Reconfinement and loss of stability in jets from active galactic nuclei
MSE490 FinalReport
1. MSE 490 Research Final Report
Effects of Pulling Velocity on Solidification of SCN-DC Eutectic System
Zhenjie Yao ycollin@umich.edu
Robert Spurney rgspurn@umich.edu
2. Introduction
A study of directional-growth of succinonitrile–(D)camphor (SCN–DC) at eutectic concentration in thin
and bulk samples using real-time observation methods had been done by S. Akamatsu et al [1]
. The results
show the growth of rod-like patterns in the bulk sample; and prove that 𝜆!
!
𝑉=constant, where 𝜆! is the
minimum-undercooling spacing and 𝑉 is the pulling velocity. Dynamical studies of solidification
microstructures require real-time observations of the solid-liquid interface during lamellar or rod growth.
This kind of observation is better performed with optical methods using transparent, nonfaceted organic
alloys, such as SCN-DC and Succinonitrile-Nitropentyl glycol (SCN-NPG). In the SCN-DC system, the
volume fraction of phases favors rod growth. However, this study is focused on thin-film growth, so the
resulting growth will be more lamellar in structure.
This research focuses on the computational simulation of the solidification behavior of a eutectic system
and compares it with experimental observation. A phase field model parameterized for a SCN-DC
eutectic system is developed to simulate the difference in solidification dynamics between speed-up and
slow-down of pulling velocity for a thin SCN-DC film on a moving temperature-gradient stage. The
effect of the changes in undercooling on the lamellar spacing and solidification behavior is discussed and
compared with experimental results for a similar SCN-NPG system.
Model
Literature Background
A solidifying eutectic system will diffuse so that the species α and β in the eutectic liquid will precipitate
out in a way that minimizes the surface energy created by lamellar formation and growth. As lamellae
grow, concentrations of species α (Cα) and species β (Cβ) in the liquid near the solid-liquid interface will
diffuse out of the way of the growing lamellae before solidifying.
Figure 1: Arbitrary phase diagram of a eutectic system
However, the ability of each species in liquid to assemble into concentrations consistent with the α and β
phases while moving towards those bulk phases, thus lowering the surface energy created upon
solidification, is hindered by the diffusion coefficient and the undercooling such that at a given growth
velocity, the species can only diffuse so far before solidifying. The resulting lamellae will be thicker for
slower growth velocities as the species can diffuse farther before solidifying. Conversely, the lamellae
will be thinner for faster growth velocities.
CβCα
3. Although the lamellar structure will not be formed when the rod minimum undercooling is lower than that
for lamellae, it is evident that the rod minimum undercooling represents a stabilizing influence on the
lamellar structure. Jackson et al proved that the steady-state solution for the diffusion equation is similar
for the rod-type eutectic system [2]
. The average curvature of the interface depends on the angles at the
phase boundaries and on the rod spacing. For the 𝛼 phase in a 𝛽 matrix, the curvature for 𝛼 phase and
𝛽 phase, respectively, is:
!
!!
=
!
!!
sin 𝜃! (1a)
!
!!
=
!!!
(!!!!!)!!!!
! sin 𝜃! (1b)
where 𝜌! and 𝜌! are the curvature, 𝑟! and 𝑟! are the radii, and the 𝜃! and 𝜃! are the interface angles
for 𝛼 phase and 𝛽 phase respectively.
Additionally, lamellar fault and termination occur during the lamellar growth and act as stabilizing
mechanisms. It has been proposed that stable lamellar growth occurs when the fault remains stationary.
The stability of an interface depends on the difference between the eutectic temperature and the local
temperature at the interface. The temperature at the interface can depart from the isotherm due to local
changes in composition, the curvature of the interface, and the chemical potential difference that is
driving the solidification [3]
. Lamellar faults form when the lamellar tip becomes too unstable and the
lamellar spacing must be adjusted.
The formation of new lamellar faults will be easier in the regions where the local lamellar spacing is
different from the average. Additionally, sudden changes in growth rate can also result in overgrowth of
one phase because of the boundary layers present at the interface. The boundary layer permits local
changes in composition for spacing stabilization during solidification. These lamellar faults and merging
events, seen in Figure 2, play an important role in controlling lamellar spacing.
Figure 2. Directional solidification of SCN-NPG.
Image shown was taken 90 s after the speed was
increased from 0.1 to 1 µm/s. Image taken at 200x [4]
.
Lamellar faults and merging events are shown.
Figure 3. Directional solidification of SCN-NPG.
