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Nucleation and Spinodal Decomposition in Ternary-Component Alloys

30 Jul 2009-

Abstract: : The Cahn-Morral System has often been used to model the dynamics of phase separation in multi-component alloys on large domains. In this paper we examine phase separation on small one-dimensional domains time independently. In particular we use AUTO to create bifurcation diagrams of equilibrium solutions for two different nonlinearities and use Matlab to observe the structure of the material at various points on the diagrams. We compare the results to determine if using different nonlinearities significantly affects the behavior of the Cahn-Morral System.
Topics: Spinodal (57%), Spinodal decomposition (55%), Nucleation (52%)

Summary (1 min read)

1. Introduction

  • Alloys are composite materials which are formed by mixing a number of pure metals together at a high temperature and then rapidly quenching or cooling the mixture to form a solid.
  • These pattern formations can be divided into two classes: Nucleation and Spinodal Decomposition.
  • This behavior is modelled mathematically by the Cahn-Moral equation.
  • G represents the set of all possible states or average mass concentrations of the ternary alloy, this follows from the conservation of mass.
  • In their research the authors used two nonlinearities.

3. Results and Discussion

  • For each of their nonlinearities the authors created Matlab code which shows which regions on the Gibbs simplex have no, one or two positive eigenvalues.
  • In the pictures below the Gibbs simplex is projected onto the plane.
  • The red area represents the nucleation region where there are no positive eigenvalues.
  • To reach the non-trivial nucleation branches the authors had to begin in the spinodal region.
  • 3.2.1. Tracing the quadratic nonlinearity.

4. Conclusions and Future Work

  • The authors would like to successfully trace alpha back into the nucleation region from several branches at the same λ value and compare bubble formations.
  • The authors hope to eventually find which bubble formations are possible at each value of λ for each nonlinearity.
  • The authors third partner James O’Beirne will be investigating a sixth degree polynomial nonlinearity and comparing the results with their investigations.
  • Eventually the research should be expanded to two and three dimensions to see if be used.
  • Both of these extensions though requiring more computing power, will make the model considerably more reliable.

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NUCLEATION AND SPINODAL DECOMPOSITION IN
TERNARY-COMPONENT ALLOYS
WILL HARDESTY AND COLLEEN ACKERMANN
Date: July 28, 2009.
Dr. Thomas Wanner and Dr. Evelyn Sander.
0

Abstract. put abstract here
NUCLEATION AND SPINODAL DECOMPOSITION IN
TERNARY-COMPONENT ALLOYS
WILL HARDESTY AND COLLEEN ACKERMANN
1. Introduction
Alloys are composite materials which are formed by mixing a number of pure
metals together at a high temperature and then rapidly quenching or cooling the
mixture to form a solid. During the process of quenching, the components undergo
a phase seperation in which they begin to form patterns. These pattern formations
can be divided into two classes: Nucleation and Spinodal Decomposition. Qualita-
tively nucleation occurs when the individual components begin to materialize from
the homogenous mixture as isolated droplets or bubbles. Spinodal decomposition
occurs when the components form connected snakelike patterns. This behavior is
modelled mathematically by the Cahn-Moral equation.
2. Background and Research Methods
2.1. Cahn-Moral system.
2.1.1. The equations. The Cahn-Morral equation is given by,
(1)
u
t
= (ε
2
u + f(u)) on
u
∂ν
=
u
∂ν
= 0 on .
Where the energy of the system is modelled by the Van der Waals free energy
functional,
E
ε
[u]=
!
"
ε
2
2
· |u|
2
+ F (u)
#
dx.
Date: July 28, 2009.
Dr. Thomas Wanner and Dr. Evelyn Sander.
1

