Atomistic Computer Simulation Analysis of Nanocrystalline
Nickel-Tungsten Alloys
MASSACHUSES INSTTE
OF TECHNOLOGY
by
FEB 0 8 2010
Alison Michelle Engwall
LIBRARIES
Submitted to the Department of Materials Science and Engineering
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Materials Science and Engineering
at the
ARCHIVES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2009
© Massachusetts Institute of Technology 2009. All rights reserved.
A
............
Author ...................................................... .........
Department of MateriaS ience and Engineering
May 22, 2009
C ertified by .................................................................................................
Christopher Schuh
Associate Professor
Thesis Advisor
Accepted by................................
..
.........
Lionel C. Kimerling
Professor of Materials Science and Engineering
Chair, Undergraduate Committee
Atomistic Computer Simulation Analysis of
Nanocrystalline Nickel-Tungsten Alloys
by
Alison Michelle Engwall
Submitted to the Department of Materials Science and Engineering
on May 8, 2009, in partial fulfillment of the
requirements for the degree of
Bachelor of Science
Abstract
Nanocrystalline nickel-tungsten alloys are harder, stronger, more resistant to degradation, and safer
to electrodeposit than chromium. Atomistic computer simulations have previously met with
success in replicating the energetic and atomic conditions of physical systems with 2-4nm grain
diameters. Here, a new model subjects a vertically thin unique volume containing 3nm or 10nm
FCC grains with aligned z axes to a Monte Carlo-type minimization to investigate the segregation
and ordering behavior of W atoms. Short-range order is also tracked with the Warren-Cowley
parameter, and energetic results are explored as well.
It was found that the Ni-W system has a very strong tendency toward SRO. The 10nm models
exhibited more robust order at low concentrations, but ordering in the 3nm model was generally
more pronounced. At the dilute limit atoms are driven to the grain boundaries, but as the
boundaries are saturated intragranular ordered formations increase and may even perpetuate over
low-angle grain boundaries. Ordering was also observed within the grain boundaries at all
concentrations for both diameters. The 10nm models were saturated at lower concentration, and
grain boundary energy was reduced by up to 93%. W atoms preferred to associate with each other
as third-nearest neighbors, but at very high concentrations formations with W atoms as second
nearest neighbors were also observed.
Thesis Supervisor: Christopher A. Schuh
Title: Associate Professor
Contents
1 Introduction
11
1.1 Background and Motivation
11
1.2 Previous Research
13
2 Modeling Procedure
14
2.1 Model Design
14
2.2 Monte Carlo Method
15
2.3 Energetics
16
2.4 Warren-Cowley Order Parameter
17
3 Results and Discussion
3.1 Solute Segregation and Ordering
19
19
3.1.1 Atom Segregation
19
3.1.2 Short Range Order
22
3.2 Energetics
25
3.2.1 Total energy
25
3.2.2 Grain boundary energy
26
3.2.3 Formation energy
27
4 Conclusions and Future Research
29
A 2D Atomistic Maps and Tungsten Concentration and Neighbor Density Plots
33
List of Figures
1-1
Grain size of electrodeposited Ni-W as a function of composition
12
2-1
2D fixed-z 3nm and 10nm model unique volumes
15
2-2
Total system energy loss vs Monte Carlo switch attempts
16
3-1
Selected atomistic and contour concentration plots of 3nm models
20
3-2
Selected atomistic and contour concentration plots of 10nm models
21
3-3
Warren-Cowley order parameter as a function of k for all 3nm models
22
3-4
Warren-Cowley order parameter as a function of k for all 10nm models
23
3-5
Warren-Cowley order parameter as a function of k for selected models
23
3-6
Total energy of models as a function of composition
25
3-7
Change in total energy of models as a function of composition
26
3-8
Grain boundary energy of models as a function of composition
27
3-9
Formation energy of models as a function of composition
28
A-1
Atomistic and contour concentration plots of 3nm models 1-3.5at%W
34
A-2
Atomistic and contour concentration plots of 3nm models 5-15at%W
35
A-3
Atomistic and contour concentration plots of 3nm and 10nm models 30at%W
36
A-4 Atomistic and contour concentration plots of 10nm models 1-3.5at%W
37
A-5
Atomistic and contour concentration plots of 10nm models 5-15at%W
38
A-6
Atomistic and tungsten neighbor density plots of 3nm models 1-3.5at%W
39
A-7
Atomistic and tungsten neighbor density plots of 3nm models 5-15at%W
40
A-8
Atomistic and tungsten neighbor density plots of 3nm and 10nm models 30at%W
41
A-9 Atomistic and tungsten neighbor density plots of 10nm models 1-3.5at%W
42
A-10 Atomistic and tungsten neighbor density plots of 3nm models 5-15at%W
43
Acknowledgments
The author would like to acknowledge Professor Christopher Schuh, Dr
Andrew Detor, Professor Craig Carter, Colin Ashe, and Nathan Holmes for
their assistance and inspiration.
