Journal of Crystallization Process and Technology, 2012, 2, 25-29
http://dx.doi.org/10.4236/jcpt.2012.21005 Published Online January 2012 (http://www.SciRP.org/journal/jcpt)
25
Qualititative Analysis of Interface Behavior under First
Phase Transition
Alex Guskov
Institute of Solid State Physics of RAS, Moscow, Russia.
Received October 25th, 2011; revised November 28th, 2011; accepted December 5th, 2011
ABSTRACT
At present there is no explanation of the nature of interface instability upon first order phase transitions. The well-
known theory of concentration overcooling under directed crystallization of solutions and Mullins-Sekerka instability
cannot account for the diversified liquid component redistribution during solid state transition. In [1-3], within the fra-
mework of the nonequilibrium mass transfer problem, it has been shown that there are regimes of the interface insta-
bility, which differ from the known ones [4-6]. Moreover, the instability theory of works [1-3] demonstrates a complete
experimental agreement of the dependence of eutectic pattern period on interface velocity. However, it is difficult to
explain interface instability within the framework of a general setting of the mass-transfer problem. This paper is de-
voted to qualitative analysis of the phenomena that are responsible for interface instability. The phenomena are con-
nected by a single equation. Qualitative analysis revealed a variety of different conditions responsible for instability of
flat interface stationary movement upon phase transition. The type of instability depends on system parameters. It is
important that interface instability in the asymptotic case of quasi-equilibrium problem setting is qualitatively different
from interface instability in the case of nonequilibrium problem setting.
Keywords: Phase Transition; Interface; Stability; Oscillation; Solution
1. Introduction
The heat-and-mass transfer problem under directed crys-
tallization is usually described by quasi-equilibrium bounda-
ry conditions [6-8]. Quasi-equilibrium problem setting has
a number of advantages. This statement was used for cal-
culation of numerous technological processes [7,8] that
are successfully used in crystal growth, met-allurgy and ma-
terials science. Yet, the quasi-equilibrium problem has its
disadvantages that in certain cases can be of principle sig-
nificance. For instance, in the problem with quasi-equoli-
brium boundary condition interface velocity cannot be con-
nected to the growth mechanism since it does not take into
account the nonequilibrium solution layer in front of the
interface. In [1-3] nonequilibrium problem setting is used
for description of the stability of the stationary regime of
directed crystallization. It is shown that the interface can
be unstable which accounts for the dynamics of component
redistribution upon phase transition. In particular, the de-
pendence of eutectic pattern period on interface velocity
was obtained. However, works [1-3] cannot explain the
obtained instability if the simple physical outline of
analysis and give a clear-cut distinction between the in-
terface dynamics of quasi-equilibrium problem setting
and that noneeqilibrium setting. This paper is intended to
make up for this deficiency.
2. Interface Stability
It is well known that first order phase transition occurs at
interface temperature which is different from equilibrium
temperature of phase transition. In particular, under crys-
tallization the temperature of crystallized liquid should
be less than the equilibrium crystallization temperature.
The difference between equilibrium temperature of phase
transition and current interface temperature is called ki-
netic overcooling. In quasi-equilibrium boundary condi-
tions kinetic overcooling is assumed to be equal to zero.
The equilibrium phase diagram is calculated subject to
the equality of phase chemical potentials. The chemical po-
tential values correspond to an infinite volume of each phase.
It is known, however, that the adsorption component for-
med on the interface is due to equality of the interface
chemical potentials. On phase transition the adsorption layer
affects interlayer component redistribution. Interface adsorp-
tion is commonly neglected in quasi-equilibrium bound-
ary conditions. The behavior of the adsorption layer upon
flat interface movement was considered by Hall [9]. Ac-
cording to Halls theory, there is relaxation time between
the component concentration in the layer and that in the

Qualititative Analysis of Interface Behavior under First Phase Transition
26
solid phase. Therefore, the interface velocity should be com-
pared to the diffusion coefficient as well as the relaxation
velocity of the component in the adsorption layer. Hall
introduced an effective distribution ratio equal to the com-
ponent solid phase-liquid solution concentration ratio. The
following expression was obtained for calculation of an
effective distribution ratio

00
=exp
ads
eff ads
V
KK KKv

 


