Qualititative Analysis of Interface Behavior under First Phase Transition

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(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

> 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

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