{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/140809"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/140809","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Phase-field simulation of freezing water droplet","abstract":"When a water droplet freezes on a cold plate, a pointy tip forms as the result of volume expansion. In this dissertation, we will introduce a quasi-compressible phase-field model that deals with this non-isothermal three-phase system involving water, ice, and air. The water-ice phase transition and the water-air fluid interface are handled by the Allen-Cahn and the Cahn-Hilliard equations, respectively. The governing equations, including the two phase-field equations, the Navier-Stokes equations, and the energy equation, are designed such that the non-negative entropy production is guaranteed. These equations are then solved by finite-element methods using the open-source deal.ii library. Our model reproduces the Gibbs-Thomson and Clausius-Clapeyron equations, which establish the dependence of the melting temperature on interface curvature and pressure, respectively. Furthermore, the built-in quasi-compressibility accurately accounts for the volume change due to the water-ice density contrast during the phase transition. With proper parameters, our simulation captures the pointy tip of the frozen droplet with good agreement with the experiment.","abstract_html":"When a water droplet freezes on a cold plate, a pointy tip forms as the result of volume expansion. In this dissertation, we will introduce a quasi-compressible phase-field model that deals with this non-isothermal three-phase system involving water, ice, and air. The water-ice phase transition and the water-air fluid interface are handled by the Allen-Cahn and the Cahn-Hilliard equations, respectively. The governing equations, including the two phase-field equations, the Navier-Stokes equations, and the energy equation, are designed such that the non-negative entropy production is guaranteed. These equations are then solved by finite-element methods using the open-source deal.ii library. Our model reproduces the Gibbs-Thomson and Clausius-Clapeyron equations, which establish the dependence of the melting temperature on interface curvature and pressure, respectively. Furthermore, the built-in quasi-compressibility accurately accounts for the volume change due to the water-ice density contrast during the phase transition. With proper parameters, our simulation captures the pointy tip of the frozen droplet with good agreement with the experiment.","abstract_has_math":false,"creators":["Li, Yichen"],"institution":"Virginia Tech","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Yue, Pengtao"],"committee_members":["Liu, Honghu","Iliescu, Traian","Borggaard, Jeffrey T."],"year":2026,"date_issued":"2026-01-14","date_published":"2026-01-14","updated_at":"2026-07-22T22:20:12Z","subjects":["phases-field method","Allen-Cahn equation","heat transfer","phase transition","finite element method"],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:45552"],"render_values":[{"text":"vt_gsexam:45552","href":null,"code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/10919/140809","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Yue, Pengtao"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Liu, Honghu","Iliescu, Traian","Borggaard, Jeffrey T."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Li, Yichen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2026-01-15T09:00:17Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2026-01-15T09:00:17Z"]},{"key":"dc:date.issued","label":"Date","values":["2026-01-14"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["phases-field method","Allen-Cahn equation","heat transfer","phase transition","finite element method"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:45552"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10919/140809"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["When a water droplet freezes on a cold plate, a pointy tip forms as the result of volume expansion. In this dissertation, we will introduce a quasi-compressible phase-field model that deals with this non-isothermal three-phase system involving water, ice, and air. The water-ice phase transition and the water-air fluid interface are handled by the Allen-Cahn and the Cahn-Hilliard equations, respectively. The governing equations, including the two phase-field equations, the Navier-Stokes equations, and the energy equation, are designed such that the non-negative entropy production is guaranteed. These equations are then solved by finite-element methods using the open-source deal.ii library. Our model reproduces the Gibbs-Thomson and Clausius-Clapeyron equations, which establish the dependence of the melting temperature on interface curvature and pressure, respectively. Furthermore, the built-in quasi-compressibility accurately accounts for the volume change due to the water-ice density contrast during the phase transition. With proper parameters, our simulation captures the pointy tip of the frozen droplet with good agreement with the experiment."