Image shown was taken 51 s after the speed was
increased from 0.1 to 1 µm/s. Image taken at 200x [4]
.
Envelopes are shown.
4. if
((i>=35
.AND.
i<46)
.OR.
(i>=115
.AND.
i<126)
.OR.
(i>=195
.AND.
i<206)
.OR.
(i>=275
.AND.
i<286))
then
phii(i,j,k)
=
0.0
phij(i,j,k)
=
0.5*(1-‐TANH(dx*(DBLE(j)
-‐
SL_int_init)/DSQRT(2.0d0))
)
else
phii(i,j,k)
=
0.5*(1-‐TANH(dx*(DBLE(j)
-‐
SL_int_init)/DSQRT(2.0d0))
)
phij(i,j,k)
=
0.0
end
if
However, when the change in undercooling becomes too extreme, a shape instability may form in the
interface. It is found that a deep groove in the center of the solidification front of the phase could lead to
splitting of the lamellae at growth rates above the extremum condition. These envelopes, as seen in Figure
3, will be at a different temperature than the lamellar tips, and will deepen as it solidifies at a slower rate
until it becomes too laterally unstable and a new phase forms within. Once formed, the lamellar spacing
and growth velocity will stabilize.
In this research, a phase-field model was chosen to simulate the eutectic solidification system. The phase-
field method is a powerful tool to study interface dynamics in areas such as solidification, which has
complex interfacial patterns [5]
. To construct these evolution equations, the physics of diffuse interfaces is
typically used as a guide, and often some qualitative physical meaning can be attached to the phase field.
In solidification, it can be interpreted as an order parameter, and its time evolution is considered to be a
relaxation towards the minimum of a free energy. Furthermore, smoothed boundary method (SBM) was
used in building the phase-field model. In contrast to the immersed interface method, which distributes a
singular source of boundary conditions to grid points near the interfaces, the SBM spreads the zero-
thickness boundary into a finite-thickness diffuse interface using a phase-field-like, continuously
transitioning domain indicator function [6]
. Therefore, boundary conditions are straightforwardly
distributed among the grid points residing within the interfacial regions in which the domain parameter
varies smoothly across a short distance. Numerical methods are added to the model to account for the
effects of using a smooth boundary versus the sharp boundary that exists in experiment. This method has
been successfully employed in simulating diffusion processes.
Improvement
Based on the established 2-dimension model, several improvements were added in order to simulate
the phenomena of SCN-DC organic eutectic solidification in this research.
a). Increase the domain range:
In the organic eutectic system, the SCN and DC phases are arranged alternately, as shown in Figure
4. In the original setup, there was only one SCN-DC alternating layer, which is not able to simulate
the potential interaction within the whole domain. In order to observe the more realistic eutectic
solidification in the model, four SCN-DC eutectic layers were recommended in this model. To
accomplish this goal, the following code was added for a system growing at 1 µm/s:
In the model, one DC rod occupied 11 grid points, while one SCN layer took 69 grid points. The
ratio of the grid points of two phases was kept the same as the ratio of mole fraction recorded from
the experiment, 8.168. For the system growing at 0.1 µm/s, thicker rods will form in the steady-state,
so 23 grid points were used for the DC phase and 140 grid points for the SCN phase. This
improvement created four SCN-DC alternative layers in the domain.
5. if(
it<
0.05*maxcycles)
then
centa=isotherm_gp*dx+it*vp*dt
centb=isotherm_gp*dx+it*vp*dt
num=it
else
centa=isotherm_gp*dx+num*vp*dt+(it-‐num)*vps*dt
centb=isotherm_gp*dx+num*vp*dt+(it-‐num)*vps*dt
endif
(a) (b)
Figure 4. SCN-DC alternative layers (a) 3-dimension overview [4]
; (b) 3-dimension cross-section view [4]
b). Change pulling velocity during the solidification process
The main purpose of this simulation model is to observe the effect of a sudden increase or decrease
in pulling velocity on the solidification behavior. To change the pulling velocity during the
solidification process in the model, a time interval was set during the simulation. When time was less
than the specific value, the solidification underwent at the initial speed; as long as the time was larger
than the period, the pulling speed automatically changed to the desired new speed. Therefore, both
initial and speed-up or slow-down solidification process could be observed in the same simulation.
