aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Diffusion-controlled phase growth on dislocations ∗ A. R. Massih
Quantum Technologies, Uppsala Science Park,SE-751 83 Uppsala, Sweden andMalm¨o University, SE-205 06 Malm¨o, Sweden (Dated: December 3, 2018)We treat the problem of diffusion of solute atoms around screw dislocations. In particular, weexpress and solve the diffusion equation, in radial symmetry, in an elastic field of a screw dislocationsubject to the flux conservation boundary condition at the interface of a new phase. We consider anincoherent second-phase precipitate growing under the action of the stress field of a screw dislocation.The second-phase growth rate as a function of the supersaturation and a strain energy parameteris evaluated in spatial dimensions d = 2 and d = 3. Our calculations show that an increase in theamplitude of dislocation force, e.g. the magnitude of the Burgers vector, enhances the second-phasegrowth in an alloy. Moreover, a relationship linking the supersaturation to the precipitate size inthe presence of the elastic field of dislocation is calculated. PACS numbers:
I. INTRODUCTION
Dislocations can alter different stages of the precipitation process in crystalline solids, which consists of nucleation,growth and coarsening [1, 2]. Distortion of the lattice in proximity of a dislocation can enhance nucleation in severalways [3, 4]. The main effect is the reduction in the volume strain energy associated with the phase transformation.Nucleation on dislocations can also be helped by solute segregation which raises the local concentration of the solutein the vicinity of a dislocation, caused by migration of solutes toward the dislocation, the Cottrell atmosphere effect.When the Cottrell atmosphere becomes supersaturated, nucleation of a new phase may occur followed by growthof nucleus. Moreover, dislocation can aid the growth of an embryo beyond its critical size by providing a diffusionpassage with a lower activation energy.Precipitation of second-phase along dislocation lines has been observed in a number of alloys [5, 6]. For example,in Al-Zn-Mg alloys, dislocations not only induce and enhance nucleation and growth of the coherent second-phaseMgZn precipitates, but also produce a spatial precipitate size gradient around them [7, 8, 9]. Cahn [10] provided thefirst quantitative model for nucleation of second-phase on dislocations in solids. In Cahn’s model, it is assumed thata cross-section of the nucleus is circular, which is strictly valid for a screw dislocation [1]. Also, it is posited that thenucleus is incoherent with the matrix so that a constant interfacial energy can be allotted to the boundary between thenew phase and the matrix. An incoherent particle interface with the matrix has a different atomic configuration thanthat of the phases. The matrix is an isotropic elastic material and the formation of the precipitate releases the elasticenergy initially stored in its volume. Moreover, the matrix energy is assumed to remain constant by precipitation. Inthis model, besides the usual volume and surface energy terms in the expression for the total free energy of formationof a nucleus of a given size, there is a term representing the strain energy of the dislocation in the region currentlyoccupied by the new phase. Cahn’s model predicts that both a larger Burgers vector and a more negative chemical freeenergy change between the precipitate and the matrix induce higher nucleation rates, in agreement with experiment[5, 6].Segregation phenomenon around dislocations, i.e. the Cottrell atmosphere effect, has been observed among othersin Fe-Al alloys doped with boron atoms [11] and in silicon containing arsenic impurities [12], in qualitative agreementwith Cottrell and Bilby’s predictions [13]. Cottrell and Bilby considered segregation of impurities to straight-edgedislocations with the Coulomb-like interaction potential of the form φ = A sin θ/r , where A contains the elasticityconstants and the Burgers vector, and ( r, θ ) are the polar coordinates. Cottrell and Bilby ignored the flow dueconcentration gradients and solved the simplified diffusion equation in the presence of the aforementioned potentialfield. The model predicts that the total number of impurity atoms removed from solution to the dislocation increaseswith time t according to N ( t ) ∼ t / , which is good agreement with the early stages of segregation of impuritiesto dislocations, e.g. in iron containing carbon and nitrogen [14]. A critical review of the Bilby-Cottrell model, itsshortcomings and its improvements are given in [15]. ∗ An extended and revised version of the paper presented in MS&T’08, October 5-9, 2008, Pittsburgh, Pennsylvania, USA.
