Atoms in Motion 6.1.Lattice Defects and Diffusion So far,when discussing the properties of materials we tacitly as- sumed that the atoms of solids remain essentially stationary.From time to time we implied,however,that the behavior of solids is af- fected by the thermally induced vibrations of atoms.The changes in properties increase even more when atoms migrate through the lattice and take new positions.In order to gain a deeper insight into many mechanical properties,we therefore need to study the "dy- namic case."It will become obvious during our endeavor that the motion of atoms through solids involves less effort (energy)when open spaces are present in a lattice,as encountered,for example, by empty lattice sites.Thus,we commence with this phenomenon. 6.1.1 Lattice We have repeatedly pointed out in previous chapters that an ideal Defects lattice is rarely found under actual conditions,that is,a lattice in which all atoms are regularly and periodically arranged over large distances.This is particularly true at high temperatures, where a substantial amount of atoms frequently and randomly change their positions leaving behind empty lattice sites,called vacancies.Even at room temperature,at which thermal motion of atoms is small,a fair number of lattice defects may still be found.The number of vacancies per unit volume,ny,increases exponentially with the absolute temperature,T,according to an equation whose generic type is commonly attributed to Arrhenius: ny=ns exp (6.1) ISvante August Arrhenius(1859-1927),Swedish chemist and founder of modern physical chemistry,received in 1903,as the first Swede,the Nobel prize in chemistry.The Arrhenius equation was originally formulated by J.J. Hood based on experiments,but Arrhenius showed that it is applicable to almost all kinds of reactions and provided a theoretical foundation for it
6 So far, when discussing the properties of materials we tacitly assumed that the atoms of solids remain essentially stationary. From time to time we implied, however, that the behavior of solids is affected by the thermally induced vibrations of atoms. The changes in properties increase even more when atoms migrate through the lattice and take new positions. In order to gain a deeper insight into many mechanical properties, we therefore need to study the “dynamic case.” It will become obvious during our endeavor that the motion of atoms through solids involves less effort (energy) when open spaces are present in a lattice, as encountered, for example, by empty lattice sites. Thus, we commence with this phenomenon. We have repeatedly pointed out in previous chapters that an ideal lattice is rarely found under actual conditions, that is, a lattice in which all atoms are regularly and periodically arranged over large distances. This is particularly true at high temperatures, where a substantial amount of atoms frequently and randomly change their positions leaving behind empty lattice sites, called vacancies. Even at room temperature, at which thermal motion of atoms is small, a fair number of lattice defects may still be found. The number of vacancies per unit volume, nv, increases exponentially with the absolute temperature, T, according to an equation whose generic type is commonly attributed to Arrhenius: 1 nv ns exp k E BT f , (6.1) 6.1.1 Lattice Defects Atoms in Motion 6.1 • Lattice Defects and Diffusion 1Svante August Arrhenius (1859–1927), Swedish chemist and founder of modern physical chemistry, received in 1903, as the first Swede, the Nobel prize in chemistry. The Arrhenius equation was originally formulated by J.J. Hood based on experiments, but Arrhenius showed that it is applicable to almost all kinds of reactions and provided a theoretical foundation for it.
