Kinetics of Materials - R. Balluff S. Allen W. Carter (Wiley 2005) WW Part 6 ppt

8 2 ATOMIC MODELS FOR DIFFUSIVITIES 181 Crystal Self- Difhion in Nonstoichiometric Materials. Nonstoichiometry of semicon- ductor oxides can be induced by the material's environment. For example, materials such as FeO (illustrated in Fig. 8.14), NO, and COO can be made metal-deficient (or 0-rich) in oxidizing environments and Ti02 and ZrOz can be made 0-deficient under reducing conditions. These induced stoichiometric variations cause large changes in point-defect concentrations and therefore affect diffusivities and electri- cal conductivities. In pure FeO, the point defects are primarily Schottky defects that satisfy mass- action and equilibrium relationships similar to those given in Eqs. 8.39 and 8.42. When FeO is oxidized through the reaction X FeO + -02 = FeOl+, (8.52) 2 each 0 atom takes two electrons from two Fe++ ions, as illustrated in Fig. 8.14~~ according to the reactions (8.53) corresponding to the combined reaction, (8.54) 1 2 2Fe++ + -02 = 2Fe"' + 0 Electrical neutrality requires that a cation vacancy be created for every 0 atom added, as in Fig. 8.14b; this, combined with site conservation, becomes (8.55) 1 2 2Fege + -02 = 2Febe + 0; + V:e te Fe+++ I o= y o= 1 o= ) Fe+++ Figure 8.14: Addition of R ncwtral 0 atom to FeO to produce 0-rich (metal-deficient) oxide. (a) An 0 atom receives two electrons from Fe" ions in the hulk material. (b) The final structure contains defects in the forni of t,wo Fc+++ ions and a cation (Fe++) vacancy. 182 CHAPTER 8 DIFFUSION IN CRYSTALS which can be written in terms of holes, h, in the valence band created by the loss of an electron from an Fe++ ion producing an Fe+ft ion, 1 -02 2 = 0; + Vge + 2h;e (8.56) hFe = FeFe - Fege Equation 8.56 predicts a relationship between the cation vacancy site fraction and the oxygen gas pressure. The equilibrium constant for this reaction is 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 pressure at constant temperature and the activation energy is (8.60) The dominance of oxidation-induced vacancies creates an additional behavior regime. The effect of this additional regime on diffusivity behavior is illustrated in Fig. 8.15. Other types of environmental effects create defects through other mechanisms and may lead to other behavior regimes. 1 IT Slope a Hcv + Hfi2 \ Figure 8.15: Arrhenius plot for self-diffusivity on the cation sublattice, *DFc, in FeO made O-rich by exposure to oxygen gas at a pressure Poz or doped with an aliovalent impurity. Three regimes of behavior are possible. each with a different activation energy. 8.3: DIFFUSIONAL ANELASTICITY (INTERNAL FRICTION) 183 8.3 DIFFUSIONAL ANELASTICITY (INTERNAL FRICTION) ‘ In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. For an ideally elastic material, the stress is linearly related to the strain by u = C& (8.61) (where the constant C represents the elastic stiffness), and conversely, the strain is linearly related to the stress by & = Su (8.62) (where the constant S = 1/C represents the compliance). For each level of stress, such a material responds immediately with a unique value of the strain. How- ever, in many real materials, stress-induced diffusional processes cause additional time-dependent anelastic strains and nonlinear behavior. This anelastic behavior degrades the mechanical work performed by the stresses into heat so that the ma- terial exhibits internal friction, which can damp out mechanical oscillations in a material. Anelasticity therefore affects the mechanical properties of materials. As seen below, its study yields unique information about a number of kinetic processes in materials, such as diffusion coefficients, especially at relatively low temperatures. 8.3.1 Anelastic behavior can be produced by the stress-induced diffusional jumping of anisotropic point defects. An example of such a process is described in Exercise 8.5, in which an f.c.c. metal contains a concentration of self-interstitial point defects hav- ing the (100) split-dumbbell configuration (see Fig. 8.5d). Each defect produces a tetragonal distortion of the crystal, elongating it preferentially along its dumbbell axis. The three types of sites in the crystal in which the interstitials can lie with their axes along [loo], [OlO], or [OOl] exist in equal numbers and will be occupied equally in the absence of any stress. However, if the crystal is suddenly stressed uni- axially