The object of our present study is the diffusion-controlled growth of a new phase, i.e., a post nucleation process inthe presence of dislocation field rather than the segregation effect. As in Cahn’s nucleation model [10], we consideran incoherent second-phase precipitate growing under the action of a screw dislocation field. This entails that thestress field due to dislocation is pure shear. The equations used for diffusion-controlled growth are radially symmetric.These equations for second-phase in a solid or from a supercooled liquid have been, in the absence of an externalfield, solved by Frank [16] and discussed by Carslaw and Jaeger [17]. The exact analytical solutions of the equationsand their various approximations thereof have been systematized and evaluated by Aaron et al. [18], which includedthe relations for growth of planar precipitates. Applications of these solutions to materials can be found in manypublications, e.g. more recent papers on growth of quasi-crystalline phase in Zr-base metallic glasses [19] and growthof Laves phase in Zircaloy [20]. We should also mention another theoretical approach to the problem of nucleationand growth of an incoherent second-phase particle in the presence of dislocation field [21]. Sundar and Hoyt [21]introduced the dislocation field, as in Cahn [10], in the nucleation part of the model, while for the growth part thesteady-state solution of the concentration field (Laplace equation) for elliptical particles was utilized.The organization of this paper as follows. The formulation of the problem, the governing equations and the formalsolutions are given in section II. Solutions of specific cases are presented in section III, where the supersaturationas a function of the growth coefficient is evaluated as well as the spatial variation of the concentration field in thepresence of dislocation. In section IV, besides a brief discourse on the issue of interaction between point defects anddislocations, we calculate the size-dependence of the concentration at the curved precipitate/matrix for the problemunder consideration. We have carried out our calculations in space dimensions d = 2 and d = 3. Some mathematicalanalyses for d = 3 are relegated to appendix A. II. FORMULATION AND GENERAL SOLUTIONS
We consider the problem of growth of the new phase, with radial symmetry (radius r ), governed by the diffusionof a single entity, u ≡ u ( r, t ), which is a function of space and time ( r, t ). u can be either matter (solvent or solute)or heat (the latent heat of formation of new phase). The diffusion in the presence of an external field obeys theSmoluchowski equation [22] of the form ∂u∂t = ∇ · J , (1) J = D ( ∇ u − β F u ) , (2)where D is the diffusivity, β = 1 /k B T , k B the Boltzmann constant, T the temperature, and F is an external field offorce. The force can be local (e.g., stresses due to dislocation cores in crystalline solids) or caused externally by anapplied field (e.g., electric field acting on charged particles). If the acting force is conservative, it can be obtained froma potential φ through F = −∇ φ . The considered geometric condition applies to the case of second-phase particlesgrowing in a solid solution under phase transformation [20] or droplets growing either from vapour or from a secondliquid [16]. A steady state is reached when J = const . = 0, resulting in u = u exp( − βφ ).Here, we suppose that the diffusion field is along the core of dislocation line and that a cross-section of the precipitate(nucleus), perpendicular to the dislocation, is circular, i.e., the precipitate surrounds the dislocation. Furthermore, wetreat the matrix and solution as linear elastic isotropic media. The elastic potential energy of a stationary dislocationof length l is given by [23, 24] φ = A ln rr , for r ≥ r (3)where A = Gb l/ π for screw dislocation, G is the elastic shear modulus of the crystal, b the magnitude of the Burgersvector, ν Poisson’s ratio, and r is the usual effective core radius. Also, we assume that the dislocation’s elastic energyis relaxed within the volume occupied by the precipitate and that the precipitate is incoherent with the matrix. Hencethe interaction energy between the elastic field of the screw dislocation and the elastic field of the solute is zero. Inthe case of an edge dislocation and coherent precipitate/matrix interface, this interaction is non-negligible.We