160 CHAPTER 7. ATOMIC MODELS FOR DIFFUSION Using the standard thermodynamic relation [~G/BP]T = V and realizing that the pressure dependence of lnv will be relatively very small, we may write to a good ap- proximation (7.57) If a plot of lnr' vs. P is now constructed using the experimental data, V" can be determined from its slope. 7.3 Consider small interstitial atoms jumping by the interstitial mechanism in b.c.c. Fe with the diffusivity D for a time T. (a) What is the most likely expected total displacement after a large number (b) What is the standard deviation of the total displacement? Solution. of diffusional jumps? (a) The expected total displacement will be zero because there is no correlation be- tween successive jumps-after a jump the interstitial loses its memory of its jump and makes its next jump randomly into any one of its nearest-neighbor sites. (b) The distribution of displacements will be Gaussian (Eq. 7.32) and the standard deviation will be the root-mean-square displacement given by Eq. 7.35 as m. 7.4 Suppose the random walking of a diffusant in a primitive orthorhombic crystal where the particle makes N1 jumps of length a1 along the XI axis, NZ jumps of length a2 along the xz axis, and N3 jumps of length a3 along the 23 axis. The three axes are orthogonal and aligned along the crystal axes of the orthorhombic unit cell and the diffusivity tensor in this axis system is Dll 0 0 .=[ : Dozz 4 (7.58) (a) Find an expression for the mean-square displacement in terms of the numbers of jumps and jump distances. (b) Find another expression for the mean-square displacement in terms of the three diffusivities in the diffusivity tensor and the diffusion time. Your answer should be analogous to Eq. 7.35, which holds for the isotropic case. Solution. (a) Using Eqs. 7.30 and 7.31, N (R2) = c r', . r', = Nla: + N2a; + N3ai (7.59) i=l (b) The diffusion equation will have the form of Eq. 4.61. By using the method of scaling described in Section 4.5 (based on the scaling relationships in Eq. 4.64), the solution can be written A c(a,m,~,t) = -exp d EXERCISES 161 where A = constant. The mean-square displacement is then S;;OS,"S;;"c(zi,22,23,t)~~d5id~2d23 (R2) = ~,oo~o~~o~c(zl,z2,z3,t)dzl~~2~~3 L -2L -L J," J," J," e-4Diit e 4D22t e 4D33t (zf + zg + z:) dzl dz2dz3 - - SoooSoooS,"C(zl1z2,X3,t)dzldz2dz3 (7.61) Equation 7.61 can be factored into standard definite integrals and the result is (R2) = 2D11t + 2022t + 2D33t (7.62) Comparison of Eqs. 7.59 and 7.62 shows that the mean-square displacement con- sists of three terms, each of which is the mean-square displacement that would be achieved in one dimension along one of the three coordinate directions. 7.5 Suppose a random walk occurs on a primitive cubic lattice and successive jumps are uncorrelated. Show explicitly that f = 1 in Eq. 7.49. Base your argument on a detailed consideration of the values that the cosOi,i+j terms assume. Solution. Because all jumps are of the same length, (7.63) 2 = 1 + - (cos 01,2 + cos 01,3 . . . + cos e2,3 + cos e2,4 ' ' + cos eN,-l,NT) NT and thus, 2 NT f= I+ -[(cosel,2) +( cosel,3)+ +(cose2,3) +(COSeN,-1,NT)1 (7.64) Any jump can be one of the six vectors: [aOO], [TiOO], [OaO], [OZO], [OOa], and [OOTi]. Each occurs with equal probability. For each pair of jump vectors, i and i+j, the six possible values of cosQz,a+3 are 1, -1,O, O,O, and 0, and these occur with equal probability. For a large number of trajectories, each mean value in Eq. 7.64 is zero and therefore f = 1. 