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
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age 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
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through paper during xerographic printing [17]. An illustrative example of such methods occurs
in the 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
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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 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 strainhardening. 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.
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Figure 1: Coherency interface with slight mismatch [14]
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
Figure 3: Concentration of dislocations
at the grain boundaries [14]
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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
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
Figure 4: Al-Zn phase diagram
parameter that distinguishes a broken symmetry 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.
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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 to quantitatively capture the
correct physics of free-boundary kinetics (i.e, solute/heat
diffusion in bulk phases + flux conservation at free
Figure 5: Order parameter
across interface
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
2
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.
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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 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
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Eq. 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:
12
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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(2001)
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Nelson Thornes Ltd. (Cheltenham, UK. 222)
[15] K.R.Elder (2003). Phase Field Modeling of Material Science Phenomena (Private
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).
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[19] N. Provatas and M. Haataja, Preprint (2004)
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