6.1.Lattice Defects and Diffusion 111 which can be solved whenever a specific set of boundary condi- tions is known.It should be noted that the diffusion coefficient in Eq.(6.10)has only a constant value if one considers self- diffusion or possibly for very dilute systems.In general,however, D varies with concentration so that Eq.(6.10)must be written in more general terms as: (6.10a) at ax The solution of Eq.(6.10),assuming D to be constant with con- centration and for "infinitely long"samples (i.e.,rods which are long enough so that the composition does not change at the outer ends),is: Ci-Cx =erf( (6.11) Ci-Co In other words the solution(6.11)is only valid when the length of a sample is larger than 10VDt.The parameters in Eq.(6.11) are as follows:Cx is the concentration of the solute at the dis- tance x;Ci is the constant concentration of the solute at the in- terface dividing materials A and B after some time,t;and Co is the initial solute concentration in material B.The initial solute (2C:-Co) Ci Cx o 0 X X Material A Material B (Solute) (Host material) FiGURE 6.6.Concentration profiles (called also penetration curves)for nonsteady-state diffusion of material A into material B for three differ- ent times.The concentration of the solute in material B at the distance x is named Cx.The solute concentration at X =0,that is,at the interface between materials A and B,is Ci.The original solute concentration in material B (or at X=)is Co.A mirror image of the diffusion of B into A can be drawn if the mutual diffusivities are identical.This is omitted for clarity.which can be solved whenever a specific set of boundary condi￾tions is known. It should be noted that the diffusion coefficient in Eq. (6.10) has only a constant value if one considers self￾diffusion or possibly for very dilute systems. In general, however, D varies with concentration so that Eq. (6.10) must be written in more general terms as:   C t
  x  D   C x  . (6.10a) The solution of Eq. (6.10), assuming D to be constant with con￾centration and for “infinitely long” samples (i.e., rods which are long enough so that the composition does not change at the outer ends), is:
erf  . (6.11) In other words the solution (6.11) is only valid when the length of a sample is larger than 10Dt. The parameters in Eq. (6.11) are as follows: Cx is the concentration of the solute at the dis￾tance x; Ci is the constant concentration of the solute at the in￾terface dividing materials A and B after some time, t; and C0 is the initial solute concentration in material B. The initial solute x 2Dt Ci  Cx Ci  C0 6.1 • Lattice Defects and Diffusion 111 X Ci Cx C0 Concentration of solute A (2Ci – C0) 0 x t 2 t 0 t 1 Material A (Solute) Material B (Host material) FIGURE 6.6. Concentration profiles (called also penetration curves) for nonsteady-state diffusion of material A into material B for three differ￾ent times. The concentration of the solute in material B at the distance x is named Cx. The solute concentration at X
0, that is, at the interface between materials A and B, is Ci. The original solute concentration in material B (or at X
") is C0. A mirror image of the diffusion of B into A can be drawn if the mutual diffusivities are identical. This is omitted for clarity.