STABILITY CONDITIONS OF THE BUILD

*Department of
Physics**, **University** of **Latvia**, Zellu str. 8, LV-1002 **Riga**, **LATVIA*

*ajakov@latnet.lv*

**Abstract:** A phenomenological
model has been developed to describe the build-up formation and to study the
stability conditions of a uniform growth of the build-up layer. Based on
suitable approximations, analytical formulae have been derived allowing to evaluate the stability region in the concentration-temperature
plane depending on the initial roughness of the growing surface.

**1.
The model**

We
consider a simple model to study the stability of a uniform build-up formation.
It is illustrated in Fig.1. The surface _{} separates the solid
build-up phase (_{}) from the molten metal (_{}). The liquid phase contains the build-up substance as an
impurity with concentration _{} - the number of particles
(molecules) per volume. It is assumed that the relative concentration
(normalized to the concentration of basic substance particles) is small. The
surface is flat except the region around _{} where a half-spherical
bulge of radius *r* is located. The
basic problem is to find the conditions at which the bulge develops or
disappears during the process of impurity deposition. We assume that the rate
of the process is limited by diffusion of the impurity through the
concentration boundary layer of thickness _{}. The liquid flow parallel to the surface is present, too.
Both the concentration and the temperature is assumed
to be constant outside of the concentration layer where the liquid flow is
strongly turbulent. The kinetics of the growth of the build-up layer is
described by means of a lattice model. In this model, one lattice site is
associated with one molecule of the impurity substance. The lattice is present
in the solid phase and can be formally continued also in the liquid phase. An
elementary volume _{} is associated with one
lattice site, where _{} is
the concentration of molecules in the solid phase, *M* is the molar mass of the build-up substance. Each molecule
on the surface can detach from the solid phase with a probability per time _{}. Each molecule which appears in an elementary volume _{} around any of the
lattice sites located in the liquid phase just at the surface _{} can attach to the
solid phase with a probability per time _{}.

*Figure 1b*. A schematic illustration of
the model

According to this model, the time development of the surface is
described by the kinetic equation

_{} . (1)

Here
_{} and _{} hold, where _{} is the linear size of
a cubic elementary cell with volume _{}. In the thermodynamic equilibrium

_{} (2)

hold, where _{} is the variation of
the system energy if one impurity particle (molecule) is subtracted from the
growing surface with curvature radius *r*. The latter relation agrees with the principle
of detailed balance [1,2]. We have assumed here that
the variation of the thermodynamic potential is equal to the variation of
system energy, which is true in the actual case of small concentration *c* at a fixed total volume of the system.
According to (1) and (2), we have

_{} . (3)

Eq.(3) should be
solved together with the diffusion equation for the impurity concentration, The
particle conservation (balance) condition at the surface reads and the boundary
conditions at the border of the concentration layer

_{} ; _{} (4,5)

_{} , (6)

where _{} and _{} are the impurity
concentration and the temperature in the bulk of molten metal. In the first
approximation we assume that the temperature on the surface _{} is constant, i.e., _{}.

Our
aim is the derivation of stability conditions for flat growing surface. If the
intensity of the surface growth at _{} exceeds that at _{}, then the bulge in Fig.1b develops, which means that the
flat surface is unstable with respect to the initial roughness characterized by
grains of radius *r*. The development of
these grains results in a distortion of flat profile of the growing surface,
i.e., in the formation of porous or dendrite-like build-up layer. In such a
way, the stability condition at which the bulge tends to dissolve reads

_{} . (7)

**2.
Estimation of the energy ****e****(r)**

For
simplicity, first let us assume that the impurity particles only interact with
each other. Then, in analogy to the classical nucleation theory [3], the
binding energy of a nucley of radius *r*
can be written as a sum of the bulk (index b) and surface (index s) terms: _{} , (8) , where _{} is the total number of
particles in the spherical nuclei, _{} is the number of
particles on the surface. For large enough *n*
we have

_{} , (9)

where _{}. According to our lattice model we have _{} and _{}, where _{} is the energy of one
bond, *l* and _{}are the average numbers of nearest impurity molecules in
solid part and on the surface, respectively. The numbers *l* and _{} depend on the lattice
symmetry and microscopic structure of the surface. Thus, Eq.(9)
becomes

_{} . (10)

In
general, where A (basic substance) and B (impurity)
particles interact, we have

_{} , (11)

where _{} is the binding energy
between two particles *X *and *Y*. Eq.(11) is consistent with a simple
counting of bonds when one impurity particle is moved from the solid phase to
the liquid phase with the same lattice structure. Besides, it is assumed that
the impurity particles have only the basic substance atoms in the neighboring
sites in the liquid phase, as consistent with our assumption that the
concentration *c* is small.

