Weight Fraction of Eutectic Microstructure Calculator
| Microconstituent | Weight Fraction | Weight % |
|---|
Calculate the Weight Fraction of the Eutectic Microstructure That Forms
Understanding phase transformations in binary alloy systems is fundamental to Materials Science and Engineering. When designing alloys for specific mechanical properties, engineers must often calculate the weight fraction of the eutectic microstructure that forms upon solidification. This calculation allows metallurgists to predict the ratio of the brittle, layered eutectic structure relative to the ductile proeutectic phases.
What is the Weight Fraction of the Eutectic Microstructure?
The eutectic microstructure is a specific physical arrangement of two solid phases that forms simultaneously from a liquid solution at a specific temperature (the eutectic temperature). In a binary phase diagram, this occurs at the Eutectic Point.
When an alloy with a composition different from the eutectic composition cools, it first forms a "primary" or proeutectic phase (either Alpha or Beta). The remaining liquid becomes richer in solute until it reaches the eutectic composition. At the eutectic temperature, this remaining liquid transforms entirely into the eutectic microstructure.
Therefore, to calculate the weight fraction of the eutectic microstructure that forms, we are essentially calculating the fraction of liquid remaining just before the final solidification event.
Who Should Use This Calculation?
- Metallurgical Engineers: To optimize alloy strength and ductility balance.
- Materials Science Students: Studying phase diagrams and the Lever Rule.
- Foundry Technicians: Controlling cooling rates and microstructure formation.
Formula and Mathematical Explanation
The calculation relies on the Inverse Lever Rule applied to the binary phase diagram just above the eutectic temperature ($T_E + \Delta T$). The formula depends on whether the alloy composition ($C_0$) is Hypoeutectic (left of eutectic point) or Hypereutectic (right of eutectic point).
The Variables
| Variable | Meaning | Typical Unit |
|---|---|---|
| $C_0$ | Overall Alloy Composition | wt% |
| $C_E$ | Eutectic Composition | wt% |
| $C_\alpha$ | Max Solubility of Phase $\alpha$ | wt% |
| $C_\beta$ | Max Solubility of Phase $\beta$ | wt% |
| $W_{eutectic}$ | Weight Fraction of Eutectic | 0.0 – 1.0 |
Case 1: Hypoeutectic Alloy ($C_\alpha < C_0 < C_E$)
In this region, the proeutectic phase is Alpha ($\alpha$). The eutectic microstructure forms from the remaining liquid.
Formula:
$$ W_{eutectic} = \frac{C_0 – C_\alpha}{C_E – C_\alpha} $$
Case 2: Hypereutectic Alloy ($C_E < C_0 < C_\beta$)
In this region, the proeutectic phase is Beta ($\beta$).
Formula:
$$ W_{eutectic} = \frac{C_\beta – C_0}{C_\beta – C_E} $$
Practical Examples
Example 1: Lead-Tin (Pb-Sn) Solder
Consider a Pb-Sn alloy used in soldering. The system has the following properties:
- Eutectic Composition ($C_E$): 61.9 wt% Sn
- Alpha Solubility ($C_\alpha$): 18.3 wt% Sn
- Beta Solubility ($C_\beta$): 97.8 wt% Sn
If we have an alloy with 40 wt% Sn ($C_0$), it is Hypoeutectic because $40 < 61.9$.
Calculation:
$$ W_{eutectic} = \frac{40 – 18.3}{61.9 – 18.3} = \frac{21.7}{43.6} \approx 0.498 $$
Result: The structure will be roughly 49.8% eutectic microconstituent and 50.2% proeutectic $\alpha$.
Example 2: Copper-Silver (Cu-Ag)
A jewelry alloy consists of Silver and Copper. Suppose the system parameters are:
- $C_E = 71.9$ wt% Ag
- $C_\alpha = 8.0$ wt% Ag
- $C_\beta = 91.2$ wt% Ag
If the alloy is 85 wt% Ag ($C_0$), it is Hypereutectic ($85 > 71.9$).
