Download The Effects of Misorientation Angle and Impurities on Grain

The Effects of Misorientation Angle and
Impurities on Grain Boundary Migration
Dynamics in Two Dimensional Bubble Raft
Group Members:
Ma Yunhan
Wang Xueqiao
Wang Weiyang
School:
Beijing National Day School
Location:
Beijing, China
Instructor:
Guan Chun
Shing-Tung Yau High School Physics Award
2015
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Abstract
The study of bubble raft configuration originated from the study of metallic structure. Bubbles in
the rafts are used to simulate the metal atoms and study the properties of metals. This research
recognizes bubbles as having unique properties and worth studying. The main goal of this research
is to study the influence misorientation angle and irregular bubbles have on grain boundary migration. The elimination mechanisms of different categories of grain boundaries are observed. Grain
boundaries tend to reduce their energy to the utmost through bursting of bubbles and slipping
of dislocations. Low angle grain boundaries reduce their angles until total elimination. High angle grain boundaries reduce their angle to 15 degrees which is of lower energy, proved by further
calculations of grain boundary energy. The existence of irregular bubbles is also a influencing factor. Grain boundaries interact with the irregulars but do not experience much influence. Irregular
bubbles with sizes much bigger than the regular bubbles tend to burst due to pressure of elastic deformation. Bubbles that are much smaller than regular bubbles become interstitial. The thesis first
considers in chapter one the works done by previous researchers, then follows a brief introduction
to the basic concepts in the research and the purpose of this research, followed by the formula and
the setup to generate bubbles in chapter two. The observations from the experiments are described
and analyzed. Theories are proposed and further calculations are used for support in the following
chapters.
Shing-Tung Yau High School Physics Award 2015
i
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Statement of Originality
The research process and result of this team are conducted and derived under
the guidance of the instructor. Other than the referenced content and the
acknowledged sources, this paper does not include any published findings by
this group or any other researchers.
Signature: Ma Yunhan, Wang Weiyang, Wang Xueqiao
Date: Sept. 20th, 2015
Shing-Tung Yau High School Physics Award 2015
ii
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Contents
Abstract
i
Statement of Originality
ii
1 Introduction
1.1 Relevance and Significance of Grain Boundary Migration . . .
1.1.1 Crystallography and Polycrystalline Solids . . . . . . .
1.1.2 Defects and Grain Boundary Migration . . . . . . . . .
1.1.3 Bubble Raft Model for Grain Boundary Migration . . .
1.2 Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Bubble Model Proposed by Lawrence Bragg and J.F.
Nye . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Studies on Grain Boundary Dynamics . . . . . . . . .
1.2.3 The Properties and Defects of Polycrystalline Structure
1.3 The Purpose of this Research . . . . . . . . . . . . . . . . . .
2 Experimental Installation Set
2.1 Apparatus Setting . . . . .
2.2 Solution Formula . . . . . .
2.2.1 Components . . . . .
2.2.2 Experimental Steps .
2.2.3 Final Formula . . . .
2.3 Air Flow Rate Controlling .
Up
. . .
. . .
. . .
. . .
. . .
. . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 Grain Boundary Migration and the Effect of Misorientation
Angles
3.1 Creating Grain Boundaries . . . . . . . . . . . . . . . . . . . .
3.2 The General Mechanism of Grain Boundary Elimination . . .
3.3 The Elimination Mechanism of High Angle Grain Boundaries
and Calculations . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 The Elimination Mechanism of Low Angle Grain Boundaries
and Calculations . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 The Behavior of Bubble Raft During GBM . . . . . . . . . . .
Shing-Tung Yau High School Physics Award 2015
1
1
1
2
2
3
3
4
5
5
7
7
9
9
10
12
12
13
13
14
17
22
25
iii
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
3.6
After the Elimination of Grain Boundaries . . . . . . . . . . .
25
4 Effect of Irregularly Sized Bubbles on GBM
26
4.1 Small Impurity Bubbles . . . . . . . . . . . . . . . . . . . . . 26
4.2 Large Impunity Bubbles . . . . . . . . . . . . . . . . . . . . . 33
4.3 Comparison with Findings in Metals . . . . . . . . . . . . . . 38
5 Conclusion
39
References
40
Acknowledgments
43
Shing-Tung Yau High School Physics Award 2015
iv
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
1
Introduction
1.1
1.1.1
Relevance and Significance of Grain Boundary Migration
Crystallography and Polycrystalline Solids
Polycrystalline solids are composed of small grains or crystallites of different sizes and orientations.
Most substances, including metals and ceramics, are polycrystalline. Characteristics of the grains,
such as the grain size and angle of misorientation, are determinants of some properties of the
material, such as strength, plasticity, thermal and electrical conductivity, magnetic properties, and
diffusivity. Correlation between grain size and strength as an important aspect of crystallography is
extensively investigated, with the result showing smaller grain size impedes slipping of dislocations,
thus increases the strength of material. Therefore, material scientists develop hard polycrystalline
solids by reducing the grain size and produce solids of high plasticity with materials of large
grain size. It is also possible to enhance the material’s strength by re-orienting the grains to a
higher angle because high angle grain boundaries are more resistant to dislocation movements
than low angel grain boundaries[1]. Other then grain boundaries, defects such as impurities can
also achieve the strengthening effect from which the solid-solution method arises. Impurity atoms
tend to interact with the dislocations and aggregate around the dislocation core to stabilize the
dislocation. Dislocations may also thwart each other when their density is too high, giving another
method to strengthening-creating dislocations[2]. However, evidence has also shown otherwise,
rendering another approach to achieve strength or plasticity. It is proposed that grain boundaries
are mobile, instead of stationary as traditionally assumed, during plastic deformation, or movement
of dislocations, thus capable of activating microstate plasticity[1]. Conclusions about the effect
of grain size on conductivity seem mostly consistent: electronic conductivity increases as grain
size is reduced[3]. Thermal conductivity depend on grain size mainly around room temperature,
with higher conductivity correlates to smaller grain sizes[4]. Angle of misorientation also exhibits
a strong influence on thermal conductivity, since isotropic, or randomly oriented, grains display
lower conductivity and antisotropic grains intensify the conductivity[5]. It is also discovered that
grain boundaries exhibit a resistance to thermal diffusion[6]. Magnetic properties are strongly and
directly related to grain size and orientation[7]. Since magnetism arises from aligned magnetic
domains, smaller grain size, larger misorientation, and higher density of defects in a polycrystalline
increase its coersivity[8].
Studies in the grain of polycrystalline solids also deepen the understanding in other fields.
For example, electromigration, or the mass transport phenomenon in microelectronic circuits, is
investigated based on grain growth and grain boundary migration[9]. The control of nano-size
materials is strongly dependent on the study of grains and grain boundaries for the grain boundaries
occupy a considerable volume in the structure[7]. Grain growth is essential to the production of
ceramics, so the studies on grains are applied to the control of grain growth in ceramic fabrication,
search for high density in sintering and reduction of the fraction of gas released by increasing the
diffusion distances to the grain boundaries in nuclear fuels[10]. It is even proposed that grain growth
is the basis of the growth of universal large scale structure, “which leads to an observed distribution
of voids in galaxy surveys”[11]. In addition to determining the structure and property of the crystal,
defects can also become reactive sites for chemical functional groups and trap different molecules
to react inside[15].
Other types of solids, particularly cellular solids, are also better understood through the study
Shing-Tung Yau High School Physics Award 2015
1
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
of grains. Although cellular solids are composed of cells with solid edges or faces, they can be represented by lattice sites in the center of each cell, therefore resembling the structure of polycrystalline
solids. Studies on the arrangement of grains help design structures similar to cellular solids such
as filters, catalyst carriers, membranes, scaffolds[12], shaving cream, polyurethane as cushions, and
thermoplastics[13].
1.1.2
Defects and Grain Boundary Migration
Polycrystalline solids are not perfectly arranged and contain various types of defects. Point defects
usually refer to vacancies, where an atom is missing but the lattice sites are not affected. Line defects
are dislocations, which are categorized into two types: edge dislocations and screw dislocations.