7.6 For the diffusion of vacancies on a face-centered cubic (f.c.c.) lattice with lattice constant a, let the probability of first- and second-nearest-neighbor jumps be p and 1 - p, respectively. At what value of p will the contributions to diffusion of first- and second-nearest-neighbor jumps be the same? Solution. There is no correlation and, using Eq. 7.29, N- (7.65) The number of first nearest-neighbor jumps is NTp and the number of second nearest- neighbor jumps is NT(l -p). Therefore, (7.66) a2 (R2) = NTp~ + NT(1 -p)a2 They make equal contributions when NTpa2/2 = NT(l -p)a2 or p = 2/3. CHAPTER 8 DIFFUSION IN CRYSTALS The driving forces necessary to induce macroscopic fluxes were introduced in Chap- ter 3 and their connection to microscopic random walks and activated processes was discussed in Chapter 7. However, for diffusion to occur, it is necessary that kinetic mechanisms be available to permit atomic transitions between adjacent locations. These mechanisms are material-dependent . In this chapter, diffusion mechanisms in metallic and ionic crystals are addressed. In crystals that are free of line and planar defects, diffusion mechanisms often involve a point defect, which may be charged in the case of ionic crystals and will interact with electric fields. Addi- tional diffusion mechanisms that occur in crystals with dislocations, free surfaces, and grain boundaries are treated in Chapter 9. 8.1 ATOMIC MECHANISMS Atom jumping in a crystal can occur by several basic mechanisms. The dominant mechanism depends on a number of factors, including the crystal structure, the nature of the bonding in the host crystal, relative differences of size and electrical charge between the host and the diffusing species, and the type of crystal site pre- ferred by the diffusing species (e.g., anion or cation, substitutional or interstitial). Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 163 Copyright @ 2005 John Wiley & Sons, Inc. 164 CHAPTER 8 DIFFUSION IN CRYSTALS 8.1.1 Ring Mechanism A substitutional atom (indicated by shading in Fig. 8.1) may jump and replace an adjacent nearest-neighbor substitutional atom. In the rang mechanasm. the substi- tutional atom exchanges places with a neighboring atom by a cooperative ringlike rotational movement. 00000 00000 OOOc90 00000 00000 00000 OOO@O 00000 Figure 8.1: Riiig riiecliaiiisiri for diffusioii of substitutioiial atorris. 8.1.2 Vacancy Mechanism A substitutional atoni can migrate to a neighboring substitutional site without co- operative motion and with a relatively small activation energy if the neighboring substitutional site is unoccupied. This is equivalent to exchange with a neighbor- ing vacancy.l In Fig. 8.2a, the vacancy is initially separated from a particular substitutional atom (again indicated by shading). In Fig. 8.2b, it has migrated by exchanging places with host atoms to a nearest-neighbor substitutional site of the shaded atom. In Fig. 8.2~ the vacancy has exchanged sites with the substitutional atom: and in Fig. 8.2d the vacancy has migrated some distance away. As a result, the particular substitutional at,om is displaced by one nearest-neighbor distance while the vacancy has undergone at least five individual displacements. The atomic environment during a vacancy-exchange mechanism can be illus- trated in a three-dimensional cubic lattice. Figure 8.3 shows an atom-vacancy exchange between two face-centered sites in an f.c.c. crystal. The migrating atom (A in Fig. 8.3) moves in a (110)-direction through a rectangular “window” framed by two cube corner atoms and two opposing face-centered atoms. The f.c.c. crys- tal is close-packed and each site has 12 equivalent nearest-neighbor sites [l]. In 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 000 0 00000 00000 Figure 8.2: Vacaiicy mecliaiiism for diffiisioii of substitutional atonis. ‘Vacancies will always exist in equilibrium in a crystal because their enthalpy of formation can always be compensated by a configurational entropy increase at finite temperatures (see the deriva- tion of Eq. 3.65). Therefore, vacancies function as a component that occupies substitutional sites. 8 1 ATOMIC MECHANISMS 165 X 4 Figure 8.3: Atom-vacancy exchange in f.c.c. cryshl. Atom init,ially at A jumps into a nearest-neighbor vacancy (dashed circle). The four nearest,-neighbor atoms common to A and t,he vacant site (joined by the bold rectangle) form a "window" 1234 through which the A atom must, pass. The A atom is centered in unit-cell face 2356. The vacancy is centered in unit-cell face 2378. a hard-sphere model. in which nearest-neighbor atoms are in contact, the atoni must "squeeze" through a window that is about 27% smaller than its diameter. The potential-energy increases required for such distortions create the energetic migration barriers discussed in Section 7.1.3. 