6.1.Lattice Defects and Diffusion 103 where ns is the number of regular lattice sites per unit volume,kB is the Boltzmann constant(see Appendix II),and Ef is the energy that is needed to form a vacant lattice site in a perfect crystal. As an example,at room temperature,n for copper is about 108 vacancies per cm3,which is equivalent to one vacancy for every 1015 lattice atoms.If copper is held instead near its melt- ing point,the vacancy concentration is about 1019 cm-3,or one vacancy for every 10,000 lattice atoms.It is possible to increase the number of vacancies at room temperature by quenching a material from high temperatures to the ambient,that is,by freez- ing-in the high temperature disorder,or to some degree also by plastic deformation. Other treatments by which a large number of vacancies can be introduced into a solid involve its bombardment with neutrons or other high energetic particles as they exist,for example,in nuclear reactors(radiation damage)or by ion implantation.These high en- ergetic particles knock out a cascade of lattice atoms from their po- sitions and deposit them between regular lattice sites (see below). It has been estimated that each fast neutron may create between 100 and 200 vacancies.At the endpoint of a primary particle,a de- pleted zone about 1 nm in diameter (several atomic distances)may be formed which is characterized by a large number of vacancies. Among other point defects are the interstitials.They involve for- eign,often smaller,atoms(such as carbon,nitrogen,hydrogen,oxy- gen)which are squeezed in between regular lattice sites.The less common self-interstitials(sometimes,and probably not correctly, called interstitialcies)are atoms of the same species as the matrix that occupy interlattice positions.Self-interstitials cause a sub- stantial distortion of the lattice.In a dumbbell,two equivalent atoms share one regular lattice site.Frenkel defects are vacancy/inter- stitial pairs.Schottky defects are formed in ionic crystals when, for example,an anion as well as a cation of the same absolute va- lency are missing (to preserve charge neutrality).Dislocations are one-dimensional defects(Figure 3.20).Two-dimensional defects are formed by grain boundaries(Figure 3.15)and free surfaces at which the continuity of the lattice and therefore the atomic bonding are disturbed.We shall elaborate on these defects when the need arises. 6.1.2 Vacancies provide,to a large extent,the basis for diffusion,that Diffusion is,the movement of atoms in materials.Specifically,an atom may move into an empty lattice site.Concomitantly,a vacancy migrates Mechanisms in the opposite direction,as depicted in Figure 6.1.The prerequi- site for the jump of an atom into a vacancy is,however,that the Diffusion by atom possesses enough energy (for example,thermal energy)to squeeze by its neighbors and thus causes the lattice to expand Vacancies momentarily and locally,involving what is called elastic strain
where ns is the number of regular lattice sites per unit volume, kB is the Boltzmann constant (see Appendix II), and Ef is the energy that is needed to form a vacant lattice site in a perfect crystal. As an example, at room temperature, nv for copper is about 108 vacancies per cm3, which is equivalent to one vacancy for every 1015 lattice atoms. If copper is held instead near its melting point, the vacancy concentration is about 1019 cm3, or one vacancy for every 10,000 lattice atoms. It is possible to increase the number of vacancies at room temperature by quenching a material from high temperatures to the ambient, that is, by freezing-in the high temperature disorder, or to some degree also by plastic deformation. Other treatments by which a large number of vacancies can be introduced into a solid involve its bombardment with neutrons or other high energetic particles as they exist, for example, in nuclear reactors (radiation damage) or by ion implantation. These high energetic particles knock out a cascade of lattice atoms from their positions and deposit them between regular lattice sites (see below). It has been estimated that each fast neutron may create between 100 and 200 vacancies. At the endpoint of a primary particle, a depleted zone about 1 nm in diameter (several atomic distances) may be formed which is characterized by a large number of vacancies. Among other point defects are the interstitials. They involve foreign, often smaller, atoms (such as carbon, nitrogen, hydrogen, oxygen) which are squeezed in between regular lattice sites. The less common self-interstitials (sometimes, and probably not correctly, called interstitialcies) are atoms of the same species as the matrix that occupy interlattice positions. Self-interstitials cause a substantial distortion of the lattice. In a dumbbell, two equivalent atoms share one regular lattice site. Frenkel defects are vacancy/interstitial pairs. Schottky defects are formed in ionic crystals when, for example, an anion as well as a cation of the same absolute valency are missing (to preserve charge neutrality). Dislocations are one-dimensional defects (Figure 3.20). Two-dimensional defects are formed by grain boundaries (Figure 3.15) and free surfaces at which the continuity of the lattice and therefore the atomic bonding are disturbed. We shall elaborate on these defects when the need arises. Vacancies provide, to a large extent, the basis for diffusion, that is, the movement of atoms in materials. Specifically, an atom may move into an empty lattice site. Concomitantly, a vacancy migrates in the opposite direction, as depicted in Figure 6.1. The prerequisite for the jump of an atom into a vacancy is, however, that the atom possesses enough energy (for example, thermal energy) to squeeze by its neighbors and thus causes the lattice to expand momentarily and locally, involving what is called elastic strain 6.1.2 Diffusion Mechanisms Diffusion by Vacancies 6.1 • Lattice Defects and Diffusion 103
