Cooling Curve Analysis to Determine Phase Fractions in Solid-State Precipitation Reactions

Abstract

This work presents a new method of cooling curve analysis to make in situ measurements of the amount of precipitate formed in solid-state phase transformations. The presented technique is based on a first-principles analysis of thermodynamics and heat flow to develop equations that relate cooling curve data to the amount transformed. The precipitation of Ag₂Al in a binary Al-Ag alloy was examined both as a practical example of this technique and to obtain metallographic measurements of the amount of precipitation for comparison purposes.

Appendices

Appendix: Limits Of Detection

Given that conducting numerous experiments to determine the minimum quantifiable heat release is impractical; instead, the following method was devised to analytically determine what the minimum detectable limit would be. The premise of this method is to determine the amount of variation between the presented mathematical model and the actual cooling behavior of the material when no transformation is occurring then determine the minimum heat release that would be distinguishable from this unwanted variation. The amount of heat released then can be converted into a minimum detectable amount of precipitate formation.

Analysis

The variation is defined as the difference between either $- {1 \mathord{\left/ {\vphantom {1 {g_{\text{pre}} }}} \right. \kern-\nulldelimiterspace} {g_{\text{pre}} }}$ or $- {1 \mathord{\left/ {\vphantom {1 {g_{\text{post}} }}} \right. \kern-\nulldelimiterspace} {g_{\text{post}} }}$ and the curve to which they are fitted $\left( {{{d\left[ {\ln \left( {T - T_{\infty } } \right)} \right]} \mathord{\left/ {\vphantom {{d\left[ {\ln \left( {T - T_{\infty } } \right)} \right]} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right),$ as described by Eq. [8]. The sources of variation in the single phase regions include thermocouple measurement noise or deviations in the cooling behavior from ideal convection. Because this analysis takes place solely in the single phase region, a generic g will be used in place of either g _pre or g _post. The variation can be described mathematically as follows:

$$ {\text{Variation}} = \left( {{\frac{{d\left[ {\ln \left( {T - T_{\infty } } \right)} \right]}}{dt}}} \right) - \left( { - \frac{1}{g}} \right) $$

(A1)

This equation can be manipulated to obtain the following:

$$ {\text{Variation}} = {\frac{{gdT + \left( {T - T_{\infty } } \right)dt}}{{g\left( {T - T_{\infty } } \right)dt}}} $$

(A2)

To convert the variation into an equivalent fraction transformed, consider Eq. [6] in the text when no transformation is occurring. Again, g _pre and g _post are replaced by a generic g, resulting in the following equation:

$$ C_{{\Updelta H}} df_{t} = gdT + \left( {T - T_{\infty } } \right)dt $$

(A3)

where df _t is not an actual transformation but an equivalent amount of transformation attributed to variation. Replacing this equation into the numerator of Eq. [A2], the variation can be related to the heat release term C _ΔH df _t as follows:

$$ {\text{Variation}} = {\frac{{C_{\Updelta H} df_{t} }}{{g\left( {T - T_{\infty } } \right)dt}}} $$

(A4)

Using the definitions of C _ΔH and g to remove the defined terms results in the following:

$$ {\text{Variation}} = {\frac{{\Updelta Hdf_{t} }}{{C_{p} \left( {T - T_{\infty } } \right)dt}}} $$

(A5)

This equation can be rearranged as follows:

$$ {\frac{{df_{t} }}{dt}} = {\frac{{\left( {\text{Variation}} \right)C_{p} \left( {T - T_{\infty } } \right)}}{\Updelta H}} $$

(A6)

This equation can be integrated to obtain an equation that equates the thermal event from the minimum detectable transformation to the thermal event causing the maximum variation between the fitted $- {1 \mathord{\left/ {\vphantom {1 {g_{\text{pre}} }}} \right. \kern-\nulldelimiterspace} {g_{\text{pre}} }}$ or $- {1 \mathord{\left/ {\vphantom {1 {g_{\text{post}} }}} \right. \kern-\nulldelimiterspace} {g_{\text{post}} }}$ and the ${{d\left[ {\ln \left( {T - T_{\infty } } \right)} \right]} \mathord{\left/ {\vphantom {{d\left[ {\ln \left( {T - T_{\infty } } \right)} \right]} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}$ data to which it is fitted.

$$ f_{t} = \int {{\frac{{\left( {\text{Variation}} \right)C_{p} \left( {T - T_{\infty } } \right)}}{\Updelta H}}dt} $$

(A7)

where the limits of integration would be two time values within the single phase cooling regime that would maximize the value f _t ; this would be done to represent the worst case scenario by capturing the largest variation. A factor X will be added to Eq. [A8]; this factor will be used to differentiate between variation caused by a transformation and error-related variation. This factor assumes only a transformation that is X times larger than the largest measured variation could be identified conclusively as a transformation and quantified. Adding this factor in, and pulling the constant terms out of the integral results in the following:

$$ f_{t} = X{\frac{{C_{p} }}{\Updelta H}}\int {\left( {\text{Variation}} \right)\left( {T - T_{\infty } } \right)dt} $$

(A8)

This equation contains constants (C _p , ΔH, and T, which is the transformation temperature) as well as the arbitrary factor X, a measurable variation, and it returns a minimum detectable fraction transformed.

Results

This minimum detection limit determination method was applied to the experiments done in this study. The values used for heat capacity and heat of transformation were the same as those used in the text; the factor X was arbitrarily set to 10, and the integration of the variation was performed individually for each cooling scenario. The resulting data (Table VI) indicate that there is a large difference in the measured variation between the as-cast sample and the solutionized sample. This is because of the experimental setup used in the as-cast sample allowing for better control of the cooling environment compared with the solutionized sample, which was allowed to cool in a busy laboratory where people walking past could create air currents that negatively would affect the variation.

Table VI Maximum Measured Variations for Each Transformation Scenario for the Al-Ag System Used in this Work*

This effect also can be observed in the minimum detectable transformation calculations that are summarized in Table VII. These calculated values range from a fraction transformed of 1.5 wt pct Ag₂Al for the well-controlled environment to slightly more than 5 wt pct Ag₂Al for the worst case scenario.

Table VII Limits of Detectability in Terms of Weight Fraction for Each of the Scenarios Used in this Work

Although it is difficult to extend the analysis of these experiments to other alloy systems that will transform at different temperatures and with different thermophysical properties, this analysis has shown that under well-controlled cooling conditions, small heat releases can be detected and differentiated from experimental error and quantified.