🔬 materials science

Analytical model for balling defects in laser melting using rivulet theory and solidification

This paper presents an analytical model based on rivulet theory and solidification competition to explain balling defects in laser melting, revealing that while fluid instabilities play a role, the discrepancy between model predictions and experimental results underscores the critical importance of fluid flows and heat transport in the process.

Original authors: Zane Taylor, Tharun Reddy, Maureen Fitzpatrick, Kwan Kim, Wei Li, Chu Lun Alex Leung, Peter D. Lee, Kaila M. Bertsch, Leora Dresselhaus-Marais

Published 2026-09-15

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Zane Taylor, Tharun Reddy, Maureen Fitzpatrick, Kwan Kim, Wei Li, Chu Lun Alex Leung, Peter D. Lee, Kaila M. Bertsch, Leora Dresselhaus-Marais

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern manufacturing, there is a technique called laser powder bed fusion that builds metal parts layer by layer. A high-powered laser scans across a bed of fine metal powder, melting it into a liquid track that solidifies almost instantly to form the next layer of an object. For this process to work well, the molten metal must flow smoothly and freeze into a uniform line. However, under certain conditions, the liquid metal refuses to behave. Instead of a straight line, it breaks apart into a string of uneven bumps and gaps, a defect known as "balling." This flaw ruins the structural integrity of the final part and limits how fast manufacturers can print. For years, scientists have tried to understand why this happens, often pointing to a classic physics principle that explains how a stream of water breaks into droplets. Yet, this traditional explanation has struggled to match the complex reality of a laser melting metal on a solid surface, where the material freezes almost as fast as it flows.

A team of researchers from Stanford University, the European Synchrotron Radiation Facility, and other institutions has taken a fresh look at this problem. They set out to create a new mathematical description of the balling defect that accounts for the fierce competition between two forces: the fluid instability trying to break the liquid track apart, and the solidification front trying to freeze it in place. To do this, they did not rely on computer simulations alone. Instead, they used powerful X-rays to watch the process happen in real time on thin samples of tantalum metal. By observing the molten metal as it was scanned by a laser, they captured images of the fluid accumulating, flowing, and being pinched off by solidifying edges. Their work suggests that the old way of thinking about this defect is incomplete because it ignores the fact that the metal is freezing while it is moving.

The researchers began by challenging the standard model used by the additive manufacturing community. That model treats the molten metal track as if it were a floating cylinder of liquid in a vacuum, similar to a stream of water falling from a tap. In that scenario, surface tension naturally causes the stream to break into droplets if it gets too long relative to its width. However, in a laser melting process, the liquid metal is not floating; it sits on top of a solid substrate, and the sides of the melt pool are supported by the unmelted material around it. The team realized that this support changes the physics significantly. They adapted a theory from fluid dynamics known as "rivulet theory," which describes how a thin stream of liquid flows down a surface, to better represent the geometry of a melt pool sitting on a solid base. This new approach allowed them to calculate how fast the instability would grow, taking into account the stabilizing effect of the substrate holding the liquid in place.

The core of their discovery lies in the timing. The researchers found that the speed at which the fluid instability tries to break the track apart is often comparable to the speed at which the metal freezes. In the past, models assumed that the liquid had plenty of time to break up before it solidified, or that the freezing happened so fast it didn't matter. The new analysis shows that these two processes are racing against each other. If the liquid freezes quickly, it can "pinch" the track before the instability has a chance to grow large, preventing balling. If the liquid stays hot longer, the instability has time to amplify, creating the characteristic hills and valleys of the defect. The team developed a way to measure this competition, defining a "balling fraction" that describes how much of the melt pool's depth contributes to the final bump. They found that deeper melt pools, which take longer to freeze, are more likely to develop large defects, even if the fluid is technically more stable in a purely liquid sense.

To test their theory, the team conducted experiments using a high-speed camera and synchrotron X-rays to image the melting of tantalum. They scanned the metal at different speeds and power levels, capturing the exact moment the liquid track fragmented. The images revealed a dynamic process where the laser pushes molten metal toward the back of the pool, where it accumulates into a hill. As the laser moves forward, the channel feeding this hill narrows and eventually solidifies, cutting off the supply and leaving behind a solid bump. The researchers measured the speed at which the solidification front moved upward and compared it to the speed at which the fluid surface moved downward. They found these speeds were of the same magnitude, confirming that neither process can be ignored. In their thin samples, where the melt pool had no side walls to support it, the fluid was highly unstable, leading to large, dramatic balling events.

