Toward developing tasks to promote these original objectives, Principal Investigator (PI) Derby visited Dr. Martin Volz at the NASA Marshall Space Flight Center on November 29, 2016, to discuss past progress and future research directions. However, after discussions with Dr. Volz and others, a shift in research focus was deemed appropriate, and this project has been redirected to address continuing opportunities for advancing the understanding of the mechanisms that determine the pushing or engulfment of foreign particles in a melt at a moving solid-liquid interface. Of particular interest and relevance is the engulfment of silicon carbide and silicon nitride particles during the solidification of multi-crystalline silicon used in photovoltaic devices.

This work has and will continue to coordinate with the ParSiWal (Partikeleinfang bei der Siliziumkristallisation im Weltall) project funded by the German DLR (Deutsches Zentrum für Luft und Raumfahrt) and involving the Fraunhofer IISB and the University of Freiburg. Dr. Martin Volz of Marshall Space Flight Center is also involved with this work on particle engulfment.

An important advance for this project was the discovery of a new scaling law that described the dependence of the critical velocity on the size of the particle for the engulfment of silicon carbide (SiC) particles during silicon solidification. Whereas all prior analytical models had predicted that the critical velocity scaled as either the inverse of particle radius or the radius raised to the -4/3 power, our finite element model showed a dependence on the radius raised to the -5/3 power.

This finding was significant, since for the first time in over a decade of research on SiC inclusions in silicon, our model was able to provide a quantitative correlation with experimental results, and furthermore allowed for the unambiguous identification of the underlying physical mechanisms that gave rise to the observed behavior of this system. In particular, we identified a significant and previously unascertained interaction between particle-induced interface deflection (originating from the thermal conductivity of the SiC particle being larger than that of the surrounding silicon liquid) and curvature-induced changes in melting temperature arising from the Gibbs-Thomson effect. For a particular range of particle sizes, the Gibbs-Thomson effect flattens the deflected solidification interface, thereby reducing drag on the particle and increasing its critical velocity for engulfment. We showed via numerical calculations and analytical reasoning that these effects gave rise to the new scaling of the critical velocity to particle radius.

The original work on this project involved the use of an in-house, research code to computationally model particle engulfment. While this code was powerful and efficient, it was unsuitable for general usage and distribution. Therefore, later effort involved the development and application of a new code, the open-source, finite-element code Goma 6.0, to describe engulfment physics. Goma 6.0 is a multiphysics finite-element code with a basis in computational fluid dynamics for problems with free and moving surfaces. It is therefore uniquely suited for the modeling of continuum transport and the representation of the many interfaces in the engulfment problem.

Significant benefits are expected through the use of Goma 6.0 for this problem. Not only will this code provide a platform to continue to analyze the engulfment problem, but, importantly, the software produced will allow a broader user community to also use this tool to study engulfment phenomena.

Finally, this project also supported one graduate student, Dr. Yuta Tao, who received his Ph.D. in materials science in December 2018. Subsequent work was carried on via a series of post-doctoral research associates, including Dr. Jeffrey H. Peterson, Dr. Benjamin Drueke, Dr. Jan Seebeck, and Dr. Chung-Husan Huang.

Publications

Derby JJ. "The synergy of modeling and novel experiments for melt crystal growth research." IOP Conf Ser: Mater Sci Eng. 2018;355:12001. https://doi.org/10.1088/1757-899X/355/1/012001

Derby JJ. "Fluid dynamics in crystal growth: The good, the bad, and the ugly." Progress in Crystal Growth and Characterization of Materials. 2016 Jun;62(2):286-301. (in Special Issue: Recent Progress on Fundamentals and Applications of Crystal Growth; The Proceedings of the 16th Summer School on Crystal Growth (ISSCG-16), Otsu, Shiga, Japan, August 1–7, 2016.) https://doi.org/10.1016/j.pcrysgrow.2016.04.015

Derby JJ, Tao Y, Reimann C, Friedrich J, Jauss T, Sorgenfrei T, Cröll A "A quantitative model with new scaling for silicon carbide particle engulfment during silicon crystal growth" J Crystal Growth 463, 100–109 (2017). https://doi.org/10.1016/j.jcrysgro.2017.02.012