Calculation:
$$ W_{eutectic} = \frac{91.2 – 85}{91.2 – 71.9} = \frac{6.2}{19.3} \approx 0.321 $$
Result: The weight fraction of the eutectic microstructure is 32.1%.
How to Use This Eutectic Calculator
- Identify System Parameters: Look up the phase diagram for your specific alloy (e.g., Fe-C, Al-Si, Pb-Sn) to find $C_E$, $C_\alpha$, and $C_\beta$.
- Enter Alloy Composition: Input your specific alloy mix ($C_0$).
- Review the Microstructure Type: The tool will automatically detect if you are in the Hypoeutectic or Hypereutectic region.
- Analyze Results: Use the generated chart and table to see the balance between the primary phase and the eutectic matrix.
Key Factors That Affect Microstructure Weight Fraction
When you calculate the weight fraction of the eutectic microstructure that forms, consider these physical factors:
- Solubility Limits: The span between $C_\alpha$ and $C_E$ determines the "lever" length. A wider solubility range means proeutectic phases form more readily.
- Cooling Rate (Non-Equilibrium): This calculator assumes equilibrium cooling (slow cooling). Rapid cooling (quenching) can shift the effective solubility lines (Scheil equation), changing the actual fractions observed.
- Impurity Elements: Ternary additions can shift the eutectic point temperature and composition, invalidating simple binary calculations.
- Under-cooling: Significant under-cooling may suppress the formation of primary phases, leading to a fully eutectic-like structure even off-composition.
- Density Differences: While this calculator provides Weight Fraction, the Volume Fraction (what you see under a microscope) depends on the densities of the respective phases.
- Diffusivity: In systems with low diffusivity, coring may occur, reducing the effective amount of eutectic liquid remaining.
Frequently Asked Questions (FAQ)
1. What if my composition is exactly the Eutectic Composition?
If $C_0 = C_E$, the weight fraction of the eutectic microstructure is 100% (or 1.0). No proeutectic phase forms.
2. What happens if $C_0 < C_\alpha$?
If the composition is less than the solubility limit, the alloy forms a single-phase solid solution upon cooling. No eutectic reaction occurs, so the eutectic fraction is 0%.
3. Can I use this for Iron-Carbon (Steel)?
Yes, but be careful with terminology. In the Fe-Fe3C system, the eutectic reaction occurs at 4.3% C. For steels (up to 2.14% C), you are typically looking at the Eutectoid reaction (Pearlite), not Eutectic. The math is identical, but the reaction is Solid $\rightarrow$ Solid + Solid.
4. Why is the Lever Rule used?
The Lever Rule is a mass balance equation derived from the conservation of mass. It geometrically determines the ratio of phase amounts based on the tie-line length in the phase diagram.
5. Does this calculator account for intermetallic compounds?
This calculator treats the end phases as generic Alpha and Beta. If your Beta phase is an intermetallic line compound (vertical line), simply set $C_\beta$ to that specific composition.
6. What is the difference between Eutectic Microstructure and Eutectic Phase?
The "Eutectic Phase" is not a standard term. The Eutectic Microstructure is a composite mixture of two phases ($\alpha + \beta$). The individual phases within it are Alpha and Beta.
7. How does this relate to mechanical properties?
Proeutectic phases are often dendritic and can be ductile (like Alpha in Pb-Sn) or hard/brittle. The eutectic structure is usually finer and stronger. Calculating the fraction helps predict the composite hardness.
8. What units should I use?
The calculator works with any unit (wt%, at%) as long as you are consistent. However, phase diagrams are traditionally provided in Weight Percent (wt%).
Related Tools and Internal Resources
- Binary Phase Diagram Generator – Visualize the full dome and solvus lines.
- Lever Rule Calculator for Tie Lines – Calculate solid vs. liquid fractions at any temperature.
- Iron-Carbon Equilibrium Diagram Guide – Specific guide for steel and cast iron microstructures.
- Solidification Time Calculator (Chvorinov's Rule) – Estimate how long it takes for the casting to freeze.
- Unit Cell Density Calculator – Convert between atomic percent and weight percent.
- Grain Size Number Calculator (ASTM) – Analyze the resulting microstructure fineness.