study the effect of the potential field (3) on diffusing atoms in solid solution using the Smoluchowski equation(1). The governing equation in spherical symmetry, in d spatial dimension, with B ≡ βA , is1 D ∂u∂t = ∂ u∂r + ( d − B ) 1 r ∂u∂r + ( d − B ur . (4)Making a usual change of variable to the dimensionless reduced radius s = r/ √ Dt , the partial differential equation(4) is reduced to an ordinary differential equation of the formd u d s + (cid:16) s d − Bs (cid:17) d u d s + ( d − B us = 0 , (5)with the boundary conditions, u ( ∞ ) = u m , and u (2 λ ) = u s , where u m is the mean (far-field) solute concentration in thematrix and u s is the concentration in the matrix at the new-phase/matrix interface determined from thermodynamicsof new phase, i.e., phase equilibrium and the capillary effect. Moreover, the conservation of flux at the interface radius R = 2 λ √ Dt gives K d R d − | J | r = R = q d V d d t , (6)where K d = 2 π d/ / Γ( d/ x ) the usual Γ-function, V d = 2 π d/ R d /d Γ( d/ q the amount of the diffusing entityejected at the boundary of the growing phase per unit volume of the latter (new phase) formed. In s -space, equation(6) is written as (cid:16) d u d s (cid:17) s =2 λ = − (cid:16) Bu s λ + qλ (cid:17) . (7)The boundary condition u (2 λ ) = u s and equation (7) will provide a relationship between u s and u m through λ .For d = 2, equation (5) is very much simplified, and we find u ( s ) = u m + ( Bu m + 2 qλ ) λ B e λ Γ( − B/ , s / − Bλ B e λ Γ( − B/ , λ ) , (8)where we utilized u ( ∞ ) = u m and equation (7). Here Γ( a, z ) is the incomplete gamma function defined by the integralΓ( a, z ) = R ∞ z t a − e − t dt [25]. The yet unknown parameter λ is found from relation (8) at u (2 λ ) = u s for a set of inputparameters u s , u m q , and B , through which the concentration field, equation (8), and the growth of second-phase( R = 2 λ √ Dt ) are determined.Let us consider the case of d = 3, that is assume that the potential in equation (3) is meaningful for a sphericallysymmetric system. In this case, for B = 0, the point z = 0 is a regular singularity of equation (5), while z = ∞ is anirregular singularity for this equation, see appendix A for further consideration. Nevertheless, for d = 3, the generalsolution of equation (5) is expressed in the form u ( s ) = 2 C F (cid:16) −
12 ; 1 + B − s (cid:17) s − + 2 B C F (cid:16) − B − B − s (cid:17) s − B , (9)where F ( a ; b ; z ) is Kummer’s confluent hypergeomtric function, sometimes denoted by M ( a, b, z ) [25]. The inte-gration constants C and C in equation (9) can be determined by invoking equation (7) and also the condition u ( ∞ ) = u m , cf. appendix A. III. COMPUTATIONS
To study the growth behavior of a second-phase in a solid solution under the action of screw dislocation field,we attempt to compute the growth rate constant as a function of the supersaturation parameter k , defined as k ≡ ( u s − u m ) /q u with q u = u p − u s , where u p is the composition of the nucleus [18]. For d = 2, i.e., a cylindricalsecond-phase platelet, equation (8) with u (2 λ ) = u s yields k = " λ + Bu m ( u p − u s ) − − Bλ B e λ Γ (cid:0) − B/ , λ (cid:1) λ B e λ Γ (cid:0) − B/ , λ (cid:1) . (10)For B = 0, the relations obtained by Frank [16] are recovered, namely u ( z ) = u m + q u λ e λ E ( z / , (11) k = λ e λ E ( λ ) , (12)where E ( x ) is the exponential integral of order one, related to the incomplete gamma function through the identity E n ( x ) = x n − Γ(1 − n, x ) [25].From equation (10), it is seen that a complete separation of the supersaturation parameter k ≡ ( u s − u m )( u p − u s ) − is not possible for B = 0. However, for u s << u p (a reasonable proviso) we write k = (cid:16) λ + B ǫ (cid:17) λ B e λ Γ (cid:0) − B/ , λ (cid:1) + O ( ǫ ) , (13)with ǫ ≡ u s /u p . For B = 1, equations (8) and (13) yield, respectively u ( z ) = u m + 2 λ e λ ( u m + 2 q u λ ) E / ( z / − e λ E / ( λ )] z , (14) k = (cid:16) λ + ǫ (cid:17) e λ E / ( λ ) + O ( ǫ ) . (15)Similarly for B = 2, we have u ( z ) = u m + 4 λ e λ ( u m + q u λ ) E ( z / − e λ E ( λ )] z , (16) k = ( λ + ǫ ) E ( λ ) + O ( ǫ ) . (17)We have plotted the growth coefficient λ = R/ √ Dt as a function of the supersaturation parameter k in figure 1and the spatial variation of the concentration field in figure 2 for d = 2 and several values of B . The computationsare performed to O ( ǫ ) with ǫ = 0 .