10
Chapter 1
Introduction
1.1 Background and Motivation
The nickel-tungsten (Ni-W) system is a unique and interesting one with highly desirable
physical properties whose chemical foundations are not well understood. Electrodeposited
nanocrystalline Ni-W coatings are harder, stronger, and more resistant to chemical corrosion and
physical degradation than chromium; they have proven to be highly resistant to stress corrosion
cracking and localized intergranular degradation, and when faced with chemically corrosive
environments a protective tungsten-rich passivation layer forms at the surface. 11 Additionally, the
precursors for the electrodeposition process have none of the hazardous effects of hexavalent
chromium compounds that have caused so much environmental damage and incurred great cost,
both economically and to human health.
Practical interest the Ni-W system is further piqued by easily attainable nanocrystalline
grain sizes which are stable at room temperature. Hall-Petch scaling subsides when grain
diameters decrease below 10 nm, but, in general, finer grains lead to stronger materials. The grain
size of Ni-W coatings can be precisely dictated by tailoring the composition, and exact control
over the composition can be attained during electrodeposition using the reverse pulse methodology
developed by Professor Schuh's group at MIT [Fig 1-1]. I2] Properties like the relatively high
solubility of tungsten in nickel and relatively low tendency to segregate compared to other binary
alloys like Ni-P result in a wide range of grain sizes available for application. However, the
atomic mechanisms that control this stabilization against Ostwald ripening are not completely
understood.
160 *
140
=
120y
120
148.17e-.766x
R2 = 0.8963
E 100 -
40
80
40
20
W
0
5
10
15
20
25
30
Composition (at%W)
Figure 1-1. Grain size of electrodepositedNi-Wasfunction of composition
There is a thermodynamic driving force in all polycrystalline materials to decrease the free
energy of the system by decreasing the intergranular surface area of grain boundaries.
Nanocrystalline Ni-W has proven to be resistant to grain growth below 500 0 C, a property which
many pure metals do not share even at room temperature.13 1 In regard to the energy, disorder
inherent in the transition across a grain from one lattice orientation to another behaves like a line
of dislocations, and this entropic energy increases with disorientation. The huge surface area of
these interfaces at tiny grain diameters causes nanocrystalline pure metals to be difficult to achieve
and too unstable at reasonable temperatures to be useful in practice. However, in binary alloys the
migration of the solute atoms to the grain boundaries lowers the surface energy, which diminishes
the driving force of Oswald ripening toward either stable or metastable conditions. Solute
stabilization of grain boundaries is common behavior and certainly present in the Ni-W system,
but less typical ordering forces are in effect in Ni-W as well.
1.2 Previous Research
Foundational modeling work was done by Detor, who used atomistic computer simulations
to model the behavior of the Ni-W system over the range of 1 to 40 at% W in polycrystalline
systems with cubic unique volume and periodic boundary conditions containing 12 FCC grains
with spherical-equivalent grain diameters of 2, 3, or 4 nanometers. [4 The grains were randomly
oriented with a disorientation distribution in agreement with the MacKenzie function describing a
purely random system of octahedral symmetry.!5 ' Identical energy minimization procedures were
utilized for those simulations and the ones investigated here. Chemical ordering of the systems
was characterized with the Warren-Cowley order parameter. Grain boundary segregation was
tracked with average normalized composition versus distance from grain center. The energies of
interface surfaces and segregation were monitored against composition and grain size. Detor found
that W segregates to the grain boundaries more strongly as the solute content decreases; the energy
required to segregate to the grain boundaries increases with solute content; as grain size decreases
so does the degree of short-range order within the grain lattices; and, ultimately, larger grains are
more stable.[4]
The present investigation extends previous work by characterizing the distribution and
segregation of W in nanocrystalline Ni for 3nm and 10nm diameter grains, and a new design of the
model is used. The 3D cube model replicated realistic conditions extremely well, but the size of
the grains possible to simulate was severely limited by the computing power required to expand
the unique volume of the cube to larger grain diameters, and the size and orientations of the grains
occluded interpretation of the intragranular behavior of tungsten atoms. The models analyzed here
simulate a vertically thin slice of a system of FCC Ni-W grains with z-axis alignment and set
horizontal disorientations.
