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ATOM PROBE ANALYSIS OF SOLUTE CLUSTERING ABOVE A MISCIBILITY GAP
P. Camus
To cite this version:
P. Camus. ATOM PROBE ANALYSIS OF SOLUTE CLUSTERING ABOVE A MISCIBILITY GAP. Journal de Physique Colloques, 1987, 48 (C6), pp.C6-331-C6-336. �10.1051/jphyscol:1987654�.
�jpa-00226859�
JOURNAL DE PHYSIQUE
Colloque C6, suppl6ment a u n O l l , Tome 48, novembre 1987
ATOM PROBE ANALYSIS O F SOLUTE CLUSTERING ABOVE A MISCIBILITY GAP
Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6376, U.S.A.
ABSTRACT
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At temperatures just above a miscibility gap, spatially small but large amplitude solute fluctuations or clusters are predicted. These clusters may be retained in the solution upon quenching to a lower temperature and may influence subsequent decomposition kinetics.A model has been developed which quantitatively predicts the shape of composition histograms as a function of temperature and composition. At high temperatures, the model predicts a binomial-shaped composition distribution indicative of a random dispersion of solute, whereas at temperatures just above the miscibility gap, a broadening of the distribution is expected signifying solute clustering. Experimental histograms measured in an Fe-45at.%Cr alloy were found to obey the predictions of the thermodynamic model.
INTRODUCTION
Most decomposition theories assume a statistically random distribution of solute deriving from Brownian motion of the atoms in a single phase material. This assumption is valid at temperatures where homogenization is generally performed. However, at solution treatment temperatures just above a miscibility gap, spatially small but large amplitude solute fluctuations or clusters are predicted. From fluctuation theory[lI, the mean squared composition deviation can be given by :
where k is Boltzmann's constant, T is the solution treatment temperature, F" is the curvature or second derivative of the free energy per unit volume with respect to composition, C, K is a gradient energy coefficient, and VC is the derivative of composition with respect to distance. The free energy or thermodynamic term, F", has the most influence upon composition deviations in most systems and can be modified significantly by composition and temperature departures from the critical values of a miscibility gap. At homogenization temperatures, the curvature of the free energy is so large that thermally induced random motion has a larger effect and results in a random distribution of solute.
However, at solution treatment temperatures approaching the miscibility gap, the curvature of the free energy is small, causing large ~ O m D ~ s i t i ~ n fluctuations or solute clustering. These fluctuations are not stable with time at the temperature, but continue to appear and decay dynamically. These fluctuations or clusters may be retained in the solution upon quenching to lower temperatures and may influence the scale and kinetics of subsequent decomposition. Therefore, characterization of the initial solute distribution is an important parameter in the understanding of subsequent decomposition.
Characterization of fluctuations in composition has typically been performed by small angle scattering techniques because of the good counting statistics through the use of a large volume of analysis and because of the inherent sensitivity of these techniques to detecting spatial correlations of atoms.
Vintaykin et a1.I21 and LaSaile and ~ c h w a r t z [ ~ ] utilized small angle neutron scattering to investigate the effect of prior solution treatment temperatures upon subsequent decomposition in Fe-Cr alloys. Both sets of results were consistent with the predictions of fluctuation theory where the scattering intensity increases as the solution treatment temperature was reduced. They concluded that larger composition gradients resulting from solute clustering increase the rate of subsequent decomposition. However, quantification of the scattering data to predict the composition gradients or spatial extent of the fluctuations was not reported.
'~ostdoctoral Fellow through Oak Ridge Associated Universities.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987654
C6-332 JOURNAL DE PHYSIQUE
A model is proposed where the shape of atom probe composition histograms may be predicted based only upon the thermodynamics of the system. This model uses the concept of the change of the free energy of a system upon formation of a solute cluster. I n the context of a sub-regular solution model, the free energy of a system may be described by the following expression :
where C, and Cb are the concentrations of solutes a and b in the solution, and AI2, AZ1, and A,, are interaction parameters that must be evaluated for each system. When a composition fluctuation forms within the original phase, the free energy of the fluctuation, G: follows the free energy curve with the appropriate increase in the system free energy, AG; shown i n k i g . 1. The probability of fluctuation formation can be related to the increase in system energy by :
where N is a normalization parameter. At the average composition, there is no change ion the free energy and the probability of formation is greatest. As the composition deviation increases, the energy increases markedly and the probability of formation decreases exponentially. A specific shape is not assumed for the predicted distribution, since the shape is determined solely by the free energy function.