Here eff
K
is the effective distribution ratio, k0 the
equilibrium distribution ratio, ads
K
the equilibrium ad-
sorption distribution ratio, ads the constant velocity of
adsorption, V the crystallization velocity. Let us consider
the physical meaning of this formula. To maintain the
composition of the adsorption layer, it is required that the
velocity of diffusion of component atoms from the melt
to the crystal exceeds the interface velocity. Hence, the
dissolved component concentration gradient might occur,
for instance, in the opposite direction and, as a result, in
front of the crystal there appears a depleted region in-
stead of the accumulation one corresponding to the equi-
librium phase diagram without regard for interface ad-
sorption. In this case the values of the effective distribu-
tion ratio will get from the region where eff
V
K
< 1 with
the equilibrium distribution ratio 0
K
< 1 over to the
region eff
K
> 1. Within the limit of high interface ve-
locities the distribution ratio will approach the adsorption
distribution ratio eff
K
= ads
K
. Interface temperature and
equilibrium phase transition temperature are differently
dependent on temperature and concentration varies of sys-
tem. For instance, equilibrium phase transition tempera-
ture is essentially dependent on component concentration
in front of the interface. Therefore, in accordance with
the phase diagram, at eff
K
< 1, equilibrium phase transi-
tion temperature decreases (considering 0
K
< 1) with in-
creasing component concentration. This results in decreased
kinetic overcooling as well as decreased interface veloc-
ity. The latter leads to decreasing concentration which
returns to its initial value. The interface is stability. De-
crease of interface velocity at eff
K
> 1 leads to further
increase of concentration which accounts for interface in-
stability.
Write kinetic overcooling in the form:

00 ke e
TT mCCT

 
Here Te∞ is the phase transition equilibrium tempera-
ture of the initial solution, Ce∞ the concentration of the
initial solution, T0 and C0 the interface temperature and
concentration of the solution, respectively. The interface
velocity is a monotonically increasing function of kinetic
overcooling . We distinguish two cases of the
solution state in the overcooled layer in front of the in-
terface: the cases of stable and instable solution [3]. For
definiteness we will consider a eutectic phase diagram for
the case when liquid solution concentration is less than
eutectic concentration, 0

k
vvT

K
< 1, the liquidus slope being
negative, m < 0. Let the flat interface in the stationary
regime move at a velocity Vs. Therefore, at eff
K
< 1, m
< 0, the interface is stable. But it is well known that this
may lead to concentration instability associated with the
so-called concentration overcooling [4]. If the tempera-
ture gradient of the liquid solution on the interface is less
than the gradient of the phase transition equilibrium tem-
perature, the interface may lose its stability. In the non-
equilibrium case it can be explained as follows. Let inter-
face temperature T0 decrease due to fluctuations. This
will involve increased kinetic overcooling and, hence, in-
creased interface velocity. There appears an asperity in the
region of concentration overcooling, i.e. an increase in the
interface equilibrium temperature of phase transition. This
involves an increase in kinetic overcooling as well as fur-
ther increase of interface velocity. Hence, the presence of
a concentration overcooling region leads to interface in-
stability. On the other hand, consideration of kinetic insta-
bility yields that increased interface velocity involves an
increase in interface component concentration. This will
decrease equilibrium temperature of phase transition and,
therefore, decrease kinetic overcooling and interface ve-
locity. These two inverse processes can be illustrated by
the following scheme:
0
0
e
k
e
Tv
TTv
CTv

   
Thus, we have obtained inverse variations of equilib-
rium temperature of phase transition. On the one hand, it
increases due to local interface movement; on the other
hand, it decreases due to edging of the component by the
interface and change of kinetic overcooling. The two in-
verse processes can either lead to or fail to lead to inter-
face instability which depends on the external conditions
and the physical parameters of the system.
3. Equation of Interface Oscillations
Let us find the relationship between temperature and con-
centration perturbations and small interface movement
zm (t) from the interface coordinate z = 0 of stationary
regime. Linearize interface movement velocity by concen-
tration and temperature perturbations [1,3]








000 0
00
,,
kkS
mm
vTTC TC
mC TV

S
S
Vm
vT

 (1)
Here
k
v
T



. As shown above, we consider a qua-
litative model of interface movement dynamics. Heat con-
duction, diffusion and phase transition heat affecting spa-

Qualititative Analysis of Interface Behavior under First Phase Transition
27
tial distribution of temperature and concentration are ne-
glected. The aim is to show the feasible reasons behind in-
terface instability using the relationship between the lump-
ed interface parameters. The interface temperature varia-
tion at distance zm (t) in the linear approximation can be
written as
where

0
00 0
d0
d
dd
,.
dd
ddS
SS
v
t
vV
Cv
CC CV
vt
t






 


)
      
00
0
mmS zmm
z
Tzt gradTzgradTzzt



 
Substitute the small temperature and concentration varia-
tions from (2) and (3) into (1). Upon elementary trans-
formations we obtain the equation
(2)
2
2
dd 0
d d
mm
m
zz
ahgz
t t

