COMPUTATIONAL MATERIALS ENGINEERING Episode 8 doc

z y x J x out ∆y ∆z ∆x P J in x FIGURE 5-6 Mass conser vation in diffusion. ∆moles = (J in x − J out x )∆y∆z · ∆t (5.18) On the other hand, the concentration c of atoms in the control volume is c = #moles ∆x∆y∆z (5.19) and the change of concentration is ∆c = ∆#moles ∆x∆y∆z (5.20) After equating (5.18) and (5.20) and using ∆J = J out x − J in x , we arrive at ∆c ∆t = − ∆J ∆x (5.21) Generalizing to three dimensions and using differentials instead of differences gives ∂c ∂t = −∇J (5.22) and inserting equation (5.17) leads to ∂c ∂t = −∇(−D∇c) (5.23) With the assumption that the diffusion coefficient D is independent of composition, we finally have ∂c ∂t = D∇ 2 c (5.24) Modeling Solid-State Diffusion 161 Equation (5.24) is known as Fick’s second law. It is a second-order linear partial differential equation describing transient diffusion processes, and it is interesting that Fick’s second law follows directly from Fick’s first law and mass conservation considerations. From this general form of the diffusion equation, Fick’s second law can also be written in planar, cylindrical, and spherical coordinates by substituting for the Laplace operator ∇ 2 . Using the symbols x, y, and z for cartesian coordinates, r, θ, and z for cylindrical coordinates, and r, θ, and φ for spherical coordinates, the diffusion equation reads ∂c ∂t = D  ∂ 2 c ∂x 2 + ∂ 2 c ∂y 2 + ∂ 2 c ∂z 2  (5.25) ∂c ∂t = D r  ∂ ∂r  r ∂c ∂r  + 1 r ∂ 2 c ∂θ 2 + r ∂ 2 c ∂z 2  (5.26) ∂c ∂t = D r 2  ∂ ∂r  r 2 ∂c ∂r  + 1 sin θ ∂ ∂θ  sin θ ∂c ∂θ  + 1 sin 2 θ ∂ 2 c ∂φ 2  (5.27) When considering diffusion in only one dimension, that is, linear diffusion with rotational sym- metry, and the spatial variable r, equations (5.25)–(5.27) reduce to ∂c ∂t = D  ∂ 2 c ∂r 2  (5.28) ∂c ∂t = D r  ∂ ∂r  r ∂c ∂r  (5.29) ∂c ∂t = D r 2  ∂ ∂r  r 2 ∂c ∂r  (5.30) Equations (5.28)– (5.30) can be further reduced when substituting λ = r/ √ t. The general form of the linear diffusion equations then becomes − λ 2 dc dλ = D λ p−1 d dλ  λ p−1 dc dλ  (5.31) which is an ordinary parabolic differential equation in the independent variable λ. The index p has the values 1, 2, and 3 for planar, cylindrical, and spherical geometry. For the case of p =1, this equation is also known as the Boltzmann diffusion equation. 5.3.2 Solutions to Fick’s Second Law In this section, we will briefly review some important solutions to Fick’s second law [equation (5.24)]. Spreading of a Diffusant from a Point Source Consider one-dimensional diffusion of a mass M along the x-direction in an infinite sample under the assumption that the entire mass is initially concentrated in a single plane. According to the law of mass conservation, we have  +∞ −∞ c(x, t)dx =1 (5.32) 162 COMPUTATIONAL MATERIALS ENGINEERING The solution to equation (5.24) with the constraint (5.32) is closely related to the problem of random walk of atoms and it is elaborated in many textbooks (e.g., ref. [Gli00]). Accordingly, for unit amount of mass, we have c(x, t)= M 2 √ πDt exp  −x 2 4Dt  (5.33) Equation (5.33) shows that, if atoms spread according to Fick’s second law, they will form a Gaussian distribution (see Figure 5-7). The general solution to a unit mass swarm of atoms spreading into an infinite sample of dimension d is c(r, t)= 1 (4πDt) d/2 exp  −r 2 4Dt  (5.34) where r is the distance from the point source. Equation (5.34) represents the probability distribution foradiffusant spreading into an infinite sample of dimension d from a general point source. This equation has also been used in the derivation of Einstein’s equation in Section 5.2.3. Diffusion into a Semi-infinite Sample Consider one-dimensional diffusion of a diffusant into a semi-infinite sample. This is a different situation now compared to the spreading diffusant from before, since we assume a continuous supply of atoms entering the sample from the outside boundary. This type of boundary condition is typical for surface treatments, such as carburization. Again, we start from Fick’s second law. The present boundary conditions are now a constant concentration of atoms at the semi-infinite boundary, that is, c(0,t)=c 0 (5.35) x (Length) 0 0.5 1.0 1.5 2.0 -3 -2 -1 01 3 2 0.025 0.05 0.1 0.25 1 c(x)/M (Length −1 ) Dt = 0 FIGURE 5-7 Solutions to Fick’s second law for a point source of diffusant in an inifinite sample. Modeling Solid-State Diffusion 163 The solution for this diffusion problem is c(x, t)=c 0 · erfc  x √ 4Dt  (5.36) with the error function, erf(z) and its complement, erfc(z), defined as erfc(z)=1− erf(z)=1− 2 √ π  z 0 e −η 2 dη (5.37) Figure 5-8 displays the evolution of the concentration profile during diffusion of solute into the semi-infinite sample. Note that all these curves degrade into one self-similar curve if the variable x/ √ 4Dt is plotted on the x-axis [compare equation (5.36)]. 