Edge dislocation refers to the phenomenon that a line of atoms stops in halfway so the adjacent
atoms are crushed to the middle, causing two rows of atoms to occupy the space of three rows and
therefore are not closely packed. Parallel dislocations, a type of edge dislocation, are formed when
a row of bubbles is not lined up with the rest. Due to the unstable state, once the limit of elastic
range is reached, bubbles will slip to release the strain. “[S]lip takes place by the bubbles in one row
moving forward over those in the next row by an amount equal to the distance between neighbors”,
as described by Bragg. Screw dislocation is caused when one side of a three dimensional grain is
“cut” in the middle and displaced while the other side remains the original arrangement. Planar
defects, or grain boundaries, are the most significant for the mechanical properties[14]. Grain
boundaries are also classified into two categories, twist boundaries and tilt boundaries. Twist
boundaries are caused by the rotation of one grain with respect to an axis parallel to the two
grains, or can be represented as a series of screw dislocations. Tilt boundaries are created by the
rotations with respect to an axis perpendicular to the two grains. Since tilt boundaries can be
achieved in a two dimensional plane, this paper is dedicated to the migration of tilt boundaries.
Boundaries can also be classified according to the angle of misorientation, giving the name of low
angle grain boundaries and high angle grain boundaries. Most studies of grain boundaries are
limited to low angle, where the boundary can be seen as an array of edge dislocations; thus high
angle grain boundaries are largely unknown[1].
Grain boundaries are constantly under motion due to shear stress[1], often caused by an atom
transferring from one grain to another (called diffusion)[16] or the slipping of dislocations. Grain
boundary migration is considered important to the structure of a crystal because it is the primary
means of recrystallization to a crystal with bigger size grains and lower energy[16].
1.1.3
Bubble Raft Model for Grain Boundary Migration
Using bubble raft to simulate atoms in polycrystalline solids is initially proposed and practiced by
Sir. William Lawrence Bragg in 1947.
Shing-Tung Yau High School Physics Award 2015
2
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Figure 1: Grain boundaries on a bubble raft, Bragg fig.5b[17]
He stated that the bubble raft model is superior to previous attempts of simulations such as
magnets and circular disks for its small fiction and large number of simulated atoms[17]. Since
bubbles follow a fcc arrangement as many polycrystalline solids, they are used as “a direct visualization and table top demonstration for control checks of theoretical predictions”[18]. Though
largely qualitative, the bubble raft clearly shows topological defects such as disclinations, dislocations and grain boundaries, “provides vivid images of the structure of defects”[18], and accounts
for some “physically important factors such as many-body forces and temperature effects”[19].
Further, bubble rafts are easy to control in creating initially perfect crystals when the dynamic
behavior of grains under shear stress is investigated[18].
1.2
Previous Studies
Numerous Studies on grain boundaries were completed by other physicists and chemists before with
a concentration on metal grains.
1.2.1
Bubble Model Proposed by Lawrence Bragg and J.F. Nye
Sir Lawrence Bragg and J.F. Nye’s study had yielded a number of important results. According to
Bragg, a bubble raft containing several grains has rather high strain due to “numerous dislocations
and other faults” , thus recrystallizes to form one regular grain. Grain boundary, as a type of fault,
contains bubbles “adhering defiantly to one crystalline arrangement or the other”.
Another type of fault, irregularly sized bubbles(impurities) disturb the uniform overall configuration of the grain. These “impurity atoms” tend to locate on grain boundaries. Grain boundaries
will move until they pass through irregularly sized bubbles.
Dislocation slipping was proposed by Orowan, Polanyi and Taylor to explain the plastic gliding
in metal. Taylor proposed this theory to explain “the mechanism of plastic deformation of crystals
considers the mutual action and equilibrium of such dislocations” in 1934.
Shing-Tung Yau High School Physics Award 2015
3
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Figure 2: A parallel dislocation, Bragg fig.6b[17]
Other types of faults are also introduced in the document, including v-shaped and triangle
shaped dislocations as well as blank spaces (vacancies) that “cannot be closed by a local readjustment” since refilling one would produce another[17].
Figure 3: Other types of faults, Bragg fig.11[17]
1.2.2
Studies on Grain Boundary Dynamics
Myrjam Winning, Anthony D. Rollett’s experiments[21] suggest that the motions of both low angle
and high angle grain boundaries are caused by external mechanical stress fields. However, low
and high angle grain boundaries have very different activation enthalpy. A sharp change in grain
boundary mobility occurs at a transition angle, which is approximately 14.4o . 14.4o is an average
value because the dislocation structure is different for every grain boundary.
Figure 4: Different grain boundaries[14]
A general model of high angle grain boundary was “diverted from a coincidence model” of
Kronberg and Wilson by D. G. Brandon[22].
Shing-Tung Yau High School Physics Award 2015
4
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Adele T. Lim[23] found that for low-angle grain boundary (LAGB), the migration speed is
“proportional to the applied stress”. The mobility is found independent of boundary misorientation.
1.2.3
The Properties and Defects of Polycrystalline Structure
According to H. Gleiter, the “migration rate of grain boundary was calculated as a function of the
orientation of two grains relative to the grain boundary and the thermodynamic constants”[24]. The
document considered grain boundaries in three dimensional present, therefore the result should be
slightly different from that of bubbles. Y. Ishuda[19] suggested the bubble raft tends to have ordered
structure. The stability varied due to the size difference and the presence or absence of vibration
(the vibration is produced to simulate atom movements in metal). Dislocations in a raft usually
“[retain] a step in the boundary”, and some of the dislocations will slip, causing the grain boundary
to migrate across the length of a step. In A. H. King’s research[25], the phenomenon of diffusion
induced grain boundary migration (DIGM) is investigated. DIGM, according to the document, is
“the sideways migration of grain boundaries accompanying the diffusion of solute along them”.The
experiment of T. J. Rupert, D. S. Gianola, Y. Gan, K. J. Hemker[1] demonstrated that grain
boundaries, instead of acting “as stationary obstacles to dislocation-based plasticity”, are actually
driven by sheer stress to migrate in a way that is consistent with recent molecular simulations. The
investigation done by M. L. Taheri, D. Molodov, G. Gottstein, A. D. Rollett[26] suggests that grain
boundary mobility is a material property thus is not affected by driving force. Apart from all the
experiments mentioned above, Timothy S. Cale and Max O. Bloomfield also developed computer
software to simulate 3D grain structure, thus acquiring full control over all the factors[27].
University of Wisconsin[28] provided a general introduction to the structure, properties and
defects of bubble rafts. It also described an apparatus used to generate uniform bubble rafts.
The apparatus consists a fish tank regulator, a plastic tube, a rubber stopper and a hypodermic needle. The former mentioned apparatus was adopted in this research. Another webpage,
Exploratorium[29], introduces a general introduction to soap bubble and soap films’ structure.
Bubble formulae for long lasting soap bubbles are listed.
1.3
The Purpose of this Research
The previous works done by generations of scientists all mainly focused on using bubbles as models
of metal atoms, this research group, however, started to appreciate bubble rafts as having unique
properties and worth studying. Bubble raft models, due to their rigidity, their uniformity and their
similarity to metal atoms, are considered useful in the study of crystallography for metal atoms.
By simulating planar and point defects in bubble rafts, this group hope to be able to propose a
theory that would not only work well on bubbles but on other uniform crystals as well.
This research intend to solve the question of how planar defects’ motion correlates to the misorientation angles and point defects in bubble rafts. Synthesizing the data from experiments, the
group hope to propose a theory that would correspond to all previous experiments and further researches. Calculation on energy is performed to examine if the proposed mechanism is energetically
favorable.
From the experiments it can be observed that the elimination of grain boundaries is unavoidable,
due to the tendency to eliminate the uneven distribution of energy. The grain boundaries disturb
the uniformity of the bubble rafts, creating strains that exert force on bubbles, causing them to
Shing-Tung Yau High School Physics Award 2015
5
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
burst. The bursting of bubbles then initiates the movement of dislocations, which is the main
impetus in pushing the migration of grain boundaries.
The mechanism of which the grain releases the grain boundaries depends on the angle of the
grain boundary. Grain boundaries are further categorized by their angle of misorientation to high
angle grain boundaries and low angle grain boundaries. Both grain boundaries reduce their angle,
with high angle grain boundaries to at most 15 degrees and low angle grain boundaries to total
elimination. High angle grain boundaries are carried by dislocations in the migration out of the
grain.
Bubbles of irregular sizes are created and inserted into the middle of the bubble raft. Irregular
bubbles do not exert much influence on the configuration of the rafts for long. Small bubbles are
compressed until they become interstitial. Much bigger bubbles burst because the strain they cause
renders greater force on them. Grain boundaries tend to migrate towards these point defects but
are not confined by the irregularities.
