112 6·Atoms in Motion -1.0 0.8 -0.6 -0.4 0.2 0.2 0.4 erf(y)0.6 FIGURE 6.7.The Gaussian er- ror function erf(y)as a func- 0.8 tion of y.Compare to Figure 6.6. 1.0 concentration in material A is 2C:-Co;see Figure 6.6.The right- hand side in Eq.(6.11)is called the probability integral or Gauss- ian error function,which is defined by: erf(y)=2 Ved (6.12) It is tabulated in handbooks similarly as trigonometric or other functions and is depicted in Figure 6.7.It can be observed that the error function(Figure 6.7)reproduces the concentration pro- file of Figure 6.6 quite well.(See also Problem 6.1.)A reasonable estimate for the distance,X,stating how far solute atoms may diffuse into a matrix during a time interval,t,can be obtained from: X=2VDt. (6.13) 6.1.8 The boundary condition for which Eq.(6.10)was solved stipu- lates,as already mentioned,that the initial compositions of ma- Interdiffusion terials A and B do not change at the free ends.Now,if the dif- fusion process is allowed to take place for long times,and if it is additionally conducted at high temperatures,a complete mixing of the two materials A and B will eventually take place.In other words,a uniform concentration of both components is then found along the entire couple,as indicated in Figure 6.6 by the horizontal line marked Ci.In this case,which is called complete interdiffusion,Eq.(6.11)is no longer a valid solution of Fick's second law.Moreover,Eq.(6.11)loses its validity long before a uniform composition is attained.concentration in material A is 2Ci–C0; see Figure 6.6. The right￾hand side in Eq. (6.11) is called the probability integral or Gauss￾ian error function, which is defined by: erf( y)
 2   y 0 e2 d. (6.12) It is tabulated in handbooks similarly as trigonometric or other functions and is depicted in Figure 6.7. It can be observed that the error function (Figure 6.7) reproduces the concentration pro￾file of Figure 6.6 quite well. (See also Problem 6.1.) A reasonable estimate for the distance, X, stating how far solute atoms may diffuse into a matrix during a time interval, t, can be obtained from: X
2Dt. (6.13) The boundary condition for which Eq. (6.10) was solved stipu￾lates, as already mentioned, that the initial compositions of ma￾terials A and B do not change at the free ends. Now, if the dif￾fusion process is allowed to take place for long times, and if it is additionally conducted at high temperatures, a complete mixing of the two materials A and B will eventually take place. In other words, a uniform concentration of both components is then found along the entire couple, as indicated in Figure 6.6 by the horizontal line marked Ci. In this case, which is called complete interdiffusion, Eq. (6.11) is no longer a valid solution of Fick’s second law. Moreover, Eq. (6.11) loses its validity long before a uniform composition is attained. 112 6 • Atoms in Motion –1.0 –0.8 –0.6 –0.4 –0.2 1 –2 –1 0 0.2 0.4 0.6 0.8 1.0 2 erf(y) Y FIGURE 6.7. The Gaussian er￾ror function erf(y) as a func￾tion of y. Compare to Figure 6.6. 6.1.8 Interdiffusion