Journal of Modern Physics, 2012, 3, 13881393 http://dx.doi.org/10.4236/jmp.2012.310175 Published Online October 2012 (http://www.SciRP.org/journal/jmp) Theoretical Evidence for Revision of Fickian First Law and New Understanding of Diffusion Problems Takahisa Okino Department of Applied Mathematics, Faculty of Engineering, Oita University, Oita , Ja pan Email: okino@oitau.ac.jp Received August 6, 2012; revised September 9, 2012; accepted September 16, 2012 ABSTRACT Based on the divergence theorem, we reveal that the Fickian first law relevant to the diffusion flux ,,,txyz in the time and space is inco mplete without an integr al constant t 0 for the integral of Fickian second law. The new dif fusion flux (NDF) taking it into account shows that we can systematically understand the problems of oneway diffu sion, impurity diffusion and selfd iffusion as a special case of the in terdiffu sion. Applying the NDF to the interdiffusion problem between metal plates, it is clarified that the Kirkenkall effect is caused by t 0 and also that the interdiffu sion coefficients in alloy can be easily obtained. The interdiffusion problems are reasonably solved regardless of the intrinsic diffusion conception. Thus the NDF to replace the Fickian first law is an essential equation in physics. Keywords: Diffusion Equation; Fickian First Law; Kirkendall Effect 1. Introduction The diffusion problems are fundamental and important in physics and/or material science, for instance some of which are the problems of Brownian motion and/or technological application to materials in metallurgy and in semiconductor science. In the present study, the fun damental problems of diffusion phenomena are discussed by investigating the interdiffusion problems between metal plates as an application example of the present diffusion theory, because they have been widely investi gated in metallurgy and a lot of dominant data have been accumulated. However, the fundamental theory discussed here is generally valid in physics. In 1855, the wellknow n Fickian first and second laws (FFL and FSL) analogous to the Fourier heat conduction equation were published [1,2]. Since then, they have been accepted as one of the most fundamental equations in physics and have been applied to diffusion problems. In the age of Fick, the existence of atoms and/or mole cules was not generally accepted. Although the molecule theory of Boltzmann was published in 1872, the exis tence of molecules was not selfevident truth [3]. In 1905, Einstein theoretically revealed that the wellknown Brownian motion depends on the existence of molecules, and it was clarified that the parabolic law is valid be tween the molecular displacement and diffusion time [4, 5]. Then, Einstein theory was experimentally confirmed by Perrin [6]. In other words, the Brownian motion re veals that the diffusion phenomena occur even in the thermal equilibrium state of material. As far as the diffu sion occurs, therefore, the diffusion flux must exist in such a state. However, the diffusion flux FFL, which is directly proportional to a concentration gradient, be comes zero in that case. It is thus inconsistent with the physical phenomena. In this stage, it was thus indicated that the FFL shou ld be reasonably modified. In 1894, Boltzmann transformed the FSL of time and space ,tx into an ordinary differential equation (B equation) of 0.5 xt 1 in accordance with the para bolic law [7]. In 1933, Matano obtained the diffusivity profile of interdiffusion between metal plates by empiri cally applying the Bequation to the experimental con centration profile [8]. The concentration dependence of diffusivity was then clarified. The Bequation has been thus widely used for the analysis of interdiffusion prob lems in metallurgy. After that, it was found that the FFL is inconsistent with the wellknown Kirkendall effect (Keffect) which occurs in the interdiffusion phenomena between metal plates [9]. This fact also suggested that the FFL should be reasonably modified. Nevertheless, it has been still accepted in physics as it is. On the contrary, the intrinsic diffusion, which is incons istent with the FSL, was devised in order to understand the Keffect [10]. In 1948, Darken thus proposed a relation between the in trinsic diffusivity and interdiffusion coefficient via the C opyright © 2012 SciRes. JMP