along [loo], an excess of dumbbells will jump to sites where they are aligned along [loo], because the crystal is elongated along the direction of the applied stress and the applied stress performs work. This principle applies to loading along the other cube directions as well. (Note that this is a good example of LeChatelier’s principle.) When the stress is released suddenly, the defects repopulate the sites in equal numbers and the crystal regains its original shape. The relaxation time for this re-population is (8.63) where r is the total jump frequency of a dumbbell (see Exercise 8.5). This process therefore causes the crystal to elongate or to contract in response to the applied Anelasticity due to Reorientation of Anisotropic Point Defects 2 r=- 317 184 CHAPTER 8. DIFFUSION IN CRYSTALS stress at a rate dependent upon the rate at which the dumbbells jump between the different types of sites. The overall response of the crystal to such a stress cycle is shown in Fig. 8.16. When the stress uo is applied suddenly, the crystal instantaneously undergoes an ideally elastic strain following Eq. 8.62. As the stress is maintained, the crystal un- dergoes further time-dependent strain due to the re-population of the interstitials. When the stress is released, the ideally elastic strain is recovered instantaneously and the remaining anelastic strain will be recovered in a time-dependent fashion as the interstitials regain their random distribution. Stress oo removed Stress t applied Time, t Figure 8.16: applied suddenly at t = 0, held constant for a period of time, and then suddenly removed. Strain vs. time for an anelastic solid during a stress cycle in which stress is General Formulation of Anelastic Behavior. Anelastic behavior where the strain is a function of both stress and time may be described by generalizing Eq. 8.62 and expressing the compliance in the more general form (8.64) The initial value of the compliance, corresponding to S(0) = su (8.65) is the unrelaxed compliance, which corresponds to ideal elastic behavior because there is no time for point-defect re-population. The value of S(t) at long times, corresponding to s(m) = SR (8.66) is the relaxed compliance, since it includes the maximum possible additional strain due to the stress-induced re-population of the defects. Clearly, SR > Su. Suppose now that the crystal is subjected to a periodic applied stress of ampli- tude uo corresponding to g = uoeiWt (8.67) The resulting strain is also periodic with the same angular frequency but generally lags behind the stress because time is required for the growth (or decay) of the anelastic strain contributed by the point-defect re-population during each cycle. The strain may therefore be written & = &oei(wt 4) (8.68) 8 3. DIFFUSIONAL ANELASTICITY (INTERNAL FRICTION) 185 where q5 is the phase angle by which the strain lags behind the stress. Note that 4 = 0 at both very high and very low frequencies. At very high frequencies, the cycling is so rapid that the point defects have insufficient time to repopulate and therefore make no contribution to the strain. At very low frequencies, there is sufficient time for the defects to re-populate (relax) at every value of the stress, and the stress and strain are therefore again in phase. To proceed with the more general intermediate case, it is convenient to write the expression for the strain, E = Ele'wt - iE2eiWt (8.69) In this formulation, the first term on the right-hand side is the component of E that is in phase with the stress, and the second term is the component that lags behind the stress by 90". Also, - E2 = tan4 (8.70) El The compliance (again the ratio of strain over stress) is then El .E2 - 1- -_ - (~1 - 2~2) eiWt S(W) = uo eiWt uo go (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 loop, AW= ode (8.72) will be dissipated (converted to heat) during each cycle. To determine AW, only the part of the strain that is out of phase with the stress must be considered. The stress and strain in Eq. 8.72 can then be represented by f u = uo coswt and E = ~2 cos(wt - 7r/2) (8.73) and 2W/T AW = -O,E~L (8.74) Figure 8.17: subjected to an oscillating stress. Hysteresis loop shown by the stress-strain curve of an anelastic solid 186 CHAPTER 8 DIFFUSION IN CRYSTALS The energy dissipated can be compared with the maximum elastic strain energy, W, which is stored in the material during the stress cycle. Because the elastic strain is