As a result, the different solidification behaviors were compared and studied. The code shown below
was used to implement the speed change during the process:
where vp and vps represented the initial and new pulling velocity respectively. centa and centb
stood for the position of the solidification front of SCN-DC eutectic layers.
c). Add shift function in the simulation to increase efficiency
The shift function was used to decrease the time for computational calculation and increase
efficiency by enabling a smaller domain size in the solidification direction to be used. After the
isotherm moves a distance of one grid point, each grid point is given data tracked by the simulation,
such as phase position and chemical potential, of the grid point in front of it. The front row of grid
points with nothing in front of them are set to the initial conditions of the eutectic liquid far from the
solidification front. As a result, every grid point is shifted back, and a larger domain size is not
needed to observe solidification far from the point of speed-change. The shift function considerably
improved the simulation efficiency. The following portion of the shifting function is a good example
of how the shift works, but only includes the shifting of the isotherm and chemical potentials:
6. Results
Two different scenarios of pulling velocity change were tested in this model: One was speed-up from
0.1𝜇𝑚 𝑠 to 1𝜇𝑚 𝑠 during the solidification process and the other was slow-down from 1𝜇𝑚 𝑠 to
0.1𝜇𝑚 𝑠. Figures 5, 6, and 7 below show the results of both experimental and simulative
observations in these two situations.
(a) (b)
Figure 5. SCN-DC eutectic solidification at initial pulling speed (a) experiment; (b) simulation
IF(it*vp*dt>=(0.5*Ny*dx-‐SL_int_init*dx)
.AND.
shiftFlag==0)
THEN
shiftInitial=real(isotherm_gp)*dx+real(it)*vp*dt
vp=vp*0.1
shiftFlag=1
ENDIF
IF((centa-‐shiftInitial)>=dx
.AND.
shiftFlag==1
.AND.
shiftSwitch==1)
THEN
centa
=
centa
-‐
dx
centb
=
centb
-‐
dx
do
k=Is3,Ie3
do
j=Is2,Ie2
do
i=Is1,Ie1
if
(j>=Ny)
then
phii(i,j,k)
=
0.0
phij(i,j,k)
=
0.0
phik(i,j,k)
=
1.0
else
phii(i,j,k)
=
phii(i,j+1,k)
phij(i,j,k)
=
phij(i,j+1,k)
phik(i,j,k)
=
phik(i,j+1,k)
endif
end
do
end
do
end
do
ENDIF
7. (a) (b) (c)
Figure 6. SCN-DC eutectic solidification in speed-up scenario (a) experiment; (b) simulation before and
(c) after nucleation of new phase
(a) (b)
Figure 7. SCN-DC eutectic solidification in slow-down scenario (a) experiment; (b) simulation
When the eutectic system solidified at the initial constant speed, the lamellar spacing between the
two layers was kept equal along the solidification direction, as shown in Figure 5(a). In the
simulation result (Figure 5(b)), the shape of the SCN matrix solidification front was convex. The
curvature could be calculated using the Eq. 1(a) and 1(b), and clearly has a sinusoidal shape in
agreement with theory. The forming of curvature in the simulation was because the solidification
speed was unmatched to the initial lamellar spacing. The initial condition used in the simulation
model was recorded from the experiment, which was not entirely stable.
When the pulling velocity increased, the spacing between the eutectic layers became smaller. The
time for solidification decreased and the SCN and DC solute did not have enough time to diffuse a
long distance. New DC phase quickly solidified in front of SCN phase, resulting in the decreased
spacing seen in Figure 6(a). In the simulation model (Figure 6(b), (c)), the same phenomenon was
observed. A shape instability formed, which eventually led to new phases forming in the envelope as
the curvature of the envelope became too large. At the initial solidification speed, the SCN matrix
8. was convex, and when the pulling velocity was increased 10 times, the convex shape became more
severe. Concentration of DC phase gathered at the convex position until nucleation of DC rods that
began growing from the SCN matrix. Here, the eutectic layer spacing is also reduced and agrees with
experiment.
When the pulling velocity decreased, the time for phases to diffuse increased and resulted in a larger
eutectic spacing, as shown in Figure 7(a). Some of the DC rods end as SCN phases merge together
over the DC phase. When the solidification speed was slowed down, the shape of SCN matrix
gradually became concave instead of convex, as seen in Figure 7(b). When the solidification process
continued experimentally, the concave shape became more severe, until eventually overgrowth
occurs after a relatively long period of time. This phenomenon also resulted in the increasing
lamellar spacing. In the simulation, a merging event was never seen. This could be because the
stabilization is a longer process after slow-down, or could be due to overstability of the system at
small spacing. For a given undercooling, there are a range of stable growth velocities and therefore a
range of stable lamellar spacings for the system [2]
. However, because the observed growth velocity will
always be at the extremum, taken to be identically the maximum growth velocity or minimum
undercooling, a given lamellar spacing is to be expected for a given undercooling. Interestingly, this
relationship does not hold for lamellar patterns at small spacing. While Jackson-Hunt theory assumes
lamellae grow perpendicular to the envelope of a solidification front, Akamatsu predicts a small amount
of lamellae also move parallel to envelope thereby overstabilizing the system even for spacings under 𝜆!