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["Have you ever noticed how a drop of water freezes into a shape with a pointy tip when it lands on a cold surface? It's a fascinating process that involves the transformation of water into ice, and it's more complex than it might seem at first glance. In our study, we explore this phenomenon using a special computer model that simulates the freezing process, including the interactions between water, ice, and air. Our model uses advanced mathematical equations to understand how water turns into ice and how the tiny boundary between water and air behaves during freezing. These equations help us ensure that our simulation follows the laws of nature, specifically the principles of entropy, which in simple terms, measures the disorder or randomness of a system. To solve these complex equations, we use a powerful computer program. Our simulations are not just theoretical exercises; they accurately predict the formation of the pointy tip on the frozen droplet, much like what we observe in real-life experiments. This success demonstrates the reliability of our model. Furthermore, our study delves into the intricate details of how the shape of the frozen droplet is influenced by various factors, including the unique conditions at the point where water, ice, and air meet, and how changes in pressure and temperature affect the freezing process. Our research provides a deeper understanding of the freezing process, which is not only fascinating from a scientific perspective but also has potential applications in various fields, such as climate studies and the development of technologies based on the properties of ice and water."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Phase-field simulation of freezing water droplet"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Yue, Pengtao"],"dc:contributor.committeemember":["Liu, Honghu","Iliescu, Traian","Borggaard, Jeffrey T."],"dc:contributor.department":["Mathematics"],"dc:creator":["Li, Yichen"],"dc:date.accessioned":["2026-01-15T09:00:17Z"],"dc:date.available":["2026-01-15T09:00:17Z"],"dc:date.issued":["2026-01-14"],"dc:description.abstract":["When a water droplet freezes on a cold plate, a pointy tip forms as the result of volume expansion. In this dissertation, we will introduce a quasi-compressible phase-field model that deals with this non-isothermal three-phase system involving water, ice, and air. The water-ice phase transition and the water-air fluid interface are handled by the Allen-Cahn and the Cahn-Hilliard equations, respectively. The governing equations, including the two phase-field equations, the Navier-Stokes equations, and the energy equation, are designed such that the non-negative entropy production is guaranteed. These equations are then solved by finite-element methods using the open-source deal.ii library. Our model reproduces the Gibbs-Thomson and Clausius-Clapeyron equations, which establish the dependence of the melting temperature on interface curvature and pressure, respectively. Furthermore, the built-in quasi-compressibility accurately accounts for the volume change due to the water-ice density contrast during the phase transition. With proper parameters, our simulation captures the pointy tip of the frozen droplet with good agreement with the experiment."],"dc:description.abstractgeneral":["Have you ever noticed how a drop of water freezes into a shape with a pointy tip when it lands on a cold surface? It's a fascinating process that involves the transformation of water into ice, and it's more complex than it might seem at first glance. In our study, we explore this phenomenon using a special computer model that simulates the freezing process, including the interactions between water, ice, and air. Our model uses advanced mathematical equations to understand how water turns into ice and how the tiny boundary between water and air behaves during freezing. These equations help us ensure that our simulation follows the laws of nature, specifically the principles of entropy, which in simple terms, measures the disorder or randomness of a system. To solve these complex equations, we use a powerful computer program. Our simulations are not just theoretical exercises; they accurately predict the formation of the pointy tip on the frozen droplet, much like what we observe in real-life experiments. This success demonstrates the reliability of our model. Furthermore, our study delves into the intricate details of how the shape of the frozen droplet is influenced by various factors, including the unique conditions at the point where water, ice, and air meet, and how changes in pressure and temperature affect the freezing process. Our research provides a deeper understanding of the freezing process, which is not only fascinating from a scientific perspective but also has potential applications in various fields, such as climate studies and the development of technologies based on the properties of ice and water."],"dc:description.degree":["Doctor of Philosophy"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:45552"],"dc:identifier.uri":["https://hdl.handle.net/10919/140809"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["phases-field method","Allen-Cahn equation","heat transfer","phase transition","finite element method"],"dc:title":["Phase-field simulation of freezing water droplet"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:12Z"}