8.1.3 lnterstitialcy Mechanism A substitutional atom can migrate to a neighboring substitutional site by the two- step process illustrated in Fig. 8.4. The first step is an exchange with an interstitial defect in which the migrating substitutional atom becomes the interstitial atom.2 The second step is to exchange the migrating atom with a neighboring substitu- tional atom. This mechanism is only possible when substitutional atoms can occupy interstitial sites. This cooperative and serpentine motion constitutes the intersti- tialcy mechanism, and when large normally substitutional atoms are involved, can occur with a much lower migration energy than the interstitial mechanism (see below). Interstitialcy migration depends on the geometry of the interstitial defect. How- ever, an a priori prediction of interstitial defect geometry is not straightforward in real materials. For an f.c.c. crystal, a variety of conceivable interstitial defect candidates are illustrated in Fig. 8.5. The lowest-energy defect will be stable and predominant. For example. in the f.c.c. metal Cu. the stable configuration is the (100) split-dumbbell configuration in Fig. 8.5d [3]. The (100) split-dumbbell defect in Fig. 8.5d, while having the lowest energy of all interstitial defects, still has a large formation energy (Ef = 2.2 eV) because of the large amount of distortion and ion-core repulsion required for its insertion into the close-packed Cu crystal. However, once the interstitial defect is present, it persists until it migrates to an interface or dislocation or annihilates with a vacancy. The 21nterstitial point defects involving normally substitutional atoms will always exist (although typically at very low concentration) at equilibrium in a crystal at finite temperatures because. as in the case of vacancies described above, their enthalpy of formation can always be compensated by a configurational entropy increase. 166 CHAPTER 8. DIFFUSION IN CRYSTALS 00000 0 0 0 0 00 00 00 00 00000 00000 00000 Figure 8.4: Substitutional diffusion by the iiiterst,itialcy mechanisrn. (a) The iiiterstit ial defect, corresponding to the interstitial atom (3) is separated from a part,iciilar siibst,itutional atom B (shaded). (b) The interstitial defect, moved adjacent t,o B when t>he previously interstitial atom (3) replaced the substit,utional atom (2). (2) then became the int,erstitial atom. (c) At.om (2) has replaced B. and B has become the interst,it,ial atom. (d) B has replaced atom (4): which has become the interstitial atom. (e) The int,erstitial defect has migrated away from B. As a result. B has completed one nearest-neighbor jump and the interstitial defect has moved at, least four times. Figure 8.5: Geonietric corifiguratioris for a self-interst,itial defect atom in an f.c.c. crystal: (a) oct,ahedral site, (b) tetrahedral site, (c) (110) crowdion, (d) (100) split, dumbbell. (e) (111) split,. (f) (110) split crowdioii [2]. activation energy for migration (Em = 0.1 eV) is small compared to Ef because little additional distortion is required for its serpentine motion, which is illustrated in Fig. 8.6. It therefore migrates relatively rapidly. 8 1 ATOMIC MECHANISMS 167 X A Z Figure 8.6: uystal. durnbbells [one of which is sliowii iii (b)] atrid into four others. creatirig [OlO] diirnbliells. Diffusiorial niigratioii of a [loo] split-duiiibbell self-iiiterstitial iii ail f.c c The durnbbell in (a) rail juirip int,o four iiearest-iieighbor sites: c,rwtiiig [OOl] 8.1.4 Interstitial Mechanism An interstitial atom can simply migrate between interstitial sites as in Fig. 8.7. The interstitial atom must attain enough energy to distort the host crystal as it migrates between substitutional sites. This mechanism is expected for small solute atoms that normally occupy interstitial sites in a host crystal of larger atoms. Diffusion by the interstitial mechanism and by the interstitialcy mechanism are quite different processes and should not be confused. Diffusion