Tao Y, Sorgenfrei T, Jauss T, Cröll A, Reimann C, Friedrich J, Derby JJ "Particle engulfment dynamics under oscillating crystal growth conditions" J Crystal Growth 468, 24–27 (2017). https://doi.org/10.1016/j.jcrysgro.2016.10.049

Friedrich J, Reimann C, Jauss T, Cröll A, Sorgenfrei T, Tao Y, Derby JJ "Engulfment and pushing of Si3N4 and SiC particles during directional solidification of silicon under microgravity conditions" J Crystal Growth 475, 33–38 (2017). https://doi.org/10.1016/j.jcrysgro.2017.05.036

Toward developing tasks to promote these original objectives, Principal Investigator (PI) Derby visited Dr. Martin Volz at the NASA Marshall Space Flight Center on November 29, 2016, to discuss past progress and future research directions. However, after discussions with Dr. Volz and others, a shift in research focus was deemed appropriate, and this project has been redirected to address continuing opportunities for advancing the understanding of the mechanisms that determine the pushing or engulfment of foreign particles in a melt at a moving solid-liquid interface. Of particular interest and relevance is the engulfment of silicon carbide and silicon nitride particles during the solidification of multi-crystalline silicon used in photovoltaic devices.

This work has and will continue to coordinate with the ParSiWal (Partikeleinfang bei der Siliziumkristallisation im Weltall) project funded by the German DLR (Deutsches Zentrum für Luft und Raumfahrt) and involving the Fraunhofer IISB and the University of Freiburg. Dr. Martin Volz of Marshall Space Flight Center is also involved with this work on particle engulfment.

An important advance for this project was the discovery of a new scaling law that described the dependence of the critical velocity for the engulfment of silicon carbide (SiC) particles by a silicon solidification front on the size of the particle. Whereas all prior analytical models had predicted that the critical velocity scaled as either the inverse of particle radius or the radius raised to the -4/3 power, the work performed using the finite element model showed a dependence on the radius raised to the -5/3 power.

Both Dr. Peterson and Dr. Drueke worked to better understand this new scaling for the engulfment of silicon carbide in silicon. Peterson performed additional calculations, and Drueke employed analytical techniques to promote a more complete understanding.

The prior work on this project involved the use of an in-house, research code to computationally model particle engulfment. While this code is powerful and efficient, it is unsuitable for general usage and distribution. Therefore, the current post-doc assigned to this project, Dr. Huang, has been tasked with modifying a new code, the open-source, finite-element code Goma 6.0, to describe engulfment physics. Goma 6.0 is a multiphysics finite-element code with a basis in computational fluid dynamics for problems with free and moving surfaces. It is therefore uniquely suited for the modeling of continuum transport and the representation of the many interfaces in the engulfment problem.

Significant benefits are expected through the use of Goma 6.0 for this problem. Not only will this code provide a platform to continue to analyze the engulfment problem, but, importantly, the software produced will allow a broader user community to also use this tool to study engulfment phenomena.

This work was to promote collaboration between research groups at the University of Minnesota (UMN) and the NASA Marshall Space Flight Center (MSFC). A research team at MSFC was to focus on experimental aspects of detached Bridgman growth, while UMN researchers were to develop and apply sophisticated mathematical models to analyze the process.

Toward developing tasks to promote these original objectives, Principal Investigator (PI) Derby visited Dr. Martin Volz at the NASA Marshall Space Flight Center on November 29, 2016, to discuss past progress and future research directions.

However, after discussions with Dr. Volz and others, a shift in research focus was deemed appropriate, and this project has been redirected to address continuing opportunities for advancing the understanding of the mechanisms that determine the dynamics of particle transport in silicon melts and the pushing or engulfment of these particles at a moving solid-liquid interface, specifically small silicon carbide and silicon nitride particles during the solidification of silicon.

This work has and will continue to coordinate with the ParSiWal (Partikeleinfang bei der Siliziumkristallisation im Weltall) project funded by the German DLR (Deutsches Zentrum für Luft- und Raumfahrt) and involving the Fraunhofer IISB and the University of Freiburg. Dr. Martin Volz of Marshall Space Flight Center is also involved with this work on particle engulfment.

An important advance for this project was the discovery of a new scaling law that described the dependence of the critical velocity for the engulfment of silicon carbide (SiC) particles by a silicon solidification front on the size of the particle. Whereas all prior analytical models had predicted that the critical velocity scaled as either the inverse of particle radius or the radius raised to the -4/3 power, our finite element model showed a dependence on the radius raised to the -5/3 power.

