Kinetics of Materials - R. Balluff_ S. Allen_ W. Carter (Wiley_ 2005) Episode 13 pptx

EXERCISES 527 20.4 Find an expression for the thickness of the growing B-rich P-phase surface layer shown in Fig. 20.3 as a function of time. Details of the growth of this layer have been discussed in Section 20.1.2, and a strategy for determining the growth rate has been outlined. Assume constant diffusivities in both phases. Solution. Solutions to the diffusion equation in the a: and p phases, which match the boundary conditions, are cp, - cp,” - erf(z/m) - erf(Az/@) Cp - Cp,” - erf(Al/m) - erf(Az/m) (20.85) where (20.86) (20.87) The Stefan conditions at interfaces 1 and 2 are given by Eqs. 20.12 and 20.17, respec- tively. Substituting the appropriate relationships from those given above into these two equations then yields equations that can be solved simultaneously for A1 and Az: Q’a cp,” - cp e-A:/(48p) A1 = J;; cp - erf (A~/-) - erf (A~/&ZF) J;; (20.88) Qa- CEO - cag00 - A: / (48- ) - cp - CEO 1 - erf (A~/&Z) Finally, the layer thickness is given by x = XI - xz = (A1 - Az)fi’. 20.5 Using the scaling method, find an expression for the diffusion-limited rate of growth of a cylindrical B-rich precipitate growing in an infinite a-phase ma- trix. Assume the same boundary conditions as in the analysis in Section 20.2.1 (Eqs. 20.37-20.39) for the growth of a spherical particle. Note that Fig. 20.6, which applied to the growth of a sphericallarticle in Section 20.2.1, will also apply. Use the scaling parameter 7 = r/(4Dat)lI2. You will need the integral (20.90) which has been tabulated [29]. Solution. Starting with the diffusion equation in cylindrical coordinates (see Eq. 5.8) and using the scaling parameter to change variables, the diffusion equation in 7-space becomes (20.91) This result, along with the boundary conditions given by Eqs. 20.37-20.39, shows that the particle will grow parabolically according to Rit) = rl~d46t (20.92) 528 CHAPTER 20: GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS Integrating Eq. 20.91 once, dcg e-72 - =a1- d17 17 where a1 = constant. Integrating again yields Determining a1 from the condition cg = c"," when 17 = VR, The Stefan condition at the interface is Finally, substituting Eq. 20.95 into Eq. 20.97, we have (20.93) (20.94) (20.95) (20.96) (20.97) (20.98) for the determination of VR. 20.6 Find an expression for the rate of thickening of a B-rich /3-phase precipitate platelet in an infinite a-phase matrix in a AIB binary system as in Fig. 20.6. Assume diffusion-limited conditions and a constant diffusivity, Ea, in the a-phase matrix. Also, assume that the atomic volume of each species is constant throughout so that there is no overall volume change and that the plate is extensive enough so that edge effects can be neglected. Use the scaling method. Solution. Let x be the distance coordinate perpendicular to the platelet. The boundary conditions will be the same as Eqs. 20.37-20.39 if T - IZ: and 77 -+ x/m. The method is basically the same as that used to obtain the solution for the sphere in Section 20.2.1. In the present case, ~(t) will be the half-thickness of the plate. 77 will be constant at the interface at the value vx = x(t)/&%, so that X(t) will increase parabolically with time according to X(t) = T)xa (20.99) We now determine qx by solving for cg and invoking the Stefan condition at the in- terface. The difFusion equation was scaled and integrated in Cartesian coordinates in Section 4.2.2 with the solution given by Eq. 4.28. When this solution is matched to the present boundary conditions, (20.100) EXERCISES 529 The Stefan condition at the interface is (20.101) Use of Eqs. 20.99 and 20.100 in Eq. 20.101 then yields the desired expression for vx, 20.7 Consider again the problem posed in Exercise 20.6, whose solution had the form of a transcendental equation. A simple and useful approximate solution can be found by using the linear approximation to the diffusion profile shown in Fig. 20.14. Find the solution based on this approximation. Cam I I I I I I I ox Z x+ Figure 20.14: diffusion-limited thickening of a slab of thickness 2x. Approximate composition vs. distance profile for determination of the Solution. The Stefan condition at the interface is (20.103) Using the linear approximation in Fig. 20.14, &/ax = (cgm - c;') /(Z - x). Also, the conservation of B atoms requires that the two shaded areas in the figure be equal. Therefore, (cg - cSm) x = (cgm - CEO) (2 - x)/2. Putting these relationships into Eq. 20.103 gives an (cy - cy)' 2 (cp - cgp) (cp - cy) x dX = dt (20.104) Therefore, (20.105) Zener has compared the approximate solution above with the exact solution found in Exercise 20.6 and finds reasonably good agreement [30]. 