594 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



where (7.15) and the Emstein relation have been used, and Do is the 

 diffusivity in the absence of pairing. 



P/Nd in (11.6) can be evaluated using (9.5) so that the coefficient pre- 

 ceding (dNo/dx) contains No as the only variable. 



In Appendix B it is shown that ion pairing itself leads to severe de- 

 partures from Fick's law.*^ In fact the diffusion flux density in the pres- 

 ence of pairing is given by 



-2 l^" - ^- + 5) 

 l/i(^° - iV. - i) + 



'-f° (11.7) 



dx 



Here again the diffusivity is specified by the factors preceding (dNo/dx) 

 and, though variable, depends only on No , the local concentration of 

 diffusant. Adding the two coefficients appearing in (11.6) and (11.7) the 

 value of the diffusivity, D, in the presence of both pairing and diffusion 

 potential is obtained. Thus 



D-^\l + 





(11.8) 



It is obvious from (11.8) that even in the absence of space charge D is 

 an extremely complicated function of Nd , and will be much more com- 

 plex if space charge needs to be considered. When Nd « A^.i (11.8) re- 

 duces to 



Comparison with equation (B15) shows that when (11.8) is true (i.e., 

 in the absence of space charge) the diffusion potential may be ignored 

 for Nd <3C Na • Comparison of (B14) with (B15) shows how much D can 

 vary with Nd when ion pairing occurs. 



The proper study of diffusion in the presence of ion pairing should be 

 augmented by a mathematical analysis, accounting for the concentra- 

 tion dependent diffusivity. Since this dependence is complicated the 

 resulting boundary value problem must be solved numerically, and this 