Problems remain that to represent metal atoms, the bubbles need to be sufficiently small and
rigid. To create capable bubbles, much more advanced apparatus needs to be applied. The bubbles
in this experiment, unfortunately, cannot satisfy this requirement. Also, the calculations in this
experiment are all very general, with many of the parameters and coefficients estimated. The
calculations are all based on a simplified model of the bubble rafts so the final results are not
exactly precise. They are sufficient enough, however, to support the theory proposed in this theory.
Bubble raft simulation is still a method that worth further investigation. This group has only
touched upon very little potential of the bubble model in the studying of crystallography. For
future research, this group hopes to improve the setup to create even smaller, more uniform and
more rigid bubbles. Computer simulation and further, more precise calculations are going to be
employed for more detailed analysis.
Shing-Tung Yau High School Physics Award 2015
6
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
2
2.1
Experimental Installation Set Up
Apparatus Setting
University of Wisconsin[28] has proposed a model to generate uniform bubble rafts.
Figure 5: Apparatus setting presented by University of Wisconsin[28]
This model was adopted as the prototype of the apparatus employed in this experiment. It has
several limitations, for instance, the air flow rate is hard to control. Therefore changes and improvements were made.
Instead of employing a combination of plastic tube and hypodermic needle, this experiment
improved the model by utilizing an injection syringe(also called infusion apparatus). An injection
syringe has multiple advantages. It contains tubes, a needle, and a velocity controller. The hypodermic needle is employed to produce monodisperse foam[30]. Velocity controller can limit the
amount of gas passing through, thus very handy in managing the bubble size and the speed of
bubbles’ generation.
The needle in this experiment was directly attached on the bottom of the water tank rather
then fixed by a rubber stopper in order to increase the depth of where bubbles are generated.
Experiments have suggested that the depth in which bubbles were generated has a great impact
on bubble size; the greater the depth, the smaller the bubble size. Smaller bubbles have greater
rigidity, which is preferred in this experiment. Greater depth also slows down the speed of generating
bubbles, thus managing the position of bubbles would be easier.
Experiments proved that a steady flow of air from below the water level can create bubbles of
uniform sizes. So an apparatus consisting of an injection syringe, an air pump, a water tank, and
tape was set up. The injection syringe was taped to the bottom of the tank using water proof
tapes, the other end of the syringe was connected to the air pump. The water tank was filled with
bubble solution mentioned in the following section. When the air pump is activated, air flows from
the pump to below the water surface. As long as the syringe is taped to the bottom, the rate of
the air flow remains constant, the size of bubbles remains the same. Hence, uniform bubbles were
created.
Shing-Tung Yau High School Physics Award 2015
7
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Figure 6: Apparatus adoped by this experiment
Cameras were employed for recording. They were placed above the water tank and its position
and focus fixed.
(a) iPad Mini
(b) Nikon D7000
Figure 7: Different Camera Settings
Shing-Tung Yau High School Physics Award 2015
8
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
2.2
Solution Formula
In order to observe grain boundaries’ movement, a bubble raft must be durable since structure
changing is a slow process.
2.2.1
Components
A stable bubble raft needs surfactants and stablelizers. The components were used following these
constrictions.
Surfactants are required for the solution to generate the film. Cetyltrimethylammonium bromide
(HDTMA) was adopted as the surfactant in this experiment.
Figure 8: Cetyltrimethylammonium bromide (HDTMA)
HDTMA is a cationic surfactant. It has a solubility of 13 g/L in 20o C water.
In order to stabilizes the bubble raft, glycerol was added.
Figure 9: Glycerol
Glycerol is soluble in water and stabilizes the film[31] by adhering to the soap film and forming
hydrogen bond with water to impede the evaporation of water molecules[32], ensuring the bubbles
last long enough for observation. It has a solubility of 500 g/L in in 20o C water.
The electrolyte sodium chloride (NaCl) was added for it reduces surface tension after it exceeds
a certain concentration (0.25%). It is also shown that the electrolyte has the effect of reducing
bubble size and increasing bubble stability through reducing the surface charge of SDS micelles
and “enhancing the structure of the adsorption monolayer and interfacial film”[33]. The solubility
of NaCl in 20o C water is 36.0 g/L.
Experiments demonstrated that HDMTA, while having a solubility of 13 g/L, can hardly dissolve
in water even though only small amount of it was added. Therefore ethanol (CH3 CH2 OH) was
added as a solubilizer, since HDTMA has a high solubility in ethanol. However, ethanol vaporizes
rapidly, which contributes to the bursting of bubbles, thus was abandoned in the final formula.
Shing-Tung Yau High School Physics Award 2015
9
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
2.2.2
Experimental Steps
Different ratios of components were tested to finalize the one that makes the longest lasting bubbles.
During each experiment, the air pump was turned on for 5 minutes, then switched off. The bubbles
generated were then carefully observed for approximately 30 minutes by eyes to test the stability of
the bubbles. Analysis of the observation helped make improvements on the solution formula. This
process was repeated to get the optimal bubble solution.
The depth of the solution was measured before each experiment and was kept in the range of
1.70cm-2.00cm. The bubbles were produced from the bottom of the tank. The size of the bubbles
was calculated by taking a photo of the bubble raft with a 1.00cm mark on the edge of the container.
With the number of bubbles produced in a given period of time counted, the speed air flow rate
for each experiment was calculated.
Figure 10: Bubble raft after air pump turned on for 5 minutes∗
Experiment Procedure:
1. Assemble the equipment as the picture shown.
2. Configure solution with the proposed formula, and put the solution into the tank.
3. Turn the air pump on for 5 minute.
4. Observe the bubble raft to determine if its stable enough.
5. Analyze the experiment to improve the formula, and repeat step 2 and 3 with the newly
proposed formula, until the solution is competent.
Environment Parameters:
• Room temperature: 26o C
• Water temperature: 25o C
∗
In this document, all figures demonstrating bubble rafts were adjusted to monochromic ones in order to increase
the clearance.
Shing-Tung Yau High School Physics Award 2015
10
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Table 1: Solution Formula Experiments
Experiment
#
1
∗
Distilled
HDMTA Glycerol Ethanol
water
(g)
(g)
(ml)
(mL)
500
0.50
48.00
0
NaCl
(g)
0.00
2
500
0.50
75.19
0
0.00
3
500
0.50
100.47
0
0.00
4
500
0.50
149.33
0
0.00
5
500
0.50
248.26
0
0.00
6
500
0.50
297.71
0
0.00
7
500
0.50
297.71
50
0.00
8
550∗
0.50
297.71
50
5.00
9
625∗
0.50
297.71
50
10.01
10
625∗
0.50
296.64
0
10.03
Observations
Bubbles burst as soon as
they are formed. Many
HDTMA particles lay insoluble on the bottom even
though its solubility is
13g/L at room temperature.
Most bubbles burst in the
first minute.
Insoluble
HDTMA particles lay on
the bottom.
Most bubbles can’t last
longer than 3 minutes. Insoluble HDTMA particles
lay on the bottom.
Bubbles bursting too fast,
but a few can last to
about 5 minutes. Insoluble HDTMA particles lay
on the bottom.
Lower bursting rate, but
most bubbles can’t last
more than 5 minutes.
Many
HDTMA
particles lay insoluble on the
bottom.
Lower bursting rate, but
most bubbles can’t last
more than 10 minutes. A
layer of insoluble glycerol
is observed at the bottom
of the tank along with the
HDTMA particles.
The HDTMA particles and
the layer of glycerol disappear. Most Bubbles can
not last for more than 5
minutes.
No insoluble substances.
Bubbles are very stable in
the first 5 minutes, but
most bubbles burst around
the 10th minute.
No insoluble substances.
Few bubbles burst in the
beginning.
After 10-15
minutes, large amount of
bubbles begin to burst.
No insoluble substances.
Most bubbles can last for
about 30 minutes; about
half of the bubbles can last
for more than 1 hour.
Analysis
Not
enough
bubble
stabilizer-glycerol
Not
enough
bubble
stabilizer-glycerol
Not
enough
bubble
stabilizer-glycerol
Not
enough
bubble
stabilizer-glycerol
Not
enough
bubble
stabilizer-glycerol,
but
the solubility of glycerol
at room temperature is
reached. We want to see
the effect of insoluble
glycerol in water.
Not
enough
bubble
stabilizer-glycerol, so we
need substance that can
increase the solubility of
glycerol in water-ethanol.
