Kinetics of Materials - R. Balluff S. Allen W. Carter (Wiley 2005) WW Part 3 potx

58 CHAPTER 3: DRIVING FORCES AND FLUXES FOR DIFFUSION Figure 3.7: An uiidulrtt,ing surface possessing rcgioiis of positive and riegat,ive curvature. The ci.irvature differences lead to diffiisiori-I)oterit,ia.l gradients t.hat ~.esiilt in surface smoothing by diffusional transport,. can be ignored, an approximation that is usually j~stifiable.~ The rate of surface smoothing can then be determined by finding expressions for the atom flux and the diffusion equation in the crystal, and then solving the diffusion equation subject to the boundary conditions at the surface. In the following section, the diffusion equa- tion and boundary conditions are established. Exercise 14.1 provides the complete solution to the problem. 3.4.1 The system contains two network-constrained components-host atoms and vacan- cies; the crystal is used as the frame for measuring the diffusional flux, and the vacancies are taken as the N,th component. Note that there is no mass flow within the crystal, so the crystal C-frame is also a V-frame. With constant temperature and no electric field, Eq. 2.21 then reduces to The Flux Equation and Diffusion Equation 4 - Jv = - JA (3.62) An expression for the coefficient LAA may be obtained by considering diffusion in a very large crystal with flat surfaces. The free energy of the system, containing NA atoms and NV vacancies (in dilute solution), can be expressed Here, pi is the free energy per atom in a vacancy-free crystal composed of only A- atoms with a flat (zero curvature) surface, GG = H; - TS;(vib) is the free energy [exclusive of that due to the mixing entropy, Sb(vib) is the vibrational entropy] to form a vacancy, and the last term is the free energy of mixing due to the entropy 'Vacancy crcation and destruction is discussed in Sections 11.1 anti 11.4 3 4: CAPILLARITY AND DIFFUSION 59 associated with the random distribution of the vacancies. Therefore, NA (NA+Nv) EP: (3.64) - + kTln 89 ~NA a9 PA =- - PV =- aNv = G; + kT In ( NATNv) = G; + kTlnXv where XV is the atom fraction of vacancies.1° may be written If pv = 0 when the vacancies are at their equilibrium fraction, XFq, Eq. 3.64 x;q = e-G:/(kT) (3.65) and Pv = kTln (s) X? = kTln (z) (3.66) Putting these expressions into Eq. 3.62 yields Using Eq. A.12, Eq. 3.67 can be written as a Fick's-law expression for the vacancy where DV is the vacancy diffusivity, the volume per site is assumed to be uniform, and the fact that CA >> cv has been incorporated. The diffusion equation for vacancies in the absence of significant dislocation sources or sinks within the crystal is then * = -V . Jv = DvV2cv (3.69) - at From Eq. 3.68, (3.70) and an expression for the atom flux can be obtained by substituting Eq. 3.70 into Eq. 3.62 to obtain (3.71) If the variations in XV throughout the crystal in Fig. 3.7 are sufficiently small, DvXv/((R)kT) can be assumed to be constant, and the conservation equation (see Eq. 1.18) may be writtenll 'ONote that Eqs. 3.64 for the chemical potentials are of the form given by Eq. 2.2. "Equations 3.71 and 3.72 can be further developed in terms of the self-diffusivity using the atomistic models for diffusion described in Chapters 7 and 8. The resulting formulation allows for simple kinetic models of processes such as dislocation climb, surface smoothing, and diffusional creep that include the operation of vacancy sources and sinks (see Eqs. 13.3, 14.48, and 16.31). 60 CHAPTER 3: DRIVING FORCES AND FLUXES FOR DIFFUSION The smoothing of a rough isotropic surface such as illustrated in Fig. 3.7 due to vacancy flow follows from Eq. 3.69 and the boundary conditions imposed on the vacancy concentration at the surface.12 In general, the surface acts as an efficient source or sink for vacancies and the equilibrium vacancy concentration will be maintained in its vicinity. The boundary condition on cv at the surface will therefore correspond to the local equilibrium concentration. Alternatively, if cv , and therefore XV, do not vary significantly throughout the crystal, smoothing can be modeled using the diffusion potential and Eq. 3.72 subject to the boundary conditions on @A at the surface and in the b~1k.l~ During surface smoothing, differences in the local equilibrium values of XV main- tained in the different regions and differences in vacancy concentration throughout the crystal will be relatively small. Assuming that the crystal has isotropic surface tension, the local equilibrium vacancy concentration at the surface is a function of the local curvature [i.e., c? = c?