Chapter 2
Modeling Procedure
2.1 Model Design
There are a few fundamental differences between the layout of a cubic model with
randomly-oriented grains and the systems studied here. Unique volume for the "2D" model is
much more extensive in the x and y directions than in the z, which is only thick enough for atoms
to not interact with themselves through the periodic boundaries [Fig 2-1]. This change has an
interesting consequence when paired with periodic boundary conditions: The grain boundaries
experience very little curvature over the minimal thickness, and so are in effect infinite vertical
planes. This means that the fraction of the volume occupied by grain boundaries, and therefore the
volume affected by grain boundaries, decreases drastically when compared with sphericalequivalent grains of comparable diameter. As a consequence, grains are effectively larger than
their set horizontal diameters would suggest. Another major alteration was locking the vertical
orientation of each of the four unique grains to facilitate visual interpretation. Fixing a rotation
axis decreases the possible grain disorientations; because FCC crystals have 90" symmetry, the
maximum possible disorientation is 45". In a perfectly random system with octahedral symmetry,
the grain disorientation distribution would follow a MacKenzie function with a maximum of 62.8"
and a mean of 40.74.[51 In the 2D fixed-z simulation type explored in this investigation, the
maximum angle is 400 and the average is 25.
__ _
__
___
_L
Figure2-1. 2Dfixed-z model unique volumes. Left: 3nm. Right: 10nm.The purple (center)grainis not
rotated,blue (sharingoppositesides of the center grain) is rotated 100, yellow (corners) 700, and green
30'. Dimensions are in Angstroms.
2.2 Monte Carlo Method
The modeling procedure was created to realistically simulate the preferred state of the NiW system, given the initial constraints, by moving individual atoms on a set lattice. Each system
was initialized as pure Ni with predetermined general atom locations defining distinct grains.
Atomic interactions were accounted for by a multi-body Finnis-Sinclair atom potential
optimization reaching agreement with experimental results.14 1 Ni atoms were randomly replaced
with W to the solute content required for each simulation with cycles of 0.01% strain in the three
primary axial directions and full system relaxation performed after each substitution to minimize
the stress of the lattice distortion caused by the larger radius of the W atoms.61 The Monte Carlo
energy minimization run on the resulting random model acted as a simulated 800 C annealing
process to find the atomic configuration with the lowest possible energy.[61
During the Monte Carlo minimization, two unlike atoms selected at random are switched
in position, and a conjugate gradient relaxation of the system is performed to reduce stress. The
total energy of the resulting structure is compared to the energy of the system before the switch. If
the new energy is lower, the alteration stays, and the next minimization step is taken using that
state as the starting point. However, if the new system has a higher total energy, it is not
necessarily discarded - higher-energy states are accepted with a probability described by an
Arrhenius function dependent on simulated temperature and the degree of energy change to
account for the randomizing effects of kinetic energy.
Eventually, further switches do nothing to decrease the system energy; a global minimum
has been achieved and the simulation completed. If there is no appreciable difference in the radius
of the species involved, it may not be necessary to execute a conjugate gradient relaxation after
each switch, but in Ni-W systems relaxation is an important step because of the amount of stress
caused by W in the lattice. [61
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0
0
1000
2000
3000
4000
5000
6000
7000
8000
0
_1
-0.02
at%W
-004
-at%W
at%W
_-0.5
at%W00
3.5
at%W
....5atW
S-
-0.02
2 at%W
3.5at%W
f
-5 at%W
-30
lat%W
15at%W
-
S-0.08
3
toW
S-0.-0.06
-0.08-
S-0.12-
0.1.