If the free energy well is narrow, the distribution will be narrow. If the energy well is flat over a wide composition range, as is the case a t temperatures just above a miscibility gap, the predicted distribution will be wide. The solute probability distribution may also be experimentally obtained from composition histogram analyses. Therefore, a quantitative comparison of measured composition histograms and predicted solute probability distributions may be performed.
The aim of this study is to test experimentally the model of solute clustering in the single phase above a miscibility gap. The Fe-Cr binary system may be ideal for this type of investigation due to the relative flatness and width of the miscibility gap. An Fe-45 at. % Cr alloy was selected because of its proximity to the center of the miscibility gap where composition fluctuations are most likely to be observed. A possible complicating effect, the formation of u phase, was not important due to the sluggish nature of the reaction[4].
EXPERIMENTAL
Three heat treatment temperatures above the critical temperature, T,, of the Fe-Cr miscibility gap and one inside the miscibility gap were chosen. The lowest solution treatment temperature of 560°C is estimated to be less than 5OC above T,[~], although T, is not known precisely. All specimens were solution treated at 1000°C for 1 h with three sets of specimens being subsequently annealed for either 2 h a t 700°C, 192 h a t 560°C, or 192 h a t 540°C. The specimens were water quenched after all stages of heat treatment. These annealing times were estimated to be long enough for the solute to arrive at equilibrium c o n f i g ~ r a t i o n s ~ ~ ] .
Atom probe analyses were performed at 80 K using an appropriate DC voltage and pulse fraction ( 18-2296 ) to obtain the correct average composition without either preferential evaporation or retention. The diameter of the area of analysis was maintained at less than approximately 1.5 nm to accurately determine the composition of the spatially small fluctuations. This resulted in approximately 10 atoms being collected per (110) plane of material removed. Since a balance must be obtained between accurate composition quantification and small number counting statistics, composition profiles were obtained based upon a volume of material containing 50 ions. Autocorrelation analyses were performed on the profiles to investigate the degree of phase separation. Composition histograms, which show the number of blocks within a composition range versus the composition, were constructed from composition profiles. The average composition of any one extended atom probe analysis was found to be between 43 and 46% Cr. The shape of all composition histograms for any particular heat treatment was found to be consistent from specimen to specimen.
As the thermodynamics of the Fe-Cr system are not fully known, the free energy of the system was approximated by equation 2. The interaction parameters were iteratively calculated to minimize the deviation between the shape of the calculated and experimentally determined miscibility gaps[6]. The interaction parameters that resulted were Alt = Aal = 4529 and A,, = -2409 cal/mole. From the master thermodynamic equation (equation 2), a coherent miscibility gap, free energy curves, and solute probability distributions were obtained.
Calculated binomial distributions based upon counting statistics were used for comparison as a limiting case. Chi-squared goodness-of-fit analyses with respect to a binomial distribution were performed on the calculated and experimental distributions. Chi-squared ratios were obtained by normalizing the measured chi-squared values by the tabular chi-squared value at the 95% confidence level. Variance measurements about the mean composition were performed on the calculated, experimental, and binomial distributions to determine the relative width of each distribution. Variance ratios for the calculated and experimental distributions were obtained by normalizing the measured values by the values obtained for the binomial distribution.
RESULTS
Field-ion micrographs for specimens heat treated at 540°C showed the characteristic two-phase contrast indicative of phase separation within the miscibility gap of Fe-Cr-X a l l 0 ~ s [ ~ - ~ 1 . In the corresponding composition profile, spatially extended, large amplitude composition deviations from the average composition are observed indicative of a two-phase mixture. Field-ion images obtained for specimens heat treated at temperatures a t and above 560°C show no contrast indicative of a second phase. In composition profiles, Fig. 2, spatially extended composition deviations were not observed in specimens heat treated at these temperatures. Autocorrelation analyses of the profiles indicated a drop to near zero correlation within a distance lag of 2, indicating the presence of only single phase material on a scale of the effective probe aperture. Thus, specimens heat treated at 540°C are phase separated within the miscibility gap, while those heat treated a t 560°C and above are single phase.