(K)], and can be found by minimizing Eq. 3.63 with respect to NV after adding in the energy required to create the vacancies directly adjacent to the surface. When a vacancy is added to the crystal at a convex region, the crystal expands by the volume AV = Rv and the surface area is increased by AA. Work must therefore be done to create the additional area. Because AA = KAV = KRV, the work is AW = YKQV (3.73) where y is the isotropic surface tension.14 When this surface work is added to the free energy in Eq. 3.63 and the sum is minimized, (3.74) When typical values are inserted into Eq. 3.74, c?(~)/cy(O) does not vary from unity by more than a few percent. Because only relatively small variations in cv occur in typical specimens un- dergoing sintering and diffusional creep (Chapter 16), we prefer to carry out the analyses of surface smoothing, sintering, and diffusional creep in terms of atom diffusion and the diffusion potential using Eq. 3.72. In this approach, the boundary conditions on @A can be expressed quite ~imp1y.l~ To solve the surface smoothing problem in Fig. 3.7, Eq. 3.72 can be simplified further by setting &A/& equal to zero because the diffusion field is, to a good approximation, in a quasi-steady state, which then reduces the problem to solving the Laplace equation v2@A = 0 (3.75) within the crystal subject to the boundary conditions on @A described below 12Methods for solving diffusion problems by setting up and solving the diffusion equation under specified boundary conditions are discussed in Chapter 5. 13The vacancy concentration far from the surface will generally be a function of the total surface curvature. In this case, the crystal can be assumed to be a large block possessing surfaces which on average have zero curvature. The vacancies in the deep interior can then be assumed to be in equilibrium with a flat surface. 14See Exercise 3.11 for further explanation. 15However, during the annealing of small dislocation loops (treated in Section 11.4.3), larger variations of the vacancy concentration occur and Eq. 3.68 must be employed. 3.5: STRESS AND DIFFUSION 61 3.4.2 Boundary Conditions The boundary conditions on the diffusion potential @A = p~ - pv are readily found using results from the preceding section. At the surface where the vacancies are maintained in equilibrium, pv = 0. The diffusion potential for the atoms is the surface work term of the form given by Eq. 3.73 plus the usual chemical term, pi: @z = pi + TKflA (3.76) Deep within the crystal, pv = 0 and p~ = p>, and therefore = pi. The diffusion potential at the convex region of the surface is greater than that at the concave region, and atoms therefore diffuse to smooth the surface as indicated in Fig. 3.7. We discuss surface smoothing in greater detail in Chapter 14. Exercise 14.1 uses Eq. 3.75 subject to the boundary condition given by Eq. 3.76 to obtain a quantitative solution for the evolution of the surface profile in Fig. 3.7. 3.5 MASS DIFFUSION IN THE PRESENCE OF STRESS Because stress affects the mobility, the diffusion potential, and the boundary con- ditions for diffusion, it both induces and influences diffusion [19]. By examining selected effects of stress in isolation, we can study the main aspects of diffusion in stressed systems. 