5.3.3 Diffusion Forces and Atomic Mobility If a force F acts on a body, according to Newtonian mechanics, this force accelerates the body along the direction of the force. Typical examples of such forces are, for instance, gravitation acting on a mass or an electric field acting on a charged particle. Generally, a force can be written as the gradient of a potential Φ and we have F ∝∇Φ (5.38) In diffusion, the force F acting on one mole of atoms is identified as the gradient of the chemical potential µ and we have F = −∇µ (5.39) On an atomistic level, the force F accelerates the atoms along the direction of the force. However, the majority of atoms will resist this force because, according to the findings of Section 5.1, they can only migrate in the presence of a vacancy. If we interpret the resistance x (Length) c(x)/c 0 0 246810 sqrt(4 Dt) = 100 2 1 0.5 0 0.2 0.4 0.6 1.0 0.8 5 20 FIGURE 5-8 Solutions to Fick’s second law for semi-inifinite sample. 164 COMPUTATIONAL MATERIALS ENGINEERING against movement in terms of friction with the coefficient of friction g f , we find the effective drift velocity u of a large number of atoms under the force F from the force balance and we can write F = g f u (5.40) With the mobility B, which is generally defined as the inverse of the friction coefficient B =1/g f ,wehave u = BF = −B∇µ (5.41) The mobility B is an intrinsic property of the atomic species in a material and it defines the response of the species to a force F , that is, the gradient in chemical potential ∇µ. The flux J of atoms passing through unit area in unit time is then J = cu = −cB∇µ = −V m XB∇µ (5.42) The unit of J is moles/m 2 s and c is the concentration in moles/m 3 . V m is the molar volume and X is the mole fraction of the atomic species. In general solutions, the chemical potential is defined in terms of the activity a [see equation (2.66)] with µ = 0 µ + RT ln a (5.43) and with the definition of the activity coefficient a = fX [compare equation (2.67)] we obtain ∇µ = RT (∇ln f + ∇ln X) (5.44) Equation (5.44) can be rearranged as ∇µ = RT X  X∇ln f ∇X +1  ∇X (5.45) or ∇µ = RT X  ∇ln f ∇ln X +1  ∇X (5.46) When substituting equation (5.46) into equation (5.42) and replacing the nabla operator by partial differentials, we obtain J = −V m XB∇µ = −BRT  1+ ∂ ln f ∂ ln X  ∇c (5.47) When comparing with the definition of Fick’s first law [equation (5.17)], we identify the relation between the diffusion coefficient D and the mobility B as D = BRT  1+ ∂ ln f ∂ ln X  (5.48) In diffusion systems without chemical interaction between the atoms, for example, in ideal solutions or in systems with radioactive isotopes, the activity coefficient f =1is constant and the term in brackets is unity. In this case, equation (5.48) reduces to D = RT B (5.49) Modeling Solid-State Diffusion 165 Equation (5.48) shows that the macroscopic diffusion coefficient D, from its purely phenomenological definition of Fick’s first law, is directly related to the microscopic mobil- ity B, which has been defined based on atomistic considerations, by a thermal factor RT and a chemical contribution, which describes the deviation from solution ideality in form of the logarithmic derivative of the activity coefficient f. An interesting consequence of the preceding considerations is that, even in the absence of chemical interactions between the species, atoms experience a force which causes a net flux of atoms. Since this force cannot be measured directly, it is often considered as a generalized force. In an ideal solution, the generalized force F is directly proportional to the concentration gradient. In nonideal solutions, the chemical contribution can be included in the thermodynamic factor φ, which is φ =1+ ∂ ln f ∂ ln X (5.50) and the relation between D and B in general solutions can be written as D = RT Bφ (5.51) Although equation (5.51) is of general validity, we have to be aware of the fact that the derivative in the expression for the thermodynamic factor [equation (5.50)] is taken with respect to the mole fraction variable X. We have recognized already in Section 2.1.3, that, in this case, the constraint  X i =1[equation (2.22)] must be taken into account and the variation ∂X i of one element can only be performed against the equivalent variation −∂X ref of some reference element. Consider a binary system A–B. From equation (5.42), the flux of atoms A is J A = −V m X A B∇µ A (5.52) Using X A + X B =1and ∇X A = −∇X B , the gradient in chemical potential ∇µ A is ∇µ A = ∂µ A ∂X A ∇X A + ∂µ A ∂X B ∇X B =  ∂µ A ∂X A − ∂µ A ∂X B  ∇X A (5.53) After multiplication with RT and rearranging, the flux of atoms A in the binary system is obtained with J A = − X A RT  ∂µ A ∂X A − ∂µ A ∂X B  RT B∇c A (5.54) Comparison of equations (5.51) and (5.54) shows that the thermodynamic factor φ in a binary system and in terms of mole fraction variables X i is φ A = X A RT  ∂µ A ∂X A − ∂µ A ∂X B  (5.55) or in a general multicomponent solution with the reference component indicated by the sub- script “ref” φ i = X i RT  ∂µ i ∂X i − ∂µ i ∂X ref  (5.56) 166 COMPUTATIONAL MATERIALS ENGINEERING The choice of reference component in equation (5.56) is somewhat arbitrary and needs further attention. If we consider the chemical contribution to the diffusion coefficient in a dilute alloy, it is reasonable to use a single reference component since the majority of all exchange processes will include this one single component. In concentrated alloys, however, the flux of component A in the laboratory frame of reference (see Section 5.3.4) is compensated by a flux of atoms of different kinds depending on the alloy composition. In this case, a weighted mean value instead of a single reference component is more appropriate, and the thermodynamic fac- tor φ can then be expressed as φ i = X i RT   ∂µ i ∂X i − 1 1 −X i  j=i X j ∂µ i ∂X j   (5.57) An interesting question that has not yet been addressed is: How can we measure the dif- fusion coefficient D in a single-component system? Analysis of concentration profiles and net diffusional fluxes is apparently impossible, if we are unable to distinguish between the individ- ual atoms and if we are unable to trace their paths through the material. A common technique for “marking” individual atoms of the same species is to use radioactive isotopes. The advan- tage of this technique is that isotopes are chemically near-identical and thus do not introduce additional chemical interactions. The concentration profiles in corresponding experiments are evaluated as a function of time and penetration depth using mechanical sectioning and radioac- tivity measurement. The diffusion coefficient is obtained from straightforward comparison with appropriate solutions to Fick’s second law [e.g., equation (5.36)]. The diffusion coefficient in a single-component system is denoted as the self-diffusion coefficient D ∗ . The self-diffusion coefficient gives a measure for the effective displacement of atoms caused by random vacancy movement. The radioactive isotope technique can likewise be used to measure “self-diffusion” coeffi- cients in solutions of multiple components. Accordingly, the specimen is prepared as a chem- ically homogeneous solution of given composition and some of the atoms are replaced by radioactive isotopes. The diffusion coefficient measured by this method is commonly denoted as the tracer or impurity diffusion coefficient. The same symbol D ∗ is commonly used for this quantity and, due to the absence of chemical interactions, the general relation between atomic mobility B and D ∗ holds 6 D ∗ = RT B (5.58) When finally looking at the thermodynamic factor in the asymptotic limit of dilute solution, we have to recall the analysis of solution thermodynamics in the dilute solution limit (Section 2.2.5). With the help of the regular solution model, we have found that the activity coefficient f approaches a constant value if the solute content goes to zero (Henry’s law). Consequently, the logarithmic derivative of the activity coefficient in the definition of the thermodynamic factor φ in equation (5.50) becomes zero and the thermodynamic factor thus approaches unity. The importance of the thermodynamic factor comes into play only in concentrated alloys, where φ accounts for the influence of solution nonideality. 