by the vacancy and interstitialcy niechanisms requires the presence of point defects in the system. whereas diffusion by the ring and interstitial mechanisms does not. 00000 00 00 .oo 00 ~00000 Figure 8.7: Interstitial mechanism for diffusion of interstitial atoms. Thr snialler shaded interstitial atom niigrates through the openiiig betweeii host atoms (1) and (2) to a neighboring interstitial site. 8.1.5 Diffusion Mechanisms in Various Materials Diffusion of relatively small atoms that normally occupy interstitial sites in the sol- vent crystal generally occurs by the interstitial mechanism. For example, hydrogen atoms are small and migrate interstitially through most crystalline materials. Car- bon is small compared to Fe and occupies the interstitial sites in b.c.c. Fe illustrated in Fig. 8.8 and migrates between neighboring interstitial sites. Migration of atoms that occupy substitutional sites may occur through a range of mechanisms involving either vacancy- or interstitial-type defects. In f.c.c., b.c.c., and hexagonal close-packed (h.c.p.) metals, self-dzffuszon occurs predominantly by the vacancy mechanism [4, 51. However, in some cases self-diffusion by the 168 CHAPTER 8 DIFFUSION IN CRYSTALS X Figure 8.8: 1nterst)itial sites for C atjoins iii b.c c Fe. (a) Tlie intrrstitial sites liwe point-group syriiirietry 4/mmm, and tlie orient,at ions of tlie fourfold axes are indicxtrti by the shorter, grey spokes on tlie symbols. (b) Noiiicnclat,ure uscd in t,he model for diffusion of interstitial atonis in b.c.c. Fe discussed in Section 8.2.1 Three different types of sit,es are present: sites 1. 2. and 3 have nearest-neighbor Fe atonis lying along 2. y. and z. rcymtiwly. interstitialcy mechanism contributes a small amount to the overall diffusion (see Section 8.2.1). In Ge, which has the less closely packed diamond-cubic structure, self-diffusion occurs by a vacancy mechanism. In Si (which like Ge has covalent bonding), self-diffusion occurs by the vacancy mechanism at low temperatures and by an interstitialcy mechanism at elevated temperatures [6-81. In ionic materials, diffusion mechanisms become more complex and varied. Self-diffusion of Ni in Ni0 occurs by a vacancy mechanism; in Cua0 the diffusion of 0 involves interstit,ial defects [9]. In the alkali halides, vacancy defects predominate and the diffusion of both anions and cations occurs by a vacancy mechanism. However: the predominant defect is not easy to predict in ionic materials. For example, vacancy-interstit,ial pairs dominate in AgBr and the smaller Ag cations diffuse by an interstitialcy mechanism (see Section 8.2.2). Solutes that normally occupy substitutional positions can migrate by a vari- ety of mechanisms. In many systems they migrate by the same mechanism as for self-diffusion of the host atoms. However, the details of migration become more complex if there is an interaction or binding energy between the solute atoms and point defects-this is described in Section 8.2.1 for vacancy-solute-atom binding. Certain solute atom can migrate by more than one mechanism. For example. while Au solute atoms in Si are mainly substitutional, under equilibrium conditions: a rel- atively small number of Au atoms occupy interstitial sites. The rate of migration of the interstitial Au atoms is orders of magnitude faster than the ratme of the substitu- tional Au atoms, and the small population of interstit,ial Au atoms therefore makes an important contribution to the overall solute-at,om diffusion rate [6. 81. The so- lute atoms transfer from substitutional sites to interstitial sites by either kick-out or dissociative mechanisms (Fig. 8.9). In the kick-out mechanism, an interstitial host atom, HI, pushes the substitutional solute atom, Ss. into an interstitial position and simultaneously takes up a substitutional position according to the reaction 8.2 ATOMIC MODELS FOR DlFFUSlVlTlES 169 0 000 00 00 00 0 00-0 (h)O00O 0000 Figure 8.9: interstitial site by (a) the kick-out rriechanisni arid (b) t,lw dissoriative mechanism. Transfer of a soliitr at,oni (filled at,orii) from a subst,it,utional site to ail In the dissociative mechanism, a substitutional solute atom enters an interstitial site. leaving a vacancy, V, behind according to the reaction ss = v, + SI (8.2) These reactions are reversible. This dual-sit,e occupancy leads to complicated solute diffusion behavior and has been described for several solut,e species in Si [4, 6, 81. There is no compelling evidence that the ring mechanism in Fig. 8.1 contributes significantly to diffusion in any material. 