proportional to the applied stress, W is equal to just half of the product of the maximum stress and strain (i.e., W = oo&1/2), and therefore E2 - = 27r- AW W El (8.75) AW/W can be measured with a torsion pendulum, in which a specimen in the form of a wire containing the point defects is made the active element and strained periodically in torsion as in Fig. 8.18. If the pendulum is put into free torsional oscillation, its amplitude will slowly decay (damp out), due to the dissipation of energy. As shown in Exercise 8.20, the maximum potential energy (the elastic energy, W) stored in the pendulum is proportional to the square of the amplitude of its oscillation, A. The amplitude of the oscillations therefore decreases according to where N is the number of the oscillation and it is realistically assumed that AW << W. The logarithmic decay of the amplitude is the logarithmic decrement, designated by 6. Therefore, (8.77) Measurements of 6 yield direct information about the magnitude of the energy dis- sipation and the phase angle. $ measures the fractional energy loss per cycle due to the anelasticity and is often termed the internal friction. According to the discus- sion above, 6 will be a function of the frequency, w; should approach zero at both low and high frequencies; and will have a maximum at some intermediate frequency. The maximum occurs at a frequency that is the reciprocal of the relaxation time for the re-population of the point defects. Specimen k Figure 8.18: to an oscillating stress. Torsion pendulum in which the specimen is in the form of a wire subjected Analog Model for Standard Anelastic Solid. To find the dependence of 6 on fre- quency, a model that relates the stress and strain and their time derivatives must 8.3: DIFFUSIONAL ANELASTICITY (INTERNAL FRICTION) 187 be constructed. Figure 8.19’s analog model for a standard anelastic solid serves this purpose; it consists of two linear springs, S1 and S2, and a dashpot, D, which is a plunger immersed in a viscous fluid. The dashpot changes length at a rate proportional to the force exerted on it. This model gives a good account of the anelastic behavior illustrated in Fig. 8.16. When a force, F, is first applied, S2 elongates instantaneously. At the same time, in the upper section of the model, F is fully supported by D and the force on S1 is zero. However, with increasing time, D extends and the force is gradually transferred to S1, which extends under its influence. Eventually, the force is fully transferred, as both S1 and S2 experience the full force while D experiences nothing. At this point, the model reaches its fullest extension. The extension remains constant until F is suddenly removed. S2 then contracts instantaneously and S1 gradually relaxes by forcing D back to its original extension and the model recovers its original state. S2 therefore accounts for the ideal elasticity of the solid, and the combination of S1 and D accounts for the anelasticity. The linear spring element S1 will undergo an extension Axsl according to Axsl = aslFs1 (8.78) where Fsl is the force on S1 and as1 is a constant. Similarly for S2, Also, for the dashpot, (8.80) where AXD is the extension of D, FD is the force on D, and aD is a constant. In addit ion, AXS~ = AXD (8.81) F = F.92 (8.82) F t F (8.83) Figure 8.19: Analog model for a standard anelastic solid. 188 CHAPTER 8: DIFFUSION IN CRYSTALS Finally, the stress, 0, and strain, E, may be expressed and (8.85) 1 a, o = -F where a, and a, are constants. By combining Eqs. 8.78-8.85, the following equa- tion, which contains three independent constants (bracketed) corresponding to the three elements in the model, can be obtained: Equation 8.86 may be solved for the time period in Fig. 8.16 during which the stress is held constant at uo. Under this condition, it reduces to (8.87) Equation 8.87’s general solution can be written The constant of integration, A, can be evaluated by recalling that at t = 0 only S2 is extended. The strain is then ~(0) = U~AXS~ = aEas2Fs2 = u,u~~F = ~,~s2~,u, (8.89) and therefore from Eq. 8.88, A = a,a,asl and (8.90) Examining the forms of ~(0) and ~(m) and comparing the results with Eqs. 8.64- 8.66 shows that a,a,asz = Su and a,a,asl = SR - Su. Also, the anelastic relaxation occurs exponentially, in agreement with the results in Exercise 8.5, and the relaxation time corresponds to r = US~/UD. Equation 8.90 then takes the simpler form (8.91) ) E(t) = a,auas2a, + a,a,as10, (1 - e-aDt’a= E(t) = suuo + (SR - SrJ)ao(l- e+) and Eq. 8.86 takes the form (8.92) Frequency Dependence of the Logarithmic Decrement. The frequency dependence of S can now be found. Putting Eqs. 8.67 and 8.69 into Eq. 8.92 and equating the real and imaginary parts yields two equations which can be solved for ~1 and ~2 in the forms 1 sR + - w2r2 El = ffo (su + (8.93) 8.3: DIFFUSIONAL ANELASTICITY (INTERNAL FRICTION) 189 Therefore, Because (SR - Su) << Su in the majority of cases, (8.94) (8.95) (8.96) The decrement 6(w) forms a Debye peak, as shown in Fig. 8.20. The maximum damping (anelasticity) occurs when the applied angular frequency is tuned to the relaxation time of the anelastic process so that wr = 1. Also, S(w) approaches zero at both high and low frequencies, as anticipated. 0 In cot Figure 8.20: exhibits a Debye peak at lnw = 0 (or w = l/~). Curve of the decrement, 6(w), according to Eq. 8.96, vs. lnur. The curve 8.3.2 Determination of Diffusivities The preceding analysis provides a powerful method for determining the diffusivities of species that produce an anelastic relaxation, such as the split-dumbbell inter- stitial point defects. A torsional pendulum can be used to find the frequency, wp, corresponding to the Debye peak. The relaxation time is then calculated using the relation r = l/wp, and the diffusivity is obtained from the known relation- ships among the relaxation time, the jump frequency, and the diffusivity. For the split-dumbbell interstitials, the relaxation time is related to the jump frequency by Eq. 8.63, and the expression for the diffusivity (i.e., D = l?a2/12), is derived in Exercise 8.6. Therefore, D = a2/18r. This method has been used to determine the diffusivities of a wide variety of interstitial species, particularly at low tem- peratures, where the jump frequency is low but still measurable through use of a torsion pendulum. A particularly important example 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 point- defects in metals. J. Phys. F, 3(2):295-321, 1973. 190 CHAPTER 8: DIFFUSION IN CRYSTALS 3. W. Schilling. Self-interstitial atoms in metals. J. Nucl. Mats., 69-70( 1-2):465-489, 1978. 4. P. Shewmon. Diffusion in Solids. The Minerals, Metals and Materials Society, War- rendale, PA, 1989. 5. G. Neumann. Diffusion mechanisms in metals. In G.E. Murch and D.J. Fischer, editors, Defect and Diffusion Forum, volume 66-69, pages 43-64, Brookfield, VT, 1990. Sci-Tech Publications. 6. W. Frank, U. Gosele, H. Mehrer, and A. Seeger. Diffusion in silicon and germanium. In G.E. Murch and A.S. Nowick, editors, Diffusion in Crystalline Solids, pages 63-142, 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):1-17, 1985. 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 121-148, Brookfield, VT, 1991. Sci- Tech Publications. 9. A. Atkinson. Interfacial diffusion. Mat. Res. SOC. Symp., 122:183-192, 1988. 10. D. Beshers. Diffusion of interstitial impurities. In Diffusion, pages 209-240, Metals 11. L.A. Girifalco. Statistical Physics of Materials. John Wiley & Sons, New York, 1973. 12. A.D. LeClaire and A.B. Lidiard. Correlation effects in diffusion in crystals. Phil. Park, OH, 1973. American Society for Metals. Mag., 1(6):518-527, 1956. 13. K. Compaan and Y. Haven. Correlation factors for diffusion in solids. Trans. Faraday 14. R.O. Simmons and R.W. Balluffi. Measurements of equilibrium vacancy concentra- tions in aluminum. Phys. Rev., 117:52-61, 1960. 15. A. Seeger. The study of point defects in metals in thermal equilibrium. I. The equi- librium concentration of point defects. Cryst. Lattice Defects, 4:221-253, 1973. 16. R.W. Balluffi. Vacancy defect mobilities and binding energies obtained from annealing studies. J. Nucl. Mats., 69-70:240-263, 1978. 17. W.D. Kingery, H.K. Bowen, and D.R. Uhlmann. Introduction to Ceramics. John Wiley & Sons, New York, 1976. 18. Y M. Chiang, D. Birnie, and W.D. Kingery. Physical Ceramics. John Wiley & Sons, New York, 1996. 19. D. Halliday and R. Resnick. Fundamentals of Physics. John Wiley & Sons, New York, 1974. SOC., 52:786-801, 1956. EXERCISES 8.1 It has sometimes been claimed that the observation of a Kirkendall effect implies that the diffusion occurred by a vacancy mechanism. However, a Kirkendall effect can be produced just as well by the interstitialcy mechanism. Explain why this is so. Solution. Substitutional atoms of type 1 may diffuse more rapidly than atoms of type 2 if they diffuse independently by the interstitialcy mechanism in Fig. 8.4. To sustain the unequal fluxes, interstitial-atom defects can be created at climbing dislocations acting [...]