2 WILL HARDESTY AND COLLEEN ACKERMANN
The term f is defined as the derivative of the double-well potential, F from the free
energy functional. The domain R = [0, 1], with imposed Neumann boundary
conditions.
2.1.2. The double well potential.
2.2. Gibbs Simplex. The gibbs simplex is defined as:
G = {(u, v, w ) R
3
: u + v + w =1,u 0 ,v 0 ,w 0}.
G represents the set of all possible states or average mass concentrations of the
ternary alloy, this follows from the conservation of mass. The Gibbs simplex can
be divided into regions corresponding to nucleation and spinodal decomposition,
which are determined through linearisation analysis.
Given a state (¯u, ¯v, ¯w), the stability of that particular state is given by computing
the eigenvalues of J
f
u, ¯v, ¯w), where J
f
is the Jacobian of f(u) [4]. The state
u, ¯v, ¯w) is in the nucleation region if J
f
has no positive eigenvalues otherwise it
lies in the spinodal region [2]. These regions can be depicted graphically with the
Gibbs Triangle, where each color represents a dierent region.
2.3. The nonlinearities. In our research we used two nonlinearities. A quadratic
nonlinearity:
F (u, v, w)=
u
2
v
2
+(u
2
+ v
2
)(w
2
)
4
And a logarithmic nonlinearity:
F (u, v, w) = 3.5(uv + uw + vw)+u ln u + v ln v + w ln w
The nonlinearities were chosen so that they were double well potentials and
symmetric. Symmetric means that each componenet contributes to the nonlinearity
in the same way making invesigations simpler.
Both of the nonlinearities have been investigated previously in both the one and
two dimensional cases. However, most of the previous research was done with time

NUCLEATION AND SPINODAL DECOMPOSITION IN T ERNARY-COMPONENT ALLOYS 3
variation. Our research diers in that we are looking at solutions at one moment
in time and comparing results from dierent nonlinearities.
3. Results and Discussion
3.1. Gibbs Triangle. For each of our nonlinearities we created Matlab code which
shows which regions on the Gibbs simplex have no, one or two positive eigenvalues.
In the pictures below the Gibbs simplex is projected onto the plane.
The red area represents the nucleation region where there are no positive eigen-
values. The light blue and dark blue represent the regions where there are one and
two positive eigenvalues respectively. Both areas are considered part of the spinodal
region.
3.2. Path following. Our ultimate goal in path following was to trace paths in
the nucleation region. However, the secondary branches in the nucleation region are
not connected to the trivial branch. To reach the non-trivial nucleation branches
we had to begin in the spinodal region. First we varied λ =
1
ε
2
. to get onto the
second secondary branch in the spinodal region. After following the second branch
and it’s bifurcations we attempted to follow in α =
¯uv
2
back into the nucleation
region. Finally we varied β =
¯u ¯v
2
in the nucleation region.
3.2.1. Tracing the quadratic nonlinearity.
3.2.2. Tracing the logarithmic nonlinearity.
3.3. Troubleshooting.
4. Conclusions and Future Work
We would like to successfully trace alpha back into the nucleation region from
several branches at the same λ value and compare bubble formations. We hope
to eventually find which bubble formations are possible at each value of λ for
each nonlinearity. Our third partner James O’Beirne will be investigating a sixth
degree polynomial nonlinearity and comparing the results with our investigations.
Eventually the research should be expanded to two and three dimensions to see if

