New Mathematical Solution for Analyzing Interdiffusion Problems

Transcription

1 Materials Transactions, Vol. 5, No. 1 (11). to 7 #11 The Jaan Institute of Metals EXPRESS REGULAR ARTICLE New Mathematical Solution for Analyzing Interiffusion Problems Takahisa Okino eartment of Alie Mathematics, Faculty of Engineering, Oita University, Oita , Jaan The Fickian secon law is wiely alicable not only to the analysis of various iffusion roblems in material science but also to that of henomena for the Brownian motion in other science fiels, such as the behavior of neurons in life science. It is thus one of the most ominant equations in science. In 1894, Boltzmann transforme it into an orinary ifferential equation alicable to the analysis of the interiffusion roblems. Since then, however, the mathematical solutions have not yet been obtaine. Here we erive two new equations suerior in calculation to the ones of Fick an Boltzmann. Using the erive integro-ifferential equation, their solutions were obtaine as analytical exressions in accorance with the results of the exerimental analysis. Hereafter, the basic equations erive here will be exceeingly useful for the analysis of the nonlinear roblems concerning the Brownian motion in various science fiels. [oi1.3/matertrans.m11137] (Receive May 9, 11 Accete Setember 5, 11 Publishe October 6, 11) Keywors iffusion, interiffusion, Fickian iffusion equation, Brownian motion, Boltzmann transformation equation 1. Introuction The artial ifferential equation of the Fickian secon law 1) of the iffusion time t an the sace coorinate ðx y zþ has been wiely alie not only to the analysis of the iffusion roblems in the material science but also to that of the various Brownian motion roblems in the other science fiels. The Fickian iffusion equation is thus one of the most ominant equations in science. The Fickian first law 1) is jrcðt x y zþi ¼ jjðt x y zþi ð1þ where, C an jji are the iffusivity, the concentration an the iffusion flux, an hrj ¼ ð@=@x@=@y@=@zþ, using the irac s bracket. The Fickian secon law x y zþ hrjrcðt x y zþi ¼ has been wiely use for the roblems of the conservation system in science. When eens on C, however, even if we try to solve the equation of the one imensional sace coorinate ðt xþ the mathematical solutions are imossible. Boltzmann transforme eq. (3) into the orinary ifferential equation of CðÞ ¼ ðþ CðÞ ð4þ in ) Here, the arabolic law of ¼ x= ffiffi t is use. As far as another relation between an C is not given, the mathematical solutions are still imossible. Then, using the exerimental C rofile in eq. (4), Matano obtaine the rofile against C in the interiffusion roblems between soli metals in ) The emirical Boltzmann-Matano (B-M) metho has been wiely alie to the analysis of the interiffusion exeriments between soli metals. Since 1894, however, the mathematical solutions of eq. (4) have not yet been obtaine for such a long time. It is hysically obvious that eens on only via C. In mathematics, this yiels the relation between C an given by @ In hysics, if we multily the both sies of eq. (5) by, it becomes the relation of the iffusion flux in the sace. Using eq. (5) as another relation mentione above, the solutions of eq. (4) are ossible. Then, the author erive the useful equation of ðþ CðÞ ¼ J ex ðþ for J ¼ ðþ CðÞ ð6þ ¼ from eq. (4). Solving eqs. (5) an (6), the mathematically an hysically reasonable solutions were obtaine as the analytical exressions in accorance with the results of the B-M metho. We can thus theoretically reict