-0.16
-0.18
-0.12
Switch attempts, 3nm
Switch attempts, 1Onm
Figure2-2. Total system energy loss vs Monte Carloswitch attempts. Left: 3nm. Right: 10nm.
2.3 Energetics
The total energy of a system is a combination of enthalpic and entropic terms: the chemical
energy of the atoms themselves, the internal surface energies of grain boundaries, the energies of
atomic mixing, bonding, and ordering, as well as any of the various other interactions in the
system. Several important energy values can be tracked from the total energy of the system. The
total energy per atom is a measure of system stability; the grain boundary energy per area relates
to the segregation energy driving solute atoms to intergranular region; the change in total energy
per atom from a system of random atom locations to an energy-minimized organization is useful
for comparing the energetic advantage of two different final states; and the formation energy per
atom is an indication of favorable or unfavorable thermodynamic conditions for a given atomic
configuration. Total system energy is found as a result of the Monte Carlo-type minimization, and
the change in total energy is the difference between that value and the initial, fully relaxed but
atomically random version of the model. Grain boundary
energy is calculated by:
y = (Edefect -Eingle xtal)/Adefect,
where E d eect is the energy of formation of the polycrystalline model, E?'e
[Equation 1]
xtal
is the formation
energy of a single-crystal model with the same composition and short-range order function, and
Adefect
is the grain boundary area.41 Adefect is calculated in the simplified geometry of the 2D fixed-z
model as linear distance multiplied by the thickness, if the boundaries are assumed to be
vertical. Energies of formation are calculated as:
Ef = Eanoy -[(1-X)*ENi + X*Ew],
[Equation 2]
where Ealloy is the energy per atom of the alloy, and ENi and Ew are the energies per atom of pure,
fully relaxed nickel and tungsten polycrystalline models with the same atomic organization as the
alloyed structure (FCC).6 I
2.4 Warren-Cowley Order Parameter
The Warren-Cowley order parameter is a simple calculation of the probability in a binary
system of an atom having similar atoms in a given neighbor shell, normalized by the probability
of finding a like atom in that neighbor shell randomly based on solute concentration:
ak =
1-Pk/Prandom],
[Equation 3]
where Pk is the probability of finding an unlike atom per neighbor shell k and the probability of an
unlike neighbor in a random system Prandom =2X(1-X) with X being the concentration of solute. 41
Neighbor shells were determined for each atom by identifying the atoms closest to it and
segregating them by euclidean distance. As the Ni-W system is FCC, the twelve nearest atoms
form the first neighbor shell, the following six the second neighbor shell, the following twentyfour the third neighbor shell, the following twelve in the fourth neighbor shell, and the following
twenty-four in the fifth neighbor shell. The lack of vacancies in the model makes any shells of
diminished number highly unlikely, and the above method proved more precise in correctly
grouping spatially symmetric shells than using predetermined radial distances to define each shell
due to local lattice distortion caused by W concentration. Periodic boundary conditions were taken
into account along all three axes to assure no edge-like effects from the borders of the unique
volume. Average probability of encountering an unlike atom in each shell was calculated for W
atoms specifically as well as across the entire system, and the Warren-Cowley order parameter was
determined using the shell probabilities and an exact composition fraction for each model.
Chapter 3
Results and Discussion
3.1 Solute Segregation and Ordering
3.1.1 Atom Segregation
The two dominant forces controlling how atoms segregate in the Ni-W system are dictated
by the enthalpic drives to minimize grain boundary energy and stress in the regular lattice by
accumulating in the intergranular regions and assuming formations of short-range order. As other
studies have found, the tendency of W to segregate to the grain boundaries is relatively weak
compared to other binary alloy systems like Ni-P. However, this behavior is still readily apparent
at very low concentrations of W [Figs 3-1 and 3-2, first row]. Atoms collect in and around the
grain boundaries, especially at large-angle boundaries and triple junctions where the disorder is
greatest. The contour plots reveal that even when the atoms may appear orderless, as in 3nm lat
%W atomistic plot, the W atoms assume formations of vertical order within the grain boundaries.
