6.1.Lattice Defects and Diffusion 109 ving forces are,for example,provided by concentration gradients in an alloy (that is,by regions in which one species is more abun- dant compared to another species).Directional diffusion can also occur as a consequence of a strong electric current (electromi- gration)or a temperature gradient (thermomigration). We learned already in Chapter 5 that concentration gradients may occur during solidification of materials(coring).These con- centration gradients need to be eliminated if a homogeneous equilibrium structure is wanted.The mechanism by which ho- mogenization can be accomplished makes use of the just-dis- cussed drift of atoms down a concentration gradient.Diffusion also plays a role in age hardening,surface oxidation,heat treat- ments,sintering,doping in microelectronic circuits,diffusion bonding,grain growth,and many other applications.Thus,we need to study this process in some detail. 6.1.6 Fick's first law describes the diffusion of atoms driven by a con- Steady-State centration gradient,aC/ax,through a cross-sectional area,A,and in a given time interval,t.The concentration,C,is given,for ex- Diffusion ample,in atoms per m3.The pertinent equation was derived in 1855 by A.Fick and reads for one-dimensional atom flow: J=-D iC (6.6) Ox where J is called the flux: J=M (6.7) measured in atoms per m2 and per second (see Figure 6.5)and D is the diffusion coefficient or diffusivity (given in m2/s).M is defined as mass or,equivalently,as the number of atoms.The negative sign indicates that the atom flux occurs towards lower concentrations,that is,in the downhill direction.The diffusivity depends,as expected,on the absolute temperature,T,and on an activation energy,Q,according to an Arrhenius-type equation: D=Do exp Q (6.8) kBT where Do is called the (temperature-independent)pre-exponen- tial diffusion constant(given in m2/s).The latter is tabulated in diffusion handbooks for many combinations of elements.A se- lection of diffusion constants is listed in Table 6.1.There exists a connection between the diffusion coefficient,D,and the jumpving forces are, for example, provided by concentration gradients in an alloy (that is, by regions in which one species is more abun￾dant compared to another species). Directional diffusion can also occur as a consequence of a strong electric current (electromi￾gration) or a temperature gradient (thermomigration). We learned already in Chapter 5 that concentration gradients may occur during solidification of materials (coring). These con￾centration gradients need to be eliminated if a homogeneous equilibrium structure is wanted. The mechanism by which ho￾mogenization can be accomplished makes use of the just-dis￾cussed drift of atoms down a concentration gradient. Diffusion also plays a role in age hardening, surface oxidation, heat treat￾ments, sintering, doping in microelectronic circuits, diffusion bonding, grain growth, and many other applications. Thus, we need to study this process in some detail. Fick’s first law describes the diffusion of atoms driven by a con￾centration gradient, C/x, through a cross-sectional area, A, and in a given time interval, t. The concentration, C, is given, for ex￾ample, in atoms per m3. The pertinent equation was derived in 1855 by A. Fick and reads for one-dimensional atom flow: J
D   C x , (6.6) where J is called the flux: J
A M t , (6.7) measured in atoms per m2 and per second (see Figure 6.5) and D is the diffusion coefficient or diffusivity (given in m2/s). M is defined as mass or, equivalently, as the number of atoms. The negative sign indicates that the atom flux occurs towards lower concentrations, that is, in the downhill direction. The diffusivity depends, as expected, on the absolute temperature, T, and on an activation energy, Q, according to an Arrhenius-type equation: D
D0 exp  k Q BT  , (6.8) where D0 is called the (temperature-independent) pre-exponen￾tial diffusion constant (given in m2/s). The latter is tabulated in diffusion handbooks for many combinations of elements. A se￾lection of diffusion constants is listed in Table 6.1. There exists a connection between the diffusion coefficient, D, and the jump 6.1.6 Steady-State Diffusion 6.1 • Lattice Defects and Diffusion 109