The Measurement of Composition During the Early Stages of the Spinodal Decomposition of Fe-Cr Alloys
- PDF / 300,686 Bytes
- 6 Pages / 420.48 x 639 pts Page_size
- 36 Downloads / 214 Views
THE MEASUREMENT OF COMPOSITION DURING THE EARLY STAGES OF THE SPINODAL DECOMPOSITION OF Fe-Cr ALLOYS M.G. Hetherington*, J.M. Hyde* and M.K. Miller"*, *Department of Materials, University of Oxford, Oxford, England "**Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6376. ABSTRACT The properties of many advanced alloys are derived from extremely fine-scale microstructures. This poses interesting questions about the measurement of composition on this scale. The phase separation of model Fe-Cr alloys has been been studied with the atom-probe. Statistical techniques have been used to estimate the composition and compare the results with the predictions of linear and non-linear theories of spinodal decomposition and the distributions obtained from Monte-Carlo calculations. EXPERIMENTAL The atom-probe is capable of measuring composition with a resolution of the order of one nanometer. Atoms are detected sequentially from random positions in a layer of atoms. The area of the specimen selected by the circular detector aperture has a radius of 1-2 nm and in a random area analysis of a material, many hundreds of layers are ionized. The position of an atom is therefore known, in principle, with the precision of the layer spacing in depth and 2 nm lateral resolution. On this scale, it is impossible to treat the composition of an alloy as a continuous variable. The determination of the coarse-grain scale (the volume over which the composition is defined) is implicit in many experimental and theoretical methods, but has to be considered explicitly in atom-probe experiments. The typical number of atoms in each experiment reported in this paper was 30,000. Thirty-three atoms were detected from each layer and the experiments were designed to keep this number constant by moving the probe aperture to ensure that a constant area was subtended on the specimen. Further experimental details of the atom-probe technique can be found in Miller and Smith [I]. SPINODAL THEORY The non-linear diffusion or Cahn-Hilliard equation which is used to describe spinodal phase separation is _x
-VFMV SL
xJ
(1)
N I1V x
where x is the composition, t is time, M is a mobility and G is the free energy,
G=-fd3r{
1 YLIV(x(r))1
2
+ VL[x(r)]
}
(2)
where YL is the gradient energy coefficient and VL is the bulk energy, and both explicitly contain the coarse-grain scaling length, L. Equation (1) is linear if VL is assumed to be a second order polynomial and has solutions of the form
x - Xm = A sin[k. r] exp[ -Dk 2t]
(3)
where xm is the mean composition, k is the wave vector and D is the thermodynamic diffusion constant. Langer, Bar-on and Miller (LBM) [2] solved the non-linear equation with some approximations. Most pertinent of these, was the assumption that the probability distribution function p(x(r)) can be written as Mat. Res. Soc. Symp. Proc. Vol. 186. 01991 Materials Research Society
210
pWxO))
2 2 2 2 /20 ] /2c0] + I, exp[-(x-l.12) ) p 2exp[-(x-pI) r--
(4)
This is a double Gaussian distribution, with one
Data Loading...