... uncorrelated, so f = 1 The vacancy diffusivity DV for this two-dimensional diffusion is rr' D v = -f 4 rr2 1000 s-' 0 5 mm2 ' = -= = 6. 25 x 4 4 m2 s-' (b) Self-diffusion of a tracer by vacancy exchange is correlated, so in this square lattice we have f E ( 2 - l ) / ( z 1) % 0 .6. The tracer self-diffusivity *D is + = 6. 25 x rn's-' x x 0 .6 = 3.8 x lo-' m's-' (c) A very simple estimate can be made by using the... times in the presence of the applied stress In view of the symmetry of Eqs 8.180, we try Cl(t) = (f ‘p> - e-k‘t + ‘ ; c2(t) = (f - c;~) e-“lt + c‘lq (8.183) which satisfy the conditions in Eq 8.182 Direct substitution shows that Eqs 8.183 indeed satisfy Eqs 8.180 when higher-order terms involving products of the small quantities U t - j / ( k T ) are neglected and k‘ = 6r’ - I +-+ -) Ul-2 3 ( 1 3kT 3kT... lowered further, the ratio of diffusivities becomes larger and short-circuit diffusion assumes even greater importance Generally, similar behavior is found in ionically bonded crystals, as shown in Fig 9.3 -1 2 N E -1 4 M -1 6 v 8 0 -1 8 -2 0 -2 2 6 8 10 1 0 4 1 (~~ - 1 ) 12 Figure 9.3: Self-diffusivities of 0 and Ni on their respective sublattices in a NiO sin le crystal free of significant line imperfections... c3 be the concentrations of interstitials occupying sites of types 1, 2, and 3, respectively Also, c1 +c2 +c3 = ctot = constant Since an interstitial CHAPTER 8: DIFFUSION IN CRYSTALS in a given type of site can jump into two sites of each other type, dci - = - 2 ( r L + rL3+ r L ) c1 + 2 (rL1 r L )c2 + 2r’3+1~t0t dt dc2 - = - 2 (rL1 rL3+ rL2)c2 + 2 ( r L - r;,2) + dt c1 + 2r $-2 ~tot (8.180) If the barrier... neglecting higher-order terms yields (8.119) Finally, identifying aAexp[-Uo/(kT)] with D and using Eq 8.117, J = dc -D- dx k dx T - DcdlC, (8.120) 8.8 Calculate the correlation factor for tracer self-diffusion by the vacancy mech- anism in the two-dimensional close-packed lattice illustrated in Fig 8.22 The tracer atom at site 7 has just exchanged with the vacancy, which is now at site 6 Following Shewmon... 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)... = 1 (case) - 3(i - p ) 1 - (case) 2 3p * D = - r(2) = - ra2 1 - p f 2 2+3p The self-diffusivity is then (8.147) (8.148) + In the random case when p = 1 /6, f = 1 In the most correlated case when p = 1, f = 0 When p = 0 and the atom cannot j u m p backward t o erase i t s previous jump, f = 3/2 and diffusion is enhanced relative t o the random, uncorrelated case 8.14 A computer simulation of diffusion... t - J / ( k T ) , + r;,3 w) + w) r ;-2 = r’ (1 = r’ (1 - h = r’ (1 2kT ) = r/ (1 - 2kT - h) r;,, 2kT r;+l = r’ (1 + W-J = r’ (1 + *) r L l = r’ (1 + w) (8.181) where I?’ is the j u m p rate between any two adjacent sites in the absence of stress Equation 8.180 is a pair o f simultaneous linear first-order equations with constant coefFicients The initial and final conditions are Cl(0) = c2(0) = Ctot -. .. case, Jnet = J(*) - J ( C ) = -TZACr, / 3 2 Using a Taylor expansion t o evaluate 3 2 - -TZBCrL ( T Z A- T Z B and ) = 'Crk(TZA 2 2 3 passing between -nB) (8.133) employing Eq 8.131, (8.134) 8.10 Show that the results obtained in Exercise 8.9 (i.e., Eqs 8.125 and 8.1 26) , can be obtained in a simpler way by using Eq 7.53 in one dimension if ( R 2 )is taken as the mean value of the squares of the jump vector... which the regular atomic structure characteristic of the crystalline state no longer exists Good bulk material is free of line or planar imperfections Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter Copyright @ 2005 John Wiley & Sons, Inc 209 210 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS Table 9.1: Notation for Short-circuit Diffusivities D D(undissoc) diffusivity . over stress) is then El .E2 - 1- -_ - (~1 - 2~2) eiWt S( W) = uo eiWt uo go (8.71) Because the strain lags behind the stress, the stress-strain curve for each cycle consists of. elastic strain following Eq. 8 .62 . As the stress is maintained, the crystal un- dergoes further time-dependent strain due to the re-population of the interstitials. When the stress is released,. different types of sites. The overall response of the crystal to such a stress cycle is shown in Fig. 8. 16. When the stress uo is applied suddenly, the crystal instantaneously undergoes an