the exerimental results if only the initial values are given. Therefore, the resent analytical metho is extremely useful for the analysis of the nonlinear roblems of the Brownian motion where the exerimentation is ifficult or imossible. Using ¼ y= ffiffi ffiffi t an ¼ z= t, we can easily exan eqs. (5) an (6) into the ones in the ð Þ sace. Then, the equation of the conservation system in the ð Þ sace is erive. The new equations erive here correson to eqs. (1) an () an they are alicable to the analysis of whatever roblems where eqs. () an (4) are alicable. Further, they are suerior in the calculation to eq. (), since the analysis in the 4 imensional time an sace ðt x y zþ is reuce to that in the 3 imensional sace ð Þ efine here. The new analytical metho to solve the nonlinear Brownian motion roblems was establishe in the resent stuy. From the new oint of view, the resent metho is wiely alicable to the analysis of henomena for the Brownian motion in various science fiels.. erivation of Basic Equations Solving analytically the nonlinear artial ifferential ð5þ

2 New Mathematical Solution for Analyzing Interiffusion Problems 1 eq. (3) is almost imossible even if the relation between C an is given. Then, after rewriting eq. (4) into ðþ ¼ 1 ðþ þ CðÞ 1 CðÞ ðþ its integral calculation yiels eq. (6). If we rewrite the righthan sie of eq. (6) into JðÞ ¼ J ex ðþ it is exresse as ðþ CðÞ ¼ JðÞ ð7þ Equation (7) is accete as the equation relevant to the iffusion flux in the sace an it corresons to the case of the one imensional sace coorinate of eq. (1). Thus, the hysical meaning of eq. (7) is obvious, although eq. (4) is not. Furthermore, the integro-ifferential eq. (7) is suerior in the aroximate calculation to eq. (4) of the secon orer ifferential equation. For examle, we can efine the effective iffusivity eff satisfying ðþ ¼ 4 eff in accorance with the characteristic of the integral calculation. By efining J an J of hjð Þj ¼ ðj J J Þ in a similar way to J yieling J ¼ J ex ð Þ for J ¼ Cð Þ ¼ ¼¼ we can exan eq. (7) into ð Þjr Cð Þi ¼ jjð Þi in the ð Þ sace, where The right-han sie of eq. (8) means the iffusion flux in the ð Þ sace. Equation (8) thus corresons to eq. (1). Here, note that eq. (8) is alicable to analyzing the iffusion roblems, since the iffusion flux jji can be exresse as the function of an ð Þ, although we cannot know such iffusion flux jji of eq. (1). In this meaning, eq. (8) is comletely ifferent from eq. (1). That is, eq. (8) is a new basic equation for the Brownian motion. For the conservation system in the hj ¼ð Þ sace, the relation of hr jr Cð Þi ¼ 1 hjr Cð Þi is vali in accorance with the mathematical theory an corresons to eq. (). When we solve eq. (9), eq. (5) shoul be rewritten as 8 @ >< ð8þ ð9þ Hereinbefore, we resente the useful equations to solve the nonlinear roblems of the Brownian motion in various science fiels. 3. Alication to Interiffusion Problems In orer to clarify the valiity of the resent metho, we alie it to the tyical interiffusion roblems where the iffusion coule between soli metals forms the comlete soli solution. The reason is as follows. The B-M metho has been wiely use for the analysis of their interiffusion roblems. The countless aers have been reorte an the useful finings have been thus accumulate. For the binary system, we efine the coorinate as x ¼ at the interface of the iffusion coule an the interiffusion area as x A x x B at the iffusion time t in the materials A an B. Using the initial values of the concentration C A an the iffusivity