20.8 Consider a binary system consisting of a thin spherical shell of (u phase embed- ded in an infinite body of /3 phase. The phase diagram is shown in Fig. 20.15. 530 CHAPTER 20: GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS Figure 20.15: Binary phase diagram. The system is at the temperature T* and the p phase is essentially pure B, while the CY phase contains a moderately low concentration of B so that Henry's law is obeyed. The average radius of the shell is (R) and the thickness of the thin shell is 6R, where 6R << (R). Assume that the diffusion rate of B in the ,8 phase is extremely slow and can be neglected in comparison to its diffusion rate in the CY phase. Find an expression for the shrinkage rate of the shell and show that the shell will shrink at a rate inversely proportional to (R). Assume that the concentration of B in the CY phase is maintained in local equilibrium with the ,B phase at both the inner and outer interphase boundaries between the shell and the p phase. Also, neglect any small volume changes that might occur. Solution. The outer interface is concave and the inner interface is convex with respect to the p phase. The concentrations of B maintained in equilibrium in the a phase at the outer interface, cE,$, and at the inner interface, cy!, are given by Eq. 15.4. The concentration difFerence across the shell is therefore Aca = c:(t - cEb 4yRc"(m) (20.106) =c"(m) 1 -crn(m) 1+- [ Z)] [ Z)] = - kT(R) The diffusion of B through the shell will be in a quasi-steady state, and since 6R << (R), the flux through the shell can be expressed to a good approximation as (20.107) The flux is therefore directed toward the outer interface, causing the spherical shell to shrink toward its center under the driving force supplied by the decrease in interfacial energy that occurs as a result of the shrinkage. (Note that ultimately the shell will shrink to form a solid sphere of a phase at the origin.) Since Aca << ca(m), the equations of continuity (for Stefan conditions, see Sec- tion 20.1.2) at the inner and outer interfaces can be expressed as winc:fi = wincp + J (20.108) EXERCISES 531 where 2)in and vout are the velocities of the inner and outer interfaces, respectively. Using these relationships and neglecting small differences between ckp, cz$, and ca(oo), the average shell velocity is then CHAPTER 21 CONCURRENT NUCLEATION AND GROWTH TRANS FO R MATI 0 N K I N ETI CS A discontinuous transformation generally occurs by the concurrent nucleation and growth of the new phase (i.e., by the nucleation of new particles and the growth of previously nucleated ones). In this chapter we present an analysis of the resulting overall rate of transformation. Time-temperature-transformation diagrams, which display the degree of overall transformation as a function of time and temperature, are introduced and interpreted in terms of a nucleation and growth model. 21.1 OVERALL RATE OF DISCONTINUOUS TRANSFORMATION Consider homogeneous nucleation in a three-dimensional system. In the simplest model, these nuclei form at random locations. The nucleation rate, J, specifies the number of nuclei forming per unit volume per unit time. These nuclei form at locations that have not already been transformed by growth of any previously formed nuclei. Once nucleated, a particle grows at a rate R = dR/dt and the untransformed volume decreases. However, no particle can grow indefinitely. A particle nucleated near a surface can grow to impingement with the surface, and the transformation at that location will cease. Similarly, two particles that nucleate near each other grow until they impinge and transformation ceases. Alternatively, growth can be driven by supersaturation and individual nuclei could have their growth limited by the decreasing supersaturation in the untransformed volume. In previous chapters we have developed models for discontinuous transforma- tions that treat nucleation and growth processes independently. However, when Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 533 Copyright @ 2005 John Wiley & Sons, Inc. 534 CHAPTER 21: CONCURRENT NUCLEATION AND GROWTH these processes occur concurrently, the overall transformation rate (i.e., the volume transformed per unit time) and microstructural characteristics such as particle or grain size depend on the interplay of nucleation and growth processes. The theory of the kinetics of concurrent nucleation and growth reactions has a rich history that includes work by Kolmogorov [l] , Johnson and Mehl[2], Avrami [3- 51, Jackson [6], and Cahn [7]. Cahn's time-cone method for treating a class of these problems is the most general of these, with the most transparent assumptions, and is presented here. The method of Johnson, Mehl, and Avrami is covered in Section 4 of Christian's text [8]. 