T. OKINO 1389 concentration. Since then, the relation has been widely used for analyzing the interdiffusion problems. However, it is considerably difficult to experimentally determine intrinsic diffusivity values, since it is necessary to inves tigate the Keffect by using multiple markers [11]. Recently, in the defined parabolic space 123 ,, for 0.5 2 t 0.5 zt ,tx ,tx 0x and 3, the general solutions of linear and/or nonlinear diffusion problems were obtained as the elegant analytical expressions [12,13]. In order to actually apply them to interdiffusion problems, we must determine initial and/or boundary diffusivity values in alloy. It was, however, difficult to experimentally deter mine them. In the present study, we revise the FFL in accordance with the divergence theorem, because the FFL is incom plete without an integral constant for the integral of FSL. The new diffusion flux (NDF) to replace the FFL is sys tematically applicable to the diffusion problems of one way diffusion, impurity diffusion and selfdiffusion as a special case of the interdiffusion. The NDF reveals that the Keffect is caused by a material source on the diffu sion boundary interface and also that the interdiffusion coefficients can be easily obtained. Using their interdif fusion coefficients for the initial and/or boundary values of the general solutions mentioned above, the interdiffu sion problems are reasonably solved regardless of the intrinsic diffusion conception. The NDF is not only essential for physics but also ex tremely useful for material science. Applying it to the interdiffusion problems between metal plates as an ex ample of the diffusion problems, its validity was con firmed in the present study. The new findings obtained here may make a fundamental change to the existin g dif fusion theory. 2. Summary of Interdiffusion Problems Since the physical essence is kept even if we investigate the diffusion problems of time and space , the FSL of is investigated in this section. It will be clari fied later that we can systematically understand the one way diffusion, the impurity diffusion and the selfdiffu sion as a special case of the interdiffusion. Therefore, we briefly summarize the interdiffusion problems for a dif fusion couple between metal plates A and B, where the plate A is the alloy composed of I atom and II atom and the plate B is the pure metal of II atom. The coordinate is then defined as at the interface between the plate A and the plate B 0x 0 AB x t. Further, the inter diffusion region at a time is defined as xx 0 n and at . x0t In the interdiffusion problems between metal plates, it is generally accepted that the deformation of specimen between diffusion beforeandafter is almost negligible. In other words, the number of total atoms on an ar bi trary crystal cross section perpendicular to axis is considered as a constant value during the diffusion proc ess. The relation III nn n n nI C II C III 1CC is thus valid, where I or II is the number of I atoms or II atoms on the same cross section. Using the normalized concentration for I atom and for II atom, the relation o f (1) is thus widely accepted in this field. In the following, the abbreviated notations of 2 22 , ,,andtxy z for I CII C . are used. The FSL for or is III txx CDC IIII II txx CDC D D , , (2) where I and II are diffusivities of I atom and II atom. Equations (1) and (2) yield III II 0 xx DD C III DD D . (3) Equation (3) sh ows D I , (4) where is the socalled interdiffusion coefficient. In this case, the diffusion flux FFL of for I atom or II for II atom is III I IIII IIII , . xx xx JDCDC DC DC III III 0 x JJ DCC (5) Equations (1) and (5) yield III 0JJ II IIIIIII for 1DCD CDCC . (6) Here, (6) shows that the number of I atoms which dif fuse from the plate A into the plate B is equal to that of II atoms which diffuse from the plate B into the plate A. The Keffect shows that must be v alid in the interdiffusion problems [9]. It is obvious that (6) is inconsistent with the Keffect. In order to solve the in consistency, Darken proposed the interdiffusion coeffi cient of III DDandDD III 0JJ (7) instead of (4), assuming inconsistent with (4) so that (6) is not valid [10]. Here, III were then designated as “intrinsic diffusion coefficient.” Equation (7) has been widely used for the analysis of interdiffusion problems. However, the author thinks that the FFL should be revised so RR is valid for the new diffusion flux R under the condition of (4). In other words, we can understand the Keffect without the in trinsic diffusion conception. In the next section, the NDF to replace the FFL will be defined, and also it will be Copyright © 2012 SciRes. JMP