A in the material A an C B an B in the material B, the initial an bounary conitions of eq. (3) are efine as 9 t an x x A < Cðx tþ ¼C A an ðx tþ ¼ A >= an > t an < x B x Cðx tþ ¼C B an ðx tþ ¼ B For eq. (4) in the sace, these are rewritten as! 1 CðÞ ¼C A an ðþ ¼ A an!1 CðÞ ¼C B an ðþ ¼ B 9 >= > ð11þ When ðþ is equal to the constant value, eq. (6) or (7) is rewritten as CðÞ ¼ C ð1þ ex for C ð1þ ¼ CðÞ 4 ¼ an its integral calculation yiels the solution of CðÞ ¼C m C erf ffiffiffiffiffi ð1þ uner the conition of eq. (11), where C m ¼ðC A þ C B Þ=, C ¼ðC A C B Þ=. The solution of eq. (1) is equal to that of eq. (3) obtaine by the comlicate calculation of the integral transformation of Lalace or Fourier. The CðÞ rofile of eq. (1) against is the S-letter curve or its reverse one with the inflection oint ð C m Þ,forC A < C B or C B < C A, resectively. Hereafter, we analyze the iffusion roblems when ðþ eens on CðÞ. The countless exerimental results always reveal that the CðÞ rofile becomes the S-letter curve or the reverse one similar to that of the eq. (1). 4) In the tyical interiffusion roblems, the B-M metho shows that the ðþ rofile is also the S-letter curve or the reverse one. These inicate that CðÞ an ðþ are exresse as the suerosition of the error functions with various inflection oints. The relation of A < B is aote in this work. In such a case, the exonential art of eq. (6) satisfies ex < ex 4 A ðþ < ex 4 B In the resent stuy, it is efine as ex ðþ ¼ ex ðþ ð13þ 4 int

3 T. Okino where int is a constant value between A < int < B an ðþ is a function to correct the error cause by int instea of ðþ. Substituting eqs. (6) an (13) into eq. (5), the relation of the iffusion flux in the sace is obtaine ¼ J e ðþ ex 4 int ð14þ For the iffusion flux, the hysical seculation rouces the relation ¼ J ðþ ð15þ where ðþ is a function of satisfying Equations (14) an (15) ¼ J e ðþ ex 4 int lim ðþ ¼.!1 e ðþ ðþ ð16þ Base on the behavior of the error function, consiering the shift arameter " cause by the eenence of ðþ on CðÞ an using the constant values of 1 an, eq. (16) is ivie into the following two equations. One is an the other is ¼ ¼ ðþ ð "Þ 4 int SðÞ ð17þ " ex ðþ " ð18þ 4 int where SðÞ ¼ex½ðÞþ " " 4 int ŠðÞ an 1 ¼ J. There is the evience of the valiity of the above ivision as shown in the following. When the relation of ðþþ " " 4 int 1 is vali, eq. (18) is aroximately rewritten as C ¼ = an its integral calculation yiels CðÞ ¼C m þ C A C B ðþ ln ffiffiffiffiffiffiffiffiffiffiffiffi ð19þ ln A ln B A B uner the conition of eq. (11). In the tyical interiffusion roblems between soli metals, eq. (19) has been wiely accete. 5 8) Since the relation of ðþ ln ðþ is the one-to-one corresonence, we efine the locus as CðzÞ ¼ f ðzþ for z ¼ ln ðþ. Using the relation of CðÞ for eq. (4), we have ¼ 1 ðþ f ðzþ ðþ z ðþ ¼ 1 ðþ ( þ ðþ f ðzþ f ðzþ ) 1 ðþ z z ðþ Equation () shows that the solutions of þ ðþ f ðzþ f ðzþ 1 ¼ z z correson to the inflection oints of the ðþ rofile. The equation is rewritten into the two equations, w ¼ an w ¼ k for k f ðzþ f ðzþ 1 ¼ z z The following analysis reveals that w ¼ an w ¼ k= intersect at ¼ an ¼, ¼ þ extremely near ¼. (1) ðþ= is consiere as the Gaussian tye function, but it has the singular oint at ¼ in the early iffusion stage because of the Heavisie s tye initial conition of ðþ. () The Heavisie s tye initial conition inicates that ðþ= is the largest value at ¼ for A < B.This means ðþ= ¼ at ¼, then f ðzþ=z ¼ is vali in eq. (), i.e., k ¼. (3) Since f ðzþ=z ¼ is vali at ¼, the