Another bubble stabilizer
(NaCl) is needed because
the newly dissolved glycerol does not exhibit a notable effect on bubble stability.
It’s proved that NaCl can
increase bubbles stability.
So higher stability could
be achieved by adding
more NaCl.
Evaporation of ethanol
may be the cause of bubble bursting, so we should
remove the ethanol.
The formula basically
satisfies our purpose.
The additional distilled water is used to dissolve the NaCl particles.
Shing-Tung Yau High School Physics Award 2015
11
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Ron Hipschman suggested adding commercial bubble solution to prolong the duration[29]. In a
experiment with commercial detergent, it is found that although it had some effect on strengthening
the bubbles, the bubble size decreased over time significantly, which is not preferred by this research.
The reason was not investigated in detail.
2.2.3
Final Formula
The final formula contains:
• Distilled Water
625ml
• HDTMA
0.50g
• NaCl
10.00g
• Glycerol
300.00g
This formula worked well in this experiment. The bubbles were long-lasting that sustained a
few observations over 1.5 hours long.
2.3
Air Flow Rate Controlling
To make sure that the bubble size is always constant during each experiment, the flow rate of air
is kept constant, measured and recorded.
To determine the rate of air flow, the injection syringe was adjusted to produce bubbles at a
suitable rate, not so fast as the bubbles stack up to form three dimension rafts, and not so slow
as the bubbles burst before a large enough grain is formed. The place of the pistol in the velocity
control was marked by a tape so the air flow is kept at constant rate in every following experiments.
The power output of the air pump is also kept at constant. A tape marked one centimeter was
taped on the edge of the tank as a length reference. The needle was taped at the bottom of the
tank and the air pump connected to an electric outlet. The device started to produce bubbles when
the switch is turned on. When the bubble produced are at the same size and at a constant rate,
the stopwatch was started and the bubbles produced starting this point gathered in a single raft
away from the other bubble rafts. The stopwatch was stopped after a few seconds and the later
produced bubbles were kept from approaching the raft. A picture of the bubble raft with the one
centimeter reference in it was taken horizontally above the liquid surface. The picture is zoomed in
and out so the reference on the picture is exactly one or two centimeters long. Then the length of
a single straight line of bubbles was measured and divided by the number of the bubbles to get the
radius of one single bubble. The number of the bubbles in the raft produced within the given time
was counted. With bubbles considered as perfect spheres, the volume of the air flow per second
can be calculated. The variation in the air flow rate and bubble size is small and acceptable for
the purpose of this research.
Table 2: Air Flow Rate Experiments
Experiment #
1
2
3
4
Depth (cm)
1.9
1.9
1.9
1.9
Bubble Numbers
60
49
39
33
Shing-Tung Yau High School Physics Award 2015
Time (s)
4.69
6
4.94
4.18
Radius (cm)
0.0656
0.06875
0.065
0.0625
Flow Rate (cm3 /s)
0.015
0.0111
0.00908
0.00807
12
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
3
3.1
Grain Boundary Migration and the Effect of Misorientation Angles
Creating Grain Boundaries
The goal of this research is to observe the movement of the grain boundaries within the bubble
rafts and its connection to the angles between the two grains and the existence of irregular bubbles.
Grain boundaries appear naturally within bubble grains. However, to gain further control over the
number of grain boundaries and the misorientation angles, artificial grain boundaries are preferred.
To create a grain boundary, the following method was adopted, as shown in Figure 11.
Two bubble rafts were created each around a stick.
Figure 11: Bubble rafts around sticks.
Because of the adhesive force of water molecules, the bubble film adheres itself to the stick,
allowing the bubble rafts to move around with the stick. External forces were applied to eliminate
the defects like grain boundaries and irregularities within the grains. Then the rafts were brought
to contact at a certain angle to form grain boundary. During the rapid recrystallization period, the
grain will recrystallize in an attempt to eliminate the grain boundary, but the first few movements
all occur rapidly and last only for seconds, as shown in Figure 12, the grain boundary moved slightly
to the left within seconds after the grains were combined.
Shing-Tung Yau High School Physics Award 2015
13
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) The grains are first brought to contact
(b) Several seconds after the grains are
brought to contact
Figure 12: Rapid recrystallizing period
After the first period of recrystallization, the shape and the position of the grain boundary are
slightly altered. The second period of recrystallization follows and spans a larger time interval.
The second period consists the major migration which this research mainly focus on. The grain
was put under a camera and observed by the naked eyes. The observation lasted until the grain
boundaries were completely gone; the overall configurations became perfectly uniform. To make
sure no external disturbance influence the natural process of migration, careful measures were taken
to prevent disturbance. The camera was placed above the water tank and its position and focus
were fixed. Environmental parameters were strictly controlled so no external forces would interfere
with the experiments. The experiment was terminated once the raft hit the side of the container.
Then the video was analyzed: the angles between the grains were measured, the mechanism of the
migration studied.
3.2
The General Mechanism of Grain Boundary Elimination
The grain boundaries can be categorized into high angle grain boundaries (HAGB) and low angle
grain boundaries (LAGB) by the misorientation angles across the boundaries. Both kinds can
be created by the rotation of one of the two grains before made into contact. High angle grain
boundaries are defined as boundaries with angles higher than 15 degrees; low angle grain boundaries
are defined as boundaries with angles lower than 10 degrees. However, this categorization is mainly
general, only as means to help clarify several different mechanisms of the grain boundary migration
(GBM).
The natural bursting of bubbles is unavoidable. The former mentioned formula is dedicated
to stabilize the bubbles so the chances of bursting is reduced to the utmost. Bubbles also burst
because they experience force. Strains in the raft exert great force on some of the bubbles, causing
them to burst, as demonstrated in Figure 13. These kind of bursting relieve the strains by creating
vacancies and dislocations, shown in Figure 14. Either way, the bursting of bubbles fuels the motion
of further movements. As experiment demonstrated, the bursting of bubbles has crucial significance
in the migration process. Usually when bubbles within the raft burst, the force generated initiates
the motion of dislocations. It is the motion of dislocations that further assists the migration of the
entire grain boundary.
Shing-Tung Yau High School Physics Award 2015
14
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) The grain on the right of the grain boundary is
closely packed
(b) A bubble on the right side of the grain boundary
bursted.
Figure 13: A bubble in the middle of raft bursted.
Shing-Tung Yau High School Physics Award 2015
15
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) Vacancies are present in the grain
(b) A bubble bursted next to the lowest vacancy
(c) The bursting of bubble initiated dislocation to move above the vacancy
(d) The movement of dislocation filled the vacancy above the bursted bubble
(e) The vacancies are filled
Figure 14: The bursting of bubbles initiates the movement of dislocations.
One basic assumption for the experiment is that the grain boundary will always be eliminated
under ideal circumstances. Infinite time interval, zero disturbance and rigid unchanging bubbles are
considered to be ideal. The assumption cannot be proved by experiments to be absolutely correct
or incorrect. However, because of the irregular configuration of bubbles at the grain boundary, the
energy of grain boundary is higher than that of any other parts within the grain. Theoretically
speaking, the tendency to become one perfect grain will drive the grain to reduce this imbalance
of energy by eliminating the grain boundary. In the experiments conducted for this research, most
of the grain boundaries indeed were totally eliminated. Few exceptions occurred when external
Shing-Tung Yau High School Physics Award 2015
16
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
disturbance was at work.
Based on the previous assumption, the grain boundary migration is actually grain boundary
elimination. In a uniform raft, the grain boundary is certain to migrate. According to our observation, the grain boundary cannot voluntarily move. The bursting of bubbles initiates the dislocations
which then move the grain boundaries. Whatever the angle is, some distinguished common features can be observed through the experiment. Because artificially created boundaries were mostly
straight. However, as they migrate, these boundaries usually adopt a zig-zag shape or the shape of
broken lines, shown in Figure 15.
(a) Grain boundary of straight shape
(b) Grain boundary of zig-zag shape
Figure 15: After five minutes grain boundary changed from straight shape to zig-zag shape
The broken line shape is postulated to form because of one of the mechanisms of GBM that
will be mentioned later. Grain boundaries do not adopt one single shape during the migration:
they keep altering their shapes and positions. Broken line still takes the dominant position in the
shapes of the boundaries. The time until the boundary is completely eliminated varies with every
experiment. If the bubbles burst at the right place at the right time, the process is very rapid,
taking only seconds to finish. If the bubbles are stable, the whole process can take up to hours to
finish. Also, the direction of which the grain boundary migrates towards depends on the bursting
of bubbles and the direction of the dislocations.