T. OKINO 1390 clarified that the NDF is mathematically and physically reasonable. 3. Divergence Theory and New Diffusion Flux For an arbitrary differentiable vector ,,, txyz V in a space closed in a surface , the divergence theo rem between the volume integral and surface integral is defined as S dd SV VJS , (8) where Dirac’s vector representation is used and a normal unit vector perpendicular to a surface element and dS ,, yz . Here, applying the relation defined as R ,,,,, 0 , txyz JtxyzJ t (9) to (8), the relation of R dd VS 0 d S VJ SJS (10) is valid, where 0 tS is defined as a vector on . In the present physical system, the first term of the right hand side of (10) means a physical quantity Q which outflows through S per unit time. On the other hand, the second term is relevant to an inflow rate of Q caused by a material source on S. The decrease rate of Q is thus ex pressed as 0d t SS d SJ SQ. (11) Substituting (11) into the righthand side of (10), the relation of Rdt V VQ (12) is generally valid. In the conventional diffusion problems, the flux correlates with the FFL of ,,, DCtxyz D ,,Ctx , (13) using the diffusivity and the concentration of the material quantity Q. Equation (13) has been used only for the diffusion problems where the concentration gradient is not zero in the initial state under the condition of no material sink or source. The substitu tion of (13) into (9) yields the NDF of ,yz R ,,,,, 0 , txyzD Ctxy zJ t ,,, dtxyzV , (14) taking the source effect into account. Further, substitut ing (14) and the relation of tt V QC into (12), the FSL is thus obtained as R,,, ,,, t DCtxyz Ctxyz, (15) where 00Jt . t In mathematics, 0 is relevant to the integral constant of d V V because of 00Jt S . In physics, it is relevant to the material inflow caused by a material source on . The present theory reveals that the FSL is applicable to the diffusion problems as it is, even if a material source in dependent of ,, yz is contained in the diffusion sys tem. 4. Application of NDF to Interdiffusion Problems The socalled Kirkendall interface is defined at K x where the number of I atoms which diffuse from the plate A into the plate B is equal to that of II atoms which diffuse from the plate B into the plate A. On the other hand, the original interface between the plate A 0x and the plate B 0x0xx at M is the socalled Matano interface. The Keffect means effKM 0xxx . In the following, we define the NDF under the condition of (4) and (6) and investigate the diffusion problems in accordance with the NDF. As is well known, the diffusion junction depth jun at a diffusion time for a material of diffusivity D is expressed as t jun Dt . (16) We use notations of jun Dt and ABA B CCC A CCIA DDA x for , at and B CC , at II B DDB x with the superscript/ subscript of IA or . For the concen tration gradient of II B AB jun Cx , the diffusion flux 0t is defined as AB 0eqABeq jun AB for , C tDJ DCCtJ x xx x (17) 0J eq where is a constant value relevant to the Brownian motion in the thermal equilibrium state, and III eq 0JJ eq must be then physically valid. Under the condition of (1), (4), (6) and (17), (9) or (14) yields IIIIII RR 00 II ABAB ,, , Jtx JtxJtJt DDCCt (18) II IIII AB AB CC CC and where Copyright © 2012 SciRes. JMP
T. OKINO 1391 Jtx III ,,0J tx t because of (1) and (6). In consid eration of the atomic migration caused by the interdiffu sion, the integral calculation of (18) with respect to correlates with the Keffect as follows: III eff0 0 0 II ABA 1d 2 . t xJtJt DDC t B t C (19) Here, note that (19) is consistent with the parabolic law. On the other hand, the experimental analysis of in terdiffusion problems also shows the parabolic law yielding eff mt mI , (20) where the slope is experimentally determined [9]. In the present diffusion system, C is defined as at B I B0C xD . In the interdiffusion problems be tween metal plates, it is widely accepted that B in the