3.5.1 Consider again the diffusion of small interstitial atoms among the interstices be- tween large host atoms in an isothermal unstressed crystal as in Section 3.1.4. According to Eqs. 3.35 and 3.42, the flux is given by Effect of Stress on Mobilities + J1 = -L11Vp1 = -M1c1Vp1 (3.77) The diffusion is isotropic and the mobility, MI, is a scalar, as assumed previously. If a general uniform stress field is imposed on a material, no force will be exerted on a diffusing interstitial because its energy is independent of position.16 Assuming no other fields, the flux remains linearly related to the gradient of the chemical potential so that = -MlclVpl. However, MI will be a tensor because the stress will cause differences in the rates of atomic migration in different directions; this general effect occurs in all types of ~rysta1s.l~ It may be understood in the following way: there will be a distortion of the host lattice when the jumping atom squeezes its way from one interstitial site to another, and work must be done during the jump against any elements of the stress field that resist this distortion. Jumps in different directions will cause different distortions in the fixed stress field, so different amounts of work, W, must be done against the stress field during these jumps. The rate of a particular jump in the absence of stress is proportional to the exponential factor exp[-Gm/(lcT)], where G" is the free-energy barrier to the 16When the stress is nonuniform and stress gradients exist, the stress will exert a force, as discussed in the following section. 17The tensor nature of the diffusivity (mobility) in anisotropic materials is discussed in Section 4.5. 62 CHAPTER 3: DRIVING FORCES AND FLUXES FOR DIFFUSION jumping process (see Chapter 7). When stress is present, the work, W, must be added to this energy barrier, and the jump rate will therefore be proportional to the factor exp[-(Gm + W)/(kT)]. For almost all cases of practical importance, W/(kT) is sufficiently small so that exp[-W/(kT)] E 1 - W/(kT), and the factor can then be written as exp[-G"/(kT)] [l - W/(kT)]. The overall interstitia1,mobility will be the result of the interstitials making numbers of different types of jumps in different directions. As just shown, each type of jump depends linearly on W, which, in turn, is a linear function of the elements of the stress tensor. The latter function depends on the direction of the jump, and it is therefore anticipated that the mobility should vary linearly with stress and be expressible as a tensor in the very general linear form (3.78) kl where the stress-dependent terms in the sum are relatively small. Similar consider- ations hold for the migration of substitutional atoms in a stress field (see Fig. 8.3), and the form of Eq. 3.78 should apply in such cases as well. These and other features of Eq. 3.78 are discussed by Larch6 and Voorhees 1191. 3.5.2 Stress as a Driving Force for Diffusion: Formation of Solute-Atom Atmosphere around Dislocations In a system containing a nonuniform stress field, a diffusing particle generally ex- periences a force in a direction that reduces its interaction energy with the stress field. Ignoring any effect of the stress on the mobility and focusing on the force stemming from the nonuniformity of the stress field, the stress-induced diffusion of interstitial solute atoms in the inhomogeneous stress field of an edge dislocation would look like Fig. 3.8. An interstitial in a host crystal is generally oversized for the space available and pushes outward, acting as a positive center of dilation and causing a volume expansion as illustrated in Fig. 3.9. To find the force exerted on an interstitial by a stress field, one must consider the entropy production in a msoDotentials - \ ,/Direction of 7t- ctrncc-inrii irnri ",I "V" I, lUUYVU force and flux dislocation Figure 3.8: Edge dislocation in an isotropic elastic body. Solid lines indicate isopotcntial cylinders for the portion of the diffusion potential of any interstitial atom present in the hydrostatic stress field of the dislocation. Dashed cylinders and tangential arrows indicate the direction of the corresponding force exerted on the interstitial atom. 3 5 STRESS AND DIFFUSION 63 Figure 3.9: out,warti displacenients of the interstit,inl's nearest neighbors. Dilation produced by an iiiterst,it,ial atoiii iii H cryst,al. Arrows iiidicate small cell embedded in the material as in Section 2.1. Suppose that the interstitial causes a pure dilation A01 and there are no