6 When measuring the diffusion coefficient based on tracer elements, this value is always smaller than the true self-diffusion coefficient, which is defined on basis of random vacancy-atom exchanges. This effect is known as the correlation effect. The factor relating correlated and uncorrelated jumps is a constant for each type of crystal lattice and it is always less than unity (e.g., f bcc =0.727 and f fcc =0.781). In this book, we will not further distinguish between the two and assume that the correlation effect is implicitly taken into account. Modeling Solid-State Diffusion 167 5.3.4 Interdiffusion and the Kirkendall Effect In Section 5.3.3, the net motion of atoms that is caused by a generalized diffusion force has been discussed. The chemical potential has been identified as the source of this driving force, and the thermodynamic factor has been introduced to account for nonideal thermodynamic behavior. In this section diffusion will be analyzed in situations where diffusion of multiple atomic species occurs simultaneously. In the 1940s, a severe controversy about the mechanism of diffusion in crystalline solids was going on. One group of scientists promoted the traditional view of diffusion, which assumed that the diffusional transport of atoms occurs on basis of an atom by atom exchange mechanism. Thereby, one atom exchanges place with another atom based on direct exchange (see Figure 5-1) or the ring-exchange mechanism, which involves four atoms that rotate simul- taneously and thus change place. The second group of scientists believed that diffusion occurs by the vacancy-exchange mechanism, that is, atoms switch place only with a neighboring empty lattice site (vacancy), and the transport of atoms occurs as a consequence of the random walk of these vacancies. In fact, the type of diffusion mechanism has substantial influence on the rate of diffusion of individual atomic species. Consequently, by careful analysis of appropriate diffusion experiments, it should be possible to identify the predominating diffusion mechanism in solid-state matter. These experiments and the corresponding theoretical analysis will now be discussed. Consider a binary diffusion couple with the pure substances A and B. After bringing A and B into contact, the A atoms will spread into the B-rich side and vice versa. If diffusion occurs by direct atomic exchange, the macroscopically observed diffusivities of the two atomic species must be identical, because one single exchange process moves the same amount of A and B atoms and diffusion of A and B occurs at the same rate. In contrast, if diffusion is carried by atom/vacancy exchange, the A and B atoms can move independently and the diffusivity of the two species can be different. Moreover, if one species diffuses from the left to the right side of the diffusion couple by vacancy/atom exchange, the flux of atoms must be accompanied by a counterflux of vacancies. If the diffusivities of A and B are identical, the two vacancy fluxes balance and annihilate. In the case where the diffusivities differ, a net flux of vacancies must occur, which “blows” through the sample. The net flux of vacancies is commonly known as vacancy wind. As a result of the vacancy wind, the lattice of the sample moves in the parallel direction of the vacancy flux. If, by some experimental technique, individual lattice planes, for example, the initial contact area of the diffusion couple, are marked, movement of the lattice planes can be recorded as a function of the difference of the diffusive fluxes of the A and B atoms. These experiments have been carried out by Ernest Kirkendall (1914–2005), an American metallurgist, between 1939 and 1947. In a series of three papers [KTU39, Kir42, SK47], Kirkendall and co-workers investigated the diffusion of copper and zinc in brass. In the experiments of the third paper, the initial contact plane of a diffusion couple between pure copper and brass with 70wt% copper/30wt% zinc has been marked with thin molybdenum wires, such that the movement of the inert markers can be observed at different stages of the experiment. Kirkendall showed that the markers moved relative to the laboratory frame of reference, which is a reference frame that is fixed to the sample surrounding. An experimentalist looking at a diffusion couple will observe the diffusion process in the laboratory frame of reference. From the results of the experiments, Kirkendall drew the following two conclusions: 1. the diffusion of zinc is much faster than the diffusion of copper, and 2. the movement of the markers is related to the difference in the diffusion coefficients. 