In the 10nm model there are also several small groups of ordered atoms in the grain centers even
at lat%W, which indicates that the smaller volume fraction of intergranular region compared to the
3nm model is already nearly saturated with W. By 5at%W, these structures are evident in the 3nm
model as well [Fig 3-1, second row].
9
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r
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i
r
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r
r rr i r
f~i"'
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i
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r r
T f-*
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,
i
i
i
'a
-
a,
-
*
I
-
S
r
a
aI
a
a
a
JJ
Figure3-1. Atomistic plots (right)and contour concentrationplots showing at%W (left) of3nm models.
Black dots on atomisticplots are tungsten. First row: lat%W Second: 5at%W Third: 15at%W
.
.........
N
II
UO
*l
W
1
31
~
1W
i.3i
lle
Figure 3-2. Atom istic plots (right) and contour concentration plots showing at%Wg(eft) of 1lOnto models.
Third:
15at%W
Black dots on atomistic plots are tungsten. First row: lat%W Second: 5at%W
witi
of 10nm models.
Figure3-2. Atomistic plots (right)and contour concentrationplots showing at%Wo(eft)
Black dots on atomisticplots are tungsten. Firstrow: lat%Wi Scond: 5at%W Third. ]at%W~
At 15at%W almost all of the W atoms are rigorously ordered, and ordering formations
even extend over a low angle (100) grain boundary [Fig 3-1, third row]. The contour plot shows
that the ordering extends through the height of the model, while several collections of W on grain
boundaries and triple junctions are confined to one layer. The primary ordering mechanism at
lower concentrations is for W atoms to relate with each other as third-nearest neighbors, but at
15at%W (especially in the 10nm model) there is evidence of a new pattern. While most of the W
in the model is still forming the locally 25at% structure, in some clusters W atoms are organizing
at second-nearest-neighbor distances in areas of locally 50at%W. The drive to order at 15%W in
both models is so strong that the grain boundary regions are comparatively depleted.
3.1.2 Short Range Order
The Warren-Cowley parameter
ak
[Equation 3] is a useful metric for quantifying the level
of order for each neighbor shell k in a binary system. Positive values of ak indicate that the
probability of an atom having atoms similar to itself as kt neighbors, while negative values
correlate with favoring unlike atoms. The Ni-W system displays strong ordering trends in both
3nm and 10nm grain diameter models [Figures 3-3, 3-4, 3-5].
0.4
0°3
Zat rw
3. at %W
-0.1
Figure3-3. Warren-Cowley orderparameteras afinction of k for all 3nm models.
0.5at %W
--
lat 'W
0.2 -4-
3 5at %W
-1-
5at %W
--
10at%W
-
15at %W
0.1 -
Figure3-4. Warren-Cowley orderparameteras afunction of kfor all 1Onm models.
0.3
at %
----
a
3nr
lat%WlOnm
0,2
10at %W3nm
0.1 --
10at %WlOnmrn
Figure3-5. Warren-Cowley orderparameteras afunction of kfor selected models.
The plots indicate that like-atom separation of W is preferred for all systems. There are
major peaks in the probability of finding a like neighbor in the k=2 shell at all concentrations,
minor peaks at k=4 with higher global solute content, and major valleys at k=1 (especially for high
concentrations) and k=3 (especially for low concentrations). The strongest preference for like
neighbors is in k=2 for both 3nm and 10nm in all concentrations of W, achieving a maximum
value of 0.389 at 10at%W for 3nm and 0.308 at 5at%W for 10nm, though the peaks at 10at%W
and 3.5at%W for 10nm are very close at 0.304 and 0.294 respectively. The difference between Xk
at k=1 and k=2 is largest for 10at%W and 15at%W in both grain diameters. The same
concentrations exhibit the longest range of order with strong k=4 peaks and k=5 valleys, though
the trends for 5at%W are comparable in the 10nm models. Very low and very high concentrations
display the strongest preferences for unlike neighbors at k=3, especially in the 3nm models.