104 6·Atoms in Motion E atom vacancy FIGURE 6.1.Schematic representation of the diffusion of an atom from its former position into a vacant lat- tice site.An activation energy for motion,Em,has to be applied which causes a momentary and local ex- distance pansion of the lattice to make room for the passage of the atom.This two-dimensional representation shows only part of the situation.Atoms above and below the depicted plane may contribute likewise to diffusion. energy.The necessary energy of motion,E to facilitate this ex- pansion is known as the activation energy for vacancy motion, which is schematically represented by an energy barrier shown in Figure 6.1.Em is in the vicinity of leV.The average thermal(ki- netic)energy of a particle,Eh,at the temperatures of interest,how- ever,is only between 0.05 to 0.1 eV,which can be calculated by making use of an equation that is borrowed from the kinetic the- ory of particles(see textbooks on thermodynamics): Et=ikgT. (6.2) This entails that for an atom to jump over an energy barrier,large fluctuations in energy need to take place until eventually enough energy has been "pooled together"in a small volume.Diffusion is therefore a statistical process. A second prerequisite for the diffusion of an atom by this mech- anism is,of course,that one or more vacancies are present in neighboring sites of the atom;see Eq.(6.1).All taken,the acti- vation energy for atomic diffusion,Q,is the sum of Ef and Em. Specifically,the activation energy for diffusion for many ele- ments is in the vicinity of 2 eV;see Table 6.1 Interstitial If atoms occupy interstitial lattice positions(see above),they may Diffusion easily diffuse by jumping from one interstitial site to the next without involving vacancies.Interstitial sites in FCC lattices are, for example,the center of a cube or the midpoints between two corner atoms.Similarly as for vacancy diffusion,the adjacent matrix must slightly and temporarily move apart to let an inter- stitial atom squeeze through.The atom is then said to have dif-
energy. The necessary energy of motion, Em, to facilitate this expansion is known as the activation energy for vacancy motion, which is schematically represented by an energy barrier shown in Figure 6.1. Em is in the vicinity of 1eV. The average thermal (kinetic) energy of a particle, Eth, at the temperatures of interest, however, is only between 0.05 to 0.1 eV, which can be calculated by making use of an equation that is borrowed from the kinetic theory of particles (see textbooks on thermodynamics): Eth 3 2 kBT. (6.2) This entails that for an atom to jump over an energy barrier, large fluctuations in energy need to take place until eventually enough energy has been “pooled together” in a small volume. Diffusion is therefore a statistical process. A second prerequisite for the diffusion of an atom by this mechanism is, of course, that one or more vacancies are present in neighboring sites of the atom; see Eq. (6.1). All taken, the activation energy for atomic diffusion, Q, is the sum of Ef and Em. Specifically, the activation energy for diffusion for many elements is in the vicinity of 2 eV; see Table 6.1 If atoms occupy interstitial lattice positions (see above), they may easily diffuse by jumping from one interstitial site to the next without involving vacancies. Interstitial sites in FCC lattices are, for example, the center of a cube or the midpoints between two corner atoms. Similarly as for vacancy diffusion, the adjacent matrix must slightly and temporarily move apart to let an interstitial atom squeeze through. The atom is then said to have difInterstitial Diffusion 104 6 • Atoms in Motion atom vacancy Em distance E FIGURE 6.1. Schematic representation of the diffusion of an atom from its former position into a vacant lattice site. An activation energy for motion, Em, has to be applied which causes a momentary and local expansion of the lattice to make room for the passage of the atom. This two-dimensional representation shows only part of the situation. Atoms above and below the depicted plane may contribute likewise to diffusion.