168 COMPUTATIONAL MATERIALS ENGINEERING The results of his experiments have been published against the strong resistance of individual researchers, in particular Robert Franklin Mehl, an American metallurgist (1898–1976). Nowa- days, the observed Kirkendall drift of the marker plane is considered to be the first striking proof of the predominance of the vacancy exchange mechanism over direct atomic exchange in diffusion. A short time after publication of Kirkendall’s papers, L. Darken [Dar48] published the first quantitative analysis of Kirkendall’s experiments, which will briefly be outlined later. Consider the two intrinsic fluxes J A and J B (i.e., the fluxes that are observed when looking at diffu- sion from a frame of reference that is fixed to an individual lattice plane, the lattice frame of reference) according to Figure 5-9. In the steady state case, using Fick’s first law, we have J A = −D A ∂c A ∂r J B = −D B ∂c B ∂r (5.59) The net flux of atoms across this lattice plane J net is given as the sum of the intrinsic fluxes of the components and we obtain J net = J A + J B = −D A ∂c A ∂r − D B ∂c B ∂r = −J Va,net (5.60) ABv J net J Va,net J Va(A) J Va(B) J A J B FIGURE 5-9 Schematic of the Kirkendall effect: Atoms A (gray circles) diffuse into the B-rich side at a higher rate than the B atoms (white circles) into the A-rich side. The net flux of atoms J net causes a shift of the position of the initial contact plane with a velocity v relative to the laboratory frame of reference, that is, the fixed corners of the specimen. The movement of the marker plane is parallel to the net flux of vacancies J Va,net . Modeling Solid-State Diffusion 169 Equation (5.60) is known as Darken’s first equation. From mass conservation, it is apparent that a net flux J net of atoms causes accumulation of matter on one side of the marker plane. If mass is conserved, this accumulation must be compensated by a shift of the marker plane into the opposite direction. With the mole fraction X i = c i V m and with the assumption that the partial molar volumes V i = V m of each element are identical, the velocity of the marker plane v is obtained with v = J Va,net V m =  D A ∂X A ∂r + D B ∂X B ∂r  (5.61) Since we have the mole fraction constraint X A + X B =1, the relation holds dX A = −dX B . Consequently, equation (5.61) can also be written in equivalent form as v =(D A − D B ) ∂X A ∂r v =(D B − D A ) ∂X B ∂r (5.62) Equations (5.59) are defined in a coordinate system that is fixed to an individual lattice plane. This frame of reference can be transformed into the laboratory frame of reference with addition of a convective term or drift term with the convective flux J conv = vc. This type of transformation is known as Galilean transformation. In the laboratory frame of reference, the flux for species A can be written as J lab A = −D A ∂c A ∂r + c A v = −D A ∂c A ∂r + c A (D A − D B ) ∂X A ∂r = −D A ∂c A ∂r + X A D A ∂c A ∂r − X A D B ∂c A ∂r = −(1 − X A )D A ∂c A ∂r − X A D B ∂c A ∂r = −[X B D A + X A D B ] ∂c A ∂r (5.63) On comparison with the flux equation (5.59) in the lattice frame of reference, the interdiffu- sion coefficient ˜ D is introduced with ˜ D = X B D A + X A D B (5.64) Equation (5.64) is known as Darken’s second equation. It is interesting to note that through an analysis of the concentration profiles in a diffusion couple experiment, only the interdiffusion coefficient ˜ D can be observed. To determine the individual intrinsic diffusion coefficients D A and D B , additional information is necessary, which is exactly the velocity or displacement of the marker plane. The interdiffusion coefficient ˜ D describes the diffusive fluxes in the laboratory frame of refer- ence, a convenient frame of reference for human experimentalists. The intrinsic diffusion coef- ficients D A and D B describe the fluxes in the lattice frame of reference, a convenient measure when operating on the atomic scale. 170 COMPUTATIONAL MATERIALS ENGINEERING [...]... AIME, 133: 186 –203, 1939 J S Kirkaldy and D J Young Diffusion in the Condensed State The Institute of Metals, London, 1 987 J Philibert Diffusion et Transport de Mati´ re dans les Solides Editions de Physique, 1 986 e P G Shewmon Diffusion in Solids The Minerals, Metals & Materials Society, Warrendale, PA, 1 986 A D Smigelskas and E O Kirkendall Zinc diffusion in alpha brass Trans AIME, 171:130–142, 1947 8 However,... 