Only small excursions from statistical scatter about the mean composition are observed in the composition profiles of the specimens aged at 700 and 1000°C. However, some large amplitude but spatially small composition excursions are observed in the profiles of material aged at 560°C. A more detailed examination of the amplitude of the fluctuations was obtained by analyzing composition histograms, Fig. 3. The histograms from the two highest temperature specimens visually fit a binomial distribution, which is expected when the solute is in random configurations. However, departures from a binomial distribution of greater than 2 standard deviations were observed in the high and low composition regions of the distribution for the specimens aged at 560°C. More detailed quantification of the distributions by chi-squared analyses indicated that the distributions a t high temperature were comparable to a binomial distribution, whereas the distribution a t 560°C was statistically different. The large variance ratio indicated that the difference between the distributions for the specimens aged at 560°C and the binomial distribution was in the width of the experimental distribution.
Free energy curves were calculated as a function of solution treatment temperature, Fig. 4. Due to approximations with only a three-parameter fit in equation 2 and analytical problems in the determination of the interaction parameters, two-phases are predicted to occur at 560°C, in variance with the experimental results. To compare single phase histograms accurately, the comparison model temperature was increased slightly to 570°C, which was found to be approximately 5OC above the miscibility gap, in reasonable agreement with the estimate of 560°C above the T, in the alloy.
Therefore, all model calculations for the lowest temperature were based upon a 570°C solution treatment temperature and may be compared directly to the 560°C experimental data. The usual trend of flattening of the free energy curves as the temperature is reduced is evident in Fig. 4. A common tangent line at the average composition was determined and the free energy increase and corresponding probability distributions were calculated as a function of composition at each temperature, Fig. 5.
At high temperatures, the calculated model distributions appear much narrower than a binomial distribution which signifies that the composition deviations are too small to be measured with the current block size and counting statistics. At temperatures slightly above the miscibility gap, the calculated distribution appears much broader than a binomial distribution because the small curvature of the free energy curve leads to a distribution with a wide composition range. Large chi-squared ratios of the calculated distributions at all temperatures signify the different forms of the distributions, Table 1. Small variance ratios are obtained at 700 and 1000°C indicating the narrowness of the free energy well, Table 1. A large variance ratio at 570°C indicates that the calculated distribution is much wider than the distribution expected for a random dispersion of solute.
JOURNAL DE PHYSIQUE
DISCUSSION
At temperatures just above the miscibility gap, the curvature of the free energy is very small, which leads to a broadening of the calculated distribution. This broadening is also observed in the measured histograms for material aged at 560°C, which indicates that the solute is not randomly distributed, but forms solute clusters in the single phase with composition deviations far from the mean. Therefore, solute clustering that is detectable on a scale of 1.5 nm is occurring in the single phase.
However, the measured width of the distribution does not match that predicted by the model at temperatures just above the miscibility gap. This discrepancy could stem from differences between the thermodynamics of the model and the actual system. Because T, is not known precisely, there is no accurate confirmation of 560°C being 5OC above T,, which precludes an accurate appraisal of the corresponding temperature for the model. It was found that a change in the temperature to 580°C for the model, which is approximately lS°C above T, or 10°C too large, would produce a probability distribution in reasonable agreement with the measured 560°C distribution. To compare the model with experiment more closely, knowledge of the exact shape and extent of the whole miscibility gap is required.
Another source of the discrepancy could arise due to the common composition quantification problem of sampling size versus feature size. If the actual fluctuation size was much smaller than the effective probe diameter, the measured composition amplitude would be markedly lower than the actual amplitude.
The damping of the composition amplitude would lead to a narrower distribution than predicted, although probably wider than expected for a random distribution of solute. Therefore, the statistically significant increase in the tail regions of the low temperature distribution, even considering damping effects, indicates the presence of composition fluctuations or solute clustering in the single phase.