21.1.1 The key to obtaining exact solutions to the transformation kinetics is to make explicit assumptions about the statistical homogeneity of nucleation and growth processes in the system. Following Cahn, we denote homogeneous nucleation on randomly dispersed sites in a volume by volume nucleation; heterogeneous nucle- ation on randomly dispersed sites on a surface or interface in a volume by surface nucleation, and heterogeneous nucleation on randomly dispersed sites along a linear feature in a volume by line nucleation. Exact solutions for transformation kinetics can be obtained when nucleation and growth rates are spatially homogeneous at any instant within the system of interest. The method is applicable to finite samples with a wide range of geometries and can yield position-dependent transformation kinetics. Statistical homogeneity is necessary for application of the time-cone method. Statistical homogeneity is not a valid assumption during precipitation from super- saturated solution because the untransformed regions develop concentration gradi- ents around growing particles and hence the nucleation rate becomes nonuniform. Also, in a material with thermal gradients undergoing a discontinuous transfor- mation (e.g. , during continuous cooling) , the nucleation rate will be nonuniform. However, the theory is applicable to discontinuous transformations in which the parent and product phases have the same composition and in which the tempera- ture is essentially uniform at any instant. Examples include recrystallization, first- order order-disorder transformations, massive transformations, and crystallization during vapor-deposition processes. Time-Cone Analysis of Concurrent Nucleation and Growth Probability that a Point r'will not be Transformed at Time t. The probability that a point Fin the sample will be untransformed at the time t is obtained by computing the probability that no nuclei had formed at any location r' and any previous time r that could have grown and led to prior transformation at r' and t. This is accomplished in three steps: i. The set of points that could possibly affect a given point grows with time; therefore, to specify this set of points, a time coordinate is required in addition to the spatial coordinates of the sample. For nucleation and growth in a sample of dimensionality 3, the augmented space has Cartesian coordinates x, y, z, and t; more generally, coordinates r' and t for any spatial dimension. Emanating from the point (.',t) and extending to earlier times is a domain that is the set of all points in the augmented space that would have caused transformation at (r', t) if nucleation had occurred at a nearby point and earlier 21.1: OVERALL RATE OF DISCONTINUOUS TRANSFORMATION 535 time (F, T). This subset of points is called V,. For the d = 2 case, the domain V, is a cone of height t; Cahn refers to this domain as the time cone. The cross section of the time cone at any time depends on the growth rate function &t). Figure 21.1 is a representation of the transformation by random nucleation along a one-dimensional sample for the constant-growth-rate case. Distance, x (4 Distance, x (b) Figure 21.1: Nucleation and growth along a one-dimensional specimen. (a) Growth cones. The apex of each cone coincides with the time and location of a nucleation event. Once nucleated, constant linear growth transforms the region surrounding a nucleus. Two nuclei have formed at tl; at tz the sample is approximately half transformed; at t3 it is fully transformed. Transforming regions impinge where growth cones intersect. (b) Time cone for the point (z, t). For the point (2, t) to be untransformed at time t, no prior nucleation can have occurred within the time cone volume, V,. 