01. Figure 1 shows that λ is an increasing function of k ; and also, as B is raised λ is elevated. This means that an increase in the amplitude of dislocation force (e.g., the magnitude of the Burgersvector) enhances second-phase growth in an alloy.Figure 2 displays the reduced concentration versus the reduced radius z = r/ √ Dt for λ = 1. The reducedconcentration is calculated via equation (8). It is seen that for z . . B , whereas for z & .
6, it is vice versa. So, for λ = 1, the crossover z -value is z c ≈ .
6. Also, as λ is reduced, z c isdecreased. −1 k λ B = 0 B = 1 B = 2 B = 3 B = 4 FIG. 1: Growth coefficient λ as a function of supersaturation k at various levels of dislocation force amplitude B for a circularplate ( d = 2) and u s = 0 . u p . For d = 3, i.e., a spherical second-phase particle in the absence of dislocation field ( B = 0), we find u ( z ) = u m + 2 q u λ e λ h e − z / z − √ π erfc( z/ i , (18) k = 2 λ h − √ π λ e λ erfc( λ ) i . (19) −2 z = r /(Dt) [ u ( z ) − u m ] / [ u p − u s ] B = 0 B = 1 B = 2 B = 3 B = 4 FIG. 2: Reduced concentration field as a function of reduced distance from the surface of the circular plate ( d = 2) at variouslevels of dislocation force amplitude B and at λ = 1. This corresponds to the results obtained by Frank [16].For d = 3 and B = 2, equation (5) is simplified and an analytical solution can be found, resulting in u ( z ) = e z / ( z + 2) h √ πλe λ (cid:16) erf( z ) − erf( λ ) (cid:17) − i + 2 λe λ z √ πλe λ erfc( λ ) − ! e − z / z u m ++ λ e λ h z − √ πe z / ( z + 2)erfc( z ) i √ πλe λ erfc( λ ) − ! e − z / z q u . (20)Putting u (2 λ ) = u s , we obtain k = 1 + 2 λ (cid:16) − √ πλ e λ erfc( λ ) (cid:17) λ (cid:16) √ πλ e λ erfc( λ ) − (cid:17) u m q u + 2 λ − (1 + 2 λ ) √ πλ e λ erfc( λ )2 (cid:16) √ πλ e λ erfc( λ ) − (cid:17) . (21)For u s << u p , we write k = − λ + √ πλ (1 + 2 λ ) e λ erfc( λ ) + (cid:16) − λ + 2 √ πλ e λ erfc( λ ) (cid:17) ǫ + O ( ǫ ) . (22)General analytical expressions of u ( z ) and k , in terms of confluent hypergeometric functions, can also be foundfor even values of B as detailed in appendix A. Furthermore, asymptotic forms of u ( z ) for large and small z can becalculated, see appendix A for analysis of z >>
1. Figure 3 compares k versus λ for d = 2 and d = 3 in the absenceof dislocation field ( B = 0). IV. DISCUSSION
The potential energy in equation (3) describes the elastic energy of the dislocation relaxed within the volumeoccupied by the second-phase precipitate [10]. It was treated here as an external field affecting the diffusion-limited −1 k λ d = 2 d = 3 FIG. 3: Growth coefficient λ as a function of supersaturation parameter k at B = 0 for a circular plate ( d = 2) versus a sphere( d = 3). growth of second-phase precipitate. The interaction energy of impurities in a crystalline with dislocations dependson the specific model or configuration of a solute atom and a matrix which is used. Commonly, it is assumed thatthe solute acts as an elastic center of dilatation. It is a fictitious sphere of radius R ′ embedded concentrically in aspherical hole of radius R cut in the matrix. If the elastic constants of the solute and matrix are the same, the workdone in inserting the atom in the presence of dislocation is w = p ∆ v , where p is the hydrostatic pressure and ∆ v isthe difference between the volume of the hole in the matrix and the sphere of the fictitious impurity. For a screwdislocation p = 0, while near an edge dislocation p = (1+ ν ) bG sin θ π (1 − ν ) r for an impurity with polar coordinates ( r, θ ) withrespect to the dislocation 0 z , hence w ∝ ∆ v sin θ/r [13]. Using a nonlinear elastic theory [26], a screw dislocationmay also interact with the spherical