An approximate analytical demonstration of the famous darken experiment
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IN 1949 L. Darken j demonstrated that the chemical potential gradient was the driving force for diffusion in an experiment where a carbon-iron-silicon diffusion couple exhibited "uphill" diffusion of carbon to establish a concentration discontinuity. To the author's knowledge, no analytical demonstration of this has been published. The development below gives an approximate analytical demonstration of the results. For the system, the composition of carbon, C, is found in OC dt
_
[
O Lc c + Lc,~ Ox ffxx Ox ]
[1]
when Lr and Lc,i are phenomenological coefficients, and/z~ and itsi are chemical potentials of carbon and silicon respectively. With Henry's law in mind, we express the chemical potential of carbon as #~
= l~ ~ +
t~L = t ~ b c 7 = c ~
/zR = t z * , C~ = C~
KTlnCy~
KTln
=
C3,
[2]
where the activity coefficient of carbon, ~,, is independent of carbon content and contains the standard state. There is 4 pct Si in one-half of the diffusion couple but not in the other. This leads to a discontinuity in the chemical potential of carbon at the weld interface initially. Below, we assume that the composition of silicon is time independent due to the slow diffusion of a substitutional solute. Thus, only carbon is assumed to move. We will therefore neglect any changes in silicon composition and chemical potential. With the above approximations in mind, we may combine Eqs. [1] and [2] assuming constant carbon diffusivity, D~ ax 2
[ a (~;/xKT)]2 +
Thus, we let Iz~/KT = /~ and 0 = D d. Then we solve the equation 1" 0~t -- ~2113"+ \/0~12 O0 Ox2 ~Ox]
[31
[41
C~
3X 2
=
C*
x
=
O, 0 L > 0 ,
OR
>0
[71
The flux continuity is somewhat more complicated. We require that J = - L V/~ is continuous. L is not continuous across the interface because L = D C, in this nomenclature, and both C and D are not continuous. To simplify the situation, we set J = -DCVI~ so that for 0 c and 0 g greater than zero, one has [8]
But we do not know C L and C R since they result from the solution. However, the ratio is known. Therefore,
[51
J. P. STARK is Professor of Mechanical Engineering, Materials Science Laboratory, University of Texas at Austin, Austin, TX 78712. Manuscript submitted February 7, 1980.
[9]
since/~ is continuous. Substituting Eq. [9] into Eq. [8] gives the final b o u n d a r y condition that at x = 0 DR V/xL = - - V/xR
"/L
OZc* -
[6]
for 0, = 0 = OR. 0 L = DLt where D L is the diffusivity to the left of the weld. Similarly OR = DRt. To assure that the chemical potential is continuous across the weld interface, we set
DL
30
x >0
CLYL = CR'~R
by solving by companion [2] equation OC*
x
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