Download 2. Objectives - McMaster Materials Science and Engineering

1. Background and Discussion of General Theory
1.1. Overview of Objectives
This thesis will model the role of dislocations on the rate of particle coarsening during
spinodal decomposition and second phase precipitation in binary alloys. Our approach will couple
the thermodynamics and kinetics of phase transformations to the evolution of dislocation density.
This will enable us to treat dislocation effects on a scale comparable to grain boundaries, where the
kinetics of phase transformations takes place.
This is inherently a multi-stage problem spanning over five orders of magnitude in length
(10-8 m for grain boundaries versus 10-4 m for crystal grains). We will tackle this problem using the
phase field model [1]. This method treats phase and/or concentration boundaries in a system through
continuous fields that generate thin boundary layers (~10-8 m) which implicitly track the location
and physics of domain boundaries. Away from grain boundaries, these fields assume values of phase
or concentration consistent with the thermodynamics built into the phase-field model. The phase
and/or concentration field(s) of the phase-field method can be coupled to recent continuum field
models [11] that describe the dynamics of dislocation densities, through strain energy relaxation
methods.
Combining a multi-scale phase-field approach with high performance computing (HPC) will
enable numerically feasible simulations to be performed, which can be related, for the first time, to
experimental situations in analogous materials.
1.1.1 Industrial Significance
Industry in the 21st century is becoming increasingly more involved in fundamental research,
to find better ways of manufacturing materials. The gains associated with utilizing new age
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materials, which often contain exotic properties (e.g. phases, structure, etc.), can be economically
significant. The current implementation methods employed by industries often involve empirical
science and trial and error approaches. The problem with using such methods is that the process of
improvement is often hit-or-miss and the costs associated with continual trials can be unnecessarily
high. The use of advanced numerical methods is emerging as a new approach that promises to help
compliment and strategically guide the direction of future experiments in materials science.
1.1.2 Relation between Microstructure and Material Properties
The relationship between the microstructure of a material and its material properties has been
well established experimentally. In the simplest terms microstructure denotes grain size and shape.
In a more complicated (and accurate) case microstructure also involves grain boundaries and
dislocation density. The properties of the microstructure are grouped into 6 categories: mechanical
(elasticity, ductility and strength), electrical (electrical conductivity), thermal (heat capacity and
thermal conductivity), magnetic (responsiveness to magnetic fields), optical and deteriorative [13].
A comparative difference in the microstructure of two materials (i.e. different grain size, dislocation
density, etc.) can result in quite different properties. Although many advances in science and
engineering have helped to explain the casual relationship between structure and property, the
mechanisms establishing these relationships is poorly understood. Through a broader and more
fundamental understanding, many desired but currently unattainable properties may in the future
become quite simply achievable.
Examples range from the use of fluid flow modeling in the optimization of stirring in steel
processing [16], finite element calculations used in the design of new structural materials, to the
emergence of new multi-scale methods that model crystal growth or the electrostatics fields through
paper during xerographic printing [17]. An illustrative example of such methods occurs in the
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manufacture of commercial paper. Paper can involve very complex microstructures that influence
the quality of contact printing. The ability to predict the microstructure associated with different
compositions of paper and different processes of production may result in better, more specific print
uniformity properties. The prediction of the microstructure control could similarly span many
industries and would most likely result in attainment of targeted properties.
An area that is of critical importance to the metals industry involves the heat treatment of
alloys. This processing is typically associated with the growth of second phase particles and particle
coarsening. The kinetics of microstructure growth as a function of thermodynamic driving forces
has been well studied [1-10]. These studies, however, often neglect the critical role of dislocation
drag on growth kinetics. Indeed there have been conflicting reports in the literature claiming that
dislocations can both enhance and retard particle coarsening times. Our work will elucidate the
mechanisms of dislocation drag on particle coarsening. We believe that our contribution will be a
first step to predicting (over relevant microstructure length and time scales), the role that dislocation
mobility has on second phase coarsening and precipitation hardening.
1.2. Review of Thermodynamics of Microstructure Growth
1.2.1 Binary Alloy Systems
We review here the various microstructures that can emerge in binary alloys, and place our
work in context of a particular binary phase diagram which we will be studying in this thesis.
a. Eutectic
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In a eutectic system, as the material is quenched (cooled) energy is required nucleate
the phases so that precipitates can be formed. Most often the quenching through a eutectic is
the transformation of a liquid into to two solid phases, arranged in plate-like lamellae. [14]
b. Classic binary –Solidification
In this system energy is required to initiate precipitation by the formation of nuclei.
