6.1.Lattice Defects and Diffusion 107 FIGURE 6.3.Schematic representation of a diffusion channel caused by an edge dislocation.(Disloca- tion-core diffusion.) ary diffusion.One should keep in mind,however,that,except for thin films,etc.,the surface area comprises only an extremely small fraction of the total number of atoms of a solid.Moreover, surfaces are often covered by oxides or other layers which have been deliberately applied or which have been formed by contact with the environment.Thus,surface diffusion represents gener- ally only a small fraction of the total diffusion. Dislocation- Finally,a dislocation core may provide a two-dimensional chan- Core nel for diffusion as shown in Figure 6.3.The cross-sectional area of this core is about 4d2,where d is the atomic diameter.Very Diffusion appropriately,the mechanism is called dislocation-core diffusion or pipe diffusion. 6.1.3 Rate The number of jumps per second which atoms perform into a Equation neighboring lattice site,that is,the rate or frequency for move- ment,f,is given again by an Arrhenius-type equation: f=fo exp (6.3) where fo is a constant that depends on the number of equivalent neighboring sites and on the vibrational frequency of atoms (about 1013 s-1).Q is again an activation energy for the process in question. We see from Eq.(6.3)that the jump rate is strongly tempera- ture-dependent.As an example,one finds for diffusion of carbon atoms in iron at room temperature(Q=0.83 ev)about one jump
ary diffusion. One should keep in mind, however, that, except for thin films, etc., the surface area comprises only an extremely small fraction of the total number of atoms of a solid. Moreover, surfaces are often covered by oxides or other layers which have been deliberately applied or which have been formed by contact with the environment. Thus, surface diffusion represents generally only a small fraction of the total diffusion. Finally, a dislocation core may provide a two-dimensional channel for diffusion as shown in Figure 6.3. The cross-sectional area of this core is about 4d2, where d is the atomic diameter. Very appropriately, the mechanism is called dislocation-core diffusion or pipe diffusion. The number of jumps per second which atoms perform into a neighboring lattice site, that is, the rate or frequency for movement, f, is given again by an Arrhenius-type equation: f f0 exp k Q BT , (6.3) where f0 is a constant that depends on the number of equivalent neighboring sites and on the vibrational frequency of atoms (about 1013 s1). Q is again an activation energy for the process in question. We see from Eq. (6.3) that the jump rate is strongly temperature-dependent. As an example, one finds for diffusion of carbon atoms in iron at room temperature (Q 0.83 eV) about one jump 6.1 • Lattice Defects and Diffusion 107 FIGURE 6.3. Schematic representation of a diffusion channel caused by an edge dislocation. (Dislocation-core diffusion.) DislocationCore Diffusion 6.1.3 Rate Equation
108 6·Atoms in Motion ←一TK volume diffusion FIGURE 6.4.Schematic representation of grain-boundary an Arrhenius diagram.Generally a loga- Inf diffusion rithmic scale (base 10)and not an In scale is used.The adjustment from In to log is a factor of 2.3.The difference be- tween volume diffusion and grain bound- ary diffusion is explained in the text.The slopes represent the respective activation energies. 1/TK-]→ in every 25 seconds.At the melting point of iron (1538C)the jump rate dramatically increases to 2 X 1011 per second. 6.1.4 Arrhenius equations are generally characterized by an exponential Arrhenius term that contains an activation energy for the process involved as well as the reciprocal of the absolute temperature.It is quite Diagrams customary to take the natural logarithm of an Arrhenius equation. For example,taking the natural logarithm of Eg.(6.3)yields: nf=a6-(份)片 (6.4) This expression has the form of an equation for a straight line, which is generically written as: y=b+mx. (6.5) Staying with the just-presented example,one then plots the ex- perimentally obtained rate using a logarithmic scale versus 1/T as depicted in Figure 6.4.The (negative)slope,m,of the straight line in an Arrhenius diagram equals QkB,from which the acti- vation energy can be calculated.The intersect of the straight line with the y-axis yields the constant fo.This procedure is widely used by scientists to calculate activation energies from a series of experimental results taken at a range of temperatures. 6.1.5 Self-diffusion is random,that is,one cannot predict in which di- Directional rection a given lattice atom will jump if it is surrounded by two or more equivalent vacancies.Indeed,an individual atom often Diffusion migrates in a haphazard zig-zag path.In order that a bias in the direction of the motion takes place,a driving force is needed.Dri-
in every 25 seconds. At the melting point of iron (1538°C) the jump rate dramatically increases to 2 1011 per second. Arrhenius equations are generally characterized by an exponential term that contains an activation energy for the process involved as well as the reciprocal of the absolute temperature. It is quite customary to take the natural logarithm of an Arrhenius equation. For example, taking the natural logarithm of Eq. (6.3) yields: ln f ln f0 k Q B T 1 . (6.4) This expression has the form of an equation for a straight line, which is generically written as: y b mx. (6.5) Staying with the just-presented example, one then plots the experimentally obtained rate using a logarithmic scale versus 1/T as depicted in Figure 6.4. The (negative) slope, m, of the straight line in an Arrhenius diagram equals Q/kB, from which the activation energy can be calculated. The intersect of the straight line with the y-axis yields the constant f0. This procedure is widely used by scientists to calculate activation energies from a series of experimental results taken at a range of temperatures. Self-diffusion is random, that is, one cannot predict in which direction a given lattice atom will jump if it is surrounded by two or more equivalent vacancies. Indeed, an individual atom often migrates in a haphazard zig-zag path. In order that a bias in the direction of the motion takes place, a driving force is needed. Dri- 108 6 • Atoms in Motion volume diffusion grain-boundary diffusion 1/T [K–1] ln f T [K] FIGURE 6.4. Schematic representation of an Arrhenius diagram. Generally a logarithmic scale (base 10) and not an ln scale is used. The adjustment from ln to log is a factor of 2.3. The difference between volume diffusion and grain boundary diffusion is explained in the text. The slopes represent the respective activation energies. 6.1.4 Arrhenius Diagrams 6.1.5 Directional Diffusion
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 jump
ving forces are, for example, provided by concentration gradients in an alloy (that is, by regions in which one species is more abundant compared to another species). Directional diffusion can also occur as a consequence of a strong electric current (electromigration) or a temperature gradient (thermomigration). We learned already in Chapter 5 that concentration gradients may occur during solidification of materials (coring). These concentration gradients need to be eliminated if a homogeneous equilibrium structure is wanted. The mechanism by which homogenization can be accomplished makes use of the just-discussed drift of atoms down a concentration gradient. Diffusion also plays a role in age hardening, surface oxidation, heat treatments, 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 concentration gradient, C/ x, through a cross-sectional area, A, and in a given time interval, t. The concentration, C, is given, for example, 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-exponential diffusion constant (given in m2/s). The latter is tabulated in diffusion handbooks for many combinations of elements. A selection 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