One of the most significant findings from this work is that the distance between the bumps is not simply equal to the length of the molten pool, as many previous theories assumed. The X-ray images showed that after a bump forms and the pool breaks, the remaining liquid pool is often too short to immediately trigger another break. The laser continues to scan, lengthening the pool until it is finally long enough to become unstable again. This creates a delay, meaning the distance between the final solid bumps is often much larger than the theoretical length of the unstable wave. This observation explains why experimental measurements of the defect spacing have historically been inconsistent with simple predictions. The researchers also noted that the shape of the melt pool matters deeply; shallow pools with a wide, flat top are far more prone to breaking apart than deep, narrow ones, because the substrate offers less support to the liquid in the shallow case.

The team's model successfully predicts a smooth transition from a perfect track to a fully balled one, rather than a sudden switch. This gradual change matches what is seen in real manufacturing, where defects appear to worsen as conditions change, rather than appearing all at once. However, the researchers are careful to note that their model is a simplified view. It focuses on the battle between fluid instability and freezing but does not fully account for the complex flows of liquid metal driven by vapor pressure or the presence of powder particles. In their thin-sample experiments, they observed fluid being pushed backward by the laser, a mechanism that also contributes to the defect. While their rivulet-based theory captures the essential role of solidification, they acknowledge that a complete picture of the defect requires understanding how these fluid flows interact with the freezing process.

Ultimately, this work provides a clearer, more realistic framework for understanding why metal parts develop these defects. By treating the melt pool as a rivulet on a surface rather than a floating cylinder, and by explicitly calculating the race between freezing and breaking, the researchers have offered a tool that can help engineers predict when balling will occur. Their findings suggest that to avoid these defects, one must manage the geometry of the melt pool and the speed of solidification together. If the metal freezes too slowly relative to the instability, the track will break. If it freezes fast enough, the track can survive. This insight moves the field beyond simple rules of thumb and toward a quantitative understanding of the delicate balance required to print high-quality metal parts.

Technical Summary: Analytical Model for Balling Defects in Laser Melting Using Rivulet Theory and Solidification

Problem Statement
In Laser Powder Bed Fusion (LPBF) and laser welding, "balling" is a critical defect characterized by the non-uniform accumulation of material along a melt track, forming periodic "hills" and discontinuities. This defect leads to poor powder spreading and lack-of-fusion issues, ultimately limiting productivity and mechanical properties. While the Plateau-Rayleigh instability (describing the breakup of free-standing fluid jets) has been widely adapted by the additive manufacturing (AM) community to explain balling, existing analytical models suffer from significant limitations. They typically assume an infinite-length fluid cylinder in isolation, neglect the stabilizing effect of the substrate, ignore fluid flows, and fail to account for the rapid timescales of solidification. Furthermore, these models often predict a binary transition (balling vs. non-balling) based on a fixed $L/W$ ratio, whereas experimental evidence suggests a gradual transition. Crucially, current literature often assumes the balling wavelength ( $\lambda$ ) is equivalent to the melt pool length ( $L$ ), an assumption this work challenges.

Methodology
The authors combine synchrotron-based X-ray radiography experiments with a novel analytical framework:

Experimental Approach: In-situ X-ray radiography was performed on thin tantalum (Ta) samples (100 µm thick) using a quad-laser system at the European Synchrotron Radiation Facility (ESRF). High-speed imaging (40 kHz) captured the dynamic evolution of the melt pool, fluid accumulation, and solidification "pinching" that leads to balling. This allowed for the direct measurement of melt pool geometry ( $W, L, D$ ), fluid surface velocities, and solidification rates.
Theoretical Framework:
- Rivulet Instability Adaptation: Instead of the standard Plateau-Rayleigh model, the authors adapt the Rivulet instability theory (specifically the model by Yang and Homsey for a fluid in a wedge) to the AM geometry. This accounts for the stabilizing influence of the substrate (the unmelted material) and the specific geometry of the melt pool (approximated as a parabolic cross-section within a wedge).
- Solidification Competition: A formalism is developed to model the competition between the growth of fluid instabilities and the solidification front. The model calculates the time ( $\tau$ ) at which the upward-moving solidification front intersects the downward-moving fluid instability (the "valley" of the perturbation).
- Balling Fraction ( $\Upsilon$ ): A normalized balling amplitude is defined as the ratio of the final balling height to the total melt pool depth ( $D+h$ ). This allows for a continuous transition from stable printing to severe balling, rather than a binary threshold. Note: The authors explicitly state that this continuous transition is derived under the assumption of neglecting fluid flows and heat transport.