6.1.Lattice Defects and Diffusion 105 TABLE 6.1 Selected diffusion constants (volume diffusion) Mechanism Do m2 Solute Host material Q [ev] Self diffusion Cu Cu 7.8×10-5 2.18 Al Al 1.7×10-5 1.40 Fe a-Fe 2.0×10-4 2.49 Si Si 32×10-4 4.25 Interstitial C a-Fe(BCC) 6.2×10-7 0.83 diffusion y-Fe (FCC) 1.0×10-5 1.40 Interdiffusion Zn Cu 3.4×10-5 1.98 Cu Al 6.5×10-5 1.40 Cu Ni 2.7×10-5 2.64 Ni Cu 2.7×10-4 2.51 Al Si 8.0×10-4 3.47 fused by an interstitial mechanism.This mechanism is quite com- mon for the diffusion of carbon in iron or hydrogen in metals but can also be observed in nonmetallic solids in which the dif fusing interstitial atoms do not distort the lattice too much.The activation energy for interstitial diffusion is generally lower than that for diffusion by a vacancy mechanism(see Table 6.1),par- ticularly if the radius of the interstitial atoms is small compared to that of the matrix atoms.Another contributing factor is that the number of empty interstitial sites is generally larger than the number of vacancies.In other words,Ef(see above)is zero in this case. Interstitialcy If the interstitial atom is of the same species as the matrix,or if Mechanism a foreign atom is of similar size compared to the matrix,then the diffusion takes place by pushing one of the nearest,regular lattice atoms into an interstitial position.As a result,the former interstitial atom occupies the regular lattice site that was previ- ously populated by the now displaced atom.Examples of this mechanism have been observed for copper in iron or silver in AgBr. Other Diffusion by an interchange mechanism,that is,the simultaneous Diffusion exchange of lattice sites involving two or more atoms,is possible but energetically not favorable.Another occasionally observed Mechanisms mechanism,the ring exchange,may occur in substitutional,body- centered cubic solid solutions that are less densely packed.In this case,four atoms are involved which jump synchronously,one po- sition at a time,around a circle.It has been calculated by Zener
fused by an interstitial mechanism. This mechanism is quite common for the diffusion of carbon in iron or hydrogen in metals but can also be observed in nonmetallic solids in which the diffusing interstitial atoms do not distort the lattice too much. The activation energy for interstitial diffusion is generally lower than that for diffusion by a vacancy mechanism (see Table 6.1), particularly if the radius of the interstitial atoms is small compared to that of the matrix atoms. Another contributing factor is that the number of empty interstitial sites is generally larger than the number of vacancies. In other words, Ef (see above) is zero in this case. If the interstitial atom is of the same species as the matrix, or if a foreign atom is of similar size compared to the matrix, then the diffusion takes place by pushing one of the nearest, regular lattice atoms into an interstitial position. As a result, the former interstitial atom occupies the regular lattice site that was previously populated by the now displaced atom. Examples of this mechanism have been observed for copper in iron or silver in AgBr. Diffusion by an interchange mechanism, that is, the simultaneous exchange of lattice sites involving two or more atoms, is possible but energetically not favorable. Another occasionally observed mechanism, the ring exchange, may occur in substitutional, bodycentered cubic solid solutions that are less densely packed. In this case, four atoms are involved which jump synchronously, one position at a time, around a circle. It has been calculated by Zener Interstitialcy Mechanism Other Diffusion Mechanisms 6.1 • Lattice Defects and Diffusion 105 TABLE 6.1 Selected diffusion constants (volume diffusion) Mechanism Solute Host material D0 m s 2 Q [eV] Self diffusion Cu Cu 7.8 105 2.18 Al Al 1.7 105 1.40 Fe -Fe 2.0 104 2.49 Si Si 32 104 4.25 Interstitial C -Fe (BCC) 6.2 107 0.83 diffusion C -Fe (FCC) 1.0 105 1.40 Interdiffusion Zn Cu 3.4 105 1.98 Cu Al 6.5 105 1.40 Cu Ni 2.7 105 2.64 Ni Cu 2.7 104 2.51 Al Si 8.0 104 3.47
106 6·Atoms in Motion that this mode requires less lattice distortions and,thus,less en- ergy than a direct interchange. Self-Diffusion Diffusion involving the jump of atoms within a material con- and Volume sisting of only one element is called self-diffusion.(Self-diffusion can be studied by observing the motion of radioactive tracer Diffusion atoms,that is,isotopes of the same element as the nonradioac- tive host substance.)Diffusion within the bulk of materials is called volume diffusion. Grain Grain boundaries are characterized by a more open structure Boundary caused by the lower packing at places where two grains meet. They can be represented by a planar channel approximately two Diffusion atoms wide,as schematically depicted in Figure 6.2.Grain boundaries therefore provide a preferred path for diffusion.The respective mechanism is appropriately called grain boundary dif- fusion.It generally has an activation energy of only one-half of that found for volume diffusion since the energy of formation of vacancies Ef(see above)is close to zero.This amounts to a dif- fusion rate that may be many orders of magnitude larger than in the bulk,depending on the temperature.However,grain boundaries represent only a small part of the crystal volume,so that the contribution of grain boundary diffusion,at least at high temperatures,is quite small.As a rule of thumb,volume diffu- sion is predominant at temperatures above one-half of the melt- ing temperature,Tm,of the material,whereas grain boundary dif- fusion predominates below 0.5 Tn. Surface Further,free surfaces provide an even easier path for migrating Diffusion atoms.This results in an activation energy for surface diffusion that is again approximately only one-half of that for grain bound- FIGURE 6.2.Schematic representation of a pla- nar diffusion channel between two grains. (Grain boundary diffu- sion.)
