Carbon segregation to grain boundaries in rapidly solidified Ni 3 Al
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- Ct~/~) = O"( C~~ -fi
O'(Crlr Crib)
[2]
where D", D ~, and D r are the diffusivities of carbon in the
different parts of the joint. The second assumption yields = c~'"f~
[31
C~'y t~ = C~'~f~
[41
c~
whereff, i f , andff are the activity coefficients of carbon in the different parts of the joint. The distances l" and It~ are approximated with the well-known expressions from random walk theory:
P = X/2D~t
[5]
l~ = ~
[6]
By combining Eqs. [1] through [6] and solving for the unknown compositions C ~ C r~~ C r/a, C ~'r we obtain after some rearrangements: C .o +
4,~(t)C ~~
C~ =
[71 1 + f?f D~DD~ ~~(t)
1 ~b~(t) = ,/'-'/--~ff
REFERENCES l~
2vy
[8]
+l
and
C oo + C ~r =
instead of C ~ = C ~~predicted by Eq. [7] as t ~ 0. However, one can easily see that Eq. [11] would yield this result under a number of conditions. For example, if D r < D ~, or if C ~~ ~ C "~ and f f >>if, Eq. [11] would yield approximately the same result as Eq. [7]. If the layer is introduced as a hindrance for the carbon diffusion, either one or both of these conditions must be fulfilled in order for the layer to be effective as a hindrance and Eqs. [7] through [10] will give reasonable results also for short times. For times longer than the relaxation time, Eqs. [7] through [10] should be good approximations in all cases, and indeed when t ---* 0o the functions 0 ~ and $~ --* 1 and Eqs. [7] and [9] approach the expressions derived previously ~'2 when the intermediate layer was neglected. As mentioned, the intermediate layer is usually introduced in order to decrease the concentration differences C ~j~ - C ~~and C ~r - C ~~ From Eqs. [7] through [10] one can deduce that this is achieved by choosing 0 ~ and q~ as small as possible. One should thus choose a meterial with a largeff and a low D r. However, Eqs. [7] through [10] also show that at long diffusion times the layer becomes gradually less important. It may be expected that an intermediate layer of an alloyed steel will be only slightly different from that of pure iron because it is usually not possible to increase f f very much by an alloying addition. If order to obtain a strong effect, it is necessary either to use a very thick intermediate layer or to use an intermediate layer of a different metal. As an example, Cu has been reported to have a very low D. It also has a very low solubility of graphite 3 and should thus have a very high f.
1. J. ,~gren: Scand. J. Metall., 1981, vol. 10, pp. 134-40. 2. J. /kgren: MetaU. Trans, A, 1983, vol. 14A, pp. 2167-70. 3. M. Hansen and K. Anderko: Constitution of Binary Alloys, McGraw-Hill, New York, NY, 1958.
O"(t)C ~~ fr
D/b~
[91
1 + ~ ~/~--~ ~'~(t) where
1 ~b"(t) =
D/~--af r
l~
2vT6
[10]
+1
C. L. BRIANT and S.C. HUANG
It is interesting to notice that when t --~ 0 the functions 0~ and ~ a --~ 0 and, consequently, C ~/r = C "~ and C ~/~ = C ~~ Although this seems a reasonable result, it cannot be generally valid. This is demonstrated as follows. For times much shorter than the relaxation time for diffusion in the 3' laye
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