4 WILL HARDESTY AND COLLEEN ACKERMANN
results vary significantly. Also more than three components will components will
be used. Both of these extensions though requiring more computing power, will
make the model considerably more reliable.
higher dimensional domains
more than 3 components
dierent nonlinearities
5. Acknowledgements
Thanks to Dr. Thomas Wanner, Dr. Evelyn Sander, James O’beirne the Depart-
ment of Defense, the National Science Foundation, and the Deparment of Mathe-
matical Sciences at George Mason University.
References
[1] Kathleen T. Alligood, Tim D. Sauer, and James A. Yorke, Chaos, An Introduction to Dynam-
ical Systems, Springer- Verlag New York, Inc., New York, New York, 1997.
[2] Jonathan P. Desi, Hanein H. Edrees, Joseph J. Price, Evelyn Sander, and Thomas Wanner,
The Dynamics of Nucleation in Stochastic Cahn-Morral Systems (May 29, 2009).
[3] Junseok Kim and Kyungkeun Kang, A numerical method for the ternary Cahn-Hilliard system
with a degenerate mobility, Applied Numerical Mathematics (May 9, 2009).
[4] Stanislaus Maier-Paape, Barbara Stoth, and Thomas Wanner, Spinodal Decomposition for
Multicomponent Cahn-Hilliard Systems, Journal of Statistical Physics (September 21, 1999).
Department of Mathematical Sciences, George Mason
E-mail address: hardes1@umbc.edu
E-mail address: cackerm@vt.edu
References
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Journal ArticleDOI
Junseok Kim1, Kyungkeun Kang2Institutions (2)
TL;DR: This work applied a second-order conservative nonlinear multigrid method for the ternary Cahn-Hilliard system with a concentration dependent degenerate mobility for a model for phase separation in a Ternary mixture and proved stability of the numerical solution for a sufficiently small time step.
Abstract: We applied a second-order conservative nonlinear multigrid method for the ternary Cahn-Hilliard system with a concentration dependent degenerate mobility for a model for phase separation in a ternary mixture. First, we used a standard finite difference approximation for spatial discretization and a Crank-Nicolson semi-implicit scheme for the temporal discretization. Then, we solved the resulting discretized equations using an efficient nonlinear multigrid method. We proved stability of the numerical solution for a sufficiently small time step. We demonstrate the second-order accuracy of the numerical scheme. We also show that our numerical solutions of the ternary Cahn-Hilliard system are consistent with the exact solutions of the linear stability analysis results in a linear regime. We demonstrate that the multigrid solver can straightforwardly deal with different boundary conditions such as Neumann, periodic, mixed, and Dirichlet. Finally, we describe numerical experiments highlighting differences of constant mobility and degenerate mobility in one, two, and three spatial dimensions.

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"Nucleation and Spinodal Decompositi..." refers background in this paper

  • ...The Gibbs triangle for the quadratic nonlinearity is shown on the left, and the triangle for the logarithmic nonlinearity is depicted on the right Both of the nonlinearities have been investigated previously in both the one and two dimensional cases [3],[4],[2]....

    [...]


Journal ArticleDOI
TL;DR: The dynamical aspects of nucleation in a stochastic version of these models using numerical simulations, concentrating on ternary, i.e., three-component, alloys on two- dimensional square domains, are studied.
Abstract: Cahn-Morral systems serve as models for several phase separation phenomena in multicomponent alloys. In this paper we study the dynamical aspects of nucleation in a stochastic version of these models using numerical simulations, concentrating on ternary, i.e., three-component, alloys on two- dimensional square domains. We perform numerical studies and give a statistical classification for the distribution of droplet types as the component structure of the alloy is varied. We relate these statistics to the low-energy equilibria of the deterministic equation.

17 citations


"Nucleation and Spinodal Decompositi..." refers background in this paper

  • ...Otherwise it lies in the spinodal region [2]....

    [...]

  • ...The Gibbs triangle for the quadratic nonlinearity is shown on the left, and the triangle for the logarithmic nonlinearity is depicted on the right Both of the nonlinearities have been investigated previously in both the one and two dimensional cases [3],[4],[2]....

    [...]


Journal ArticleDOI
Abstract: We consider the initial-stage phase separation process in multicomponent Cahn–Hilliard systems through spinodal decomposition. Relying on recent work of Maier-Paape and Wanner, we establish the existence of certain dominating subspaces determining the behavior of most solutions originating near a spatially homogeneous state. It turns out that, depending on the initial concentrations of the alloy components, several distinct phenomena can be observed. For ternary alloys we observe the following two phenomena: If the initial concentrations of the three components are almost equal, the dominating subspace consists of two copies of the finite-dimensional dominating subspace from the binary alloy case. For all other initial concentrations, only one copy of the binary dominating subspace determines the behavior. Thus, in the latter case we observe a strong mutual coupling of the concentrations in the alloy during the initial separation process.

17 citations


"Nucleation and Spinodal Decompositi..." refers background or methods in this paper

  • ...Given a state (ū, v̄, w̄) ∈ G, the stability of that particular state is given by computing the eigenvalues of Jf (ū, v̄, w̄), where Jf is the Jacobian of f(u) [4]....

    [...]

  • ...The Gibbs triangle for the quadratic nonlinearity is shown on the left, and the triangle for the logarithmic nonlinearity is depicted on the right Both of the nonlinearities have been investigated previously in both the one and two dimensional cases [3],[4],[2]....

    [...]