impurity with the interaction energy w ∝ ∆ v/r . Moreover, accounting forthe differences in the elastic constants of a solute and a matrix, the solute will relieve shear strain energy as wellas dilatation energy, which will also interact with a screw dislocation with a potential w ∝ ∆ v/r [24]. Indeed,Friedel [24] has formulated that by introducing a dislocation into a solid solution of uniform concentration c , theinteraction energy between the dislocation and solute atoms can be written as w ⋍ w ( b/δ ) n f ( θ ), where δ is thedistance between the two defects, w the binding energy when δ = b , and f ( θ ) accounts for the angular dependenceof the interaction along the dislocation. Also, n = 1 for size effects and n = 2 for effects due to differences in elasticconstants. The discussed model for the interaction energy between solute atoms and dislocations has been used tostudy the precipitation process on dislocations by number of workers in the past [27, 28] and thoroughly reviewed in[15]. These studies concern primarily the overall phase transformation (precipitation of a new phase) rather than thegrowth of a new phase considered in our note. That is, they used different boundary conditions as compared to theones used here.Let us now link the supersaturation parameter k to an experimental situation. For this purpose, the values of u s ,i.e. the concentration at the interface between the second-phase and matrix should be known. The capillary effectleads to a relationship between u s and the equilibrium composition u eq (solubility line in a phase diagram). To obtainthis relationship, we consider an incoherent nucleation of second-phase on a dislocation `a la Cahn [10]. A Burgers looparound the dislocation in the matrix material around the incoherent second-phase (circular plate) will have a closuremismatch equal to b . Following Cahn, on forming the incoherent plate of radius R , the total free energy change perunit length is G = − πR ∆ g v + 2 πγR − A ′ ln( R/r ) , (23)where ∆ g v is the volume free energy of formation, γ the interfacial energy and the last term is the dislocation energy, A ′ = Gb / π for screw dislocations, cf. equation (3). Setting d G / d R = 0, yields R = γ g v (cid:16) ± √ − α (cid:17) , (24)where α = 2 A ′ ∆ g v /πγ . So, if α >
1, the nucleation is barrierless, i.e., the phase transition kinetics is only governedby growth kinetics, which is the subject of our investigation here. If, however, α <
1, there is an energy barrier andthe local minimum of G at R = R , which corresponds to the negative sign in equation (24), ensued by a maximum at R = R ∗ corresponding to the positive sign in this equation. The local minimum corresponds to a subcritcal metastableparticle of the second-phase surrounding the dislocation line, and it is similar to the Cottrell atmosphere of soluteatoms in a segregation problem. When α = 0, corresponding to B = 0, the two phases are in equilibrium and themaximum in G is infinite, as for homogeneous nucleation.For a dilute regular solution, ∆ g v = ( k B T /V p ) ln( u s /u eq ), where V p is the atomic volume of the precipitate com-pound, u s is the concentration of the matrix at a curved particle/matrix interface and u eq that of a flat interface,which is in equilibrium with the solute concentration in the matrix. Equation (24) gives ∆ g v = γ/R − A ′ / πR .Hence, for a dilute regular solution, we write u s = u eq exp h ζR (cid:16) − ηR (cid:17)i , (25)where ζ = βV p γ , β = 1 /k B T and η = A ′ / πγ . Subsequently, the supersaturation parameter is expressed by k = u eq exp[ ζR (1 − ηR )] − u m u p − u eq exp[ ζR (1 − ηR )] . (26)Taking the following typical values: γ = 0 . − , G = 40 GPa, and b = 0 .