Key Contributions

Integration of Solidification and Instability: The paper presents an analytical formalism that explicitly couples fluid instability growth rates with solidification rates. This addresses the gap in literature where these competing phenomena were often treated in isolation.
Rivulet Theory for AM: The work adapts Rivulet instability theory to the specific geometry of laser melting (a rivulet on a substrate), providing a more physically grounded stability criterion than the isolated jet assumption of Plateau-Rayleigh.
Decoupling of Wavelength and Melt Pool Length: The authors demonstrate experimentally and theoretically that the balling wavelength ( $\lambda$ ) is not necessarily equivalent to the melt pool length ( $L$ ). They propose a mechanism where the melt pool length oscillates (growing and fragmenting) between balling events, leading to observed wavelengths that can differ significantly from the theoretical instability wavelength.
Continuous Transition Model: By introducing the balling fraction ( $\Upsilon$ ), the model describes a gradual progression from unevenness to swelling, humping, and complete balling, dependent on the ratio of instability growth to solidification rates. Crucially, the authors clarify that this continuous transition is a result of their specific analytical formalism which neglects fluid flows and heat transport; they note that deviations between these model predictions and experimental results demonstrate the importance of including those factors in the balling process.

Results

Experimental Observations: Radiography revealed that fluid accumulation is driven by flow reversal and vapor depression, followed by solidification "pinching" the melt pool. The measured solidification velocities (0.16–0.26 m/s) were found to be comparable in magnitude to fluid surface velocities, confirming that neither can be neglected.
Stability Diagrams: The adapted Rivulet model predicts that melt pools with smaller depth-to-width ratios ( $D/W$ ) are less stable (higher instability growth rates) due to reduced substrate stabilization. However, the paper highlights a competing effect: while deeper pools have more stable instability growth rates ( $\omega$ ), they take longer to solidify for a given solidification rate ( $R$ ). Consequently, deeper melt pools can favor balling because the extended fluid state allows more time for instabilities to grow.
Model vs. Experiment:
- The model successfully predicts the trend that shallower melt pools (low $D/W$ ) are more prone to balling in terms of instability growth rates, though the absolute balling amplitude is also influenced by the total melt pool depth and solidification time.
- However, a discrepancy was observed: experimentally measured wavelengths were often larger than the maximum unstable wavelength predicted by the Rivulet theory. The authors attribute this to the "wavelength enhancement" caused by the melt pool growing in length between fragmentation events, a dynamic not captured by static instability models.
- In the limit of vanishing $D/W$ (thin-wall experiments), the Rivulet model's mathematical formalism begins to diverge, indicating a breakdown of the model for extremely shallow pools without side-wall support.

Significance and Claims
The authors claim that this work provides the "most realistic analytical discussion" of balling driven purely by fluid instabilities, explicitly accounting for the competition with solidification. The primary significance lies in:

Challenging the $L/W > \pi$ Threshold: The paper argues that simple geometric thresholds are insufficient because balling is a dynamic competition between timescales of instability growth and solidification.
Refining the Mechanism: By showing that $\lambda \neq L$ and that solidification rates are comparable to fluid flow rates, the work highlights the need for comprehensive models that include fluid dynamics and heat transport, rather than relying solely on instability theories.
Framework for Future Work: The authors position their analytical framework as a foundation. They acknowledge that while their model neglects internal fluid flows and powder effects (focusing on the instability-solidification competition), it offers a necessary step toward understanding the multifaceted nature of balling. They suggest that future efforts should substitute more precise physical models for each component within this established framework.

The paper concludes that while fluid instabilities are a primary driver, the interplay with solidification determines the timescale and extent of balling, and that current process optimization relies on empirical identification of these regimes because a single mechanism does not fully explain the phenomenon.

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