 (4)
In harmonic analysis of interface stability the function
Tm (z) was found as the solution of a common differential
equation [2] and depended on time frequency and wave
number. Here we do not consider spatial distribution of
temperature and concentration; rather follow small time
variations of the interface temperature and concentration.
Thus, in (2) the spatial gradient of small temperature per-
turbation is neglected. To connect concentration perturba-
tion with interface movement, let us use the known rela-
tion between pulse variation of interface velocity and in-
terface variation of concentration upon stationary move-
ment of the interface [4]. If the distribution ratio is <1,
the velocity increase leads to increase of concentration
which then returns to its original value. If the distribution
ratio is >1, the velocity increase first leads to decrease of
concentration which then returns to its stationary value.
Thus, interface concentration is dependent on velocity va-
riation. Consider also variation of concentration at the mo-
vement of the interface zm (t) owing to the spatial varia-
tion of the concentration of the stationary regime. For the
same conditions as in the case of (2) the linear approxi-
mation for concentration variation yields
where
   
1,
SS
g
gradCzgrad Tz
m
 1
hm 

.
The parameters α and g are dependent on stationary
interface velocity. The parameter h > 0 is independent of
the movement interface regime. It involves only parame-
ters dependent on the nature of the solution. The stability
and form of the nonsimple point

m
Z
t = 0 depend on
the relationship between the coefficients of Equation (4).
This is the well-known linear oscillator equation and its
solutions are thoroughly studied in [10].
4. Nonsimple Point Type of System
  
00 0
0
d()
d
Sm
z
v
CtCagrad Cs zzTCC
t
 
0Sm
To describe stability of stationary interface movement upon
perturbations of temperature and concentration, let us di-
vide the regimes into groups by the characteristics shown
in Table 1. The sign of α indicates whether the compo-
nent is edged or captured by the solid phase. The sign of
the stationary concentration distribution gradient gradCS
together with α indicates whether the nonequilibrium so-
lution layer in front of the interface is in the stable or me-
tastable state. The discriminant sign of the characteristic
equation indicates whether at given parameters Equation
(3)
Table 1. The nonsimple point zm (t) = 0 type at the vary values of the Equation (4) parameters.
α gradCS h g det zm = 0
Equilibrium α > 0 gradCS = 0 h > 0 g = 0 det = h2 − g = h2 stable exponent
gradTS ≠ 0 α > 0 gradCS = 0 h > 0 g > 0 det > 0 stable node
gradTS ≠ 0 α > 0 gradCS = 0 h > 0 g > 0 det > 0 stable focus
1 α > 0 gradCS < 0 h > 0 g > 0 det > 0 stable node
2 α > 0 gradCS < 0 h > 0 g > 0 det < 0 stable focus
3 α > 0 gradCS > 0 h > 0 g > 0 det > 0 stable node
4 α > 0 gradCS > 0 h > 0 g > 0 det > 0 stable focus
5 α > 0 gradCS < 0 h > 0 g < 0 det > 0 saddle
6 α < 0 gradCS > 0 h < 0 g < 0 det > 0 saddle
7 α < 0 gradCS < 0 h < 0 g < 0 det > 0 saddle
8 α < 0 gradCS < 0 h < 0 g > 0 det < 0 nonstable node
9 α < 0 gradCS < 0 h < 0 g > 0 det < 0 nonstable focus