**3.
Estimation of the critical radius in the case of v=0**

At
large times *t*, the impurity
concentration *c* is given by the
stationary solution of Eq.(4), if the motion of the surface
is relatively slow. The latter condition is satisfied in the actually
considered limit of small concentrations. At zero velocity along the surface,
i.e. * _{}*, the linear stationary concentration profile along

_{} . (12)

Thus,
at the actual *x* and *y* values, the kinetic equation (3) and the
balance condition (5) at the surface read

_{} . (13,14)

Here
_{} and _{}. If the bulge is small as compared to the thickness of the
concentration layer, i.e. _{}, then we may assume _{}, (15) where _{} is the thickness of an
unperturbed (i.e., totally flat) layer. Since a very small bulge consisting of
few atoms is not of interest, we have _{}, which means that the exponent in (14) can be approximated
as

_{} . (16)

The
diffusion coefficient can be represented as

_{}, where n is the
characteristic frequency of atom oscillations, i.e., the frequency of attempts
to overcame the potential (diffusion) barrier _{}. The attachment frequency _{}can be represented in a similar way as

_{}, where _{} is the barrier of the
attachment reaction. One may assume that _{} is smaller or
comparable with _{}, since both the diffusion and the reaction are related to a similar
spatial rearrangement of particles, but in the case of the attachment reaction
the attraction forces between particles in the crystalline phase tend to
decrease the potential barrier. According to this,

_{} is
a very small quantity of order _{}. Therefore, the stability condition (7) at _{} reads

_{}, (17) where _{} is the supersaturation
and

_{} , (18)

is the equilibrium concentration for a flat surface.

Assuming _{}*mm* and _{}, at _{} we have roughly _{}, for _{} (_{}J, _{}) *k _{i}*

The
stability region _{} for both these cases,
depending on the relative oversaturation _{}, is shown in Fig.2. The same, but depending on the content
of the impurity in liquid Fe expressed in mass percents, at two different
temperatures _{} and _{} is
shown in Fig.3.

**4.
Estimation of the critical radius in the case of **_{}

The
linear approximation (12) is the stationary solution of the diffusion equation
(4) at _{} for _{} and _{}, as well as at _{} in a special case of flat
homogeneous surface, i.e., this approximation in any case makes sense far from
the bulge at _{}. Thus, according to (12) and (13), the concentrations _{} and _{} on the surface at _{} and _{}, respectively, are given by

_{} , _{} , (19, 20)

where _{}, defined by Eq.(2), is the equilibrium concentration of a
curved surface with radius *r*. In any case,
we can write _{} (21), where the factor *f* depends on *v*. Obviously, we have _{} at _{}. At _{}, _{} is expected due to the
strong flow along the surface. From (3) and (19) to (21) we obtain an equation
for the critical radius where _{} values are equal at _{} and _{}, i.e.,

_{},(22)

The
factor _{} (23) is dimensionless,
therefore it should be a function of a dimensionless argument composed of
physical quantities responsible for the concentration difference
_{}. On the one hand, the diffusion causes the mass transport
perpendicular to the surface which tends to maintain some difference between
the concentrations at _{} and _{}. On the other hand, the flow along the surface tends to
smear out the concentration profile _{}. According to these arguments, *f* should be a function of the ratio of parallel
and perpendicular fluxes at a characteristic distance _{} from the surface,
i.e., a function of the dimensionless argument _{}. The linear approximation _{} or _{} provides a
qualitatively correct behavior of the function *f*. Taking into account also that _{}, we obtain a simplified equation

_{}; _{};_{}. (24)

At _{} Eq.(24)
gives (17). An estimation of the order of magnitude shows that in a typical
case of the induction channel furnaces _{} holds (_{} or even larger), so that the transport parallel to the
surface changes the result (17) rather strongly. Taking into account that _{}, we obtain an approximate solution of Eq.(24)

_{} , (25)

where _{} is the velocity at the
border of the laminar sublayer of thickness _{} and _{}. An approximation _{} has been used. This
makes sense if _{} is remarkably smaller
than _{} and _{}.

To
estimate the order of magnitude of _{} expected in our
application, we have set _{},_{},_{}, and _{}. This yields at _{}: _{} for _{} (_{}) *g _{i} *=
3.07, for

In
Fig.4 we have shown the corresponding stability region in the _{}and concentration (the content in mass percents) plane at two
different temperatures - _{} and _{}.

**5. Conclusions**

1.
Based on suitable
approximations, analytical formulae have been derived allowing to evaluate the stability region in the
concentration-temperature plane depending on the initial roughness of the
growing surface.

2.
The further problem is to
test numerically our analytical approximations, particularly the validity of
our description by the phenomenological function *f*, and to make a quantitative estimation of the parameters _{} and _{}.

**6. References**

[1] Rolov, B.; Ivin, V.; Kuzovkov, V.:
Statistics and Kinetics of Phase Transitions in Solid.

[2] Nicolis, G.; Prigogine,

[3] Lifshitz, **19**, 35 (1961).