deviatoric strains associated with the interstitial; then the supplemental work term which must be added to the right side of Eq. 2.4 is dw = -PARIdcl (3.79) where P is the hydrostatic pressure. For the case of an edge dislocation in an isotropic elastic material - - ffxx + ffyy + ffzz - ff1.7. + 066 + ffzz p=- (3.80) 3 3 p(1+ v)b sine - p(1 + v)b y - - - 3~(1 - U) T 3~(l - V) x2 + y2 where p and v are the elastic shear modulus and Poisson's ratio: respectively, and b is the magnitude of the Burgers vector [20]. When this work term is added to the chemical potential term, pldcl, and the procedure leading to Eq. 2.11 is followed: the force is $1 = -V (p1 + AR1P) (3.81) The diffusion potential is therefore an "elastochemical" type of potential corre- sponding to18 a1 = p1+ ARlP (3.82) Therefore, using Eqs. 2.16, 3.37, 3.43, and 3.82, clAR1 J1 = L11F1 = -L11V@1 = -L11V (p1 + AQlP) = -D1 VCl + - - f (3.83) The flux has two components: t'he first results from the concentration gradient and the second from the gradient in hydrostatic stress.19 The solid circles (cylinders 18The general diffusion potential for stress and chemical effects is = 1-11 + Ae,,oi,cl, where Aczj is the local strain associated with the migrating species. "Several typically negligible effects have been neglected in the derivation of Eq. 3.83: including (1) interactions between the interstitials, (2) effects of the interstitials on the local elastic constants, (3) quadratic terms in the elastic energy, and (4) nonlinear stress-strain behavior. A more complete treatment, applicable to the present problem, takes into account many of these effects and has been presented by Larch6 and Cahn 1211. ( kT 64 CHAPTER 3: DRIVING FORCES AND FLUXES FOR DIFFUSION in three dimensions) in Fig. 3.8 are isopotential lines for the portion of the diffusion potential due to hydrostatic stress. They were obtained by setting P equal to constant values in Eq. 3.80. Tangents to the dashed circles indicate the directions of the corresponding diffusive force arising from the dislocation stress field (this is treated in Exercise 3.6). Because AR1 is generally positive, this force is directed away from the compressive region (y > 0) and toward the tensile region (y < 0) of the dislocation, as shown. In the case where an edge dislocation is suddenly introduced into a region of uni- form interstitial concentration, solute atoms will immediately begin diffusing toward the tensile region of the dislocation due to the pressure gradient alone (treated in Exercise 3.7). However, opposing concentration gradients build up, and eventually a steady-state equilibrium solute atmosphere, known as a Cottrell atmosphere, is created where the composition-gradient term cancels the stress-gradient term of Eq. 3.83 (this is demonstrated in Exercise 3.8). From these considerations, Cottrell demonstrated that the rate at which solute atoms diffuse to dislocations and subsequently pin them in place is proportional to time2/3 (this time dependence is derived by an approximate method in Exercise 3.9). This provided the first quantifiable theory for the strain aging caused by solute pinning of dislocations [22]. 3.5.3 Influence of Stress on the Boundary Conditions for Diffusion: Diffusional Creep In a process termed dz~usional creep, the applied stress establishes different diffu- sion potentials at various sources and sinks for atoms in the material. Diffusion currents between these sources and sinks are then generated which transport atoms between them in a manner that changes the specimen shape in response to the applied stress. A particularly simple example of this type of stress-induced diffusional trans- port is illustrated in Fig. 3.10, where a polycrystalline wire specimen possessing a “bamboo” grain structure is subjected to an applied tensile force, $app. This force subjects the transverse grain boundaries to a normal tensile stress and therefore reduces the diffusion potential at these boundaries. On the other hand, the applied stress has no normal component acting on the cylindrical specimen surface and, to first order, the diffusion potential maintained