Qualititative Analysis of Interface Behavior under First Phase Transition
28
(4) has real or complex characteristic numbers. Let us give
a brief description of the regimes presented.
In the equilibrium state g = 0, α > 0 (k = k0). Equation
(4) gives stable solutions. In the equilibrium regime any
perturbation involves finite movement of the interface. The
system asymptotically approaches the equilibrium state
at large values of t. Two regimes with gradTS ≠ 0 and a
motionless interface represent the initial state of the sys-
tem. These regimes are introduced to emphasize that even
with a motionless interface the system is in the nonequi-
librium state. If in the equilibrium state the coordinate
point (a, g) is on the a axis, then at VS = 0 the coordinate
point can be anywhere within the interval (a > 0, g > 0).
Depending on the sign of det, the point of the parameter
values can be either a stable node or a stable focus. Re-
gimes 1 and 2 exhibit a qualitatively identical temperature
and concentration distribution. The nonsimple point is sta-
ble. The sign of det determines whether it will be a node
or a focus. It should be noted that with changing velocity
the nonsimple point can have either node-focus or focus-
node bifurcation. Phase portrait on Figure 1 shows regions
with different types of nonsimple points. The figures in
the circles indicate the range of system parameter values
corresponding to the row number in Table 1. Regimes 3
and 4 are different from 1 and 2 only by the sign of the
concentration gradient. This corresponds to solution insta-
bility in the kinetic overcooling region .The change of the
concentration sign gradient leads to the fact that g > 0 at
any stationary interface velocity. Regime 5 is the bifurca-
tion of the nonsimple point of Regimes 1 and 2 when g be-
comes negative. This sign reversal indicates transition to
the concentration overcooling region. It should be empha-
sized that this regime requires a well-developed concen-
tration profile and, hence, a fairly high interface velocity.
Regimes 6 - 9 are different from 1 - 5 regimes by the sign
of parameter a. These regimes can be implemented pro-
vided the interface velocity of the stationary regime is suf-
ficiently large for the distribution ratio to exceed unity due
to interface component adsorption. Regimes 1 and 2 can
change to Regime 6 which is actually transition of stable
solution from a regime with a > 0 to that with a < 0 at an
Figure 1. Phase portrait of nonequilibrium system.
interface velocity sufficient for interface component ad-
sorption. By transition between stable regimes with dif-
ferent values of stationary interface velocity we mean a slow
change of stationary interface velocity without failure of
the flat interface surface. As the stable regime changes to an
instable one, the interface in experiments can remain flat
only to the stability boundary. Further interface dynamics is
determined by the domain of attraction of some steady-state
regime of the system. Transient and steady-state regimes
cannot be considered in this qualitative analysis. As it
follows from the analysis, depending on parameter gradCS
and a with increasing velocity of the stationary regime,
stable solutions 1 and 2 can formally change into one of
the two instable solutions: Regime 5 related to transition
of the stable regime to the concentration over-cooling re-
gime or Regime 6 related to transition of the distribution
ratio to the region with eff
K
> 1 owing to interface com-
ponent adsorption. Regimes 3 and 4 can change to re-
gime 7 since they all have instable solution in front of the
interface. Regimes 3 and 4 can also change to Regime 8
and 9 provided the sign of parameter α is changed, i.e.
due to adsorption, the effective distribution ration be-
comes greater than unity and changes the sign of g and the
system gets over to the concentration overcooling region.
Now let us consider quasiequilibrium problem setting.
In the equation obtained the kinetic coefficient 1/Λ stands
for a dissipative term. The smaller 1/Λ, the larger Λ and,
hence, the higher the interface velocity at the same kine-
tic overcooling. In the limit Λ = ∞ (h = 0) at any infinite-
simal kinetic overcooling the interface immediately shifts
to the equilibrium phase transition surface. In this case ki-
netic overcooling tends to zero whereas the equilibrium
temperature tends to equilibrium phase transition tempe-
rature, i.e. the regime corresponds to quasiequilibrium set-
ting of the directed crystallization problem. Within the li-
mit Λ = ∞ Equation (4) transforms to a harmonic oscilla-
tor equation which yields two variants of solution, namely,
solutions with imaginary characteristic numbers α > 0, g
> 0 or, which is formally the same, α < 0, g < 0. Here the
nonsimple point is the center, And solutions with real
characteristic equal sign numbers α > 0, g < 0 or, which
is formally the same, α < 0, g > 0, the nonsimple point
being a saddle. Figure 2 shows a phase portrait of non-
simple points for Equation (4) at h = 0. In the general case
the phase portrait of the system should be considered in
the three- dimensional system (α, g, h). If the coefficient
h of the oscillation equation is taken as a control pa-
rameter, the value of h = 0 is presented as a special case
of fixed physical parameters of the system and the phase
portrait of nonsimple points shown in Figure 2 belongs
to the case of a noncoarse dynamic system. The value h =
0 corresponds to the zero value of kinetic overcooling.
Strictly speaking, in this case the interface velocity is equal
to zero which does not imply equilibrium. The condition

Qualititative Analysis of Interface Behavior under First Phase Transition 29
Figure 2. Phase portrait of quasi-equilibrium system.
h → 0 is equivalent to the assumption that the interface
coincides with the geometric surface whose temperature
is equal to that of phase transition. Such problem does not
involve an overcooled solution layer in front of the inter-
face. Hence, there are no corresponding interface dynamics
regimes in the vicinity of nonsimple points. There are only
two types of nonsimple points: a center and a saddle.
5. Conclusions
1) Interaction of kinetic overcooling, growth mecha-
nism and interface adsorption can lead up to interface
instability under first order phase transition.
2) Iinterface dynamics in quasiequilibrium problem set-
ting is qualitatively different from that in the nonequilib-
rium case.
6. Acknowledgements
This work was supported by the Russian Foundation for
Basic Research, Grant N 11-03-01259.
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