3.3
The Elimination Mechanism of High Angle Grain Boundaries and Calculations
The high angle grain boundaries generally migrate faster than low angle grain boundaries. The
basic movement of high angle grain boundaries is steady until they reach the edge of the grain,
then completely disappear.
Shing-Tung Yau High School Physics Award 2015
17
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) Grain boundary is in the middle of the raft
(b) Grain boundary moves to the upper left corner
(c) Grain boundary is still in the upper left corner
(d) Grain boundary moves closer to the edge
Figure 16: Grain boundary migrates to the edge of the raft
From a macroscopic view, one of the grains in the raft twisted and shifted, oriented around
the grain boundary until the orientation of the two grains fitted perfectly, eliminating the grain
boundary, as shown in Figure 16. This phenomenon usually occurs when a series of bubbles burst
around the grain boundary within a very short time, and the process usually last for only a few
seconds. During further analysis, it is found that because of the bursting of bubbles, dislocations
are initiated from the grain boundary radiating towards every directions. Every time a dislocation
moves, it changes the configuration of a little grain of bubbles nearby, making this grain change its
orientation to that of the other grain. Each dislocation changes a little grain in the bigger grain,
hence moving part of the grain boundary further into the other grain.
The zig-zag and broken line pattern can also be explained. The dislocations caused by the series
of bubbles bursting change little grains one at a time. After the period of rapid bubbles bursting
is terminated, the grain boundary had already migrated a considerable distance. If the process
is strong enough, the whole boundary may be eliminated. If not, it appears that the previously
straight boundary is broken to a zig-zag shape.
Without the rapid bursting period, the process for high angle grain boundary elimination is
Shing-Tung Yau High School Physics Award 2015
18
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
basically the same. The process is also fueled by the bursting of bubbles, but at a slower rate. Fewer
bubbles burst at longer time intervals. However, the steady bursting of bubbles also accumulate
dislocations to change the position and the shape of the grain boundary. From the macroscopic view,
the grain boundary is migrating steadily across the entire grain until it moves to the edge of the grain
and finally disappears. Most of the time the migration of high angle grain boundaries lasts only
for a short time. The grain quickly readjusts itself to be perfect again. It is also found that during
the migration, the angles of the grain boundaries reduce to around 15 degrees(shown in Figure 17),
close to the critical angle of low angle grain boundary, which corresponding to previous research
by Rollett[21]. Grain boundaries need to reduce their energy. The greater the misorientation, the
greater the boundary energy. As a result, to reduce the energy of grain boundaries and to make the
task of GBM easier, grain boundaries will first reduce their angle to around 15 degrees. The energy
of grain boundaries reduces as they migrate towards elimination. This postulation is supported by
further calculation.
(a) Before
(b) After
Figure 17: The angle of the grain boundary reduces to around 15 degrees
Shing-Tung Yau High School Physics Award 2015
19
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
The following equations are used in our calculation to support the observations. The representation of each letter is listed as well. These equations were obtained from F. J. Humphreys and M.
Hatherly[34]; Dr. Helmut Föll[35].
θ ≈
γs =
γ0 =
A =
γ =
b ≈
ν ≈
b
= tan α
h
γ0 θ(A − ln θ) (For LAGB)
Gb
4π(1 − ν)
b
1 + ln
2πr0
θ
θ
γ0
1 − ln
(For HAGB)
θm
θm
r0 †
1‡
3
θ ≡ The ratio of the Burgers vector to the[14]
spacing of the dislocation core, or the
tangent value of the misorientation anlge
b ≡ Burgers Vector
h ≡ Spacing of the dislocation core
γs ≡ Energy of the grain boundary
G ≡ Shear modulus
ν ≡ Poisson’s ratio
r0 ≡ Radius of dislocation core
θm ≡ Normalized parameter
α ≡ Misorientation angle
d ≡ Diameter of a bubble
According to Dr. Helmut Föll, “[w]e equate r0 with the magnitude of the Burgers vector, |~b|. This makes sense
because the Burgers vector is a direct measure of the ”strength” of a dislocation, i.e. the strength of the displacement
in the core region.”
‡
According to Dr. Helmut Föll, ν ≈ 31
†
Shing-Tung Yau High School Physics Award 2015
20
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Specific moments from videos are selected.
At the beginning of the experiment:
α = 20o
b = 3d
θ
θ Gb
1 − ln
4π(1 − ν) θm
θm b=3d,
γ =
tan 20o
ν= 13 , θ=tan α, θm =tan 20o
tan 20o
G × 3d
×
1 − ln
tan 15o
4π(1 − 13 ) tan 15o
= 0.3374 · d · G
=
A random moment in the middle of the experiment:
α = 15o
b = 2d
b
Gb
θ(1 + ln
4π(1 − ν)
2πr0
γ =
− ln θ)
b=r0 =2d, ν= 31 , θ=tan α
G × 2d
2d
o
=
1 × tan 15 × (1 + ln 2π × 2d
4π(1 − 3 )
= 0.003064 · d · G
− ln(tan 15o ))
Right before the grain boundary disappears:
α = 15o
b = 1d
= d
Gb
b
θ(1 + ln
4π(1 − ν)
2πr0
γ =
− ln θ)
b=r0 =d, ν= 13 , θ=tan α
G×d
d
o
=
1 × tan 15 × (1 + ln 2π × d
4π(1 − 3 )
= 0.001532 · d · G
− ln(tan 15o ))
Thus proved our hypothesis of high angle grain boundary reduces the angle to approximately
15o to reduce energy, which a crucial step in eliminating the grain boundary.
Shing-Tung Yau High School Physics Award 2015
21
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
3.4
The Elimination Mechanism of Low Angle Grain Boundaries and Calculations
For grain boundaries of lower angles, the movement is much slower. Low angle grain boundaries
have boundary angles of lower than 15 degrees, and can be seen as a series of dislocations and
strain, shown in Figure 18.
Figure 18: Low angle grain boundary consists only of dislocations
The dislocations that made up the boundary are all connected. This strain and dislocation
grain boundaries undergo GBM in a way similar to that of high angle grain boundaries. The basic
elimination method is approximately the same, with bubbles bursting and dislocations running
around, altering the shape and the position of the boundary. The angle is reduced slowly and
finally towards disappearance, as shown in Figure 19.
Shing-Tung Yau High School Physics Award 2015
22
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) The angle across the grain boundary is approximately 10 degrees
(b) The angle is reduced to about 5 degrees
Figure 19: The angle of LAGB is reduced
For low angle grain boundaries with angles lower than 10 degrees, the features of grain boundary
are not as obvious. This kind of grain boundaries are considered as strains, as shown in Figure
20, with no distinguished sighs of misorientations and dislocations. To relieve the strains, the
bubbles at which the strains are the greatest burst under stress. The vacancies created give room
to movement. The energy released from the bursted bubbles initiates the movements of dislocations,
which run along the rows filling vacancies and relieving strains. If the bubbles burst at the right
place, the strains can be completely released and the boundary completely eliminated very quickly.
Figure 20: Grain boundary consists only of strains
Shing-Tung Yau High School Physics Award 2015
23
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Although different in appearances, the two types of low angle grain boundaries both follow the
same elimination mechanism, by which they reduce their angle to the utmost. Energy is reduced
in the process, as proved by further calculation.
In an experiment, for example, in the beginning:
α = 10o
b = 3d
b
Gb
θ(1 + ln
4π(1 − ν)
2πr0
γ =
− ln θ)
b=r0 =3d, ν= 31 , θ=tan α
3d
G × 3d
o
=
1 × tan 10 × (1 + ln 2π × 3d
4π(1 − 3 )
= 0.178 · d · G
− ln(tan 10o ))
Before the elimination of grain boundary:
α = 5o
b = 2d
Gb
b
θ(1 + ln
4π(1 − ν)
2πr0
γ =
− ln θ)
b=r0 =2d, ν= 13 , θ=tan α
G × 2d
2d
o
1 × tan 5 × (1 + ln 2π × 2d
4π(1 − 3 )
= 0.104 · d · G
=
− ln(tan 5o ))
The reduction in energy supports the mechanism of low angle grain boundary elimination quantitatively.