present diffusion system can be approximately replaced by the impurity diffusivity near B I imp D x A D because of (4). Therefore, the interdiffusion coefficient at A x is obtained as II Aimp A DDmC 10 Zn 0.3C , (21) by using (19) and (20). For example, substituting and A in Ref. [9] and imp near 80.5 m s 14 21 10m s 4.9m Zn 1.D26 B x14 2 10ms in Cu plate into (21), the interdiffusion coefficient 1 A7.6D 1058 D D III ,,1C tx D D is obtained at the absolute temperature T be cause of in the present case. Therefore, var ious A values are obtained through the interdiffusion experimentation by using diffusion couples between the plate A (various compositio n rates of I and II atoms) and the plate B (the pure metal of II atom) for various com binations of I and II ato ms. After a large number of A data were thus accumulated, using those A data at the same for the initial and/or boundary values of the general solutions of Refs. [12,13], the solutions of inter diffusion problems between alloy plates are possible. On the contrary, using those A data for (19), we can also predict the behavior of Keffect. As can be seen from the above discussion, the intrinsic diffusion conception in consistent with the FSL is not unnecessary for under standing the interdiffusion problems. Zn Aimp D D T II DD In the present diffusion system, using the NDF of (14) as an additional condition equation for the FSL, the problems of interdiffusion, oneway diffusion, impurity diffusio n an d self d if fu sion are systematicall y und er stood as follows: 1) Interdiffusion: For and , Ctx I III RA ,, xIII ABeq ,DCtxDCCtJ IIII IIIIIIII RBABeq ,, x txDCtxD CCtJ 2) Oneway diffusion: tx . III AABeq II II Req , ,. III R ,, x txDC txD CCt J JtxJ The oneway diffusion corresponds to II I DD a DD in the above 1). In this case, the plate B is 3) Impu nd BA considered as a solvent material. rity diffusion: II II , II II eqReq ,, R , x txDC txJJtxJ . The impurity diffusion also corresponds to II I DD and DD in the above 1), and I1C and B0C I BA A ered as a nt materi in (17). In this case, the plate B is also consid solveal. 4) Selfdiffusion: For, III ,,1CtxCtx and III DD IIII II A CC BA B CC, I ,, III Req IIII IIII Req , ,,. x x txDC txJ txDC txJ The selfdiffusion is considered as a special interdiffu sion where the concentration gradient is zero in the initial state. In consideration of the NDF in time and space ,tx, (15) becomes ,,d Rt txCtx x . ) (22 Further, the selfdiffusion behavio follows. For convenience, a pure material is divi tw r is understood as ded into o regions 0and 0xx at 0t in the present diffusion system. The diffusion region is between AB xx ahe bory condition of t a time t. Tunda III0 ,,Ctx Ctx C is then used for A x or B x in the present diffusion system, where 00.5C is valid because of III ,,1txCtx. Fure al between AB Cor a p materi xx , (22) yields II II ,o , xx DCtxD CtxJ I IIII eq eq rJ (23) ,0 tCtxbecause of . In this case, even if III RR 0JJ is valid, the random movement of a le occurs an3) in the present diff n atom or a molecud it is governed by (2 usion system. Equation (23) thus shows the correlation between D and eq , where D is relevant to a jump frequency of an atom or a molecule and eq is relevant to its thermal motio The integral calculation of (23) gives n. I andCx xCC 0 for , x xxx II 0 AB xC (24) where II IIII eq eq DJD for III DD and I eq J . Equation (24) shows that I Cx and II Cx II eq 0J Copyright © 2012 SciRes. JMP
T. OKINO 1392 depend on and that x Cis valid AB 0xxxx . If we pay attention to only one side of I C or II C, thtion occurs via the ran the other hand, if we pay at tention to boI C and II C, III0 ,,Ctx Ctx C is actually valid, because we cannot know in a pure mrial whher a AB III x be atomic migra etn atom between C e movement. On th sides of ate tween dom xx concentratio in th e is one of se e early stage, we I C or II C in the initial state. That (24) is independent of t means the timeaveraged n profile caud bye Brownian motion. Therefore, the diffusion occus as a result of the random movement even in a pure material. Equation (24) thus gives the evid ence of the Brownian motion in a pure ma terial. From the historical point of view, if the FFL had been revised th r might have understood the bhavior of Brownian motion before the Einstein