Obtaining of micro-spheres in plasma: theoretical model

Materials Science and Engineering B77 (2000) 293 – 296 www.elsevier.com/locate/mseb

Letter

Obtaining of micro-spheres in plasma: theoretical model Ioan Bica Faculty of Physics, West Uni6ersity of Timisoara, Bd. V. Par6an no. 4, 1900 Timisoara, Romania Received 1 December 1999; received in revised form 31 May 2000; accepted 12 June 2000

Abstract The paper presents the mechanism of formation of iron and glass micro-spheres in plasma. We developed a set of equations for computing the inner and outer micro-sphere radii, and for describing their dependence on technological process parameters. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Micro-spheres; Plasma; Glass

1. Introduction To know the dependency of dimensional characteristics of micrographies on technological parameters of particle obtaining, is a problem of high practical importance. In paper [1] a model of micro-sphere formation, by using the sol– gel method, is proposed. In this paper, the diameter and the thickness of the wall of micro-sphere, as a function on the gas–vapor system and sub-cooling and on the iron or glass vapor quantity respectively, will be determined, when plasma method is used. The technological process description of micro-sphere obtaining in plasma, is presented in [2]. 2. Theoretical model The production of micro-spheres with pre-established physical characteristics is the result of experimental research. The elaboration of a theoretical physical model, which takes into account all parameters that contribute to the production of the micro-spheres, is very difficult. Although, in the following, our goal is to describe the mechanisms that intervene in the process of producing micro-spheres in the plasma jet, but without exhausting the subject. At temperatures T ] 3000 K [3], the materials of interest (in the present case, the steel rods and the glass

powder, respectively) are in vapor state. Let’s assume that at temperature T the vapors partial pressure is pv, and the total pressure of the vapor– plasma gas mixture is p. Then, X=

pv p

(1)

is the molar ratio of the vapors. The molar density Cm of the vapor–plasma gas mixture can be computed (for low vapor concentrations) from the law of perfect gases, Cm =

p RT

(2)

where R is the universal constant of the perfect gases, and p is the total pressure of the mixture. Let’s consider that the vapors mixed with the plasma gas occupy the volume of a sphere with the radius r0,





3 m 1 r0 = 4p m C0

1 3

(3)

where m is the mass of vapors, m is the molar (atomic) mass of p the vapors, and C0 = v (4) RT is the molar concentration of the vapors. In places having temperatures T1 B T, the vapors in the peripheral areas of the sphere condense. In this way, a sphere with the radius r1 forms (r1 B r0), having

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I. Bica / Materials Science and Engineering B77 (2000) 293–296

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Fig. 1. Referring to the forming of micro-spheres, C0, the initial concentration of vapors; Cv, the critical concentration of germs in liquid phase; r1, the exterior radius of the micro-sphere; r2, the interior radius of the micro-sphere; d, the thickness of the wall of the micro-sphere; 1, melt membrane; 2, vapors mixed with gas; t1, the time corresponding to the formation of membrane; t2, the time corresponding to the formation of the wall of the micro-sphere.

kg of vapors, particles with a radius r1 = 9.5 mm are obtained. The particle diameter increases with the mass of vapors. For m= 700× 10 − 15 kg, the particle diameter reaches the value of 220 mm. The glass particles used in our study contain 93% SiO2; 6.2% Al2O3; 0.1% Na2O; 0.2% CaO; 0.5% Fe2O3. The molar mass of the glass vapors is considered to be m = 67 kg·kmol − 1, and the pressure of the vapors pv = 4× 107 N·m − 2 at T= 3000 K [5]. The condensation temperature of the vapors is T1 = 2000 K. For the glass particles, the dependence of r0 and r1, respectively, on the quantity of vapors m is presented in Fig. 3. It can be seen in Fig. 3 that the value of the radius r1 is dependent on the mass m of the vapors. For example, for m= 10× 10 − 15 kg of vapors, one gets r1 =2.5 mm, while for m=2100× 10 − 15 kg of vapors, one gets r1 = 30 mm. We consider that, inside the sphere of radius r1 (Fig. 1a), a vapor element comes in contact with the liquid membrane, for a certain time interval. During this time interval, the vapor diffusion occurs, in a non-stationary regime. At the end of this time period, this vapor volume is replaced with another one. Following the vapor diffusion, the growth of germs on the interior surface of the sphere occurs. At the end of the diffusion process a micro-sphere forms. The thickness of the micro-sphere wall is d (Fig. 1b). It consists of condensation germs. Their critical concentration is,

 

CC = C0 exp

DE RDT

(6)

where DE is the diffusion activation energy, and DT = T− T2 is the sub-cooling. For short contact times between the vapor volume and the liquid membrane, the vapor volume can be assumed semi-infinite. In these conditions we can apply the second Fick’s law. Projected on the Or axis in Fig. 1, it can be written,

Fig. 2. The dependence of the radii r0 and r1 on the quantity m of iron vapors.

as surface a liquid membrane (Fig. 1a). The molar concentration of the vapors in the sphere of radius r1 is C0. Assuming the same pressure of the vapor–plasma gas mixture, it results,

'

r1 =r0

3

T1 T

(5)

If we consider for iron, pv =8 × 104 N·m − 2 at T= 3000 K [13], m= 56 kg·kmol − 1, and T1 =1500 K, we obtain the dependence of r0 and r1 on the vapors quantity m, as presented in Fig. 2. For m =5 × 10 − 15

Fig. 3. The dependence of the radii r0 and r1 on the quantity m of vapors, in producing the glass micro-spheres.

