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.)