one of f ðzþ=z < or f ðzþ=z > is vali near ¼. In relation to ðþ= < for A < B, it is natural that we aot f ðzþ=z < for C A < C B. Inversely, f ðzþ=z > is aote for C A > C B. (4) j f ðzþ=z j or jkj is an extremely small value with reference to the curvature of the aroximate eq. (19). Base on the above-mentione, w ¼ k= is exresse as 8 ¼ jkj for < >< w ¼ for ¼ > ¼jkj for >. As shown in Fig. 1, w ¼ an w ¼ k= intersect at ¼ an ¼, ¼ þ extremely near ¼. In other wors, the ðþ rofile has the inflection oints at ¼, ¼ an ¼ þ in the extremely narrow area, although it seems as if it has the one inflection oint at ¼ on the orinary scale. Even then we relace these 3 inflection oints with the stationary-inflection oint at ¼ ¼, the ðþ rofile seems as if it has the one inflection oint at ¼ on the orinary scale. It is thus aroximately accetable that the ðþ rofile has the stationary-inflection oint at ¼ ¼ on an extremely enlarge scale. The situation mentione here is shown in the schematic Fig.. Hereafter, the aroximation, i.e., ðþ= ¼ ðþ= ¼ at ¼ ¼ is thus aote as a technical roceure of the alie mathematics in the resent work, although ξ+ k ξ ( ξ ) w = k ξ ξ w w = ξ ( ξ ) w = k ξ ξ ξ + ξ < < ξ + Fig. 1 The behavior of the inflection oints of ¼ ðþ. Uner the conition of k!, w ¼ an w ¼ k= intersect at ¼ an ¼, ¼ þ extremely near ¼. As shown in the extene illustration A of Fig., these values correson to the inflection oints of ¼ ðþ. ξ

4 New Mathematical Solution for Analyzing Interiffusion Problems 3 4. Comarison between the Present Metho an the B-M Metho ðþ= ¼ is never vali. The ðþ rofile is thus shown as the ouble S-letter curves smoothly connecte at ¼ for A < B on an extremely enlarge scale, although it seems the single S-letter curve on the orinary scale. Therefore, we accet that the effective interiffusion coefficient int is the Heavisie s tye function of int ¼ intþ for > an int ¼ int for <, similar to the initial conition of ðþ ¼ B for ¼1an ðþ ¼ A for ¼ 1. From the characteristic of the stationary-inflection oint ð Þ on the ð ðþþ lane, it is foun that SðÞ in eq. (17) must satisfy the relation of lim SðÞ ¼ Sð Þ ¼ex ð "Þ!1 4 int an ξ B ξ Sð Þ = ( ξ ) k Fig. The aroximate metho in alie mathematics. The stationaryinflection oint of ¼ ðþ, ¼ near ¼, is shown in the extene illustration B. On the extene scale, the illustration B is comletely ifferent from the illustration A. On the orinary scale, however, we cannot istinguish between them. Therefore, the resent calculation is carrie out in accorance with the illustration B as a technical roceure in alie mathematics. ¼ " ex ð "Þ int 4 int since ðþ= ¼ ðþ= ¼ must be vali at ¼ ¼. As shown in the Aenix A, the solution of eq. (17), i.e., the solution ðþ of eq. (4) or (7) is ossible as the suerosition of the error functions, an the solution CðÞ of eq. (4) or (7) is also obtaine by the similar metho using the characteristic of the inflection oint ð IN C IN Þ on the ð CðÞÞ lane. Their analytical solutions are thus obtaine as ðþ ¼ m erf ffiffiffiffiffiffiffi int ffiffiffiffiffiffiffiffiffiffi intþ þ erf 1 m ð1þ where m ¼ð A þ B Þ= an ¼ð A B Þ=, an CðÞ ¼C m C erf ffiffiffiffiffiffiffi IN int ffiffiffiffiffiffiffiffiffiffi int þ erf 1 C m C IN ðþ C ξ ξ ξ + ξ A The valiity of the resent analytical metho is numerically confirme by the comarison with the results of the B-M metho. In the analysis of the tyical interiffusion roblems where the iffusion coule between soli metals forms the comlete soli solution, it is wiely accete that the B-M metho results in eq. (19) with a goo aroximation. 