ISSN 10637850, Technical Physics Letters, 2014, Vol. 40, No. 6, pp. 460–461. © Pleiades Publishing, Ltd., 2014. Original Russian Text © V.I. Bochegov, A.S. Parakhin, 2014, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2014, Vol. 40, No. 11, pp. 17–22.

Limiting Impurity Distribution during Zone Refining V. I. Bochegov* and A. S. Parakhin Herzen State Pedagogical University of Russia, St. Petersburg, 191186 Russia Kurgan State University, Kurgan, 640669 Russia *email: [email protected] Received November 25, 2013

Abstract—The problem of the limiting distribution of the impurity removed using the zonerefining tech nique is considered for different distribution factors and molten zone sizes. An analytical solution is suggested and some examples are discussed in the approximation of a long finitelength ingot with a uniform cross sec tion and constant moltenzone size. DOI: 10.1134/S1063785014060029

Interest in the theory and practice of zone refining has lately rekindled [1–3]. Unlike these papers, as well as the major work of Pfann [4], in which the issue of the limiting distribution has either been completely ignored or calcu lation of the limiting distribution was provided for the approximation of a semiinfinite ingot, this Letter offers a direct analytical calculation of the limiting distribution (corresponding to the infinite amount of zone passages) in the refined long finitelength ingot with uniform cross section and constant molten zone size. The suggested solution procedure requires no individual successive problem solution for each zone passage and provides an exacter distribution in the normal crystallization region at the ingot end. Knowledge of the function of this distri bution allows estimating the required (reasonable) amount of zone passages in the course of the zone refin ing of the given material as dependent on the given impu rity. In addition, the solution of this problem, together with the technique from [3], allowed us to develop soft ware not requiring large computer resources and provid ing the user with an illustrative estimation of the amount of zone passages in case of zone refining [5]. To find directly the limiting distribution in a finite length ingot, a differential equation must be solved that is fulfilled for the region of 0 ≤ x ≤ l0 – 2l: dc s ( x )  = k ( c s ( x + l ) – c s ( x ) ). (1) dx l Here, l0 is the length of the whole ingot, x is the coor dinate of the crystallization front relative to the origin of the ingot, k is the impurity distribution factor, cs(x) is the impurity concentration function on coordinate x in the solid phase, and l is the moltenzone length. As opposed to work [3], both addends in this differ ential equation on the right of the equals sign are the same soughtfor function, which corresponds to the limiting distribution. In addition, the ingot is divided lengthwise into only three regions, irrespective of the amount of passages: (1) 0 ≤ x ≤ l0 – 2l, (2) l0 – 2l < x ≤ l0 – l, and (3) l0 – l < x ≤ l0. The solution of Eq. (1) can

be obtained analytically in the form of the following expression: Bx

(2) c sI ( x ) = Ae , and it is true, as stated above, only in the first region. Here, A and B are some constants, csI(x) is the impurity distribution function in the first region. Constant B must satisfy simultaneously conditions (1) and (2), which corresponds to the equation Bl B = k ( e – 1 ). (3) l It can be solved only numerically for B. Thus, Bl = 1.25643121 for k = 0.5 and Bl = –1.59362426 for k = 2. Constant A is determined on the basis of the nor malization condition: l 0 – 2l

c sr l 0 =

∫

l0 – l

c sI ( x ) dx +

∫

l0

c sII ( x ) dx +

∫c

sIII ( x ) dx,

(4) 0 l 0 – 2l l0 – l where csr is the initial (average along the ingot length) impurity concentration and csII and csIII are the distri butions of the concentration in the second and third region, respectively. The third region must satisfy the following equa tion: dc sIII 1 – k c , (5)   =  sIII dx l0 – x corresponding to normal crystallization. The solution of this equation is a function of k–1 c sIII ( x ) = D ( l 0 – x ) , (6) where D is a constant obtained on the basis of the fol lowing condition: c sII ( l 0 – l ) = c sIII ( l 0 – l ). (7) The second region must satisfy the following equa tion: dc sII k–1 k – c sII ( x ) ),  =  ( D ( l 0 – x – l ) (8) dx l