8.2 ATOMIC MODELS FOR DlFFUSlVlTlES Atomic models for the diffusivity can be constructed when the diffusion occurs by a specified mechanism in various crystalline materials. A number of cases are considered below. 8.2.1 Metals Diffusion of Solute Atoms by the Interstitial Mechanism in the B.C.C. Structure. The general expression that connects the jump rate. I?. the intersite jump distance, r, and the correlation factor, Eq. 7.52. then takes the form (8.3) Because each interstitial site has four nearest-neighbors, the jump rate. r. is given by 4r', where I" has the form of Eq. 7.25.3 If a is the lattice constant for the b.c.c. unit cell in b.c.c. Fe, then r = a12 and Eq. 8.3 yields (8.4) ,'The quantity r', introduced in Section 7.1.1, is the jump rate of an atom from one specified site to a specified neighboring site. r is the total jump rate of the atom in the material. If the atom is diffusing among equivalent sites in a crystal where each site has z equivalent nearest-neighbors, then r = zr'. 170 CHAPTER 8 DIFFUSION IN CRYSTALS where the weakly temperature-dependent terms have been collected into D,”. Be- cause D; is relatively temperature independent, the Arrhenius form of Eq. 8.4 indicates a thermally activated process. The enthalpy of migration, Hm, is the activation energy, E, for the interstitial diffusion. For C in Fe, 0; = 0.004 cm2 s-l and Hm= 80.1 kJ mol-’ [lo]. This experimental value of D; is consistent with the value predicted by Eq. 8.4 for a = 2.9 x m, I/ = 1013 s-l, and S” = lk/atom. The relationship between jump rate and diffusivity in Eq. 8.3 can be obtained by an alternate method that considers the local concentration gradient and the number of site-pairs that can contribute to flux across a crystal plane. A concentration gradient of C along the y-axis in Fig. 8.8b results in a flux of C atoms from three distinguishable types of interstitial sites in the cy plane (labeled 1, 2, and 3 in Fig. 8.8). The sites are assumed to be occupied at random with small relative populations of C atoms that can migrate between nearest-neighbor interstitial sites. If c’ is the number of C atoms in the cy plane per unit area, the carbon concentration on each type of site is c’/3. Carbon atoms on the types 1 and 3 sites jump from plane Q to plane at the rate (c1/3)I”. The jump rate from type-2 sites in plane Q to plane p is zero. The contribution to the flux from all three site types is If c is the number of C atoms per unit volume, c = 2c’/a, and therefore ar’c 3 Ja+P = - The reverse flux can be obtained by using a first-order expansion of the concentra- tion in the p plane, so that Therefore, the net flux is Comparison of Eq. 8.8 with the Fick’s law expression, Droduces (8.9) (8.10) The total jump frequency for a given C atom is r = 4r’, and therefore a2r r2r 24 6 DI=-=- (8.11) which is identical to Eq. 8.3. The same result would have been obtained with the cy and /3 planes chosen at any arbitrary inclination in the Fe crystal because DI is isotropic in all cubic crystals (see Exercise 4.6). [...]... factors for diffusion in solids Trans Faraday SOC. ,52 :78 6-8 01, 1 956 14 R.O Simmons and R.W Balluffi Measurements of equilibrium vacancy concentrations in aluminum Phys Rev., 117 :5 2-6 1, 1960 15 A Seeger The study of point defects in metals in thermal equilibrium I The equilibrium concentration of point defects Cryst Lattice Defects, 4:22 1-2 53 , 1973 16 R.W Balluffi Vacancy defect mobilities and binding... is the determination of the diffusivity of C in b.c.c Fe, which is taken up in Exercise 8.22 Bibliography 1 S.M Allen and E.L Thomas The Structure of Materials John Wiley & Sons, New York, 1999 2 R.A Johnson Empirical potentials and their use in calculation of energies of pointdefects in metals J Phys F , 3(2):29 5- 3 21, 1973 190 CHAPTER 8: DIFFUSION IN CRYSTALS 3 W Schilling Self-interstitial atoms... vacancy self-diffusion mechanism in many metals, experimental values of *Do approximately 0. 