5 8) Inversely, the exerimental CðÞ rofile is rerouce, if we numerically solve eq. (4) using eq. (19) as another relation. In orer to secify the solutions of eqs. (1) an (),,, IN, C IN an int must be etermine from the initial values. In relation to eq. (), ¼ is aote in the resent metho. Substituting eqs. (1) an () into eqs. (4) an (19) an using the mathematical characteristic at the inflection oint, the others were etermine through the consierably comlicate aroximate calculations as follows. ¼ð A B Þ=ðln A ln B Þ IN ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi A B ð A BÞ=ð A þ BÞ C IN ¼ C m C ð m Þ= intþ ¼ð A þ B Þ= an int ¼ ffiffiffiffiffiffiffiffiffiffiffiffi A B Here, note that intþ an int are equivalent to the arithmetical mean an the geometrical mean, resectively. When the iffusivity oes not een on the concentration, we can set A ¼ B ¼. Then, the above relations yiel int ¼ intþ ¼ int ¼, m ¼ ¼, ¼, ¼ IN ¼ an C IN ¼ C m. Then, eqs. (1) an () agree with ðþ ¼ an eq. (1), resectively. Thus, the analytical solutions of eqs. (1) an () are the generalize ones. The hysical quantities excet C IN een on only A an B. The erivation of these hysical quantities is briefly given in the Aenix B. Using the above hysical quantities for various cases of the initial values, ð A B Þ an ðc A C B Þ, the rofiles of ðþ an CðÞ against were investigate in comarison with those of the B-M metho. As a result, it was foun that the investigate ifferences between the resent solutions an those of the B-M metho are almost similar levels. In the resent stuy, therefore, the ðþ an CðÞ rofiles for three cases are shown in Figs. 3 an 4. For the three cases of ð A B Þ in eq. (1), the case 1 of ð Þ, the case of ð Þ an the case 3 of ð Þ, the ðþ rofiles against are resente in Fig. 3. The results of resent metho an the ones of the B-M metho are shown by the soli lines an the notation, resectively. Using the same values of ð A B Þ in Fig. 3, the CðÞ rofiles are resente in Fig. 4. The three cases of ðc A C B Þ in eq. (), ð5 1Þ, ð 1Þ an ð 5Þ, are shown corresoning to the case 1, the case an the case 3 in Fig. 3, resectively. The results of resent metho an the ones of the B-M metho are shown by the soli lines an the notation, resectively. As can be seen from Figs. 3 an 4, there is the slight ifference between the results of the B-M metho an those of the resent analytical metho. Uner the reconition of eq. (19), the results of the B-M metho are obtaine. On the other han, the resent results are obtaine by the aroximate roceure. At resent, therefore, we cannot estimate

5 4 T. Okino iffusivity, (ξ) / m s Case 1 Case Case 3 Concentration, C( ) 1.5 Case 1 Case Case ξ / m s -.5 Fig. 3 The behavior of the iffusivity against. The soli curves an the notations enote the ðþ rofile against obtaine by the resent analytical metho an the B-M metho, resectively. The re, green an blue colors are use for the initial values of ð A B Þ¼ ð Þ m s 1, ðð A B Þ¼ð ÞÞ m s 1 an ð A B Þ¼ð Þ m s 1, resectively /m s -.5 Fig. 4 The behavior of the concentration against. The soli curves, the notations an the colors use here correson to ones of Fig. 3. The re, green an blue colors are use for the normalize initial concentration of ðc A C B Þ¼ð5 1Þ, ðc A C B Þ¼ð 1Þ an ðc A C B Þ¼ ð 5Þ, resectively. Fickian first law of eq.(1) Fickian secon law of eq.() C (t, x, y, z) = J (t, x, y, z) C (t, x, y, z) = C (t, x, y, z)/ t Boltzmann transformation of eq.(4) ξ C ( ξ) C ( ξ) = ξ ξ ξ New equation of eq.(8) New equation of eq.