observations on the remaining g(r) plots generated through re-running the liquid clusters obtained previously (Figures 3.1 b to 3.9 b) found the maintained disappearance of atoms at positions √ 2r1 and √ 5r1 that are characteristic of face-centered cubic and hexagonal close-packed crystal lattices (Figures 4.1-4.9), suggesting the cluster failed to achieve a crystalline state. More importantly for classifying the pair distribution plots however was the overall loss of long
- 54. Chapter 4. Amorphous46 range order, with a splitting of the second peak, demonstrating greater short range order (Figures 4.1-4.9) than in a liquid state g(r). Thus it is concluded that the liquid clusters produced initially by running the Monte Carlo simulation at high temperatures and then re-running them at 20 ◦C achieved the characteristic amorphous metal properties. Now contrasting the obtained results to the DRPHS theory, comparisons of the first peak position r1 with the first split peak ∆r2 = √ 3r1 − r2 and the second split peak ∆r3 = 2r1 − r3 across nearly all simulations, the results demonstrated a slight narrowing of the distance between the two split peaks as ∆r2 < 0 and ∆r3 > 0 (Tables 4.1-4.3). As this narrowing and slight deviation from expected positioning was found to be between 0.2588% at its lowest and 3.9562% at its highest from theoretical distances, it is concluded that the simulated amorphous clusters are in good agreement with the DRPHS theory. Therefore it was found that the method of generating first liquid clusters through the Monte Carlo with the three, two-body potential energy models, and then "quenching" the resulting liquids by re-running them at 20 ◦C, the Monte Carlo was able to produce g(r) plots characteristic for amorphous metals. Further conclusions demonstrated these simulated amorphous metals were in agreement with the Dense Random Packing of Hard Spheres theory, whereby the splitting of the second peak into two sub-peaks is due to the abundance of two nearly co-planar hard-bonded equilateral triangles sharing common sides, and of two second neighbors in contact with an intervening sphere in a nearly co-planar linear array.
- 55. Chapter 5. Conclusion47 Chapter 5 Conclusion Utilizing the Monte Carlo to compare the abilities of the various two-body po- tential energy models in simulating metallic clusters, it was revealed that the Erkoc demonstrated liquid state properties closest to real experimental results, followed closely by the Morse potential and far removed was the Lennard-Jones. It was found that this can be attributed to the intrinsic considerations on crys- talline properties, dimer, and small micro clusters when calculating parameters in the Erkoc model, whereas the Morse and Lennard-Jones models only consider crystalline properties. As such, it was concluded that in the event of a loss of long range order due to the transition from solid to liquid state in real experiments, a proper description of the liquid state more greatly necessitates considerations on the complicated interactions between more distant neighboring atoms. Additional considerations on the well depth of the potential energy mod- els, and in connection to simulating characteristic liquid properties closest to real experimental results demonstrated a direct link between well depth within each potential. An ordering by which lowest initial temperature necessary to demonstrate liquid properties were achieved was thus attributed to shallower well depths within each model. It was thus concluded that deeper wells would
- 56. Chapter 5. Conclusion48 contribute to hindering the atomic cluster from making the necessary movements to simulate the short-range ordering of a liquid in the pair distribution plots g(r). When taking the generated liquid clusters and simulating a fast "quenching" of the melts to 20 ◦C, it was found that all simulations maintained an absence of long range order, with nearly all demonstrating a split second peak character- istic of amorphous metals. These results reflected a failure to reach properties attributed to a solid that would experimentally come from gradual cooling of a liquid melt. It was noted for metallic copper in the Morse and Erkoc potential however, that this second split peak was not observed. With gold and silver in the Morse and Erkoc models showing very prominent split second peaks, it was concluded that this unexplained event was independent of the potentials used and warrants further investigation. Contrasting the simulated results to the DRPHS theory, comparisons of the split sub-peaks for all simulations demonstrated a narrowing of the peak gap with ∆r2 < 0 and ∆r3 > 0 predicted. As this narrowing and slight deviation from the expected positioning was found to be between 0.2588% at its lowest and 3.9562% at its highest from theoretical distances, conclusions on the simu- lated amorphous metals were in agreement with DRPHS. Therefore splitting of the second peak into two sub-peaks attributed to the abundance of two nearly co-planar hard-bonded equilateral triangles sharing common sides, and of two second neighbors in contact with an intervening sphere in a nearly co-planar lin- ear array in experimental amorphous metals was achieved in this Monte Carlo.
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