25 nm, then A ′ ≈ . × − N and η = 0 .
16 nm. Figure 4 depicts u s /u eq , from equation (25), as a function of scaled radius R/ζ for V p = 1 . × − m , η = 0 and η = 0 .
16 nm at T = 600 K. Equation (25) is analogous to the Gibbs-Thomson-Freundlich relationship[4] comprising a dislocation defect.Recalling now the values used for the interaction parameter B in the computations presented in the foregoingsection, we note that for B = 2 and the above numerical values for G and b at T = 1000 K, we find l ≈ .
14 nm,which is close to the calculated value of η .In Cahn’s model, the assumption that all the strain energy of the dislocation within the volume occupied by thenucleus can be relaxed to zero demands that the nucleus is incoherent. For a coherent nucleus forming on or inproximity of dislocations, this supposition is not true. Instead, it is necessary to calculate the elastic interactionenergy between the nucleus and the matrix, which for an edge dislocation is in the form Gb / [4 π (1 − ν ) r ] for theenergy density per unit length [29]. In the same manner, to extend our calculations for growth of coherent precipitate,we must employ this kind of potential energy, i.e. the potential energy of the form φ ( r ) = − A ln( r/r ) + C sin θ/r , inthe governing kinetic equation rather than relation (3). APPENDIX A: EVALUATION OF SOLUTIONS OF EQUATION (5) FOR d = 3 For an ordinary second-order differential equation with a regular singularity, the Frobenius method can be used toobtain power series solution. On the other hand, when singularity is irregular, no convergent solution may be found;nevertheless, albeit divergent, the solution can be asymptotic. Let us write equation (5) for d = 3 in a generic form u ′′ + p ( z ) u ′ + q ( z ) u = 0 , (A1)where primes denote differentiation with respect to z , p ( z ) = z/ B ) /z and q ( z ) = B/z . Since we haveimposed the boundary condition u ( ∞ ) = u m , it is worthwhile to explore the behavior of the solution as z → ∞ .But, first let us put equation (A1) in a more convenient form by setting u ( z ) = e u ( z ) exp[ − R p ( z ) dz ], which gives e u ′′ + ( q − p ′ / p ) e u = 0. Here, without loss of generality, we consider u ′′ + f ( z ) u = 0 , (A2)where f ( z ) = − z − B + 34 − B ( B − z . (A3)Since f ( z ) is not O ( z − ) as z → ∞ , then the point at infinity is an irregular singularity for u ( z ). We now look forsolutions of (A2) by considering u ( z ) ∼ exp (cid:16) ∞ X n =0 ψ n ( z ) (cid:17) , (A4) R/ ζ u s / u eq η = 0 η = 0.16 nm FIG. 4: The size dependence of the concentration at the curved precipitate/matrix interface u s relative to that of the flatinterface u eq for a set of parameter values given in the text, cf. eq. (25).TABLE I: Solutions to equation (A5).Sequence Solution 1 Solution 2 ψ z / − z / ψ ψ ( B +22 ) ln z − ( B +42 ) ln zψ ψ Bz − − B + 2) z − where { ψ n ( z ) } , n = 0 , , . . . , is an asymptotic sequence as z → ∞ . Substituting (A4) into equation (A2) ψ ′′ + ψ ′′ + · · · + ( ψ ′ + ψ ′ + . . . ) + f ( z ) ∼ , (A5)where we have tacitly assumed that ψ n ( z ) is (twice) differentiable and the resulting series are