(9) σ C ( ξ, ψ, ζ ) = J ( ξ, ψ, ζ ) C ( ξ, ψ, ζ ) = θ C ( ξ, ψ, ζ ) / σ σ σ Fig. 5 Correlation between iffusion equations. whether the exerimental results agree well with the results of the B-M metho or those of the resent metho. However, the resent results agree aroximately with those of the B-M metho. The figures thus show that the resent metho is accetable an at the same time the hysical quantities,,, IN, C IN, an int are reasonable. The rofiles of ðtþ an CðtÞ against t at an arbitrary x ¼ x M an also ðxþ an CðxÞ against x at an arbitrary t ¼ t N are ossible. However, they are neglecte in the resent stuy. 5. iscussion an Conclusion The author erive the two new eqs. (8) an (9) alicable to the analysis of whatever roblems where eqs. () an (4) are alicable. The new analytical metho to solve the nonlinear roblems of the Brownian motion was establishe. Figure 5 shows the correlation between the new equation I an/or II an the Fickian first an secon laws through the Boltzmann transformation equation. The iffusion flux jjð Þi of the new equation I is exresse as a function of ð Þ an ð Þ, while we cannot know the functional form of jjðt x y zþi of the Fickian first law. It is thus alicable to the analysis of the various roblems of the Brownian motion in science, although we cannot use the Fickian first law for the analysis. Further, it excels the Boltzmann transformation equation in the analysis, since the integro-ifferential equation is suerior in the aroximate calculation to the secon orer ifferential equation. In articular, it is very simle to solve the linear roblems where the iffusivity oes not een on the concentration. Then, the one imensional case is shown in the following. When the iffusivity ðþ is equal to the constant value, eq. (7) is rewritten as CðÞ ¼ C ð1þ ðþ ex ð3þ 4 where C ð1þ ¼ CðÞ=j ¼. We can use eq. (3) for the analysis of whatever roblems where ðþ oes not een on CðÞ. The general solution of eq. (3) is CðÞ ¼A þ B erf ffiffiffiffiffiffi

6 New Mathematical Solution for Analyzing Interiffusion Problems 5 where A an B are etermine from the given initial values. For the interiffusion roblem uner the conition of CðÞ ¼ C A for ¼ 1an CðÞ ¼C B for ¼1, an for the oneorientation iffusion roblem uner the conition of CðÞ ¼ C for ¼ an CðÞ ¼ for ¼1, A an B become A ¼ C m, B ¼ C an A ¼ C, B ¼ C, resectively. Their solutions thus correson to eq. (1) an the wellknown solution of CðÞ ¼C 1 erf ffiffiffiffiffiffi Further, when C ð1þ in eq. (3) eens on the iffusion time ð¼ tþ, the thin film iffusion roblem is consiere. Then, using the well-known irac s -function an relacing C ð1þ by ðþ, eq. (3) is rewritten as CðÞ ¼ ðþð Þ ex 4 an the solution is obtaine as 1 CðÞ ¼ ðþ ð Þ ex 1 4 ð4þ ¼ ðþ ex 4 Base on the Boltzmann transformation of ¼ t an ¼ x ffi t, an using the total iffusion material quantity M given by M ¼ 1 1 Cðt xþx eq. (4) is rewritten as the well-known exression of M Cðt xþ ¼ ffiffiffiffiffiffiffiffiffiffi ex x t 4 t The well-known solutions are simly obtaine here by using eq. (7), although they have been reviously obtaine by using the integral transformation of Lalace or Fourier. The new equation II is also suerior in the calculation to the Fickian secon law, since the analysis in the 4 imensional time an sace of ðt x y zþ is reuce to that in the 3 imensional sace of ð Þ. By alying eq. (7) to the tyical interiffusion roblem between soli metals, which has not yet been mathematically solve for a long time, the mathematically an hysically reasonable solutions ðþ an CðÞ of