In general, very short-range ordering trends are more pronounced for 3nm grain diameter
models, which has both higher peaks and lower valleys. However, models of lower concentrations
display significantly higher unlike-preference peaks in the k=4 shell for 10nm grains [Fig 3-5]. At
low concentrations, atoms are segregating to the grain boundaries, where they order strongly
through k=3. In larger grains there are purely intragranular collections of SRO in addition to the
clusters around the grain boundaries, and these increase the average order through k=4. It is
expected for larger volume fraction of grain boundary regions to lead to less significant ordering in
systems with finer grains due to increased interference from internal surfaces, but that is not the
overall result observed here. Increased ordering through k=4 and k=5 for higher concentrations in
the 3nm models may be an effect of the lower-than-random average grain disorientation. Ordering
through the boundaries is more difficult at higher angles, and W atoms face less resistance
assuming vertical order when the grain boundaries are aligned in that direction.
Another significant trend is that from 10at%W to 30 at%W, the probability of finding an
unlike atom in the k=1 and k=3 shells decreases, but so does the probability of finding one in the
k=2 shell. For most concentrations, the dominant ordering pattern is for each W to be surrounded
by Ni in the first and second neighbor shells in local concentrations of 25at%W. At high global
concentrations, the W starts to assemble in groups with like atoms in the second neighbor shell,
creating areas with a local concentration of 50at%W [Fig 3-2, bottom]. The new pattern drives the
k=2 peaks downward in the Warren-Cowley ordering system, but could be interpreted as a
transition in ordering rather than solely a decrease.
1
--"
3.2 Energetics
3.2.1 Total energy
The total energy per atom [Fig 3-6] is a measure of system stability. Several trends may be
observed in this plot. The first is that the total energy per atom of the 10nm models is consistently
less than that of the 3nm models, which is a result of the much lower fraction of the unique
volume space being within the area of effect of the grain boundaries. Lacking any grain boundary
effects, the energy per atom of the single crystals is lower still. All of the energies scale linearly
with W content because the chemical energy of W is much greater than that of Ni; a pure Ni single
crystal model was calculated to have -4.43 eV/atom, while pure W was almost twice that at -8.75
eV/atom.
-4
25
3 nm initial
o 10nm Inital
* 3nm Final
* 1Onm Final
SSingle Xal with 3nm SRO
* Single Xtal
with Onm SR
-42-4.4
-4.6
30
"eg
"P
.g-42-45
± -5.2
-52
m
-5.4.
-5.6
-5.8
-6Global Composition (at%W)
Figure3-6. Total energy of models as afunction of composition.
The change in total energy is plotted separately [Fig 3-7] to emphasize another important
result: For every concentration above lat%W, the change of energy per atom from the atomically
random initial state to the final state is greatest for 3nm diameter models. The effects of global
composition and grain boundaries are factors in the initialized, atomically random cell, so the
change in energy is dominated by the local effects of W movement in the lattice. This is also
reflected in the comparison of minimized single crystals in the total energy plot; the single crystal
with 3nm SRO is lower energy than the single crystal with 10nm SRO at 15at%W. The 3nm
diameter model experiences a greater enthalpic benefit per atom from solute segregation and
ordering.
0
5
10
15
20
25
*3nm
3
35
30
*10
nm
-0.04 0
T -0.08 -
-0.1 -
a,
-0.12
-0.14
-0.16
i
Global Composition (at% W)
Figure3-7. Change in total energy of models as afunction of composition.
3.2.2 Grain boundary energy
The grain boundary energy per area [Fig 3-8] is indicative of the force driving solute atoms
to intergranular regions. Lower values are typical for larger grain diameters because the same
global concentration of solute is drawn to a lower volume fraction of boundary region, and this
trend is observed in all the 2D fixed-z models. Actual values are lower than those noted in 3D
cubic models, but that is a result of the dfference in model design causing the grain boundary area
to be a function of grain perimeter rather
than 3D surface area for a given diameter. Smaller
disorientation angles and narrower angle distribution also decrease the average segregation energy,
I~
-'
-
-
--
because lower angle grains do less to interrupt lattice ordering.
1.800
-
1.600
* 3 nm
-
-
-
* 10 nm
1.400
1.200
0.800
i
o 0.600
0.400
T
0.200
0.000
0
5
10
15
20
25
30
lobal ComposMion at%W)
Figure3-8. Grain boundary energy of models as afunction of composition.