1-1 .0 cm2 s-l, which correspond to physically reasonable are values of the quantities in *Do according to Eq 8.19: f x 1, a x 3 .5 x lo-’’ m, v x 1013 s-l, and (S& SF) x 2 k In metals, as in many classes of materials, the + 7 A calculation of f in a two-dimensional lattice that takes into account multiple return vacancy... important for oxygen-sensing materials: (8 .57 ) For the regime in which the dominant charged defects are the oxidation-induced cation vacancies and their associated holes, the electrical neutrality condition is [Gel =2 [Gel (8 .58 ) Therefore, inserting Eq 8 .58 into Eq 8 .57 and solving for [Vke] yields (8 .59 ) The cation self-diffusivity due to the vacancy mechanism varies as the one-sixth power of the oxygen... W Schilling Self-interstitial atoms in metals J Nucl Mats., 6 9-7 0( 1-2 ):46 5- 4 89, 1978 4 P Shewmon Diffusion in Solids The Minerals, Metals and Materials Society, Warrendale, PA, 1989 5 G Neumann Diffusion mechanisms in metals In G.E Murch and D.J Fischer, editors, Defect and Diffusion Forum, volume 6 6-6 9, pages 4 3-6 4, Brookfield, VT, 1990 Sci-Tech Publications 6 W Frank, U Gosele, H Mehrer, and A... pages 6 3-1 42, Orlando, Florida, 1984 Academic Press 7 T.Y Tan and U Gosele Point-defects, diffusion processes, and swirl defect formation in silicon Appl Phys A , 37(1): 1-1 7, 19 85 8 W Frank The interplay of solute and self-diffusion-A key for revealing diffusion mechanisms in silicon and germanium In D Gupta, H Jain, and R.W Siegel, editors, Defect and Diffusion Forum, volume 75, pages 12 1-1 48, Brookfield,... the component that lags behind the stress by 90" Also, - =tan4 E2 (8.70) El The compliance (again the ratio of strain over stress) is then S(W) = ( ~ - 2 ~ 2 eiWt 1 ) uoeiWt - _l -E uo E2 - 1- g o (8.71) Because the strain lags behind the stress, the stress-strain curve for each cycle consists of a hysteresis loop, as in Fig 8.17, and an amount of mechanical work, given by the area enclosed by the hysteresis... therefore r& Let and be the frequencies for type 1 + 2 (A-type) and type 1 -+ 3 (B-type) jumps, respectively, in Fig 8.8b Then, because there are four nearest-neighbors for A-type jumps and eight next-nearest-neighbors for B-type jumps, the frequencies for A-type and B-type jumps are = 4 r a and FB = 8FL, respectively The mean-square displacement during time 7 is then (8.107) and Therefore, (8.109)... defects of opposite charge that can contribute to the diffusivity or electronic conductivity The addition of aliovalent solute (impurity) atoms to an initially pure ionic solid therefore creates extrinsic defects.'O For example, the self-diffusivity of K in KC1 depends on the population of both extrinsic and intrinsic cation-site vacancies Extrinsic cation-site vacancies can be created by incorporation of. .. fraction of the extrinsic Ca++ impurity, [Cakl+ [Vt,l = [v’,] (8.49) The mass-action relationship in Eq 8.40 for the product of the cation and anion vacancy site fractions combined with Eq 8.49 yields (8 .50 ) The last term on the right-hand side of Eq 8 .50 is the square of the cation vacancy site fraction in pure (intrinsic) KC1 Solving the quadratic equation for the cation vacancy site fraction yields (8 .51 ) . the probability of first- and second-nearest-neighbor jumps be p and 1 - p, respectively. At what value of p will the contributions to diffusion of first- and second-nearest-neighbor. necessary to induce macroscopic fluxes were introduced in Chap- ter 3 and their connection to microscopic random walks and activated processes was discussed in Chapter 7. However, for diffusion. the case of ionic crystals and will interact with electric fields. Addi- tional diffusion mechanisms that occur in crystals with dislocations, free surfaces, and grain boundaries are treated