there is unaffected by the applied stress. When gaPp is sufficiently large that the diffusion potential at the transverse boundaries becomes lower than that at the surface, atoms will diffuse from the surface (acting as an atom source) to the transverse boundaries (acting as sinks), thereby causing the specimen to lengthen in response to the applied stress.20 A similar phenomenon would occur in a single-crystal wire containing disloca- tions possessing Burgers vectors inclined at various angles to the stress axis. The diffusion potential at dislocations (each acting as sources or sinks) varies with each dislocation’s inclination. Vacancy fluxes develop in response to gradients in diffu- sion potential and cause the edge dislocations to climb, and as a result, the wire lengthens in the applied tensile stress direction. The problem of determining the elongation rate in both cases is therefore reduced to a boundary-value diffusion problem where the boundary conditions at the sources 20Surface sources and grain boundary sinks for atoms are considered in Sections 12.2 and 13.2. 3.5: STRESS AND DIFFUSION 65 t EPP t EPP Figure 3.10: Polycrystalline wire specimen with bamb2o grain structure subjected to uniaxial tensile stress, uzz, arising from the applied force, Fapp. The bulk crystal-diffusion fluxes shown in (a) and grain-boundary and surface-diffusion fluxes shown in (b) cause diffusional elongation. (c) Enlarged view at the junction of the grain boundary with the surface. and sinks are determined by the inclination of the sources and sinks relative to the applied stress and the magnitude of the applied stress. In the following we outline the procedure for obtaining the elongation rate of the polycrystalline wire shown in Fig. 3.10 for the case where the material is a pure cubic metal and the diffusion occurs through the grains as in Fig. 3.10a by a vacancy exchange mechanism. The diffusional creep rate of a single crystal containing various types of dislocations is treated in Chapter 16. Flux and diffusion equations. During diffusional creep, the stresses are relatively small, so variations in the vacancy concentration throughout the specimen will generally be small and can be ignored. The flux equation and diffusion equation in the grains are then given by Eqs. 3.71 and 3.75 (with @A = p~ - pv), which were derived for diffusion in a crystal during surface smoothing. In both cases, quasi-steady-state diffusion may be assumed, and any creation or destruction of vacancies at dislocations within the grains can be neglected. Boundary conditions. The cylindrical wire surface is a source and sink for vacancies, and the condition pv = 0 is therefore maintained there. The diffusion potential at the curved surface, a;, is given by Eq. 3.76. At the grain boundaries, the condition pv = 0 should also hold. The boundaries will be under a traction, unn = fiT.cr.fi, and when an atom is inserted, the tractions will be displaced as the grain expands by the volume CIA. For the case in Fig. 3.10, the boundary is oriented so that its normal is parallel to the z-axis and therefore unn = urz. This displacement contributes work, unnCI~ = ~,,QA, and reduces the potential energy of the system by a corresponding amount. This term must be added to the chemical term, p;, and therefore the diffusion potential along the 66 CHAPTER 3: DRIVING FORCES AND FLUXES FOR DIFFUSION grain boundary is2' @: decreases as the stress increases; an increase in the applied force increases onn, and when onn is sufficiently large so that @: < @:, atoms will diffuse from the surface to the boundaries at a quasi-steady rate. The bamboo wire behaves like a viscous material, due to the quasi-steady-state diffusional transport.22 Complete solutions for the elongation rates due to the grain boundary and surface diffusion fluxes shown in Fig. 3.10a and b are presented in Sections 16.1.1 and 16.1.3. 