theory. As can be seen from the above 1)  4), the diffusivity depends on the concentration when 0 t depends on t. The concentration dependence of diffusivity is thus caused by the material source on the dion boundary terface at A iffus in x or B x. In consideration of the above 1)  4), the FFL is obviously incomplete without 0 t for FSversal. Hereinbefore, it was clarified that we can revise the o it is applicable to various diffu L and it is not u sions with the conserva ni F wFL s lu sion problem divergence theorem sh tion law. On the other s. It as also shown that the NDF plays an extremely impor tant role to understan d diffusion problems. Hereafter, the diffusion problems should be analyzed by using the NDF for a diffusion system as an additional condition equation of (15). 5. Conc The FSL consistent be exactly valid as aould hand, the FFL should be replaced by the NDF which is not only exactly valid in mathematics but also extremely useful for physics. The obtained novel results in the present study are as follows: 1) Even if a material source independent of the space ,, yz is contained in the diffusion system, the FSL is applicable to analyzing the diffusion problems as it is. terial source 0 The ma t plays an extremely impor tant role in the diffusion problems. 2) A law must be universal. The NDF is systematically applicable to the problems of interdiffusion, oneway n coefficients can be ob ion study. Equation (21) is dominant in the techno lo [1] J. B. J. Fourier la Chaleur,” Chez Firmin Didot, , 1855, pp. 3139. hte, Vol. 66, rte Bewegung von in Ruhenden Flus diffusion, impurity diffusion and selfdiffusion. Further, the NDF of (14) or (22) is applicable to analyzing diffu sion problems, for instance as seen from the derivations of (21) and (24). However, the FFL has not ever been used for analyzing the diffusion problems because of its incompleteness. The NDF is thus universal to the diffu sion problems, but the FFL is not. 3) The NDF reveals that the Keffect is reasonably obtained as (19). The interdiffusio tained by applying (21) to the experimental results. On the contrary, (19) can predict the behavior of Keffect using various combinations of the obtained diffusivities. As a result, the intrinsic diffusion conception inconsistent with the FSL is thus not only unnecessary but also un real. The NDF derived here is a fundamental equation in the diffus gical material science, since the atomic diffusivity val ues in alloy are obtained by using it for the interdiffusion experimentation. Although the present study was dis cussed in relation to diffusion problems between metal plates, the results obtained here are also applicable to various material problems described by the FSL. Hereaf ter, the new findings obtained here may make a funda mental change to the existing diffusi on theory. REFERENCES , “Theorie Analytique de Paris, 1822. [2] A. Fick, “On Liquid Diffusion,” Philosophical Magazine Journal of Science, Vol. 10 [3] L. Boltzmann, “Weitere Studien uber das Warmegleich gewicht unter GasmolekÄulen,” Wiener Beric 1872, pp. 275370. [4] A. Einstein, “Die von der Molekularkinetischen Theorie der Warme Geforde siigkeiten Suspendierten Teilchen,” Annalen der Physik, Vol. 18, No. 8, 1905, pp. 549560. doi:10.1002/andp.19053220806 [5] R. Brown, “A Brief Account of Mi tions Made in the Months of Junecroscopical Observa , July and August, 1827, e Physique, Vol. 18, No. 8, 1909, eln Diffusionscoefficienten,” Annual Review e 171, 1947, Energy in Binary Metallic System,” Trans “Shift of Multiple Markers and Intrinsic Diffusion in on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies,” Philosophical Magazine, Vol. 4, 1828, pp. 161173. [6] J. Perrin, “Mouvement Brownien et Realite Moleculare,” Annales de chimie et d pp. 5114. [7] L. 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T. OKINO Copyright © 2012 SciRes. JMP 1393 tters, Vol.Gold Iron Alloys,” Philosophical Magazine Le 74, No. 6, 1996, pp. 423428. doi:10.1080/095008396179959 [12] T. Okino, “New Mathematical Solution for Analyzing Interdiffusion Problems,” Materials Transactions, Vol. 52, No. 12, 2011, pp. 22202227. doi:10.2320/matertrans.M2011137 rabolic Space,” Jour [13] T. Okino, “Brownian Motion in Pa nal of Modern Physics, Vol. 3, No. 3, 2012, pp. 255259. doi:10.4236/jmp.2012.33034