This finding was significant, since for the first time in over a decade of research on SiC inclusions in silicon, our model was able to provide a quantitative correlation with experimental results, and furthermore allowed for the unambiguous identification of the underlying physical mechanisms that gave rise to the observed behavior of this system. In particular, we identified a significant and previously unascertained interaction between particle-induced interface deflection (originating from the thermal conductivity of the SiC particle being larger than that of the surrounding silicon liquid) and curvature-induced changes in melting temperature arising from the Gibbs-Thomson effect. For a particular range of particle sizes, the Gibbs-Thomson effect flattens the deflected solidification interface, thereby reducing drag on the particle and increasing its critical velocity for engulfment. We showed via numerical calculations and analytical reasoning that these effects gave rise to the new scaling of the critical velocity to particle radius.

The primary goal of this research project is to understand detached Bridgman growth in both microgravity and terrestrial environments. Detached Bridgman growth is a potentially transformative method to grow large electronic and photonic crystals. It originated from the serendipitous observation of unusual behavior exhibited by early, space-based melt crystal growth experiments, where the melt dewetted the wall, allowing the crystal to pull away and grow detached from the ampoule. The detached growth mode eliminated deleterious interactions between the growing crystal and the ampoule wall, producing crystals of heretofore unprecedented levels of structural perfection. However, what was so readily attained in space is far more difficult on Earth.

There is an opportunity to continue to advance our understanding of detached Bridgman growth under terrestrial conditions by a combination of ground-based modeling coupled with microgravity and terrestrial experiments. Specifically, this project will develop and apply rigorous thermal-capillary models employing finite-element methods and coupled with furnace-scale heat transfer models to describe detached solidification. These models will be applied to assess the onset, stability, and termination of detached growth and to further ascertain the details about why this growth is so much more easily attained in microgravity environments than on Earth. Of great importance is understanding how larger-scale, Earth-based systems may be designed and operated successfully.

This work will promote a collaboration between research groups at the University of Minnesota (UMN) and the NASA Marshall Space Flight Center (MSFC). A research team at MSFC will focus on experimental aspects of detached Bridgman growth, while UMN researchers will develop and apply sophisticated mathematical models to analyze the process. Experiments and theory will be conducted with complementary goals; the synergy between the two approaches will promote progress much faster than pursuing either path alone. The proposed research teams two of the foremost crystal growth groups in the world to leverage their strongly complementary skills toward the realization of the great potential of detached solidification.

Detached Bridgman growth is a potentially transformative method to grow large electronic and photonic crystals. It originated from the serendipitous observation of unusual behavior exhibited by early, space-based melt crystal growth experiments, where the melt dewetted the wall, allowing the crystal to pull away and grow detached from the ampoule. The detached growth mode eliminated deleterious interactions between the growing crystal and the ampoule wall, producing crystals of heretofore unprecedented levels of structural perfection. However, what was so readily attained in space is far more difficult on Earth. There is an opportunity to continue to advance our understanding of detached Bridgman growth under terrestrial conditions by a combination of graund-based modeling coupled with microgravity experiments. Specifically, this project will develop and apply rigorous thermal-capillary models employing finite-element methods and coupled with furnace-scale heat transfer models to describe detached solidification. These models will be applied to assess the onset, stability, and termination of detached growth and to further ascertain the details about why this growth is so much more easily attained in microgravity environments than on Earth. Of great importance is understanding how larger-scale, Earth-based systems may be designed and operated successfully.

This work will promote a collaboration between research groups at the University of Minnesota (UMN) and the NASA Marshall Space Flight Center (MSFC). A research team at MSFC will focus on experimental aspects of detached Bridgman growth, while UMN researchers will develop and apply sophisticated mathematical models to analyze the process. Experiments and theory will be conducted with complementary goals; the synergy between the two approaches will promote progress much faster than pursuing either path alone. The proposed research teams two of the foremost crystal growth groups in the world to leverage their strongly complementary skills toward the realization of the great potential of detached solidification.

NOTE (February 2016): Continuation of "Modeling of Particle Transport in the Melt and Its Interaction with the Solid-Liquid Interface," grant NNX10AR70G, with the same Principal Investigator.