3.5.4 Summary of Diffusion Potentials The diffusion potential is the generalized thermodynamic driving force that pro- duces fluxes of atomic or molecular species. The diffusion potential reflects the change in energy that results from the motion of a species; therefore, it includes energy-storage mechanisms and any constraints on motion. @j = pj: For chemical interactions and entropic effects with no other constraint (e.g., interstitial diffusion). Section 3.1.4. @j = pj - pv: Reflecting the additional network constraint when sites are con- served (e.g., vacancy substitution). Section 3.1.1. @j = pj + qj4: When the diffusing species has an associated charge qj in an elec- trostatic potential #J (e.g., interstitial Li ions in a separator between an anode and a cathode). Section 3.2.1. @j = pj + RjP: Accounting for the work against a hydrostatic pressure, P, to move a species with volume Rj (e.g., interstitial diffusion in response to hydrostatic stress gradients). Section 3.5.2. @j = pj + 7~R.j: Accounting for the work against capillary pressure TK to move a species with volume Rj to an isotropic surface (e.g., surface diffusion in response to a curvature gradient). Section 3.4.2. @j = pj + K~R~: Accounting for the anisotropic equivalent to capillary pressure. K~, the weighted mean curvature, is the rate of energy increase with volume addition (e.g., surface diffusion on a faceted surface). Section 14.2.2. @j = pj - a,,Rj: Accounting for the work against an applied normal traction onn = fiT - (a fi) as an atom with volume Rj is added to an interface with normal fi; fiT is the transpose of fi (e.g., diffusion along an incoherent grain boundary in response to gradients in applied stress). Section 3.5.3. @j= pj+Rj{[(P.a) x~].((x~)}/{[((x~) xi].;}: Accountingforthechange in energy as a dislocation with Burgers vector b' and unit tangent ( climbs 21Again, as in the derivation of Eq. 3.82, quadratic terms in the elastic energy, which are of lower order in importance, have been neglected (see Larch6 and Cahn [21]). 22For an ideally viscous material, the strain rate is linearly related to the applied stress u by the relation = (l/a)o, where 17 is the viscosity. 3.5: STRESS AND DIFFUSION 67 with stress CT due to applied loads and other stress sources (i.e., other defects) for each added volume Rj (e.g., diffusion to a climbing dislocation by the substitutional mechanism). Section 13.3.2.23 Cpj = d2 fhom/acj2 - 2K,V2cj: Accounting for the gradient-energy term in the dif- fuse interface model for conserved order parameters (e.g., “uphill” diffusion during spinodal decomposition). Section 18.3.1. Bibliography 1. J.S. Kirkaldy and D.J. Young. Diffusion in the Condensed State. Institute of Metals, London, 1987. 2. J.G. Kirkwood, R.L. Baldwin, P.J. Dunlap, L.J. Gosting, and G. Kegeles. Flow equations and frames of reference for isothermal diffusion in liquids. J. Chem. Phys., 3. J. Bardeen and C. Herring. Diffusion in alloys and the Kirkendall effect. In J.H. Hol- lomon, editor, Atom Movements, pages 87-111. American Society for Metals, Cleve- land, OH, 1951. 4. A.D. Smigelskas and E.O. Kirkendall. Zinc diffusion in alpha brass. Pans. AIME, 5. R.W. Balluffi and B.H. Alexander. Dimensional changes normal to the direction of diffusion. J. Appl. Phys., 23:953-956, 1952. 6. L.S. Darken. Diffusion, mobility and their interrelation through free energy in binary metallic systems. Pans. AIME, 175:184-201, 1948. 7. J. Crank. Oxford University Press, Oxford, 2nd edition, 1975. 8. R.W. Balluffi. The supersaturation and precipitation of vacancies during diffusion. Acta Metall., 2(2):194-202, 1954. 9. R.F. Sekerka, C.L. Jeanfils, and R.W. Heckel. The moving boundary problem. In H.I. Aaronson, editor, Lectures on the Theory of Phase Transformations, pages 117-169. AIME, New York, 1975. 10. R.W. Balluffi. On the determination of diffusion coefficients in chemical diffusion. Acta Metall., 8(12):871-873, 1960. 11. R.W. Balluffi and B.H. Alexander. Development of porosity during diffusion in sub- stitutional solid solutions. J. Appl. Phys., 23(11):1237-1244, 1952. 12. R.W. Balluffi. Polygonization during diffusion. J. Appl. Phys., 23(12):1407-1408, 1952. 13. V.Y. Doo and R.W. Balluffi. Structural changes in single crystal copper-alpha-brass diffusion couples. Acta Metall., 6(6):428-438, 1959. 