11. iii. The nucleation rate can be integrated over the time cone to obtain the number of nuclei expected in V,, denoted by (N),. The untransformed fraction 1 -C is obtained from the stochastic independence of the nucleation events-that is, any particular nucleation event in untrans- formed material is not influenced by any other nucleation event. Under these conditions, the theory can be formulated using the Poisson probability equa- tion [9], which states that if p = the mean rate at which events occur, then the probability, p(k), that exactly Ic events occur in time t is (21.1) The probability that exactly zero events occurs is given by Eq. 21.1 as p(0) = e-p' (21.2) and the probability that at least one event that leads to some transformation is p(k 2 1) = 1 - P(o) = 1 - e-p' (21.3) In the context of nucleation and growth kinetics, the quantity p corresponds to the nucleation rate, J; the product pt corresponds to the number of nucleation events, (N),, in the time cone; and the probability that exactly zero nucleation will have occurred within the time cone is equal to the untransformed volume fraction at time t, 1 - C, giving the relation 1 - C = e-w, (21.4) 536 CHAPTER 21: CONCURRENT NUCLEATION AND GROWTH Equation 21.4 forms the basis of the theory for the kinetics of concurrent nu- cleation and growth transformations. Specific cases can be formulated by deriving appropriate expressions for the quantity (N),. Time Cone V, for Isotropic, Time-Dependent Growth Rate R(t). The time cone's geometry is given by simple relations. For isotropic (i.e., radial) growth, at time t the radius of a transformed region nucleated at an earlier time T is given by t R (t, T) = R (t') dt' (21.5) For the transformation to have occurred at ?prior to time t, the radius of a trans- formed region nucleated earlier at r' must be larger than the distance between r' and 7, [R(~,T)]' - I?- JI2 2 0 (21.6) The time cone is the set of points Eq. 21.6. Number of Nuclei Expected in the Time Cone, (N),. For time-dependent nucleation rates J (t) and isotropic growth rates R (t) (such as in nonisothermal transforma- tions under conditions in which thermal gradients can be neglected), the number of nuclei in V, is given for the d = 3 case as t (N), (t) = $ / J (t') [R (t, t')I3 dt' 0 (21.7) Expressions for Transformation Rate when Nucleation and Growth Rates are Constant. If the growth velocity R is isotropic and constant, Eq. 21.5 can be integrated and the time cone is the set of points ?that obey 2 R2 (t - T)2 - I?- Jl 2 0 (21.8) The radius of the time cone, I?- Jl, is linear in time, and hence the time cone will be a right circular cone of height t - T. The volume of the time cone, which Cahn calls the nucleation volume V,, for transformation in a system of dimensionality d is given by B(t - r) d+1 v, = (21.9) where B is an appropriate measure of the base of the cone. Taking T = 0 as the earliest time that nucleation becomes possible, the following expressions are obtained for systems of dimensionalities 1, 2, and 3: B = 2Rt V, = Rt2 (d = 1) B = nR2t2 V, = 4jR2t3 (d = 2) (2 1.10) B= TR 47r '3 t 3 V - "k3t4 (d=3) c- 3 Equation 21.7 takes a particularly simple form when the nucleation rate is a constant: (N), is equal to the product of J and V, for the d = 3 case. This result 21.1: OVERALL RATE OF DISCONTINUOUS TRANSFORMATION 537 can be generalized to systems of arbitrary dimensionality, d, by use of Eq. 21.9, giving JBt (WC = d+l (21.11) Substituting the appropriate factors from Eq. 21.10 into Eqs. 21.11 and 21.4 gives expressions for the fraction transformed in one, two, and three dimensions for the case of constant nucleation and growth rates J and R. The resulting expressions for the untransformed volume are 1 - < = ,-JRt2 1 - < = e-(77/3)Jk3t4 (d = 1) (d = 3) 1 - < = e-(T/3)JA2t3 (d = 2) (21.12) The function <(t) in Eq. 21.12 has a characteristic sigmoidal shape with a maximum rate of transformation at intermediate times. Examples are shown in Fig. 21.2. The d = 3 form of Eq. 21.12 is commonly known as the Johnson-Mehl-Avrami equation. ;- 0.8 E 0.6 9 - 0.4 L C c 0 c 8 0.2 lt 0 I 0 0.5 1 1.5 2 2.5 3 Time, t Figure 21.2: Comparison of volume fraction transformed, 6, in the interior of a semi- infinite thin-film specimen and at the specimen edge, s = 0. Calculations for J = 1, R = 1. 21.1.2 Transformations near the Edge of a Thin Semi-Infinite Plate Consider a semi-infinite thin plate that is effectively two-dimensional, lying in the zy-plane, with a single edge along x = 0. It is assumed that there is no heteroge- neous line nucleation at the edge of the sheet. For constant-volume nucleation and growth rates and isotropic growth, points lying within the near-edge region x < Rt require special consideration. In the bulk of the plate away from this region, the time cone will be a right circular cone of height t, and the fraction transformed will be given by the d = 2 form of Eq. 21.12. Close to the edge, the time cone will be truncated by the plane x = 0, its volume will be less than in the bulk, and the number of nuclei (N)c contained in the truncated cone will decrease as x + 0.l The transformation rate in the near-edge region will thus be slower than in the 'Nuclei cannot form outside the sample and hence cannot influence the transformation anywhere inside it. [...]