Abstract

Concentration distributions produced from the combined effect of a unidirectional electric field and diffusion are presented for a slab and for a hollow cylinder. A field in the direction such that slow ions follow fast ones results in a stationary distribution that moves with a constant or nearly constant velocity. If the field is in the opposite direction, it tends to mix the ions, and an entirely different distribution that never reaches a steady state is obtained. These results are discussed in the context of the production of graded-index optical materials.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Nishizawa, in Digest of Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper TuB3.
  2. P. K. Gupta, A. R. Cooper, Physica 54, 39 (1971).
    [CrossRef]
  3. N. Weber, U.S. Patent3,218,220 (1965).
  4. M. A. el-Leil, A. R. Cooper, J. Am. Ceram. Soc. 62, 390 (1979).
    [CrossRef]
  5. K. S. Spiegler, C. D. Coryell, J. Phys. Chem. 56, 106 (1952).
    [CrossRef]
  6. M. Chemla, Ann. Phys. (Leipzig) 1, 959 (1956).
  7. T. Kaneko, H. Yamamoto, in Proceedings, Tenth International Congress on Glass (Ceramic Society of Japan, Tokyo, 1974), 8-19-86.
  8. H. Ohta, M. Hara, Rep. Res. Lab Asahi Glass 20 (1), 14 (1970).
  9. F. Baucke, Mater. Sci. Res. 9, 337 (1974).
  10. G. H. Chartier, P. Jaussaud, A. D. Oliveira, O. Parriaux, Electron. Lett. 14, 132 (1978).
    [CrossRef]
  11. D. E. Day, Amorphous Materials (Wiley, New York, 1972).
  12. M. Abou el-Leil, Ph.D. Thesis, Case Western Reserve University, Cleveland, Ohio (1978).

1979 (1)

M. A. el-Leil, A. R. Cooper, J. Am. Ceram. Soc. 62, 390 (1979).
[CrossRef]

1978 (1)

G. H. Chartier, P. Jaussaud, A. D. Oliveira, O. Parriaux, Electron. Lett. 14, 132 (1978).
[CrossRef]

1974 (1)

F. Baucke, Mater. Sci. Res. 9, 337 (1974).

1971 (1)

P. K. Gupta, A. R. Cooper, Physica 54, 39 (1971).
[CrossRef]

1970 (1)

H. Ohta, M. Hara, Rep. Res. Lab Asahi Glass 20 (1), 14 (1970).

1956 (1)

M. Chemla, Ann. Phys. (Leipzig) 1, 959 (1956).

1952 (1)

K. S. Spiegler, C. D. Coryell, J. Phys. Chem. 56, 106 (1952).
[CrossRef]

Abou el-Leil, M.

M. Abou el-Leil, Ph.D. Thesis, Case Western Reserve University, Cleveland, Ohio (1978).

Baucke, F.

F. Baucke, Mater. Sci. Res. 9, 337 (1974).

Chartier, G. H.

G. H. Chartier, P. Jaussaud, A. D. Oliveira, O. Parriaux, Electron. Lett. 14, 132 (1978).
[CrossRef]

Chemla, M.

M. Chemla, Ann. Phys. (Leipzig) 1, 959 (1956).

Cooper, A. R.

M. A. el-Leil, A. R. Cooper, J. Am. Ceram. Soc. 62, 390 (1979).
[CrossRef]

P. K. Gupta, A. R. Cooper, Physica 54, 39 (1971).
[CrossRef]

Coryell, C. D.

K. S. Spiegler, C. D. Coryell, J. Phys. Chem. 56, 106 (1952).
[CrossRef]

Day, D. E.

D. E. Day, Amorphous Materials (Wiley, New York, 1972).

el-Leil, M. A.

M. A. el-Leil, A. R. Cooper, J. Am. Ceram. Soc. 62, 390 (1979).
[CrossRef]

Gupta, P. K.

P. K. Gupta, A. R. Cooper, Physica 54, 39 (1971).
[CrossRef]

Hara, M.

H. Ohta, M. Hara, Rep. Res. Lab Asahi Glass 20 (1), 14 (1970).

Jaussaud, P.

G. H. Chartier, P. Jaussaud, A. D. Oliveira, O. Parriaux, Electron. Lett. 14, 132 (1978).
[CrossRef]

Kaneko, T.

T. Kaneko, H. Yamamoto, in Proceedings, Tenth International Congress on Glass (Ceramic Society of Japan, Tokyo, 1974), 8-19-86.

Nishizawa, K.

K. Nishizawa, in Digest of Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper TuB3.

Ohta, H.

H. Ohta, M. Hara, Rep. Res. Lab Asahi Glass 20 (1), 14 (1970).

Oliveira, A. D.

G. H. Chartier, P. Jaussaud, A. D. Oliveira, O. Parriaux, Electron. Lett. 14, 132 (1978).
[CrossRef]

Parriaux, O.

G. H. Chartier, P. Jaussaud, A. D. Oliveira, O. Parriaux, Electron. Lett. 14, 132 (1978).
[CrossRef]

Spiegler, K. S.

K. S. Spiegler, C. D. Coryell, J. Phys. Chem. 56, 106 (1952).
[CrossRef]

Weber, N.

N. Weber, U.S. Patent3,218,220 (1965).

Yamamoto, H.

T. Kaneko, H. Yamamoto, in Proceedings, Tenth International Congress on Glass (Ceramic Society of Japan, Tokyo, 1974), 8-19-86.