14. R.W. Cahn. Recovery and recrystallization. In R.W. Cahn and P. Haasen, editors, Physical Metallurgy, pages 1595-1671. North-Holland, Amsterdam, 1983. 15. C. Robinson. Diffusion and swelling of high polymers. 11. The orientation of polymer molecules which accompanies unidirectional diffusion. Pans. Faraday Soc., 42B: 12- 17, 1946. 16. D.R. Gaskell. Introduction to Metallurgical- Thermodynamics. McGraw-Hill, New York, 2nd edition, 1981. 33(5):1505-1513, 1960. 171:130-142, 1947. The Mathematics of Diffusion. 23The expression for this diffusion potential is derived in Exercise 13.3 [...]... diffusion What role does the heat of transport play in this phenomenon? Solution The basic force-flux relations are - 1 J1 = - L 1 1 V p i - L i g - V T T TQ= - L ~ 1 O p 1 - LQQ-VT 1 T (3. 85) Under isothermal conditions J; = -L11Vp1 TQ= - L ~ l V p i (3. 86) Therefore, using Eqs 3. 61 and 3. 86, (3. 87) The heat flux consists o f t w o parts The first is the heat flux due t o the flux of entropy, which is carried... o f a fixed plane in the V-frame due t o a loss o f interstitials is (3. 90) In the V-frame this must be compensated for by a gain of volume due t o a gain o f host atoms so that - + - = odV2 dV1 (3. 91) dt dt where dVz/dt is the rate o f volume gain due t o the gain o f host atoms corresponding to (3. 92) Substituting Eqs 3. 90 and 3. 92 into Eq 3. 91 and using Eq A.lO, (3. 93) 3. 3 In a classic diffusion... thermal conductivity is K1 = 35 5 J m-' s-' K-* , nearly that 1 of carbon-diamond Between the graphite layers, where the bonding is very weak, the conductivity is much lower, K l = 89 .3 J m - l s - l K-l Figure 4.6 is therefore representative of single-crystal graphite, where the basal plane is parallel to the layers shown In general, the properties of crystals and other types of materials, such as composites,... magnitude than the heat of transport Solution The local C concentration will be coupled t o the local temperature by Ea 3. 102 and therefore dcl dci dT - - - AH dT I dx dT dx kT2 dx Substitution of Eq 3. 1 03 into Eq 3. 60 then yields Jl kT2 Because ( A H observed dz = -D1cl ( A H + Qtrans) dT (3. 1 03) (3. 104) + Qtrans)is positive, the C atoms will be swept toward the cold end, as 3. 6 Show that the forces... out of the Si-containing alloy Si is a large substitutional atom, so the Fe and Si remained essentially immobile during the 6 0.6 0 C 0.5 e c u + a, 2 % 0.4 E-l 0 - 0 .3 -2 0 -1 0 0 I l 10 l Distance from weld (mm) 1 20 Figure 3. 11: Nonuniform concentration of C produced by diffusion from an initially uniform distribution Carbon migrated from the Fe-Si-C (left) to the Fe-C alloy (right) From Darken [ 23] ... cf(T2) ci(Ti), { [ Ac1 = cl(T1) exp 1 -8 4000 (Ti - T2) NokTiTz -l> (3. 101) 3. 5 Suppose that a two-phase system consists of a fine dispersion of a carbide phase in a matrix The carbide particles are in equilibrium with C dissolved interstitially in the matrix phase, with the equilibrium solubility given by c1 = c,e - A H / ( k T ) o (3. 102) If a bar-shaped specimen of this material is subjected to a steep... D ) Let elements of such a transformed system be identified by a “hat.” Then B,, 0 0 (4.60) =[ i3] ; The diagonal elements of b are the eigenvalues of D, and the coordinate system of b defines the principal axes 21,22, 23 (the eigensystem) In the principal axes coordinate system, the diffusion equation then has the relatively simple form det Dll D12 Dl3 Dl2 022 0 23 X Dl3 0 23 033 - =0 (4.62) 90 CHAPTER... keeping dA = 27rhdh (3. 126) (3. 127) Therefore, using Eqs 3. 125, 3. 126, and 3. 127, and the fact that h2/r2 . the free-energy barrier to the 16When the stress is nonuniform and stress gradients exist, the stress will exert a force, as discussed in the following section. 17The tensor nature of the. applied stress. A particularly simple example of this type of stress-induced diffusional trans- port is illustrated in Fig. 3. 10, where a polycrystalline wire specimen possessing a “bamboo”. atmosphere, known as a Cottrell atmosphere, is created where the composition-gradient term cancels the stress-gradient term of Eq. 3. 83 (this is demonstrated in Exercise 3. 8). From these