In the solidification process crystals are formed and as cooling and diffusion continues many
more crystals are produced. The end process involves the fusion of these crystals, where the
larger crystals/grains grow at the expense of the smaller ones, though an as yet not
understood co-operative process. [2,4-7,14]
c. Classic binary-Spinodal decomposition
The spinodal decomposition system involves the transformation of a solid into two
other solid phases. A paradigm is the Al-Zn system. This system contains both a second
order transition (at the critical point) and first order transitions (off the critical point), the
letter which requires nucleation of precipitates before the reaction starts. This type of phase
transformation involves what in called uphill diffusion. This diffusion occurs in alloys with
miscibility gaps. As the alloy is quenched it becomes unstable and thus small fluctuations in
composition will decrease the free energy of the system. The diffusion continues until the
final compositions of the phases are reached. [14]]. The thermodynamics and kinetics of this
alloy system will be the main focus of the Thesis work
1.2.2 Refinement of microstructure
Quite often obtaining the desired microstructure requires thermal and/or mechanical
treatments. These treatments help to refine the microstructure through the process of diffusion in
which atoms within the material are transported by atomic motion. Diffusion alters the arrangement
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of the atoms so as to change the properties of a material. Thermal or heat treatments, such as
annealing, increase the ductility of a material that was previously strain-hardening. Strain-hardening
is an example of a mechanical treatment, which is done to increase the strength of the material.
Either of these two treatments can be used for the improvement of a material. Process times for
these treatments depend critically on the diffusion times involved in each process. [13].
1.2.3 Effect of dislocations and stress
Grain growth involves the larger grains becoming larger and the smaller grains becoming
smaller until they completely eliminated. This process serves to decrease the total surface areas.
This decrease in the boundary area results in the decrease in total energy of the alloy and thus the
driving force for grain growth [10,11,13,14]. As the grains grow into each other, planes of atoms
within each grain on the boundaries meet. In order to meet “noble” state configurations (assuming
similar crystal structure) the atoms from different grains tend to bond to one another. For the lattice
planes to join exactly (achieve coherency) the two grains have to have the same atomic
configuration. Since distances between two adjacent atoms in one phase are different than that in
another, strain occurs. This is what is called coherency strains and it depicted in figure 1.
Figure 1: Coherency interface with slight mismatch [14]
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These strains increase the total energy of the system [14] and once the strains become high enough it
becomes energetically favorable for dislocations to form. The addition of these dislocations changes
the interface from coherent to semi-coherent (see Figure 2).
Figure 2: Semi-coherent interface. [14]
These dislocations will change the surface tension properties of the interface and thus alter
grain growth kinetics. [14].
In addition to the dislocations that emerge due to coherency strains, dislocations also emerge
when a metal (e.g. Al-Zn) is strain-hardened. The resultant stress causes strain within the alloy. The
appearance of dislocations within the alloy can
themselves cause strain due to incoherency of
lattice
planes In order to further decrease the strain the
dislocations migrate to interfaces of the phase or
the grain
boundaries through glide and climb. [14]. Figure
3 shows
the concentration of the dislocations at the grain
boundaries. A study of the mobility of these
Figure 3: Concentration of dislocations
at the grain boundaries [14]
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dislocations and their role in phase separation is the major focus of this project.
1.3 Phase Field Method
This section of the paper will serve as a brief outline of the phase-field method, a methodology
that will play a key role in the models we will be
using in
this thesis.
The phase field model originated as a means of
tracking
the movement of the interfaces between the two
phases
while its evolution couples to the physics of free
surface
kinetics and concentration (and/or heat) diffusion.
In general “a phase field is a local order
parameter that distinguishes a broken symmetry
Figure 4: Al-Zn phase diagram
between two distinct phases”. In the case of solidification case mentioned above the phase field
tracks the physics of liquid/solid or solid/solid phase boundary motions as a quench proceeds into
the solid liquid phase region of the phase diagram (See Figure 4- line 1). In areas where
precipitation (Figure 4- line 2) or spinodal decomposition occurs (Figure 4-line 3), the phase field
tracks differences in concentration between two phases.
Variois researchers have aplied the phase field
model
to made advancements of our understanding of the
kinetics
of solidification and eutectic phase transformations [1-9].
A
recent breakthrough in this methodology [7,18,19] allows
parameters of the phase-field model can now be set so as
quantitatively capture the correct physics of free-
to

Figure 5: Order parameter
across interface
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boundary kinetics (i.e, solute/heat diffusion in bulk phases + flux conservation at free boundaries +
Gibb’s Thomson curvature effects at free surfaces).
In case of alloy transformations, the starting point of the phase field model is the development
of a phenomenological free energy of the form
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F   {Wc C  W   f ( , C , T )}dV
2
(1)
The function f is designed to capture the bulk thermodynamics of the system. The gradient terms
capture the surface tension energy created between phases of different order (liquid/solid) or
different concentration Equations of motion for the phase, c concentration are derived by a suitable
dissipational dynamics derived from Eq. (1).
Ref [10] was the first to extend the phase field concept to examine the phase separation
process in the presence of elastic strains (coherent strains), which are produced by quenching
through the spinodal. Ref[11] extends this work by examining the role of mobile dislocations on the
coarsening process in spinodal decomposition. The model of Ref [11] is of particular interest to this
thesis since we believe that it is hypothesized that it can also describe precipitation of second phase
particles.