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LIMITING IMPURITY DISTRIBUTION DURING ZONE REFINING

461

Taking into account the above, we obtain the distri bution in all regions. The first region corresponds to formula (2). The second region corresponds to

C/Csr 4.0 3.5

B ( l – 2l )

Ae 0 c sII ( x ) =  S ( x ) k–1 k S ( l 0 – 2l ) + l e

(a)

3.0

B ( l 0 – 2l )

Ae +  l k–1 k S ( l 0 – 2l ) + l e

2.5 2.0

k ( l0 – l – x )  l k–1

e

(11) .

In the third region, B ( l – 2l )

1.5

k–1 Ae 0 c sIII ( x ) =  (l – x) . k–1 k 0 S ( l 0 – 2l ) + l e

1.0

S is determined according to the following expression:

0.5 0 C/Csr 4.0

k S(x) = – l

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/l0

∞

∑

n=0

k+n

n ( l0 – x – l ) ⎛ k ⎞  . ⎝ l ⎠ k ( k + 1 ) ( k + 2 )… ( k + n )

(13)

where n is the amount of zone passages. We showed that series (13) is convergent at any k. Figure 1 shows examples of the limiting impurity dis tribution for k = 2 (k > 1) and k = 0.5 (k < 1). As already stated, we developed user software [5] for calculation of the distribution at any amount of zone passages, including the limiting distribution, for any combinations of the distribution factor and mol tenzone size. The software allows optimizing the refining process in terms of energy and time efficiency. Thus, e.g., as seen from the presented plots (see Figs. 1a, 1b), in the case of l = 0.2 and k = 2, the dis tribution of the impurity to be removed becomes prac tically limiting after ten zone passages, while, in the case of l = 0.2 and k = 0.5, practical approximation to the limiting distribution requires at least seven or eight zone passages. The presented software calculates not only the limiting distribution according to the above procedure (shown in the plots as a thick line), but also the impurity distribution after each zone passage, in accordance with the procedure from [3] (thinner lines in the plot).

(b) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

(12)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X/l0

Fig. 1. Limiting distribution (thick line) for the case of moltenzone size (in fractions of ingot length) l = 0.2 and impurity distribution factor (a) k = 2 and (b) k = 0.5. The vertical axis corresponds to the ratio C/Csr of the impurity concentration to its initial (average) concentration, and the horizontal axis corresponds to the coordinate relative to the origin of the ingot (also in ingot length fractions X/l0). Thinner lines correspond to the distribution with a finite amount of passages with their numbers increasing as the lines approach the limiting distribution.

REFERENCES 1. T. Cheung, N. Cheung, C. M. T. Tobar, R. Caram, and A. Garcia, Mater. Manuf. Processes, 26 (3), 493 (2011). 2. T. Cheung, N. Cheung, and A. Garcia, J. Electron. Mater. 39 (1), 49 (2010). 3. V. I. Bochegov and A. S. Parakhin, Vestn. Kurgan. Gos. Univ., Ser. Estestv. Nauki 3 (5), 83 (2012).

its solution being kx

kx

kx

–   –  k–1 l l l c sII ( x ) = k De e ( l 0 – x – l ) dx + Fe , (9) l where F is a constant determined on the basis of the following condition: c sI ( l 0 – 2l ) = c sll ( l 0 – 2l ). (10)

∫

TECHNICAL PHYSICS LETTERS

Vol. 40

No. 6

2014

4. W.G. Pfann, Zone Melting (Wiley, New York, 1966). 5. V. I. Bochegov and A. S. Parakhin, http://infima.kgsu. ru/index.php?option=com_content&view=article&id =102:20130726061929&catid=34:2013072605 5426&Itemid=38

Translated by M. Ehrenburg