Shing-Tung Yau High School Physics Award 2015
24
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
3.5
The Behavior of Bubble Raft During GBM
During the grain boundary migration process, the raft as a whole also experiences a dynamic
change. Because of the movement of dislocations, strain form across the raft. Most of the time the
strains are released by bubbles bursting and dislocations running. Because the size of the grain is
rather small, the grain rotates due to the stress, as shown in Figure 21.
(a) Before
(b) After
Figure 21: The rotation of bubble raft
The grain rotates to relieve the strains. When the grain boundary is stable and no strains exist,
the bubble raft stops rotating and stands still.
3.6
After the Elimination of Grain Boundaries
The bubble raft becomes perfectly uniform after the GBM process. Both strains and grain boundaries are eliminated at this point. The unavoidable bursting of random bubbles creates vacancies
within the raft, causing bubbles in nearby rows to fill the vacancies, creating dislocations. The
irregularities adjust themselves rather quickly. After the elimination the greatest defect – grain
boundary, point and linear defects emerge randomly and disappear quickly.
Shing-Tung Yau High School Physics Award 2015
25
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
4
Effect of Irregularly Sized Bubbles on GBM
The study of the relationship between irregular size bubbles and crystal defects in bubble raft originated from Bragg and Nye, who stated that irregular bubbles generally fit into grain boundaries[17].
However, this is not usually observed in the experiment.
To study the effect of impurity bubble on grain boundary migration, the impurity bubble is
created separately and implanted into an ordered grain before the grain is combined with another
grain. A larger irregular bubble is produced by a hypodermic syringe and a smaller one is created
by violent disturbance of a bubble raft. The irregular bubble is then moved to a perfect grain
with tweezers while the newly produced bubbles adhere around it and incorporate it into the grain.
Pre-connecting disturbance is employed to control the variables and ensure that the point defect
of the irregular bubble and the planar defect grain boundary are the only defects present in the
whole raft.
4.1
Small Impurity Bubbles
†
The small impurity bubble can be either interstitial or substitutional. Interstitial impurity bubble
is so small that it is squeezed into the interstitial void among the regular bubbles. Substitutional
impurity bubble, despite its different size, simply takes the place of a regular bubble in the grain[36].
Since smaller bubbles are generally more rigid[37], they usually do not burst, and the experiment
proves that the final destination of small bubbles is to become interstitial. Smaller bubbles sometimes fit in the grain boundary, but only in an intermediate state, while the final equilibrium state
of lowest energy is obtained when small bubbles are stabilized in interstitial voids. In a short period
of time, when the bubble size can be assumed to be constant, point defect may interact with other
defects, such as linear defect and planar defect to lower Gibbs energy[38].
The segregation of impurity bubbles at grain boundary, that is, interaction between point and
planar defects, is a common phenomenon. For example, in the following experiment, the small
bubble is initially interstitially implanted near the high angle grain boundary (about 20o ) while
still deforming the uniform structure of the grain as shown in Figure 22(a). In the first 5 minutes,
the shape of the boundary transforms to accommodate the small bubble in it. Figure 22(b) shows
the impurity on the grain boundary at the 5th minute. From the 5th to the 20th minute, as in
Figure 22(c), the grain boundary adjusts its shape while keeping the small bubble on it. However,
after the 20th minute, the grain boundary migrates away from the impurity bubble because the
diameter of the small bubble has shrunk from half of that of the normal bubble to 0.4. At the 55th
minute in Figure 22(d), the grain boundary completely leaves the impurity bubble, whose diameter
is already 0.2 of the regular ones. At this moment, the impurity bubble fits in the interstitial void
perfectly, only slightly deforming the bubbles beside it and not disrupting the fcc crystal structure.
From the 55th minute, the grain boundary migrates continuously towards the other end of the raft
through slipping of dislocations while maintaining the high angle, with the impurity bubble locked
in its place and shrinking to a size incomparable to the size of regular bubbles(Figure 22(e)).
†
The figures in this section were magnified to present the small impurities more obviously.
Shing-Tung Yau High School Physics Award 2015
26
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) t=0min, the small impurity bubble is interstitially implanted near the grain boundary.
(b) t=5min, the grain boundary migrates to accomodate the small impurity.
Shing-Tung Yau High School Physics Award 2015
27
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(c) t=20min, the grain boundary changes shape while keeping the small impurity on it.
(d) t=55min, the grain boundary leaves the small impurity and the impurity has reduced
its diameter significantly to 0.2 of the diameter of the regular bubbles.
Shing-Tung Yau High School Physics Award 2015
28
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(e) t=65min, the small impurity perfectly fits in the interstice and does
not affect the fcc structure. The high angle grain boundary slowly
migrates out away from the impurity.
Figure 22
Interaction between point defects and linear defects is also observed as an intermediate state. In
the orderly aligned grain, the impurity bubble interacts with the nearby vacancy and temporarily
gets loose inside the vacancy, acquiring a state of relatively low energy. Then the impurity enclosing
vacancy further interacts with several vacancies. In an exemplary experiment, in the first 20
minutes, the separate point defects and dislocations evolve into an array of dislocations, that is,
a low angle grain boundary while the original, artificially-created low angle grain boundary is
eliminated. This process is recorded in Figure 23 (a) through (c). This state is non-equilibrium
grain boundary segregation, in which the impurity bubble interacts with an excess of vacancies
around the boundary[38]. This interaction drives dislocations, which moves the small bubble once
more into the interstice, where the gas leak becomes prominent owing to the strong elastic force
of the deformation of closely packed bubbles. As the diameter of the small bubble reduces from
about 5/9 to 2/7 of that of the regular bubbles, the small bubble no longer affects the alignment
and the regular bubbles form an ordered grain, as in Figure 23(d).
Shing-Tung Yau High School Physics Award 2015
29
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) t=0min, The small impurity interacts with the nearby vacancy.
(b) t=10min, the impurity temporarily gets freed in the vacant
space, which is a relatively stable state.
Shing-Tung Yau High School Physics Award 2015
30
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(c) t=20min, the impurity interacts with vacancies on a line to form an array of
dislocations.
(d) t=65min, further slipping of dislocations positions the impurity in an interstice and compresses it to a size 2/7 of the regular bubbles, making the structure
ordered with an interstitial point defect.
Figure 23
Shing-Tung Yau High School Physics Award 2015
31
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
The originally substitutional impurity also seeks a low energy state in the interstice. In the
beginning of an experiment, for example, as in Figure 24(a), the substitutional small bubble is
initially about 5/7 of the size of regular bubbles in diameter and causes a slight dislocation around
it. Through extensive slipping, the small bubble is finally squeezed into the interstitial void at the
65th minute, which is shown in Figure 24(b), with a diameter about 0.4 of the normal ones. After
it is fit into the interstice, slipping process becomes more rapid and the grain quickly adjusts to
the perfect alignment.
Gas leaking of small impurity bubble is a notable phenomenon in all the experiments. The
surrounding regular bubbles experience elastic deformation and thus exert pressure on the impurity, which is responsible for its shrinkage. The shrinking process releases energy, verifying that
interstitial point defect is a state of lower energy than substitutional point defect or grain boundary
segregation of impurities. This state possesses the lowest energy for smaller impurities, which is
much lower than the boundary energy so that grain boundary migration is not affected by interstitial point defect.
(a) t=0min, the substitutional impurity causes a dislocation in the bubble raft.
Shing-Tung Yau High School Physics Award 2015
32
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(b) t=65 min, the originally substitutional impurity is compressed and
fit into a interstitial void, allowing the structure to be quickly adjusted
to an ordered state via slipping.
Figure 24
4.2
Large Impunity Bubbles
Large impurity bubbles can only be substitutional. Similar to small impurities, big bubbles also
interact with the grain boundary and nearby vacancies.
In an example of the interaction between large impurity and grain boundary, the grain boundary
migrates toward the big bubble and keeps the impurity on it while further transforming its shape
during the interaction, which is displayed in Figure 25 (a) through (c). The frequent slipping of
dislocations and constantly changing shape of the grain boundary exert a great pressure on the big
bubble. At the 2nd minute in Figure 25(d), when the big bubble is tightly packed in the regular
bubbles, it bursts and creates considerable air flow that causes the bursting of nearby bubbles.
The huge vacancy initiates a rapid recrystallization process, fueling the subsequent rapid slipping
and grain boundary migration. After 10 minutes, the grain boundary is completely eliminated by
slipping of dislocations as shown in Figure 25(e). .
