Let's solve each part step by step using the provided **Zr-C-B ternary phase diagram**. --- ## a) **Propose a Suitable Composition (Zr, ZrC, ZrB₂, ≤30 at.% Liquid)** ### **Step 1: Identify Phase Regions** - You want a mixture of **Zr, ZrC, ZrB₂** with up to **30% liquid** during sintering. - On the diagram: - **Zr**: bottom-left corner - **ZrC**: bottom-right (labeled 344°C) - **ZrB₂**: upper left (labeled 3245°C) ### **Step 2: Locate Suitable Region** - For all three solids to coexist, select a composition near the triangle connecting Zr, ZrC, and ZrB₂. - Find where this triangle lies just inside (or near) the liquid phase field for up to 30% liquid. ### **Step 3: Choose a Composition** - Example: A point near Zr (say 40 at.%), ZrC (30 at.%), ZrB₂ (30 at.%). - Check the diagram: This point is near the lower center, inside the 260–280°C isotherms, close to the boundary where a liquid phase appears. - At slightly above 260°C, a small amount of liquid can form. ### **Step 4: Justify** - This composition (Zr 40, ZrC 30, ZrB₂ 30 at.%) will: - Be in the three-phase solid region (Zr, ZrC, ZrB₂) just before melting. - On heating, up to 30% liquid will form around 260–270°C. --- ## b) **30C-30B-40Zr (at.%) Sintering at 260°C: First Liquid Phase?** ### **Step 1: Locate Composition** - Find point: 40% Zr, 30% C, 30% B. - This is close to the center of the triangle. ### **Step 2: Find Liquidus Temperature** - Find the **lowest temperature contour line** that passes through this point. - The point is between the **260°C and 280°C** contours, but closer to 260°C. **→ The first liquid appears at ~260°C.** ### **Step 3: Amount of First Liquid** - At the temperature where liquid first appears (solidus), the amount of liquid is **just barely above zero**. - To find the amount at a specific higher temperature (e.g., 260°C), use the lever rule between the solidus and liquidus lines. - Since your point is almost exactly on the 260°C contour, the amount of liquid at 260°C is **almost zero** (incipient melting). - As temperature increases above 260°C, the liquid fraction increases. --- ## c) **Can You Use 5C-80B-15Zr (at.%) at 240°C? (≤10 at.% Liquid Allowed)** ### **Step 1: Locate Composition** - 5% C, 80% B, 15% Zr. - This point is near the top of the triangle (near B). ### **Step 2: Check Isotherms** - The **240°C isotherm** passes below this point. - This means at 240°C, the composition is **above the solidus** and **inside the liquid phase region**. ### **Step 3: Estimate Liquid Fraction** - Since the point is clearly above the solidus at 240°C, a significant amount of liquid will form. - Likely, **much more than 10 at.% liquid** will be present at 240°C. **→ It is **not possible** to use this UHTC at 240°C if you require ≤10 at.% liquid.** --- ## **Final Answers Summary** ### a) **Suitable composition:** - **Zr 40 at.%, ZrC 30 at.%, ZrB₂ 30 at.%** (Ensures coexistence of three solids with up to 30% liquid during sintering.) ### b) **30C-30B-40Zr at 260°C:** - **First liquid forms at:** ~**260°C** - **Amount of first liquid at 260°C:** **Almost zero** (incipient melting; increases above 260°C) ### c) **5C-80B-15Zr at 240°C:** - **Not possible** to use this composition at 240°C if liquid phase must be ≤10 at.%, because a much larger liquid fraction will be present. --- **If you need detailed lever rule calculations or further phase fraction estimates, let me know!**