the nonlinear eq. (4) were obtaine as the analytical exressions. We can thus reict the exerimental results if only the initial values are given. Therefore, the resent metho is extremely useful for the analysis of the roblems of the Brownian motion where the exerimentation is ifficult or imossible. As a necessary conition of the locus erive from the solutions ðþ an CðÞ of eq. (4), the relationshi of f ðzþ z ¼ must be vali at ¼ in relation to eq. (). Here, note that eq. (19) satisfies the relationshi as a secial case. However, the interiffusion roblems between soli metals are comlicate in the actual case. There are the results of the B-M metho where this conition is not satisfie. This reason is as follows. The well-known Kirkenall effect reveals that the atoms in the metal crystal iffuse via vacancies. 9) The atomic iffusion roblem in the metal crystal is base on the reconition that their vacancies are homogeneous an are in the thermal equilibrium state. It is, however, ossible that the reconition is not vali, when the iffusion coule oes not form the comlete soli solution. Therefore, we nee consier an alication limit of eq. (4), cause by various factors uring the iffusion rocess, to the interiffusion roblems between soli metals. In such a case, eq. (7) shoul be rewritten as ðþ CðÞ ¼ JðÞ J est ðþ where J est ðþ is the term to estimate the ifference from the tyical case. Hereafter, from the new oint of view, the stuy on the nonlinear roblems of the Brownian motion will be romote, an the basic equations erive here will be thus more an more useful for the analysis of their nonlinear roblems in science an technology. Acknowlegment The author woul like to thank Emeritus Professor Ohnishi, Professor Shimozaki an Emeritus Professor Iijima for their useful comments on the Boltzmann-Matano metho. REFERENCES 1) A. Fick Phil. Mag. 1 (1855) ) L. Boltzmann Ann. Phys. Chem. 53 (1894) ) C. Matano Jn. J. Phys. 8 (1933) ) For examle, P. G. Shewmon iffusion In Solis, (McGraw-Hill, 1963) ) M. Onishi, T. Ikea, Y. Wakamatu an T. Shimozaki Trans. JIM 9 (1988) ) L. C. C. a Silva an R. F. Mehl Trans. AIME 191 (1951) ) K. Hoshino, Y. Iijima an K. Hirano Trans. JIM 1 (198) ) M. Onishi an H. Miura Trans. JIM 18 (1977) ) A. Smigelskas an E. Kirkenall Trans. AIME 171 (1947) 13. Aenix A The aroximate calculation to obtain the solutions of eqs. (1) an () from eq. (17) is carrie out in the following. The solution ðþ is hysically consiere as the suerosition of the error functions, an ðþ= ¼ ðþ= ¼ must hol true at the stationary-inflection oint ¼ in the resent metho. Such SðÞ in eq. (17) may be exresse as 1 SðÞ ¼ N þ N n þ 1 XN ði þ 1Þ ex ðði þ 1Þ i "Þ þ i ex ði ðiþ1þ þ "Þ ða1þ i¼n 4 int 4 int

7 6 T. Okino where n an N (N n ) are arbitrary ositive integers. Using eqs. (11) an (A 1), the integral calculation of eq. (17) is given as ( ðþ ¼ m G ðn NÞ erf ffiffiffiffiffiffiffi þ erf 1 m int 1 X N ði þ 1Þð Þ erf N þ N n þ 1 i¼n ffiffiffiffiffiffiffi þ erf 1 m ið Þ þ erf int ffiffiffiffiffiffiffi erf 1 m ) int ðaþ where G ðn NÞ ¼ N þ N n þ 1 N n þ n 1 an " ¼ ffiffiffiffiffiffiffi int erf 1 m G ðn NÞ Uner the conition of N ¼ n, eq. (A ) is rewritten as n þ 1 ðþ ¼ m erf n 1 ffiffiffiffiffiffiffi þ erf 1 m int 1 ðn þ 1Þð Þ erf n þ 1 ffiffiffiffiffiffiffi þ erf 1 m nð Þ þ erf int ffiffiffiffiffiffiffi erf 1 m ða3þ int For a large value n, the first error function in f g of eq. (A 3) mainly contributes to the left-han sie. In such a case, the aroximate solution is equal to eq. (1). Substituting eqs. (13) an (1) into