Shing-Tung Yau High School Physics Award 2015
33
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
(a) t=0s, the grain boundary starts to migrate towards (b) t=7s, the gran boundary passes through the imputhe big impurity.
rity.
(c) The grain boundary stays on the impurity for about (d) t=2min 5s, the big impurity bursts under the pres2 minutes while changing its shape.
sure of the transforming grain boundary, creating air
flow that forces several bubbles to burst. A big hole is
created at the grain boundary.
(e) t=10min 1s, the bursting initiates extensive slipping
motion. The raft finally gains an ordered structure.
Figure 25
Shing-Tung Yau High School Physics Award 2015
34
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
In some cases, the interaction between impurity and grain boundary drives the boundary away
from the impurity. In an example, while several bubbles burst around the impurity immediately
after the implantation, slipping occurs and moves the surrounding bubbles to positions in fcc
structure. The tendency of the tightly packed bubbles to resume their positions deforms the big
bubble, finally causing it to burst. The busting and the quick recrystallization form the perfect
alignment around the original position of the impurity. Meanwhile, the high angle grain boundary
slowly and continuously migrates toward the other end during the whole process in Figure 26 (a)
through (c). Grain boundary migration is not influenced by the bursting of impurity bubble and
finally migrates out of the other end of the raft through slipping while maintaining its high angle.
(a) The big impurity bubble is inserted and (b) The grain boundary is further away
the grain boundary is repelled towards the from the irregular bubble, which is
other direction.
tightly surrounded by regular bubbles
trying to resume the fcc structure.
(c) The big bubble bursts under the
pressure and the grain boundary finally
migrates out of the raft with its high
angle maintained.
Figure 26
Shing-Tung Yau High School Physics Award 2015
35
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Interaction between impurity and vacancy is a common step in obtaining a relatively low energy
state. In an experiment, for instance, a vacancy emerges beside the impurity bubble, starting a
slipping that transfers the vacant space to the impurity bubble(Figure27(a)). In Figure 27(b), the
vacant space gives room for the big bubble to resume its original shape, and the alleviation of
deformation brings a relatively stable state for the impurity bubble. However, the lowest possible
energy state is achieved when the normal bubbles display fcc alignment, so further slipping caused
the impurity to be surrounded by tightly packed bubbles in Figure 28(c). The stable state is finally
obtained through bursting of the big bubble, which sends dislocations that transform the grain
boundary to dislocations and strains and finally eliminate it as in Figure 28(d).
(a) A bubble bursts just beside the big irregular bubble, (b) Slipping positions the big impurity inside the vawhich creates a vacancy for the big bubble to interact cancy, temporarily alleviating the deformation.
with.
(c) Dislocation movement once again fits the impurity (d) After the big bubble bursts, slipping of dislocations
inside tightly packed bubbles that exert pressure to fi- adjusts the structure to fcc arrangement.
nally make it burst.
Figure 27
Shing-Tung Yau High School Physics Award 2015
36
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Big impurity bubbles often experience an intermediate state of relative stability when the big
impurity substitutes an exact number of normal bubbles. In the following experiment, through
movement of dislocations, the normal bubbles are positioned tightly around the irregular bubble,
slightly deforming the big bubble to make it replace exactly three regular bubbles, which is shown in
Figure 28(a). This arrangement lasts for more than 30 minutes, which is a relatively long period of
time, indicating its state of low energy and stability. However, given enough time, the big bubble
eventually bursts under pressure, which occurs in this experiment at the 31st minute in Figure
28(b). The grain boundary consistently migrates out of the raft and disappear in Figure 28(c).
It can be concluded that big bubbles generally does not interfere with the migration of grain
boundary. Unlike small impurities, the big bubbles are less rigid and thus easy to deform. When
the deformation reaches the film’s elastic limit, the big bubble bursts, which is inevitable since
the lowest energy state is obtained with perfect fcc structure. The bursting of big impurity is an
essential step of recrystallization, since the bursting blows up other normal bubbles nearby, leaving
a big vacancy that initiates extensive dislocations throughout the raft. Usually, the recrystallization
process becomes more rapid after the big impurity is eliminated.
(a) t=0min, the relatively stable state where the (b) t=31min 51sec, the big impurity
big impurity replaces exactly three regular bubbles bursts and the grain boundary is migrating out of the raft.
(c) The bubble raft is adjusted to
fcc structure.
Figure 28
Shing-Tung Yau High School Physics Award 2015
37
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
4.3
Comparison with Findings in Metals
Some of our conclusions are compatible with observations on impurity-introduced metals. The
stable state of interstitial point defects is proved to be true by cohesive energy calculations, which
points out that substitutional point defects are on a higher energy level[39]. The phenomenon
that diffusion of substitutional impurities in metals augments transport at grain boundaries is
also observed in our experiment[40]. These observations, however, are not completely consistent
with studies on polycrystalline materials such as metals and alloys. Most studies on the grain
boundary segregation of metals agree that the impurity density at the grain boundary tend to
increase[40][41][42] and impurities are immobilized at the grain boundaries[40]. Interstitial diffusion is not observed in this experiments while the interstitial transport of hydrogen across grain
boundaries is prominent in metals[40]. However, it is also pointed out that the degree of diffusion
is dependent on the physical and chemical properties of the impurity and its bonding with the solvent, with some more soluble and transportable and others more insoluble and immobile[40][42][43].
Calculations have demonstrated that the tendency of grain boundary segregation increases with
decreasing cohesive energy of the solute[42], which possibly explains the lack of segregation phenomenon with the bubble raft.
Shing-Tung Yau High School Physics Award 2015
38
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
5
Conclusion
Grain boundary migration is a complex process affected by many factors. In an ideal raft of
no defects other than the grain boundaries, vacancies and dislocations emerge and disappear to
alleviate strain and promote the movement of grain boundaries. For all types of boundaries, slipping
of dislocations caused by the bursting of stressed bubbles is the propellant for grain rotation and
boundary migration. As a dislocation moves, disturbance is sent through the whole raft, whose small
sections rotate slightly one by one to adjust position of bubbles, angle of misorientation, and shape
of grain boundary. High angle grain boundaries tend to reduce their angle of misorientation while
still maintaining above 15o as they migrate. Low angle grain boundaries, or arrays of dislocations
and strain are easily driven out by the bursting of a compressed bubble and slipping that rotates
the whole grain into fcc arrangement. Whatever the angles of grain boundaries, they all tend to
reduce their angles, while HAGB still remain HAGB and LAGB remain LAGB, which is proved to
be energetically favorable by calculation.
Irregular bubbles generally do not interfere with the migration of grain boundaries although
they may interact with vancancies, dislocations and boundaries as intermediate steps to obtain a
relatively low energy state. Overall, the diffusion at the boundary and the dislocation movement
exert great pressure on the impurities, finally positioning small impurities in interstitial voids and
compressing big bubbles to burst. The bubble raft gains a stable state of equilibrium with extremely small impurities as interstitial point defects while not disturbing the fcc alignment because
interstitial impurities occupy the lowest energy state of impurites.
In this experiment, only the simplest cases of defects are investigated, where there are only two
grains and one irregular bubble within the whole raft. The scope of the research greatly limits
its application to polycrystalline solids because various types of defects interact with one another
in a mutual existing environment, which is present in most materials. The bubble raft model
is also unrealistic in the bubbles’ ability to deform, diffuse gas, and burst, making it impossible
to investigate the effect of impurities with non-changing shape and volume while only general
conclusions, such as interstitial sites are favored over substitutional sites, can be made.
Still, the bubble raft model is proved to be useful in the simple cases. For further investigation,
more complex factors such as the combination of defects can be incorporated in a bubble raft.
Computer simulation, measurement of elastic forces, and more comprehensive calculation of energy
and entropy can bring more accurate data and allow the model to become more applicable.
Shing-Tung Yau High School Physics Award 2015
39
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
References
[1] T. J. Rupert, D. S. Gianola, Y. Gan, and K. J. Hemker, Experimental Observations of StressDriven Grain Boundary Migration. Science, December 18 2009, vol. 326, no. 1686-1690.
[2] Chapter 7: Dislocations and Strengthening Mechanisms.
Materials Science and Engineering.
University of Tennessee, Dept. of
[3] A. Tschope, E. Sommer, and R. Birringer, Grain Size-Dependent Electrical Conductivity of
Polycrystalline Cerium Oxide I. Experiments. FB Physik, Gebäude 43, Universität des Saarlandes, 66041 Saarbrücken, Germany: Elsevier Science, February 2001, vol. 139, no. 255-265.