... Avrami Kinetics of phase change-I J Chem Phys., 7(12):110 3-1 112, 1939 4 M Avrami Kinetics of phase change-11 Transformation-time relations for random distribution of nuclei J Chem Phys., 8(2):21 2-2 24, 1940 5 M Avrami Kinetics of phase change-111 Granulation, phase change, and microstructure J Chem Phys., 9(2):17 7-1 84, 1941 6 J.L Jackson Dynamics of expanding inhibitory fields Science, 183(4123):44 6-4 47,... according to L(x) = l ( 0 ) - k‘x (22.15) Solution (a) If the zone moves by dz, the amount o f solute that is lost from the zone is d S ( z )= -C SL (z) dz = - k / c L (z) d x = -k - S(X)dz 1 (22.16) This relation may be integrated in the form (22.17) so that S ( X )= S(O)e-”l”/l (22.18) Also, IC’s(Z) P ( z )= k /c L(z) = - e -k -k‘x/l 1 1 - k p ( 0 ) e - - k ’ 4 1 = csL(0)e-k/“/l 551 EXERCISES Now... GTTm/A - 1 GpTm / A The nose of the C-curve in this case will therefore occur at T,,,, , lies between T and T,/2 (21.16) which typically Bibliography 1 A.N Kolmogorov Statistical theory of metal crystallization (in Russian) Izv Akad Nauk SSSR, 1:35 5-3 59, 1937 2 W.A Johnson and R.F Mehl Reaction kinetics in processes of nucleation and growth Trans AIME, 135 :41 6-4 42, 1939 See also discussion on pp 44 2-4 58... 2Rtx - x2 (21.19) 2 R 2R so that p ( n 2 1, A z ) = 1 - e J A 2 or , ~ -+ (21.20) Case 3: Very short wire or long times There is interference from both boundaries The condition for this case is L < Rt (or x < Rt and x < L - k t ) The area of the time cone, As, is A1 minus the area where X > L: A3 = A1 - Rt - (L- z) Rt - (L- x) - 2(LRt 2 R + Lz )- (L2 + 222) 2R (21.21) 542 CHAPTER 21: CONCURRENT NUCLEATION... transformed is independent of z when Rt < z < L/2: p ( n 21, A ~= 1 - eJRt2 ) (21.18) Case 2: Near the end of a finite wire There is interference only from the boundary a t x = 0 The condition for this case is L > Rt and z < Rt (or, in a slightly different form, x < Rt and z < L - k t ) The area of the time cone, A2, is A1 minus the area where z < 0: Rt - x A2 = A1 - Rt - x -= R2t2 2Rtx - x2 (21.19) 2 R 2R.. . transformations Metall Trans., 24A(2):241276, 1993 11 F.K LeGoues and H.I Aaronson Influence of crystallography upon critical nucleus shapes and kinetics of homogeneous f.c.c.-f.c.c nucleation-IV Comparisons between theory and experiment in Cu-Co alloys Acta Metall., 32(10):185 5-1 864, 1984 12 H.I Aaronson and F.K LeGoues An assessment of studies on homogeneous diffusional nucleation kinetics in binary metallic... binary alloy initially of uniform composition co is placed in a bar-shaped crucible of length L The bar is progressively cooled from one end, so it solidifies from one end to the other Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter Copyright @ 2005 John Wiley & Sons, Inc 543 544 CHAPTER 22: SOLIDIFICATION Figure 22.1: (a) Bar-shaped specimen after plane-front solidification... 1 - exp (-( N),) (21.25) CHAPTER 22 S0 L IDIFICAT I0 N The mechanisms of the motion of liquid/crystal interfaces during solidification were discussed in Section 12.3, and aspects of the heat-conduction-controlled motion of liquid/solid interfaces and their morphological stability under various solidification conditions were treated in Chapter 20 This sets the stage for considering the entire process of. .. equilibrium will prevail at the liquid/solid interface and growth will stop A calculation of this efFect in the solidification of a AI/Cu alloy is plotted in Fig 22.8 104 I I 102 1 0-2 1 n-4 I" 1 0-6 1 0-4 1 0-2 Tip radius R (mm) 1 Figure 22.8: Dendrite tip velocity vs tip radius for an Al/Cu alloy The diffusion-limit portion of the curve is unaffected by capillarity The capillarity limit indicates the point... a two-phase mixture of lower energy The process occurs by the nucleation and growth of particles (precipitates) of the new phase embedded in the original phase The form of the precipitation may vary widely depending upon factors such as the degree of coherency between the precipitates and matrix, the degree of supersaturation, and the availability of heterogeneous nucleation sites Basic aspects of nucleation . uniform at any instant. Examples include recrystallization, first- order order-disorder transformations, massive transformations, and crystallization during vapor-deposition processes. Time-Cone. previous chapters we have developed models for discontinuous transforma- tions that treat nucleation and growth processes independently. However, when Kinetics of Materials. By Robert W. . Society Symposia Proceedings, pages 42 5-4 38, Pittsburgh, PA, 1996. Materials Research Society. J .W. Christian. The Theory of Transformations zn Metals and Alloys. Pergamon Press, Oxford,