JAKOVICS Andris, KAUPUZS Jevgenijs

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

NACKE Bernard

Institute for Electrothermal Processes, University of Hannover, Wilhelm-Busch-Str. 4,

D-30167 Hanover, GERMANY


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


Text Box:  

Figure 1a. Thin layer with compact and thick layer with fractal structure of build-up in the channel of ICF by melting of ferrous materials
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


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 z axis is the solution of Eq.(4) at  (due to the symmetry) as well as (infinitely) far away from the bulge at  where we have

 .                                                                                  (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)


Text Box:  

Figure 2.  The stability and unstability regions depending on the relative oversaturation   and radius r for the growth of a flat build-up layer (solid line -  , dashed line -  ;  ).
Text Box:  
Figure 3. The stability and unstability regions for the growth of a flat build-up layer depending on the radius  and the concentration (in mass percents) (solid line -  , dashed line -  ;  ).






















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, )  ki = 0.00172 and for  (J, ) ki = 0.00126.

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.,


Text Box:  
Figure 4.  The same as in Fig.3, but with nonzero flow along the surface  .

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  () gi = 3.07, for  () - gi = 5.68.

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. Riga: 1979.

[2] Nicolis, G.; Prigogine, I.: Emploring Complexity. New York: Freeman 1989

[3] Lifshitz, I.; Slyozov, V.: J. Phys. Chem. Solids 19, 35 (1961).