2∆x (5 .88 ) The difference expression (5 .88 ) thus represents the linearized concentration gradient at point xi determined from the concentrations in the neighboring points xi−1 and xi+1 7 The distances are assumed to be equidistant for simplicity here The finite difference methodology can be applied to variable discretization intervals as well; however, the expressions become more involved then 174 COMPUTATIONAL. .. which calculates multicomponent thermodynamic potentials based on the CALPHAD methodology (see also Section 2.2 .8) Other software that can be used is the package MatCalc (http://matcalc.tugraz.at) Bibliography [AHJA90] [Cra75] [Dar 48] [Ein05] [Fic55] [Gli00] [Kir42] [KTU39] [KY87] [Phi86] [She86] [SK47] ¨ J O Andersson, L Hoglund, B J¨ nsson, and J Agren Computer Simulation of Multicomponent Diffuo sional... − Ji (5.77) j=1 and v = JVa Vm 172 COMPUTATIONAL MATERIALS ENGINEERING (5. 78) ˜ The flux Ji in the laboratory frame of reference can then be expressed by the sum of the intrinsic intr conv flux Ji and the convective flux Ji with intr conv ˜ J i = Ji + J i (5.79) With the intrinsic flux given by equation (5.69) and the convective flux given as n conv Ji = ci v = −Xi (5 .80 ) Ji j=1 the flux of component i reads...                =    t+∆t c0,t c1,t c2,t c3,t         (5. 98) t Note that the first row in the coefficient matrix differs from the other rows The factor of 2 in the second coefficient of this row is due to the assumption that symmetric boundary conditions 176 COMPUTATIONAL MATERIALS ENGINEERING hold at the left boundary of the sample with the index 0 This kind of boundary... element n and insertion of equation (5.73) into (5 .81 ) leads to ˜ Ji = − n−1 ∂cj + Xi ∂r (Dij − Din ) j=1 n−1 n−1 (Dkj − Dkn ) k=1 j=1 ∂cj ∂r (5 .84 ) After some algebra, the independent fluxes in the laboratory frame of reference can be written as ˜ Ji = − n−1 j=1 ˜ ∂cj Dij ∂r (5 .85 ) with the diffusion coefficient matrix ˜ Dij = n−1 n−1 (δjs − Xj )(δmi − Xi )Dms (5 .86 ) m=1 s=1 Finally, the relations between the... + Xi ∂r Dij j=1 n n Dkj k=1 j=1 ∂cj ∂r (5 .81 ) After expansion of the first term, introduction of the Kronecker delta δij , with the usual meaning of δij = 1 if i = j and δij = 0 if i = j , and rearrangement of terms, equation (5 .81 ) becomes ˜ Ji = − n n δik Dkj k=1 j=1 ∂cj + Xi ∂r n n Dkj k=1 j=1 ∂cj ∂r (5 .82 ) and ˜ Ji = − n n (δik − Xi )Dkj k=1 j=1 ∂cj ∂r (5 .83 ) Again, the constraint (5.72) applies,... austenite to ferrite transformation, optical microscopy) and precipitates in a multiphase alloy (bottom, γ ’-precipitates in nickel-base superalloy UDIMET 720, scanning electron microscopy) 180 COMPUTATIONAL MATERIALS ENGINEERING volume concept The second section of this chapter deals with nucleation theory, followed by modeling of precipitation processes and a description of the kinetics of precipitation... obtain ˙ dNτ = Nτ Vtot dτ (6.2) where Vtot is the total volume of the transforming sample The total increase in extended volume within the time interval dτ is given as dV e = v e · dNτ = 182 COMPUTATIONAL MATERIALS ENGINEERING 4π ˙ 3 ˙ G (t − τ )3 Nτ Vtot dτ 3 (6.3) The total extended volume, which is created by all nucleation events, can be obtained by ˙ ˙ integration of equation (6.3), which, for... exponent n can be characteristic for multiple types of transformation Nevertheless, some values for n are summarized in Table 6-1, which have originally been compiled in reference [Chr02] 184 COMPUTATIONAL MATERIALS ENGINEERING TABLE 6-1 Interpretation of the Avrami Exponent in Equation (6.10)a Value of n Interpretation Polymorphic changes, discontinuous precipitation, eutectoid reactions, interface-controlled . (Length) c(x)/c 0 0 24 681 0 sqrt(4 Dt) = 100 2 1 0.5 0 0.2 0.4 0.6 1.0 0 .8 5 20 FIGURE 5 -8 Solutions to Fick’s second law for semi-inifinite sample. 164 COMPUTATIONAL MATERIALS ENGINEERING against. movement of the markers is related to the difference in the diffusion coefficients. 1 68 COMPUTATIONAL MATERIALS ENGINEERING The results of his experiments have been published against the strong resistance. vacancies, which can be calculated from J Va = − n  j=1 J i (5.77) and v = J Va V m (5. 78) 172 COMPUTATIONAL MATERIALS ENGINEERING The flux ˜ J i in the laboratory frame of reference can then be expressed