Grain boundary energies stabilize for both the 3nm and 10nm models, which indicates that
above a certain concentration, it is no longer advantageous for the systems to reduce energy by
moving W to occupy intergranular sites. The 10Onm model reaches critical saturation at a global
concentration near 2at%W, while for the 3nm it is closer to 3.5at%W. Grain boundary energies
were reduced by up to 93% for the 10nm system and 47% for the 3nm system, but as they never
reached 0 true thermodynamic boundary stability is not achieved.
3.2.3 Formationenergy
The formation energy per atom [Fig 3-9] is an indication of favorable or unfavorable
thermodynamic conditions for the system to assemble into a given configuration. Positive
formation energy means that energy would have to be added to the system, while negative
formation energy means that it could spontaneously form. However, both systems do achieve a
formation energy of less than 0 above a critical concentration; for 10nm it is 5at%W, and for 3nm
it is 10at%W. From [Fig 1] it may be estimated that experimentally, 3nm grains are likely to form
--
"F~ebBslp------ __ I------s----sl-LI~ --1------------
-
_ _
1
CI311111~
-L-t~-- m~~-1~~11I~ --
U~V-
II*UI--IE
at 21 at%W and 10nm grains at 16at%W, but this discrepancy is another result of the model layout
causing effective grain size to be larger than the set diameters.
0 .1 ....
. .........
............
......
......
* 3 nm
S10 nm
o Single Xtal,3nm SRO
o Single Xtal, 10nm SRO
0.06 -i
0.04
0.02
o()
4 -0.02
0.04
-0.04-
-0.06
-
-0.08 -
15
20
25
30
Global Composition (at% W)
Figure3-9. Formationenergy ofmodels as afjnction of composition.
The formation energy of the model with 10nm grain diameter is always less than the
formation energy of the 3nm model configuration, so it is more advantageous for all
concentrations to form with 10nm grain diameters at the conditions simulated here, another
indication that the 3nm diameter models are not thermodynamically stable. However, systems with
a formation energy below 0 may be metastable and therefore resistant to Oswald ripening.
Chapter 4
Conclusions and Future Research
The 2D fixed-z modeling technique presents clear and interesting trends in the segregation
and ordering behavior of W in Ni-W binary alloys, though a non-random grain disorientation
distribution does prevent rigorously realistic energy results. At the dilute limit, it is beneficial for
W atoms to segregate to the grain boundary regions until they are saturated, which occurs at lower
concentrations in larger grain sizes. The saturation of grain boundary regions is also signaled by a
stabilization in the grain boundary energy and an increase in the persistence of short-range order.
At high concentrations grain boundaries are saturated at well below the global concentration, so
instead of retaining the bulk of the solute they are relatively depleted.
Atoms will associate with each other in a regular fashion within the grain boundaries, and
assemble into formations of short-range order within the grains. This aggressive tendency for the
W to order competes with grain boundary segregation and can perpetuate across low-angle grain
boundaries. The ordering is most universal at compositions slightly above the grain boundary
saturation level, and the most common ordering system has W atoms associating with each other
as third nearest neighbors in regions with a local concentration of 25at%W. At higher
concentrations a second ordering system develops with W atoms as second nearest neighbors, in
regions with a local concentration of 50at%W, which impinges on the length to which either
system can extend. Grain boundaries also do interfere with the long-range ordering capability of a
system, but they assist small collections of order.
Neither the 10nm nor the 3nm 2D fixed-z system attained thermodynamically stable
grains. However, grain boundary energies were reduced by up to 93% for the 10nm system and
47% for the 3nm system and negative formation energies were achieved. At every concentration
modeled it was more energetically advantageous for the system to have large grain diameters.
Though the 3nm system has more robust ordering at very short ranges, the energetic effects are
overwhelmed by the penalties for greater grain boundary volume.
In the future it would be interesting to have an in-depth analysis of the two ordering
patterns observed. In addition to more models of the 2D fixed-z variety in conjunction with 3D
random-grain systems at more concentrations and larger grain diameters, models of the lattice
stress of each configuration as well as the interaction of the formations with grain boundaries
could prove enlightening. Correlating model calculations with experimental results is also
important work, as it is logical that the behavior of these ordered regions is the key to truly
understanding Ni-W binary alloys.