Through the coupling of composition and dislocations it has been shown [11] that
dislocation can lead to many growth regimes. The dislocations were shown to move to the grain
boundaries and this process caused an increased effective surface tension of the interfaces. The
result of the increased interface tension is the accelerated phase separation of the alloy. This effect,
however, is not seen until later times due to the lack of dislocations at the interfaces (limited
interface tension) at earlier times and the drag produced due to limited mobility (decreases phase
separation) at intermediate times. Therefore depending on the mobility of dislocations relative to the
solute diffusion, various coarsening regimes may be identified. It should be noted that the issue of
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what role dislocations play in particle coarsening has been controversial in the scienctific literature
[20-23].
The Model of Ref. [11] is expressed by the following two equations:
c
  2 [c  c 3   2 c   2  d ]
t
,
(2)
2
2
bx
 (m g  x  mc  y )[ y  d    dr G(r , r ) 2r  c  bx ]
t
,
(3)

and a similar equation for ∂tby, where c is concentration, b  (bx , b y ) is the Burger’s
vector density,
χd is the Airy stress due to dislocation strain fields,
m g & mc are the
mobilities for dislocation glide and climb, respectively,  &  are constants that
depend on material propertesm and G is a so-called green’s function describing the
elastic influence of a point source strain in a an infinite domain. Figure 6 (main
figure) shows some typical simulated results of average spinodal size Vs. time for
different dislocation mobility (climb and glide have been assumed equal here).
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Figure 6: Graph of Characteristic length versus time [24]
A simplification of the model in Eqs.2&3 can be obtained neglecting the energies of
dislocations cores [11]. This simplification leads to an analytically tractable model
that can be integrated analytically, yielding a simple ordinary differential equation for
the growth of the average spinodal size. This equation is given by,
2
 mt
dR  0 1   (1  e )

dt
4R 2
1
1
 2 0 1   2 (1  e  mt ) (mte mt  1  e  mt )
(7)
4mR
Results of this equation are shown in the inset of Fig. 6 for three mobility values
(different from those in the main figure). While very approximate, the attraction of the
simplified Eq. 7 is it potential simplicity of use (compared with the model of Eq.
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3&3), once the parameters and range of its validity of the equation are determined.
These objectives will be part of the focus of this Thesis.
2. Objectives
The objectives of this thesis are as follows:
1. To create a program to simulate the simplified model explained
earlier.
2. To compare the data generated from the full model of Eqs. 2&3 to
determine the range of validity of Eq. 7. The goal of this step will be
to determine regimes where data generated from the simplified model
will be a good approximation to that of full model.
3. To develop a similarity solution that predicts the coarseing rate in
spinodal decomposition using a similarity solution that depends only
on a dimensionless combination of variable involving time, mass
transfer and dislocation mobility. A powerful example of a similarity
solution is found in the carburization of many alloys, which can be
uniquely described at any point in space or time merely by knowing
the similarity variable x / Dt , where D is the constant describing
diffusion in a metal.
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4. To examine the role that different ‘freezed-in” dislocation density has
on the subsequent precipitation coarsening. Dislocation mobility will
be held constant by maintaining a constant quench temperature. This
procedure has many parallels to examining how different degrees of
cold work influence the subsequent particle coarsening during
spinodal decomposition and/or precipitation.
(Due to time
limitations, this objective will likely be continued to completion in the
sequel to this thesis work by a new graduate student working in the
research group of Dr. Provatas).
5. To compare our results against published experimental results on
particle coarsening and spinodal decomposition as a function of cold
work. In At this time, a series of new experiments has also been
proposed to Alcan International, which will examine the role of strain
hardening of particle coarsening in aluminum alloys. The results of
this proposal are pending. (As with objective 4, the proposal of new
experiments will likely comprise an extension of this Thesis)
3. Progress to date:
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1
Completion of preliminary literature review.
2. finite difference Fortran90 program has been written to simulate the model of Eq.
7. Results and their analysis is pending.
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[2] A. Karma and W-J. Rappel, Phys. Rev. E 53 3017 (1996)
[3] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1997)
[4] R. Kobayashi, Physica D 63, 410 (1993).
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[6] N. Provatas, Nigel Goldenfeld, and Jonathan Dantzig, J. Comp. Phys. vol 148, 265 (1999).
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[11] Mikko Haataja and Francois Leonard, Preprint (2003)
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ed.
Nelson Thornes Ltd. (Cheltenham, UK. 222)
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Communication)
[16] Gu, L. and Irons, G.A., 2001, Physical and Mathematical Modeling of Bottom Stirring and Gas
Evolution in Electric Arc Furnaces, Trans. ISS, Jan, pp. 57-68.
[17] A. Cassidy, M. Grant, and N. Provatas, Modeling and Simulations in Materials Science and
Engineering (MSMSE), Accepted (2003).
[18] A. Karma, Phys. Rev. Lett., 87, 115701 (2001)
[19] N. Provatas and M. Haataja, Preprint (2004)
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