that this mode requires less lattice distortions and, thus, less energy than a direct interchange. Diffusion involving the jump of atoms within a material consisting of only one element is called self-diffusion. (Self-diffusion can be studied by observing the motion of radioactive tracer atoms, that is, isotopes of the same element as the nonradioactive host substance.) Diffusion within the bulk of materials is called volume diffusion. Grain boundaries are characterized by a more open structure caused by the lower packing at places where two grains meet. They can be represented by a planar channel approximately two atoms wide, as schematically depicted in Figure 6.2. Grain boundaries therefore provide a preferred path for diffusion. The respective mechanism is appropriately called grain boundary diffusion. It generally has an activation energy of only one-half of that found for volume diffusion since the energy of formation of vacancies Ef (see above) is close to zero. This amounts to a diffusion rate that may be many orders of magnitude larger than in the bulk, depending on the temperature. However, grain boundaries represent only a small part of the crystal volume, so that the contribution of grain boundary diffusion, at least at high temperatures, is quite small. As a rule of thumb, volume diffusion is predominant at temperatures above one-half of the melting temperature, Tm, of the material, whereas grain boundary diffusion predominates below 0.5 Tm. Further, free surfaces provide an even easier path for migrating atoms. This results in an activation energy for surface diffusion that is again approximately only one-half of that for grain boundSelf-Diffusion and Volume Diffusion Grain Boundary Diffusion Surface Diffusion 106 6 • Atoms in Motion FIGURE 6.2. Schematic representation of a planar diffusion channel between two grains. (Grain boundary diffusion.)
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
110 6·Atoms in Motion (a) FIGURE 6.5.(a)Steady- state diffusion through a slab.(b)Linear con- centration gradient which stays constant with time by constantly supplying solute atoms on the left and remov- ing the same number of atoms on the right side (b) of the slab.Ca and CB are two assumed sur- face concentrations, where Ca>CB. frequency of the atoms,f,which we introduced above [Equation (6.3)].This interrelationship reads: D=名A2f, (6.9) where A is the diffusion jump distance which is,in isotropic sys- tems,identical with the interatomic distance. Fick's first law [Eq.(6.6)]may be used to describe steady-state flow.One assumes for this case an infinite source and an infi- nite sink,respectively,on the opposite ends of a plate of metal that causes a constant flow of solute atoms through a given area.Steady-state flow is observed,for example,when gases (such as hydrogen or oxygen)diffuse through metals as a con- sequence of a constant (but different)gas pressure on each side of a plate.This requires a supply of gas atoms on one side and a removal of the same amount of gas atoms on the other side. As an example,hydrogen-filled gas tanks for fuel-cell-propelled automobiles are empty after about two months.(Hand tools in university labs seem to disappear by a similar mechanism.) 6.1.7 Fick's second law deals with the common case for which the con- Nonsteady- centration gradient of a diffusing species,A,in the host material, B,changes gradually with time (Figure 6.6).The nonsteady-state State (or dynamic)case is governed by the partial differential equation: Diffusion =D℃ (6.10) at
frequency of the atoms, f, which we introduced above [Equation (6.3)]. This interrelationship reads: D 1 6 2f, (6.9) where is the diffusion jump distance which is, in isotropic systems, identical with the interatomic distance. Fick’s first law [Eq. (6.6)] may be used to describe steady-state flow. One assumes for this case an infinite source and an infinite sink, respectively, on the opposite ends of a plate of metal that causes a constant flow of solute atoms through a given area. Steady-state flow is observed, for example, when gases (such as hydrogen or oxygen) diffuse through metals as a consequence of a constant (but different) gas pressure on each side of a plate. This requires a supply of gas atoms on one side and a removal of the same amount of gas atoms on the other side. As an example, hydrogen-filled gas tanks for fuel-cell-propelled automobiles are empty after about two months. (Hand tools in university labs seem to disappear by a similar mechanism.) Fick’s second law deals with the common case for which the concentration gradient of a diffusing species, A, in the host material, B, changes gradually with time (Figure 6.6). The nonsteady-state (or dynamic) case is governed by the partial differential equation: C t D 2 x C 2 , (6.10) 6.1.7 NonsteadyState Diffusion 110 6 • Atoms in Motion (b) (a) C C X A J FIGURE 6.5. (a) Steadystate diffusion through a slab. (b) Linear concentration gradient which stays constant with time by constantly supplying solute atoms on the left and removing the same number of atoms on the right side of the slab. C and C are two assumed surface concentrations, where C ! C.
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 conditions is known. It should be noted that the diffusion coefficient in Eq. (6.10) has only a constant value if one considers selfdiffusion 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 concentration 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 distance x; Ci is the constant concentration of the solute at the interface 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 different 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.