eq. (6), CðÞ is obtaine as CðÞ ¼ C ð1þ ð1 f ðþþ 1 ex ðþ þ C A m 1 4 int where f ðþ ¼ erf m ffiffiffiffiffiffiffi int ffiffiffiffiffiffiffiffiffiffi þerf 1 m intþ Then, using the first law of the average value in the integral calculation, the above equation for f ðþ < 1 is aroximately rewritten as CðÞ ¼ C ð1þ ð1 þ f ðþþ ex ðþ þ C A ða4þ m 1 4 int where 1 <<. Taking account of the shift arameter to form the error functions in the integral calculation, the integran ex½ =4 int ðþš is written as ex ð Þ ðþ ¼ ex TðÞ ða5þ 4 int 4 int where TðÞ is the function to correct the error cause by the aroximation. After substituting eq. (A 5) into eq. (A 4), in orer to satisfy the relation of C= ¼ at ¼ IN, TðÞ may satisfy the following relations, 6 X M TðÞ ¼ i ex ðið INÞ IN þ Þ M 3 þ 3M þ M m 3 þ 3m m i¼m 4 int an Tð IN Þ ¼ IN ex ð IN Þ int 4 int where m an M (M m 1) are arbitrary ositive integers. From TðÞ an eqs. (11) an (A 5), eq. (A 4) is rewritten as CðÞ ¼C m C G C ðm MÞ erf ffiffiffiffiffiffiffi þ F C ðþ ða6þ int In eq. (A 6), the notations are M 3 þ 3M þ M m 3 þ 3m m G C ðm MÞ ¼ M 3 þ 3M þ 7M m 3 þ 3m 7m þ 6 6 X M ið IN Þ IN þ F C ðþ ¼ erf M 3 þ 3M þ M m 3 þ 3m m i¼m ffiffiffiffiffiffiffi int

8 an New Mathematical Solution for Analyzing Interiffusion Problems 7 ¼ IN ffiffiffiffiffiffiffi int erf 1 C m C IN C H C ðm MÞ H C ðm MÞ ¼ M3 þ 3M 5M m 3 þ 3m þ 5m 6 M 3 þ 3M þ 7M m 3 þ 3m 7m þ 6 Uner the conition of M ¼ m, eq. (A 6) is aroximately equal to eq. () for a large value m. Aenix B The hysical quantities,,, IN, C IN, intþ an int are aroximately erive from the mathematical characteristic of the inflection oints of eqs. (1) an (). Using the aroximate equations, the aroximate calculations are carrie out to obtain the hysical quantities. Here, the rimary aim is to obtain their exressions as simle as ossible. In the text, ¼ is alreay obtaine in relation to eq. (). Then, we etermine as follows. Using the relation of an erf 1 ðxþ ¼ X1 k¼ erfðxþ ¼ X k 1 m¼ ffiffiffi X1 ð 1Þ n x nþ1 n!ðn þ 1Þ n¼! a m a k 1 m 1 ðm þ 1Þðm þ 1Þ k þ 1 ffiffiffi x kþ1 for a ¼ 1 in eqs. (1) an (), the aroximate equations of Cð Þ¼C IN þ s ffiffiffi C an ð IN Þ ¼ s ffiffiffi ðb1þ are obtaine, where s ¼ IN = ffiffiffiffiffiffiffiffiffiffi int. The value of eq. (19) at ¼ an the one at ¼ IN yiel ðc C IN Þ=ðC A C B Þ ¼ðln ln IN Þ=ðln A ln B Þ ðbþ The aroximate calculation of eqs. (B 1) an (B ) yiels ¼ð A B Þ=ðln A ln B Þ Using equation (B 1) an the obtaine, eq. (B ) is aroximately rewritten as! s ffiffiffi ¼ ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C m C IN ðb3þ m C After substituting eqs. (1) an () into eq. (19), ifferentiating it yiels ¼ ex s þ erf 1 m IN erf 1 C m C IN C at ¼. The equation is further aroximately rewritten as s ffiffiffi ¼ ffiffiffi s ffiffiffi s þ m 4 þ m C m C IN ðb4þ C When the term ð m Þ = ðc m C IN Þ =C in f g of eq. (B 4) is as small as negligible, we have two equations, C IN ¼ C m C ð m Þ= an ¼ ffiffiffi s þ m ðb5þ 4 Substituting eq. (1) into = þ ðþ= ¼ vali at ¼ IN in eq. (4) yiels s ¼ ffiffiffiffiffiffiffiffiffiffi 1 ð IN Þ int ¼ 1 ffiffiffi " ex s þ erf 1 m # ðb6þ int The aroximate calculation for a system of eqs. (B 3), (B 5) an (B 6) yiels int ¼ ffiffiffiffiffiffiffiffiffiffiffiffi A B an IN ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi A B ð A BÞ=ð A þ BÞ Equation (4) at ¼ satisfies the relation of C þ C ¼ ðb 7Þ After substituting the erivative value at ¼þfor eq. (1) an the erivative value at ¼ for eq. () into eq. (B 7), the aroximate calculation yiels intþ ¼ð A þ B Þ=