In this study, the average composition deviation from the mean, as determined from the variance measurements, is approximately 1.5 times that expected for a random distribution of solute. All measurements indicate that the scale of the composition deviations is no larger than 1.5 nm, possibly smaller. The increased average composition deviation coupled with the small spatial extent of the fluctuations would lead to steeper concentration gradients in the solution, consistent with the small angle scattering result^[^^^].
CONCLUSIONS
The top of the miscibility gap in the Fe-45 at. % Cr alloy was determined to be between 540°C and 560°C. A thermodynamic model was developed which predicts the shape of composition histograms in the single phase region as a function of solution treatment temperature and alloy composition. As the miscibility gap was approached from above, the width of composition histograms was observed to be greater than that predicted for a random distribution of solute. This observation indicates the presence of solute clustering in the single phase above the miscibility gap.
Acknowledgements
The APFIM experiments were performed at the University of Pittsburgh in partial fulfillment of the requirements for a doctoral degree at the University of Pittsburgh. Research sponsored through the National Science Foundation under University/Industrial Cooperative grant DMR-80-22225 and by the Division of Materials Sciences, U.S. Department of Energy, under contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc.
REFERENCES
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L.D. Landau, E.M. Lifshitz, "Statistical Physics" (Addison-Wesley Pub. Co., 1958) 344.2
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Y.Z. Vintaykin, V.B. Dmitriyev, V.Y. Kolontsov, Fiz. Metal. Metalloved., 27 (1969) 1131.3
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R.O. Williams, Trans. AIME, 212 (1958) 497.5
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M.K. Miller and S. Spooner, to be published.6
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S.S. Brenner, M.K. Miller, W.A. Soffa, Scripta Met., 16 (1982) 831.7
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M.K. Miller, S.S. Brenner, P.P. Camus, J. Piller, W.A. Soffa, Proc. 29th International Field Emission Symposium (ed. H. Norden and H-0. Andren, Alqvist and Wiksell Intl., Stockholm, 1982) 489.8
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P.P. Camus, Ph.D. Dissertation, University of Pittsburgh, 1986.Fig. 1. Increase in free energy, A G ~ due to composition fluctuation, C$ The curve @(c) is the free energy of a single phase material. The average composition of the alloy is C,.
c o
c t,
COMPOSITION
0 100 2 0 0 0 100 200
Distance ( 50 ion blocks )
Fig. 2. Composition profiles for Fe-45 at. % Cr specimens. Dashed lines indicate the average composition of the data. The profile for material aged at 540°C indicates a two phase microstructure whereas the other profiles indicate a single phase.
80 1000 20 40 80 00 100
COMPOSlTlON (at. %)
Fig. 3. Experimentally determined composition histograms with 20 error bars and calculated binomial distributions (dashed curves). Note the good fit to the binomials in (a) and (b), whereas in (c) the histogram shows increased counts in the tail regions. Phase separation is shown for comparison in (d).
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0 2 0 4 0 6 0 8 0 100
COMPOSITION (at. %)
Fig. 4. Calculated free energy curves for Fe-Cr binary alloys as a function of solution treatment temperature. Note ihe small energy increase over a wide composition range at the low temperature due to the flattening of the free energy curve.
0 2 0 4 0 6 0 8 0 100
COMPOSlTiON (at. %)
Fig. 5. Calculated and binomial distributions for Fe-45 at. % Cr as a function of solution treatment temperature. Note the increase in width of the distributions as the temperature is reduced.
TABLE 1 Cumulative Statistical Analyses of Histograms
TemD Number Variance Variance Ratio Chi-Sauared $ Ratio (oC) Blocks Measured Binomial (Measured/BinomialyMeasured ( ~ e a s u r e d / x i , ~ ~ )
Experimental
1000 1254 4.14 3.21 1.29 66.5 16.9 3.9
700 550 3.85 3.15 1.22 14.7 14.1 1.05
560 1111 5.55 3.18 1.75 234.5 15.5 15.1
Calculated
1000 1254 0.17 3.21 0.05 2300 3.8 600
700 550 0.7 3.20 0.2 360 6.0 60
570 1111 8.5 3.21 2.6 2500 16.9 150