Bibliography
I11 Sriraman KR., Raman GS, Seshadri SK. "Corrosion behavior of electrodeposited
nanocrystalline Ni-W and Ni-Fe-W alloys." Materials Science and Engineering A
2007;460-461:39
[2]Detor AJ,
Schuh CA. "Tailoring and patterning the grain size of nanocrystalline alloys." Acta
Mater 2007;55:371
E31Lund AC, Schuh CA. "Driven Alloys in the Athermal Limit." Physical Review Letters
2003;91:235505
141Detor AJ, Schuh CA. "Grain boundary segregation, chemical ordering and stability of
nanocrystalline alloys: Atomistic computer simulations in the Ni-W system." Acta Mater
2007;55:4221
15
sMorawiec A. "Misorientation-Angle Distribution of Randomly Oriented Symmetric Objects."
Journal of Applied Crystallography 2995;28:289
161Detor
AJ, Miller MK.,Schuh, CA. "Solute distribution in nanocrystalline Ni-W alloys examined
through atom probe tomography." Philosophical Magazine 2006 ;86:28,4459 - 4475
32
Appendix A
Atomistic plots (on the right) are top-down views of each system. Different colors of
small, light dots represent different grains, and the black dots are tungsten. On the left, the
concentration plots show at%W and the density plots the number of tungsten atoms within each
atom's first and second neighbor shells.
__ _.
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30
V1
FigureA-1. Atomistic plots (right)and contour concentrationplots (left) of3nm models.
Firstrow: iat%W Second: 2at%W Third: 3.5at%W
W
U
."
9'
**
a
1
9
.+
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Figure~~~~I A-.Aoitcpos(ih)adcnor ocnrto
lt
lf)o
Firs ro: 5aW.Secnd:
~atW. hird l~tW
46
mmdl
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"i
A-3: Atomistic plots (right)and contour concentrationplots (left) of 30at%W models.
First row: 3nm. Second: lOnm.
'..
.
..
e:
"-
.
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:NUj
1~1
A
oti
m
u~
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tri
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Figure A-4. Atomistic plots (right) and contour concentration plots (left) of lOnm models.
Firstrow: 0.at%W Second: lat%W Third: 3.5at%W
I :
I+
I*
+
4W
4*
F Tt
IV
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__
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Figure
plt+rgt
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First~~~~~~~~~
eod ~to
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oes
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6
to
AS
A,
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5
a''
10
,
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11
3
03
Figure A-6. Atomistic plots (right) and tungsten neighbor density plots (left) of3nm models.
First row: lat%W Second: 2at%W Third: 3.5at%W
41-
.
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-
4
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41
.4
FigureA-7. Atomistic plots (right) and tungsten neighbor density plots (left) of3nm models.
First row: Sat%W Second: 1Oat%W Third: 30at%W
40
I-a
V
9 9
..
.
r
9*I
C -*
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r
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r
ra
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n
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l
a
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a
a
91
ia
* .
a
4
9W
IV
_
i
9
1
i'9
j
Figure A-8. Atomistic plots (right) and tungsten neighbor density plots (left) of30at%W models.
First row: 3nm. Second: 10nm.
a
,
_ _
__~_ _ ___
......
.........
i ii i ii i" i
ed
42
i !ip
V
i
'
,
jiiii:i
IIij:
iiiii~
i= =
lll
t +~
*9W
I~i!
ii iii ii
i!i
i~iI
>iiilll!
iiilb~lii iiiiii!
iU ii
1l!! i
*
.
:II
density plots (left) of l Onm models.
Figure A-9. Atomistic plots (right) and tungsten neighbor
FigyureA-9. Atomistic plots (right)and tungsten neighbor density plots (left) of 10nm models.
3.5at%W
Third:
row: O.5at%W Second: lat%W
First
Firstrow: 0.5at%W Second: l at%W Third: 3.5at%W
42
^as-
IF
~It
'
ICA
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4
V4.
_
M
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*
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;I* t''r
*
}ji
13k,
V
4\
+
Ito
NA*
'Vo
4
X(~4
44
f
FigureA-JO. Atomistic plots (right) and tungsten neighbor density plots (left) of IOnm models.
Firstrow: 5atoW Second: lOat%W Third: 3Oat%W
Y