I. Bica / Materials Science and Engineering B77 (2000) 293–296

(X ( 2X −D 2 =0 (t (r

for 0 5r B

and

and then Eq. (8) becomes,

0 5t 5t2 (7)

where D is the vapor diffusion coefficient in the sphere of radius r1. The coefficient D is assumed constant in time. The solution to Eq. (7) is,



X =A+ Berfc

r −r1

2 Dt



(8)

where A and B are integration constants, and erfc[(r − r1)/2 Dt] is the complementary to the error function. The constants A and B can be evaluated from the following initial and boundary conditions, for t = 0 and r B r1,

X =X0

for t= t2 and r = r2,

X =XC

295

?

X= X0 +



XC − X0 r− r1 erfc erfc(d/b) 2 Dt



(10)

where d = r2 − r1, and b= 2 Dt2 is a notation. The mean value of the relative molar flux of the vapors at the inner surface of the membrane (r=r1) in the time interval 05 t5t2 can be computed using Eq. (10) and the first Fick’s law. Hence, Ja=

2(XC − X0)Cm erfc(d/b)

'

D pt2

(11)

Assuming a concentration CC and the time value t2 for the formation of the wall of a thickness d, then, J a = 2pd

CC t2

(12)

Using Eqs. (11) and (12), and taking into account that C0 = CmX0, C= CmX and CC = CmXC, where Cm is the molar density of the mixture, it results,

 

C0 d = 1−2p p CC b

1− erf

 n d b

(13)

Taking into account Eq. (6), Eq. (13) becomes,

 

exp

DE 1 C = = C RDT 1−2p p(d/b)[1−erf(d/b)] C0

(14)

One can notice from Eq. (14) that, given a substance, the formation of the micro-sphere wall of thickness d, depends only on the sub-cooling DT, for constant diffusion coefficients. Let us denote, Fig. 4. The dependence of the non-dimensional parameter Y on Z, in the micro-spheres production process.

Fig. 5. The diffusion time t2 vs. the thickness d of the micro-sphere wall, (t2)Si-diffusion time for the glass micro-spheres; (t2)Fe-diffusion time for the iron micro-spheres.

Y=

DE RDT

and

Z=

d 2 Dt2

(15)

Eq. (14) gives a dependence of Y on Z, whose shape is presented in Fig. 4. For iron we have DE = 40×106 J·kmol − 1; D=8× 10 − 8 m2·s − 1 [4], and for simple ionic melted materials (glass, for instance) we have DE =30× 106 J kmol − 1; D= 2 × 10 − 7 m2 s − 1 [4]. In this case, from Fig. 4 we obtain the dependence of the time t2 on the thickness d of the wall, having the sub-cooling DT as a parameter (Fig. 5). It can be seen in Fig. 5 that the speed of the formation of the wall having a thickness d, at the same sub-cooling (DT = 2000 K) is approximately 30 times larger for the glass micro-spheres, in comparison to the iron micro-spheres. If applying the mass conservation law, corresponding to the first and last stages in the formation of a micro-sphere, one obtains,

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I. Bica / Materials Science and Engineering B77 (2000) 293–296

  

r0 3 DE r2 = r1 1− exp − r1 RDT

n

1 3

(17)

where r2 is the interior radius of the micro-sphere. After some simple calculations, the interior radius of the micro-sphere can be obtained from Eq. (17),

  

r 3 DE r2 = r1 − d=r1 1− 0 exp − r1 RDT

Fig. 6. The modification of the exterior radius r1 and of the interior radius r2 of the iron micro-sphere vs. the vapors mass m.

n

1 3

(18)

Taking into account r0 and r1 from Fig. 2 and from Fig. 3, respectively, we obtain the dependence of the interior radius r2 of the micro-sphere on the mass m of vapors (Figs. 6 and 7). It can be noticed from Figs. 6 and 7 that the thickness d of the micro-spheres wall can be controlled through the vapors quantity m.

3. Conclusions The theoretical model presented allows to determine, the mass of vapors necessary for the micro-sphere formation; the diameter and the thickness of the wall of micro-spheres depending on the sub-cooling DT of the gas–vapors system and, respectively, on the mass m of the vapors.

References Fig. 7. The modification of the exterior radius r1 and of the interior radius r2 of the glass micro-sphere vs. the vapors mass m.

4p 4p C0r 30 = CC(r 31 −r 32) 3 3

(16)

Using Eqs. (6) and (16) we obtain,

.

[1] G.J. Liu, L. David, Sr Wilcox, J. Mater. Res. 10 (1995) 84. [2] I. Bica, Obtaining of micro-spheres in plasma: experimental devices, Rev. Mexicana de Fisica (to be published). [3] I. Bica, The use of rotating electric arc for spherical particle production, Rev. Metal Madrid (2000) (to be published). [4] F. Oprea, Metallurgic Processes Theory, Didactica si Pedagogica, (Ed.), Bucuresti, (1984) (in Romanian). [5] I. Bica, Rev. de Sold. (Madrid) 27 (1997) 17.