still asymptotic.Equation (A5) is used to determine the { ψ n ( z ) } , n = 0 , , . . . by successively applying the asymptotic limit z → ∞ .The results for the first few terms are shown in table I. Hence, we write for z → ∞ : u + ( z ) ∼ A e z / z (1+ B/ h Bz + O ( z − ) i , (A6) u − ( z ) ∼ A e − z / z − ( B/ h − B + 2) z + O ( z − ) i , (A7)where A and A are arbitrary constants. Note that the solution (A6) is divergent for large z , whereas (A7) isconvergent and thus is physically admissible. Considering u ( ∞ ) = u m , we write u − ( z ) ∼ u m + A e − z / z − B/ − h − B + 2) z + O ( z − ) i , as z → ∞ . (A8)Let us now evaluate the general solution to equation (5) for d = 3 as expressed by equation (9). We apply the fluxconservation relation (6) to obtain C , and then substitute C in equation (9) to write u ( z ) = K ( z, B ) C + K ( z, B ) q, (A9) TABLE II: Special cases of F ( a ; b ; z ). F ( − ; − z ) = 1 + 2 z F ( − ; ; − z ) = e − z + √ πz erf( z ) F ( − ; ; − z ) = e − z + √ π (1 + 2 z )erf( z ) z − F ( ; ; − z ) = √ π erf( z ) z − F ( ; ; − z ) = “ ze − z + √ π (2 z − z ) ” z − where K ( z, B ) = (cid:16) z (cid:17) B F (cid:16) − B − B − z (cid:17) ++ Bλ − B F (cid:16) − B ; − B ; − λ (cid:17) B − K ( z, B ) , (A10) K ( z, B ) = 4( B + 1) λ F ( − ; B ; − z ) /z ( B − F ( − ; B ; − λ ) + 2 λ F ( ; B ; − λ ) , (A11) K ( z,
0) = 2 λ e λ h e − z / z + √ π erf( z/ i . (A12)Here, F ( a ; b ; z ) is the confluent hypergeometric function. If a <
0, and either b > b < a , this function canbe expressed as a polynomial with finite number of terms. If, however, b = 0 or a negative integer, then F ( a ; b ; z )itself is infinite. Thus, relations (A11)-(A12) become singular for B = 1 , , , , . . . , making the solutions meaningless.Some useful relations for computations are listed in table II. Additional relations and properties for F ( a ; b ; z ) canbe found in [25].Next, we utilize the remote boundary condition u ( ∞ ) = u m to determine C , then we formally write u ( z ) = K ( z, B ) K ( ∞ , B ) u m + (cid:16) K ( z, B ) − K ( z, B ) K ( ∞ , B ) K ( ∞ , B ) (cid:17) q. (A13)In computations of K ( ∞ , B ) prudence must be exercised, i.e., first evaluate this quantity for a given value of B , thentake the limit z → ∞ . Note also that K ( z,
0) = 1 ∀ z and K ( ∞ ,
0) = 2 √ πλ e λ .Furthermore, we may calculate a relation for the supersaturation parameter, k = ( u s − u m ) / ( u p − u s ), defined inthe main text by using the condition u (2 λ ) = u s on equation (A9), which gives k = K (2 λ, B ) K ( ∞ , B ) − ! u m q + K (2 λ, B ) − K (2 λ, B ) K ( ∞ , B ) K ( ∞ , B ) . (A14)For dilute alloys, u s << u p ; so with ǫ ≡ u s /u p , we write k = − K ( ∞ , B ) K (2 λ, B ) ! ǫ + K (2 λ, B ) K (2 λ, B ) K ( ∞ , B ) − K ( ∞ , B ) + O ( ǫ ) . (A15) [1] F. C. Larch´e, in Dislocations in Solids , edited by F. R. N. Nabarro (North-Holland Publishing Company, Amsterdam,Holland, 1979), vol. 4.[2] R. Wagner and R. Kampmann, in
Phase Transformation in Materials , edited by E. K. R.W. Cahn, P. Haasen (VCH,Weinheim, Germany, 1991), vol. 5 of
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