[7]
. Perturbations to the model may be necessary to see any merging events.
The phase-field model is based on tracking the order parameter of each phase in every grid in the
domain. The sum of order parameter for the SCN phase, DC phase, and liquid should be 1 at any
place on the grid. Figure 8 plots the order parameter of the SCN and DC phase traversing the
solidified section of the domain, perpendicular to the growth direction. When the position was totally
located in one of the phases, the order parameter of that phase is 1; at the interface, the order
parameter of both phase changed simultaneously, but the sum is still kept to 1. The transition
between phases can be seen as a curve on the plot of the order parameter, instead of a step-wise
function, and agrees with the smooth-boundary method employed in this model. It could also be
concluded from the figure that the volume fraction of the SCN phase was larger, as the order
parameter is kept as 1 for a longer distance.
Figure 8. Order parameter for the SCN (black) and DC (red) phase, plotted behind the
solidification front in the direction perpendicular to the growth direction
9. A problem encountered with this new phase-field model was oscillation of the boundary between
SCN and DC phases during solidification process, as seen in the Figure 9.
(a) (b)
Figure 9. (a) A cut-line along the interphase between solidified phases and (b) plot of the oscillating DC order
parameter along the cut-line that drops off in the liquid
The reason for this unexpected oscillating boundary in the model was due to the smaller time-step for
the system. The initial model setup prevented the diffusion in the solid, but uses a numerical anti-
trapping function that makes sure the concentrations within the smooth boundary diffuse thoroughly
before solidifying as they would in a sharp-interface system. The cutoff value related to the order
parameters of the solid phases, below which anti-trapping will not occur, was initially set to be 10!6
.
However, our smaller time-step resulted in lower parameter values along the edge of the interface,
especially near trijunctions, so an even smaller cutoff value of 10!9
was selected to keep the anti-
trapping term active at lower values. After this change, the oscillating boundary phenomenon was
successfully eliminated.
Conclusion
Changes in pulling velocity have strong effects on the solidification behavior of the SCN-DC eutectic
systems. The increasing pulling velocity will decrease the lamellar spacing and create new DC rods
in the eutectic system; while decreasing pulling velocity will slowly increase the lamellar spacing
and redistribute the DC rods. The phase field model successfully simulates these solidification
behaviors as observed in the experiment. For the slow-down scenario, the model needs much more
time to simulate solidification behavior, which supports the experimental observations. The shift
function is useful in this case because it can help improve the simulation efficiency and calculate
much more cycles in the same period of time. Therefore, applying the shift function to the model is
important for the further observation of slow-down solidification behavior. In addition, mathematical
instability is another major concern in phase field model and affects the simulation results. Further
investigations are needed to simulate the 3-dimensional solidification process of the eutectic system
to reveal different behaviors before and after pulling velocity change. The increased space in a 3D
model will be able to simulate more rod-like behavior and could contribute to the ability of faults to
occur.
10. References
[1] S. Akamatsu, et al. Real-time study of thin and bulk eutectic growth in succinonitrile–(D)camphor alloys.
Journal of Crystal Growth, 2007, 299, 418–428.
[2] K.A.Jackson, and J.D.Hunt. Lamellar and Rod Eutectic Growth. Transactions of the Metallurgical Society of
AIME,1996, 236, 1129-1142
[3] F. D. Lemkey, R. W. Hertzberg, and J. A. Ford: Trans. Met. Soc. AIME, 1965, vol. 233, p. 334.
[4] Vladislava Tomeckova, Halloran Lab group, University of Michigan, 2015
[5] R. Folch, and M. Plapp. Quantitative phase-field modeling of two-phase solidification. 2007
[6] Hui-Chia Yu, Hsun-Yi Chen and K Thornton. Extended smoothed boundary method for solving partial
differential equations with general boundary conditions on complex boundaries. Modelling Simul. Mater. Sci. Eng.,
2012, 20, (41pp)
[7] S. Akamatsu, M. Plapp, G. Faivre, A. Karma. Overstability of Lamellar Eutectic Growth Below the Minimum
Undercooling Spacing. Metallurgical and Materials Transactions A, 2004, 35A, 1815-1828.