[4] F. Suriano, M. Ferri, F. Moscatelli, F. Mancarella, L. Belsito, S. Solmi, A. Roncaglia, S. Frabboni, G. Gazzadi, and D. Narducci, Influence of Grain Size on the Thermoelectric Properties
of Polycrystalline Silicon Nanowires. Journal of Electronic Materials, January 2015, vol. 44,
no. 371-376.
[5] D. Li, Y. Li, S. Hu, X. Sun, and M. Khaleel, Predicting Thermal Conductivity Evolution of
Polycrystalline Materials Under Irradiation Using Multiscale Approach. Metallurgical and
Materials Transactions A, March 2012, vol. 43, no. 1060-1069.
[6] A. M. Limarga and D. R. Clarke, The Grain Size and Temperature Dependence of the Thermal
Conductivity of Polycrystalline, Tetragonal Yttria-Stabilized Zirconia. American Institute of
Physics, 2011, vol. 98, no. 211096.
[7] L. Zhou, X. Q. Wei, N. G. Zhou, and D. G. Li, Nanosie Effect in Grain Boundary Migration
of Copper. Acta Materialia, February 2004, vol. 17, no. 1.
[8] L. Liu, J. P. Liu, J. Zhang, W. Xia, J. Du, A. Yan, W. Li, and Z. Guo, The microstructure and
magnetic properties of anisotropic polycrystalline Nd2Fe14B nanoflakes prepared by surfactantassisted cryomilling. Materials Research Express, February 24th 2014, vol. 1, no. 1.
[9] J. A. Sethian and J. Wilkening, A Numerical Model of Stress Driven Grain Boundary Diffusion.
Journal of Computational Physics, March 20th 2003.
[10] M. S. Veshchunov, Modelling of Grain Growth Kinetics in Porous Ceramic Materials under
Normal and Irradiation Conditions. Materials, 2009, vol. 2, no. 1252-1287.
[11] A. A. de Laix and T. Vachaspati, Random Bubble Lattices. Physical Reviev D, January 25th
1999, vol. 59, no. 045017.
[12] M. F. Ashby, The Properties of Foams and Lattices.
no. 15-30.
Phil. Trans. R. Soc. A, 2006, vol. 364,
[13] A. Kraynik, D. Reinelt, F. van Swol, and S. Hilgenfeldt, “Foam structure and rheology: The
shape and feel of random soap froth “foam microrheology”,” Sandia National Labtoratories.
[14] F. S. University. Grain boundary lecture notes. [Online]. Available: www.eng.fsu.edu/∼kalu/
ema4225/lec notes/W16.ppt
Shing-Tung Yau High School Physics Award 2015
40
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
[15] A. Azizi, X. Zou, P. Ercius, Z. Zhang, A. L. E. as, N. stor Perea-Lo pez, G. Stone, M. Terrones,
B. I. Yakobson, and N. Alem, Dislocation Motion and Grain Boundary Migration in TwoDimensional Tungsten Disulphide. Nature Communications, September 9th 2014, vol. 5, no.
4867.
[16] U. of Maine, “Gbm theory.” [Online]. Available: http://www.geology.um.maine.edu/
geodynamics/Microdynamics/ellemodules/ELLE%20GBM%20webpage/HTML/theory.html
[17] L. Bragg and J. F. Nye, A Dynamical Model of a Crystal Structure, ser. A. Jstor, September
9th 1947, vol. 190, no. 1023.
[18] M. J. Bowick, L. Giomi, H. Shin, and C. K. Thomas, Bubble-Raft Model for a Paraboloidal
Crystal. Physics Review, 2008, vol. 77, no. 021602.
[19] Y. ISHIDA, Order Structures and Dislocations in Bubble Raft Grain Boundary.
Material Seience, 1972, vol. 7, no. 72-83.
Journal of
[20] M. Meier, “The bragg bubble raft film.”
[21] M. Winning and A. D. Rollett, Transition between low and high angle grain boundaries.
sevier, April 18th 2005, vol. 53, no. 2901–2907.
El-
[22] D. G. Brandon, The Structure of High-Angle Grain Boundaries. Acta Materialia, 1966, vol. 14,
no. 1479-1484.
[23] A. T. Lim, “Migration and mobility of low-angle grain boundaries,” p. 212, September 2012.
[24] H. Gleiter, Theory of Grain Boundary Migration Rate.
853-862.
Acta Materialia, 1969, vol. 17, no.
[25] A. H. King, Diffusion Induced Grain Boundary Migration.
1987, vol. 32, no. 4.
International Materials Reviews,
[26] M. L. Taheri, D. Molodov, G. Gottstein, and A. D. Rollett, Grain Boundary Mobility Under a
Stored-Energy Driving Force: a Comparison to Curvature-Driven Boundary Migration. Carl
Hanser Verlag, 2005, vol. 10.
[27] T. S. Cale and M. O. Bloomfield, Grain Boundary Migration: An Introduction.
Eng., 2007, vol. 76, no. 1-4.
Microelect.
[28] U. of Wisconsin. [Online]. Available: http://homepages.cae.wisc.edu/∼stone/bubble%20raft%
20movies.htm
[29] R. Hipschman. Bubbles. [Online]. Available: http://www.exploratorium.edu/ronh/bubbles/
bubbles.html
[30] A. van der Net, W. Drenckhan, D. Weaire, and S. Hutzler., The Crystal Structure of Bubbles
in the Wet Foam Limit. Soft Matter, 2006, no. 129-134.
[31] Glycerine basics and faq. [Online]. Available: http://soapbubble.wikia.com/wiki/Glycerine
Basics and FAQ
Shing-Tung Yau High School Physics Award 2015
41
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
[32] Glycerin in soap bubble mixtures. [Online]. Available: http://soapbubble.dk/english/science/
glycerin-in-soap-bubble-mixtures/
[33] Q. Xu, M. Nakajima, S. Ichikawa, N. Nakamura, P. Roy, H. Okadome, and T. Shiina, Effects
of Surfactant and Electrolyte Concentration on Bubble Formation and Stabilization. Journal
of Colloid and Interface Science, April 1st 2009, vol. 322, no. 208-214.
[34] F. J. Humphreys and M. Hatherly, Recrystallization and Related Annealing Phenomena, Second Edition. Pergamon, February 2nd 2004.
[35] 5.2.3 energy of a dislocation. [Online]. Available: http://www.tf.uni-kiel.de/matwis/amat/
def en/kap 5/backbone/r5 2 3.html
[36] M. J. Rahman, “Lecture 7: Defects in solids: Point defects and line defects,” no. 39-51.
[37] O. Busaryev, T. K. Dey, H. Wang, and Z. Ren, “Animating bubble interactions in a liquid
foam.”
[38] L. Pavel, Grain Boundary Segregation in Metals.
Springer Series in Materials Science, 2010.
[39] H. H. Kart and T. Cagin, The Effects of Boron Impurity Atoms on Nickel 5 (012) Grain
Boundary by First Principles Calculations. Journal of Achievements in Materials and Manufacturing Engineering, January 2008, vol. 30, no. 177-181.
[40] K. D. Hammond, L. Hu, D. Maroudas, and B. D. Wirth, Helium Impurity Transport on Grain
Boundaries: Enhanced or Inhibited? Europhysics Letters, June 17th 2015, vol. 110, no. 5.
[41] W. Thomas and B. Chalmers, The Segregation of Impurities to Grain Boundaries.
Metallurgica, January 1955, vol. 3, no. 17-21.
Acta
[42] V. Vitek and G. Wang, Atomic Structure of Grain Boundaries and Intergranular Segration.
Journal de Physique Colloques, 1982, vol. 43, no. 147-161.
[43] L. Heatherly and E. George, Grain-Boundary Segregation of Impurities in Iridium and Effects
on Mechanical Properties. Acta Materialia, January 22nd 2011, vol. 49, no. 289–298.
Shing-Tung Yau High School Physics Award 2015
42
The Effects of Misorientation Angle and Impurities on GBM Dynamics in 2-D Bubble Raft
Acknowledgments
The authors would like to thank Sir William Lawrence Bragg for his pioneer
research on bubble rafts that inspired this research, Mr. Guan Chun for his
guidance, and Mr. Chen Long for advice on the bubble solution formula.
Shing-Tung Yau High School Physics Award 2015
43