Ann. Phys. (Leipzig) (1)

M. Chemla, Ann. Phys. (Leipzig) 1, 959 (1956).

Electron. Lett. (1)

G. H. Chartier, P. Jaussaud, A. D. Oliveira, O. Parriaux, Electron. Lett. 14, 132 (1978).
[CrossRef]

J. Am. Ceram. Soc. (1)

M. A. el-Leil, A. R. Cooper, J. Am. Ceram. Soc. 62, 390 (1979).
[CrossRef]

J. Phys. Chem. (1)

K. S. Spiegler, C. D. Coryell, J. Phys. Chem. 56, 106 (1952).
[CrossRef]

Mater. Sci. Res. (1)

F. Baucke, Mater. Sci. Res. 9, 337 (1974).

Physica (1)

P. K. Gupta, A. R. Cooper, Physica 54, 39 (1971).
[CrossRef]

Rep. Res. Lab Asahi Glass (1)

H. Ohta, M. Hara, Rep. Res. Lab Asahi Glass 20 (1), 14 (1970).

Other (5)

K. Nishizawa, in Digest of Topical Meeting on Gradient Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1979), paper TuB3.

T. Kaneko, H. Yamamoto, in Proceedings, Tenth International Congress on Glass (Ceramic Society of Japan, Tokyo, 1974), 8-19-86.

N. Weber, U.S. Patent3,218,220 (1965).

D. E. Day, Amorphous Materials (Wiley, New York, 1972).

M. Abou el-Leil, Ph.D. Thesis, Case Western Reserve University, Cleveland, Ohio (1978).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Schematic of process for field-assisted ion exchange of tubes.

Fig. 2
Fig. 2

Concentration distribution achieved after application of current j for time t on an initial step function in concentration at x = 0 when the fast ion follows the slow ion. (Horizontal lines represent the concentration of the original step function.)

Fig. 3
Fig. 3

Concentration distribution achieved after application of current j for time t on an initial step function in concentration at x = 0 when the slow ion follows the faster ion.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

j A = D A C A x + E C A D A / kT .
C t = 1 [ C 0 + ( M 1 ) C ] 2 ( { C 0 [ C 0 + ( M 1 ) C ] D } _ 2 C x 2 { C 0 _ + ( C 0 C ) C ln M C } MJ C x { ( M 1 ) _ + CM ln M C [ C 0 + ( M 1 ) C ] × ln D C } D C 0 ( C x ) 2 ) ,
C t = Mj C 0 [ C 0 + ( M 1 ) C ] 2 C x .
d C d t = ( C x ) t dx dt + ( C t ) x .
( C t ) x + ( x t ) c ( C x ) t = 0 .
Mj C 0 [ C 0 + ( M 1 ) C ] 2 = ( x t ) C ,
Δ X ( C ) = Mj C 0 t [ C 0 + ( M 1 ) C ] 2 .
C = f { x Mj C 0 t [ C 0 + ( M 1 ) C ] 2 } ,
x < Mj C 0 t [ M C s + ( C 0 C s ) ] 2 C = C s x > Mjt C 0 [ M C g + ( C 0 C g ) ] 2 C = C = C g } ,
C = ( Mj C 0 t / x ) 1 / 2 C 0 M 1 ,
( C C g ) / ( C s C g ) = [ 1 + exp υ ( 1 M ) ( C s C g ) ( x υ t ) D C 0 ] 1 ,
υ = Mj C 0 [ C 0 + ( M 1 ) C g ] [ C 0 + ( M 1 ) C s ] .
C / C 0 = { 1 + exp [ ( j / C 0 D ) ( 1 M ) ( x j t / C 0 ) ] } 1 .
υ = Mj ( C 0 C s ) + C s M .
c t = 1 [ C 0 + ( M 1 ) C ] 2 ( C 0 [ C 0 + ( M 1 ) C ] D 2 C r 2 + { D r [ C 0 2 + ( M 1 ) C C 0 ] MJ C 0 r } × C r C 0 ( M 1 ) D ( C r ) 2 ) .
C t 1 [ C 0 + ( M 1 ) C ] 2 { C 0 [ C 0 + ( M 1 ) C ] × D 2 C r 2 MJ C 0 r ( C r ) C 0 ( M 1 ) D ( C r ) 2 } .
( r t ) C = MJ C 0 r [ C 0 + ( M 1 ) C ] 2 .
Δ r 2 ( C ) = 2 MJ C 0 t [ C 0 + ( M 1 ) C ] 2 .
C = g { r 2 R 1 2 2 MJ C 0 t [ C 0 + ( M 1 ) C ] 2 } ,
( r 2 R 1 2 ) < 2 MJ t C 0 [ M C s + ( C 0 C s ) ] 2 C = C s ( r 2 R 1 2 ) > 2 MJ t C 0 [ M C g + ( C 0 C g ) ] 2 C = C g } ,
C = [ 2 MJ C 0 t / ( r 2 R 1 2 ) ] 1 / 2 C 0 M 1 .
( R 2 2 R 1 2 ) < 2 MJt C 0 / ( M C g + C 0 C g ) 2 .
t > ( R 2 2 R 1 2 ) [ M C s + ( C 0 C s ) ] 2 2 MJ C 0 .
C C g C s C g = [ 1 + exp υ ( 1 M ) ( C s C g ) ( r R 1 υ t ) D C 0 ] 1 ,
υ = ( 1 υ t / R 1 ) MJ C 0 R 1 [ C 0 + ( M 1 ) C g ] [ C 0 + ( M 1 ) C s ] .

Metrics